Download - Iso Calculation of Gear Rating
CLASSIFICATION NOTES NO 412
DET NORSKE VERITAS Veritasveien 1 N-1322 Hoslashvik Norway Tel +47 67 57 99 00 Fax +47 67 57 99 11
CALCULATION OF GEAR RATING FOR MARINE TRANSMISSIONS
MAY 2003
copy Det Norske Veritas 2003 Data processed and typeset by Det Norske Veritas Printed in Norway 04072003 313 PM - CN412doc If any person suffers loss or damage which is proved to have been caused by any negligent act or omission of Det Norske Veritas then Det Norske Veritas shall pay compensation to such person for his proved direct loss or damage However the compensation shall not exceed an amount equal to ten times the fee charged for the service in question provided that the maximum compensation shall never exceed USD 2 million In this provision ldquoDet Norske Veritasrdquo shall mean the Foundation Det Norske Veritas as well as all its subsidiaries directors officers employees agents and any other acting on behalf of Det Norske Veritas
FOREWORD DET NORSKE VERITAS is an autonomous and independent Foundation with the objective of safeguarding life property and the environment at sea and ashore
DET NORSKE VERITAS AS is a fully owned subsidiary Society of the Foundation It undertakes classification and certifica-tion of ships mobile offshore units fixed offshore structures facilities and systems for shipping and other industries The So-ciety also carries out research and development associated with these functions
DET NORSKE VERITAS operates a worldwide network of survey stations and is authorised by more than 120 national ad-ministrations to carry out surveys and in most cases issue certificates on their behalf
Classification Notes
Classification Notes are publications that give practical information on classification of ships and other objects Examples of design solutions calculation methods specifications of test procedures as well as acceptable repair methods for some compo-nents are given as interpretations of the more general rule requirements
A list of Classification Notes is found in the latest edition of the Introduction booklets to the rdquoRules for Classification of Shipsrdquo and the rdquoRules for Classification of High Speed Light Craft and Naval Surface Craftrdquo In ldquoRules for Classification of Fixed Offshore Installationsrdquo only those Classification Notes that are relevant for this type of structure have been listed
The list of Classification Notes is also included in the current ldquoClassification Services ndash Publicationsrdquo issued by the Society which is available on request All publications may be ordered from the Societyrsquos Web site httpexchangednvcom
Provisions
It is assumed that the execution of the provisions of this Classification Note is entrusted to appropriately qualified and experienced people for whose use it has been prepared
DET NORSKE VERITAS
CONTENTS
1 Basic Principles and General Influence Factors 4 11 Scope and Basic Principles 4 12 Symbols Nomenclature and Units 4 13 Geometrical Definitions5 14 Bevel Gear Conversion Formulae and Specific
Formulae6 15 Nominal Tangential Load Ft Fbt Fmt and Fmbt 6 16 Application Factors KA and KAP 7 161 KA 7 162 KAP7 163 Frequent overloads8 17 Load Sharing Factor Kγ 8 171 General method8 172 Simplified method 8 18 Dynamic Factor Kv 8 181 Single resonance method 8 182 Multi-resonance method 10 19 Face Load Factors KHβ and KFβ 10 191 Relations between KHβ and KFβ10 192 Measurement of face load factors 10 193 Theoretical determination of KHβ11 194 Determination of fsh 12 195 Determination of fdefl13 196 Determination of fbe 13 197 Determination of fma 13 198 Comments to various gear types 13 199 Determination of KHβ for bevel gears 13 110 Transversal Load Distribution Factors KHα and KFα14 111 Tooth Stiffness Constants cacute and cγ 14 112 Running-in Allowances 15 2 Calculation of Surface Durability17 21 Scope and General Remarks 17 22 Basic Equations 17 221 Contact stress 17 222 Permissible contact stress 17 23 Zone Factors ZH ZBD and ZM18 231 Zone factor ZH 18 232 Zone factors ZBD 18 233 Zone factor ZM 18 234 Inner contact point 18 24 Elasticity Factor ZE 18 25 Contact Ratio Factor Zε18 26 Helix Angle Factor Zβ18 27 Bevel Gear Factor ZK18 28 Values of Endurance Limit σHlim and Static
Strength 5H10σ 3H10σ 18 29 Life Factor ZN 19 210 Influence Factors on Lubrication Film ZL ZV and
ZR 19 211 Work Hardening Factor ZW 20 212 Size Factor ZX20 213 Subsurface Fatigue20 3 Calculation of Tooth Strength 22 31 Scope and General Remarks 22 32 Tooth Root Stresses 22 321 Local tooth root stress22 322 Permissible tooth root stress 22
33 Tooth Form Factors YF YFa 23 331 Determination of parameters 23 332 Gearing with εαn gt 2 24 34 Stress Correction Factors YS YSa 24 35 Contact Ratio Factor Yε 25 36 Helix Angle Factor Yβ 25 37 Values of Endurance Limit σFE 25 38 Mean stress influence Factor YM 26 381 For idlers planets and PTO with ice class 26 382 For gears with periodical change of rotational
direction 26 383 For gears with shrinkage stresses and unidirectional
load 26 384 For shrink-fitted idlers and planets 26 385 Additional requirements for peak loads 27 39 Life Factor YN 27 310 Relative Notch Sensitivity Factor YδrelT 28 311 Relative Surface Condition Factor YRrelT 28 312 Size Factor YX 28 313 Case Depth Factor YC 28 314 Thin rim factor YB 29 315 Stresses in Thin Rims 29 3151 General 29 3152 Stress concentration factors at the 75ordm tangents 30 3153 Nominal rim stresses 30 3154 Root fillet stresses 30 316 Permissible Stresses in Thin Rims 31 3161 General 31 3162 For gt3middot106 cycles 31 3163 For le 103 cycles 31 3164 For 103 lt cycles lt 3106 32 4 Calculation of Scuffing Load Capacity 33 41 Introduction 33 42 General Criteria 33 43 Influence Factors 34 431 Coefficient of friction 34 432 Effective tip relief Ceff 34 433 Tip relief and extension 34 434 Bulk temperature 35 44 The Flash Temperature flaϑ 35 441 Basic formula 35 442 Geometrical relations 35 443 Load sharing factor XΓ 36 Appendix A Fatigue Damage Accumulation 40 A1 Stress Spectrum 40 A2 σminusN-curve 40 A3 Damage accumulation 40 Appendix B Application Factors for Diesel Driven
Gears 41 B1 Definitions 41 B2 Determination of decisive load 41 B3 Simplified procedure 41 Appendix C Calculation of Pinion-Rack 42 C1 Pinion tooth root stresses 42 C2 Rack tooth root stresses 42 C3 Surface hardened pinions 42
4 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
1 Basic Principles and General Influence Factors
11 Scope and Basic Principles The gear rating procedures given in this Classification Note are mainly based on the ISO-6336 Part 1-5 (cylindrical gears) and partly on ISO 10300 Part 1-3 (bevel gears) and ISO Technical Reports on Scuffing and Fatigue Damage Accumulation but especially applied for marine purposes such as marine propulsion and important auxiliaries onboard ships and mobile offshore units
The calculation procedures cover gear rating as limited by contact stresses (pitting spalling or case crushing) tooth root stresses (fatigue breakage or overload breakage) and scuff-ing resistance Even though no calculation procedures for other damages such as wear grey staining (micropitting) etc are given such damages may limit the gear rating
The Classification Note applies to enclosed parallel shaft gears epicyclic gears and bevel gears (with intersecting axis) However open gear trains may be considered with regard to tooth strength ie part 1 and 3 may apply Even pinion-rack tooth strength may be considered but since such gear trains often are designed with non-involute pinions the calculation procedure of pinion-racks is described in Appen-dix C
Steel is the only material considered
The methods applied throughout this document are only valid for a transverse contact ratio 1 lt εα lt 2 If εα gt 2 either special considerations are to be made or suggested simplifi-cation may be used
All influence factors are defined regarding their physical interpretation Some of the influence factors are determined by the gear geometry or have been established by conven-tions These factors are to be calculated in accordance with the equations provided Other factors are approximations which are clearly stated in the text by terms as laquomay be cal-culated asraquo These approximations are substitutes for exact evaluations where such are lacking or too extensive for prac-tical purposes or factors based on experience In principle any suitable method may replace these approximations
Bevel gears are calculated on basis of virtual (equivalent) cylindrical gears using the geometry of the midsection The virtual (helical) cylindrical gear is to be calculated by using all the factors as a real cylindrical gear with some exceptions These exceptions are mentioned in connection with the ap-plicable factors Wherever a factor or calculation procedure has no reference to either cylindrical gears or bevel gears it is generally valid ie combined for both cylindrical and bevel
In order to minimise the volume of this Classification Note such combinations are widely used and everywhere it is necessary to distinguish it is clearly pointed out by local headings such as
Cylindrical gears
Bevel gears
The permissible contact stresses tooth root stresses and scuffing load capacity depend on the safety factors as re-quired in the respective Rule sections
Terms as endurance limit and static strength are used throughout this Classification Note
Endurance limit is to be understood as the fatigue strength in the range of cycles beyond the lower knee of the σndashN curves regardless if it is constant or drops with higher number of cycles
Static strength is to be understood as the fatigue strength in the range of cycles less than at the upper knee of the σndashN curves
For gears that are subjected to a limited number of cycles at different load levels a cumulative fatigue calculation applies Information on this is given in Appendix A
When the term infinite life is used it means number of cy-cles in the range 108ndash1010
12 Symbols Nomenclature and Units The symbols are generally from ISO 701 ISOR31 and ISO 1328 with a few additional symbols Only SI units are used
The main symbols as influence factors (K Z Y and X with indeces) etc are presented in their respective headings Symbols which are not explained in their respective Secs are as follows
a = centre distance (mm) b = facewidth (mm) d = reference diameter (mm) da = tip diameter (mm) db = base diameter (mm) dw = working pitch diameter (mm) ha = addendum (mm) ha0 = addendum of tool ref to mn hfp = dedendum of basic rack ref to mn (= ha0) hFe = bending moment arm (mm) for tooth root
stresses for application of load at the outer point of single tooth pair contact
hFa = bending moment arm (mm) for tooth root stresses for application of load at tooth tip
HB = Brinell hardness HV = Vickers hardness HRC = Rockwell C hardness mn = normal module n = rev per minute NL = number of load cycles qs = notch parameter Ra = average roughness value (microm) Ry = peak to valley roughness (microm) Rz = mean peak to valley roughness (microm) san = tooth top land thickness (mm) sat = transverse top land thickness (mm) sFn = tooth root chord (mm) in the critical section spr = protuberance value of tool minus grinding stock
equal residual undercut of basic rack ref to mn T = torque (Nm) u = gear ratio (per stage) v = linear speed (ms) at reference diameter
Classification Notes- No 412 5 May 2003
DET NORSKE VERITAS
x = addendum modification coefficient z = number of teeth zn = virtual number of spur teeth αn = normal pressure angle at ref cylinder αt = transverse pressure angle at ref cylinder αa = transverse pressure angle at tip cylinder αwt = transverse pressure angle at pitch cylinder β = helix angle at ref cylinder βb = helix angle at base cylinder βa = helix angle at tip cylinder εα = transverse contact ratio εβ = overlap ratio εγ = total contact ratio ρa0 = tip radius of tool ref to mn ρfp = root radius of basic rack ref to mn ( = ρa0) ρC = effective radius (mm) of curvature at pitch point ρF = root fillet radius (mm) in the critical section σB = ultimate tensile strength (Nmm2) σy = yield strength resp 02 proof stress (Nmm2)
Index 1 refers to the pinion 2 to the wheel
Index n refers to normal section or virtual spur gear of a heli-cal gear
Index w refers to pitch point
Special additional symbols for bevel gears are as follows
Σ = angle between intersection axis Kϑ = angle modification (Klingelnberg)
m0 = tool module (Klingelnberg) δ = pitch cone angle xsm = tooth thickness modification coefficient (mid-
face) R = pitch cone distance (mm)
Index v refers to the virtual (equivalent) helical cylindrical gear
Index m refers to the midsection of the bevel gear
13 Geometrical Definitions For internal gearing z2 a da2 dw2 d2 and db2 are negative x2 is positive if da2 is increased ie the numeric value is de-creased
The pinion has the smaller number of teeth ie
11
2 ge=zz
u
For calculation of surface durability b is the common face-width on pitch diameter
For tooth strength calculations b1 or b2 are facewidths at the respective tooth roots If b1 or b2 differ much from b above they are not to be taken more than 1 module on either side of b
Cylindrical gears
tan αt = tan αn cos β tan βb = tan β cos αt
tan βa = tan β da d
cos αa = dbda
d = z mn cos β mt = mn cos β db = d cos αt = dw cos αwt
a = 05 (dw1 + dw2)
dw1dw2 = z1 z2
inv α = tan α - α (radians)
inv αwt = inv αt + 2 tan αn (x1 + x2)(z1 + z2)
zn = z (cos2 βb cos β)
1
aw1fw1α T
ξξε
+=
where ξfw1 is to be taken as the smaller of
bull wtfw1 αtanξ =
bull soi1
b1wtfw1 d
dacostan -tanαξ =
bull 1
2wt
a2
b2fw1 z
ztanα
d
dacostan ξ
minus=
and
2
1fw2aw1 z
zξξ = where ξfw2 is calculated as ξfw1
substituting the values for the wheel by the values for the pinion and visa versa
11 z
2πT =
( ) +
sdot+minusminussdot= minus
2sinαρρxhm
2d2d nfpfp1fpnsoi1
21
t
nfpfplfpn2
tanα)sinαρρx(hm
sdot+minusminus
nmsinbπ
β=εβ
(for double helix b is to be taken as the width of one helix)
εy = βα εε +
ρC = ( )2
b
wt
u1βcos
αsinua
+
v = 311 10dn
60π minus
6 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
pbt = βcos
αcosmπ tn
sat =
minus++
at
ninvααinv
z
αtanx22
π
d a
san = aβcossat
14 Bevel Gear Conversion Formulae and Specific Formulae Conversion of bevel gears to virtual equivalent helical cylin-drical gears is based on the bevel gear midsection The con-version formulae are
Number of teeth
zv12 = z12 cos δ12
(δ1 + δ2 = Σ)
Gear ratio
uv = 1v
2vzz
tan αvt = tan αn cos βm
tan βbm = tan βm cos αvt
Base pitch
pbtm = m
vtnmβcos
αcosmπ
Reference pitch diameters
dv12 = 21
21m
cosdδ
Centre distance
av = 05 (dv1 + dv2)
Tip diameters
dva 12 = dv 12 + 2 ham 12
Addenda
for gears with constant addenda (Klingelnberg)
ham 12 = mmn (1 + xm 12)
for gears with variable addenda (Gleason)
ham 12 = ha 12 ndash b2 tan (δa 12 ndash δ12)
(when ha is addendum at outer end and δa is the outer cone angle)
Addendum modification coefficients
xm 12 = mn
12am21am
m2
hh minus
Base circle
dvb 12 = dv 12 cos αvt
Transverse contact ratio)
εα = btmP
αsinadd05dd05 vtv2
2vb2
2va2
1vb2
1va minusminus+minus
Overlap ratio) (theoretical value for bevel gears with no crowning but used as approximations in the calculation pro-cedures)
εβ = nm
mmπβsinb
Total contact ratio)
εγ = 2β
2α εε +
( Note that index laquovraquo is left out in order to combine formu-lae for cylindrical and bevel gears)
Tangential speed at midsection
vmt = 3m11 10dn
60π minus
Effective radius of curvature (normal section)
ρvc = ( )2vbm
vtvv
u1βcosαsinua
+
Length of line of contact
lb = ( )( )( )
1εifε
ε1ε2ε
βcosεb
β2γ
2βα
2γ
bm
α ltminusminusminus
lb = 1εifβcosε
εbβ
bmγ
α ge
15 Nominal Tangential Load Ft Fbt Fmt and Fmbt The nominal tangential load (tangential to the reference cyl-inder with diameter d and perpendicular to an axial plane) is calculated from the nominal (rated) torque T transmitted by the gear set
Cylindrical gears
dT2000
Ft = t
tbt αcos
FF =
Bevel gears
mmt d
T2000F =
vt
mtmbt αcos
FF =
Classification Notes- No 412 7 May 2003
DET NORSKE VERITAS
16 Application Factors KA and KAP The application factor KA accounts for dynamic overloads from sources external to the gearing
It is distinguished between the influence of repetitive cyclic torques KA (161) and the influence of temporary occasional peak torques KAP (162)
Calculations are always to be made with KA In certain cases additional calculations with KAP may be necessary
For gears with a defined load spectrum the calculation with a KA may be replaced by a fatigue damage calculation as given in Appendix A
161 KA For gears designed for long or infinite life at nominal rated torque KA is defined as the ratio between the maximum re-petitive cyclic torque applied to the gear set and nominal rated torque
This definition is suitable for main propulsion gears and most of the auxiliary gears
KA can be determined by measurements or system analysis or may be ruled by conventions (ice classes) (For the pur-pose of a preliminary (but not binding) calculation before KA is determined it is advised to apply either the max values mentioned below or values known from similar plants)
a) For main propulsion gears KA can be taken from the (mandatory) torsional vibration analysis thereby con-sidering all permissible driving conditions) Unless specially agreed the rules do not allow KA in excess of 135 for diesel propulsion) With turbine or electric propulsion KA would normally not exceed 12 However special attention should be given to thrusters that are arranged in such a way that heavy vessel movements andor manoeuvring can cause severe load fluctuations This means eg thrusters positioned far from the rolling axis of vessels that could be susceptible to rolling If leading to propeller air suction the condi-tions may be even worse The above mentioned movements or manoeuvring will result in increased propeller excitation If the thruster is driven by a diesel engine the engine mean torque is limited to 100 However thrusters driven by electric motors can suffer temporary mean torque much above 100 unless a suitable load control system (limiting available e-motor torque) is provided
b) For main propulsion gears with ice class notation (see Rules Pt5 Ch1 Sec J500) KA ice has to be taken as the higher value of the applicable (rule defined) ice shock torque referred to nominal rated torque and the value under a) The Baltic ice class notations refer to a few millions ice shock loads Thus the life factors may be put YN=1 and ZN=12 (except for nitrided gears where ZN = 1 applies) Additionally the calculations with the normal KA (no ice class) are to fulfil the normal requirements For polar ice class notations KA ice applies to all criteria and for long or infinite life
c) For a power take off (PTO) branch from a main propul-sion gear with ice class ice shocks result in negative torques It is assumed that the PTO branch is unloaded when the ice shock load occurs The influence of these reverse shock loads may be taken into account as follows The negative torque (reversed load) expressed by means of an application factor based on rated forward load (T or Ft) is KAreverse = KA ice ndash1 (the minus 1 be-cause no mean torque assumed) KAice to be calculated as in the ice class rules This KAreverse should be used for back flank considera-tions such as pitting and scuffing The influence on tooth bending strength (forward di-rection) may be simplified by using the factor YM = 1 minus 03 middot KAreverse KA
d) For diesel driven auxiliaries KA can be taken from the torsional vibration analysis if available For units where no vibration analysis is required (lt 200 kW) or available it is advised to apply KA as the upper allow-able value 135)
e) For turbine or electro driven auxiliaries the same as for c) applies however the practical upper value is 12
) For diesel driven gears more information on KA for mis-firing and normal driving is given in Appendix B
162 KAP The peak overload factor KAP is defined as the ratio between the temporary occasional peak overload torque and the nominal rated torque
For plants where high temporary occasional peak torques can occur (ie in excess of the above mentioned KA) the gearing (if nitrided) has to be checked with regard to static strength Unless otherwise specified the same safety factors as for infinite life apply
The scuffing safety is to be specially considered whereby the KA applies in connection with the bulk temperature and the KAP applies for the flash temperature calculation and should replace KA in the formulae in 431 432 and 441
KAP can be evaluated from the torsional impact vibration calculation (as required by the rules)
If the overloads have a duration corresponding to several revolutions of the shafts the scuffing safety has to be consid-ered on basis of this overload both with respect to bulk and flash temperature This applies to plants with ice class nota-tions (Baltic and polar) and to plants with prime movers which have high temporary overload capacity such as eg electric motors (provided the driven member can have a con-siderable increase in demand torque as eg azimuth thrusters during manoeuvring)
For plants without additional ice class notation KAP should normally not exceed 15
8 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
163 Frequent overloads For plants where high overloads or shock loads occur regu-larly the influence of this is to be considered by means of cumulative fatigue (see Appendix A)
17 Load Sharing Factor Kγ The load sharing factor Kγ accounts for the maldistribution of load in multiple-path transmissions (dual tandem epicyclic double helix etc) Kγ is defined as the ratio between the max load through an actual path and the evenly shared load
171 General method Kγ mainly depends on accuracy and flexibility of the branches (eg quill shaft planet support external forces etc) and should be considered on basis of measurements or of relevant analysis as eg
Kγ = δ+δ f
δ = total compliance of a branch under full load (assuming even load share) referred to gear mesh
f = minusminusminusminus+++ 23
22
21 fff where f1 f2 etc are the
main individual errors that may contribute to a maldis-tribution between the branches Eg tooth pitch errors planet carrier pitch errors bearing clearance influences etc Compensating effects should also be considered
For double helical gears
An external axial force Fext applied from sources outside the actual gearing (eg thrust via or from a tooth coupling) will cause a maldistribution of forces between the two helices Expressed by a load sharing factor the
βsdot
plusmn=γ tanFF1K
t
ext
If the direction of Fext is known the calculation should be carried out separately for each helix and with the tangential force corrected with the pertinent Kγ If the direction of Fext is unknown both combinations are to be calculated and the higher σH or σF to be used
172 Simplified method If no relevant analysis is available the following may apply
For epicyclic gears
Kγ = 32501 minus+ pln
where npl = number of planets ( gt3 )
For multistage gears with locked paths and gear stages sepa-rated by quill shafts (see figure below)
Figure 10 Locked paths gear
Kγ = ( )φ201+
where φ = quill shaft twist (degrees) under full load
18 Dynamic Factor Kv The dynamic factor Kv accounts for the internally generated dynamic loads
Kv is defined as the ratio between the maximum load that dynamically acts on the tooth flanks and the maximum ex-ternally applied load Ft KA Kγ
In the following 2 different methods (181 and 182) are described In case of controversy between the methods the next following is decisive ie the methods are listed with increasing priority
It is important to observe the limitations for the method in 181 In particular the influence of lateral stiffness of shafts is often underestimated and resonances occur at considerably lower speed than determined in 1811
However for low speed gears with vmiddotz1 lt 300 calculations may be omitted and the dynamic factor simplified to Kv=105
181 Single resonance method For a single stage gear Kv may be determined on basis of the relative proximity (or resonance ratio) N between actual speed n1 and the lowest resonance speed nE1
N = 1E
1
nn
Note that for epicyclic gears n is the relative speed ie the speed that multiplied with z gives the mesh frequency
1811 Determination of critical speed It is not advised to apply this method for multimesh gears for N gt 085 as the influence of higher modes has to be consid-ered see 182 In case of significant lateral shaft flexibility (eg overhung mounted bevel gears) the influence of cou-pled bending and torsional vibrations between pinion and wheel should be considered if N ge 075 see 182
nE1 = redmγc
1zπ
31030 sdot
where
cγ is the actual mesh stiffness per unit facewidth see 111
Classification Notes- No 412 9 May 2003
DET NORSKE VERITAS
For gears with inactive ends of the facewidth as eg due to high crowning or end relief such as often applied for bevel gears the use of cγ in connection with determination of natu-ral frequencies may need correction cγ is defined as stiffness per unit facewidth but when used in connection with the total mesh stiffness it is not as simple as cγmiddotb as only a part of the facewidth is active Such corrections are given in 111
mred is the reduced mass of the gear pair per unit facewidth and referred to the plane of contact
For a single gear stage where no significant inertias are closely connected to neither pinion nor wheel mred is calcu-lated as
mred = 21
21mm
mm+
The individual masses per unit facewidth are calculated as
m12 = 221b
21
)2d(b
I
where I is the polar moment of inertia (kgmm2)
The inertia of bevel gears may be approximated as discs with diameter equal the midface pitch diameter and width equal to b However if the shape of the pinion or wheel body differs much from this idealised cylinder the inertia should be cor-rected accordingly
For all kind of gears if a significant inertia (eg a clutch) is very rigidly connected to the pinion or wheel it should be added to that particular inertia (pinion or wheel) If there is a shaft piece between these inertias the torsional shaft stiffness alters the system into a 3-mass (or more) system This can be calculated as in 182 but also simplified as a 2-mass system calculated with only pinion and wheel masses
1812 Factors used for determination of Kv Non-dimensional gear accuracy dependent parameters
( )bKKF
yfcB
At
pptp
γ
minus=
( )bKKF
yFcBAt
ff
γ
α minus=
Non-dimensional tip relief parameter
bKKFcC
1BγAt
ak sdotsdot
sdotminus=
For gears of quality grade (ISO 1328) Q = 7 or coarser Bk = 1
For gears with Q le 6 and excessive tip relief Bk is limited to max 1
For gears (all quality grades) with tip relief of more than 2middotCeff (see 432) the reduction of αε has to be considered (see 443)
where
fpt = the single pitch deviation (ISO 1328) max of pinion or wheel
Fα = the total profile form deviation (ISO 1328) max of pinion or wheel (Note Fα is pt not available for bevel gears thus use Fα = fpt)
yp and yf = the respective running-in allowances and may be calculated similarly to yα in 112 ie the value of fpt is replaced by Fα for yf
cacute = the single tooth stiffness see 111
Ca = the amount of tip relief see 433 In case of different tip relief on pinion and wheel the value that results in the greater value of Bk is to be used If Ca is zero by design the value of running-in tip relief Cay (see 112) may be used in the above formula
1813 Kv in the subcritical range Cylindrical gears N le 085
Bevel gears N le 075
Kv = 1 + N K
K = Cv1 Bp + Cv2 Bf + Cv3 Bk
Cv1 accounts for the pitch error influence
Cv1 = 032
Cv2 accounts for profile error influence
Cv2 = 034 for γε le 2
Cv2 = 03ε
057
γ minus for γε gt 2
Cv3 accounts for the cyclic mesh stiffness variation
Cv3 = 023 for γε le 2
Cv3 = 156ε
0096
γ minus for γε gt 2
1814 Kv in the main resonance range Cylindrical gears 085 lt N le 115
Bevel gears 075 lt N le 125
Running in this range should preferably be avoided and is only allowed for high precision gears
Kv = 1 + Cv1 Bp + Cv2 Bf + Cv4 Bk
Cv4 accounts for the resonance condition with the cyclic mesh stiffness variation
Cv4 = 090 for γε le 2
Cv4 = 144ε
ε005057
γ
γ
minus
minus for γε gt 2
10 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
1815 Kv in the supercritical range Cylindrical gears N ge 15
Bevel gears N ge 15
Special care should be taken as to influence of higher vibra-tion modes andor influence of coupled bending (ie lateral shaft vibrations) and torsional vibrations between pinion and wheel These influences are not covered by the following approach
Kv = Cv5 Bp + Cv6 Bf + Cv7
Cv5 accounts for the pitch error influence
Cv5 = 047
Cv6 accounts for the profile error influence
Cv6 = 047 for εγ le 2
Cv6 = 174ε
012
γ minus for εγ gt 2
Cv7 relates the maximum externally applied tooth loading to the maximum tooth loading of ideal accurate gears operating in the supercritical speed sector when the circumferential vibration becomes very soft
Cv7 = 075 for εγ le 15
Cv7 = [ ] 8750)2(sin 0125 +minusβεπ for 15 lt εγ le 25
Cv7 = 10 for εγ gt 25
1816 Kv in the intermediate range Cylindrical gears 115 lt N lt 15
Bevel gears 125 lt N lt 15
Comments raised in 1814 and 1815 should be observed
Kv is determined by linear interpolation between Kv for N = 115 respectively 125 and N = 15 as
Cylindrical gears
( ) ( ) ( )[ ]51Nv151Nv51Nvv KK350
N51KK === minussdot
minus
+=
Bevel gears
( ) ( ) ( )[ ]51Nv251Nv51Nvv KK250
N51KK === minussdot
minus
+=
182 Multi-resonance method For high speed gear (vgt40 ms) for multimesh medium speed gears for gears with significant lateral shaft flexibility etc it is advised to determine Kv on basis of relevant dy-namic analysis
Incorporating lateral shaft compliance requires transforma-tion of even a simple pinion-wheel system into a lumped multi-mass system It is advised to incorporate all relevant inertias and torsional shaft stiffnesses into an equivalent (to pinion speed) system Thereby the mesh stiffness appears as an equivalent torsional stiffness
cγ b (db12)2 (Nmrad)
The natural frequencies are found by solving the set of dif-ferential equations (one equation per inertia) Note that for a gear put on a laterally flexible shaft the coupling bending-torsionals is arranged by introducing the gear mass and the lateral stiffness with its relation to the torsional displacement and torque in that shaft
Only the natural frequency (ies) having high relative dis-placement and relative torque through the actual pinion-wheel flexible element need(s) to be considered as critical frequency (ies)
Kv may be determined by means of the method mentioned in 181 thereby using N as the least favourable ratio (in case of more than one pinion-wheel dominated natural frequency) Ie the N-ratio that results in the highest Kv has to be consid-ered
The level of the dynamic factor may also be determined on basis of simulation technique using numeric time integration with relevant tooth stiffness variation and pitchprofile er-rors
19 Face Load Factors KHβ and KFβ The face load factors KHβ for contact stresses and for scuff-ing KFβ for tooth root stresses account for non-uniform load distribution across the facewidth
KHβ is defined as the ratio between the maximum load per unit facewidth and the mean load per unit facewidth
KFβ is defined as the ratio between the maximum tooth root stress per unit facewidth and the mean tooth root stress per unit facewidth The mean tooth root stress relates to the con-sidered facewidth b1 respectively b2
Note that facewidth in this context is the design facewidth b even if the ends are unloaded as often applies to eg bevel gears
The plane of contact is considered
191 Relations between KHβ and KFβ
KFβ = expβHK 2(hb)hb1
1exp++
=
where hb is the ratio tooth heightfacewidth The maximum of h1b1 and h2b2 is to be used but not higher than 13 For double helical gears use only the facewidth of one helix
If the tooth root facewidth (b1 or b2) is considerably wider than b the value of KFβ(1or2) is to be specially considered as it may even exceed KHβ
Eg in pinion-rack lifting systems for jack up rigs where b = b2 asymp mn and b1 asymp 3 mn the typical KHβ asymp KFβ2 asymp 1 and KFβ1 asymp 13
192 Measurement of face load factors Primarily
KFβ may be determined by a number of strain gauges distrib-uted over the facewidth Such strain gauges must be put in
Classification Notes- No 412 11 May 2003
DET NORSKE VERITAS
exactly the same position relative to the root fillet Relations in 191 apply for conversion to KHβ
Secondarily
KHβ may be evaluated by observed contact patterns on vari-ous defined load levels It is imperative that the various test loads are well defined Usually it is also necessary to evalu-ate the elastic deflections Some teeth at each 90 degrees are to be painted with a suitable lacquer Always consider the poorest of the contact patterns
After having run the gear for a suitable time at test load 1 (the lowest) observe the contact pattern with respect to ex-tension over the facewidth Evaluate that KHβ by means of the methods mentioned in this section Proceed in the same way for the next higher test load etc until there is a full face contact pattern From these data the initial mesh misalign-ment (ie without elastic deflections) can be found by ex-trapolation and then also the KHβ at design load can be found by calculation and extrapolation See example
Figure 11 Example of experimental determination of KHβ
It must be considered that inaccurate gears may accumulate a larger observed contact pattern than the actual single mesh to mesh contact patterns This is particularly important for lapped bevel gears Ground or hard metal hobbed bevel gears are assumed to present an accumulated contact pattern that is practically equal the actual single mesh to mesh con-tact patterns As a rough guidance the (observed) accumu-lated contact pattern of lapped bevel gears may be reduced by 10 in order to assess the single mesh to mesh contact pattern which is used in 199
193 Theoretical determination of KHβ The methods described in 193 to 198 may be used for cy-lindrical gears The principles may to some extent also be used for bevel gears but a more practical approach is given in 199
General For gears where the tooth contact pattern cannot be verified during assembly or under load all assumptions are to be well on the safe side
KHβ is to be determined in the plane of contact
The influence parameters considered in this method are
bull mean mesh stiffness cγ (see 111) (if necessary also variable stiffness over b)
bull mean unit load Fmb = Fbt KA Kγ Kvb (for double helical gears see 17 for use of Kγ)
bull misalignment fsh due to elastic deflections of shafts and gear bodies (both pinion and wheel)
bull misalignment fdefl due to elastic deflections of and work-ing positions in bearings
bull misalignment fbe due to bearing clearance tolerances bull misalignment fma due to manufacturing tolerances bull helix modifications as crowning end relief helix correc-
tion bull running in amount yβ (see 112)
In practice several other parameters such as centrifugal ex-pansion thermal expansion housing deflection etc contrib-ute to KHβ However these parameters are not taken into account unless in special cases when being considered as particularly important
When all or most of the am parameters are to be considered the most practical way to determine KHβ is by means of a graphical approach described in 1931
If cγ can be considered constant over the facewidth and no helix modifications apply KHβ can be determined analyti-cally as described in 1932
1931 Graphical method The graphical method utilises the superposition principle and is as follows
bull Calculate the mean mesh deflection Mδ as a function of
γm c and bF see 111
bull Draw a base line with length b and draw up a rectangu-lar with height δM (The area δM b is proportional to the transmitted force)
bull Calculate the elastic deflection fsh in the plane of contact Balance this deflection curve around a zero line so that the areas above and below this zero line are equal
Figure 12 fsh balanced around zero line
bull Superimpose these ordinates of the fsh curve to the previ-ous load distribution curve (The area under this new load distribution curve is still δM b)
bull Calculate the bearing deflections andor working posi-tions in the bearings and evaluate the influence fdefl in the plane of contact This is a straight line and is bal-anced around a zero line as indicated in Fig 14 but with one distinct direction Superimpose these ordinates to the previous load distribution curve
12 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
bull The amount of crowning end relief or helix correction (defined in the plane of contact) is to be balanced around a zero line similarly to fsh
Figure 13 Crowning Cc balanced aound zero line
bull Superimpose these ordinates to the previous load distri-bution curve In case of high crowning etc as eg often applied to bevel gears the new load distribution curve may cross the base line (the real zero line) The result is areas with negative load that is not real as the load in those areas should be zero Thus corrective actions must be made but for practical reasons it may be postponed to after next operation
bull The amount of initial mesh misalignment bema ff + (defined in the plane of contact) is to be bal-
anced around a zero line If the direction of bema ff + is known (due to initial contact check) or if the direction of fbe is known due to design (eg overhang bevel pin-ion) this should be taken into account If direction un-known the influence of bema ff + in both directions as well as equal zero should be considered
Figure 14 fma+fbe in both directions balanced around zero line
Superimpose these ordinates to the previous load distri-bution curve This results in up to 3 different curves of which the one with the highest peak is to be chosen for further evaluation
bull If the chosen load distribution curve crosses the base line (ie mathematically negative load) the curve is to be corrected by adding the negative areas and dividing this with the active facewidth The (constant) ordinates of this rectangular correction area are to be subtracted from the positive part of the load distribution curve It is advisable to check that the area covered under this new load distribution curve is still equal δM b
bull If cγ cannot be considered as constant over b then cor-rect the ordinates of the load distribution curve with the local (on various positions over the facewidth) ratio between local mesh stiffness and average mesh stiffness cγ (average over the active facewidth only) Note that the result is to be a curve that covers the same area δM b as before
bull The influence of running in yβ is to be determined as in 112 whereby the value for Fβx is to be taken as twice the distance between the peak of the load distribution curve and δM
bull Determine
KHβ = M
β
δycurveofpeak minus
1932 Simplified analytical method for cylindrical gears The analytical approach is similar to 1931 but has a more limited application as cγ is assumed constant over the face-width and no helix modification applies
bull Calculate the elastic deflection fsh in the plane of contact Balance this deflection curve around a zero line so that the area above and below this zero line are equal see Fig 12 The max positive ordinate is frac12∆fsh
bull Calculate the initial mesh alignment as Fβx= deflbemash ffff plusmnplusmnplusmn∆ The negative signs may only be used if this is justified andor verified by a contact pattern test Otherwise al-ways use positive signs If a negative sign is justified the value of Fβx is not to be taken less than the largest of each of these elements
bull Calculate the effective mesh misalignment as Fβy = Fβx - yβ (yβ see 112)
bull Determine
2KforF2
bFc1K βH
m
βγγH le+=β
or
2KforF
bFc2K Hβ
m
βγγH gt=β
where cγ as used here is the effective mesh stiffness see 111
194 Determination of fsh fsh is the mesh misalignment due to elastic deflections Usu-ally it is sufficient to consider the combined mesh deflection of the pinion body and shaft and the wheel shaft The calcu-lation is to be made in the plane of contact (of the considered gear mesh) and to consider all forces (incl axial) acting on the shafts Forces from other meshes can be parted into components parallel respectively vertical to the considered plane of contact Forces vertical to this plane of contact have no influence on fsh
It is advised to use following diameters for toothed elements
d + 2 x mn for bending and shear deflection
d + 2 mn (x ndash ha0 + 02) for torsional deflection
Usually fsh is calculated on basis of an evenly distributed load If the analysis of KHβ shows a considerable maldis-
Classification Notes- No 412 13 May 2003
DET NORSKE VERITAS
tribution in term of hard end contact or if it is known by other reasons that there exists a hard end contact the load should be correspondingly distributed when calculating fsh In fact the whole KHβ procedure can be used iteratively 2-3 iterations will be enough even for almost triangular load distributions
195 Determination of fdefl fdefl is the mesh misalignment in the plane of contact due to bearing deflections and working positions (housing deflec-tion may be included if determined)
First the journal working positions in the bearings are to be determined The influence of external moments and forces must be considered This is of special importance for twin pinion single output gears with all 3 shafts in one plane
For rolling bearings fdefl is further determined on basis of the elastic deflection of the bearings An elastic bearing deflec-tion depends on the bearing load and size and number of rolling elements Note that the bearing clearance tolerances are not included here
For fluid film bearings fdefl is further determined on basis of the lift and angular shift of the shafts due to lubrication oil film thickness Note that fbe takes into account the influence of the bearing clearance tolerance
When working positions bearing deflections and oil film lift are combined for all bearings the angular misalignment as projected into the plane of the contact is to be determined fdefl is this angular misalignment (radians) times the face-width
196 Determination of fbe fbe is the mesh misalignment in the plane of contact due to tolerances in bearing clearances In principle fbe and fdefl could be combined But as fdefl can be determined by analy-sis and has a distinct direction and fbe is dependent on toler-ances and in most cases has no distinct direction (ie + toler-ance) it is practicable to separate these two influences
Due to different bearing clearance tolerances in both pinion and wheel shafts the two shaft axis will have an angular mis-alignment in the plane of contact that is superimposed to the working positions determined in 195 fbe is the facewidth times this angular misalignment Note that fbe may have a distinct direction or be given as a + tolerance or a combina-tion of both For combination of + tolerance it is adviced to use
fbe= 22be
21be ff ++plusmn
fbe is particularly important for overhang designs for gears with widely different kinds of bearings on each side and when the bearings have wide tolerances on clearances In general it shall be possible to replace standard bearings with-out causing the real load distribution to exceed the design premises For slow speed gears with journal bearings the expected wear should also be considered
197 Determination of fma fma is the mesh misalignment due to manufacturing toler-ances (helix slope deviation) of pinion fHβ1 wheel fHβ2 and housing bore
For gear without specifically approved requirements to as-sembly control the value of fma is to be determined as
fma= 22Hβ
21Hβ ff +
For gears with specially approved assembly control the value of fma will depend on those specific requirements
198 Comments to various gear types For double helical gears KHβ is to be determined for both helices Usually an even load share between the helices can be assumed If not the calculation is to be made as de-scribed in 171
For planetary gears the free floating sun pinion suffers only twist no bending It must be noted that the total twist is the sum of the twist due to each mesh If the value of 1K γ ne this must be taken into account when calculating the total sun pinion twist (ie twist calculated with the force per mesh without Kγ and multiplied with the number of planets)
When planets are mounted on spherical bearings the mesh misalignments sun-planet respectively planet-annulus will be balanced Ie the misalignment will be the average between the two theoretical individual misalignments The faceload distribution on the flanks of the planets can take full advan-tage of this However as the sun and annulus mesh with several planets with possibly different lead errors the sun and annulus cannot obtain the above mentioned advantage to the full extent
199 Determination of KHβ for bevel gears If a theoretical approach similar to 193-198 is not docu-mented the following may be used
testeff
H Kb
b851851K sdot
minussdot=β
beff b represents the relative active facewidth (regarding lapped gears see 192 last part)
Higher values than beff b = 090 are normally not to be used in the formula
For dual directional gears it may be difficult to obtain a high beff b in both directions In that case the smaller beff b is to be used
Ktest represents the influence of the bearing arrangement shaft stiffness bearing stiffness housing stiffness etc on the faceload distribution and the verification thereof Expected variations in length- and height-wise tooth profile is also accounted for to some extent
a) Ktest = 1 For ground or hard metal hobbed gears with the specified contact pattern verified at full rating or at full torque slow turning at a condition representative for the thermal expansion at normal operation
14 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
It also applies when the bearing arrangementsupport has insignificant elastic deflections and thermal axial expansion However each initial mesh contact must be verified to be within acceptance criteria that are cali-brated against a type test at full load Reproduction of the gear tooth length- and height-wise profile must also be verified This can be made through 3D measurements or by initial contact movements caused by defined axial offsets of the pinion (tolerances to be agreed upon)
b) Ktest = 1 + 04middot(beffbndash06) For designs with possible influence of thermal expansion in the axial direction of the pinion The initial mesh contact verified with low load or spin test where the acceptance criteria are calibrated against a type test at full load
c) Ktest = 12 if mesh is only checked by toolmakerrsquos blue or by spin test contact For gears in this category beffb gt 085 is not to be used in the calculation
110 Transversal Load Distribution Factors KHα and KFα The transverse load distribution factors KHα for contact stresses and for scuffing KFα for tooth root stresses account for the effects of pitch and profile errors on the transversal load distribution between 2 or more pairs of teeth in mesh
The following relations may be used
Cylindrical gears
( )
minus+==
tH
αptγγHαFα F
byfc0409
2ε
KK
valid for 2εγ le
( ) ( )tH
αptγ
γ
γHαFα F
byfcε
1ε20409KK
minusminus+==
valid for 2ε γ gt
where
FtH = Ft KA Kγ Kv KHβ
cγ = See 111
γα = See 112
fpt = Maximum single pitch deviation (microm) of pinion or wheel or maximum total profile form devia-tion Fα of pinion or wheel if this is larger than the maximum single pitch deviation
Note In case of adequate equivalent tip relief adapted to the load half of the above mentioned fpt can be introduced A tip relief is considered adequate when the average of Ca1 and Ca2 is within plusmn40 of the value of Ceff in 432
Limitations of KHα and KFα
If the calculated values for
KFα = KHα lt 1 use KFα = KHα = 10
If the calculated value of KHα gt 2εα
γ
Zε
ε use KHα = 2
εα
γ
Zε
ε
If the calculated value of KFα gt εα
γ
Yεε
use KFα = εα
γ
Yεε
where Yε = αnε
075025+ (for εαn see 331c)
Bevel gears
For ground or hard metal hobbed gears KFα = KHα = 1
For lapped gears KFα = KHα = 11
111 Tooth Stiffness Constants cacute and cγ The tooth stiffness is defined as the load which is necessary to deform one or several meshing gear teeth having 1 mm facewidth by an amount of 1 microm in the plane of contact cacute is the maximum stiffness of a single pair of teeth cγ is the mean value of the mesh stiffness in a transverse plane (brief term mesh stiffness)
Both valid for high unit load (Unit load = Ft middot KA middot Kγb)
Cylindrical gears The real stiffness is a combination of the progressive Hertzian contact stiffness and the linear tooth bending stiff-nesses For high unit loads the Hertzian stiffness has little importance and can be disregarded This approach is on the safe side for determination of KHβ and KHα However for moderate or low loads Kv may be underestimated due to determination of a too high resonance speed
The linear approach is described in A
An optional approach for inclusion of the non-linear stiffness is described in B
A The linear approach
BRCCq
βcos08acutec =
and
( )025ε075cc αγ +prime=
where
( )[ ]n02a01a
B α2000212
hh12051C minusminus
+minus+=
12n1n
x000635z
025791z
015551004723q minus++=
12
2n
22
1n
1 x005290z
x024188x000193z
x011654+minusminusminus
+ 000182 x22
Classification Notes- No 412 15 May 2003
DET NORSKE VERITAS
(for internal gears use zn2 equal infinite and x2 = 0) ha0 = hfp for all practical purposes CR considers the increased flexibility of the wheel teeth if the wheel is not a solid disc and may be calculated as
( )( )nR m5s
sR e5
bbln1C +=
where
bs = thickness of a central web
sR = average thickness of rim (net value from tooth root to inside of rim)
The formula is valid for bs b ge 02 and sRmn ge 1 Outside this range of validity and if the web is not centrally posi-tioned CR has to be specially considered
Note CR is the ratio between the average mesh stiffness over the facewidth and the mesh stiffness of a gear pair of solid discs The local mesh stiffness in way of the web corresponds to the mesh stiffness with CR = 1 The local mesh stiffness where there is no web support will be less than calculated with CR above Thus eg a centrally positioned web will have an effect corresponding to a longitudinal crowning of the teeth See also 1931 regarding KHβ
B The non-linear approach
In the following an example is given on how to consider the non-linearity
The relation between unit load Fb as a function of mesh deflection δ is assumed to be a progressive curve up to 500 Nmm and from there on a straight line This straight line when extended to the baseline is assumed to intersect at 10microm
With these assumptions the unit force Fb as a function of mesh deflection δ can be expressed as
( )10δKbF
minus= for 500bFgt
minus=
500Fb10δK
bF for 500
bFlt
with γAt KK
bF
bF
sdotsdot= etc (Nmm) ie unit load incorporat-
ing the relevant factors as
KA middot Kγ for determination of Kv
KA middot Kγ middot Kv for determination of KHβ
KA middot Kγ middot Kv middot KHβ for determination of KHα
δ = mesh deflection (microm)
K = applicable stiffness (c or cγ)
Use of stiffnesses for KV KHβ and KHα
For calculation of Kv and KHα the stiffness is calculated as follows
When Fb lt 500
the stiffness is determined as δ∆
∆ bF
where the increment is chosen as eg ∆ Fb = 10 and thus
50010Fb10
K10Fb∆δ +
++
=
When F b gt 500 the stiffness is c or cγ For calculation of KHβ the mesh deflection δ is used directly
or an equivalent stiffness determined as δsdotb
F
Bevel gears
In lack of more detailed relationship between stiffness and geometry the following may be used
b085
b13cacute eff=
b085b
16c effγ =
beff not to be used in excess of 085 b in these formulae
Bevel gears with heightwise and lengthwise crowning have progressive mesh stiffness The values mentioned above are only valid for high loads They should not be used for de-termination of Ceff (see 432) or KHβ (see 1931)
112 Running-in Allowances The running-in allowances account for the influence of run-ning-in wear on the various error elements
yα respectively yβ are the running-in amounts which reduce the influence of pitch and profile errors respectively influ-ence of localised faceload
Cay is defined as the running-in amount that compensates for lack of tip relief
The following relations may be used
For not surface hardened steel
ptHlim
α fσ160y =
yβ = βxlimH
f320σ
with the following upper limits
V lt 5 ms 5-10 ms gt 10 ms
yα max none
limH
12800σ limH
6400σ
yβ max none
limH
25600σ limH
12800σ
For surface hardened steel
yα = 0075 fpt but not more than 3 for any speed
yβ = 015 Fβx but not more than 6 for any speed
16 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
For all kinds of steel
5145189718
1C2
limHay +
minusσ
=
When pinion and wheel material differ the following ap-plies
bull Use the larger of fpt1 - yα1 and fpt2 - yα2 to replace fpt - yα in the calculation of KHα and Kv
bull Use ( )2β1ββ yy21y += in the calculation of KHβ
bull Use ( )2ay1aya CC21C += in the calculation of Kv
bull Use ( )2ay1ay2a1a CC21CC +== in the scuffing
calculation if no design tip relief is foreseen
Classification Notes- No 412 17 May 2003
DET NORSKE VERITAS
2 Calculation of Surface Durability
21 Scope and General Remarks Part 2 includes the calculations of flank surface durability as limited by pitting spalling case crushing and subsurface yielding Endurance and time limited flank surface fatigue is calculated by means of 22 ndash 212 In a way also tooth frac-tures starting from the flank due to subsurface fatigue is in-cluded through the criteria in 213
Pitting itself is not considered as a critical damage for slow speed gears However pits can create a severe notch effect that may result in tooth breakage This is particularly im-portant for surface hardened teeth but also for high strength through hardened teeth For high-speed gears pitting is not permitted
Spalling and case crushing are considered similar to pitting but may have a more severe effect on tooth breakage due to the larger material breakouts initiated below the surface Subsurface fatigue is considered in 213
For jacking gears (self-elevating offshore units) or similar slow speed gears designed for very limited life the max static (or very slow running) surface load for surface hard-ened flanks is limited by the subsurface yield strength
For case hardened gears operating with relatively thin lubri-cation oil films grey staining (micropitting) may be the lim-iting criterion for the gear rating Specific calculation meth-ods for this purpose are not given here but are under consid-eration for future revisions Thus depending on experience with similar gear designs limitations on surface durability rating other than those according to 22 - 213 may be ap-plied
22 Basic Equations Calculation of surface durability (pitting) for spur gears is based on the contact stress at the inner point of single pair contact or the contact at the pitch point whichever is greater
Calculation of surface durability for helical gears is based on the contact stress at the pitch point
For helical gears with 0 lt εβ lt 1 a linear interpolation be-tween the above mentioned applies
Calculation of surface durability for spiral bevel gears is based on the contact stress at the midpoint of the zone of contact
Alternatively for bevel gears the contact stress may be cal-culated with the program ldquoBECALrdquo In that case KA and Kv are to be included in the applied tooth force but not KHβ and KHα The calculated (real) Hertzian stresses are to be multi-plied with ZK in order to be comparable with the permissible contact stresses
The contact stresses calculated with the method in part 2 are based on the Hertzian theory but do not always represent the real Hertzian stresses
The corresponding permissible contact stresses σHP are to be calculated for both pinion and wheel
221 Contact stress Cylindrical gears
( )HαHβvγA
1
tβεEHDBH KKKKK
bud1uF
ZZZZZσ+
=
where
ZBD = Zone factor for inner point of single pair contact for pinion resp wheel (see 232)
ZH = Zone factor for pitch point (see 231)
ZE = Elasticity factor (see 24)
Zε = Contact ratio factor (see 25)
Zβ = Helix angle factor (see 26)
Ft KA Kγ Kv KHβ KHα see 15 ndash 110
d1 b u see 12 ndash 15
Bevel gears
( )HαHβvγA
v1v
vmtKEMH KKKKK
bud1uF
ZZZ105σ+
sdot=
where
105 is a correlation factor to reach real Hertzian stresses (when ZK = 1)
ZE KA etc see above
ZM = mid-zone factor see 233
ZK = bevel gear factor see 27
Fmt dv1 uv see 12 ndash 15
It is assumed that the heightwise crowning is chosen so as to result in the maximum contact stresses at or near the mid-point of the flanks
222 Permissible contact stress
XWRvLH
NHlimHP ZZZZZ
SZσ
σ =
where
σH lim = Endurance limit for contact stresses (see 28)
ZN = Life factor for contact stresses (see 29)
SH = Required safety factor according to the rules
ZLZvZR = Oil film influence factors (see 210)
ZW = Work hardening factor (see 211)
ZX = Size factor (see 212)
18 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
23 Zone Factors ZH ZBD and ZM
231 Zone factor ZH The zone factor ZH accounts for the influence on contact stresses of the tooth flank curvature at the pitch point and converts the tangential force at the reference cylinder to the normal force at the pitch cylinder
wtt2
wtbH sinααcos
cosαcosβ2Z =
232 Zone factors ZBD The zone factors ZBD account for the influence on contact stresses of the tooth flank curvature at the inner point of sin-gle pair contact in relation to ZH Index B refers to pinion D to wheel
For εβ ge 1 ZBD = 1
For internal gears ZD = 1
For εβ = 0 (spur gears)
( )
π
minusεminusminus
π
minusminus
α=
α2
2
2b
2a
1
2
1b
1a
wtB
z211
dd
z21
dd
tanZ
( )
π
minusεminusminus
π
minusminus
α=
α1
2
1b
1a
2
2
2b
2a
wtD
z211
dd
z21
dd
tanZ
If ZB lt 1 use ZB = 1
If ZD lt 1 use ZD = 1
For 0 lt εβ lt 1
ZBD = ZBD (for spur gears) ndash εβ (ZBD (for spur gears) ndash 1)
233 Zone factor ZM The mid-zone factor ZM accounts for the influence of the contact stress at the mid point of the flank and applies to spi-ral bevel gears
εminusminus
εminusminus
αβ=
αα btm2
2vb2
2vabtm2
1vb2val
2v1vvtbmM
pddpdd
ddtancos2Z
This factor is the product of ZH and ZM-B in ISO 10300 with the condition that the heightwise crowning is sufficient to move the peak load towards the midpoint
234 Inner contact point For cylindrical or bevel gears with very low number of teeth the inner contact point (A) may be close to the base circle In order to avoid a wear edge near A it is required to have suitable tip relief on the wheel
24 Elasticity Factor ZE The elasticity factor ZE accounts for the influence of the material properties as modulus of elasticity and Poissonrsquos ratio on the contact stresses
For steel against steel ZE = 1898
25 Contact Ratio Factor Zε The contact ratio factor Zε accounts for the influence of the transverse contact ratio εα and the overlap ratio εβ on the contact stresses
αε1Z =ε for 1εβ ge
( )α
ββ
αε ε
εε1
3ε4
Z +minusminus
= for εβ lt 1
26 Helix Angle Factor Zβ The helix angle factor Zβ accounts for the influence of helix angle (independent of its influence on Zε) on the surface du-rability
cosβZβ =
27 Bevel Gear Factor ZK The bevel gear factor accounts for the difference between the real Hertzian stresses in spiral bevel gears and the contact stresses assumed responsible for surface fatigue (pitting) ZK adjusts the contact stresses in such a way that the same per-missible stresses as for cylindrical gears may apply
The following may be used ZK = 080
28 Values of Endurance Limit σHlim and Static Strength 5H10σ 3H10σ
σHlim is the limit of contact stress that may be sustained for 5middot107 cycles without the occurrence of progressive pitting
For most materials 5middot107 cycles are considered to be the be-ginning of the endurance strength range or lower knee of the σ-N curve (See also Life Factor ZN) However for nitrided steels 2middot106 apply
For this purpose pitting is defined by
bull for not surface hardened gears pitted area ge 2 of total active flank area
bull for surface hardened gears pitted area ge 05 of total active flank area or ge 4 of one particular tooth flank area 510Hσ and 310Hσ are the contact stresses which the given
material can withstand for 105 respectively 103 cycles without subsurface yielding or flank damages as pitting spalling or case crushing when adequate case depth applies
The following listed values for σHlim 510Hσ and 310Hσ may only be used for materials subjected to a quality control as
Classification Notes- No 412 19 May 2003
DET NORSKE VERITAS
the one referred to in the rules Results of approved fatigue tests may also be used as the basis for establishing these values The defined survival probability is 99
σHlim σH105 σH10
3
Alloyed case hardened steels (surface hardness 58-63 HRC) - of specially approved high grade - of normal grade
1650 1500
2500 2400
3100 3100
Nitrided steel of approved grade gas nitrided (surface hardness 700-800 HV) 1250 13 σHlim 13 σHlim
Alloyed quenched and tempered steel bath or gas nitrided (surface hardness 500-700 HV)
1000
13 σHlim
13 σHlim
Alloyed flame or induction hardened steel (surface hardness 500-650 HV) 075 HV + 750 16 σHlim 45 HV
Alloyed quenched and tempered steel 14 HV + 350 16 σHlim 45 HV
Carbon steel 15 HV + 250 16 σHlim 16 σHlim
These values refer to forged or hot rolled steel For cast steel the values for σHlim are to be reduced by 15
29 Life Factor ZN The life factor ZN takes account of a higher permissible contact stress if only limited life (number of cycles NL) is demanded or lower permissible contact stress if very high number of cycles apply
If this is not documented by approved fatigue tests the fol-lowing method may be used
For all steels except nitrided
7L 105N sdotge ZN = 1 or
01570
L
7
N N105Z
sdot=
Ie ZN = 092 for 1010 cycles
The ZN = 1 from 5middot107 on may only be used when the mate-rial cleanliness is of approved high grade (see Rules Pt4 Ch2) and the lubrication is optimised by a specially approved filtering process
105 lt NL lt 5middot107 510N
logZ037
L
7
N N105Z
sdot=
NL = 105 WXRVLHlim
WstX10H1010NN ZZZZZσ
ZZσZZ
55
5=
103 lt NL lt 105 )Z(Zlog05
L
5
10NN
5N103N10
5N10ZZ
==
3L 10N le
WXRVLlimH
Wst10X10H10NN ZZZZZ
ZZZZ
33
3σ
σ==
(but not less than ZN105)
For nitrided steels
6L 102N sdotge ZN = 1 or
00980
L
6
N N102Z
sdot=
Ie ZN = 092 for 1010 cycles
The ZN = 1 from 2middot106 on may only be used when the mate-rial cleanliness is of approved high grade (see Rules Pt4 Ch2) and the lubrication is optimised by a specially approved filtering process
105 lt NL lt 2middot106 510N
Zlog76860
L
6
N N102Z
sdot=
5L 10N le
XWRVL
10XWst10NN ZZZZZ
ZZ13ZZ
55 ==
Note that when no index indicating number of cycles is used the factors are valid for 5middot107 (respectively 2middot106 for nitrid-ing) cycles
210 Influence Factors on Lubrication Film ZL ZV and ZR The lubricant factor ZL accounts for the influence of the type of lubricant and its viscosity the speed factor ZV ac-counts for the influence of the pitch line velocity and the roughness factor ZR accounts for influence of the surface roughness on the surface endurance capacity
The following methods may be applied in connection with the endurance limit
20 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
Surface hardened steels Not surface hardened steels
ZL ( )24013421
360910ν+
+
( )24013421
680830ν+
+
ZV ( )v3280
140930+
+ ( )v3280300850
++
ZR
080
ZrelR3
150
ZrelR3
where
ν40 = Kinematic oil viscosity at 40ordmC (mm2s) For case hardened steels the influence of a high bulk temperature (see 4 Scuffing) should be con-sidered Eg bulk temperatures in excess of 120ordmC for long periods may cause reduced flank surface endurance limits
For values of ν40 gt 500 use ν40 = 500
RZrel = The mean roughness between pinion and wheel (after running in) relative to an equivalent radius of curvature at the pitch point ρc = 10mm
RZrel
= ( ) 31
cZ2Z1 ρ
10RR50
+
RZ = Mean peak to valley roughness (microm) (DIN defini-tion) (roughly RZ = 6 Ra)
For 5
L 10N le ZL ZV ZR = 10
211 Work Hardening Factor ZW The work hardening factor ZW accounts for the increase of surface durability of a soft steel gear when meshing the soft steel gear with a surface hardened or substantially harder gear with a smooth surface
The following approximation may be used for the endurance limit
Surface hardened steel against not surface hardened steel 150
ZeqW R
31700
130HB21Z
minus
minus=
where HB = the Brinell hardness of the soft member For HB gt 470 use HB = 470 For HB lt 130 use HB = 130
RZeq = equivalent roughness
033
c40
066
ZS
ZHZHZEQ ρvν
15000RRRR
=
If RZeq gt 16 then use RZeq = 16
If RZeq lt 15 then use RZeq = 15
where
RZH = surface roughness of the hard member before run in
RZS = surface roughness of the soft member before run in
ν40 = see 210
If values of ZW lt 1 are evaluated ZW = 1 should be used for flank endurance However the low value for ZW may indi-cate a potential wear problem
Through hardened pinion against softer wheel
( )
minussdotminus+= 000829
HBHB0008981u1Z
2
1W
For 21HBHB
2
1 le use ZW = 1
For 71HBHB
2
1 gt use 17HBHB
2
1 =
For u gt 20 use u = 20
For static strength (lt 105 cycles)
Surface hardened against not surface hardened
ZWst = 105
Through hardened pinion against softer wheel
ZWst = 1
212 Size Factor ZX The size factor accounts for statistics indicating that the stress levels at which fatigue damage occurs decrease with an increase of component size as a consequence of the influ-ence on subsurface defects combined with small stress gradi-ents and of the influence of size on material quality
ZX may be taken unity provided that subsurface fatigue for surface hardened pinions and wheels is considered eg as in the following subsection 213
213 Subsurface Fatigue This is only applicable to surface hardened pinions and wheels The main objective is to have a subsurface safety against fatigue (endurance limit) or deformation (static strength) which is at least as high as the safety SH required for the surface The following method may be used as an approximation unless otherwise documented
The high cycle fatigue (gt3106 cycles) is assumed to mainly depend on the orthogonal shear stresses Static strength (lt103 cycles) is assumed to depend mainly on equivalent
Classification Notes- No 412 21 May 2003
DET NORSKE VERITAS
stresses (von Mises) Both are influenced by residual stresses but this is only considered roughly and empirically
The subsurface working stresses at depths inside the peak of the orthogonal shear stresses respectively the equivalent stresses are only dependent on the (real) Hertzian stresses Surface related conditions as expressed by ZL ZV and ZR are assumed to have a negligible influence
The real Hertzian stresses σHR are determined as
For helical gears with εβ gt 1
σHR = σH
For helical gears with εβ lt 1 and spur gears
ε
α
ββ ε
ε+εminus
sdotσ=σZ
1
HHR
For bevel gears
KHHR Z
1σσ sdot=
The necessary hardness HV is given as a function of the net depth tz (net = after grinding or hard metal hobbing and per-pendicular to the flank)
The coordinates tz and HV are to be compared with the de-sign specification such as
bull for flame and induction hardening tHVmin HVmin bull for nitriding t400min HV = 400 bull for case hardening t550min HV = 550 t400min HV = 400
and t300min HV = 300 (the latter only if the core hardness lt 300 If the core hardness gt 400 the t400 is to be re-placed by a fictive t400 = 16 t550)
In addition the specified surface hardness is not to be less than the max necessary hardness (at tz = 05aH) This applies to all hardening methods
For high cycle fatigue (gt3 106 cycles) the following applies
sdot+
minussdotsdotsdot= o90
05azt
05azt
cosSσ04HV
H
HHHR
applicable to 05a
zt
Hge
For 05at
H
z lt the value for 05at
H
z = applies
56300ρSσ12a cHHR
Hsdotsdot
sdot=
Where aH is half the hertzian contact width multiplied by an empirical factor of 12 that takes into account the possible influence of reduced compressive residual stresses (or even tensile residual stresses) on the local fatigue strength
If any of the specified hardness depths including the surface hardness is below the curve described by HV = f (tz) the actual safety factor against subsurface fatigue is determined as follows
reduce SH stepwise in the formula for HV and aH until all specified hardness depths and surface hardness balance with the corrected curve The safety factor obtained through this method is the safety against subsurface fatigue
For static strength (lt103 cycles) the following applies
sdot+
minussdotsdotsdot= o90
07at
06a
zt
cosSσ019HV
Hst
z
HstHHR
applicable to 06at
Hst
z ge
56300ρSσa cHHR
Hstsdotsdot
=
In the case of insufficient specified hardness depths the same procedure for determination of the actual safety factor as above applies
For limited life fatigue (103 lt cycles lt 3106 )
For this purpose it is necessary to extend the correction of safety factors to include also higher values than required Ie in the case of more than sufficient hardness and depths the safety factor in the formulae for both high cycle fatigue and static strength are to be increased until necessary and speci-fied values balance
The actual safety factor for a given number of cycles N be-tween 103 and 3106 is found by linear interpolation in a dou-ble logarithmic diagram
minussdotminus
= sdot logN3477
logSlogSlogS
310H6103HHN
36 10H103H Slog86281Slog86280 sdot+sdot sdot
22 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
3 Calculation of Tooth Strength
31 Scope and General Remarks Part 3 include the calculation of tooth root strength as limited by tooth root cracking (surface or subsurface initiated) and yielding
For rim thickness sR ge 35middotmn the strength is calculated by means of 32 ndash 313 For cylindrical gears the calculation is based on the assumption that the highest tooth root tensile stress arises by application of the force at the outer point of single tooth pair contact of the virtual spur gears The method has however a few limitations that are mentioned in 36
For bevel gears the calculation is based on force application at the tooth tip of the virtual cylindrical gear Subsequently the stress is converted to load application at the mid point of the flank due to the heightwise crowning
Bevel gears may also be calculated with the program BECAL In that case KA and Kv are to be included in the applied tooth force but not KFβ and KFα
In case of a thin annulus or a thin gear rim etc radial crack-ing can occur rather than tangential cracking (from root fillet to root fillet) Cracking can also start from the compression fillet rather than the tension fillet For rim thickness sR lt 35middotmn a special calculation procedure is given in 315 and 316 and a simplified procedure in 314
A tooth breakage is often the end of the life of a gear trans-mission Therefore a high safety SF against breakage is re-quired
It should be noted that this part 3 does not cover fractures caused by
bull oil holes in the tooth root space bull wear steps on the flank bull flank surface distress such as pits spalls or grey staining
Especially the latter is known to cause oblique fractures starting from the active flank predominately in spiral bevel gears but also sometimes in cylindrical gears
Specific calculation methods for these purposes are not given here but are under consideration for future revisions Thus depending on experience with similar gear designs limita-tions other than those outlined in part 3 may be applied
32 Tooth Root Stresses The local tooth root stress is defined as the max principal stress in the tooth root caused by application of the tooth force Ie the stress ratio R = 0 Other stress ratios such as for eg idler gears (R asymp -12) shrunk on gear rims (R gt 0) etc are considered by correcting the permissible stress level
321 Local tooth root stress The local tooth root stress for pinion and wheel may be as-sessed by strain gauge measurements or FE calculations or similar For both measurements and calculations all details are to be agreed in advance
Normally the stresses for pinion and wheel are calculated as
Cylindrical gears
FαFβvγAβSFn
tF K K K K K Y Y Y
m bFσ =
where
YF = Tooth form factor (see 33)
YS = Stress correction factor (see 34)
Yβ = Helix angle factor (see 36)
Ft KA Kγ Kv KFβ KFα see 15 ndash 110
b see 13
Bevel gears
FαFβvγASaFamn
mtF K K K K K Y Y Y
m bFσ ε=
where
YFa = Tooth form factor see 33
YSa = Stress correction factor see 34
Yε = Contact ratio factor see 35
Fmt KA etc see 15 ndash 110
b see 13
322 Permissible tooth root stress The permissible local tooth root stress for pinion respectively wheel for a given number of cycles N is
CXRrelTrelTF
NMFEFP YYYY
SYY
δσ
=σ
Note that all these factors YM etc are applicable to 3middot106 cycles when used in this formula for σFP The influence of other number of cycles on these factors is covered by the calculation of YN
where
σFE = Local tooth root bending endurance limit of reference test gear (see 37)
YM = Mean stress influence factor which accounts for other loads than constant load direction eg idler gears temporary change of load di-rection pre-stress due to shrinkage etc (see 38)
YN = Life factor for tooth root stresses related to reference test gear dimensions (see 39)
SF = Required safety factor according to the rules
YδrelT = Relative notch sensitivity factor of the gear to be determined related to the reference test gear (see 310)
YRrelT = Relative (root fillet) surface condition factor of the gear to be determined related to the reference test gear (see 311)
Classification Notes- No 412 23 May 2003
DET NORSKE VERITAS
YX = Size factor (see 312)
YC = Case depth factor (see 313)
33 Tooth Form Factors YF YFa The tooth form factors YF and YFa take into account the in-fluence of the tooth form on the nominal bending stress
YF applies to load application at the outer point of single tooth pair contact of the virtual spur gear pair and is used for cylindrical gears
YFa applies to load application at the tooth tip and is used for bevel gears
Both YF and YFa are based on the distance between the con-tact points of the 30˚-tangents at the root fillet of the tooth profile for external gears respectively 60˚ tangents for inter-nal gears
Figure 31 External tooth in normal section
Figure 32 Internal tooth in normal section
Definitions
n
2
n
Fn
enFn
Fe
F
α cosms
α cosmh6
Y
=
n
2
n
Fn
anFn
Fa
Fa
α cosms
α cosmh6
Y
=
In the case of helical gears YF and YFa are determined in the normal section ie for a virtual number of teeth
YFa differs from YF by the bending moment arm hFa and αFan and can be determined by the same procedure as YF with exception of hFe and αFan For hFa and αFan all indices e will change to a (tip)
The following formulae apply to cylindrical gears but may also be used for bevel gears when replacing
mn with mnm
zn with zvn
αt with αvt
β with βm
with undercut without undercut
Fig 33 Dimensions and basic rack profile of the teeth (finished profile)
Tool and basic rack data such as hfP ρfp and spr etc are re-ferred to mn ie dimensionless
331 Determination of parameters
( )n
n
prnfPnfP m
α cossαsin 1ρ
αtan h4πE
minusminusminusminus=
For external gears fPfP ρρ =
For internal gears ( )0z
195fPfP0
fPfP 10363156ρhxρρ
sdotminus+
+=
where
z0 = number of teeth of pinion cutter
x0 = addendum modification coefficient of pinion cutter
hfP = addendum of pinion cutter
ρfP = tip radius of pinion cutter
xhρG fpfp +minus=
24 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
τmE
2π
z2H
nnminus
minus=
with
3πτ = for external gears
6πτ = for internal gears
Htan zG2
nminusϑ=ϑ
(to be solved iteratively suitable start value6π
=ϑ for exter-
nal gears and 3π for internal gears)
a) Tooth root chord sFn For external gears
minus
ϑ+
ϑminus= ρ
cosG3
3πsinz
ms
fpnn
Fn
For bevel gears with a tooth thickness modification
xsm affects mainly sFn but also hFe and αFen The total influence of xsm on YFa Ysa can be approximated by only adding 2 xsm to sFn mn
For internal gears
minus
ϑ+
ϑminus= ρ
cosG
6πsinz
ms
fPnn
Fn
b) Root fillet radius ρF at 30ordm tangent
( )G2coszcosG2ρ
mρ
2n
2
fpn
F
minusϑϑ+=
c) Determination of bending moment arm hF dn = zn mn
b2α
αn βcosε
ε =
dan = dn + 2 ha
pbn = π mn cos αn
dbn = dn cos αn
( )4
d1εpzz
2dd
zz2d
2bn
2
αnbn
2bn
2an
en +
minusminus
minus=
en
bnen d
dcos arcα =
ennnn
e α invα invα x tan 22π
z1
minus+
+=γ
αFen = αen ndash γe
For external gears
( )
minus=
n
enFenee
n
Fe
mdαtan sin γ γcos
21
mh
]ρcos
G3πcosz fpn +
ϑminus
ϑminusminus
For internal gears
( ) minus
sdotsdotminus=
n
enFenee
n
Fe
mdα tanγ sinγ cos
21
mh
minus
ϑminus
ϑminussdot ρ
cosG3
6πcosz fPn
332 Gearing with εαn gt 2 For deep tooth form gearing ( )25ε2 αn lele produced with a verified grade of accuracy of 4 or better and with applied profile modification to obtain a trapezoidal load distribution along the path of contact the YF may be corrected by the factor YDT as
250ε205for 0666ε2366Y αnαnDT leleminus=
205εfor 10Y αnDT lt=
34 Stress Correction Factors YS YSa The stress correction factors YS and YSa take into account the conversion of the nominal bending stress to the local tooth root stress Thereby YS and YSa cover the stress increasing effect of the notch (fillet) and the fact that not only bending stresses arise at the root A part of the local stress is inde-pendent of the bending moment arm This part increases the more the decisive point of load application approaches the critical tooth root section
Therefore in addition to its dependence on the notch radius the stress correction is also dependent on the position of the load application ie the size of the bending moment arm
YS applies to the load application at the outer point of single tooth pair contact YSa to the load application at tooth tip
YS can be determined as follows
( )
+
+=L23121
1
sS qL01312Y
where Fe
Fn
hsL = and
F
Fns ρ2
sq = (see 33)
YSa can be calculated by replacing hFe with hFa in the above formulae
Note a) Range of validity 1 lt qs lt 8
In case of sharper root radii (ie produced with tools having too sharp tip radii) YS resp YSa must be specially considered
b) In case of grinding notches (due to insufficient protuberance of the hob) YS resp YSa can rise considerably and must be multiplied with
Classification Notes- No 412 25 May 2003
DET NORSKE VERITAS
g
g
ρt
0613
13
minus
where
tg = depth of the grinding notch
ρg = radius of the grinding notch
c) The formulae for YS resp YSa are only valid for αn = 20˚ However the same formulae can be used as a safe approximation for other pressure angles
35 Contact Ratio Factor Yε The contact ratio factor Yε covers the conversion from load application at the tooth tip to the load application at the mid point of the flank (heightwise) for bevel gears
The following may be used
Yε = 0625
36 Helix Angle Factor Yβ The helix angle factor Yβ takes into account the difference between the helical gear and the virtual spur gear in the nor-mal section on which the calculation is based in the first step In this way it is accounted for that the conditions for tooth root stresses are more favourable because the lines of contact are sloping over the flank
The following may be used (β input in degrees)
Yβ = 1 ndash εβ β120
When εβ gt 1 use εβ = 1 and when β gt 30deg use β = 30deg in the formula
However the above equation for Yβ may only be used for gears with β gt 25deg if adequate tip relief is applied to both pinion and wheel (adequate = at least 05 middot Ceff see 432)
37 Values of Endurance Limit σFE σFE is the local tooth root stress (max principal) which the material can endure permanently with 99 survival prob-ability 3106 load cycles is regarded as the beginning of the endurance limit or the lower knee of the σ ndash N curve σFE is defined as the unidirectional pulsating stress with a minimum stress of zero (disregarding residual stresses due to heat treatment) Other stress conditions such as alternating or pre-stressed etc are covered by the conversion factor YM
σFE can be found by pulsating tests or gear running tests for any material in any condition If the approval of the gear is to be based on the results of such tests all details on the testing conditions have to be approved by the Society Fur-ther the tests may have to be made under the Societys su-pervision
If no fatigue tests are available the following listed values for σFE may be used for materials subjected to a quality con-trol as the one referred to in the rules
σFE
Alloyed case hardened steels 1) (fillet surface hardness 58 ndash 63 HRC)
bull of specially approved high grade
1050
bull of normal grade
minus CrNiMo steels with approved process
1000
minus CrNi and CrNiMo steels generally 920 minus MnCr steels generally 850
Nitriding steel of approved grade quenched tempered and gas nitrided (surface hardness 700 ndash 800 HV)
840
Alloyed quenched and tempered steel bath or gas nitrided (surface hardness 500 ndash 700 HV)
720
Alloyed quenched and tempered steel flame or induction hardened 2) (incl entire root fillet) (fillet surface hardness 500 ndash 650 HV)
07 HV + 300
Alloyed quenched and tempered steel flame or induction hardened (excl entire root fillet) (σB = uts of base material)
025 σB + 125
Alloyed quenched and tempered steel 04 σB + 200
Carbon steel 025 σB + 250
Note All numbers given above are valid for separate forgings and for blanks cut from bars forged according to a qualified procedure see Pt 4 Ch 2 Sec 3 For rolled steel the values are to be reduced with 10 For blanks cut from forged bars that are not qualified as mentioned above the values are to be reduced with 20 For cast steel reduce with 40
1) These values are valid for a root radius
bull being unground If however any grinding is made in the root fillet area in such a way that the residual stresses may be affected σFE is to be reduced by 20 (If the grinding also leaves a notch see 34)
bull with fillet surface hardness 58 ndash 63 HRC In case of lower surface hardness than 58 HRC σFE is to be reduced with 20(58 ndash HRC) where HRC is the detected hardness (This may lead to a permissible tooth root stress that varies along the facewidth If so the actual tooth root stresses may also be considered along facewidth)
bull not being shot peened In case of approved shot peening σFE may be increased by 200 for gears where σFE is reduced by 20 due to root grinding Otherwise σFE may be increased by 100 for
6nm le and 100 ndash 5 (mn - 6) for mn gt 6 However the possible adverse influence on the flanks regarding grey staining should be considered and if necessary the flanks should be masked
2) The fillet is not to be ground after surface hardening Regarding possible root grinding see 1)
26 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
38 Mean stress influence Factor YM The mean stress influence factor YM takes into account the influence of other working stress conditions than pure pulsa-tions (R = 0) such as eg load reversals idler gears planets and shrink-fitted gears
YM (YMst) are defined as the ratio between the endurance (or static) strength with a stress ratio R ne 0 and the endurance (or static) strength with R = 0
YM and YMst apply only to a calculation method that assesses the positive (tensile) stresses and is therefore suitable for comparison between the calculated (positive) working stress σF and the permissible stress σFP calculated with YM or YMst
For thin rings (annulus) in epicyclic gears where the com-pression fillet may be decisive special considerations apply see 316
The following method may be used within a stress ratio ndash12 lt R lt 05
381 For idlers planets and PTO with ice class
M1M1R1
1Yor Y MstM
+minus
minus=
where
R = stress ratio = min stress divided by max stress
For designs with the same force applied on both forward- and back-flank R may be assumed to ndash 12
For designs with considerably different forces on forward- and back-flank such as eg a marine propulsion wheel with a power take off pinion R may be assessed as
branchmain theoffacewidth unit per forcepto offacewidt unit per force21minus
For a power take off (PTO) with ice class see 161 c
M considers the mean stress influence on the endurance (or static) strength amplitudes
M is defined as the reduction of the endurance strength am-plitude for a certain increase of the mean stress divided by that increase of the mean stress
Following M values may be used
Endurance limit
Static strength
Case hardened 08 ndash 015 Ys 1) 07
If shot peened 04 06
Nitrided 03 03
Induction or flame hardened
04 06
Not surface hardened steel 03 05
Cast steels 04 06 1)For bevel gears use Ys=2 for determination of M
The listed M values for the endurance limit are independent of the fillet shape (Ys) except for case hardening In princi-ple there is a dependency but wide variations usually only occur for case hardening eg smooth semicircular fillets versus grinding notches
382 For gears with periodical change of rotational direction For case hardened gears with full load applied periodically in both directions such as side thrusters the same formula for YM as for idlers (with R = ndash 12) may be used together with the M values for endurance limit This simplified approach is valid when the number of changes of direction exceeds 100 and the total number of load cycles exceeds 3middot106
For gears of other materials YM will normally be higher than for a pure idler provided the number of changes of direction is below 3middot106 A linear interpolation in a diagram with loga-rithmic number of changes of direction may be used ie from YM = 09 with one change to YM (idler) for 3middot106 changes This is applicable to YM for endurance limit For static strength use YM as for idlers
For gears with occasional full load in reversed direction such as the main wheel in a reversing gear box YM = 09 may be used
383 For gears with shrinkage stresses and unidirectional load For endurance strength
FE
fitM σ
σM1
M21Y+
minus=
σFE is the endurance limit for R = 0
For static strength YMst = 1 and σfit accounted for in 39b
σfit is the shrinkage stress in the fillet (30˚ tangent) and may be found by multiplying the nominal tangential (hoop) stress with a stress concentration factor
n
Ffit m
ρ215scf minus=
384 For shrink-fitted idlers and planets When combined conditions apply such as idlers with shrink-age stresses the design factor for endurance strength is
( ) ( ) FE
fitM σ
σR1M1
M2
M1M1R1
1Yminussdot+
minus
+minus
minus=
Symbols as above but note that the stress ratio R in this par-ticular connection should disregard the influence of σfit ie R normally equal ndash 12
For static strength
M1M1R1
1YMst
+minus
minus=
The effect of σfit is accounted for in 39 b
Classification Notes- No 412 27 May 2003
DET NORSKE VERITAS
385 Additional requirements for peak loads The total stress range (σmax ndash σmin) in a tooth root fillet is not to exceed
F
y
Sσ225
for not surface hardened fillets
FSHV5 for surface hardened fillets
39 Life Factor YN The life factor YN takes into account that in the case of limited life (number of cycles) a higher tooth root stress can be permitted and that lower stresses may apply for very high number of cycles
Decisive for the strength at limited life is the σ ndash N ndash curve of the respective material for given hardening module fillet radius roughness in the tooth root etc Ie the factors YδrelT YRelT YX and YM have an influence on YN
If no σ ndash N ndash curve for the actual material and hardening etc is available the following method may be used
Determination of the σ ndash N ndash curve
a) Calculate the permissible stress σFP for the beginning of the endurance limit (3middot106 cycles) including the influ-ence of all relevant factors as SF YδrelT YRelT YX YM and YC ie σFP = σFE middotYM middotYδrelT middotYRelT middotYX middotYC SF
b) Calculate the permissible laquostaticraquo stress (le 103 load cy-cles) including the influence of all relevant factors as SFst YδrelTst YMst and YCst ( )fitσCstYδrelTstYMstYFstσ
FstS1
FPstσ minussdotsdotsdot=
where σFst is the local tooth root stress which the material can resist without cracking (surface hardened materials) or unacceptable deformation (not surface hardened materials) with 99 survival probability
σFst
Alloyed case hardened steel 1) 2300
Nitriding steel quenched tempered and gas nitrided (surface hardness 700 ndash 800 HV)
1250
Alloyed quenched and tempered steel bath or gas nitrided (surface hardness 500 ndash 700 HV)
1050
Alloyed quenched and tempered steel flame or induction hardened (fillet surface hardness 500 ndash 650 HV)
18 HV + 800
Steel with not surface hardened fillets the smaller value of 2)
18 σB or 225 σy
1) This is valid for a fillet surface hardness of 58 ndash 63 HRC In case of lower fillet surface hardness than 58 HRC σFst is to be reduced with 30(58 ndash HRC) where HRC is the actual hardness Shot peening or grinding notches are not considered to have any significant influence on σFst 2) Actual stresses exceeding the yield point (σy or σ02) will alter the residual stresses locally in the ldquotensionrdquo fillet re-spectively ldquocompressionrdquo fillet This is only to be utilised for gears that are not later loaded with a high number of cycles at lower loads that could cause fatigue in the ldquocompressionrdquo fillet
c) Calculate YN as
NL gt 3middot106
YN = 1 or 001
L
6
N N103Y
sdot= ie Yn = 092 for 1010
The YN = 1 from 3middot106 on may only be used when special material cleanness applies see rules Pt4 Ch2
103ltNLlt3middot106exp
L
6
N N103Y
sdot=
cycles6103forσcycles310forσlog02876exp
FP
FPst
sdot=
NL lt 103 cycles6103forσcycles310forσ
NYFP
FPst
sdot=
or simply use σFPst as mentioned in b) directly
Guidance on number of load cycles NL for various applica-tions
bull For propulsion purpose normally NL = 1010 at full load (yachts etc may have lower values)
bull For auxiliary gears driving generators that normally operate with 70-90 of rated power NL = 108 with rated power may be applied
28 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
310 Relative Notch Sensitivity Factor YδrelT The dynamic (respectively static) relative notch sensitivity factor YδrelT (YδrelTst) indicate to which extent the theoreti-cally concentrated stress lies above the endurance limits (re-spectively static strengths) in the case of fatigue (respectively overload) breakage
YδrelT is a function of the material and the relative stress gra-dient It differs for static strength and endurance limit
The following method may be used
For endurance limit
for not surface hardened fillets
( )
024
s024
relT σ103133q21σ1012201351
Ysdotsdotminus
+sdotsdotminus+= minus
minus
δ
for all surface hardened fillets except nitrided
( )
106q21002451
Y srelT
++=δ
for nitrided fillets
( )
1347q2101421
Y srelT
++=δ
For static strength
for not surface hardened fillets1)
( )( )( )025
02
02502s
relTst σ3000821σ300 1Y0821Y
+
minus+=δ
for surface hardened fillets except nitrided
YδrelTst = 044 YS + 012
for nitrided fillets
YδrelTst = 06 + 02 YS 1) These values are only valid if the local stresses do not
exceed the yield point and thereby alter the residual stress level See also 39b footnote 2
311 Relative Surface Condition Factor YRrelT The relative surface condition factor YRrelT takes into ac-count the dependence of the tooth root strength on the sur-face condition in the tooth root fillet mainly the dependence on the peak to valley surface roughness
YRrelT differs for endurance limit and static strength
The following method may be used
For endurance limit
YRrelT = 1675 ndash 053 (Ry + 1)01 for surface hardened steels and alloyed quenched and tempered steels except nitrided
YRrelT = 53 ndash 42 (Ry + 1)001 for carbon steels
YδrelT = 43 ndash 326 (Ry + 1)0005
for nitrided steels
For static strength
YRrelTst = 1 for all Ry and all materials
For a fillet without any longitudinal machining trace Ry asymp Rz
312 Size Factor YX The size factor YX takes into account the decrease of the strength with increasing size YX differs for endurance limit and static strength
The following may be used
For endurance limit
YX = 1 for mn le 5 generally
YX = 103 ndash 0006 mn for 5 lt mn lt 30
YX = 085 for mn ge 30
for not surface hard-ened steels
YX = 105 ndash 001 mn for 5 lt mn ge 25
YX = 08 for mn ge 25
for surface hardened steels
For static strength
YXst = 1 for all mn and all materials
313 Case Depth Factor YC The case depth factor YC takes into account the influence of hardening depth on tooth root strength
YC applies only to surface hardened tooth roots and is dif-ferent for endurance limit and static strength
In case of insufficient hardening depth fatigue cracks can develop in the transition zone between the hardened layer and the core For static strength yielding shall not occur in the transition zone as this would alter the surface residual stresses and therewith also the fatigue strength
The major parameters are case depth stress gradient permis-sible surface respectively subsurface stresses and subsurface residual stresses
The following simplified method for YC may be used
YC consists of a ratio between permissible subsurface stress (incl influence of expected residual stresses) and permissible surface stress This ratio is multiplied with a bracket con-taining the influence of case depth and stress gradient (The empirical numbers in the bracket are based on a high number of teeth and are somewhat on the safe side for low number of teeth)
YC and YCst may be calculated as given below but calculated values above 10 are to be put equal 10
For endurance limit
+
+=nFFE
C m02ρt31
σconstY
Classification Notes- No 412 29 May 2003
DET NORSKE VERITAS
For static strength
+
+=nFFst
Cst m02ρt31
σconstY
where const and t are connected as
Hardening process
t = endurance limit const =
static strength const =
t550 640 1900
t400 500 1200 Case hard-ening
t300 380 800
Nitriding t400 500 1200
Induction- or flame hardening
tHVmin 11 HVmin 25 HVmin
For symbols see 213
In addition to these requirements to minimum case depths for endurance limit some upper limitations apply to case hard-ened gears
The max depth to 550 HV should not exceed
1) 13 of the top land thickness san unless adequate tip relief is applied (see 110)
2) 025 mn If this is exceeded the following applies addi-tionally in connection with endurance limit
minusminus= 025
mt
1Yn
max550C
314 Thin rim factor YB Where the rim thickness is not sufficient to provide full sup-port for the tooth root the location of a bending failure may be through the gear rim rather than from root fillet to root fillet
YB is not a factor used to convert calculated root stresses at the 30deg tangent to actual stresses in a thin rim tension fillet Actually the compression fillet can be more susceptible to fatigue
YB is a simplified empirical factor used to de-rate thin rim gears (external as well as internal) when no detailed calcula-tion of stresses in both tension and compression fillets are available
Figure 34 Examples on thin rims
YB is applicable in the range 175 lt sRmn lt 35
YB = 115 middot ln (8324 middot mnsR)
(for sRmn ge 35 YB = 1)
(for sRmn le 175 use 315)
σF as calculated in 321 is to be multiplied with YB when sRmn lt 35 Thus YB is used for both high and low cycle fatigue
Note This method is considered to be on the safe side for external gear rims However for internal gear rims without any flange or web stiffeners the method may not be on the safe side and it is advised to check with the method in 315316
315 Stresses in Thin Rims For rim thickness sR lt 35 mn the safety against rim cracking has to be checked
The following method may be used
3151 General The stresses in the standardised 30ordm tangent section tension side are slightly reduced due to decreasing stress correction factor with decreasing relative rim thickness sRmn On the other hand during the complete stress cycle of that fillet a certain amount of compression stresses are also introduced The complete stress range remains approximately constant Therefore the standardised calculation of stresses at the 30ordm tangent may be retained for thin rims as one of the necessary criteria
The maximum stress range for thin rims usually occurs at the 60ordm ndash 80ordm tangents both for laquotensionraquo and laquocompressionraquo side The following method assumes the 75ordm tangent to be the decisive Therefore in addition to the am criterion ap-plied at the 30ordm tangent it is necessary to evaluate the max and min stresses at the 75ordm tangent for both laquotensionraquo (loaded flank) fillet and laquocompressionraquo (back-flank) fillet For this purpose the whole stress cycle of each fillet should be considered but usually the following simplification is justified
Figure 35 Nomenclature of fillets
Index laquoTraquo means laquotensileraquo fillet laquoCraquo means laquocompressionraquo fillet
σFTmin and σFCmax are determined on basis of the nominal rim stresses times the stress concentration factor Y75
σFCmin and σFTmax are determined on basis of superposition of nominal rim stresses times Y75 plus the tooth bending stresses at 75ordm tangent
30 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
3152 Stress concentration factors at the 75ordm tangents The nominal rim stress consists of bending stresses due to local bending moments tangential stresses due to the tan-gential force Ft and radial shear stresses due to Fr
The major influence is given by the bending stresses The influence of the tangential stresses is minor and even though its stress concentration factor is slightly higher than for bending it is considered to be safe enough when the sum of these nominal stresses are combined with the stress concen-tration factor for bending The influence of the radial shear stress is neglected
The stress concentration factor relating nominal rim stresses to local fillet stresses at the 75ordm tangent may be calculated as
75n
R75
n
R
75
ρmsρ185
ms3
Y+
sdot=
where ρ75 is the root radius at the 75ordm tangent ref to mn Usually ρ75 is closed to the tool radius ρao and
ρ75 = ρao
is a safe approximation compensating for the am simplifi-cations to the laquounsaferaquo side
The tooth root stresses of the loaded tooth are decreasing with decreasing relative rim thickness approximately with the empirical factor
15
R
ncorr s
m311Y
minus=
3153 Nominal rim stresses The bending moment applied to the rim consists of a part of the tooth tilt moment Ft (hF + 05 sR) and the bending caused by the radial force Fr
The sectional modulus (first moment of area) which is used for determination of the nominal bending stresses is not nec-essarily the same for the 2 am bending moments If flanges webs etc outside the toothed section contribute to stiffening the rim against various deflections the influence of these stiffeners should be considered Eg an end flange will have an almost negligible influence on the effective sectional modulus for the stresses due to tooth tilt as the deflection caused by the tilt moment is rather small and would not much involve the flange On the other hand the radial forces as for instance from the meshes in an annulus would cause considerable radial deflections that the flange might restrict to a substantial amount When considering the stiff-ening of such flanges or webs on basis of simplified models it is advised to use an effective rim thickness sR = sR + 02 mn for the first moment of area of the rim (toothed part) cross section
For a high number of rim teeth it may be assumed that the rim bending moments in the fillets adjacent to the loaded tooth are of the same magnitude as right under the applied
force This assumption is reasonable for an annulus but rather much on the safe side for a hollow pinion
The influence of Ft on the nominal tangential stress is simpli-fied by half of it for compressive stresses (σ1) and the other half for tensile stresses (σ2) Applying these assumptions the nominal rim stresses adjacent to the loaded tooth are
( ) ( )A2
FW
f RFW
s05hF05 t
R
r
T
RFt1 minus
ϑminus
+minus=σ
( ) ( )A2
FW
f RFW
s05hF05 t
R
r
T
RFt2 +
ϑminus
+=σ
where σ1 σ2 see Fig 35
A = minimum area of cross section (usually bR sR)
WT = the sectional modulus of rim with respect to tooth tilt moment (usually 6sb 2
rR )
WR = the sectional modulus of the rim including the influence of stiffeners as flanges webs etc (WR ge WT)
bR = the width of the rim
R = the radius of the neutral axis in the rim ie from wheel centre to midpoint of rim
( )ϑf = a function for bending moment distribution around the rim For a rim (pinion) with one mesh only the ( )ϑf at the position of load application is 024 For an annulus with 3 or more meshes ( )ϑf at the position of each load application is approx 3 planets ( )ϑf = 019 4 planets ( )ϑf = 014 5 planets ( )ϑf = 011 6 planets ( )ϑf = 009
It must be checked if the max (tensile) stress in the compres-sion fillet really occurs when the fillet is adjacent to the loaded tooth In principle the stress variation through a complete rotation should be considered and the max value used The max value is usually never less than 0 For an annulus eg the tilt moment is zero in the mid position be-tween the planet meshes whilst the bending moment due to the radial forces is half of that at the mesh but with opposite sign
If these formulae are applied to idler gears as eg planets the influence of nominal tangential stresses must be cor-rected by deleting Ft(2 A) for σ1 and using FtA for σ2 Further the influence of Fr on the nominal bending stresses is usually negligible due to the planet bearing support
3154 Root fillet stresses Determination of min and max stresses in the laquotensionraquo fillet
Classification Notes- No 412 31 May 2003
DET NORSKE VERITAS
Minimum stress
751FTmin YσK σ =
Maximum stress
corrF752 Yσ03Yσ KσFTmax +=
where
03 is an empirical factor relating the tension stresses (σF) at the 30ordm tangent to the part of the tension stresses at the 75ordm tangent which add to the rim related stresses (03 also takes into account that full superposition of nominal stresses times stress concentration factors from both laquosides of the corner filletraquo would result in too high stresses)
αβγ sdotsdotsdotsdot= FFvA KKKKKK
Determination of min and max stresses in the laquocompres-sionraquo fillet
Minimum stress
corrF751FCmin Yσ036Yσ Kσ minus=
Maximum stress
752FCmax Yσ Kσ =
where
036 is an empirical factor relating the tension stresses (σF) at the 30ordm tangent to the part of the compression stresses at the 75ordm tangent which add to the rim related stresses
For gears with reversed loads as idler gears and planets there is no distinct laquotensionraquo or laquocompressionraquo fillet The mini-mum stress σFmin is the minimum of σFTmin and σFCmin (usually the latter is decisive) The maximum stress σFTmax is the maximum of σFTmax and σFCmax (usually the former is deci-sive)
316 Permissible Stresses in Thin Rims
3161 General The safety against fatigue fracture respectively overload fracture is to be at least at the same level as for solid gears The laquoordinaryraquo criteria at the 30ordm tangent apply as given in 31 through 313
Additionally the following criteria at the 75ordm tangent may apply
3162 For gt3middot106 cycles The permissible stresses for the laquotensionraquo fillets and for the laquocompressionraquo fillets are determined by means of a relevant fatigue diagram
If the actual tooth root stress (tensile or compressive) ex-ceeds the yield strength to the material the induced residual stresses are to be taken into account
For determination of permissible stresses the following is defined
R = stress ratio ie FTmax
FTminσσ respectively
FCmax
FCmin
σσ
∆σ = stress range ie σFTmax ndash σFTmin resp σFCmax ndash σFCmin
(For idler gears and planets Fmax
FminσσR =
and ∆σ = σFmax minus σFmin)
The permissible stress range ∆σρ for the laquotensionraquo respec-tively laquocompressionraquo fillets can be calculated as
For R gt minus1 FPp σ
R1R1031
13∆σ
minus+
+=
For minus infin lt R lt minus1 FPp σ
R1R10151
13∆σ
minus+
+=
where
σFP = see 32 determined for unidirectional stresses (YM = 1)
If the yield strength σy is exceeded in either tension or com-pression residual stresses are induced This may be consid-ered by correcting the stress ratio R for the respective fillets (tension or compression)
Eg if yFCmin σσ gt (ie exceeded in compression) the
difference yFCmin σσ∆ minus= affects the stress ratio as
∆σ
σ∆σ∆σR
FCmax
y
FCmax
FCmin
+
minus=
++
=
Similarly the stress ratio in the tension fillet may require cor-rection
If the yield strength is exceeded in tension σFTmax gt σy the difference ∆ = σFTmax ndash σy affects the stress ratio as
y
FTmin
FTmax
FTmin
σ∆σ
∆σ∆σR minus=
minusminus
=
Checking for possible exceeding of the yield strength has to be made with the highest torque and with the KAP if this ex-ceeds KA
3163 Forle 103 cycles The permissible stress range ∆σp is not to exceed
For R gt minus1 FPstpst σ
R1R1051
15∆σ
minus+
+=
For ndash infin lt R lt minus1 FPstpst σ
R1R10251
15∆σ
minus+
+=
For all values of R ∆σpst is limited by
not surface hardened F
y
Sσ225
32 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
surface hardened CF
YSHV5
Definition of ∆σ and R see 3162 with particular attention to possible correction of R if the yield strength is exceeded
σFPst see 32 determined for unidirectional stresses (YM = 1) and lt 103 cycles
3164 For 103 lt cycles lt 3106 ∆σp is to be determined by linear interpolation a log-log dia-gram
∆σp at NL load cycles is
6L 103p
exp
L
6
N p ∆σN103∆σ sdot
sdot=
6
3
103p
10p
∆σ
∆σlog28760exp
sdot
=
Classification Notes- No 412 33 May 2003
DET NORSKE VERITAS
4 Calculation of Scuffing Load Capacity
41 Introduction High surface temperatures due to high loads and sliding ve-locities can cause lubricant films to break down This sei-zure or welding together of areas of tooth surface is termed scuffing
In contrast to pitting and fatigue breakage which show a dis-tinct incubation period a single short overloading can lead to a scuffing failure In the ISO-TR13989 two criteria are mentioned The method used in this Classification Note is based on the principles of the flash temperature criterion
Note Bulk temperature in excess of 120ordmC for long periods may have an adverse effect on the surface durability see 211
42 General Criteria In no point along the path of contact the local contact tem-perature may exceed the permissible temperature ie
oilS
oilSB S
ϑ+ϑminusϑ
leϑ
50SB minusϑleϑ
where
Bϑ = max contact temperature along the path of contact
maxflaMBB ϑ+ϑ=ϑ
MBϑ = bulk temperature see 434
maxflaϑ = max flash temperature along the path of con-tact see 44
Sϑ = scuffing temperature see below
oilϑ = oil temperature before it reaches the mesh (max applicable for the actual load case to be used ie normally alarm temperature except for ice classes where a max expected tem-perature applies)
SS = required safety factor according to the Rules
The scuffing temperature Sϑ may be calculated as
L2
wrelT
002
40S XFZGX
ν100112085780
sdot
sdot++=ϑ
where
XwrelT = relative welding factor
XwrelT
Through hardened steel 10
Phosphated steel 125
Copper-plated steel 150
Nitrided steel 150
Less than 10 retained austenite
115
10 ndash 20 retained austenite 10
Casehardened steel
20 ndash 30 retained austenite 085
Austenitic steel 045
FZG = load stage according to FZG-Test A8390 (Note This is the load stage where scuffing oc-curs However due to scatter in test results cal-culations are to be made with one load stage less than the specification)
XL = lubricant factor
= 10 for mineral oils
= 08 for polyalfaolefins
= 07 for non-water-soluble polyglycols
= 06 for water-soluble polyglycols
= 15 for traction fluids
= 13 for phosphate esters
ν40 = kinematic oil viscosity at 40˚C (mm2s)
Application of other test methods such as the Ryder the FZG-Ryder R46574 and the FZG L-42 Test 141195110 may be specially considered
For high speed gears with very short time of contact Sϑ may be increased as follows provided use of EP-oils
Addition to the calculated scuffing temperature Sϑ
If micros 18tc ge no addition
If micros 18tc lt add ( )cwrelT t18X18 minussdotsdot
where
ct = contact time ( micros ) which is the time needed to cross the Hertzian contact width
( ) [ ]microsu1cosβn
uσ340tb1
Hc +sdotsdot
sdot=
σH as calculated in 221
For bevel gears use vu in stead of u
34 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
43 Influence Factors
431 Coefficient of friction The following coefficient of friction may apply
L025a
005oil
02
redC ΣC
Bt X R ηρv
w0048micro minus
=
where
wBt = specific tooth load (Nmm)
vΣC = sum of tangential velocities at pitch point At pitch line velocities gt 50 ms the limiting value of vΣC at v = 50 ms is to be used
ρredC = relative radius of curvature (transversal plane) at the pitch point
Cylindrical gears
HαHβvγAbt
Bt KKKKKb
Fw sdotsdotsdotsdotsdot= (see 1)
twΣC αsin v2v =
bCredC β cos ρρ =
Bevel gears
HαHβvγAtmb
Bt KKKKKb
Fw sdotsdotsdotsdotsdot= (see 1)
vtmtΣC αsin v2v =
bmvCredC β cos ρρ =
ηoil = dynamic viscosity (mPa s) at oilϑ calculated as
1000
ρ νη oiloil =
where ρ in kgm3 approximated as
( ) 70 15oil15 minusϑminusρ=ρ and νoil is kinematic viscosity at oilϑ and may be calculated by means of the following equation
( ) ( )++=+ 08νloglog08νloglog 100oil ( )
sdotminus
ϑ+minus313log373log
273log373log oil
( ) ( )( )08νloglog08νloglog 10040 +minus+
Ra = composite arithmetic mean roughness (micron) of pinion an wheel calculated as ( )2a1aa RR 05R +=This is defined as the roughness on the new flanks ie as manufactured
XL = see 42
432 Effective tip relief Ceff Ceff is the effective tip relief that amount of tip relief which compensates for the elastic deformation of the gear mesh ie
zero load at the tooth tip It is assumed (simplified) to be equal for both pinion and wheel
Cylindrical gears
for helical
γ
γ=cb
KKFC Abt
eff (see 1)
For spur
cbKKF
C Abteff
γ= (see 1)
Alternatively for spur and helical gears the non-linear ap-proach in 111 may be used (taking Ceff=δ)
Bevel gears
bcKFC Ambt
effγ
= (see 1)
where
γ
α
ε2ε44c
+sdot
=γ
433 Tip relief and extension Cylindrical gears
The extension of the tip relief is not to result in an effective contact ratio 1ltεα when the gear is unloaded (exceptions to this may only apply for applications where the gear is not to run at light loads) This means that the unrelieved part of the path of contact is to be minimum pbt It is further assumed that this unrelieved part is placed centrally on the path of contact
If root relief applies it has to be calculated as equivalent tip relief Ie pinion root relief (at mesh position A) is added to Ca2 and wheel root relief (at mesh position E) is added to Ca1 If no design tip relief or root relief on the mating gear is specified (ie if Ca1+Croot2 = 0 and visa versa) use the run-ning in amount see 112
Bevel gears
Bevel gears are to have heightwise crowning ie no distinct relievedunrelieved area This may be treated as tip and root relief For calculation purposes the root relief is combined with the tip relief of the mating member into an equivalent tip relief If no resulting tip relieves are specified the equivalent tip relives may be calculated as an approxima-tion based on the tool crowning Ca tool (per mille of tool module m0) as follows
2root1aeq1a CCC +=
1root2aeq2a CCC +=
where
Classification Notes- No 412 35 May 2003
DET NORSKE VERITAS
2
0
n0101atoolal m
)m2(mA1middotmCC
minus+minussdot=
( )
sdotminusminussdotsdotminus+= vtvb1
2vb1
21vavt1n1 αtan dddαsin 05x1mA
2
0
n020atool22a m
)mm(2A1middotmCC
minus+minussdot=
( )
sdotminusminussdotsdotminus+= vtvb2
2vb2
2va2vt2n2 αtan dddαsin 05x1mA
2
0
20atool1root1 m
A1mCC
minussdotsdot=
2
0
10atool2root2 m
A1mCC
minussdotsdot=
If the pressure angles of the cutter blades are modified (and verified) to balance the tip relieves the following may be assumed
( )eq2aeq1aeq2aeq1a C andC calculated of sum50CC sdot==
Throughout the following Ca1 and Ca2 mean the equivalent tip relieves Ca1eq and Ca2eq
434 Bulk temperature The bulk temperature may be calculated as
flaaveragempsoilMB X X 50 ϑ+ϑ=ϑ
where
Xs = lubrication factor
= 12 for spray lubrication
= 10 for dip lubrication (provided both pinion and wheel are dip lubricated and tip speed lt 5 ms)
= 10 for spray lubrication with additional cooling spray (spray on both pinion and wheel or spray on pinion and dip lubrication of wheel)
= 02 for meshes fully submerged in oil
Xmp = contact factor ( )pmp nX += 150
np = number of mesh contact on the pinion (for small gear ratii the number of wheel meshes should be used if higher)
flaaverageϑ
= average of the integrated flash temperature (see 44) along the path of contact
( )
AE
E
Ayyyfla
flaaverage
d
ΓminusΓ
ΓΓϑ=ϑint =
For high speed gears (v gt 50 ms) it may be necessary to assess the bulk temperature on the basis of a thermal rating of the entire gear transmission
44 The Flash Temperature flaϑ
441 Basic formula The local flash temperature flaϑ may be calculated as
( ) 41redy
y2ly
211
43Btcorrfla
unXwX3250
y ρ
ρminusρ
micro=ϑ Γ
(For bevel gears replace u with uv)
and is to be calculated stepwise along the path of contact from A to E
where
micro = coefficient of friction see 431
Xcorr = correction factor taking empirically into account the increased scuffing risk in the beginning of the approach path due to mesh starting without any previously built up oil film and possible shuffling away oil before meshing if insufficient tip relief
( )
3
AD
yaeffcorr 50
CC1X
ΓminusΓε
Γminus+=
α
Ca = tip relief of driven member Xcorr is only applicable in the approach path and if Ca lt Ceff otherwise 10
wBt = unit load see 431
yXΓ = load sharing factor see 443
n1 = pinion rpm
Γy ρly etc see 442
442 Geometrical relations The various radii of flank curvature (transversal plane) are
ρ1y = pinion flank radius at mesh point y
ρ2y = wheel flank radius at mesh point y
ρredy = equivalent radius of curvature at mesh point y
y2y1
y2y1redy
ρ+ρ
ρρ=ρ
Cylindrical gears
twy
1y αsin au1Γ1
ρ+
+=
twy
2y αsin au1Γu
ρ+
minus=
Note that for internal gears a and u are negative
36 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
Bevel gears
vtvv
y1y αsin a
u1Γ1
ρ+
+=
vtvv
yv2y αsin a
u1Γu
ρ+
minus=
Γ is the parameter on the path of contact and y is any point between A end E
At the respective ends Γ has the following values
Root piniontip wheel
Cylindrical gears
( )
minus
minusminus= 1
αtan 1dd
zzΓ
tw
2b2a2
1
2A
Bevel gears
( )
minus
minusminus= 1
αtan 1dd
uΓvt
2vb2va2
vA
Tip pinionroot wheel
Cylindrical gears
( )
1αtan
1ddΓ
tw
2b1a1
E minusminus
=
Bevel gears
( )
1αtan
1ddΓ
vt
2vb1va1
E minusminus
=
At inner point of single pair contact
Cylindrical gears
tw1
EB α tan zπ2ΓΓ minus=
Bevel gears
vtv1
EB α tan zπ2ΓΓ minus=
At outer point of single pair contact
Cylindrical gears
tw1AD α tan z
π2ΓΓ +=
Bevel gears
vt v1
AD αtan z π2ΓΓ +=
At pitch point ΓC = 0
The points F and G (only applicable to cylindrical gears) limiting the extension of tip relief (so as to maintain a mini-mum contact ratio of unity for unloaded gears) are at
2
BAF
Γ+Γ=Γ
2
EDG
Γ+Γ=Γ
443 Load sharing factor XΓ The load sharing factor XΓ accounts for the load sharing be-tween the various pairs of teeth in mesh along the path of contact
XΓ is to be calculated stepwise from A to E using the pa-rameter Γy
4431 Cylindrical gears with β = 0 and no tip relief
Figure 41
ByAAB
Ay for31
31X
yΓltΓleΓ
ΓminusΓ
ΓminusΓ+=Γ
DyB for1Xy
ΓltΓleΓ=Γ
EyDDE
yE for31
31X
yΓleΓltΓ
ΓminusΓ
ΓminusΓ+=Γ
4432 Cylindrical gears with β = 0 and tip relief Tip relief on the pinion reduces XΓ in the range G ndash E and increases correspondingly XΓ in the range F ndash B
Tip relief on the wheel reduces XΓ in the range A ndash F and increases correspondingly XΓ in the range D ndash G
Following remains generally
DyB for1Xy
ΓleΓleΓ=Γ
21XXGF== ΓΓ
In the following it must be distinguished between Ca lt Ceff respectively Ca gt Ceff This is shown by an example below where Ca1 lt Ceff and Ca2 gt Ceff
Classification Notes- No 412 37 May 2003
DET NORSKE VERITAS
Figure 42
Note When Ca gt Ceff the path of contact is shortened by A ndash A respectively E ndash E The single pair contact path is extended into B respectively D If this shift is significant it is necessary to consider the negative effect on surface durability (B) and bending stresses (B and D)
Range A - F
For effa2 CC le
minus=Γ
eff
2a
CC1
31X
A
FyAeff
2a
AF
Ay for C3
C61XX
AyΓleΓleΓ
sdot
+ΓminusΓ
ΓminusΓ+= ΓΓ
For effa2 CC ge
AyAfor0Xy
ΓleΓleΓ=Γ
with ( )21
CC
1CC
eff
2a
eff
2a
AFAAminus
minusΓminusΓ+Γ=Γ
FyAeff
2a
AF
Ay
eff
2a for 21
CC
CC1X
yΓleΓleΓ
minus
ΓminusΓ
ΓminusΓ+minus=Γ
Range F - B
For effa1 CC le
ByFeff
1a
FB
Fy forC3
C61
21X
yΓleΓleΓ
sdot
+ΓminusΓ
ΓminusΓ+=Γ
For effa1 CC ge
ByFeff
1a
FB
Fy for21
CC
21X
yΓleΓleΓ
minus
ΓminusΓ
ΓminusΓ+=Γ
ByB for1Xy
ΓleΓleΓ=Γ
with 1
CC2
eff
1a
FBFB
minus
ΓminusΓ+Γ=Γ
Range D ndash G
For effa2 CC le
GyDeff
2a
DG
Dy
eff
2a for C 3C
61
C 3C
32X
yΓleΓleΓ
+
ΓminusΓ
ΓminusΓminus+=Γ F
or effa2 CC ge
DyD for1Xy
ΓleΓleΓ=Γ
with ( )21
CC
1CC
eff
2a
eff
2a
DGDDminus
minusΓminusΓ+Γ=Γ
GyDeff
2a
DG
Dy
eff
2a for21
CC
CCX ΓleΓleΓ
minus
ΓminusΓ
ΓminusΓminus=Γ
Range G - E
For effa1 CC le
minus=Γ
eff
1a
CC1
31X
E
EyGeff
1a
GE
Gy forC3
C61
21X
yΓleΓleΓ
sdot
+ΓminusΓ
ΓminusΓminus=Γ
For effa1 CC gt
EyGeff
1a
GE
Gy for21
CC
21X
yΓleΓleΓ
minus
ΓminusΓ
ΓminusΓminus=Γ
EyE for0Xy
ΓleΓleΓ=Γ
with 1
CC2
eff
1a
GEGE
minus
ΓminusΓ+Γ=Γ
4433 Gears with β gt 0 buttressing Due to oblique contact lines over the flanks a certain but-tressing may occur near A and E
This applies to both cylindrical and bevel gears with tip relief lt Ceff The buttressing Xbutt is simplified as a linear function within the ranges A ndash H respectively I ndash E
Figure 43
1εwhen13X βbutt EAge=
1εwhenε031X ββbutt EAlt+=
38 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
Cylindrical gears
bIEAH βsin 02ΓΓΓΓ =minus=minus
Bevel gears
bmIEAH βsin 02ΓΓΓΓ =minus=minus
4434 Cylindrical gears with 2εγ le and no tip relief
yXΓ is obtained by multiplication of
yXΓ in 4431 with
Xbutt in 4433
4435 Gears with 2εγ gt and no tip relief
Applicable to both cylindrical and bevel gears
IyH for1Xy
ΓleΓleΓε
=α
Γ
IyHybutt and forX1Xy
ΓgtΓΓltΓε
=α
Γ
Figure 44
4436 Cylindrical gears with 2εγ le and tip relief
yXΓ is obtained by multiplication of
yXΓ in 4432 with Xbutt
in 4433
4437 Cylindrical gears with 2εγ gt and tip relief
Tip relief on the pinion (respectively wheel) reduces XГ in the range G ndash E (respectively A ndash F) and increases XГ in the range F ndash G
yXΓ is obtained by multiplication of
yXΓ as described below
with Xbutt in 4433
In the XГ example below the influence of tip relief is shown (without the influence of Xbutt) by means of
eff2aeff1a CC and CC ltgt
Tip relief gt Ceff causes new end points A respectively E of the path of contact
Figure 45
Range A ndash F
( ) ( )( ) eff
a2αa1α
AF
Ay
eff
2aeff
C 12C 13εC 1ε
CCCX
y +εε++minus
ΓminusΓ
ΓminusΓ+
εminus
=ααα
Γ
eff2aFyA CC if for leΓleΓleΓ
eff2aFyA CifCfor and geΓleΓleΓ
eff2aAyA CC iffor0Xy
gtΓleΓleΓ=Γ
( ) ( ) ( )( ) ( ) a2αa1α
αeffa2AFAA C 1ε 3C 1ε
1ε 2 CC with++minus+minus
ΓminusΓ+Γ=Γ
Range F ndash G
( )( )( ) GyF
eff
2a1a forC 1 2CC 11X
yΓleΓleΓ
+εε+minusε
+ε
=αα
α
αΓ
Range G ndash E
( ) ( )( ) eff
2a1a
GE
Gy
C 1 2C 1C 1 3XX
GFy +εεminusε++ε
ΓminusΓ
ΓminusΓminus=
αα
ααΓΓ minus
effa1EyG CC if for leΓleΓleΓ
effa1EyG CC if Γfor and geΓleΓle
eff1aEyE CC if for0Xy
geΓleΓleΓ=Γ
( ) ( ) ( )( ) ( ) 2a1a
eff1aGEEE C 1C 1 3
1 2 CC withminusε++ε+εminus
ΓminusΓminusΓ=Γαα
α
4438 Bevel gears with εγ more than approx 18 and heightwise crowning For Ca1 = Ca2 = Ceff the following applies
Figure 46
Classification Notes- No 412 39 May 2003
DET NORSKE VERITAS
( )AEM 50 Γ+Γ=Γ
( )( )2AD
3
2My651X
y ΓminusΓε
ΓminusΓminus
ε=
ααΓ
For tip relief lt Ceff yXΓ is found by linear interpolation be-
tween ( )effay CCX =Γ and ( )0CayX =Γ as in 4435
The interpolation is to be made stepwise from A to M with the influence of Ca2 and from M to E with the influence of Ca1 (For 2a1a CC ne there is a discontinuity at M)
Eg with Ca1 = 04 Ceff and Ca2 = 055 Ceff then
Range A ndash M
)CC()0C( eff2ay2ayyX 550X 450X =Γ=ΓΓ +=
Range M ndash E
)CC()0C( eff1ay1ayyX 40X 06X =Γ=ΓΓ +=
For tip relief gt Ceff the new end points A and E are found as
( )
minusΓminusΓ
ε+Γ=Γ α 1
CC
6 eff
2aADAA
( )
minusΓminusΓ
ε+Γ=Γ α 1
CC
6 eff
1aADEE
Range A ndash A
0Xy=Γ
Range A ndash M
( )( )
ΓminusΓ
ΓminusΓminus
minusε=
αΓ 2
MA
2My
eff
2a1
CC4
351Xy
Range M ndash E
( )( )
ΓminusΓ
ΓminusΓminus
minusε=
αΓ 2
ME
2My
eff
1a1
CC4
351Xy
Range E ndash E
0Xy=Γ
40 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
Appendix A Fatigue Damage Accumulation
The Palmgren-Miner cumulative damage calculation princi-ple is used The procedure may be applied as follows
A1 Stress Spectrum From the individual torque classes the torques (Ti) at the peak values of class intervals and the associated number of cycles (NLi) for both pinion and wheel are to be listed from the highest to the lowest torque
(In case of a cyclic torque variation within the torque classes it is advised to use the peak torque If the cyclic variation is such that the same teeth will repeatedly suffer the peak torque this is a must)
The stress spectra for tooth roots and flanks (σFi σHi) with all relevant factors (except KA) are to be calculated on the basis of the torque spectrum The load dependent K-factors are to be determined for each torque class
A2 σminusN-curve The stress versus load cycle curves for tooth roots and flanks (both pinion and wheel) are to be drawn on the basis of per-missible stresses (ie including the demanded minimum safety factors) as determined in 2 respectively 3 If different safety levels for high cycle fatigue and low cycle fatigue are desired this may be expressed by different demand safety factors applied at the endurance limit respectively at static strength
A3 Damage accumulation The individual damage ratio Di at ith stress level is defined as
Fi
Lii N
ND =
where
NLi = The number of applied cycles at ith stress
NFi = The number of cycles to failure at ith stress
Basically stresses σi below the permissible stress level for infinite life (if a constant ZN or YN is accepted) do not con-tribute to the damage sum However calculating the actual safety factor Sact as described below all the σi for which the product S σi is bigger than or equal to the permissible stress level for infinite life contribute to the damage sum and thus to the determination of Sact The final value of S is decisive
(NFi can be found mathematically by putting the permissible stress σpi equal the actual stress σi thereby finding the actual life factor This life factor can be solved with regard to load cycles ie NFi)
The damage sum ΣDi is not to exceed unity
If ΣDi ne 1 the safety against cumulative fatigue damage is different from the applied demand safety factor For deter-mination of this theoretical safety factor an iteration proce-dure is required as described in the following flowchart
S is correction factor with which the actual safety factor Sact can be found
Sact is the demand safety factor (used in determination of the permissible stresses in the σ ndash N ndash curve) times the correc-tion factor S
The full procedure is to be applied for pinion and wheel tooth roots and flanks
Note If alternating stresses occur in a spectrum of mainly pulsating stresses the alternating stresses may be replaced by equiva-lent pulsating stresses ie by means of division with the ac-tual mean stress influence factor YM
Classification Notes- No 412 41 May 2003
DET NORSKE VERITAS
Appendix B Application Factors for Diesel Driven Gears
For diesel driven gears the application factor KA depends on torsional vibrations Both normal operation and misfiring conditions have to be considered
Normally these two running conditions can be covered by only one calculation
B1 Definitions
Normal operation KAnorm = 0
normv0
TTT +
where
T0 = rated nominal torque
Tv norm = vibratory torque amplitude for normal opera-tion (see rules Pt4 Ch3 Sec1 G301 for defi-nition of ldquonormalrdquo irregularity)
Misfiring operation KA misf = 0
misfv
TTT +
where
T = remaining nominal torque when one cylinder out of action
Tv misf = vibratory torque amplitude in misfiring con-dition This refers to a permissible misfiring condition ie a condition that does not require automatic or immediate corrective actions as speed or pitch reduction
The normal operation is assumed to last for a very high num-ber of cycles such as 1010
The misfiring operation is assumed to last for a limited dura-tion such as 107 cycles
B2 Determination of decisive load Assuming life factor at 1010 cycles as YN = ZN = 092 which usually is relevant the calculation may be performed only once with the combination having the highest value of appli-cation factorlife factor
For bending stresses and scuffing the higher value of
092
K normA and 098
K misfA
For contact stresses the higher value of
092K normA and
113K misfA (but
097K misfA for nitrided gears)
B3 Simplified procedure Note that this is only a guidance and is not a binding con-vention
T = oTZ
1Zsdot
minus where Z = number of cylinders
( ) ideal Videal Vmisf Vnorm V TTT24ZT +minus=
where TV ideal = vibratory torque with all cylinders perfectly equal See also rules Pt4 Ch3 Sec1 G300
When using trends from torsional vibration analysis and measurements the following may be used
TV ideal To is close to zero for engines with few cylinders and using a suitable elastic coupling and increases with relative coupling stiffness and number of cylinders
TV misf To may be high for engines with few cylinders and decreases with number of cylinders
This can be indicated as
200Z
TT
ο
idealV asymp and 80Z04
TT
o
misfV minusasymp
Inserting this into the formulae for the two application fac-tors the following guidance can be given
118112K misfA minusasymp
115110K normA minusasymp
Since KA norm is to be combined with the lower life factors the decisive load condition will be the normal one and a KA of 115 will cover most relevant cases when a suitable elas-tic coupling is chosen
42 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
Appendix C Calculation of Pinion-Rack
Pinion-racks used for elevating of mobile offshore units are open gears that are subjected to wear and tear With normal specifications such as only a few hundred total operation cycles (site to site) the tooth bending stresses for static strength will be decisive for the lay out however with ex-ception of surface hardened pinions where case crushing has to be considered
In the following the use of part 1 and 3 for pinion-racks is shown including relevant simplifications
C1 Pinion tooth root stresses Since the load spectrum normally is dominated by high torques and few load cycles (in the range up to 1-2000) the static strength is decisive
The actual stress is calculated as
β1FSaFan1
tF1 KYY
mbFσ sdotsdotsdotsdot
=
b1 is limited to b2 + 2middotmn
YFa and YSa replace YF and YS because load application at tooth tip has to be assumed for such inaccurate gears
Pinions often use a non-involute profile in the dedendum part of the flank eg a constant radius equal the radius of curva-ture at reference circle For such pinions sFn and hFa are to be measured directly on a sectional drawing of the pinion tooth
Due to high loads and narrow facewidths it may be assumed that KFβ2 = KHβ = 10 However when b1 gt b2 then KFβ1 gt10
If no detailed documentation of KFβ1 is available the fol-lowing may be used
KFβ1 = 1 + 015middot(b1b2 ndash 1)
The permissible stress (not surface hardened) is calculated as
δrelTstF
Fst1FP1 Y
Sσσ sdot=
The mean stress influence due to leg lifting may be disre-garded
The actual and permissible stresses should be calculated for the relevant loads as given in the rules
C2 Rack tooth root stresses The actual stress is calculated as
SaFan2
tF2 YY
mbFσ sdotsdotsdot
=
See C1 for details
The permissible stress is calculated as
δrelTstF
Fst2FP2 Y
Sσσ sdot=
For alloyed steels (Ni Cr Mo) with high toughness and duc-tility the value of YδrelTst may be put equal to YSa
C3 Surface hardened pinions For surface hardened pinions the maximum load is not to cause crushing of the hardened layer of the flank
In principle the calculation described in 213 may be used but when the theoretical Hertzian stress exceeds the ap-proximately 18 times the yield strength of the rack material plastic deformation will occur This will limit the peak Hertzian stress but increases the contact width and thus the penetration of stresses into the depth
An approximation may be based on an assessment of contact width determined by means of equal areas under the theoreti-cal (elastic range) Hertzian contact and the elasto-plastic contact stress (limited to 18 middot σy) with the unknown width
copy Det Norske Veritas 2003 Data processed and typeset by Det Norske Veritas Printed in Norway 04072003 313 PM - CN412doc If any person suffers loss or damage which is proved to have been caused by any negligent act or omission of Det Norske Veritas then Det Norske Veritas shall pay compensation to such person for his proved direct loss or damage However the compensation shall not exceed an amount equal to ten times the fee charged for the service in question provided that the maximum compensation shall never exceed USD 2 million In this provision ldquoDet Norske Veritasrdquo shall mean the Foundation Det Norske Veritas as well as all its subsidiaries directors officers employees agents and any other acting on behalf of Det Norske Veritas
FOREWORD DET NORSKE VERITAS is an autonomous and independent Foundation with the objective of safeguarding life property and the environment at sea and ashore
DET NORSKE VERITAS AS is a fully owned subsidiary Society of the Foundation It undertakes classification and certifica-tion of ships mobile offshore units fixed offshore structures facilities and systems for shipping and other industries The So-ciety also carries out research and development associated with these functions
DET NORSKE VERITAS operates a worldwide network of survey stations and is authorised by more than 120 national ad-ministrations to carry out surveys and in most cases issue certificates on their behalf
Classification Notes
Classification Notes are publications that give practical information on classification of ships and other objects Examples of design solutions calculation methods specifications of test procedures as well as acceptable repair methods for some compo-nents are given as interpretations of the more general rule requirements
A list of Classification Notes is found in the latest edition of the Introduction booklets to the rdquoRules for Classification of Shipsrdquo and the rdquoRules for Classification of High Speed Light Craft and Naval Surface Craftrdquo In ldquoRules for Classification of Fixed Offshore Installationsrdquo only those Classification Notes that are relevant for this type of structure have been listed
The list of Classification Notes is also included in the current ldquoClassification Services ndash Publicationsrdquo issued by the Society which is available on request All publications may be ordered from the Societyrsquos Web site httpexchangednvcom
Provisions
It is assumed that the execution of the provisions of this Classification Note is entrusted to appropriately qualified and experienced people for whose use it has been prepared
DET NORSKE VERITAS
CONTENTS
1 Basic Principles and General Influence Factors 4 11 Scope and Basic Principles 4 12 Symbols Nomenclature and Units 4 13 Geometrical Definitions5 14 Bevel Gear Conversion Formulae and Specific
Formulae6 15 Nominal Tangential Load Ft Fbt Fmt and Fmbt 6 16 Application Factors KA and KAP 7 161 KA 7 162 KAP7 163 Frequent overloads8 17 Load Sharing Factor Kγ 8 171 General method8 172 Simplified method 8 18 Dynamic Factor Kv 8 181 Single resonance method 8 182 Multi-resonance method 10 19 Face Load Factors KHβ and KFβ 10 191 Relations between KHβ and KFβ10 192 Measurement of face load factors 10 193 Theoretical determination of KHβ11 194 Determination of fsh 12 195 Determination of fdefl13 196 Determination of fbe 13 197 Determination of fma 13 198 Comments to various gear types 13 199 Determination of KHβ for bevel gears 13 110 Transversal Load Distribution Factors KHα and KFα14 111 Tooth Stiffness Constants cacute and cγ 14 112 Running-in Allowances 15 2 Calculation of Surface Durability17 21 Scope and General Remarks 17 22 Basic Equations 17 221 Contact stress 17 222 Permissible contact stress 17 23 Zone Factors ZH ZBD and ZM18 231 Zone factor ZH 18 232 Zone factors ZBD 18 233 Zone factor ZM 18 234 Inner contact point 18 24 Elasticity Factor ZE 18 25 Contact Ratio Factor Zε18 26 Helix Angle Factor Zβ18 27 Bevel Gear Factor ZK18 28 Values of Endurance Limit σHlim and Static
Strength 5H10σ 3H10σ 18 29 Life Factor ZN 19 210 Influence Factors on Lubrication Film ZL ZV and
ZR 19 211 Work Hardening Factor ZW 20 212 Size Factor ZX20 213 Subsurface Fatigue20 3 Calculation of Tooth Strength 22 31 Scope and General Remarks 22 32 Tooth Root Stresses 22 321 Local tooth root stress22 322 Permissible tooth root stress 22
33 Tooth Form Factors YF YFa 23 331 Determination of parameters 23 332 Gearing with εαn gt 2 24 34 Stress Correction Factors YS YSa 24 35 Contact Ratio Factor Yε 25 36 Helix Angle Factor Yβ 25 37 Values of Endurance Limit σFE 25 38 Mean stress influence Factor YM 26 381 For idlers planets and PTO with ice class 26 382 For gears with periodical change of rotational
direction 26 383 For gears with shrinkage stresses and unidirectional
load 26 384 For shrink-fitted idlers and planets 26 385 Additional requirements for peak loads 27 39 Life Factor YN 27 310 Relative Notch Sensitivity Factor YδrelT 28 311 Relative Surface Condition Factor YRrelT 28 312 Size Factor YX 28 313 Case Depth Factor YC 28 314 Thin rim factor YB 29 315 Stresses in Thin Rims 29 3151 General 29 3152 Stress concentration factors at the 75ordm tangents 30 3153 Nominal rim stresses 30 3154 Root fillet stresses 30 316 Permissible Stresses in Thin Rims 31 3161 General 31 3162 For gt3middot106 cycles 31 3163 For le 103 cycles 31 3164 For 103 lt cycles lt 3106 32 4 Calculation of Scuffing Load Capacity 33 41 Introduction 33 42 General Criteria 33 43 Influence Factors 34 431 Coefficient of friction 34 432 Effective tip relief Ceff 34 433 Tip relief and extension 34 434 Bulk temperature 35 44 The Flash Temperature flaϑ 35 441 Basic formula 35 442 Geometrical relations 35 443 Load sharing factor XΓ 36 Appendix A Fatigue Damage Accumulation 40 A1 Stress Spectrum 40 A2 σminusN-curve 40 A3 Damage accumulation 40 Appendix B Application Factors for Diesel Driven
Gears 41 B1 Definitions 41 B2 Determination of decisive load 41 B3 Simplified procedure 41 Appendix C Calculation of Pinion-Rack 42 C1 Pinion tooth root stresses 42 C2 Rack tooth root stresses 42 C3 Surface hardened pinions 42
4 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
1 Basic Principles and General Influence Factors
11 Scope and Basic Principles The gear rating procedures given in this Classification Note are mainly based on the ISO-6336 Part 1-5 (cylindrical gears) and partly on ISO 10300 Part 1-3 (bevel gears) and ISO Technical Reports on Scuffing and Fatigue Damage Accumulation but especially applied for marine purposes such as marine propulsion and important auxiliaries onboard ships and mobile offshore units
The calculation procedures cover gear rating as limited by contact stresses (pitting spalling or case crushing) tooth root stresses (fatigue breakage or overload breakage) and scuff-ing resistance Even though no calculation procedures for other damages such as wear grey staining (micropitting) etc are given such damages may limit the gear rating
The Classification Note applies to enclosed parallel shaft gears epicyclic gears and bevel gears (with intersecting axis) However open gear trains may be considered with regard to tooth strength ie part 1 and 3 may apply Even pinion-rack tooth strength may be considered but since such gear trains often are designed with non-involute pinions the calculation procedure of pinion-racks is described in Appen-dix C
Steel is the only material considered
The methods applied throughout this document are only valid for a transverse contact ratio 1 lt εα lt 2 If εα gt 2 either special considerations are to be made or suggested simplifi-cation may be used
All influence factors are defined regarding their physical interpretation Some of the influence factors are determined by the gear geometry or have been established by conven-tions These factors are to be calculated in accordance with the equations provided Other factors are approximations which are clearly stated in the text by terms as laquomay be cal-culated asraquo These approximations are substitutes for exact evaluations where such are lacking or too extensive for prac-tical purposes or factors based on experience In principle any suitable method may replace these approximations
Bevel gears are calculated on basis of virtual (equivalent) cylindrical gears using the geometry of the midsection The virtual (helical) cylindrical gear is to be calculated by using all the factors as a real cylindrical gear with some exceptions These exceptions are mentioned in connection with the ap-plicable factors Wherever a factor or calculation procedure has no reference to either cylindrical gears or bevel gears it is generally valid ie combined for both cylindrical and bevel
In order to minimise the volume of this Classification Note such combinations are widely used and everywhere it is necessary to distinguish it is clearly pointed out by local headings such as
Cylindrical gears
Bevel gears
The permissible contact stresses tooth root stresses and scuffing load capacity depend on the safety factors as re-quired in the respective Rule sections
Terms as endurance limit and static strength are used throughout this Classification Note
Endurance limit is to be understood as the fatigue strength in the range of cycles beyond the lower knee of the σndashN curves regardless if it is constant or drops with higher number of cycles
Static strength is to be understood as the fatigue strength in the range of cycles less than at the upper knee of the σndashN curves
For gears that are subjected to a limited number of cycles at different load levels a cumulative fatigue calculation applies Information on this is given in Appendix A
When the term infinite life is used it means number of cy-cles in the range 108ndash1010
12 Symbols Nomenclature and Units The symbols are generally from ISO 701 ISOR31 and ISO 1328 with a few additional symbols Only SI units are used
The main symbols as influence factors (K Z Y and X with indeces) etc are presented in their respective headings Symbols which are not explained in their respective Secs are as follows
a = centre distance (mm) b = facewidth (mm) d = reference diameter (mm) da = tip diameter (mm) db = base diameter (mm) dw = working pitch diameter (mm) ha = addendum (mm) ha0 = addendum of tool ref to mn hfp = dedendum of basic rack ref to mn (= ha0) hFe = bending moment arm (mm) for tooth root
stresses for application of load at the outer point of single tooth pair contact
hFa = bending moment arm (mm) for tooth root stresses for application of load at tooth tip
HB = Brinell hardness HV = Vickers hardness HRC = Rockwell C hardness mn = normal module n = rev per minute NL = number of load cycles qs = notch parameter Ra = average roughness value (microm) Ry = peak to valley roughness (microm) Rz = mean peak to valley roughness (microm) san = tooth top land thickness (mm) sat = transverse top land thickness (mm) sFn = tooth root chord (mm) in the critical section spr = protuberance value of tool minus grinding stock
equal residual undercut of basic rack ref to mn T = torque (Nm) u = gear ratio (per stage) v = linear speed (ms) at reference diameter
Classification Notes- No 412 5 May 2003
DET NORSKE VERITAS
x = addendum modification coefficient z = number of teeth zn = virtual number of spur teeth αn = normal pressure angle at ref cylinder αt = transverse pressure angle at ref cylinder αa = transverse pressure angle at tip cylinder αwt = transverse pressure angle at pitch cylinder β = helix angle at ref cylinder βb = helix angle at base cylinder βa = helix angle at tip cylinder εα = transverse contact ratio εβ = overlap ratio εγ = total contact ratio ρa0 = tip radius of tool ref to mn ρfp = root radius of basic rack ref to mn ( = ρa0) ρC = effective radius (mm) of curvature at pitch point ρF = root fillet radius (mm) in the critical section σB = ultimate tensile strength (Nmm2) σy = yield strength resp 02 proof stress (Nmm2)
Index 1 refers to the pinion 2 to the wheel
Index n refers to normal section or virtual spur gear of a heli-cal gear
Index w refers to pitch point
Special additional symbols for bevel gears are as follows
Σ = angle between intersection axis Kϑ = angle modification (Klingelnberg)
m0 = tool module (Klingelnberg) δ = pitch cone angle xsm = tooth thickness modification coefficient (mid-
face) R = pitch cone distance (mm)
Index v refers to the virtual (equivalent) helical cylindrical gear
Index m refers to the midsection of the bevel gear
13 Geometrical Definitions For internal gearing z2 a da2 dw2 d2 and db2 are negative x2 is positive if da2 is increased ie the numeric value is de-creased
The pinion has the smaller number of teeth ie
11
2 ge=zz
u
For calculation of surface durability b is the common face-width on pitch diameter
For tooth strength calculations b1 or b2 are facewidths at the respective tooth roots If b1 or b2 differ much from b above they are not to be taken more than 1 module on either side of b
Cylindrical gears
tan αt = tan αn cos β tan βb = tan β cos αt
tan βa = tan β da d
cos αa = dbda
d = z mn cos β mt = mn cos β db = d cos αt = dw cos αwt
a = 05 (dw1 + dw2)
dw1dw2 = z1 z2
inv α = tan α - α (radians)
inv αwt = inv αt + 2 tan αn (x1 + x2)(z1 + z2)
zn = z (cos2 βb cos β)
1
aw1fw1α T
ξξε
+=
where ξfw1 is to be taken as the smaller of
bull wtfw1 αtanξ =
bull soi1
b1wtfw1 d
dacostan -tanαξ =
bull 1
2wt
a2
b2fw1 z
ztanα
d
dacostan ξ
minus=
and
2
1fw2aw1 z
zξξ = where ξfw2 is calculated as ξfw1
substituting the values for the wheel by the values for the pinion and visa versa
11 z
2πT =
( ) +
sdot+minusminussdot= minus
2sinαρρxhm
2d2d nfpfp1fpnsoi1
21
t
nfpfplfpn2
tanα)sinαρρx(hm
sdot+minusminus
nmsinbπ
β=εβ
(for double helix b is to be taken as the width of one helix)
εy = βα εε +
ρC = ( )2
b
wt
u1βcos
αsinua
+
v = 311 10dn
60π minus
6 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
pbt = βcos
αcosmπ tn
sat =
minus++
at
ninvααinv
z
αtanx22
π
d a
san = aβcossat
14 Bevel Gear Conversion Formulae and Specific Formulae Conversion of bevel gears to virtual equivalent helical cylin-drical gears is based on the bevel gear midsection The con-version formulae are
Number of teeth
zv12 = z12 cos δ12
(δ1 + δ2 = Σ)
Gear ratio
uv = 1v
2vzz
tan αvt = tan αn cos βm
tan βbm = tan βm cos αvt
Base pitch
pbtm = m
vtnmβcos
αcosmπ
Reference pitch diameters
dv12 = 21
21m
cosdδ
Centre distance
av = 05 (dv1 + dv2)
Tip diameters
dva 12 = dv 12 + 2 ham 12
Addenda
for gears with constant addenda (Klingelnberg)
ham 12 = mmn (1 + xm 12)
for gears with variable addenda (Gleason)
ham 12 = ha 12 ndash b2 tan (δa 12 ndash δ12)
(when ha is addendum at outer end and δa is the outer cone angle)
Addendum modification coefficients
xm 12 = mn
12am21am
m2
hh minus
Base circle
dvb 12 = dv 12 cos αvt
Transverse contact ratio)
εα = btmP
αsinadd05dd05 vtv2
2vb2
2va2
1vb2
1va minusminus+minus
Overlap ratio) (theoretical value for bevel gears with no crowning but used as approximations in the calculation pro-cedures)
εβ = nm
mmπβsinb
Total contact ratio)
εγ = 2β
2α εε +
( Note that index laquovraquo is left out in order to combine formu-lae for cylindrical and bevel gears)
Tangential speed at midsection
vmt = 3m11 10dn
60π minus
Effective radius of curvature (normal section)
ρvc = ( )2vbm
vtvv
u1βcosαsinua
+
Length of line of contact
lb = ( )( )( )
1εifε
ε1ε2ε
βcosεb
β2γ
2βα
2γ
bm
α ltminusminusminus
lb = 1εifβcosε
εbβ
bmγ
α ge
15 Nominal Tangential Load Ft Fbt Fmt and Fmbt The nominal tangential load (tangential to the reference cyl-inder with diameter d and perpendicular to an axial plane) is calculated from the nominal (rated) torque T transmitted by the gear set
Cylindrical gears
dT2000
Ft = t
tbt αcos
FF =
Bevel gears
mmt d
T2000F =
vt
mtmbt αcos
FF =
Classification Notes- No 412 7 May 2003
DET NORSKE VERITAS
16 Application Factors KA and KAP The application factor KA accounts for dynamic overloads from sources external to the gearing
It is distinguished between the influence of repetitive cyclic torques KA (161) and the influence of temporary occasional peak torques KAP (162)
Calculations are always to be made with KA In certain cases additional calculations with KAP may be necessary
For gears with a defined load spectrum the calculation with a KA may be replaced by a fatigue damage calculation as given in Appendix A
161 KA For gears designed for long or infinite life at nominal rated torque KA is defined as the ratio between the maximum re-petitive cyclic torque applied to the gear set and nominal rated torque
This definition is suitable for main propulsion gears and most of the auxiliary gears
KA can be determined by measurements or system analysis or may be ruled by conventions (ice classes) (For the pur-pose of a preliminary (but not binding) calculation before KA is determined it is advised to apply either the max values mentioned below or values known from similar plants)
a) For main propulsion gears KA can be taken from the (mandatory) torsional vibration analysis thereby con-sidering all permissible driving conditions) Unless specially agreed the rules do not allow KA in excess of 135 for diesel propulsion) With turbine or electric propulsion KA would normally not exceed 12 However special attention should be given to thrusters that are arranged in such a way that heavy vessel movements andor manoeuvring can cause severe load fluctuations This means eg thrusters positioned far from the rolling axis of vessels that could be susceptible to rolling If leading to propeller air suction the condi-tions may be even worse The above mentioned movements or manoeuvring will result in increased propeller excitation If the thruster is driven by a diesel engine the engine mean torque is limited to 100 However thrusters driven by electric motors can suffer temporary mean torque much above 100 unless a suitable load control system (limiting available e-motor torque) is provided
b) For main propulsion gears with ice class notation (see Rules Pt5 Ch1 Sec J500) KA ice has to be taken as the higher value of the applicable (rule defined) ice shock torque referred to nominal rated torque and the value under a) The Baltic ice class notations refer to a few millions ice shock loads Thus the life factors may be put YN=1 and ZN=12 (except for nitrided gears where ZN = 1 applies) Additionally the calculations with the normal KA (no ice class) are to fulfil the normal requirements For polar ice class notations KA ice applies to all criteria and for long or infinite life
c) For a power take off (PTO) branch from a main propul-sion gear with ice class ice shocks result in negative torques It is assumed that the PTO branch is unloaded when the ice shock load occurs The influence of these reverse shock loads may be taken into account as follows The negative torque (reversed load) expressed by means of an application factor based on rated forward load (T or Ft) is KAreverse = KA ice ndash1 (the minus 1 be-cause no mean torque assumed) KAice to be calculated as in the ice class rules This KAreverse should be used for back flank considera-tions such as pitting and scuffing The influence on tooth bending strength (forward di-rection) may be simplified by using the factor YM = 1 minus 03 middot KAreverse KA
d) For diesel driven auxiliaries KA can be taken from the torsional vibration analysis if available For units where no vibration analysis is required (lt 200 kW) or available it is advised to apply KA as the upper allow-able value 135)
e) For turbine or electro driven auxiliaries the same as for c) applies however the practical upper value is 12
) For diesel driven gears more information on KA for mis-firing and normal driving is given in Appendix B
162 KAP The peak overload factor KAP is defined as the ratio between the temporary occasional peak overload torque and the nominal rated torque
For plants where high temporary occasional peak torques can occur (ie in excess of the above mentioned KA) the gearing (if nitrided) has to be checked with regard to static strength Unless otherwise specified the same safety factors as for infinite life apply
The scuffing safety is to be specially considered whereby the KA applies in connection with the bulk temperature and the KAP applies for the flash temperature calculation and should replace KA in the formulae in 431 432 and 441
KAP can be evaluated from the torsional impact vibration calculation (as required by the rules)
If the overloads have a duration corresponding to several revolutions of the shafts the scuffing safety has to be consid-ered on basis of this overload both with respect to bulk and flash temperature This applies to plants with ice class nota-tions (Baltic and polar) and to plants with prime movers which have high temporary overload capacity such as eg electric motors (provided the driven member can have a con-siderable increase in demand torque as eg azimuth thrusters during manoeuvring)
For plants without additional ice class notation KAP should normally not exceed 15
8 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
163 Frequent overloads For plants where high overloads or shock loads occur regu-larly the influence of this is to be considered by means of cumulative fatigue (see Appendix A)
17 Load Sharing Factor Kγ The load sharing factor Kγ accounts for the maldistribution of load in multiple-path transmissions (dual tandem epicyclic double helix etc) Kγ is defined as the ratio between the max load through an actual path and the evenly shared load
171 General method Kγ mainly depends on accuracy and flexibility of the branches (eg quill shaft planet support external forces etc) and should be considered on basis of measurements or of relevant analysis as eg
Kγ = δ+δ f
δ = total compliance of a branch under full load (assuming even load share) referred to gear mesh
f = minusminusminusminus+++ 23
22
21 fff where f1 f2 etc are the
main individual errors that may contribute to a maldis-tribution between the branches Eg tooth pitch errors planet carrier pitch errors bearing clearance influences etc Compensating effects should also be considered
For double helical gears
An external axial force Fext applied from sources outside the actual gearing (eg thrust via or from a tooth coupling) will cause a maldistribution of forces between the two helices Expressed by a load sharing factor the
βsdot
plusmn=γ tanFF1K
t
ext
If the direction of Fext is known the calculation should be carried out separately for each helix and with the tangential force corrected with the pertinent Kγ If the direction of Fext is unknown both combinations are to be calculated and the higher σH or σF to be used
172 Simplified method If no relevant analysis is available the following may apply
For epicyclic gears
Kγ = 32501 minus+ pln
where npl = number of planets ( gt3 )
For multistage gears with locked paths and gear stages sepa-rated by quill shafts (see figure below)
Figure 10 Locked paths gear
Kγ = ( )φ201+
where φ = quill shaft twist (degrees) under full load
18 Dynamic Factor Kv The dynamic factor Kv accounts for the internally generated dynamic loads
Kv is defined as the ratio between the maximum load that dynamically acts on the tooth flanks and the maximum ex-ternally applied load Ft KA Kγ
In the following 2 different methods (181 and 182) are described In case of controversy between the methods the next following is decisive ie the methods are listed with increasing priority
It is important to observe the limitations for the method in 181 In particular the influence of lateral stiffness of shafts is often underestimated and resonances occur at considerably lower speed than determined in 1811
However for low speed gears with vmiddotz1 lt 300 calculations may be omitted and the dynamic factor simplified to Kv=105
181 Single resonance method For a single stage gear Kv may be determined on basis of the relative proximity (or resonance ratio) N between actual speed n1 and the lowest resonance speed nE1
N = 1E
1
nn
Note that for epicyclic gears n is the relative speed ie the speed that multiplied with z gives the mesh frequency
1811 Determination of critical speed It is not advised to apply this method for multimesh gears for N gt 085 as the influence of higher modes has to be consid-ered see 182 In case of significant lateral shaft flexibility (eg overhung mounted bevel gears) the influence of cou-pled bending and torsional vibrations between pinion and wheel should be considered if N ge 075 see 182
nE1 = redmγc
1zπ
31030 sdot
where
cγ is the actual mesh stiffness per unit facewidth see 111
Classification Notes- No 412 9 May 2003
DET NORSKE VERITAS
For gears with inactive ends of the facewidth as eg due to high crowning or end relief such as often applied for bevel gears the use of cγ in connection with determination of natu-ral frequencies may need correction cγ is defined as stiffness per unit facewidth but when used in connection with the total mesh stiffness it is not as simple as cγmiddotb as only a part of the facewidth is active Such corrections are given in 111
mred is the reduced mass of the gear pair per unit facewidth and referred to the plane of contact
For a single gear stage where no significant inertias are closely connected to neither pinion nor wheel mred is calcu-lated as
mred = 21
21mm
mm+
The individual masses per unit facewidth are calculated as
m12 = 221b
21
)2d(b
I
where I is the polar moment of inertia (kgmm2)
The inertia of bevel gears may be approximated as discs with diameter equal the midface pitch diameter and width equal to b However if the shape of the pinion or wheel body differs much from this idealised cylinder the inertia should be cor-rected accordingly
For all kind of gears if a significant inertia (eg a clutch) is very rigidly connected to the pinion or wheel it should be added to that particular inertia (pinion or wheel) If there is a shaft piece between these inertias the torsional shaft stiffness alters the system into a 3-mass (or more) system This can be calculated as in 182 but also simplified as a 2-mass system calculated with only pinion and wheel masses
1812 Factors used for determination of Kv Non-dimensional gear accuracy dependent parameters
( )bKKF
yfcB
At
pptp
γ
minus=
( )bKKF
yFcBAt
ff
γ
α minus=
Non-dimensional tip relief parameter
bKKFcC
1BγAt
ak sdotsdot
sdotminus=
For gears of quality grade (ISO 1328) Q = 7 or coarser Bk = 1
For gears with Q le 6 and excessive tip relief Bk is limited to max 1
For gears (all quality grades) with tip relief of more than 2middotCeff (see 432) the reduction of αε has to be considered (see 443)
where
fpt = the single pitch deviation (ISO 1328) max of pinion or wheel
Fα = the total profile form deviation (ISO 1328) max of pinion or wheel (Note Fα is pt not available for bevel gears thus use Fα = fpt)
yp and yf = the respective running-in allowances and may be calculated similarly to yα in 112 ie the value of fpt is replaced by Fα for yf
cacute = the single tooth stiffness see 111
Ca = the amount of tip relief see 433 In case of different tip relief on pinion and wheel the value that results in the greater value of Bk is to be used If Ca is zero by design the value of running-in tip relief Cay (see 112) may be used in the above formula
1813 Kv in the subcritical range Cylindrical gears N le 085
Bevel gears N le 075
Kv = 1 + N K
K = Cv1 Bp + Cv2 Bf + Cv3 Bk
Cv1 accounts for the pitch error influence
Cv1 = 032
Cv2 accounts for profile error influence
Cv2 = 034 for γε le 2
Cv2 = 03ε
057
γ minus for γε gt 2
Cv3 accounts for the cyclic mesh stiffness variation
Cv3 = 023 for γε le 2
Cv3 = 156ε
0096
γ minus for γε gt 2
1814 Kv in the main resonance range Cylindrical gears 085 lt N le 115
Bevel gears 075 lt N le 125
Running in this range should preferably be avoided and is only allowed for high precision gears
Kv = 1 + Cv1 Bp + Cv2 Bf + Cv4 Bk
Cv4 accounts for the resonance condition with the cyclic mesh stiffness variation
Cv4 = 090 for γε le 2
Cv4 = 144ε
ε005057
γ
γ
minus
minus for γε gt 2
10 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
1815 Kv in the supercritical range Cylindrical gears N ge 15
Bevel gears N ge 15
Special care should be taken as to influence of higher vibra-tion modes andor influence of coupled bending (ie lateral shaft vibrations) and torsional vibrations between pinion and wheel These influences are not covered by the following approach
Kv = Cv5 Bp + Cv6 Bf + Cv7
Cv5 accounts for the pitch error influence
Cv5 = 047
Cv6 accounts for the profile error influence
Cv6 = 047 for εγ le 2
Cv6 = 174ε
012
γ minus for εγ gt 2
Cv7 relates the maximum externally applied tooth loading to the maximum tooth loading of ideal accurate gears operating in the supercritical speed sector when the circumferential vibration becomes very soft
Cv7 = 075 for εγ le 15
Cv7 = [ ] 8750)2(sin 0125 +minusβεπ for 15 lt εγ le 25
Cv7 = 10 for εγ gt 25
1816 Kv in the intermediate range Cylindrical gears 115 lt N lt 15
Bevel gears 125 lt N lt 15
Comments raised in 1814 and 1815 should be observed
Kv is determined by linear interpolation between Kv for N = 115 respectively 125 and N = 15 as
Cylindrical gears
( ) ( ) ( )[ ]51Nv151Nv51Nvv KK350
N51KK === minussdot
minus
+=
Bevel gears
( ) ( ) ( )[ ]51Nv251Nv51Nvv KK250
N51KK === minussdot
minus
+=
182 Multi-resonance method For high speed gear (vgt40 ms) for multimesh medium speed gears for gears with significant lateral shaft flexibility etc it is advised to determine Kv on basis of relevant dy-namic analysis
Incorporating lateral shaft compliance requires transforma-tion of even a simple pinion-wheel system into a lumped multi-mass system It is advised to incorporate all relevant inertias and torsional shaft stiffnesses into an equivalent (to pinion speed) system Thereby the mesh stiffness appears as an equivalent torsional stiffness
cγ b (db12)2 (Nmrad)
The natural frequencies are found by solving the set of dif-ferential equations (one equation per inertia) Note that for a gear put on a laterally flexible shaft the coupling bending-torsionals is arranged by introducing the gear mass and the lateral stiffness with its relation to the torsional displacement and torque in that shaft
Only the natural frequency (ies) having high relative dis-placement and relative torque through the actual pinion-wheel flexible element need(s) to be considered as critical frequency (ies)
Kv may be determined by means of the method mentioned in 181 thereby using N as the least favourable ratio (in case of more than one pinion-wheel dominated natural frequency) Ie the N-ratio that results in the highest Kv has to be consid-ered
The level of the dynamic factor may also be determined on basis of simulation technique using numeric time integration with relevant tooth stiffness variation and pitchprofile er-rors
19 Face Load Factors KHβ and KFβ The face load factors KHβ for contact stresses and for scuff-ing KFβ for tooth root stresses account for non-uniform load distribution across the facewidth
KHβ is defined as the ratio between the maximum load per unit facewidth and the mean load per unit facewidth
KFβ is defined as the ratio between the maximum tooth root stress per unit facewidth and the mean tooth root stress per unit facewidth The mean tooth root stress relates to the con-sidered facewidth b1 respectively b2
Note that facewidth in this context is the design facewidth b even if the ends are unloaded as often applies to eg bevel gears
The plane of contact is considered
191 Relations between KHβ and KFβ
KFβ = expβHK 2(hb)hb1
1exp++
=
where hb is the ratio tooth heightfacewidth The maximum of h1b1 and h2b2 is to be used but not higher than 13 For double helical gears use only the facewidth of one helix
If the tooth root facewidth (b1 or b2) is considerably wider than b the value of KFβ(1or2) is to be specially considered as it may even exceed KHβ
Eg in pinion-rack lifting systems for jack up rigs where b = b2 asymp mn and b1 asymp 3 mn the typical KHβ asymp KFβ2 asymp 1 and KFβ1 asymp 13
192 Measurement of face load factors Primarily
KFβ may be determined by a number of strain gauges distrib-uted over the facewidth Such strain gauges must be put in
Classification Notes- No 412 11 May 2003
DET NORSKE VERITAS
exactly the same position relative to the root fillet Relations in 191 apply for conversion to KHβ
Secondarily
KHβ may be evaluated by observed contact patterns on vari-ous defined load levels It is imperative that the various test loads are well defined Usually it is also necessary to evalu-ate the elastic deflections Some teeth at each 90 degrees are to be painted with a suitable lacquer Always consider the poorest of the contact patterns
After having run the gear for a suitable time at test load 1 (the lowest) observe the contact pattern with respect to ex-tension over the facewidth Evaluate that KHβ by means of the methods mentioned in this section Proceed in the same way for the next higher test load etc until there is a full face contact pattern From these data the initial mesh misalign-ment (ie without elastic deflections) can be found by ex-trapolation and then also the KHβ at design load can be found by calculation and extrapolation See example
Figure 11 Example of experimental determination of KHβ
It must be considered that inaccurate gears may accumulate a larger observed contact pattern than the actual single mesh to mesh contact patterns This is particularly important for lapped bevel gears Ground or hard metal hobbed bevel gears are assumed to present an accumulated contact pattern that is practically equal the actual single mesh to mesh con-tact patterns As a rough guidance the (observed) accumu-lated contact pattern of lapped bevel gears may be reduced by 10 in order to assess the single mesh to mesh contact pattern which is used in 199
193 Theoretical determination of KHβ The methods described in 193 to 198 may be used for cy-lindrical gears The principles may to some extent also be used for bevel gears but a more practical approach is given in 199
General For gears where the tooth contact pattern cannot be verified during assembly or under load all assumptions are to be well on the safe side
KHβ is to be determined in the plane of contact
The influence parameters considered in this method are
bull mean mesh stiffness cγ (see 111) (if necessary also variable stiffness over b)
bull mean unit load Fmb = Fbt KA Kγ Kvb (for double helical gears see 17 for use of Kγ)
bull misalignment fsh due to elastic deflections of shafts and gear bodies (both pinion and wheel)
bull misalignment fdefl due to elastic deflections of and work-ing positions in bearings
bull misalignment fbe due to bearing clearance tolerances bull misalignment fma due to manufacturing tolerances bull helix modifications as crowning end relief helix correc-
tion bull running in amount yβ (see 112)
In practice several other parameters such as centrifugal ex-pansion thermal expansion housing deflection etc contrib-ute to KHβ However these parameters are not taken into account unless in special cases when being considered as particularly important
When all or most of the am parameters are to be considered the most practical way to determine KHβ is by means of a graphical approach described in 1931
If cγ can be considered constant over the facewidth and no helix modifications apply KHβ can be determined analyti-cally as described in 1932
1931 Graphical method The graphical method utilises the superposition principle and is as follows
bull Calculate the mean mesh deflection Mδ as a function of
γm c and bF see 111
bull Draw a base line with length b and draw up a rectangu-lar with height δM (The area δM b is proportional to the transmitted force)
bull Calculate the elastic deflection fsh in the plane of contact Balance this deflection curve around a zero line so that the areas above and below this zero line are equal
Figure 12 fsh balanced around zero line
bull Superimpose these ordinates of the fsh curve to the previ-ous load distribution curve (The area under this new load distribution curve is still δM b)
bull Calculate the bearing deflections andor working posi-tions in the bearings and evaluate the influence fdefl in the plane of contact This is a straight line and is bal-anced around a zero line as indicated in Fig 14 but with one distinct direction Superimpose these ordinates to the previous load distribution curve
12 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
bull The amount of crowning end relief or helix correction (defined in the plane of contact) is to be balanced around a zero line similarly to fsh
Figure 13 Crowning Cc balanced aound zero line
bull Superimpose these ordinates to the previous load distri-bution curve In case of high crowning etc as eg often applied to bevel gears the new load distribution curve may cross the base line (the real zero line) The result is areas with negative load that is not real as the load in those areas should be zero Thus corrective actions must be made but for practical reasons it may be postponed to after next operation
bull The amount of initial mesh misalignment bema ff + (defined in the plane of contact) is to be bal-
anced around a zero line If the direction of bema ff + is known (due to initial contact check) or if the direction of fbe is known due to design (eg overhang bevel pin-ion) this should be taken into account If direction un-known the influence of bema ff + in both directions as well as equal zero should be considered
Figure 14 fma+fbe in both directions balanced around zero line
Superimpose these ordinates to the previous load distri-bution curve This results in up to 3 different curves of which the one with the highest peak is to be chosen for further evaluation
bull If the chosen load distribution curve crosses the base line (ie mathematically negative load) the curve is to be corrected by adding the negative areas and dividing this with the active facewidth The (constant) ordinates of this rectangular correction area are to be subtracted from the positive part of the load distribution curve It is advisable to check that the area covered under this new load distribution curve is still equal δM b
bull If cγ cannot be considered as constant over b then cor-rect the ordinates of the load distribution curve with the local (on various positions over the facewidth) ratio between local mesh stiffness and average mesh stiffness cγ (average over the active facewidth only) Note that the result is to be a curve that covers the same area δM b as before
bull The influence of running in yβ is to be determined as in 112 whereby the value for Fβx is to be taken as twice the distance between the peak of the load distribution curve and δM
bull Determine
KHβ = M
β
δycurveofpeak minus
1932 Simplified analytical method for cylindrical gears The analytical approach is similar to 1931 but has a more limited application as cγ is assumed constant over the face-width and no helix modification applies
bull Calculate the elastic deflection fsh in the plane of contact Balance this deflection curve around a zero line so that the area above and below this zero line are equal see Fig 12 The max positive ordinate is frac12∆fsh
bull Calculate the initial mesh alignment as Fβx= deflbemash ffff plusmnplusmnplusmn∆ The negative signs may only be used if this is justified andor verified by a contact pattern test Otherwise al-ways use positive signs If a negative sign is justified the value of Fβx is not to be taken less than the largest of each of these elements
bull Calculate the effective mesh misalignment as Fβy = Fβx - yβ (yβ see 112)
bull Determine
2KforF2
bFc1K βH
m
βγγH le+=β
or
2KforF
bFc2K Hβ
m
βγγH gt=β
where cγ as used here is the effective mesh stiffness see 111
194 Determination of fsh fsh is the mesh misalignment due to elastic deflections Usu-ally it is sufficient to consider the combined mesh deflection of the pinion body and shaft and the wheel shaft The calcu-lation is to be made in the plane of contact (of the considered gear mesh) and to consider all forces (incl axial) acting on the shafts Forces from other meshes can be parted into components parallel respectively vertical to the considered plane of contact Forces vertical to this plane of contact have no influence on fsh
It is advised to use following diameters for toothed elements
d + 2 x mn for bending and shear deflection
d + 2 mn (x ndash ha0 + 02) for torsional deflection
Usually fsh is calculated on basis of an evenly distributed load If the analysis of KHβ shows a considerable maldis-
Classification Notes- No 412 13 May 2003
DET NORSKE VERITAS
tribution in term of hard end contact or if it is known by other reasons that there exists a hard end contact the load should be correspondingly distributed when calculating fsh In fact the whole KHβ procedure can be used iteratively 2-3 iterations will be enough even for almost triangular load distributions
195 Determination of fdefl fdefl is the mesh misalignment in the plane of contact due to bearing deflections and working positions (housing deflec-tion may be included if determined)
First the journal working positions in the bearings are to be determined The influence of external moments and forces must be considered This is of special importance for twin pinion single output gears with all 3 shafts in one plane
For rolling bearings fdefl is further determined on basis of the elastic deflection of the bearings An elastic bearing deflec-tion depends on the bearing load and size and number of rolling elements Note that the bearing clearance tolerances are not included here
For fluid film bearings fdefl is further determined on basis of the lift and angular shift of the shafts due to lubrication oil film thickness Note that fbe takes into account the influence of the bearing clearance tolerance
When working positions bearing deflections and oil film lift are combined for all bearings the angular misalignment as projected into the plane of the contact is to be determined fdefl is this angular misalignment (radians) times the face-width
196 Determination of fbe fbe is the mesh misalignment in the plane of contact due to tolerances in bearing clearances In principle fbe and fdefl could be combined But as fdefl can be determined by analy-sis and has a distinct direction and fbe is dependent on toler-ances and in most cases has no distinct direction (ie + toler-ance) it is practicable to separate these two influences
Due to different bearing clearance tolerances in both pinion and wheel shafts the two shaft axis will have an angular mis-alignment in the plane of contact that is superimposed to the working positions determined in 195 fbe is the facewidth times this angular misalignment Note that fbe may have a distinct direction or be given as a + tolerance or a combina-tion of both For combination of + tolerance it is adviced to use
fbe= 22be
21be ff ++plusmn
fbe is particularly important for overhang designs for gears with widely different kinds of bearings on each side and when the bearings have wide tolerances on clearances In general it shall be possible to replace standard bearings with-out causing the real load distribution to exceed the design premises For slow speed gears with journal bearings the expected wear should also be considered
197 Determination of fma fma is the mesh misalignment due to manufacturing toler-ances (helix slope deviation) of pinion fHβ1 wheel fHβ2 and housing bore
For gear without specifically approved requirements to as-sembly control the value of fma is to be determined as
fma= 22Hβ
21Hβ ff +
For gears with specially approved assembly control the value of fma will depend on those specific requirements
198 Comments to various gear types For double helical gears KHβ is to be determined for both helices Usually an even load share between the helices can be assumed If not the calculation is to be made as de-scribed in 171
For planetary gears the free floating sun pinion suffers only twist no bending It must be noted that the total twist is the sum of the twist due to each mesh If the value of 1K γ ne this must be taken into account when calculating the total sun pinion twist (ie twist calculated with the force per mesh without Kγ and multiplied with the number of planets)
When planets are mounted on spherical bearings the mesh misalignments sun-planet respectively planet-annulus will be balanced Ie the misalignment will be the average between the two theoretical individual misalignments The faceload distribution on the flanks of the planets can take full advan-tage of this However as the sun and annulus mesh with several planets with possibly different lead errors the sun and annulus cannot obtain the above mentioned advantage to the full extent
199 Determination of KHβ for bevel gears If a theoretical approach similar to 193-198 is not docu-mented the following may be used
testeff
H Kb
b851851K sdot
minussdot=β
beff b represents the relative active facewidth (regarding lapped gears see 192 last part)
Higher values than beff b = 090 are normally not to be used in the formula
For dual directional gears it may be difficult to obtain a high beff b in both directions In that case the smaller beff b is to be used
Ktest represents the influence of the bearing arrangement shaft stiffness bearing stiffness housing stiffness etc on the faceload distribution and the verification thereof Expected variations in length- and height-wise tooth profile is also accounted for to some extent
a) Ktest = 1 For ground or hard metal hobbed gears with the specified contact pattern verified at full rating or at full torque slow turning at a condition representative for the thermal expansion at normal operation
14 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
It also applies when the bearing arrangementsupport has insignificant elastic deflections and thermal axial expansion However each initial mesh contact must be verified to be within acceptance criteria that are cali-brated against a type test at full load Reproduction of the gear tooth length- and height-wise profile must also be verified This can be made through 3D measurements or by initial contact movements caused by defined axial offsets of the pinion (tolerances to be agreed upon)
b) Ktest = 1 + 04middot(beffbndash06) For designs with possible influence of thermal expansion in the axial direction of the pinion The initial mesh contact verified with low load or spin test where the acceptance criteria are calibrated against a type test at full load
c) Ktest = 12 if mesh is only checked by toolmakerrsquos blue or by spin test contact For gears in this category beffb gt 085 is not to be used in the calculation
110 Transversal Load Distribution Factors KHα and KFα The transverse load distribution factors KHα for contact stresses and for scuffing KFα for tooth root stresses account for the effects of pitch and profile errors on the transversal load distribution between 2 or more pairs of teeth in mesh
The following relations may be used
Cylindrical gears
( )
minus+==
tH
αptγγHαFα F
byfc0409
2ε
KK
valid for 2εγ le
( ) ( )tH
αptγ
γ
γHαFα F
byfcε
1ε20409KK
minusminus+==
valid for 2ε γ gt
where
FtH = Ft KA Kγ Kv KHβ
cγ = See 111
γα = See 112
fpt = Maximum single pitch deviation (microm) of pinion or wheel or maximum total profile form devia-tion Fα of pinion or wheel if this is larger than the maximum single pitch deviation
Note In case of adequate equivalent tip relief adapted to the load half of the above mentioned fpt can be introduced A tip relief is considered adequate when the average of Ca1 and Ca2 is within plusmn40 of the value of Ceff in 432
Limitations of KHα and KFα
If the calculated values for
KFα = KHα lt 1 use KFα = KHα = 10
If the calculated value of KHα gt 2εα
γ
Zε
ε use KHα = 2
εα
γ
Zε
ε
If the calculated value of KFα gt εα
γ
Yεε
use KFα = εα
γ
Yεε
where Yε = αnε
075025+ (for εαn see 331c)
Bevel gears
For ground or hard metal hobbed gears KFα = KHα = 1
For lapped gears KFα = KHα = 11
111 Tooth Stiffness Constants cacute and cγ The tooth stiffness is defined as the load which is necessary to deform one or several meshing gear teeth having 1 mm facewidth by an amount of 1 microm in the plane of contact cacute is the maximum stiffness of a single pair of teeth cγ is the mean value of the mesh stiffness in a transverse plane (brief term mesh stiffness)
Both valid for high unit load (Unit load = Ft middot KA middot Kγb)
Cylindrical gears The real stiffness is a combination of the progressive Hertzian contact stiffness and the linear tooth bending stiff-nesses For high unit loads the Hertzian stiffness has little importance and can be disregarded This approach is on the safe side for determination of KHβ and KHα However for moderate or low loads Kv may be underestimated due to determination of a too high resonance speed
The linear approach is described in A
An optional approach for inclusion of the non-linear stiffness is described in B
A The linear approach
BRCCq
βcos08acutec =
and
( )025ε075cc αγ +prime=
where
( )[ ]n02a01a
B α2000212
hh12051C minusminus
+minus+=
12n1n
x000635z
025791z
015551004723q minus++=
12
2n
22
1n
1 x005290z
x024188x000193z
x011654+minusminusminus
+ 000182 x22
Classification Notes- No 412 15 May 2003
DET NORSKE VERITAS
(for internal gears use zn2 equal infinite and x2 = 0) ha0 = hfp for all practical purposes CR considers the increased flexibility of the wheel teeth if the wheel is not a solid disc and may be calculated as
( )( )nR m5s
sR e5
bbln1C +=
where
bs = thickness of a central web
sR = average thickness of rim (net value from tooth root to inside of rim)
The formula is valid for bs b ge 02 and sRmn ge 1 Outside this range of validity and if the web is not centrally posi-tioned CR has to be specially considered
Note CR is the ratio between the average mesh stiffness over the facewidth and the mesh stiffness of a gear pair of solid discs The local mesh stiffness in way of the web corresponds to the mesh stiffness with CR = 1 The local mesh stiffness where there is no web support will be less than calculated with CR above Thus eg a centrally positioned web will have an effect corresponding to a longitudinal crowning of the teeth See also 1931 regarding KHβ
B The non-linear approach
In the following an example is given on how to consider the non-linearity
The relation between unit load Fb as a function of mesh deflection δ is assumed to be a progressive curve up to 500 Nmm and from there on a straight line This straight line when extended to the baseline is assumed to intersect at 10microm
With these assumptions the unit force Fb as a function of mesh deflection δ can be expressed as
( )10δKbF
minus= for 500bFgt
minus=
500Fb10δK
bF for 500
bFlt
with γAt KK
bF
bF
sdotsdot= etc (Nmm) ie unit load incorporat-
ing the relevant factors as
KA middot Kγ for determination of Kv
KA middot Kγ middot Kv for determination of KHβ
KA middot Kγ middot Kv middot KHβ for determination of KHα
δ = mesh deflection (microm)
K = applicable stiffness (c or cγ)
Use of stiffnesses for KV KHβ and KHα
For calculation of Kv and KHα the stiffness is calculated as follows
When Fb lt 500
the stiffness is determined as δ∆
∆ bF
where the increment is chosen as eg ∆ Fb = 10 and thus
50010Fb10
K10Fb∆δ +
++
=
When F b gt 500 the stiffness is c or cγ For calculation of KHβ the mesh deflection δ is used directly
or an equivalent stiffness determined as δsdotb
F
Bevel gears
In lack of more detailed relationship between stiffness and geometry the following may be used
b085
b13cacute eff=
b085b
16c effγ =
beff not to be used in excess of 085 b in these formulae
Bevel gears with heightwise and lengthwise crowning have progressive mesh stiffness The values mentioned above are only valid for high loads They should not be used for de-termination of Ceff (see 432) or KHβ (see 1931)
112 Running-in Allowances The running-in allowances account for the influence of run-ning-in wear on the various error elements
yα respectively yβ are the running-in amounts which reduce the influence of pitch and profile errors respectively influ-ence of localised faceload
Cay is defined as the running-in amount that compensates for lack of tip relief
The following relations may be used
For not surface hardened steel
ptHlim
α fσ160y =
yβ = βxlimH
f320σ
with the following upper limits
V lt 5 ms 5-10 ms gt 10 ms
yα max none
limH
12800σ limH
6400σ
yβ max none
limH
25600σ limH
12800σ
For surface hardened steel
yα = 0075 fpt but not more than 3 for any speed
yβ = 015 Fβx but not more than 6 for any speed
16 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
For all kinds of steel
5145189718
1C2
limHay +
minusσ
=
When pinion and wheel material differ the following ap-plies
bull Use the larger of fpt1 - yα1 and fpt2 - yα2 to replace fpt - yα in the calculation of KHα and Kv
bull Use ( )2β1ββ yy21y += in the calculation of KHβ
bull Use ( )2ay1aya CC21C += in the calculation of Kv
bull Use ( )2ay1ay2a1a CC21CC +== in the scuffing
calculation if no design tip relief is foreseen
Classification Notes- No 412 17 May 2003
DET NORSKE VERITAS
2 Calculation of Surface Durability
21 Scope and General Remarks Part 2 includes the calculations of flank surface durability as limited by pitting spalling case crushing and subsurface yielding Endurance and time limited flank surface fatigue is calculated by means of 22 ndash 212 In a way also tooth frac-tures starting from the flank due to subsurface fatigue is in-cluded through the criteria in 213
Pitting itself is not considered as a critical damage for slow speed gears However pits can create a severe notch effect that may result in tooth breakage This is particularly im-portant for surface hardened teeth but also for high strength through hardened teeth For high-speed gears pitting is not permitted
Spalling and case crushing are considered similar to pitting but may have a more severe effect on tooth breakage due to the larger material breakouts initiated below the surface Subsurface fatigue is considered in 213
For jacking gears (self-elevating offshore units) or similar slow speed gears designed for very limited life the max static (or very slow running) surface load for surface hard-ened flanks is limited by the subsurface yield strength
For case hardened gears operating with relatively thin lubri-cation oil films grey staining (micropitting) may be the lim-iting criterion for the gear rating Specific calculation meth-ods for this purpose are not given here but are under consid-eration for future revisions Thus depending on experience with similar gear designs limitations on surface durability rating other than those according to 22 - 213 may be ap-plied
22 Basic Equations Calculation of surface durability (pitting) for spur gears is based on the contact stress at the inner point of single pair contact or the contact at the pitch point whichever is greater
Calculation of surface durability for helical gears is based on the contact stress at the pitch point
For helical gears with 0 lt εβ lt 1 a linear interpolation be-tween the above mentioned applies
Calculation of surface durability for spiral bevel gears is based on the contact stress at the midpoint of the zone of contact
Alternatively for bevel gears the contact stress may be cal-culated with the program ldquoBECALrdquo In that case KA and Kv are to be included in the applied tooth force but not KHβ and KHα The calculated (real) Hertzian stresses are to be multi-plied with ZK in order to be comparable with the permissible contact stresses
The contact stresses calculated with the method in part 2 are based on the Hertzian theory but do not always represent the real Hertzian stresses
The corresponding permissible contact stresses σHP are to be calculated for both pinion and wheel
221 Contact stress Cylindrical gears
( )HαHβvγA
1
tβεEHDBH KKKKK
bud1uF
ZZZZZσ+
=
where
ZBD = Zone factor for inner point of single pair contact for pinion resp wheel (see 232)
ZH = Zone factor for pitch point (see 231)
ZE = Elasticity factor (see 24)
Zε = Contact ratio factor (see 25)
Zβ = Helix angle factor (see 26)
Ft KA Kγ Kv KHβ KHα see 15 ndash 110
d1 b u see 12 ndash 15
Bevel gears
( )HαHβvγA
v1v
vmtKEMH KKKKK
bud1uF
ZZZ105σ+
sdot=
where
105 is a correlation factor to reach real Hertzian stresses (when ZK = 1)
ZE KA etc see above
ZM = mid-zone factor see 233
ZK = bevel gear factor see 27
Fmt dv1 uv see 12 ndash 15
It is assumed that the heightwise crowning is chosen so as to result in the maximum contact stresses at or near the mid-point of the flanks
222 Permissible contact stress
XWRvLH
NHlimHP ZZZZZ
SZσ
σ =
where
σH lim = Endurance limit for contact stresses (see 28)
ZN = Life factor for contact stresses (see 29)
SH = Required safety factor according to the rules
ZLZvZR = Oil film influence factors (see 210)
ZW = Work hardening factor (see 211)
ZX = Size factor (see 212)
18 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
23 Zone Factors ZH ZBD and ZM
231 Zone factor ZH The zone factor ZH accounts for the influence on contact stresses of the tooth flank curvature at the pitch point and converts the tangential force at the reference cylinder to the normal force at the pitch cylinder
wtt2
wtbH sinααcos
cosαcosβ2Z =
232 Zone factors ZBD The zone factors ZBD account for the influence on contact stresses of the tooth flank curvature at the inner point of sin-gle pair contact in relation to ZH Index B refers to pinion D to wheel
For εβ ge 1 ZBD = 1
For internal gears ZD = 1
For εβ = 0 (spur gears)
( )
π
minusεminusminus
π
minusminus
α=
α2
2
2b
2a
1
2
1b
1a
wtB
z211
dd
z21
dd
tanZ
( )
π
minusεminusminus
π
minusminus
α=
α1
2
1b
1a
2
2
2b
2a
wtD
z211
dd
z21
dd
tanZ
If ZB lt 1 use ZB = 1
If ZD lt 1 use ZD = 1
For 0 lt εβ lt 1
ZBD = ZBD (for spur gears) ndash εβ (ZBD (for spur gears) ndash 1)
233 Zone factor ZM The mid-zone factor ZM accounts for the influence of the contact stress at the mid point of the flank and applies to spi-ral bevel gears
εminusminus
εminusminus
αβ=
αα btm2
2vb2
2vabtm2
1vb2val
2v1vvtbmM
pddpdd
ddtancos2Z
This factor is the product of ZH and ZM-B in ISO 10300 with the condition that the heightwise crowning is sufficient to move the peak load towards the midpoint
234 Inner contact point For cylindrical or bevel gears with very low number of teeth the inner contact point (A) may be close to the base circle In order to avoid a wear edge near A it is required to have suitable tip relief on the wheel
24 Elasticity Factor ZE The elasticity factor ZE accounts for the influence of the material properties as modulus of elasticity and Poissonrsquos ratio on the contact stresses
For steel against steel ZE = 1898
25 Contact Ratio Factor Zε The contact ratio factor Zε accounts for the influence of the transverse contact ratio εα and the overlap ratio εβ on the contact stresses
αε1Z =ε for 1εβ ge
( )α
ββ
αε ε
εε1
3ε4
Z +minusminus
= for εβ lt 1
26 Helix Angle Factor Zβ The helix angle factor Zβ accounts for the influence of helix angle (independent of its influence on Zε) on the surface du-rability
cosβZβ =
27 Bevel Gear Factor ZK The bevel gear factor accounts for the difference between the real Hertzian stresses in spiral bevel gears and the contact stresses assumed responsible for surface fatigue (pitting) ZK adjusts the contact stresses in such a way that the same per-missible stresses as for cylindrical gears may apply
The following may be used ZK = 080
28 Values of Endurance Limit σHlim and Static Strength 5H10σ 3H10σ
σHlim is the limit of contact stress that may be sustained for 5middot107 cycles without the occurrence of progressive pitting
For most materials 5middot107 cycles are considered to be the be-ginning of the endurance strength range or lower knee of the σ-N curve (See also Life Factor ZN) However for nitrided steels 2middot106 apply
For this purpose pitting is defined by
bull for not surface hardened gears pitted area ge 2 of total active flank area
bull for surface hardened gears pitted area ge 05 of total active flank area or ge 4 of one particular tooth flank area 510Hσ and 310Hσ are the contact stresses which the given
material can withstand for 105 respectively 103 cycles without subsurface yielding or flank damages as pitting spalling or case crushing when adequate case depth applies
The following listed values for σHlim 510Hσ and 310Hσ may only be used for materials subjected to a quality control as
Classification Notes- No 412 19 May 2003
DET NORSKE VERITAS
the one referred to in the rules Results of approved fatigue tests may also be used as the basis for establishing these values The defined survival probability is 99
σHlim σH105 σH10
3
Alloyed case hardened steels (surface hardness 58-63 HRC) - of specially approved high grade - of normal grade
1650 1500
2500 2400
3100 3100
Nitrided steel of approved grade gas nitrided (surface hardness 700-800 HV) 1250 13 σHlim 13 σHlim
Alloyed quenched and tempered steel bath or gas nitrided (surface hardness 500-700 HV)
1000
13 σHlim
13 σHlim
Alloyed flame or induction hardened steel (surface hardness 500-650 HV) 075 HV + 750 16 σHlim 45 HV
Alloyed quenched and tempered steel 14 HV + 350 16 σHlim 45 HV
Carbon steel 15 HV + 250 16 σHlim 16 σHlim
These values refer to forged or hot rolled steel For cast steel the values for σHlim are to be reduced by 15
29 Life Factor ZN The life factor ZN takes account of a higher permissible contact stress if only limited life (number of cycles NL) is demanded or lower permissible contact stress if very high number of cycles apply
If this is not documented by approved fatigue tests the fol-lowing method may be used
For all steels except nitrided
7L 105N sdotge ZN = 1 or
01570
L
7
N N105Z
sdot=
Ie ZN = 092 for 1010 cycles
The ZN = 1 from 5middot107 on may only be used when the mate-rial cleanliness is of approved high grade (see Rules Pt4 Ch2) and the lubrication is optimised by a specially approved filtering process
105 lt NL lt 5middot107 510N
logZ037
L
7
N N105Z
sdot=
NL = 105 WXRVLHlim
WstX10H1010NN ZZZZZσ
ZZσZZ
55
5=
103 lt NL lt 105 )Z(Zlog05
L
5
10NN
5N103N10
5N10ZZ
==
3L 10N le
WXRVLlimH
Wst10X10H10NN ZZZZZ
ZZZZ
33
3σ
σ==
(but not less than ZN105)
For nitrided steels
6L 102N sdotge ZN = 1 or
00980
L
6
N N102Z
sdot=
Ie ZN = 092 for 1010 cycles
The ZN = 1 from 2middot106 on may only be used when the mate-rial cleanliness is of approved high grade (see Rules Pt4 Ch2) and the lubrication is optimised by a specially approved filtering process
105 lt NL lt 2middot106 510N
Zlog76860
L
6
N N102Z
sdot=
5L 10N le
XWRVL
10XWst10NN ZZZZZ
ZZ13ZZ
55 ==
Note that when no index indicating number of cycles is used the factors are valid for 5middot107 (respectively 2middot106 for nitrid-ing) cycles
210 Influence Factors on Lubrication Film ZL ZV and ZR The lubricant factor ZL accounts for the influence of the type of lubricant and its viscosity the speed factor ZV ac-counts for the influence of the pitch line velocity and the roughness factor ZR accounts for influence of the surface roughness on the surface endurance capacity
The following methods may be applied in connection with the endurance limit
20 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
Surface hardened steels Not surface hardened steels
ZL ( )24013421
360910ν+
+
( )24013421
680830ν+
+
ZV ( )v3280
140930+
+ ( )v3280300850
++
ZR
080
ZrelR3
150
ZrelR3
where
ν40 = Kinematic oil viscosity at 40ordmC (mm2s) For case hardened steels the influence of a high bulk temperature (see 4 Scuffing) should be con-sidered Eg bulk temperatures in excess of 120ordmC for long periods may cause reduced flank surface endurance limits
For values of ν40 gt 500 use ν40 = 500
RZrel = The mean roughness between pinion and wheel (after running in) relative to an equivalent radius of curvature at the pitch point ρc = 10mm
RZrel
= ( ) 31
cZ2Z1 ρ
10RR50
+
RZ = Mean peak to valley roughness (microm) (DIN defini-tion) (roughly RZ = 6 Ra)
For 5
L 10N le ZL ZV ZR = 10
211 Work Hardening Factor ZW The work hardening factor ZW accounts for the increase of surface durability of a soft steel gear when meshing the soft steel gear with a surface hardened or substantially harder gear with a smooth surface
The following approximation may be used for the endurance limit
Surface hardened steel against not surface hardened steel 150
ZeqW R
31700
130HB21Z
minus
minus=
where HB = the Brinell hardness of the soft member For HB gt 470 use HB = 470 For HB lt 130 use HB = 130
RZeq = equivalent roughness
033
c40
066
ZS
ZHZHZEQ ρvν
15000RRRR
=
If RZeq gt 16 then use RZeq = 16
If RZeq lt 15 then use RZeq = 15
where
RZH = surface roughness of the hard member before run in
RZS = surface roughness of the soft member before run in
ν40 = see 210
If values of ZW lt 1 are evaluated ZW = 1 should be used for flank endurance However the low value for ZW may indi-cate a potential wear problem
Through hardened pinion against softer wheel
( )
minussdotminus+= 000829
HBHB0008981u1Z
2
1W
For 21HBHB
2
1 le use ZW = 1
For 71HBHB
2
1 gt use 17HBHB
2
1 =
For u gt 20 use u = 20
For static strength (lt 105 cycles)
Surface hardened against not surface hardened
ZWst = 105
Through hardened pinion against softer wheel
ZWst = 1
212 Size Factor ZX The size factor accounts for statistics indicating that the stress levels at which fatigue damage occurs decrease with an increase of component size as a consequence of the influ-ence on subsurface defects combined with small stress gradi-ents and of the influence of size on material quality
ZX may be taken unity provided that subsurface fatigue for surface hardened pinions and wheels is considered eg as in the following subsection 213
213 Subsurface Fatigue This is only applicable to surface hardened pinions and wheels The main objective is to have a subsurface safety against fatigue (endurance limit) or deformation (static strength) which is at least as high as the safety SH required for the surface The following method may be used as an approximation unless otherwise documented
The high cycle fatigue (gt3106 cycles) is assumed to mainly depend on the orthogonal shear stresses Static strength (lt103 cycles) is assumed to depend mainly on equivalent
Classification Notes- No 412 21 May 2003
DET NORSKE VERITAS
stresses (von Mises) Both are influenced by residual stresses but this is only considered roughly and empirically
The subsurface working stresses at depths inside the peak of the orthogonal shear stresses respectively the equivalent stresses are only dependent on the (real) Hertzian stresses Surface related conditions as expressed by ZL ZV and ZR are assumed to have a negligible influence
The real Hertzian stresses σHR are determined as
For helical gears with εβ gt 1
σHR = σH
For helical gears with εβ lt 1 and spur gears
ε
α
ββ ε
ε+εminus
sdotσ=σZ
1
HHR
For bevel gears
KHHR Z
1σσ sdot=
The necessary hardness HV is given as a function of the net depth tz (net = after grinding or hard metal hobbing and per-pendicular to the flank)
The coordinates tz and HV are to be compared with the de-sign specification such as
bull for flame and induction hardening tHVmin HVmin bull for nitriding t400min HV = 400 bull for case hardening t550min HV = 550 t400min HV = 400
and t300min HV = 300 (the latter only if the core hardness lt 300 If the core hardness gt 400 the t400 is to be re-placed by a fictive t400 = 16 t550)
In addition the specified surface hardness is not to be less than the max necessary hardness (at tz = 05aH) This applies to all hardening methods
For high cycle fatigue (gt3 106 cycles) the following applies
sdot+
minussdotsdotsdot= o90
05azt
05azt
cosSσ04HV
H
HHHR
applicable to 05a
zt
Hge
For 05at
H
z lt the value for 05at
H
z = applies
56300ρSσ12a cHHR
Hsdotsdot
sdot=
Where aH is half the hertzian contact width multiplied by an empirical factor of 12 that takes into account the possible influence of reduced compressive residual stresses (or even tensile residual stresses) on the local fatigue strength
If any of the specified hardness depths including the surface hardness is below the curve described by HV = f (tz) the actual safety factor against subsurface fatigue is determined as follows
reduce SH stepwise in the formula for HV and aH until all specified hardness depths and surface hardness balance with the corrected curve The safety factor obtained through this method is the safety against subsurface fatigue
For static strength (lt103 cycles) the following applies
sdot+
minussdotsdotsdot= o90
07at
06a
zt
cosSσ019HV
Hst
z
HstHHR
applicable to 06at
Hst
z ge
56300ρSσa cHHR
Hstsdotsdot
=
In the case of insufficient specified hardness depths the same procedure for determination of the actual safety factor as above applies
For limited life fatigue (103 lt cycles lt 3106 )
For this purpose it is necessary to extend the correction of safety factors to include also higher values than required Ie in the case of more than sufficient hardness and depths the safety factor in the formulae for both high cycle fatigue and static strength are to be increased until necessary and speci-fied values balance
The actual safety factor for a given number of cycles N be-tween 103 and 3106 is found by linear interpolation in a dou-ble logarithmic diagram
minussdotminus
= sdot logN3477
logSlogSlogS
310H6103HHN
36 10H103H Slog86281Slog86280 sdot+sdot sdot
22 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
3 Calculation of Tooth Strength
31 Scope and General Remarks Part 3 include the calculation of tooth root strength as limited by tooth root cracking (surface or subsurface initiated) and yielding
For rim thickness sR ge 35middotmn the strength is calculated by means of 32 ndash 313 For cylindrical gears the calculation is based on the assumption that the highest tooth root tensile stress arises by application of the force at the outer point of single tooth pair contact of the virtual spur gears The method has however a few limitations that are mentioned in 36
For bevel gears the calculation is based on force application at the tooth tip of the virtual cylindrical gear Subsequently the stress is converted to load application at the mid point of the flank due to the heightwise crowning
Bevel gears may also be calculated with the program BECAL In that case KA and Kv are to be included in the applied tooth force but not KFβ and KFα
In case of a thin annulus or a thin gear rim etc radial crack-ing can occur rather than tangential cracking (from root fillet to root fillet) Cracking can also start from the compression fillet rather than the tension fillet For rim thickness sR lt 35middotmn a special calculation procedure is given in 315 and 316 and a simplified procedure in 314
A tooth breakage is often the end of the life of a gear trans-mission Therefore a high safety SF against breakage is re-quired
It should be noted that this part 3 does not cover fractures caused by
bull oil holes in the tooth root space bull wear steps on the flank bull flank surface distress such as pits spalls or grey staining
Especially the latter is known to cause oblique fractures starting from the active flank predominately in spiral bevel gears but also sometimes in cylindrical gears
Specific calculation methods for these purposes are not given here but are under consideration for future revisions Thus depending on experience with similar gear designs limita-tions other than those outlined in part 3 may be applied
32 Tooth Root Stresses The local tooth root stress is defined as the max principal stress in the tooth root caused by application of the tooth force Ie the stress ratio R = 0 Other stress ratios such as for eg idler gears (R asymp -12) shrunk on gear rims (R gt 0) etc are considered by correcting the permissible stress level
321 Local tooth root stress The local tooth root stress for pinion and wheel may be as-sessed by strain gauge measurements or FE calculations or similar For both measurements and calculations all details are to be agreed in advance
Normally the stresses for pinion and wheel are calculated as
Cylindrical gears
FαFβvγAβSFn
tF K K K K K Y Y Y
m bFσ =
where
YF = Tooth form factor (see 33)
YS = Stress correction factor (see 34)
Yβ = Helix angle factor (see 36)
Ft KA Kγ Kv KFβ KFα see 15 ndash 110
b see 13
Bevel gears
FαFβvγASaFamn
mtF K K K K K Y Y Y
m bFσ ε=
where
YFa = Tooth form factor see 33
YSa = Stress correction factor see 34
Yε = Contact ratio factor see 35
Fmt KA etc see 15 ndash 110
b see 13
322 Permissible tooth root stress The permissible local tooth root stress for pinion respectively wheel for a given number of cycles N is
CXRrelTrelTF
NMFEFP YYYY
SYY
δσ
=σ
Note that all these factors YM etc are applicable to 3middot106 cycles when used in this formula for σFP The influence of other number of cycles on these factors is covered by the calculation of YN
where
σFE = Local tooth root bending endurance limit of reference test gear (see 37)
YM = Mean stress influence factor which accounts for other loads than constant load direction eg idler gears temporary change of load di-rection pre-stress due to shrinkage etc (see 38)
YN = Life factor for tooth root stresses related to reference test gear dimensions (see 39)
SF = Required safety factor according to the rules
YδrelT = Relative notch sensitivity factor of the gear to be determined related to the reference test gear (see 310)
YRrelT = Relative (root fillet) surface condition factor of the gear to be determined related to the reference test gear (see 311)
Classification Notes- No 412 23 May 2003
DET NORSKE VERITAS
YX = Size factor (see 312)
YC = Case depth factor (see 313)
33 Tooth Form Factors YF YFa The tooth form factors YF and YFa take into account the in-fluence of the tooth form on the nominal bending stress
YF applies to load application at the outer point of single tooth pair contact of the virtual spur gear pair and is used for cylindrical gears
YFa applies to load application at the tooth tip and is used for bevel gears
Both YF and YFa are based on the distance between the con-tact points of the 30˚-tangents at the root fillet of the tooth profile for external gears respectively 60˚ tangents for inter-nal gears
Figure 31 External tooth in normal section
Figure 32 Internal tooth in normal section
Definitions
n
2
n
Fn
enFn
Fe
F
α cosms
α cosmh6
Y
=
n
2
n
Fn
anFn
Fa
Fa
α cosms
α cosmh6
Y
=
In the case of helical gears YF and YFa are determined in the normal section ie for a virtual number of teeth
YFa differs from YF by the bending moment arm hFa and αFan and can be determined by the same procedure as YF with exception of hFe and αFan For hFa and αFan all indices e will change to a (tip)
The following formulae apply to cylindrical gears but may also be used for bevel gears when replacing
mn with mnm
zn with zvn
αt with αvt
β with βm
with undercut without undercut
Fig 33 Dimensions and basic rack profile of the teeth (finished profile)
Tool and basic rack data such as hfP ρfp and spr etc are re-ferred to mn ie dimensionless
331 Determination of parameters
( )n
n
prnfPnfP m
α cossαsin 1ρ
αtan h4πE
minusminusminusminus=
For external gears fPfP ρρ =
For internal gears ( )0z
195fPfP0
fPfP 10363156ρhxρρ
sdotminus+
+=
where
z0 = number of teeth of pinion cutter
x0 = addendum modification coefficient of pinion cutter
hfP = addendum of pinion cutter
ρfP = tip radius of pinion cutter
xhρG fpfp +minus=
24 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
τmE
2π
z2H
nnminus
minus=
with
3πτ = for external gears
6πτ = for internal gears
Htan zG2
nminusϑ=ϑ
(to be solved iteratively suitable start value6π
=ϑ for exter-
nal gears and 3π for internal gears)
a) Tooth root chord sFn For external gears
minus
ϑ+
ϑminus= ρ
cosG3
3πsinz
ms
fpnn
Fn
For bevel gears with a tooth thickness modification
xsm affects mainly sFn but also hFe and αFen The total influence of xsm on YFa Ysa can be approximated by only adding 2 xsm to sFn mn
For internal gears
minus
ϑ+
ϑminus= ρ
cosG
6πsinz
ms
fPnn
Fn
b) Root fillet radius ρF at 30ordm tangent
( )G2coszcosG2ρ
mρ
2n
2
fpn
F
minusϑϑ+=
c) Determination of bending moment arm hF dn = zn mn
b2α
αn βcosε
ε =
dan = dn + 2 ha
pbn = π mn cos αn
dbn = dn cos αn
( )4
d1εpzz
2dd
zz2d
2bn
2
αnbn
2bn
2an
en +
minusminus
minus=
en
bnen d
dcos arcα =
ennnn
e α invα invα x tan 22π
z1
minus+
+=γ
αFen = αen ndash γe
For external gears
( )
minus=
n
enFenee
n
Fe
mdαtan sin γ γcos
21
mh
]ρcos
G3πcosz fpn +
ϑminus
ϑminusminus
For internal gears
( ) minus
sdotsdotminus=
n
enFenee
n
Fe
mdα tanγ sinγ cos
21
mh
minus
ϑminus
ϑminussdot ρ
cosG3
6πcosz fPn
332 Gearing with εαn gt 2 For deep tooth form gearing ( )25ε2 αn lele produced with a verified grade of accuracy of 4 or better and with applied profile modification to obtain a trapezoidal load distribution along the path of contact the YF may be corrected by the factor YDT as
250ε205for 0666ε2366Y αnαnDT leleminus=
205εfor 10Y αnDT lt=
34 Stress Correction Factors YS YSa The stress correction factors YS and YSa take into account the conversion of the nominal bending stress to the local tooth root stress Thereby YS and YSa cover the stress increasing effect of the notch (fillet) and the fact that not only bending stresses arise at the root A part of the local stress is inde-pendent of the bending moment arm This part increases the more the decisive point of load application approaches the critical tooth root section
Therefore in addition to its dependence on the notch radius the stress correction is also dependent on the position of the load application ie the size of the bending moment arm
YS applies to the load application at the outer point of single tooth pair contact YSa to the load application at tooth tip
YS can be determined as follows
( )
+
+=L23121
1
sS qL01312Y
where Fe
Fn
hsL = and
F
Fns ρ2
sq = (see 33)
YSa can be calculated by replacing hFe with hFa in the above formulae
Note a) Range of validity 1 lt qs lt 8
In case of sharper root radii (ie produced with tools having too sharp tip radii) YS resp YSa must be specially considered
b) In case of grinding notches (due to insufficient protuberance of the hob) YS resp YSa can rise considerably and must be multiplied with
Classification Notes- No 412 25 May 2003
DET NORSKE VERITAS
g
g
ρt
0613
13
minus
where
tg = depth of the grinding notch
ρg = radius of the grinding notch
c) The formulae for YS resp YSa are only valid for αn = 20˚ However the same formulae can be used as a safe approximation for other pressure angles
35 Contact Ratio Factor Yε The contact ratio factor Yε covers the conversion from load application at the tooth tip to the load application at the mid point of the flank (heightwise) for bevel gears
The following may be used
Yε = 0625
36 Helix Angle Factor Yβ The helix angle factor Yβ takes into account the difference between the helical gear and the virtual spur gear in the nor-mal section on which the calculation is based in the first step In this way it is accounted for that the conditions for tooth root stresses are more favourable because the lines of contact are sloping over the flank
The following may be used (β input in degrees)
Yβ = 1 ndash εβ β120
When εβ gt 1 use εβ = 1 and when β gt 30deg use β = 30deg in the formula
However the above equation for Yβ may only be used for gears with β gt 25deg if adequate tip relief is applied to both pinion and wheel (adequate = at least 05 middot Ceff see 432)
37 Values of Endurance Limit σFE σFE is the local tooth root stress (max principal) which the material can endure permanently with 99 survival prob-ability 3106 load cycles is regarded as the beginning of the endurance limit or the lower knee of the σ ndash N curve σFE is defined as the unidirectional pulsating stress with a minimum stress of zero (disregarding residual stresses due to heat treatment) Other stress conditions such as alternating or pre-stressed etc are covered by the conversion factor YM
σFE can be found by pulsating tests or gear running tests for any material in any condition If the approval of the gear is to be based on the results of such tests all details on the testing conditions have to be approved by the Society Fur-ther the tests may have to be made under the Societys su-pervision
If no fatigue tests are available the following listed values for σFE may be used for materials subjected to a quality con-trol as the one referred to in the rules
σFE
Alloyed case hardened steels 1) (fillet surface hardness 58 ndash 63 HRC)
bull of specially approved high grade
1050
bull of normal grade
minus CrNiMo steels with approved process
1000
minus CrNi and CrNiMo steels generally 920 minus MnCr steels generally 850
Nitriding steel of approved grade quenched tempered and gas nitrided (surface hardness 700 ndash 800 HV)
840
Alloyed quenched and tempered steel bath or gas nitrided (surface hardness 500 ndash 700 HV)
720
Alloyed quenched and tempered steel flame or induction hardened 2) (incl entire root fillet) (fillet surface hardness 500 ndash 650 HV)
07 HV + 300
Alloyed quenched and tempered steel flame or induction hardened (excl entire root fillet) (σB = uts of base material)
025 σB + 125
Alloyed quenched and tempered steel 04 σB + 200
Carbon steel 025 σB + 250
Note All numbers given above are valid for separate forgings and for blanks cut from bars forged according to a qualified procedure see Pt 4 Ch 2 Sec 3 For rolled steel the values are to be reduced with 10 For blanks cut from forged bars that are not qualified as mentioned above the values are to be reduced with 20 For cast steel reduce with 40
1) These values are valid for a root radius
bull being unground If however any grinding is made in the root fillet area in such a way that the residual stresses may be affected σFE is to be reduced by 20 (If the grinding also leaves a notch see 34)
bull with fillet surface hardness 58 ndash 63 HRC In case of lower surface hardness than 58 HRC σFE is to be reduced with 20(58 ndash HRC) where HRC is the detected hardness (This may lead to a permissible tooth root stress that varies along the facewidth If so the actual tooth root stresses may also be considered along facewidth)
bull not being shot peened In case of approved shot peening σFE may be increased by 200 for gears where σFE is reduced by 20 due to root grinding Otherwise σFE may be increased by 100 for
6nm le and 100 ndash 5 (mn - 6) for mn gt 6 However the possible adverse influence on the flanks regarding grey staining should be considered and if necessary the flanks should be masked
2) The fillet is not to be ground after surface hardening Regarding possible root grinding see 1)
26 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
38 Mean stress influence Factor YM The mean stress influence factor YM takes into account the influence of other working stress conditions than pure pulsa-tions (R = 0) such as eg load reversals idler gears planets and shrink-fitted gears
YM (YMst) are defined as the ratio between the endurance (or static) strength with a stress ratio R ne 0 and the endurance (or static) strength with R = 0
YM and YMst apply only to a calculation method that assesses the positive (tensile) stresses and is therefore suitable for comparison between the calculated (positive) working stress σF and the permissible stress σFP calculated with YM or YMst
For thin rings (annulus) in epicyclic gears where the com-pression fillet may be decisive special considerations apply see 316
The following method may be used within a stress ratio ndash12 lt R lt 05
381 For idlers planets and PTO with ice class
M1M1R1
1Yor Y MstM
+minus
minus=
where
R = stress ratio = min stress divided by max stress
For designs with the same force applied on both forward- and back-flank R may be assumed to ndash 12
For designs with considerably different forces on forward- and back-flank such as eg a marine propulsion wheel with a power take off pinion R may be assessed as
branchmain theoffacewidth unit per forcepto offacewidt unit per force21minus
For a power take off (PTO) with ice class see 161 c
M considers the mean stress influence on the endurance (or static) strength amplitudes
M is defined as the reduction of the endurance strength am-plitude for a certain increase of the mean stress divided by that increase of the mean stress
Following M values may be used
Endurance limit
Static strength
Case hardened 08 ndash 015 Ys 1) 07
If shot peened 04 06
Nitrided 03 03
Induction or flame hardened
04 06
Not surface hardened steel 03 05
Cast steels 04 06 1)For bevel gears use Ys=2 for determination of M
The listed M values for the endurance limit are independent of the fillet shape (Ys) except for case hardening In princi-ple there is a dependency but wide variations usually only occur for case hardening eg smooth semicircular fillets versus grinding notches
382 For gears with periodical change of rotational direction For case hardened gears with full load applied periodically in both directions such as side thrusters the same formula for YM as for idlers (with R = ndash 12) may be used together with the M values for endurance limit This simplified approach is valid when the number of changes of direction exceeds 100 and the total number of load cycles exceeds 3middot106
For gears of other materials YM will normally be higher than for a pure idler provided the number of changes of direction is below 3middot106 A linear interpolation in a diagram with loga-rithmic number of changes of direction may be used ie from YM = 09 with one change to YM (idler) for 3middot106 changes This is applicable to YM for endurance limit For static strength use YM as for idlers
For gears with occasional full load in reversed direction such as the main wheel in a reversing gear box YM = 09 may be used
383 For gears with shrinkage stresses and unidirectional load For endurance strength
FE
fitM σ
σM1
M21Y+
minus=
σFE is the endurance limit for R = 0
For static strength YMst = 1 and σfit accounted for in 39b
σfit is the shrinkage stress in the fillet (30˚ tangent) and may be found by multiplying the nominal tangential (hoop) stress with a stress concentration factor
n
Ffit m
ρ215scf minus=
384 For shrink-fitted idlers and planets When combined conditions apply such as idlers with shrink-age stresses the design factor for endurance strength is
( ) ( ) FE
fitM σ
σR1M1
M2
M1M1R1
1Yminussdot+
minus
+minus
minus=
Symbols as above but note that the stress ratio R in this par-ticular connection should disregard the influence of σfit ie R normally equal ndash 12
For static strength
M1M1R1
1YMst
+minus
minus=
The effect of σfit is accounted for in 39 b
Classification Notes- No 412 27 May 2003
DET NORSKE VERITAS
385 Additional requirements for peak loads The total stress range (σmax ndash σmin) in a tooth root fillet is not to exceed
F
y
Sσ225
for not surface hardened fillets
FSHV5 for surface hardened fillets
39 Life Factor YN The life factor YN takes into account that in the case of limited life (number of cycles) a higher tooth root stress can be permitted and that lower stresses may apply for very high number of cycles
Decisive for the strength at limited life is the σ ndash N ndash curve of the respective material for given hardening module fillet radius roughness in the tooth root etc Ie the factors YδrelT YRelT YX and YM have an influence on YN
If no σ ndash N ndash curve for the actual material and hardening etc is available the following method may be used
Determination of the σ ndash N ndash curve
a) Calculate the permissible stress σFP for the beginning of the endurance limit (3middot106 cycles) including the influ-ence of all relevant factors as SF YδrelT YRelT YX YM and YC ie σFP = σFE middotYM middotYδrelT middotYRelT middotYX middotYC SF
b) Calculate the permissible laquostaticraquo stress (le 103 load cy-cles) including the influence of all relevant factors as SFst YδrelTst YMst and YCst ( )fitσCstYδrelTstYMstYFstσ
FstS1
FPstσ minussdotsdotsdot=
where σFst is the local tooth root stress which the material can resist without cracking (surface hardened materials) or unacceptable deformation (not surface hardened materials) with 99 survival probability
σFst
Alloyed case hardened steel 1) 2300
Nitriding steel quenched tempered and gas nitrided (surface hardness 700 ndash 800 HV)
1250
Alloyed quenched and tempered steel bath or gas nitrided (surface hardness 500 ndash 700 HV)
1050
Alloyed quenched and tempered steel flame or induction hardened (fillet surface hardness 500 ndash 650 HV)
18 HV + 800
Steel with not surface hardened fillets the smaller value of 2)
18 σB or 225 σy
1) This is valid for a fillet surface hardness of 58 ndash 63 HRC In case of lower fillet surface hardness than 58 HRC σFst is to be reduced with 30(58 ndash HRC) where HRC is the actual hardness Shot peening or grinding notches are not considered to have any significant influence on σFst 2) Actual stresses exceeding the yield point (σy or σ02) will alter the residual stresses locally in the ldquotensionrdquo fillet re-spectively ldquocompressionrdquo fillet This is only to be utilised for gears that are not later loaded with a high number of cycles at lower loads that could cause fatigue in the ldquocompressionrdquo fillet
c) Calculate YN as
NL gt 3middot106
YN = 1 or 001
L
6
N N103Y
sdot= ie Yn = 092 for 1010
The YN = 1 from 3middot106 on may only be used when special material cleanness applies see rules Pt4 Ch2
103ltNLlt3middot106exp
L
6
N N103Y
sdot=
cycles6103forσcycles310forσlog02876exp
FP
FPst
sdot=
NL lt 103 cycles6103forσcycles310forσ
NYFP
FPst
sdot=
or simply use σFPst as mentioned in b) directly
Guidance on number of load cycles NL for various applica-tions
bull For propulsion purpose normally NL = 1010 at full load (yachts etc may have lower values)
bull For auxiliary gears driving generators that normally operate with 70-90 of rated power NL = 108 with rated power may be applied
28 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
310 Relative Notch Sensitivity Factor YδrelT The dynamic (respectively static) relative notch sensitivity factor YδrelT (YδrelTst) indicate to which extent the theoreti-cally concentrated stress lies above the endurance limits (re-spectively static strengths) in the case of fatigue (respectively overload) breakage
YδrelT is a function of the material and the relative stress gra-dient It differs for static strength and endurance limit
The following method may be used
For endurance limit
for not surface hardened fillets
( )
024
s024
relT σ103133q21σ1012201351
Ysdotsdotminus
+sdotsdotminus+= minus
minus
δ
for all surface hardened fillets except nitrided
( )
106q21002451
Y srelT
++=δ
for nitrided fillets
( )
1347q2101421
Y srelT
++=δ
For static strength
for not surface hardened fillets1)
( )( )( )025
02
02502s
relTst σ3000821σ300 1Y0821Y
+
minus+=δ
for surface hardened fillets except nitrided
YδrelTst = 044 YS + 012
for nitrided fillets
YδrelTst = 06 + 02 YS 1) These values are only valid if the local stresses do not
exceed the yield point and thereby alter the residual stress level See also 39b footnote 2
311 Relative Surface Condition Factor YRrelT The relative surface condition factor YRrelT takes into ac-count the dependence of the tooth root strength on the sur-face condition in the tooth root fillet mainly the dependence on the peak to valley surface roughness
YRrelT differs for endurance limit and static strength
The following method may be used
For endurance limit
YRrelT = 1675 ndash 053 (Ry + 1)01 for surface hardened steels and alloyed quenched and tempered steels except nitrided
YRrelT = 53 ndash 42 (Ry + 1)001 for carbon steels
YδrelT = 43 ndash 326 (Ry + 1)0005
for nitrided steels
For static strength
YRrelTst = 1 for all Ry and all materials
For a fillet without any longitudinal machining trace Ry asymp Rz
312 Size Factor YX The size factor YX takes into account the decrease of the strength with increasing size YX differs for endurance limit and static strength
The following may be used
For endurance limit
YX = 1 for mn le 5 generally
YX = 103 ndash 0006 mn for 5 lt mn lt 30
YX = 085 for mn ge 30
for not surface hard-ened steels
YX = 105 ndash 001 mn for 5 lt mn ge 25
YX = 08 for mn ge 25
for surface hardened steels
For static strength
YXst = 1 for all mn and all materials
313 Case Depth Factor YC The case depth factor YC takes into account the influence of hardening depth on tooth root strength
YC applies only to surface hardened tooth roots and is dif-ferent for endurance limit and static strength
In case of insufficient hardening depth fatigue cracks can develop in the transition zone between the hardened layer and the core For static strength yielding shall not occur in the transition zone as this would alter the surface residual stresses and therewith also the fatigue strength
The major parameters are case depth stress gradient permis-sible surface respectively subsurface stresses and subsurface residual stresses
The following simplified method for YC may be used
YC consists of a ratio between permissible subsurface stress (incl influence of expected residual stresses) and permissible surface stress This ratio is multiplied with a bracket con-taining the influence of case depth and stress gradient (The empirical numbers in the bracket are based on a high number of teeth and are somewhat on the safe side for low number of teeth)
YC and YCst may be calculated as given below but calculated values above 10 are to be put equal 10
For endurance limit
+
+=nFFE
C m02ρt31
σconstY
Classification Notes- No 412 29 May 2003
DET NORSKE VERITAS
For static strength
+
+=nFFst
Cst m02ρt31
σconstY
where const and t are connected as
Hardening process
t = endurance limit const =
static strength const =
t550 640 1900
t400 500 1200 Case hard-ening
t300 380 800
Nitriding t400 500 1200
Induction- or flame hardening
tHVmin 11 HVmin 25 HVmin
For symbols see 213
In addition to these requirements to minimum case depths for endurance limit some upper limitations apply to case hard-ened gears
The max depth to 550 HV should not exceed
1) 13 of the top land thickness san unless adequate tip relief is applied (see 110)
2) 025 mn If this is exceeded the following applies addi-tionally in connection with endurance limit
minusminus= 025
mt
1Yn
max550C
314 Thin rim factor YB Where the rim thickness is not sufficient to provide full sup-port for the tooth root the location of a bending failure may be through the gear rim rather than from root fillet to root fillet
YB is not a factor used to convert calculated root stresses at the 30deg tangent to actual stresses in a thin rim tension fillet Actually the compression fillet can be more susceptible to fatigue
YB is a simplified empirical factor used to de-rate thin rim gears (external as well as internal) when no detailed calcula-tion of stresses in both tension and compression fillets are available
Figure 34 Examples on thin rims
YB is applicable in the range 175 lt sRmn lt 35
YB = 115 middot ln (8324 middot mnsR)
(for sRmn ge 35 YB = 1)
(for sRmn le 175 use 315)
σF as calculated in 321 is to be multiplied with YB when sRmn lt 35 Thus YB is used for both high and low cycle fatigue
Note This method is considered to be on the safe side for external gear rims However for internal gear rims without any flange or web stiffeners the method may not be on the safe side and it is advised to check with the method in 315316
315 Stresses in Thin Rims For rim thickness sR lt 35 mn the safety against rim cracking has to be checked
The following method may be used
3151 General The stresses in the standardised 30ordm tangent section tension side are slightly reduced due to decreasing stress correction factor with decreasing relative rim thickness sRmn On the other hand during the complete stress cycle of that fillet a certain amount of compression stresses are also introduced The complete stress range remains approximately constant Therefore the standardised calculation of stresses at the 30ordm tangent may be retained for thin rims as one of the necessary criteria
The maximum stress range for thin rims usually occurs at the 60ordm ndash 80ordm tangents both for laquotensionraquo and laquocompressionraquo side The following method assumes the 75ordm tangent to be the decisive Therefore in addition to the am criterion ap-plied at the 30ordm tangent it is necessary to evaluate the max and min stresses at the 75ordm tangent for both laquotensionraquo (loaded flank) fillet and laquocompressionraquo (back-flank) fillet For this purpose the whole stress cycle of each fillet should be considered but usually the following simplification is justified
Figure 35 Nomenclature of fillets
Index laquoTraquo means laquotensileraquo fillet laquoCraquo means laquocompressionraquo fillet
σFTmin and σFCmax are determined on basis of the nominal rim stresses times the stress concentration factor Y75
σFCmin and σFTmax are determined on basis of superposition of nominal rim stresses times Y75 plus the tooth bending stresses at 75ordm tangent
30 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
3152 Stress concentration factors at the 75ordm tangents The nominal rim stress consists of bending stresses due to local bending moments tangential stresses due to the tan-gential force Ft and radial shear stresses due to Fr
The major influence is given by the bending stresses The influence of the tangential stresses is minor and even though its stress concentration factor is slightly higher than for bending it is considered to be safe enough when the sum of these nominal stresses are combined with the stress concen-tration factor for bending The influence of the radial shear stress is neglected
The stress concentration factor relating nominal rim stresses to local fillet stresses at the 75ordm tangent may be calculated as
75n
R75
n
R
75
ρmsρ185
ms3
Y+
sdot=
where ρ75 is the root radius at the 75ordm tangent ref to mn Usually ρ75 is closed to the tool radius ρao and
ρ75 = ρao
is a safe approximation compensating for the am simplifi-cations to the laquounsaferaquo side
The tooth root stresses of the loaded tooth are decreasing with decreasing relative rim thickness approximately with the empirical factor
15
R
ncorr s
m311Y
minus=
3153 Nominal rim stresses The bending moment applied to the rim consists of a part of the tooth tilt moment Ft (hF + 05 sR) and the bending caused by the radial force Fr
The sectional modulus (first moment of area) which is used for determination of the nominal bending stresses is not nec-essarily the same for the 2 am bending moments If flanges webs etc outside the toothed section contribute to stiffening the rim against various deflections the influence of these stiffeners should be considered Eg an end flange will have an almost negligible influence on the effective sectional modulus for the stresses due to tooth tilt as the deflection caused by the tilt moment is rather small and would not much involve the flange On the other hand the radial forces as for instance from the meshes in an annulus would cause considerable radial deflections that the flange might restrict to a substantial amount When considering the stiff-ening of such flanges or webs on basis of simplified models it is advised to use an effective rim thickness sR = sR + 02 mn for the first moment of area of the rim (toothed part) cross section
For a high number of rim teeth it may be assumed that the rim bending moments in the fillets adjacent to the loaded tooth are of the same magnitude as right under the applied
force This assumption is reasonable for an annulus but rather much on the safe side for a hollow pinion
The influence of Ft on the nominal tangential stress is simpli-fied by half of it for compressive stresses (σ1) and the other half for tensile stresses (σ2) Applying these assumptions the nominal rim stresses adjacent to the loaded tooth are
( ) ( )A2
FW
f RFW
s05hF05 t
R
r
T
RFt1 minus
ϑminus
+minus=σ
( ) ( )A2
FW
f RFW
s05hF05 t
R
r
T
RFt2 +
ϑminus
+=σ
where σ1 σ2 see Fig 35
A = minimum area of cross section (usually bR sR)
WT = the sectional modulus of rim with respect to tooth tilt moment (usually 6sb 2
rR )
WR = the sectional modulus of the rim including the influence of stiffeners as flanges webs etc (WR ge WT)
bR = the width of the rim
R = the radius of the neutral axis in the rim ie from wheel centre to midpoint of rim
( )ϑf = a function for bending moment distribution around the rim For a rim (pinion) with one mesh only the ( )ϑf at the position of load application is 024 For an annulus with 3 or more meshes ( )ϑf at the position of each load application is approx 3 planets ( )ϑf = 019 4 planets ( )ϑf = 014 5 planets ( )ϑf = 011 6 planets ( )ϑf = 009
It must be checked if the max (tensile) stress in the compres-sion fillet really occurs when the fillet is adjacent to the loaded tooth In principle the stress variation through a complete rotation should be considered and the max value used The max value is usually never less than 0 For an annulus eg the tilt moment is zero in the mid position be-tween the planet meshes whilst the bending moment due to the radial forces is half of that at the mesh but with opposite sign
If these formulae are applied to idler gears as eg planets the influence of nominal tangential stresses must be cor-rected by deleting Ft(2 A) for σ1 and using FtA for σ2 Further the influence of Fr on the nominal bending stresses is usually negligible due to the planet bearing support
3154 Root fillet stresses Determination of min and max stresses in the laquotensionraquo fillet
Classification Notes- No 412 31 May 2003
DET NORSKE VERITAS
Minimum stress
751FTmin YσK σ =
Maximum stress
corrF752 Yσ03Yσ KσFTmax +=
where
03 is an empirical factor relating the tension stresses (σF) at the 30ordm tangent to the part of the tension stresses at the 75ordm tangent which add to the rim related stresses (03 also takes into account that full superposition of nominal stresses times stress concentration factors from both laquosides of the corner filletraquo would result in too high stresses)
αβγ sdotsdotsdotsdot= FFvA KKKKKK
Determination of min and max stresses in the laquocompres-sionraquo fillet
Minimum stress
corrF751FCmin Yσ036Yσ Kσ minus=
Maximum stress
752FCmax Yσ Kσ =
where
036 is an empirical factor relating the tension stresses (σF) at the 30ordm tangent to the part of the compression stresses at the 75ordm tangent which add to the rim related stresses
For gears with reversed loads as idler gears and planets there is no distinct laquotensionraquo or laquocompressionraquo fillet The mini-mum stress σFmin is the minimum of σFTmin and σFCmin (usually the latter is decisive) The maximum stress σFTmax is the maximum of σFTmax and σFCmax (usually the former is deci-sive)
316 Permissible Stresses in Thin Rims
3161 General The safety against fatigue fracture respectively overload fracture is to be at least at the same level as for solid gears The laquoordinaryraquo criteria at the 30ordm tangent apply as given in 31 through 313
Additionally the following criteria at the 75ordm tangent may apply
3162 For gt3middot106 cycles The permissible stresses for the laquotensionraquo fillets and for the laquocompressionraquo fillets are determined by means of a relevant fatigue diagram
If the actual tooth root stress (tensile or compressive) ex-ceeds the yield strength to the material the induced residual stresses are to be taken into account
For determination of permissible stresses the following is defined
R = stress ratio ie FTmax
FTminσσ respectively
FCmax
FCmin
σσ
∆σ = stress range ie σFTmax ndash σFTmin resp σFCmax ndash σFCmin
(For idler gears and planets Fmax
FminσσR =
and ∆σ = σFmax minus σFmin)
The permissible stress range ∆σρ for the laquotensionraquo respec-tively laquocompressionraquo fillets can be calculated as
For R gt minus1 FPp σ
R1R1031
13∆σ
minus+
+=
For minus infin lt R lt minus1 FPp σ
R1R10151
13∆σ
minus+
+=
where
σFP = see 32 determined for unidirectional stresses (YM = 1)
If the yield strength σy is exceeded in either tension or com-pression residual stresses are induced This may be consid-ered by correcting the stress ratio R for the respective fillets (tension or compression)
Eg if yFCmin σσ gt (ie exceeded in compression) the
difference yFCmin σσ∆ minus= affects the stress ratio as
∆σ
σ∆σ∆σR
FCmax
y
FCmax
FCmin
+
minus=
++
=
Similarly the stress ratio in the tension fillet may require cor-rection
If the yield strength is exceeded in tension σFTmax gt σy the difference ∆ = σFTmax ndash σy affects the stress ratio as
y
FTmin
FTmax
FTmin
σ∆σ
∆σ∆σR minus=
minusminus
=
Checking for possible exceeding of the yield strength has to be made with the highest torque and with the KAP if this ex-ceeds KA
3163 Forle 103 cycles The permissible stress range ∆σp is not to exceed
For R gt minus1 FPstpst σ
R1R1051
15∆σ
minus+
+=
For ndash infin lt R lt minus1 FPstpst σ
R1R10251
15∆σ
minus+
+=
For all values of R ∆σpst is limited by
not surface hardened F
y
Sσ225
32 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
surface hardened CF
YSHV5
Definition of ∆σ and R see 3162 with particular attention to possible correction of R if the yield strength is exceeded
σFPst see 32 determined for unidirectional stresses (YM = 1) and lt 103 cycles
3164 For 103 lt cycles lt 3106 ∆σp is to be determined by linear interpolation a log-log dia-gram
∆σp at NL load cycles is
6L 103p
exp
L
6
N p ∆σN103∆σ sdot
sdot=
6
3
103p
10p
∆σ
∆σlog28760exp
sdot
=
Classification Notes- No 412 33 May 2003
DET NORSKE VERITAS
4 Calculation of Scuffing Load Capacity
41 Introduction High surface temperatures due to high loads and sliding ve-locities can cause lubricant films to break down This sei-zure or welding together of areas of tooth surface is termed scuffing
In contrast to pitting and fatigue breakage which show a dis-tinct incubation period a single short overloading can lead to a scuffing failure In the ISO-TR13989 two criteria are mentioned The method used in this Classification Note is based on the principles of the flash temperature criterion
Note Bulk temperature in excess of 120ordmC for long periods may have an adverse effect on the surface durability see 211
42 General Criteria In no point along the path of contact the local contact tem-perature may exceed the permissible temperature ie
oilS
oilSB S
ϑ+ϑminusϑ
leϑ
50SB minusϑleϑ
where
Bϑ = max contact temperature along the path of contact
maxflaMBB ϑ+ϑ=ϑ
MBϑ = bulk temperature see 434
maxflaϑ = max flash temperature along the path of con-tact see 44
Sϑ = scuffing temperature see below
oilϑ = oil temperature before it reaches the mesh (max applicable for the actual load case to be used ie normally alarm temperature except for ice classes where a max expected tem-perature applies)
SS = required safety factor according to the Rules
The scuffing temperature Sϑ may be calculated as
L2
wrelT
002
40S XFZGX
ν100112085780
sdot
sdot++=ϑ
where
XwrelT = relative welding factor
XwrelT
Through hardened steel 10
Phosphated steel 125
Copper-plated steel 150
Nitrided steel 150
Less than 10 retained austenite
115
10 ndash 20 retained austenite 10
Casehardened steel
20 ndash 30 retained austenite 085
Austenitic steel 045
FZG = load stage according to FZG-Test A8390 (Note This is the load stage where scuffing oc-curs However due to scatter in test results cal-culations are to be made with one load stage less than the specification)
XL = lubricant factor
= 10 for mineral oils
= 08 for polyalfaolefins
= 07 for non-water-soluble polyglycols
= 06 for water-soluble polyglycols
= 15 for traction fluids
= 13 for phosphate esters
ν40 = kinematic oil viscosity at 40˚C (mm2s)
Application of other test methods such as the Ryder the FZG-Ryder R46574 and the FZG L-42 Test 141195110 may be specially considered
For high speed gears with very short time of contact Sϑ may be increased as follows provided use of EP-oils
Addition to the calculated scuffing temperature Sϑ
If micros 18tc ge no addition
If micros 18tc lt add ( )cwrelT t18X18 minussdotsdot
where
ct = contact time ( micros ) which is the time needed to cross the Hertzian contact width
( ) [ ]microsu1cosβn
uσ340tb1
Hc +sdotsdot
sdot=
σH as calculated in 221
For bevel gears use vu in stead of u
34 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
43 Influence Factors
431 Coefficient of friction The following coefficient of friction may apply
L025a
005oil
02
redC ΣC
Bt X R ηρv
w0048micro minus
=
where
wBt = specific tooth load (Nmm)
vΣC = sum of tangential velocities at pitch point At pitch line velocities gt 50 ms the limiting value of vΣC at v = 50 ms is to be used
ρredC = relative radius of curvature (transversal plane) at the pitch point
Cylindrical gears
HαHβvγAbt
Bt KKKKKb
Fw sdotsdotsdotsdotsdot= (see 1)
twΣC αsin v2v =
bCredC β cos ρρ =
Bevel gears
HαHβvγAtmb
Bt KKKKKb
Fw sdotsdotsdotsdotsdot= (see 1)
vtmtΣC αsin v2v =
bmvCredC β cos ρρ =
ηoil = dynamic viscosity (mPa s) at oilϑ calculated as
1000
ρ νη oiloil =
where ρ in kgm3 approximated as
( ) 70 15oil15 minusϑminusρ=ρ and νoil is kinematic viscosity at oilϑ and may be calculated by means of the following equation
( ) ( )++=+ 08νloglog08νloglog 100oil ( )
sdotminus
ϑ+minus313log373log
273log373log oil
( ) ( )( )08νloglog08νloglog 10040 +minus+
Ra = composite arithmetic mean roughness (micron) of pinion an wheel calculated as ( )2a1aa RR 05R +=This is defined as the roughness on the new flanks ie as manufactured
XL = see 42
432 Effective tip relief Ceff Ceff is the effective tip relief that amount of tip relief which compensates for the elastic deformation of the gear mesh ie
zero load at the tooth tip It is assumed (simplified) to be equal for both pinion and wheel
Cylindrical gears
for helical
γ
γ=cb
KKFC Abt
eff (see 1)
For spur
cbKKF
C Abteff
γ= (see 1)
Alternatively for spur and helical gears the non-linear ap-proach in 111 may be used (taking Ceff=δ)
Bevel gears
bcKFC Ambt
effγ
= (see 1)
where
γ
α
ε2ε44c
+sdot
=γ
433 Tip relief and extension Cylindrical gears
The extension of the tip relief is not to result in an effective contact ratio 1ltεα when the gear is unloaded (exceptions to this may only apply for applications where the gear is not to run at light loads) This means that the unrelieved part of the path of contact is to be minimum pbt It is further assumed that this unrelieved part is placed centrally on the path of contact
If root relief applies it has to be calculated as equivalent tip relief Ie pinion root relief (at mesh position A) is added to Ca2 and wheel root relief (at mesh position E) is added to Ca1 If no design tip relief or root relief on the mating gear is specified (ie if Ca1+Croot2 = 0 and visa versa) use the run-ning in amount see 112
Bevel gears
Bevel gears are to have heightwise crowning ie no distinct relievedunrelieved area This may be treated as tip and root relief For calculation purposes the root relief is combined with the tip relief of the mating member into an equivalent tip relief If no resulting tip relieves are specified the equivalent tip relives may be calculated as an approxima-tion based on the tool crowning Ca tool (per mille of tool module m0) as follows
2root1aeq1a CCC +=
1root2aeq2a CCC +=
where
Classification Notes- No 412 35 May 2003
DET NORSKE VERITAS
2
0
n0101atoolal m
)m2(mA1middotmCC
minus+minussdot=
( )
sdotminusminussdotsdotminus+= vtvb1
2vb1
21vavt1n1 αtan dddαsin 05x1mA
2
0
n020atool22a m
)mm(2A1middotmCC
minus+minussdot=
( )
sdotminusminussdotsdotminus+= vtvb2
2vb2
2va2vt2n2 αtan dddαsin 05x1mA
2
0
20atool1root1 m
A1mCC
minussdotsdot=
2
0
10atool2root2 m
A1mCC
minussdotsdot=
If the pressure angles of the cutter blades are modified (and verified) to balance the tip relieves the following may be assumed
( )eq2aeq1aeq2aeq1a C andC calculated of sum50CC sdot==
Throughout the following Ca1 and Ca2 mean the equivalent tip relieves Ca1eq and Ca2eq
434 Bulk temperature The bulk temperature may be calculated as
flaaveragempsoilMB X X 50 ϑ+ϑ=ϑ
where
Xs = lubrication factor
= 12 for spray lubrication
= 10 for dip lubrication (provided both pinion and wheel are dip lubricated and tip speed lt 5 ms)
= 10 for spray lubrication with additional cooling spray (spray on both pinion and wheel or spray on pinion and dip lubrication of wheel)
= 02 for meshes fully submerged in oil
Xmp = contact factor ( )pmp nX += 150
np = number of mesh contact on the pinion (for small gear ratii the number of wheel meshes should be used if higher)
flaaverageϑ
= average of the integrated flash temperature (see 44) along the path of contact
( )
AE
E
Ayyyfla
flaaverage
d
ΓminusΓ
ΓΓϑ=ϑint =
For high speed gears (v gt 50 ms) it may be necessary to assess the bulk temperature on the basis of a thermal rating of the entire gear transmission
44 The Flash Temperature flaϑ
441 Basic formula The local flash temperature flaϑ may be calculated as
( ) 41redy
y2ly
211
43Btcorrfla
unXwX3250
y ρ
ρminusρ
micro=ϑ Γ
(For bevel gears replace u with uv)
and is to be calculated stepwise along the path of contact from A to E
where
micro = coefficient of friction see 431
Xcorr = correction factor taking empirically into account the increased scuffing risk in the beginning of the approach path due to mesh starting without any previously built up oil film and possible shuffling away oil before meshing if insufficient tip relief
( )
3
AD
yaeffcorr 50
CC1X
ΓminusΓε
Γminus+=
α
Ca = tip relief of driven member Xcorr is only applicable in the approach path and if Ca lt Ceff otherwise 10
wBt = unit load see 431
yXΓ = load sharing factor see 443
n1 = pinion rpm
Γy ρly etc see 442
442 Geometrical relations The various radii of flank curvature (transversal plane) are
ρ1y = pinion flank radius at mesh point y
ρ2y = wheel flank radius at mesh point y
ρredy = equivalent radius of curvature at mesh point y
y2y1
y2y1redy
ρ+ρ
ρρ=ρ
Cylindrical gears
twy
1y αsin au1Γ1
ρ+
+=
twy
2y αsin au1Γu
ρ+
minus=
Note that for internal gears a and u are negative
36 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
Bevel gears
vtvv
y1y αsin a
u1Γ1
ρ+
+=
vtvv
yv2y αsin a
u1Γu
ρ+
minus=
Γ is the parameter on the path of contact and y is any point between A end E
At the respective ends Γ has the following values
Root piniontip wheel
Cylindrical gears
( )
minus
minusminus= 1
αtan 1dd
zzΓ
tw
2b2a2
1
2A
Bevel gears
( )
minus
minusminus= 1
αtan 1dd
uΓvt
2vb2va2
vA
Tip pinionroot wheel
Cylindrical gears
( )
1αtan
1ddΓ
tw
2b1a1
E minusminus
=
Bevel gears
( )
1αtan
1ddΓ
vt
2vb1va1
E minusminus
=
At inner point of single pair contact
Cylindrical gears
tw1
EB α tan zπ2ΓΓ minus=
Bevel gears
vtv1
EB α tan zπ2ΓΓ minus=
At outer point of single pair contact
Cylindrical gears
tw1AD α tan z
π2ΓΓ +=
Bevel gears
vt v1
AD αtan z π2ΓΓ +=
At pitch point ΓC = 0
The points F and G (only applicable to cylindrical gears) limiting the extension of tip relief (so as to maintain a mini-mum contact ratio of unity for unloaded gears) are at
2
BAF
Γ+Γ=Γ
2
EDG
Γ+Γ=Γ
443 Load sharing factor XΓ The load sharing factor XΓ accounts for the load sharing be-tween the various pairs of teeth in mesh along the path of contact
XΓ is to be calculated stepwise from A to E using the pa-rameter Γy
4431 Cylindrical gears with β = 0 and no tip relief
Figure 41
ByAAB
Ay for31
31X
yΓltΓleΓ
ΓminusΓ
ΓminusΓ+=Γ
DyB for1Xy
ΓltΓleΓ=Γ
EyDDE
yE for31
31X
yΓleΓltΓ
ΓminusΓ
ΓminusΓ+=Γ
4432 Cylindrical gears with β = 0 and tip relief Tip relief on the pinion reduces XΓ in the range G ndash E and increases correspondingly XΓ in the range F ndash B
Tip relief on the wheel reduces XΓ in the range A ndash F and increases correspondingly XΓ in the range D ndash G
Following remains generally
DyB for1Xy
ΓleΓleΓ=Γ
21XXGF== ΓΓ
In the following it must be distinguished between Ca lt Ceff respectively Ca gt Ceff This is shown by an example below where Ca1 lt Ceff and Ca2 gt Ceff
Classification Notes- No 412 37 May 2003
DET NORSKE VERITAS
Figure 42
Note When Ca gt Ceff the path of contact is shortened by A ndash A respectively E ndash E The single pair contact path is extended into B respectively D If this shift is significant it is necessary to consider the negative effect on surface durability (B) and bending stresses (B and D)
Range A - F
For effa2 CC le
minus=Γ
eff
2a
CC1
31X
A
FyAeff
2a
AF
Ay for C3
C61XX
AyΓleΓleΓ
sdot
+ΓminusΓ
ΓminusΓ+= ΓΓ
For effa2 CC ge
AyAfor0Xy
ΓleΓleΓ=Γ
with ( )21
CC
1CC
eff
2a
eff
2a
AFAAminus
minusΓminusΓ+Γ=Γ
FyAeff
2a
AF
Ay
eff
2a for 21
CC
CC1X
yΓleΓleΓ
minus
ΓminusΓ
ΓminusΓ+minus=Γ
Range F - B
For effa1 CC le
ByFeff
1a
FB
Fy forC3
C61
21X
yΓleΓleΓ
sdot
+ΓminusΓ
ΓminusΓ+=Γ
For effa1 CC ge
ByFeff
1a
FB
Fy for21
CC
21X
yΓleΓleΓ
minus
ΓminusΓ
ΓminusΓ+=Γ
ByB for1Xy
ΓleΓleΓ=Γ
with 1
CC2
eff
1a
FBFB
minus
ΓminusΓ+Γ=Γ
Range D ndash G
For effa2 CC le
GyDeff
2a
DG
Dy
eff
2a for C 3C
61
C 3C
32X
yΓleΓleΓ
+
ΓminusΓ
ΓminusΓminus+=Γ F
or effa2 CC ge
DyD for1Xy
ΓleΓleΓ=Γ
with ( )21
CC
1CC
eff
2a
eff
2a
DGDDminus
minusΓminusΓ+Γ=Γ
GyDeff
2a
DG
Dy
eff
2a for21
CC
CCX ΓleΓleΓ
minus
ΓminusΓ
ΓminusΓminus=Γ
Range G - E
For effa1 CC le
minus=Γ
eff
1a
CC1
31X
E
EyGeff
1a
GE
Gy forC3
C61
21X
yΓleΓleΓ
sdot
+ΓminusΓ
ΓminusΓminus=Γ
For effa1 CC gt
EyGeff
1a
GE
Gy for21
CC
21X
yΓleΓleΓ
minus
ΓminusΓ
ΓminusΓminus=Γ
EyE for0Xy
ΓleΓleΓ=Γ
with 1
CC2
eff
1a
GEGE
minus
ΓminusΓ+Γ=Γ
4433 Gears with β gt 0 buttressing Due to oblique contact lines over the flanks a certain but-tressing may occur near A and E
This applies to both cylindrical and bevel gears with tip relief lt Ceff The buttressing Xbutt is simplified as a linear function within the ranges A ndash H respectively I ndash E
Figure 43
1εwhen13X βbutt EAge=
1εwhenε031X ββbutt EAlt+=
38 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
Cylindrical gears
bIEAH βsin 02ΓΓΓΓ =minus=minus
Bevel gears
bmIEAH βsin 02ΓΓΓΓ =minus=minus
4434 Cylindrical gears with 2εγ le and no tip relief
yXΓ is obtained by multiplication of
yXΓ in 4431 with
Xbutt in 4433
4435 Gears with 2εγ gt and no tip relief
Applicable to both cylindrical and bevel gears
IyH for1Xy
ΓleΓleΓε
=α
Γ
IyHybutt and forX1Xy
ΓgtΓΓltΓε
=α
Γ
Figure 44
4436 Cylindrical gears with 2εγ le and tip relief
yXΓ is obtained by multiplication of
yXΓ in 4432 with Xbutt
in 4433
4437 Cylindrical gears with 2εγ gt and tip relief
Tip relief on the pinion (respectively wheel) reduces XГ in the range G ndash E (respectively A ndash F) and increases XГ in the range F ndash G
yXΓ is obtained by multiplication of
yXΓ as described below
with Xbutt in 4433
In the XГ example below the influence of tip relief is shown (without the influence of Xbutt) by means of
eff2aeff1a CC and CC ltgt
Tip relief gt Ceff causes new end points A respectively E of the path of contact
Figure 45
Range A ndash F
( ) ( )( ) eff
a2αa1α
AF
Ay
eff
2aeff
C 12C 13εC 1ε
CCCX
y +εε++minus
ΓminusΓ
ΓminusΓ+
εminus
=ααα
Γ
eff2aFyA CC if for leΓleΓleΓ
eff2aFyA CifCfor and geΓleΓleΓ
eff2aAyA CC iffor0Xy
gtΓleΓleΓ=Γ
( ) ( ) ( )( ) ( ) a2αa1α
αeffa2AFAA C 1ε 3C 1ε
1ε 2 CC with++minus+minus
ΓminusΓ+Γ=Γ
Range F ndash G
( )( )( ) GyF
eff
2a1a forC 1 2CC 11X
yΓleΓleΓ
+εε+minusε
+ε
=αα
α
αΓ
Range G ndash E
( ) ( )( ) eff
2a1a
GE
Gy
C 1 2C 1C 1 3XX
GFy +εεminusε++ε
ΓminusΓ
ΓminusΓminus=
αα
ααΓΓ minus
effa1EyG CC if for leΓleΓleΓ
effa1EyG CC if Γfor and geΓleΓle
eff1aEyE CC if for0Xy
geΓleΓleΓ=Γ
( ) ( ) ( )( ) ( ) 2a1a
eff1aGEEE C 1C 1 3
1 2 CC withminusε++ε+εminus
ΓminusΓminusΓ=Γαα
α
4438 Bevel gears with εγ more than approx 18 and heightwise crowning For Ca1 = Ca2 = Ceff the following applies
Figure 46
Classification Notes- No 412 39 May 2003
DET NORSKE VERITAS
( )AEM 50 Γ+Γ=Γ
( )( )2AD
3
2My651X
y ΓminusΓε
ΓminusΓminus
ε=
ααΓ
For tip relief lt Ceff yXΓ is found by linear interpolation be-
tween ( )effay CCX =Γ and ( )0CayX =Γ as in 4435
The interpolation is to be made stepwise from A to M with the influence of Ca2 and from M to E with the influence of Ca1 (For 2a1a CC ne there is a discontinuity at M)
Eg with Ca1 = 04 Ceff and Ca2 = 055 Ceff then
Range A ndash M
)CC()0C( eff2ay2ayyX 550X 450X =Γ=ΓΓ +=
Range M ndash E
)CC()0C( eff1ay1ayyX 40X 06X =Γ=ΓΓ +=
For tip relief gt Ceff the new end points A and E are found as
( )
minusΓminusΓ
ε+Γ=Γ α 1
CC
6 eff
2aADAA
( )
minusΓminusΓ
ε+Γ=Γ α 1
CC
6 eff
1aADEE
Range A ndash A
0Xy=Γ
Range A ndash M
( )( )
ΓminusΓ
ΓminusΓminus
minusε=
αΓ 2
MA
2My
eff
2a1
CC4
351Xy
Range M ndash E
( )( )
ΓminusΓ
ΓminusΓminus
minusε=
αΓ 2
ME
2My
eff
1a1
CC4
351Xy
Range E ndash E
0Xy=Γ
40 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
Appendix A Fatigue Damage Accumulation
The Palmgren-Miner cumulative damage calculation princi-ple is used The procedure may be applied as follows
A1 Stress Spectrum From the individual torque classes the torques (Ti) at the peak values of class intervals and the associated number of cycles (NLi) for both pinion and wheel are to be listed from the highest to the lowest torque
(In case of a cyclic torque variation within the torque classes it is advised to use the peak torque If the cyclic variation is such that the same teeth will repeatedly suffer the peak torque this is a must)
The stress spectra for tooth roots and flanks (σFi σHi) with all relevant factors (except KA) are to be calculated on the basis of the torque spectrum The load dependent K-factors are to be determined for each torque class
A2 σminusN-curve The stress versus load cycle curves for tooth roots and flanks (both pinion and wheel) are to be drawn on the basis of per-missible stresses (ie including the demanded minimum safety factors) as determined in 2 respectively 3 If different safety levels for high cycle fatigue and low cycle fatigue are desired this may be expressed by different demand safety factors applied at the endurance limit respectively at static strength
A3 Damage accumulation The individual damage ratio Di at ith stress level is defined as
Fi
Lii N
ND =
where
NLi = The number of applied cycles at ith stress
NFi = The number of cycles to failure at ith stress
Basically stresses σi below the permissible stress level for infinite life (if a constant ZN or YN is accepted) do not con-tribute to the damage sum However calculating the actual safety factor Sact as described below all the σi for which the product S σi is bigger than or equal to the permissible stress level for infinite life contribute to the damage sum and thus to the determination of Sact The final value of S is decisive
(NFi can be found mathematically by putting the permissible stress σpi equal the actual stress σi thereby finding the actual life factor This life factor can be solved with regard to load cycles ie NFi)
The damage sum ΣDi is not to exceed unity
If ΣDi ne 1 the safety against cumulative fatigue damage is different from the applied demand safety factor For deter-mination of this theoretical safety factor an iteration proce-dure is required as described in the following flowchart
S is correction factor with which the actual safety factor Sact can be found
Sact is the demand safety factor (used in determination of the permissible stresses in the σ ndash N ndash curve) times the correc-tion factor S
The full procedure is to be applied for pinion and wheel tooth roots and flanks
Note If alternating stresses occur in a spectrum of mainly pulsating stresses the alternating stresses may be replaced by equiva-lent pulsating stresses ie by means of division with the ac-tual mean stress influence factor YM
Classification Notes- No 412 41 May 2003
DET NORSKE VERITAS
Appendix B Application Factors for Diesel Driven Gears
For diesel driven gears the application factor KA depends on torsional vibrations Both normal operation and misfiring conditions have to be considered
Normally these two running conditions can be covered by only one calculation
B1 Definitions
Normal operation KAnorm = 0
normv0
TTT +
where
T0 = rated nominal torque
Tv norm = vibratory torque amplitude for normal opera-tion (see rules Pt4 Ch3 Sec1 G301 for defi-nition of ldquonormalrdquo irregularity)
Misfiring operation KA misf = 0
misfv
TTT +
where
T = remaining nominal torque when one cylinder out of action
Tv misf = vibratory torque amplitude in misfiring con-dition This refers to a permissible misfiring condition ie a condition that does not require automatic or immediate corrective actions as speed or pitch reduction
The normal operation is assumed to last for a very high num-ber of cycles such as 1010
The misfiring operation is assumed to last for a limited dura-tion such as 107 cycles
B2 Determination of decisive load Assuming life factor at 1010 cycles as YN = ZN = 092 which usually is relevant the calculation may be performed only once with the combination having the highest value of appli-cation factorlife factor
For bending stresses and scuffing the higher value of
092
K normA and 098
K misfA
For contact stresses the higher value of
092K normA and
113K misfA (but
097K misfA for nitrided gears)
B3 Simplified procedure Note that this is only a guidance and is not a binding con-vention
T = oTZ
1Zsdot
minus where Z = number of cylinders
( ) ideal Videal Vmisf Vnorm V TTT24ZT +minus=
where TV ideal = vibratory torque with all cylinders perfectly equal See also rules Pt4 Ch3 Sec1 G300
When using trends from torsional vibration analysis and measurements the following may be used
TV ideal To is close to zero for engines with few cylinders and using a suitable elastic coupling and increases with relative coupling stiffness and number of cylinders
TV misf To may be high for engines with few cylinders and decreases with number of cylinders
This can be indicated as
200Z
TT
ο
idealV asymp and 80Z04
TT
o
misfV minusasymp
Inserting this into the formulae for the two application fac-tors the following guidance can be given
118112K misfA minusasymp
115110K normA minusasymp
Since KA norm is to be combined with the lower life factors the decisive load condition will be the normal one and a KA of 115 will cover most relevant cases when a suitable elas-tic coupling is chosen
42 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
Appendix C Calculation of Pinion-Rack
Pinion-racks used for elevating of mobile offshore units are open gears that are subjected to wear and tear With normal specifications such as only a few hundred total operation cycles (site to site) the tooth bending stresses for static strength will be decisive for the lay out however with ex-ception of surface hardened pinions where case crushing has to be considered
In the following the use of part 1 and 3 for pinion-racks is shown including relevant simplifications
C1 Pinion tooth root stresses Since the load spectrum normally is dominated by high torques and few load cycles (in the range up to 1-2000) the static strength is decisive
The actual stress is calculated as
β1FSaFan1
tF1 KYY
mbFσ sdotsdotsdotsdot
=
b1 is limited to b2 + 2middotmn
YFa and YSa replace YF and YS because load application at tooth tip has to be assumed for such inaccurate gears
Pinions often use a non-involute profile in the dedendum part of the flank eg a constant radius equal the radius of curva-ture at reference circle For such pinions sFn and hFa are to be measured directly on a sectional drawing of the pinion tooth
Due to high loads and narrow facewidths it may be assumed that KFβ2 = KHβ = 10 However when b1 gt b2 then KFβ1 gt10
If no detailed documentation of KFβ1 is available the fol-lowing may be used
KFβ1 = 1 + 015middot(b1b2 ndash 1)
The permissible stress (not surface hardened) is calculated as
δrelTstF
Fst1FP1 Y
Sσσ sdot=
The mean stress influence due to leg lifting may be disre-garded
The actual and permissible stresses should be calculated for the relevant loads as given in the rules
C2 Rack tooth root stresses The actual stress is calculated as
SaFan2
tF2 YY
mbFσ sdotsdotsdot
=
See C1 for details
The permissible stress is calculated as
δrelTstF
Fst2FP2 Y
Sσσ sdot=
For alloyed steels (Ni Cr Mo) with high toughness and duc-tility the value of YδrelTst may be put equal to YSa
C3 Surface hardened pinions For surface hardened pinions the maximum load is not to cause crushing of the hardened layer of the flank
In principle the calculation described in 213 may be used but when the theoretical Hertzian stress exceeds the ap-proximately 18 times the yield strength of the rack material plastic deformation will occur This will limit the peak Hertzian stress but increases the contact width and thus the penetration of stresses into the depth
An approximation may be based on an assessment of contact width determined by means of equal areas under the theoreti-cal (elastic range) Hertzian contact and the elasto-plastic contact stress (limited to 18 middot σy) with the unknown width
DET NORSKE VERITAS
CONTENTS
1 Basic Principles and General Influence Factors 4 11 Scope and Basic Principles 4 12 Symbols Nomenclature and Units 4 13 Geometrical Definitions5 14 Bevel Gear Conversion Formulae and Specific
Formulae6 15 Nominal Tangential Load Ft Fbt Fmt and Fmbt 6 16 Application Factors KA and KAP 7 161 KA 7 162 KAP7 163 Frequent overloads8 17 Load Sharing Factor Kγ 8 171 General method8 172 Simplified method 8 18 Dynamic Factor Kv 8 181 Single resonance method 8 182 Multi-resonance method 10 19 Face Load Factors KHβ and KFβ 10 191 Relations between KHβ and KFβ10 192 Measurement of face load factors 10 193 Theoretical determination of KHβ11 194 Determination of fsh 12 195 Determination of fdefl13 196 Determination of fbe 13 197 Determination of fma 13 198 Comments to various gear types 13 199 Determination of KHβ for bevel gears 13 110 Transversal Load Distribution Factors KHα and KFα14 111 Tooth Stiffness Constants cacute and cγ 14 112 Running-in Allowances 15 2 Calculation of Surface Durability17 21 Scope and General Remarks 17 22 Basic Equations 17 221 Contact stress 17 222 Permissible contact stress 17 23 Zone Factors ZH ZBD and ZM18 231 Zone factor ZH 18 232 Zone factors ZBD 18 233 Zone factor ZM 18 234 Inner contact point 18 24 Elasticity Factor ZE 18 25 Contact Ratio Factor Zε18 26 Helix Angle Factor Zβ18 27 Bevel Gear Factor ZK18 28 Values of Endurance Limit σHlim and Static
Strength 5H10σ 3H10σ 18 29 Life Factor ZN 19 210 Influence Factors on Lubrication Film ZL ZV and
ZR 19 211 Work Hardening Factor ZW 20 212 Size Factor ZX20 213 Subsurface Fatigue20 3 Calculation of Tooth Strength 22 31 Scope and General Remarks 22 32 Tooth Root Stresses 22 321 Local tooth root stress22 322 Permissible tooth root stress 22
33 Tooth Form Factors YF YFa 23 331 Determination of parameters 23 332 Gearing with εαn gt 2 24 34 Stress Correction Factors YS YSa 24 35 Contact Ratio Factor Yε 25 36 Helix Angle Factor Yβ 25 37 Values of Endurance Limit σFE 25 38 Mean stress influence Factor YM 26 381 For idlers planets and PTO with ice class 26 382 For gears with periodical change of rotational
direction 26 383 For gears with shrinkage stresses and unidirectional
load 26 384 For shrink-fitted idlers and planets 26 385 Additional requirements for peak loads 27 39 Life Factor YN 27 310 Relative Notch Sensitivity Factor YδrelT 28 311 Relative Surface Condition Factor YRrelT 28 312 Size Factor YX 28 313 Case Depth Factor YC 28 314 Thin rim factor YB 29 315 Stresses in Thin Rims 29 3151 General 29 3152 Stress concentration factors at the 75ordm tangents 30 3153 Nominal rim stresses 30 3154 Root fillet stresses 30 316 Permissible Stresses in Thin Rims 31 3161 General 31 3162 For gt3middot106 cycles 31 3163 For le 103 cycles 31 3164 For 103 lt cycles lt 3106 32 4 Calculation of Scuffing Load Capacity 33 41 Introduction 33 42 General Criteria 33 43 Influence Factors 34 431 Coefficient of friction 34 432 Effective tip relief Ceff 34 433 Tip relief and extension 34 434 Bulk temperature 35 44 The Flash Temperature flaϑ 35 441 Basic formula 35 442 Geometrical relations 35 443 Load sharing factor XΓ 36 Appendix A Fatigue Damage Accumulation 40 A1 Stress Spectrum 40 A2 σminusN-curve 40 A3 Damage accumulation 40 Appendix B Application Factors for Diesel Driven
Gears 41 B1 Definitions 41 B2 Determination of decisive load 41 B3 Simplified procedure 41 Appendix C Calculation of Pinion-Rack 42 C1 Pinion tooth root stresses 42 C2 Rack tooth root stresses 42 C3 Surface hardened pinions 42
4 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
1 Basic Principles and General Influence Factors
11 Scope and Basic Principles The gear rating procedures given in this Classification Note are mainly based on the ISO-6336 Part 1-5 (cylindrical gears) and partly on ISO 10300 Part 1-3 (bevel gears) and ISO Technical Reports on Scuffing and Fatigue Damage Accumulation but especially applied for marine purposes such as marine propulsion and important auxiliaries onboard ships and mobile offshore units
The calculation procedures cover gear rating as limited by contact stresses (pitting spalling or case crushing) tooth root stresses (fatigue breakage or overload breakage) and scuff-ing resistance Even though no calculation procedures for other damages such as wear grey staining (micropitting) etc are given such damages may limit the gear rating
The Classification Note applies to enclosed parallel shaft gears epicyclic gears and bevel gears (with intersecting axis) However open gear trains may be considered with regard to tooth strength ie part 1 and 3 may apply Even pinion-rack tooth strength may be considered but since such gear trains often are designed with non-involute pinions the calculation procedure of pinion-racks is described in Appen-dix C
Steel is the only material considered
The methods applied throughout this document are only valid for a transverse contact ratio 1 lt εα lt 2 If εα gt 2 either special considerations are to be made or suggested simplifi-cation may be used
All influence factors are defined regarding their physical interpretation Some of the influence factors are determined by the gear geometry or have been established by conven-tions These factors are to be calculated in accordance with the equations provided Other factors are approximations which are clearly stated in the text by terms as laquomay be cal-culated asraquo These approximations are substitutes for exact evaluations where such are lacking or too extensive for prac-tical purposes or factors based on experience In principle any suitable method may replace these approximations
Bevel gears are calculated on basis of virtual (equivalent) cylindrical gears using the geometry of the midsection The virtual (helical) cylindrical gear is to be calculated by using all the factors as a real cylindrical gear with some exceptions These exceptions are mentioned in connection with the ap-plicable factors Wherever a factor or calculation procedure has no reference to either cylindrical gears or bevel gears it is generally valid ie combined for both cylindrical and bevel
In order to minimise the volume of this Classification Note such combinations are widely used and everywhere it is necessary to distinguish it is clearly pointed out by local headings such as
Cylindrical gears
Bevel gears
The permissible contact stresses tooth root stresses and scuffing load capacity depend on the safety factors as re-quired in the respective Rule sections
Terms as endurance limit and static strength are used throughout this Classification Note
Endurance limit is to be understood as the fatigue strength in the range of cycles beyond the lower knee of the σndashN curves regardless if it is constant or drops with higher number of cycles
Static strength is to be understood as the fatigue strength in the range of cycles less than at the upper knee of the σndashN curves
For gears that are subjected to a limited number of cycles at different load levels a cumulative fatigue calculation applies Information on this is given in Appendix A
When the term infinite life is used it means number of cy-cles in the range 108ndash1010
12 Symbols Nomenclature and Units The symbols are generally from ISO 701 ISOR31 and ISO 1328 with a few additional symbols Only SI units are used
The main symbols as influence factors (K Z Y and X with indeces) etc are presented in their respective headings Symbols which are not explained in their respective Secs are as follows
a = centre distance (mm) b = facewidth (mm) d = reference diameter (mm) da = tip diameter (mm) db = base diameter (mm) dw = working pitch diameter (mm) ha = addendum (mm) ha0 = addendum of tool ref to mn hfp = dedendum of basic rack ref to mn (= ha0) hFe = bending moment arm (mm) for tooth root
stresses for application of load at the outer point of single tooth pair contact
hFa = bending moment arm (mm) for tooth root stresses for application of load at tooth tip
HB = Brinell hardness HV = Vickers hardness HRC = Rockwell C hardness mn = normal module n = rev per minute NL = number of load cycles qs = notch parameter Ra = average roughness value (microm) Ry = peak to valley roughness (microm) Rz = mean peak to valley roughness (microm) san = tooth top land thickness (mm) sat = transverse top land thickness (mm) sFn = tooth root chord (mm) in the critical section spr = protuberance value of tool minus grinding stock
equal residual undercut of basic rack ref to mn T = torque (Nm) u = gear ratio (per stage) v = linear speed (ms) at reference diameter
Classification Notes- No 412 5 May 2003
DET NORSKE VERITAS
x = addendum modification coefficient z = number of teeth zn = virtual number of spur teeth αn = normal pressure angle at ref cylinder αt = transverse pressure angle at ref cylinder αa = transverse pressure angle at tip cylinder αwt = transverse pressure angle at pitch cylinder β = helix angle at ref cylinder βb = helix angle at base cylinder βa = helix angle at tip cylinder εα = transverse contact ratio εβ = overlap ratio εγ = total contact ratio ρa0 = tip radius of tool ref to mn ρfp = root radius of basic rack ref to mn ( = ρa0) ρC = effective radius (mm) of curvature at pitch point ρF = root fillet radius (mm) in the critical section σB = ultimate tensile strength (Nmm2) σy = yield strength resp 02 proof stress (Nmm2)
Index 1 refers to the pinion 2 to the wheel
Index n refers to normal section or virtual spur gear of a heli-cal gear
Index w refers to pitch point
Special additional symbols for bevel gears are as follows
Σ = angle between intersection axis Kϑ = angle modification (Klingelnberg)
m0 = tool module (Klingelnberg) δ = pitch cone angle xsm = tooth thickness modification coefficient (mid-
face) R = pitch cone distance (mm)
Index v refers to the virtual (equivalent) helical cylindrical gear
Index m refers to the midsection of the bevel gear
13 Geometrical Definitions For internal gearing z2 a da2 dw2 d2 and db2 are negative x2 is positive if da2 is increased ie the numeric value is de-creased
The pinion has the smaller number of teeth ie
11
2 ge=zz
u
For calculation of surface durability b is the common face-width on pitch diameter
For tooth strength calculations b1 or b2 are facewidths at the respective tooth roots If b1 or b2 differ much from b above they are not to be taken more than 1 module on either side of b
Cylindrical gears
tan αt = tan αn cos β tan βb = tan β cos αt
tan βa = tan β da d
cos αa = dbda
d = z mn cos β mt = mn cos β db = d cos αt = dw cos αwt
a = 05 (dw1 + dw2)
dw1dw2 = z1 z2
inv α = tan α - α (radians)
inv αwt = inv αt + 2 tan αn (x1 + x2)(z1 + z2)
zn = z (cos2 βb cos β)
1
aw1fw1α T
ξξε
+=
where ξfw1 is to be taken as the smaller of
bull wtfw1 αtanξ =
bull soi1
b1wtfw1 d
dacostan -tanαξ =
bull 1
2wt
a2
b2fw1 z
ztanα
d
dacostan ξ
minus=
and
2
1fw2aw1 z
zξξ = where ξfw2 is calculated as ξfw1
substituting the values for the wheel by the values for the pinion and visa versa
11 z
2πT =
( ) +
sdot+minusminussdot= minus
2sinαρρxhm
2d2d nfpfp1fpnsoi1
21
t
nfpfplfpn2
tanα)sinαρρx(hm
sdot+minusminus
nmsinbπ
β=εβ
(for double helix b is to be taken as the width of one helix)
εy = βα εε +
ρC = ( )2
b
wt
u1βcos
αsinua
+
v = 311 10dn
60π minus
6 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
pbt = βcos
αcosmπ tn
sat =
minus++
at
ninvααinv
z
αtanx22
π
d a
san = aβcossat
14 Bevel Gear Conversion Formulae and Specific Formulae Conversion of bevel gears to virtual equivalent helical cylin-drical gears is based on the bevel gear midsection The con-version formulae are
Number of teeth
zv12 = z12 cos δ12
(δ1 + δ2 = Σ)
Gear ratio
uv = 1v
2vzz
tan αvt = tan αn cos βm
tan βbm = tan βm cos αvt
Base pitch
pbtm = m
vtnmβcos
αcosmπ
Reference pitch diameters
dv12 = 21
21m
cosdδ
Centre distance
av = 05 (dv1 + dv2)
Tip diameters
dva 12 = dv 12 + 2 ham 12
Addenda
for gears with constant addenda (Klingelnberg)
ham 12 = mmn (1 + xm 12)
for gears with variable addenda (Gleason)
ham 12 = ha 12 ndash b2 tan (δa 12 ndash δ12)
(when ha is addendum at outer end and δa is the outer cone angle)
Addendum modification coefficients
xm 12 = mn
12am21am
m2
hh minus
Base circle
dvb 12 = dv 12 cos αvt
Transverse contact ratio)
εα = btmP
αsinadd05dd05 vtv2
2vb2
2va2
1vb2
1va minusminus+minus
Overlap ratio) (theoretical value for bevel gears with no crowning but used as approximations in the calculation pro-cedures)
εβ = nm
mmπβsinb
Total contact ratio)
εγ = 2β
2α εε +
( Note that index laquovraquo is left out in order to combine formu-lae for cylindrical and bevel gears)
Tangential speed at midsection
vmt = 3m11 10dn
60π minus
Effective radius of curvature (normal section)
ρvc = ( )2vbm
vtvv
u1βcosαsinua
+
Length of line of contact
lb = ( )( )( )
1εifε
ε1ε2ε
βcosεb
β2γ
2βα
2γ
bm
α ltminusminusminus
lb = 1εifβcosε
εbβ
bmγ
α ge
15 Nominal Tangential Load Ft Fbt Fmt and Fmbt The nominal tangential load (tangential to the reference cyl-inder with diameter d and perpendicular to an axial plane) is calculated from the nominal (rated) torque T transmitted by the gear set
Cylindrical gears
dT2000
Ft = t
tbt αcos
FF =
Bevel gears
mmt d
T2000F =
vt
mtmbt αcos
FF =
Classification Notes- No 412 7 May 2003
DET NORSKE VERITAS
16 Application Factors KA and KAP The application factor KA accounts for dynamic overloads from sources external to the gearing
It is distinguished between the influence of repetitive cyclic torques KA (161) and the influence of temporary occasional peak torques KAP (162)
Calculations are always to be made with KA In certain cases additional calculations with KAP may be necessary
For gears with a defined load spectrum the calculation with a KA may be replaced by a fatigue damage calculation as given in Appendix A
161 KA For gears designed for long or infinite life at nominal rated torque KA is defined as the ratio between the maximum re-petitive cyclic torque applied to the gear set and nominal rated torque
This definition is suitable for main propulsion gears and most of the auxiliary gears
KA can be determined by measurements or system analysis or may be ruled by conventions (ice classes) (For the pur-pose of a preliminary (but not binding) calculation before KA is determined it is advised to apply either the max values mentioned below or values known from similar plants)
a) For main propulsion gears KA can be taken from the (mandatory) torsional vibration analysis thereby con-sidering all permissible driving conditions) Unless specially agreed the rules do not allow KA in excess of 135 for diesel propulsion) With turbine or electric propulsion KA would normally not exceed 12 However special attention should be given to thrusters that are arranged in such a way that heavy vessel movements andor manoeuvring can cause severe load fluctuations This means eg thrusters positioned far from the rolling axis of vessels that could be susceptible to rolling If leading to propeller air suction the condi-tions may be even worse The above mentioned movements or manoeuvring will result in increased propeller excitation If the thruster is driven by a diesel engine the engine mean torque is limited to 100 However thrusters driven by electric motors can suffer temporary mean torque much above 100 unless a suitable load control system (limiting available e-motor torque) is provided
b) For main propulsion gears with ice class notation (see Rules Pt5 Ch1 Sec J500) KA ice has to be taken as the higher value of the applicable (rule defined) ice shock torque referred to nominal rated torque and the value under a) The Baltic ice class notations refer to a few millions ice shock loads Thus the life factors may be put YN=1 and ZN=12 (except for nitrided gears where ZN = 1 applies) Additionally the calculations with the normal KA (no ice class) are to fulfil the normal requirements For polar ice class notations KA ice applies to all criteria and for long or infinite life
c) For a power take off (PTO) branch from a main propul-sion gear with ice class ice shocks result in negative torques It is assumed that the PTO branch is unloaded when the ice shock load occurs The influence of these reverse shock loads may be taken into account as follows The negative torque (reversed load) expressed by means of an application factor based on rated forward load (T or Ft) is KAreverse = KA ice ndash1 (the minus 1 be-cause no mean torque assumed) KAice to be calculated as in the ice class rules This KAreverse should be used for back flank considera-tions such as pitting and scuffing The influence on tooth bending strength (forward di-rection) may be simplified by using the factor YM = 1 minus 03 middot KAreverse KA
d) For diesel driven auxiliaries KA can be taken from the torsional vibration analysis if available For units where no vibration analysis is required (lt 200 kW) or available it is advised to apply KA as the upper allow-able value 135)
e) For turbine or electro driven auxiliaries the same as for c) applies however the practical upper value is 12
) For diesel driven gears more information on KA for mis-firing and normal driving is given in Appendix B
162 KAP The peak overload factor KAP is defined as the ratio between the temporary occasional peak overload torque and the nominal rated torque
For plants where high temporary occasional peak torques can occur (ie in excess of the above mentioned KA) the gearing (if nitrided) has to be checked with regard to static strength Unless otherwise specified the same safety factors as for infinite life apply
The scuffing safety is to be specially considered whereby the KA applies in connection with the bulk temperature and the KAP applies for the flash temperature calculation and should replace KA in the formulae in 431 432 and 441
KAP can be evaluated from the torsional impact vibration calculation (as required by the rules)
If the overloads have a duration corresponding to several revolutions of the shafts the scuffing safety has to be consid-ered on basis of this overload both with respect to bulk and flash temperature This applies to plants with ice class nota-tions (Baltic and polar) and to plants with prime movers which have high temporary overload capacity such as eg electric motors (provided the driven member can have a con-siderable increase in demand torque as eg azimuth thrusters during manoeuvring)
For plants without additional ice class notation KAP should normally not exceed 15
8 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
163 Frequent overloads For plants where high overloads or shock loads occur regu-larly the influence of this is to be considered by means of cumulative fatigue (see Appendix A)
17 Load Sharing Factor Kγ The load sharing factor Kγ accounts for the maldistribution of load in multiple-path transmissions (dual tandem epicyclic double helix etc) Kγ is defined as the ratio between the max load through an actual path and the evenly shared load
171 General method Kγ mainly depends on accuracy and flexibility of the branches (eg quill shaft planet support external forces etc) and should be considered on basis of measurements or of relevant analysis as eg
Kγ = δ+δ f
δ = total compliance of a branch under full load (assuming even load share) referred to gear mesh
f = minusminusminusminus+++ 23
22
21 fff where f1 f2 etc are the
main individual errors that may contribute to a maldis-tribution between the branches Eg tooth pitch errors planet carrier pitch errors bearing clearance influences etc Compensating effects should also be considered
For double helical gears
An external axial force Fext applied from sources outside the actual gearing (eg thrust via or from a tooth coupling) will cause a maldistribution of forces between the two helices Expressed by a load sharing factor the
βsdot
plusmn=γ tanFF1K
t
ext
If the direction of Fext is known the calculation should be carried out separately for each helix and with the tangential force corrected with the pertinent Kγ If the direction of Fext is unknown both combinations are to be calculated and the higher σH or σF to be used
172 Simplified method If no relevant analysis is available the following may apply
For epicyclic gears
Kγ = 32501 minus+ pln
where npl = number of planets ( gt3 )
For multistage gears with locked paths and gear stages sepa-rated by quill shafts (see figure below)
Figure 10 Locked paths gear
Kγ = ( )φ201+
where φ = quill shaft twist (degrees) under full load
18 Dynamic Factor Kv The dynamic factor Kv accounts for the internally generated dynamic loads
Kv is defined as the ratio between the maximum load that dynamically acts on the tooth flanks and the maximum ex-ternally applied load Ft KA Kγ
In the following 2 different methods (181 and 182) are described In case of controversy between the methods the next following is decisive ie the methods are listed with increasing priority
It is important to observe the limitations for the method in 181 In particular the influence of lateral stiffness of shafts is often underestimated and resonances occur at considerably lower speed than determined in 1811
However for low speed gears with vmiddotz1 lt 300 calculations may be omitted and the dynamic factor simplified to Kv=105
181 Single resonance method For a single stage gear Kv may be determined on basis of the relative proximity (or resonance ratio) N between actual speed n1 and the lowest resonance speed nE1
N = 1E
1
nn
Note that for epicyclic gears n is the relative speed ie the speed that multiplied with z gives the mesh frequency
1811 Determination of critical speed It is not advised to apply this method for multimesh gears for N gt 085 as the influence of higher modes has to be consid-ered see 182 In case of significant lateral shaft flexibility (eg overhung mounted bevel gears) the influence of cou-pled bending and torsional vibrations between pinion and wheel should be considered if N ge 075 see 182
nE1 = redmγc
1zπ
31030 sdot
where
cγ is the actual mesh stiffness per unit facewidth see 111
Classification Notes- No 412 9 May 2003
DET NORSKE VERITAS
For gears with inactive ends of the facewidth as eg due to high crowning or end relief such as often applied for bevel gears the use of cγ in connection with determination of natu-ral frequencies may need correction cγ is defined as stiffness per unit facewidth but when used in connection with the total mesh stiffness it is not as simple as cγmiddotb as only a part of the facewidth is active Such corrections are given in 111
mred is the reduced mass of the gear pair per unit facewidth and referred to the plane of contact
For a single gear stage where no significant inertias are closely connected to neither pinion nor wheel mred is calcu-lated as
mred = 21
21mm
mm+
The individual masses per unit facewidth are calculated as
m12 = 221b
21
)2d(b
I
where I is the polar moment of inertia (kgmm2)
The inertia of bevel gears may be approximated as discs with diameter equal the midface pitch diameter and width equal to b However if the shape of the pinion or wheel body differs much from this idealised cylinder the inertia should be cor-rected accordingly
For all kind of gears if a significant inertia (eg a clutch) is very rigidly connected to the pinion or wheel it should be added to that particular inertia (pinion or wheel) If there is a shaft piece between these inertias the torsional shaft stiffness alters the system into a 3-mass (or more) system This can be calculated as in 182 but also simplified as a 2-mass system calculated with only pinion and wheel masses
1812 Factors used for determination of Kv Non-dimensional gear accuracy dependent parameters
( )bKKF
yfcB
At
pptp
γ
minus=
( )bKKF
yFcBAt
ff
γ
α minus=
Non-dimensional tip relief parameter
bKKFcC
1BγAt
ak sdotsdot
sdotminus=
For gears of quality grade (ISO 1328) Q = 7 or coarser Bk = 1
For gears with Q le 6 and excessive tip relief Bk is limited to max 1
For gears (all quality grades) with tip relief of more than 2middotCeff (see 432) the reduction of αε has to be considered (see 443)
where
fpt = the single pitch deviation (ISO 1328) max of pinion or wheel
Fα = the total profile form deviation (ISO 1328) max of pinion or wheel (Note Fα is pt not available for bevel gears thus use Fα = fpt)
yp and yf = the respective running-in allowances and may be calculated similarly to yα in 112 ie the value of fpt is replaced by Fα for yf
cacute = the single tooth stiffness see 111
Ca = the amount of tip relief see 433 In case of different tip relief on pinion and wheel the value that results in the greater value of Bk is to be used If Ca is zero by design the value of running-in tip relief Cay (see 112) may be used in the above formula
1813 Kv in the subcritical range Cylindrical gears N le 085
Bevel gears N le 075
Kv = 1 + N K
K = Cv1 Bp + Cv2 Bf + Cv3 Bk
Cv1 accounts for the pitch error influence
Cv1 = 032
Cv2 accounts for profile error influence
Cv2 = 034 for γε le 2
Cv2 = 03ε
057
γ minus for γε gt 2
Cv3 accounts for the cyclic mesh stiffness variation
Cv3 = 023 for γε le 2
Cv3 = 156ε
0096
γ minus for γε gt 2
1814 Kv in the main resonance range Cylindrical gears 085 lt N le 115
Bevel gears 075 lt N le 125
Running in this range should preferably be avoided and is only allowed for high precision gears
Kv = 1 + Cv1 Bp + Cv2 Bf + Cv4 Bk
Cv4 accounts for the resonance condition with the cyclic mesh stiffness variation
Cv4 = 090 for γε le 2
Cv4 = 144ε
ε005057
γ
γ
minus
minus for γε gt 2
10 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
1815 Kv in the supercritical range Cylindrical gears N ge 15
Bevel gears N ge 15
Special care should be taken as to influence of higher vibra-tion modes andor influence of coupled bending (ie lateral shaft vibrations) and torsional vibrations between pinion and wheel These influences are not covered by the following approach
Kv = Cv5 Bp + Cv6 Bf + Cv7
Cv5 accounts for the pitch error influence
Cv5 = 047
Cv6 accounts for the profile error influence
Cv6 = 047 for εγ le 2
Cv6 = 174ε
012
γ minus for εγ gt 2
Cv7 relates the maximum externally applied tooth loading to the maximum tooth loading of ideal accurate gears operating in the supercritical speed sector when the circumferential vibration becomes very soft
Cv7 = 075 for εγ le 15
Cv7 = [ ] 8750)2(sin 0125 +minusβεπ for 15 lt εγ le 25
Cv7 = 10 for εγ gt 25
1816 Kv in the intermediate range Cylindrical gears 115 lt N lt 15
Bevel gears 125 lt N lt 15
Comments raised in 1814 and 1815 should be observed
Kv is determined by linear interpolation between Kv for N = 115 respectively 125 and N = 15 as
Cylindrical gears
( ) ( ) ( )[ ]51Nv151Nv51Nvv KK350
N51KK === minussdot
minus
+=
Bevel gears
( ) ( ) ( )[ ]51Nv251Nv51Nvv KK250
N51KK === minussdot
minus
+=
182 Multi-resonance method For high speed gear (vgt40 ms) for multimesh medium speed gears for gears with significant lateral shaft flexibility etc it is advised to determine Kv on basis of relevant dy-namic analysis
Incorporating lateral shaft compliance requires transforma-tion of even a simple pinion-wheel system into a lumped multi-mass system It is advised to incorporate all relevant inertias and torsional shaft stiffnesses into an equivalent (to pinion speed) system Thereby the mesh stiffness appears as an equivalent torsional stiffness
cγ b (db12)2 (Nmrad)
The natural frequencies are found by solving the set of dif-ferential equations (one equation per inertia) Note that for a gear put on a laterally flexible shaft the coupling bending-torsionals is arranged by introducing the gear mass and the lateral stiffness with its relation to the torsional displacement and torque in that shaft
Only the natural frequency (ies) having high relative dis-placement and relative torque through the actual pinion-wheel flexible element need(s) to be considered as critical frequency (ies)
Kv may be determined by means of the method mentioned in 181 thereby using N as the least favourable ratio (in case of more than one pinion-wheel dominated natural frequency) Ie the N-ratio that results in the highest Kv has to be consid-ered
The level of the dynamic factor may also be determined on basis of simulation technique using numeric time integration with relevant tooth stiffness variation and pitchprofile er-rors
19 Face Load Factors KHβ and KFβ The face load factors KHβ for contact stresses and for scuff-ing KFβ for tooth root stresses account for non-uniform load distribution across the facewidth
KHβ is defined as the ratio between the maximum load per unit facewidth and the mean load per unit facewidth
KFβ is defined as the ratio between the maximum tooth root stress per unit facewidth and the mean tooth root stress per unit facewidth The mean tooth root stress relates to the con-sidered facewidth b1 respectively b2
Note that facewidth in this context is the design facewidth b even if the ends are unloaded as often applies to eg bevel gears
The plane of contact is considered
191 Relations between KHβ and KFβ
KFβ = expβHK 2(hb)hb1
1exp++
=
where hb is the ratio tooth heightfacewidth The maximum of h1b1 and h2b2 is to be used but not higher than 13 For double helical gears use only the facewidth of one helix
If the tooth root facewidth (b1 or b2) is considerably wider than b the value of KFβ(1or2) is to be specially considered as it may even exceed KHβ
Eg in pinion-rack lifting systems for jack up rigs where b = b2 asymp mn and b1 asymp 3 mn the typical KHβ asymp KFβ2 asymp 1 and KFβ1 asymp 13
192 Measurement of face load factors Primarily
KFβ may be determined by a number of strain gauges distrib-uted over the facewidth Such strain gauges must be put in
Classification Notes- No 412 11 May 2003
DET NORSKE VERITAS
exactly the same position relative to the root fillet Relations in 191 apply for conversion to KHβ
Secondarily
KHβ may be evaluated by observed contact patterns on vari-ous defined load levels It is imperative that the various test loads are well defined Usually it is also necessary to evalu-ate the elastic deflections Some teeth at each 90 degrees are to be painted with a suitable lacquer Always consider the poorest of the contact patterns
After having run the gear for a suitable time at test load 1 (the lowest) observe the contact pattern with respect to ex-tension over the facewidth Evaluate that KHβ by means of the methods mentioned in this section Proceed in the same way for the next higher test load etc until there is a full face contact pattern From these data the initial mesh misalign-ment (ie without elastic deflections) can be found by ex-trapolation and then also the KHβ at design load can be found by calculation and extrapolation See example
Figure 11 Example of experimental determination of KHβ
It must be considered that inaccurate gears may accumulate a larger observed contact pattern than the actual single mesh to mesh contact patterns This is particularly important for lapped bevel gears Ground or hard metal hobbed bevel gears are assumed to present an accumulated contact pattern that is practically equal the actual single mesh to mesh con-tact patterns As a rough guidance the (observed) accumu-lated contact pattern of lapped bevel gears may be reduced by 10 in order to assess the single mesh to mesh contact pattern which is used in 199
193 Theoretical determination of KHβ The methods described in 193 to 198 may be used for cy-lindrical gears The principles may to some extent also be used for bevel gears but a more practical approach is given in 199
General For gears where the tooth contact pattern cannot be verified during assembly or under load all assumptions are to be well on the safe side
KHβ is to be determined in the plane of contact
The influence parameters considered in this method are
bull mean mesh stiffness cγ (see 111) (if necessary also variable stiffness over b)
bull mean unit load Fmb = Fbt KA Kγ Kvb (for double helical gears see 17 for use of Kγ)
bull misalignment fsh due to elastic deflections of shafts and gear bodies (both pinion and wheel)
bull misalignment fdefl due to elastic deflections of and work-ing positions in bearings
bull misalignment fbe due to bearing clearance tolerances bull misalignment fma due to manufacturing tolerances bull helix modifications as crowning end relief helix correc-
tion bull running in amount yβ (see 112)
In practice several other parameters such as centrifugal ex-pansion thermal expansion housing deflection etc contrib-ute to KHβ However these parameters are not taken into account unless in special cases when being considered as particularly important
When all or most of the am parameters are to be considered the most practical way to determine KHβ is by means of a graphical approach described in 1931
If cγ can be considered constant over the facewidth and no helix modifications apply KHβ can be determined analyti-cally as described in 1932
1931 Graphical method The graphical method utilises the superposition principle and is as follows
bull Calculate the mean mesh deflection Mδ as a function of
γm c and bF see 111
bull Draw a base line with length b and draw up a rectangu-lar with height δM (The area δM b is proportional to the transmitted force)
bull Calculate the elastic deflection fsh in the plane of contact Balance this deflection curve around a zero line so that the areas above and below this zero line are equal
Figure 12 fsh balanced around zero line
bull Superimpose these ordinates of the fsh curve to the previ-ous load distribution curve (The area under this new load distribution curve is still δM b)
bull Calculate the bearing deflections andor working posi-tions in the bearings and evaluate the influence fdefl in the plane of contact This is a straight line and is bal-anced around a zero line as indicated in Fig 14 but with one distinct direction Superimpose these ordinates to the previous load distribution curve
12 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
bull The amount of crowning end relief or helix correction (defined in the plane of contact) is to be balanced around a zero line similarly to fsh
Figure 13 Crowning Cc balanced aound zero line
bull Superimpose these ordinates to the previous load distri-bution curve In case of high crowning etc as eg often applied to bevel gears the new load distribution curve may cross the base line (the real zero line) The result is areas with negative load that is not real as the load in those areas should be zero Thus corrective actions must be made but for practical reasons it may be postponed to after next operation
bull The amount of initial mesh misalignment bema ff + (defined in the plane of contact) is to be bal-
anced around a zero line If the direction of bema ff + is known (due to initial contact check) or if the direction of fbe is known due to design (eg overhang bevel pin-ion) this should be taken into account If direction un-known the influence of bema ff + in both directions as well as equal zero should be considered
Figure 14 fma+fbe in both directions balanced around zero line
Superimpose these ordinates to the previous load distri-bution curve This results in up to 3 different curves of which the one with the highest peak is to be chosen for further evaluation
bull If the chosen load distribution curve crosses the base line (ie mathematically negative load) the curve is to be corrected by adding the negative areas and dividing this with the active facewidth The (constant) ordinates of this rectangular correction area are to be subtracted from the positive part of the load distribution curve It is advisable to check that the area covered under this new load distribution curve is still equal δM b
bull If cγ cannot be considered as constant over b then cor-rect the ordinates of the load distribution curve with the local (on various positions over the facewidth) ratio between local mesh stiffness and average mesh stiffness cγ (average over the active facewidth only) Note that the result is to be a curve that covers the same area δM b as before
bull The influence of running in yβ is to be determined as in 112 whereby the value for Fβx is to be taken as twice the distance between the peak of the load distribution curve and δM
bull Determine
KHβ = M
β
δycurveofpeak minus
1932 Simplified analytical method for cylindrical gears The analytical approach is similar to 1931 but has a more limited application as cγ is assumed constant over the face-width and no helix modification applies
bull Calculate the elastic deflection fsh in the plane of contact Balance this deflection curve around a zero line so that the area above and below this zero line are equal see Fig 12 The max positive ordinate is frac12∆fsh
bull Calculate the initial mesh alignment as Fβx= deflbemash ffff plusmnplusmnplusmn∆ The negative signs may only be used if this is justified andor verified by a contact pattern test Otherwise al-ways use positive signs If a negative sign is justified the value of Fβx is not to be taken less than the largest of each of these elements
bull Calculate the effective mesh misalignment as Fβy = Fβx - yβ (yβ see 112)
bull Determine
2KforF2
bFc1K βH
m
βγγH le+=β
or
2KforF
bFc2K Hβ
m
βγγH gt=β
where cγ as used here is the effective mesh stiffness see 111
194 Determination of fsh fsh is the mesh misalignment due to elastic deflections Usu-ally it is sufficient to consider the combined mesh deflection of the pinion body and shaft and the wheel shaft The calcu-lation is to be made in the plane of contact (of the considered gear mesh) and to consider all forces (incl axial) acting on the shafts Forces from other meshes can be parted into components parallel respectively vertical to the considered plane of contact Forces vertical to this plane of contact have no influence on fsh
It is advised to use following diameters for toothed elements
d + 2 x mn for bending and shear deflection
d + 2 mn (x ndash ha0 + 02) for torsional deflection
Usually fsh is calculated on basis of an evenly distributed load If the analysis of KHβ shows a considerable maldis-
Classification Notes- No 412 13 May 2003
DET NORSKE VERITAS
tribution in term of hard end contact or if it is known by other reasons that there exists a hard end contact the load should be correspondingly distributed when calculating fsh In fact the whole KHβ procedure can be used iteratively 2-3 iterations will be enough even for almost triangular load distributions
195 Determination of fdefl fdefl is the mesh misalignment in the plane of contact due to bearing deflections and working positions (housing deflec-tion may be included if determined)
First the journal working positions in the bearings are to be determined The influence of external moments and forces must be considered This is of special importance for twin pinion single output gears with all 3 shafts in one plane
For rolling bearings fdefl is further determined on basis of the elastic deflection of the bearings An elastic bearing deflec-tion depends on the bearing load and size and number of rolling elements Note that the bearing clearance tolerances are not included here
For fluid film bearings fdefl is further determined on basis of the lift and angular shift of the shafts due to lubrication oil film thickness Note that fbe takes into account the influence of the bearing clearance tolerance
When working positions bearing deflections and oil film lift are combined for all bearings the angular misalignment as projected into the plane of the contact is to be determined fdefl is this angular misalignment (radians) times the face-width
196 Determination of fbe fbe is the mesh misalignment in the plane of contact due to tolerances in bearing clearances In principle fbe and fdefl could be combined But as fdefl can be determined by analy-sis and has a distinct direction and fbe is dependent on toler-ances and in most cases has no distinct direction (ie + toler-ance) it is practicable to separate these two influences
Due to different bearing clearance tolerances in both pinion and wheel shafts the two shaft axis will have an angular mis-alignment in the plane of contact that is superimposed to the working positions determined in 195 fbe is the facewidth times this angular misalignment Note that fbe may have a distinct direction or be given as a + tolerance or a combina-tion of both For combination of + tolerance it is adviced to use
fbe= 22be
21be ff ++plusmn
fbe is particularly important for overhang designs for gears with widely different kinds of bearings on each side and when the bearings have wide tolerances on clearances In general it shall be possible to replace standard bearings with-out causing the real load distribution to exceed the design premises For slow speed gears with journal bearings the expected wear should also be considered
197 Determination of fma fma is the mesh misalignment due to manufacturing toler-ances (helix slope deviation) of pinion fHβ1 wheel fHβ2 and housing bore
For gear without specifically approved requirements to as-sembly control the value of fma is to be determined as
fma= 22Hβ
21Hβ ff +
For gears with specially approved assembly control the value of fma will depend on those specific requirements
198 Comments to various gear types For double helical gears KHβ is to be determined for both helices Usually an even load share between the helices can be assumed If not the calculation is to be made as de-scribed in 171
For planetary gears the free floating sun pinion suffers only twist no bending It must be noted that the total twist is the sum of the twist due to each mesh If the value of 1K γ ne this must be taken into account when calculating the total sun pinion twist (ie twist calculated with the force per mesh without Kγ and multiplied with the number of planets)
When planets are mounted on spherical bearings the mesh misalignments sun-planet respectively planet-annulus will be balanced Ie the misalignment will be the average between the two theoretical individual misalignments The faceload distribution on the flanks of the planets can take full advan-tage of this However as the sun and annulus mesh with several planets with possibly different lead errors the sun and annulus cannot obtain the above mentioned advantage to the full extent
199 Determination of KHβ for bevel gears If a theoretical approach similar to 193-198 is not docu-mented the following may be used
testeff
H Kb
b851851K sdot
minussdot=β
beff b represents the relative active facewidth (regarding lapped gears see 192 last part)
Higher values than beff b = 090 are normally not to be used in the formula
For dual directional gears it may be difficult to obtain a high beff b in both directions In that case the smaller beff b is to be used
Ktest represents the influence of the bearing arrangement shaft stiffness bearing stiffness housing stiffness etc on the faceload distribution and the verification thereof Expected variations in length- and height-wise tooth profile is also accounted for to some extent
a) Ktest = 1 For ground or hard metal hobbed gears with the specified contact pattern verified at full rating or at full torque slow turning at a condition representative for the thermal expansion at normal operation
14 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
It also applies when the bearing arrangementsupport has insignificant elastic deflections and thermal axial expansion However each initial mesh contact must be verified to be within acceptance criteria that are cali-brated against a type test at full load Reproduction of the gear tooth length- and height-wise profile must also be verified This can be made through 3D measurements or by initial contact movements caused by defined axial offsets of the pinion (tolerances to be agreed upon)
b) Ktest = 1 + 04middot(beffbndash06) For designs with possible influence of thermal expansion in the axial direction of the pinion The initial mesh contact verified with low load or spin test where the acceptance criteria are calibrated against a type test at full load
c) Ktest = 12 if mesh is only checked by toolmakerrsquos blue or by spin test contact For gears in this category beffb gt 085 is not to be used in the calculation
110 Transversal Load Distribution Factors KHα and KFα The transverse load distribution factors KHα for contact stresses and for scuffing KFα for tooth root stresses account for the effects of pitch and profile errors on the transversal load distribution between 2 or more pairs of teeth in mesh
The following relations may be used
Cylindrical gears
( )
minus+==
tH
αptγγHαFα F
byfc0409
2ε
KK
valid for 2εγ le
( ) ( )tH
αptγ
γ
γHαFα F
byfcε
1ε20409KK
minusminus+==
valid for 2ε γ gt
where
FtH = Ft KA Kγ Kv KHβ
cγ = See 111
γα = See 112
fpt = Maximum single pitch deviation (microm) of pinion or wheel or maximum total profile form devia-tion Fα of pinion or wheel if this is larger than the maximum single pitch deviation
Note In case of adequate equivalent tip relief adapted to the load half of the above mentioned fpt can be introduced A tip relief is considered adequate when the average of Ca1 and Ca2 is within plusmn40 of the value of Ceff in 432
Limitations of KHα and KFα
If the calculated values for
KFα = KHα lt 1 use KFα = KHα = 10
If the calculated value of KHα gt 2εα
γ
Zε
ε use KHα = 2
εα
γ
Zε
ε
If the calculated value of KFα gt εα
γ
Yεε
use KFα = εα
γ
Yεε
where Yε = αnε
075025+ (for εαn see 331c)
Bevel gears
For ground or hard metal hobbed gears KFα = KHα = 1
For lapped gears KFα = KHα = 11
111 Tooth Stiffness Constants cacute and cγ The tooth stiffness is defined as the load which is necessary to deform one or several meshing gear teeth having 1 mm facewidth by an amount of 1 microm in the plane of contact cacute is the maximum stiffness of a single pair of teeth cγ is the mean value of the mesh stiffness in a transverse plane (brief term mesh stiffness)
Both valid for high unit load (Unit load = Ft middot KA middot Kγb)
Cylindrical gears The real stiffness is a combination of the progressive Hertzian contact stiffness and the linear tooth bending stiff-nesses For high unit loads the Hertzian stiffness has little importance and can be disregarded This approach is on the safe side for determination of KHβ and KHα However for moderate or low loads Kv may be underestimated due to determination of a too high resonance speed
The linear approach is described in A
An optional approach for inclusion of the non-linear stiffness is described in B
A The linear approach
BRCCq
βcos08acutec =
and
( )025ε075cc αγ +prime=
where
( )[ ]n02a01a
B α2000212
hh12051C minusminus
+minus+=
12n1n
x000635z
025791z
015551004723q minus++=
12
2n
22
1n
1 x005290z
x024188x000193z
x011654+minusminusminus
+ 000182 x22
Classification Notes- No 412 15 May 2003
DET NORSKE VERITAS
(for internal gears use zn2 equal infinite and x2 = 0) ha0 = hfp for all practical purposes CR considers the increased flexibility of the wheel teeth if the wheel is not a solid disc and may be calculated as
( )( )nR m5s
sR e5
bbln1C +=
where
bs = thickness of a central web
sR = average thickness of rim (net value from tooth root to inside of rim)
The formula is valid for bs b ge 02 and sRmn ge 1 Outside this range of validity and if the web is not centrally posi-tioned CR has to be specially considered
Note CR is the ratio between the average mesh stiffness over the facewidth and the mesh stiffness of a gear pair of solid discs The local mesh stiffness in way of the web corresponds to the mesh stiffness with CR = 1 The local mesh stiffness where there is no web support will be less than calculated with CR above Thus eg a centrally positioned web will have an effect corresponding to a longitudinal crowning of the teeth See also 1931 regarding KHβ
B The non-linear approach
In the following an example is given on how to consider the non-linearity
The relation between unit load Fb as a function of mesh deflection δ is assumed to be a progressive curve up to 500 Nmm and from there on a straight line This straight line when extended to the baseline is assumed to intersect at 10microm
With these assumptions the unit force Fb as a function of mesh deflection δ can be expressed as
( )10δKbF
minus= for 500bFgt
minus=
500Fb10δK
bF for 500
bFlt
with γAt KK
bF
bF
sdotsdot= etc (Nmm) ie unit load incorporat-
ing the relevant factors as
KA middot Kγ for determination of Kv
KA middot Kγ middot Kv for determination of KHβ
KA middot Kγ middot Kv middot KHβ for determination of KHα
δ = mesh deflection (microm)
K = applicable stiffness (c or cγ)
Use of stiffnesses for KV KHβ and KHα
For calculation of Kv and KHα the stiffness is calculated as follows
When Fb lt 500
the stiffness is determined as δ∆
∆ bF
where the increment is chosen as eg ∆ Fb = 10 and thus
50010Fb10
K10Fb∆δ +
++
=
When F b gt 500 the stiffness is c or cγ For calculation of KHβ the mesh deflection δ is used directly
or an equivalent stiffness determined as δsdotb
F
Bevel gears
In lack of more detailed relationship between stiffness and geometry the following may be used
b085
b13cacute eff=
b085b
16c effγ =
beff not to be used in excess of 085 b in these formulae
Bevel gears with heightwise and lengthwise crowning have progressive mesh stiffness The values mentioned above are only valid for high loads They should not be used for de-termination of Ceff (see 432) or KHβ (see 1931)
112 Running-in Allowances The running-in allowances account for the influence of run-ning-in wear on the various error elements
yα respectively yβ are the running-in amounts which reduce the influence of pitch and profile errors respectively influ-ence of localised faceload
Cay is defined as the running-in amount that compensates for lack of tip relief
The following relations may be used
For not surface hardened steel
ptHlim
α fσ160y =
yβ = βxlimH
f320σ
with the following upper limits
V lt 5 ms 5-10 ms gt 10 ms
yα max none
limH
12800σ limH
6400σ
yβ max none
limH
25600σ limH
12800σ
For surface hardened steel
yα = 0075 fpt but not more than 3 for any speed
yβ = 015 Fβx but not more than 6 for any speed
16 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
For all kinds of steel
5145189718
1C2
limHay +
minusσ
=
When pinion and wheel material differ the following ap-plies
bull Use the larger of fpt1 - yα1 and fpt2 - yα2 to replace fpt - yα in the calculation of KHα and Kv
bull Use ( )2β1ββ yy21y += in the calculation of KHβ
bull Use ( )2ay1aya CC21C += in the calculation of Kv
bull Use ( )2ay1ay2a1a CC21CC +== in the scuffing
calculation if no design tip relief is foreseen
Classification Notes- No 412 17 May 2003
DET NORSKE VERITAS
2 Calculation of Surface Durability
21 Scope and General Remarks Part 2 includes the calculations of flank surface durability as limited by pitting spalling case crushing and subsurface yielding Endurance and time limited flank surface fatigue is calculated by means of 22 ndash 212 In a way also tooth frac-tures starting from the flank due to subsurface fatigue is in-cluded through the criteria in 213
Pitting itself is not considered as a critical damage for slow speed gears However pits can create a severe notch effect that may result in tooth breakage This is particularly im-portant for surface hardened teeth but also for high strength through hardened teeth For high-speed gears pitting is not permitted
Spalling and case crushing are considered similar to pitting but may have a more severe effect on tooth breakage due to the larger material breakouts initiated below the surface Subsurface fatigue is considered in 213
For jacking gears (self-elevating offshore units) or similar slow speed gears designed for very limited life the max static (or very slow running) surface load for surface hard-ened flanks is limited by the subsurface yield strength
For case hardened gears operating with relatively thin lubri-cation oil films grey staining (micropitting) may be the lim-iting criterion for the gear rating Specific calculation meth-ods for this purpose are not given here but are under consid-eration for future revisions Thus depending on experience with similar gear designs limitations on surface durability rating other than those according to 22 - 213 may be ap-plied
22 Basic Equations Calculation of surface durability (pitting) for spur gears is based on the contact stress at the inner point of single pair contact or the contact at the pitch point whichever is greater
Calculation of surface durability for helical gears is based on the contact stress at the pitch point
For helical gears with 0 lt εβ lt 1 a linear interpolation be-tween the above mentioned applies
Calculation of surface durability for spiral bevel gears is based on the contact stress at the midpoint of the zone of contact
Alternatively for bevel gears the contact stress may be cal-culated with the program ldquoBECALrdquo In that case KA and Kv are to be included in the applied tooth force but not KHβ and KHα The calculated (real) Hertzian stresses are to be multi-plied with ZK in order to be comparable with the permissible contact stresses
The contact stresses calculated with the method in part 2 are based on the Hertzian theory but do not always represent the real Hertzian stresses
The corresponding permissible contact stresses σHP are to be calculated for both pinion and wheel
221 Contact stress Cylindrical gears
( )HαHβvγA
1
tβεEHDBH KKKKK
bud1uF
ZZZZZσ+
=
where
ZBD = Zone factor for inner point of single pair contact for pinion resp wheel (see 232)
ZH = Zone factor for pitch point (see 231)
ZE = Elasticity factor (see 24)
Zε = Contact ratio factor (see 25)
Zβ = Helix angle factor (see 26)
Ft KA Kγ Kv KHβ KHα see 15 ndash 110
d1 b u see 12 ndash 15
Bevel gears
( )HαHβvγA
v1v
vmtKEMH KKKKK
bud1uF
ZZZ105σ+
sdot=
where
105 is a correlation factor to reach real Hertzian stresses (when ZK = 1)
ZE KA etc see above
ZM = mid-zone factor see 233
ZK = bevel gear factor see 27
Fmt dv1 uv see 12 ndash 15
It is assumed that the heightwise crowning is chosen so as to result in the maximum contact stresses at or near the mid-point of the flanks
222 Permissible contact stress
XWRvLH
NHlimHP ZZZZZ
SZσ
σ =
where
σH lim = Endurance limit for contact stresses (see 28)
ZN = Life factor for contact stresses (see 29)
SH = Required safety factor according to the rules
ZLZvZR = Oil film influence factors (see 210)
ZW = Work hardening factor (see 211)
ZX = Size factor (see 212)
18 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
23 Zone Factors ZH ZBD and ZM
231 Zone factor ZH The zone factor ZH accounts for the influence on contact stresses of the tooth flank curvature at the pitch point and converts the tangential force at the reference cylinder to the normal force at the pitch cylinder
wtt2
wtbH sinααcos
cosαcosβ2Z =
232 Zone factors ZBD The zone factors ZBD account for the influence on contact stresses of the tooth flank curvature at the inner point of sin-gle pair contact in relation to ZH Index B refers to pinion D to wheel
For εβ ge 1 ZBD = 1
For internal gears ZD = 1
For εβ = 0 (spur gears)
( )
π
minusεminusminus
π
minusminus
α=
α2
2
2b
2a
1
2
1b
1a
wtB
z211
dd
z21
dd
tanZ
( )
π
minusεminusminus
π
minusminus
α=
α1
2
1b
1a
2
2
2b
2a
wtD
z211
dd
z21
dd
tanZ
If ZB lt 1 use ZB = 1
If ZD lt 1 use ZD = 1
For 0 lt εβ lt 1
ZBD = ZBD (for spur gears) ndash εβ (ZBD (for spur gears) ndash 1)
233 Zone factor ZM The mid-zone factor ZM accounts for the influence of the contact stress at the mid point of the flank and applies to spi-ral bevel gears
εminusminus
εminusminus
αβ=
αα btm2
2vb2
2vabtm2
1vb2val
2v1vvtbmM
pddpdd
ddtancos2Z
This factor is the product of ZH and ZM-B in ISO 10300 with the condition that the heightwise crowning is sufficient to move the peak load towards the midpoint
234 Inner contact point For cylindrical or bevel gears with very low number of teeth the inner contact point (A) may be close to the base circle In order to avoid a wear edge near A it is required to have suitable tip relief on the wheel
24 Elasticity Factor ZE The elasticity factor ZE accounts for the influence of the material properties as modulus of elasticity and Poissonrsquos ratio on the contact stresses
For steel against steel ZE = 1898
25 Contact Ratio Factor Zε The contact ratio factor Zε accounts for the influence of the transverse contact ratio εα and the overlap ratio εβ on the contact stresses
αε1Z =ε for 1εβ ge
( )α
ββ
αε ε
εε1
3ε4
Z +minusminus
= for εβ lt 1
26 Helix Angle Factor Zβ The helix angle factor Zβ accounts for the influence of helix angle (independent of its influence on Zε) on the surface du-rability
cosβZβ =
27 Bevel Gear Factor ZK The bevel gear factor accounts for the difference between the real Hertzian stresses in spiral bevel gears and the contact stresses assumed responsible for surface fatigue (pitting) ZK adjusts the contact stresses in such a way that the same per-missible stresses as for cylindrical gears may apply
The following may be used ZK = 080
28 Values of Endurance Limit σHlim and Static Strength 5H10σ 3H10σ
σHlim is the limit of contact stress that may be sustained for 5middot107 cycles without the occurrence of progressive pitting
For most materials 5middot107 cycles are considered to be the be-ginning of the endurance strength range or lower knee of the σ-N curve (See also Life Factor ZN) However for nitrided steels 2middot106 apply
For this purpose pitting is defined by
bull for not surface hardened gears pitted area ge 2 of total active flank area
bull for surface hardened gears pitted area ge 05 of total active flank area or ge 4 of one particular tooth flank area 510Hσ and 310Hσ are the contact stresses which the given
material can withstand for 105 respectively 103 cycles without subsurface yielding or flank damages as pitting spalling or case crushing when adequate case depth applies
The following listed values for σHlim 510Hσ and 310Hσ may only be used for materials subjected to a quality control as
Classification Notes- No 412 19 May 2003
DET NORSKE VERITAS
the one referred to in the rules Results of approved fatigue tests may also be used as the basis for establishing these values The defined survival probability is 99
σHlim σH105 σH10
3
Alloyed case hardened steels (surface hardness 58-63 HRC) - of specially approved high grade - of normal grade
1650 1500
2500 2400
3100 3100
Nitrided steel of approved grade gas nitrided (surface hardness 700-800 HV) 1250 13 σHlim 13 σHlim
Alloyed quenched and tempered steel bath or gas nitrided (surface hardness 500-700 HV)
1000
13 σHlim
13 σHlim
Alloyed flame or induction hardened steel (surface hardness 500-650 HV) 075 HV + 750 16 σHlim 45 HV
Alloyed quenched and tempered steel 14 HV + 350 16 σHlim 45 HV
Carbon steel 15 HV + 250 16 σHlim 16 σHlim
These values refer to forged or hot rolled steel For cast steel the values for σHlim are to be reduced by 15
29 Life Factor ZN The life factor ZN takes account of a higher permissible contact stress if only limited life (number of cycles NL) is demanded or lower permissible contact stress if very high number of cycles apply
If this is not documented by approved fatigue tests the fol-lowing method may be used
For all steels except nitrided
7L 105N sdotge ZN = 1 or
01570
L
7
N N105Z
sdot=
Ie ZN = 092 for 1010 cycles
The ZN = 1 from 5middot107 on may only be used when the mate-rial cleanliness is of approved high grade (see Rules Pt4 Ch2) and the lubrication is optimised by a specially approved filtering process
105 lt NL lt 5middot107 510N
logZ037
L
7
N N105Z
sdot=
NL = 105 WXRVLHlim
WstX10H1010NN ZZZZZσ
ZZσZZ
55
5=
103 lt NL lt 105 )Z(Zlog05
L
5
10NN
5N103N10
5N10ZZ
==
3L 10N le
WXRVLlimH
Wst10X10H10NN ZZZZZ
ZZZZ
33
3σ
σ==
(but not less than ZN105)
For nitrided steels
6L 102N sdotge ZN = 1 or
00980
L
6
N N102Z
sdot=
Ie ZN = 092 for 1010 cycles
The ZN = 1 from 2middot106 on may only be used when the mate-rial cleanliness is of approved high grade (see Rules Pt4 Ch2) and the lubrication is optimised by a specially approved filtering process
105 lt NL lt 2middot106 510N
Zlog76860
L
6
N N102Z
sdot=
5L 10N le
XWRVL
10XWst10NN ZZZZZ
ZZ13ZZ
55 ==
Note that when no index indicating number of cycles is used the factors are valid for 5middot107 (respectively 2middot106 for nitrid-ing) cycles
210 Influence Factors on Lubrication Film ZL ZV and ZR The lubricant factor ZL accounts for the influence of the type of lubricant and its viscosity the speed factor ZV ac-counts for the influence of the pitch line velocity and the roughness factor ZR accounts for influence of the surface roughness on the surface endurance capacity
The following methods may be applied in connection with the endurance limit
20 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
Surface hardened steels Not surface hardened steels
ZL ( )24013421
360910ν+
+
( )24013421
680830ν+
+
ZV ( )v3280
140930+
+ ( )v3280300850
++
ZR
080
ZrelR3
150
ZrelR3
where
ν40 = Kinematic oil viscosity at 40ordmC (mm2s) For case hardened steels the influence of a high bulk temperature (see 4 Scuffing) should be con-sidered Eg bulk temperatures in excess of 120ordmC for long periods may cause reduced flank surface endurance limits
For values of ν40 gt 500 use ν40 = 500
RZrel = The mean roughness between pinion and wheel (after running in) relative to an equivalent radius of curvature at the pitch point ρc = 10mm
RZrel
= ( ) 31
cZ2Z1 ρ
10RR50
+
RZ = Mean peak to valley roughness (microm) (DIN defini-tion) (roughly RZ = 6 Ra)
For 5
L 10N le ZL ZV ZR = 10
211 Work Hardening Factor ZW The work hardening factor ZW accounts for the increase of surface durability of a soft steel gear when meshing the soft steel gear with a surface hardened or substantially harder gear with a smooth surface
The following approximation may be used for the endurance limit
Surface hardened steel against not surface hardened steel 150
ZeqW R
31700
130HB21Z
minus
minus=
where HB = the Brinell hardness of the soft member For HB gt 470 use HB = 470 For HB lt 130 use HB = 130
RZeq = equivalent roughness
033
c40
066
ZS
ZHZHZEQ ρvν
15000RRRR
=
If RZeq gt 16 then use RZeq = 16
If RZeq lt 15 then use RZeq = 15
where
RZH = surface roughness of the hard member before run in
RZS = surface roughness of the soft member before run in
ν40 = see 210
If values of ZW lt 1 are evaluated ZW = 1 should be used for flank endurance However the low value for ZW may indi-cate a potential wear problem
Through hardened pinion against softer wheel
( )
minussdotminus+= 000829
HBHB0008981u1Z
2
1W
For 21HBHB
2
1 le use ZW = 1
For 71HBHB
2
1 gt use 17HBHB
2
1 =
For u gt 20 use u = 20
For static strength (lt 105 cycles)
Surface hardened against not surface hardened
ZWst = 105
Through hardened pinion against softer wheel
ZWst = 1
212 Size Factor ZX The size factor accounts for statistics indicating that the stress levels at which fatigue damage occurs decrease with an increase of component size as a consequence of the influ-ence on subsurface defects combined with small stress gradi-ents and of the influence of size on material quality
ZX may be taken unity provided that subsurface fatigue for surface hardened pinions and wheels is considered eg as in the following subsection 213
213 Subsurface Fatigue This is only applicable to surface hardened pinions and wheels The main objective is to have a subsurface safety against fatigue (endurance limit) or deformation (static strength) which is at least as high as the safety SH required for the surface The following method may be used as an approximation unless otherwise documented
The high cycle fatigue (gt3106 cycles) is assumed to mainly depend on the orthogonal shear stresses Static strength (lt103 cycles) is assumed to depend mainly on equivalent
Classification Notes- No 412 21 May 2003
DET NORSKE VERITAS
stresses (von Mises) Both are influenced by residual stresses but this is only considered roughly and empirically
The subsurface working stresses at depths inside the peak of the orthogonal shear stresses respectively the equivalent stresses are only dependent on the (real) Hertzian stresses Surface related conditions as expressed by ZL ZV and ZR are assumed to have a negligible influence
The real Hertzian stresses σHR are determined as
For helical gears with εβ gt 1
σHR = σH
For helical gears with εβ lt 1 and spur gears
ε
α
ββ ε
ε+εminus
sdotσ=σZ
1
HHR
For bevel gears
KHHR Z
1σσ sdot=
The necessary hardness HV is given as a function of the net depth tz (net = after grinding or hard metal hobbing and per-pendicular to the flank)
The coordinates tz and HV are to be compared with the de-sign specification such as
bull for flame and induction hardening tHVmin HVmin bull for nitriding t400min HV = 400 bull for case hardening t550min HV = 550 t400min HV = 400
and t300min HV = 300 (the latter only if the core hardness lt 300 If the core hardness gt 400 the t400 is to be re-placed by a fictive t400 = 16 t550)
In addition the specified surface hardness is not to be less than the max necessary hardness (at tz = 05aH) This applies to all hardening methods
For high cycle fatigue (gt3 106 cycles) the following applies
sdot+
minussdotsdotsdot= o90
05azt
05azt
cosSσ04HV
H
HHHR
applicable to 05a
zt
Hge
For 05at
H
z lt the value for 05at
H
z = applies
56300ρSσ12a cHHR
Hsdotsdot
sdot=
Where aH is half the hertzian contact width multiplied by an empirical factor of 12 that takes into account the possible influence of reduced compressive residual stresses (or even tensile residual stresses) on the local fatigue strength
If any of the specified hardness depths including the surface hardness is below the curve described by HV = f (tz) the actual safety factor against subsurface fatigue is determined as follows
reduce SH stepwise in the formula for HV and aH until all specified hardness depths and surface hardness balance with the corrected curve The safety factor obtained through this method is the safety against subsurface fatigue
For static strength (lt103 cycles) the following applies
sdot+
minussdotsdotsdot= o90
07at
06a
zt
cosSσ019HV
Hst
z
HstHHR
applicable to 06at
Hst
z ge
56300ρSσa cHHR
Hstsdotsdot
=
In the case of insufficient specified hardness depths the same procedure for determination of the actual safety factor as above applies
For limited life fatigue (103 lt cycles lt 3106 )
For this purpose it is necessary to extend the correction of safety factors to include also higher values than required Ie in the case of more than sufficient hardness and depths the safety factor in the formulae for both high cycle fatigue and static strength are to be increased until necessary and speci-fied values balance
The actual safety factor for a given number of cycles N be-tween 103 and 3106 is found by linear interpolation in a dou-ble logarithmic diagram
minussdotminus
= sdot logN3477
logSlogSlogS
310H6103HHN
36 10H103H Slog86281Slog86280 sdot+sdot sdot
22 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
3 Calculation of Tooth Strength
31 Scope and General Remarks Part 3 include the calculation of tooth root strength as limited by tooth root cracking (surface or subsurface initiated) and yielding
For rim thickness sR ge 35middotmn the strength is calculated by means of 32 ndash 313 For cylindrical gears the calculation is based on the assumption that the highest tooth root tensile stress arises by application of the force at the outer point of single tooth pair contact of the virtual spur gears The method has however a few limitations that are mentioned in 36
For bevel gears the calculation is based on force application at the tooth tip of the virtual cylindrical gear Subsequently the stress is converted to load application at the mid point of the flank due to the heightwise crowning
Bevel gears may also be calculated with the program BECAL In that case KA and Kv are to be included in the applied tooth force but not KFβ and KFα
In case of a thin annulus or a thin gear rim etc radial crack-ing can occur rather than tangential cracking (from root fillet to root fillet) Cracking can also start from the compression fillet rather than the tension fillet For rim thickness sR lt 35middotmn a special calculation procedure is given in 315 and 316 and a simplified procedure in 314
A tooth breakage is often the end of the life of a gear trans-mission Therefore a high safety SF against breakage is re-quired
It should be noted that this part 3 does not cover fractures caused by
bull oil holes in the tooth root space bull wear steps on the flank bull flank surface distress such as pits spalls or grey staining
Especially the latter is known to cause oblique fractures starting from the active flank predominately in spiral bevel gears but also sometimes in cylindrical gears
Specific calculation methods for these purposes are not given here but are under consideration for future revisions Thus depending on experience with similar gear designs limita-tions other than those outlined in part 3 may be applied
32 Tooth Root Stresses The local tooth root stress is defined as the max principal stress in the tooth root caused by application of the tooth force Ie the stress ratio R = 0 Other stress ratios such as for eg idler gears (R asymp -12) shrunk on gear rims (R gt 0) etc are considered by correcting the permissible stress level
321 Local tooth root stress The local tooth root stress for pinion and wheel may be as-sessed by strain gauge measurements or FE calculations or similar For both measurements and calculations all details are to be agreed in advance
Normally the stresses for pinion and wheel are calculated as
Cylindrical gears
FαFβvγAβSFn
tF K K K K K Y Y Y
m bFσ =
where
YF = Tooth form factor (see 33)
YS = Stress correction factor (see 34)
Yβ = Helix angle factor (see 36)
Ft KA Kγ Kv KFβ KFα see 15 ndash 110
b see 13
Bevel gears
FαFβvγASaFamn
mtF K K K K K Y Y Y
m bFσ ε=
where
YFa = Tooth form factor see 33
YSa = Stress correction factor see 34
Yε = Contact ratio factor see 35
Fmt KA etc see 15 ndash 110
b see 13
322 Permissible tooth root stress The permissible local tooth root stress for pinion respectively wheel for a given number of cycles N is
CXRrelTrelTF
NMFEFP YYYY
SYY
δσ
=σ
Note that all these factors YM etc are applicable to 3middot106 cycles when used in this formula for σFP The influence of other number of cycles on these factors is covered by the calculation of YN
where
σFE = Local tooth root bending endurance limit of reference test gear (see 37)
YM = Mean stress influence factor which accounts for other loads than constant load direction eg idler gears temporary change of load di-rection pre-stress due to shrinkage etc (see 38)
YN = Life factor for tooth root stresses related to reference test gear dimensions (see 39)
SF = Required safety factor according to the rules
YδrelT = Relative notch sensitivity factor of the gear to be determined related to the reference test gear (see 310)
YRrelT = Relative (root fillet) surface condition factor of the gear to be determined related to the reference test gear (see 311)
Classification Notes- No 412 23 May 2003
DET NORSKE VERITAS
YX = Size factor (see 312)
YC = Case depth factor (see 313)
33 Tooth Form Factors YF YFa The tooth form factors YF and YFa take into account the in-fluence of the tooth form on the nominal bending stress
YF applies to load application at the outer point of single tooth pair contact of the virtual spur gear pair and is used for cylindrical gears
YFa applies to load application at the tooth tip and is used for bevel gears
Both YF and YFa are based on the distance between the con-tact points of the 30˚-tangents at the root fillet of the tooth profile for external gears respectively 60˚ tangents for inter-nal gears
Figure 31 External tooth in normal section
Figure 32 Internal tooth in normal section
Definitions
n
2
n
Fn
enFn
Fe
F
α cosms
α cosmh6
Y
=
n
2
n
Fn
anFn
Fa
Fa
α cosms
α cosmh6
Y
=
In the case of helical gears YF and YFa are determined in the normal section ie for a virtual number of teeth
YFa differs from YF by the bending moment arm hFa and αFan and can be determined by the same procedure as YF with exception of hFe and αFan For hFa and αFan all indices e will change to a (tip)
The following formulae apply to cylindrical gears but may also be used for bevel gears when replacing
mn with mnm
zn with zvn
αt with αvt
β with βm
with undercut without undercut
Fig 33 Dimensions and basic rack profile of the teeth (finished profile)
Tool and basic rack data such as hfP ρfp and spr etc are re-ferred to mn ie dimensionless
331 Determination of parameters
( )n
n
prnfPnfP m
α cossαsin 1ρ
αtan h4πE
minusminusminusminus=
For external gears fPfP ρρ =
For internal gears ( )0z
195fPfP0
fPfP 10363156ρhxρρ
sdotminus+
+=
where
z0 = number of teeth of pinion cutter
x0 = addendum modification coefficient of pinion cutter
hfP = addendum of pinion cutter
ρfP = tip radius of pinion cutter
xhρG fpfp +minus=
24 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
τmE
2π
z2H
nnminus
minus=
with
3πτ = for external gears
6πτ = for internal gears
Htan zG2
nminusϑ=ϑ
(to be solved iteratively suitable start value6π
=ϑ for exter-
nal gears and 3π for internal gears)
a) Tooth root chord sFn For external gears
minus
ϑ+
ϑminus= ρ
cosG3
3πsinz
ms
fpnn
Fn
For bevel gears with a tooth thickness modification
xsm affects mainly sFn but also hFe and αFen The total influence of xsm on YFa Ysa can be approximated by only adding 2 xsm to sFn mn
For internal gears
minus
ϑ+
ϑminus= ρ
cosG
6πsinz
ms
fPnn
Fn
b) Root fillet radius ρF at 30ordm tangent
( )G2coszcosG2ρ
mρ
2n
2
fpn
F
minusϑϑ+=
c) Determination of bending moment arm hF dn = zn mn
b2α
αn βcosε
ε =
dan = dn + 2 ha
pbn = π mn cos αn
dbn = dn cos αn
( )4
d1εpzz
2dd
zz2d
2bn
2
αnbn
2bn
2an
en +
minusminus
minus=
en
bnen d
dcos arcα =
ennnn
e α invα invα x tan 22π
z1
minus+
+=γ
αFen = αen ndash γe
For external gears
( )
minus=
n
enFenee
n
Fe
mdαtan sin γ γcos
21
mh
]ρcos
G3πcosz fpn +
ϑminus
ϑminusminus
For internal gears
( ) minus
sdotsdotminus=
n
enFenee
n
Fe
mdα tanγ sinγ cos
21
mh
minus
ϑminus
ϑminussdot ρ
cosG3
6πcosz fPn
332 Gearing with εαn gt 2 For deep tooth form gearing ( )25ε2 αn lele produced with a verified grade of accuracy of 4 or better and with applied profile modification to obtain a trapezoidal load distribution along the path of contact the YF may be corrected by the factor YDT as
250ε205for 0666ε2366Y αnαnDT leleminus=
205εfor 10Y αnDT lt=
34 Stress Correction Factors YS YSa The stress correction factors YS and YSa take into account the conversion of the nominal bending stress to the local tooth root stress Thereby YS and YSa cover the stress increasing effect of the notch (fillet) and the fact that not only bending stresses arise at the root A part of the local stress is inde-pendent of the bending moment arm This part increases the more the decisive point of load application approaches the critical tooth root section
Therefore in addition to its dependence on the notch radius the stress correction is also dependent on the position of the load application ie the size of the bending moment arm
YS applies to the load application at the outer point of single tooth pair contact YSa to the load application at tooth tip
YS can be determined as follows
( )
+
+=L23121
1
sS qL01312Y
where Fe
Fn
hsL = and
F
Fns ρ2
sq = (see 33)
YSa can be calculated by replacing hFe with hFa in the above formulae
Note a) Range of validity 1 lt qs lt 8
In case of sharper root radii (ie produced with tools having too sharp tip radii) YS resp YSa must be specially considered
b) In case of grinding notches (due to insufficient protuberance of the hob) YS resp YSa can rise considerably and must be multiplied with
Classification Notes- No 412 25 May 2003
DET NORSKE VERITAS
g
g
ρt
0613
13
minus
where
tg = depth of the grinding notch
ρg = radius of the grinding notch
c) The formulae for YS resp YSa are only valid for αn = 20˚ However the same formulae can be used as a safe approximation for other pressure angles
35 Contact Ratio Factor Yε The contact ratio factor Yε covers the conversion from load application at the tooth tip to the load application at the mid point of the flank (heightwise) for bevel gears
The following may be used
Yε = 0625
36 Helix Angle Factor Yβ The helix angle factor Yβ takes into account the difference between the helical gear and the virtual spur gear in the nor-mal section on which the calculation is based in the first step In this way it is accounted for that the conditions for tooth root stresses are more favourable because the lines of contact are sloping over the flank
The following may be used (β input in degrees)
Yβ = 1 ndash εβ β120
When εβ gt 1 use εβ = 1 and when β gt 30deg use β = 30deg in the formula
However the above equation for Yβ may only be used for gears with β gt 25deg if adequate tip relief is applied to both pinion and wheel (adequate = at least 05 middot Ceff see 432)
37 Values of Endurance Limit σFE σFE is the local tooth root stress (max principal) which the material can endure permanently with 99 survival prob-ability 3106 load cycles is regarded as the beginning of the endurance limit or the lower knee of the σ ndash N curve σFE is defined as the unidirectional pulsating stress with a minimum stress of zero (disregarding residual stresses due to heat treatment) Other stress conditions such as alternating or pre-stressed etc are covered by the conversion factor YM
σFE can be found by pulsating tests or gear running tests for any material in any condition If the approval of the gear is to be based on the results of such tests all details on the testing conditions have to be approved by the Society Fur-ther the tests may have to be made under the Societys su-pervision
If no fatigue tests are available the following listed values for σFE may be used for materials subjected to a quality con-trol as the one referred to in the rules
σFE
Alloyed case hardened steels 1) (fillet surface hardness 58 ndash 63 HRC)
bull of specially approved high grade
1050
bull of normal grade
minus CrNiMo steels with approved process
1000
minus CrNi and CrNiMo steels generally 920 minus MnCr steels generally 850
Nitriding steel of approved grade quenched tempered and gas nitrided (surface hardness 700 ndash 800 HV)
840
Alloyed quenched and tempered steel bath or gas nitrided (surface hardness 500 ndash 700 HV)
720
Alloyed quenched and tempered steel flame or induction hardened 2) (incl entire root fillet) (fillet surface hardness 500 ndash 650 HV)
07 HV + 300
Alloyed quenched and tempered steel flame or induction hardened (excl entire root fillet) (σB = uts of base material)
025 σB + 125
Alloyed quenched and tempered steel 04 σB + 200
Carbon steel 025 σB + 250
Note All numbers given above are valid for separate forgings and for blanks cut from bars forged according to a qualified procedure see Pt 4 Ch 2 Sec 3 For rolled steel the values are to be reduced with 10 For blanks cut from forged bars that are not qualified as mentioned above the values are to be reduced with 20 For cast steel reduce with 40
1) These values are valid for a root radius
bull being unground If however any grinding is made in the root fillet area in such a way that the residual stresses may be affected σFE is to be reduced by 20 (If the grinding also leaves a notch see 34)
bull with fillet surface hardness 58 ndash 63 HRC In case of lower surface hardness than 58 HRC σFE is to be reduced with 20(58 ndash HRC) where HRC is the detected hardness (This may lead to a permissible tooth root stress that varies along the facewidth If so the actual tooth root stresses may also be considered along facewidth)
bull not being shot peened In case of approved shot peening σFE may be increased by 200 for gears where σFE is reduced by 20 due to root grinding Otherwise σFE may be increased by 100 for
6nm le and 100 ndash 5 (mn - 6) for mn gt 6 However the possible adverse influence on the flanks regarding grey staining should be considered and if necessary the flanks should be masked
2) The fillet is not to be ground after surface hardening Regarding possible root grinding see 1)
26 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
38 Mean stress influence Factor YM The mean stress influence factor YM takes into account the influence of other working stress conditions than pure pulsa-tions (R = 0) such as eg load reversals idler gears planets and shrink-fitted gears
YM (YMst) are defined as the ratio between the endurance (or static) strength with a stress ratio R ne 0 and the endurance (or static) strength with R = 0
YM and YMst apply only to a calculation method that assesses the positive (tensile) stresses and is therefore suitable for comparison between the calculated (positive) working stress σF and the permissible stress σFP calculated with YM or YMst
For thin rings (annulus) in epicyclic gears where the com-pression fillet may be decisive special considerations apply see 316
The following method may be used within a stress ratio ndash12 lt R lt 05
381 For idlers planets and PTO with ice class
M1M1R1
1Yor Y MstM
+minus
minus=
where
R = stress ratio = min stress divided by max stress
For designs with the same force applied on both forward- and back-flank R may be assumed to ndash 12
For designs with considerably different forces on forward- and back-flank such as eg a marine propulsion wheel with a power take off pinion R may be assessed as
branchmain theoffacewidth unit per forcepto offacewidt unit per force21minus
For a power take off (PTO) with ice class see 161 c
M considers the mean stress influence on the endurance (or static) strength amplitudes
M is defined as the reduction of the endurance strength am-plitude for a certain increase of the mean stress divided by that increase of the mean stress
Following M values may be used
Endurance limit
Static strength
Case hardened 08 ndash 015 Ys 1) 07
If shot peened 04 06
Nitrided 03 03
Induction or flame hardened
04 06
Not surface hardened steel 03 05
Cast steels 04 06 1)For bevel gears use Ys=2 for determination of M
The listed M values for the endurance limit are independent of the fillet shape (Ys) except for case hardening In princi-ple there is a dependency but wide variations usually only occur for case hardening eg smooth semicircular fillets versus grinding notches
382 For gears with periodical change of rotational direction For case hardened gears with full load applied periodically in both directions such as side thrusters the same formula for YM as for idlers (with R = ndash 12) may be used together with the M values for endurance limit This simplified approach is valid when the number of changes of direction exceeds 100 and the total number of load cycles exceeds 3middot106
For gears of other materials YM will normally be higher than for a pure idler provided the number of changes of direction is below 3middot106 A linear interpolation in a diagram with loga-rithmic number of changes of direction may be used ie from YM = 09 with one change to YM (idler) for 3middot106 changes This is applicable to YM for endurance limit For static strength use YM as for idlers
For gears with occasional full load in reversed direction such as the main wheel in a reversing gear box YM = 09 may be used
383 For gears with shrinkage stresses and unidirectional load For endurance strength
FE
fitM σ
σM1
M21Y+
minus=
σFE is the endurance limit for R = 0
For static strength YMst = 1 and σfit accounted for in 39b
σfit is the shrinkage stress in the fillet (30˚ tangent) and may be found by multiplying the nominal tangential (hoop) stress with a stress concentration factor
n
Ffit m
ρ215scf minus=
384 For shrink-fitted idlers and planets When combined conditions apply such as idlers with shrink-age stresses the design factor for endurance strength is
( ) ( ) FE
fitM σ
σR1M1
M2
M1M1R1
1Yminussdot+
minus
+minus
minus=
Symbols as above but note that the stress ratio R in this par-ticular connection should disregard the influence of σfit ie R normally equal ndash 12
For static strength
M1M1R1
1YMst
+minus
minus=
The effect of σfit is accounted for in 39 b
Classification Notes- No 412 27 May 2003
DET NORSKE VERITAS
385 Additional requirements for peak loads The total stress range (σmax ndash σmin) in a tooth root fillet is not to exceed
F
y
Sσ225
for not surface hardened fillets
FSHV5 for surface hardened fillets
39 Life Factor YN The life factor YN takes into account that in the case of limited life (number of cycles) a higher tooth root stress can be permitted and that lower stresses may apply for very high number of cycles
Decisive for the strength at limited life is the σ ndash N ndash curve of the respective material for given hardening module fillet radius roughness in the tooth root etc Ie the factors YδrelT YRelT YX and YM have an influence on YN
If no σ ndash N ndash curve for the actual material and hardening etc is available the following method may be used
Determination of the σ ndash N ndash curve
a) Calculate the permissible stress σFP for the beginning of the endurance limit (3middot106 cycles) including the influ-ence of all relevant factors as SF YδrelT YRelT YX YM and YC ie σFP = σFE middotYM middotYδrelT middotYRelT middotYX middotYC SF
b) Calculate the permissible laquostaticraquo stress (le 103 load cy-cles) including the influence of all relevant factors as SFst YδrelTst YMst and YCst ( )fitσCstYδrelTstYMstYFstσ
FstS1
FPstσ minussdotsdotsdot=
where σFst is the local tooth root stress which the material can resist without cracking (surface hardened materials) or unacceptable deformation (not surface hardened materials) with 99 survival probability
σFst
Alloyed case hardened steel 1) 2300
Nitriding steel quenched tempered and gas nitrided (surface hardness 700 ndash 800 HV)
1250
Alloyed quenched and tempered steel bath or gas nitrided (surface hardness 500 ndash 700 HV)
1050
Alloyed quenched and tempered steel flame or induction hardened (fillet surface hardness 500 ndash 650 HV)
18 HV + 800
Steel with not surface hardened fillets the smaller value of 2)
18 σB or 225 σy
1) This is valid for a fillet surface hardness of 58 ndash 63 HRC In case of lower fillet surface hardness than 58 HRC σFst is to be reduced with 30(58 ndash HRC) where HRC is the actual hardness Shot peening or grinding notches are not considered to have any significant influence on σFst 2) Actual stresses exceeding the yield point (σy or σ02) will alter the residual stresses locally in the ldquotensionrdquo fillet re-spectively ldquocompressionrdquo fillet This is only to be utilised for gears that are not later loaded with a high number of cycles at lower loads that could cause fatigue in the ldquocompressionrdquo fillet
c) Calculate YN as
NL gt 3middot106
YN = 1 or 001
L
6
N N103Y
sdot= ie Yn = 092 for 1010
The YN = 1 from 3middot106 on may only be used when special material cleanness applies see rules Pt4 Ch2
103ltNLlt3middot106exp
L
6
N N103Y
sdot=
cycles6103forσcycles310forσlog02876exp
FP
FPst
sdot=
NL lt 103 cycles6103forσcycles310forσ
NYFP
FPst
sdot=
or simply use σFPst as mentioned in b) directly
Guidance on number of load cycles NL for various applica-tions
bull For propulsion purpose normally NL = 1010 at full load (yachts etc may have lower values)
bull For auxiliary gears driving generators that normally operate with 70-90 of rated power NL = 108 with rated power may be applied
28 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
310 Relative Notch Sensitivity Factor YδrelT The dynamic (respectively static) relative notch sensitivity factor YδrelT (YδrelTst) indicate to which extent the theoreti-cally concentrated stress lies above the endurance limits (re-spectively static strengths) in the case of fatigue (respectively overload) breakage
YδrelT is a function of the material and the relative stress gra-dient It differs for static strength and endurance limit
The following method may be used
For endurance limit
for not surface hardened fillets
( )
024
s024
relT σ103133q21σ1012201351
Ysdotsdotminus
+sdotsdotminus+= minus
minus
δ
for all surface hardened fillets except nitrided
( )
106q21002451
Y srelT
++=δ
for nitrided fillets
( )
1347q2101421
Y srelT
++=δ
For static strength
for not surface hardened fillets1)
( )( )( )025
02
02502s
relTst σ3000821σ300 1Y0821Y
+
minus+=δ
for surface hardened fillets except nitrided
YδrelTst = 044 YS + 012
for nitrided fillets
YδrelTst = 06 + 02 YS 1) These values are only valid if the local stresses do not
exceed the yield point and thereby alter the residual stress level See also 39b footnote 2
311 Relative Surface Condition Factor YRrelT The relative surface condition factor YRrelT takes into ac-count the dependence of the tooth root strength on the sur-face condition in the tooth root fillet mainly the dependence on the peak to valley surface roughness
YRrelT differs for endurance limit and static strength
The following method may be used
For endurance limit
YRrelT = 1675 ndash 053 (Ry + 1)01 for surface hardened steels and alloyed quenched and tempered steels except nitrided
YRrelT = 53 ndash 42 (Ry + 1)001 for carbon steels
YδrelT = 43 ndash 326 (Ry + 1)0005
for nitrided steels
For static strength
YRrelTst = 1 for all Ry and all materials
For a fillet without any longitudinal machining trace Ry asymp Rz
312 Size Factor YX The size factor YX takes into account the decrease of the strength with increasing size YX differs for endurance limit and static strength
The following may be used
For endurance limit
YX = 1 for mn le 5 generally
YX = 103 ndash 0006 mn for 5 lt mn lt 30
YX = 085 for mn ge 30
for not surface hard-ened steels
YX = 105 ndash 001 mn for 5 lt mn ge 25
YX = 08 for mn ge 25
for surface hardened steels
For static strength
YXst = 1 for all mn and all materials
313 Case Depth Factor YC The case depth factor YC takes into account the influence of hardening depth on tooth root strength
YC applies only to surface hardened tooth roots and is dif-ferent for endurance limit and static strength
In case of insufficient hardening depth fatigue cracks can develop in the transition zone between the hardened layer and the core For static strength yielding shall not occur in the transition zone as this would alter the surface residual stresses and therewith also the fatigue strength
The major parameters are case depth stress gradient permis-sible surface respectively subsurface stresses and subsurface residual stresses
The following simplified method for YC may be used
YC consists of a ratio between permissible subsurface stress (incl influence of expected residual stresses) and permissible surface stress This ratio is multiplied with a bracket con-taining the influence of case depth and stress gradient (The empirical numbers in the bracket are based on a high number of teeth and are somewhat on the safe side for low number of teeth)
YC and YCst may be calculated as given below but calculated values above 10 are to be put equal 10
For endurance limit
+
+=nFFE
C m02ρt31
σconstY
Classification Notes- No 412 29 May 2003
DET NORSKE VERITAS
For static strength
+
+=nFFst
Cst m02ρt31
σconstY
where const and t are connected as
Hardening process
t = endurance limit const =
static strength const =
t550 640 1900
t400 500 1200 Case hard-ening
t300 380 800
Nitriding t400 500 1200
Induction- or flame hardening
tHVmin 11 HVmin 25 HVmin
For symbols see 213
In addition to these requirements to minimum case depths for endurance limit some upper limitations apply to case hard-ened gears
The max depth to 550 HV should not exceed
1) 13 of the top land thickness san unless adequate tip relief is applied (see 110)
2) 025 mn If this is exceeded the following applies addi-tionally in connection with endurance limit
minusminus= 025
mt
1Yn
max550C
314 Thin rim factor YB Where the rim thickness is not sufficient to provide full sup-port for the tooth root the location of a bending failure may be through the gear rim rather than from root fillet to root fillet
YB is not a factor used to convert calculated root stresses at the 30deg tangent to actual stresses in a thin rim tension fillet Actually the compression fillet can be more susceptible to fatigue
YB is a simplified empirical factor used to de-rate thin rim gears (external as well as internal) when no detailed calcula-tion of stresses in both tension and compression fillets are available
Figure 34 Examples on thin rims
YB is applicable in the range 175 lt sRmn lt 35
YB = 115 middot ln (8324 middot mnsR)
(for sRmn ge 35 YB = 1)
(for sRmn le 175 use 315)
σF as calculated in 321 is to be multiplied with YB when sRmn lt 35 Thus YB is used for both high and low cycle fatigue
Note This method is considered to be on the safe side for external gear rims However for internal gear rims without any flange or web stiffeners the method may not be on the safe side and it is advised to check with the method in 315316
315 Stresses in Thin Rims For rim thickness sR lt 35 mn the safety against rim cracking has to be checked
The following method may be used
3151 General The stresses in the standardised 30ordm tangent section tension side are slightly reduced due to decreasing stress correction factor with decreasing relative rim thickness sRmn On the other hand during the complete stress cycle of that fillet a certain amount of compression stresses are also introduced The complete stress range remains approximately constant Therefore the standardised calculation of stresses at the 30ordm tangent may be retained for thin rims as one of the necessary criteria
The maximum stress range for thin rims usually occurs at the 60ordm ndash 80ordm tangents both for laquotensionraquo and laquocompressionraquo side The following method assumes the 75ordm tangent to be the decisive Therefore in addition to the am criterion ap-plied at the 30ordm tangent it is necessary to evaluate the max and min stresses at the 75ordm tangent for both laquotensionraquo (loaded flank) fillet and laquocompressionraquo (back-flank) fillet For this purpose the whole stress cycle of each fillet should be considered but usually the following simplification is justified
Figure 35 Nomenclature of fillets
Index laquoTraquo means laquotensileraquo fillet laquoCraquo means laquocompressionraquo fillet
σFTmin and σFCmax are determined on basis of the nominal rim stresses times the stress concentration factor Y75
σFCmin and σFTmax are determined on basis of superposition of nominal rim stresses times Y75 plus the tooth bending stresses at 75ordm tangent
30 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
3152 Stress concentration factors at the 75ordm tangents The nominal rim stress consists of bending stresses due to local bending moments tangential stresses due to the tan-gential force Ft and radial shear stresses due to Fr
The major influence is given by the bending stresses The influence of the tangential stresses is minor and even though its stress concentration factor is slightly higher than for bending it is considered to be safe enough when the sum of these nominal stresses are combined with the stress concen-tration factor for bending The influence of the radial shear stress is neglected
The stress concentration factor relating nominal rim stresses to local fillet stresses at the 75ordm tangent may be calculated as
75n
R75
n
R
75
ρmsρ185
ms3
Y+
sdot=
where ρ75 is the root radius at the 75ordm tangent ref to mn Usually ρ75 is closed to the tool radius ρao and
ρ75 = ρao
is a safe approximation compensating for the am simplifi-cations to the laquounsaferaquo side
The tooth root stresses of the loaded tooth are decreasing with decreasing relative rim thickness approximately with the empirical factor
15
R
ncorr s
m311Y
minus=
3153 Nominal rim stresses The bending moment applied to the rim consists of a part of the tooth tilt moment Ft (hF + 05 sR) and the bending caused by the radial force Fr
The sectional modulus (first moment of area) which is used for determination of the nominal bending stresses is not nec-essarily the same for the 2 am bending moments If flanges webs etc outside the toothed section contribute to stiffening the rim against various deflections the influence of these stiffeners should be considered Eg an end flange will have an almost negligible influence on the effective sectional modulus for the stresses due to tooth tilt as the deflection caused by the tilt moment is rather small and would not much involve the flange On the other hand the radial forces as for instance from the meshes in an annulus would cause considerable radial deflections that the flange might restrict to a substantial amount When considering the stiff-ening of such flanges or webs on basis of simplified models it is advised to use an effective rim thickness sR = sR + 02 mn for the first moment of area of the rim (toothed part) cross section
For a high number of rim teeth it may be assumed that the rim bending moments in the fillets adjacent to the loaded tooth are of the same magnitude as right under the applied
force This assumption is reasonable for an annulus but rather much on the safe side for a hollow pinion
The influence of Ft on the nominal tangential stress is simpli-fied by half of it for compressive stresses (σ1) and the other half for tensile stresses (σ2) Applying these assumptions the nominal rim stresses adjacent to the loaded tooth are
( ) ( )A2
FW
f RFW
s05hF05 t
R
r
T
RFt1 minus
ϑminus
+minus=σ
( ) ( )A2
FW
f RFW
s05hF05 t
R
r
T
RFt2 +
ϑminus
+=σ
where σ1 σ2 see Fig 35
A = minimum area of cross section (usually bR sR)
WT = the sectional modulus of rim with respect to tooth tilt moment (usually 6sb 2
rR )
WR = the sectional modulus of the rim including the influence of stiffeners as flanges webs etc (WR ge WT)
bR = the width of the rim
R = the radius of the neutral axis in the rim ie from wheel centre to midpoint of rim
( )ϑf = a function for bending moment distribution around the rim For a rim (pinion) with one mesh only the ( )ϑf at the position of load application is 024 For an annulus with 3 or more meshes ( )ϑf at the position of each load application is approx 3 planets ( )ϑf = 019 4 planets ( )ϑf = 014 5 planets ( )ϑf = 011 6 planets ( )ϑf = 009
It must be checked if the max (tensile) stress in the compres-sion fillet really occurs when the fillet is adjacent to the loaded tooth In principle the stress variation through a complete rotation should be considered and the max value used The max value is usually never less than 0 For an annulus eg the tilt moment is zero in the mid position be-tween the planet meshes whilst the bending moment due to the radial forces is half of that at the mesh but with opposite sign
If these formulae are applied to idler gears as eg planets the influence of nominal tangential stresses must be cor-rected by deleting Ft(2 A) for σ1 and using FtA for σ2 Further the influence of Fr on the nominal bending stresses is usually negligible due to the planet bearing support
3154 Root fillet stresses Determination of min and max stresses in the laquotensionraquo fillet
Classification Notes- No 412 31 May 2003
DET NORSKE VERITAS
Minimum stress
751FTmin YσK σ =
Maximum stress
corrF752 Yσ03Yσ KσFTmax +=
where
03 is an empirical factor relating the tension stresses (σF) at the 30ordm tangent to the part of the tension stresses at the 75ordm tangent which add to the rim related stresses (03 also takes into account that full superposition of nominal stresses times stress concentration factors from both laquosides of the corner filletraquo would result in too high stresses)
αβγ sdotsdotsdotsdot= FFvA KKKKKK
Determination of min and max stresses in the laquocompres-sionraquo fillet
Minimum stress
corrF751FCmin Yσ036Yσ Kσ minus=
Maximum stress
752FCmax Yσ Kσ =
where
036 is an empirical factor relating the tension stresses (σF) at the 30ordm tangent to the part of the compression stresses at the 75ordm tangent which add to the rim related stresses
For gears with reversed loads as idler gears and planets there is no distinct laquotensionraquo or laquocompressionraquo fillet The mini-mum stress σFmin is the minimum of σFTmin and σFCmin (usually the latter is decisive) The maximum stress σFTmax is the maximum of σFTmax and σFCmax (usually the former is deci-sive)
316 Permissible Stresses in Thin Rims
3161 General The safety against fatigue fracture respectively overload fracture is to be at least at the same level as for solid gears The laquoordinaryraquo criteria at the 30ordm tangent apply as given in 31 through 313
Additionally the following criteria at the 75ordm tangent may apply
3162 For gt3middot106 cycles The permissible stresses for the laquotensionraquo fillets and for the laquocompressionraquo fillets are determined by means of a relevant fatigue diagram
If the actual tooth root stress (tensile or compressive) ex-ceeds the yield strength to the material the induced residual stresses are to be taken into account
For determination of permissible stresses the following is defined
R = stress ratio ie FTmax
FTminσσ respectively
FCmax
FCmin
σσ
∆σ = stress range ie σFTmax ndash σFTmin resp σFCmax ndash σFCmin
(For idler gears and planets Fmax
FminσσR =
and ∆σ = σFmax minus σFmin)
The permissible stress range ∆σρ for the laquotensionraquo respec-tively laquocompressionraquo fillets can be calculated as
For R gt minus1 FPp σ
R1R1031
13∆σ
minus+
+=
For minus infin lt R lt minus1 FPp σ
R1R10151
13∆σ
minus+
+=
where
σFP = see 32 determined for unidirectional stresses (YM = 1)
If the yield strength σy is exceeded in either tension or com-pression residual stresses are induced This may be consid-ered by correcting the stress ratio R for the respective fillets (tension or compression)
Eg if yFCmin σσ gt (ie exceeded in compression) the
difference yFCmin σσ∆ minus= affects the stress ratio as
∆σ
σ∆σ∆σR
FCmax
y
FCmax
FCmin
+
minus=
++
=
Similarly the stress ratio in the tension fillet may require cor-rection
If the yield strength is exceeded in tension σFTmax gt σy the difference ∆ = σFTmax ndash σy affects the stress ratio as
y
FTmin
FTmax
FTmin
σ∆σ
∆σ∆σR minus=
minusminus
=
Checking for possible exceeding of the yield strength has to be made with the highest torque and with the KAP if this ex-ceeds KA
3163 Forle 103 cycles The permissible stress range ∆σp is not to exceed
For R gt minus1 FPstpst σ
R1R1051
15∆σ
minus+
+=
For ndash infin lt R lt minus1 FPstpst σ
R1R10251
15∆σ
minus+
+=
For all values of R ∆σpst is limited by
not surface hardened F
y
Sσ225
32 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
surface hardened CF
YSHV5
Definition of ∆σ and R see 3162 with particular attention to possible correction of R if the yield strength is exceeded
σFPst see 32 determined for unidirectional stresses (YM = 1) and lt 103 cycles
3164 For 103 lt cycles lt 3106 ∆σp is to be determined by linear interpolation a log-log dia-gram
∆σp at NL load cycles is
6L 103p
exp
L
6
N p ∆σN103∆σ sdot
sdot=
6
3
103p
10p
∆σ
∆σlog28760exp
sdot
=
Classification Notes- No 412 33 May 2003
DET NORSKE VERITAS
4 Calculation of Scuffing Load Capacity
41 Introduction High surface temperatures due to high loads and sliding ve-locities can cause lubricant films to break down This sei-zure or welding together of areas of tooth surface is termed scuffing
In contrast to pitting and fatigue breakage which show a dis-tinct incubation period a single short overloading can lead to a scuffing failure In the ISO-TR13989 two criteria are mentioned The method used in this Classification Note is based on the principles of the flash temperature criterion
Note Bulk temperature in excess of 120ordmC for long periods may have an adverse effect on the surface durability see 211
42 General Criteria In no point along the path of contact the local contact tem-perature may exceed the permissible temperature ie
oilS
oilSB S
ϑ+ϑminusϑ
leϑ
50SB minusϑleϑ
where
Bϑ = max contact temperature along the path of contact
maxflaMBB ϑ+ϑ=ϑ
MBϑ = bulk temperature see 434
maxflaϑ = max flash temperature along the path of con-tact see 44
Sϑ = scuffing temperature see below
oilϑ = oil temperature before it reaches the mesh (max applicable for the actual load case to be used ie normally alarm temperature except for ice classes where a max expected tem-perature applies)
SS = required safety factor according to the Rules
The scuffing temperature Sϑ may be calculated as
L2
wrelT
002
40S XFZGX
ν100112085780
sdot
sdot++=ϑ
where
XwrelT = relative welding factor
XwrelT
Through hardened steel 10
Phosphated steel 125
Copper-plated steel 150
Nitrided steel 150
Less than 10 retained austenite
115
10 ndash 20 retained austenite 10
Casehardened steel
20 ndash 30 retained austenite 085
Austenitic steel 045
FZG = load stage according to FZG-Test A8390 (Note This is the load stage where scuffing oc-curs However due to scatter in test results cal-culations are to be made with one load stage less than the specification)
XL = lubricant factor
= 10 for mineral oils
= 08 for polyalfaolefins
= 07 for non-water-soluble polyglycols
= 06 for water-soluble polyglycols
= 15 for traction fluids
= 13 for phosphate esters
ν40 = kinematic oil viscosity at 40˚C (mm2s)
Application of other test methods such as the Ryder the FZG-Ryder R46574 and the FZG L-42 Test 141195110 may be specially considered
For high speed gears with very short time of contact Sϑ may be increased as follows provided use of EP-oils
Addition to the calculated scuffing temperature Sϑ
If micros 18tc ge no addition
If micros 18tc lt add ( )cwrelT t18X18 minussdotsdot
where
ct = contact time ( micros ) which is the time needed to cross the Hertzian contact width
( ) [ ]microsu1cosβn
uσ340tb1
Hc +sdotsdot
sdot=
σH as calculated in 221
For bevel gears use vu in stead of u
34 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
43 Influence Factors
431 Coefficient of friction The following coefficient of friction may apply
L025a
005oil
02
redC ΣC
Bt X R ηρv
w0048micro minus
=
where
wBt = specific tooth load (Nmm)
vΣC = sum of tangential velocities at pitch point At pitch line velocities gt 50 ms the limiting value of vΣC at v = 50 ms is to be used
ρredC = relative radius of curvature (transversal plane) at the pitch point
Cylindrical gears
HαHβvγAbt
Bt KKKKKb
Fw sdotsdotsdotsdotsdot= (see 1)
twΣC αsin v2v =
bCredC β cos ρρ =
Bevel gears
HαHβvγAtmb
Bt KKKKKb
Fw sdotsdotsdotsdotsdot= (see 1)
vtmtΣC αsin v2v =
bmvCredC β cos ρρ =
ηoil = dynamic viscosity (mPa s) at oilϑ calculated as
1000
ρ νη oiloil =
where ρ in kgm3 approximated as
( ) 70 15oil15 minusϑminusρ=ρ and νoil is kinematic viscosity at oilϑ and may be calculated by means of the following equation
( ) ( )++=+ 08νloglog08νloglog 100oil ( )
sdotminus
ϑ+minus313log373log
273log373log oil
( ) ( )( )08νloglog08νloglog 10040 +minus+
Ra = composite arithmetic mean roughness (micron) of pinion an wheel calculated as ( )2a1aa RR 05R +=This is defined as the roughness on the new flanks ie as manufactured
XL = see 42
432 Effective tip relief Ceff Ceff is the effective tip relief that amount of tip relief which compensates for the elastic deformation of the gear mesh ie
zero load at the tooth tip It is assumed (simplified) to be equal for both pinion and wheel
Cylindrical gears
for helical
γ
γ=cb
KKFC Abt
eff (see 1)
For spur
cbKKF
C Abteff
γ= (see 1)
Alternatively for spur and helical gears the non-linear ap-proach in 111 may be used (taking Ceff=δ)
Bevel gears
bcKFC Ambt
effγ
= (see 1)
where
γ
α
ε2ε44c
+sdot
=γ
433 Tip relief and extension Cylindrical gears
The extension of the tip relief is not to result in an effective contact ratio 1ltεα when the gear is unloaded (exceptions to this may only apply for applications where the gear is not to run at light loads) This means that the unrelieved part of the path of contact is to be minimum pbt It is further assumed that this unrelieved part is placed centrally on the path of contact
If root relief applies it has to be calculated as equivalent tip relief Ie pinion root relief (at mesh position A) is added to Ca2 and wheel root relief (at mesh position E) is added to Ca1 If no design tip relief or root relief on the mating gear is specified (ie if Ca1+Croot2 = 0 and visa versa) use the run-ning in amount see 112
Bevel gears
Bevel gears are to have heightwise crowning ie no distinct relievedunrelieved area This may be treated as tip and root relief For calculation purposes the root relief is combined with the tip relief of the mating member into an equivalent tip relief If no resulting tip relieves are specified the equivalent tip relives may be calculated as an approxima-tion based on the tool crowning Ca tool (per mille of tool module m0) as follows
2root1aeq1a CCC +=
1root2aeq2a CCC +=
where
Classification Notes- No 412 35 May 2003
DET NORSKE VERITAS
2
0
n0101atoolal m
)m2(mA1middotmCC
minus+minussdot=
( )
sdotminusminussdotsdotminus+= vtvb1
2vb1
21vavt1n1 αtan dddαsin 05x1mA
2
0
n020atool22a m
)mm(2A1middotmCC
minus+minussdot=
( )
sdotminusminussdotsdotminus+= vtvb2
2vb2
2va2vt2n2 αtan dddαsin 05x1mA
2
0
20atool1root1 m
A1mCC
minussdotsdot=
2
0
10atool2root2 m
A1mCC
minussdotsdot=
If the pressure angles of the cutter blades are modified (and verified) to balance the tip relieves the following may be assumed
( )eq2aeq1aeq2aeq1a C andC calculated of sum50CC sdot==
Throughout the following Ca1 and Ca2 mean the equivalent tip relieves Ca1eq and Ca2eq
434 Bulk temperature The bulk temperature may be calculated as
flaaveragempsoilMB X X 50 ϑ+ϑ=ϑ
where
Xs = lubrication factor
= 12 for spray lubrication
= 10 for dip lubrication (provided both pinion and wheel are dip lubricated and tip speed lt 5 ms)
= 10 for spray lubrication with additional cooling spray (spray on both pinion and wheel or spray on pinion and dip lubrication of wheel)
= 02 for meshes fully submerged in oil
Xmp = contact factor ( )pmp nX += 150
np = number of mesh contact on the pinion (for small gear ratii the number of wheel meshes should be used if higher)
flaaverageϑ
= average of the integrated flash temperature (see 44) along the path of contact
( )
AE
E
Ayyyfla
flaaverage
d
ΓminusΓ
ΓΓϑ=ϑint =
For high speed gears (v gt 50 ms) it may be necessary to assess the bulk temperature on the basis of a thermal rating of the entire gear transmission
44 The Flash Temperature flaϑ
441 Basic formula The local flash temperature flaϑ may be calculated as
( ) 41redy
y2ly
211
43Btcorrfla
unXwX3250
y ρ
ρminusρ
micro=ϑ Γ
(For bevel gears replace u with uv)
and is to be calculated stepwise along the path of contact from A to E
where
micro = coefficient of friction see 431
Xcorr = correction factor taking empirically into account the increased scuffing risk in the beginning of the approach path due to mesh starting without any previously built up oil film and possible shuffling away oil before meshing if insufficient tip relief
( )
3
AD
yaeffcorr 50
CC1X
ΓminusΓε
Γminus+=
α
Ca = tip relief of driven member Xcorr is only applicable in the approach path and if Ca lt Ceff otherwise 10
wBt = unit load see 431
yXΓ = load sharing factor see 443
n1 = pinion rpm
Γy ρly etc see 442
442 Geometrical relations The various radii of flank curvature (transversal plane) are
ρ1y = pinion flank radius at mesh point y
ρ2y = wheel flank radius at mesh point y
ρredy = equivalent radius of curvature at mesh point y
y2y1
y2y1redy
ρ+ρ
ρρ=ρ
Cylindrical gears
twy
1y αsin au1Γ1
ρ+
+=
twy
2y αsin au1Γu
ρ+
minus=
Note that for internal gears a and u are negative
36 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
Bevel gears
vtvv
y1y αsin a
u1Γ1
ρ+
+=
vtvv
yv2y αsin a
u1Γu
ρ+
minus=
Γ is the parameter on the path of contact and y is any point between A end E
At the respective ends Γ has the following values
Root piniontip wheel
Cylindrical gears
( )
minus
minusminus= 1
αtan 1dd
zzΓ
tw
2b2a2
1
2A
Bevel gears
( )
minus
minusminus= 1
αtan 1dd
uΓvt
2vb2va2
vA
Tip pinionroot wheel
Cylindrical gears
( )
1αtan
1ddΓ
tw
2b1a1
E minusminus
=
Bevel gears
( )
1αtan
1ddΓ
vt
2vb1va1
E minusminus
=
At inner point of single pair contact
Cylindrical gears
tw1
EB α tan zπ2ΓΓ minus=
Bevel gears
vtv1
EB α tan zπ2ΓΓ minus=
At outer point of single pair contact
Cylindrical gears
tw1AD α tan z
π2ΓΓ +=
Bevel gears
vt v1
AD αtan z π2ΓΓ +=
At pitch point ΓC = 0
The points F and G (only applicable to cylindrical gears) limiting the extension of tip relief (so as to maintain a mini-mum contact ratio of unity for unloaded gears) are at
2
BAF
Γ+Γ=Γ
2
EDG
Γ+Γ=Γ
443 Load sharing factor XΓ The load sharing factor XΓ accounts for the load sharing be-tween the various pairs of teeth in mesh along the path of contact
XΓ is to be calculated stepwise from A to E using the pa-rameter Γy
4431 Cylindrical gears with β = 0 and no tip relief
Figure 41
ByAAB
Ay for31
31X
yΓltΓleΓ
ΓminusΓ
ΓminusΓ+=Γ
DyB for1Xy
ΓltΓleΓ=Γ
EyDDE
yE for31
31X
yΓleΓltΓ
ΓminusΓ
ΓminusΓ+=Γ
4432 Cylindrical gears with β = 0 and tip relief Tip relief on the pinion reduces XΓ in the range G ndash E and increases correspondingly XΓ in the range F ndash B
Tip relief on the wheel reduces XΓ in the range A ndash F and increases correspondingly XΓ in the range D ndash G
Following remains generally
DyB for1Xy
ΓleΓleΓ=Γ
21XXGF== ΓΓ
In the following it must be distinguished between Ca lt Ceff respectively Ca gt Ceff This is shown by an example below where Ca1 lt Ceff and Ca2 gt Ceff
Classification Notes- No 412 37 May 2003
DET NORSKE VERITAS
Figure 42
Note When Ca gt Ceff the path of contact is shortened by A ndash A respectively E ndash E The single pair contact path is extended into B respectively D If this shift is significant it is necessary to consider the negative effect on surface durability (B) and bending stresses (B and D)
Range A - F
For effa2 CC le
minus=Γ
eff
2a
CC1
31X
A
FyAeff
2a
AF
Ay for C3
C61XX
AyΓleΓleΓ
sdot
+ΓminusΓ
ΓminusΓ+= ΓΓ
For effa2 CC ge
AyAfor0Xy
ΓleΓleΓ=Γ
with ( )21
CC
1CC
eff
2a
eff
2a
AFAAminus
minusΓminusΓ+Γ=Γ
FyAeff
2a
AF
Ay
eff
2a for 21
CC
CC1X
yΓleΓleΓ
minus
ΓminusΓ
ΓminusΓ+minus=Γ
Range F - B
For effa1 CC le
ByFeff
1a
FB
Fy forC3
C61
21X
yΓleΓleΓ
sdot
+ΓminusΓ
ΓminusΓ+=Γ
For effa1 CC ge
ByFeff
1a
FB
Fy for21
CC
21X
yΓleΓleΓ
minus
ΓminusΓ
ΓminusΓ+=Γ
ByB for1Xy
ΓleΓleΓ=Γ
with 1
CC2
eff
1a
FBFB
minus
ΓminusΓ+Γ=Γ
Range D ndash G
For effa2 CC le
GyDeff
2a
DG
Dy
eff
2a for C 3C
61
C 3C
32X
yΓleΓleΓ
+
ΓminusΓ
ΓminusΓminus+=Γ F
or effa2 CC ge
DyD for1Xy
ΓleΓleΓ=Γ
with ( )21
CC
1CC
eff
2a
eff
2a
DGDDminus
minusΓminusΓ+Γ=Γ
GyDeff
2a
DG
Dy
eff
2a for21
CC
CCX ΓleΓleΓ
minus
ΓminusΓ
ΓminusΓminus=Γ
Range G - E
For effa1 CC le
minus=Γ
eff
1a
CC1
31X
E
EyGeff
1a
GE
Gy forC3
C61
21X
yΓleΓleΓ
sdot
+ΓminusΓ
ΓminusΓminus=Γ
For effa1 CC gt
EyGeff
1a
GE
Gy for21
CC
21X
yΓleΓleΓ
minus
ΓminusΓ
ΓminusΓminus=Γ
EyE for0Xy
ΓleΓleΓ=Γ
with 1
CC2
eff
1a
GEGE
minus
ΓminusΓ+Γ=Γ
4433 Gears with β gt 0 buttressing Due to oblique contact lines over the flanks a certain but-tressing may occur near A and E
This applies to both cylindrical and bevel gears with tip relief lt Ceff The buttressing Xbutt is simplified as a linear function within the ranges A ndash H respectively I ndash E
Figure 43
1εwhen13X βbutt EAge=
1εwhenε031X ββbutt EAlt+=
38 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
Cylindrical gears
bIEAH βsin 02ΓΓΓΓ =minus=minus
Bevel gears
bmIEAH βsin 02ΓΓΓΓ =minus=minus
4434 Cylindrical gears with 2εγ le and no tip relief
yXΓ is obtained by multiplication of
yXΓ in 4431 with
Xbutt in 4433
4435 Gears with 2εγ gt and no tip relief
Applicable to both cylindrical and bevel gears
IyH for1Xy
ΓleΓleΓε
=α
Γ
IyHybutt and forX1Xy
ΓgtΓΓltΓε
=α
Γ
Figure 44
4436 Cylindrical gears with 2εγ le and tip relief
yXΓ is obtained by multiplication of
yXΓ in 4432 with Xbutt
in 4433
4437 Cylindrical gears with 2εγ gt and tip relief
Tip relief on the pinion (respectively wheel) reduces XГ in the range G ndash E (respectively A ndash F) and increases XГ in the range F ndash G
yXΓ is obtained by multiplication of
yXΓ as described below
with Xbutt in 4433
In the XГ example below the influence of tip relief is shown (without the influence of Xbutt) by means of
eff2aeff1a CC and CC ltgt
Tip relief gt Ceff causes new end points A respectively E of the path of contact
Figure 45
Range A ndash F
( ) ( )( ) eff
a2αa1α
AF
Ay
eff
2aeff
C 12C 13εC 1ε
CCCX
y +εε++minus
ΓminusΓ
ΓminusΓ+
εminus
=ααα
Γ
eff2aFyA CC if for leΓleΓleΓ
eff2aFyA CifCfor and geΓleΓleΓ
eff2aAyA CC iffor0Xy
gtΓleΓleΓ=Γ
( ) ( ) ( )( ) ( ) a2αa1α
αeffa2AFAA C 1ε 3C 1ε
1ε 2 CC with++minus+minus
ΓminusΓ+Γ=Γ
Range F ndash G
( )( )( ) GyF
eff
2a1a forC 1 2CC 11X
yΓleΓleΓ
+εε+minusε
+ε
=αα
α
αΓ
Range G ndash E
( ) ( )( ) eff
2a1a
GE
Gy
C 1 2C 1C 1 3XX
GFy +εεminusε++ε
ΓminusΓ
ΓminusΓminus=
αα
ααΓΓ minus
effa1EyG CC if for leΓleΓleΓ
effa1EyG CC if Γfor and geΓleΓle
eff1aEyE CC if for0Xy
geΓleΓleΓ=Γ
( ) ( ) ( )( ) ( ) 2a1a
eff1aGEEE C 1C 1 3
1 2 CC withminusε++ε+εminus
ΓminusΓminusΓ=Γαα
α
4438 Bevel gears with εγ more than approx 18 and heightwise crowning For Ca1 = Ca2 = Ceff the following applies
Figure 46
Classification Notes- No 412 39 May 2003
DET NORSKE VERITAS
( )AEM 50 Γ+Γ=Γ
( )( )2AD
3
2My651X
y ΓminusΓε
ΓminusΓminus
ε=
ααΓ
For tip relief lt Ceff yXΓ is found by linear interpolation be-
tween ( )effay CCX =Γ and ( )0CayX =Γ as in 4435
The interpolation is to be made stepwise from A to M with the influence of Ca2 and from M to E with the influence of Ca1 (For 2a1a CC ne there is a discontinuity at M)
Eg with Ca1 = 04 Ceff and Ca2 = 055 Ceff then
Range A ndash M
)CC()0C( eff2ay2ayyX 550X 450X =Γ=ΓΓ +=
Range M ndash E
)CC()0C( eff1ay1ayyX 40X 06X =Γ=ΓΓ +=
For tip relief gt Ceff the new end points A and E are found as
( )
minusΓminusΓ
ε+Γ=Γ α 1
CC
6 eff
2aADAA
( )
minusΓminusΓ
ε+Γ=Γ α 1
CC
6 eff
1aADEE
Range A ndash A
0Xy=Γ
Range A ndash M
( )( )
ΓminusΓ
ΓminusΓminus
minusε=
αΓ 2
MA
2My
eff
2a1
CC4
351Xy
Range M ndash E
( )( )
ΓminusΓ
ΓminusΓminus
minusε=
αΓ 2
ME
2My
eff
1a1
CC4
351Xy
Range E ndash E
0Xy=Γ
40 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
Appendix A Fatigue Damage Accumulation
The Palmgren-Miner cumulative damage calculation princi-ple is used The procedure may be applied as follows
A1 Stress Spectrum From the individual torque classes the torques (Ti) at the peak values of class intervals and the associated number of cycles (NLi) for both pinion and wheel are to be listed from the highest to the lowest torque
(In case of a cyclic torque variation within the torque classes it is advised to use the peak torque If the cyclic variation is such that the same teeth will repeatedly suffer the peak torque this is a must)
The stress spectra for tooth roots and flanks (σFi σHi) with all relevant factors (except KA) are to be calculated on the basis of the torque spectrum The load dependent K-factors are to be determined for each torque class
A2 σminusN-curve The stress versus load cycle curves for tooth roots and flanks (both pinion and wheel) are to be drawn on the basis of per-missible stresses (ie including the demanded minimum safety factors) as determined in 2 respectively 3 If different safety levels for high cycle fatigue and low cycle fatigue are desired this may be expressed by different demand safety factors applied at the endurance limit respectively at static strength
A3 Damage accumulation The individual damage ratio Di at ith stress level is defined as
Fi
Lii N
ND =
where
NLi = The number of applied cycles at ith stress
NFi = The number of cycles to failure at ith stress
Basically stresses σi below the permissible stress level for infinite life (if a constant ZN or YN is accepted) do not con-tribute to the damage sum However calculating the actual safety factor Sact as described below all the σi for which the product S σi is bigger than or equal to the permissible stress level for infinite life contribute to the damage sum and thus to the determination of Sact The final value of S is decisive
(NFi can be found mathematically by putting the permissible stress σpi equal the actual stress σi thereby finding the actual life factor This life factor can be solved with regard to load cycles ie NFi)
The damage sum ΣDi is not to exceed unity
If ΣDi ne 1 the safety against cumulative fatigue damage is different from the applied demand safety factor For deter-mination of this theoretical safety factor an iteration proce-dure is required as described in the following flowchart
S is correction factor with which the actual safety factor Sact can be found
Sact is the demand safety factor (used in determination of the permissible stresses in the σ ndash N ndash curve) times the correc-tion factor S
The full procedure is to be applied for pinion and wheel tooth roots and flanks
Note If alternating stresses occur in a spectrum of mainly pulsating stresses the alternating stresses may be replaced by equiva-lent pulsating stresses ie by means of division with the ac-tual mean stress influence factor YM
Classification Notes- No 412 41 May 2003
DET NORSKE VERITAS
Appendix B Application Factors for Diesel Driven Gears
For diesel driven gears the application factor KA depends on torsional vibrations Both normal operation and misfiring conditions have to be considered
Normally these two running conditions can be covered by only one calculation
B1 Definitions
Normal operation KAnorm = 0
normv0
TTT +
where
T0 = rated nominal torque
Tv norm = vibratory torque amplitude for normal opera-tion (see rules Pt4 Ch3 Sec1 G301 for defi-nition of ldquonormalrdquo irregularity)
Misfiring operation KA misf = 0
misfv
TTT +
where
T = remaining nominal torque when one cylinder out of action
Tv misf = vibratory torque amplitude in misfiring con-dition This refers to a permissible misfiring condition ie a condition that does not require automatic or immediate corrective actions as speed or pitch reduction
The normal operation is assumed to last for a very high num-ber of cycles such as 1010
The misfiring operation is assumed to last for a limited dura-tion such as 107 cycles
B2 Determination of decisive load Assuming life factor at 1010 cycles as YN = ZN = 092 which usually is relevant the calculation may be performed only once with the combination having the highest value of appli-cation factorlife factor
For bending stresses and scuffing the higher value of
092
K normA and 098
K misfA
For contact stresses the higher value of
092K normA and
113K misfA (but
097K misfA for nitrided gears)
B3 Simplified procedure Note that this is only a guidance and is not a binding con-vention
T = oTZ
1Zsdot
minus where Z = number of cylinders
( ) ideal Videal Vmisf Vnorm V TTT24ZT +minus=
where TV ideal = vibratory torque with all cylinders perfectly equal See also rules Pt4 Ch3 Sec1 G300
When using trends from torsional vibration analysis and measurements the following may be used
TV ideal To is close to zero for engines with few cylinders and using a suitable elastic coupling and increases with relative coupling stiffness and number of cylinders
TV misf To may be high for engines with few cylinders and decreases with number of cylinders
This can be indicated as
200Z
TT
ο
idealV asymp and 80Z04
TT
o
misfV minusasymp
Inserting this into the formulae for the two application fac-tors the following guidance can be given
118112K misfA minusasymp
115110K normA minusasymp
Since KA norm is to be combined with the lower life factors the decisive load condition will be the normal one and a KA of 115 will cover most relevant cases when a suitable elas-tic coupling is chosen
42 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
Appendix C Calculation of Pinion-Rack
Pinion-racks used for elevating of mobile offshore units are open gears that are subjected to wear and tear With normal specifications such as only a few hundred total operation cycles (site to site) the tooth bending stresses for static strength will be decisive for the lay out however with ex-ception of surface hardened pinions where case crushing has to be considered
In the following the use of part 1 and 3 for pinion-racks is shown including relevant simplifications
C1 Pinion tooth root stresses Since the load spectrum normally is dominated by high torques and few load cycles (in the range up to 1-2000) the static strength is decisive
The actual stress is calculated as
β1FSaFan1
tF1 KYY
mbFσ sdotsdotsdotsdot
=
b1 is limited to b2 + 2middotmn
YFa and YSa replace YF and YS because load application at tooth tip has to be assumed for such inaccurate gears
Pinions often use a non-involute profile in the dedendum part of the flank eg a constant radius equal the radius of curva-ture at reference circle For such pinions sFn and hFa are to be measured directly on a sectional drawing of the pinion tooth
Due to high loads and narrow facewidths it may be assumed that KFβ2 = KHβ = 10 However when b1 gt b2 then KFβ1 gt10
If no detailed documentation of KFβ1 is available the fol-lowing may be used
KFβ1 = 1 + 015middot(b1b2 ndash 1)
The permissible stress (not surface hardened) is calculated as
δrelTstF
Fst1FP1 Y
Sσσ sdot=
The mean stress influence due to leg lifting may be disre-garded
The actual and permissible stresses should be calculated for the relevant loads as given in the rules
C2 Rack tooth root stresses The actual stress is calculated as
SaFan2
tF2 YY
mbFσ sdotsdotsdot
=
See C1 for details
The permissible stress is calculated as
δrelTstF
Fst2FP2 Y
Sσσ sdot=
For alloyed steels (Ni Cr Mo) with high toughness and duc-tility the value of YδrelTst may be put equal to YSa
C3 Surface hardened pinions For surface hardened pinions the maximum load is not to cause crushing of the hardened layer of the flank
In principle the calculation described in 213 may be used but when the theoretical Hertzian stress exceeds the ap-proximately 18 times the yield strength of the rack material plastic deformation will occur This will limit the peak Hertzian stress but increases the contact width and thus the penetration of stresses into the depth
An approximation may be based on an assessment of contact width determined by means of equal areas under the theoreti-cal (elastic range) Hertzian contact and the elasto-plastic contact stress (limited to 18 middot σy) with the unknown width
4 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
1 Basic Principles and General Influence Factors
11 Scope and Basic Principles The gear rating procedures given in this Classification Note are mainly based on the ISO-6336 Part 1-5 (cylindrical gears) and partly on ISO 10300 Part 1-3 (bevel gears) and ISO Technical Reports on Scuffing and Fatigue Damage Accumulation but especially applied for marine purposes such as marine propulsion and important auxiliaries onboard ships and mobile offshore units
The calculation procedures cover gear rating as limited by contact stresses (pitting spalling or case crushing) tooth root stresses (fatigue breakage or overload breakage) and scuff-ing resistance Even though no calculation procedures for other damages such as wear grey staining (micropitting) etc are given such damages may limit the gear rating
The Classification Note applies to enclosed parallel shaft gears epicyclic gears and bevel gears (with intersecting axis) However open gear trains may be considered with regard to tooth strength ie part 1 and 3 may apply Even pinion-rack tooth strength may be considered but since such gear trains often are designed with non-involute pinions the calculation procedure of pinion-racks is described in Appen-dix C
Steel is the only material considered
The methods applied throughout this document are only valid for a transverse contact ratio 1 lt εα lt 2 If εα gt 2 either special considerations are to be made or suggested simplifi-cation may be used
All influence factors are defined regarding their physical interpretation Some of the influence factors are determined by the gear geometry or have been established by conven-tions These factors are to be calculated in accordance with the equations provided Other factors are approximations which are clearly stated in the text by terms as laquomay be cal-culated asraquo These approximations are substitutes for exact evaluations where such are lacking or too extensive for prac-tical purposes or factors based on experience In principle any suitable method may replace these approximations
Bevel gears are calculated on basis of virtual (equivalent) cylindrical gears using the geometry of the midsection The virtual (helical) cylindrical gear is to be calculated by using all the factors as a real cylindrical gear with some exceptions These exceptions are mentioned in connection with the ap-plicable factors Wherever a factor or calculation procedure has no reference to either cylindrical gears or bevel gears it is generally valid ie combined for both cylindrical and bevel
In order to minimise the volume of this Classification Note such combinations are widely used and everywhere it is necessary to distinguish it is clearly pointed out by local headings such as
Cylindrical gears
Bevel gears
The permissible contact stresses tooth root stresses and scuffing load capacity depend on the safety factors as re-quired in the respective Rule sections
Terms as endurance limit and static strength are used throughout this Classification Note
Endurance limit is to be understood as the fatigue strength in the range of cycles beyond the lower knee of the σndashN curves regardless if it is constant or drops with higher number of cycles
Static strength is to be understood as the fatigue strength in the range of cycles less than at the upper knee of the σndashN curves
For gears that are subjected to a limited number of cycles at different load levels a cumulative fatigue calculation applies Information on this is given in Appendix A
When the term infinite life is used it means number of cy-cles in the range 108ndash1010
12 Symbols Nomenclature and Units The symbols are generally from ISO 701 ISOR31 and ISO 1328 with a few additional symbols Only SI units are used
The main symbols as influence factors (K Z Y and X with indeces) etc are presented in their respective headings Symbols which are not explained in their respective Secs are as follows
a = centre distance (mm) b = facewidth (mm) d = reference diameter (mm) da = tip diameter (mm) db = base diameter (mm) dw = working pitch diameter (mm) ha = addendum (mm) ha0 = addendum of tool ref to mn hfp = dedendum of basic rack ref to mn (= ha0) hFe = bending moment arm (mm) for tooth root
stresses for application of load at the outer point of single tooth pair contact
hFa = bending moment arm (mm) for tooth root stresses for application of load at tooth tip
HB = Brinell hardness HV = Vickers hardness HRC = Rockwell C hardness mn = normal module n = rev per minute NL = number of load cycles qs = notch parameter Ra = average roughness value (microm) Ry = peak to valley roughness (microm) Rz = mean peak to valley roughness (microm) san = tooth top land thickness (mm) sat = transverse top land thickness (mm) sFn = tooth root chord (mm) in the critical section spr = protuberance value of tool minus grinding stock
equal residual undercut of basic rack ref to mn T = torque (Nm) u = gear ratio (per stage) v = linear speed (ms) at reference diameter
Classification Notes- No 412 5 May 2003
DET NORSKE VERITAS
x = addendum modification coefficient z = number of teeth zn = virtual number of spur teeth αn = normal pressure angle at ref cylinder αt = transverse pressure angle at ref cylinder αa = transverse pressure angle at tip cylinder αwt = transverse pressure angle at pitch cylinder β = helix angle at ref cylinder βb = helix angle at base cylinder βa = helix angle at tip cylinder εα = transverse contact ratio εβ = overlap ratio εγ = total contact ratio ρa0 = tip radius of tool ref to mn ρfp = root radius of basic rack ref to mn ( = ρa0) ρC = effective radius (mm) of curvature at pitch point ρF = root fillet radius (mm) in the critical section σB = ultimate tensile strength (Nmm2) σy = yield strength resp 02 proof stress (Nmm2)
Index 1 refers to the pinion 2 to the wheel
Index n refers to normal section or virtual spur gear of a heli-cal gear
Index w refers to pitch point
Special additional symbols for bevel gears are as follows
Σ = angle between intersection axis Kϑ = angle modification (Klingelnberg)
m0 = tool module (Klingelnberg) δ = pitch cone angle xsm = tooth thickness modification coefficient (mid-
face) R = pitch cone distance (mm)
Index v refers to the virtual (equivalent) helical cylindrical gear
Index m refers to the midsection of the bevel gear
13 Geometrical Definitions For internal gearing z2 a da2 dw2 d2 and db2 are negative x2 is positive if da2 is increased ie the numeric value is de-creased
The pinion has the smaller number of teeth ie
11
2 ge=zz
u
For calculation of surface durability b is the common face-width on pitch diameter
For tooth strength calculations b1 or b2 are facewidths at the respective tooth roots If b1 or b2 differ much from b above they are not to be taken more than 1 module on either side of b
Cylindrical gears
tan αt = tan αn cos β tan βb = tan β cos αt
tan βa = tan β da d
cos αa = dbda
d = z mn cos β mt = mn cos β db = d cos αt = dw cos αwt
a = 05 (dw1 + dw2)
dw1dw2 = z1 z2
inv α = tan α - α (radians)
inv αwt = inv αt + 2 tan αn (x1 + x2)(z1 + z2)
zn = z (cos2 βb cos β)
1
aw1fw1α T
ξξε
+=
where ξfw1 is to be taken as the smaller of
bull wtfw1 αtanξ =
bull soi1
b1wtfw1 d
dacostan -tanαξ =
bull 1
2wt
a2
b2fw1 z
ztanα
d
dacostan ξ
minus=
and
2
1fw2aw1 z
zξξ = where ξfw2 is calculated as ξfw1
substituting the values for the wheel by the values for the pinion and visa versa
11 z
2πT =
( ) +
sdot+minusminussdot= minus
2sinαρρxhm
2d2d nfpfp1fpnsoi1
21
t
nfpfplfpn2
tanα)sinαρρx(hm
sdot+minusminus
nmsinbπ
β=εβ
(for double helix b is to be taken as the width of one helix)
εy = βα εε +
ρC = ( )2
b
wt
u1βcos
αsinua
+
v = 311 10dn
60π minus
6 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
pbt = βcos
αcosmπ tn
sat =
minus++
at
ninvααinv
z
αtanx22
π
d a
san = aβcossat
14 Bevel Gear Conversion Formulae and Specific Formulae Conversion of bevel gears to virtual equivalent helical cylin-drical gears is based on the bevel gear midsection The con-version formulae are
Number of teeth
zv12 = z12 cos δ12
(δ1 + δ2 = Σ)
Gear ratio
uv = 1v
2vzz
tan αvt = tan αn cos βm
tan βbm = tan βm cos αvt
Base pitch
pbtm = m
vtnmβcos
αcosmπ
Reference pitch diameters
dv12 = 21
21m
cosdδ
Centre distance
av = 05 (dv1 + dv2)
Tip diameters
dva 12 = dv 12 + 2 ham 12
Addenda
for gears with constant addenda (Klingelnberg)
ham 12 = mmn (1 + xm 12)
for gears with variable addenda (Gleason)
ham 12 = ha 12 ndash b2 tan (δa 12 ndash δ12)
(when ha is addendum at outer end and δa is the outer cone angle)
Addendum modification coefficients
xm 12 = mn
12am21am
m2
hh minus
Base circle
dvb 12 = dv 12 cos αvt
Transverse contact ratio)
εα = btmP
αsinadd05dd05 vtv2
2vb2
2va2
1vb2
1va minusminus+minus
Overlap ratio) (theoretical value for bevel gears with no crowning but used as approximations in the calculation pro-cedures)
εβ = nm
mmπβsinb
Total contact ratio)
εγ = 2β
2α εε +
( Note that index laquovraquo is left out in order to combine formu-lae for cylindrical and bevel gears)
Tangential speed at midsection
vmt = 3m11 10dn
60π minus
Effective radius of curvature (normal section)
ρvc = ( )2vbm
vtvv
u1βcosαsinua
+
Length of line of contact
lb = ( )( )( )
1εifε
ε1ε2ε
βcosεb
β2γ
2βα
2γ
bm
α ltminusminusminus
lb = 1εifβcosε
εbβ
bmγ
α ge
15 Nominal Tangential Load Ft Fbt Fmt and Fmbt The nominal tangential load (tangential to the reference cyl-inder with diameter d and perpendicular to an axial plane) is calculated from the nominal (rated) torque T transmitted by the gear set
Cylindrical gears
dT2000
Ft = t
tbt αcos
FF =
Bevel gears
mmt d
T2000F =
vt
mtmbt αcos
FF =
Classification Notes- No 412 7 May 2003
DET NORSKE VERITAS
16 Application Factors KA and KAP The application factor KA accounts for dynamic overloads from sources external to the gearing
It is distinguished between the influence of repetitive cyclic torques KA (161) and the influence of temporary occasional peak torques KAP (162)
Calculations are always to be made with KA In certain cases additional calculations with KAP may be necessary
For gears with a defined load spectrum the calculation with a KA may be replaced by a fatigue damage calculation as given in Appendix A
161 KA For gears designed for long or infinite life at nominal rated torque KA is defined as the ratio between the maximum re-petitive cyclic torque applied to the gear set and nominal rated torque
This definition is suitable for main propulsion gears and most of the auxiliary gears
KA can be determined by measurements or system analysis or may be ruled by conventions (ice classes) (For the pur-pose of a preliminary (but not binding) calculation before KA is determined it is advised to apply either the max values mentioned below or values known from similar plants)
a) For main propulsion gears KA can be taken from the (mandatory) torsional vibration analysis thereby con-sidering all permissible driving conditions) Unless specially agreed the rules do not allow KA in excess of 135 for diesel propulsion) With turbine or electric propulsion KA would normally not exceed 12 However special attention should be given to thrusters that are arranged in such a way that heavy vessel movements andor manoeuvring can cause severe load fluctuations This means eg thrusters positioned far from the rolling axis of vessels that could be susceptible to rolling If leading to propeller air suction the condi-tions may be even worse The above mentioned movements or manoeuvring will result in increased propeller excitation If the thruster is driven by a diesel engine the engine mean torque is limited to 100 However thrusters driven by electric motors can suffer temporary mean torque much above 100 unless a suitable load control system (limiting available e-motor torque) is provided
b) For main propulsion gears with ice class notation (see Rules Pt5 Ch1 Sec J500) KA ice has to be taken as the higher value of the applicable (rule defined) ice shock torque referred to nominal rated torque and the value under a) The Baltic ice class notations refer to a few millions ice shock loads Thus the life factors may be put YN=1 and ZN=12 (except for nitrided gears where ZN = 1 applies) Additionally the calculations with the normal KA (no ice class) are to fulfil the normal requirements For polar ice class notations KA ice applies to all criteria and for long or infinite life
c) For a power take off (PTO) branch from a main propul-sion gear with ice class ice shocks result in negative torques It is assumed that the PTO branch is unloaded when the ice shock load occurs The influence of these reverse shock loads may be taken into account as follows The negative torque (reversed load) expressed by means of an application factor based on rated forward load (T or Ft) is KAreverse = KA ice ndash1 (the minus 1 be-cause no mean torque assumed) KAice to be calculated as in the ice class rules This KAreverse should be used for back flank considera-tions such as pitting and scuffing The influence on tooth bending strength (forward di-rection) may be simplified by using the factor YM = 1 minus 03 middot KAreverse KA
d) For diesel driven auxiliaries KA can be taken from the torsional vibration analysis if available For units where no vibration analysis is required (lt 200 kW) or available it is advised to apply KA as the upper allow-able value 135)
e) For turbine or electro driven auxiliaries the same as for c) applies however the practical upper value is 12
) For diesel driven gears more information on KA for mis-firing and normal driving is given in Appendix B
162 KAP The peak overload factor KAP is defined as the ratio between the temporary occasional peak overload torque and the nominal rated torque
For plants where high temporary occasional peak torques can occur (ie in excess of the above mentioned KA) the gearing (if nitrided) has to be checked with regard to static strength Unless otherwise specified the same safety factors as for infinite life apply
The scuffing safety is to be specially considered whereby the KA applies in connection with the bulk temperature and the KAP applies for the flash temperature calculation and should replace KA in the formulae in 431 432 and 441
KAP can be evaluated from the torsional impact vibration calculation (as required by the rules)
If the overloads have a duration corresponding to several revolutions of the shafts the scuffing safety has to be consid-ered on basis of this overload both with respect to bulk and flash temperature This applies to plants with ice class nota-tions (Baltic and polar) and to plants with prime movers which have high temporary overload capacity such as eg electric motors (provided the driven member can have a con-siderable increase in demand torque as eg azimuth thrusters during manoeuvring)
For plants without additional ice class notation KAP should normally not exceed 15
8 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
163 Frequent overloads For plants where high overloads or shock loads occur regu-larly the influence of this is to be considered by means of cumulative fatigue (see Appendix A)
17 Load Sharing Factor Kγ The load sharing factor Kγ accounts for the maldistribution of load in multiple-path transmissions (dual tandem epicyclic double helix etc) Kγ is defined as the ratio between the max load through an actual path and the evenly shared load
171 General method Kγ mainly depends on accuracy and flexibility of the branches (eg quill shaft planet support external forces etc) and should be considered on basis of measurements or of relevant analysis as eg
Kγ = δ+δ f
δ = total compliance of a branch under full load (assuming even load share) referred to gear mesh
f = minusminusminusminus+++ 23
22
21 fff where f1 f2 etc are the
main individual errors that may contribute to a maldis-tribution between the branches Eg tooth pitch errors planet carrier pitch errors bearing clearance influences etc Compensating effects should also be considered
For double helical gears
An external axial force Fext applied from sources outside the actual gearing (eg thrust via or from a tooth coupling) will cause a maldistribution of forces between the two helices Expressed by a load sharing factor the
βsdot
plusmn=γ tanFF1K
t
ext
If the direction of Fext is known the calculation should be carried out separately for each helix and with the tangential force corrected with the pertinent Kγ If the direction of Fext is unknown both combinations are to be calculated and the higher σH or σF to be used
172 Simplified method If no relevant analysis is available the following may apply
For epicyclic gears
Kγ = 32501 minus+ pln
where npl = number of planets ( gt3 )
For multistage gears with locked paths and gear stages sepa-rated by quill shafts (see figure below)
Figure 10 Locked paths gear
Kγ = ( )φ201+
where φ = quill shaft twist (degrees) under full load
18 Dynamic Factor Kv The dynamic factor Kv accounts for the internally generated dynamic loads
Kv is defined as the ratio between the maximum load that dynamically acts on the tooth flanks and the maximum ex-ternally applied load Ft KA Kγ
In the following 2 different methods (181 and 182) are described In case of controversy between the methods the next following is decisive ie the methods are listed with increasing priority
It is important to observe the limitations for the method in 181 In particular the influence of lateral stiffness of shafts is often underestimated and resonances occur at considerably lower speed than determined in 1811
However for low speed gears with vmiddotz1 lt 300 calculations may be omitted and the dynamic factor simplified to Kv=105
181 Single resonance method For a single stage gear Kv may be determined on basis of the relative proximity (or resonance ratio) N between actual speed n1 and the lowest resonance speed nE1
N = 1E
1
nn
Note that for epicyclic gears n is the relative speed ie the speed that multiplied with z gives the mesh frequency
1811 Determination of critical speed It is not advised to apply this method for multimesh gears for N gt 085 as the influence of higher modes has to be consid-ered see 182 In case of significant lateral shaft flexibility (eg overhung mounted bevel gears) the influence of cou-pled bending and torsional vibrations between pinion and wheel should be considered if N ge 075 see 182
nE1 = redmγc
1zπ
31030 sdot
where
cγ is the actual mesh stiffness per unit facewidth see 111
Classification Notes- No 412 9 May 2003
DET NORSKE VERITAS
For gears with inactive ends of the facewidth as eg due to high crowning or end relief such as often applied for bevel gears the use of cγ in connection with determination of natu-ral frequencies may need correction cγ is defined as stiffness per unit facewidth but when used in connection with the total mesh stiffness it is not as simple as cγmiddotb as only a part of the facewidth is active Such corrections are given in 111
mred is the reduced mass of the gear pair per unit facewidth and referred to the plane of contact
For a single gear stage where no significant inertias are closely connected to neither pinion nor wheel mred is calcu-lated as
mred = 21
21mm
mm+
The individual masses per unit facewidth are calculated as
m12 = 221b
21
)2d(b
I
where I is the polar moment of inertia (kgmm2)
The inertia of bevel gears may be approximated as discs with diameter equal the midface pitch diameter and width equal to b However if the shape of the pinion or wheel body differs much from this idealised cylinder the inertia should be cor-rected accordingly
For all kind of gears if a significant inertia (eg a clutch) is very rigidly connected to the pinion or wheel it should be added to that particular inertia (pinion or wheel) If there is a shaft piece between these inertias the torsional shaft stiffness alters the system into a 3-mass (or more) system This can be calculated as in 182 but also simplified as a 2-mass system calculated with only pinion and wheel masses
1812 Factors used for determination of Kv Non-dimensional gear accuracy dependent parameters
( )bKKF
yfcB
At
pptp
γ
minus=
( )bKKF
yFcBAt
ff
γ
α minus=
Non-dimensional tip relief parameter
bKKFcC
1BγAt
ak sdotsdot
sdotminus=
For gears of quality grade (ISO 1328) Q = 7 or coarser Bk = 1
For gears with Q le 6 and excessive tip relief Bk is limited to max 1
For gears (all quality grades) with tip relief of more than 2middotCeff (see 432) the reduction of αε has to be considered (see 443)
where
fpt = the single pitch deviation (ISO 1328) max of pinion or wheel
Fα = the total profile form deviation (ISO 1328) max of pinion or wheel (Note Fα is pt not available for bevel gears thus use Fα = fpt)
yp and yf = the respective running-in allowances and may be calculated similarly to yα in 112 ie the value of fpt is replaced by Fα for yf
cacute = the single tooth stiffness see 111
Ca = the amount of tip relief see 433 In case of different tip relief on pinion and wheel the value that results in the greater value of Bk is to be used If Ca is zero by design the value of running-in tip relief Cay (see 112) may be used in the above formula
1813 Kv in the subcritical range Cylindrical gears N le 085
Bevel gears N le 075
Kv = 1 + N K
K = Cv1 Bp + Cv2 Bf + Cv3 Bk
Cv1 accounts for the pitch error influence
Cv1 = 032
Cv2 accounts for profile error influence
Cv2 = 034 for γε le 2
Cv2 = 03ε
057
γ minus for γε gt 2
Cv3 accounts for the cyclic mesh stiffness variation
Cv3 = 023 for γε le 2
Cv3 = 156ε
0096
γ minus for γε gt 2
1814 Kv in the main resonance range Cylindrical gears 085 lt N le 115
Bevel gears 075 lt N le 125
Running in this range should preferably be avoided and is only allowed for high precision gears
Kv = 1 + Cv1 Bp + Cv2 Bf + Cv4 Bk
Cv4 accounts for the resonance condition with the cyclic mesh stiffness variation
Cv4 = 090 for γε le 2
Cv4 = 144ε
ε005057
γ
γ
minus
minus for γε gt 2
10 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
1815 Kv in the supercritical range Cylindrical gears N ge 15
Bevel gears N ge 15
Special care should be taken as to influence of higher vibra-tion modes andor influence of coupled bending (ie lateral shaft vibrations) and torsional vibrations between pinion and wheel These influences are not covered by the following approach
Kv = Cv5 Bp + Cv6 Bf + Cv7
Cv5 accounts for the pitch error influence
Cv5 = 047
Cv6 accounts for the profile error influence
Cv6 = 047 for εγ le 2
Cv6 = 174ε
012
γ minus for εγ gt 2
Cv7 relates the maximum externally applied tooth loading to the maximum tooth loading of ideal accurate gears operating in the supercritical speed sector when the circumferential vibration becomes very soft
Cv7 = 075 for εγ le 15
Cv7 = [ ] 8750)2(sin 0125 +minusβεπ for 15 lt εγ le 25
Cv7 = 10 for εγ gt 25
1816 Kv in the intermediate range Cylindrical gears 115 lt N lt 15
Bevel gears 125 lt N lt 15
Comments raised in 1814 and 1815 should be observed
Kv is determined by linear interpolation between Kv for N = 115 respectively 125 and N = 15 as
Cylindrical gears
( ) ( ) ( )[ ]51Nv151Nv51Nvv KK350
N51KK === minussdot
minus
+=
Bevel gears
( ) ( ) ( )[ ]51Nv251Nv51Nvv KK250
N51KK === minussdot
minus
+=
182 Multi-resonance method For high speed gear (vgt40 ms) for multimesh medium speed gears for gears with significant lateral shaft flexibility etc it is advised to determine Kv on basis of relevant dy-namic analysis
Incorporating lateral shaft compliance requires transforma-tion of even a simple pinion-wheel system into a lumped multi-mass system It is advised to incorporate all relevant inertias and torsional shaft stiffnesses into an equivalent (to pinion speed) system Thereby the mesh stiffness appears as an equivalent torsional stiffness
cγ b (db12)2 (Nmrad)
The natural frequencies are found by solving the set of dif-ferential equations (one equation per inertia) Note that for a gear put on a laterally flexible shaft the coupling bending-torsionals is arranged by introducing the gear mass and the lateral stiffness with its relation to the torsional displacement and torque in that shaft
Only the natural frequency (ies) having high relative dis-placement and relative torque through the actual pinion-wheel flexible element need(s) to be considered as critical frequency (ies)
Kv may be determined by means of the method mentioned in 181 thereby using N as the least favourable ratio (in case of more than one pinion-wheel dominated natural frequency) Ie the N-ratio that results in the highest Kv has to be consid-ered
The level of the dynamic factor may also be determined on basis of simulation technique using numeric time integration with relevant tooth stiffness variation and pitchprofile er-rors
19 Face Load Factors KHβ and KFβ The face load factors KHβ for contact stresses and for scuff-ing KFβ for tooth root stresses account for non-uniform load distribution across the facewidth
KHβ is defined as the ratio between the maximum load per unit facewidth and the mean load per unit facewidth
KFβ is defined as the ratio between the maximum tooth root stress per unit facewidth and the mean tooth root stress per unit facewidth The mean tooth root stress relates to the con-sidered facewidth b1 respectively b2
Note that facewidth in this context is the design facewidth b even if the ends are unloaded as often applies to eg bevel gears
The plane of contact is considered
191 Relations between KHβ and KFβ
KFβ = expβHK 2(hb)hb1
1exp++
=
where hb is the ratio tooth heightfacewidth The maximum of h1b1 and h2b2 is to be used but not higher than 13 For double helical gears use only the facewidth of one helix
If the tooth root facewidth (b1 or b2) is considerably wider than b the value of KFβ(1or2) is to be specially considered as it may even exceed KHβ
Eg in pinion-rack lifting systems for jack up rigs where b = b2 asymp mn and b1 asymp 3 mn the typical KHβ asymp KFβ2 asymp 1 and KFβ1 asymp 13
192 Measurement of face load factors Primarily
KFβ may be determined by a number of strain gauges distrib-uted over the facewidth Such strain gauges must be put in
Classification Notes- No 412 11 May 2003
DET NORSKE VERITAS
exactly the same position relative to the root fillet Relations in 191 apply for conversion to KHβ
Secondarily
KHβ may be evaluated by observed contact patterns on vari-ous defined load levels It is imperative that the various test loads are well defined Usually it is also necessary to evalu-ate the elastic deflections Some teeth at each 90 degrees are to be painted with a suitable lacquer Always consider the poorest of the contact patterns
After having run the gear for a suitable time at test load 1 (the lowest) observe the contact pattern with respect to ex-tension over the facewidth Evaluate that KHβ by means of the methods mentioned in this section Proceed in the same way for the next higher test load etc until there is a full face contact pattern From these data the initial mesh misalign-ment (ie without elastic deflections) can be found by ex-trapolation and then also the KHβ at design load can be found by calculation and extrapolation See example
Figure 11 Example of experimental determination of KHβ
It must be considered that inaccurate gears may accumulate a larger observed contact pattern than the actual single mesh to mesh contact patterns This is particularly important for lapped bevel gears Ground or hard metal hobbed bevel gears are assumed to present an accumulated contact pattern that is practically equal the actual single mesh to mesh con-tact patterns As a rough guidance the (observed) accumu-lated contact pattern of lapped bevel gears may be reduced by 10 in order to assess the single mesh to mesh contact pattern which is used in 199
193 Theoretical determination of KHβ The methods described in 193 to 198 may be used for cy-lindrical gears The principles may to some extent also be used for bevel gears but a more practical approach is given in 199
General For gears where the tooth contact pattern cannot be verified during assembly or under load all assumptions are to be well on the safe side
KHβ is to be determined in the plane of contact
The influence parameters considered in this method are
bull mean mesh stiffness cγ (see 111) (if necessary also variable stiffness over b)
bull mean unit load Fmb = Fbt KA Kγ Kvb (for double helical gears see 17 for use of Kγ)
bull misalignment fsh due to elastic deflections of shafts and gear bodies (both pinion and wheel)
bull misalignment fdefl due to elastic deflections of and work-ing positions in bearings
bull misalignment fbe due to bearing clearance tolerances bull misalignment fma due to manufacturing tolerances bull helix modifications as crowning end relief helix correc-
tion bull running in amount yβ (see 112)
In practice several other parameters such as centrifugal ex-pansion thermal expansion housing deflection etc contrib-ute to KHβ However these parameters are not taken into account unless in special cases when being considered as particularly important
When all or most of the am parameters are to be considered the most practical way to determine KHβ is by means of a graphical approach described in 1931
If cγ can be considered constant over the facewidth and no helix modifications apply KHβ can be determined analyti-cally as described in 1932
1931 Graphical method The graphical method utilises the superposition principle and is as follows
bull Calculate the mean mesh deflection Mδ as a function of
γm c and bF see 111
bull Draw a base line with length b and draw up a rectangu-lar with height δM (The area δM b is proportional to the transmitted force)
bull Calculate the elastic deflection fsh in the plane of contact Balance this deflection curve around a zero line so that the areas above and below this zero line are equal
Figure 12 fsh balanced around zero line
bull Superimpose these ordinates of the fsh curve to the previ-ous load distribution curve (The area under this new load distribution curve is still δM b)
bull Calculate the bearing deflections andor working posi-tions in the bearings and evaluate the influence fdefl in the plane of contact This is a straight line and is bal-anced around a zero line as indicated in Fig 14 but with one distinct direction Superimpose these ordinates to the previous load distribution curve
12 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
bull The amount of crowning end relief or helix correction (defined in the plane of contact) is to be balanced around a zero line similarly to fsh
Figure 13 Crowning Cc balanced aound zero line
bull Superimpose these ordinates to the previous load distri-bution curve In case of high crowning etc as eg often applied to bevel gears the new load distribution curve may cross the base line (the real zero line) The result is areas with negative load that is not real as the load in those areas should be zero Thus corrective actions must be made but for practical reasons it may be postponed to after next operation
bull The amount of initial mesh misalignment bema ff + (defined in the plane of contact) is to be bal-
anced around a zero line If the direction of bema ff + is known (due to initial contact check) or if the direction of fbe is known due to design (eg overhang bevel pin-ion) this should be taken into account If direction un-known the influence of bema ff + in both directions as well as equal zero should be considered
Figure 14 fma+fbe in both directions balanced around zero line
Superimpose these ordinates to the previous load distri-bution curve This results in up to 3 different curves of which the one with the highest peak is to be chosen for further evaluation
bull If the chosen load distribution curve crosses the base line (ie mathematically negative load) the curve is to be corrected by adding the negative areas and dividing this with the active facewidth The (constant) ordinates of this rectangular correction area are to be subtracted from the positive part of the load distribution curve It is advisable to check that the area covered under this new load distribution curve is still equal δM b
bull If cγ cannot be considered as constant over b then cor-rect the ordinates of the load distribution curve with the local (on various positions over the facewidth) ratio between local mesh stiffness and average mesh stiffness cγ (average over the active facewidth only) Note that the result is to be a curve that covers the same area δM b as before
bull The influence of running in yβ is to be determined as in 112 whereby the value for Fβx is to be taken as twice the distance between the peak of the load distribution curve and δM
bull Determine
KHβ = M
β
δycurveofpeak minus
1932 Simplified analytical method for cylindrical gears The analytical approach is similar to 1931 but has a more limited application as cγ is assumed constant over the face-width and no helix modification applies
bull Calculate the elastic deflection fsh in the plane of contact Balance this deflection curve around a zero line so that the area above and below this zero line are equal see Fig 12 The max positive ordinate is frac12∆fsh
bull Calculate the initial mesh alignment as Fβx= deflbemash ffff plusmnplusmnplusmn∆ The negative signs may only be used if this is justified andor verified by a contact pattern test Otherwise al-ways use positive signs If a negative sign is justified the value of Fβx is not to be taken less than the largest of each of these elements
bull Calculate the effective mesh misalignment as Fβy = Fβx - yβ (yβ see 112)
bull Determine
2KforF2
bFc1K βH
m
βγγH le+=β
or
2KforF
bFc2K Hβ
m
βγγH gt=β
where cγ as used here is the effective mesh stiffness see 111
194 Determination of fsh fsh is the mesh misalignment due to elastic deflections Usu-ally it is sufficient to consider the combined mesh deflection of the pinion body and shaft and the wheel shaft The calcu-lation is to be made in the plane of contact (of the considered gear mesh) and to consider all forces (incl axial) acting on the shafts Forces from other meshes can be parted into components parallel respectively vertical to the considered plane of contact Forces vertical to this plane of contact have no influence on fsh
It is advised to use following diameters for toothed elements
d + 2 x mn for bending and shear deflection
d + 2 mn (x ndash ha0 + 02) for torsional deflection
Usually fsh is calculated on basis of an evenly distributed load If the analysis of KHβ shows a considerable maldis-
Classification Notes- No 412 13 May 2003
DET NORSKE VERITAS
tribution in term of hard end contact or if it is known by other reasons that there exists a hard end contact the load should be correspondingly distributed when calculating fsh In fact the whole KHβ procedure can be used iteratively 2-3 iterations will be enough even for almost triangular load distributions
195 Determination of fdefl fdefl is the mesh misalignment in the plane of contact due to bearing deflections and working positions (housing deflec-tion may be included if determined)
First the journal working positions in the bearings are to be determined The influence of external moments and forces must be considered This is of special importance for twin pinion single output gears with all 3 shafts in one plane
For rolling bearings fdefl is further determined on basis of the elastic deflection of the bearings An elastic bearing deflec-tion depends on the bearing load and size and number of rolling elements Note that the bearing clearance tolerances are not included here
For fluid film bearings fdefl is further determined on basis of the lift and angular shift of the shafts due to lubrication oil film thickness Note that fbe takes into account the influence of the bearing clearance tolerance
When working positions bearing deflections and oil film lift are combined for all bearings the angular misalignment as projected into the plane of the contact is to be determined fdefl is this angular misalignment (radians) times the face-width
196 Determination of fbe fbe is the mesh misalignment in the plane of contact due to tolerances in bearing clearances In principle fbe and fdefl could be combined But as fdefl can be determined by analy-sis and has a distinct direction and fbe is dependent on toler-ances and in most cases has no distinct direction (ie + toler-ance) it is practicable to separate these two influences
Due to different bearing clearance tolerances in both pinion and wheel shafts the two shaft axis will have an angular mis-alignment in the plane of contact that is superimposed to the working positions determined in 195 fbe is the facewidth times this angular misalignment Note that fbe may have a distinct direction or be given as a + tolerance or a combina-tion of both For combination of + tolerance it is adviced to use
fbe= 22be
21be ff ++plusmn
fbe is particularly important for overhang designs for gears with widely different kinds of bearings on each side and when the bearings have wide tolerances on clearances In general it shall be possible to replace standard bearings with-out causing the real load distribution to exceed the design premises For slow speed gears with journal bearings the expected wear should also be considered
197 Determination of fma fma is the mesh misalignment due to manufacturing toler-ances (helix slope deviation) of pinion fHβ1 wheel fHβ2 and housing bore
For gear without specifically approved requirements to as-sembly control the value of fma is to be determined as
fma= 22Hβ
21Hβ ff +
For gears with specially approved assembly control the value of fma will depend on those specific requirements
198 Comments to various gear types For double helical gears KHβ is to be determined for both helices Usually an even load share between the helices can be assumed If not the calculation is to be made as de-scribed in 171
For planetary gears the free floating sun pinion suffers only twist no bending It must be noted that the total twist is the sum of the twist due to each mesh If the value of 1K γ ne this must be taken into account when calculating the total sun pinion twist (ie twist calculated with the force per mesh without Kγ and multiplied with the number of planets)
When planets are mounted on spherical bearings the mesh misalignments sun-planet respectively planet-annulus will be balanced Ie the misalignment will be the average between the two theoretical individual misalignments The faceload distribution on the flanks of the planets can take full advan-tage of this However as the sun and annulus mesh with several planets with possibly different lead errors the sun and annulus cannot obtain the above mentioned advantage to the full extent
199 Determination of KHβ for bevel gears If a theoretical approach similar to 193-198 is not docu-mented the following may be used
testeff
H Kb
b851851K sdot
minussdot=β
beff b represents the relative active facewidth (regarding lapped gears see 192 last part)
Higher values than beff b = 090 are normally not to be used in the formula
For dual directional gears it may be difficult to obtain a high beff b in both directions In that case the smaller beff b is to be used
Ktest represents the influence of the bearing arrangement shaft stiffness bearing stiffness housing stiffness etc on the faceload distribution and the verification thereof Expected variations in length- and height-wise tooth profile is also accounted for to some extent
a) Ktest = 1 For ground or hard metal hobbed gears with the specified contact pattern verified at full rating or at full torque slow turning at a condition representative for the thermal expansion at normal operation
14 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
It also applies when the bearing arrangementsupport has insignificant elastic deflections and thermal axial expansion However each initial mesh contact must be verified to be within acceptance criteria that are cali-brated against a type test at full load Reproduction of the gear tooth length- and height-wise profile must also be verified This can be made through 3D measurements or by initial contact movements caused by defined axial offsets of the pinion (tolerances to be agreed upon)
b) Ktest = 1 + 04middot(beffbndash06) For designs with possible influence of thermal expansion in the axial direction of the pinion The initial mesh contact verified with low load or spin test where the acceptance criteria are calibrated against a type test at full load
c) Ktest = 12 if mesh is only checked by toolmakerrsquos blue or by spin test contact For gears in this category beffb gt 085 is not to be used in the calculation
110 Transversal Load Distribution Factors KHα and KFα The transverse load distribution factors KHα for contact stresses and for scuffing KFα for tooth root stresses account for the effects of pitch and profile errors on the transversal load distribution between 2 or more pairs of teeth in mesh
The following relations may be used
Cylindrical gears
( )
minus+==
tH
αptγγHαFα F
byfc0409
2ε
KK
valid for 2εγ le
( ) ( )tH
αptγ
γ
γHαFα F
byfcε
1ε20409KK
minusminus+==
valid for 2ε γ gt
where
FtH = Ft KA Kγ Kv KHβ
cγ = See 111
γα = See 112
fpt = Maximum single pitch deviation (microm) of pinion or wheel or maximum total profile form devia-tion Fα of pinion or wheel if this is larger than the maximum single pitch deviation
Note In case of adequate equivalent tip relief adapted to the load half of the above mentioned fpt can be introduced A tip relief is considered adequate when the average of Ca1 and Ca2 is within plusmn40 of the value of Ceff in 432
Limitations of KHα and KFα
If the calculated values for
KFα = KHα lt 1 use KFα = KHα = 10
If the calculated value of KHα gt 2εα
γ
Zε
ε use KHα = 2
εα
γ
Zε
ε
If the calculated value of KFα gt εα
γ
Yεε
use KFα = εα
γ
Yεε
where Yε = αnε
075025+ (for εαn see 331c)
Bevel gears
For ground or hard metal hobbed gears KFα = KHα = 1
For lapped gears KFα = KHα = 11
111 Tooth Stiffness Constants cacute and cγ The tooth stiffness is defined as the load which is necessary to deform one or several meshing gear teeth having 1 mm facewidth by an amount of 1 microm in the plane of contact cacute is the maximum stiffness of a single pair of teeth cγ is the mean value of the mesh stiffness in a transverse plane (brief term mesh stiffness)
Both valid for high unit load (Unit load = Ft middot KA middot Kγb)
Cylindrical gears The real stiffness is a combination of the progressive Hertzian contact stiffness and the linear tooth bending stiff-nesses For high unit loads the Hertzian stiffness has little importance and can be disregarded This approach is on the safe side for determination of KHβ and KHα However for moderate or low loads Kv may be underestimated due to determination of a too high resonance speed
The linear approach is described in A
An optional approach for inclusion of the non-linear stiffness is described in B
A The linear approach
BRCCq
βcos08acutec =
and
( )025ε075cc αγ +prime=
where
( )[ ]n02a01a
B α2000212
hh12051C minusminus
+minus+=
12n1n
x000635z
025791z
015551004723q minus++=
12
2n
22
1n
1 x005290z
x024188x000193z
x011654+minusminusminus
+ 000182 x22
Classification Notes- No 412 15 May 2003
DET NORSKE VERITAS
(for internal gears use zn2 equal infinite and x2 = 0) ha0 = hfp for all practical purposes CR considers the increased flexibility of the wheel teeth if the wheel is not a solid disc and may be calculated as
( )( )nR m5s
sR e5
bbln1C +=
where
bs = thickness of a central web
sR = average thickness of rim (net value from tooth root to inside of rim)
The formula is valid for bs b ge 02 and sRmn ge 1 Outside this range of validity and if the web is not centrally posi-tioned CR has to be specially considered
Note CR is the ratio between the average mesh stiffness over the facewidth and the mesh stiffness of a gear pair of solid discs The local mesh stiffness in way of the web corresponds to the mesh stiffness with CR = 1 The local mesh stiffness where there is no web support will be less than calculated with CR above Thus eg a centrally positioned web will have an effect corresponding to a longitudinal crowning of the teeth See also 1931 regarding KHβ
B The non-linear approach
In the following an example is given on how to consider the non-linearity
The relation between unit load Fb as a function of mesh deflection δ is assumed to be a progressive curve up to 500 Nmm and from there on a straight line This straight line when extended to the baseline is assumed to intersect at 10microm
With these assumptions the unit force Fb as a function of mesh deflection δ can be expressed as
( )10δKbF
minus= for 500bFgt
minus=
500Fb10δK
bF for 500
bFlt
with γAt KK
bF
bF
sdotsdot= etc (Nmm) ie unit load incorporat-
ing the relevant factors as
KA middot Kγ for determination of Kv
KA middot Kγ middot Kv for determination of KHβ
KA middot Kγ middot Kv middot KHβ for determination of KHα
δ = mesh deflection (microm)
K = applicable stiffness (c or cγ)
Use of stiffnesses for KV KHβ and KHα
For calculation of Kv and KHα the stiffness is calculated as follows
When Fb lt 500
the stiffness is determined as δ∆
∆ bF
where the increment is chosen as eg ∆ Fb = 10 and thus
50010Fb10
K10Fb∆δ +
++
=
When F b gt 500 the stiffness is c or cγ For calculation of KHβ the mesh deflection δ is used directly
or an equivalent stiffness determined as δsdotb
F
Bevel gears
In lack of more detailed relationship between stiffness and geometry the following may be used
b085
b13cacute eff=
b085b
16c effγ =
beff not to be used in excess of 085 b in these formulae
Bevel gears with heightwise and lengthwise crowning have progressive mesh stiffness The values mentioned above are only valid for high loads They should not be used for de-termination of Ceff (see 432) or KHβ (see 1931)
112 Running-in Allowances The running-in allowances account for the influence of run-ning-in wear on the various error elements
yα respectively yβ are the running-in amounts which reduce the influence of pitch and profile errors respectively influ-ence of localised faceload
Cay is defined as the running-in amount that compensates for lack of tip relief
The following relations may be used
For not surface hardened steel
ptHlim
α fσ160y =
yβ = βxlimH
f320σ
with the following upper limits
V lt 5 ms 5-10 ms gt 10 ms
yα max none
limH
12800σ limH
6400σ
yβ max none
limH
25600σ limH
12800σ
For surface hardened steel
yα = 0075 fpt but not more than 3 for any speed
yβ = 015 Fβx but not more than 6 for any speed
16 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
For all kinds of steel
5145189718
1C2
limHay +
minusσ
=
When pinion and wheel material differ the following ap-plies
bull Use the larger of fpt1 - yα1 and fpt2 - yα2 to replace fpt - yα in the calculation of KHα and Kv
bull Use ( )2β1ββ yy21y += in the calculation of KHβ
bull Use ( )2ay1aya CC21C += in the calculation of Kv
bull Use ( )2ay1ay2a1a CC21CC +== in the scuffing
calculation if no design tip relief is foreseen
Classification Notes- No 412 17 May 2003
DET NORSKE VERITAS
2 Calculation of Surface Durability
21 Scope and General Remarks Part 2 includes the calculations of flank surface durability as limited by pitting spalling case crushing and subsurface yielding Endurance and time limited flank surface fatigue is calculated by means of 22 ndash 212 In a way also tooth frac-tures starting from the flank due to subsurface fatigue is in-cluded through the criteria in 213
Pitting itself is not considered as a critical damage for slow speed gears However pits can create a severe notch effect that may result in tooth breakage This is particularly im-portant for surface hardened teeth but also for high strength through hardened teeth For high-speed gears pitting is not permitted
Spalling and case crushing are considered similar to pitting but may have a more severe effect on tooth breakage due to the larger material breakouts initiated below the surface Subsurface fatigue is considered in 213
For jacking gears (self-elevating offshore units) or similar slow speed gears designed for very limited life the max static (or very slow running) surface load for surface hard-ened flanks is limited by the subsurface yield strength
For case hardened gears operating with relatively thin lubri-cation oil films grey staining (micropitting) may be the lim-iting criterion for the gear rating Specific calculation meth-ods for this purpose are not given here but are under consid-eration for future revisions Thus depending on experience with similar gear designs limitations on surface durability rating other than those according to 22 - 213 may be ap-plied
22 Basic Equations Calculation of surface durability (pitting) for spur gears is based on the contact stress at the inner point of single pair contact or the contact at the pitch point whichever is greater
Calculation of surface durability for helical gears is based on the contact stress at the pitch point
For helical gears with 0 lt εβ lt 1 a linear interpolation be-tween the above mentioned applies
Calculation of surface durability for spiral bevel gears is based on the contact stress at the midpoint of the zone of contact
Alternatively for bevel gears the contact stress may be cal-culated with the program ldquoBECALrdquo In that case KA and Kv are to be included in the applied tooth force but not KHβ and KHα The calculated (real) Hertzian stresses are to be multi-plied with ZK in order to be comparable with the permissible contact stresses
The contact stresses calculated with the method in part 2 are based on the Hertzian theory but do not always represent the real Hertzian stresses
The corresponding permissible contact stresses σHP are to be calculated for both pinion and wheel
221 Contact stress Cylindrical gears
( )HαHβvγA
1
tβεEHDBH KKKKK
bud1uF
ZZZZZσ+
=
where
ZBD = Zone factor for inner point of single pair contact for pinion resp wheel (see 232)
ZH = Zone factor for pitch point (see 231)
ZE = Elasticity factor (see 24)
Zε = Contact ratio factor (see 25)
Zβ = Helix angle factor (see 26)
Ft KA Kγ Kv KHβ KHα see 15 ndash 110
d1 b u see 12 ndash 15
Bevel gears
( )HαHβvγA
v1v
vmtKEMH KKKKK
bud1uF
ZZZ105σ+
sdot=
where
105 is a correlation factor to reach real Hertzian stresses (when ZK = 1)
ZE KA etc see above
ZM = mid-zone factor see 233
ZK = bevel gear factor see 27
Fmt dv1 uv see 12 ndash 15
It is assumed that the heightwise crowning is chosen so as to result in the maximum contact stresses at or near the mid-point of the flanks
222 Permissible contact stress
XWRvLH
NHlimHP ZZZZZ
SZσ
σ =
where
σH lim = Endurance limit for contact stresses (see 28)
ZN = Life factor for contact stresses (see 29)
SH = Required safety factor according to the rules
ZLZvZR = Oil film influence factors (see 210)
ZW = Work hardening factor (see 211)
ZX = Size factor (see 212)
18 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
23 Zone Factors ZH ZBD and ZM
231 Zone factor ZH The zone factor ZH accounts for the influence on contact stresses of the tooth flank curvature at the pitch point and converts the tangential force at the reference cylinder to the normal force at the pitch cylinder
wtt2
wtbH sinααcos
cosαcosβ2Z =
232 Zone factors ZBD The zone factors ZBD account for the influence on contact stresses of the tooth flank curvature at the inner point of sin-gle pair contact in relation to ZH Index B refers to pinion D to wheel
For εβ ge 1 ZBD = 1
For internal gears ZD = 1
For εβ = 0 (spur gears)
( )
π
minusεminusminus
π
minusminus
α=
α2
2
2b
2a
1
2
1b
1a
wtB
z211
dd
z21
dd
tanZ
( )
π
minusεminusminus
π
minusminus
α=
α1
2
1b
1a
2
2
2b
2a
wtD
z211
dd
z21
dd
tanZ
If ZB lt 1 use ZB = 1
If ZD lt 1 use ZD = 1
For 0 lt εβ lt 1
ZBD = ZBD (for spur gears) ndash εβ (ZBD (for spur gears) ndash 1)
233 Zone factor ZM The mid-zone factor ZM accounts for the influence of the contact stress at the mid point of the flank and applies to spi-ral bevel gears
εminusminus
εminusminus
αβ=
αα btm2
2vb2
2vabtm2
1vb2val
2v1vvtbmM
pddpdd
ddtancos2Z
This factor is the product of ZH and ZM-B in ISO 10300 with the condition that the heightwise crowning is sufficient to move the peak load towards the midpoint
234 Inner contact point For cylindrical or bevel gears with very low number of teeth the inner contact point (A) may be close to the base circle In order to avoid a wear edge near A it is required to have suitable tip relief on the wheel
24 Elasticity Factor ZE The elasticity factor ZE accounts for the influence of the material properties as modulus of elasticity and Poissonrsquos ratio on the contact stresses
For steel against steel ZE = 1898
25 Contact Ratio Factor Zε The contact ratio factor Zε accounts for the influence of the transverse contact ratio εα and the overlap ratio εβ on the contact stresses
αε1Z =ε for 1εβ ge
( )α
ββ
αε ε
εε1
3ε4
Z +minusminus
= for εβ lt 1
26 Helix Angle Factor Zβ The helix angle factor Zβ accounts for the influence of helix angle (independent of its influence on Zε) on the surface du-rability
cosβZβ =
27 Bevel Gear Factor ZK The bevel gear factor accounts for the difference between the real Hertzian stresses in spiral bevel gears and the contact stresses assumed responsible for surface fatigue (pitting) ZK adjusts the contact stresses in such a way that the same per-missible stresses as for cylindrical gears may apply
The following may be used ZK = 080
28 Values of Endurance Limit σHlim and Static Strength 5H10σ 3H10σ
σHlim is the limit of contact stress that may be sustained for 5middot107 cycles without the occurrence of progressive pitting
For most materials 5middot107 cycles are considered to be the be-ginning of the endurance strength range or lower knee of the σ-N curve (See also Life Factor ZN) However for nitrided steels 2middot106 apply
For this purpose pitting is defined by
bull for not surface hardened gears pitted area ge 2 of total active flank area
bull for surface hardened gears pitted area ge 05 of total active flank area or ge 4 of one particular tooth flank area 510Hσ and 310Hσ are the contact stresses which the given
material can withstand for 105 respectively 103 cycles without subsurface yielding or flank damages as pitting spalling or case crushing when adequate case depth applies
The following listed values for σHlim 510Hσ and 310Hσ may only be used for materials subjected to a quality control as
Classification Notes- No 412 19 May 2003
DET NORSKE VERITAS
the one referred to in the rules Results of approved fatigue tests may also be used as the basis for establishing these values The defined survival probability is 99
σHlim σH105 σH10
3
Alloyed case hardened steels (surface hardness 58-63 HRC) - of specially approved high grade - of normal grade
1650 1500
2500 2400
3100 3100
Nitrided steel of approved grade gas nitrided (surface hardness 700-800 HV) 1250 13 σHlim 13 σHlim
Alloyed quenched and tempered steel bath or gas nitrided (surface hardness 500-700 HV)
1000
13 σHlim
13 σHlim
Alloyed flame or induction hardened steel (surface hardness 500-650 HV) 075 HV + 750 16 σHlim 45 HV
Alloyed quenched and tempered steel 14 HV + 350 16 σHlim 45 HV
Carbon steel 15 HV + 250 16 σHlim 16 σHlim
These values refer to forged or hot rolled steel For cast steel the values for σHlim are to be reduced by 15
29 Life Factor ZN The life factor ZN takes account of a higher permissible contact stress if only limited life (number of cycles NL) is demanded or lower permissible contact stress if very high number of cycles apply
If this is not documented by approved fatigue tests the fol-lowing method may be used
For all steels except nitrided
7L 105N sdotge ZN = 1 or
01570
L
7
N N105Z
sdot=
Ie ZN = 092 for 1010 cycles
The ZN = 1 from 5middot107 on may only be used when the mate-rial cleanliness is of approved high grade (see Rules Pt4 Ch2) and the lubrication is optimised by a specially approved filtering process
105 lt NL lt 5middot107 510N
logZ037
L
7
N N105Z
sdot=
NL = 105 WXRVLHlim
WstX10H1010NN ZZZZZσ
ZZσZZ
55
5=
103 lt NL lt 105 )Z(Zlog05
L
5
10NN
5N103N10
5N10ZZ
==
3L 10N le
WXRVLlimH
Wst10X10H10NN ZZZZZ
ZZZZ
33
3σ
σ==
(but not less than ZN105)
For nitrided steels
6L 102N sdotge ZN = 1 or
00980
L
6
N N102Z
sdot=
Ie ZN = 092 for 1010 cycles
The ZN = 1 from 2middot106 on may only be used when the mate-rial cleanliness is of approved high grade (see Rules Pt4 Ch2) and the lubrication is optimised by a specially approved filtering process
105 lt NL lt 2middot106 510N
Zlog76860
L
6
N N102Z
sdot=
5L 10N le
XWRVL
10XWst10NN ZZZZZ
ZZ13ZZ
55 ==
Note that when no index indicating number of cycles is used the factors are valid for 5middot107 (respectively 2middot106 for nitrid-ing) cycles
210 Influence Factors on Lubrication Film ZL ZV and ZR The lubricant factor ZL accounts for the influence of the type of lubricant and its viscosity the speed factor ZV ac-counts for the influence of the pitch line velocity and the roughness factor ZR accounts for influence of the surface roughness on the surface endurance capacity
The following methods may be applied in connection with the endurance limit
20 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
Surface hardened steels Not surface hardened steels
ZL ( )24013421
360910ν+
+
( )24013421
680830ν+
+
ZV ( )v3280
140930+
+ ( )v3280300850
++
ZR
080
ZrelR3
150
ZrelR3
where
ν40 = Kinematic oil viscosity at 40ordmC (mm2s) For case hardened steels the influence of a high bulk temperature (see 4 Scuffing) should be con-sidered Eg bulk temperatures in excess of 120ordmC for long periods may cause reduced flank surface endurance limits
For values of ν40 gt 500 use ν40 = 500
RZrel = The mean roughness between pinion and wheel (after running in) relative to an equivalent radius of curvature at the pitch point ρc = 10mm
RZrel
= ( ) 31
cZ2Z1 ρ
10RR50
+
RZ = Mean peak to valley roughness (microm) (DIN defini-tion) (roughly RZ = 6 Ra)
For 5
L 10N le ZL ZV ZR = 10
211 Work Hardening Factor ZW The work hardening factor ZW accounts for the increase of surface durability of a soft steel gear when meshing the soft steel gear with a surface hardened or substantially harder gear with a smooth surface
The following approximation may be used for the endurance limit
Surface hardened steel against not surface hardened steel 150
ZeqW R
31700
130HB21Z
minus
minus=
where HB = the Brinell hardness of the soft member For HB gt 470 use HB = 470 For HB lt 130 use HB = 130
RZeq = equivalent roughness
033
c40
066
ZS
ZHZHZEQ ρvν
15000RRRR
=
If RZeq gt 16 then use RZeq = 16
If RZeq lt 15 then use RZeq = 15
where
RZH = surface roughness of the hard member before run in
RZS = surface roughness of the soft member before run in
ν40 = see 210
If values of ZW lt 1 are evaluated ZW = 1 should be used for flank endurance However the low value for ZW may indi-cate a potential wear problem
Through hardened pinion against softer wheel
( )
minussdotminus+= 000829
HBHB0008981u1Z
2
1W
For 21HBHB
2
1 le use ZW = 1
For 71HBHB
2
1 gt use 17HBHB
2
1 =
For u gt 20 use u = 20
For static strength (lt 105 cycles)
Surface hardened against not surface hardened
ZWst = 105
Through hardened pinion against softer wheel
ZWst = 1
212 Size Factor ZX The size factor accounts for statistics indicating that the stress levels at which fatigue damage occurs decrease with an increase of component size as a consequence of the influ-ence on subsurface defects combined with small stress gradi-ents and of the influence of size on material quality
ZX may be taken unity provided that subsurface fatigue for surface hardened pinions and wheels is considered eg as in the following subsection 213
213 Subsurface Fatigue This is only applicable to surface hardened pinions and wheels The main objective is to have a subsurface safety against fatigue (endurance limit) or deformation (static strength) which is at least as high as the safety SH required for the surface The following method may be used as an approximation unless otherwise documented
The high cycle fatigue (gt3106 cycles) is assumed to mainly depend on the orthogonal shear stresses Static strength (lt103 cycles) is assumed to depend mainly on equivalent
Classification Notes- No 412 21 May 2003
DET NORSKE VERITAS
stresses (von Mises) Both are influenced by residual stresses but this is only considered roughly and empirically
The subsurface working stresses at depths inside the peak of the orthogonal shear stresses respectively the equivalent stresses are only dependent on the (real) Hertzian stresses Surface related conditions as expressed by ZL ZV and ZR are assumed to have a negligible influence
The real Hertzian stresses σHR are determined as
For helical gears with εβ gt 1
σHR = σH
For helical gears with εβ lt 1 and spur gears
ε
α
ββ ε
ε+εminus
sdotσ=σZ
1
HHR
For bevel gears
KHHR Z
1σσ sdot=
The necessary hardness HV is given as a function of the net depth tz (net = after grinding or hard metal hobbing and per-pendicular to the flank)
The coordinates tz and HV are to be compared with the de-sign specification such as
bull for flame and induction hardening tHVmin HVmin bull for nitriding t400min HV = 400 bull for case hardening t550min HV = 550 t400min HV = 400
and t300min HV = 300 (the latter only if the core hardness lt 300 If the core hardness gt 400 the t400 is to be re-placed by a fictive t400 = 16 t550)
In addition the specified surface hardness is not to be less than the max necessary hardness (at tz = 05aH) This applies to all hardening methods
For high cycle fatigue (gt3 106 cycles) the following applies
sdot+
minussdotsdotsdot= o90
05azt
05azt
cosSσ04HV
H
HHHR
applicable to 05a
zt
Hge
For 05at
H
z lt the value for 05at
H
z = applies
56300ρSσ12a cHHR
Hsdotsdot
sdot=
Where aH is half the hertzian contact width multiplied by an empirical factor of 12 that takes into account the possible influence of reduced compressive residual stresses (or even tensile residual stresses) on the local fatigue strength
If any of the specified hardness depths including the surface hardness is below the curve described by HV = f (tz) the actual safety factor against subsurface fatigue is determined as follows
reduce SH stepwise in the formula for HV and aH until all specified hardness depths and surface hardness balance with the corrected curve The safety factor obtained through this method is the safety against subsurface fatigue
For static strength (lt103 cycles) the following applies
sdot+
minussdotsdotsdot= o90
07at
06a
zt
cosSσ019HV
Hst
z
HstHHR
applicable to 06at
Hst
z ge
56300ρSσa cHHR
Hstsdotsdot
=
In the case of insufficient specified hardness depths the same procedure for determination of the actual safety factor as above applies
For limited life fatigue (103 lt cycles lt 3106 )
For this purpose it is necessary to extend the correction of safety factors to include also higher values than required Ie in the case of more than sufficient hardness and depths the safety factor in the formulae for both high cycle fatigue and static strength are to be increased until necessary and speci-fied values balance
The actual safety factor for a given number of cycles N be-tween 103 and 3106 is found by linear interpolation in a dou-ble logarithmic diagram
minussdotminus
= sdot logN3477
logSlogSlogS
310H6103HHN
36 10H103H Slog86281Slog86280 sdot+sdot sdot
22 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
3 Calculation of Tooth Strength
31 Scope and General Remarks Part 3 include the calculation of tooth root strength as limited by tooth root cracking (surface or subsurface initiated) and yielding
For rim thickness sR ge 35middotmn the strength is calculated by means of 32 ndash 313 For cylindrical gears the calculation is based on the assumption that the highest tooth root tensile stress arises by application of the force at the outer point of single tooth pair contact of the virtual spur gears The method has however a few limitations that are mentioned in 36
For bevel gears the calculation is based on force application at the tooth tip of the virtual cylindrical gear Subsequently the stress is converted to load application at the mid point of the flank due to the heightwise crowning
Bevel gears may also be calculated with the program BECAL In that case KA and Kv are to be included in the applied tooth force but not KFβ and KFα
In case of a thin annulus or a thin gear rim etc radial crack-ing can occur rather than tangential cracking (from root fillet to root fillet) Cracking can also start from the compression fillet rather than the tension fillet For rim thickness sR lt 35middotmn a special calculation procedure is given in 315 and 316 and a simplified procedure in 314
A tooth breakage is often the end of the life of a gear trans-mission Therefore a high safety SF against breakage is re-quired
It should be noted that this part 3 does not cover fractures caused by
bull oil holes in the tooth root space bull wear steps on the flank bull flank surface distress such as pits spalls or grey staining
Especially the latter is known to cause oblique fractures starting from the active flank predominately in spiral bevel gears but also sometimes in cylindrical gears
Specific calculation methods for these purposes are not given here but are under consideration for future revisions Thus depending on experience with similar gear designs limita-tions other than those outlined in part 3 may be applied
32 Tooth Root Stresses The local tooth root stress is defined as the max principal stress in the tooth root caused by application of the tooth force Ie the stress ratio R = 0 Other stress ratios such as for eg idler gears (R asymp -12) shrunk on gear rims (R gt 0) etc are considered by correcting the permissible stress level
321 Local tooth root stress The local tooth root stress for pinion and wheel may be as-sessed by strain gauge measurements or FE calculations or similar For both measurements and calculations all details are to be agreed in advance
Normally the stresses for pinion and wheel are calculated as
Cylindrical gears
FαFβvγAβSFn
tF K K K K K Y Y Y
m bFσ =
where
YF = Tooth form factor (see 33)
YS = Stress correction factor (see 34)
Yβ = Helix angle factor (see 36)
Ft KA Kγ Kv KFβ KFα see 15 ndash 110
b see 13
Bevel gears
FαFβvγASaFamn
mtF K K K K K Y Y Y
m bFσ ε=
where
YFa = Tooth form factor see 33
YSa = Stress correction factor see 34
Yε = Contact ratio factor see 35
Fmt KA etc see 15 ndash 110
b see 13
322 Permissible tooth root stress The permissible local tooth root stress for pinion respectively wheel for a given number of cycles N is
CXRrelTrelTF
NMFEFP YYYY
SYY
δσ
=σ
Note that all these factors YM etc are applicable to 3middot106 cycles when used in this formula for σFP The influence of other number of cycles on these factors is covered by the calculation of YN
where
σFE = Local tooth root bending endurance limit of reference test gear (see 37)
YM = Mean stress influence factor which accounts for other loads than constant load direction eg idler gears temporary change of load di-rection pre-stress due to shrinkage etc (see 38)
YN = Life factor for tooth root stresses related to reference test gear dimensions (see 39)
SF = Required safety factor according to the rules
YδrelT = Relative notch sensitivity factor of the gear to be determined related to the reference test gear (see 310)
YRrelT = Relative (root fillet) surface condition factor of the gear to be determined related to the reference test gear (see 311)
Classification Notes- No 412 23 May 2003
DET NORSKE VERITAS
YX = Size factor (see 312)
YC = Case depth factor (see 313)
33 Tooth Form Factors YF YFa The tooth form factors YF and YFa take into account the in-fluence of the tooth form on the nominal bending stress
YF applies to load application at the outer point of single tooth pair contact of the virtual spur gear pair and is used for cylindrical gears
YFa applies to load application at the tooth tip and is used for bevel gears
Both YF and YFa are based on the distance between the con-tact points of the 30˚-tangents at the root fillet of the tooth profile for external gears respectively 60˚ tangents for inter-nal gears
Figure 31 External tooth in normal section
Figure 32 Internal tooth in normal section
Definitions
n
2
n
Fn
enFn
Fe
F
α cosms
α cosmh6
Y
=
n
2
n
Fn
anFn
Fa
Fa
α cosms
α cosmh6
Y
=
In the case of helical gears YF and YFa are determined in the normal section ie for a virtual number of teeth
YFa differs from YF by the bending moment arm hFa and αFan and can be determined by the same procedure as YF with exception of hFe and αFan For hFa and αFan all indices e will change to a (tip)
The following formulae apply to cylindrical gears but may also be used for bevel gears when replacing
mn with mnm
zn with zvn
αt with αvt
β with βm
with undercut without undercut
Fig 33 Dimensions and basic rack profile of the teeth (finished profile)
Tool and basic rack data such as hfP ρfp and spr etc are re-ferred to mn ie dimensionless
331 Determination of parameters
( )n
n
prnfPnfP m
α cossαsin 1ρ
αtan h4πE
minusminusminusminus=
For external gears fPfP ρρ =
For internal gears ( )0z
195fPfP0
fPfP 10363156ρhxρρ
sdotminus+
+=
where
z0 = number of teeth of pinion cutter
x0 = addendum modification coefficient of pinion cutter
hfP = addendum of pinion cutter
ρfP = tip radius of pinion cutter
xhρG fpfp +minus=
24 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
τmE
2π
z2H
nnminus
minus=
with
3πτ = for external gears
6πτ = for internal gears
Htan zG2
nminusϑ=ϑ
(to be solved iteratively suitable start value6π
=ϑ for exter-
nal gears and 3π for internal gears)
a) Tooth root chord sFn For external gears
minus
ϑ+
ϑminus= ρ
cosG3
3πsinz
ms
fpnn
Fn
For bevel gears with a tooth thickness modification
xsm affects mainly sFn but also hFe and αFen The total influence of xsm on YFa Ysa can be approximated by only adding 2 xsm to sFn mn
For internal gears
minus
ϑ+
ϑminus= ρ
cosG
6πsinz
ms
fPnn
Fn
b) Root fillet radius ρF at 30ordm tangent
( )G2coszcosG2ρ
mρ
2n
2
fpn
F
minusϑϑ+=
c) Determination of bending moment arm hF dn = zn mn
b2α
αn βcosε
ε =
dan = dn + 2 ha
pbn = π mn cos αn
dbn = dn cos αn
( )4
d1εpzz
2dd
zz2d
2bn
2
αnbn
2bn
2an
en +
minusminus
minus=
en
bnen d
dcos arcα =
ennnn
e α invα invα x tan 22π
z1
minus+
+=γ
αFen = αen ndash γe
For external gears
( )
minus=
n
enFenee
n
Fe
mdαtan sin γ γcos
21
mh
]ρcos
G3πcosz fpn +
ϑminus
ϑminusminus
For internal gears
( ) minus
sdotsdotminus=
n
enFenee
n
Fe
mdα tanγ sinγ cos
21
mh
minus
ϑminus
ϑminussdot ρ
cosG3
6πcosz fPn
332 Gearing with εαn gt 2 For deep tooth form gearing ( )25ε2 αn lele produced with a verified grade of accuracy of 4 or better and with applied profile modification to obtain a trapezoidal load distribution along the path of contact the YF may be corrected by the factor YDT as
250ε205for 0666ε2366Y αnαnDT leleminus=
205εfor 10Y αnDT lt=
34 Stress Correction Factors YS YSa The stress correction factors YS and YSa take into account the conversion of the nominal bending stress to the local tooth root stress Thereby YS and YSa cover the stress increasing effect of the notch (fillet) and the fact that not only bending stresses arise at the root A part of the local stress is inde-pendent of the bending moment arm This part increases the more the decisive point of load application approaches the critical tooth root section
Therefore in addition to its dependence on the notch radius the stress correction is also dependent on the position of the load application ie the size of the bending moment arm
YS applies to the load application at the outer point of single tooth pair contact YSa to the load application at tooth tip
YS can be determined as follows
( )
+
+=L23121
1
sS qL01312Y
where Fe
Fn
hsL = and
F
Fns ρ2
sq = (see 33)
YSa can be calculated by replacing hFe with hFa in the above formulae
Note a) Range of validity 1 lt qs lt 8
In case of sharper root radii (ie produced with tools having too sharp tip radii) YS resp YSa must be specially considered
b) In case of grinding notches (due to insufficient protuberance of the hob) YS resp YSa can rise considerably and must be multiplied with
Classification Notes- No 412 25 May 2003
DET NORSKE VERITAS
g
g
ρt
0613
13
minus
where
tg = depth of the grinding notch
ρg = radius of the grinding notch
c) The formulae for YS resp YSa are only valid for αn = 20˚ However the same formulae can be used as a safe approximation for other pressure angles
35 Contact Ratio Factor Yε The contact ratio factor Yε covers the conversion from load application at the tooth tip to the load application at the mid point of the flank (heightwise) for bevel gears
The following may be used
Yε = 0625
36 Helix Angle Factor Yβ The helix angle factor Yβ takes into account the difference between the helical gear and the virtual spur gear in the nor-mal section on which the calculation is based in the first step In this way it is accounted for that the conditions for tooth root stresses are more favourable because the lines of contact are sloping over the flank
The following may be used (β input in degrees)
Yβ = 1 ndash εβ β120
When εβ gt 1 use εβ = 1 and when β gt 30deg use β = 30deg in the formula
However the above equation for Yβ may only be used for gears with β gt 25deg if adequate tip relief is applied to both pinion and wheel (adequate = at least 05 middot Ceff see 432)
37 Values of Endurance Limit σFE σFE is the local tooth root stress (max principal) which the material can endure permanently with 99 survival prob-ability 3106 load cycles is regarded as the beginning of the endurance limit or the lower knee of the σ ndash N curve σFE is defined as the unidirectional pulsating stress with a minimum stress of zero (disregarding residual stresses due to heat treatment) Other stress conditions such as alternating or pre-stressed etc are covered by the conversion factor YM
σFE can be found by pulsating tests or gear running tests for any material in any condition If the approval of the gear is to be based on the results of such tests all details on the testing conditions have to be approved by the Society Fur-ther the tests may have to be made under the Societys su-pervision
If no fatigue tests are available the following listed values for σFE may be used for materials subjected to a quality con-trol as the one referred to in the rules
σFE
Alloyed case hardened steels 1) (fillet surface hardness 58 ndash 63 HRC)
bull of specially approved high grade
1050
bull of normal grade
minus CrNiMo steels with approved process
1000
minus CrNi and CrNiMo steels generally 920 minus MnCr steels generally 850
Nitriding steel of approved grade quenched tempered and gas nitrided (surface hardness 700 ndash 800 HV)
840
Alloyed quenched and tempered steel bath or gas nitrided (surface hardness 500 ndash 700 HV)
720
Alloyed quenched and tempered steel flame or induction hardened 2) (incl entire root fillet) (fillet surface hardness 500 ndash 650 HV)
07 HV + 300
Alloyed quenched and tempered steel flame or induction hardened (excl entire root fillet) (σB = uts of base material)
025 σB + 125
Alloyed quenched and tempered steel 04 σB + 200
Carbon steel 025 σB + 250
Note All numbers given above are valid for separate forgings and for blanks cut from bars forged according to a qualified procedure see Pt 4 Ch 2 Sec 3 For rolled steel the values are to be reduced with 10 For blanks cut from forged bars that are not qualified as mentioned above the values are to be reduced with 20 For cast steel reduce with 40
1) These values are valid for a root radius
bull being unground If however any grinding is made in the root fillet area in such a way that the residual stresses may be affected σFE is to be reduced by 20 (If the grinding also leaves a notch see 34)
bull with fillet surface hardness 58 ndash 63 HRC In case of lower surface hardness than 58 HRC σFE is to be reduced with 20(58 ndash HRC) where HRC is the detected hardness (This may lead to a permissible tooth root stress that varies along the facewidth If so the actual tooth root stresses may also be considered along facewidth)
bull not being shot peened In case of approved shot peening σFE may be increased by 200 for gears where σFE is reduced by 20 due to root grinding Otherwise σFE may be increased by 100 for
6nm le and 100 ndash 5 (mn - 6) for mn gt 6 However the possible adverse influence on the flanks regarding grey staining should be considered and if necessary the flanks should be masked
2) The fillet is not to be ground after surface hardening Regarding possible root grinding see 1)
26 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
38 Mean stress influence Factor YM The mean stress influence factor YM takes into account the influence of other working stress conditions than pure pulsa-tions (R = 0) such as eg load reversals idler gears planets and shrink-fitted gears
YM (YMst) are defined as the ratio between the endurance (or static) strength with a stress ratio R ne 0 and the endurance (or static) strength with R = 0
YM and YMst apply only to a calculation method that assesses the positive (tensile) stresses and is therefore suitable for comparison between the calculated (positive) working stress σF and the permissible stress σFP calculated with YM or YMst
For thin rings (annulus) in epicyclic gears where the com-pression fillet may be decisive special considerations apply see 316
The following method may be used within a stress ratio ndash12 lt R lt 05
381 For idlers planets and PTO with ice class
M1M1R1
1Yor Y MstM
+minus
minus=
where
R = stress ratio = min stress divided by max stress
For designs with the same force applied on both forward- and back-flank R may be assumed to ndash 12
For designs with considerably different forces on forward- and back-flank such as eg a marine propulsion wheel with a power take off pinion R may be assessed as
branchmain theoffacewidth unit per forcepto offacewidt unit per force21minus
For a power take off (PTO) with ice class see 161 c
M considers the mean stress influence on the endurance (or static) strength amplitudes
M is defined as the reduction of the endurance strength am-plitude for a certain increase of the mean stress divided by that increase of the mean stress
Following M values may be used
Endurance limit
Static strength
Case hardened 08 ndash 015 Ys 1) 07
If shot peened 04 06
Nitrided 03 03
Induction or flame hardened
04 06
Not surface hardened steel 03 05
Cast steels 04 06 1)For bevel gears use Ys=2 for determination of M
The listed M values for the endurance limit are independent of the fillet shape (Ys) except for case hardening In princi-ple there is a dependency but wide variations usually only occur for case hardening eg smooth semicircular fillets versus grinding notches
382 For gears with periodical change of rotational direction For case hardened gears with full load applied periodically in both directions such as side thrusters the same formula for YM as for idlers (with R = ndash 12) may be used together with the M values for endurance limit This simplified approach is valid when the number of changes of direction exceeds 100 and the total number of load cycles exceeds 3middot106
For gears of other materials YM will normally be higher than for a pure idler provided the number of changes of direction is below 3middot106 A linear interpolation in a diagram with loga-rithmic number of changes of direction may be used ie from YM = 09 with one change to YM (idler) for 3middot106 changes This is applicable to YM for endurance limit For static strength use YM as for idlers
For gears with occasional full load in reversed direction such as the main wheel in a reversing gear box YM = 09 may be used
383 For gears with shrinkage stresses and unidirectional load For endurance strength
FE
fitM σ
σM1
M21Y+
minus=
σFE is the endurance limit for R = 0
For static strength YMst = 1 and σfit accounted for in 39b
σfit is the shrinkage stress in the fillet (30˚ tangent) and may be found by multiplying the nominal tangential (hoop) stress with a stress concentration factor
n
Ffit m
ρ215scf minus=
384 For shrink-fitted idlers and planets When combined conditions apply such as idlers with shrink-age stresses the design factor for endurance strength is
( ) ( ) FE
fitM σ
σR1M1
M2
M1M1R1
1Yminussdot+
minus
+minus
minus=
Symbols as above but note that the stress ratio R in this par-ticular connection should disregard the influence of σfit ie R normally equal ndash 12
For static strength
M1M1R1
1YMst
+minus
minus=
The effect of σfit is accounted for in 39 b
Classification Notes- No 412 27 May 2003
DET NORSKE VERITAS
385 Additional requirements for peak loads The total stress range (σmax ndash σmin) in a tooth root fillet is not to exceed
F
y
Sσ225
for not surface hardened fillets
FSHV5 for surface hardened fillets
39 Life Factor YN The life factor YN takes into account that in the case of limited life (number of cycles) a higher tooth root stress can be permitted and that lower stresses may apply for very high number of cycles
Decisive for the strength at limited life is the σ ndash N ndash curve of the respective material for given hardening module fillet radius roughness in the tooth root etc Ie the factors YδrelT YRelT YX and YM have an influence on YN
If no σ ndash N ndash curve for the actual material and hardening etc is available the following method may be used
Determination of the σ ndash N ndash curve
a) Calculate the permissible stress σFP for the beginning of the endurance limit (3middot106 cycles) including the influ-ence of all relevant factors as SF YδrelT YRelT YX YM and YC ie σFP = σFE middotYM middotYδrelT middotYRelT middotYX middotYC SF
b) Calculate the permissible laquostaticraquo stress (le 103 load cy-cles) including the influence of all relevant factors as SFst YδrelTst YMst and YCst ( )fitσCstYδrelTstYMstYFstσ
FstS1
FPstσ minussdotsdotsdot=
where σFst is the local tooth root stress which the material can resist without cracking (surface hardened materials) or unacceptable deformation (not surface hardened materials) with 99 survival probability
σFst
Alloyed case hardened steel 1) 2300
Nitriding steel quenched tempered and gas nitrided (surface hardness 700 ndash 800 HV)
1250
Alloyed quenched and tempered steel bath or gas nitrided (surface hardness 500 ndash 700 HV)
1050
Alloyed quenched and tempered steel flame or induction hardened (fillet surface hardness 500 ndash 650 HV)
18 HV + 800
Steel with not surface hardened fillets the smaller value of 2)
18 σB or 225 σy
1) This is valid for a fillet surface hardness of 58 ndash 63 HRC In case of lower fillet surface hardness than 58 HRC σFst is to be reduced with 30(58 ndash HRC) where HRC is the actual hardness Shot peening or grinding notches are not considered to have any significant influence on σFst 2) Actual stresses exceeding the yield point (σy or σ02) will alter the residual stresses locally in the ldquotensionrdquo fillet re-spectively ldquocompressionrdquo fillet This is only to be utilised for gears that are not later loaded with a high number of cycles at lower loads that could cause fatigue in the ldquocompressionrdquo fillet
c) Calculate YN as
NL gt 3middot106
YN = 1 or 001
L
6
N N103Y
sdot= ie Yn = 092 for 1010
The YN = 1 from 3middot106 on may only be used when special material cleanness applies see rules Pt4 Ch2
103ltNLlt3middot106exp
L
6
N N103Y
sdot=
cycles6103forσcycles310forσlog02876exp
FP
FPst
sdot=
NL lt 103 cycles6103forσcycles310forσ
NYFP
FPst
sdot=
or simply use σFPst as mentioned in b) directly
Guidance on number of load cycles NL for various applica-tions
bull For propulsion purpose normally NL = 1010 at full load (yachts etc may have lower values)
bull For auxiliary gears driving generators that normally operate with 70-90 of rated power NL = 108 with rated power may be applied
28 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
310 Relative Notch Sensitivity Factor YδrelT The dynamic (respectively static) relative notch sensitivity factor YδrelT (YδrelTst) indicate to which extent the theoreti-cally concentrated stress lies above the endurance limits (re-spectively static strengths) in the case of fatigue (respectively overload) breakage
YδrelT is a function of the material and the relative stress gra-dient It differs for static strength and endurance limit
The following method may be used
For endurance limit
for not surface hardened fillets
( )
024
s024
relT σ103133q21σ1012201351
Ysdotsdotminus
+sdotsdotminus+= minus
minus
δ
for all surface hardened fillets except nitrided
( )
106q21002451
Y srelT
++=δ
for nitrided fillets
( )
1347q2101421
Y srelT
++=δ
For static strength
for not surface hardened fillets1)
( )( )( )025
02
02502s
relTst σ3000821σ300 1Y0821Y
+
minus+=δ
for surface hardened fillets except nitrided
YδrelTst = 044 YS + 012
for nitrided fillets
YδrelTst = 06 + 02 YS 1) These values are only valid if the local stresses do not
exceed the yield point and thereby alter the residual stress level See also 39b footnote 2
311 Relative Surface Condition Factor YRrelT The relative surface condition factor YRrelT takes into ac-count the dependence of the tooth root strength on the sur-face condition in the tooth root fillet mainly the dependence on the peak to valley surface roughness
YRrelT differs for endurance limit and static strength
The following method may be used
For endurance limit
YRrelT = 1675 ndash 053 (Ry + 1)01 for surface hardened steels and alloyed quenched and tempered steels except nitrided
YRrelT = 53 ndash 42 (Ry + 1)001 for carbon steels
YδrelT = 43 ndash 326 (Ry + 1)0005
for nitrided steels
For static strength
YRrelTst = 1 for all Ry and all materials
For a fillet without any longitudinal machining trace Ry asymp Rz
312 Size Factor YX The size factor YX takes into account the decrease of the strength with increasing size YX differs for endurance limit and static strength
The following may be used
For endurance limit
YX = 1 for mn le 5 generally
YX = 103 ndash 0006 mn for 5 lt mn lt 30
YX = 085 for mn ge 30
for not surface hard-ened steels
YX = 105 ndash 001 mn for 5 lt mn ge 25
YX = 08 for mn ge 25
for surface hardened steels
For static strength
YXst = 1 for all mn and all materials
313 Case Depth Factor YC The case depth factor YC takes into account the influence of hardening depth on tooth root strength
YC applies only to surface hardened tooth roots and is dif-ferent for endurance limit and static strength
In case of insufficient hardening depth fatigue cracks can develop in the transition zone between the hardened layer and the core For static strength yielding shall not occur in the transition zone as this would alter the surface residual stresses and therewith also the fatigue strength
The major parameters are case depth stress gradient permis-sible surface respectively subsurface stresses and subsurface residual stresses
The following simplified method for YC may be used
YC consists of a ratio between permissible subsurface stress (incl influence of expected residual stresses) and permissible surface stress This ratio is multiplied with a bracket con-taining the influence of case depth and stress gradient (The empirical numbers in the bracket are based on a high number of teeth and are somewhat on the safe side for low number of teeth)
YC and YCst may be calculated as given below but calculated values above 10 are to be put equal 10
For endurance limit
+
+=nFFE
C m02ρt31
σconstY
Classification Notes- No 412 29 May 2003
DET NORSKE VERITAS
For static strength
+
+=nFFst
Cst m02ρt31
σconstY
where const and t are connected as
Hardening process
t = endurance limit const =
static strength const =
t550 640 1900
t400 500 1200 Case hard-ening
t300 380 800
Nitriding t400 500 1200
Induction- or flame hardening
tHVmin 11 HVmin 25 HVmin
For symbols see 213
In addition to these requirements to minimum case depths for endurance limit some upper limitations apply to case hard-ened gears
The max depth to 550 HV should not exceed
1) 13 of the top land thickness san unless adequate tip relief is applied (see 110)
2) 025 mn If this is exceeded the following applies addi-tionally in connection with endurance limit
minusminus= 025
mt
1Yn
max550C
314 Thin rim factor YB Where the rim thickness is not sufficient to provide full sup-port for the tooth root the location of a bending failure may be through the gear rim rather than from root fillet to root fillet
YB is not a factor used to convert calculated root stresses at the 30deg tangent to actual stresses in a thin rim tension fillet Actually the compression fillet can be more susceptible to fatigue
YB is a simplified empirical factor used to de-rate thin rim gears (external as well as internal) when no detailed calcula-tion of stresses in both tension and compression fillets are available
Figure 34 Examples on thin rims
YB is applicable in the range 175 lt sRmn lt 35
YB = 115 middot ln (8324 middot mnsR)
(for sRmn ge 35 YB = 1)
(for sRmn le 175 use 315)
σF as calculated in 321 is to be multiplied with YB when sRmn lt 35 Thus YB is used for both high and low cycle fatigue
Note This method is considered to be on the safe side for external gear rims However for internal gear rims without any flange or web stiffeners the method may not be on the safe side and it is advised to check with the method in 315316
315 Stresses in Thin Rims For rim thickness sR lt 35 mn the safety against rim cracking has to be checked
The following method may be used
3151 General The stresses in the standardised 30ordm tangent section tension side are slightly reduced due to decreasing stress correction factor with decreasing relative rim thickness sRmn On the other hand during the complete stress cycle of that fillet a certain amount of compression stresses are also introduced The complete stress range remains approximately constant Therefore the standardised calculation of stresses at the 30ordm tangent may be retained for thin rims as one of the necessary criteria
The maximum stress range for thin rims usually occurs at the 60ordm ndash 80ordm tangents both for laquotensionraquo and laquocompressionraquo side The following method assumes the 75ordm tangent to be the decisive Therefore in addition to the am criterion ap-plied at the 30ordm tangent it is necessary to evaluate the max and min stresses at the 75ordm tangent for both laquotensionraquo (loaded flank) fillet and laquocompressionraquo (back-flank) fillet For this purpose the whole stress cycle of each fillet should be considered but usually the following simplification is justified
Figure 35 Nomenclature of fillets
Index laquoTraquo means laquotensileraquo fillet laquoCraquo means laquocompressionraquo fillet
σFTmin and σFCmax are determined on basis of the nominal rim stresses times the stress concentration factor Y75
σFCmin and σFTmax are determined on basis of superposition of nominal rim stresses times Y75 plus the tooth bending stresses at 75ordm tangent
30 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
3152 Stress concentration factors at the 75ordm tangents The nominal rim stress consists of bending stresses due to local bending moments tangential stresses due to the tan-gential force Ft and radial shear stresses due to Fr
The major influence is given by the bending stresses The influence of the tangential stresses is minor and even though its stress concentration factor is slightly higher than for bending it is considered to be safe enough when the sum of these nominal stresses are combined with the stress concen-tration factor for bending The influence of the radial shear stress is neglected
The stress concentration factor relating nominal rim stresses to local fillet stresses at the 75ordm tangent may be calculated as
75n
R75
n
R
75
ρmsρ185
ms3
Y+
sdot=
where ρ75 is the root radius at the 75ordm tangent ref to mn Usually ρ75 is closed to the tool radius ρao and
ρ75 = ρao
is a safe approximation compensating for the am simplifi-cations to the laquounsaferaquo side
The tooth root stresses of the loaded tooth are decreasing with decreasing relative rim thickness approximately with the empirical factor
15
R
ncorr s
m311Y
minus=
3153 Nominal rim stresses The bending moment applied to the rim consists of a part of the tooth tilt moment Ft (hF + 05 sR) and the bending caused by the radial force Fr
The sectional modulus (first moment of area) which is used for determination of the nominal bending stresses is not nec-essarily the same for the 2 am bending moments If flanges webs etc outside the toothed section contribute to stiffening the rim against various deflections the influence of these stiffeners should be considered Eg an end flange will have an almost negligible influence on the effective sectional modulus for the stresses due to tooth tilt as the deflection caused by the tilt moment is rather small and would not much involve the flange On the other hand the radial forces as for instance from the meshes in an annulus would cause considerable radial deflections that the flange might restrict to a substantial amount When considering the stiff-ening of such flanges or webs on basis of simplified models it is advised to use an effective rim thickness sR = sR + 02 mn for the first moment of area of the rim (toothed part) cross section
For a high number of rim teeth it may be assumed that the rim bending moments in the fillets adjacent to the loaded tooth are of the same magnitude as right under the applied
force This assumption is reasonable for an annulus but rather much on the safe side for a hollow pinion
The influence of Ft on the nominal tangential stress is simpli-fied by half of it for compressive stresses (σ1) and the other half for tensile stresses (σ2) Applying these assumptions the nominal rim stresses adjacent to the loaded tooth are
( ) ( )A2
FW
f RFW
s05hF05 t
R
r
T
RFt1 minus
ϑminus
+minus=σ
( ) ( )A2
FW
f RFW
s05hF05 t
R
r
T
RFt2 +
ϑminus
+=σ
where σ1 σ2 see Fig 35
A = minimum area of cross section (usually bR sR)
WT = the sectional modulus of rim with respect to tooth tilt moment (usually 6sb 2
rR )
WR = the sectional modulus of the rim including the influence of stiffeners as flanges webs etc (WR ge WT)
bR = the width of the rim
R = the radius of the neutral axis in the rim ie from wheel centre to midpoint of rim
( )ϑf = a function for bending moment distribution around the rim For a rim (pinion) with one mesh only the ( )ϑf at the position of load application is 024 For an annulus with 3 or more meshes ( )ϑf at the position of each load application is approx 3 planets ( )ϑf = 019 4 planets ( )ϑf = 014 5 planets ( )ϑf = 011 6 planets ( )ϑf = 009
It must be checked if the max (tensile) stress in the compres-sion fillet really occurs when the fillet is adjacent to the loaded tooth In principle the stress variation through a complete rotation should be considered and the max value used The max value is usually never less than 0 For an annulus eg the tilt moment is zero in the mid position be-tween the planet meshes whilst the bending moment due to the radial forces is half of that at the mesh but with opposite sign
If these formulae are applied to idler gears as eg planets the influence of nominal tangential stresses must be cor-rected by deleting Ft(2 A) for σ1 and using FtA for σ2 Further the influence of Fr on the nominal bending stresses is usually negligible due to the planet bearing support
3154 Root fillet stresses Determination of min and max stresses in the laquotensionraquo fillet
Classification Notes- No 412 31 May 2003
DET NORSKE VERITAS
Minimum stress
751FTmin YσK σ =
Maximum stress
corrF752 Yσ03Yσ KσFTmax +=
where
03 is an empirical factor relating the tension stresses (σF) at the 30ordm tangent to the part of the tension stresses at the 75ordm tangent which add to the rim related stresses (03 also takes into account that full superposition of nominal stresses times stress concentration factors from both laquosides of the corner filletraquo would result in too high stresses)
αβγ sdotsdotsdotsdot= FFvA KKKKKK
Determination of min and max stresses in the laquocompres-sionraquo fillet
Minimum stress
corrF751FCmin Yσ036Yσ Kσ minus=
Maximum stress
752FCmax Yσ Kσ =
where
036 is an empirical factor relating the tension stresses (σF) at the 30ordm tangent to the part of the compression stresses at the 75ordm tangent which add to the rim related stresses
For gears with reversed loads as idler gears and planets there is no distinct laquotensionraquo or laquocompressionraquo fillet The mini-mum stress σFmin is the minimum of σFTmin and σFCmin (usually the latter is decisive) The maximum stress σFTmax is the maximum of σFTmax and σFCmax (usually the former is deci-sive)
316 Permissible Stresses in Thin Rims
3161 General The safety against fatigue fracture respectively overload fracture is to be at least at the same level as for solid gears The laquoordinaryraquo criteria at the 30ordm tangent apply as given in 31 through 313
Additionally the following criteria at the 75ordm tangent may apply
3162 For gt3middot106 cycles The permissible stresses for the laquotensionraquo fillets and for the laquocompressionraquo fillets are determined by means of a relevant fatigue diagram
If the actual tooth root stress (tensile or compressive) ex-ceeds the yield strength to the material the induced residual stresses are to be taken into account
For determination of permissible stresses the following is defined
R = stress ratio ie FTmax
FTminσσ respectively
FCmax
FCmin
σσ
∆σ = stress range ie σFTmax ndash σFTmin resp σFCmax ndash σFCmin
(For idler gears and planets Fmax
FminσσR =
and ∆σ = σFmax minus σFmin)
The permissible stress range ∆σρ for the laquotensionraquo respec-tively laquocompressionraquo fillets can be calculated as
For R gt minus1 FPp σ
R1R1031
13∆σ
minus+
+=
For minus infin lt R lt minus1 FPp σ
R1R10151
13∆σ
minus+
+=
where
σFP = see 32 determined for unidirectional stresses (YM = 1)
If the yield strength σy is exceeded in either tension or com-pression residual stresses are induced This may be consid-ered by correcting the stress ratio R for the respective fillets (tension or compression)
Eg if yFCmin σσ gt (ie exceeded in compression) the
difference yFCmin σσ∆ minus= affects the stress ratio as
∆σ
σ∆σ∆σR
FCmax
y
FCmax
FCmin
+
minus=
++
=
Similarly the stress ratio in the tension fillet may require cor-rection
If the yield strength is exceeded in tension σFTmax gt σy the difference ∆ = σFTmax ndash σy affects the stress ratio as
y
FTmin
FTmax
FTmin
σ∆σ
∆σ∆σR minus=
minusminus
=
Checking for possible exceeding of the yield strength has to be made with the highest torque and with the KAP if this ex-ceeds KA
3163 Forle 103 cycles The permissible stress range ∆σp is not to exceed
For R gt minus1 FPstpst σ
R1R1051
15∆σ
minus+
+=
For ndash infin lt R lt minus1 FPstpst σ
R1R10251
15∆σ
minus+
+=
For all values of R ∆σpst is limited by
not surface hardened F
y
Sσ225
32 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
surface hardened CF
YSHV5
Definition of ∆σ and R see 3162 with particular attention to possible correction of R if the yield strength is exceeded
σFPst see 32 determined for unidirectional stresses (YM = 1) and lt 103 cycles
3164 For 103 lt cycles lt 3106 ∆σp is to be determined by linear interpolation a log-log dia-gram
∆σp at NL load cycles is
6L 103p
exp
L
6
N p ∆σN103∆σ sdot
sdot=
6
3
103p
10p
∆σ
∆σlog28760exp
sdot
=
Classification Notes- No 412 33 May 2003
DET NORSKE VERITAS
4 Calculation of Scuffing Load Capacity
41 Introduction High surface temperatures due to high loads and sliding ve-locities can cause lubricant films to break down This sei-zure or welding together of areas of tooth surface is termed scuffing
In contrast to pitting and fatigue breakage which show a dis-tinct incubation period a single short overloading can lead to a scuffing failure In the ISO-TR13989 two criteria are mentioned The method used in this Classification Note is based on the principles of the flash temperature criterion
Note Bulk temperature in excess of 120ordmC for long periods may have an adverse effect on the surface durability see 211
42 General Criteria In no point along the path of contact the local contact tem-perature may exceed the permissible temperature ie
oilS
oilSB S
ϑ+ϑminusϑ
leϑ
50SB minusϑleϑ
where
Bϑ = max contact temperature along the path of contact
maxflaMBB ϑ+ϑ=ϑ
MBϑ = bulk temperature see 434
maxflaϑ = max flash temperature along the path of con-tact see 44
Sϑ = scuffing temperature see below
oilϑ = oil temperature before it reaches the mesh (max applicable for the actual load case to be used ie normally alarm temperature except for ice classes where a max expected tem-perature applies)
SS = required safety factor according to the Rules
The scuffing temperature Sϑ may be calculated as
L2
wrelT
002
40S XFZGX
ν100112085780
sdot
sdot++=ϑ
where
XwrelT = relative welding factor
XwrelT
Through hardened steel 10
Phosphated steel 125
Copper-plated steel 150
Nitrided steel 150
Less than 10 retained austenite
115
10 ndash 20 retained austenite 10
Casehardened steel
20 ndash 30 retained austenite 085
Austenitic steel 045
FZG = load stage according to FZG-Test A8390 (Note This is the load stage where scuffing oc-curs However due to scatter in test results cal-culations are to be made with one load stage less than the specification)
XL = lubricant factor
= 10 for mineral oils
= 08 for polyalfaolefins
= 07 for non-water-soluble polyglycols
= 06 for water-soluble polyglycols
= 15 for traction fluids
= 13 for phosphate esters
ν40 = kinematic oil viscosity at 40˚C (mm2s)
Application of other test methods such as the Ryder the FZG-Ryder R46574 and the FZG L-42 Test 141195110 may be specially considered
For high speed gears with very short time of contact Sϑ may be increased as follows provided use of EP-oils
Addition to the calculated scuffing temperature Sϑ
If micros 18tc ge no addition
If micros 18tc lt add ( )cwrelT t18X18 minussdotsdot
where
ct = contact time ( micros ) which is the time needed to cross the Hertzian contact width
( ) [ ]microsu1cosβn
uσ340tb1
Hc +sdotsdot
sdot=
σH as calculated in 221
For bevel gears use vu in stead of u
34 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
43 Influence Factors
431 Coefficient of friction The following coefficient of friction may apply
L025a
005oil
02
redC ΣC
Bt X R ηρv
w0048micro minus
=
where
wBt = specific tooth load (Nmm)
vΣC = sum of tangential velocities at pitch point At pitch line velocities gt 50 ms the limiting value of vΣC at v = 50 ms is to be used
ρredC = relative radius of curvature (transversal plane) at the pitch point
Cylindrical gears
HαHβvγAbt
Bt KKKKKb
Fw sdotsdotsdotsdotsdot= (see 1)
twΣC αsin v2v =
bCredC β cos ρρ =
Bevel gears
HαHβvγAtmb
Bt KKKKKb
Fw sdotsdotsdotsdotsdot= (see 1)
vtmtΣC αsin v2v =
bmvCredC β cos ρρ =
ηoil = dynamic viscosity (mPa s) at oilϑ calculated as
1000
ρ νη oiloil =
where ρ in kgm3 approximated as
( ) 70 15oil15 minusϑminusρ=ρ and νoil is kinematic viscosity at oilϑ and may be calculated by means of the following equation
( ) ( )++=+ 08νloglog08νloglog 100oil ( )
sdotminus
ϑ+minus313log373log
273log373log oil
( ) ( )( )08νloglog08νloglog 10040 +minus+
Ra = composite arithmetic mean roughness (micron) of pinion an wheel calculated as ( )2a1aa RR 05R +=This is defined as the roughness on the new flanks ie as manufactured
XL = see 42
432 Effective tip relief Ceff Ceff is the effective tip relief that amount of tip relief which compensates for the elastic deformation of the gear mesh ie
zero load at the tooth tip It is assumed (simplified) to be equal for both pinion and wheel
Cylindrical gears
for helical
γ
γ=cb
KKFC Abt
eff (see 1)
For spur
cbKKF
C Abteff
γ= (see 1)
Alternatively for spur and helical gears the non-linear ap-proach in 111 may be used (taking Ceff=δ)
Bevel gears
bcKFC Ambt
effγ
= (see 1)
where
γ
α
ε2ε44c
+sdot
=γ
433 Tip relief and extension Cylindrical gears
The extension of the tip relief is not to result in an effective contact ratio 1ltεα when the gear is unloaded (exceptions to this may only apply for applications where the gear is not to run at light loads) This means that the unrelieved part of the path of contact is to be minimum pbt It is further assumed that this unrelieved part is placed centrally on the path of contact
If root relief applies it has to be calculated as equivalent tip relief Ie pinion root relief (at mesh position A) is added to Ca2 and wheel root relief (at mesh position E) is added to Ca1 If no design tip relief or root relief on the mating gear is specified (ie if Ca1+Croot2 = 0 and visa versa) use the run-ning in amount see 112
Bevel gears
Bevel gears are to have heightwise crowning ie no distinct relievedunrelieved area This may be treated as tip and root relief For calculation purposes the root relief is combined with the tip relief of the mating member into an equivalent tip relief If no resulting tip relieves are specified the equivalent tip relives may be calculated as an approxima-tion based on the tool crowning Ca tool (per mille of tool module m0) as follows
2root1aeq1a CCC +=
1root2aeq2a CCC +=
where
Classification Notes- No 412 35 May 2003
DET NORSKE VERITAS
2
0
n0101atoolal m
)m2(mA1middotmCC
minus+minussdot=
( )
sdotminusminussdotsdotminus+= vtvb1
2vb1
21vavt1n1 αtan dddαsin 05x1mA
2
0
n020atool22a m
)mm(2A1middotmCC
minus+minussdot=
( )
sdotminusminussdotsdotminus+= vtvb2
2vb2
2va2vt2n2 αtan dddαsin 05x1mA
2
0
20atool1root1 m
A1mCC
minussdotsdot=
2
0
10atool2root2 m
A1mCC
minussdotsdot=
If the pressure angles of the cutter blades are modified (and verified) to balance the tip relieves the following may be assumed
( )eq2aeq1aeq2aeq1a C andC calculated of sum50CC sdot==
Throughout the following Ca1 and Ca2 mean the equivalent tip relieves Ca1eq and Ca2eq
434 Bulk temperature The bulk temperature may be calculated as
flaaveragempsoilMB X X 50 ϑ+ϑ=ϑ
where
Xs = lubrication factor
= 12 for spray lubrication
= 10 for dip lubrication (provided both pinion and wheel are dip lubricated and tip speed lt 5 ms)
= 10 for spray lubrication with additional cooling spray (spray on both pinion and wheel or spray on pinion and dip lubrication of wheel)
= 02 for meshes fully submerged in oil
Xmp = contact factor ( )pmp nX += 150
np = number of mesh contact on the pinion (for small gear ratii the number of wheel meshes should be used if higher)
flaaverageϑ
= average of the integrated flash temperature (see 44) along the path of contact
( )
AE
E
Ayyyfla
flaaverage
d
ΓminusΓ
ΓΓϑ=ϑint =
For high speed gears (v gt 50 ms) it may be necessary to assess the bulk temperature on the basis of a thermal rating of the entire gear transmission
44 The Flash Temperature flaϑ
441 Basic formula The local flash temperature flaϑ may be calculated as
( ) 41redy
y2ly
211
43Btcorrfla
unXwX3250
y ρ
ρminusρ
micro=ϑ Γ
(For bevel gears replace u with uv)
and is to be calculated stepwise along the path of contact from A to E
where
micro = coefficient of friction see 431
Xcorr = correction factor taking empirically into account the increased scuffing risk in the beginning of the approach path due to mesh starting without any previously built up oil film and possible shuffling away oil before meshing if insufficient tip relief
( )
3
AD
yaeffcorr 50
CC1X
ΓminusΓε
Γminus+=
α
Ca = tip relief of driven member Xcorr is only applicable in the approach path and if Ca lt Ceff otherwise 10
wBt = unit load see 431
yXΓ = load sharing factor see 443
n1 = pinion rpm
Γy ρly etc see 442
442 Geometrical relations The various radii of flank curvature (transversal plane) are
ρ1y = pinion flank radius at mesh point y
ρ2y = wheel flank radius at mesh point y
ρredy = equivalent radius of curvature at mesh point y
y2y1
y2y1redy
ρ+ρ
ρρ=ρ
Cylindrical gears
twy
1y αsin au1Γ1
ρ+
+=
twy
2y αsin au1Γu
ρ+
minus=
Note that for internal gears a and u are negative
36 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
Bevel gears
vtvv
y1y αsin a
u1Γ1
ρ+
+=
vtvv
yv2y αsin a
u1Γu
ρ+
minus=
Γ is the parameter on the path of contact and y is any point between A end E
At the respective ends Γ has the following values
Root piniontip wheel
Cylindrical gears
( )
minus
minusminus= 1
αtan 1dd
zzΓ
tw
2b2a2
1
2A
Bevel gears
( )
minus
minusminus= 1
αtan 1dd
uΓvt
2vb2va2
vA
Tip pinionroot wheel
Cylindrical gears
( )
1αtan
1ddΓ
tw
2b1a1
E minusminus
=
Bevel gears
( )
1αtan
1ddΓ
vt
2vb1va1
E minusminus
=
At inner point of single pair contact
Cylindrical gears
tw1
EB α tan zπ2ΓΓ minus=
Bevel gears
vtv1
EB α tan zπ2ΓΓ minus=
At outer point of single pair contact
Cylindrical gears
tw1AD α tan z
π2ΓΓ +=
Bevel gears
vt v1
AD αtan z π2ΓΓ +=
At pitch point ΓC = 0
The points F and G (only applicable to cylindrical gears) limiting the extension of tip relief (so as to maintain a mini-mum contact ratio of unity for unloaded gears) are at
2
BAF
Γ+Γ=Γ
2
EDG
Γ+Γ=Γ
443 Load sharing factor XΓ The load sharing factor XΓ accounts for the load sharing be-tween the various pairs of teeth in mesh along the path of contact
XΓ is to be calculated stepwise from A to E using the pa-rameter Γy
4431 Cylindrical gears with β = 0 and no tip relief
Figure 41
ByAAB
Ay for31
31X
yΓltΓleΓ
ΓminusΓ
ΓminusΓ+=Γ
DyB for1Xy
ΓltΓleΓ=Γ
EyDDE
yE for31
31X
yΓleΓltΓ
ΓminusΓ
ΓminusΓ+=Γ
4432 Cylindrical gears with β = 0 and tip relief Tip relief on the pinion reduces XΓ in the range G ndash E and increases correspondingly XΓ in the range F ndash B
Tip relief on the wheel reduces XΓ in the range A ndash F and increases correspondingly XΓ in the range D ndash G
Following remains generally
DyB for1Xy
ΓleΓleΓ=Γ
21XXGF== ΓΓ
In the following it must be distinguished between Ca lt Ceff respectively Ca gt Ceff This is shown by an example below where Ca1 lt Ceff and Ca2 gt Ceff
Classification Notes- No 412 37 May 2003
DET NORSKE VERITAS
Figure 42
Note When Ca gt Ceff the path of contact is shortened by A ndash A respectively E ndash E The single pair contact path is extended into B respectively D If this shift is significant it is necessary to consider the negative effect on surface durability (B) and bending stresses (B and D)
Range A - F
For effa2 CC le
minus=Γ
eff
2a
CC1
31X
A
FyAeff
2a
AF
Ay for C3
C61XX
AyΓleΓleΓ
sdot
+ΓminusΓ
ΓminusΓ+= ΓΓ
For effa2 CC ge
AyAfor0Xy
ΓleΓleΓ=Γ
with ( )21
CC
1CC
eff
2a
eff
2a
AFAAminus
minusΓminusΓ+Γ=Γ
FyAeff
2a
AF
Ay
eff
2a for 21
CC
CC1X
yΓleΓleΓ
minus
ΓminusΓ
ΓminusΓ+minus=Γ
Range F - B
For effa1 CC le
ByFeff
1a
FB
Fy forC3
C61
21X
yΓleΓleΓ
sdot
+ΓminusΓ
ΓminusΓ+=Γ
For effa1 CC ge
ByFeff
1a
FB
Fy for21
CC
21X
yΓleΓleΓ
minus
ΓminusΓ
ΓminusΓ+=Γ
ByB for1Xy
ΓleΓleΓ=Γ
with 1
CC2
eff
1a
FBFB
minus
ΓminusΓ+Γ=Γ
Range D ndash G
For effa2 CC le
GyDeff
2a
DG
Dy
eff
2a for C 3C
61
C 3C
32X
yΓleΓleΓ
+
ΓminusΓ
ΓminusΓminus+=Γ F
or effa2 CC ge
DyD for1Xy
ΓleΓleΓ=Γ
with ( )21
CC
1CC
eff
2a
eff
2a
DGDDminus
minusΓminusΓ+Γ=Γ
GyDeff
2a
DG
Dy
eff
2a for21
CC
CCX ΓleΓleΓ
minus
ΓminusΓ
ΓminusΓminus=Γ
Range G - E
For effa1 CC le
minus=Γ
eff
1a
CC1
31X
E
EyGeff
1a
GE
Gy forC3
C61
21X
yΓleΓleΓ
sdot
+ΓminusΓ
ΓminusΓminus=Γ
For effa1 CC gt
EyGeff
1a
GE
Gy for21
CC
21X
yΓleΓleΓ
minus
ΓminusΓ
ΓminusΓminus=Γ
EyE for0Xy
ΓleΓleΓ=Γ
with 1
CC2
eff
1a
GEGE
minus
ΓminusΓ+Γ=Γ
4433 Gears with β gt 0 buttressing Due to oblique contact lines over the flanks a certain but-tressing may occur near A and E
This applies to both cylindrical and bevel gears with tip relief lt Ceff The buttressing Xbutt is simplified as a linear function within the ranges A ndash H respectively I ndash E
Figure 43
1εwhen13X βbutt EAge=
1εwhenε031X ββbutt EAlt+=
38 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
Cylindrical gears
bIEAH βsin 02ΓΓΓΓ =minus=minus
Bevel gears
bmIEAH βsin 02ΓΓΓΓ =minus=minus
4434 Cylindrical gears with 2εγ le and no tip relief
yXΓ is obtained by multiplication of
yXΓ in 4431 with
Xbutt in 4433
4435 Gears with 2εγ gt and no tip relief
Applicable to both cylindrical and bevel gears
IyH for1Xy
ΓleΓleΓε
=α
Γ
IyHybutt and forX1Xy
ΓgtΓΓltΓε
=α
Γ
Figure 44
4436 Cylindrical gears with 2εγ le and tip relief
yXΓ is obtained by multiplication of
yXΓ in 4432 with Xbutt
in 4433
4437 Cylindrical gears with 2εγ gt and tip relief
Tip relief on the pinion (respectively wheel) reduces XГ in the range G ndash E (respectively A ndash F) and increases XГ in the range F ndash G
yXΓ is obtained by multiplication of
yXΓ as described below
with Xbutt in 4433
In the XГ example below the influence of tip relief is shown (without the influence of Xbutt) by means of
eff2aeff1a CC and CC ltgt
Tip relief gt Ceff causes new end points A respectively E of the path of contact
Figure 45
Range A ndash F
( ) ( )( ) eff
a2αa1α
AF
Ay
eff
2aeff
C 12C 13εC 1ε
CCCX
y +εε++minus
ΓminusΓ
ΓminusΓ+
εminus
=ααα
Γ
eff2aFyA CC if for leΓleΓleΓ
eff2aFyA CifCfor and geΓleΓleΓ
eff2aAyA CC iffor0Xy
gtΓleΓleΓ=Γ
( ) ( ) ( )( ) ( ) a2αa1α
αeffa2AFAA C 1ε 3C 1ε
1ε 2 CC with++minus+minus
ΓminusΓ+Γ=Γ
Range F ndash G
( )( )( ) GyF
eff
2a1a forC 1 2CC 11X
yΓleΓleΓ
+εε+minusε
+ε
=αα
α
αΓ
Range G ndash E
( ) ( )( ) eff
2a1a
GE
Gy
C 1 2C 1C 1 3XX
GFy +εεminusε++ε
ΓminusΓ
ΓminusΓminus=
αα
ααΓΓ minus
effa1EyG CC if for leΓleΓleΓ
effa1EyG CC if Γfor and geΓleΓle
eff1aEyE CC if for0Xy
geΓleΓleΓ=Γ
( ) ( ) ( )( ) ( ) 2a1a
eff1aGEEE C 1C 1 3
1 2 CC withminusε++ε+εminus
ΓminusΓminusΓ=Γαα
α
4438 Bevel gears with εγ more than approx 18 and heightwise crowning For Ca1 = Ca2 = Ceff the following applies
Figure 46
Classification Notes- No 412 39 May 2003
DET NORSKE VERITAS
( )AEM 50 Γ+Γ=Γ
( )( )2AD
3
2My651X
y ΓminusΓε
ΓminusΓminus
ε=
ααΓ
For tip relief lt Ceff yXΓ is found by linear interpolation be-
tween ( )effay CCX =Γ and ( )0CayX =Γ as in 4435
The interpolation is to be made stepwise from A to M with the influence of Ca2 and from M to E with the influence of Ca1 (For 2a1a CC ne there is a discontinuity at M)
Eg with Ca1 = 04 Ceff and Ca2 = 055 Ceff then
Range A ndash M
)CC()0C( eff2ay2ayyX 550X 450X =Γ=ΓΓ +=
Range M ndash E
)CC()0C( eff1ay1ayyX 40X 06X =Γ=ΓΓ +=
For tip relief gt Ceff the new end points A and E are found as
( )
minusΓminusΓ
ε+Γ=Γ α 1
CC
6 eff
2aADAA
( )
minusΓminusΓ
ε+Γ=Γ α 1
CC
6 eff
1aADEE
Range A ndash A
0Xy=Γ
Range A ndash M
( )( )
ΓminusΓ
ΓminusΓminus
minusε=
αΓ 2
MA
2My
eff
2a1
CC4
351Xy
Range M ndash E
( )( )
ΓminusΓ
ΓminusΓminus
minusε=
αΓ 2
ME
2My
eff
1a1
CC4
351Xy
Range E ndash E
0Xy=Γ
40 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
Appendix A Fatigue Damage Accumulation
The Palmgren-Miner cumulative damage calculation princi-ple is used The procedure may be applied as follows
A1 Stress Spectrum From the individual torque classes the torques (Ti) at the peak values of class intervals and the associated number of cycles (NLi) for both pinion and wheel are to be listed from the highest to the lowest torque
(In case of a cyclic torque variation within the torque classes it is advised to use the peak torque If the cyclic variation is such that the same teeth will repeatedly suffer the peak torque this is a must)
The stress spectra for tooth roots and flanks (σFi σHi) with all relevant factors (except KA) are to be calculated on the basis of the torque spectrum The load dependent K-factors are to be determined for each torque class
A2 σminusN-curve The stress versus load cycle curves for tooth roots and flanks (both pinion and wheel) are to be drawn on the basis of per-missible stresses (ie including the demanded minimum safety factors) as determined in 2 respectively 3 If different safety levels for high cycle fatigue and low cycle fatigue are desired this may be expressed by different demand safety factors applied at the endurance limit respectively at static strength
A3 Damage accumulation The individual damage ratio Di at ith stress level is defined as
Fi
Lii N
ND =
where
NLi = The number of applied cycles at ith stress
NFi = The number of cycles to failure at ith stress
Basically stresses σi below the permissible stress level for infinite life (if a constant ZN or YN is accepted) do not con-tribute to the damage sum However calculating the actual safety factor Sact as described below all the σi for which the product S σi is bigger than or equal to the permissible stress level for infinite life contribute to the damage sum and thus to the determination of Sact The final value of S is decisive
(NFi can be found mathematically by putting the permissible stress σpi equal the actual stress σi thereby finding the actual life factor This life factor can be solved with regard to load cycles ie NFi)
The damage sum ΣDi is not to exceed unity
If ΣDi ne 1 the safety against cumulative fatigue damage is different from the applied demand safety factor For deter-mination of this theoretical safety factor an iteration proce-dure is required as described in the following flowchart
S is correction factor with which the actual safety factor Sact can be found
Sact is the demand safety factor (used in determination of the permissible stresses in the σ ndash N ndash curve) times the correc-tion factor S
The full procedure is to be applied for pinion and wheel tooth roots and flanks
Note If alternating stresses occur in a spectrum of mainly pulsating stresses the alternating stresses may be replaced by equiva-lent pulsating stresses ie by means of division with the ac-tual mean stress influence factor YM
Classification Notes- No 412 41 May 2003
DET NORSKE VERITAS
Appendix B Application Factors for Diesel Driven Gears
For diesel driven gears the application factor KA depends on torsional vibrations Both normal operation and misfiring conditions have to be considered
Normally these two running conditions can be covered by only one calculation
B1 Definitions
Normal operation KAnorm = 0
normv0
TTT +
where
T0 = rated nominal torque
Tv norm = vibratory torque amplitude for normal opera-tion (see rules Pt4 Ch3 Sec1 G301 for defi-nition of ldquonormalrdquo irregularity)
Misfiring operation KA misf = 0
misfv
TTT +
where
T = remaining nominal torque when one cylinder out of action
Tv misf = vibratory torque amplitude in misfiring con-dition This refers to a permissible misfiring condition ie a condition that does not require automatic or immediate corrective actions as speed or pitch reduction
The normal operation is assumed to last for a very high num-ber of cycles such as 1010
The misfiring operation is assumed to last for a limited dura-tion such as 107 cycles
B2 Determination of decisive load Assuming life factor at 1010 cycles as YN = ZN = 092 which usually is relevant the calculation may be performed only once with the combination having the highest value of appli-cation factorlife factor
For bending stresses and scuffing the higher value of
092
K normA and 098
K misfA
For contact stresses the higher value of
092K normA and
113K misfA (but
097K misfA for nitrided gears)
B3 Simplified procedure Note that this is only a guidance and is not a binding con-vention
T = oTZ
1Zsdot
minus where Z = number of cylinders
( ) ideal Videal Vmisf Vnorm V TTT24ZT +minus=
where TV ideal = vibratory torque with all cylinders perfectly equal See also rules Pt4 Ch3 Sec1 G300
When using trends from torsional vibration analysis and measurements the following may be used
TV ideal To is close to zero for engines with few cylinders and using a suitable elastic coupling and increases with relative coupling stiffness and number of cylinders
TV misf To may be high for engines with few cylinders and decreases with number of cylinders
This can be indicated as
200Z
TT
ο
idealV asymp and 80Z04
TT
o
misfV minusasymp
Inserting this into the formulae for the two application fac-tors the following guidance can be given
118112K misfA minusasymp
115110K normA minusasymp
Since KA norm is to be combined with the lower life factors the decisive load condition will be the normal one and a KA of 115 will cover most relevant cases when a suitable elas-tic coupling is chosen
42 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
Appendix C Calculation of Pinion-Rack
Pinion-racks used for elevating of mobile offshore units are open gears that are subjected to wear and tear With normal specifications such as only a few hundred total operation cycles (site to site) the tooth bending stresses for static strength will be decisive for the lay out however with ex-ception of surface hardened pinions where case crushing has to be considered
In the following the use of part 1 and 3 for pinion-racks is shown including relevant simplifications
C1 Pinion tooth root stresses Since the load spectrum normally is dominated by high torques and few load cycles (in the range up to 1-2000) the static strength is decisive
The actual stress is calculated as
β1FSaFan1
tF1 KYY
mbFσ sdotsdotsdotsdot
=
b1 is limited to b2 + 2middotmn
YFa and YSa replace YF and YS because load application at tooth tip has to be assumed for such inaccurate gears
Pinions often use a non-involute profile in the dedendum part of the flank eg a constant radius equal the radius of curva-ture at reference circle For such pinions sFn and hFa are to be measured directly on a sectional drawing of the pinion tooth
Due to high loads and narrow facewidths it may be assumed that KFβ2 = KHβ = 10 However when b1 gt b2 then KFβ1 gt10
If no detailed documentation of KFβ1 is available the fol-lowing may be used
KFβ1 = 1 + 015middot(b1b2 ndash 1)
The permissible stress (not surface hardened) is calculated as
δrelTstF
Fst1FP1 Y
Sσσ sdot=
The mean stress influence due to leg lifting may be disre-garded
The actual and permissible stresses should be calculated for the relevant loads as given in the rules
C2 Rack tooth root stresses The actual stress is calculated as
SaFan2
tF2 YY
mbFσ sdotsdotsdot
=
See C1 for details
The permissible stress is calculated as
δrelTstF
Fst2FP2 Y
Sσσ sdot=
For alloyed steels (Ni Cr Mo) with high toughness and duc-tility the value of YδrelTst may be put equal to YSa
C3 Surface hardened pinions For surface hardened pinions the maximum load is not to cause crushing of the hardened layer of the flank
In principle the calculation described in 213 may be used but when the theoretical Hertzian stress exceeds the ap-proximately 18 times the yield strength of the rack material plastic deformation will occur This will limit the peak Hertzian stress but increases the contact width and thus the penetration of stresses into the depth
An approximation may be based on an assessment of contact width determined by means of equal areas under the theoreti-cal (elastic range) Hertzian contact and the elasto-plastic contact stress (limited to 18 middot σy) with the unknown width
Classification Notes- No 412 5 May 2003
DET NORSKE VERITAS
x = addendum modification coefficient z = number of teeth zn = virtual number of spur teeth αn = normal pressure angle at ref cylinder αt = transverse pressure angle at ref cylinder αa = transverse pressure angle at tip cylinder αwt = transverse pressure angle at pitch cylinder β = helix angle at ref cylinder βb = helix angle at base cylinder βa = helix angle at tip cylinder εα = transverse contact ratio εβ = overlap ratio εγ = total contact ratio ρa0 = tip radius of tool ref to mn ρfp = root radius of basic rack ref to mn ( = ρa0) ρC = effective radius (mm) of curvature at pitch point ρF = root fillet radius (mm) in the critical section σB = ultimate tensile strength (Nmm2) σy = yield strength resp 02 proof stress (Nmm2)
Index 1 refers to the pinion 2 to the wheel
Index n refers to normal section or virtual spur gear of a heli-cal gear
Index w refers to pitch point
Special additional symbols for bevel gears are as follows
Σ = angle between intersection axis Kϑ = angle modification (Klingelnberg)
m0 = tool module (Klingelnberg) δ = pitch cone angle xsm = tooth thickness modification coefficient (mid-
face) R = pitch cone distance (mm)
Index v refers to the virtual (equivalent) helical cylindrical gear
Index m refers to the midsection of the bevel gear
13 Geometrical Definitions For internal gearing z2 a da2 dw2 d2 and db2 are negative x2 is positive if da2 is increased ie the numeric value is de-creased
The pinion has the smaller number of teeth ie
11
2 ge=zz
u
For calculation of surface durability b is the common face-width on pitch diameter
For tooth strength calculations b1 or b2 are facewidths at the respective tooth roots If b1 or b2 differ much from b above they are not to be taken more than 1 module on either side of b
Cylindrical gears
tan αt = tan αn cos β tan βb = tan β cos αt
tan βa = tan β da d
cos αa = dbda
d = z mn cos β mt = mn cos β db = d cos αt = dw cos αwt
a = 05 (dw1 + dw2)
dw1dw2 = z1 z2
inv α = tan α - α (radians)
inv αwt = inv αt + 2 tan αn (x1 + x2)(z1 + z2)
zn = z (cos2 βb cos β)
1
aw1fw1α T
ξξε
+=
where ξfw1 is to be taken as the smaller of
bull wtfw1 αtanξ =
bull soi1
b1wtfw1 d
dacostan -tanαξ =
bull 1
2wt
a2
b2fw1 z
ztanα
d
dacostan ξ
minus=
and
2
1fw2aw1 z
zξξ = where ξfw2 is calculated as ξfw1
substituting the values for the wheel by the values for the pinion and visa versa
11 z
2πT =
( ) +
sdot+minusminussdot= minus
2sinαρρxhm
2d2d nfpfp1fpnsoi1
21
t
nfpfplfpn2
tanα)sinαρρx(hm
sdot+minusminus
nmsinbπ
β=εβ
(for double helix b is to be taken as the width of one helix)
εy = βα εε +
ρC = ( )2
b
wt
u1βcos
αsinua
+
v = 311 10dn
60π minus
6 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
pbt = βcos
αcosmπ tn
sat =
minus++
at
ninvααinv
z
αtanx22
π
d a
san = aβcossat
14 Bevel Gear Conversion Formulae and Specific Formulae Conversion of bevel gears to virtual equivalent helical cylin-drical gears is based on the bevel gear midsection The con-version formulae are
Number of teeth
zv12 = z12 cos δ12
(δ1 + δ2 = Σ)
Gear ratio
uv = 1v
2vzz
tan αvt = tan αn cos βm
tan βbm = tan βm cos αvt
Base pitch
pbtm = m
vtnmβcos
αcosmπ
Reference pitch diameters
dv12 = 21
21m
cosdδ
Centre distance
av = 05 (dv1 + dv2)
Tip diameters
dva 12 = dv 12 + 2 ham 12
Addenda
for gears with constant addenda (Klingelnberg)
ham 12 = mmn (1 + xm 12)
for gears with variable addenda (Gleason)
ham 12 = ha 12 ndash b2 tan (δa 12 ndash δ12)
(when ha is addendum at outer end and δa is the outer cone angle)
Addendum modification coefficients
xm 12 = mn
12am21am
m2
hh minus
Base circle
dvb 12 = dv 12 cos αvt
Transverse contact ratio)
εα = btmP
αsinadd05dd05 vtv2
2vb2
2va2
1vb2
1va minusminus+minus
Overlap ratio) (theoretical value for bevel gears with no crowning but used as approximations in the calculation pro-cedures)
εβ = nm
mmπβsinb
Total contact ratio)
εγ = 2β
2α εε +
( Note that index laquovraquo is left out in order to combine formu-lae for cylindrical and bevel gears)
Tangential speed at midsection
vmt = 3m11 10dn
60π minus
Effective radius of curvature (normal section)
ρvc = ( )2vbm
vtvv
u1βcosαsinua
+
Length of line of contact
lb = ( )( )( )
1εifε
ε1ε2ε
βcosεb
β2γ
2βα
2γ
bm
α ltminusminusminus
lb = 1εifβcosε
εbβ
bmγ
α ge
15 Nominal Tangential Load Ft Fbt Fmt and Fmbt The nominal tangential load (tangential to the reference cyl-inder with diameter d and perpendicular to an axial plane) is calculated from the nominal (rated) torque T transmitted by the gear set
Cylindrical gears
dT2000
Ft = t
tbt αcos
FF =
Bevel gears
mmt d
T2000F =
vt
mtmbt αcos
FF =
Classification Notes- No 412 7 May 2003
DET NORSKE VERITAS
16 Application Factors KA and KAP The application factor KA accounts for dynamic overloads from sources external to the gearing
It is distinguished between the influence of repetitive cyclic torques KA (161) and the influence of temporary occasional peak torques KAP (162)
Calculations are always to be made with KA In certain cases additional calculations with KAP may be necessary
For gears with a defined load spectrum the calculation with a KA may be replaced by a fatigue damage calculation as given in Appendix A
161 KA For gears designed for long or infinite life at nominal rated torque KA is defined as the ratio between the maximum re-petitive cyclic torque applied to the gear set and nominal rated torque
This definition is suitable for main propulsion gears and most of the auxiliary gears
KA can be determined by measurements or system analysis or may be ruled by conventions (ice classes) (For the pur-pose of a preliminary (but not binding) calculation before KA is determined it is advised to apply either the max values mentioned below or values known from similar plants)
a) For main propulsion gears KA can be taken from the (mandatory) torsional vibration analysis thereby con-sidering all permissible driving conditions) Unless specially agreed the rules do not allow KA in excess of 135 for diesel propulsion) With turbine or electric propulsion KA would normally not exceed 12 However special attention should be given to thrusters that are arranged in such a way that heavy vessel movements andor manoeuvring can cause severe load fluctuations This means eg thrusters positioned far from the rolling axis of vessels that could be susceptible to rolling If leading to propeller air suction the condi-tions may be even worse The above mentioned movements or manoeuvring will result in increased propeller excitation If the thruster is driven by a diesel engine the engine mean torque is limited to 100 However thrusters driven by electric motors can suffer temporary mean torque much above 100 unless a suitable load control system (limiting available e-motor torque) is provided
b) For main propulsion gears with ice class notation (see Rules Pt5 Ch1 Sec J500) KA ice has to be taken as the higher value of the applicable (rule defined) ice shock torque referred to nominal rated torque and the value under a) The Baltic ice class notations refer to a few millions ice shock loads Thus the life factors may be put YN=1 and ZN=12 (except for nitrided gears where ZN = 1 applies) Additionally the calculations with the normal KA (no ice class) are to fulfil the normal requirements For polar ice class notations KA ice applies to all criteria and for long or infinite life
c) For a power take off (PTO) branch from a main propul-sion gear with ice class ice shocks result in negative torques It is assumed that the PTO branch is unloaded when the ice shock load occurs The influence of these reverse shock loads may be taken into account as follows The negative torque (reversed load) expressed by means of an application factor based on rated forward load (T or Ft) is KAreverse = KA ice ndash1 (the minus 1 be-cause no mean torque assumed) KAice to be calculated as in the ice class rules This KAreverse should be used for back flank considera-tions such as pitting and scuffing The influence on tooth bending strength (forward di-rection) may be simplified by using the factor YM = 1 minus 03 middot KAreverse KA
d) For diesel driven auxiliaries KA can be taken from the torsional vibration analysis if available For units where no vibration analysis is required (lt 200 kW) or available it is advised to apply KA as the upper allow-able value 135)
e) For turbine or electro driven auxiliaries the same as for c) applies however the practical upper value is 12
) For diesel driven gears more information on KA for mis-firing and normal driving is given in Appendix B
162 KAP The peak overload factor KAP is defined as the ratio between the temporary occasional peak overload torque and the nominal rated torque
For plants where high temporary occasional peak torques can occur (ie in excess of the above mentioned KA) the gearing (if nitrided) has to be checked with regard to static strength Unless otherwise specified the same safety factors as for infinite life apply
The scuffing safety is to be specially considered whereby the KA applies in connection with the bulk temperature and the KAP applies for the flash temperature calculation and should replace KA in the formulae in 431 432 and 441
KAP can be evaluated from the torsional impact vibration calculation (as required by the rules)
If the overloads have a duration corresponding to several revolutions of the shafts the scuffing safety has to be consid-ered on basis of this overload both with respect to bulk and flash temperature This applies to plants with ice class nota-tions (Baltic and polar) and to plants with prime movers which have high temporary overload capacity such as eg electric motors (provided the driven member can have a con-siderable increase in demand torque as eg azimuth thrusters during manoeuvring)
For plants without additional ice class notation KAP should normally not exceed 15
8 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
163 Frequent overloads For plants where high overloads or shock loads occur regu-larly the influence of this is to be considered by means of cumulative fatigue (see Appendix A)
17 Load Sharing Factor Kγ The load sharing factor Kγ accounts for the maldistribution of load in multiple-path transmissions (dual tandem epicyclic double helix etc) Kγ is defined as the ratio between the max load through an actual path and the evenly shared load
171 General method Kγ mainly depends on accuracy and flexibility of the branches (eg quill shaft planet support external forces etc) and should be considered on basis of measurements or of relevant analysis as eg
Kγ = δ+δ f
δ = total compliance of a branch under full load (assuming even load share) referred to gear mesh
f = minusminusminusminus+++ 23
22
21 fff where f1 f2 etc are the
main individual errors that may contribute to a maldis-tribution between the branches Eg tooth pitch errors planet carrier pitch errors bearing clearance influences etc Compensating effects should also be considered
For double helical gears
An external axial force Fext applied from sources outside the actual gearing (eg thrust via or from a tooth coupling) will cause a maldistribution of forces between the two helices Expressed by a load sharing factor the
βsdot
plusmn=γ tanFF1K
t
ext
If the direction of Fext is known the calculation should be carried out separately for each helix and with the tangential force corrected with the pertinent Kγ If the direction of Fext is unknown both combinations are to be calculated and the higher σH or σF to be used
172 Simplified method If no relevant analysis is available the following may apply
For epicyclic gears
Kγ = 32501 minus+ pln
where npl = number of planets ( gt3 )
For multistage gears with locked paths and gear stages sepa-rated by quill shafts (see figure below)
Figure 10 Locked paths gear
Kγ = ( )φ201+
where φ = quill shaft twist (degrees) under full load
18 Dynamic Factor Kv The dynamic factor Kv accounts for the internally generated dynamic loads
Kv is defined as the ratio between the maximum load that dynamically acts on the tooth flanks and the maximum ex-ternally applied load Ft KA Kγ
In the following 2 different methods (181 and 182) are described In case of controversy between the methods the next following is decisive ie the methods are listed with increasing priority
It is important to observe the limitations for the method in 181 In particular the influence of lateral stiffness of shafts is often underestimated and resonances occur at considerably lower speed than determined in 1811
However for low speed gears with vmiddotz1 lt 300 calculations may be omitted and the dynamic factor simplified to Kv=105
181 Single resonance method For a single stage gear Kv may be determined on basis of the relative proximity (or resonance ratio) N between actual speed n1 and the lowest resonance speed nE1
N = 1E
1
nn
Note that for epicyclic gears n is the relative speed ie the speed that multiplied with z gives the mesh frequency
1811 Determination of critical speed It is not advised to apply this method for multimesh gears for N gt 085 as the influence of higher modes has to be consid-ered see 182 In case of significant lateral shaft flexibility (eg overhung mounted bevel gears) the influence of cou-pled bending and torsional vibrations between pinion and wheel should be considered if N ge 075 see 182
nE1 = redmγc
1zπ
31030 sdot
where
cγ is the actual mesh stiffness per unit facewidth see 111
Classification Notes- No 412 9 May 2003
DET NORSKE VERITAS
For gears with inactive ends of the facewidth as eg due to high crowning or end relief such as often applied for bevel gears the use of cγ in connection with determination of natu-ral frequencies may need correction cγ is defined as stiffness per unit facewidth but when used in connection with the total mesh stiffness it is not as simple as cγmiddotb as only a part of the facewidth is active Such corrections are given in 111
mred is the reduced mass of the gear pair per unit facewidth and referred to the plane of contact
For a single gear stage where no significant inertias are closely connected to neither pinion nor wheel mred is calcu-lated as
mred = 21
21mm
mm+
The individual masses per unit facewidth are calculated as
m12 = 221b
21
)2d(b
I
where I is the polar moment of inertia (kgmm2)
The inertia of bevel gears may be approximated as discs with diameter equal the midface pitch diameter and width equal to b However if the shape of the pinion or wheel body differs much from this idealised cylinder the inertia should be cor-rected accordingly
For all kind of gears if a significant inertia (eg a clutch) is very rigidly connected to the pinion or wheel it should be added to that particular inertia (pinion or wheel) If there is a shaft piece between these inertias the torsional shaft stiffness alters the system into a 3-mass (or more) system This can be calculated as in 182 but also simplified as a 2-mass system calculated with only pinion and wheel masses
1812 Factors used for determination of Kv Non-dimensional gear accuracy dependent parameters
( )bKKF
yfcB
At
pptp
γ
minus=
( )bKKF
yFcBAt
ff
γ
α minus=
Non-dimensional tip relief parameter
bKKFcC
1BγAt
ak sdotsdot
sdotminus=
For gears of quality grade (ISO 1328) Q = 7 or coarser Bk = 1
For gears with Q le 6 and excessive tip relief Bk is limited to max 1
For gears (all quality grades) with tip relief of more than 2middotCeff (see 432) the reduction of αε has to be considered (see 443)
where
fpt = the single pitch deviation (ISO 1328) max of pinion or wheel
Fα = the total profile form deviation (ISO 1328) max of pinion or wheel (Note Fα is pt not available for bevel gears thus use Fα = fpt)
yp and yf = the respective running-in allowances and may be calculated similarly to yα in 112 ie the value of fpt is replaced by Fα for yf
cacute = the single tooth stiffness see 111
Ca = the amount of tip relief see 433 In case of different tip relief on pinion and wheel the value that results in the greater value of Bk is to be used If Ca is zero by design the value of running-in tip relief Cay (see 112) may be used in the above formula
1813 Kv in the subcritical range Cylindrical gears N le 085
Bevel gears N le 075
Kv = 1 + N K
K = Cv1 Bp + Cv2 Bf + Cv3 Bk
Cv1 accounts for the pitch error influence
Cv1 = 032
Cv2 accounts for profile error influence
Cv2 = 034 for γε le 2
Cv2 = 03ε
057
γ minus for γε gt 2
Cv3 accounts for the cyclic mesh stiffness variation
Cv3 = 023 for γε le 2
Cv3 = 156ε
0096
γ minus for γε gt 2
1814 Kv in the main resonance range Cylindrical gears 085 lt N le 115
Bevel gears 075 lt N le 125
Running in this range should preferably be avoided and is only allowed for high precision gears
Kv = 1 + Cv1 Bp + Cv2 Bf + Cv4 Bk
Cv4 accounts for the resonance condition with the cyclic mesh stiffness variation
Cv4 = 090 for γε le 2
Cv4 = 144ε
ε005057
γ
γ
minus
minus for γε gt 2
10 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
1815 Kv in the supercritical range Cylindrical gears N ge 15
Bevel gears N ge 15
Special care should be taken as to influence of higher vibra-tion modes andor influence of coupled bending (ie lateral shaft vibrations) and torsional vibrations between pinion and wheel These influences are not covered by the following approach
Kv = Cv5 Bp + Cv6 Bf + Cv7
Cv5 accounts for the pitch error influence
Cv5 = 047
Cv6 accounts for the profile error influence
Cv6 = 047 for εγ le 2
Cv6 = 174ε
012
γ minus for εγ gt 2
Cv7 relates the maximum externally applied tooth loading to the maximum tooth loading of ideal accurate gears operating in the supercritical speed sector when the circumferential vibration becomes very soft
Cv7 = 075 for εγ le 15
Cv7 = [ ] 8750)2(sin 0125 +minusβεπ for 15 lt εγ le 25
Cv7 = 10 for εγ gt 25
1816 Kv in the intermediate range Cylindrical gears 115 lt N lt 15
Bevel gears 125 lt N lt 15
Comments raised in 1814 and 1815 should be observed
Kv is determined by linear interpolation between Kv for N = 115 respectively 125 and N = 15 as
Cylindrical gears
( ) ( ) ( )[ ]51Nv151Nv51Nvv KK350
N51KK === minussdot
minus
+=
Bevel gears
( ) ( ) ( )[ ]51Nv251Nv51Nvv KK250
N51KK === minussdot
minus
+=
182 Multi-resonance method For high speed gear (vgt40 ms) for multimesh medium speed gears for gears with significant lateral shaft flexibility etc it is advised to determine Kv on basis of relevant dy-namic analysis
Incorporating lateral shaft compliance requires transforma-tion of even a simple pinion-wheel system into a lumped multi-mass system It is advised to incorporate all relevant inertias and torsional shaft stiffnesses into an equivalent (to pinion speed) system Thereby the mesh stiffness appears as an equivalent torsional stiffness
cγ b (db12)2 (Nmrad)
The natural frequencies are found by solving the set of dif-ferential equations (one equation per inertia) Note that for a gear put on a laterally flexible shaft the coupling bending-torsionals is arranged by introducing the gear mass and the lateral stiffness with its relation to the torsional displacement and torque in that shaft
Only the natural frequency (ies) having high relative dis-placement and relative torque through the actual pinion-wheel flexible element need(s) to be considered as critical frequency (ies)
Kv may be determined by means of the method mentioned in 181 thereby using N as the least favourable ratio (in case of more than one pinion-wheel dominated natural frequency) Ie the N-ratio that results in the highest Kv has to be consid-ered
The level of the dynamic factor may also be determined on basis of simulation technique using numeric time integration with relevant tooth stiffness variation and pitchprofile er-rors
19 Face Load Factors KHβ and KFβ The face load factors KHβ for contact stresses and for scuff-ing KFβ for tooth root stresses account for non-uniform load distribution across the facewidth
KHβ is defined as the ratio between the maximum load per unit facewidth and the mean load per unit facewidth
KFβ is defined as the ratio between the maximum tooth root stress per unit facewidth and the mean tooth root stress per unit facewidth The mean tooth root stress relates to the con-sidered facewidth b1 respectively b2
Note that facewidth in this context is the design facewidth b even if the ends are unloaded as often applies to eg bevel gears
The plane of contact is considered
191 Relations between KHβ and KFβ
KFβ = expβHK 2(hb)hb1
1exp++
=
where hb is the ratio tooth heightfacewidth The maximum of h1b1 and h2b2 is to be used but not higher than 13 For double helical gears use only the facewidth of one helix
If the tooth root facewidth (b1 or b2) is considerably wider than b the value of KFβ(1or2) is to be specially considered as it may even exceed KHβ
Eg in pinion-rack lifting systems for jack up rigs where b = b2 asymp mn and b1 asymp 3 mn the typical KHβ asymp KFβ2 asymp 1 and KFβ1 asymp 13
192 Measurement of face load factors Primarily
KFβ may be determined by a number of strain gauges distrib-uted over the facewidth Such strain gauges must be put in
Classification Notes- No 412 11 May 2003
DET NORSKE VERITAS
exactly the same position relative to the root fillet Relations in 191 apply for conversion to KHβ
Secondarily
KHβ may be evaluated by observed contact patterns on vari-ous defined load levels It is imperative that the various test loads are well defined Usually it is also necessary to evalu-ate the elastic deflections Some teeth at each 90 degrees are to be painted with a suitable lacquer Always consider the poorest of the contact patterns
After having run the gear for a suitable time at test load 1 (the lowest) observe the contact pattern with respect to ex-tension over the facewidth Evaluate that KHβ by means of the methods mentioned in this section Proceed in the same way for the next higher test load etc until there is a full face contact pattern From these data the initial mesh misalign-ment (ie without elastic deflections) can be found by ex-trapolation and then also the KHβ at design load can be found by calculation and extrapolation See example
Figure 11 Example of experimental determination of KHβ
It must be considered that inaccurate gears may accumulate a larger observed contact pattern than the actual single mesh to mesh contact patterns This is particularly important for lapped bevel gears Ground or hard metal hobbed bevel gears are assumed to present an accumulated contact pattern that is practically equal the actual single mesh to mesh con-tact patterns As a rough guidance the (observed) accumu-lated contact pattern of lapped bevel gears may be reduced by 10 in order to assess the single mesh to mesh contact pattern which is used in 199
193 Theoretical determination of KHβ The methods described in 193 to 198 may be used for cy-lindrical gears The principles may to some extent also be used for bevel gears but a more practical approach is given in 199
General For gears where the tooth contact pattern cannot be verified during assembly or under load all assumptions are to be well on the safe side
KHβ is to be determined in the plane of contact
The influence parameters considered in this method are
bull mean mesh stiffness cγ (see 111) (if necessary also variable stiffness over b)
bull mean unit load Fmb = Fbt KA Kγ Kvb (for double helical gears see 17 for use of Kγ)
bull misalignment fsh due to elastic deflections of shafts and gear bodies (both pinion and wheel)
bull misalignment fdefl due to elastic deflections of and work-ing positions in bearings
bull misalignment fbe due to bearing clearance tolerances bull misalignment fma due to manufacturing tolerances bull helix modifications as crowning end relief helix correc-
tion bull running in amount yβ (see 112)
In practice several other parameters such as centrifugal ex-pansion thermal expansion housing deflection etc contrib-ute to KHβ However these parameters are not taken into account unless in special cases when being considered as particularly important
When all or most of the am parameters are to be considered the most practical way to determine KHβ is by means of a graphical approach described in 1931
If cγ can be considered constant over the facewidth and no helix modifications apply KHβ can be determined analyti-cally as described in 1932
1931 Graphical method The graphical method utilises the superposition principle and is as follows
bull Calculate the mean mesh deflection Mδ as a function of
γm c and bF see 111
bull Draw a base line with length b and draw up a rectangu-lar with height δM (The area δM b is proportional to the transmitted force)
bull Calculate the elastic deflection fsh in the plane of contact Balance this deflection curve around a zero line so that the areas above and below this zero line are equal
Figure 12 fsh balanced around zero line
bull Superimpose these ordinates of the fsh curve to the previ-ous load distribution curve (The area under this new load distribution curve is still δM b)
bull Calculate the bearing deflections andor working posi-tions in the bearings and evaluate the influence fdefl in the plane of contact This is a straight line and is bal-anced around a zero line as indicated in Fig 14 but with one distinct direction Superimpose these ordinates to the previous load distribution curve
12 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
bull The amount of crowning end relief or helix correction (defined in the plane of contact) is to be balanced around a zero line similarly to fsh
Figure 13 Crowning Cc balanced aound zero line
bull Superimpose these ordinates to the previous load distri-bution curve In case of high crowning etc as eg often applied to bevel gears the new load distribution curve may cross the base line (the real zero line) The result is areas with negative load that is not real as the load in those areas should be zero Thus corrective actions must be made but for practical reasons it may be postponed to after next operation
bull The amount of initial mesh misalignment bema ff + (defined in the plane of contact) is to be bal-
anced around a zero line If the direction of bema ff + is known (due to initial contact check) or if the direction of fbe is known due to design (eg overhang bevel pin-ion) this should be taken into account If direction un-known the influence of bema ff + in both directions as well as equal zero should be considered
Figure 14 fma+fbe in both directions balanced around zero line
Superimpose these ordinates to the previous load distri-bution curve This results in up to 3 different curves of which the one with the highest peak is to be chosen for further evaluation
bull If the chosen load distribution curve crosses the base line (ie mathematically negative load) the curve is to be corrected by adding the negative areas and dividing this with the active facewidth The (constant) ordinates of this rectangular correction area are to be subtracted from the positive part of the load distribution curve It is advisable to check that the area covered under this new load distribution curve is still equal δM b
bull If cγ cannot be considered as constant over b then cor-rect the ordinates of the load distribution curve with the local (on various positions over the facewidth) ratio between local mesh stiffness and average mesh stiffness cγ (average over the active facewidth only) Note that the result is to be a curve that covers the same area δM b as before
bull The influence of running in yβ is to be determined as in 112 whereby the value for Fβx is to be taken as twice the distance between the peak of the load distribution curve and δM
bull Determine
KHβ = M
β
δycurveofpeak minus
1932 Simplified analytical method for cylindrical gears The analytical approach is similar to 1931 but has a more limited application as cγ is assumed constant over the face-width and no helix modification applies
bull Calculate the elastic deflection fsh in the plane of contact Balance this deflection curve around a zero line so that the area above and below this zero line are equal see Fig 12 The max positive ordinate is frac12∆fsh
bull Calculate the initial mesh alignment as Fβx= deflbemash ffff plusmnplusmnplusmn∆ The negative signs may only be used if this is justified andor verified by a contact pattern test Otherwise al-ways use positive signs If a negative sign is justified the value of Fβx is not to be taken less than the largest of each of these elements
bull Calculate the effective mesh misalignment as Fβy = Fβx - yβ (yβ see 112)
bull Determine
2KforF2
bFc1K βH
m
βγγH le+=β
or
2KforF
bFc2K Hβ
m
βγγH gt=β
where cγ as used here is the effective mesh stiffness see 111
194 Determination of fsh fsh is the mesh misalignment due to elastic deflections Usu-ally it is sufficient to consider the combined mesh deflection of the pinion body and shaft and the wheel shaft The calcu-lation is to be made in the plane of contact (of the considered gear mesh) and to consider all forces (incl axial) acting on the shafts Forces from other meshes can be parted into components parallel respectively vertical to the considered plane of contact Forces vertical to this plane of contact have no influence on fsh
It is advised to use following diameters for toothed elements
d + 2 x mn for bending and shear deflection
d + 2 mn (x ndash ha0 + 02) for torsional deflection
Usually fsh is calculated on basis of an evenly distributed load If the analysis of KHβ shows a considerable maldis-
Classification Notes- No 412 13 May 2003
DET NORSKE VERITAS
tribution in term of hard end contact or if it is known by other reasons that there exists a hard end contact the load should be correspondingly distributed when calculating fsh In fact the whole KHβ procedure can be used iteratively 2-3 iterations will be enough even for almost triangular load distributions
195 Determination of fdefl fdefl is the mesh misalignment in the plane of contact due to bearing deflections and working positions (housing deflec-tion may be included if determined)
First the journal working positions in the bearings are to be determined The influence of external moments and forces must be considered This is of special importance for twin pinion single output gears with all 3 shafts in one plane
For rolling bearings fdefl is further determined on basis of the elastic deflection of the bearings An elastic bearing deflec-tion depends on the bearing load and size and number of rolling elements Note that the bearing clearance tolerances are not included here
For fluid film bearings fdefl is further determined on basis of the lift and angular shift of the shafts due to lubrication oil film thickness Note that fbe takes into account the influence of the bearing clearance tolerance
When working positions bearing deflections and oil film lift are combined for all bearings the angular misalignment as projected into the plane of the contact is to be determined fdefl is this angular misalignment (radians) times the face-width
196 Determination of fbe fbe is the mesh misalignment in the plane of contact due to tolerances in bearing clearances In principle fbe and fdefl could be combined But as fdefl can be determined by analy-sis and has a distinct direction and fbe is dependent on toler-ances and in most cases has no distinct direction (ie + toler-ance) it is practicable to separate these two influences
Due to different bearing clearance tolerances in both pinion and wheel shafts the two shaft axis will have an angular mis-alignment in the plane of contact that is superimposed to the working positions determined in 195 fbe is the facewidth times this angular misalignment Note that fbe may have a distinct direction or be given as a + tolerance or a combina-tion of both For combination of + tolerance it is adviced to use
fbe= 22be
21be ff ++plusmn
fbe is particularly important for overhang designs for gears with widely different kinds of bearings on each side and when the bearings have wide tolerances on clearances In general it shall be possible to replace standard bearings with-out causing the real load distribution to exceed the design premises For slow speed gears with journal bearings the expected wear should also be considered
197 Determination of fma fma is the mesh misalignment due to manufacturing toler-ances (helix slope deviation) of pinion fHβ1 wheel fHβ2 and housing bore
For gear without specifically approved requirements to as-sembly control the value of fma is to be determined as
fma= 22Hβ
21Hβ ff +
For gears with specially approved assembly control the value of fma will depend on those specific requirements
198 Comments to various gear types For double helical gears KHβ is to be determined for both helices Usually an even load share between the helices can be assumed If not the calculation is to be made as de-scribed in 171
For planetary gears the free floating sun pinion suffers only twist no bending It must be noted that the total twist is the sum of the twist due to each mesh If the value of 1K γ ne this must be taken into account when calculating the total sun pinion twist (ie twist calculated with the force per mesh without Kγ and multiplied with the number of planets)
When planets are mounted on spherical bearings the mesh misalignments sun-planet respectively planet-annulus will be balanced Ie the misalignment will be the average between the two theoretical individual misalignments The faceload distribution on the flanks of the planets can take full advan-tage of this However as the sun and annulus mesh with several planets with possibly different lead errors the sun and annulus cannot obtain the above mentioned advantage to the full extent
199 Determination of KHβ for bevel gears If a theoretical approach similar to 193-198 is not docu-mented the following may be used
testeff
H Kb
b851851K sdot
minussdot=β
beff b represents the relative active facewidth (regarding lapped gears see 192 last part)
Higher values than beff b = 090 are normally not to be used in the formula
For dual directional gears it may be difficult to obtain a high beff b in both directions In that case the smaller beff b is to be used
Ktest represents the influence of the bearing arrangement shaft stiffness bearing stiffness housing stiffness etc on the faceload distribution and the verification thereof Expected variations in length- and height-wise tooth profile is also accounted for to some extent
a) Ktest = 1 For ground or hard metal hobbed gears with the specified contact pattern verified at full rating or at full torque slow turning at a condition representative for the thermal expansion at normal operation
14 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
It also applies when the bearing arrangementsupport has insignificant elastic deflections and thermal axial expansion However each initial mesh contact must be verified to be within acceptance criteria that are cali-brated against a type test at full load Reproduction of the gear tooth length- and height-wise profile must also be verified This can be made through 3D measurements or by initial contact movements caused by defined axial offsets of the pinion (tolerances to be agreed upon)
b) Ktest = 1 + 04middot(beffbndash06) For designs with possible influence of thermal expansion in the axial direction of the pinion The initial mesh contact verified with low load or spin test where the acceptance criteria are calibrated against a type test at full load
c) Ktest = 12 if mesh is only checked by toolmakerrsquos blue or by spin test contact For gears in this category beffb gt 085 is not to be used in the calculation
110 Transversal Load Distribution Factors KHα and KFα The transverse load distribution factors KHα for contact stresses and for scuffing KFα for tooth root stresses account for the effects of pitch and profile errors on the transversal load distribution between 2 or more pairs of teeth in mesh
The following relations may be used
Cylindrical gears
( )
minus+==
tH
αptγγHαFα F
byfc0409
2ε
KK
valid for 2εγ le
( ) ( )tH
αptγ
γ
γHαFα F
byfcε
1ε20409KK
minusminus+==
valid for 2ε γ gt
where
FtH = Ft KA Kγ Kv KHβ
cγ = See 111
γα = See 112
fpt = Maximum single pitch deviation (microm) of pinion or wheel or maximum total profile form devia-tion Fα of pinion or wheel if this is larger than the maximum single pitch deviation
Note In case of adequate equivalent tip relief adapted to the load half of the above mentioned fpt can be introduced A tip relief is considered adequate when the average of Ca1 and Ca2 is within plusmn40 of the value of Ceff in 432
Limitations of KHα and KFα
If the calculated values for
KFα = KHα lt 1 use KFα = KHα = 10
If the calculated value of KHα gt 2εα
γ
Zε
ε use KHα = 2
εα
γ
Zε
ε
If the calculated value of KFα gt εα
γ
Yεε
use KFα = εα
γ
Yεε
where Yε = αnε
075025+ (for εαn see 331c)
Bevel gears
For ground or hard metal hobbed gears KFα = KHα = 1
For lapped gears KFα = KHα = 11
111 Tooth Stiffness Constants cacute and cγ The tooth stiffness is defined as the load which is necessary to deform one or several meshing gear teeth having 1 mm facewidth by an amount of 1 microm in the plane of contact cacute is the maximum stiffness of a single pair of teeth cγ is the mean value of the mesh stiffness in a transverse plane (brief term mesh stiffness)
Both valid for high unit load (Unit load = Ft middot KA middot Kγb)
Cylindrical gears The real stiffness is a combination of the progressive Hertzian contact stiffness and the linear tooth bending stiff-nesses For high unit loads the Hertzian stiffness has little importance and can be disregarded This approach is on the safe side for determination of KHβ and KHα However for moderate or low loads Kv may be underestimated due to determination of a too high resonance speed
The linear approach is described in A
An optional approach for inclusion of the non-linear stiffness is described in B
A The linear approach
BRCCq
βcos08acutec =
and
( )025ε075cc αγ +prime=
where
( )[ ]n02a01a
B α2000212
hh12051C minusminus
+minus+=
12n1n
x000635z
025791z
015551004723q minus++=
12
2n
22
1n
1 x005290z
x024188x000193z
x011654+minusminusminus
+ 000182 x22
Classification Notes- No 412 15 May 2003
DET NORSKE VERITAS
(for internal gears use zn2 equal infinite and x2 = 0) ha0 = hfp for all practical purposes CR considers the increased flexibility of the wheel teeth if the wheel is not a solid disc and may be calculated as
( )( )nR m5s
sR e5
bbln1C +=
where
bs = thickness of a central web
sR = average thickness of rim (net value from tooth root to inside of rim)
The formula is valid for bs b ge 02 and sRmn ge 1 Outside this range of validity and if the web is not centrally posi-tioned CR has to be specially considered
Note CR is the ratio between the average mesh stiffness over the facewidth and the mesh stiffness of a gear pair of solid discs The local mesh stiffness in way of the web corresponds to the mesh stiffness with CR = 1 The local mesh stiffness where there is no web support will be less than calculated with CR above Thus eg a centrally positioned web will have an effect corresponding to a longitudinal crowning of the teeth See also 1931 regarding KHβ
B The non-linear approach
In the following an example is given on how to consider the non-linearity
The relation between unit load Fb as a function of mesh deflection δ is assumed to be a progressive curve up to 500 Nmm and from there on a straight line This straight line when extended to the baseline is assumed to intersect at 10microm
With these assumptions the unit force Fb as a function of mesh deflection δ can be expressed as
( )10δKbF
minus= for 500bFgt
minus=
500Fb10δK
bF for 500
bFlt
with γAt KK
bF
bF
sdotsdot= etc (Nmm) ie unit load incorporat-
ing the relevant factors as
KA middot Kγ for determination of Kv
KA middot Kγ middot Kv for determination of KHβ
KA middot Kγ middot Kv middot KHβ for determination of KHα
δ = mesh deflection (microm)
K = applicable stiffness (c or cγ)
Use of stiffnesses for KV KHβ and KHα
For calculation of Kv and KHα the stiffness is calculated as follows
When Fb lt 500
the stiffness is determined as δ∆
∆ bF
where the increment is chosen as eg ∆ Fb = 10 and thus
50010Fb10
K10Fb∆δ +
++
=
When F b gt 500 the stiffness is c or cγ For calculation of KHβ the mesh deflection δ is used directly
or an equivalent stiffness determined as δsdotb
F
Bevel gears
In lack of more detailed relationship between stiffness and geometry the following may be used
b085
b13cacute eff=
b085b
16c effγ =
beff not to be used in excess of 085 b in these formulae
Bevel gears with heightwise and lengthwise crowning have progressive mesh stiffness The values mentioned above are only valid for high loads They should not be used for de-termination of Ceff (see 432) or KHβ (see 1931)
112 Running-in Allowances The running-in allowances account for the influence of run-ning-in wear on the various error elements
yα respectively yβ are the running-in amounts which reduce the influence of pitch and profile errors respectively influ-ence of localised faceload
Cay is defined as the running-in amount that compensates for lack of tip relief
The following relations may be used
For not surface hardened steel
ptHlim
α fσ160y =
yβ = βxlimH
f320σ
with the following upper limits
V lt 5 ms 5-10 ms gt 10 ms
yα max none
limH
12800σ limH
6400σ
yβ max none
limH
25600σ limH
12800σ
For surface hardened steel
yα = 0075 fpt but not more than 3 for any speed
yβ = 015 Fβx but not more than 6 for any speed
16 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
For all kinds of steel
5145189718
1C2
limHay +
minusσ
=
When pinion and wheel material differ the following ap-plies
bull Use the larger of fpt1 - yα1 and fpt2 - yα2 to replace fpt - yα in the calculation of KHα and Kv
bull Use ( )2β1ββ yy21y += in the calculation of KHβ
bull Use ( )2ay1aya CC21C += in the calculation of Kv
bull Use ( )2ay1ay2a1a CC21CC +== in the scuffing
calculation if no design tip relief is foreseen
Classification Notes- No 412 17 May 2003
DET NORSKE VERITAS
2 Calculation of Surface Durability
21 Scope and General Remarks Part 2 includes the calculations of flank surface durability as limited by pitting spalling case crushing and subsurface yielding Endurance and time limited flank surface fatigue is calculated by means of 22 ndash 212 In a way also tooth frac-tures starting from the flank due to subsurface fatigue is in-cluded through the criteria in 213
Pitting itself is not considered as a critical damage for slow speed gears However pits can create a severe notch effect that may result in tooth breakage This is particularly im-portant for surface hardened teeth but also for high strength through hardened teeth For high-speed gears pitting is not permitted
Spalling and case crushing are considered similar to pitting but may have a more severe effect on tooth breakage due to the larger material breakouts initiated below the surface Subsurface fatigue is considered in 213
For jacking gears (self-elevating offshore units) or similar slow speed gears designed for very limited life the max static (or very slow running) surface load for surface hard-ened flanks is limited by the subsurface yield strength
For case hardened gears operating with relatively thin lubri-cation oil films grey staining (micropitting) may be the lim-iting criterion for the gear rating Specific calculation meth-ods for this purpose are not given here but are under consid-eration for future revisions Thus depending on experience with similar gear designs limitations on surface durability rating other than those according to 22 - 213 may be ap-plied
22 Basic Equations Calculation of surface durability (pitting) for spur gears is based on the contact stress at the inner point of single pair contact or the contact at the pitch point whichever is greater
Calculation of surface durability for helical gears is based on the contact stress at the pitch point
For helical gears with 0 lt εβ lt 1 a linear interpolation be-tween the above mentioned applies
Calculation of surface durability for spiral bevel gears is based on the contact stress at the midpoint of the zone of contact
Alternatively for bevel gears the contact stress may be cal-culated with the program ldquoBECALrdquo In that case KA and Kv are to be included in the applied tooth force but not KHβ and KHα The calculated (real) Hertzian stresses are to be multi-plied with ZK in order to be comparable with the permissible contact stresses
The contact stresses calculated with the method in part 2 are based on the Hertzian theory but do not always represent the real Hertzian stresses
The corresponding permissible contact stresses σHP are to be calculated for both pinion and wheel
221 Contact stress Cylindrical gears
( )HαHβvγA
1
tβεEHDBH KKKKK
bud1uF
ZZZZZσ+
=
where
ZBD = Zone factor for inner point of single pair contact for pinion resp wheel (see 232)
ZH = Zone factor for pitch point (see 231)
ZE = Elasticity factor (see 24)
Zε = Contact ratio factor (see 25)
Zβ = Helix angle factor (see 26)
Ft KA Kγ Kv KHβ KHα see 15 ndash 110
d1 b u see 12 ndash 15
Bevel gears
( )HαHβvγA
v1v
vmtKEMH KKKKK
bud1uF
ZZZ105σ+
sdot=
where
105 is a correlation factor to reach real Hertzian stresses (when ZK = 1)
ZE KA etc see above
ZM = mid-zone factor see 233
ZK = bevel gear factor see 27
Fmt dv1 uv see 12 ndash 15
It is assumed that the heightwise crowning is chosen so as to result in the maximum contact stresses at or near the mid-point of the flanks
222 Permissible contact stress
XWRvLH
NHlimHP ZZZZZ
SZσ
σ =
where
σH lim = Endurance limit for contact stresses (see 28)
ZN = Life factor for contact stresses (see 29)
SH = Required safety factor according to the rules
ZLZvZR = Oil film influence factors (see 210)
ZW = Work hardening factor (see 211)
ZX = Size factor (see 212)
18 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
23 Zone Factors ZH ZBD and ZM
231 Zone factor ZH The zone factor ZH accounts for the influence on contact stresses of the tooth flank curvature at the pitch point and converts the tangential force at the reference cylinder to the normal force at the pitch cylinder
wtt2
wtbH sinααcos
cosαcosβ2Z =
232 Zone factors ZBD The zone factors ZBD account for the influence on contact stresses of the tooth flank curvature at the inner point of sin-gle pair contact in relation to ZH Index B refers to pinion D to wheel
For εβ ge 1 ZBD = 1
For internal gears ZD = 1
For εβ = 0 (spur gears)
( )
π
minusεminusminus
π
minusminus
α=
α2
2
2b
2a
1
2
1b
1a
wtB
z211
dd
z21
dd
tanZ
( )
π
minusεminusminus
π
minusminus
α=
α1
2
1b
1a
2
2
2b
2a
wtD
z211
dd
z21
dd
tanZ
If ZB lt 1 use ZB = 1
If ZD lt 1 use ZD = 1
For 0 lt εβ lt 1
ZBD = ZBD (for spur gears) ndash εβ (ZBD (for spur gears) ndash 1)
233 Zone factor ZM The mid-zone factor ZM accounts for the influence of the contact stress at the mid point of the flank and applies to spi-ral bevel gears
εminusminus
εminusminus
αβ=
αα btm2
2vb2
2vabtm2
1vb2val
2v1vvtbmM
pddpdd
ddtancos2Z
This factor is the product of ZH and ZM-B in ISO 10300 with the condition that the heightwise crowning is sufficient to move the peak load towards the midpoint
234 Inner contact point For cylindrical or bevel gears with very low number of teeth the inner contact point (A) may be close to the base circle In order to avoid a wear edge near A it is required to have suitable tip relief on the wheel
24 Elasticity Factor ZE The elasticity factor ZE accounts for the influence of the material properties as modulus of elasticity and Poissonrsquos ratio on the contact stresses
For steel against steel ZE = 1898
25 Contact Ratio Factor Zε The contact ratio factor Zε accounts for the influence of the transverse contact ratio εα and the overlap ratio εβ on the contact stresses
αε1Z =ε for 1εβ ge
( )α
ββ
αε ε
εε1
3ε4
Z +minusminus
= for εβ lt 1
26 Helix Angle Factor Zβ The helix angle factor Zβ accounts for the influence of helix angle (independent of its influence on Zε) on the surface du-rability
cosβZβ =
27 Bevel Gear Factor ZK The bevel gear factor accounts for the difference between the real Hertzian stresses in spiral bevel gears and the contact stresses assumed responsible for surface fatigue (pitting) ZK adjusts the contact stresses in such a way that the same per-missible stresses as for cylindrical gears may apply
The following may be used ZK = 080
28 Values of Endurance Limit σHlim and Static Strength 5H10σ 3H10σ
σHlim is the limit of contact stress that may be sustained for 5middot107 cycles without the occurrence of progressive pitting
For most materials 5middot107 cycles are considered to be the be-ginning of the endurance strength range or lower knee of the σ-N curve (See also Life Factor ZN) However for nitrided steels 2middot106 apply
For this purpose pitting is defined by
bull for not surface hardened gears pitted area ge 2 of total active flank area
bull for surface hardened gears pitted area ge 05 of total active flank area or ge 4 of one particular tooth flank area 510Hσ and 310Hσ are the contact stresses which the given
material can withstand for 105 respectively 103 cycles without subsurface yielding or flank damages as pitting spalling or case crushing when adequate case depth applies
The following listed values for σHlim 510Hσ and 310Hσ may only be used for materials subjected to a quality control as
Classification Notes- No 412 19 May 2003
DET NORSKE VERITAS
the one referred to in the rules Results of approved fatigue tests may also be used as the basis for establishing these values The defined survival probability is 99
σHlim σH105 σH10
3
Alloyed case hardened steels (surface hardness 58-63 HRC) - of specially approved high grade - of normal grade
1650 1500
2500 2400
3100 3100
Nitrided steel of approved grade gas nitrided (surface hardness 700-800 HV) 1250 13 σHlim 13 σHlim
Alloyed quenched and tempered steel bath or gas nitrided (surface hardness 500-700 HV)
1000
13 σHlim
13 σHlim
Alloyed flame or induction hardened steel (surface hardness 500-650 HV) 075 HV + 750 16 σHlim 45 HV
Alloyed quenched and tempered steel 14 HV + 350 16 σHlim 45 HV
Carbon steel 15 HV + 250 16 σHlim 16 σHlim
These values refer to forged or hot rolled steel For cast steel the values for σHlim are to be reduced by 15
29 Life Factor ZN The life factor ZN takes account of a higher permissible contact stress if only limited life (number of cycles NL) is demanded or lower permissible contact stress if very high number of cycles apply
If this is not documented by approved fatigue tests the fol-lowing method may be used
For all steels except nitrided
7L 105N sdotge ZN = 1 or
01570
L
7
N N105Z
sdot=
Ie ZN = 092 for 1010 cycles
The ZN = 1 from 5middot107 on may only be used when the mate-rial cleanliness is of approved high grade (see Rules Pt4 Ch2) and the lubrication is optimised by a specially approved filtering process
105 lt NL lt 5middot107 510N
logZ037
L
7
N N105Z
sdot=
NL = 105 WXRVLHlim
WstX10H1010NN ZZZZZσ
ZZσZZ
55
5=
103 lt NL lt 105 )Z(Zlog05
L
5
10NN
5N103N10
5N10ZZ
==
3L 10N le
WXRVLlimH
Wst10X10H10NN ZZZZZ
ZZZZ
33
3σ
σ==
(but not less than ZN105)
For nitrided steels
6L 102N sdotge ZN = 1 or
00980
L
6
N N102Z
sdot=
Ie ZN = 092 for 1010 cycles
The ZN = 1 from 2middot106 on may only be used when the mate-rial cleanliness is of approved high grade (see Rules Pt4 Ch2) and the lubrication is optimised by a specially approved filtering process
105 lt NL lt 2middot106 510N
Zlog76860
L
6
N N102Z
sdot=
5L 10N le
XWRVL
10XWst10NN ZZZZZ
ZZ13ZZ
55 ==
Note that when no index indicating number of cycles is used the factors are valid for 5middot107 (respectively 2middot106 for nitrid-ing) cycles
210 Influence Factors on Lubrication Film ZL ZV and ZR The lubricant factor ZL accounts for the influence of the type of lubricant and its viscosity the speed factor ZV ac-counts for the influence of the pitch line velocity and the roughness factor ZR accounts for influence of the surface roughness on the surface endurance capacity
The following methods may be applied in connection with the endurance limit
20 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
Surface hardened steels Not surface hardened steels
ZL ( )24013421
360910ν+
+
( )24013421
680830ν+
+
ZV ( )v3280
140930+
+ ( )v3280300850
++
ZR
080
ZrelR3
150
ZrelR3
where
ν40 = Kinematic oil viscosity at 40ordmC (mm2s) For case hardened steels the influence of a high bulk temperature (see 4 Scuffing) should be con-sidered Eg bulk temperatures in excess of 120ordmC for long periods may cause reduced flank surface endurance limits
For values of ν40 gt 500 use ν40 = 500
RZrel = The mean roughness between pinion and wheel (after running in) relative to an equivalent radius of curvature at the pitch point ρc = 10mm
RZrel
= ( ) 31
cZ2Z1 ρ
10RR50
+
RZ = Mean peak to valley roughness (microm) (DIN defini-tion) (roughly RZ = 6 Ra)
For 5
L 10N le ZL ZV ZR = 10
211 Work Hardening Factor ZW The work hardening factor ZW accounts for the increase of surface durability of a soft steel gear when meshing the soft steel gear with a surface hardened or substantially harder gear with a smooth surface
The following approximation may be used for the endurance limit
Surface hardened steel against not surface hardened steel 150
ZeqW R
31700
130HB21Z
minus
minus=
where HB = the Brinell hardness of the soft member For HB gt 470 use HB = 470 For HB lt 130 use HB = 130
RZeq = equivalent roughness
033
c40
066
ZS
ZHZHZEQ ρvν
15000RRRR
=
If RZeq gt 16 then use RZeq = 16
If RZeq lt 15 then use RZeq = 15
where
RZH = surface roughness of the hard member before run in
RZS = surface roughness of the soft member before run in
ν40 = see 210
If values of ZW lt 1 are evaluated ZW = 1 should be used for flank endurance However the low value for ZW may indi-cate a potential wear problem
Through hardened pinion against softer wheel
( )
minussdotminus+= 000829
HBHB0008981u1Z
2
1W
For 21HBHB
2
1 le use ZW = 1
For 71HBHB
2
1 gt use 17HBHB
2
1 =
For u gt 20 use u = 20
For static strength (lt 105 cycles)
Surface hardened against not surface hardened
ZWst = 105
Through hardened pinion against softer wheel
ZWst = 1
212 Size Factor ZX The size factor accounts for statistics indicating that the stress levels at which fatigue damage occurs decrease with an increase of component size as a consequence of the influ-ence on subsurface defects combined with small stress gradi-ents and of the influence of size on material quality
ZX may be taken unity provided that subsurface fatigue for surface hardened pinions and wheels is considered eg as in the following subsection 213
213 Subsurface Fatigue This is only applicable to surface hardened pinions and wheels The main objective is to have a subsurface safety against fatigue (endurance limit) or deformation (static strength) which is at least as high as the safety SH required for the surface The following method may be used as an approximation unless otherwise documented
The high cycle fatigue (gt3106 cycles) is assumed to mainly depend on the orthogonal shear stresses Static strength (lt103 cycles) is assumed to depend mainly on equivalent
Classification Notes- No 412 21 May 2003
DET NORSKE VERITAS
stresses (von Mises) Both are influenced by residual stresses but this is only considered roughly and empirically
The subsurface working stresses at depths inside the peak of the orthogonal shear stresses respectively the equivalent stresses are only dependent on the (real) Hertzian stresses Surface related conditions as expressed by ZL ZV and ZR are assumed to have a negligible influence
The real Hertzian stresses σHR are determined as
For helical gears with εβ gt 1
σHR = σH
For helical gears with εβ lt 1 and spur gears
ε
α
ββ ε
ε+εminus
sdotσ=σZ
1
HHR
For bevel gears
KHHR Z
1σσ sdot=
The necessary hardness HV is given as a function of the net depth tz (net = after grinding or hard metal hobbing and per-pendicular to the flank)
The coordinates tz and HV are to be compared with the de-sign specification such as
bull for flame and induction hardening tHVmin HVmin bull for nitriding t400min HV = 400 bull for case hardening t550min HV = 550 t400min HV = 400
and t300min HV = 300 (the latter only if the core hardness lt 300 If the core hardness gt 400 the t400 is to be re-placed by a fictive t400 = 16 t550)
In addition the specified surface hardness is not to be less than the max necessary hardness (at tz = 05aH) This applies to all hardening methods
For high cycle fatigue (gt3 106 cycles) the following applies
sdot+
minussdotsdotsdot= o90
05azt
05azt
cosSσ04HV
H
HHHR
applicable to 05a
zt
Hge
For 05at
H
z lt the value for 05at
H
z = applies
56300ρSσ12a cHHR
Hsdotsdot
sdot=
Where aH is half the hertzian contact width multiplied by an empirical factor of 12 that takes into account the possible influence of reduced compressive residual stresses (or even tensile residual stresses) on the local fatigue strength
If any of the specified hardness depths including the surface hardness is below the curve described by HV = f (tz) the actual safety factor against subsurface fatigue is determined as follows
reduce SH stepwise in the formula for HV and aH until all specified hardness depths and surface hardness balance with the corrected curve The safety factor obtained through this method is the safety against subsurface fatigue
For static strength (lt103 cycles) the following applies
sdot+
minussdotsdotsdot= o90
07at
06a
zt
cosSσ019HV
Hst
z
HstHHR
applicable to 06at
Hst
z ge
56300ρSσa cHHR
Hstsdotsdot
=
In the case of insufficient specified hardness depths the same procedure for determination of the actual safety factor as above applies
For limited life fatigue (103 lt cycles lt 3106 )
For this purpose it is necessary to extend the correction of safety factors to include also higher values than required Ie in the case of more than sufficient hardness and depths the safety factor in the formulae for both high cycle fatigue and static strength are to be increased until necessary and speci-fied values balance
The actual safety factor for a given number of cycles N be-tween 103 and 3106 is found by linear interpolation in a dou-ble logarithmic diagram
minussdotminus
= sdot logN3477
logSlogSlogS
310H6103HHN
36 10H103H Slog86281Slog86280 sdot+sdot sdot
22 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
3 Calculation of Tooth Strength
31 Scope and General Remarks Part 3 include the calculation of tooth root strength as limited by tooth root cracking (surface or subsurface initiated) and yielding
For rim thickness sR ge 35middotmn the strength is calculated by means of 32 ndash 313 For cylindrical gears the calculation is based on the assumption that the highest tooth root tensile stress arises by application of the force at the outer point of single tooth pair contact of the virtual spur gears The method has however a few limitations that are mentioned in 36
For bevel gears the calculation is based on force application at the tooth tip of the virtual cylindrical gear Subsequently the stress is converted to load application at the mid point of the flank due to the heightwise crowning
Bevel gears may also be calculated with the program BECAL In that case KA and Kv are to be included in the applied tooth force but not KFβ and KFα
In case of a thin annulus or a thin gear rim etc radial crack-ing can occur rather than tangential cracking (from root fillet to root fillet) Cracking can also start from the compression fillet rather than the tension fillet For rim thickness sR lt 35middotmn a special calculation procedure is given in 315 and 316 and a simplified procedure in 314
A tooth breakage is often the end of the life of a gear trans-mission Therefore a high safety SF against breakage is re-quired
It should be noted that this part 3 does not cover fractures caused by
bull oil holes in the tooth root space bull wear steps on the flank bull flank surface distress such as pits spalls or grey staining
Especially the latter is known to cause oblique fractures starting from the active flank predominately in spiral bevel gears but also sometimes in cylindrical gears
Specific calculation methods for these purposes are not given here but are under consideration for future revisions Thus depending on experience with similar gear designs limita-tions other than those outlined in part 3 may be applied
32 Tooth Root Stresses The local tooth root stress is defined as the max principal stress in the tooth root caused by application of the tooth force Ie the stress ratio R = 0 Other stress ratios such as for eg idler gears (R asymp -12) shrunk on gear rims (R gt 0) etc are considered by correcting the permissible stress level
321 Local tooth root stress The local tooth root stress for pinion and wheel may be as-sessed by strain gauge measurements or FE calculations or similar For both measurements and calculations all details are to be agreed in advance
Normally the stresses for pinion and wheel are calculated as
Cylindrical gears
FαFβvγAβSFn
tF K K K K K Y Y Y
m bFσ =
where
YF = Tooth form factor (see 33)
YS = Stress correction factor (see 34)
Yβ = Helix angle factor (see 36)
Ft KA Kγ Kv KFβ KFα see 15 ndash 110
b see 13
Bevel gears
FαFβvγASaFamn
mtF K K K K K Y Y Y
m bFσ ε=
where
YFa = Tooth form factor see 33
YSa = Stress correction factor see 34
Yε = Contact ratio factor see 35
Fmt KA etc see 15 ndash 110
b see 13
322 Permissible tooth root stress The permissible local tooth root stress for pinion respectively wheel for a given number of cycles N is
CXRrelTrelTF
NMFEFP YYYY
SYY
δσ
=σ
Note that all these factors YM etc are applicable to 3middot106 cycles when used in this formula for σFP The influence of other number of cycles on these factors is covered by the calculation of YN
where
σFE = Local tooth root bending endurance limit of reference test gear (see 37)
YM = Mean stress influence factor which accounts for other loads than constant load direction eg idler gears temporary change of load di-rection pre-stress due to shrinkage etc (see 38)
YN = Life factor for tooth root stresses related to reference test gear dimensions (see 39)
SF = Required safety factor according to the rules
YδrelT = Relative notch sensitivity factor of the gear to be determined related to the reference test gear (see 310)
YRrelT = Relative (root fillet) surface condition factor of the gear to be determined related to the reference test gear (see 311)
Classification Notes- No 412 23 May 2003
DET NORSKE VERITAS
YX = Size factor (see 312)
YC = Case depth factor (see 313)
33 Tooth Form Factors YF YFa The tooth form factors YF and YFa take into account the in-fluence of the tooth form on the nominal bending stress
YF applies to load application at the outer point of single tooth pair contact of the virtual spur gear pair and is used for cylindrical gears
YFa applies to load application at the tooth tip and is used for bevel gears
Both YF and YFa are based on the distance between the con-tact points of the 30˚-tangents at the root fillet of the tooth profile for external gears respectively 60˚ tangents for inter-nal gears
Figure 31 External tooth in normal section
Figure 32 Internal tooth in normal section
Definitions
n
2
n
Fn
enFn
Fe
F
α cosms
α cosmh6
Y
=
n
2
n
Fn
anFn
Fa
Fa
α cosms
α cosmh6
Y
=
In the case of helical gears YF and YFa are determined in the normal section ie for a virtual number of teeth
YFa differs from YF by the bending moment arm hFa and αFan and can be determined by the same procedure as YF with exception of hFe and αFan For hFa and αFan all indices e will change to a (tip)
The following formulae apply to cylindrical gears but may also be used for bevel gears when replacing
mn with mnm
zn with zvn
αt with αvt
β with βm
with undercut without undercut
Fig 33 Dimensions and basic rack profile of the teeth (finished profile)
Tool and basic rack data such as hfP ρfp and spr etc are re-ferred to mn ie dimensionless
331 Determination of parameters
( )n
n
prnfPnfP m
α cossαsin 1ρ
αtan h4πE
minusminusminusminus=
For external gears fPfP ρρ =
For internal gears ( )0z
195fPfP0
fPfP 10363156ρhxρρ
sdotminus+
+=
where
z0 = number of teeth of pinion cutter
x0 = addendum modification coefficient of pinion cutter
hfP = addendum of pinion cutter
ρfP = tip radius of pinion cutter
xhρG fpfp +minus=
24 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
τmE
2π
z2H
nnminus
minus=
with
3πτ = for external gears
6πτ = for internal gears
Htan zG2
nminusϑ=ϑ
(to be solved iteratively suitable start value6π
=ϑ for exter-
nal gears and 3π for internal gears)
a) Tooth root chord sFn For external gears
minus
ϑ+
ϑminus= ρ
cosG3
3πsinz
ms
fpnn
Fn
For bevel gears with a tooth thickness modification
xsm affects mainly sFn but also hFe and αFen The total influence of xsm on YFa Ysa can be approximated by only adding 2 xsm to sFn mn
For internal gears
minus
ϑ+
ϑminus= ρ
cosG
6πsinz
ms
fPnn
Fn
b) Root fillet radius ρF at 30ordm tangent
( )G2coszcosG2ρ
mρ
2n
2
fpn
F
minusϑϑ+=
c) Determination of bending moment arm hF dn = zn mn
b2α
αn βcosε
ε =
dan = dn + 2 ha
pbn = π mn cos αn
dbn = dn cos αn
( )4
d1εpzz
2dd
zz2d
2bn
2
αnbn
2bn
2an
en +
minusminus
minus=
en
bnen d
dcos arcα =
ennnn
e α invα invα x tan 22π
z1
minus+
+=γ
αFen = αen ndash γe
For external gears
( )
minus=
n
enFenee
n
Fe
mdαtan sin γ γcos
21
mh
]ρcos
G3πcosz fpn +
ϑminus
ϑminusminus
For internal gears
( ) minus
sdotsdotminus=
n
enFenee
n
Fe
mdα tanγ sinγ cos
21
mh
minus
ϑminus
ϑminussdot ρ
cosG3
6πcosz fPn
332 Gearing with εαn gt 2 For deep tooth form gearing ( )25ε2 αn lele produced with a verified grade of accuracy of 4 or better and with applied profile modification to obtain a trapezoidal load distribution along the path of contact the YF may be corrected by the factor YDT as
250ε205for 0666ε2366Y αnαnDT leleminus=
205εfor 10Y αnDT lt=
34 Stress Correction Factors YS YSa The stress correction factors YS and YSa take into account the conversion of the nominal bending stress to the local tooth root stress Thereby YS and YSa cover the stress increasing effect of the notch (fillet) and the fact that not only bending stresses arise at the root A part of the local stress is inde-pendent of the bending moment arm This part increases the more the decisive point of load application approaches the critical tooth root section
Therefore in addition to its dependence on the notch radius the stress correction is also dependent on the position of the load application ie the size of the bending moment arm
YS applies to the load application at the outer point of single tooth pair contact YSa to the load application at tooth tip
YS can be determined as follows
( )
+
+=L23121
1
sS qL01312Y
where Fe
Fn
hsL = and
F
Fns ρ2
sq = (see 33)
YSa can be calculated by replacing hFe with hFa in the above formulae
Note a) Range of validity 1 lt qs lt 8
In case of sharper root radii (ie produced with tools having too sharp tip radii) YS resp YSa must be specially considered
b) In case of grinding notches (due to insufficient protuberance of the hob) YS resp YSa can rise considerably and must be multiplied with
Classification Notes- No 412 25 May 2003
DET NORSKE VERITAS
g
g
ρt
0613
13
minus
where
tg = depth of the grinding notch
ρg = radius of the grinding notch
c) The formulae for YS resp YSa are only valid for αn = 20˚ However the same formulae can be used as a safe approximation for other pressure angles
35 Contact Ratio Factor Yε The contact ratio factor Yε covers the conversion from load application at the tooth tip to the load application at the mid point of the flank (heightwise) for bevel gears
The following may be used
Yε = 0625
36 Helix Angle Factor Yβ The helix angle factor Yβ takes into account the difference between the helical gear and the virtual spur gear in the nor-mal section on which the calculation is based in the first step In this way it is accounted for that the conditions for tooth root stresses are more favourable because the lines of contact are sloping over the flank
The following may be used (β input in degrees)
Yβ = 1 ndash εβ β120
When εβ gt 1 use εβ = 1 and when β gt 30deg use β = 30deg in the formula
However the above equation for Yβ may only be used for gears with β gt 25deg if adequate tip relief is applied to both pinion and wheel (adequate = at least 05 middot Ceff see 432)
37 Values of Endurance Limit σFE σFE is the local tooth root stress (max principal) which the material can endure permanently with 99 survival prob-ability 3106 load cycles is regarded as the beginning of the endurance limit or the lower knee of the σ ndash N curve σFE is defined as the unidirectional pulsating stress with a minimum stress of zero (disregarding residual stresses due to heat treatment) Other stress conditions such as alternating or pre-stressed etc are covered by the conversion factor YM
σFE can be found by pulsating tests or gear running tests for any material in any condition If the approval of the gear is to be based on the results of such tests all details on the testing conditions have to be approved by the Society Fur-ther the tests may have to be made under the Societys su-pervision
If no fatigue tests are available the following listed values for σFE may be used for materials subjected to a quality con-trol as the one referred to in the rules
σFE
Alloyed case hardened steels 1) (fillet surface hardness 58 ndash 63 HRC)
bull of specially approved high grade
1050
bull of normal grade
minus CrNiMo steels with approved process
1000
minus CrNi and CrNiMo steels generally 920 minus MnCr steels generally 850
Nitriding steel of approved grade quenched tempered and gas nitrided (surface hardness 700 ndash 800 HV)
840
Alloyed quenched and tempered steel bath or gas nitrided (surface hardness 500 ndash 700 HV)
720
Alloyed quenched and tempered steel flame or induction hardened 2) (incl entire root fillet) (fillet surface hardness 500 ndash 650 HV)
07 HV + 300
Alloyed quenched and tempered steel flame or induction hardened (excl entire root fillet) (σB = uts of base material)
025 σB + 125
Alloyed quenched and tempered steel 04 σB + 200
Carbon steel 025 σB + 250
Note All numbers given above are valid for separate forgings and for blanks cut from bars forged according to a qualified procedure see Pt 4 Ch 2 Sec 3 For rolled steel the values are to be reduced with 10 For blanks cut from forged bars that are not qualified as mentioned above the values are to be reduced with 20 For cast steel reduce with 40
1) These values are valid for a root radius
bull being unground If however any grinding is made in the root fillet area in such a way that the residual stresses may be affected σFE is to be reduced by 20 (If the grinding also leaves a notch see 34)
bull with fillet surface hardness 58 ndash 63 HRC In case of lower surface hardness than 58 HRC σFE is to be reduced with 20(58 ndash HRC) where HRC is the detected hardness (This may lead to a permissible tooth root stress that varies along the facewidth If so the actual tooth root stresses may also be considered along facewidth)
bull not being shot peened In case of approved shot peening σFE may be increased by 200 for gears where σFE is reduced by 20 due to root grinding Otherwise σFE may be increased by 100 for
6nm le and 100 ndash 5 (mn - 6) for mn gt 6 However the possible adverse influence on the flanks regarding grey staining should be considered and if necessary the flanks should be masked
2) The fillet is not to be ground after surface hardening Regarding possible root grinding see 1)
26 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
38 Mean stress influence Factor YM The mean stress influence factor YM takes into account the influence of other working stress conditions than pure pulsa-tions (R = 0) such as eg load reversals idler gears planets and shrink-fitted gears
YM (YMst) are defined as the ratio between the endurance (or static) strength with a stress ratio R ne 0 and the endurance (or static) strength with R = 0
YM and YMst apply only to a calculation method that assesses the positive (tensile) stresses and is therefore suitable for comparison between the calculated (positive) working stress σF and the permissible stress σFP calculated with YM or YMst
For thin rings (annulus) in epicyclic gears where the com-pression fillet may be decisive special considerations apply see 316
The following method may be used within a stress ratio ndash12 lt R lt 05
381 For idlers planets and PTO with ice class
M1M1R1
1Yor Y MstM
+minus
minus=
where
R = stress ratio = min stress divided by max stress
For designs with the same force applied on both forward- and back-flank R may be assumed to ndash 12
For designs with considerably different forces on forward- and back-flank such as eg a marine propulsion wheel with a power take off pinion R may be assessed as
branchmain theoffacewidth unit per forcepto offacewidt unit per force21minus
For a power take off (PTO) with ice class see 161 c
M considers the mean stress influence on the endurance (or static) strength amplitudes
M is defined as the reduction of the endurance strength am-plitude for a certain increase of the mean stress divided by that increase of the mean stress
Following M values may be used
Endurance limit
Static strength
Case hardened 08 ndash 015 Ys 1) 07
If shot peened 04 06
Nitrided 03 03
Induction or flame hardened
04 06
Not surface hardened steel 03 05
Cast steels 04 06 1)For bevel gears use Ys=2 for determination of M
The listed M values for the endurance limit are independent of the fillet shape (Ys) except for case hardening In princi-ple there is a dependency but wide variations usually only occur for case hardening eg smooth semicircular fillets versus grinding notches
382 For gears with periodical change of rotational direction For case hardened gears with full load applied periodically in both directions such as side thrusters the same formula for YM as for idlers (with R = ndash 12) may be used together with the M values for endurance limit This simplified approach is valid when the number of changes of direction exceeds 100 and the total number of load cycles exceeds 3middot106
For gears of other materials YM will normally be higher than for a pure idler provided the number of changes of direction is below 3middot106 A linear interpolation in a diagram with loga-rithmic number of changes of direction may be used ie from YM = 09 with one change to YM (idler) for 3middot106 changes This is applicable to YM for endurance limit For static strength use YM as for idlers
For gears with occasional full load in reversed direction such as the main wheel in a reversing gear box YM = 09 may be used
383 For gears with shrinkage stresses and unidirectional load For endurance strength
FE
fitM σ
σM1
M21Y+
minus=
σFE is the endurance limit for R = 0
For static strength YMst = 1 and σfit accounted for in 39b
σfit is the shrinkage stress in the fillet (30˚ tangent) and may be found by multiplying the nominal tangential (hoop) stress with a stress concentration factor
n
Ffit m
ρ215scf minus=
384 For shrink-fitted idlers and planets When combined conditions apply such as idlers with shrink-age stresses the design factor for endurance strength is
( ) ( ) FE
fitM σ
σR1M1
M2
M1M1R1
1Yminussdot+
minus
+minus
minus=
Symbols as above but note that the stress ratio R in this par-ticular connection should disregard the influence of σfit ie R normally equal ndash 12
For static strength
M1M1R1
1YMst
+minus
minus=
The effect of σfit is accounted for in 39 b
Classification Notes- No 412 27 May 2003
DET NORSKE VERITAS
385 Additional requirements for peak loads The total stress range (σmax ndash σmin) in a tooth root fillet is not to exceed
F
y
Sσ225
for not surface hardened fillets
FSHV5 for surface hardened fillets
39 Life Factor YN The life factor YN takes into account that in the case of limited life (number of cycles) a higher tooth root stress can be permitted and that lower stresses may apply for very high number of cycles
Decisive for the strength at limited life is the σ ndash N ndash curve of the respective material for given hardening module fillet radius roughness in the tooth root etc Ie the factors YδrelT YRelT YX and YM have an influence on YN
If no σ ndash N ndash curve for the actual material and hardening etc is available the following method may be used
Determination of the σ ndash N ndash curve
a) Calculate the permissible stress σFP for the beginning of the endurance limit (3middot106 cycles) including the influ-ence of all relevant factors as SF YδrelT YRelT YX YM and YC ie σFP = σFE middotYM middotYδrelT middotYRelT middotYX middotYC SF
b) Calculate the permissible laquostaticraquo stress (le 103 load cy-cles) including the influence of all relevant factors as SFst YδrelTst YMst and YCst ( )fitσCstYδrelTstYMstYFstσ
FstS1
FPstσ minussdotsdotsdot=
where σFst is the local tooth root stress which the material can resist without cracking (surface hardened materials) or unacceptable deformation (not surface hardened materials) with 99 survival probability
σFst
Alloyed case hardened steel 1) 2300
Nitriding steel quenched tempered and gas nitrided (surface hardness 700 ndash 800 HV)
1250
Alloyed quenched and tempered steel bath or gas nitrided (surface hardness 500 ndash 700 HV)
1050
Alloyed quenched and tempered steel flame or induction hardened (fillet surface hardness 500 ndash 650 HV)
18 HV + 800
Steel with not surface hardened fillets the smaller value of 2)
18 σB or 225 σy
1) This is valid for a fillet surface hardness of 58 ndash 63 HRC In case of lower fillet surface hardness than 58 HRC σFst is to be reduced with 30(58 ndash HRC) where HRC is the actual hardness Shot peening or grinding notches are not considered to have any significant influence on σFst 2) Actual stresses exceeding the yield point (σy or σ02) will alter the residual stresses locally in the ldquotensionrdquo fillet re-spectively ldquocompressionrdquo fillet This is only to be utilised for gears that are not later loaded with a high number of cycles at lower loads that could cause fatigue in the ldquocompressionrdquo fillet
c) Calculate YN as
NL gt 3middot106
YN = 1 or 001
L
6
N N103Y
sdot= ie Yn = 092 for 1010
The YN = 1 from 3middot106 on may only be used when special material cleanness applies see rules Pt4 Ch2
103ltNLlt3middot106exp
L
6
N N103Y
sdot=
cycles6103forσcycles310forσlog02876exp
FP
FPst
sdot=
NL lt 103 cycles6103forσcycles310forσ
NYFP
FPst
sdot=
or simply use σFPst as mentioned in b) directly
Guidance on number of load cycles NL for various applica-tions
bull For propulsion purpose normally NL = 1010 at full load (yachts etc may have lower values)
bull For auxiliary gears driving generators that normally operate with 70-90 of rated power NL = 108 with rated power may be applied
28 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
310 Relative Notch Sensitivity Factor YδrelT The dynamic (respectively static) relative notch sensitivity factor YδrelT (YδrelTst) indicate to which extent the theoreti-cally concentrated stress lies above the endurance limits (re-spectively static strengths) in the case of fatigue (respectively overload) breakage
YδrelT is a function of the material and the relative stress gra-dient It differs for static strength and endurance limit
The following method may be used
For endurance limit
for not surface hardened fillets
( )
024
s024
relT σ103133q21σ1012201351
Ysdotsdotminus
+sdotsdotminus+= minus
minus
δ
for all surface hardened fillets except nitrided
( )
106q21002451
Y srelT
++=δ
for nitrided fillets
( )
1347q2101421
Y srelT
++=δ
For static strength
for not surface hardened fillets1)
( )( )( )025
02
02502s
relTst σ3000821σ300 1Y0821Y
+
minus+=δ
for surface hardened fillets except nitrided
YδrelTst = 044 YS + 012
for nitrided fillets
YδrelTst = 06 + 02 YS 1) These values are only valid if the local stresses do not
exceed the yield point and thereby alter the residual stress level See also 39b footnote 2
311 Relative Surface Condition Factor YRrelT The relative surface condition factor YRrelT takes into ac-count the dependence of the tooth root strength on the sur-face condition in the tooth root fillet mainly the dependence on the peak to valley surface roughness
YRrelT differs for endurance limit and static strength
The following method may be used
For endurance limit
YRrelT = 1675 ndash 053 (Ry + 1)01 for surface hardened steels and alloyed quenched and tempered steels except nitrided
YRrelT = 53 ndash 42 (Ry + 1)001 for carbon steels
YδrelT = 43 ndash 326 (Ry + 1)0005
for nitrided steels
For static strength
YRrelTst = 1 for all Ry and all materials
For a fillet without any longitudinal machining trace Ry asymp Rz
312 Size Factor YX The size factor YX takes into account the decrease of the strength with increasing size YX differs for endurance limit and static strength
The following may be used
For endurance limit
YX = 1 for mn le 5 generally
YX = 103 ndash 0006 mn for 5 lt mn lt 30
YX = 085 for mn ge 30
for not surface hard-ened steels
YX = 105 ndash 001 mn for 5 lt mn ge 25
YX = 08 for mn ge 25
for surface hardened steels
For static strength
YXst = 1 for all mn and all materials
313 Case Depth Factor YC The case depth factor YC takes into account the influence of hardening depth on tooth root strength
YC applies only to surface hardened tooth roots and is dif-ferent for endurance limit and static strength
In case of insufficient hardening depth fatigue cracks can develop in the transition zone between the hardened layer and the core For static strength yielding shall not occur in the transition zone as this would alter the surface residual stresses and therewith also the fatigue strength
The major parameters are case depth stress gradient permis-sible surface respectively subsurface stresses and subsurface residual stresses
The following simplified method for YC may be used
YC consists of a ratio between permissible subsurface stress (incl influence of expected residual stresses) and permissible surface stress This ratio is multiplied with a bracket con-taining the influence of case depth and stress gradient (The empirical numbers in the bracket are based on a high number of teeth and are somewhat on the safe side for low number of teeth)
YC and YCst may be calculated as given below but calculated values above 10 are to be put equal 10
For endurance limit
+
+=nFFE
C m02ρt31
σconstY
Classification Notes- No 412 29 May 2003
DET NORSKE VERITAS
For static strength
+
+=nFFst
Cst m02ρt31
σconstY
where const and t are connected as
Hardening process
t = endurance limit const =
static strength const =
t550 640 1900
t400 500 1200 Case hard-ening
t300 380 800
Nitriding t400 500 1200
Induction- or flame hardening
tHVmin 11 HVmin 25 HVmin
For symbols see 213
In addition to these requirements to minimum case depths for endurance limit some upper limitations apply to case hard-ened gears
The max depth to 550 HV should not exceed
1) 13 of the top land thickness san unless adequate tip relief is applied (see 110)
2) 025 mn If this is exceeded the following applies addi-tionally in connection with endurance limit
minusminus= 025
mt
1Yn
max550C
314 Thin rim factor YB Where the rim thickness is not sufficient to provide full sup-port for the tooth root the location of a bending failure may be through the gear rim rather than from root fillet to root fillet
YB is not a factor used to convert calculated root stresses at the 30deg tangent to actual stresses in a thin rim tension fillet Actually the compression fillet can be more susceptible to fatigue
YB is a simplified empirical factor used to de-rate thin rim gears (external as well as internal) when no detailed calcula-tion of stresses in both tension and compression fillets are available
Figure 34 Examples on thin rims
YB is applicable in the range 175 lt sRmn lt 35
YB = 115 middot ln (8324 middot mnsR)
(for sRmn ge 35 YB = 1)
(for sRmn le 175 use 315)
σF as calculated in 321 is to be multiplied with YB when sRmn lt 35 Thus YB is used for both high and low cycle fatigue
Note This method is considered to be on the safe side for external gear rims However for internal gear rims without any flange or web stiffeners the method may not be on the safe side and it is advised to check with the method in 315316
315 Stresses in Thin Rims For rim thickness sR lt 35 mn the safety against rim cracking has to be checked
The following method may be used
3151 General The stresses in the standardised 30ordm tangent section tension side are slightly reduced due to decreasing stress correction factor with decreasing relative rim thickness sRmn On the other hand during the complete stress cycle of that fillet a certain amount of compression stresses are also introduced The complete stress range remains approximately constant Therefore the standardised calculation of stresses at the 30ordm tangent may be retained for thin rims as one of the necessary criteria
The maximum stress range for thin rims usually occurs at the 60ordm ndash 80ordm tangents both for laquotensionraquo and laquocompressionraquo side The following method assumes the 75ordm tangent to be the decisive Therefore in addition to the am criterion ap-plied at the 30ordm tangent it is necessary to evaluate the max and min stresses at the 75ordm tangent for both laquotensionraquo (loaded flank) fillet and laquocompressionraquo (back-flank) fillet For this purpose the whole stress cycle of each fillet should be considered but usually the following simplification is justified
Figure 35 Nomenclature of fillets
Index laquoTraquo means laquotensileraquo fillet laquoCraquo means laquocompressionraquo fillet
σFTmin and σFCmax are determined on basis of the nominal rim stresses times the stress concentration factor Y75
σFCmin and σFTmax are determined on basis of superposition of nominal rim stresses times Y75 plus the tooth bending stresses at 75ordm tangent
30 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
3152 Stress concentration factors at the 75ordm tangents The nominal rim stress consists of bending stresses due to local bending moments tangential stresses due to the tan-gential force Ft and radial shear stresses due to Fr
The major influence is given by the bending stresses The influence of the tangential stresses is minor and even though its stress concentration factor is slightly higher than for bending it is considered to be safe enough when the sum of these nominal stresses are combined with the stress concen-tration factor for bending The influence of the radial shear stress is neglected
The stress concentration factor relating nominal rim stresses to local fillet stresses at the 75ordm tangent may be calculated as
75n
R75
n
R
75
ρmsρ185
ms3
Y+
sdot=
where ρ75 is the root radius at the 75ordm tangent ref to mn Usually ρ75 is closed to the tool radius ρao and
ρ75 = ρao
is a safe approximation compensating for the am simplifi-cations to the laquounsaferaquo side
The tooth root stresses of the loaded tooth are decreasing with decreasing relative rim thickness approximately with the empirical factor
15
R
ncorr s
m311Y
minus=
3153 Nominal rim stresses The bending moment applied to the rim consists of a part of the tooth tilt moment Ft (hF + 05 sR) and the bending caused by the radial force Fr
The sectional modulus (first moment of area) which is used for determination of the nominal bending stresses is not nec-essarily the same for the 2 am bending moments If flanges webs etc outside the toothed section contribute to stiffening the rim against various deflections the influence of these stiffeners should be considered Eg an end flange will have an almost negligible influence on the effective sectional modulus for the stresses due to tooth tilt as the deflection caused by the tilt moment is rather small and would not much involve the flange On the other hand the radial forces as for instance from the meshes in an annulus would cause considerable radial deflections that the flange might restrict to a substantial amount When considering the stiff-ening of such flanges or webs on basis of simplified models it is advised to use an effective rim thickness sR = sR + 02 mn for the first moment of area of the rim (toothed part) cross section
For a high number of rim teeth it may be assumed that the rim bending moments in the fillets adjacent to the loaded tooth are of the same magnitude as right under the applied
force This assumption is reasonable for an annulus but rather much on the safe side for a hollow pinion
The influence of Ft on the nominal tangential stress is simpli-fied by half of it for compressive stresses (σ1) and the other half for tensile stresses (σ2) Applying these assumptions the nominal rim stresses adjacent to the loaded tooth are
( ) ( )A2
FW
f RFW
s05hF05 t
R
r
T
RFt1 minus
ϑminus
+minus=σ
( ) ( )A2
FW
f RFW
s05hF05 t
R
r
T
RFt2 +
ϑminus
+=σ
where σ1 σ2 see Fig 35
A = minimum area of cross section (usually bR sR)
WT = the sectional modulus of rim with respect to tooth tilt moment (usually 6sb 2
rR )
WR = the sectional modulus of the rim including the influence of stiffeners as flanges webs etc (WR ge WT)
bR = the width of the rim
R = the radius of the neutral axis in the rim ie from wheel centre to midpoint of rim
( )ϑf = a function for bending moment distribution around the rim For a rim (pinion) with one mesh only the ( )ϑf at the position of load application is 024 For an annulus with 3 or more meshes ( )ϑf at the position of each load application is approx 3 planets ( )ϑf = 019 4 planets ( )ϑf = 014 5 planets ( )ϑf = 011 6 planets ( )ϑf = 009
It must be checked if the max (tensile) stress in the compres-sion fillet really occurs when the fillet is adjacent to the loaded tooth In principle the stress variation through a complete rotation should be considered and the max value used The max value is usually never less than 0 For an annulus eg the tilt moment is zero in the mid position be-tween the planet meshes whilst the bending moment due to the radial forces is half of that at the mesh but with opposite sign
If these formulae are applied to idler gears as eg planets the influence of nominal tangential stresses must be cor-rected by deleting Ft(2 A) for σ1 and using FtA for σ2 Further the influence of Fr on the nominal bending stresses is usually negligible due to the planet bearing support
3154 Root fillet stresses Determination of min and max stresses in the laquotensionraquo fillet
Classification Notes- No 412 31 May 2003
DET NORSKE VERITAS
Minimum stress
751FTmin YσK σ =
Maximum stress
corrF752 Yσ03Yσ KσFTmax +=
where
03 is an empirical factor relating the tension stresses (σF) at the 30ordm tangent to the part of the tension stresses at the 75ordm tangent which add to the rim related stresses (03 also takes into account that full superposition of nominal stresses times stress concentration factors from both laquosides of the corner filletraquo would result in too high stresses)
αβγ sdotsdotsdotsdot= FFvA KKKKKK
Determination of min and max stresses in the laquocompres-sionraquo fillet
Minimum stress
corrF751FCmin Yσ036Yσ Kσ minus=
Maximum stress
752FCmax Yσ Kσ =
where
036 is an empirical factor relating the tension stresses (σF) at the 30ordm tangent to the part of the compression stresses at the 75ordm tangent which add to the rim related stresses
For gears with reversed loads as idler gears and planets there is no distinct laquotensionraquo or laquocompressionraquo fillet The mini-mum stress σFmin is the minimum of σFTmin and σFCmin (usually the latter is decisive) The maximum stress σFTmax is the maximum of σFTmax and σFCmax (usually the former is deci-sive)
316 Permissible Stresses in Thin Rims
3161 General The safety against fatigue fracture respectively overload fracture is to be at least at the same level as for solid gears The laquoordinaryraquo criteria at the 30ordm tangent apply as given in 31 through 313
Additionally the following criteria at the 75ordm tangent may apply
3162 For gt3middot106 cycles The permissible stresses for the laquotensionraquo fillets and for the laquocompressionraquo fillets are determined by means of a relevant fatigue diagram
If the actual tooth root stress (tensile or compressive) ex-ceeds the yield strength to the material the induced residual stresses are to be taken into account
For determination of permissible stresses the following is defined
R = stress ratio ie FTmax
FTminσσ respectively
FCmax
FCmin
σσ
∆σ = stress range ie σFTmax ndash σFTmin resp σFCmax ndash σFCmin
(For idler gears and planets Fmax
FminσσR =
and ∆σ = σFmax minus σFmin)
The permissible stress range ∆σρ for the laquotensionraquo respec-tively laquocompressionraquo fillets can be calculated as
For R gt minus1 FPp σ
R1R1031
13∆σ
minus+
+=
For minus infin lt R lt minus1 FPp σ
R1R10151
13∆σ
minus+
+=
where
σFP = see 32 determined for unidirectional stresses (YM = 1)
If the yield strength σy is exceeded in either tension or com-pression residual stresses are induced This may be consid-ered by correcting the stress ratio R for the respective fillets (tension or compression)
Eg if yFCmin σσ gt (ie exceeded in compression) the
difference yFCmin σσ∆ minus= affects the stress ratio as
∆σ
σ∆σ∆σR
FCmax
y
FCmax
FCmin
+
minus=
++
=
Similarly the stress ratio in the tension fillet may require cor-rection
If the yield strength is exceeded in tension σFTmax gt σy the difference ∆ = σFTmax ndash σy affects the stress ratio as
y
FTmin
FTmax
FTmin
σ∆σ
∆σ∆σR minus=
minusminus
=
Checking for possible exceeding of the yield strength has to be made with the highest torque and with the KAP if this ex-ceeds KA
3163 Forle 103 cycles The permissible stress range ∆σp is not to exceed
For R gt minus1 FPstpst σ
R1R1051
15∆σ
minus+
+=
For ndash infin lt R lt minus1 FPstpst σ
R1R10251
15∆σ
minus+
+=
For all values of R ∆σpst is limited by
not surface hardened F
y
Sσ225
32 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
surface hardened CF
YSHV5
Definition of ∆σ and R see 3162 with particular attention to possible correction of R if the yield strength is exceeded
σFPst see 32 determined for unidirectional stresses (YM = 1) and lt 103 cycles
3164 For 103 lt cycles lt 3106 ∆σp is to be determined by linear interpolation a log-log dia-gram
∆σp at NL load cycles is
6L 103p
exp
L
6
N p ∆σN103∆σ sdot
sdot=
6
3
103p
10p
∆σ
∆σlog28760exp
sdot
=
Classification Notes- No 412 33 May 2003
DET NORSKE VERITAS
4 Calculation of Scuffing Load Capacity
41 Introduction High surface temperatures due to high loads and sliding ve-locities can cause lubricant films to break down This sei-zure or welding together of areas of tooth surface is termed scuffing
In contrast to pitting and fatigue breakage which show a dis-tinct incubation period a single short overloading can lead to a scuffing failure In the ISO-TR13989 two criteria are mentioned The method used in this Classification Note is based on the principles of the flash temperature criterion
Note Bulk temperature in excess of 120ordmC for long periods may have an adverse effect on the surface durability see 211
42 General Criteria In no point along the path of contact the local contact tem-perature may exceed the permissible temperature ie
oilS
oilSB S
ϑ+ϑminusϑ
leϑ
50SB minusϑleϑ
where
Bϑ = max contact temperature along the path of contact
maxflaMBB ϑ+ϑ=ϑ
MBϑ = bulk temperature see 434
maxflaϑ = max flash temperature along the path of con-tact see 44
Sϑ = scuffing temperature see below
oilϑ = oil temperature before it reaches the mesh (max applicable for the actual load case to be used ie normally alarm temperature except for ice classes where a max expected tem-perature applies)
SS = required safety factor according to the Rules
The scuffing temperature Sϑ may be calculated as
L2
wrelT
002
40S XFZGX
ν100112085780
sdot
sdot++=ϑ
where
XwrelT = relative welding factor
XwrelT
Through hardened steel 10
Phosphated steel 125
Copper-plated steel 150
Nitrided steel 150
Less than 10 retained austenite
115
10 ndash 20 retained austenite 10
Casehardened steel
20 ndash 30 retained austenite 085
Austenitic steel 045
FZG = load stage according to FZG-Test A8390 (Note This is the load stage where scuffing oc-curs However due to scatter in test results cal-culations are to be made with one load stage less than the specification)
XL = lubricant factor
= 10 for mineral oils
= 08 for polyalfaolefins
= 07 for non-water-soluble polyglycols
= 06 for water-soluble polyglycols
= 15 for traction fluids
= 13 for phosphate esters
ν40 = kinematic oil viscosity at 40˚C (mm2s)
Application of other test methods such as the Ryder the FZG-Ryder R46574 and the FZG L-42 Test 141195110 may be specially considered
For high speed gears with very short time of contact Sϑ may be increased as follows provided use of EP-oils
Addition to the calculated scuffing temperature Sϑ
If micros 18tc ge no addition
If micros 18tc lt add ( )cwrelT t18X18 minussdotsdot
where
ct = contact time ( micros ) which is the time needed to cross the Hertzian contact width
( ) [ ]microsu1cosβn
uσ340tb1
Hc +sdotsdot
sdot=
σH as calculated in 221
For bevel gears use vu in stead of u
34 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
43 Influence Factors
431 Coefficient of friction The following coefficient of friction may apply
L025a
005oil
02
redC ΣC
Bt X R ηρv
w0048micro minus
=
where
wBt = specific tooth load (Nmm)
vΣC = sum of tangential velocities at pitch point At pitch line velocities gt 50 ms the limiting value of vΣC at v = 50 ms is to be used
ρredC = relative radius of curvature (transversal plane) at the pitch point
Cylindrical gears
HαHβvγAbt
Bt KKKKKb
Fw sdotsdotsdotsdotsdot= (see 1)
twΣC αsin v2v =
bCredC β cos ρρ =
Bevel gears
HαHβvγAtmb
Bt KKKKKb
Fw sdotsdotsdotsdotsdot= (see 1)
vtmtΣC αsin v2v =
bmvCredC β cos ρρ =
ηoil = dynamic viscosity (mPa s) at oilϑ calculated as
1000
ρ νη oiloil =
where ρ in kgm3 approximated as
( ) 70 15oil15 minusϑminusρ=ρ and νoil is kinematic viscosity at oilϑ and may be calculated by means of the following equation
( ) ( )++=+ 08νloglog08νloglog 100oil ( )
sdotminus
ϑ+minus313log373log
273log373log oil
( ) ( )( )08νloglog08νloglog 10040 +minus+
Ra = composite arithmetic mean roughness (micron) of pinion an wheel calculated as ( )2a1aa RR 05R +=This is defined as the roughness on the new flanks ie as manufactured
XL = see 42
432 Effective tip relief Ceff Ceff is the effective tip relief that amount of tip relief which compensates for the elastic deformation of the gear mesh ie
zero load at the tooth tip It is assumed (simplified) to be equal for both pinion and wheel
Cylindrical gears
for helical
γ
γ=cb
KKFC Abt
eff (see 1)
For spur
cbKKF
C Abteff
γ= (see 1)
Alternatively for spur and helical gears the non-linear ap-proach in 111 may be used (taking Ceff=δ)
Bevel gears
bcKFC Ambt
effγ
= (see 1)
where
γ
α
ε2ε44c
+sdot
=γ
433 Tip relief and extension Cylindrical gears
The extension of the tip relief is not to result in an effective contact ratio 1ltεα when the gear is unloaded (exceptions to this may only apply for applications where the gear is not to run at light loads) This means that the unrelieved part of the path of contact is to be minimum pbt It is further assumed that this unrelieved part is placed centrally on the path of contact
If root relief applies it has to be calculated as equivalent tip relief Ie pinion root relief (at mesh position A) is added to Ca2 and wheel root relief (at mesh position E) is added to Ca1 If no design tip relief or root relief on the mating gear is specified (ie if Ca1+Croot2 = 0 and visa versa) use the run-ning in amount see 112
Bevel gears
Bevel gears are to have heightwise crowning ie no distinct relievedunrelieved area This may be treated as tip and root relief For calculation purposes the root relief is combined with the tip relief of the mating member into an equivalent tip relief If no resulting tip relieves are specified the equivalent tip relives may be calculated as an approxima-tion based on the tool crowning Ca tool (per mille of tool module m0) as follows
2root1aeq1a CCC +=
1root2aeq2a CCC +=
where
Classification Notes- No 412 35 May 2003
DET NORSKE VERITAS
2
0
n0101atoolal m
)m2(mA1middotmCC
minus+minussdot=
( )
sdotminusminussdotsdotminus+= vtvb1
2vb1
21vavt1n1 αtan dddαsin 05x1mA
2
0
n020atool22a m
)mm(2A1middotmCC
minus+minussdot=
( )
sdotminusminussdotsdotminus+= vtvb2
2vb2
2va2vt2n2 αtan dddαsin 05x1mA
2
0
20atool1root1 m
A1mCC
minussdotsdot=
2
0
10atool2root2 m
A1mCC
minussdotsdot=
If the pressure angles of the cutter blades are modified (and verified) to balance the tip relieves the following may be assumed
( )eq2aeq1aeq2aeq1a C andC calculated of sum50CC sdot==
Throughout the following Ca1 and Ca2 mean the equivalent tip relieves Ca1eq and Ca2eq
434 Bulk temperature The bulk temperature may be calculated as
flaaveragempsoilMB X X 50 ϑ+ϑ=ϑ
where
Xs = lubrication factor
= 12 for spray lubrication
= 10 for dip lubrication (provided both pinion and wheel are dip lubricated and tip speed lt 5 ms)
= 10 for spray lubrication with additional cooling spray (spray on both pinion and wheel or spray on pinion and dip lubrication of wheel)
= 02 for meshes fully submerged in oil
Xmp = contact factor ( )pmp nX += 150
np = number of mesh contact on the pinion (for small gear ratii the number of wheel meshes should be used if higher)
flaaverageϑ
= average of the integrated flash temperature (see 44) along the path of contact
( )
AE
E
Ayyyfla
flaaverage
d
ΓminusΓ
ΓΓϑ=ϑint =
For high speed gears (v gt 50 ms) it may be necessary to assess the bulk temperature on the basis of a thermal rating of the entire gear transmission
44 The Flash Temperature flaϑ
441 Basic formula The local flash temperature flaϑ may be calculated as
( ) 41redy
y2ly
211
43Btcorrfla
unXwX3250
y ρ
ρminusρ
micro=ϑ Γ
(For bevel gears replace u with uv)
and is to be calculated stepwise along the path of contact from A to E
where
micro = coefficient of friction see 431
Xcorr = correction factor taking empirically into account the increased scuffing risk in the beginning of the approach path due to mesh starting without any previously built up oil film and possible shuffling away oil before meshing if insufficient tip relief
( )
3
AD
yaeffcorr 50
CC1X
ΓminusΓε
Γminus+=
α
Ca = tip relief of driven member Xcorr is only applicable in the approach path and if Ca lt Ceff otherwise 10
wBt = unit load see 431
yXΓ = load sharing factor see 443
n1 = pinion rpm
Γy ρly etc see 442
442 Geometrical relations The various radii of flank curvature (transversal plane) are
ρ1y = pinion flank radius at mesh point y
ρ2y = wheel flank radius at mesh point y
ρredy = equivalent radius of curvature at mesh point y
y2y1
y2y1redy
ρ+ρ
ρρ=ρ
Cylindrical gears
twy
1y αsin au1Γ1
ρ+
+=
twy
2y αsin au1Γu
ρ+
minus=
Note that for internal gears a and u are negative
36 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
Bevel gears
vtvv
y1y αsin a
u1Γ1
ρ+
+=
vtvv
yv2y αsin a
u1Γu
ρ+
minus=
Γ is the parameter on the path of contact and y is any point between A end E
At the respective ends Γ has the following values
Root piniontip wheel
Cylindrical gears
( )
minus
minusminus= 1
αtan 1dd
zzΓ
tw
2b2a2
1
2A
Bevel gears
( )
minus
minusminus= 1
αtan 1dd
uΓvt
2vb2va2
vA
Tip pinionroot wheel
Cylindrical gears
( )
1αtan
1ddΓ
tw
2b1a1
E minusminus
=
Bevel gears
( )
1αtan
1ddΓ
vt
2vb1va1
E minusminus
=
At inner point of single pair contact
Cylindrical gears
tw1
EB α tan zπ2ΓΓ minus=
Bevel gears
vtv1
EB α tan zπ2ΓΓ minus=
At outer point of single pair contact
Cylindrical gears
tw1AD α tan z
π2ΓΓ +=
Bevel gears
vt v1
AD αtan z π2ΓΓ +=
At pitch point ΓC = 0
The points F and G (only applicable to cylindrical gears) limiting the extension of tip relief (so as to maintain a mini-mum contact ratio of unity for unloaded gears) are at
2
BAF
Γ+Γ=Γ
2
EDG
Γ+Γ=Γ
443 Load sharing factor XΓ The load sharing factor XΓ accounts for the load sharing be-tween the various pairs of teeth in mesh along the path of contact
XΓ is to be calculated stepwise from A to E using the pa-rameter Γy
4431 Cylindrical gears with β = 0 and no tip relief
Figure 41
ByAAB
Ay for31
31X
yΓltΓleΓ
ΓminusΓ
ΓminusΓ+=Γ
DyB for1Xy
ΓltΓleΓ=Γ
EyDDE
yE for31
31X
yΓleΓltΓ
ΓminusΓ
ΓminusΓ+=Γ
4432 Cylindrical gears with β = 0 and tip relief Tip relief on the pinion reduces XΓ in the range G ndash E and increases correspondingly XΓ in the range F ndash B
Tip relief on the wheel reduces XΓ in the range A ndash F and increases correspondingly XΓ in the range D ndash G
Following remains generally
DyB for1Xy
ΓleΓleΓ=Γ
21XXGF== ΓΓ
In the following it must be distinguished between Ca lt Ceff respectively Ca gt Ceff This is shown by an example below where Ca1 lt Ceff and Ca2 gt Ceff
Classification Notes- No 412 37 May 2003
DET NORSKE VERITAS
Figure 42
Note When Ca gt Ceff the path of contact is shortened by A ndash A respectively E ndash E The single pair contact path is extended into B respectively D If this shift is significant it is necessary to consider the negative effect on surface durability (B) and bending stresses (B and D)
Range A - F
For effa2 CC le
minus=Γ
eff
2a
CC1
31X
A
FyAeff
2a
AF
Ay for C3
C61XX
AyΓleΓleΓ
sdot
+ΓminusΓ
ΓminusΓ+= ΓΓ
For effa2 CC ge
AyAfor0Xy
ΓleΓleΓ=Γ
with ( )21
CC
1CC
eff
2a
eff
2a
AFAAminus
minusΓminusΓ+Γ=Γ
FyAeff
2a
AF
Ay
eff
2a for 21
CC
CC1X
yΓleΓleΓ
minus
ΓminusΓ
ΓminusΓ+minus=Γ
Range F - B
For effa1 CC le
ByFeff
1a
FB
Fy forC3
C61
21X
yΓleΓleΓ
sdot
+ΓminusΓ
ΓminusΓ+=Γ
For effa1 CC ge
ByFeff
1a
FB
Fy for21
CC
21X
yΓleΓleΓ
minus
ΓminusΓ
ΓminusΓ+=Γ
ByB for1Xy
ΓleΓleΓ=Γ
with 1
CC2
eff
1a
FBFB
minus
ΓminusΓ+Γ=Γ
Range D ndash G
For effa2 CC le
GyDeff
2a
DG
Dy
eff
2a for C 3C
61
C 3C
32X
yΓleΓleΓ
+
ΓminusΓ
ΓminusΓminus+=Γ F
or effa2 CC ge
DyD for1Xy
ΓleΓleΓ=Γ
with ( )21
CC
1CC
eff
2a
eff
2a
DGDDminus
minusΓminusΓ+Γ=Γ
GyDeff
2a
DG
Dy
eff
2a for21
CC
CCX ΓleΓleΓ
minus
ΓminusΓ
ΓminusΓminus=Γ
Range G - E
For effa1 CC le
minus=Γ
eff
1a
CC1
31X
E
EyGeff
1a
GE
Gy forC3
C61
21X
yΓleΓleΓ
sdot
+ΓminusΓ
ΓminusΓminus=Γ
For effa1 CC gt
EyGeff
1a
GE
Gy for21
CC
21X
yΓleΓleΓ
minus
ΓminusΓ
ΓminusΓminus=Γ
EyE for0Xy
ΓleΓleΓ=Γ
with 1
CC2
eff
1a
GEGE
minus
ΓminusΓ+Γ=Γ
4433 Gears with β gt 0 buttressing Due to oblique contact lines over the flanks a certain but-tressing may occur near A and E
This applies to both cylindrical and bevel gears with tip relief lt Ceff The buttressing Xbutt is simplified as a linear function within the ranges A ndash H respectively I ndash E
Figure 43
1εwhen13X βbutt EAge=
1εwhenε031X ββbutt EAlt+=
38 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
Cylindrical gears
bIEAH βsin 02ΓΓΓΓ =minus=minus
Bevel gears
bmIEAH βsin 02ΓΓΓΓ =minus=minus
4434 Cylindrical gears with 2εγ le and no tip relief
yXΓ is obtained by multiplication of
yXΓ in 4431 with
Xbutt in 4433
4435 Gears with 2εγ gt and no tip relief
Applicable to both cylindrical and bevel gears
IyH for1Xy
ΓleΓleΓε
=α
Γ
IyHybutt and forX1Xy
ΓgtΓΓltΓε
=α
Γ
Figure 44
4436 Cylindrical gears with 2εγ le and tip relief
yXΓ is obtained by multiplication of
yXΓ in 4432 with Xbutt
in 4433
4437 Cylindrical gears with 2εγ gt and tip relief
Tip relief on the pinion (respectively wheel) reduces XГ in the range G ndash E (respectively A ndash F) and increases XГ in the range F ndash G
yXΓ is obtained by multiplication of
yXΓ as described below
with Xbutt in 4433
In the XГ example below the influence of tip relief is shown (without the influence of Xbutt) by means of
eff2aeff1a CC and CC ltgt
Tip relief gt Ceff causes new end points A respectively E of the path of contact
Figure 45
Range A ndash F
( ) ( )( ) eff
a2αa1α
AF
Ay
eff
2aeff
C 12C 13εC 1ε
CCCX
y +εε++minus
ΓminusΓ
ΓminusΓ+
εminus
=ααα
Γ
eff2aFyA CC if for leΓleΓleΓ
eff2aFyA CifCfor and geΓleΓleΓ
eff2aAyA CC iffor0Xy
gtΓleΓleΓ=Γ
( ) ( ) ( )( ) ( ) a2αa1α
αeffa2AFAA C 1ε 3C 1ε
1ε 2 CC with++minus+minus
ΓminusΓ+Γ=Γ
Range F ndash G
( )( )( ) GyF
eff
2a1a forC 1 2CC 11X
yΓleΓleΓ
+εε+minusε
+ε
=αα
α
αΓ
Range G ndash E
( ) ( )( ) eff
2a1a
GE
Gy
C 1 2C 1C 1 3XX
GFy +εεminusε++ε
ΓminusΓ
ΓminusΓminus=
αα
ααΓΓ minus
effa1EyG CC if for leΓleΓleΓ
effa1EyG CC if Γfor and geΓleΓle
eff1aEyE CC if for0Xy
geΓleΓleΓ=Γ
( ) ( ) ( )( ) ( ) 2a1a
eff1aGEEE C 1C 1 3
1 2 CC withminusε++ε+εminus
ΓminusΓminusΓ=Γαα
α
4438 Bevel gears with εγ more than approx 18 and heightwise crowning For Ca1 = Ca2 = Ceff the following applies
Figure 46
Classification Notes- No 412 39 May 2003
DET NORSKE VERITAS
( )AEM 50 Γ+Γ=Γ
( )( )2AD
3
2My651X
y ΓminusΓε
ΓminusΓminus
ε=
ααΓ
For tip relief lt Ceff yXΓ is found by linear interpolation be-
tween ( )effay CCX =Γ and ( )0CayX =Γ as in 4435
The interpolation is to be made stepwise from A to M with the influence of Ca2 and from M to E with the influence of Ca1 (For 2a1a CC ne there is a discontinuity at M)
Eg with Ca1 = 04 Ceff and Ca2 = 055 Ceff then
Range A ndash M
)CC()0C( eff2ay2ayyX 550X 450X =Γ=ΓΓ +=
Range M ndash E
)CC()0C( eff1ay1ayyX 40X 06X =Γ=ΓΓ +=
For tip relief gt Ceff the new end points A and E are found as
( )
minusΓminusΓ
ε+Γ=Γ α 1
CC
6 eff
2aADAA
( )
minusΓminusΓ
ε+Γ=Γ α 1
CC
6 eff
1aADEE
Range A ndash A
0Xy=Γ
Range A ndash M
( )( )
ΓminusΓ
ΓminusΓminus
minusε=
αΓ 2
MA
2My
eff
2a1
CC4
351Xy
Range M ndash E
( )( )
ΓminusΓ
ΓminusΓminus
minusε=
αΓ 2
ME
2My
eff
1a1
CC4
351Xy
Range E ndash E
0Xy=Γ
40 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
Appendix A Fatigue Damage Accumulation
The Palmgren-Miner cumulative damage calculation princi-ple is used The procedure may be applied as follows
A1 Stress Spectrum From the individual torque classes the torques (Ti) at the peak values of class intervals and the associated number of cycles (NLi) for both pinion and wheel are to be listed from the highest to the lowest torque
(In case of a cyclic torque variation within the torque classes it is advised to use the peak torque If the cyclic variation is such that the same teeth will repeatedly suffer the peak torque this is a must)
The stress spectra for tooth roots and flanks (σFi σHi) with all relevant factors (except KA) are to be calculated on the basis of the torque spectrum The load dependent K-factors are to be determined for each torque class
A2 σminusN-curve The stress versus load cycle curves for tooth roots and flanks (both pinion and wheel) are to be drawn on the basis of per-missible stresses (ie including the demanded minimum safety factors) as determined in 2 respectively 3 If different safety levels for high cycle fatigue and low cycle fatigue are desired this may be expressed by different demand safety factors applied at the endurance limit respectively at static strength
A3 Damage accumulation The individual damage ratio Di at ith stress level is defined as
Fi
Lii N
ND =
where
NLi = The number of applied cycles at ith stress
NFi = The number of cycles to failure at ith stress
Basically stresses σi below the permissible stress level for infinite life (if a constant ZN or YN is accepted) do not con-tribute to the damage sum However calculating the actual safety factor Sact as described below all the σi for which the product S σi is bigger than or equal to the permissible stress level for infinite life contribute to the damage sum and thus to the determination of Sact The final value of S is decisive
(NFi can be found mathematically by putting the permissible stress σpi equal the actual stress σi thereby finding the actual life factor This life factor can be solved with regard to load cycles ie NFi)
The damage sum ΣDi is not to exceed unity
If ΣDi ne 1 the safety against cumulative fatigue damage is different from the applied demand safety factor For deter-mination of this theoretical safety factor an iteration proce-dure is required as described in the following flowchart
S is correction factor with which the actual safety factor Sact can be found
Sact is the demand safety factor (used in determination of the permissible stresses in the σ ndash N ndash curve) times the correc-tion factor S
The full procedure is to be applied for pinion and wheel tooth roots and flanks
Note If alternating stresses occur in a spectrum of mainly pulsating stresses the alternating stresses may be replaced by equiva-lent pulsating stresses ie by means of division with the ac-tual mean stress influence factor YM
Classification Notes- No 412 41 May 2003
DET NORSKE VERITAS
Appendix B Application Factors for Diesel Driven Gears
For diesel driven gears the application factor KA depends on torsional vibrations Both normal operation and misfiring conditions have to be considered
Normally these two running conditions can be covered by only one calculation
B1 Definitions
Normal operation KAnorm = 0
normv0
TTT +
where
T0 = rated nominal torque
Tv norm = vibratory torque amplitude for normal opera-tion (see rules Pt4 Ch3 Sec1 G301 for defi-nition of ldquonormalrdquo irregularity)
Misfiring operation KA misf = 0
misfv
TTT +
where
T = remaining nominal torque when one cylinder out of action
Tv misf = vibratory torque amplitude in misfiring con-dition This refers to a permissible misfiring condition ie a condition that does not require automatic or immediate corrective actions as speed or pitch reduction
The normal operation is assumed to last for a very high num-ber of cycles such as 1010
The misfiring operation is assumed to last for a limited dura-tion such as 107 cycles
B2 Determination of decisive load Assuming life factor at 1010 cycles as YN = ZN = 092 which usually is relevant the calculation may be performed only once with the combination having the highest value of appli-cation factorlife factor
For bending stresses and scuffing the higher value of
092
K normA and 098
K misfA
For contact stresses the higher value of
092K normA and
113K misfA (but
097K misfA for nitrided gears)
B3 Simplified procedure Note that this is only a guidance and is not a binding con-vention
T = oTZ
1Zsdot
minus where Z = number of cylinders
( ) ideal Videal Vmisf Vnorm V TTT24ZT +minus=
where TV ideal = vibratory torque with all cylinders perfectly equal See also rules Pt4 Ch3 Sec1 G300
When using trends from torsional vibration analysis and measurements the following may be used
TV ideal To is close to zero for engines with few cylinders and using a suitable elastic coupling and increases with relative coupling stiffness and number of cylinders
TV misf To may be high for engines with few cylinders and decreases with number of cylinders
This can be indicated as
200Z
TT
ο
idealV asymp and 80Z04
TT
o
misfV minusasymp
Inserting this into the formulae for the two application fac-tors the following guidance can be given
118112K misfA minusasymp
115110K normA minusasymp
Since KA norm is to be combined with the lower life factors the decisive load condition will be the normal one and a KA of 115 will cover most relevant cases when a suitable elas-tic coupling is chosen
42 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
Appendix C Calculation of Pinion-Rack
Pinion-racks used for elevating of mobile offshore units are open gears that are subjected to wear and tear With normal specifications such as only a few hundred total operation cycles (site to site) the tooth bending stresses for static strength will be decisive for the lay out however with ex-ception of surface hardened pinions where case crushing has to be considered
In the following the use of part 1 and 3 for pinion-racks is shown including relevant simplifications
C1 Pinion tooth root stresses Since the load spectrum normally is dominated by high torques and few load cycles (in the range up to 1-2000) the static strength is decisive
The actual stress is calculated as
β1FSaFan1
tF1 KYY
mbFσ sdotsdotsdotsdot
=
b1 is limited to b2 + 2middotmn
YFa and YSa replace YF and YS because load application at tooth tip has to be assumed for such inaccurate gears
Pinions often use a non-involute profile in the dedendum part of the flank eg a constant radius equal the radius of curva-ture at reference circle For such pinions sFn and hFa are to be measured directly on a sectional drawing of the pinion tooth
Due to high loads and narrow facewidths it may be assumed that KFβ2 = KHβ = 10 However when b1 gt b2 then KFβ1 gt10
If no detailed documentation of KFβ1 is available the fol-lowing may be used
KFβ1 = 1 + 015middot(b1b2 ndash 1)
The permissible stress (not surface hardened) is calculated as
δrelTstF
Fst1FP1 Y
Sσσ sdot=
The mean stress influence due to leg lifting may be disre-garded
The actual and permissible stresses should be calculated for the relevant loads as given in the rules
C2 Rack tooth root stresses The actual stress is calculated as
SaFan2
tF2 YY
mbFσ sdotsdotsdot
=
See C1 for details
The permissible stress is calculated as
δrelTstF
Fst2FP2 Y
Sσσ sdot=
For alloyed steels (Ni Cr Mo) with high toughness and duc-tility the value of YδrelTst may be put equal to YSa
C3 Surface hardened pinions For surface hardened pinions the maximum load is not to cause crushing of the hardened layer of the flank
In principle the calculation described in 213 may be used but when the theoretical Hertzian stress exceeds the ap-proximately 18 times the yield strength of the rack material plastic deformation will occur This will limit the peak Hertzian stress but increases the contact width and thus the penetration of stresses into the depth
An approximation may be based on an assessment of contact width determined by means of equal areas under the theoreti-cal (elastic range) Hertzian contact and the elasto-plastic contact stress (limited to 18 middot σy) with the unknown width
6 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
pbt = βcos
αcosmπ tn
sat =
minus++
at
ninvααinv
z
αtanx22
π
d a
san = aβcossat
14 Bevel Gear Conversion Formulae and Specific Formulae Conversion of bevel gears to virtual equivalent helical cylin-drical gears is based on the bevel gear midsection The con-version formulae are
Number of teeth
zv12 = z12 cos δ12
(δ1 + δ2 = Σ)
Gear ratio
uv = 1v
2vzz
tan αvt = tan αn cos βm
tan βbm = tan βm cos αvt
Base pitch
pbtm = m
vtnmβcos
αcosmπ
Reference pitch diameters
dv12 = 21
21m
cosdδ
Centre distance
av = 05 (dv1 + dv2)
Tip diameters
dva 12 = dv 12 + 2 ham 12
Addenda
for gears with constant addenda (Klingelnberg)
ham 12 = mmn (1 + xm 12)
for gears with variable addenda (Gleason)
ham 12 = ha 12 ndash b2 tan (δa 12 ndash δ12)
(when ha is addendum at outer end and δa is the outer cone angle)
Addendum modification coefficients
xm 12 = mn
12am21am
m2
hh minus
Base circle
dvb 12 = dv 12 cos αvt
Transverse contact ratio)
εα = btmP
αsinadd05dd05 vtv2
2vb2
2va2
1vb2
1va minusminus+minus
Overlap ratio) (theoretical value for bevel gears with no crowning but used as approximations in the calculation pro-cedures)
εβ = nm
mmπβsinb
Total contact ratio)
εγ = 2β
2α εε +
( Note that index laquovraquo is left out in order to combine formu-lae for cylindrical and bevel gears)
Tangential speed at midsection
vmt = 3m11 10dn
60π minus
Effective radius of curvature (normal section)
ρvc = ( )2vbm
vtvv
u1βcosαsinua
+
Length of line of contact
lb = ( )( )( )
1εifε
ε1ε2ε
βcosεb
β2γ
2βα
2γ
bm
α ltminusminusminus
lb = 1εifβcosε
εbβ
bmγ
α ge
15 Nominal Tangential Load Ft Fbt Fmt and Fmbt The nominal tangential load (tangential to the reference cyl-inder with diameter d and perpendicular to an axial plane) is calculated from the nominal (rated) torque T transmitted by the gear set
Cylindrical gears
dT2000
Ft = t
tbt αcos
FF =
Bevel gears
mmt d
T2000F =
vt
mtmbt αcos
FF =
Classification Notes- No 412 7 May 2003
DET NORSKE VERITAS
16 Application Factors KA and KAP The application factor KA accounts for dynamic overloads from sources external to the gearing
It is distinguished between the influence of repetitive cyclic torques KA (161) and the influence of temporary occasional peak torques KAP (162)
Calculations are always to be made with KA In certain cases additional calculations with KAP may be necessary
For gears with a defined load spectrum the calculation with a KA may be replaced by a fatigue damage calculation as given in Appendix A
161 KA For gears designed for long or infinite life at nominal rated torque KA is defined as the ratio between the maximum re-petitive cyclic torque applied to the gear set and nominal rated torque
This definition is suitable for main propulsion gears and most of the auxiliary gears
KA can be determined by measurements or system analysis or may be ruled by conventions (ice classes) (For the pur-pose of a preliminary (but not binding) calculation before KA is determined it is advised to apply either the max values mentioned below or values known from similar plants)
a) For main propulsion gears KA can be taken from the (mandatory) torsional vibration analysis thereby con-sidering all permissible driving conditions) Unless specially agreed the rules do not allow KA in excess of 135 for diesel propulsion) With turbine or electric propulsion KA would normally not exceed 12 However special attention should be given to thrusters that are arranged in such a way that heavy vessel movements andor manoeuvring can cause severe load fluctuations This means eg thrusters positioned far from the rolling axis of vessels that could be susceptible to rolling If leading to propeller air suction the condi-tions may be even worse The above mentioned movements or manoeuvring will result in increased propeller excitation If the thruster is driven by a diesel engine the engine mean torque is limited to 100 However thrusters driven by electric motors can suffer temporary mean torque much above 100 unless a suitable load control system (limiting available e-motor torque) is provided
b) For main propulsion gears with ice class notation (see Rules Pt5 Ch1 Sec J500) KA ice has to be taken as the higher value of the applicable (rule defined) ice shock torque referred to nominal rated torque and the value under a) The Baltic ice class notations refer to a few millions ice shock loads Thus the life factors may be put YN=1 and ZN=12 (except for nitrided gears where ZN = 1 applies) Additionally the calculations with the normal KA (no ice class) are to fulfil the normal requirements For polar ice class notations KA ice applies to all criteria and for long or infinite life
c) For a power take off (PTO) branch from a main propul-sion gear with ice class ice shocks result in negative torques It is assumed that the PTO branch is unloaded when the ice shock load occurs The influence of these reverse shock loads may be taken into account as follows The negative torque (reversed load) expressed by means of an application factor based on rated forward load (T or Ft) is KAreverse = KA ice ndash1 (the minus 1 be-cause no mean torque assumed) KAice to be calculated as in the ice class rules This KAreverse should be used for back flank considera-tions such as pitting and scuffing The influence on tooth bending strength (forward di-rection) may be simplified by using the factor YM = 1 minus 03 middot KAreverse KA
d) For diesel driven auxiliaries KA can be taken from the torsional vibration analysis if available For units where no vibration analysis is required (lt 200 kW) or available it is advised to apply KA as the upper allow-able value 135)
e) For turbine or electro driven auxiliaries the same as for c) applies however the practical upper value is 12
) For diesel driven gears more information on KA for mis-firing and normal driving is given in Appendix B
162 KAP The peak overload factor KAP is defined as the ratio between the temporary occasional peak overload torque and the nominal rated torque
For plants where high temporary occasional peak torques can occur (ie in excess of the above mentioned KA) the gearing (if nitrided) has to be checked with regard to static strength Unless otherwise specified the same safety factors as for infinite life apply
The scuffing safety is to be specially considered whereby the KA applies in connection with the bulk temperature and the KAP applies for the flash temperature calculation and should replace KA in the formulae in 431 432 and 441
KAP can be evaluated from the torsional impact vibration calculation (as required by the rules)
If the overloads have a duration corresponding to several revolutions of the shafts the scuffing safety has to be consid-ered on basis of this overload both with respect to bulk and flash temperature This applies to plants with ice class nota-tions (Baltic and polar) and to plants with prime movers which have high temporary overload capacity such as eg electric motors (provided the driven member can have a con-siderable increase in demand torque as eg azimuth thrusters during manoeuvring)
For plants without additional ice class notation KAP should normally not exceed 15
8 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
163 Frequent overloads For plants where high overloads or shock loads occur regu-larly the influence of this is to be considered by means of cumulative fatigue (see Appendix A)
17 Load Sharing Factor Kγ The load sharing factor Kγ accounts for the maldistribution of load in multiple-path transmissions (dual tandem epicyclic double helix etc) Kγ is defined as the ratio between the max load through an actual path and the evenly shared load
171 General method Kγ mainly depends on accuracy and flexibility of the branches (eg quill shaft planet support external forces etc) and should be considered on basis of measurements or of relevant analysis as eg
Kγ = δ+δ f
δ = total compliance of a branch under full load (assuming even load share) referred to gear mesh
f = minusminusminusminus+++ 23
22
21 fff where f1 f2 etc are the
main individual errors that may contribute to a maldis-tribution between the branches Eg tooth pitch errors planet carrier pitch errors bearing clearance influences etc Compensating effects should also be considered
For double helical gears
An external axial force Fext applied from sources outside the actual gearing (eg thrust via or from a tooth coupling) will cause a maldistribution of forces between the two helices Expressed by a load sharing factor the
βsdot
plusmn=γ tanFF1K
t
ext
If the direction of Fext is known the calculation should be carried out separately for each helix and with the tangential force corrected with the pertinent Kγ If the direction of Fext is unknown both combinations are to be calculated and the higher σH or σF to be used
172 Simplified method If no relevant analysis is available the following may apply
For epicyclic gears
Kγ = 32501 minus+ pln
where npl = number of planets ( gt3 )
For multistage gears with locked paths and gear stages sepa-rated by quill shafts (see figure below)
Figure 10 Locked paths gear
Kγ = ( )φ201+
where φ = quill shaft twist (degrees) under full load
18 Dynamic Factor Kv The dynamic factor Kv accounts for the internally generated dynamic loads
Kv is defined as the ratio between the maximum load that dynamically acts on the tooth flanks and the maximum ex-ternally applied load Ft KA Kγ
In the following 2 different methods (181 and 182) are described In case of controversy between the methods the next following is decisive ie the methods are listed with increasing priority
It is important to observe the limitations for the method in 181 In particular the influence of lateral stiffness of shafts is often underestimated and resonances occur at considerably lower speed than determined in 1811
However for low speed gears with vmiddotz1 lt 300 calculations may be omitted and the dynamic factor simplified to Kv=105
181 Single resonance method For a single stage gear Kv may be determined on basis of the relative proximity (or resonance ratio) N between actual speed n1 and the lowest resonance speed nE1
N = 1E
1
nn
Note that for epicyclic gears n is the relative speed ie the speed that multiplied with z gives the mesh frequency
1811 Determination of critical speed It is not advised to apply this method for multimesh gears for N gt 085 as the influence of higher modes has to be consid-ered see 182 In case of significant lateral shaft flexibility (eg overhung mounted bevel gears) the influence of cou-pled bending and torsional vibrations between pinion and wheel should be considered if N ge 075 see 182
nE1 = redmγc
1zπ
31030 sdot
where
cγ is the actual mesh stiffness per unit facewidth see 111
Classification Notes- No 412 9 May 2003
DET NORSKE VERITAS
For gears with inactive ends of the facewidth as eg due to high crowning or end relief such as often applied for bevel gears the use of cγ in connection with determination of natu-ral frequencies may need correction cγ is defined as stiffness per unit facewidth but when used in connection with the total mesh stiffness it is not as simple as cγmiddotb as only a part of the facewidth is active Such corrections are given in 111
mred is the reduced mass of the gear pair per unit facewidth and referred to the plane of contact
For a single gear stage where no significant inertias are closely connected to neither pinion nor wheel mred is calcu-lated as
mred = 21
21mm
mm+
The individual masses per unit facewidth are calculated as
m12 = 221b
21
)2d(b
I
where I is the polar moment of inertia (kgmm2)
The inertia of bevel gears may be approximated as discs with diameter equal the midface pitch diameter and width equal to b However if the shape of the pinion or wheel body differs much from this idealised cylinder the inertia should be cor-rected accordingly
For all kind of gears if a significant inertia (eg a clutch) is very rigidly connected to the pinion or wheel it should be added to that particular inertia (pinion or wheel) If there is a shaft piece between these inertias the torsional shaft stiffness alters the system into a 3-mass (or more) system This can be calculated as in 182 but also simplified as a 2-mass system calculated with only pinion and wheel masses
1812 Factors used for determination of Kv Non-dimensional gear accuracy dependent parameters
( )bKKF
yfcB
At
pptp
γ
minus=
( )bKKF
yFcBAt
ff
γ
α minus=
Non-dimensional tip relief parameter
bKKFcC
1BγAt
ak sdotsdot
sdotminus=
For gears of quality grade (ISO 1328) Q = 7 or coarser Bk = 1
For gears with Q le 6 and excessive tip relief Bk is limited to max 1
For gears (all quality grades) with tip relief of more than 2middotCeff (see 432) the reduction of αε has to be considered (see 443)
where
fpt = the single pitch deviation (ISO 1328) max of pinion or wheel
Fα = the total profile form deviation (ISO 1328) max of pinion or wheel (Note Fα is pt not available for bevel gears thus use Fα = fpt)
yp and yf = the respective running-in allowances and may be calculated similarly to yα in 112 ie the value of fpt is replaced by Fα for yf
cacute = the single tooth stiffness see 111
Ca = the amount of tip relief see 433 In case of different tip relief on pinion and wheel the value that results in the greater value of Bk is to be used If Ca is zero by design the value of running-in tip relief Cay (see 112) may be used in the above formula
1813 Kv in the subcritical range Cylindrical gears N le 085
Bevel gears N le 075
Kv = 1 + N K
K = Cv1 Bp + Cv2 Bf + Cv3 Bk
Cv1 accounts for the pitch error influence
Cv1 = 032
Cv2 accounts for profile error influence
Cv2 = 034 for γε le 2
Cv2 = 03ε
057
γ minus for γε gt 2
Cv3 accounts for the cyclic mesh stiffness variation
Cv3 = 023 for γε le 2
Cv3 = 156ε
0096
γ minus for γε gt 2
1814 Kv in the main resonance range Cylindrical gears 085 lt N le 115
Bevel gears 075 lt N le 125
Running in this range should preferably be avoided and is only allowed for high precision gears
Kv = 1 + Cv1 Bp + Cv2 Bf + Cv4 Bk
Cv4 accounts for the resonance condition with the cyclic mesh stiffness variation
Cv4 = 090 for γε le 2
Cv4 = 144ε
ε005057
γ
γ
minus
minus for γε gt 2
10 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
1815 Kv in the supercritical range Cylindrical gears N ge 15
Bevel gears N ge 15
Special care should be taken as to influence of higher vibra-tion modes andor influence of coupled bending (ie lateral shaft vibrations) and torsional vibrations between pinion and wheel These influences are not covered by the following approach
Kv = Cv5 Bp + Cv6 Bf + Cv7
Cv5 accounts for the pitch error influence
Cv5 = 047
Cv6 accounts for the profile error influence
Cv6 = 047 for εγ le 2
Cv6 = 174ε
012
γ minus for εγ gt 2
Cv7 relates the maximum externally applied tooth loading to the maximum tooth loading of ideal accurate gears operating in the supercritical speed sector when the circumferential vibration becomes very soft
Cv7 = 075 for εγ le 15
Cv7 = [ ] 8750)2(sin 0125 +minusβεπ for 15 lt εγ le 25
Cv7 = 10 for εγ gt 25
1816 Kv in the intermediate range Cylindrical gears 115 lt N lt 15
Bevel gears 125 lt N lt 15
Comments raised in 1814 and 1815 should be observed
Kv is determined by linear interpolation between Kv for N = 115 respectively 125 and N = 15 as
Cylindrical gears
( ) ( ) ( )[ ]51Nv151Nv51Nvv KK350
N51KK === minussdot
minus
+=
Bevel gears
( ) ( ) ( )[ ]51Nv251Nv51Nvv KK250
N51KK === minussdot
minus
+=
182 Multi-resonance method For high speed gear (vgt40 ms) for multimesh medium speed gears for gears with significant lateral shaft flexibility etc it is advised to determine Kv on basis of relevant dy-namic analysis
Incorporating lateral shaft compliance requires transforma-tion of even a simple pinion-wheel system into a lumped multi-mass system It is advised to incorporate all relevant inertias and torsional shaft stiffnesses into an equivalent (to pinion speed) system Thereby the mesh stiffness appears as an equivalent torsional stiffness
cγ b (db12)2 (Nmrad)
The natural frequencies are found by solving the set of dif-ferential equations (one equation per inertia) Note that for a gear put on a laterally flexible shaft the coupling bending-torsionals is arranged by introducing the gear mass and the lateral stiffness with its relation to the torsional displacement and torque in that shaft
Only the natural frequency (ies) having high relative dis-placement and relative torque through the actual pinion-wheel flexible element need(s) to be considered as critical frequency (ies)
Kv may be determined by means of the method mentioned in 181 thereby using N as the least favourable ratio (in case of more than one pinion-wheel dominated natural frequency) Ie the N-ratio that results in the highest Kv has to be consid-ered
The level of the dynamic factor may also be determined on basis of simulation technique using numeric time integration with relevant tooth stiffness variation and pitchprofile er-rors
19 Face Load Factors KHβ and KFβ The face load factors KHβ for contact stresses and for scuff-ing KFβ for tooth root stresses account for non-uniform load distribution across the facewidth
KHβ is defined as the ratio between the maximum load per unit facewidth and the mean load per unit facewidth
KFβ is defined as the ratio between the maximum tooth root stress per unit facewidth and the mean tooth root stress per unit facewidth The mean tooth root stress relates to the con-sidered facewidth b1 respectively b2
Note that facewidth in this context is the design facewidth b even if the ends are unloaded as often applies to eg bevel gears
The plane of contact is considered
191 Relations between KHβ and KFβ
KFβ = expβHK 2(hb)hb1
1exp++
=
where hb is the ratio tooth heightfacewidth The maximum of h1b1 and h2b2 is to be used but not higher than 13 For double helical gears use only the facewidth of one helix
If the tooth root facewidth (b1 or b2) is considerably wider than b the value of KFβ(1or2) is to be specially considered as it may even exceed KHβ
Eg in pinion-rack lifting systems for jack up rigs where b = b2 asymp mn and b1 asymp 3 mn the typical KHβ asymp KFβ2 asymp 1 and KFβ1 asymp 13
192 Measurement of face load factors Primarily
KFβ may be determined by a number of strain gauges distrib-uted over the facewidth Such strain gauges must be put in
Classification Notes- No 412 11 May 2003
DET NORSKE VERITAS
exactly the same position relative to the root fillet Relations in 191 apply for conversion to KHβ
Secondarily
KHβ may be evaluated by observed contact patterns on vari-ous defined load levels It is imperative that the various test loads are well defined Usually it is also necessary to evalu-ate the elastic deflections Some teeth at each 90 degrees are to be painted with a suitable lacquer Always consider the poorest of the contact patterns
After having run the gear for a suitable time at test load 1 (the lowest) observe the contact pattern with respect to ex-tension over the facewidth Evaluate that KHβ by means of the methods mentioned in this section Proceed in the same way for the next higher test load etc until there is a full face contact pattern From these data the initial mesh misalign-ment (ie without elastic deflections) can be found by ex-trapolation and then also the KHβ at design load can be found by calculation and extrapolation See example
Figure 11 Example of experimental determination of KHβ
It must be considered that inaccurate gears may accumulate a larger observed contact pattern than the actual single mesh to mesh contact patterns This is particularly important for lapped bevel gears Ground or hard metal hobbed bevel gears are assumed to present an accumulated contact pattern that is practically equal the actual single mesh to mesh con-tact patterns As a rough guidance the (observed) accumu-lated contact pattern of lapped bevel gears may be reduced by 10 in order to assess the single mesh to mesh contact pattern which is used in 199
193 Theoretical determination of KHβ The methods described in 193 to 198 may be used for cy-lindrical gears The principles may to some extent also be used for bevel gears but a more practical approach is given in 199
General For gears where the tooth contact pattern cannot be verified during assembly or under load all assumptions are to be well on the safe side
KHβ is to be determined in the plane of contact
The influence parameters considered in this method are
bull mean mesh stiffness cγ (see 111) (if necessary also variable stiffness over b)
bull mean unit load Fmb = Fbt KA Kγ Kvb (for double helical gears see 17 for use of Kγ)
bull misalignment fsh due to elastic deflections of shafts and gear bodies (both pinion and wheel)
bull misalignment fdefl due to elastic deflections of and work-ing positions in bearings
bull misalignment fbe due to bearing clearance tolerances bull misalignment fma due to manufacturing tolerances bull helix modifications as crowning end relief helix correc-
tion bull running in amount yβ (see 112)
In practice several other parameters such as centrifugal ex-pansion thermal expansion housing deflection etc contrib-ute to KHβ However these parameters are not taken into account unless in special cases when being considered as particularly important
When all or most of the am parameters are to be considered the most practical way to determine KHβ is by means of a graphical approach described in 1931
If cγ can be considered constant over the facewidth and no helix modifications apply KHβ can be determined analyti-cally as described in 1932
1931 Graphical method The graphical method utilises the superposition principle and is as follows
bull Calculate the mean mesh deflection Mδ as a function of
γm c and bF see 111
bull Draw a base line with length b and draw up a rectangu-lar with height δM (The area δM b is proportional to the transmitted force)
bull Calculate the elastic deflection fsh in the plane of contact Balance this deflection curve around a zero line so that the areas above and below this zero line are equal
Figure 12 fsh balanced around zero line
bull Superimpose these ordinates of the fsh curve to the previ-ous load distribution curve (The area under this new load distribution curve is still δM b)
bull Calculate the bearing deflections andor working posi-tions in the bearings and evaluate the influence fdefl in the plane of contact This is a straight line and is bal-anced around a zero line as indicated in Fig 14 but with one distinct direction Superimpose these ordinates to the previous load distribution curve
12 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
bull The amount of crowning end relief or helix correction (defined in the plane of contact) is to be balanced around a zero line similarly to fsh
Figure 13 Crowning Cc balanced aound zero line
bull Superimpose these ordinates to the previous load distri-bution curve In case of high crowning etc as eg often applied to bevel gears the new load distribution curve may cross the base line (the real zero line) The result is areas with negative load that is not real as the load in those areas should be zero Thus corrective actions must be made but for practical reasons it may be postponed to after next operation
bull The amount of initial mesh misalignment bema ff + (defined in the plane of contact) is to be bal-
anced around a zero line If the direction of bema ff + is known (due to initial contact check) or if the direction of fbe is known due to design (eg overhang bevel pin-ion) this should be taken into account If direction un-known the influence of bema ff + in both directions as well as equal zero should be considered
Figure 14 fma+fbe in both directions balanced around zero line
Superimpose these ordinates to the previous load distri-bution curve This results in up to 3 different curves of which the one with the highest peak is to be chosen for further evaluation
bull If the chosen load distribution curve crosses the base line (ie mathematically negative load) the curve is to be corrected by adding the negative areas and dividing this with the active facewidth The (constant) ordinates of this rectangular correction area are to be subtracted from the positive part of the load distribution curve It is advisable to check that the area covered under this new load distribution curve is still equal δM b
bull If cγ cannot be considered as constant over b then cor-rect the ordinates of the load distribution curve with the local (on various positions over the facewidth) ratio between local mesh stiffness and average mesh stiffness cγ (average over the active facewidth only) Note that the result is to be a curve that covers the same area δM b as before
bull The influence of running in yβ is to be determined as in 112 whereby the value for Fβx is to be taken as twice the distance between the peak of the load distribution curve and δM
bull Determine
KHβ = M
β
δycurveofpeak minus
1932 Simplified analytical method for cylindrical gears The analytical approach is similar to 1931 but has a more limited application as cγ is assumed constant over the face-width and no helix modification applies
bull Calculate the elastic deflection fsh in the plane of contact Balance this deflection curve around a zero line so that the area above and below this zero line are equal see Fig 12 The max positive ordinate is frac12∆fsh
bull Calculate the initial mesh alignment as Fβx= deflbemash ffff plusmnplusmnplusmn∆ The negative signs may only be used if this is justified andor verified by a contact pattern test Otherwise al-ways use positive signs If a negative sign is justified the value of Fβx is not to be taken less than the largest of each of these elements
bull Calculate the effective mesh misalignment as Fβy = Fβx - yβ (yβ see 112)
bull Determine
2KforF2
bFc1K βH
m
βγγH le+=β
or
2KforF
bFc2K Hβ
m
βγγH gt=β
where cγ as used here is the effective mesh stiffness see 111
194 Determination of fsh fsh is the mesh misalignment due to elastic deflections Usu-ally it is sufficient to consider the combined mesh deflection of the pinion body and shaft and the wheel shaft The calcu-lation is to be made in the plane of contact (of the considered gear mesh) and to consider all forces (incl axial) acting on the shafts Forces from other meshes can be parted into components parallel respectively vertical to the considered plane of contact Forces vertical to this plane of contact have no influence on fsh
It is advised to use following diameters for toothed elements
d + 2 x mn for bending and shear deflection
d + 2 mn (x ndash ha0 + 02) for torsional deflection
Usually fsh is calculated on basis of an evenly distributed load If the analysis of KHβ shows a considerable maldis-
Classification Notes- No 412 13 May 2003
DET NORSKE VERITAS
tribution in term of hard end contact or if it is known by other reasons that there exists a hard end contact the load should be correspondingly distributed when calculating fsh In fact the whole KHβ procedure can be used iteratively 2-3 iterations will be enough even for almost triangular load distributions
195 Determination of fdefl fdefl is the mesh misalignment in the plane of contact due to bearing deflections and working positions (housing deflec-tion may be included if determined)
First the journal working positions in the bearings are to be determined The influence of external moments and forces must be considered This is of special importance for twin pinion single output gears with all 3 shafts in one plane
For rolling bearings fdefl is further determined on basis of the elastic deflection of the bearings An elastic bearing deflec-tion depends on the bearing load and size and number of rolling elements Note that the bearing clearance tolerances are not included here
For fluid film bearings fdefl is further determined on basis of the lift and angular shift of the shafts due to lubrication oil film thickness Note that fbe takes into account the influence of the bearing clearance tolerance
When working positions bearing deflections and oil film lift are combined for all bearings the angular misalignment as projected into the plane of the contact is to be determined fdefl is this angular misalignment (radians) times the face-width
196 Determination of fbe fbe is the mesh misalignment in the plane of contact due to tolerances in bearing clearances In principle fbe and fdefl could be combined But as fdefl can be determined by analy-sis and has a distinct direction and fbe is dependent on toler-ances and in most cases has no distinct direction (ie + toler-ance) it is practicable to separate these two influences
Due to different bearing clearance tolerances in both pinion and wheel shafts the two shaft axis will have an angular mis-alignment in the plane of contact that is superimposed to the working positions determined in 195 fbe is the facewidth times this angular misalignment Note that fbe may have a distinct direction or be given as a + tolerance or a combina-tion of both For combination of + tolerance it is adviced to use
fbe= 22be
21be ff ++plusmn
fbe is particularly important for overhang designs for gears with widely different kinds of bearings on each side and when the bearings have wide tolerances on clearances In general it shall be possible to replace standard bearings with-out causing the real load distribution to exceed the design premises For slow speed gears with journal bearings the expected wear should also be considered
197 Determination of fma fma is the mesh misalignment due to manufacturing toler-ances (helix slope deviation) of pinion fHβ1 wheel fHβ2 and housing bore
For gear without specifically approved requirements to as-sembly control the value of fma is to be determined as
fma= 22Hβ
21Hβ ff +
For gears with specially approved assembly control the value of fma will depend on those specific requirements
198 Comments to various gear types For double helical gears KHβ is to be determined for both helices Usually an even load share between the helices can be assumed If not the calculation is to be made as de-scribed in 171
For planetary gears the free floating sun pinion suffers only twist no bending It must be noted that the total twist is the sum of the twist due to each mesh If the value of 1K γ ne this must be taken into account when calculating the total sun pinion twist (ie twist calculated with the force per mesh without Kγ and multiplied with the number of planets)
When planets are mounted on spherical bearings the mesh misalignments sun-planet respectively planet-annulus will be balanced Ie the misalignment will be the average between the two theoretical individual misalignments The faceload distribution on the flanks of the planets can take full advan-tage of this However as the sun and annulus mesh with several planets with possibly different lead errors the sun and annulus cannot obtain the above mentioned advantage to the full extent
199 Determination of KHβ for bevel gears If a theoretical approach similar to 193-198 is not docu-mented the following may be used
testeff
H Kb
b851851K sdot
minussdot=β
beff b represents the relative active facewidth (regarding lapped gears see 192 last part)
Higher values than beff b = 090 are normally not to be used in the formula
For dual directional gears it may be difficult to obtain a high beff b in both directions In that case the smaller beff b is to be used
Ktest represents the influence of the bearing arrangement shaft stiffness bearing stiffness housing stiffness etc on the faceload distribution and the verification thereof Expected variations in length- and height-wise tooth profile is also accounted for to some extent
a) Ktest = 1 For ground or hard metal hobbed gears with the specified contact pattern verified at full rating or at full torque slow turning at a condition representative for the thermal expansion at normal operation
14 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
It also applies when the bearing arrangementsupport has insignificant elastic deflections and thermal axial expansion However each initial mesh contact must be verified to be within acceptance criteria that are cali-brated against a type test at full load Reproduction of the gear tooth length- and height-wise profile must also be verified This can be made through 3D measurements or by initial contact movements caused by defined axial offsets of the pinion (tolerances to be agreed upon)
b) Ktest = 1 + 04middot(beffbndash06) For designs with possible influence of thermal expansion in the axial direction of the pinion The initial mesh contact verified with low load or spin test where the acceptance criteria are calibrated against a type test at full load
c) Ktest = 12 if mesh is only checked by toolmakerrsquos blue or by spin test contact For gears in this category beffb gt 085 is not to be used in the calculation
110 Transversal Load Distribution Factors KHα and KFα The transverse load distribution factors KHα for contact stresses and for scuffing KFα for tooth root stresses account for the effects of pitch and profile errors on the transversal load distribution between 2 or more pairs of teeth in mesh
The following relations may be used
Cylindrical gears
( )
minus+==
tH
αptγγHαFα F
byfc0409
2ε
KK
valid for 2εγ le
( ) ( )tH
αptγ
γ
γHαFα F
byfcε
1ε20409KK
minusminus+==
valid for 2ε γ gt
where
FtH = Ft KA Kγ Kv KHβ
cγ = See 111
γα = See 112
fpt = Maximum single pitch deviation (microm) of pinion or wheel or maximum total profile form devia-tion Fα of pinion or wheel if this is larger than the maximum single pitch deviation
Note In case of adequate equivalent tip relief adapted to the load half of the above mentioned fpt can be introduced A tip relief is considered adequate when the average of Ca1 and Ca2 is within plusmn40 of the value of Ceff in 432
Limitations of KHα and KFα
If the calculated values for
KFα = KHα lt 1 use KFα = KHα = 10
If the calculated value of KHα gt 2εα
γ
Zε
ε use KHα = 2
εα
γ
Zε
ε
If the calculated value of KFα gt εα
γ
Yεε
use KFα = εα
γ
Yεε
where Yε = αnε
075025+ (for εαn see 331c)
Bevel gears
For ground or hard metal hobbed gears KFα = KHα = 1
For lapped gears KFα = KHα = 11
111 Tooth Stiffness Constants cacute and cγ The tooth stiffness is defined as the load which is necessary to deform one or several meshing gear teeth having 1 mm facewidth by an amount of 1 microm in the plane of contact cacute is the maximum stiffness of a single pair of teeth cγ is the mean value of the mesh stiffness in a transverse plane (brief term mesh stiffness)
Both valid for high unit load (Unit load = Ft middot KA middot Kγb)
Cylindrical gears The real stiffness is a combination of the progressive Hertzian contact stiffness and the linear tooth bending stiff-nesses For high unit loads the Hertzian stiffness has little importance and can be disregarded This approach is on the safe side for determination of KHβ and KHα However for moderate or low loads Kv may be underestimated due to determination of a too high resonance speed
The linear approach is described in A
An optional approach for inclusion of the non-linear stiffness is described in B
A The linear approach
BRCCq
βcos08acutec =
and
( )025ε075cc αγ +prime=
where
( )[ ]n02a01a
B α2000212
hh12051C minusminus
+minus+=
12n1n
x000635z
025791z
015551004723q minus++=
12
2n
22
1n
1 x005290z
x024188x000193z
x011654+minusminusminus
+ 000182 x22
Classification Notes- No 412 15 May 2003
DET NORSKE VERITAS
(for internal gears use zn2 equal infinite and x2 = 0) ha0 = hfp for all practical purposes CR considers the increased flexibility of the wheel teeth if the wheel is not a solid disc and may be calculated as
( )( )nR m5s
sR e5
bbln1C +=
where
bs = thickness of a central web
sR = average thickness of rim (net value from tooth root to inside of rim)
The formula is valid for bs b ge 02 and sRmn ge 1 Outside this range of validity and if the web is not centrally posi-tioned CR has to be specially considered
Note CR is the ratio between the average mesh stiffness over the facewidth and the mesh stiffness of a gear pair of solid discs The local mesh stiffness in way of the web corresponds to the mesh stiffness with CR = 1 The local mesh stiffness where there is no web support will be less than calculated with CR above Thus eg a centrally positioned web will have an effect corresponding to a longitudinal crowning of the teeth See also 1931 regarding KHβ
B The non-linear approach
In the following an example is given on how to consider the non-linearity
The relation between unit load Fb as a function of mesh deflection δ is assumed to be a progressive curve up to 500 Nmm and from there on a straight line This straight line when extended to the baseline is assumed to intersect at 10microm
With these assumptions the unit force Fb as a function of mesh deflection δ can be expressed as
( )10δKbF
minus= for 500bFgt
minus=
500Fb10δK
bF for 500
bFlt
with γAt KK
bF
bF
sdotsdot= etc (Nmm) ie unit load incorporat-
ing the relevant factors as
KA middot Kγ for determination of Kv
KA middot Kγ middot Kv for determination of KHβ
KA middot Kγ middot Kv middot KHβ for determination of KHα
δ = mesh deflection (microm)
K = applicable stiffness (c or cγ)
Use of stiffnesses for KV KHβ and KHα
For calculation of Kv and KHα the stiffness is calculated as follows
When Fb lt 500
the stiffness is determined as δ∆
∆ bF
where the increment is chosen as eg ∆ Fb = 10 and thus
50010Fb10
K10Fb∆δ +
++
=
When F b gt 500 the stiffness is c or cγ For calculation of KHβ the mesh deflection δ is used directly
or an equivalent stiffness determined as δsdotb
F
Bevel gears
In lack of more detailed relationship between stiffness and geometry the following may be used
b085
b13cacute eff=
b085b
16c effγ =
beff not to be used in excess of 085 b in these formulae
Bevel gears with heightwise and lengthwise crowning have progressive mesh stiffness The values mentioned above are only valid for high loads They should not be used for de-termination of Ceff (see 432) or KHβ (see 1931)
112 Running-in Allowances The running-in allowances account for the influence of run-ning-in wear on the various error elements
yα respectively yβ are the running-in amounts which reduce the influence of pitch and profile errors respectively influ-ence of localised faceload
Cay is defined as the running-in amount that compensates for lack of tip relief
The following relations may be used
For not surface hardened steel
ptHlim
α fσ160y =
yβ = βxlimH
f320σ
with the following upper limits
V lt 5 ms 5-10 ms gt 10 ms
yα max none
limH
12800σ limH
6400σ
yβ max none
limH
25600σ limH
12800σ
For surface hardened steel
yα = 0075 fpt but not more than 3 for any speed
yβ = 015 Fβx but not more than 6 for any speed
16 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
For all kinds of steel
5145189718
1C2
limHay +
minusσ
=
When pinion and wheel material differ the following ap-plies
bull Use the larger of fpt1 - yα1 and fpt2 - yα2 to replace fpt - yα in the calculation of KHα and Kv
bull Use ( )2β1ββ yy21y += in the calculation of KHβ
bull Use ( )2ay1aya CC21C += in the calculation of Kv
bull Use ( )2ay1ay2a1a CC21CC +== in the scuffing
calculation if no design tip relief is foreseen
Classification Notes- No 412 17 May 2003
DET NORSKE VERITAS
2 Calculation of Surface Durability
21 Scope and General Remarks Part 2 includes the calculations of flank surface durability as limited by pitting spalling case crushing and subsurface yielding Endurance and time limited flank surface fatigue is calculated by means of 22 ndash 212 In a way also tooth frac-tures starting from the flank due to subsurface fatigue is in-cluded through the criteria in 213
Pitting itself is not considered as a critical damage for slow speed gears However pits can create a severe notch effect that may result in tooth breakage This is particularly im-portant for surface hardened teeth but also for high strength through hardened teeth For high-speed gears pitting is not permitted
Spalling and case crushing are considered similar to pitting but may have a more severe effect on tooth breakage due to the larger material breakouts initiated below the surface Subsurface fatigue is considered in 213
For jacking gears (self-elevating offshore units) or similar slow speed gears designed for very limited life the max static (or very slow running) surface load for surface hard-ened flanks is limited by the subsurface yield strength
For case hardened gears operating with relatively thin lubri-cation oil films grey staining (micropitting) may be the lim-iting criterion for the gear rating Specific calculation meth-ods for this purpose are not given here but are under consid-eration for future revisions Thus depending on experience with similar gear designs limitations on surface durability rating other than those according to 22 - 213 may be ap-plied
22 Basic Equations Calculation of surface durability (pitting) for spur gears is based on the contact stress at the inner point of single pair contact or the contact at the pitch point whichever is greater
Calculation of surface durability for helical gears is based on the contact stress at the pitch point
For helical gears with 0 lt εβ lt 1 a linear interpolation be-tween the above mentioned applies
Calculation of surface durability for spiral bevel gears is based on the contact stress at the midpoint of the zone of contact
Alternatively for bevel gears the contact stress may be cal-culated with the program ldquoBECALrdquo In that case KA and Kv are to be included in the applied tooth force but not KHβ and KHα The calculated (real) Hertzian stresses are to be multi-plied with ZK in order to be comparable with the permissible contact stresses
The contact stresses calculated with the method in part 2 are based on the Hertzian theory but do not always represent the real Hertzian stresses
The corresponding permissible contact stresses σHP are to be calculated for both pinion and wheel
221 Contact stress Cylindrical gears
( )HαHβvγA
1
tβεEHDBH KKKKK
bud1uF
ZZZZZσ+
=
where
ZBD = Zone factor for inner point of single pair contact for pinion resp wheel (see 232)
ZH = Zone factor for pitch point (see 231)
ZE = Elasticity factor (see 24)
Zε = Contact ratio factor (see 25)
Zβ = Helix angle factor (see 26)
Ft KA Kγ Kv KHβ KHα see 15 ndash 110
d1 b u see 12 ndash 15
Bevel gears
( )HαHβvγA
v1v
vmtKEMH KKKKK
bud1uF
ZZZ105σ+
sdot=
where
105 is a correlation factor to reach real Hertzian stresses (when ZK = 1)
ZE KA etc see above
ZM = mid-zone factor see 233
ZK = bevel gear factor see 27
Fmt dv1 uv see 12 ndash 15
It is assumed that the heightwise crowning is chosen so as to result in the maximum contact stresses at or near the mid-point of the flanks
222 Permissible contact stress
XWRvLH
NHlimHP ZZZZZ
SZσ
σ =
where
σH lim = Endurance limit for contact stresses (see 28)
ZN = Life factor for contact stresses (see 29)
SH = Required safety factor according to the rules
ZLZvZR = Oil film influence factors (see 210)
ZW = Work hardening factor (see 211)
ZX = Size factor (see 212)
18 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
23 Zone Factors ZH ZBD and ZM
231 Zone factor ZH The zone factor ZH accounts for the influence on contact stresses of the tooth flank curvature at the pitch point and converts the tangential force at the reference cylinder to the normal force at the pitch cylinder
wtt2
wtbH sinααcos
cosαcosβ2Z =
232 Zone factors ZBD The zone factors ZBD account for the influence on contact stresses of the tooth flank curvature at the inner point of sin-gle pair contact in relation to ZH Index B refers to pinion D to wheel
For εβ ge 1 ZBD = 1
For internal gears ZD = 1
For εβ = 0 (spur gears)
( )
π
minusεminusminus
π
minusminus
α=
α2
2
2b
2a
1
2
1b
1a
wtB
z211
dd
z21
dd
tanZ
( )
π
minusεminusminus
π
minusminus
α=
α1
2
1b
1a
2
2
2b
2a
wtD
z211
dd
z21
dd
tanZ
If ZB lt 1 use ZB = 1
If ZD lt 1 use ZD = 1
For 0 lt εβ lt 1
ZBD = ZBD (for spur gears) ndash εβ (ZBD (for spur gears) ndash 1)
233 Zone factor ZM The mid-zone factor ZM accounts for the influence of the contact stress at the mid point of the flank and applies to spi-ral bevel gears
εminusminus
εminusminus
αβ=
αα btm2
2vb2
2vabtm2
1vb2val
2v1vvtbmM
pddpdd
ddtancos2Z
This factor is the product of ZH and ZM-B in ISO 10300 with the condition that the heightwise crowning is sufficient to move the peak load towards the midpoint
234 Inner contact point For cylindrical or bevel gears with very low number of teeth the inner contact point (A) may be close to the base circle In order to avoid a wear edge near A it is required to have suitable tip relief on the wheel
24 Elasticity Factor ZE The elasticity factor ZE accounts for the influence of the material properties as modulus of elasticity and Poissonrsquos ratio on the contact stresses
For steel against steel ZE = 1898
25 Contact Ratio Factor Zε The contact ratio factor Zε accounts for the influence of the transverse contact ratio εα and the overlap ratio εβ on the contact stresses
αε1Z =ε for 1εβ ge
( )α
ββ
αε ε
εε1
3ε4
Z +minusminus
= for εβ lt 1
26 Helix Angle Factor Zβ The helix angle factor Zβ accounts for the influence of helix angle (independent of its influence on Zε) on the surface du-rability
cosβZβ =
27 Bevel Gear Factor ZK The bevel gear factor accounts for the difference between the real Hertzian stresses in spiral bevel gears and the contact stresses assumed responsible for surface fatigue (pitting) ZK adjusts the contact stresses in such a way that the same per-missible stresses as for cylindrical gears may apply
The following may be used ZK = 080
28 Values of Endurance Limit σHlim and Static Strength 5H10σ 3H10σ
σHlim is the limit of contact stress that may be sustained for 5middot107 cycles without the occurrence of progressive pitting
For most materials 5middot107 cycles are considered to be the be-ginning of the endurance strength range or lower knee of the σ-N curve (See also Life Factor ZN) However for nitrided steels 2middot106 apply
For this purpose pitting is defined by
bull for not surface hardened gears pitted area ge 2 of total active flank area
bull for surface hardened gears pitted area ge 05 of total active flank area or ge 4 of one particular tooth flank area 510Hσ and 310Hσ are the contact stresses which the given
material can withstand for 105 respectively 103 cycles without subsurface yielding or flank damages as pitting spalling or case crushing when adequate case depth applies
The following listed values for σHlim 510Hσ and 310Hσ may only be used for materials subjected to a quality control as
Classification Notes- No 412 19 May 2003
DET NORSKE VERITAS
the one referred to in the rules Results of approved fatigue tests may also be used as the basis for establishing these values The defined survival probability is 99
σHlim σH105 σH10
3
Alloyed case hardened steels (surface hardness 58-63 HRC) - of specially approved high grade - of normal grade
1650 1500
2500 2400
3100 3100
Nitrided steel of approved grade gas nitrided (surface hardness 700-800 HV) 1250 13 σHlim 13 σHlim
Alloyed quenched and tempered steel bath or gas nitrided (surface hardness 500-700 HV)
1000
13 σHlim
13 σHlim
Alloyed flame or induction hardened steel (surface hardness 500-650 HV) 075 HV + 750 16 σHlim 45 HV
Alloyed quenched and tempered steel 14 HV + 350 16 σHlim 45 HV
Carbon steel 15 HV + 250 16 σHlim 16 σHlim
These values refer to forged or hot rolled steel For cast steel the values for σHlim are to be reduced by 15
29 Life Factor ZN The life factor ZN takes account of a higher permissible contact stress if only limited life (number of cycles NL) is demanded or lower permissible contact stress if very high number of cycles apply
If this is not documented by approved fatigue tests the fol-lowing method may be used
For all steels except nitrided
7L 105N sdotge ZN = 1 or
01570
L
7
N N105Z
sdot=
Ie ZN = 092 for 1010 cycles
The ZN = 1 from 5middot107 on may only be used when the mate-rial cleanliness is of approved high grade (see Rules Pt4 Ch2) and the lubrication is optimised by a specially approved filtering process
105 lt NL lt 5middot107 510N
logZ037
L
7
N N105Z
sdot=
NL = 105 WXRVLHlim
WstX10H1010NN ZZZZZσ
ZZσZZ
55
5=
103 lt NL lt 105 )Z(Zlog05
L
5
10NN
5N103N10
5N10ZZ
==
3L 10N le
WXRVLlimH
Wst10X10H10NN ZZZZZ
ZZZZ
33
3σ
σ==
(but not less than ZN105)
For nitrided steels
6L 102N sdotge ZN = 1 or
00980
L
6
N N102Z
sdot=
Ie ZN = 092 for 1010 cycles
The ZN = 1 from 2middot106 on may only be used when the mate-rial cleanliness is of approved high grade (see Rules Pt4 Ch2) and the lubrication is optimised by a specially approved filtering process
105 lt NL lt 2middot106 510N
Zlog76860
L
6
N N102Z
sdot=
5L 10N le
XWRVL
10XWst10NN ZZZZZ
ZZ13ZZ
55 ==
Note that when no index indicating number of cycles is used the factors are valid for 5middot107 (respectively 2middot106 for nitrid-ing) cycles
210 Influence Factors on Lubrication Film ZL ZV and ZR The lubricant factor ZL accounts for the influence of the type of lubricant and its viscosity the speed factor ZV ac-counts for the influence of the pitch line velocity and the roughness factor ZR accounts for influence of the surface roughness on the surface endurance capacity
The following methods may be applied in connection with the endurance limit
20 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
Surface hardened steels Not surface hardened steels
ZL ( )24013421
360910ν+
+
( )24013421
680830ν+
+
ZV ( )v3280
140930+
+ ( )v3280300850
++
ZR
080
ZrelR3
150
ZrelR3
where
ν40 = Kinematic oil viscosity at 40ordmC (mm2s) For case hardened steels the influence of a high bulk temperature (see 4 Scuffing) should be con-sidered Eg bulk temperatures in excess of 120ordmC for long periods may cause reduced flank surface endurance limits
For values of ν40 gt 500 use ν40 = 500
RZrel = The mean roughness between pinion and wheel (after running in) relative to an equivalent radius of curvature at the pitch point ρc = 10mm
RZrel
= ( ) 31
cZ2Z1 ρ
10RR50
+
RZ = Mean peak to valley roughness (microm) (DIN defini-tion) (roughly RZ = 6 Ra)
For 5
L 10N le ZL ZV ZR = 10
211 Work Hardening Factor ZW The work hardening factor ZW accounts for the increase of surface durability of a soft steel gear when meshing the soft steel gear with a surface hardened or substantially harder gear with a smooth surface
The following approximation may be used for the endurance limit
Surface hardened steel against not surface hardened steel 150
ZeqW R
31700
130HB21Z
minus
minus=
where HB = the Brinell hardness of the soft member For HB gt 470 use HB = 470 For HB lt 130 use HB = 130
RZeq = equivalent roughness
033
c40
066
ZS
ZHZHZEQ ρvν
15000RRRR
=
If RZeq gt 16 then use RZeq = 16
If RZeq lt 15 then use RZeq = 15
where
RZH = surface roughness of the hard member before run in
RZS = surface roughness of the soft member before run in
ν40 = see 210
If values of ZW lt 1 are evaluated ZW = 1 should be used for flank endurance However the low value for ZW may indi-cate a potential wear problem
Through hardened pinion against softer wheel
( )
minussdotminus+= 000829
HBHB0008981u1Z
2
1W
For 21HBHB
2
1 le use ZW = 1
For 71HBHB
2
1 gt use 17HBHB
2
1 =
For u gt 20 use u = 20
For static strength (lt 105 cycles)
Surface hardened against not surface hardened
ZWst = 105
Through hardened pinion against softer wheel
ZWst = 1
212 Size Factor ZX The size factor accounts for statistics indicating that the stress levels at which fatigue damage occurs decrease with an increase of component size as a consequence of the influ-ence on subsurface defects combined with small stress gradi-ents and of the influence of size on material quality
ZX may be taken unity provided that subsurface fatigue for surface hardened pinions and wheels is considered eg as in the following subsection 213
213 Subsurface Fatigue This is only applicable to surface hardened pinions and wheels The main objective is to have a subsurface safety against fatigue (endurance limit) or deformation (static strength) which is at least as high as the safety SH required for the surface The following method may be used as an approximation unless otherwise documented
The high cycle fatigue (gt3106 cycles) is assumed to mainly depend on the orthogonal shear stresses Static strength (lt103 cycles) is assumed to depend mainly on equivalent
Classification Notes- No 412 21 May 2003
DET NORSKE VERITAS
stresses (von Mises) Both are influenced by residual stresses but this is only considered roughly and empirically
The subsurface working stresses at depths inside the peak of the orthogonal shear stresses respectively the equivalent stresses are only dependent on the (real) Hertzian stresses Surface related conditions as expressed by ZL ZV and ZR are assumed to have a negligible influence
The real Hertzian stresses σHR are determined as
For helical gears with εβ gt 1
σHR = σH
For helical gears with εβ lt 1 and spur gears
ε
α
ββ ε
ε+εminus
sdotσ=σZ
1
HHR
For bevel gears
KHHR Z
1σσ sdot=
The necessary hardness HV is given as a function of the net depth tz (net = after grinding or hard metal hobbing and per-pendicular to the flank)
The coordinates tz and HV are to be compared with the de-sign specification such as
bull for flame and induction hardening tHVmin HVmin bull for nitriding t400min HV = 400 bull for case hardening t550min HV = 550 t400min HV = 400
and t300min HV = 300 (the latter only if the core hardness lt 300 If the core hardness gt 400 the t400 is to be re-placed by a fictive t400 = 16 t550)
In addition the specified surface hardness is not to be less than the max necessary hardness (at tz = 05aH) This applies to all hardening methods
For high cycle fatigue (gt3 106 cycles) the following applies
sdot+
minussdotsdotsdot= o90
05azt
05azt
cosSσ04HV
H
HHHR
applicable to 05a
zt
Hge
For 05at
H
z lt the value for 05at
H
z = applies
56300ρSσ12a cHHR
Hsdotsdot
sdot=
Where aH is half the hertzian contact width multiplied by an empirical factor of 12 that takes into account the possible influence of reduced compressive residual stresses (or even tensile residual stresses) on the local fatigue strength
If any of the specified hardness depths including the surface hardness is below the curve described by HV = f (tz) the actual safety factor against subsurface fatigue is determined as follows
reduce SH stepwise in the formula for HV and aH until all specified hardness depths and surface hardness balance with the corrected curve The safety factor obtained through this method is the safety against subsurface fatigue
For static strength (lt103 cycles) the following applies
sdot+
minussdotsdotsdot= o90
07at
06a
zt
cosSσ019HV
Hst
z
HstHHR
applicable to 06at
Hst
z ge
56300ρSσa cHHR
Hstsdotsdot
=
In the case of insufficient specified hardness depths the same procedure for determination of the actual safety factor as above applies
For limited life fatigue (103 lt cycles lt 3106 )
For this purpose it is necessary to extend the correction of safety factors to include also higher values than required Ie in the case of more than sufficient hardness and depths the safety factor in the formulae for both high cycle fatigue and static strength are to be increased until necessary and speci-fied values balance
The actual safety factor for a given number of cycles N be-tween 103 and 3106 is found by linear interpolation in a dou-ble logarithmic diagram
minussdotminus
= sdot logN3477
logSlogSlogS
310H6103HHN
36 10H103H Slog86281Slog86280 sdot+sdot sdot
22 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
3 Calculation of Tooth Strength
31 Scope and General Remarks Part 3 include the calculation of tooth root strength as limited by tooth root cracking (surface or subsurface initiated) and yielding
For rim thickness sR ge 35middotmn the strength is calculated by means of 32 ndash 313 For cylindrical gears the calculation is based on the assumption that the highest tooth root tensile stress arises by application of the force at the outer point of single tooth pair contact of the virtual spur gears The method has however a few limitations that are mentioned in 36
For bevel gears the calculation is based on force application at the tooth tip of the virtual cylindrical gear Subsequently the stress is converted to load application at the mid point of the flank due to the heightwise crowning
Bevel gears may also be calculated with the program BECAL In that case KA and Kv are to be included in the applied tooth force but not KFβ and KFα
In case of a thin annulus or a thin gear rim etc radial crack-ing can occur rather than tangential cracking (from root fillet to root fillet) Cracking can also start from the compression fillet rather than the tension fillet For rim thickness sR lt 35middotmn a special calculation procedure is given in 315 and 316 and a simplified procedure in 314
A tooth breakage is often the end of the life of a gear trans-mission Therefore a high safety SF against breakage is re-quired
It should be noted that this part 3 does not cover fractures caused by
bull oil holes in the tooth root space bull wear steps on the flank bull flank surface distress such as pits spalls or grey staining
Especially the latter is known to cause oblique fractures starting from the active flank predominately in spiral bevel gears but also sometimes in cylindrical gears
Specific calculation methods for these purposes are not given here but are under consideration for future revisions Thus depending on experience with similar gear designs limita-tions other than those outlined in part 3 may be applied
32 Tooth Root Stresses The local tooth root stress is defined as the max principal stress in the tooth root caused by application of the tooth force Ie the stress ratio R = 0 Other stress ratios such as for eg idler gears (R asymp -12) shrunk on gear rims (R gt 0) etc are considered by correcting the permissible stress level
321 Local tooth root stress The local tooth root stress for pinion and wheel may be as-sessed by strain gauge measurements or FE calculations or similar For both measurements and calculations all details are to be agreed in advance
Normally the stresses for pinion and wheel are calculated as
Cylindrical gears
FαFβvγAβSFn
tF K K K K K Y Y Y
m bFσ =
where
YF = Tooth form factor (see 33)
YS = Stress correction factor (see 34)
Yβ = Helix angle factor (see 36)
Ft KA Kγ Kv KFβ KFα see 15 ndash 110
b see 13
Bevel gears
FαFβvγASaFamn
mtF K K K K K Y Y Y
m bFσ ε=
where
YFa = Tooth form factor see 33
YSa = Stress correction factor see 34
Yε = Contact ratio factor see 35
Fmt KA etc see 15 ndash 110
b see 13
322 Permissible tooth root stress The permissible local tooth root stress for pinion respectively wheel for a given number of cycles N is
CXRrelTrelTF
NMFEFP YYYY
SYY
δσ
=σ
Note that all these factors YM etc are applicable to 3middot106 cycles when used in this formula for σFP The influence of other number of cycles on these factors is covered by the calculation of YN
where
σFE = Local tooth root bending endurance limit of reference test gear (see 37)
YM = Mean stress influence factor which accounts for other loads than constant load direction eg idler gears temporary change of load di-rection pre-stress due to shrinkage etc (see 38)
YN = Life factor for tooth root stresses related to reference test gear dimensions (see 39)
SF = Required safety factor according to the rules
YδrelT = Relative notch sensitivity factor of the gear to be determined related to the reference test gear (see 310)
YRrelT = Relative (root fillet) surface condition factor of the gear to be determined related to the reference test gear (see 311)
Classification Notes- No 412 23 May 2003
DET NORSKE VERITAS
YX = Size factor (see 312)
YC = Case depth factor (see 313)
33 Tooth Form Factors YF YFa The tooth form factors YF and YFa take into account the in-fluence of the tooth form on the nominal bending stress
YF applies to load application at the outer point of single tooth pair contact of the virtual spur gear pair and is used for cylindrical gears
YFa applies to load application at the tooth tip and is used for bevel gears
Both YF and YFa are based on the distance between the con-tact points of the 30˚-tangents at the root fillet of the tooth profile for external gears respectively 60˚ tangents for inter-nal gears
Figure 31 External tooth in normal section
Figure 32 Internal tooth in normal section
Definitions
n
2
n
Fn
enFn
Fe
F
α cosms
α cosmh6
Y
=
n
2
n
Fn
anFn
Fa
Fa
α cosms
α cosmh6
Y
=
In the case of helical gears YF and YFa are determined in the normal section ie for a virtual number of teeth
YFa differs from YF by the bending moment arm hFa and αFan and can be determined by the same procedure as YF with exception of hFe and αFan For hFa and αFan all indices e will change to a (tip)
The following formulae apply to cylindrical gears but may also be used for bevel gears when replacing
mn with mnm
zn with zvn
αt with αvt
β with βm
with undercut without undercut
Fig 33 Dimensions and basic rack profile of the teeth (finished profile)
Tool and basic rack data such as hfP ρfp and spr etc are re-ferred to mn ie dimensionless
331 Determination of parameters
( )n
n
prnfPnfP m
α cossαsin 1ρ
αtan h4πE
minusminusminusminus=
For external gears fPfP ρρ =
For internal gears ( )0z
195fPfP0
fPfP 10363156ρhxρρ
sdotminus+
+=
where
z0 = number of teeth of pinion cutter
x0 = addendum modification coefficient of pinion cutter
hfP = addendum of pinion cutter
ρfP = tip radius of pinion cutter
xhρG fpfp +minus=
24 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
τmE
2π
z2H
nnminus
minus=
with
3πτ = for external gears
6πτ = for internal gears
Htan zG2
nminusϑ=ϑ
(to be solved iteratively suitable start value6π
=ϑ for exter-
nal gears and 3π for internal gears)
a) Tooth root chord sFn For external gears
minus
ϑ+
ϑminus= ρ
cosG3
3πsinz
ms
fpnn
Fn
For bevel gears with a tooth thickness modification
xsm affects mainly sFn but also hFe and αFen The total influence of xsm on YFa Ysa can be approximated by only adding 2 xsm to sFn mn
For internal gears
minus
ϑ+
ϑminus= ρ
cosG
6πsinz
ms
fPnn
Fn
b) Root fillet radius ρF at 30ordm tangent
( )G2coszcosG2ρ
mρ
2n
2
fpn
F
minusϑϑ+=
c) Determination of bending moment arm hF dn = zn mn
b2α
αn βcosε
ε =
dan = dn + 2 ha
pbn = π mn cos αn
dbn = dn cos αn
( )4
d1εpzz
2dd
zz2d
2bn
2
αnbn
2bn
2an
en +
minusminus
minus=
en
bnen d
dcos arcα =
ennnn
e α invα invα x tan 22π
z1
minus+
+=γ
αFen = αen ndash γe
For external gears
( )
minus=
n
enFenee
n
Fe
mdαtan sin γ γcos
21
mh
]ρcos
G3πcosz fpn +
ϑminus
ϑminusminus
For internal gears
( ) minus
sdotsdotminus=
n
enFenee
n
Fe
mdα tanγ sinγ cos
21
mh
minus
ϑminus
ϑminussdot ρ
cosG3
6πcosz fPn
332 Gearing with εαn gt 2 For deep tooth form gearing ( )25ε2 αn lele produced with a verified grade of accuracy of 4 or better and with applied profile modification to obtain a trapezoidal load distribution along the path of contact the YF may be corrected by the factor YDT as
250ε205for 0666ε2366Y αnαnDT leleminus=
205εfor 10Y αnDT lt=
34 Stress Correction Factors YS YSa The stress correction factors YS and YSa take into account the conversion of the nominal bending stress to the local tooth root stress Thereby YS and YSa cover the stress increasing effect of the notch (fillet) and the fact that not only bending stresses arise at the root A part of the local stress is inde-pendent of the bending moment arm This part increases the more the decisive point of load application approaches the critical tooth root section
Therefore in addition to its dependence on the notch radius the stress correction is also dependent on the position of the load application ie the size of the bending moment arm
YS applies to the load application at the outer point of single tooth pair contact YSa to the load application at tooth tip
YS can be determined as follows
( )
+
+=L23121
1
sS qL01312Y
where Fe
Fn
hsL = and
F
Fns ρ2
sq = (see 33)
YSa can be calculated by replacing hFe with hFa in the above formulae
Note a) Range of validity 1 lt qs lt 8
In case of sharper root radii (ie produced with tools having too sharp tip radii) YS resp YSa must be specially considered
b) In case of grinding notches (due to insufficient protuberance of the hob) YS resp YSa can rise considerably and must be multiplied with
Classification Notes- No 412 25 May 2003
DET NORSKE VERITAS
g
g
ρt
0613
13
minus
where
tg = depth of the grinding notch
ρg = radius of the grinding notch
c) The formulae for YS resp YSa are only valid for αn = 20˚ However the same formulae can be used as a safe approximation for other pressure angles
35 Contact Ratio Factor Yε The contact ratio factor Yε covers the conversion from load application at the tooth tip to the load application at the mid point of the flank (heightwise) for bevel gears
The following may be used
Yε = 0625
36 Helix Angle Factor Yβ The helix angle factor Yβ takes into account the difference between the helical gear and the virtual spur gear in the nor-mal section on which the calculation is based in the first step In this way it is accounted for that the conditions for tooth root stresses are more favourable because the lines of contact are sloping over the flank
The following may be used (β input in degrees)
Yβ = 1 ndash εβ β120
When εβ gt 1 use εβ = 1 and when β gt 30deg use β = 30deg in the formula
However the above equation for Yβ may only be used for gears with β gt 25deg if adequate tip relief is applied to both pinion and wheel (adequate = at least 05 middot Ceff see 432)
37 Values of Endurance Limit σFE σFE is the local tooth root stress (max principal) which the material can endure permanently with 99 survival prob-ability 3106 load cycles is regarded as the beginning of the endurance limit or the lower knee of the σ ndash N curve σFE is defined as the unidirectional pulsating stress with a minimum stress of zero (disregarding residual stresses due to heat treatment) Other stress conditions such as alternating or pre-stressed etc are covered by the conversion factor YM
σFE can be found by pulsating tests or gear running tests for any material in any condition If the approval of the gear is to be based on the results of such tests all details on the testing conditions have to be approved by the Society Fur-ther the tests may have to be made under the Societys su-pervision
If no fatigue tests are available the following listed values for σFE may be used for materials subjected to a quality con-trol as the one referred to in the rules
σFE
Alloyed case hardened steels 1) (fillet surface hardness 58 ndash 63 HRC)
bull of specially approved high grade
1050
bull of normal grade
minus CrNiMo steels with approved process
1000
minus CrNi and CrNiMo steels generally 920 minus MnCr steels generally 850
Nitriding steel of approved grade quenched tempered and gas nitrided (surface hardness 700 ndash 800 HV)
840
Alloyed quenched and tempered steel bath or gas nitrided (surface hardness 500 ndash 700 HV)
720
Alloyed quenched and tempered steel flame or induction hardened 2) (incl entire root fillet) (fillet surface hardness 500 ndash 650 HV)
07 HV + 300
Alloyed quenched and tempered steel flame or induction hardened (excl entire root fillet) (σB = uts of base material)
025 σB + 125
Alloyed quenched and tempered steel 04 σB + 200
Carbon steel 025 σB + 250
Note All numbers given above are valid for separate forgings and for blanks cut from bars forged according to a qualified procedure see Pt 4 Ch 2 Sec 3 For rolled steel the values are to be reduced with 10 For blanks cut from forged bars that are not qualified as mentioned above the values are to be reduced with 20 For cast steel reduce with 40
1) These values are valid for a root radius
bull being unground If however any grinding is made in the root fillet area in such a way that the residual stresses may be affected σFE is to be reduced by 20 (If the grinding also leaves a notch see 34)
bull with fillet surface hardness 58 ndash 63 HRC In case of lower surface hardness than 58 HRC σFE is to be reduced with 20(58 ndash HRC) where HRC is the detected hardness (This may lead to a permissible tooth root stress that varies along the facewidth If so the actual tooth root stresses may also be considered along facewidth)
bull not being shot peened In case of approved shot peening σFE may be increased by 200 for gears where σFE is reduced by 20 due to root grinding Otherwise σFE may be increased by 100 for
6nm le and 100 ndash 5 (mn - 6) for mn gt 6 However the possible adverse influence on the flanks regarding grey staining should be considered and if necessary the flanks should be masked
2) The fillet is not to be ground after surface hardening Regarding possible root grinding see 1)
26 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
38 Mean stress influence Factor YM The mean stress influence factor YM takes into account the influence of other working stress conditions than pure pulsa-tions (R = 0) such as eg load reversals idler gears planets and shrink-fitted gears
YM (YMst) are defined as the ratio between the endurance (or static) strength with a stress ratio R ne 0 and the endurance (or static) strength with R = 0
YM and YMst apply only to a calculation method that assesses the positive (tensile) stresses and is therefore suitable for comparison between the calculated (positive) working stress σF and the permissible stress σFP calculated with YM or YMst
For thin rings (annulus) in epicyclic gears where the com-pression fillet may be decisive special considerations apply see 316
The following method may be used within a stress ratio ndash12 lt R lt 05
381 For idlers planets and PTO with ice class
M1M1R1
1Yor Y MstM
+minus
minus=
where
R = stress ratio = min stress divided by max stress
For designs with the same force applied on both forward- and back-flank R may be assumed to ndash 12
For designs with considerably different forces on forward- and back-flank such as eg a marine propulsion wheel with a power take off pinion R may be assessed as
branchmain theoffacewidth unit per forcepto offacewidt unit per force21minus
For a power take off (PTO) with ice class see 161 c
M considers the mean stress influence on the endurance (or static) strength amplitudes
M is defined as the reduction of the endurance strength am-plitude for a certain increase of the mean stress divided by that increase of the mean stress
Following M values may be used
Endurance limit
Static strength
Case hardened 08 ndash 015 Ys 1) 07
If shot peened 04 06
Nitrided 03 03
Induction or flame hardened
04 06
Not surface hardened steel 03 05
Cast steels 04 06 1)For bevel gears use Ys=2 for determination of M
The listed M values for the endurance limit are independent of the fillet shape (Ys) except for case hardening In princi-ple there is a dependency but wide variations usually only occur for case hardening eg smooth semicircular fillets versus grinding notches
382 For gears with periodical change of rotational direction For case hardened gears with full load applied periodically in both directions such as side thrusters the same formula for YM as for idlers (with R = ndash 12) may be used together with the M values for endurance limit This simplified approach is valid when the number of changes of direction exceeds 100 and the total number of load cycles exceeds 3middot106
For gears of other materials YM will normally be higher than for a pure idler provided the number of changes of direction is below 3middot106 A linear interpolation in a diagram with loga-rithmic number of changes of direction may be used ie from YM = 09 with one change to YM (idler) for 3middot106 changes This is applicable to YM for endurance limit For static strength use YM as for idlers
For gears with occasional full load in reversed direction such as the main wheel in a reversing gear box YM = 09 may be used
383 For gears with shrinkage stresses and unidirectional load For endurance strength
FE
fitM σ
σM1
M21Y+
minus=
σFE is the endurance limit for R = 0
For static strength YMst = 1 and σfit accounted for in 39b
σfit is the shrinkage stress in the fillet (30˚ tangent) and may be found by multiplying the nominal tangential (hoop) stress with a stress concentration factor
n
Ffit m
ρ215scf minus=
384 For shrink-fitted idlers and planets When combined conditions apply such as idlers with shrink-age stresses the design factor for endurance strength is
( ) ( ) FE
fitM σ
σR1M1
M2
M1M1R1
1Yminussdot+
minus
+minus
minus=
Symbols as above but note that the stress ratio R in this par-ticular connection should disregard the influence of σfit ie R normally equal ndash 12
For static strength
M1M1R1
1YMst
+minus
minus=
The effect of σfit is accounted for in 39 b
Classification Notes- No 412 27 May 2003
DET NORSKE VERITAS
385 Additional requirements for peak loads The total stress range (σmax ndash σmin) in a tooth root fillet is not to exceed
F
y
Sσ225
for not surface hardened fillets
FSHV5 for surface hardened fillets
39 Life Factor YN The life factor YN takes into account that in the case of limited life (number of cycles) a higher tooth root stress can be permitted and that lower stresses may apply for very high number of cycles
Decisive for the strength at limited life is the σ ndash N ndash curve of the respective material for given hardening module fillet radius roughness in the tooth root etc Ie the factors YδrelT YRelT YX and YM have an influence on YN
If no σ ndash N ndash curve for the actual material and hardening etc is available the following method may be used
Determination of the σ ndash N ndash curve
a) Calculate the permissible stress σFP for the beginning of the endurance limit (3middot106 cycles) including the influ-ence of all relevant factors as SF YδrelT YRelT YX YM and YC ie σFP = σFE middotYM middotYδrelT middotYRelT middotYX middotYC SF
b) Calculate the permissible laquostaticraquo stress (le 103 load cy-cles) including the influence of all relevant factors as SFst YδrelTst YMst and YCst ( )fitσCstYδrelTstYMstYFstσ
FstS1
FPstσ minussdotsdotsdot=
where σFst is the local tooth root stress which the material can resist without cracking (surface hardened materials) or unacceptable deformation (not surface hardened materials) with 99 survival probability
σFst
Alloyed case hardened steel 1) 2300
Nitriding steel quenched tempered and gas nitrided (surface hardness 700 ndash 800 HV)
1250
Alloyed quenched and tempered steel bath or gas nitrided (surface hardness 500 ndash 700 HV)
1050
Alloyed quenched and tempered steel flame or induction hardened (fillet surface hardness 500 ndash 650 HV)
18 HV + 800
Steel with not surface hardened fillets the smaller value of 2)
18 σB or 225 σy
1) This is valid for a fillet surface hardness of 58 ndash 63 HRC In case of lower fillet surface hardness than 58 HRC σFst is to be reduced with 30(58 ndash HRC) where HRC is the actual hardness Shot peening or grinding notches are not considered to have any significant influence on σFst 2) Actual stresses exceeding the yield point (σy or σ02) will alter the residual stresses locally in the ldquotensionrdquo fillet re-spectively ldquocompressionrdquo fillet This is only to be utilised for gears that are not later loaded with a high number of cycles at lower loads that could cause fatigue in the ldquocompressionrdquo fillet
c) Calculate YN as
NL gt 3middot106
YN = 1 or 001
L
6
N N103Y
sdot= ie Yn = 092 for 1010
The YN = 1 from 3middot106 on may only be used when special material cleanness applies see rules Pt4 Ch2
103ltNLlt3middot106exp
L
6
N N103Y
sdot=
cycles6103forσcycles310forσlog02876exp
FP
FPst
sdot=
NL lt 103 cycles6103forσcycles310forσ
NYFP
FPst
sdot=
or simply use σFPst as mentioned in b) directly
Guidance on number of load cycles NL for various applica-tions
bull For propulsion purpose normally NL = 1010 at full load (yachts etc may have lower values)
bull For auxiliary gears driving generators that normally operate with 70-90 of rated power NL = 108 with rated power may be applied
28 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
310 Relative Notch Sensitivity Factor YδrelT The dynamic (respectively static) relative notch sensitivity factor YδrelT (YδrelTst) indicate to which extent the theoreti-cally concentrated stress lies above the endurance limits (re-spectively static strengths) in the case of fatigue (respectively overload) breakage
YδrelT is a function of the material and the relative stress gra-dient It differs for static strength and endurance limit
The following method may be used
For endurance limit
for not surface hardened fillets
( )
024
s024
relT σ103133q21σ1012201351
Ysdotsdotminus
+sdotsdotminus+= minus
minus
δ
for all surface hardened fillets except nitrided
( )
106q21002451
Y srelT
++=δ
for nitrided fillets
( )
1347q2101421
Y srelT
++=δ
For static strength
for not surface hardened fillets1)
( )( )( )025
02
02502s
relTst σ3000821σ300 1Y0821Y
+
minus+=δ
for surface hardened fillets except nitrided
YδrelTst = 044 YS + 012
for nitrided fillets
YδrelTst = 06 + 02 YS 1) These values are only valid if the local stresses do not
exceed the yield point and thereby alter the residual stress level See also 39b footnote 2
311 Relative Surface Condition Factor YRrelT The relative surface condition factor YRrelT takes into ac-count the dependence of the tooth root strength on the sur-face condition in the tooth root fillet mainly the dependence on the peak to valley surface roughness
YRrelT differs for endurance limit and static strength
The following method may be used
For endurance limit
YRrelT = 1675 ndash 053 (Ry + 1)01 for surface hardened steels and alloyed quenched and tempered steels except nitrided
YRrelT = 53 ndash 42 (Ry + 1)001 for carbon steels
YδrelT = 43 ndash 326 (Ry + 1)0005
for nitrided steels
For static strength
YRrelTst = 1 for all Ry and all materials
For a fillet without any longitudinal machining trace Ry asymp Rz
312 Size Factor YX The size factor YX takes into account the decrease of the strength with increasing size YX differs for endurance limit and static strength
The following may be used
For endurance limit
YX = 1 for mn le 5 generally
YX = 103 ndash 0006 mn for 5 lt mn lt 30
YX = 085 for mn ge 30
for not surface hard-ened steels
YX = 105 ndash 001 mn for 5 lt mn ge 25
YX = 08 for mn ge 25
for surface hardened steels
For static strength
YXst = 1 for all mn and all materials
313 Case Depth Factor YC The case depth factor YC takes into account the influence of hardening depth on tooth root strength
YC applies only to surface hardened tooth roots and is dif-ferent for endurance limit and static strength
In case of insufficient hardening depth fatigue cracks can develop in the transition zone between the hardened layer and the core For static strength yielding shall not occur in the transition zone as this would alter the surface residual stresses and therewith also the fatigue strength
The major parameters are case depth stress gradient permis-sible surface respectively subsurface stresses and subsurface residual stresses
The following simplified method for YC may be used
YC consists of a ratio between permissible subsurface stress (incl influence of expected residual stresses) and permissible surface stress This ratio is multiplied with a bracket con-taining the influence of case depth and stress gradient (The empirical numbers in the bracket are based on a high number of teeth and are somewhat on the safe side for low number of teeth)
YC and YCst may be calculated as given below but calculated values above 10 are to be put equal 10
For endurance limit
+
+=nFFE
C m02ρt31
σconstY
Classification Notes- No 412 29 May 2003
DET NORSKE VERITAS
For static strength
+
+=nFFst
Cst m02ρt31
σconstY
where const and t are connected as
Hardening process
t = endurance limit const =
static strength const =
t550 640 1900
t400 500 1200 Case hard-ening
t300 380 800
Nitriding t400 500 1200
Induction- or flame hardening
tHVmin 11 HVmin 25 HVmin
For symbols see 213
In addition to these requirements to minimum case depths for endurance limit some upper limitations apply to case hard-ened gears
The max depth to 550 HV should not exceed
1) 13 of the top land thickness san unless adequate tip relief is applied (see 110)
2) 025 mn If this is exceeded the following applies addi-tionally in connection with endurance limit
minusminus= 025
mt
1Yn
max550C
314 Thin rim factor YB Where the rim thickness is not sufficient to provide full sup-port for the tooth root the location of a bending failure may be through the gear rim rather than from root fillet to root fillet
YB is not a factor used to convert calculated root stresses at the 30deg tangent to actual stresses in a thin rim tension fillet Actually the compression fillet can be more susceptible to fatigue
YB is a simplified empirical factor used to de-rate thin rim gears (external as well as internal) when no detailed calcula-tion of stresses in both tension and compression fillets are available
Figure 34 Examples on thin rims
YB is applicable in the range 175 lt sRmn lt 35
YB = 115 middot ln (8324 middot mnsR)
(for sRmn ge 35 YB = 1)
(for sRmn le 175 use 315)
σF as calculated in 321 is to be multiplied with YB when sRmn lt 35 Thus YB is used for both high and low cycle fatigue
Note This method is considered to be on the safe side for external gear rims However for internal gear rims without any flange or web stiffeners the method may not be on the safe side and it is advised to check with the method in 315316
315 Stresses in Thin Rims For rim thickness sR lt 35 mn the safety against rim cracking has to be checked
The following method may be used
3151 General The stresses in the standardised 30ordm tangent section tension side are slightly reduced due to decreasing stress correction factor with decreasing relative rim thickness sRmn On the other hand during the complete stress cycle of that fillet a certain amount of compression stresses are also introduced The complete stress range remains approximately constant Therefore the standardised calculation of stresses at the 30ordm tangent may be retained for thin rims as one of the necessary criteria
The maximum stress range for thin rims usually occurs at the 60ordm ndash 80ordm tangents both for laquotensionraquo and laquocompressionraquo side The following method assumes the 75ordm tangent to be the decisive Therefore in addition to the am criterion ap-plied at the 30ordm tangent it is necessary to evaluate the max and min stresses at the 75ordm tangent for both laquotensionraquo (loaded flank) fillet and laquocompressionraquo (back-flank) fillet For this purpose the whole stress cycle of each fillet should be considered but usually the following simplification is justified
Figure 35 Nomenclature of fillets
Index laquoTraquo means laquotensileraquo fillet laquoCraquo means laquocompressionraquo fillet
σFTmin and σFCmax are determined on basis of the nominal rim stresses times the stress concentration factor Y75
σFCmin and σFTmax are determined on basis of superposition of nominal rim stresses times Y75 plus the tooth bending stresses at 75ordm tangent
30 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
3152 Stress concentration factors at the 75ordm tangents The nominal rim stress consists of bending stresses due to local bending moments tangential stresses due to the tan-gential force Ft and radial shear stresses due to Fr
The major influence is given by the bending stresses The influence of the tangential stresses is minor and even though its stress concentration factor is slightly higher than for bending it is considered to be safe enough when the sum of these nominal stresses are combined with the stress concen-tration factor for bending The influence of the radial shear stress is neglected
The stress concentration factor relating nominal rim stresses to local fillet stresses at the 75ordm tangent may be calculated as
75n
R75
n
R
75
ρmsρ185
ms3
Y+
sdot=
where ρ75 is the root radius at the 75ordm tangent ref to mn Usually ρ75 is closed to the tool radius ρao and
ρ75 = ρao
is a safe approximation compensating for the am simplifi-cations to the laquounsaferaquo side
The tooth root stresses of the loaded tooth are decreasing with decreasing relative rim thickness approximately with the empirical factor
15
R
ncorr s
m311Y
minus=
3153 Nominal rim stresses The bending moment applied to the rim consists of a part of the tooth tilt moment Ft (hF + 05 sR) and the bending caused by the radial force Fr
The sectional modulus (first moment of area) which is used for determination of the nominal bending stresses is not nec-essarily the same for the 2 am bending moments If flanges webs etc outside the toothed section contribute to stiffening the rim against various deflections the influence of these stiffeners should be considered Eg an end flange will have an almost negligible influence on the effective sectional modulus for the stresses due to tooth tilt as the deflection caused by the tilt moment is rather small and would not much involve the flange On the other hand the radial forces as for instance from the meshes in an annulus would cause considerable radial deflections that the flange might restrict to a substantial amount When considering the stiff-ening of such flanges or webs on basis of simplified models it is advised to use an effective rim thickness sR = sR + 02 mn for the first moment of area of the rim (toothed part) cross section
For a high number of rim teeth it may be assumed that the rim bending moments in the fillets adjacent to the loaded tooth are of the same magnitude as right under the applied
force This assumption is reasonable for an annulus but rather much on the safe side for a hollow pinion
The influence of Ft on the nominal tangential stress is simpli-fied by half of it for compressive stresses (σ1) and the other half for tensile stresses (σ2) Applying these assumptions the nominal rim stresses adjacent to the loaded tooth are
( ) ( )A2
FW
f RFW
s05hF05 t
R
r
T
RFt1 minus
ϑminus
+minus=σ
( ) ( )A2
FW
f RFW
s05hF05 t
R
r
T
RFt2 +
ϑminus
+=σ
where σ1 σ2 see Fig 35
A = minimum area of cross section (usually bR sR)
WT = the sectional modulus of rim with respect to tooth tilt moment (usually 6sb 2
rR )
WR = the sectional modulus of the rim including the influence of stiffeners as flanges webs etc (WR ge WT)
bR = the width of the rim
R = the radius of the neutral axis in the rim ie from wheel centre to midpoint of rim
( )ϑf = a function for bending moment distribution around the rim For a rim (pinion) with one mesh only the ( )ϑf at the position of load application is 024 For an annulus with 3 or more meshes ( )ϑf at the position of each load application is approx 3 planets ( )ϑf = 019 4 planets ( )ϑf = 014 5 planets ( )ϑf = 011 6 planets ( )ϑf = 009
It must be checked if the max (tensile) stress in the compres-sion fillet really occurs when the fillet is adjacent to the loaded tooth In principle the stress variation through a complete rotation should be considered and the max value used The max value is usually never less than 0 For an annulus eg the tilt moment is zero in the mid position be-tween the planet meshes whilst the bending moment due to the radial forces is half of that at the mesh but with opposite sign
If these formulae are applied to idler gears as eg planets the influence of nominal tangential stresses must be cor-rected by deleting Ft(2 A) for σ1 and using FtA for σ2 Further the influence of Fr on the nominal bending stresses is usually negligible due to the planet bearing support
3154 Root fillet stresses Determination of min and max stresses in the laquotensionraquo fillet
Classification Notes- No 412 31 May 2003
DET NORSKE VERITAS
Minimum stress
751FTmin YσK σ =
Maximum stress
corrF752 Yσ03Yσ KσFTmax +=
where
03 is an empirical factor relating the tension stresses (σF) at the 30ordm tangent to the part of the tension stresses at the 75ordm tangent which add to the rim related stresses (03 also takes into account that full superposition of nominal stresses times stress concentration factors from both laquosides of the corner filletraquo would result in too high stresses)
αβγ sdotsdotsdotsdot= FFvA KKKKKK
Determination of min and max stresses in the laquocompres-sionraquo fillet
Minimum stress
corrF751FCmin Yσ036Yσ Kσ minus=
Maximum stress
752FCmax Yσ Kσ =
where
036 is an empirical factor relating the tension stresses (σF) at the 30ordm tangent to the part of the compression stresses at the 75ordm tangent which add to the rim related stresses
For gears with reversed loads as idler gears and planets there is no distinct laquotensionraquo or laquocompressionraquo fillet The mini-mum stress σFmin is the minimum of σFTmin and σFCmin (usually the latter is decisive) The maximum stress σFTmax is the maximum of σFTmax and σFCmax (usually the former is deci-sive)
316 Permissible Stresses in Thin Rims
3161 General The safety against fatigue fracture respectively overload fracture is to be at least at the same level as for solid gears The laquoordinaryraquo criteria at the 30ordm tangent apply as given in 31 through 313
Additionally the following criteria at the 75ordm tangent may apply
3162 For gt3middot106 cycles The permissible stresses for the laquotensionraquo fillets and for the laquocompressionraquo fillets are determined by means of a relevant fatigue diagram
If the actual tooth root stress (tensile or compressive) ex-ceeds the yield strength to the material the induced residual stresses are to be taken into account
For determination of permissible stresses the following is defined
R = stress ratio ie FTmax
FTminσσ respectively
FCmax
FCmin
σσ
∆σ = stress range ie σFTmax ndash σFTmin resp σFCmax ndash σFCmin
(For idler gears and planets Fmax
FminσσR =
and ∆σ = σFmax minus σFmin)
The permissible stress range ∆σρ for the laquotensionraquo respec-tively laquocompressionraquo fillets can be calculated as
For R gt minus1 FPp σ
R1R1031
13∆σ
minus+
+=
For minus infin lt R lt minus1 FPp σ
R1R10151
13∆σ
minus+
+=
where
σFP = see 32 determined for unidirectional stresses (YM = 1)
If the yield strength σy is exceeded in either tension or com-pression residual stresses are induced This may be consid-ered by correcting the stress ratio R for the respective fillets (tension or compression)
Eg if yFCmin σσ gt (ie exceeded in compression) the
difference yFCmin σσ∆ minus= affects the stress ratio as
∆σ
σ∆σ∆σR
FCmax
y
FCmax
FCmin
+
minus=
++
=
Similarly the stress ratio in the tension fillet may require cor-rection
If the yield strength is exceeded in tension σFTmax gt σy the difference ∆ = σFTmax ndash σy affects the stress ratio as
y
FTmin
FTmax
FTmin
σ∆σ
∆σ∆σR minus=
minusminus
=
Checking for possible exceeding of the yield strength has to be made with the highest torque and with the KAP if this ex-ceeds KA
3163 Forle 103 cycles The permissible stress range ∆σp is not to exceed
For R gt minus1 FPstpst σ
R1R1051
15∆σ
minus+
+=
For ndash infin lt R lt minus1 FPstpst σ
R1R10251
15∆σ
minus+
+=
For all values of R ∆σpst is limited by
not surface hardened F
y
Sσ225
32 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
surface hardened CF
YSHV5
Definition of ∆σ and R see 3162 with particular attention to possible correction of R if the yield strength is exceeded
σFPst see 32 determined for unidirectional stresses (YM = 1) and lt 103 cycles
3164 For 103 lt cycles lt 3106 ∆σp is to be determined by linear interpolation a log-log dia-gram
∆σp at NL load cycles is
6L 103p
exp
L
6
N p ∆σN103∆σ sdot
sdot=
6
3
103p
10p
∆σ
∆σlog28760exp
sdot
=
Classification Notes- No 412 33 May 2003
DET NORSKE VERITAS
4 Calculation of Scuffing Load Capacity
41 Introduction High surface temperatures due to high loads and sliding ve-locities can cause lubricant films to break down This sei-zure or welding together of areas of tooth surface is termed scuffing
In contrast to pitting and fatigue breakage which show a dis-tinct incubation period a single short overloading can lead to a scuffing failure In the ISO-TR13989 two criteria are mentioned The method used in this Classification Note is based on the principles of the flash temperature criterion
Note Bulk temperature in excess of 120ordmC for long periods may have an adverse effect on the surface durability see 211
42 General Criteria In no point along the path of contact the local contact tem-perature may exceed the permissible temperature ie
oilS
oilSB S
ϑ+ϑminusϑ
leϑ
50SB minusϑleϑ
where
Bϑ = max contact temperature along the path of contact
maxflaMBB ϑ+ϑ=ϑ
MBϑ = bulk temperature see 434
maxflaϑ = max flash temperature along the path of con-tact see 44
Sϑ = scuffing temperature see below
oilϑ = oil temperature before it reaches the mesh (max applicable for the actual load case to be used ie normally alarm temperature except for ice classes where a max expected tem-perature applies)
SS = required safety factor according to the Rules
The scuffing temperature Sϑ may be calculated as
L2
wrelT
002
40S XFZGX
ν100112085780
sdot
sdot++=ϑ
where
XwrelT = relative welding factor
XwrelT
Through hardened steel 10
Phosphated steel 125
Copper-plated steel 150
Nitrided steel 150
Less than 10 retained austenite
115
10 ndash 20 retained austenite 10
Casehardened steel
20 ndash 30 retained austenite 085
Austenitic steel 045
FZG = load stage according to FZG-Test A8390 (Note This is the load stage where scuffing oc-curs However due to scatter in test results cal-culations are to be made with one load stage less than the specification)
XL = lubricant factor
= 10 for mineral oils
= 08 for polyalfaolefins
= 07 for non-water-soluble polyglycols
= 06 for water-soluble polyglycols
= 15 for traction fluids
= 13 for phosphate esters
ν40 = kinematic oil viscosity at 40˚C (mm2s)
Application of other test methods such as the Ryder the FZG-Ryder R46574 and the FZG L-42 Test 141195110 may be specially considered
For high speed gears with very short time of contact Sϑ may be increased as follows provided use of EP-oils
Addition to the calculated scuffing temperature Sϑ
If micros 18tc ge no addition
If micros 18tc lt add ( )cwrelT t18X18 minussdotsdot
where
ct = contact time ( micros ) which is the time needed to cross the Hertzian contact width
( ) [ ]microsu1cosβn
uσ340tb1
Hc +sdotsdot
sdot=
σH as calculated in 221
For bevel gears use vu in stead of u
34 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
43 Influence Factors
431 Coefficient of friction The following coefficient of friction may apply
L025a
005oil
02
redC ΣC
Bt X R ηρv
w0048micro minus
=
where
wBt = specific tooth load (Nmm)
vΣC = sum of tangential velocities at pitch point At pitch line velocities gt 50 ms the limiting value of vΣC at v = 50 ms is to be used
ρredC = relative radius of curvature (transversal plane) at the pitch point
Cylindrical gears
HαHβvγAbt
Bt KKKKKb
Fw sdotsdotsdotsdotsdot= (see 1)
twΣC αsin v2v =
bCredC β cos ρρ =
Bevel gears
HαHβvγAtmb
Bt KKKKKb
Fw sdotsdotsdotsdotsdot= (see 1)
vtmtΣC αsin v2v =
bmvCredC β cos ρρ =
ηoil = dynamic viscosity (mPa s) at oilϑ calculated as
1000
ρ νη oiloil =
where ρ in kgm3 approximated as
( ) 70 15oil15 minusϑminusρ=ρ and νoil is kinematic viscosity at oilϑ and may be calculated by means of the following equation
( ) ( )++=+ 08νloglog08νloglog 100oil ( )
sdotminus
ϑ+minus313log373log
273log373log oil
( ) ( )( )08νloglog08νloglog 10040 +minus+
Ra = composite arithmetic mean roughness (micron) of pinion an wheel calculated as ( )2a1aa RR 05R +=This is defined as the roughness on the new flanks ie as manufactured
XL = see 42
432 Effective tip relief Ceff Ceff is the effective tip relief that amount of tip relief which compensates for the elastic deformation of the gear mesh ie
zero load at the tooth tip It is assumed (simplified) to be equal for both pinion and wheel
Cylindrical gears
for helical
γ
γ=cb
KKFC Abt
eff (see 1)
For spur
cbKKF
C Abteff
γ= (see 1)
Alternatively for spur and helical gears the non-linear ap-proach in 111 may be used (taking Ceff=δ)
Bevel gears
bcKFC Ambt
effγ
= (see 1)
where
γ
α
ε2ε44c
+sdot
=γ
433 Tip relief and extension Cylindrical gears
The extension of the tip relief is not to result in an effective contact ratio 1ltεα when the gear is unloaded (exceptions to this may only apply for applications where the gear is not to run at light loads) This means that the unrelieved part of the path of contact is to be minimum pbt It is further assumed that this unrelieved part is placed centrally on the path of contact
If root relief applies it has to be calculated as equivalent tip relief Ie pinion root relief (at mesh position A) is added to Ca2 and wheel root relief (at mesh position E) is added to Ca1 If no design tip relief or root relief on the mating gear is specified (ie if Ca1+Croot2 = 0 and visa versa) use the run-ning in amount see 112
Bevel gears
Bevel gears are to have heightwise crowning ie no distinct relievedunrelieved area This may be treated as tip and root relief For calculation purposes the root relief is combined with the tip relief of the mating member into an equivalent tip relief If no resulting tip relieves are specified the equivalent tip relives may be calculated as an approxima-tion based on the tool crowning Ca tool (per mille of tool module m0) as follows
2root1aeq1a CCC +=
1root2aeq2a CCC +=
where
Classification Notes- No 412 35 May 2003
DET NORSKE VERITAS
2
0
n0101atoolal m
)m2(mA1middotmCC
minus+minussdot=
( )
sdotminusminussdotsdotminus+= vtvb1
2vb1
21vavt1n1 αtan dddαsin 05x1mA
2
0
n020atool22a m
)mm(2A1middotmCC
minus+minussdot=
( )
sdotminusminussdotsdotminus+= vtvb2
2vb2
2va2vt2n2 αtan dddαsin 05x1mA
2
0
20atool1root1 m
A1mCC
minussdotsdot=
2
0
10atool2root2 m
A1mCC
minussdotsdot=
If the pressure angles of the cutter blades are modified (and verified) to balance the tip relieves the following may be assumed
( )eq2aeq1aeq2aeq1a C andC calculated of sum50CC sdot==
Throughout the following Ca1 and Ca2 mean the equivalent tip relieves Ca1eq and Ca2eq
434 Bulk temperature The bulk temperature may be calculated as
flaaveragempsoilMB X X 50 ϑ+ϑ=ϑ
where
Xs = lubrication factor
= 12 for spray lubrication
= 10 for dip lubrication (provided both pinion and wheel are dip lubricated and tip speed lt 5 ms)
= 10 for spray lubrication with additional cooling spray (spray on both pinion and wheel or spray on pinion and dip lubrication of wheel)
= 02 for meshes fully submerged in oil
Xmp = contact factor ( )pmp nX += 150
np = number of mesh contact on the pinion (for small gear ratii the number of wheel meshes should be used if higher)
flaaverageϑ
= average of the integrated flash temperature (see 44) along the path of contact
( )
AE
E
Ayyyfla
flaaverage
d
ΓminusΓ
ΓΓϑ=ϑint =
For high speed gears (v gt 50 ms) it may be necessary to assess the bulk temperature on the basis of a thermal rating of the entire gear transmission
44 The Flash Temperature flaϑ
441 Basic formula The local flash temperature flaϑ may be calculated as
( ) 41redy
y2ly
211
43Btcorrfla
unXwX3250
y ρ
ρminusρ
micro=ϑ Γ
(For bevel gears replace u with uv)
and is to be calculated stepwise along the path of contact from A to E
where
micro = coefficient of friction see 431
Xcorr = correction factor taking empirically into account the increased scuffing risk in the beginning of the approach path due to mesh starting without any previously built up oil film and possible shuffling away oil before meshing if insufficient tip relief
( )
3
AD
yaeffcorr 50
CC1X
ΓminusΓε
Γminus+=
α
Ca = tip relief of driven member Xcorr is only applicable in the approach path and if Ca lt Ceff otherwise 10
wBt = unit load see 431
yXΓ = load sharing factor see 443
n1 = pinion rpm
Γy ρly etc see 442
442 Geometrical relations The various radii of flank curvature (transversal plane) are
ρ1y = pinion flank radius at mesh point y
ρ2y = wheel flank radius at mesh point y
ρredy = equivalent radius of curvature at mesh point y
y2y1
y2y1redy
ρ+ρ
ρρ=ρ
Cylindrical gears
twy
1y αsin au1Γ1
ρ+
+=
twy
2y αsin au1Γu
ρ+
minus=
Note that for internal gears a and u are negative
36 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
Bevel gears
vtvv
y1y αsin a
u1Γ1
ρ+
+=
vtvv
yv2y αsin a
u1Γu
ρ+
minus=
Γ is the parameter on the path of contact and y is any point between A end E
At the respective ends Γ has the following values
Root piniontip wheel
Cylindrical gears
( )
minus
minusminus= 1
αtan 1dd
zzΓ
tw
2b2a2
1
2A
Bevel gears
( )
minus
minusminus= 1
αtan 1dd
uΓvt
2vb2va2
vA
Tip pinionroot wheel
Cylindrical gears
( )
1αtan
1ddΓ
tw
2b1a1
E minusminus
=
Bevel gears
( )
1αtan
1ddΓ
vt
2vb1va1
E minusminus
=
At inner point of single pair contact
Cylindrical gears
tw1
EB α tan zπ2ΓΓ minus=
Bevel gears
vtv1
EB α tan zπ2ΓΓ minus=
At outer point of single pair contact
Cylindrical gears
tw1AD α tan z
π2ΓΓ +=
Bevel gears
vt v1
AD αtan z π2ΓΓ +=
At pitch point ΓC = 0
The points F and G (only applicable to cylindrical gears) limiting the extension of tip relief (so as to maintain a mini-mum contact ratio of unity for unloaded gears) are at
2
BAF
Γ+Γ=Γ
2
EDG
Γ+Γ=Γ
443 Load sharing factor XΓ The load sharing factor XΓ accounts for the load sharing be-tween the various pairs of teeth in mesh along the path of contact
XΓ is to be calculated stepwise from A to E using the pa-rameter Γy
4431 Cylindrical gears with β = 0 and no tip relief
Figure 41
ByAAB
Ay for31
31X
yΓltΓleΓ
ΓminusΓ
ΓminusΓ+=Γ
DyB for1Xy
ΓltΓleΓ=Γ
EyDDE
yE for31
31X
yΓleΓltΓ
ΓminusΓ
ΓminusΓ+=Γ
4432 Cylindrical gears with β = 0 and tip relief Tip relief on the pinion reduces XΓ in the range G ndash E and increases correspondingly XΓ in the range F ndash B
Tip relief on the wheel reduces XΓ in the range A ndash F and increases correspondingly XΓ in the range D ndash G
Following remains generally
DyB for1Xy
ΓleΓleΓ=Γ
21XXGF== ΓΓ
In the following it must be distinguished between Ca lt Ceff respectively Ca gt Ceff This is shown by an example below where Ca1 lt Ceff and Ca2 gt Ceff
Classification Notes- No 412 37 May 2003
DET NORSKE VERITAS
Figure 42
Note When Ca gt Ceff the path of contact is shortened by A ndash A respectively E ndash E The single pair contact path is extended into B respectively D If this shift is significant it is necessary to consider the negative effect on surface durability (B) and bending stresses (B and D)
Range A - F
For effa2 CC le
minus=Γ
eff
2a
CC1
31X
A
FyAeff
2a
AF
Ay for C3
C61XX
AyΓleΓleΓ
sdot
+ΓminusΓ
ΓminusΓ+= ΓΓ
For effa2 CC ge
AyAfor0Xy
ΓleΓleΓ=Γ
with ( )21
CC
1CC
eff
2a
eff
2a
AFAAminus
minusΓminusΓ+Γ=Γ
FyAeff
2a
AF
Ay
eff
2a for 21
CC
CC1X
yΓleΓleΓ
minus
ΓminusΓ
ΓminusΓ+minus=Γ
Range F - B
For effa1 CC le
ByFeff
1a
FB
Fy forC3
C61
21X
yΓleΓleΓ
sdot
+ΓminusΓ
ΓminusΓ+=Γ
For effa1 CC ge
ByFeff
1a
FB
Fy for21
CC
21X
yΓleΓleΓ
minus
ΓminusΓ
ΓminusΓ+=Γ
ByB for1Xy
ΓleΓleΓ=Γ
with 1
CC2
eff
1a
FBFB
minus
ΓminusΓ+Γ=Γ
Range D ndash G
For effa2 CC le
GyDeff
2a
DG
Dy
eff
2a for C 3C
61
C 3C
32X
yΓleΓleΓ
+
ΓminusΓ
ΓminusΓminus+=Γ F
or effa2 CC ge
DyD for1Xy
ΓleΓleΓ=Γ
with ( )21
CC
1CC
eff
2a
eff
2a
DGDDminus
minusΓminusΓ+Γ=Γ
GyDeff
2a
DG
Dy
eff
2a for21
CC
CCX ΓleΓleΓ
minus
ΓminusΓ
ΓminusΓminus=Γ
Range G - E
For effa1 CC le
minus=Γ
eff
1a
CC1
31X
E
EyGeff
1a
GE
Gy forC3
C61
21X
yΓleΓleΓ
sdot
+ΓminusΓ
ΓminusΓminus=Γ
For effa1 CC gt
EyGeff
1a
GE
Gy for21
CC
21X
yΓleΓleΓ
minus
ΓminusΓ
ΓminusΓminus=Γ
EyE for0Xy
ΓleΓleΓ=Γ
with 1
CC2
eff
1a
GEGE
minus
ΓminusΓ+Γ=Γ
4433 Gears with β gt 0 buttressing Due to oblique contact lines over the flanks a certain but-tressing may occur near A and E
This applies to both cylindrical and bevel gears with tip relief lt Ceff The buttressing Xbutt is simplified as a linear function within the ranges A ndash H respectively I ndash E
Figure 43
1εwhen13X βbutt EAge=
1εwhenε031X ββbutt EAlt+=
38 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
Cylindrical gears
bIEAH βsin 02ΓΓΓΓ =minus=minus
Bevel gears
bmIEAH βsin 02ΓΓΓΓ =minus=minus
4434 Cylindrical gears with 2εγ le and no tip relief
yXΓ is obtained by multiplication of
yXΓ in 4431 with
Xbutt in 4433
4435 Gears with 2εγ gt and no tip relief
Applicable to both cylindrical and bevel gears
IyH for1Xy
ΓleΓleΓε
=α
Γ
IyHybutt and forX1Xy
ΓgtΓΓltΓε
=α
Γ
Figure 44
4436 Cylindrical gears with 2εγ le and tip relief
yXΓ is obtained by multiplication of
yXΓ in 4432 with Xbutt
in 4433
4437 Cylindrical gears with 2εγ gt and tip relief
Tip relief on the pinion (respectively wheel) reduces XГ in the range G ndash E (respectively A ndash F) and increases XГ in the range F ndash G
yXΓ is obtained by multiplication of
yXΓ as described below
with Xbutt in 4433
In the XГ example below the influence of tip relief is shown (without the influence of Xbutt) by means of
eff2aeff1a CC and CC ltgt
Tip relief gt Ceff causes new end points A respectively E of the path of contact
Figure 45
Range A ndash F
( ) ( )( ) eff
a2αa1α
AF
Ay
eff
2aeff
C 12C 13εC 1ε
CCCX
y +εε++minus
ΓminusΓ
ΓminusΓ+
εminus
=ααα
Γ
eff2aFyA CC if for leΓleΓleΓ
eff2aFyA CifCfor and geΓleΓleΓ
eff2aAyA CC iffor0Xy
gtΓleΓleΓ=Γ
( ) ( ) ( )( ) ( ) a2αa1α
αeffa2AFAA C 1ε 3C 1ε
1ε 2 CC with++minus+minus
ΓminusΓ+Γ=Γ
Range F ndash G
( )( )( ) GyF
eff
2a1a forC 1 2CC 11X
yΓleΓleΓ
+εε+minusε
+ε
=αα
α
αΓ
Range G ndash E
( ) ( )( ) eff
2a1a
GE
Gy
C 1 2C 1C 1 3XX
GFy +εεminusε++ε
ΓminusΓ
ΓminusΓminus=
αα
ααΓΓ minus
effa1EyG CC if for leΓleΓleΓ
effa1EyG CC if Γfor and geΓleΓle
eff1aEyE CC if for0Xy
geΓleΓleΓ=Γ
( ) ( ) ( )( ) ( ) 2a1a
eff1aGEEE C 1C 1 3
1 2 CC withminusε++ε+εminus
ΓminusΓminusΓ=Γαα
α
4438 Bevel gears with εγ more than approx 18 and heightwise crowning For Ca1 = Ca2 = Ceff the following applies
Figure 46
Classification Notes- No 412 39 May 2003
DET NORSKE VERITAS
( )AEM 50 Γ+Γ=Γ
( )( )2AD
3
2My651X
y ΓminusΓε
ΓminusΓminus
ε=
ααΓ
For tip relief lt Ceff yXΓ is found by linear interpolation be-
tween ( )effay CCX =Γ and ( )0CayX =Γ as in 4435
The interpolation is to be made stepwise from A to M with the influence of Ca2 and from M to E with the influence of Ca1 (For 2a1a CC ne there is a discontinuity at M)
Eg with Ca1 = 04 Ceff and Ca2 = 055 Ceff then
Range A ndash M
)CC()0C( eff2ay2ayyX 550X 450X =Γ=ΓΓ +=
Range M ndash E
)CC()0C( eff1ay1ayyX 40X 06X =Γ=ΓΓ +=
For tip relief gt Ceff the new end points A and E are found as
( )
minusΓminusΓ
ε+Γ=Γ α 1
CC
6 eff
2aADAA
( )
minusΓminusΓ
ε+Γ=Γ α 1
CC
6 eff
1aADEE
Range A ndash A
0Xy=Γ
Range A ndash M
( )( )
ΓminusΓ
ΓminusΓminus
minusε=
αΓ 2
MA
2My
eff
2a1
CC4
351Xy
Range M ndash E
( )( )
ΓminusΓ
ΓminusΓminus
minusε=
αΓ 2
ME
2My
eff
1a1
CC4
351Xy
Range E ndash E
0Xy=Γ
40 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
Appendix A Fatigue Damage Accumulation
The Palmgren-Miner cumulative damage calculation princi-ple is used The procedure may be applied as follows
A1 Stress Spectrum From the individual torque classes the torques (Ti) at the peak values of class intervals and the associated number of cycles (NLi) for both pinion and wheel are to be listed from the highest to the lowest torque
(In case of a cyclic torque variation within the torque classes it is advised to use the peak torque If the cyclic variation is such that the same teeth will repeatedly suffer the peak torque this is a must)
The stress spectra for tooth roots and flanks (σFi σHi) with all relevant factors (except KA) are to be calculated on the basis of the torque spectrum The load dependent K-factors are to be determined for each torque class
A2 σminusN-curve The stress versus load cycle curves for tooth roots and flanks (both pinion and wheel) are to be drawn on the basis of per-missible stresses (ie including the demanded minimum safety factors) as determined in 2 respectively 3 If different safety levels for high cycle fatigue and low cycle fatigue are desired this may be expressed by different demand safety factors applied at the endurance limit respectively at static strength
A3 Damage accumulation The individual damage ratio Di at ith stress level is defined as
Fi
Lii N
ND =
where
NLi = The number of applied cycles at ith stress
NFi = The number of cycles to failure at ith stress
Basically stresses σi below the permissible stress level for infinite life (if a constant ZN or YN is accepted) do not con-tribute to the damage sum However calculating the actual safety factor Sact as described below all the σi for which the product S σi is bigger than or equal to the permissible stress level for infinite life contribute to the damage sum and thus to the determination of Sact The final value of S is decisive
(NFi can be found mathematically by putting the permissible stress σpi equal the actual stress σi thereby finding the actual life factor This life factor can be solved with regard to load cycles ie NFi)
The damage sum ΣDi is not to exceed unity
If ΣDi ne 1 the safety against cumulative fatigue damage is different from the applied demand safety factor For deter-mination of this theoretical safety factor an iteration proce-dure is required as described in the following flowchart
S is correction factor with which the actual safety factor Sact can be found
Sact is the demand safety factor (used in determination of the permissible stresses in the σ ndash N ndash curve) times the correc-tion factor S
The full procedure is to be applied for pinion and wheel tooth roots and flanks
Note If alternating stresses occur in a spectrum of mainly pulsating stresses the alternating stresses may be replaced by equiva-lent pulsating stresses ie by means of division with the ac-tual mean stress influence factor YM
Classification Notes- No 412 41 May 2003
DET NORSKE VERITAS
Appendix B Application Factors for Diesel Driven Gears
For diesel driven gears the application factor KA depends on torsional vibrations Both normal operation and misfiring conditions have to be considered
Normally these two running conditions can be covered by only one calculation
B1 Definitions
Normal operation KAnorm = 0
normv0
TTT +
where
T0 = rated nominal torque
Tv norm = vibratory torque amplitude for normal opera-tion (see rules Pt4 Ch3 Sec1 G301 for defi-nition of ldquonormalrdquo irregularity)
Misfiring operation KA misf = 0
misfv
TTT +
where
T = remaining nominal torque when one cylinder out of action
Tv misf = vibratory torque amplitude in misfiring con-dition This refers to a permissible misfiring condition ie a condition that does not require automatic or immediate corrective actions as speed or pitch reduction
The normal operation is assumed to last for a very high num-ber of cycles such as 1010
The misfiring operation is assumed to last for a limited dura-tion such as 107 cycles
B2 Determination of decisive load Assuming life factor at 1010 cycles as YN = ZN = 092 which usually is relevant the calculation may be performed only once with the combination having the highest value of appli-cation factorlife factor
For bending stresses and scuffing the higher value of
092
K normA and 098
K misfA
For contact stresses the higher value of
092K normA and
113K misfA (but
097K misfA for nitrided gears)
B3 Simplified procedure Note that this is only a guidance and is not a binding con-vention
T = oTZ
1Zsdot
minus where Z = number of cylinders
( ) ideal Videal Vmisf Vnorm V TTT24ZT +minus=
where TV ideal = vibratory torque with all cylinders perfectly equal See also rules Pt4 Ch3 Sec1 G300
When using trends from torsional vibration analysis and measurements the following may be used
TV ideal To is close to zero for engines with few cylinders and using a suitable elastic coupling and increases with relative coupling stiffness and number of cylinders
TV misf To may be high for engines with few cylinders and decreases with number of cylinders
This can be indicated as
200Z
TT
ο
idealV asymp and 80Z04
TT
o
misfV minusasymp
Inserting this into the formulae for the two application fac-tors the following guidance can be given
118112K misfA minusasymp
115110K normA minusasymp
Since KA norm is to be combined with the lower life factors the decisive load condition will be the normal one and a KA of 115 will cover most relevant cases when a suitable elas-tic coupling is chosen
42 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
Appendix C Calculation of Pinion-Rack
Pinion-racks used for elevating of mobile offshore units are open gears that are subjected to wear and tear With normal specifications such as only a few hundred total operation cycles (site to site) the tooth bending stresses for static strength will be decisive for the lay out however with ex-ception of surface hardened pinions where case crushing has to be considered
In the following the use of part 1 and 3 for pinion-racks is shown including relevant simplifications
C1 Pinion tooth root stresses Since the load spectrum normally is dominated by high torques and few load cycles (in the range up to 1-2000) the static strength is decisive
The actual stress is calculated as
β1FSaFan1
tF1 KYY
mbFσ sdotsdotsdotsdot
=
b1 is limited to b2 + 2middotmn
YFa and YSa replace YF and YS because load application at tooth tip has to be assumed for such inaccurate gears
Pinions often use a non-involute profile in the dedendum part of the flank eg a constant radius equal the radius of curva-ture at reference circle For such pinions sFn and hFa are to be measured directly on a sectional drawing of the pinion tooth
Due to high loads and narrow facewidths it may be assumed that KFβ2 = KHβ = 10 However when b1 gt b2 then KFβ1 gt10
If no detailed documentation of KFβ1 is available the fol-lowing may be used
KFβ1 = 1 + 015middot(b1b2 ndash 1)
The permissible stress (not surface hardened) is calculated as
δrelTstF
Fst1FP1 Y
Sσσ sdot=
The mean stress influence due to leg lifting may be disre-garded
The actual and permissible stresses should be calculated for the relevant loads as given in the rules
C2 Rack tooth root stresses The actual stress is calculated as
SaFan2
tF2 YY
mbFσ sdotsdotsdot
=
See C1 for details
The permissible stress is calculated as
δrelTstF
Fst2FP2 Y
Sσσ sdot=
For alloyed steels (Ni Cr Mo) with high toughness and duc-tility the value of YδrelTst may be put equal to YSa
C3 Surface hardened pinions For surface hardened pinions the maximum load is not to cause crushing of the hardened layer of the flank
In principle the calculation described in 213 may be used but when the theoretical Hertzian stress exceeds the ap-proximately 18 times the yield strength of the rack material plastic deformation will occur This will limit the peak Hertzian stress but increases the contact width and thus the penetration of stresses into the depth
An approximation may be based on an assessment of contact width determined by means of equal areas under the theoreti-cal (elastic range) Hertzian contact and the elasto-plastic contact stress (limited to 18 middot σy) with the unknown width
Classification Notes- No 412 7 May 2003
DET NORSKE VERITAS
16 Application Factors KA and KAP The application factor KA accounts for dynamic overloads from sources external to the gearing
It is distinguished between the influence of repetitive cyclic torques KA (161) and the influence of temporary occasional peak torques KAP (162)
Calculations are always to be made with KA In certain cases additional calculations with KAP may be necessary
For gears with a defined load spectrum the calculation with a KA may be replaced by a fatigue damage calculation as given in Appendix A
161 KA For gears designed for long or infinite life at nominal rated torque KA is defined as the ratio between the maximum re-petitive cyclic torque applied to the gear set and nominal rated torque
This definition is suitable for main propulsion gears and most of the auxiliary gears
KA can be determined by measurements or system analysis or may be ruled by conventions (ice classes) (For the pur-pose of a preliminary (but not binding) calculation before KA is determined it is advised to apply either the max values mentioned below or values known from similar plants)
a) For main propulsion gears KA can be taken from the (mandatory) torsional vibration analysis thereby con-sidering all permissible driving conditions) Unless specially agreed the rules do not allow KA in excess of 135 for diesel propulsion) With turbine or electric propulsion KA would normally not exceed 12 However special attention should be given to thrusters that are arranged in such a way that heavy vessel movements andor manoeuvring can cause severe load fluctuations This means eg thrusters positioned far from the rolling axis of vessels that could be susceptible to rolling If leading to propeller air suction the condi-tions may be even worse The above mentioned movements or manoeuvring will result in increased propeller excitation If the thruster is driven by a diesel engine the engine mean torque is limited to 100 However thrusters driven by electric motors can suffer temporary mean torque much above 100 unless a suitable load control system (limiting available e-motor torque) is provided
b) For main propulsion gears with ice class notation (see Rules Pt5 Ch1 Sec J500) KA ice has to be taken as the higher value of the applicable (rule defined) ice shock torque referred to nominal rated torque and the value under a) The Baltic ice class notations refer to a few millions ice shock loads Thus the life factors may be put YN=1 and ZN=12 (except for nitrided gears where ZN = 1 applies) Additionally the calculations with the normal KA (no ice class) are to fulfil the normal requirements For polar ice class notations KA ice applies to all criteria and for long or infinite life
c) For a power take off (PTO) branch from a main propul-sion gear with ice class ice shocks result in negative torques It is assumed that the PTO branch is unloaded when the ice shock load occurs The influence of these reverse shock loads may be taken into account as follows The negative torque (reversed load) expressed by means of an application factor based on rated forward load (T or Ft) is KAreverse = KA ice ndash1 (the minus 1 be-cause no mean torque assumed) KAice to be calculated as in the ice class rules This KAreverse should be used for back flank considera-tions such as pitting and scuffing The influence on tooth bending strength (forward di-rection) may be simplified by using the factor YM = 1 minus 03 middot KAreverse KA
d) For diesel driven auxiliaries KA can be taken from the torsional vibration analysis if available For units where no vibration analysis is required (lt 200 kW) or available it is advised to apply KA as the upper allow-able value 135)
e) For turbine or electro driven auxiliaries the same as for c) applies however the practical upper value is 12
) For diesel driven gears more information on KA for mis-firing and normal driving is given in Appendix B
162 KAP The peak overload factor KAP is defined as the ratio between the temporary occasional peak overload torque and the nominal rated torque
For plants where high temporary occasional peak torques can occur (ie in excess of the above mentioned KA) the gearing (if nitrided) has to be checked with regard to static strength Unless otherwise specified the same safety factors as for infinite life apply
The scuffing safety is to be specially considered whereby the KA applies in connection with the bulk temperature and the KAP applies for the flash temperature calculation and should replace KA in the formulae in 431 432 and 441
KAP can be evaluated from the torsional impact vibration calculation (as required by the rules)
If the overloads have a duration corresponding to several revolutions of the shafts the scuffing safety has to be consid-ered on basis of this overload both with respect to bulk and flash temperature This applies to plants with ice class nota-tions (Baltic and polar) and to plants with prime movers which have high temporary overload capacity such as eg electric motors (provided the driven member can have a con-siderable increase in demand torque as eg azimuth thrusters during manoeuvring)
For plants without additional ice class notation KAP should normally not exceed 15
8 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
163 Frequent overloads For plants where high overloads or shock loads occur regu-larly the influence of this is to be considered by means of cumulative fatigue (see Appendix A)
17 Load Sharing Factor Kγ The load sharing factor Kγ accounts for the maldistribution of load in multiple-path transmissions (dual tandem epicyclic double helix etc) Kγ is defined as the ratio between the max load through an actual path and the evenly shared load
171 General method Kγ mainly depends on accuracy and flexibility of the branches (eg quill shaft planet support external forces etc) and should be considered on basis of measurements or of relevant analysis as eg
Kγ = δ+δ f
δ = total compliance of a branch under full load (assuming even load share) referred to gear mesh
f = minusminusminusminus+++ 23
22
21 fff where f1 f2 etc are the
main individual errors that may contribute to a maldis-tribution between the branches Eg tooth pitch errors planet carrier pitch errors bearing clearance influences etc Compensating effects should also be considered
For double helical gears
An external axial force Fext applied from sources outside the actual gearing (eg thrust via or from a tooth coupling) will cause a maldistribution of forces between the two helices Expressed by a load sharing factor the
βsdot
plusmn=γ tanFF1K
t
ext
If the direction of Fext is known the calculation should be carried out separately for each helix and with the tangential force corrected with the pertinent Kγ If the direction of Fext is unknown both combinations are to be calculated and the higher σH or σF to be used
172 Simplified method If no relevant analysis is available the following may apply
For epicyclic gears
Kγ = 32501 minus+ pln
where npl = number of planets ( gt3 )
For multistage gears with locked paths and gear stages sepa-rated by quill shafts (see figure below)
Figure 10 Locked paths gear
Kγ = ( )φ201+
where φ = quill shaft twist (degrees) under full load
18 Dynamic Factor Kv The dynamic factor Kv accounts for the internally generated dynamic loads
Kv is defined as the ratio between the maximum load that dynamically acts on the tooth flanks and the maximum ex-ternally applied load Ft KA Kγ
In the following 2 different methods (181 and 182) are described In case of controversy between the methods the next following is decisive ie the methods are listed with increasing priority
It is important to observe the limitations for the method in 181 In particular the influence of lateral stiffness of shafts is often underestimated and resonances occur at considerably lower speed than determined in 1811
However for low speed gears with vmiddotz1 lt 300 calculations may be omitted and the dynamic factor simplified to Kv=105
181 Single resonance method For a single stage gear Kv may be determined on basis of the relative proximity (or resonance ratio) N between actual speed n1 and the lowest resonance speed nE1
N = 1E
1
nn
Note that for epicyclic gears n is the relative speed ie the speed that multiplied with z gives the mesh frequency
1811 Determination of critical speed It is not advised to apply this method for multimesh gears for N gt 085 as the influence of higher modes has to be consid-ered see 182 In case of significant lateral shaft flexibility (eg overhung mounted bevel gears) the influence of cou-pled bending and torsional vibrations between pinion and wheel should be considered if N ge 075 see 182
nE1 = redmγc
1zπ
31030 sdot
where
cγ is the actual mesh stiffness per unit facewidth see 111
Classification Notes- No 412 9 May 2003
DET NORSKE VERITAS
For gears with inactive ends of the facewidth as eg due to high crowning or end relief such as often applied for bevel gears the use of cγ in connection with determination of natu-ral frequencies may need correction cγ is defined as stiffness per unit facewidth but when used in connection with the total mesh stiffness it is not as simple as cγmiddotb as only a part of the facewidth is active Such corrections are given in 111
mred is the reduced mass of the gear pair per unit facewidth and referred to the plane of contact
For a single gear stage where no significant inertias are closely connected to neither pinion nor wheel mred is calcu-lated as
mred = 21
21mm
mm+
The individual masses per unit facewidth are calculated as
m12 = 221b
21
)2d(b
I
where I is the polar moment of inertia (kgmm2)
The inertia of bevel gears may be approximated as discs with diameter equal the midface pitch diameter and width equal to b However if the shape of the pinion or wheel body differs much from this idealised cylinder the inertia should be cor-rected accordingly
For all kind of gears if a significant inertia (eg a clutch) is very rigidly connected to the pinion or wheel it should be added to that particular inertia (pinion or wheel) If there is a shaft piece between these inertias the torsional shaft stiffness alters the system into a 3-mass (or more) system This can be calculated as in 182 but also simplified as a 2-mass system calculated with only pinion and wheel masses
1812 Factors used for determination of Kv Non-dimensional gear accuracy dependent parameters
( )bKKF
yfcB
At
pptp
γ
minus=
( )bKKF
yFcBAt
ff
γ
α minus=
Non-dimensional tip relief parameter
bKKFcC
1BγAt
ak sdotsdot
sdotminus=
For gears of quality grade (ISO 1328) Q = 7 or coarser Bk = 1
For gears with Q le 6 and excessive tip relief Bk is limited to max 1
For gears (all quality grades) with tip relief of more than 2middotCeff (see 432) the reduction of αε has to be considered (see 443)
where
fpt = the single pitch deviation (ISO 1328) max of pinion or wheel
Fα = the total profile form deviation (ISO 1328) max of pinion or wheel (Note Fα is pt not available for bevel gears thus use Fα = fpt)
yp and yf = the respective running-in allowances and may be calculated similarly to yα in 112 ie the value of fpt is replaced by Fα for yf
cacute = the single tooth stiffness see 111
Ca = the amount of tip relief see 433 In case of different tip relief on pinion and wheel the value that results in the greater value of Bk is to be used If Ca is zero by design the value of running-in tip relief Cay (see 112) may be used in the above formula
1813 Kv in the subcritical range Cylindrical gears N le 085
Bevel gears N le 075
Kv = 1 + N K
K = Cv1 Bp + Cv2 Bf + Cv3 Bk
Cv1 accounts for the pitch error influence
Cv1 = 032
Cv2 accounts for profile error influence
Cv2 = 034 for γε le 2
Cv2 = 03ε
057
γ minus for γε gt 2
Cv3 accounts for the cyclic mesh stiffness variation
Cv3 = 023 for γε le 2
Cv3 = 156ε
0096
γ minus for γε gt 2
1814 Kv in the main resonance range Cylindrical gears 085 lt N le 115
Bevel gears 075 lt N le 125
Running in this range should preferably be avoided and is only allowed for high precision gears
Kv = 1 + Cv1 Bp + Cv2 Bf + Cv4 Bk
Cv4 accounts for the resonance condition with the cyclic mesh stiffness variation
Cv4 = 090 for γε le 2
Cv4 = 144ε
ε005057
γ
γ
minus
minus for γε gt 2
10 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
1815 Kv in the supercritical range Cylindrical gears N ge 15
Bevel gears N ge 15
Special care should be taken as to influence of higher vibra-tion modes andor influence of coupled bending (ie lateral shaft vibrations) and torsional vibrations between pinion and wheel These influences are not covered by the following approach
Kv = Cv5 Bp + Cv6 Bf + Cv7
Cv5 accounts for the pitch error influence
Cv5 = 047
Cv6 accounts for the profile error influence
Cv6 = 047 for εγ le 2
Cv6 = 174ε
012
γ minus for εγ gt 2
Cv7 relates the maximum externally applied tooth loading to the maximum tooth loading of ideal accurate gears operating in the supercritical speed sector when the circumferential vibration becomes very soft
Cv7 = 075 for εγ le 15
Cv7 = [ ] 8750)2(sin 0125 +minusβεπ for 15 lt εγ le 25
Cv7 = 10 for εγ gt 25
1816 Kv in the intermediate range Cylindrical gears 115 lt N lt 15
Bevel gears 125 lt N lt 15
Comments raised in 1814 and 1815 should be observed
Kv is determined by linear interpolation between Kv for N = 115 respectively 125 and N = 15 as
Cylindrical gears
( ) ( ) ( )[ ]51Nv151Nv51Nvv KK350
N51KK === minussdot
minus
+=
Bevel gears
( ) ( ) ( )[ ]51Nv251Nv51Nvv KK250
N51KK === minussdot
minus
+=
182 Multi-resonance method For high speed gear (vgt40 ms) for multimesh medium speed gears for gears with significant lateral shaft flexibility etc it is advised to determine Kv on basis of relevant dy-namic analysis
Incorporating lateral shaft compliance requires transforma-tion of even a simple pinion-wheel system into a lumped multi-mass system It is advised to incorporate all relevant inertias and torsional shaft stiffnesses into an equivalent (to pinion speed) system Thereby the mesh stiffness appears as an equivalent torsional stiffness
cγ b (db12)2 (Nmrad)
The natural frequencies are found by solving the set of dif-ferential equations (one equation per inertia) Note that for a gear put on a laterally flexible shaft the coupling bending-torsionals is arranged by introducing the gear mass and the lateral stiffness with its relation to the torsional displacement and torque in that shaft
Only the natural frequency (ies) having high relative dis-placement and relative torque through the actual pinion-wheel flexible element need(s) to be considered as critical frequency (ies)
Kv may be determined by means of the method mentioned in 181 thereby using N as the least favourable ratio (in case of more than one pinion-wheel dominated natural frequency) Ie the N-ratio that results in the highest Kv has to be consid-ered
The level of the dynamic factor may also be determined on basis of simulation technique using numeric time integration with relevant tooth stiffness variation and pitchprofile er-rors
19 Face Load Factors KHβ and KFβ The face load factors KHβ for contact stresses and for scuff-ing KFβ for tooth root stresses account for non-uniform load distribution across the facewidth
KHβ is defined as the ratio between the maximum load per unit facewidth and the mean load per unit facewidth
KFβ is defined as the ratio between the maximum tooth root stress per unit facewidth and the mean tooth root stress per unit facewidth The mean tooth root stress relates to the con-sidered facewidth b1 respectively b2
Note that facewidth in this context is the design facewidth b even if the ends are unloaded as often applies to eg bevel gears
The plane of contact is considered
191 Relations between KHβ and KFβ
KFβ = expβHK 2(hb)hb1
1exp++
=
where hb is the ratio tooth heightfacewidth The maximum of h1b1 and h2b2 is to be used but not higher than 13 For double helical gears use only the facewidth of one helix
If the tooth root facewidth (b1 or b2) is considerably wider than b the value of KFβ(1or2) is to be specially considered as it may even exceed KHβ
Eg in pinion-rack lifting systems for jack up rigs where b = b2 asymp mn and b1 asymp 3 mn the typical KHβ asymp KFβ2 asymp 1 and KFβ1 asymp 13
192 Measurement of face load factors Primarily
KFβ may be determined by a number of strain gauges distrib-uted over the facewidth Such strain gauges must be put in
Classification Notes- No 412 11 May 2003
DET NORSKE VERITAS
exactly the same position relative to the root fillet Relations in 191 apply for conversion to KHβ
Secondarily
KHβ may be evaluated by observed contact patterns on vari-ous defined load levels It is imperative that the various test loads are well defined Usually it is also necessary to evalu-ate the elastic deflections Some teeth at each 90 degrees are to be painted with a suitable lacquer Always consider the poorest of the contact patterns
After having run the gear for a suitable time at test load 1 (the lowest) observe the contact pattern with respect to ex-tension over the facewidth Evaluate that KHβ by means of the methods mentioned in this section Proceed in the same way for the next higher test load etc until there is a full face contact pattern From these data the initial mesh misalign-ment (ie without elastic deflections) can be found by ex-trapolation and then also the KHβ at design load can be found by calculation and extrapolation See example
Figure 11 Example of experimental determination of KHβ
It must be considered that inaccurate gears may accumulate a larger observed contact pattern than the actual single mesh to mesh contact patterns This is particularly important for lapped bevel gears Ground or hard metal hobbed bevel gears are assumed to present an accumulated contact pattern that is practically equal the actual single mesh to mesh con-tact patterns As a rough guidance the (observed) accumu-lated contact pattern of lapped bevel gears may be reduced by 10 in order to assess the single mesh to mesh contact pattern which is used in 199
193 Theoretical determination of KHβ The methods described in 193 to 198 may be used for cy-lindrical gears The principles may to some extent also be used for bevel gears but a more practical approach is given in 199
General For gears where the tooth contact pattern cannot be verified during assembly or under load all assumptions are to be well on the safe side
KHβ is to be determined in the plane of contact
The influence parameters considered in this method are
bull mean mesh stiffness cγ (see 111) (if necessary also variable stiffness over b)
bull mean unit load Fmb = Fbt KA Kγ Kvb (for double helical gears see 17 for use of Kγ)
bull misalignment fsh due to elastic deflections of shafts and gear bodies (both pinion and wheel)
bull misalignment fdefl due to elastic deflections of and work-ing positions in bearings
bull misalignment fbe due to bearing clearance tolerances bull misalignment fma due to manufacturing tolerances bull helix modifications as crowning end relief helix correc-
tion bull running in amount yβ (see 112)
In practice several other parameters such as centrifugal ex-pansion thermal expansion housing deflection etc contrib-ute to KHβ However these parameters are not taken into account unless in special cases when being considered as particularly important
When all or most of the am parameters are to be considered the most practical way to determine KHβ is by means of a graphical approach described in 1931
If cγ can be considered constant over the facewidth and no helix modifications apply KHβ can be determined analyti-cally as described in 1932
1931 Graphical method The graphical method utilises the superposition principle and is as follows
bull Calculate the mean mesh deflection Mδ as a function of
γm c and bF see 111
bull Draw a base line with length b and draw up a rectangu-lar with height δM (The area δM b is proportional to the transmitted force)
bull Calculate the elastic deflection fsh in the plane of contact Balance this deflection curve around a zero line so that the areas above and below this zero line are equal
Figure 12 fsh balanced around zero line
bull Superimpose these ordinates of the fsh curve to the previ-ous load distribution curve (The area under this new load distribution curve is still δM b)
bull Calculate the bearing deflections andor working posi-tions in the bearings and evaluate the influence fdefl in the plane of contact This is a straight line and is bal-anced around a zero line as indicated in Fig 14 but with one distinct direction Superimpose these ordinates to the previous load distribution curve
12 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
bull The amount of crowning end relief or helix correction (defined in the plane of contact) is to be balanced around a zero line similarly to fsh
Figure 13 Crowning Cc balanced aound zero line
bull Superimpose these ordinates to the previous load distri-bution curve In case of high crowning etc as eg often applied to bevel gears the new load distribution curve may cross the base line (the real zero line) The result is areas with negative load that is not real as the load in those areas should be zero Thus corrective actions must be made but for practical reasons it may be postponed to after next operation
bull The amount of initial mesh misalignment bema ff + (defined in the plane of contact) is to be bal-
anced around a zero line If the direction of bema ff + is known (due to initial contact check) or if the direction of fbe is known due to design (eg overhang bevel pin-ion) this should be taken into account If direction un-known the influence of bema ff + in both directions as well as equal zero should be considered
Figure 14 fma+fbe in both directions balanced around zero line
Superimpose these ordinates to the previous load distri-bution curve This results in up to 3 different curves of which the one with the highest peak is to be chosen for further evaluation
bull If the chosen load distribution curve crosses the base line (ie mathematically negative load) the curve is to be corrected by adding the negative areas and dividing this with the active facewidth The (constant) ordinates of this rectangular correction area are to be subtracted from the positive part of the load distribution curve It is advisable to check that the area covered under this new load distribution curve is still equal δM b
bull If cγ cannot be considered as constant over b then cor-rect the ordinates of the load distribution curve with the local (on various positions over the facewidth) ratio between local mesh stiffness and average mesh stiffness cγ (average over the active facewidth only) Note that the result is to be a curve that covers the same area δM b as before
bull The influence of running in yβ is to be determined as in 112 whereby the value for Fβx is to be taken as twice the distance between the peak of the load distribution curve and δM
bull Determine
KHβ = M
β
δycurveofpeak minus
1932 Simplified analytical method for cylindrical gears The analytical approach is similar to 1931 but has a more limited application as cγ is assumed constant over the face-width and no helix modification applies
bull Calculate the elastic deflection fsh in the plane of contact Balance this deflection curve around a zero line so that the area above and below this zero line are equal see Fig 12 The max positive ordinate is frac12∆fsh
bull Calculate the initial mesh alignment as Fβx= deflbemash ffff plusmnplusmnplusmn∆ The negative signs may only be used if this is justified andor verified by a contact pattern test Otherwise al-ways use positive signs If a negative sign is justified the value of Fβx is not to be taken less than the largest of each of these elements
bull Calculate the effective mesh misalignment as Fβy = Fβx - yβ (yβ see 112)
bull Determine
2KforF2
bFc1K βH
m
βγγH le+=β
or
2KforF
bFc2K Hβ
m
βγγH gt=β
where cγ as used here is the effective mesh stiffness see 111
194 Determination of fsh fsh is the mesh misalignment due to elastic deflections Usu-ally it is sufficient to consider the combined mesh deflection of the pinion body and shaft and the wheel shaft The calcu-lation is to be made in the plane of contact (of the considered gear mesh) and to consider all forces (incl axial) acting on the shafts Forces from other meshes can be parted into components parallel respectively vertical to the considered plane of contact Forces vertical to this plane of contact have no influence on fsh
It is advised to use following diameters for toothed elements
d + 2 x mn for bending and shear deflection
d + 2 mn (x ndash ha0 + 02) for torsional deflection
Usually fsh is calculated on basis of an evenly distributed load If the analysis of KHβ shows a considerable maldis-
Classification Notes- No 412 13 May 2003
DET NORSKE VERITAS
tribution in term of hard end contact or if it is known by other reasons that there exists a hard end contact the load should be correspondingly distributed when calculating fsh In fact the whole KHβ procedure can be used iteratively 2-3 iterations will be enough even for almost triangular load distributions
195 Determination of fdefl fdefl is the mesh misalignment in the plane of contact due to bearing deflections and working positions (housing deflec-tion may be included if determined)
First the journal working positions in the bearings are to be determined The influence of external moments and forces must be considered This is of special importance for twin pinion single output gears with all 3 shafts in one plane
For rolling bearings fdefl is further determined on basis of the elastic deflection of the bearings An elastic bearing deflec-tion depends on the bearing load and size and number of rolling elements Note that the bearing clearance tolerances are not included here
For fluid film bearings fdefl is further determined on basis of the lift and angular shift of the shafts due to lubrication oil film thickness Note that fbe takes into account the influence of the bearing clearance tolerance
When working positions bearing deflections and oil film lift are combined for all bearings the angular misalignment as projected into the plane of the contact is to be determined fdefl is this angular misalignment (radians) times the face-width
196 Determination of fbe fbe is the mesh misalignment in the plane of contact due to tolerances in bearing clearances In principle fbe and fdefl could be combined But as fdefl can be determined by analy-sis and has a distinct direction and fbe is dependent on toler-ances and in most cases has no distinct direction (ie + toler-ance) it is practicable to separate these two influences
Due to different bearing clearance tolerances in both pinion and wheel shafts the two shaft axis will have an angular mis-alignment in the plane of contact that is superimposed to the working positions determined in 195 fbe is the facewidth times this angular misalignment Note that fbe may have a distinct direction or be given as a + tolerance or a combina-tion of both For combination of + tolerance it is adviced to use
fbe= 22be
21be ff ++plusmn
fbe is particularly important for overhang designs for gears with widely different kinds of bearings on each side and when the bearings have wide tolerances on clearances In general it shall be possible to replace standard bearings with-out causing the real load distribution to exceed the design premises For slow speed gears with journal bearings the expected wear should also be considered
197 Determination of fma fma is the mesh misalignment due to manufacturing toler-ances (helix slope deviation) of pinion fHβ1 wheel fHβ2 and housing bore
For gear without specifically approved requirements to as-sembly control the value of fma is to be determined as
fma= 22Hβ
21Hβ ff +
For gears with specially approved assembly control the value of fma will depend on those specific requirements
198 Comments to various gear types For double helical gears KHβ is to be determined for both helices Usually an even load share between the helices can be assumed If not the calculation is to be made as de-scribed in 171
For planetary gears the free floating sun pinion suffers only twist no bending It must be noted that the total twist is the sum of the twist due to each mesh If the value of 1K γ ne this must be taken into account when calculating the total sun pinion twist (ie twist calculated with the force per mesh without Kγ and multiplied with the number of planets)
When planets are mounted on spherical bearings the mesh misalignments sun-planet respectively planet-annulus will be balanced Ie the misalignment will be the average between the two theoretical individual misalignments The faceload distribution on the flanks of the planets can take full advan-tage of this However as the sun and annulus mesh with several planets with possibly different lead errors the sun and annulus cannot obtain the above mentioned advantage to the full extent
199 Determination of KHβ for bevel gears If a theoretical approach similar to 193-198 is not docu-mented the following may be used
testeff
H Kb
b851851K sdot
minussdot=β
beff b represents the relative active facewidth (regarding lapped gears see 192 last part)
Higher values than beff b = 090 are normally not to be used in the formula
For dual directional gears it may be difficult to obtain a high beff b in both directions In that case the smaller beff b is to be used
Ktest represents the influence of the bearing arrangement shaft stiffness bearing stiffness housing stiffness etc on the faceload distribution and the verification thereof Expected variations in length- and height-wise tooth profile is also accounted for to some extent
a) Ktest = 1 For ground or hard metal hobbed gears with the specified contact pattern verified at full rating or at full torque slow turning at a condition representative for the thermal expansion at normal operation
14 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
It also applies when the bearing arrangementsupport has insignificant elastic deflections and thermal axial expansion However each initial mesh contact must be verified to be within acceptance criteria that are cali-brated against a type test at full load Reproduction of the gear tooth length- and height-wise profile must also be verified This can be made through 3D measurements or by initial contact movements caused by defined axial offsets of the pinion (tolerances to be agreed upon)
b) Ktest = 1 + 04middot(beffbndash06) For designs with possible influence of thermal expansion in the axial direction of the pinion The initial mesh contact verified with low load or spin test where the acceptance criteria are calibrated against a type test at full load
c) Ktest = 12 if mesh is only checked by toolmakerrsquos blue or by spin test contact For gears in this category beffb gt 085 is not to be used in the calculation
110 Transversal Load Distribution Factors KHα and KFα The transverse load distribution factors KHα for contact stresses and for scuffing KFα for tooth root stresses account for the effects of pitch and profile errors on the transversal load distribution between 2 or more pairs of teeth in mesh
The following relations may be used
Cylindrical gears
( )
minus+==
tH
αptγγHαFα F
byfc0409
2ε
KK
valid for 2εγ le
( ) ( )tH
αptγ
γ
γHαFα F
byfcε
1ε20409KK
minusminus+==
valid for 2ε γ gt
where
FtH = Ft KA Kγ Kv KHβ
cγ = See 111
γα = See 112
fpt = Maximum single pitch deviation (microm) of pinion or wheel or maximum total profile form devia-tion Fα of pinion or wheel if this is larger than the maximum single pitch deviation
Note In case of adequate equivalent tip relief adapted to the load half of the above mentioned fpt can be introduced A tip relief is considered adequate when the average of Ca1 and Ca2 is within plusmn40 of the value of Ceff in 432
Limitations of KHα and KFα
If the calculated values for
KFα = KHα lt 1 use KFα = KHα = 10
If the calculated value of KHα gt 2εα
γ
Zε
ε use KHα = 2
εα
γ
Zε
ε
If the calculated value of KFα gt εα
γ
Yεε
use KFα = εα
γ
Yεε
where Yε = αnε
075025+ (for εαn see 331c)
Bevel gears
For ground or hard metal hobbed gears KFα = KHα = 1
For lapped gears KFα = KHα = 11
111 Tooth Stiffness Constants cacute and cγ The tooth stiffness is defined as the load which is necessary to deform one or several meshing gear teeth having 1 mm facewidth by an amount of 1 microm in the plane of contact cacute is the maximum stiffness of a single pair of teeth cγ is the mean value of the mesh stiffness in a transverse plane (brief term mesh stiffness)
Both valid for high unit load (Unit load = Ft middot KA middot Kγb)
Cylindrical gears The real stiffness is a combination of the progressive Hertzian contact stiffness and the linear tooth bending stiff-nesses For high unit loads the Hertzian stiffness has little importance and can be disregarded This approach is on the safe side for determination of KHβ and KHα However for moderate or low loads Kv may be underestimated due to determination of a too high resonance speed
The linear approach is described in A
An optional approach for inclusion of the non-linear stiffness is described in B
A The linear approach
BRCCq
βcos08acutec =
and
( )025ε075cc αγ +prime=
where
( )[ ]n02a01a
B α2000212
hh12051C minusminus
+minus+=
12n1n
x000635z
025791z
015551004723q minus++=
12
2n
22
1n
1 x005290z
x024188x000193z
x011654+minusminusminus
+ 000182 x22
Classification Notes- No 412 15 May 2003
DET NORSKE VERITAS
(for internal gears use zn2 equal infinite and x2 = 0) ha0 = hfp for all practical purposes CR considers the increased flexibility of the wheel teeth if the wheel is not a solid disc and may be calculated as
( )( )nR m5s
sR e5
bbln1C +=
where
bs = thickness of a central web
sR = average thickness of rim (net value from tooth root to inside of rim)
The formula is valid for bs b ge 02 and sRmn ge 1 Outside this range of validity and if the web is not centrally posi-tioned CR has to be specially considered
Note CR is the ratio between the average mesh stiffness over the facewidth and the mesh stiffness of a gear pair of solid discs The local mesh stiffness in way of the web corresponds to the mesh stiffness with CR = 1 The local mesh stiffness where there is no web support will be less than calculated with CR above Thus eg a centrally positioned web will have an effect corresponding to a longitudinal crowning of the teeth See also 1931 regarding KHβ
B The non-linear approach
In the following an example is given on how to consider the non-linearity
The relation between unit load Fb as a function of mesh deflection δ is assumed to be a progressive curve up to 500 Nmm and from there on a straight line This straight line when extended to the baseline is assumed to intersect at 10microm
With these assumptions the unit force Fb as a function of mesh deflection δ can be expressed as
( )10δKbF
minus= for 500bFgt
minus=
500Fb10δK
bF for 500
bFlt
with γAt KK
bF
bF
sdotsdot= etc (Nmm) ie unit load incorporat-
ing the relevant factors as
KA middot Kγ for determination of Kv
KA middot Kγ middot Kv for determination of KHβ
KA middot Kγ middot Kv middot KHβ for determination of KHα
δ = mesh deflection (microm)
K = applicable stiffness (c or cγ)
Use of stiffnesses for KV KHβ and KHα
For calculation of Kv and KHα the stiffness is calculated as follows
When Fb lt 500
the stiffness is determined as δ∆
∆ bF
where the increment is chosen as eg ∆ Fb = 10 and thus
50010Fb10
K10Fb∆δ +
++
=
When F b gt 500 the stiffness is c or cγ For calculation of KHβ the mesh deflection δ is used directly
or an equivalent stiffness determined as δsdotb
F
Bevel gears
In lack of more detailed relationship between stiffness and geometry the following may be used
b085
b13cacute eff=
b085b
16c effγ =
beff not to be used in excess of 085 b in these formulae
Bevel gears with heightwise and lengthwise crowning have progressive mesh stiffness The values mentioned above are only valid for high loads They should not be used for de-termination of Ceff (see 432) or KHβ (see 1931)
112 Running-in Allowances The running-in allowances account for the influence of run-ning-in wear on the various error elements
yα respectively yβ are the running-in amounts which reduce the influence of pitch and profile errors respectively influ-ence of localised faceload
Cay is defined as the running-in amount that compensates for lack of tip relief
The following relations may be used
For not surface hardened steel
ptHlim
α fσ160y =
yβ = βxlimH
f320σ
with the following upper limits
V lt 5 ms 5-10 ms gt 10 ms
yα max none
limH
12800σ limH
6400σ
yβ max none
limH
25600σ limH
12800σ
For surface hardened steel
yα = 0075 fpt but not more than 3 for any speed
yβ = 015 Fβx but not more than 6 for any speed
16 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
For all kinds of steel
5145189718
1C2
limHay +
minusσ
=
When pinion and wheel material differ the following ap-plies
bull Use the larger of fpt1 - yα1 and fpt2 - yα2 to replace fpt - yα in the calculation of KHα and Kv
bull Use ( )2β1ββ yy21y += in the calculation of KHβ
bull Use ( )2ay1aya CC21C += in the calculation of Kv
bull Use ( )2ay1ay2a1a CC21CC +== in the scuffing
calculation if no design tip relief is foreseen
Classification Notes- No 412 17 May 2003
DET NORSKE VERITAS
2 Calculation of Surface Durability
21 Scope and General Remarks Part 2 includes the calculations of flank surface durability as limited by pitting spalling case crushing and subsurface yielding Endurance and time limited flank surface fatigue is calculated by means of 22 ndash 212 In a way also tooth frac-tures starting from the flank due to subsurface fatigue is in-cluded through the criteria in 213
Pitting itself is not considered as a critical damage for slow speed gears However pits can create a severe notch effect that may result in tooth breakage This is particularly im-portant for surface hardened teeth but also for high strength through hardened teeth For high-speed gears pitting is not permitted
Spalling and case crushing are considered similar to pitting but may have a more severe effect on tooth breakage due to the larger material breakouts initiated below the surface Subsurface fatigue is considered in 213
For jacking gears (self-elevating offshore units) or similar slow speed gears designed for very limited life the max static (or very slow running) surface load for surface hard-ened flanks is limited by the subsurface yield strength
For case hardened gears operating with relatively thin lubri-cation oil films grey staining (micropitting) may be the lim-iting criterion for the gear rating Specific calculation meth-ods for this purpose are not given here but are under consid-eration for future revisions Thus depending on experience with similar gear designs limitations on surface durability rating other than those according to 22 - 213 may be ap-plied
22 Basic Equations Calculation of surface durability (pitting) for spur gears is based on the contact stress at the inner point of single pair contact or the contact at the pitch point whichever is greater
Calculation of surface durability for helical gears is based on the contact stress at the pitch point
For helical gears with 0 lt εβ lt 1 a linear interpolation be-tween the above mentioned applies
Calculation of surface durability for spiral bevel gears is based on the contact stress at the midpoint of the zone of contact
Alternatively for bevel gears the contact stress may be cal-culated with the program ldquoBECALrdquo In that case KA and Kv are to be included in the applied tooth force but not KHβ and KHα The calculated (real) Hertzian stresses are to be multi-plied with ZK in order to be comparable with the permissible contact stresses
The contact stresses calculated with the method in part 2 are based on the Hertzian theory but do not always represent the real Hertzian stresses
The corresponding permissible contact stresses σHP are to be calculated for both pinion and wheel
221 Contact stress Cylindrical gears
( )HαHβvγA
1
tβεEHDBH KKKKK
bud1uF
ZZZZZσ+
=
where
ZBD = Zone factor for inner point of single pair contact for pinion resp wheel (see 232)
ZH = Zone factor for pitch point (see 231)
ZE = Elasticity factor (see 24)
Zε = Contact ratio factor (see 25)
Zβ = Helix angle factor (see 26)
Ft KA Kγ Kv KHβ KHα see 15 ndash 110
d1 b u see 12 ndash 15
Bevel gears
( )HαHβvγA
v1v
vmtKEMH KKKKK
bud1uF
ZZZ105σ+
sdot=
where
105 is a correlation factor to reach real Hertzian stresses (when ZK = 1)
ZE KA etc see above
ZM = mid-zone factor see 233
ZK = bevel gear factor see 27
Fmt dv1 uv see 12 ndash 15
It is assumed that the heightwise crowning is chosen so as to result in the maximum contact stresses at or near the mid-point of the flanks
222 Permissible contact stress
XWRvLH
NHlimHP ZZZZZ
SZσ
σ =
where
σH lim = Endurance limit for contact stresses (see 28)
ZN = Life factor for contact stresses (see 29)
SH = Required safety factor according to the rules
ZLZvZR = Oil film influence factors (see 210)
ZW = Work hardening factor (see 211)
ZX = Size factor (see 212)
18 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
23 Zone Factors ZH ZBD and ZM
231 Zone factor ZH The zone factor ZH accounts for the influence on contact stresses of the tooth flank curvature at the pitch point and converts the tangential force at the reference cylinder to the normal force at the pitch cylinder
wtt2
wtbH sinααcos
cosαcosβ2Z =
232 Zone factors ZBD The zone factors ZBD account for the influence on contact stresses of the tooth flank curvature at the inner point of sin-gle pair contact in relation to ZH Index B refers to pinion D to wheel
For εβ ge 1 ZBD = 1
For internal gears ZD = 1
For εβ = 0 (spur gears)
( )
π
minusεminusminus
π
minusminus
α=
α2
2
2b
2a
1
2
1b
1a
wtB
z211
dd
z21
dd
tanZ
( )
π
minusεminusminus
π
minusminus
α=
α1
2
1b
1a
2
2
2b
2a
wtD
z211
dd
z21
dd
tanZ
If ZB lt 1 use ZB = 1
If ZD lt 1 use ZD = 1
For 0 lt εβ lt 1
ZBD = ZBD (for spur gears) ndash εβ (ZBD (for spur gears) ndash 1)
233 Zone factor ZM The mid-zone factor ZM accounts for the influence of the contact stress at the mid point of the flank and applies to spi-ral bevel gears
εminusminus
εminusminus
αβ=
αα btm2
2vb2
2vabtm2
1vb2val
2v1vvtbmM
pddpdd
ddtancos2Z
This factor is the product of ZH and ZM-B in ISO 10300 with the condition that the heightwise crowning is sufficient to move the peak load towards the midpoint
234 Inner contact point For cylindrical or bevel gears with very low number of teeth the inner contact point (A) may be close to the base circle In order to avoid a wear edge near A it is required to have suitable tip relief on the wheel
24 Elasticity Factor ZE The elasticity factor ZE accounts for the influence of the material properties as modulus of elasticity and Poissonrsquos ratio on the contact stresses
For steel against steel ZE = 1898
25 Contact Ratio Factor Zε The contact ratio factor Zε accounts for the influence of the transverse contact ratio εα and the overlap ratio εβ on the contact stresses
αε1Z =ε for 1εβ ge
( )α
ββ
αε ε
εε1
3ε4
Z +minusminus
= for εβ lt 1
26 Helix Angle Factor Zβ The helix angle factor Zβ accounts for the influence of helix angle (independent of its influence on Zε) on the surface du-rability
cosβZβ =
27 Bevel Gear Factor ZK The bevel gear factor accounts for the difference between the real Hertzian stresses in spiral bevel gears and the contact stresses assumed responsible for surface fatigue (pitting) ZK adjusts the contact stresses in such a way that the same per-missible stresses as for cylindrical gears may apply
The following may be used ZK = 080
28 Values of Endurance Limit σHlim and Static Strength 5H10σ 3H10σ
σHlim is the limit of contact stress that may be sustained for 5middot107 cycles without the occurrence of progressive pitting
For most materials 5middot107 cycles are considered to be the be-ginning of the endurance strength range or lower knee of the σ-N curve (See also Life Factor ZN) However for nitrided steels 2middot106 apply
For this purpose pitting is defined by
bull for not surface hardened gears pitted area ge 2 of total active flank area
bull for surface hardened gears pitted area ge 05 of total active flank area or ge 4 of one particular tooth flank area 510Hσ and 310Hσ are the contact stresses which the given
material can withstand for 105 respectively 103 cycles without subsurface yielding or flank damages as pitting spalling or case crushing when adequate case depth applies
The following listed values for σHlim 510Hσ and 310Hσ may only be used for materials subjected to a quality control as
Classification Notes- No 412 19 May 2003
DET NORSKE VERITAS
the one referred to in the rules Results of approved fatigue tests may also be used as the basis for establishing these values The defined survival probability is 99
σHlim σH105 σH10
3
Alloyed case hardened steels (surface hardness 58-63 HRC) - of specially approved high grade - of normal grade
1650 1500
2500 2400
3100 3100
Nitrided steel of approved grade gas nitrided (surface hardness 700-800 HV) 1250 13 σHlim 13 σHlim
Alloyed quenched and tempered steel bath or gas nitrided (surface hardness 500-700 HV)
1000
13 σHlim
13 σHlim
Alloyed flame or induction hardened steel (surface hardness 500-650 HV) 075 HV + 750 16 σHlim 45 HV
Alloyed quenched and tempered steel 14 HV + 350 16 σHlim 45 HV
Carbon steel 15 HV + 250 16 σHlim 16 σHlim
These values refer to forged or hot rolled steel For cast steel the values for σHlim are to be reduced by 15
29 Life Factor ZN The life factor ZN takes account of a higher permissible contact stress if only limited life (number of cycles NL) is demanded or lower permissible contact stress if very high number of cycles apply
If this is not documented by approved fatigue tests the fol-lowing method may be used
For all steels except nitrided
7L 105N sdotge ZN = 1 or
01570
L
7
N N105Z
sdot=
Ie ZN = 092 for 1010 cycles
The ZN = 1 from 5middot107 on may only be used when the mate-rial cleanliness is of approved high grade (see Rules Pt4 Ch2) and the lubrication is optimised by a specially approved filtering process
105 lt NL lt 5middot107 510N
logZ037
L
7
N N105Z
sdot=
NL = 105 WXRVLHlim
WstX10H1010NN ZZZZZσ
ZZσZZ
55
5=
103 lt NL lt 105 )Z(Zlog05
L
5
10NN
5N103N10
5N10ZZ
==
3L 10N le
WXRVLlimH
Wst10X10H10NN ZZZZZ
ZZZZ
33
3σ
σ==
(but not less than ZN105)
For nitrided steels
6L 102N sdotge ZN = 1 or
00980
L
6
N N102Z
sdot=
Ie ZN = 092 for 1010 cycles
The ZN = 1 from 2middot106 on may only be used when the mate-rial cleanliness is of approved high grade (see Rules Pt4 Ch2) and the lubrication is optimised by a specially approved filtering process
105 lt NL lt 2middot106 510N
Zlog76860
L
6
N N102Z
sdot=
5L 10N le
XWRVL
10XWst10NN ZZZZZ
ZZ13ZZ
55 ==
Note that when no index indicating number of cycles is used the factors are valid for 5middot107 (respectively 2middot106 for nitrid-ing) cycles
210 Influence Factors on Lubrication Film ZL ZV and ZR The lubricant factor ZL accounts for the influence of the type of lubricant and its viscosity the speed factor ZV ac-counts for the influence of the pitch line velocity and the roughness factor ZR accounts for influence of the surface roughness on the surface endurance capacity
The following methods may be applied in connection with the endurance limit
20 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
Surface hardened steels Not surface hardened steels
ZL ( )24013421
360910ν+
+
( )24013421
680830ν+
+
ZV ( )v3280
140930+
+ ( )v3280300850
++
ZR
080
ZrelR3
150
ZrelR3
where
ν40 = Kinematic oil viscosity at 40ordmC (mm2s) For case hardened steels the influence of a high bulk temperature (see 4 Scuffing) should be con-sidered Eg bulk temperatures in excess of 120ordmC for long periods may cause reduced flank surface endurance limits
For values of ν40 gt 500 use ν40 = 500
RZrel = The mean roughness between pinion and wheel (after running in) relative to an equivalent radius of curvature at the pitch point ρc = 10mm
RZrel
= ( ) 31
cZ2Z1 ρ
10RR50
+
RZ = Mean peak to valley roughness (microm) (DIN defini-tion) (roughly RZ = 6 Ra)
For 5
L 10N le ZL ZV ZR = 10
211 Work Hardening Factor ZW The work hardening factor ZW accounts for the increase of surface durability of a soft steel gear when meshing the soft steel gear with a surface hardened or substantially harder gear with a smooth surface
The following approximation may be used for the endurance limit
Surface hardened steel against not surface hardened steel 150
ZeqW R
31700
130HB21Z
minus
minus=
where HB = the Brinell hardness of the soft member For HB gt 470 use HB = 470 For HB lt 130 use HB = 130
RZeq = equivalent roughness
033
c40
066
ZS
ZHZHZEQ ρvν
15000RRRR
=
If RZeq gt 16 then use RZeq = 16
If RZeq lt 15 then use RZeq = 15
where
RZH = surface roughness of the hard member before run in
RZS = surface roughness of the soft member before run in
ν40 = see 210
If values of ZW lt 1 are evaluated ZW = 1 should be used for flank endurance However the low value for ZW may indi-cate a potential wear problem
Through hardened pinion against softer wheel
( )
minussdotminus+= 000829
HBHB0008981u1Z
2
1W
For 21HBHB
2
1 le use ZW = 1
For 71HBHB
2
1 gt use 17HBHB
2
1 =
For u gt 20 use u = 20
For static strength (lt 105 cycles)
Surface hardened against not surface hardened
ZWst = 105
Through hardened pinion against softer wheel
ZWst = 1
212 Size Factor ZX The size factor accounts for statistics indicating that the stress levels at which fatigue damage occurs decrease with an increase of component size as a consequence of the influ-ence on subsurface defects combined with small stress gradi-ents and of the influence of size on material quality
ZX may be taken unity provided that subsurface fatigue for surface hardened pinions and wheels is considered eg as in the following subsection 213
213 Subsurface Fatigue This is only applicable to surface hardened pinions and wheels The main objective is to have a subsurface safety against fatigue (endurance limit) or deformation (static strength) which is at least as high as the safety SH required for the surface The following method may be used as an approximation unless otherwise documented
The high cycle fatigue (gt3106 cycles) is assumed to mainly depend on the orthogonal shear stresses Static strength (lt103 cycles) is assumed to depend mainly on equivalent
Classification Notes- No 412 21 May 2003
DET NORSKE VERITAS
stresses (von Mises) Both are influenced by residual stresses but this is only considered roughly and empirically
The subsurface working stresses at depths inside the peak of the orthogonal shear stresses respectively the equivalent stresses are only dependent on the (real) Hertzian stresses Surface related conditions as expressed by ZL ZV and ZR are assumed to have a negligible influence
The real Hertzian stresses σHR are determined as
For helical gears with εβ gt 1
σHR = σH
For helical gears with εβ lt 1 and spur gears
ε
α
ββ ε
ε+εminus
sdotσ=σZ
1
HHR
For bevel gears
KHHR Z
1σσ sdot=
The necessary hardness HV is given as a function of the net depth tz (net = after grinding or hard metal hobbing and per-pendicular to the flank)
The coordinates tz and HV are to be compared with the de-sign specification such as
bull for flame and induction hardening tHVmin HVmin bull for nitriding t400min HV = 400 bull for case hardening t550min HV = 550 t400min HV = 400
and t300min HV = 300 (the latter only if the core hardness lt 300 If the core hardness gt 400 the t400 is to be re-placed by a fictive t400 = 16 t550)
In addition the specified surface hardness is not to be less than the max necessary hardness (at tz = 05aH) This applies to all hardening methods
For high cycle fatigue (gt3 106 cycles) the following applies
sdot+
minussdotsdotsdot= o90
05azt
05azt
cosSσ04HV
H
HHHR
applicable to 05a
zt
Hge
For 05at
H
z lt the value for 05at
H
z = applies
56300ρSσ12a cHHR
Hsdotsdot
sdot=
Where aH is half the hertzian contact width multiplied by an empirical factor of 12 that takes into account the possible influence of reduced compressive residual stresses (or even tensile residual stresses) on the local fatigue strength
If any of the specified hardness depths including the surface hardness is below the curve described by HV = f (tz) the actual safety factor against subsurface fatigue is determined as follows
reduce SH stepwise in the formula for HV and aH until all specified hardness depths and surface hardness balance with the corrected curve The safety factor obtained through this method is the safety against subsurface fatigue
For static strength (lt103 cycles) the following applies
sdot+
minussdotsdotsdot= o90
07at
06a
zt
cosSσ019HV
Hst
z
HstHHR
applicable to 06at
Hst
z ge
56300ρSσa cHHR
Hstsdotsdot
=
In the case of insufficient specified hardness depths the same procedure for determination of the actual safety factor as above applies
For limited life fatigue (103 lt cycles lt 3106 )
For this purpose it is necessary to extend the correction of safety factors to include also higher values than required Ie in the case of more than sufficient hardness and depths the safety factor in the formulae for both high cycle fatigue and static strength are to be increased until necessary and speci-fied values balance
The actual safety factor for a given number of cycles N be-tween 103 and 3106 is found by linear interpolation in a dou-ble logarithmic diagram
minussdotminus
= sdot logN3477
logSlogSlogS
310H6103HHN
36 10H103H Slog86281Slog86280 sdot+sdot sdot
22 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
3 Calculation of Tooth Strength
31 Scope and General Remarks Part 3 include the calculation of tooth root strength as limited by tooth root cracking (surface or subsurface initiated) and yielding
For rim thickness sR ge 35middotmn the strength is calculated by means of 32 ndash 313 For cylindrical gears the calculation is based on the assumption that the highest tooth root tensile stress arises by application of the force at the outer point of single tooth pair contact of the virtual spur gears The method has however a few limitations that are mentioned in 36
For bevel gears the calculation is based on force application at the tooth tip of the virtual cylindrical gear Subsequently the stress is converted to load application at the mid point of the flank due to the heightwise crowning
Bevel gears may also be calculated with the program BECAL In that case KA and Kv are to be included in the applied tooth force but not KFβ and KFα
In case of a thin annulus or a thin gear rim etc radial crack-ing can occur rather than tangential cracking (from root fillet to root fillet) Cracking can also start from the compression fillet rather than the tension fillet For rim thickness sR lt 35middotmn a special calculation procedure is given in 315 and 316 and a simplified procedure in 314
A tooth breakage is often the end of the life of a gear trans-mission Therefore a high safety SF against breakage is re-quired
It should be noted that this part 3 does not cover fractures caused by
bull oil holes in the tooth root space bull wear steps on the flank bull flank surface distress such as pits spalls or grey staining
Especially the latter is known to cause oblique fractures starting from the active flank predominately in spiral bevel gears but also sometimes in cylindrical gears
Specific calculation methods for these purposes are not given here but are under consideration for future revisions Thus depending on experience with similar gear designs limita-tions other than those outlined in part 3 may be applied
32 Tooth Root Stresses The local tooth root stress is defined as the max principal stress in the tooth root caused by application of the tooth force Ie the stress ratio R = 0 Other stress ratios such as for eg idler gears (R asymp -12) shrunk on gear rims (R gt 0) etc are considered by correcting the permissible stress level
321 Local tooth root stress The local tooth root stress for pinion and wheel may be as-sessed by strain gauge measurements or FE calculations or similar For both measurements and calculations all details are to be agreed in advance
Normally the stresses for pinion and wheel are calculated as
Cylindrical gears
FαFβvγAβSFn
tF K K K K K Y Y Y
m bFσ =
where
YF = Tooth form factor (see 33)
YS = Stress correction factor (see 34)
Yβ = Helix angle factor (see 36)
Ft KA Kγ Kv KFβ KFα see 15 ndash 110
b see 13
Bevel gears
FαFβvγASaFamn
mtF K K K K K Y Y Y
m bFσ ε=
where
YFa = Tooth form factor see 33
YSa = Stress correction factor see 34
Yε = Contact ratio factor see 35
Fmt KA etc see 15 ndash 110
b see 13
322 Permissible tooth root stress The permissible local tooth root stress for pinion respectively wheel for a given number of cycles N is
CXRrelTrelTF
NMFEFP YYYY
SYY
δσ
=σ
Note that all these factors YM etc are applicable to 3middot106 cycles when used in this formula for σFP The influence of other number of cycles on these factors is covered by the calculation of YN
where
σFE = Local tooth root bending endurance limit of reference test gear (see 37)
YM = Mean stress influence factor which accounts for other loads than constant load direction eg idler gears temporary change of load di-rection pre-stress due to shrinkage etc (see 38)
YN = Life factor for tooth root stresses related to reference test gear dimensions (see 39)
SF = Required safety factor according to the rules
YδrelT = Relative notch sensitivity factor of the gear to be determined related to the reference test gear (see 310)
YRrelT = Relative (root fillet) surface condition factor of the gear to be determined related to the reference test gear (see 311)
Classification Notes- No 412 23 May 2003
DET NORSKE VERITAS
YX = Size factor (see 312)
YC = Case depth factor (see 313)
33 Tooth Form Factors YF YFa The tooth form factors YF and YFa take into account the in-fluence of the tooth form on the nominal bending stress
YF applies to load application at the outer point of single tooth pair contact of the virtual spur gear pair and is used for cylindrical gears
YFa applies to load application at the tooth tip and is used for bevel gears
Both YF and YFa are based on the distance between the con-tact points of the 30˚-tangents at the root fillet of the tooth profile for external gears respectively 60˚ tangents for inter-nal gears
Figure 31 External tooth in normal section
Figure 32 Internal tooth in normal section
Definitions
n
2
n
Fn
enFn
Fe
F
α cosms
α cosmh6
Y
=
n
2
n
Fn
anFn
Fa
Fa
α cosms
α cosmh6
Y
=
In the case of helical gears YF and YFa are determined in the normal section ie for a virtual number of teeth
YFa differs from YF by the bending moment arm hFa and αFan and can be determined by the same procedure as YF with exception of hFe and αFan For hFa and αFan all indices e will change to a (tip)
The following formulae apply to cylindrical gears but may also be used for bevel gears when replacing
mn with mnm
zn with zvn
αt with αvt
β with βm
with undercut without undercut
Fig 33 Dimensions and basic rack profile of the teeth (finished profile)
Tool and basic rack data such as hfP ρfp and spr etc are re-ferred to mn ie dimensionless
331 Determination of parameters
( )n
n
prnfPnfP m
α cossαsin 1ρ
αtan h4πE
minusminusminusminus=
For external gears fPfP ρρ =
For internal gears ( )0z
195fPfP0
fPfP 10363156ρhxρρ
sdotminus+
+=
where
z0 = number of teeth of pinion cutter
x0 = addendum modification coefficient of pinion cutter
hfP = addendum of pinion cutter
ρfP = tip radius of pinion cutter
xhρG fpfp +minus=
24 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
τmE
2π
z2H
nnminus
minus=
with
3πτ = for external gears
6πτ = for internal gears
Htan zG2
nminusϑ=ϑ
(to be solved iteratively suitable start value6π
=ϑ for exter-
nal gears and 3π for internal gears)
a) Tooth root chord sFn For external gears
minus
ϑ+
ϑminus= ρ
cosG3
3πsinz
ms
fpnn
Fn
For bevel gears with a tooth thickness modification
xsm affects mainly sFn but also hFe and αFen The total influence of xsm on YFa Ysa can be approximated by only adding 2 xsm to sFn mn
For internal gears
minus
ϑ+
ϑminus= ρ
cosG
6πsinz
ms
fPnn
Fn
b) Root fillet radius ρF at 30ordm tangent
( )G2coszcosG2ρ
mρ
2n
2
fpn
F
minusϑϑ+=
c) Determination of bending moment arm hF dn = zn mn
b2α
αn βcosε
ε =
dan = dn + 2 ha
pbn = π mn cos αn
dbn = dn cos αn
( )4
d1εpzz
2dd
zz2d
2bn
2
αnbn
2bn
2an
en +
minusminus
minus=
en
bnen d
dcos arcα =
ennnn
e α invα invα x tan 22π
z1
minus+
+=γ
αFen = αen ndash γe
For external gears
( )
minus=
n
enFenee
n
Fe
mdαtan sin γ γcos
21
mh
]ρcos
G3πcosz fpn +
ϑminus
ϑminusminus
For internal gears
( ) minus
sdotsdotminus=
n
enFenee
n
Fe
mdα tanγ sinγ cos
21
mh
minus
ϑminus
ϑminussdot ρ
cosG3
6πcosz fPn
332 Gearing with εαn gt 2 For deep tooth form gearing ( )25ε2 αn lele produced with a verified grade of accuracy of 4 or better and with applied profile modification to obtain a trapezoidal load distribution along the path of contact the YF may be corrected by the factor YDT as
250ε205for 0666ε2366Y αnαnDT leleminus=
205εfor 10Y αnDT lt=
34 Stress Correction Factors YS YSa The stress correction factors YS and YSa take into account the conversion of the nominal bending stress to the local tooth root stress Thereby YS and YSa cover the stress increasing effect of the notch (fillet) and the fact that not only bending stresses arise at the root A part of the local stress is inde-pendent of the bending moment arm This part increases the more the decisive point of load application approaches the critical tooth root section
Therefore in addition to its dependence on the notch radius the stress correction is also dependent on the position of the load application ie the size of the bending moment arm
YS applies to the load application at the outer point of single tooth pair contact YSa to the load application at tooth tip
YS can be determined as follows
( )
+
+=L23121
1
sS qL01312Y
where Fe
Fn
hsL = and
F
Fns ρ2
sq = (see 33)
YSa can be calculated by replacing hFe with hFa in the above formulae
Note a) Range of validity 1 lt qs lt 8
In case of sharper root radii (ie produced with tools having too sharp tip radii) YS resp YSa must be specially considered
b) In case of grinding notches (due to insufficient protuberance of the hob) YS resp YSa can rise considerably and must be multiplied with
Classification Notes- No 412 25 May 2003
DET NORSKE VERITAS
g
g
ρt
0613
13
minus
where
tg = depth of the grinding notch
ρg = radius of the grinding notch
c) The formulae for YS resp YSa are only valid for αn = 20˚ However the same formulae can be used as a safe approximation for other pressure angles
35 Contact Ratio Factor Yε The contact ratio factor Yε covers the conversion from load application at the tooth tip to the load application at the mid point of the flank (heightwise) for bevel gears
The following may be used
Yε = 0625
36 Helix Angle Factor Yβ The helix angle factor Yβ takes into account the difference between the helical gear and the virtual spur gear in the nor-mal section on which the calculation is based in the first step In this way it is accounted for that the conditions for tooth root stresses are more favourable because the lines of contact are sloping over the flank
The following may be used (β input in degrees)
Yβ = 1 ndash εβ β120
When εβ gt 1 use εβ = 1 and when β gt 30deg use β = 30deg in the formula
However the above equation for Yβ may only be used for gears with β gt 25deg if adequate tip relief is applied to both pinion and wheel (adequate = at least 05 middot Ceff see 432)
37 Values of Endurance Limit σFE σFE is the local tooth root stress (max principal) which the material can endure permanently with 99 survival prob-ability 3106 load cycles is regarded as the beginning of the endurance limit or the lower knee of the σ ndash N curve σFE is defined as the unidirectional pulsating stress with a minimum stress of zero (disregarding residual stresses due to heat treatment) Other stress conditions such as alternating or pre-stressed etc are covered by the conversion factor YM
σFE can be found by pulsating tests or gear running tests for any material in any condition If the approval of the gear is to be based on the results of such tests all details on the testing conditions have to be approved by the Society Fur-ther the tests may have to be made under the Societys su-pervision
If no fatigue tests are available the following listed values for σFE may be used for materials subjected to a quality con-trol as the one referred to in the rules
σFE
Alloyed case hardened steels 1) (fillet surface hardness 58 ndash 63 HRC)
bull of specially approved high grade
1050
bull of normal grade
minus CrNiMo steels with approved process
1000
minus CrNi and CrNiMo steels generally 920 minus MnCr steels generally 850
Nitriding steel of approved grade quenched tempered and gas nitrided (surface hardness 700 ndash 800 HV)
840
Alloyed quenched and tempered steel bath or gas nitrided (surface hardness 500 ndash 700 HV)
720
Alloyed quenched and tempered steel flame or induction hardened 2) (incl entire root fillet) (fillet surface hardness 500 ndash 650 HV)
07 HV + 300
Alloyed quenched and tempered steel flame or induction hardened (excl entire root fillet) (σB = uts of base material)
025 σB + 125
Alloyed quenched and tempered steel 04 σB + 200
Carbon steel 025 σB + 250
Note All numbers given above are valid for separate forgings and for blanks cut from bars forged according to a qualified procedure see Pt 4 Ch 2 Sec 3 For rolled steel the values are to be reduced with 10 For blanks cut from forged bars that are not qualified as mentioned above the values are to be reduced with 20 For cast steel reduce with 40
1) These values are valid for a root radius
bull being unground If however any grinding is made in the root fillet area in such a way that the residual stresses may be affected σFE is to be reduced by 20 (If the grinding also leaves a notch see 34)
bull with fillet surface hardness 58 ndash 63 HRC In case of lower surface hardness than 58 HRC σFE is to be reduced with 20(58 ndash HRC) where HRC is the detected hardness (This may lead to a permissible tooth root stress that varies along the facewidth If so the actual tooth root stresses may also be considered along facewidth)
bull not being shot peened In case of approved shot peening σFE may be increased by 200 for gears where σFE is reduced by 20 due to root grinding Otherwise σFE may be increased by 100 for
6nm le and 100 ndash 5 (mn - 6) for mn gt 6 However the possible adverse influence on the flanks regarding grey staining should be considered and if necessary the flanks should be masked
2) The fillet is not to be ground after surface hardening Regarding possible root grinding see 1)
26 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
38 Mean stress influence Factor YM The mean stress influence factor YM takes into account the influence of other working stress conditions than pure pulsa-tions (R = 0) such as eg load reversals idler gears planets and shrink-fitted gears
YM (YMst) are defined as the ratio between the endurance (or static) strength with a stress ratio R ne 0 and the endurance (or static) strength with R = 0
YM and YMst apply only to a calculation method that assesses the positive (tensile) stresses and is therefore suitable for comparison between the calculated (positive) working stress σF and the permissible stress σFP calculated with YM or YMst
For thin rings (annulus) in epicyclic gears where the com-pression fillet may be decisive special considerations apply see 316
The following method may be used within a stress ratio ndash12 lt R lt 05
381 For idlers planets and PTO with ice class
M1M1R1
1Yor Y MstM
+minus
minus=
where
R = stress ratio = min stress divided by max stress
For designs with the same force applied on both forward- and back-flank R may be assumed to ndash 12
For designs with considerably different forces on forward- and back-flank such as eg a marine propulsion wheel with a power take off pinion R may be assessed as
branchmain theoffacewidth unit per forcepto offacewidt unit per force21minus
For a power take off (PTO) with ice class see 161 c
M considers the mean stress influence on the endurance (or static) strength amplitudes
M is defined as the reduction of the endurance strength am-plitude for a certain increase of the mean stress divided by that increase of the mean stress
Following M values may be used
Endurance limit
Static strength
Case hardened 08 ndash 015 Ys 1) 07
If shot peened 04 06
Nitrided 03 03
Induction or flame hardened
04 06
Not surface hardened steel 03 05
Cast steels 04 06 1)For bevel gears use Ys=2 for determination of M
The listed M values for the endurance limit are independent of the fillet shape (Ys) except for case hardening In princi-ple there is a dependency but wide variations usually only occur for case hardening eg smooth semicircular fillets versus grinding notches
382 For gears with periodical change of rotational direction For case hardened gears with full load applied periodically in both directions such as side thrusters the same formula for YM as for idlers (with R = ndash 12) may be used together with the M values for endurance limit This simplified approach is valid when the number of changes of direction exceeds 100 and the total number of load cycles exceeds 3middot106
For gears of other materials YM will normally be higher than for a pure idler provided the number of changes of direction is below 3middot106 A linear interpolation in a diagram with loga-rithmic number of changes of direction may be used ie from YM = 09 with one change to YM (idler) for 3middot106 changes This is applicable to YM for endurance limit For static strength use YM as for idlers
For gears with occasional full load in reversed direction such as the main wheel in a reversing gear box YM = 09 may be used
383 For gears with shrinkage stresses and unidirectional load For endurance strength
FE
fitM σ
σM1
M21Y+
minus=
σFE is the endurance limit for R = 0
For static strength YMst = 1 and σfit accounted for in 39b
σfit is the shrinkage stress in the fillet (30˚ tangent) and may be found by multiplying the nominal tangential (hoop) stress with a stress concentration factor
n
Ffit m
ρ215scf minus=
384 For shrink-fitted idlers and planets When combined conditions apply such as idlers with shrink-age stresses the design factor for endurance strength is
( ) ( ) FE
fitM σ
σR1M1
M2
M1M1R1
1Yminussdot+
minus
+minus
minus=
Symbols as above but note that the stress ratio R in this par-ticular connection should disregard the influence of σfit ie R normally equal ndash 12
For static strength
M1M1R1
1YMst
+minus
minus=
The effect of σfit is accounted for in 39 b
Classification Notes- No 412 27 May 2003
DET NORSKE VERITAS
385 Additional requirements for peak loads The total stress range (σmax ndash σmin) in a tooth root fillet is not to exceed
F
y
Sσ225
for not surface hardened fillets
FSHV5 for surface hardened fillets
39 Life Factor YN The life factor YN takes into account that in the case of limited life (number of cycles) a higher tooth root stress can be permitted and that lower stresses may apply for very high number of cycles
Decisive for the strength at limited life is the σ ndash N ndash curve of the respective material for given hardening module fillet radius roughness in the tooth root etc Ie the factors YδrelT YRelT YX and YM have an influence on YN
If no σ ndash N ndash curve for the actual material and hardening etc is available the following method may be used
Determination of the σ ndash N ndash curve
a) Calculate the permissible stress σFP for the beginning of the endurance limit (3middot106 cycles) including the influ-ence of all relevant factors as SF YδrelT YRelT YX YM and YC ie σFP = σFE middotYM middotYδrelT middotYRelT middotYX middotYC SF
b) Calculate the permissible laquostaticraquo stress (le 103 load cy-cles) including the influence of all relevant factors as SFst YδrelTst YMst and YCst ( )fitσCstYδrelTstYMstYFstσ
FstS1
FPstσ minussdotsdotsdot=
where σFst is the local tooth root stress which the material can resist without cracking (surface hardened materials) or unacceptable deformation (not surface hardened materials) with 99 survival probability
σFst
Alloyed case hardened steel 1) 2300
Nitriding steel quenched tempered and gas nitrided (surface hardness 700 ndash 800 HV)
1250
Alloyed quenched and tempered steel bath or gas nitrided (surface hardness 500 ndash 700 HV)
1050
Alloyed quenched and tempered steel flame or induction hardened (fillet surface hardness 500 ndash 650 HV)
18 HV + 800
Steel with not surface hardened fillets the smaller value of 2)
18 σB or 225 σy
1) This is valid for a fillet surface hardness of 58 ndash 63 HRC In case of lower fillet surface hardness than 58 HRC σFst is to be reduced with 30(58 ndash HRC) where HRC is the actual hardness Shot peening or grinding notches are not considered to have any significant influence on σFst 2) Actual stresses exceeding the yield point (σy or σ02) will alter the residual stresses locally in the ldquotensionrdquo fillet re-spectively ldquocompressionrdquo fillet This is only to be utilised for gears that are not later loaded with a high number of cycles at lower loads that could cause fatigue in the ldquocompressionrdquo fillet
c) Calculate YN as
NL gt 3middot106
YN = 1 or 001
L
6
N N103Y
sdot= ie Yn = 092 for 1010
The YN = 1 from 3middot106 on may only be used when special material cleanness applies see rules Pt4 Ch2
103ltNLlt3middot106exp
L
6
N N103Y
sdot=
cycles6103forσcycles310forσlog02876exp
FP
FPst
sdot=
NL lt 103 cycles6103forσcycles310forσ
NYFP
FPst
sdot=
or simply use σFPst as mentioned in b) directly
Guidance on number of load cycles NL for various applica-tions
bull For propulsion purpose normally NL = 1010 at full load (yachts etc may have lower values)
bull For auxiliary gears driving generators that normally operate with 70-90 of rated power NL = 108 with rated power may be applied
28 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
310 Relative Notch Sensitivity Factor YδrelT The dynamic (respectively static) relative notch sensitivity factor YδrelT (YδrelTst) indicate to which extent the theoreti-cally concentrated stress lies above the endurance limits (re-spectively static strengths) in the case of fatigue (respectively overload) breakage
YδrelT is a function of the material and the relative stress gra-dient It differs for static strength and endurance limit
The following method may be used
For endurance limit
for not surface hardened fillets
( )
024
s024
relT σ103133q21σ1012201351
Ysdotsdotminus
+sdotsdotminus+= minus
minus
δ
for all surface hardened fillets except nitrided
( )
106q21002451
Y srelT
++=δ
for nitrided fillets
( )
1347q2101421
Y srelT
++=δ
For static strength
for not surface hardened fillets1)
( )( )( )025
02
02502s
relTst σ3000821σ300 1Y0821Y
+
minus+=δ
for surface hardened fillets except nitrided
YδrelTst = 044 YS + 012
for nitrided fillets
YδrelTst = 06 + 02 YS 1) These values are only valid if the local stresses do not
exceed the yield point and thereby alter the residual stress level See also 39b footnote 2
311 Relative Surface Condition Factor YRrelT The relative surface condition factor YRrelT takes into ac-count the dependence of the tooth root strength on the sur-face condition in the tooth root fillet mainly the dependence on the peak to valley surface roughness
YRrelT differs for endurance limit and static strength
The following method may be used
For endurance limit
YRrelT = 1675 ndash 053 (Ry + 1)01 for surface hardened steels and alloyed quenched and tempered steels except nitrided
YRrelT = 53 ndash 42 (Ry + 1)001 for carbon steels
YδrelT = 43 ndash 326 (Ry + 1)0005
for nitrided steels
For static strength
YRrelTst = 1 for all Ry and all materials
For a fillet without any longitudinal machining trace Ry asymp Rz
312 Size Factor YX The size factor YX takes into account the decrease of the strength with increasing size YX differs for endurance limit and static strength
The following may be used
For endurance limit
YX = 1 for mn le 5 generally
YX = 103 ndash 0006 mn for 5 lt mn lt 30
YX = 085 for mn ge 30
for not surface hard-ened steels
YX = 105 ndash 001 mn for 5 lt mn ge 25
YX = 08 for mn ge 25
for surface hardened steels
For static strength
YXst = 1 for all mn and all materials
313 Case Depth Factor YC The case depth factor YC takes into account the influence of hardening depth on tooth root strength
YC applies only to surface hardened tooth roots and is dif-ferent for endurance limit and static strength
In case of insufficient hardening depth fatigue cracks can develop in the transition zone between the hardened layer and the core For static strength yielding shall not occur in the transition zone as this would alter the surface residual stresses and therewith also the fatigue strength
The major parameters are case depth stress gradient permis-sible surface respectively subsurface stresses and subsurface residual stresses
The following simplified method for YC may be used
YC consists of a ratio between permissible subsurface stress (incl influence of expected residual stresses) and permissible surface stress This ratio is multiplied with a bracket con-taining the influence of case depth and stress gradient (The empirical numbers in the bracket are based on a high number of teeth and are somewhat on the safe side for low number of teeth)
YC and YCst may be calculated as given below but calculated values above 10 are to be put equal 10
For endurance limit
+
+=nFFE
C m02ρt31
σconstY
Classification Notes- No 412 29 May 2003
DET NORSKE VERITAS
For static strength
+
+=nFFst
Cst m02ρt31
σconstY
where const and t are connected as
Hardening process
t = endurance limit const =
static strength const =
t550 640 1900
t400 500 1200 Case hard-ening
t300 380 800
Nitriding t400 500 1200
Induction- or flame hardening
tHVmin 11 HVmin 25 HVmin
For symbols see 213
In addition to these requirements to minimum case depths for endurance limit some upper limitations apply to case hard-ened gears
The max depth to 550 HV should not exceed
1) 13 of the top land thickness san unless adequate tip relief is applied (see 110)
2) 025 mn If this is exceeded the following applies addi-tionally in connection with endurance limit
minusminus= 025
mt
1Yn
max550C
314 Thin rim factor YB Where the rim thickness is not sufficient to provide full sup-port for the tooth root the location of a bending failure may be through the gear rim rather than from root fillet to root fillet
YB is not a factor used to convert calculated root stresses at the 30deg tangent to actual stresses in a thin rim tension fillet Actually the compression fillet can be more susceptible to fatigue
YB is a simplified empirical factor used to de-rate thin rim gears (external as well as internal) when no detailed calcula-tion of stresses in both tension and compression fillets are available
Figure 34 Examples on thin rims
YB is applicable in the range 175 lt sRmn lt 35
YB = 115 middot ln (8324 middot mnsR)
(for sRmn ge 35 YB = 1)
(for sRmn le 175 use 315)
σF as calculated in 321 is to be multiplied with YB when sRmn lt 35 Thus YB is used for both high and low cycle fatigue
Note This method is considered to be on the safe side for external gear rims However for internal gear rims without any flange or web stiffeners the method may not be on the safe side and it is advised to check with the method in 315316
315 Stresses in Thin Rims For rim thickness sR lt 35 mn the safety against rim cracking has to be checked
The following method may be used
3151 General The stresses in the standardised 30ordm tangent section tension side are slightly reduced due to decreasing stress correction factor with decreasing relative rim thickness sRmn On the other hand during the complete stress cycle of that fillet a certain amount of compression stresses are also introduced The complete stress range remains approximately constant Therefore the standardised calculation of stresses at the 30ordm tangent may be retained for thin rims as one of the necessary criteria
The maximum stress range for thin rims usually occurs at the 60ordm ndash 80ordm tangents both for laquotensionraquo and laquocompressionraquo side The following method assumes the 75ordm tangent to be the decisive Therefore in addition to the am criterion ap-plied at the 30ordm tangent it is necessary to evaluate the max and min stresses at the 75ordm tangent for both laquotensionraquo (loaded flank) fillet and laquocompressionraquo (back-flank) fillet For this purpose the whole stress cycle of each fillet should be considered but usually the following simplification is justified
Figure 35 Nomenclature of fillets
Index laquoTraquo means laquotensileraquo fillet laquoCraquo means laquocompressionraquo fillet
σFTmin and σFCmax are determined on basis of the nominal rim stresses times the stress concentration factor Y75
σFCmin and σFTmax are determined on basis of superposition of nominal rim stresses times Y75 plus the tooth bending stresses at 75ordm tangent
30 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
3152 Stress concentration factors at the 75ordm tangents The nominal rim stress consists of bending stresses due to local bending moments tangential stresses due to the tan-gential force Ft and radial shear stresses due to Fr
The major influence is given by the bending stresses The influence of the tangential stresses is minor and even though its stress concentration factor is slightly higher than for bending it is considered to be safe enough when the sum of these nominal stresses are combined with the stress concen-tration factor for bending The influence of the radial shear stress is neglected
The stress concentration factor relating nominal rim stresses to local fillet stresses at the 75ordm tangent may be calculated as
75n
R75
n
R
75
ρmsρ185
ms3
Y+
sdot=
where ρ75 is the root radius at the 75ordm tangent ref to mn Usually ρ75 is closed to the tool radius ρao and
ρ75 = ρao
is a safe approximation compensating for the am simplifi-cations to the laquounsaferaquo side
The tooth root stresses of the loaded tooth are decreasing with decreasing relative rim thickness approximately with the empirical factor
15
R
ncorr s
m311Y
minus=
3153 Nominal rim stresses The bending moment applied to the rim consists of a part of the tooth tilt moment Ft (hF + 05 sR) and the bending caused by the radial force Fr
The sectional modulus (first moment of area) which is used for determination of the nominal bending stresses is not nec-essarily the same for the 2 am bending moments If flanges webs etc outside the toothed section contribute to stiffening the rim against various deflections the influence of these stiffeners should be considered Eg an end flange will have an almost negligible influence on the effective sectional modulus for the stresses due to tooth tilt as the deflection caused by the tilt moment is rather small and would not much involve the flange On the other hand the radial forces as for instance from the meshes in an annulus would cause considerable radial deflections that the flange might restrict to a substantial amount When considering the stiff-ening of such flanges or webs on basis of simplified models it is advised to use an effective rim thickness sR = sR + 02 mn for the first moment of area of the rim (toothed part) cross section
For a high number of rim teeth it may be assumed that the rim bending moments in the fillets adjacent to the loaded tooth are of the same magnitude as right under the applied
force This assumption is reasonable for an annulus but rather much on the safe side for a hollow pinion
The influence of Ft on the nominal tangential stress is simpli-fied by half of it for compressive stresses (σ1) and the other half for tensile stresses (σ2) Applying these assumptions the nominal rim stresses adjacent to the loaded tooth are
( ) ( )A2
FW
f RFW
s05hF05 t
R
r
T
RFt1 minus
ϑminus
+minus=σ
( ) ( )A2
FW
f RFW
s05hF05 t
R
r
T
RFt2 +
ϑminus
+=σ
where σ1 σ2 see Fig 35
A = minimum area of cross section (usually bR sR)
WT = the sectional modulus of rim with respect to tooth tilt moment (usually 6sb 2
rR )
WR = the sectional modulus of the rim including the influence of stiffeners as flanges webs etc (WR ge WT)
bR = the width of the rim
R = the radius of the neutral axis in the rim ie from wheel centre to midpoint of rim
( )ϑf = a function for bending moment distribution around the rim For a rim (pinion) with one mesh only the ( )ϑf at the position of load application is 024 For an annulus with 3 or more meshes ( )ϑf at the position of each load application is approx 3 planets ( )ϑf = 019 4 planets ( )ϑf = 014 5 planets ( )ϑf = 011 6 planets ( )ϑf = 009
It must be checked if the max (tensile) stress in the compres-sion fillet really occurs when the fillet is adjacent to the loaded tooth In principle the stress variation through a complete rotation should be considered and the max value used The max value is usually never less than 0 For an annulus eg the tilt moment is zero in the mid position be-tween the planet meshes whilst the bending moment due to the radial forces is half of that at the mesh but with opposite sign
If these formulae are applied to idler gears as eg planets the influence of nominal tangential stresses must be cor-rected by deleting Ft(2 A) for σ1 and using FtA for σ2 Further the influence of Fr on the nominal bending stresses is usually negligible due to the planet bearing support
3154 Root fillet stresses Determination of min and max stresses in the laquotensionraquo fillet
Classification Notes- No 412 31 May 2003
DET NORSKE VERITAS
Minimum stress
751FTmin YσK σ =
Maximum stress
corrF752 Yσ03Yσ KσFTmax +=
where
03 is an empirical factor relating the tension stresses (σF) at the 30ordm tangent to the part of the tension stresses at the 75ordm tangent which add to the rim related stresses (03 also takes into account that full superposition of nominal stresses times stress concentration factors from both laquosides of the corner filletraquo would result in too high stresses)
αβγ sdotsdotsdotsdot= FFvA KKKKKK
Determination of min and max stresses in the laquocompres-sionraquo fillet
Minimum stress
corrF751FCmin Yσ036Yσ Kσ minus=
Maximum stress
752FCmax Yσ Kσ =
where
036 is an empirical factor relating the tension stresses (σF) at the 30ordm tangent to the part of the compression stresses at the 75ordm tangent which add to the rim related stresses
For gears with reversed loads as idler gears and planets there is no distinct laquotensionraquo or laquocompressionraquo fillet The mini-mum stress σFmin is the minimum of σFTmin and σFCmin (usually the latter is decisive) The maximum stress σFTmax is the maximum of σFTmax and σFCmax (usually the former is deci-sive)
316 Permissible Stresses in Thin Rims
3161 General The safety against fatigue fracture respectively overload fracture is to be at least at the same level as for solid gears The laquoordinaryraquo criteria at the 30ordm tangent apply as given in 31 through 313
Additionally the following criteria at the 75ordm tangent may apply
3162 For gt3middot106 cycles The permissible stresses for the laquotensionraquo fillets and for the laquocompressionraquo fillets are determined by means of a relevant fatigue diagram
If the actual tooth root stress (tensile or compressive) ex-ceeds the yield strength to the material the induced residual stresses are to be taken into account
For determination of permissible stresses the following is defined
R = stress ratio ie FTmax
FTminσσ respectively
FCmax
FCmin
σσ
∆σ = stress range ie σFTmax ndash σFTmin resp σFCmax ndash σFCmin
(For idler gears and planets Fmax
FminσσR =
and ∆σ = σFmax minus σFmin)
The permissible stress range ∆σρ for the laquotensionraquo respec-tively laquocompressionraquo fillets can be calculated as
For R gt minus1 FPp σ
R1R1031
13∆σ
minus+
+=
For minus infin lt R lt minus1 FPp σ
R1R10151
13∆σ
minus+
+=
where
σFP = see 32 determined for unidirectional stresses (YM = 1)
If the yield strength σy is exceeded in either tension or com-pression residual stresses are induced This may be consid-ered by correcting the stress ratio R for the respective fillets (tension or compression)
Eg if yFCmin σσ gt (ie exceeded in compression) the
difference yFCmin σσ∆ minus= affects the stress ratio as
∆σ
σ∆σ∆σR
FCmax
y
FCmax
FCmin
+
minus=
++
=
Similarly the stress ratio in the tension fillet may require cor-rection
If the yield strength is exceeded in tension σFTmax gt σy the difference ∆ = σFTmax ndash σy affects the stress ratio as
y
FTmin
FTmax
FTmin
σ∆σ
∆σ∆σR minus=
minusminus
=
Checking for possible exceeding of the yield strength has to be made with the highest torque and with the KAP if this ex-ceeds KA
3163 Forle 103 cycles The permissible stress range ∆σp is not to exceed
For R gt minus1 FPstpst σ
R1R1051
15∆σ
minus+
+=
For ndash infin lt R lt minus1 FPstpst σ
R1R10251
15∆σ
minus+
+=
For all values of R ∆σpst is limited by
not surface hardened F
y
Sσ225
32 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
surface hardened CF
YSHV5
Definition of ∆σ and R see 3162 with particular attention to possible correction of R if the yield strength is exceeded
σFPst see 32 determined for unidirectional stresses (YM = 1) and lt 103 cycles
3164 For 103 lt cycles lt 3106 ∆σp is to be determined by linear interpolation a log-log dia-gram
∆σp at NL load cycles is
6L 103p
exp
L
6
N p ∆σN103∆σ sdot
sdot=
6
3
103p
10p
∆σ
∆σlog28760exp
sdot
=
Classification Notes- No 412 33 May 2003
DET NORSKE VERITAS
4 Calculation of Scuffing Load Capacity
41 Introduction High surface temperatures due to high loads and sliding ve-locities can cause lubricant films to break down This sei-zure or welding together of areas of tooth surface is termed scuffing
In contrast to pitting and fatigue breakage which show a dis-tinct incubation period a single short overloading can lead to a scuffing failure In the ISO-TR13989 two criteria are mentioned The method used in this Classification Note is based on the principles of the flash temperature criterion
Note Bulk temperature in excess of 120ordmC for long periods may have an adverse effect on the surface durability see 211
42 General Criteria In no point along the path of contact the local contact tem-perature may exceed the permissible temperature ie
oilS
oilSB S
ϑ+ϑminusϑ
leϑ
50SB minusϑleϑ
where
Bϑ = max contact temperature along the path of contact
maxflaMBB ϑ+ϑ=ϑ
MBϑ = bulk temperature see 434
maxflaϑ = max flash temperature along the path of con-tact see 44
Sϑ = scuffing temperature see below
oilϑ = oil temperature before it reaches the mesh (max applicable for the actual load case to be used ie normally alarm temperature except for ice classes where a max expected tem-perature applies)
SS = required safety factor according to the Rules
The scuffing temperature Sϑ may be calculated as
L2
wrelT
002
40S XFZGX
ν100112085780
sdot
sdot++=ϑ
where
XwrelT = relative welding factor
XwrelT
Through hardened steel 10
Phosphated steel 125
Copper-plated steel 150
Nitrided steel 150
Less than 10 retained austenite
115
10 ndash 20 retained austenite 10
Casehardened steel
20 ndash 30 retained austenite 085
Austenitic steel 045
FZG = load stage according to FZG-Test A8390 (Note This is the load stage where scuffing oc-curs However due to scatter in test results cal-culations are to be made with one load stage less than the specification)
XL = lubricant factor
= 10 for mineral oils
= 08 for polyalfaolefins
= 07 for non-water-soluble polyglycols
= 06 for water-soluble polyglycols
= 15 for traction fluids
= 13 for phosphate esters
ν40 = kinematic oil viscosity at 40˚C (mm2s)
Application of other test methods such as the Ryder the FZG-Ryder R46574 and the FZG L-42 Test 141195110 may be specially considered
For high speed gears with very short time of contact Sϑ may be increased as follows provided use of EP-oils
Addition to the calculated scuffing temperature Sϑ
If micros 18tc ge no addition
If micros 18tc lt add ( )cwrelT t18X18 minussdotsdot
where
ct = contact time ( micros ) which is the time needed to cross the Hertzian contact width
( ) [ ]microsu1cosβn
uσ340tb1
Hc +sdotsdot
sdot=
σH as calculated in 221
For bevel gears use vu in stead of u
34 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
43 Influence Factors
431 Coefficient of friction The following coefficient of friction may apply
L025a
005oil
02
redC ΣC
Bt X R ηρv
w0048micro minus
=
where
wBt = specific tooth load (Nmm)
vΣC = sum of tangential velocities at pitch point At pitch line velocities gt 50 ms the limiting value of vΣC at v = 50 ms is to be used
ρredC = relative radius of curvature (transversal plane) at the pitch point
Cylindrical gears
HαHβvγAbt
Bt KKKKKb
Fw sdotsdotsdotsdotsdot= (see 1)
twΣC αsin v2v =
bCredC β cos ρρ =
Bevel gears
HαHβvγAtmb
Bt KKKKKb
Fw sdotsdotsdotsdotsdot= (see 1)
vtmtΣC αsin v2v =
bmvCredC β cos ρρ =
ηoil = dynamic viscosity (mPa s) at oilϑ calculated as
1000
ρ νη oiloil =
where ρ in kgm3 approximated as
( ) 70 15oil15 minusϑminusρ=ρ and νoil is kinematic viscosity at oilϑ and may be calculated by means of the following equation
( ) ( )++=+ 08νloglog08νloglog 100oil ( )
sdotminus
ϑ+minus313log373log
273log373log oil
( ) ( )( )08νloglog08νloglog 10040 +minus+
Ra = composite arithmetic mean roughness (micron) of pinion an wheel calculated as ( )2a1aa RR 05R +=This is defined as the roughness on the new flanks ie as manufactured
XL = see 42
432 Effective tip relief Ceff Ceff is the effective tip relief that amount of tip relief which compensates for the elastic deformation of the gear mesh ie
zero load at the tooth tip It is assumed (simplified) to be equal for both pinion and wheel
Cylindrical gears
for helical
γ
γ=cb
KKFC Abt
eff (see 1)
For spur
cbKKF
C Abteff
γ= (see 1)
Alternatively for spur and helical gears the non-linear ap-proach in 111 may be used (taking Ceff=δ)
Bevel gears
bcKFC Ambt
effγ
= (see 1)
where
γ
α
ε2ε44c
+sdot
=γ
433 Tip relief and extension Cylindrical gears
The extension of the tip relief is not to result in an effective contact ratio 1ltεα when the gear is unloaded (exceptions to this may only apply for applications where the gear is not to run at light loads) This means that the unrelieved part of the path of contact is to be minimum pbt It is further assumed that this unrelieved part is placed centrally on the path of contact
If root relief applies it has to be calculated as equivalent tip relief Ie pinion root relief (at mesh position A) is added to Ca2 and wheel root relief (at mesh position E) is added to Ca1 If no design tip relief or root relief on the mating gear is specified (ie if Ca1+Croot2 = 0 and visa versa) use the run-ning in amount see 112
Bevel gears
Bevel gears are to have heightwise crowning ie no distinct relievedunrelieved area This may be treated as tip and root relief For calculation purposes the root relief is combined with the tip relief of the mating member into an equivalent tip relief If no resulting tip relieves are specified the equivalent tip relives may be calculated as an approxima-tion based on the tool crowning Ca tool (per mille of tool module m0) as follows
2root1aeq1a CCC +=
1root2aeq2a CCC +=
where
Classification Notes- No 412 35 May 2003
DET NORSKE VERITAS
2
0
n0101atoolal m
)m2(mA1middotmCC
minus+minussdot=
( )
sdotminusminussdotsdotminus+= vtvb1
2vb1
21vavt1n1 αtan dddαsin 05x1mA
2
0
n020atool22a m
)mm(2A1middotmCC
minus+minussdot=
( )
sdotminusminussdotsdotminus+= vtvb2
2vb2
2va2vt2n2 αtan dddαsin 05x1mA
2
0
20atool1root1 m
A1mCC
minussdotsdot=
2
0
10atool2root2 m
A1mCC
minussdotsdot=
If the pressure angles of the cutter blades are modified (and verified) to balance the tip relieves the following may be assumed
( )eq2aeq1aeq2aeq1a C andC calculated of sum50CC sdot==
Throughout the following Ca1 and Ca2 mean the equivalent tip relieves Ca1eq and Ca2eq
434 Bulk temperature The bulk temperature may be calculated as
flaaveragempsoilMB X X 50 ϑ+ϑ=ϑ
where
Xs = lubrication factor
= 12 for spray lubrication
= 10 for dip lubrication (provided both pinion and wheel are dip lubricated and tip speed lt 5 ms)
= 10 for spray lubrication with additional cooling spray (spray on both pinion and wheel or spray on pinion and dip lubrication of wheel)
= 02 for meshes fully submerged in oil
Xmp = contact factor ( )pmp nX += 150
np = number of mesh contact on the pinion (for small gear ratii the number of wheel meshes should be used if higher)
flaaverageϑ
= average of the integrated flash temperature (see 44) along the path of contact
( )
AE
E
Ayyyfla
flaaverage
d
ΓminusΓ
ΓΓϑ=ϑint =
For high speed gears (v gt 50 ms) it may be necessary to assess the bulk temperature on the basis of a thermal rating of the entire gear transmission
44 The Flash Temperature flaϑ
441 Basic formula The local flash temperature flaϑ may be calculated as
( ) 41redy
y2ly
211
43Btcorrfla
unXwX3250
y ρ
ρminusρ
micro=ϑ Γ
(For bevel gears replace u with uv)
and is to be calculated stepwise along the path of contact from A to E
where
micro = coefficient of friction see 431
Xcorr = correction factor taking empirically into account the increased scuffing risk in the beginning of the approach path due to mesh starting without any previously built up oil film and possible shuffling away oil before meshing if insufficient tip relief
( )
3
AD
yaeffcorr 50
CC1X
ΓminusΓε
Γminus+=
α
Ca = tip relief of driven member Xcorr is only applicable in the approach path and if Ca lt Ceff otherwise 10
wBt = unit load see 431
yXΓ = load sharing factor see 443
n1 = pinion rpm
Γy ρly etc see 442
442 Geometrical relations The various radii of flank curvature (transversal plane) are
ρ1y = pinion flank radius at mesh point y
ρ2y = wheel flank radius at mesh point y
ρredy = equivalent radius of curvature at mesh point y
y2y1
y2y1redy
ρ+ρ
ρρ=ρ
Cylindrical gears
twy
1y αsin au1Γ1
ρ+
+=
twy
2y αsin au1Γu
ρ+
minus=
Note that for internal gears a and u are negative
36 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
Bevel gears
vtvv
y1y αsin a
u1Γ1
ρ+
+=
vtvv
yv2y αsin a
u1Γu
ρ+
minus=
Γ is the parameter on the path of contact and y is any point between A end E
At the respective ends Γ has the following values
Root piniontip wheel
Cylindrical gears
( )
minus
minusminus= 1
αtan 1dd
zzΓ
tw
2b2a2
1
2A
Bevel gears
( )
minus
minusminus= 1
αtan 1dd
uΓvt
2vb2va2
vA
Tip pinionroot wheel
Cylindrical gears
( )
1αtan
1ddΓ
tw
2b1a1
E minusminus
=
Bevel gears
( )
1αtan
1ddΓ
vt
2vb1va1
E minusminus
=
At inner point of single pair contact
Cylindrical gears
tw1
EB α tan zπ2ΓΓ minus=
Bevel gears
vtv1
EB α tan zπ2ΓΓ minus=
At outer point of single pair contact
Cylindrical gears
tw1AD α tan z
π2ΓΓ +=
Bevel gears
vt v1
AD αtan z π2ΓΓ +=
At pitch point ΓC = 0
The points F and G (only applicable to cylindrical gears) limiting the extension of tip relief (so as to maintain a mini-mum contact ratio of unity for unloaded gears) are at
2
BAF
Γ+Γ=Γ
2
EDG
Γ+Γ=Γ
443 Load sharing factor XΓ The load sharing factor XΓ accounts for the load sharing be-tween the various pairs of teeth in mesh along the path of contact
XΓ is to be calculated stepwise from A to E using the pa-rameter Γy
4431 Cylindrical gears with β = 0 and no tip relief
Figure 41
ByAAB
Ay for31
31X
yΓltΓleΓ
ΓminusΓ
ΓminusΓ+=Γ
DyB for1Xy
ΓltΓleΓ=Γ
EyDDE
yE for31
31X
yΓleΓltΓ
ΓminusΓ
ΓminusΓ+=Γ
4432 Cylindrical gears with β = 0 and tip relief Tip relief on the pinion reduces XΓ in the range G ndash E and increases correspondingly XΓ in the range F ndash B
Tip relief on the wheel reduces XΓ in the range A ndash F and increases correspondingly XΓ in the range D ndash G
Following remains generally
DyB for1Xy
ΓleΓleΓ=Γ
21XXGF== ΓΓ
In the following it must be distinguished between Ca lt Ceff respectively Ca gt Ceff This is shown by an example below where Ca1 lt Ceff and Ca2 gt Ceff
Classification Notes- No 412 37 May 2003
DET NORSKE VERITAS
Figure 42
Note When Ca gt Ceff the path of contact is shortened by A ndash A respectively E ndash E The single pair contact path is extended into B respectively D If this shift is significant it is necessary to consider the negative effect on surface durability (B) and bending stresses (B and D)
Range A - F
For effa2 CC le
minus=Γ
eff
2a
CC1
31X
A
FyAeff
2a
AF
Ay for C3
C61XX
AyΓleΓleΓ
sdot
+ΓminusΓ
ΓminusΓ+= ΓΓ
For effa2 CC ge
AyAfor0Xy
ΓleΓleΓ=Γ
with ( )21
CC
1CC
eff
2a
eff
2a
AFAAminus
minusΓminusΓ+Γ=Γ
FyAeff
2a
AF
Ay
eff
2a for 21
CC
CC1X
yΓleΓleΓ
minus
ΓminusΓ
ΓminusΓ+minus=Γ
Range F - B
For effa1 CC le
ByFeff
1a
FB
Fy forC3
C61
21X
yΓleΓleΓ
sdot
+ΓminusΓ
ΓminusΓ+=Γ
For effa1 CC ge
ByFeff
1a
FB
Fy for21
CC
21X
yΓleΓleΓ
minus
ΓminusΓ
ΓminusΓ+=Γ
ByB for1Xy
ΓleΓleΓ=Γ
with 1
CC2
eff
1a
FBFB
minus
ΓminusΓ+Γ=Γ
Range D ndash G
For effa2 CC le
GyDeff
2a
DG
Dy
eff
2a for C 3C
61
C 3C
32X
yΓleΓleΓ
+
ΓminusΓ
ΓminusΓminus+=Γ F
or effa2 CC ge
DyD for1Xy
ΓleΓleΓ=Γ
with ( )21
CC
1CC
eff
2a
eff
2a
DGDDminus
minusΓminusΓ+Γ=Γ
GyDeff
2a
DG
Dy
eff
2a for21
CC
CCX ΓleΓleΓ
minus
ΓminusΓ
ΓminusΓminus=Γ
Range G - E
For effa1 CC le
minus=Γ
eff
1a
CC1
31X
E
EyGeff
1a
GE
Gy forC3
C61
21X
yΓleΓleΓ
sdot
+ΓminusΓ
ΓminusΓminus=Γ
For effa1 CC gt
EyGeff
1a
GE
Gy for21
CC
21X
yΓleΓleΓ
minus
ΓminusΓ
ΓminusΓminus=Γ
EyE for0Xy
ΓleΓleΓ=Γ
with 1
CC2
eff
1a
GEGE
minus
ΓminusΓ+Γ=Γ
4433 Gears with β gt 0 buttressing Due to oblique contact lines over the flanks a certain but-tressing may occur near A and E
This applies to both cylindrical and bevel gears with tip relief lt Ceff The buttressing Xbutt is simplified as a linear function within the ranges A ndash H respectively I ndash E
Figure 43
1εwhen13X βbutt EAge=
1εwhenε031X ββbutt EAlt+=
38 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
Cylindrical gears
bIEAH βsin 02ΓΓΓΓ =minus=minus
Bevel gears
bmIEAH βsin 02ΓΓΓΓ =minus=minus
4434 Cylindrical gears with 2εγ le and no tip relief
yXΓ is obtained by multiplication of
yXΓ in 4431 with
Xbutt in 4433
4435 Gears with 2εγ gt and no tip relief
Applicable to both cylindrical and bevel gears
IyH for1Xy
ΓleΓleΓε
=α
Γ
IyHybutt and forX1Xy
ΓgtΓΓltΓε
=α
Γ
Figure 44
4436 Cylindrical gears with 2εγ le and tip relief
yXΓ is obtained by multiplication of
yXΓ in 4432 with Xbutt
in 4433
4437 Cylindrical gears with 2εγ gt and tip relief
Tip relief on the pinion (respectively wheel) reduces XГ in the range G ndash E (respectively A ndash F) and increases XГ in the range F ndash G
yXΓ is obtained by multiplication of
yXΓ as described below
with Xbutt in 4433
In the XГ example below the influence of tip relief is shown (without the influence of Xbutt) by means of
eff2aeff1a CC and CC ltgt
Tip relief gt Ceff causes new end points A respectively E of the path of contact
Figure 45
Range A ndash F
( ) ( )( ) eff
a2αa1α
AF
Ay
eff
2aeff
C 12C 13εC 1ε
CCCX
y +εε++minus
ΓminusΓ
ΓminusΓ+
εminus
=ααα
Γ
eff2aFyA CC if for leΓleΓleΓ
eff2aFyA CifCfor and geΓleΓleΓ
eff2aAyA CC iffor0Xy
gtΓleΓleΓ=Γ
( ) ( ) ( )( ) ( ) a2αa1α
αeffa2AFAA C 1ε 3C 1ε
1ε 2 CC with++minus+minus
ΓminusΓ+Γ=Γ
Range F ndash G
( )( )( ) GyF
eff
2a1a forC 1 2CC 11X
yΓleΓleΓ
+εε+minusε
+ε
=αα
α
αΓ
Range G ndash E
( ) ( )( ) eff
2a1a
GE
Gy
C 1 2C 1C 1 3XX
GFy +εεminusε++ε
ΓminusΓ
ΓminusΓminus=
αα
ααΓΓ minus
effa1EyG CC if for leΓleΓleΓ
effa1EyG CC if Γfor and geΓleΓle
eff1aEyE CC if for0Xy
geΓleΓleΓ=Γ
( ) ( ) ( )( ) ( ) 2a1a
eff1aGEEE C 1C 1 3
1 2 CC withminusε++ε+εminus
ΓminusΓminusΓ=Γαα
α
4438 Bevel gears with εγ more than approx 18 and heightwise crowning For Ca1 = Ca2 = Ceff the following applies
Figure 46
Classification Notes- No 412 39 May 2003
DET NORSKE VERITAS
( )AEM 50 Γ+Γ=Γ
( )( )2AD
3
2My651X
y ΓminusΓε
ΓminusΓminus
ε=
ααΓ
For tip relief lt Ceff yXΓ is found by linear interpolation be-
tween ( )effay CCX =Γ and ( )0CayX =Γ as in 4435
The interpolation is to be made stepwise from A to M with the influence of Ca2 and from M to E with the influence of Ca1 (For 2a1a CC ne there is a discontinuity at M)
Eg with Ca1 = 04 Ceff and Ca2 = 055 Ceff then
Range A ndash M
)CC()0C( eff2ay2ayyX 550X 450X =Γ=ΓΓ +=
Range M ndash E
)CC()0C( eff1ay1ayyX 40X 06X =Γ=ΓΓ +=
For tip relief gt Ceff the new end points A and E are found as
( )
minusΓminusΓ
ε+Γ=Γ α 1
CC
6 eff
2aADAA
( )
minusΓminusΓ
ε+Γ=Γ α 1
CC
6 eff
1aADEE
Range A ndash A
0Xy=Γ
Range A ndash M
( )( )
ΓminusΓ
ΓminusΓminus
minusε=
αΓ 2
MA
2My
eff
2a1
CC4
351Xy
Range M ndash E
( )( )
ΓminusΓ
ΓminusΓminus
minusε=
αΓ 2
ME
2My
eff
1a1
CC4
351Xy
Range E ndash E
0Xy=Γ
40 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
Appendix A Fatigue Damage Accumulation
The Palmgren-Miner cumulative damage calculation princi-ple is used The procedure may be applied as follows
A1 Stress Spectrum From the individual torque classes the torques (Ti) at the peak values of class intervals and the associated number of cycles (NLi) for both pinion and wheel are to be listed from the highest to the lowest torque
(In case of a cyclic torque variation within the torque classes it is advised to use the peak torque If the cyclic variation is such that the same teeth will repeatedly suffer the peak torque this is a must)
The stress spectra for tooth roots and flanks (σFi σHi) with all relevant factors (except KA) are to be calculated on the basis of the torque spectrum The load dependent K-factors are to be determined for each torque class
A2 σminusN-curve The stress versus load cycle curves for tooth roots and flanks (both pinion and wheel) are to be drawn on the basis of per-missible stresses (ie including the demanded minimum safety factors) as determined in 2 respectively 3 If different safety levels for high cycle fatigue and low cycle fatigue are desired this may be expressed by different demand safety factors applied at the endurance limit respectively at static strength
A3 Damage accumulation The individual damage ratio Di at ith stress level is defined as
Fi
Lii N
ND =
where
NLi = The number of applied cycles at ith stress
NFi = The number of cycles to failure at ith stress
Basically stresses σi below the permissible stress level for infinite life (if a constant ZN or YN is accepted) do not con-tribute to the damage sum However calculating the actual safety factor Sact as described below all the σi for which the product S σi is bigger than or equal to the permissible stress level for infinite life contribute to the damage sum and thus to the determination of Sact The final value of S is decisive
(NFi can be found mathematically by putting the permissible stress σpi equal the actual stress σi thereby finding the actual life factor This life factor can be solved with regard to load cycles ie NFi)
The damage sum ΣDi is not to exceed unity
If ΣDi ne 1 the safety against cumulative fatigue damage is different from the applied demand safety factor For deter-mination of this theoretical safety factor an iteration proce-dure is required as described in the following flowchart
S is correction factor with which the actual safety factor Sact can be found
Sact is the demand safety factor (used in determination of the permissible stresses in the σ ndash N ndash curve) times the correc-tion factor S
The full procedure is to be applied for pinion and wheel tooth roots and flanks
Note If alternating stresses occur in a spectrum of mainly pulsating stresses the alternating stresses may be replaced by equiva-lent pulsating stresses ie by means of division with the ac-tual mean stress influence factor YM
Classification Notes- No 412 41 May 2003
DET NORSKE VERITAS
Appendix B Application Factors for Diesel Driven Gears
For diesel driven gears the application factor KA depends on torsional vibrations Both normal operation and misfiring conditions have to be considered
Normally these two running conditions can be covered by only one calculation
B1 Definitions
Normal operation KAnorm = 0
normv0
TTT +
where
T0 = rated nominal torque
Tv norm = vibratory torque amplitude for normal opera-tion (see rules Pt4 Ch3 Sec1 G301 for defi-nition of ldquonormalrdquo irregularity)
Misfiring operation KA misf = 0
misfv
TTT +
where
T = remaining nominal torque when one cylinder out of action
Tv misf = vibratory torque amplitude in misfiring con-dition This refers to a permissible misfiring condition ie a condition that does not require automatic or immediate corrective actions as speed or pitch reduction
The normal operation is assumed to last for a very high num-ber of cycles such as 1010
The misfiring operation is assumed to last for a limited dura-tion such as 107 cycles
B2 Determination of decisive load Assuming life factor at 1010 cycles as YN = ZN = 092 which usually is relevant the calculation may be performed only once with the combination having the highest value of appli-cation factorlife factor
For bending stresses and scuffing the higher value of
092
K normA and 098
K misfA
For contact stresses the higher value of
092K normA and
113K misfA (but
097K misfA for nitrided gears)
B3 Simplified procedure Note that this is only a guidance and is not a binding con-vention
T = oTZ
1Zsdot
minus where Z = number of cylinders
( ) ideal Videal Vmisf Vnorm V TTT24ZT +minus=
where TV ideal = vibratory torque with all cylinders perfectly equal See also rules Pt4 Ch3 Sec1 G300
When using trends from torsional vibration analysis and measurements the following may be used
TV ideal To is close to zero for engines with few cylinders and using a suitable elastic coupling and increases with relative coupling stiffness and number of cylinders
TV misf To may be high for engines with few cylinders and decreases with number of cylinders
This can be indicated as
200Z
TT
ο
idealV asymp and 80Z04
TT
o
misfV minusasymp
Inserting this into the formulae for the two application fac-tors the following guidance can be given
118112K misfA minusasymp
115110K normA minusasymp
Since KA norm is to be combined with the lower life factors the decisive load condition will be the normal one and a KA of 115 will cover most relevant cases when a suitable elas-tic coupling is chosen
42 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
Appendix C Calculation of Pinion-Rack
Pinion-racks used for elevating of mobile offshore units are open gears that are subjected to wear and tear With normal specifications such as only a few hundred total operation cycles (site to site) the tooth bending stresses for static strength will be decisive for the lay out however with ex-ception of surface hardened pinions where case crushing has to be considered
In the following the use of part 1 and 3 for pinion-racks is shown including relevant simplifications
C1 Pinion tooth root stresses Since the load spectrum normally is dominated by high torques and few load cycles (in the range up to 1-2000) the static strength is decisive
The actual stress is calculated as
β1FSaFan1
tF1 KYY
mbFσ sdotsdotsdotsdot
=
b1 is limited to b2 + 2middotmn
YFa and YSa replace YF and YS because load application at tooth tip has to be assumed for such inaccurate gears
Pinions often use a non-involute profile in the dedendum part of the flank eg a constant radius equal the radius of curva-ture at reference circle For such pinions sFn and hFa are to be measured directly on a sectional drawing of the pinion tooth
Due to high loads and narrow facewidths it may be assumed that KFβ2 = KHβ = 10 However when b1 gt b2 then KFβ1 gt10
If no detailed documentation of KFβ1 is available the fol-lowing may be used
KFβ1 = 1 + 015middot(b1b2 ndash 1)
The permissible stress (not surface hardened) is calculated as
δrelTstF
Fst1FP1 Y
Sσσ sdot=
The mean stress influence due to leg lifting may be disre-garded
The actual and permissible stresses should be calculated for the relevant loads as given in the rules
C2 Rack tooth root stresses The actual stress is calculated as
SaFan2
tF2 YY
mbFσ sdotsdotsdot
=
See C1 for details
The permissible stress is calculated as
δrelTstF
Fst2FP2 Y
Sσσ sdot=
For alloyed steels (Ni Cr Mo) with high toughness and duc-tility the value of YδrelTst may be put equal to YSa
C3 Surface hardened pinions For surface hardened pinions the maximum load is not to cause crushing of the hardened layer of the flank
In principle the calculation described in 213 may be used but when the theoretical Hertzian stress exceeds the ap-proximately 18 times the yield strength of the rack material plastic deformation will occur This will limit the peak Hertzian stress but increases the contact width and thus the penetration of stresses into the depth
An approximation may be based on an assessment of contact width determined by means of equal areas under the theoreti-cal (elastic range) Hertzian contact and the elasto-plastic contact stress (limited to 18 middot σy) with the unknown width
8 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
163 Frequent overloads For plants where high overloads or shock loads occur regu-larly the influence of this is to be considered by means of cumulative fatigue (see Appendix A)
17 Load Sharing Factor Kγ The load sharing factor Kγ accounts for the maldistribution of load in multiple-path transmissions (dual tandem epicyclic double helix etc) Kγ is defined as the ratio between the max load through an actual path and the evenly shared load
171 General method Kγ mainly depends on accuracy and flexibility of the branches (eg quill shaft planet support external forces etc) and should be considered on basis of measurements or of relevant analysis as eg
Kγ = δ+δ f
δ = total compliance of a branch under full load (assuming even load share) referred to gear mesh
f = minusminusminusminus+++ 23
22
21 fff where f1 f2 etc are the
main individual errors that may contribute to a maldis-tribution between the branches Eg tooth pitch errors planet carrier pitch errors bearing clearance influences etc Compensating effects should also be considered
For double helical gears
An external axial force Fext applied from sources outside the actual gearing (eg thrust via or from a tooth coupling) will cause a maldistribution of forces between the two helices Expressed by a load sharing factor the
βsdot
plusmn=γ tanFF1K
t
ext
If the direction of Fext is known the calculation should be carried out separately for each helix and with the tangential force corrected with the pertinent Kγ If the direction of Fext is unknown both combinations are to be calculated and the higher σH or σF to be used
172 Simplified method If no relevant analysis is available the following may apply
For epicyclic gears
Kγ = 32501 minus+ pln
where npl = number of planets ( gt3 )
For multistage gears with locked paths and gear stages sepa-rated by quill shafts (see figure below)
Figure 10 Locked paths gear
Kγ = ( )φ201+
where φ = quill shaft twist (degrees) under full load
18 Dynamic Factor Kv The dynamic factor Kv accounts for the internally generated dynamic loads
Kv is defined as the ratio between the maximum load that dynamically acts on the tooth flanks and the maximum ex-ternally applied load Ft KA Kγ
In the following 2 different methods (181 and 182) are described In case of controversy between the methods the next following is decisive ie the methods are listed with increasing priority
It is important to observe the limitations for the method in 181 In particular the influence of lateral stiffness of shafts is often underestimated and resonances occur at considerably lower speed than determined in 1811
However for low speed gears with vmiddotz1 lt 300 calculations may be omitted and the dynamic factor simplified to Kv=105
181 Single resonance method For a single stage gear Kv may be determined on basis of the relative proximity (or resonance ratio) N between actual speed n1 and the lowest resonance speed nE1
N = 1E
1
nn
Note that for epicyclic gears n is the relative speed ie the speed that multiplied with z gives the mesh frequency
1811 Determination of critical speed It is not advised to apply this method for multimesh gears for N gt 085 as the influence of higher modes has to be consid-ered see 182 In case of significant lateral shaft flexibility (eg overhung mounted bevel gears) the influence of cou-pled bending and torsional vibrations between pinion and wheel should be considered if N ge 075 see 182
nE1 = redmγc
1zπ
31030 sdot
where
cγ is the actual mesh stiffness per unit facewidth see 111
Classification Notes- No 412 9 May 2003
DET NORSKE VERITAS
For gears with inactive ends of the facewidth as eg due to high crowning or end relief such as often applied for bevel gears the use of cγ in connection with determination of natu-ral frequencies may need correction cγ is defined as stiffness per unit facewidth but when used in connection with the total mesh stiffness it is not as simple as cγmiddotb as only a part of the facewidth is active Such corrections are given in 111
mred is the reduced mass of the gear pair per unit facewidth and referred to the plane of contact
For a single gear stage where no significant inertias are closely connected to neither pinion nor wheel mred is calcu-lated as
mred = 21
21mm
mm+
The individual masses per unit facewidth are calculated as
m12 = 221b
21
)2d(b
I
where I is the polar moment of inertia (kgmm2)
The inertia of bevel gears may be approximated as discs with diameter equal the midface pitch diameter and width equal to b However if the shape of the pinion or wheel body differs much from this idealised cylinder the inertia should be cor-rected accordingly
For all kind of gears if a significant inertia (eg a clutch) is very rigidly connected to the pinion or wheel it should be added to that particular inertia (pinion or wheel) If there is a shaft piece between these inertias the torsional shaft stiffness alters the system into a 3-mass (or more) system This can be calculated as in 182 but also simplified as a 2-mass system calculated with only pinion and wheel masses
1812 Factors used for determination of Kv Non-dimensional gear accuracy dependent parameters
( )bKKF
yfcB
At
pptp
γ
minus=
( )bKKF
yFcBAt
ff
γ
α minus=
Non-dimensional tip relief parameter
bKKFcC
1BγAt
ak sdotsdot
sdotminus=
For gears of quality grade (ISO 1328) Q = 7 or coarser Bk = 1
For gears with Q le 6 and excessive tip relief Bk is limited to max 1
For gears (all quality grades) with tip relief of more than 2middotCeff (see 432) the reduction of αε has to be considered (see 443)
where
fpt = the single pitch deviation (ISO 1328) max of pinion or wheel
Fα = the total profile form deviation (ISO 1328) max of pinion or wheel (Note Fα is pt not available for bevel gears thus use Fα = fpt)
yp and yf = the respective running-in allowances and may be calculated similarly to yα in 112 ie the value of fpt is replaced by Fα for yf
cacute = the single tooth stiffness see 111
Ca = the amount of tip relief see 433 In case of different tip relief on pinion and wheel the value that results in the greater value of Bk is to be used If Ca is zero by design the value of running-in tip relief Cay (see 112) may be used in the above formula
1813 Kv in the subcritical range Cylindrical gears N le 085
Bevel gears N le 075
Kv = 1 + N K
K = Cv1 Bp + Cv2 Bf + Cv3 Bk
Cv1 accounts for the pitch error influence
Cv1 = 032
Cv2 accounts for profile error influence
Cv2 = 034 for γε le 2
Cv2 = 03ε
057
γ minus for γε gt 2
Cv3 accounts for the cyclic mesh stiffness variation
Cv3 = 023 for γε le 2
Cv3 = 156ε
0096
γ minus for γε gt 2
1814 Kv in the main resonance range Cylindrical gears 085 lt N le 115
Bevel gears 075 lt N le 125
Running in this range should preferably be avoided and is only allowed for high precision gears
Kv = 1 + Cv1 Bp + Cv2 Bf + Cv4 Bk
Cv4 accounts for the resonance condition with the cyclic mesh stiffness variation
Cv4 = 090 for γε le 2
Cv4 = 144ε
ε005057
γ
γ
minus
minus for γε gt 2
10 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
1815 Kv in the supercritical range Cylindrical gears N ge 15
Bevel gears N ge 15
Special care should be taken as to influence of higher vibra-tion modes andor influence of coupled bending (ie lateral shaft vibrations) and torsional vibrations between pinion and wheel These influences are not covered by the following approach
Kv = Cv5 Bp + Cv6 Bf + Cv7
Cv5 accounts for the pitch error influence
Cv5 = 047
Cv6 accounts for the profile error influence
Cv6 = 047 for εγ le 2
Cv6 = 174ε
012
γ minus for εγ gt 2
Cv7 relates the maximum externally applied tooth loading to the maximum tooth loading of ideal accurate gears operating in the supercritical speed sector when the circumferential vibration becomes very soft
Cv7 = 075 for εγ le 15
Cv7 = [ ] 8750)2(sin 0125 +minusβεπ for 15 lt εγ le 25
Cv7 = 10 for εγ gt 25
1816 Kv in the intermediate range Cylindrical gears 115 lt N lt 15
Bevel gears 125 lt N lt 15
Comments raised in 1814 and 1815 should be observed
Kv is determined by linear interpolation between Kv for N = 115 respectively 125 and N = 15 as
Cylindrical gears
( ) ( ) ( )[ ]51Nv151Nv51Nvv KK350
N51KK === minussdot
minus
+=
Bevel gears
( ) ( ) ( )[ ]51Nv251Nv51Nvv KK250
N51KK === minussdot
minus
+=
182 Multi-resonance method For high speed gear (vgt40 ms) for multimesh medium speed gears for gears with significant lateral shaft flexibility etc it is advised to determine Kv on basis of relevant dy-namic analysis
Incorporating lateral shaft compliance requires transforma-tion of even a simple pinion-wheel system into a lumped multi-mass system It is advised to incorporate all relevant inertias and torsional shaft stiffnesses into an equivalent (to pinion speed) system Thereby the mesh stiffness appears as an equivalent torsional stiffness
cγ b (db12)2 (Nmrad)
The natural frequencies are found by solving the set of dif-ferential equations (one equation per inertia) Note that for a gear put on a laterally flexible shaft the coupling bending-torsionals is arranged by introducing the gear mass and the lateral stiffness with its relation to the torsional displacement and torque in that shaft
Only the natural frequency (ies) having high relative dis-placement and relative torque through the actual pinion-wheel flexible element need(s) to be considered as critical frequency (ies)
Kv may be determined by means of the method mentioned in 181 thereby using N as the least favourable ratio (in case of more than one pinion-wheel dominated natural frequency) Ie the N-ratio that results in the highest Kv has to be consid-ered
The level of the dynamic factor may also be determined on basis of simulation technique using numeric time integration with relevant tooth stiffness variation and pitchprofile er-rors
19 Face Load Factors KHβ and KFβ The face load factors KHβ for contact stresses and for scuff-ing KFβ for tooth root stresses account for non-uniform load distribution across the facewidth
KHβ is defined as the ratio between the maximum load per unit facewidth and the mean load per unit facewidth
KFβ is defined as the ratio between the maximum tooth root stress per unit facewidth and the mean tooth root stress per unit facewidth The mean tooth root stress relates to the con-sidered facewidth b1 respectively b2
Note that facewidth in this context is the design facewidth b even if the ends are unloaded as often applies to eg bevel gears
The plane of contact is considered
191 Relations between KHβ and KFβ
KFβ = expβHK 2(hb)hb1
1exp++
=
where hb is the ratio tooth heightfacewidth The maximum of h1b1 and h2b2 is to be used but not higher than 13 For double helical gears use only the facewidth of one helix
If the tooth root facewidth (b1 or b2) is considerably wider than b the value of KFβ(1or2) is to be specially considered as it may even exceed KHβ
Eg in pinion-rack lifting systems for jack up rigs where b = b2 asymp mn and b1 asymp 3 mn the typical KHβ asymp KFβ2 asymp 1 and KFβ1 asymp 13
192 Measurement of face load factors Primarily
KFβ may be determined by a number of strain gauges distrib-uted over the facewidth Such strain gauges must be put in
Classification Notes- No 412 11 May 2003
DET NORSKE VERITAS
exactly the same position relative to the root fillet Relations in 191 apply for conversion to KHβ
Secondarily
KHβ may be evaluated by observed contact patterns on vari-ous defined load levels It is imperative that the various test loads are well defined Usually it is also necessary to evalu-ate the elastic deflections Some teeth at each 90 degrees are to be painted with a suitable lacquer Always consider the poorest of the contact patterns
After having run the gear for a suitable time at test load 1 (the lowest) observe the contact pattern with respect to ex-tension over the facewidth Evaluate that KHβ by means of the methods mentioned in this section Proceed in the same way for the next higher test load etc until there is a full face contact pattern From these data the initial mesh misalign-ment (ie without elastic deflections) can be found by ex-trapolation and then also the KHβ at design load can be found by calculation and extrapolation See example
Figure 11 Example of experimental determination of KHβ
It must be considered that inaccurate gears may accumulate a larger observed contact pattern than the actual single mesh to mesh contact patterns This is particularly important for lapped bevel gears Ground or hard metal hobbed bevel gears are assumed to present an accumulated contact pattern that is practically equal the actual single mesh to mesh con-tact patterns As a rough guidance the (observed) accumu-lated contact pattern of lapped bevel gears may be reduced by 10 in order to assess the single mesh to mesh contact pattern which is used in 199
193 Theoretical determination of KHβ The methods described in 193 to 198 may be used for cy-lindrical gears The principles may to some extent also be used for bevel gears but a more practical approach is given in 199
General For gears where the tooth contact pattern cannot be verified during assembly or under load all assumptions are to be well on the safe side
KHβ is to be determined in the plane of contact
The influence parameters considered in this method are
bull mean mesh stiffness cγ (see 111) (if necessary also variable stiffness over b)
bull mean unit load Fmb = Fbt KA Kγ Kvb (for double helical gears see 17 for use of Kγ)
bull misalignment fsh due to elastic deflections of shafts and gear bodies (both pinion and wheel)
bull misalignment fdefl due to elastic deflections of and work-ing positions in bearings
bull misalignment fbe due to bearing clearance tolerances bull misalignment fma due to manufacturing tolerances bull helix modifications as crowning end relief helix correc-
tion bull running in amount yβ (see 112)
In practice several other parameters such as centrifugal ex-pansion thermal expansion housing deflection etc contrib-ute to KHβ However these parameters are not taken into account unless in special cases when being considered as particularly important
When all or most of the am parameters are to be considered the most practical way to determine KHβ is by means of a graphical approach described in 1931
If cγ can be considered constant over the facewidth and no helix modifications apply KHβ can be determined analyti-cally as described in 1932
1931 Graphical method The graphical method utilises the superposition principle and is as follows
bull Calculate the mean mesh deflection Mδ as a function of
γm c and bF see 111
bull Draw a base line with length b and draw up a rectangu-lar with height δM (The area δM b is proportional to the transmitted force)
bull Calculate the elastic deflection fsh in the plane of contact Balance this deflection curve around a zero line so that the areas above and below this zero line are equal
Figure 12 fsh balanced around zero line
bull Superimpose these ordinates of the fsh curve to the previ-ous load distribution curve (The area under this new load distribution curve is still δM b)
bull Calculate the bearing deflections andor working posi-tions in the bearings and evaluate the influence fdefl in the plane of contact This is a straight line and is bal-anced around a zero line as indicated in Fig 14 but with one distinct direction Superimpose these ordinates to the previous load distribution curve
12 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
bull The amount of crowning end relief or helix correction (defined in the plane of contact) is to be balanced around a zero line similarly to fsh
Figure 13 Crowning Cc balanced aound zero line
bull Superimpose these ordinates to the previous load distri-bution curve In case of high crowning etc as eg often applied to bevel gears the new load distribution curve may cross the base line (the real zero line) The result is areas with negative load that is not real as the load in those areas should be zero Thus corrective actions must be made but for practical reasons it may be postponed to after next operation
bull The amount of initial mesh misalignment bema ff + (defined in the plane of contact) is to be bal-
anced around a zero line If the direction of bema ff + is known (due to initial contact check) or if the direction of fbe is known due to design (eg overhang bevel pin-ion) this should be taken into account If direction un-known the influence of bema ff + in both directions as well as equal zero should be considered
Figure 14 fma+fbe in both directions balanced around zero line
Superimpose these ordinates to the previous load distri-bution curve This results in up to 3 different curves of which the one with the highest peak is to be chosen for further evaluation
bull If the chosen load distribution curve crosses the base line (ie mathematically negative load) the curve is to be corrected by adding the negative areas and dividing this with the active facewidth The (constant) ordinates of this rectangular correction area are to be subtracted from the positive part of the load distribution curve It is advisable to check that the area covered under this new load distribution curve is still equal δM b
bull If cγ cannot be considered as constant over b then cor-rect the ordinates of the load distribution curve with the local (on various positions over the facewidth) ratio between local mesh stiffness and average mesh stiffness cγ (average over the active facewidth only) Note that the result is to be a curve that covers the same area δM b as before
bull The influence of running in yβ is to be determined as in 112 whereby the value for Fβx is to be taken as twice the distance between the peak of the load distribution curve and δM
bull Determine
KHβ = M
β
δycurveofpeak minus
1932 Simplified analytical method for cylindrical gears The analytical approach is similar to 1931 but has a more limited application as cγ is assumed constant over the face-width and no helix modification applies
bull Calculate the elastic deflection fsh in the plane of contact Balance this deflection curve around a zero line so that the area above and below this zero line are equal see Fig 12 The max positive ordinate is frac12∆fsh
bull Calculate the initial mesh alignment as Fβx= deflbemash ffff plusmnplusmnplusmn∆ The negative signs may only be used if this is justified andor verified by a contact pattern test Otherwise al-ways use positive signs If a negative sign is justified the value of Fβx is not to be taken less than the largest of each of these elements
bull Calculate the effective mesh misalignment as Fβy = Fβx - yβ (yβ see 112)
bull Determine
2KforF2
bFc1K βH
m
βγγH le+=β
or
2KforF
bFc2K Hβ
m
βγγH gt=β
where cγ as used here is the effective mesh stiffness see 111
194 Determination of fsh fsh is the mesh misalignment due to elastic deflections Usu-ally it is sufficient to consider the combined mesh deflection of the pinion body and shaft and the wheel shaft The calcu-lation is to be made in the plane of contact (of the considered gear mesh) and to consider all forces (incl axial) acting on the shafts Forces from other meshes can be parted into components parallel respectively vertical to the considered plane of contact Forces vertical to this plane of contact have no influence on fsh
It is advised to use following diameters for toothed elements
d + 2 x mn for bending and shear deflection
d + 2 mn (x ndash ha0 + 02) for torsional deflection
Usually fsh is calculated on basis of an evenly distributed load If the analysis of KHβ shows a considerable maldis-
Classification Notes- No 412 13 May 2003
DET NORSKE VERITAS
tribution in term of hard end contact or if it is known by other reasons that there exists a hard end contact the load should be correspondingly distributed when calculating fsh In fact the whole KHβ procedure can be used iteratively 2-3 iterations will be enough even for almost triangular load distributions
195 Determination of fdefl fdefl is the mesh misalignment in the plane of contact due to bearing deflections and working positions (housing deflec-tion may be included if determined)
First the journal working positions in the bearings are to be determined The influence of external moments and forces must be considered This is of special importance for twin pinion single output gears with all 3 shafts in one plane
For rolling bearings fdefl is further determined on basis of the elastic deflection of the bearings An elastic bearing deflec-tion depends on the bearing load and size and number of rolling elements Note that the bearing clearance tolerances are not included here
For fluid film bearings fdefl is further determined on basis of the lift and angular shift of the shafts due to lubrication oil film thickness Note that fbe takes into account the influence of the bearing clearance tolerance
When working positions bearing deflections and oil film lift are combined for all bearings the angular misalignment as projected into the plane of the contact is to be determined fdefl is this angular misalignment (radians) times the face-width
196 Determination of fbe fbe is the mesh misalignment in the plane of contact due to tolerances in bearing clearances In principle fbe and fdefl could be combined But as fdefl can be determined by analy-sis and has a distinct direction and fbe is dependent on toler-ances and in most cases has no distinct direction (ie + toler-ance) it is practicable to separate these two influences
Due to different bearing clearance tolerances in both pinion and wheel shafts the two shaft axis will have an angular mis-alignment in the plane of contact that is superimposed to the working positions determined in 195 fbe is the facewidth times this angular misalignment Note that fbe may have a distinct direction or be given as a + tolerance or a combina-tion of both For combination of + tolerance it is adviced to use
fbe= 22be
21be ff ++plusmn
fbe is particularly important for overhang designs for gears with widely different kinds of bearings on each side and when the bearings have wide tolerances on clearances In general it shall be possible to replace standard bearings with-out causing the real load distribution to exceed the design premises For slow speed gears with journal bearings the expected wear should also be considered
197 Determination of fma fma is the mesh misalignment due to manufacturing toler-ances (helix slope deviation) of pinion fHβ1 wheel fHβ2 and housing bore
For gear without specifically approved requirements to as-sembly control the value of fma is to be determined as
fma= 22Hβ
21Hβ ff +
For gears with specially approved assembly control the value of fma will depend on those specific requirements
198 Comments to various gear types For double helical gears KHβ is to be determined for both helices Usually an even load share between the helices can be assumed If not the calculation is to be made as de-scribed in 171
For planetary gears the free floating sun pinion suffers only twist no bending It must be noted that the total twist is the sum of the twist due to each mesh If the value of 1K γ ne this must be taken into account when calculating the total sun pinion twist (ie twist calculated with the force per mesh without Kγ and multiplied with the number of planets)
When planets are mounted on spherical bearings the mesh misalignments sun-planet respectively planet-annulus will be balanced Ie the misalignment will be the average between the two theoretical individual misalignments The faceload distribution on the flanks of the planets can take full advan-tage of this However as the sun and annulus mesh with several planets with possibly different lead errors the sun and annulus cannot obtain the above mentioned advantage to the full extent
199 Determination of KHβ for bevel gears If a theoretical approach similar to 193-198 is not docu-mented the following may be used
testeff
H Kb
b851851K sdot
minussdot=β
beff b represents the relative active facewidth (regarding lapped gears see 192 last part)
Higher values than beff b = 090 are normally not to be used in the formula
For dual directional gears it may be difficult to obtain a high beff b in both directions In that case the smaller beff b is to be used
Ktest represents the influence of the bearing arrangement shaft stiffness bearing stiffness housing stiffness etc on the faceload distribution and the verification thereof Expected variations in length- and height-wise tooth profile is also accounted for to some extent
a) Ktest = 1 For ground or hard metal hobbed gears with the specified contact pattern verified at full rating or at full torque slow turning at a condition representative for the thermal expansion at normal operation
14 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
It also applies when the bearing arrangementsupport has insignificant elastic deflections and thermal axial expansion However each initial mesh contact must be verified to be within acceptance criteria that are cali-brated against a type test at full load Reproduction of the gear tooth length- and height-wise profile must also be verified This can be made through 3D measurements or by initial contact movements caused by defined axial offsets of the pinion (tolerances to be agreed upon)
b) Ktest = 1 + 04middot(beffbndash06) For designs with possible influence of thermal expansion in the axial direction of the pinion The initial mesh contact verified with low load or spin test where the acceptance criteria are calibrated against a type test at full load
c) Ktest = 12 if mesh is only checked by toolmakerrsquos blue or by spin test contact For gears in this category beffb gt 085 is not to be used in the calculation
110 Transversal Load Distribution Factors KHα and KFα The transverse load distribution factors KHα for contact stresses and for scuffing KFα for tooth root stresses account for the effects of pitch and profile errors on the transversal load distribution between 2 or more pairs of teeth in mesh
The following relations may be used
Cylindrical gears
( )
minus+==
tH
αptγγHαFα F
byfc0409
2ε
KK
valid for 2εγ le
( ) ( )tH
αptγ
γ
γHαFα F
byfcε
1ε20409KK
minusminus+==
valid for 2ε γ gt
where
FtH = Ft KA Kγ Kv KHβ
cγ = See 111
γα = See 112
fpt = Maximum single pitch deviation (microm) of pinion or wheel or maximum total profile form devia-tion Fα of pinion or wheel if this is larger than the maximum single pitch deviation
Note In case of adequate equivalent tip relief adapted to the load half of the above mentioned fpt can be introduced A tip relief is considered adequate when the average of Ca1 and Ca2 is within plusmn40 of the value of Ceff in 432
Limitations of KHα and KFα
If the calculated values for
KFα = KHα lt 1 use KFα = KHα = 10
If the calculated value of KHα gt 2εα
γ
Zε
ε use KHα = 2
εα
γ
Zε
ε
If the calculated value of KFα gt εα
γ
Yεε
use KFα = εα
γ
Yεε
where Yε = αnε
075025+ (for εαn see 331c)
Bevel gears
For ground or hard metal hobbed gears KFα = KHα = 1
For lapped gears KFα = KHα = 11
111 Tooth Stiffness Constants cacute and cγ The tooth stiffness is defined as the load which is necessary to deform one or several meshing gear teeth having 1 mm facewidth by an amount of 1 microm in the plane of contact cacute is the maximum stiffness of a single pair of teeth cγ is the mean value of the mesh stiffness in a transverse plane (brief term mesh stiffness)
Both valid for high unit load (Unit load = Ft middot KA middot Kγb)
Cylindrical gears The real stiffness is a combination of the progressive Hertzian contact stiffness and the linear tooth bending stiff-nesses For high unit loads the Hertzian stiffness has little importance and can be disregarded This approach is on the safe side for determination of KHβ and KHα However for moderate or low loads Kv may be underestimated due to determination of a too high resonance speed
The linear approach is described in A
An optional approach for inclusion of the non-linear stiffness is described in B
A The linear approach
BRCCq
βcos08acutec =
and
( )025ε075cc αγ +prime=
where
( )[ ]n02a01a
B α2000212
hh12051C minusminus
+minus+=
12n1n
x000635z
025791z
015551004723q minus++=
12
2n
22
1n
1 x005290z
x024188x000193z
x011654+minusminusminus
+ 000182 x22
Classification Notes- No 412 15 May 2003
DET NORSKE VERITAS
(for internal gears use zn2 equal infinite and x2 = 0) ha0 = hfp for all practical purposes CR considers the increased flexibility of the wheel teeth if the wheel is not a solid disc and may be calculated as
( )( )nR m5s
sR e5
bbln1C +=
where
bs = thickness of a central web
sR = average thickness of rim (net value from tooth root to inside of rim)
The formula is valid for bs b ge 02 and sRmn ge 1 Outside this range of validity and if the web is not centrally posi-tioned CR has to be specially considered
Note CR is the ratio between the average mesh stiffness over the facewidth and the mesh stiffness of a gear pair of solid discs The local mesh stiffness in way of the web corresponds to the mesh stiffness with CR = 1 The local mesh stiffness where there is no web support will be less than calculated with CR above Thus eg a centrally positioned web will have an effect corresponding to a longitudinal crowning of the teeth See also 1931 regarding KHβ
B The non-linear approach
In the following an example is given on how to consider the non-linearity
The relation between unit load Fb as a function of mesh deflection δ is assumed to be a progressive curve up to 500 Nmm and from there on a straight line This straight line when extended to the baseline is assumed to intersect at 10microm
With these assumptions the unit force Fb as a function of mesh deflection δ can be expressed as
( )10δKbF
minus= for 500bFgt
minus=
500Fb10δK
bF for 500
bFlt
with γAt KK
bF
bF
sdotsdot= etc (Nmm) ie unit load incorporat-
ing the relevant factors as
KA middot Kγ for determination of Kv
KA middot Kγ middot Kv for determination of KHβ
KA middot Kγ middot Kv middot KHβ for determination of KHα
δ = mesh deflection (microm)
K = applicable stiffness (c or cγ)
Use of stiffnesses for KV KHβ and KHα
For calculation of Kv and KHα the stiffness is calculated as follows
When Fb lt 500
the stiffness is determined as δ∆
∆ bF
where the increment is chosen as eg ∆ Fb = 10 and thus
50010Fb10
K10Fb∆δ +
++
=
When F b gt 500 the stiffness is c or cγ For calculation of KHβ the mesh deflection δ is used directly
or an equivalent stiffness determined as δsdotb
F
Bevel gears
In lack of more detailed relationship between stiffness and geometry the following may be used
b085
b13cacute eff=
b085b
16c effγ =
beff not to be used in excess of 085 b in these formulae
Bevel gears with heightwise and lengthwise crowning have progressive mesh stiffness The values mentioned above are only valid for high loads They should not be used for de-termination of Ceff (see 432) or KHβ (see 1931)
112 Running-in Allowances The running-in allowances account for the influence of run-ning-in wear on the various error elements
yα respectively yβ are the running-in amounts which reduce the influence of pitch and profile errors respectively influ-ence of localised faceload
Cay is defined as the running-in amount that compensates for lack of tip relief
The following relations may be used
For not surface hardened steel
ptHlim
α fσ160y =
yβ = βxlimH
f320σ
with the following upper limits
V lt 5 ms 5-10 ms gt 10 ms
yα max none
limH
12800σ limH
6400σ
yβ max none
limH
25600σ limH
12800σ
For surface hardened steel
yα = 0075 fpt but not more than 3 for any speed
yβ = 015 Fβx but not more than 6 for any speed
16 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
For all kinds of steel
5145189718
1C2
limHay +
minusσ
=
When pinion and wheel material differ the following ap-plies
bull Use the larger of fpt1 - yα1 and fpt2 - yα2 to replace fpt - yα in the calculation of KHα and Kv
bull Use ( )2β1ββ yy21y += in the calculation of KHβ
bull Use ( )2ay1aya CC21C += in the calculation of Kv
bull Use ( )2ay1ay2a1a CC21CC +== in the scuffing
calculation if no design tip relief is foreseen
Classification Notes- No 412 17 May 2003
DET NORSKE VERITAS
2 Calculation of Surface Durability
21 Scope and General Remarks Part 2 includes the calculations of flank surface durability as limited by pitting spalling case crushing and subsurface yielding Endurance and time limited flank surface fatigue is calculated by means of 22 ndash 212 In a way also tooth frac-tures starting from the flank due to subsurface fatigue is in-cluded through the criteria in 213
Pitting itself is not considered as a critical damage for slow speed gears However pits can create a severe notch effect that may result in tooth breakage This is particularly im-portant for surface hardened teeth but also for high strength through hardened teeth For high-speed gears pitting is not permitted
Spalling and case crushing are considered similar to pitting but may have a more severe effect on tooth breakage due to the larger material breakouts initiated below the surface Subsurface fatigue is considered in 213
For jacking gears (self-elevating offshore units) or similar slow speed gears designed for very limited life the max static (or very slow running) surface load for surface hard-ened flanks is limited by the subsurface yield strength
For case hardened gears operating with relatively thin lubri-cation oil films grey staining (micropitting) may be the lim-iting criterion for the gear rating Specific calculation meth-ods for this purpose are not given here but are under consid-eration for future revisions Thus depending on experience with similar gear designs limitations on surface durability rating other than those according to 22 - 213 may be ap-plied
22 Basic Equations Calculation of surface durability (pitting) for spur gears is based on the contact stress at the inner point of single pair contact or the contact at the pitch point whichever is greater
Calculation of surface durability for helical gears is based on the contact stress at the pitch point
For helical gears with 0 lt εβ lt 1 a linear interpolation be-tween the above mentioned applies
Calculation of surface durability for spiral bevel gears is based on the contact stress at the midpoint of the zone of contact
Alternatively for bevel gears the contact stress may be cal-culated with the program ldquoBECALrdquo In that case KA and Kv are to be included in the applied tooth force but not KHβ and KHα The calculated (real) Hertzian stresses are to be multi-plied with ZK in order to be comparable with the permissible contact stresses
The contact stresses calculated with the method in part 2 are based on the Hertzian theory but do not always represent the real Hertzian stresses
The corresponding permissible contact stresses σHP are to be calculated for both pinion and wheel
221 Contact stress Cylindrical gears
( )HαHβvγA
1
tβεEHDBH KKKKK
bud1uF
ZZZZZσ+
=
where
ZBD = Zone factor for inner point of single pair contact for pinion resp wheel (see 232)
ZH = Zone factor for pitch point (see 231)
ZE = Elasticity factor (see 24)
Zε = Contact ratio factor (see 25)
Zβ = Helix angle factor (see 26)
Ft KA Kγ Kv KHβ KHα see 15 ndash 110
d1 b u see 12 ndash 15
Bevel gears
( )HαHβvγA
v1v
vmtKEMH KKKKK
bud1uF
ZZZ105σ+
sdot=
where
105 is a correlation factor to reach real Hertzian stresses (when ZK = 1)
ZE KA etc see above
ZM = mid-zone factor see 233
ZK = bevel gear factor see 27
Fmt dv1 uv see 12 ndash 15
It is assumed that the heightwise crowning is chosen so as to result in the maximum contact stresses at or near the mid-point of the flanks
222 Permissible contact stress
XWRvLH
NHlimHP ZZZZZ
SZσ
σ =
where
σH lim = Endurance limit for contact stresses (see 28)
ZN = Life factor for contact stresses (see 29)
SH = Required safety factor according to the rules
ZLZvZR = Oil film influence factors (see 210)
ZW = Work hardening factor (see 211)
ZX = Size factor (see 212)
18 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
23 Zone Factors ZH ZBD and ZM
231 Zone factor ZH The zone factor ZH accounts for the influence on contact stresses of the tooth flank curvature at the pitch point and converts the tangential force at the reference cylinder to the normal force at the pitch cylinder
wtt2
wtbH sinααcos
cosαcosβ2Z =
232 Zone factors ZBD The zone factors ZBD account for the influence on contact stresses of the tooth flank curvature at the inner point of sin-gle pair contact in relation to ZH Index B refers to pinion D to wheel
For εβ ge 1 ZBD = 1
For internal gears ZD = 1
For εβ = 0 (spur gears)
( )
π
minusεminusminus
π
minusminus
α=
α2
2
2b
2a
1
2
1b
1a
wtB
z211
dd
z21
dd
tanZ
( )
π
minusεminusminus
π
minusminus
α=
α1
2
1b
1a
2
2
2b
2a
wtD
z211
dd
z21
dd
tanZ
If ZB lt 1 use ZB = 1
If ZD lt 1 use ZD = 1
For 0 lt εβ lt 1
ZBD = ZBD (for spur gears) ndash εβ (ZBD (for spur gears) ndash 1)
233 Zone factor ZM The mid-zone factor ZM accounts for the influence of the contact stress at the mid point of the flank and applies to spi-ral bevel gears
εminusminus
εminusminus
αβ=
αα btm2
2vb2
2vabtm2
1vb2val
2v1vvtbmM
pddpdd
ddtancos2Z
This factor is the product of ZH and ZM-B in ISO 10300 with the condition that the heightwise crowning is sufficient to move the peak load towards the midpoint
234 Inner contact point For cylindrical or bevel gears with very low number of teeth the inner contact point (A) may be close to the base circle In order to avoid a wear edge near A it is required to have suitable tip relief on the wheel
24 Elasticity Factor ZE The elasticity factor ZE accounts for the influence of the material properties as modulus of elasticity and Poissonrsquos ratio on the contact stresses
For steel against steel ZE = 1898
25 Contact Ratio Factor Zε The contact ratio factor Zε accounts for the influence of the transverse contact ratio εα and the overlap ratio εβ on the contact stresses
αε1Z =ε for 1εβ ge
( )α
ββ
αε ε
εε1
3ε4
Z +minusminus
= for εβ lt 1
26 Helix Angle Factor Zβ The helix angle factor Zβ accounts for the influence of helix angle (independent of its influence on Zε) on the surface du-rability
cosβZβ =
27 Bevel Gear Factor ZK The bevel gear factor accounts for the difference between the real Hertzian stresses in spiral bevel gears and the contact stresses assumed responsible for surface fatigue (pitting) ZK adjusts the contact stresses in such a way that the same per-missible stresses as for cylindrical gears may apply
The following may be used ZK = 080
28 Values of Endurance Limit σHlim and Static Strength 5H10σ 3H10σ
σHlim is the limit of contact stress that may be sustained for 5middot107 cycles without the occurrence of progressive pitting
For most materials 5middot107 cycles are considered to be the be-ginning of the endurance strength range or lower knee of the σ-N curve (See also Life Factor ZN) However for nitrided steels 2middot106 apply
For this purpose pitting is defined by
bull for not surface hardened gears pitted area ge 2 of total active flank area
bull for surface hardened gears pitted area ge 05 of total active flank area or ge 4 of one particular tooth flank area 510Hσ and 310Hσ are the contact stresses which the given
material can withstand for 105 respectively 103 cycles without subsurface yielding or flank damages as pitting spalling or case crushing when adequate case depth applies
The following listed values for σHlim 510Hσ and 310Hσ may only be used for materials subjected to a quality control as
Classification Notes- No 412 19 May 2003
DET NORSKE VERITAS
the one referred to in the rules Results of approved fatigue tests may also be used as the basis for establishing these values The defined survival probability is 99
σHlim σH105 σH10
3
Alloyed case hardened steels (surface hardness 58-63 HRC) - of specially approved high grade - of normal grade
1650 1500
2500 2400
3100 3100
Nitrided steel of approved grade gas nitrided (surface hardness 700-800 HV) 1250 13 σHlim 13 σHlim
Alloyed quenched and tempered steel bath or gas nitrided (surface hardness 500-700 HV)
1000
13 σHlim
13 σHlim
Alloyed flame or induction hardened steel (surface hardness 500-650 HV) 075 HV + 750 16 σHlim 45 HV
Alloyed quenched and tempered steel 14 HV + 350 16 σHlim 45 HV
Carbon steel 15 HV + 250 16 σHlim 16 σHlim
These values refer to forged or hot rolled steel For cast steel the values for σHlim are to be reduced by 15
29 Life Factor ZN The life factor ZN takes account of a higher permissible contact stress if only limited life (number of cycles NL) is demanded or lower permissible contact stress if very high number of cycles apply
If this is not documented by approved fatigue tests the fol-lowing method may be used
For all steels except nitrided
7L 105N sdotge ZN = 1 or
01570
L
7
N N105Z
sdot=
Ie ZN = 092 for 1010 cycles
The ZN = 1 from 5middot107 on may only be used when the mate-rial cleanliness is of approved high grade (see Rules Pt4 Ch2) and the lubrication is optimised by a specially approved filtering process
105 lt NL lt 5middot107 510N
logZ037
L
7
N N105Z
sdot=
NL = 105 WXRVLHlim
WstX10H1010NN ZZZZZσ
ZZσZZ
55
5=
103 lt NL lt 105 )Z(Zlog05
L
5
10NN
5N103N10
5N10ZZ
==
3L 10N le
WXRVLlimH
Wst10X10H10NN ZZZZZ
ZZZZ
33
3σ
σ==
(but not less than ZN105)
For nitrided steels
6L 102N sdotge ZN = 1 or
00980
L
6
N N102Z
sdot=
Ie ZN = 092 for 1010 cycles
The ZN = 1 from 2middot106 on may only be used when the mate-rial cleanliness is of approved high grade (see Rules Pt4 Ch2) and the lubrication is optimised by a specially approved filtering process
105 lt NL lt 2middot106 510N
Zlog76860
L
6
N N102Z
sdot=
5L 10N le
XWRVL
10XWst10NN ZZZZZ
ZZ13ZZ
55 ==
Note that when no index indicating number of cycles is used the factors are valid for 5middot107 (respectively 2middot106 for nitrid-ing) cycles
210 Influence Factors on Lubrication Film ZL ZV and ZR The lubricant factor ZL accounts for the influence of the type of lubricant and its viscosity the speed factor ZV ac-counts for the influence of the pitch line velocity and the roughness factor ZR accounts for influence of the surface roughness on the surface endurance capacity
The following methods may be applied in connection with the endurance limit
20 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
Surface hardened steels Not surface hardened steels
ZL ( )24013421
360910ν+
+
( )24013421
680830ν+
+
ZV ( )v3280
140930+
+ ( )v3280300850
++
ZR
080
ZrelR3
150
ZrelR3
where
ν40 = Kinematic oil viscosity at 40ordmC (mm2s) For case hardened steels the influence of a high bulk temperature (see 4 Scuffing) should be con-sidered Eg bulk temperatures in excess of 120ordmC for long periods may cause reduced flank surface endurance limits
For values of ν40 gt 500 use ν40 = 500
RZrel = The mean roughness between pinion and wheel (after running in) relative to an equivalent radius of curvature at the pitch point ρc = 10mm
RZrel
= ( ) 31
cZ2Z1 ρ
10RR50
+
RZ = Mean peak to valley roughness (microm) (DIN defini-tion) (roughly RZ = 6 Ra)
For 5
L 10N le ZL ZV ZR = 10
211 Work Hardening Factor ZW The work hardening factor ZW accounts for the increase of surface durability of a soft steel gear when meshing the soft steel gear with a surface hardened or substantially harder gear with a smooth surface
The following approximation may be used for the endurance limit
Surface hardened steel against not surface hardened steel 150
ZeqW R
31700
130HB21Z
minus
minus=
where HB = the Brinell hardness of the soft member For HB gt 470 use HB = 470 For HB lt 130 use HB = 130
RZeq = equivalent roughness
033
c40
066
ZS
ZHZHZEQ ρvν
15000RRRR
=
If RZeq gt 16 then use RZeq = 16
If RZeq lt 15 then use RZeq = 15
where
RZH = surface roughness of the hard member before run in
RZS = surface roughness of the soft member before run in
ν40 = see 210
If values of ZW lt 1 are evaluated ZW = 1 should be used for flank endurance However the low value for ZW may indi-cate a potential wear problem
Through hardened pinion against softer wheel
( )
minussdotminus+= 000829
HBHB0008981u1Z
2
1W
For 21HBHB
2
1 le use ZW = 1
For 71HBHB
2
1 gt use 17HBHB
2
1 =
For u gt 20 use u = 20
For static strength (lt 105 cycles)
Surface hardened against not surface hardened
ZWst = 105
Through hardened pinion against softer wheel
ZWst = 1
212 Size Factor ZX The size factor accounts for statistics indicating that the stress levels at which fatigue damage occurs decrease with an increase of component size as a consequence of the influ-ence on subsurface defects combined with small stress gradi-ents and of the influence of size on material quality
ZX may be taken unity provided that subsurface fatigue for surface hardened pinions and wheels is considered eg as in the following subsection 213
213 Subsurface Fatigue This is only applicable to surface hardened pinions and wheels The main objective is to have a subsurface safety against fatigue (endurance limit) or deformation (static strength) which is at least as high as the safety SH required for the surface The following method may be used as an approximation unless otherwise documented
The high cycle fatigue (gt3106 cycles) is assumed to mainly depend on the orthogonal shear stresses Static strength (lt103 cycles) is assumed to depend mainly on equivalent
Classification Notes- No 412 21 May 2003
DET NORSKE VERITAS
stresses (von Mises) Both are influenced by residual stresses but this is only considered roughly and empirically
The subsurface working stresses at depths inside the peak of the orthogonal shear stresses respectively the equivalent stresses are only dependent on the (real) Hertzian stresses Surface related conditions as expressed by ZL ZV and ZR are assumed to have a negligible influence
The real Hertzian stresses σHR are determined as
For helical gears with εβ gt 1
σHR = σH
For helical gears with εβ lt 1 and spur gears
ε
α
ββ ε
ε+εminus
sdotσ=σZ
1
HHR
For bevel gears
KHHR Z
1σσ sdot=
The necessary hardness HV is given as a function of the net depth tz (net = after grinding or hard metal hobbing and per-pendicular to the flank)
The coordinates tz and HV are to be compared with the de-sign specification such as
bull for flame and induction hardening tHVmin HVmin bull for nitriding t400min HV = 400 bull for case hardening t550min HV = 550 t400min HV = 400
and t300min HV = 300 (the latter only if the core hardness lt 300 If the core hardness gt 400 the t400 is to be re-placed by a fictive t400 = 16 t550)
In addition the specified surface hardness is not to be less than the max necessary hardness (at tz = 05aH) This applies to all hardening methods
For high cycle fatigue (gt3 106 cycles) the following applies
sdot+
minussdotsdotsdot= o90
05azt
05azt
cosSσ04HV
H
HHHR
applicable to 05a
zt
Hge
For 05at
H
z lt the value for 05at
H
z = applies
56300ρSσ12a cHHR
Hsdotsdot
sdot=
Where aH is half the hertzian contact width multiplied by an empirical factor of 12 that takes into account the possible influence of reduced compressive residual stresses (or even tensile residual stresses) on the local fatigue strength
If any of the specified hardness depths including the surface hardness is below the curve described by HV = f (tz) the actual safety factor against subsurface fatigue is determined as follows
reduce SH stepwise in the formula for HV and aH until all specified hardness depths and surface hardness balance with the corrected curve The safety factor obtained through this method is the safety against subsurface fatigue
For static strength (lt103 cycles) the following applies
sdot+
minussdotsdotsdot= o90
07at
06a
zt
cosSσ019HV
Hst
z
HstHHR
applicable to 06at
Hst
z ge
56300ρSσa cHHR
Hstsdotsdot
=
In the case of insufficient specified hardness depths the same procedure for determination of the actual safety factor as above applies
For limited life fatigue (103 lt cycles lt 3106 )
For this purpose it is necessary to extend the correction of safety factors to include also higher values than required Ie in the case of more than sufficient hardness and depths the safety factor in the formulae for both high cycle fatigue and static strength are to be increased until necessary and speci-fied values balance
The actual safety factor for a given number of cycles N be-tween 103 and 3106 is found by linear interpolation in a dou-ble logarithmic diagram
minussdotminus
= sdot logN3477
logSlogSlogS
310H6103HHN
36 10H103H Slog86281Slog86280 sdot+sdot sdot
22 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
3 Calculation of Tooth Strength
31 Scope and General Remarks Part 3 include the calculation of tooth root strength as limited by tooth root cracking (surface or subsurface initiated) and yielding
For rim thickness sR ge 35middotmn the strength is calculated by means of 32 ndash 313 For cylindrical gears the calculation is based on the assumption that the highest tooth root tensile stress arises by application of the force at the outer point of single tooth pair contact of the virtual spur gears The method has however a few limitations that are mentioned in 36
For bevel gears the calculation is based on force application at the tooth tip of the virtual cylindrical gear Subsequently the stress is converted to load application at the mid point of the flank due to the heightwise crowning
Bevel gears may also be calculated with the program BECAL In that case KA and Kv are to be included in the applied tooth force but not KFβ and KFα
In case of a thin annulus or a thin gear rim etc radial crack-ing can occur rather than tangential cracking (from root fillet to root fillet) Cracking can also start from the compression fillet rather than the tension fillet For rim thickness sR lt 35middotmn a special calculation procedure is given in 315 and 316 and a simplified procedure in 314
A tooth breakage is often the end of the life of a gear trans-mission Therefore a high safety SF against breakage is re-quired
It should be noted that this part 3 does not cover fractures caused by
bull oil holes in the tooth root space bull wear steps on the flank bull flank surface distress such as pits spalls or grey staining
Especially the latter is known to cause oblique fractures starting from the active flank predominately in spiral bevel gears but also sometimes in cylindrical gears
Specific calculation methods for these purposes are not given here but are under consideration for future revisions Thus depending on experience with similar gear designs limita-tions other than those outlined in part 3 may be applied
32 Tooth Root Stresses The local tooth root stress is defined as the max principal stress in the tooth root caused by application of the tooth force Ie the stress ratio R = 0 Other stress ratios such as for eg idler gears (R asymp -12) shrunk on gear rims (R gt 0) etc are considered by correcting the permissible stress level
321 Local tooth root stress The local tooth root stress for pinion and wheel may be as-sessed by strain gauge measurements or FE calculations or similar For both measurements and calculations all details are to be agreed in advance
Normally the stresses for pinion and wheel are calculated as
Cylindrical gears
FαFβvγAβSFn
tF K K K K K Y Y Y
m bFσ =
where
YF = Tooth form factor (see 33)
YS = Stress correction factor (see 34)
Yβ = Helix angle factor (see 36)
Ft KA Kγ Kv KFβ KFα see 15 ndash 110
b see 13
Bevel gears
FαFβvγASaFamn
mtF K K K K K Y Y Y
m bFσ ε=
where
YFa = Tooth form factor see 33
YSa = Stress correction factor see 34
Yε = Contact ratio factor see 35
Fmt KA etc see 15 ndash 110
b see 13
322 Permissible tooth root stress The permissible local tooth root stress for pinion respectively wheel for a given number of cycles N is
CXRrelTrelTF
NMFEFP YYYY
SYY
δσ
=σ
Note that all these factors YM etc are applicable to 3middot106 cycles when used in this formula for σFP The influence of other number of cycles on these factors is covered by the calculation of YN
where
σFE = Local tooth root bending endurance limit of reference test gear (see 37)
YM = Mean stress influence factor which accounts for other loads than constant load direction eg idler gears temporary change of load di-rection pre-stress due to shrinkage etc (see 38)
YN = Life factor for tooth root stresses related to reference test gear dimensions (see 39)
SF = Required safety factor according to the rules
YδrelT = Relative notch sensitivity factor of the gear to be determined related to the reference test gear (see 310)
YRrelT = Relative (root fillet) surface condition factor of the gear to be determined related to the reference test gear (see 311)
Classification Notes- No 412 23 May 2003
DET NORSKE VERITAS
YX = Size factor (see 312)
YC = Case depth factor (see 313)
33 Tooth Form Factors YF YFa The tooth form factors YF and YFa take into account the in-fluence of the tooth form on the nominal bending stress
YF applies to load application at the outer point of single tooth pair contact of the virtual spur gear pair and is used for cylindrical gears
YFa applies to load application at the tooth tip and is used for bevel gears
Both YF and YFa are based on the distance between the con-tact points of the 30˚-tangents at the root fillet of the tooth profile for external gears respectively 60˚ tangents for inter-nal gears
Figure 31 External tooth in normal section
Figure 32 Internal tooth in normal section
Definitions
n
2
n
Fn
enFn
Fe
F
α cosms
α cosmh6
Y
=
n
2
n
Fn
anFn
Fa
Fa
α cosms
α cosmh6
Y
=
In the case of helical gears YF and YFa are determined in the normal section ie for a virtual number of teeth
YFa differs from YF by the bending moment arm hFa and αFan and can be determined by the same procedure as YF with exception of hFe and αFan For hFa and αFan all indices e will change to a (tip)
The following formulae apply to cylindrical gears but may also be used for bevel gears when replacing
mn with mnm
zn with zvn
αt with αvt
β with βm
with undercut without undercut
Fig 33 Dimensions and basic rack profile of the teeth (finished profile)
Tool and basic rack data such as hfP ρfp and spr etc are re-ferred to mn ie dimensionless
331 Determination of parameters
( )n
n
prnfPnfP m
α cossαsin 1ρ
αtan h4πE
minusminusminusminus=
For external gears fPfP ρρ =
For internal gears ( )0z
195fPfP0
fPfP 10363156ρhxρρ
sdotminus+
+=
where
z0 = number of teeth of pinion cutter
x0 = addendum modification coefficient of pinion cutter
hfP = addendum of pinion cutter
ρfP = tip radius of pinion cutter
xhρG fpfp +minus=
24 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
τmE
2π
z2H
nnminus
minus=
with
3πτ = for external gears
6πτ = for internal gears
Htan zG2
nminusϑ=ϑ
(to be solved iteratively suitable start value6π
=ϑ for exter-
nal gears and 3π for internal gears)
a) Tooth root chord sFn For external gears
minus
ϑ+
ϑminus= ρ
cosG3
3πsinz
ms
fpnn
Fn
For bevel gears with a tooth thickness modification
xsm affects mainly sFn but also hFe and αFen The total influence of xsm on YFa Ysa can be approximated by only adding 2 xsm to sFn mn
For internal gears
minus
ϑ+
ϑminus= ρ
cosG
6πsinz
ms
fPnn
Fn
b) Root fillet radius ρF at 30ordm tangent
( )G2coszcosG2ρ
mρ
2n
2
fpn
F
minusϑϑ+=
c) Determination of bending moment arm hF dn = zn mn
b2α
αn βcosε
ε =
dan = dn + 2 ha
pbn = π mn cos αn
dbn = dn cos αn
( )4
d1εpzz
2dd
zz2d
2bn
2
αnbn
2bn
2an
en +
minusminus
minus=
en
bnen d
dcos arcα =
ennnn
e α invα invα x tan 22π
z1
minus+
+=γ
αFen = αen ndash γe
For external gears
( )
minus=
n
enFenee
n
Fe
mdαtan sin γ γcos
21
mh
]ρcos
G3πcosz fpn +
ϑminus
ϑminusminus
For internal gears
( ) minus
sdotsdotminus=
n
enFenee
n
Fe
mdα tanγ sinγ cos
21
mh
minus
ϑminus
ϑminussdot ρ
cosG3
6πcosz fPn
332 Gearing with εαn gt 2 For deep tooth form gearing ( )25ε2 αn lele produced with a verified grade of accuracy of 4 or better and with applied profile modification to obtain a trapezoidal load distribution along the path of contact the YF may be corrected by the factor YDT as
250ε205for 0666ε2366Y αnαnDT leleminus=
205εfor 10Y αnDT lt=
34 Stress Correction Factors YS YSa The stress correction factors YS and YSa take into account the conversion of the nominal bending stress to the local tooth root stress Thereby YS and YSa cover the stress increasing effect of the notch (fillet) and the fact that not only bending stresses arise at the root A part of the local stress is inde-pendent of the bending moment arm This part increases the more the decisive point of load application approaches the critical tooth root section
Therefore in addition to its dependence on the notch radius the stress correction is also dependent on the position of the load application ie the size of the bending moment arm
YS applies to the load application at the outer point of single tooth pair contact YSa to the load application at tooth tip
YS can be determined as follows
( )
+
+=L23121
1
sS qL01312Y
where Fe
Fn
hsL = and
F
Fns ρ2
sq = (see 33)
YSa can be calculated by replacing hFe with hFa in the above formulae
Note a) Range of validity 1 lt qs lt 8
In case of sharper root radii (ie produced with tools having too sharp tip radii) YS resp YSa must be specially considered
b) In case of grinding notches (due to insufficient protuberance of the hob) YS resp YSa can rise considerably and must be multiplied with
Classification Notes- No 412 25 May 2003
DET NORSKE VERITAS
g
g
ρt
0613
13
minus
where
tg = depth of the grinding notch
ρg = radius of the grinding notch
c) The formulae for YS resp YSa are only valid for αn = 20˚ However the same formulae can be used as a safe approximation for other pressure angles
35 Contact Ratio Factor Yε The contact ratio factor Yε covers the conversion from load application at the tooth tip to the load application at the mid point of the flank (heightwise) for bevel gears
The following may be used
Yε = 0625
36 Helix Angle Factor Yβ The helix angle factor Yβ takes into account the difference between the helical gear and the virtual spur gear in the nor-mal section on which the calculation is based in the first step In this way it is accounted for that the conditions for tooth root stresses are more favourable because the lines of contact are sloping over the flank
The following may be used (β input in degrees)
Yβ = 1 ndash εβ β120
When εβ gt 1 use εβ = 1 and when β gt 30deg use β = 30deg in the formula
However the above equation for Yβ may only be used for gears with β gt 25deg if adequate tip relief is applied to both pinion and wheel (adequate = at least 05 middot Ceff see 432)
37 Values of Endurance Limit σFE σFE is the local tooth root stress (max principal) which the material can endure permanently with 99 survival prob-ability 3106 load cycles is regarded as the beginning of the endurance limit or the lower knee of the σ ndash N curve σFE is defined as the unidirectional pulsating stress with a minimum stress of zero (disregarding residual stresses due to heat treatment) Other stress conditions such as alternating or pre-stressed etc are covered by the conversion factor YM
σFE can be found by pulsating tests or gear running tests for any material in any condition If the approval of the gear is to be based on the results of such tests all details on the testing conditions have to be approved by the Society Fur-ther the tests may have to be made under the Societys su-pervision
If no fatigue tests are available the following listed values for σFE may be used for materials subjected to a quality con-trol as the one referred to in the rules
σFE
Alloyed case hardened steels 1) (fillet surface hardness 58 ndash 63 HRC)
bull of specially approved high grade
1050
bull of normal grade
minus CrNiMo steels with approved process
1000
minus CrNi and CrNiMo steels generally 920 minus MnCr steels generally 850
Nitriding steel of approved grade quenched tempered and gas nitrided (surface hardness 700 ndash 800 HV)
840
Alloyed quenched and tempered steel bath or gas nitrided (surface hardness 500 ndash 700 HV)
720
Alloyed quenched and tempered steel flame or induction hardened 2) (incl entire root fillet) (fillet surface hardness 500 ndash 650 HV)
07 HV + 300
Alloyed quenched and tempered steel flame or induction hardened (excl entire root fillet) (σB = uts of base material)
025 σB + 125
Alloyed quenched and tempered steel 04 σB + 200
Carbon steel 025 σB + 250
Note All numbers given above are valid for separate forgings and for blanks cut from bars forged according to a qualified procedure see Pt 4 Ch 2 Sec 3 For rolled steel the values are to be reduced with 10 For blanks cut from forged bars that are not qualified as mentioned above the values are to be reduced with 20 For cast steel reduce with 40
1) These values are valid for a root radius
bull being unground If however any grinding is made in the root fillet area in such a way that the residual stresses may be affected σFE is to be reduced by 20 (If the grinding also leaves a notch see 34)
bull with fillet surface hardness 58 ndash 63 HRC In case of lower surface hardness than 58 HRC σFE is to be reduced with 20(58 ndash HRC) where HRC is the detected hardness (This may lead to a permissible tooth root stress that varies along the facewidth If so the actual tooth root stresses may also be considered along facewidth)
bull not being shot peened In case of approved shot peening σFE may be increased by 200 for gears where σFE is reduced by 20 due to root grinding Otherwise σFE may be increased by 100 for
6nm le and 100 ndash 5 (mn - 6) for mn gt 6 However the possible adverse influence on the flanks regarding grey staining should be considered and if necessary the flanks should be masked
2) The fillet is not to be ground after surface hardening Regarding possible root grinding see 1)
26 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
38 Mean stress influence Factor YM The mean stress influence factor YM takes into account the influence of other working stress conditions than pure pulsa-tions (R = 0) such as eg load reversals idler gears planets and shrink-fitted gears
YM (YMst) are defined as the ratio between the endurance (or static) strength with a stress ratio R ne 0 and the endurance (or static) strength with R = 0
YM and YMst apply only to a calculation method that assesses the positive (tensile) stresses and is therefore suitable for comparison between the calculated (positive) working stress σF and the permissible stress σFP calculated with YM or YMst
For thin rings (annulus) in epicyclic gears where the com-pression fillet may be decisive special considerations apply see 316
The following method may be used within a stress ratio ndash12 lt R lt 05
381 For idlers planets and PTO with ice class
M1M1R1
1Yor Y MstM
+minus
minus=
where
R = stress ratio = min stress divided by max stress
For designs with the same force applied on both forward- and back-flank R may be assumed to ndash 12
For designs with considerably different forces on forward- and back-flank such as eg a marine propulsion wheel with a power take off pinion R may be assessed as
branchmain theoffacewidth unit per forcepto offacewidt unit per force21minus
For a power take off (PTO) with ice class see 161 c
M considers the mean stress influence on the endurance (or static) strength amplitudes
M is defined as the reduction of the endurance strength am-plitude for a certain increase of the mean stress divided by that increase of the mean stress
Following M values may be used
Endurance limit
Static strength
Case hardened 08 ndash 015 Ys 1) 07
If shot peened 04 06
Nitrided 03 03
Induction or flame hardened
04 06
Not surface hardened steel 03 05
Cast steels 04 06 1)For bevel gears use Ys=2 for determination of M
The listed M values for the endurance limit are independent of the fillet shape (Ys) except for case hardening In princi-ple there is a dependency but wide variations usually only occur for case hardening eg smooth semicircular fillets versus grinding notches
382 For gears with periodical change of rotational direction For case hardened gears with full load applied periodically in both directions such as side thrusters the same formula for YM as for idlers (with R = ndash 12) may be used together with the M values for endurance limit This simplified approach is valid when the number of changes of direction exceeds 100 and the total number of load cycles exceeds 3middot106
For gears of other materials YM will normally be higher than for a pure idler provided the number of changes of direction is below 3middot106 A linear interpolation in a diagram with loga-rithmic number of changes of direction may be used ie from YM = 09 with one change to YM (idler) for 3middot106 changes This is applicable to YM for endurance limit For static strength use YM as for idlers
For gears with occasional full load in reversed direction such as the main wheel in a reversing gear box YM = 09 may be used
383 For gears with shrinkage stresses and unidirectional load For endurance strength
FE
fitM σ
σM1
M21Y+
minus=
σFE is the endurance limit for R = 0
For static strength YMst = 1 and σfit accounted for in 39b
σfit is the shrinkage stress in the fillet (30˚ tangent) and may be found by multiplying the nominal tangential (hoop) stress with a stress concentration factor
n
Ffit m
ρ215scf minus=
384 For shrink-fitted idlers and planets When combined conditions apply such as idlers with shrink-age stresses the design factor for endurance strength is
( ) ( ) FE
fitM σ
σR1M1
M2
M1M1R1
1Yminussdot+
minus
+minus
minus=
Symbols as above but note that the stress ratio R in this par-ticular connection should disregard the influence of σfit ie R normally equal ndash 12
For static strength
M1M1R1
1YMst
+minus
minus=
The effect of σfit is accounted for in 39 b
Classification Notes- No 412 27 May 2003
DET NORSKE VERITAS
385 Additional requirements for peak loads The total stress range (σmax ndash σmin) in a tooth root fillet is not to exceed
F
y
Sσ225
for not surface hardened fillets
FSHV5 for surface hardened fillets
39 Life Factor YN The life factor YN takes into account that in the case of limited life (number of cycles) a higher tooth root stress can be permitted and that lower stresses may apply for very high number of cycles
Decisive for the strength at limited life is the σ ndash N ndash curve of the respective material for given hardening module fillet radius roughness in the tooth root etc Ie the factors YδrelT YRelT YX and YM have an influence on YN
If no σ ndash N ndash curve for the actual material and hardening etc is available the following method may be used
Determination of the σ ndash N ndash curve
a) Calculate the permissible stress σFP for the beginning of the endurance limit (3middot106 cycles) including the influ-ence of all relevant factors as SF YδrelT YRelT YX YM and YC ie σFP = σFE middotYM middotYδrelT middotYRelT middotYX middotYC SF
b) Calculate the permissible laquostaticraquo stress (le 103 load cy-cles) including the influence of all relevant factors as SFst YδrelTst YMst and YCst ( )fitσCstYδrelTstYMstYFstσ
FstS1
FPstσ minussdotsdotsdot=
where σFst is the local tooth root stress which the material can resist without cracking (surface hardened materials) or unacceptable deformation (not surface hardened materials) with 99 survival probability
σFst
Alloyed case hardened steel 1) 2300
Nitriding steel quenched tempered and gas nitrided (surface hardness 700 ndash 800 HV)
1250
Alloyed quenched and tempered steel bath or gas nitrided (surface hardness 500 ndash 700 HV)
1050
Alloyed quenched and tempered steel flame or induction hardened (fillet surface hardness 500 ndash 650 HV)
18 HV + 800
Steel with not surface hardened fillets the smaller value of 2)
18 σB or 225 σy
1) This is valid for a fillet surface hardness of 58 ndash 63 HRC In case of lower fillet surface hardness than 58 HRC σFst is to be reduced with 30(58 ndash HRC) where HRC is the actual hardness Shot peening or grinding notches are not considered to have any significant influence on σFst 2) Actual stresses exceeding the yield point (σy or σ02) will alter the residual stresses locally in the ldquotensionrdquo fillet re-spectively ldquocompressionrdquo fillet This is only to be utilised for gears that are not later loaded with a high number of cycles at lower loads that could cause fatigue in the ldquocompressionrdquo fillet
c) Calculate YN as
NL gt 3middot106
YN = 1 or 001
L
6
N N103Y
sdot= ie Yn = 092 for 1010
The YN = 1 from 3middot106 on may only be used when special material cleanness applies see rules Pt4 Ch2
103ltNLlt3middot106exp
L
6
N N103Y
sdot=
cycles6103forσcycles310forσlog02876exp
FP
FPst
sdot=
NL lt 103 cycles6103forσcycles310forσ
NYFP
FPst
sdot=
or simply use σFPst as mentioned in b) directly
Guidance on number of load cycles NL for various applica-tions
bull For propulsion purpose normally NL = 1010 at full load (yachts etc may have lower values)
bull For auxiliary gears driving generators that normally operate with 70-90 of rated power NL = 108 with rated power may be applied
28 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
310 Relative Notch Sensitivity Factor YδrelT The dynamic (respectively static) relative notch sensitivity factor YδrelT (YδrelTst) indicate to which extent the theoreti-cally concentrated stress lies above the endurance limits (re-spectively static strengths) in the case of fatigue (respectively overload) breakage
YδrelT is a function of the material and the relative stress gra-dient It differs for static strength and endurance limit
The following method may be used
For endurance limit
for not surface hardened fillets
( )
024
s024
relT σ103133q21σ1012201351
Ysdotsdotminus
+sdotsdotminus+= minus
minus
δ
for all surface hardened fillets except nitrided
( )
106q21002451
Y srelT
++=δ
for nitrided fillets
( )
1347q2101421
Y srelT
++=δ
For static strength
for not surface hardened fillets1)
( )( )( )025
02
02502s
relTst σ3000821σ300 1Y0821Y
+
minus+=δ
for surface hardened fillets except nitrided
YδrelTst = 044 YS + 012
for nitrided fillets
YδrelTst = 06 + 02 YS 1) These values are only valid if the local stresses do not
exceed the yield point and thereby alter the residual stress level See also 39b footnote 2
311 Relative Surface Condition Factor YRrelT The relative surface condition factor YRrelT takes into ac-count the dependence of the tooth root strength on the sur-face condition in the tooth root fillet mainly the dependence on the peak to valley surface roughness
YRrelT differs for endurance limit and static strength
The following method may be used
For endurance limit
YRrelT = 1675 ndash 053 (Ry + 1)01 for surface hardened steels and alloyed quenched and tempered steels except nitrided
YRrelT = 53 ndash 42 (Ry + 1)001 for carbon steels
YδrelT = 43 ndash 326 (Ry + 1)0005
for nitrided steels
For static strength
YRrelTst = 1 for all Ry and all materials
For a fillet without any longitudinal machining trace Ry asymp Rz
312 Size Factor YX The size factor YX takes into account the decrease of the strength with increasing size YX differs for endurance limit and static strength
The following may be used
For endurance limit
YX = 1 for mn le 5 generally
YX = 103 ndash 0006 mn for 5 lt mn lt 30
YX = 085 for mn ge 30
for not surface hard-ened steels
YX = 105 ndash 001 mn for 5 lt mn ge 25
YX = 08 for mn ge 25
for surface hardened steels
For static strength
YXst = 1 for all mn and all materials
313 Case Depth Factor YC The case depth factor YC takes into account the influence of hardening depth on tooth root strength
YC applies only to surface hardened tooth roots and is dif-ferent for endurance limit and static strength
In case of insufficient hardening depth fatigue cracks can develop in the transition zone between the hardened layer and the core For static strength yielding shall not occur in the transition zone as this would alter the surface residual stresses and therewith also the fatigue strength
The major parameters are case depth stress gradient permis-sible surface respectively subsurface stresses and subsurface residual stresses
The following simplified method for YC may be used
YC consists of a ratio between permissible subsurface stress (incl influence of expected residual stresses) and permissible surface stress This ratio is multiplied with a bracket con-taining the influence of case depth and stress gradient (The empirical numbers in the bracket are based on a high number of teeth and are somewhat on the safe side for low number of teeth)
YC and YCst may be calculated as given below but calculated values above 10 are to be put equal 10
For endurance limit
+
+=nFFE
C m02ρt31
σconstY
Classification Notes- No 412 29 May 2003
DET NORSKE VERITAS
For static strength
+
+=nFFst
Cst m02ρt31
σconstY
where const and t are connected as
Hardening process
t = endurance limit const =
static strength const =
t550 640 1900
t400 500 1200 Case hard-ening
t300 380 800
Nitriding t400 500 1200
Induction- or flame hardening
tHVmin 11 HVmin 25 HVmin
For symbols see 213
In addition to these requirements to minimum case depths for endurance limit some upper limitations apply to case hard-ened gears
The max depth to 550 HV should not exceed
1) 13 of the top land thickness san unless adequate tip relief is applied (see 110)
2) 025 mn If this is exceeded the following applies addi-tionally in connection with endurance limit
minusminus= 025
mt
1Yn
max550C
314 Thin rim factor YB Where the rim thickness is not sufficient to provide full sup-port for the tooth root the location of a bending failure may be through the gear rim rather than from root fillet to root fillet
YB is not a factor used to convert calculated root stresses at the 30deg tangent to actual stresses in a thin rim tension fillet Actually the compression fillet can be more susceptible to fatigue
YB is a simplified empirical factor used to de-rate thin rim gears (external as well as internal) when no detailed calcula-tion of stresses in both tension and compression fillets are available
Figure 34 Examples on thin rims
YB is applicable in the range 175 lt sRmn lt 35
YB = 115 middot ln (8324 middot mnsR)
(for sRmn ge 35 YB = 1)
(for sRmn le 175 use 315)
σF as calculated in 321 is to be multiplied with YB when sRmn lt 35 Thus YB is used for both high and low cycle fatigue
Note This method is considered to be on the safe side for external gear rims However for internal gear rims without any flange or web stiffeners the method may not be on the safe side and it is advised to check with the method in 315316
315 Stresses in Thin Rims For rim thickness sR lt 35 mn the safety against rim cracking has to be checked
The following method may be used
3151 General The stresses in the standardised 30ordm tangent section tension side are slightly reduced due to decreasing stress correction factor with decreasing relative rim thickness sRmn On the other hand during the complete stress cycle of that fillet a certain amount of compression stresses are also introduced The complete stress range remains approximately constant Therefore the standardised calculation of stresses at the 30ordm tangent may be retained for thin rims as one of the necessary criteria
The maximum stress range for thin rims usually occurs at the 60ordm ndash 80ordm tangents both for laquotensionraquo and laquocompressionraquo side The following method assumes the 75ordm tangent to be the decisive Therefore in addition to the am criterion ap-plied at the 30ordm tangent it is necessary to evaluate the max and min stresses at the 75ordm tangent for both laquotensionraquo (loaded flank) fillet and laquocompressionraquo (back-flank) fillet For this purpose the whole stress cycle of each fillet should be considered but usually the following simplification is justified
Figure 35 Nomenclature of fillets
Index laquoTraquo means laquotensileraquo fillet laquoCraquo means laquocompressionraquo fillet
σFTmin and σFCmax are determined on basis of the nominal rim stresses times the stress concentration factor Y75
σFCmin and σFTmax are determined on basis of superposition of nominal rim stresses times Y75 plus the tooth bending stresses at 75ordm tangent
30 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
3152 Stress concentration factors at the 75ordm tangents The nominal rim stress consists of bending stresses due to local bending moments tangential stresses due to the tan-gential force Ft and radial shear stresses due to Fr
The major influence is given by the bending stresses The influence of the tangential stresses is minor and even though its stress concentration factor is slightly higher than for bending it is considered to be safe enough when the sum of these nominal stresses are combined with the stress concen-tration factor for bending The influence of the radial shear stress is neglected
The stress concentration factor relating nominal rim stresses to local fillet stresses at the 75ordm tangent may be calculated as
75n
R75
n
R
75
ρmsρ185
ms3
Y+
sdot=
where ρ75 is the root radius at the 75ordm tangent ref to mn Usually ρ75 is closed to the tool radius ρao and
ρ75 = ρao
is a safe approximation compensating for the am simplifi-cations to the laquounsaferaquo side
The tooth root stresses of the loaded tooth are decreasing with decreasing relative rim thickness approximately with the empirical factor
15
R
ncorr s
m311Y
minus=
3153 Nominal rim stresses The bending moment applied to the rim consists of a part of the tooth tilt moment Ft (hF + 05 sR) and the bending caused by the radial force Fr
The sectional modulus (first moment of area) which is used for determination of the nominal bending stresses is not nec-essarily the same for the 2 am bending moments If flanges webs etc outside the toothed section contribute to stiffening the rim against various deflections the influence of these stiffeners should be considered Eg an end flange will have an almost negligible influence on the effective sectional modulus for the stresses due to tooth tilt as the deflection caused by the tilt moment is rather small and would not much involve the flange On the other hand the radial forces as for instance from the meshes in an annulus would cause considerable radial deflections that the flange might restrict to a substantial amount When considering the stiff-ening of such flanges or webs on basis of simplified models it is advised to use an effective rim thickness sR = sR + 02 mn for the first moment of area of the rim (toothed part) cross section
For a high number of rim teeth it may be assumed that the rim bending moments in the fillets adjacent to the loaded tooth are of the same magnitude as right under the applied
force This assumption is reasonable for an annulus but rather much on the safe side for a hollow pinion
The influence of Ft on the nominal tangential stress is simpli-fied by half of it for compressive stresses (σ1) and the other half for tensile stresses (σ2) Applying these assumptions the nominal rim stresses adjacent to the loaded tooth are
( ) ( )A2
FW
f RFW
s05hF05 t
R
r
T
RFt1 minus
ϑminus
+minus=σ
( ) ( )A2
FW
f RFW
s05hF05 t
R
r
T
RFt2 +
ϑminus
+=σ
where σ1 σ2 see Fig 35
A = minimum area of cross section (usually bR sR)
WT = the sectional modulus of rim with respect to tooth tilt moment (usually 6sb 2
rR )
WR = the sectional modulus of the rim including the influence of stiffeners as flanges webs etc (WR ge WT)
bR = the width of the rim
R = the radius of the neutral axis in the rim ie from wheel centre to midpoint of rim
( )ϑf = a function for bending moment distribution around the rim For a rim (pinion) with one mesh only the ( )ϑf at the position of load application is 024 For an annulus with 3 or more meshes ( )ϑf at the position of each load application is approx 3 planets ( )ϑf = 019 4 planets ( )ϑf = 014 5 planets ( )ϑf = 011 6 planets ( )ϑf = 009
It must be checked if the max (tensile) stress in the compres-sion fillet really occurs when the fillet is adjacent to the loaded tooth In principle the stress variation through a complete rotation should be considered and the max value used The max value is usually never less than 0 For an annulus eg the tilt moment is zero in the mid position be-tween the planet meshes whilst the bending moment due to the radial forces is half of that at the mesh but with opposite sign
If these formulae are applied to idler gears as eg planets the influence of nominal tangential stresses must be cor-rected by deleting Ft(2 A) for σ1 and using FtA for σ2 Further the influence of Fr on the nominal bending stresses is usually negligible due to the planet bearing support
3154 Root fillet stresses Determination of min and max stresses in the laquotensionraquo fillet
Classification Notes- No 412 31 May 2003
DET NORSKE VERITAS
Minimum stress
751FTmin YσK σ =
Maximum stress
corrF752 Yσ03Yσ KσFTmax +=
where
03 is an empirical factor relating the tension stresses (σF) at the 30ordm tangent to the part of the tension stresses at the 75ordm tangent which add to the rim related stresses (03 also takes into account that full superposition of nominal stresses times stress concentration factors from both laquosides of the corner filletraquo would result in too high stresses)
αβγ sdotsdotsdotsdot= FFvA KKKKKK
Determination of min and max stresses in the laquocompres-sionraquo fillet
Minimum stress
corrF751FCmin Yσ036Yσ Kσ minus=
Maximum stress
752FCmax Yσ Kσ =
where
036 is an empirical factor relating the tension stresses (σF) at the 30ordm tangent to the part of the compression stresses at the 75ordm tangent which add to the rim related stresses
For gears with reversed loads as idler gears and planets there is no distinct laquotensionraquo or laquocompressionraquo fillet The mini-mum stress σFmin is the minimum of σFTmin and σFCmin (usually the latter is decisive) The maximum stress σFTmax is the maximum of σFTmax and σFCmax (usually the former is deci-sive)
316 Permissible Stresses in Thin Rims
3161 General The safety against fatigue fracture respectively overload fracture is to be at least at the same level as for solid gears The laquoordinaryraquo criteria at the 30ordm tangent apply as given in 31 through 313
Additionally the following criteria at the 75ordm tangent may apply
3162 For gt3middot106 cycles The permissible stresses for the laquotensionraquo fillets and for the laquocompressionraquo fillets are determined by means of a relevant fatigue diagram
If the actual tooth root stress (tensile or compressive) ex-ceeds the yield strength to the material the induced residual stresses are to be taken into account
For determination of permissible stresses the following is defined
R = stress ratio ie FTmax
FTminσσ respectively
FCmax
FCmin
σσ
∆σ = stress range ie σFTmax ndash σFTmin resp σFCmax ndash σFCmin
(For idler gears and planets Fmax
FminσσR =
and ∆σ = σFmax minus σFmin)
The permissible stress range ∆σρ for the laquotensionraquo respec-tively laquocompressionraquo fillets can be calculated as
For R gt minus1 FPp σ
R1R1031
13∆σ
minus+
+=
For minus infin lt R lt minus1 FPp σ
R1R10151
13∆σ
minus+
+=
where
σFP = see 32 determined for unidirectional stresses (YM = 1)
If the yield strength σy is exceeded in either tension or com-pression residual stresses are induced This may be consid-ered by correcting the stress ratio R for the respective fillets (tension or compression)
Eg if yFCmin σσ gt (ie exceeded in compression) the
difference yFCmin σσ∆ minus= affects the stress ratio as
∆σ
σ∆σ∆σR
FCmax
y
FCmax
FCmin
+
minus=
++
=
Similarly the stress ratio in the tension fillet may require cor-rection
If the yield strength is exceeded in tension σFTmax gt σy the difference ∆ = σFTmax ndash σy affects the stress ratio as
y
FTmin
FTmax
FTmin
σ∆σ
∆σ∆σR minus=
minusminus
=
Checking for possible exceeding of the yield strength has to be made with the highest torque and with the KAP if this ex-ceeds KA
3163 Forle 103 cycles The permissible stress range ∆σp is not to exceed
For R gt minus1 FPstpst σ
R1R1051
15∆σ
minus+
+=
For ndash infin lt R lt minus1 FPstpst σ
R1R10251
15∆σ
minus+
+=
For all values of R ∆σpst is limited by
not surface hardened F
y
Sσ225
32 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
surface hardened CF
YSHV5
Definition of ∆σ and R see 3162 with particular attention to possible correction of R if the yield strength is exceeded
σFPst see 32 determined for unidirectional stresses (YM = 1) and lt 103 cycles
3164 For 103 lt cycles lt 3106 ∆σp is to be determined by linear interpolation a log-log dia-gram
∆σp at NL load cycles is
6L 103p
exp
L
6
N p ∆σN103∆σ sdot
sdot=
6
3
103p
10p
∆σ
∆σlog28760exp
sdot
=
Classification Notes- No 412 33 May 2003
DET NORSKE VERITAS
4 Calculation of Scuffing Load Capacity
41 Introduction High surface temperatures due to high loads and sliding ve-locities can cause lubricant films to break down This sei-zure or welding together of areas of tooth surface is termed scuffing
In contrast to pitting and fatigue breakage which show a dis-tinct incubation period a single short overloading can lead to a scuffing failure In the ISO-TR13989 two criteria are mentioned The method used in this Classification Note is based on the principles of the flash temperature criterion
Note Bulk temperature in excess of 120ordmC for long periods may have an adverse effect on the surface durability see 211
42 General Criteria In no point along the path of contact the local contact tem-perature may exceed the permissible temperature ie
oilS
oilSB S
ϑ+ϑminusϑ
leϑ
50SB minusϑleϑ
where
Bϑ = max contact temperature along the path of contact
maxflaMBB ϑ+ϑ=ϑ
MBϑ = bulk temperature see 434
maxflaϑ = max flash temperature along the path of con-tact see 44
Sϑ = scuffing temperature see below
oilϑ = oil temperature before it reaches the mesh (max applicable for the actual load case to be used ie normally alarm temperature except for ice classes where a max expected tem-perature applies)
SS = required safety factor according to the Rules
The scuffing temperature Sϑ may be calculated as
L2
wrelT
002
40S XFZGX
ν100112085780
sdot
sdot++=ϑ
where
XwrelT = relative welding factor
XwrelT
Through hardened steel 10
Phosphated steel 125
Copper-plated steel 150
Nitrided steel 150
Less than 10 retained austenite
115
10 ndash 20 retained austenite 10
Casehardened steel
20 ndash 30 retained austenite 085
Austenitic steel 045
FZG = load stage according to FZG-Test A8390 (Note This is the load stage where scuffing oc-curs However due to scatter in test results cal-culations are to be made with one load stage less than the specification)
XL = lubricant factor
= 10 for mineral oils
= 08 for polyalfaolefins
= 07 for non-water-soluble polyglycols
= 06 for water-soluble polyglycols
= 15 for traction fluids
= 13 for phosphate esters
ν40 = kinematic oil viscosity at 40˚C (mm2s)
Application of other test methods such as the Ryder the FZG-Ryder R46574 and the FZG L-42 Test 141195110 may be specially considered
For high speed gears with very short time of contact Sϑ may be increased as follows provided use of EP-oils
Addition to the calculated scuffing temperature Sϑ
If micros 18tc ge no addition
If micros 18tc lt add ( )cwrelT t18X18 minussdotsdot
where
ct = contact time ( micros ) which is the time needed to cross the Hertzian contact width
( ) [ ]microsu1cosβn
uσ340tb1
Hc +sdotsdot
sdot=
σH as calculated in 221
For bevel gears use vu in stead of u
34 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
43 Influence Factors
431 Coefficient of friction The following coefficient of friction may apply
L025a
005oil
02
redC ΣC
Bt X R ηρv
w0048micro minus
=
where
wBt = specific tooth load (Nmm)
vΣC = sum of tangential velocities at pitch point At pitch line velocities gt 50 ms the limiting value of vΣC at v = 50 ms is to be used
ρredC = relative radius of curvature (transversal plane) at the pitch point
Cylindrical gears
HαHβvγAbt
Bt KKKKKb
Fw sdotsdotsdotsdotsdot= (see 1)
twΣC αsin v2v =
bCredC β cos ρρ =
Bevel gears
HαHβvγAtmb
Bt KKKKKb
Fw sdotsdotsdotsdotsdot= (see 1)
vtmtΣC αsin v2v =
bmvCredC β cos ρρ =
ηoil = dynamic viscosity (mPa s) at oilϑ calculated as
1000
ρ νη oiloil =
where ρ in kgm3 approximated as
( ) 70 15oil15 minusϑminusρ=ρ and νoil is kinematic viscosity at oilϑ and may be calculated by means of the following equation
( ) ( )++=+ 08νloglog08νloglog 100oil ( )
sdotminus
ϑ+minus313log373log
273log373log oil
( ) ( )( )08νloglog08νloglog 10040 +minus+
Ra = composite arithmetic mean roughness (micron) of pinion an wheel calculated as ( )2a1aa RR 05R +=This is defined as the roughness on the new flanks ie as manufactured
XL = see 42
432 Effective tip relief Ceff Ceff is the effective tip relief that amount of tip relief which compensates for the elastic deformation of the gear mesh ie
zero load at the tooth tip It is assumed (simplified) to be equal for both pinion and wheel
Cylindrical gears
for helical
γ
γ=cb
KKFC Abt
eff (see 1)
For spur
cbKKF
C Abteff
γ= (see 1)
Alternatively for spur and helical gears the non-linear ap-proach in 111 may be used (taking Ceff=δ)
Bevel gears
bcKFC Ambt
effγ
= (see 1)
where
γ
α
ε2ε44c
+sdot
=γ
433 Tip relief and extension Cylindrical gears
The extension of the tip relief is not to result in an effective contact ratio 1ltεα when the gear is unloaded (exceptions to this may only apply for applications where the gear is not to run at light loads) This means that the unrelieved part of the path of contact is to be minimum pbt It is further assumed that this unrelieved part is placed centrally on the path of contact
If root relief applies it has to be calculated as equivalent tip relief Ie pinion root relief (at mesh position A) is added to Ca2 and wheel root relief (at mesh position E) is added to Ca1 If no design tip relief or root relief on the mating gear is specified (ie if Ca1+Croot2 = 0 and visa versa) use the run-ning in amount see 112
Bevel gears
Bevel gears are to have heightwise crowning ie no distinct relievedunrelieved area This may be treated as tip and root relief For calculation purposes the root relief is combined with the tip relief of the mating member into an equivalent tip relief If no resulting tip relieves are specified the equivalent tip relives may be calculated as an approxima-tion based on the tool crowning Ca tool (per mille of tool module m0) as follows
2root1aeq1a CCC +=
1root2aeq2a CCC +=
where
Classification Notes- No 412 35 May 2003
DET NORSKE VERITAS
2
0
n0101atoolal m
)m2(mA1middotmCC
minus+minussdot=
( )
sdotminusminussdotsdotminus+= vtvb1
2vb1
21vavt1n1 αtan dddαsin 05x1mA
2
0
n020atool22a m
)mm(2A1middotmCC
minus+minussdot=
( )
sdotminusminussdotsdotminus+= vtvb2
2vb2
2va2vt2n2 αtan dddαsin 05x1mA
2
0
20atool1root1 m
A1mCC
minussdotsdot=
2
0
10atool2root2 m
A1mCC
minussdotsdot=
If the pressure angles of the cutter blades are modified (and verified) to balance the tip relieves the following may be assumed
( )eq2aeq1aeq2aeq1a C andC calculated of sum50CC sdot==
Throughout the following Ca1 and Ca2 mean the equivalent tip relieves Ca1eq and Ca2eq
434 Bulk temperature The bulk temperature may be calculated as
flaaveragempsoilMB X X 50 ϑ+ϑ=ϑ
where
Xs = lubrication factor
= 12 for spray lubrication
= 10 for dip lubrication (provided both pinion and wheel are dip lubricated and tip speed lt 5 ms)
= 10 for spray lubrication with additional cooling spray (spray on both pinion and wheel or spray on pinion and dip lubrication of wheel)
= 02 for meshes fully submerged in oil
Xmp = contact factor ( )pmp nX += 150
np = number of mesh contact on the pinion (for small gear ratii the number of wheel meshes should be used if higher)
flaaverageϑ
= average of the integrated flash temperature (see 44) along the path of contact
( )
AE
E
Ayyyfla
flaaverage
d
ΓminusΓ
ΓΓϑ=ϑint =
For high speed gears (v gt 50 ms) it may be necessary to assess the bulk temperature on the basis of a thermal rating of the entire gear transmission
44 The Flash Temperature flaϑ
441 Basic formula The local flash temperature flaϑ may be calculated as
( ) 41redy
y2ly
211
43Btcorrfla
unXwX3250
y ρ
ρminusρ
micro=ϑ Γ
(For bevel gears replace u with uv)
and is to be calculated stepwise along the path of contact from A to E
where
micro = coefficient of friction see 431
Xcorr = correction factor taking empirically into account the increased scuffing risk in the beginning of the approach path due to mesh starting without any previously built up oil film and possible shuffling away oil before meshing if insufficient tip relief
( )
3
AD
yaeffcorr 50
CC1X
ΓminusΓε
Γminus+=
α
Ca = tip relief of driven member Xcorr is only applicable in the approach path and if Ca lt Ceff otherwise 10
wBt = unit load see 431
yXΓ = load sharing factor see 443
n1 = pinion rpm
Γy ρly etc see 442
442 Geometrical relations The various radii of flank curvature (transversal plane) are
ρ1y = pinion flank radius at mesh point y
ρ2y = wheel flank radius at mesh point y
ρredy = equivalent radius of curvature at mesh point y
y2y1
y2y1redy
ρ+ρ
ρρ=ρ
Cylindrical gears
twy
1y αsin au1Γ1
ρ+
+=
twy
2y αsin au1Γu
ρ+
minus=
Note that for internal gears a and u are negative
36 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
Bevel gears
vtvv
y1y αsin a
u1Γ1
ρ+
+=
vtvv
yv2y αsin a
u1Γu
ρ+
minus=
Γ is the parameter on the path of contact and y is any point between A end E
At the respective ends Γ has the following values
Root piniontip wheel
Cylindrical gears
( )
minus
minusminus= 1
αtan 1dd
zzΓ
tw
2b2a2
1
2A
Bevel gears
( )
minus
minusminus= 1
αtan 1dd
uΓvt
2vb2va2
vA
Tip pinionroot wheel
Cylindrical gears
( )
1αtan
1ddΓ
tw
2b1a1
E minusminus
=
Bevel gears
( )
1αtan
1ddΓ
vt
2vb1va1
E minusminus
=
At inner point of single pair contact
Cylindrical gears
tw1
EB α tan zπ2ΓΓ minus=
Bevel gears
vtv1
EB α tan zπ2ΓΓ minus=
At outer point of single pair contact
Cylindrical gears
tw1AD α tan z
π2ΓΓ +=
Bevel gears
vt v1
AD αtan z π2ΓΓ +=
At pitch point ΓC = 0
The points F and G (only applicable to cylindrical gears) limiting the extension of tip relief (so as to maintain a mini-mum contact ratio of unity for unloaded gears) are at
2
BAF
Γ+Γ=Γ
2
EDG
Γ+Γ=Γ
443 Load sharing factor XΓ The load sharing factor XΓ accounts for the load sharing be-tween the various pairs of teeth in mesh along the path of contact
XΓ is to be calculated stepwise from A to E using the pa-rameter Γy
4431 Cylindrical gears with β = 0 and no tip relief
Figure 41
ByAAB
Ay for31
31X
yΓltΓleΓ
ΓminusΓ
ΓminusΓ+=Γ
DyB for1Xy
ΓltΓleΓ=Γ
EyDDE
yE for31
31X
yΓleΓltΓ
ΓminusΓ
ΓminusΓ+=Γ
4432 Cylindrical gears with β = 0 and tip relief Tip relief on the pinion reduces XΓ in the range G ndash E and increases correspondingly XΓ in the range F ndash B
Tip relief on the wheel reduces XΓ in the range A ndash F and increases correspondingly XΓ in the range D ndash G
Following remains generally
DyB for1Xy
ΓleΓleΓ=Γ
21XXGF== ΓΓ
In the following it must be distinguished between Ca lt Ceff respectively Ca gt Ceff This is shown by an example below where Ca1 lt Ceff and Ca2 gt Ceff
Classification Notes- No 412 37 May 2003
DET NORSKE VERITAS
Figure 42
Note When Ca gt Ceff the path of contact is shortened by A ndash A respectively E ndash E The single pair contact path is extended into B respectively D If this shift is significant it is necessary to consider the negative effect on surface durability (B) and bending stresses (B and D)
Range A - F
For effa2 CC le
minus=Γ
eff
2a
CC1
31X
A
FyAeff
2a
AF
Ay for C3
C61XX
AyΓleΓleΓ
sdot
+ΓminusΓ
ΓminusΓ+= ΓΓ
For effa2 CC ge
AyAfor0Xy
ΓleΓleΓ=Γ
with ( )21
CC
1CC
eff
2a
eff
2a
AFAAminus
minusΓminusΓ+Γ=Γ
FyAeff
2a
AF
Ay
eff
2a for 21
CC
CC1X
yΓleΓleΓ
minus
ΓminusΓ
ΓminusΓ+minus=Γ
Range F - B
For effa1 CC le
ByFeff
1a
FB
Fy forC3
C61
21X
yΓleΓleΓ
sdot
+ΓminusΓ
ΓminusΓ+=Γ
For effa1 CC ge
ByFeff
1a
FB
Fy for21
CC
21X
yΓleΓleΓ
minus
ΓminusΓ
ΓminusΓ+=Γ
ByB for1Xy
ΓleΓleΓ=Γ
with 1
CC2
eff
1a
FBFB
minus
ΓminusΓ+Γ=Γ
Range D ndash G
For effa2 CC le
GyDeff
2a
DG
Dy
eff
2a for C 3C
61
C 3C
32X
yΓleΓleΓ
+
ΓminusΓ
ΓminusΓminus+=Γ F
or effa2 CC ge
DyD for1Xy
ΓleΓleΓ=Γ
with ( )21
CC
1CC
eff
2a
eff
2a
DGDDminus
minusΓminusΓ+Γ=Γ
GyDeff
2a
DG
Dy
eff
2a for21
CC
CCX ΓleΓleΓ
minus
ΓminusΓ
ΓminusΓminus=Γ
Range G - E
For effa1 CC le
minus=Γ
eff
1a
CC1
31X
E
EyGeff
1a
GE
Gy forC3
C61
21X
yΓleΓleΓ
sdot
+ΓminusΓ
ΓminusΓminus=Γ
For effa1 CC gt
EyGeff
1a
GE
Gy for21
CC
21X
yΓleΓleΓ
minus
ΓminusΓ
ΓminusΓminus=Γ
EyE for0Xy
ΓleΓleΓ=Γ
with 1
CC2
eff
1a
GEGE
minus
ΓminusΓ+Γ=Γ
4433 Gears with β gt 0 buttressing Due to oblique contact lines over the flanks a certain but-tressing may occur near A and E
This applies to both cylindrical and bevel gears with tip relief lt Ceff The buttressing Xbutt is simplified as a linear function within the ranges A ndash H respectively I ndash E
Figure 43
1εwhen13X βbutt EAge=
1εwhenε031X ββbutt EAlt+=
38 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
Cylindrical gears
bIEAH βsin 02ΓΓΓΓ =minus=minus
Bevel gears
bmIEAH βsin 02ΓΓΓΓ =minus=minus
4434 Cylindrical gears with 2εγ le and no tip relief
yXΓ is obtained by multiplication of
yXΓ in 4431 with
Xbutt in 4433
4435 Gears with 2εγ gt and no tip relief
Applicable to both cylindrical and bevel gears
IyH for1Xy
ΓleΓleΓε
=α
Γ
IyHybutt and forX1Xy
ΓgtΓΓltΓε
=α
Γ
Figure 44
4436 Cylindrical gears with 2εγ le and tip relief
yXΓ is obtained by multiplication of
yXΓ in 4432 with Xbutt
in 4433
4437 Cylindrical gears with 2εγ gt and tip relief
Tip relief on the pinion (respectively wheel) reduces XГ in the range G ndash E (respectively A ndash F) and increases XГ in the range F ndash G
yXΓ is obtained by multiplication of
yXΓ as described below
with Xbutt in 4433
In the XГ example below the influence of tip relief is shown (without the influence of Xbutt) by means of
eff2aeff1a CC and CC ltgt
Tip relief gt Ceff causes new end points A respectively E of the path of contact
Figure 45
Range A ndash F
( ) ( )( ) eff
a2αa1α
AF
Ay
eff
2aeff
C 12C 13εC 1ε
CCCX
y +εε++minus
ΓminusΓ
ΓminusΓ+
εminus
=ααα
Γ
eff2aFyA CC if for leΓleΓleΓ
eff2aFyA CifCfor and geΓleΓleΓ
eff2aAyA CC iffor0Xy
gtΓleΓleΓ=Γ
( ) ( ) ( )( ) ( ) a2αa1α
αeffa2AFAA C 1ε 3C 1ε
1ε 2 CC with++minus+minus
ΓminusΓ+Γ=Γ
Range F ndash G
( )( )( ) GyF
eff
2a1a forC 1 2CC 11X
yΓleΓleΓ
+εε+minusε
+ε
=αα
α
αΓ
Range G ndash E
( ) ( )( ) eff
2a1a
GE
Gy
C 1 2C 1C 1 3XX
GFy +εεminusε++ε
ΓminusΓ
ΓminusΓminus=
αα
ααΓΓ minus
effa1EyG CC if for leΓleΓleΓ
effa1EyG CC if Γfor and geΓleΓle
eff1aEyE CC if for0Xy
geΓleΓleΓ=Γ
( ) ( ) ( )( ) ( ) 2a1a
eff1aGEEE C 1C 1 3
1 2 CC withminusε++ε+εminus
ΓminusΓminusΓ=Γαα
α
4438 Bevel gears with εγ more than approx 18 and heightwise crowning For Ca1 = Ca2 = Ceff the following applies
Figure 46
Classification Notes- No 412 39 May 2003
DET NORSKE VERITAS
( )AEM 50 Γ+Γ=Γ
( )( )2AD
3
2My651X
y ΓminusΓε
ΓminusΓminus
ε=
ααΓ
For tip relief lt Ceff yXΓ is found by linear interpolation be-
tween ( )effay CCX =Γ and ( )0CayX =Γ as in 4435
The interpolation is to be made stepwise from A to M with the influence of Ca2 and from M to E with the influence of Ca1 (For 2a1a CC ne there is a discontinuity at M)
Eg with Ca1 = 04 Ceff and Ca2 = 055 Ceff then
Range A ndash M
)CC()0C( eff2ay2ayyX 550X 450X =Γ=ΓΓ +=
Range M ndash E
)CC()0C( eff1ay1ayyX 40X 06X =Γ=ΓΓ +=
For tip relief gt Ceff the new end points A and E are found as
( )
minusΓminusΓ
ε+Γ=Γ α 1
CC
6 eff
2aADAA
( )
minusΓminusΓ
ε+Γ=Γ α 1
CC
6 eff
1aADEE
Range A ndash A
0Xy=Γ
Range A ndash M
( )( )
ΓminusΓ
ΓminusΓminus
minusε=
αΓ 2
MA
2My
eff
2a1
CC4
351Xy
Range M ndash E
( )( )
ΓminusΓ
ΓminusΓminus
minusε=
αΓ 2
ME
2My
eff
1a1
CC4
351Xy
Range E ndash E
0Xy=Γ
40 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
Appendix A Fatigue Damage Accumulation
The Palmgren-Miner cumulative damage calculation princi-ple is used The procedure may be applied as follows
A1 Stress Spectrum From the individual torque classes the torques (Ti) at the peak values of class intervals and the associated number of cycles (NLi) for both pinion and wheel are to be listed from the highest to the lowest torque
(In case of a cyclic torque variation within the torque classes it is advised to use the peak torque If the cyclic variation is such that the same teeth will repeatedly suffer the peak torque this is a must)
The stress spectra for tooth roots and flanks (σFi σHi) with all relevant factors (except KA) are to be calculated on the basis of the torque spectrum The load dependent K-factors are to be determined for each torque class
A2 σminusN-curve The stress versus load cycle curves for tooth roots and flanks (both pinion and wheel) are to be drawn on the basis of per-missible stresses (ie including the demanded minimum safety factors) as determined in 2 respectively 3 If different safety levels for high cycle fatigue and low cycle fatigue are desired this may be expressed by different demand safety factors applied at the endurance limit respectively at static strength
A3 Damage accumulation The individual damage ratio Di at ith stress level is defined as
Fi
Lii N
ND =
where
NLi = The number of applied cycles at ith stress
NFi = The number of cycles to failure at ith stress
Basically stresses σi below the permissible stress level for infinite life (if a constant ZN or YN is accepted) do not con-tribute to the damage sum However calculating the actual safety factor Sact as described below all the σi for which the product S σi is bigger than or equal to the permissible stress level for infinite life contribute to the damage sum and thus to the determination of Sact The final value of S is decisive
(NFi can be found mathematically by putting the permissible stress σpi equal the actual stress σi thereby finding the actual life factor This life factor can be solved with regard to load cycles ie NFi)
The damage sum ΣDi is not to exceed unity
If ΣDi ne 1 the safety against cumulative fatigue damage is different from the applied demand safety factor For deter-mination of this theoretical safety factor an iteration proce-dure is required as described in the following flowchart
S is correction factor with which the actual safety factor Sact can be found
Sact is the demand safety factor (used in determination of the permissible stresses in the σ ndash N ndash curve) times the correc-tion factor S
The full procedure is to be applied for pinion and wheel tooth roots and flanks
Note If alternating stresses occur in a spectrum of mainly pulsating stresses the alternating stresses may be replaced by equiva-lent pulsating stresses ie by means of division with the ac-tual mean stress influence factor YM
Classification Notes- No 412 41 May 2003
DET NORSKE VERITAS
Appendix B Application Factors for Diesel Driven Gears
For diesel driven gears the application factor KA depends on torsional vibrations Both normal operation and misfiring conditions have to be considered
Normally these two running conditions can be covered by only one calculation
B1 Definitions
Normal operation KAnorm = 0
normv0
TTT +
where
T0 = rated nominal torque
Tv norm = vibratory torque amplitude for normal opera-tion (see rules Pt4 Ch3 Sec1 G301 for defi-nition of ldquonormalrdquo irregularity)
Misfiring operation KA misf = 0
misfv
TTT +
where
T = remaining nominal torque when one cylinder out of action
Tv misf = vibratory torque amplitude in misfiring con-dition This refers to a permissible misfiring condition ie a condition that does not require automatic or immediate corrective actions as speed or pitch reduction
The normal operation is assumed to last for a very high num-ber of cycles such as 1010
The misfiring operation is assumed to last for a limited dura-tion such as 107 cycles
B2 Determination of decisive load Assuming life factor at 1010 cycles as YN = ZN = 092 which usually is relevant the calculation may be performed only once with the combination having the highest value of appli-cation factorlife factor
For bending stresses and scuffing the higher value of
092
K normA and 098
K misfA
For contact stresses the higher value of
092K normA and
113K misfA (but
097K misfA for nitrided gears)
B3 Simplified procedure Note that this is only a guidance and is not a binding con-vention
T = oTZ
1Zsdot
minus where Z = number of cylinders
( ) ideal Videal Vmisf Vnorm V TTT24ZT +minus=
where TV ideal = vibratory torque with all cylinders perfectly equal See also rules Pt4 Ch3 Sec1 G300
When using trends from torsional vibration analysis and measurements the following may be used
TV ideal To is close to zero for engines with few cylinders and using a suitable elastic coupling and increases with relative coupling stiffness and number of cylinders
TV misf To may be high for engines with few cylinders and decreases with number of cylinders
This can be indicated as
200Z
TT
ο
idealV asymp and 80Z04
TT
o
misfV minusasymp
Inserting this into the formulae for the two application fac-tors the following guidance can be given
118112K misfA minusasymp
115110K normA minusasymp
Since KA norm is to be combined with the lower life factors the decisive load condition will be the normal one and a KA of 115 will cover most relevant cases when a suitable elas-tic coupling is chosen
42 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
Appendix C Calculation of Pinion-Rack
Pinion-racks used for elevating of mobile offshore units are open gears that are subjected to wear and tear With normal specifications such as only a few hundred total operation cycles (site to site) the tooth bending stresses for static strength will be decisive for the lay out however with ex-ception of surface hardened pinions where case crushing has to be considered
In the following the use of part 1 and 3 for pinion-racks is shown including relevant simplifications
C1 Pinion tooth root stresses Since the load spectrum normally is dominated by high torques and few load cycles (in the range up to 1-2000) the static strength is decisive
The actual stress is calculated as
β1FSaFan1
tF1 KYY
mbFσ sdotsdotsdotsdot
=
b1 is limited to b2 + 2middotmn
YFa and YSa replace YF and YS because load application at tooth tip has to be assumed for such inaccurate gears
Pinions often use a non-involute profile in the dedendum part of the flank eg a constant radius equal the radius of curva-ture at reference circle For such pinions sFn and hFa are to be measured directly on a sectional drawing of the pinion tooth
Due to high loads and narrow facewidths it may be assumed that KFβ2 = KHβ = 10 However when b1 gt b2 then KFβ1 gt10
If no detailed documentation of KFβ1 is available the fol-lowing may be used
KFβ1 = 1 + 015middot(b1b2 ndash 1)
The permissible stress (not surface hardened) is calculated as
δrelTstF
Fst1FP1 Y
Sσσ sdot=
The mean stress influence due to leg lifting may be disre-garded
The actual and permissible stresses should be calculated for the relevant loads as given in the rules
C2 Rack tooth root stresses The actual stress is calculated as
SaFan2
tF2 YY
mbFσ sdotsdotsdot
=
See C1 for details
The permissible stress is calculated as
δrelTstF
Fst2FP2 Y
Sσσ sdot=
For alloyed steels (Ni Cr Mo) with high toughness and duc-tility the value of YδrelTst may be put equal to YSa
C3 Surface hardened pinions For surface hardened pinions the maximum load is not to cause crushing of the hardened layer of the flank
In principle the calculation described in 213 may be used but when the theoretical Hertzian stress exceeds the ap-proximately 18 times the yield strength of the rack material plastic deformation will occur This will limit the peak Hertzian stress but increases the contact width and thus the penetration of stresses into the depth
An approximation may be based on an assessment of contact width determined by means of equal areas under the theoreti-cal (elastic range) Hertzian contact and the elasto-plastic contact stress (limited to 18 middot σy) with the unknown width
Classification Notes- No 412 9 May 2003
DET NORSKE VERITAS
For gears with inactive ends of the facewidth as eg due to high crowning or end relief such as often applied for bevel gears the use of cγ in connection with determination of natu-ral frequencies may need correction cγ is defined as stiffness per unit facewidth but when used in connection with the total mesh stiffness it is not as simple as cγmiddotb as only a part of the facewidth is active Such corrections are given in 111
mred is the reduced mass of the gear pair per unit facewidth and referred to the plane of contact
For a single gear stage where no significant inertias are closely connected to neither pinion nor wheel mred is calcu-lated as
mred = 21
21mm
mm+
The individual masses per unit facewidth are calculated as
m12 = 221b
21
)2d(b
I
where I is the polar moment of inertia (kgmm2)
The inertia of bevel gears may be approximated as discs with diameter equal the midface pitch diameter and width equal to b However if the shape of the pinion or wheel body differs much from this idealised cylinder the inertia should be cor-rected accordingly
For all kind of gears if a significant inertia (eg a clutch) is very rigidly connected to the pinion or wheel it should be added to that particular inertia (pinion or wheel) If there is a shaft piece between these inertias the torsional shaft stiffness alters the system into a 3-mass (or more) system This can be calculated as in 182 but also simplified as a 2-mass system calculated with only pinion and wheel masses
1812 Factors used for determination of Kv Non-dimensional gear accuracy dependent parameters
( )bKKF
yfcB
At
pptp
γ
minus=
( )bKKF
yFcBAt
ff
γ
α minus=
Non-dimensional tip relief parameter
bKKFcC
1BγAt
ak sdotsdot
sdotminus=
For gears of quality grade (ISO 1328) Q = 7 or coarser Bk = 1
For gears with Q le 6 and excessive tip relief Bk is limited to max 1
For gears (all quality grades) with tip relief of more than 2middotCeff (see 432) the reduction of αε has to be considered (see 443)
where
fpt = the single pitch deviation (ISO 1328) max of pinion or wheel
Fα = the total profile form deviation (ISO 1328) max of pinion or wheel (Note Fα is pt not available for bevel gears thus use Fα = fpt)
yp and yf = the respective running-in allowances and may be calculated similarly to yα in 112 ie the value of fpt is replaced by Fα for yf
cacute = the single tooth stiffness see 111
Ca = the amount of tip relief see 433 In case of different tip relief on pinion and wheel the value that results in the greater value of Bk is to be used If Ca is zero by design the value of running-in tip relief Cay (see 112) may be used in the above formula
1813 Kv in the subcritical range Cylindrical gears N le 085
Bevel gears N le 075
Kv = 1 + N K
K = Cv1 Bp + Cv2 Bf + Cv3 Bk
Cv1 accounts for the pitch error influence
Cv1 = 032
Cv2 accounts for profile error influence
Cv2 = 034 for γε le 2
Cv2 = 03ε
057
γ minus for γε gt 2
Cv3 accounts for the cyclic mesh stiffness variation
Cv3 = 023 for γε le 2
Cv3 = 156ε
0096
γ minus for γε gt 2
1814 Kv in the main resonance range Cylindrical gears 085 lt N le 115
Bevel gears 075 lt N le 125
Running in this range should preferably be avoided and is only allowed for high precision gears
Kv = 1 + Cv1 Bp + Cv2 Bf + Cv4 Bk
Cv4 accounts for the resonance condition with the cyclic mesh stiffness variation
Cv4 = 090 for γε le 2
Cv4 = 144ε
ε005057
γ
γ
minus
minus for γε gt 2
10 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
1815 Kv in the supercritical range Cylindrical gears N ge 15
Bevel gears N ge 15
Special care should be taken as to influence of higher vibra-tion modes andor influence of coupled bending (ie lateral shaft vibrations) and torsional vibrations between pinion and wheel These influences are not covered by the following approach
Kv = Cv5 Bp + Cv6 Bf + Cv7
Cv5 accounts for the pitch error influence
Cv5 = 047
Cv6 accounts for the profile error influence
Cv6 = 047 for εγ le 2
Cv6 = 174ε
012
γ minus for εγ gt 2
Cv7 relates the maximum externally applied tooth loading to the maximum tooth loading of ideal accurate gears operating in the supercritical speed sector when the circumferential vibration becomes very soft
Cv7 = 075 for εγ le 15
Cv7 = [ ] 8750)2(sin 0125 +minusβεπ for 15 lt εγ le 25
Cv7 = 10 for εγ gt 25
1816 Kv in the intermediate range Cylindrical gears 115 lt N lt 15
Bevel gears 125 lt N lt 15
Comments raised in 1814 and 1815 should be observed
Kv is determined by linear interpolation between Kv for N = 115 respectively 125 and N = 15 as
Cylindrical gears
( ) ( ) ( )[ ]51Nv151Nv51Nvv KK350
N51KK === minussdot
minus
+=
Bevel gears
( ) ( ) ( )[ ]51Nv251Nv51Nvv KK250
N51KK === minussdot
minus
+=
182 Multi-resonance method For high speed gear (vgt40 ms) for multimesh medium speed gears for gears with significant lateral shaft flexibility etc it is advised to determine Kv on basis of relevant dy-namic analysis
Incorporating lateral shaft compliance requires transforma-tion of even a simple pinion-wheel system into a lumped multi-mass system It is advised to incorporate all relevant inertias and torsional shaft stiffnesses into an equivalent (to pinion speed) system Thereby the mesh stiffness appears as an equivalent torsional stiffness
cγ b (db12)2 (Nmrad)
The natural frequencies are found by solving the set of dif-ferential equations (one equation per inertia) Note that for a gear put on a laterally flexible shaft the coupling bending-torsionals is arranged by introducing the gear mass and the lateral stiffness with its relation to the torsional displacement and torque in that shaft
Only the natural frequency (ies) having high relative dis-placement and relative torque through the actual pinion-wheel flexible element need(s) to be considered as critical frequency (ies)
Kv may be determined by means of the method mentioned in 181 thereby using N as the least favourable ratio (in case of more than one pinion-wheel dominated natural frequency) Ie the N-ratio that results in the highest Kv has to be consid-ered
The level of the dynamic factor may also be determined on basis of simulation technique using numeric time integration with relevant tooth stiffness variation and pitchprofile er-rors
19 Face Load Factors KHβ and KFβ The face load factors KHβ for contact stresses and for scuff-ing KFβ for tooth root stresses account for non-uniform load distribution across the facewidth
KHβ is defined as the ratio between the maximum load per unit facewidth and the mean load per unit facewidth
KFβ is defined as the ratio between the maximum tooth root stress per unit facewidth and the mean tooth root stress per unit facewidth The mean tooth root stress relates to the con-sidered facewidth b1 respectively b2
Note that facewidth in this context is the design facewidth b even if the ends are unloaded as often applies to eg bevel gears
The plane of contact is considered
191 Relations between KHβ and KFβ
KFβ = expβHK 2(hb)hb1
1exp++
=
where hb is the ratio tooth heightfacewidth The maximum of h1b1 and h2b2 is to be used but not higher than 13 For double helical gears use only the facewidth of one helix
If the tooth root facewidth (b1 or b2) is considerably wider than b the value of KFβ(1or2) is to be specially considered as it may even exceed KHβ
Eg in pinion-rack lifting systems for jack up rigs where b = b2 asymp mn and b1 asymp 3 mn the typical KHβ asymp KFβ2 asymp 1 and KFβ1 asymp 13
192 Measurement of face load factors Primarily
KFβ may be determined by a number of strain gauges distrib-uted over the facewidth Such strain gauges must be put in
Classification Notes- No 412 11 May 2003
DET NORSKE VERITAS
exactly the same position relative to the root fillet Relations in 191 apply for conversion to KHβ
Secondarily
KHβ may be evaluated by observed contact patterns on vari-ous defined load levels It is imperative that the various test loads are well defined Usually it is also necessary to evalu-ate the elastic deflections Some teeth at each 90 degrees are to be painted with a suitable lacquer Always consider the poorest of the contact patterns
After having run the gear for a suitable time at test load 1 (the lowest) observe the contact pattern with respect to ex-tension over the facewidth Evaluate that KHβ by means of the methods mentioned in this section Proceed in the same way for the next higher test load etc until there is a full face contact pattern From these data the initial mesh misalign-ment (ie without elastic deflections) can be found by ex-trapolation and then also the KHβ at design load can be found by calculation and extrapolation See example
Figure 11 Example of experimental determination of KHβ
It must be considered that inaccurate gears may accumulate a larger observed contact pattern than the actual single mesh to mesh contact patterns This is particularly important for lapped bevel gears Ground or hard metal hobbed bevel gears are assumed to present an accumulated contact pattern that is practically equal the actual single mesh to mesh con-tact patterns As a rough guidance the (observed) accumu-lated contact pattern of lapped bevel gears may be reduced by 10 in order to assess the single mesh to mesh contact pattern which is used in 199
193 Theoretical determination of KHβ The methods described in 193 to 198 may be used for cy-lindrical gears The principles may to some extent also be used for bevel gears but a more practical approach is given in 199
General For gears where the tooth contact pattern cannot be verified during assembly or under load all assumptions are to be well on the safe side
KHβ is to be determined in the plane of contact
The influence parameters considered in this method are
bull mean mesh stiffness cγ (see 111) (if necessary also variable stiffness over b)
bull mean unit load Fmb = Fbt KA Kγ Kvb (for double helical gears see 17 for use of Kγ)
bull misalignment fsh due to elastic deflections of shafts and gear bodies (both pinion and wheel)
bull misalignment fdefl due to elastic deflections of and work-ing positions in bearings
bull misalignment fbe due to bearing clearance tolerances bull misalignment fma due to manufacturing tolerances bull helix modifications as crowning end relief helix correc-
tion bull running in amount yβ (see 112)
In practice several other parameters such as centrifugal ex-pansion thermal expansion housing deflection etc contrib-ute to KHβ However these parameters are not taken into account unless in special cases when being considered as particularly important
When all or most of the am parameters are to be considered the most practical way to determine KHβ is by means of a graphical approach described in 1931
If cγ can be considered constant over the facewidth and no helix modifications apply KHβ can be determined analyti-cally as described in 1932
1931 Graphical method The graphical method utilises the superposition principle and is as follows
bull Calculate the mean mesh deflection Mδ as a function of
γm c and bF see 111
bull Draw a base line with length b and draw up a rectangu-lar with height δM (The area δM b is proportional to the transmitted force)
bull Calculate the elastic deflection fsh in the plane of contact Balance this deflection curve around a zero line so that the areas above and below this zero line are equal
Figure 12 fsh balanced around zero line
bull Superimpose these ordinates of the fsh curve to the previ-ous load distribution curve (The area under this new load distribution curve is still δM b)
bull Calculate the bearing deflections andor working posi-tions in the bearings and evaluate the influence fdefl in the plane of contact This is a straight line and is bal-anced around a zero line as indicated in Fig 14 but with one distinct direction Superimpose these ordinates to the previous load distribution curve
12 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
bull The amount of crowning end relief or helix correction (defined in the plane of contact) is to be balanced around a zero line similarly to fsh
Figure 13 Crowning Cc balanced aound zero line
bull Superimpose these ordinates to the previous load distri-bution curve In case of high crowning etc as eg often applied to bevel gears the new load distribution curve may cross the base line (the real zero line) The result is areas with negative load that is not real as the load in those areas should be zero Thus corrective actions must be made but for practical reasons it may be postponed to after next operation
bull The amount of initial mesh misalignment bema ff + (defined in the plane of contact) is to be bal-
anced around a zero line If the direction of bema ff + is known (due to initial contact check) or if the direction of fbe is known due to design (eg overhang bevel pin-ion) this should be taken into account If direction un-known the influence of bema ff + in both directions as well as equal zero should be considered
Figure 14 fma+fbe in both directions balanced around zero line
Superimpose these ordinates to the previous load distri-bution curve This results in up to 3 different curves of which the one with the highest peak is to be chosen for further evaluation
bull If the chosen load distribution curve crosses the base line (ie mathematically negative load) the curve is to be corrected by adding the negative areas and dividing this with the active facewidth The (constant) ordinates of this rectangular correction area are to be subtracted from the positive part of the load distribution curve It is advisable to check that the area covered under this new load distribution curve is still equal δM b
bull If cγ cannot be considered as constant over b then cor-rect the ordinates of the load distribution curve with the local (on various positions over the facewidth) ratio between local mesh stiffness and average mesh stiffness cγ (average over the active facewidth only) Note that the result is to be a curve that covers the same area δM b as before
bull The influence of running in yβ is to be determined as in 112 whereby the value for Fβx is to be taken as twice the distance between the peak of the load distribution curve and δM
bull Determine
KHβ = M
β
δycurveofpeak minus
1932 Simplified analytical method for cylindrical gears The analytical approach is similar to 1931 but has a more limited application as cγ is assumed constant over the face-width and no helix modification applies
bull Calculate the elastic deflection fsh in the plane of contact Balance this deflection curve around a zero line so that the area above and below this zero line are equal see Fig 12 The max positive ordinate is frac12∆fsh
bull Calculate the initial mesh alignment as Fβx= deflbemash ffff plusmnplusmnplusmn∆ The negative signs may only be used if this is justified andor verified by a contact pattern test Otherwise al-ways use positive signs If a negative sign is justified the value of Fβx is not to be taken less than the largest of each of these elements
bull Calculate the effective mesh misalignment as Fβy = Fβx - yβ (yβ see 112)
bull Determine
2KforF2
bFc1K βH
m
βγγH le+=β
or
2KforF
bFc2K Hβ
m
βγγH gt=β
where cγ as used here is the effective mesh stiffness see 111
194 Determination of fsh fsh is the mesh misalignment due to elastic deflections Usu-ally it is sufficient to consider the combined mesh deflection of the pinion body and shaft and the wheel shaft The calcu-lation is to be made in the plane of contact (of the considered gear mesh) and to consider all forces (incl axial) acting on the shafts Forces from other meshes can be parted into components parallel respectively vertical to the considered plane of contact Forces vertical to this plane of contact have no influence on fsh
It is advised to use following diameters for toothed elements
d + 2 x mn for bending and shear deflection
d + 2 mn (x ndash ha0 + 02) for torsional deflection
Usually fsh is calculated on basis of an evenly distributed load If the analysis of KHβ shows a considerable maldis-
Classification Notes- No 412 13 May 2003
DET NORSKE VERITAS
tribution in term of hard end contact or if it is known by other reasons that there exists a hard end contact the load should be correspondingly distributed when calculating fsh In fact the whole KHβ procedure can be used iteratively 2-3 iterations will be enough even for almost triangular load distributions
195 Determination of fdefl fdefl is the mesh misalignment in the plane of contact due to bearing deflections and working positions (housing deflec-tion may be included if determined)
First the journal working positions in the bearings are to be determined The influence of external moments and forces must be considered This is of special importance for twin pinion single output gears with all 3 shafts in one plane
For rolling bearings fdefl is further determined on basis of the elastic deflection of the bearings An elastic bearing deflec-tion depends on the bearing load and size and number of rolling elements Note that the bearing clearance tolerances are not included here
For fluid film bearings fdefl is further determined on basis of the lift and angular shift of the shafts due to lubrication oil film thickness Note that fbe takes into account the influence of the bearing clearance tolerance
When working positions bearing deflections and oil film lift are combined for all bearings the angular misalignment as projected into the plane of the contact is to be determined fdefl is this angular misalignment (radians) times the face-width
196 Determination of fbe fbe is the mesh misalignment in the plane of contact due to tolerances in bearing clearances In principle fbe and fdefl could be combined But as fdefl can be determined by analy-sis and has a distinct direction and fbe is dependent on toler-ances and in most cases has no distinct direction (ie + toler-ance) it is practicable to separate these two influences
Due to different bearing clearance tolerances in both pinion and wheel shafts the two shaft axis will have an angular mis-alignment in the plane of contact that is superimposed to the working positions determined in 195 fbe is the facewidth times this angular misalignment Note that fbe may have a distinct direction or be given as a + tolerance or a combina-tion of both For combination of + tolerance it is adviced to use
fbe= 22be
21be ff ++plusmn
fbe is particularly important for overhang designs for gears with widely different kinds of bearings on each side and when the bearings have wide tolerances on clearances In general it shall be possible to replace standard bearings with-out causing the real load distribution to exceed the design premises For slow speed gears with journal bearings the expected wear should also be considered
197 Determination of fma fma is the mesh misalignment due to manufacturing toler-ances (helix slope deviation) of pinion fHβ1 wheel fHβ2 and housing bore
For gear without specifically approved requirements to as-sembly control the value of fma is to be determined as
fma= 22Hβ
21Hβ ff +
For gears with specially approved assembly control the value of fma will depend on those specific requirements
198 Comments to various gear types For double helical gears KHβ is to be determined for both helices Usually an even load share between the helices can be assumed If not the calculation is to be made as de-scribed in 171
For planetary gears the free floating sun pinion suffers only twist no bending It must be noted that the total twist is the sum of the twist due to each mesh If the value of 1K γ ne this must be taken into account when calculating the total sun pinion twist (ie twist calculated with the force per mesh without Kγ and multiplied with the number of planets)
When planets are mounted on spherical bearings the mesh misalignments sun-planet respectively planet-annulus will be balanced Ie the misalignment will be the average between the two theoretical individual misalignments The faceload distribution on the flanks of the planets can take full advan-tage of this However as the sun and annulus mesh with several planets with possibly different lead errors the sun and annulus cannot obtain the above mentioned advantage to the full extent
199 Determination of KHβ for bevel gears If a theoretical approach similar to 193-198 is not docu-mented the following may be used
testeff
H Kb
b851851K sdot
minussdot=β
beff b represents the relative active facewidth (regarding lapped gears see 192 last part)
Higher values than beff b = 090 are normally not to be used in the formula
For dual directional gears it may be difficult to obtain a high beff b in both directions In that case the smaller beff b is to be used
Ktest represents the influence of the bearing arrangement shaft stiffness bearing stiffness housing stiffness etc on the faceload distribution and the verification thereof Expected variations in length- and height-wise tooth profile is also accounted for to some extent
a) Ktest = 1 For ground or hard metal hobbed gears with the specified contact pattern verified at full rating or at full torque slow turning at a condition representative for the thermal expansion at normal operation
14 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
It also applies when the bearing arrangementsupport has insignificant elastic deflections and thermal axial expansion However each initial mesh contact must be verified to be within acceptance criteria that are cali-brated against a type test at full load Reproduction of the gear tooth length- and height-wise profile must also be verified This can be made through 3D measurements or by initial contact movements caused by defined axial offsets of the pinion (tolerances to be agreed upon)
b) Ktest = 1 + 04middot(beffbndash06) For designs with possible influence of thermal expansion in the axial direction of the pinion The initial mesh contact verified with low load or spin test where the acceptance criteria are calibrated against a type test at full load
c) Ktest = 12 if mesh is only checked by toolmakerrsquos blue or by spin test contact For gears in this category beffb gt 085 is not to be used in the calculation
110 Transversal Load Distribution Factors KHα and KFα The transverse load distribution factors KHα for contact stresses and for scuffing KFα for tooth root stresses account for the effects of pitch and profile errors on the transversal load distribution between 2 or more pairs of teeth in mesh
The following relations may be used
Cylindrical gears
( )
minus+==
tH
αptγγHαFα F
byfc0409
2ε
KK
valid for 2εγ le
( ) ( )tH
αptγ
γ
γHαFα F
byfcε
1ε20409KK
minusminus+==
valid for 2ε γ gt
where
FtH = Ft KA Kγ Kv KHβ
cγ = See 111
γα = See 112
fpt = Maximum single pitch deviation (microm) of pinion or wheel or maximum total profile form devia-tion Fα of pinion or wheel if this is larger than the maximum single pitch deviation
Note In case of adequate equivalent tip relief adapted to the load half of the above mentioned fpt can be introduced A tip relief is considered adequate when the average of Ca1 and Ca2 is within plusmn40 of the value of Ceff in 432
Limitations of KHα and KFα
If the calculated values for
KFα = KHα lt 1 use KFα = KHα = 10
If the calculated value of KHα gt 2εα
γ
Zε
ε use KHα = 2
εα
γ
Zε
ε
If the calculated value of KFα gt εα
γ
Yεε
use KFα = εα
γ
Yεε
where Yε = αnε
075025+ (for εαn see 331c)
Bevel gears
For ground or hard metal hobbed gears KFα = KHα = 1
For lapped gears KFα = KHα = 11
111 Tooth Stiffness Constants cacute and cγ The tooth stiffness is defined as the load which is necessary to deform one or several meshing gear teeth having 1 mm facewidth by an amount of 1 microm in the plane of contact cacute is the maximum stiffness of a single pair of teeth cγ is the mean value of the mesh stiffness in a transverse plane (brief term mesh stiffness)
Both valid for high unit load (Unit load = Ft middot KA middot Kγb)
Cylindrical gears The real stiffness is a combination of the progressive Hertzian contact stiffness and the linear tooth bending stiff-nesses For high unit loads the Hertzian stiffness has little importance and can be disregarded This approach is on the safe side for determination of KHβ and KHα However for moderate or low loads Kv may be underestimated due to determination of a too high resonance speed
The linear approach is described in A
An optional approach for inclusion of the non-linear stiffness is described in B
A The linear approach
BRCCq
βcos08acutec =
and
( )025ε075cc αγ +prime=
where
( )[ ]n02a01a
B α2000212
hh12051C minusminus
+minus+=
12n1n
x000635z
025791z
015551004723q minus++=
12
2n
22
1n
1 x005290z
x024188x000193z
x011654+minusminusminus
+ 000182 x22
Classification Notes- No 412 15 May 2003
DET NORSKE VERITAS
(for internal gears use zn2 equal infinite and x2 = 0) ha0 = hfp for all practical purposes CR considers the increased flexibility of the wheel teeth if the wheel is not a solid disc and may be calculated as
( )( )nR m5s
sR e5
bbln1C +=
where
bs = thickness of a central web
sR = average thickness of rim (net value from tooth root to inside of rim)
The formula is valid for bs b ge 02 and sRmn ge 1 Outside this range of validity and if the web is not centrally posi-tioned CR has to be specially considered
Note CR is the ratio between the average mesh stiffness over the facewidth and the mesh stiffness of a gear pair of solid discs The local mesh stiffness in way of the web corresponds to the mesh stiffness with CR = 1 The local mesh stiffness where there is no web support will be less than calculated with CR above Thus eg a centrally positioned web will have an effect corresponding to a longitudinal crowning of the teeth See also 1931 regarding KHβ
B The non-linear approach
In the following an example is given on how to consider the non-linearity
The relation between unit load Fb as a function of mesh deflection δ is assumed to be a progressive curve up to 500 Nmm and from there on a straight line This straight line when extended to the baseline is assumed to intersect at 10microm
With these assumptions the unit force Fb as a function of mesh deflection δ can be expressed as
( )10δKbF
minus= for 500bFgt
minus=
500Fb10δK
bF for 500
bFlt
with γAt KK
bF
bF
sdotsdot= etc (Nmm) ie unit load incorporat-
ing the relevant factors as
KA middot Kγ for determination of Kv
KA middot Kγ middot Kv for determination of KHβ
KA middot Kγ middot Kv middot KHβ for determination of KHα
δ = mesh deflection (microm)
K = applicable stiffness (c or cγ)
Use of stiffnesses for KV KHβ and KHα
For calculation of Kv and KHα the stiffness is calculated as follows
When Fb lt 500
the stiffness is determined as δ∆
∆ bF
where the increment is chosen as eg ∆ Fb = 10 and thus
50010Fb10
K10Fb∆δ +
++
=
When F b gt 500 the stiffness is c or cγ For calculation of KHβ the mesh deflection δ is used directly
or an equivalent stiffness determined as δsdotb
F
Bevel gears
In lack of more detailed relationship between stiffness and geometry the following may be used
b085
b13cacute eff=
b085b
16c effγ =
beff not to be used in excess of 085 b in these formulae
Bevel gears with heightwise and lengthwise crowning have progressive mesh stiffness The values mentioned above are only valid for high loads They should not be used for de-termination of Ceff (see 432) or KHβ (see 1931)
112 Running-in Allowances The running-in allowances account for the influence of run-ning-in wear on the various error elements
yα respectively yβ are the running-in amounts which reduce the influence of pitch and profile errors respectively influ-ence of localised faceload
Cay is defined as the running-in amount that compensates for lack of tip relief
The following relations may be used
For not surface hardened steel
ptHlim
α fσ160y =
yβ = βxlimH
f320σ
with the following upper limits
V lt 5 ms 5-10 ms gt 10 ms
yα max none
limH
12800σ limH
6400σ
yβ max none
limH
25600σ limH
12800σ
For surface hardened steel
yα = 0075 fpt but not more than 3 for any speed
yβ = 015 Fβx but not more than 6 for any speed
16 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
For all kinds of steel
5145189718
1C2
limHay +
minusσ
=
When pinion and wheel material differ the following ap-plies
bull Use the larger of fpt1 - yα1 and fpt2 - yα2 to replace fpt - yα in the calculation of KHα and Kv
bull Use ( )2β1ββ yy21y += in the calculation of KHβ
bull Use ( )2ay1aya CC21C += in the calculation of Kv
bull Use ( )2ay1ay2a1a CC21CC +== in the scuffing
calculation if no design tip relief is foreseen
Classification Notes- No 412 17 May 2003
DET NORSKE VERITAS
2 Calculation of Surface Durability
21 Scope and General Remarks Part 2 includes the calculations of flank surface durability as limited by pitting spalling case crushing and subsurface yielding Endurance and time limited flank surface fatigue is calculated by means of 22 ndash 212 In a way also tooth frac-tures starting from the flank due to subsurface fatigue is in-cluded through the criteria in 213
Pitting itself is not considered as a critical damage for slow speed gears However pits can create a severe notch effect that may result in tooth breakage This is particularly im-portant for surface hardened teeth but also for high strength through hardened teeth For high-speed gears pitting is not permitted
Spalling and case crushing are considered similar to pitting but may have a more severe effect on tooth breakage due to the larger material breakouts initiated below the surface Subsurface fatigue is considered in 213
For jacking gears (self-elevating offshore units) or similar slow speed gears designed for very limited life the max static (or very slow running) surface load for surface hard-ened flanks is limited by the subsurface yield strength
For case hardened gears operating with relatively thin lubri-cation oil films grey staining (micropitting) may be the lim-iting criterion for the gear rating Specific calculation meth-ods for this purpose are not given here but are under consid-eration for future revisions Thus depending on experience with similar gear designs limitations on surface durability rating other than those according to 22 - 213 may be ap-plied
22 Basic Equations Calculation of surface durability (pitting) for spur gears is based on the contact stress at the inner point of single pair contact or the contact at the pitch point whichever is greater
Calculation of surface durability for helical gears is based on the contact stress at the pitch point
For helical gears with 0 lt εβ lt 1 a linear interpolation be-tween the above mentioned applies
Calculation of surface durability for spiral bevel gears is based on the contact stress at the midpoint of the zone of contact
Alternatively for bevel gears the contact stress may be cal-culated with the program ldquoBECALrdquo In that case KA and Kv are to be included in the applied tooth force but not KHβ and KHα The calculated (real) Hertzian stresses are to be multi-plied with ZK in order to be comparable with the permissible contact stresses
The contact stresses calculated with the method in part 2 are based on the Hertzian theory but do not always represent the real Hertzian stresses
The corresponding permissible contact stresses σHP are to be calculated for both pinion and wheel
221 Contact stress Cylindrical gears
( )HαHβvγA
1
tβεEHDBH KKKKK
bud1uF
ZZZZZσ+
=
where
ZBD = Zone factor for inner point of single pair contact for pinion resp wheel (see 232)
ZH = Zone factor for pitch point (see 231)
ZE = Elasticity factor (see 24)
Zε = Contact ratio factor (see 25)
Zβ = Helix angle factor (see 26)
Ft KA Kγ Kv KHβ KHα see 15 ndash 110
d1 b u see 12 ndash 15
Bevel gears
( )HαHβvγA
v1v
vmtKEMH KKKKK
bud1uF
ZZZ105σ+
sdot=
where
105 is a correlation factor to reach real Hertzian stresses (when ZK = 1)
ZE KA etc see above
ZM = mid-zone factor see 233
ZK = bevel gear factor see 27
Fmt dv1 uv see 12 ndash 15
It is assumed that the heightwise crowning is chosen so as to result in the maximum contact stresses at or near the mid-point of the flanks
222 Permissible contact stress
XWRvLH
NHlimHP ZZZZZ
SZσ
σ =
where
σH lim = Endurance limit for contact stresses (see 28)
ZN = Life factor for contact stresses (see 29)
SH = Required safety factor according to the rules
ZLZvZR = Oil film influence factors (see 210)
ZW = Work hardening factor (see 211)
ZX = Size factor (see 212)
18 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
23 Zone Factors ZH ZBD and ZM
231 Zone factor ZH The zone factor ZH accounts for the influence on contact stresses of the tooth flank curvature at the pitch point and converts the tangential force at the reference cylinder to the normal force at the pitch cylinder
wtt2
wtbH sinααcos
cosαcosβ2Z =
232 Zone factors ZBD The zone factors ZBD account for the influence on contact stresses of the tooth flank curvature at the inner point of sin-gle pair contact in relation to ZH Index B refers to pinion D to wheel
For εβ ge 1 ZBD = 1
For internal gears ZD = 1
For εβ = 0 (spur gears)
( )
π
minusεminusminus
π
minusminus
α=
α2
2
2b
2a
1
2
1b
1a
wtB
z211
dd
z21
dd
tanZ
( )
π
minusεminusminus
π
minusminus
α=
α1
2
1b
1a
2
2
2b
2a
wtD
z211
dd
z21
dd
tanZ
If ZB lt 1 use ZB = 1
If ZD lt 1 use ZD = 1
For 0 lt εβ lt 1
ZBD = ZBD (for spur gears) ndash εβ (ZBD (for spur gears) ndash 1)
233 Zone factor ZM The mid-zone factor ZM accounts for the influence of the contact stress at the mid point of the flank and applies to spi-ral bevel gears
εminusminus
εminusminus
αβ=
αα btm2
2vb2
2vabtm2
1vb2val
2v1vvtbmM
pddpdd
ddtancos2Z
This factor is the product of ZH and ZM-B in ISO 10300 with the condition that the heightwise crowning is sufficient to move the peak load towards the midpoint
234 Inner contact point For cylindrical or bevel gears with very low number of teeth the inner contact point (A) may be close to the base circle In order to avoid a wear edge near A it is required to have suitable tip relief on the wheel
24 Elasticity Factor ZE The elasticity factor ZE accounts for the influence of the material properties as modulus of elasticity and Poissonrsquos ratio on the contact stresses
For steel against steel ZE = 1898
25 Contact Ratio Factor Zε The contact ratio factor Zε accounts for the influence of the transverse contact ratio εα and the overlap ratio εβ on the contact stresses
αε1Z =ε for 1εβ ge
( )α
ββ
αε ε
εε1
3ε4
Z +minusminus
= for εβ lt 1
26 Helix Angle Factor Zβ The helix angle factor Zβ accounts for the influence of helix angle (independent of its influence on Zε) on the surface du-rability
cosβZβ =
27 Bevel Gear Factor ZK The bevel gear factor accounts for the difference between the real Hertzian stresses in spiral bevel gears and the contact stresses assumed responsible for surface fatigue (pitting) ZK adjusts the contact stresses in such a way that the same per-missible stresses as for cylindrical gears may apply
The following may be used ZK = 080
28 Values of Endurance Limit σHlim and Static Strength 5H10σ 3H10σ
σHlim is the limit of contact stress that may be sustained for 5middot107 cycles without the occurrence of progressive pitting
For most materials 5middot107 cycles are considered to be the be-ginning of the endurance strength range or lower knee of the σ-N curve (See also Life Factor ZN) However for nitrided steels 2middot106 apply
For this purpose pitting is defined by
bull for not surface hardened gears pitted area ge 2 of total active flank area
bull for surface hardened gears pitted area ge 05 of total active flank area or ge 4 of one particular tooth flank area 510Hσ and 310Hσ are the contact stresses which the given
material can withstand for 105 respectively 103 cycles without subsurface yielding or flank damages as pitting spalling or case crushing when adequate case depth applies
The following listed values for σHlim 510Hσ and 310Hσ may only be used for materials subjected to a quality control as
Classification Notes- No 412 19 May 2003
DET NORSKE VERITAS
the one referred to in the rules Results of approved fatigue tests may also be used as the basis for establishing these values The defined survival probability is 99
σHlim σH105 σH10
3
Alloyed case hardened steels (surface hardness 58-63 HRC) - of specially approved high grade - of normal grade
1650 1500
2500 2400
3100 3100
Nitrided steel of approved grade gas nitrided (surface hardness 700-800 HV) 1250 13 σHlim 13 σHlim
Alloyed quenched and tempered steel bath or gas nitrided (surface hardness 500-700 HV)
1000
13 σHlim
13 σHlim
Alloyed flame or induction hardened steel (surface hardness 500-650 HV) 075 HV + 750 16 σHlim 45 HV
Alloyed quenched and tempered steel 14 HV + 350 16 σHlim 45 HV
Carbon steel 15 HV + 250 16 σHlim 16 σHlim
These values refer to forged or hot rolled steel For cast steel the values for σHlim are to be reduced by 15
29 Life Factor ZN The life factor ZN takes account of a higher permissible contact stress if only limited life (number of cycles NL) is demanded or lower permissible contact stress if very high number of cycles apply
If this is not documented by approved fatigue tests the fol-lowing method may be used
For all steels except nitrided
7L 105N sdotge ZN = 1 or
01570
L
7
N N105Z
sdot=
Ie ZN = 092 for 1010 cycles
The ZN = 1 from 5middot107 on may only be used when the mate-rial cleanliness is of approved high grade (see Rules Pt4 Ch2) and the lubrication is optimised by a specially approved filtering process
105 lt NL lt 5middot107 510N
logZ037
L
7
N N105Z
sdot=
NL = 105 WXRVLHlim
WstX10H1010NN ZZZZZσ
ZZσZZ
55
5=
103 lt NL lt 105 )Z(Zlog05
L
5
10NN
5N103N10
5N10ZZ
==
3L 10N le
WXRVLlimH
Wst10X10H10NN ZZZZZ
ZZZZ
33
3σ
σ==
(but not less than ZN105)
For nitrided steels
6L 102N sdotge ZN = 1 or
00980
L
6
N N102Z
sdot=
Ie ZN = 092 for 1010 cycles
The ZN = 1 from 2middot106 on may only be used when the mate-rial cleanliness is of approved high grade (see Rules Pt4 Ch2) and the lubrication is optimised by a specially approved filtering process
105 lt NL lt 2middot106 510N
Zlog76860
L
6
N N102Z
sdot=
5L 10N le
XWRVL
10XWst10NN ZZZZZ
ZZ13ZZ
55 ==
Note that when no index indicating number of cycles is used the factors are valid for 5middot107 (respectively 2middot106 for nitrid-ing) cycles
210 Influence Factors on Lubrication Film ZL ZV and ZR The lubricant factor ZL accounts for the influence of the type of lubricant and its viscosity the speed factor ZV ac-counts for the influence of the pitch line velocity and the roughness factor ZR accounts for influence of the surface roughness on the surface endurance capacity
The following methods may be applied in connection with the endurance limit
20 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
Surface hardened steels Not surface hardened steels
ZL ( )24013421
360910ν+
+
( )24013421
680830ν+
+
ZV ( )v3280
140930+
+ ( )v3280300850
++
ZR
080
ZrelR3
150
ZrelR3
where
ν40 = Kinematic oil viscosity at 40ordmC (mm2s) For case hardened steels the influence of a high bulk temperature (see 4 Scuffing) should be con-sidered Eg bulk temperatures in excess of 120ordmC for long periods may cause reduced flank surface endurance limits
For values of ν40 gt 500 use ν40 = 500
RZrel = The mean roughness between pinion and wheel (after running in) relative to an equivalent radius of curvature at the pitch point ρc = 10mm
RZrel
= ( ) 31
cZ2Z1 ρ
10RR50
+
RZ = Mean peak to valley roughness (microm) (DIN defini-tion) (roughly RZ = 6 Ra)
For 5
L 10N le ZL ZV ZR = 10
211 Work Hardening Factor ZW The work hardening factor ZW accounts for the increase of surface durability of a soft steel gear when meshing the soft steel gear with a surface hardened or substantially harder gear with a smooth surface
The following approximation may be used for the endurance limit
Surface hardened steel against not surface hardened steel 150
ZeqW R
31700
130HB21Z
minus
minus=
where HB = the Brinell hardness of the soft member For HB gt 470 use HB = 470 For HB lt 130 use HB = 130
RZeq = equivalent roughness
033
c40
066
ZS
ZHZHZEQ ρvν
15000RRRR
=
If RZeq gt 16 then use RZeq = 16
If RZeq lt 15 then use RZeq = 15
where
RZH = surface roughness of the hard member before run in
RZS = surface roughness of the soft member before run in
ν40 = see 210
If values of ZW lt 1 are evaluated ZW = 1 should be used for flank endurance However the low value for ZW may indi-cate a potential wear problem
Through hardened pinion against softer wheel
( )
minussdotminus+= 000829
HBHB0008981u1Z
2
1W
For 21HBHB
2
1 le use ZW = 1
For 71HBHB
2
1 gt use 17HBHB
2
1 =
For u gt 20 use u = 20
For static strength (lt 105 cycles)
Surface hardened against not surface hardened
ZWst = 105
Through hardened pinion against softer wheel
ZWst = 1
212 Size Factor ZX The size factor accounts for statistics indicating that the stress levels at which fatigue damage occurs decrease with an increase of component size as a consequence of the influ-ence on subsurface defects combined with small stress gradi-ents and of the influence of size on material quality
ZX may be taken unity provided that subsurface fatigue for surface hardened pinions and wheels is considered eg as in the following subsection 213
213 Subsurface Fatigue This is only applicable to surface hardened pinions and wheels The main objective is to have a subsurface safety against fatigue (endurance limit) or deformation (static strength) which is at least as high as the safety SH required for the surface The following method may be used as an approximation unless otherwise documented
The high cycle fatigue (gt3106 cycles) is assumed to mainly depend on the orthogonal shear stresses Static strength (lt103 cycles) is assumed to depend mainly on equivalent
Classification Notes- No 412 21 May 2003
DET NORSKE VERITAS
stresses (von Mises) Both are influenced by residual stresses but this is only considered roughly and empirically
The subsurface working stresses at depths inside the peak of the orthogonal shear stresses respectively the equivalent stresses are only dependent on the (real) Hertzian stresses Surface related conditions as expressed by ZL ZV and ZR are assumed to have a negligible influence
The real Hertzian stresses σHR are determined as
For helical gears with εβ gt 1
σHR = σH
For helical gears with εβ lt 1 and spur gears
ε
α
ββ ε
ε+εminus
sdotσ=σZ
1
HHR
For bevel gears
KHHR Z
1σσ sdot=
The necessary hardness HV is given as a function of the net depth tz (net = after grinding or hard metal hobbing and per-pendicular to the flank)
The coordinates tz and HV are to be compared with the de-sign specification such as
bull for flame and induction hardening tHVmin HVmin bull for nitriding t400min HV = 400 bull for case hardening t550min HV = 550 t400min HV = 400
and t300min HV = 300 (the latter only if the core hardness lt 300 If the core hardness gt 400 the t400 is to be re-placed by a fictive t400 = 16 t550)
In addition the specified surface hardness is not to be less than the max necessary hardness (at tz = 05aH) This applies to all hardening methods
For high cycle fatigue (gt3 106 cycles) the following applies
sdot+
minussdotsdotsdot= o90
05azt
05azt
cosSσ04HV
H
HHHR
applicable to 05a
zt
Hge
For 05at
H
z lt the value for 05at
H
z = applies
56300ρSσ12a cHHR
Hsdotsdot
sdot=
Where aH is half the hertzian contact width multiplied by an empirical factor of 12 that takes into account the possible influence of reduced compressive residual stresses (or even tensile residual stresses) on the local fatigue strength
If any of the specified hardness depths including the surface hardness is below the curve described by HV = f (tz) the actual safety factor against subsurface fatigue is determined as follows
reduce SH stepwise in the formula for HV and aH until all specified hardness depths and surface hardness balance with the corrected curve The safety factor obtained through this method is the safety against subsurface fatigue
For static strength (lt103 cycles) the following applies
sdot+
minussdotsdotsdot= o90
07at
06a
zt
cosSσ019HV
Hst
z
HstHHR
applicable to 06at
Hst
z ge
56300ρSσa cHHR
Hstsdotsdot
=
In the case of insufficient specified hardness depths the same procedure for determination of the actual safety factor as above applies
For limited life fatigue (103 lt cycles lt 3106 )
For this purpose it is necessary to extend the correction of safety factors to include also higher values than required Ie in the case of more than sufficient hardness and depths the safety factor in the formulae for both high cycle fatigue and static strength are to be increased until necessary and speci-fied values balance
The actual safety factor for a given number of cycles N be-tween 103 and 3106 is found by linear interpolation in a dou-ble logarithmic diagram
minussdotminus
= sdot logN3477
logSlogSlogS
310H6103HHN
36 10H103H Slog86281Slog86280 sdot+sdot sdot
22 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
3 Calculation of Tooth Strength
31 Scope and General Remarks Part 3 include the calculation of tooth root strength as limited by tooth root cracking (surface or subsurface initiated) and yielding
For rim thickness sR ge 35middotmn the strength is calculated by means of 32 ndash 313 For cylindrical gears the calculation is based on the assumption that the highest tooth root tensile stress arises by application of the force at the outer point of single tooth pair contact of the virtual spur gears The method has however a few limitations that are mentioned in 36
For bevel gears the calculation is based on force application at the tooth tip of the virtual cylindrical gear Subsequently the stress is converted to load application at the mid point of the flank due to the heightwise crowning
Bevel gears may also be calculated with the program BECAL In that case KA and Kv are to be included in the applied tooth force but not KFβ and KFα
In case of a thin annulus or a thin gear rim etc radial crack-ing can occur rather than tangential cracking (from root fillet to root fillet) Cracking can also start from the compression fillet rather than the tension fillet For rim thickness sR lt 35middotmn a special calculation procedure is given in 315 and 316 and a simplified procedure in 314
A tooth breakage is often the end of the life of a gear trans-mission Therefore a high safety SF against breakage is re-quired
It should be noted that this part 3 does not cover fractures caused by
bull oil holes in the tooth root space bull wear steps on the flank bull flank surface distress such as pits spalls or grey staining
Especially the latter is known to cause oblique fractures starting from the active flank predominately in spiral bevel gears but also sometimes in cylindrical gears
Specific calculation methods for these purposes are not given here but are under consideration for future revisions Thus depending on experience with similar gear designs limita-tions other than those outlined in part 3 may be applied
32 Tooth Root Stresses The local tooth root stress is defined as the max principal stress in the tooth root caused by application of the tooth force Ie the stress ratio R = 0 Other stress ratios such as for eg idler gears (R asymp -12) shrunk on gear rims (R gt 0) etc are considered by correcting the permissible stress level
321 Local tooth root stress The local tooth root stress for pinion and wheel may be as-sessed by strain gauge measurements or FE calculations or similar For both measurements and calculations all details are to be agreed in advance
Normally the stresses for pinion and wheel are calculated as
Cylindrical gears
FαFβvγAβSFn
tF K K K K K Y Y Y
m bFσ =
where
YF = Tooth form factor (see 33)
YS = Stress correction factor (see 34)
Yβ = Helix angle factor (see 36)
Ft KA Kγ Kv KFβ KFα see 15 ndash 110
b see 13
Bevel gears
FαFβvγASaFamn
mtF K K K K K Y Y Y
m bFσ ε=
where
YFa = Tooth form factor see 33
YSa = Stress correction factor see 34
Yε = Contact ratio factor see 35
Fmt KA etc see 15 ndash 110
b see 13
322 Permissible tooth root stress The permissible local tooth root stress for pinion respectively wheel for a given number of cycles N is
CXRrelTrelTF
NMFEFP YYYY
SYY
δσ
=σ
Note that all these factors YM etc are applicable to 3middot106 cycles when used in this formula for σFP The influence of other number of cycles on these factors is covered by the calculation of YN
where
σFE = Local tooth root bending endurance limit of reference test gear (see 37)
YM = Mean stress influence factor which accounts for other loads than constant load direction eg idler gears temporary change of load di-rection pre-stress due to shrinkage etc (see 38)
YN = Life factor for tooth root stresses related to reference test gear dimensions (see 39)
SF = Required safety factor according to the rules
YδrelT = Relative notch sensitivity factor of the gear to be determined related to the reference test gear (see 310)
YRrelT = Relative (root fillet) surface condition factor of the gear to be determined related to the reference test gear (see 311)
Classification Notes- No 412 23 May 2003
DET NORSKE VERITAS
YX = Size factor (see 312)
YC = Case depth factor (see 313)
33 Tooth Form Factors YF YFa The tooth form factors YF and YFa take into account the in-fluence of the tooth form on the nominal bending stress
YF applies to load application at the outer point of single tooth pair contact of the virtual spur gear pair and is used for cylindrical gears
YFa applies to load application at the tooth tip and is used for bevel gears
Both YF and YFa are based on the distance between the con-tact points of the 30˚-tangents at the root fillet of the tooth profile for external gears respectively 60˚ tangents for inter-nal gears
Figure 31 External tooth in normal section
Figure 32 Internal tooth in normal section
Definitions
n
2
n
Fn
enFn
Fe
F
α cosms
α cosmh6
Y
=
n
2
n
Fn
anFn
Fa
Fa
α cosms
α cosmh6
Y
=
In the case of helical gears YF and YFa are determined in the normal section ie for a virtual number of teeth
YFa differs from YF by the bending moment arm hFa and αFan and can be determined by the same procedure as YF with exception of hFe and αFan For hFa and αFan all indices e will change to a (tip)
The following formulae apply to cylindrical gears but may also be used for bevel gears when replacing
mn with mnm
zn with zvn
αt with αvt
β with βm
with undercut without undercut
Fig 33 Dimensions and basic rack profile of the teeth (finished profile)
Tool and basic rack data such as hfP ρfp and spr etc are re-ferred to mn ie dimensionless
331 Determination of parameters
( )n
n
prnfPnfP m
α cossαsin 1ρ
αtan h4πE
minusminusminusminus=
For external gears fPfP ρρ =
For internal gears ( )0z
195fPfP0
fPfP 10363156ρhxρρ
sdotminus+
+=
where
z0 = number of teeth of pinion cutter
x0 = addendum modification coefficient of pinion cutter
hfP = addendum of pinion cutter
ρfP = tip radius of pinion cutter
xhρG fpfp +minus=
24 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
τmE
2π
z2H
nnminus
minus=
with
3πτ = for external gears
6πτ = for internal gears
Htan zG2
nminusϑ=ϑ
(to be solved iteratively suitable start value6π
=ϑ for exter-
nal gears and 3π for internal gears)
a) Tooth root chord sFn For external gears
minus
ϑ+
ϑminus= ρ
cosG3
3πsinz
ms
fpnn
Fn
For bevel gears with a tooth thickness modification
xsm affects mainly sFn but also hFe and αFen The total influence of xsm on YFa Ysa can be approximated by only adding 2 xsm to sFn mn
For internal gears
minus
ϑ+
ϑminus= ρ
cosG
6πsinz
ms
fPnn
Fn
b) Root fillet radius ρF at 30ordm tangent
( )G2coszcosG2ρ
mρ
2n
2
fpn
F
minusϑϑ+=
c) Determination of bending moment arm hF dn = zn mn
b2α
αn βcosε
ε =
dan = dn + 2 ha
pbn = π mn cos αn
dbn = dn cos αn
( )4
d1εpzz
2dd
zz2d
2bn
2
αnbn
2bn
2an
en +
minusminus
minus=
en
bnen d
dcos arcα =
ennnn
e α invα invα x tan 22π
z1
minus+
+=γ
αFen = αen ndash γe
For external gears
( )
minus=
n
enFenee
n
Fe
mdαtan sin γ γcos
21
mh
]ρcos
G3πcosz fpn +
ϑminus
ϑminusminus
For internal gears
( ) minus
sdotsdotminus=
n
enFenee
n
Fe
mdα tanγ sinγ cos
21
mh
minus
ϑminus
ϑminussdot ρ
cosG3
6πcosz fPn
332 Gearing with εαn gt 2 For deep tooth form gearing ( )25ε2 αn lele produced with a verified grade of accuracy of 4 or better and with applied profile modification to obtain a trapezoidal load distribution along the path of contact the YF may be corrected by the factor YDT as
250ε205for 0666ε2366Y αnαnDT leleminus=
205εfor 10Y αnDT lt=
34 Stress Correction Factors YS YSa The stress correction factors YS and YSa take into account the conversion of the nominal bending stress to the local tooth root stress Thereby YS and YSa cover the stress increasing effect of the notch (fillet) and the fact that not only bending stresses arise at the root A part of the local stress is inde-pendent of the bending moment arm This part increases the more the decisive point of load application approaches the critical tooth root section
Therefore in addition to its dependence on the notch radius the stress correction is also dependent on the position of the load application ie the size of the bending moment arm
YS applies to the load application at the outer point of single tooth pair contact YSa to the load application at tooth tip
YS can be determined as follows
( )
+
+=L23121
1
sS qL01312Y
where Fe
Fn
hsL = and
F
Fns ρ2
sq = (see 33)
YSa can be calculated by replacing hFe with hFa in the above formulae
Note a) Range of validity 1 lt qs lt 8
In case of sharper root radii (ie produced with tools having too sharp tip radii) YS resp YSa must be specially considered
b) In case of grinding notches (due to insufficient protuberance of the hob) YS resp YSa can rise considerably and must be multiplied with
Classification Notes- No 412 25 May 2003
DET NORSKE VERITAS
g
g
ρt
0613
13
minus
where
tg = depth of the grinding notch
ρg = radius of the grinding notch
c) The formulae for YS resp YSa are only valid for αn = 20˚ However the same formulae can be used as a safe approximation for other pressure angles
35 Contact Ratio Factor Yε The contact ratio factor Yε covers the conversion from load application at the tooth tip to the load application at the mid point of the flank (heightwise) for bevel gears
The following may be used
Yε = 0625
36 Helix Angle Factor Yβ The helix angle factor Yβ takes into account the difference between the helical gear and the virtual spur gear in the nor-mal section on which the calculation is based in the first step In this way it is accounted for that the conditions for tooth root stresses are more favourable because the lines of contact are sloping over the flank
The following may be used (β input in degrees)
Yβ = 1 ndash εβ β120
When εβ gt 1 use εβ = 1 and when β gt 30deg use β = 30deg in the formula
However the above equation for Yβ may only be used for gears with β gt 25deg if adequate tip relief is applied to both pinion and wheel (adequate = at least 05 middot Ceff see 432)
37 Values of Endurance Limit σFE σFE is the local tooth root stress (max principal) which the material can endure permanently with 99 survival prob-ability 3106 load cycles is regarded as the beginning of the endurance limit or the lower knee of the σ ndash N curve σFE is defined as the unidirectional pulsating stress with a minimum stress of zero (disregarding residual stresses due to heat treatment) Other stress conditions such as alternating or pre-stressed etc are covered by the conversion factor YM
σFE can be found by pulsating tests or gear running tests for any material in any condition If the approval of the gear is to be based on the results of such tests all details on the testing conditions have to be approved by the Society Fur-ther the tests may have to be made under the Societys su-pervision
If no fatigue tests are available the following listed values for σFE may be used for materials subjected to a quality con-trol as the one referred to in the rules
σFE
Alloyed case hardened steels 1) (fillet surface hardness 58 ndash 63 HRC)
bull of specially approved high grade
1050
bull of normal grade
minus CrNiMo steels with approved process
1000
minus CrNi and CrNiMo steels generally 920 minus MnCr steels generally 850
Nitriding steel of approved grade quenched tempered and gas nitrided (surface hardness 700 ndash 800 HV)
840
Alloyed quenched and tempered steel bath or gas nitrided (surface hardness 500 ndash 700 HV)
720
Alloyed quenched and tempered steel flame or induction hardened 2) (incl entire root fillet) (fillet surface hardness 500 ndash 650 HV)
07 HV + 300
Alloyed quenched and tempered steel flame or induction hardened (excl entire root fillet) (σB = uts of base material)
025 σB + 125
Alloyed quenched and tempered steel 04 σB + 200
Carbon steel 025 σB + 250
Note All numbers given above are valid for separate forgings and for blanks cut from bars forged according to a qualified procedure see Pt 4 Ch 2 Sec 3 For rolled steel the values are to be reduced with 10 For blanks cut from forged bars that are not qualified as mentioned above the values are to be reduced with 20 For cast steel reduce with 40
1) These values are valid for a root radius
bull being unground If however any grinding is made in the root fillet area in such a way that the residual stresses may be affected σFE is to be reduced by 20 (If the grinding also leaves a notch see 34)
bull with fillet surface hardness 58 ndash 63 HRC In case of lower surface hardness than 58 HRC σFE is to be reduced with 20(58 ndash HRC) where HRC is the detected hardness (This may lead to a permissible tooth root stress that varies along the facewidth If so the actual tooth root stresses may also be considered along facewidth)
bull not being shot peened In case of approved shot peening σFE may be increased by 200 for gears where σFE is reduced by 20 due to root grinding Otherwise σFE may be increased by 100 for
6nm le and 100 ndash 5 (mn - 6) for mn gt 6 However the possible adverse influence on the flanks regarding grey staining should be considered and if necessary the flanks should be masked
2) The fillet is not to be ground after surface hardening Regarding possible root grinding see 1)
26 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
38 Mean stress influence Factor YM The mean stress influence factor YM takes into account the influence of other working stress conditions than pure pulsa-tions (R = 0) such as eg load reversals idler gears planets and shrink-fitted gears
YM (YMst) are defined as the ratio between the endurance (or static) strength with a stress ratio R ne 0 and the endurance (or static) strength with R = 0
YM and YMst apply only to a calculation method that assesses the positive (tensile) stresses and is therefore suitable for comparison between the calculated (positive) working stress σF and the permissible stress σFP calculated with YM or YMst
For thin rings (annulus) in epicyclic gears where the com-pression fillet may be decisive special considerations apply see 316
The following method may be used within a stress ratio ndash12 lt R lt 05
381 For idlers planets and PTO with ice class
M1M1R1
1Yor Y MstM
+minus
minus=
where
R = stress ratio = min stress divided by max stress
For designs with the same force applied on both forward- and back-flank R may be assumed to ndash 12
For designs with considerably different forces on forward- and back-flank such as eg a marine propulsion wheel with a power take off pinion R may be assessed as
branchmain theoffacewidth unit per forcepto offacewidt unit per force21minus
For a power take off (PTO) with ice class see 161 c
M considers the mean stress influence on the endurance (or static) strength amplitudes
M is defined as the reduction of the endurance strength am-plitude for a certain increase of the mean stress divided by that increase of the mean stress
Following M values may be used
Endurance limit
Static strength
Case hardened 08 ndash 015 Ys 1) 07
If shot peened 04 06
Nitrided 03 03
Induction or flame hardened
04 06
Not surface hardened steel 03 05
Cast steels 04 06 1)For bevel gears use Ys=2 for determination of M
The listed M values for the endurance limit are independent of the fillet shape (Ys) except for case hardening In princi-ple there is a dependency but wide variations usually only occur for case hardening eg smooth semicircular fillets versus grinding notches
382 For gears with periodical change of rotational direction For case hardened gears with full load applied periodically in both directions such as side thrusters the same formula for YM as for idlers (with R = ndash 12) may be used together with the M values for endurance limit This simplified approach is valid when the number of changes of direction exceeds 100 and the total number of load cycles exceeds 3middot106
For gears of other materials YM will normally be higher than for a pure idler provided the number of changes of direction is below 3middot106 A linear interpolation in a diagram with loga-rithmic number of changes of direction may be used ie from YM = 09 with one change to YM (idler) for 3middot106 changes This is applicable to YM for endurance limit For static strength use YM as for idlers
For gears with occasional full load in reversed direction such as the main wheel in a reversing gear box YM = 09 may be used
383 For gears with shrinkage stresses and unidirectional load For endurance strength
FE
fitM σ
σM1
M21Y+
minus=
σFE is the endurance limit for R = 0
For static strength YMst = 1 and σfit accounted for in 39b
σfit is the shrinkage stress in the fillet (30˚ tangent) and may be found by multiplying the nominal tangential (hoop) stress with a stress concentration factor
n
Ffit m
ρ215scf minus=
384 For shrink-fitted idlers and planets When combined conditions apply such as idlers with shrink-age stresses the design factor for endurance strength is
( ) ( ) FE
fitM σ
σR1M1
M2
M1M1R1
1Yminussdot+
minus
+minus
minus=
Symbols as above but note that the stress ratio R in this par-ticular connection should disregard the influence of σfit ie R normally equal ndash 12
For static strength
M1M1R1
1YMst
+minus
minus=
The effect of σfit is accounted for in 39 b
Classification Notes- No 412 27 May 2003
DET NORSKE VERITAS
385 Additional requirements for peak loads The total stress range (σmax ndash σmin) in a tooth root fillet is not to exceed
F
y
Sσ225
for not surface hardened fillets
FSHV5 for surface hardened fillets
39 Life Factor YN The life factor YN takes into account that in the case of limited life (number of cycles) a higher tooth root stress can be permitted and that lower stresses may apply for very high number of cycles
Decisive for the strength at limited life is the σ ndash N ndash curve of the respective material for given hardening module fillet radius roughness in the tooth root etc Ie the factors YδrelT YRelT YX and YM have an influence on YN
If no σ ndash N ndash curve for the actual material and hardening etc is available the following method may be used
Determination of the σ ndash N ndash curve
a) Calculate the permissible stress σFP for the beginning of the endurance limit (3middot106 cycles) including the influ-ence of all relevant factors as SF YδrelT YRelT YX YM and YC ie σFP = σFE middotYM middotYδrelT middotYRelT middotYX middotYC SF
b) Calculate the permissible laquostaticraquo stress (le 103 load cy-cles) including the influence of all relevant factors as SFst YδrelTst YMst and YCst ( )fitσCstYδrelTstYMstYFstσ
FstS1
FPstσ minussdotsdotsdot=
where σFst is the local tooth root stress which the material can resist without cracking (surface hardened materials) or unacceptable deformation (not surface hardened materials) with 99 survival probability
σFst
Alloyed case hardened steel 1) 2300
Nitriding steel quenched tempered and gas nitrided (surface hardness 700 ndash 800 HV)
1250
Alloyed quenched and tempered steel bath or gas nitrided (surface hardness 500 ndash 700 HV)
1050
Alloyed quenched and tempered steel flame or induction hardened (fillet surface hardness 500 ndash 650 HV)
18 HV + 800
Steel with not surface hardened fillets the smaller value of 2)
18 σB or 225 σy
1) This is valid for a fillet surface hardness of 58 ndash 63 HRC In case of lower fillet surface hardness than 58 HRC σFst is to be reduced with 30(58 ndash HRC) where HRC is the actual hardness Shot peening or grinding notches are not considered to have any significant influence on σFst 2) Actual stresses exceeding the yield point (σy or σ02) will alter the residual stresses locally in the ldquotensionrdquo fillet re-spectively ldquocompressionrdquo fillet This is only to be utilised for gears that are not later loaded with a high number of cycles at lower loads that could cause fatigue in the ldquocompressionrdquo fillet
c) Calculate YN as
NL gt 3middot106
YN = 1 or 001
L
6
N N103Y
sdot= ie Yn = 092 for 1010
The YN = 1 from 3middot106 on may only be used when special material cleanness applies see rules Pt4 Ch2
103ltNLlt3middot106exp
L
6
N N103Y
sdot=
cycles6103forσcycles310forσlog02876exp
FP
FPst
sdot=
NL lt 103 cycles6103forσcycles310forσ
NYFP
FPst
sdot=
or simply use σFPst as mentioned in b) directly
Guidance on number of load cycles NL for various applica-tions
bull For propulsion purpose normally NL = 1010 at full load (yachts etc may have lower values)
bull For auxiliary gears driving generators that normally operate with 70-90 of rated power NL = 108 with rated power may be applied
28 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
310 Relative Notch Sensitivity Factor YδrelT The dynamic (respectively static) relative notch sensitivity factor YδrelT (YδrelTst) indicate to which extent the theoreti-cally concentrated stress lies above the endurance limits (re-spectively static strengths) in the case of fatigue (respectively overload) breakage
YδrelT is a function of the material and the relative stress gra-dient It differs for static strength and endurance limit
The following method may be used
For endurance limit
for not surface hardened fillets
( )
024
s024
relT σ103133q21σ1012201351
Ysdotsdotminus
+sdotsdotminus+= minus
minus
δ
for all surface hardened fillets except nitrided
( )
106q21002451
Y srelT
++=δ
for nitrided fillets
( )
1347q2101421
Y srelT
++=δ
For static strength
for not surface hardened fillets1)
( )( )( )025
02
02502s
relTst σ3000821σ300 1Y0821Y
+
minus+=δ
for surface hardened fillets except nitrided
YδrelTst = 044 YS + 012
for nitrided fillets
YδrelTst = 06 + 02 YS 1) These values are only valid if the local stresses do not
exceed the yield point and thereby alter the residual stress level See also 39b footnote 2
311 Relative Surface Condition Factor YRrelT The relative surface condition factor YRrelT takes into ac-count the dependence of the tooth root strength on the sur-face condition in the tooth root fillet mainly the dependence on the peak to valley surface roughness
YRrelT differs for endurance limit and static strength
The following method may be used
For endurance limit
YRrelT = 1675 ndash 053 (Ry + 1)01 for surface hardened steels and alloyed quenched and tempered steels except nitrided
YRrelT = 53 ndash 42 (Ry + 1)001 for carbon steels
YδrelT = 43 ndash 326 (Ry + 1)0005
for nitrided steels
For static strength
YRrelTst = 1 for all Ry and all materials
For a fillet without any longitudinal machining trace Ry asymp Rz
312 Size Factor YX The size factor YX takes into account the decrease of the strength with increasing size YX differs for endurance limit and static strength
The following may be used
For endurance limit
YX = 1 for mn le 5 generally
YX = 103 ndash 0006 mn for 5 lt mn lt 30
YX = 085 for mn ge 30
for not surface hard-ened steels
YX = 105 ndash 001 mn for 5 lt mn ge 25
YX = 08 for mn ge 25
for surface hardened steels
For static strength
YXst = 1 for all mn and all materials
313 Case Depth Factor YC The case depth factor YC takes into account the influence of hardening depth on tooth root strength
YC applies only to surface hardened tooth roots and is dif-ferent for endurance limit and static strength
In case of insufficient hardening depth fatigue cracks can develop in the transition zone between the hardened layer and the core For static strength yielding shall not occur in the transition zone as this would alter the surface residual stresses and therewith also the fatigue strength
The major parameters are case depth stress gradient permis-sible surface respectively subsurface stresses and subsurface residual stresses
The following simplified method for YC may be used
YC consists of a ratio between permissible subsurface stress (incl influence of expected residual stresses) and permissible surface stress This ratio is multiplied with a bracket con-taining the influence of case depth and stress gradient (The empirical numbers in the bracket are based on a high number of teeth and are somewhat on the safe side for low number of teeth)
YC and YCst may be calculated as given below but calculated values above 10 are to be put equal 10
For endurance limit
+
+=nFFE
C m02ρt31
σconstY
Classification Notes- No 412 29 May 2003
DET NORSKE VERITAS
For static strength
+
+=nFFst
Cst m02ρt31
σconstY
where const and t are connected as
Hardening process
t = endurance limit const =
static strength const =
t550 640 1900
t400 500 1200 Case hard-ening
t300 380 800
Nitriding t400 500 1200
Induction- or flame hardening
tHVmin 11 HVmin 25 HVmin
For symbols see 213
In addition to these requirements to minimum case depths for endurance limit some upper limitations apply to case hard-ened gears
The max depth to 550 HV should not exceed
1) 13 of the top land thickness san unless adequate tip relief is applied (see 110)
2) 025 mn If this is exceeded the following applies addi-tionally in connection with endurance limit
minusminus= 025
mt
1Yn
max550C
314 Thin rim factor YB Where the rim thickness is not sufficient to provide full sup-port for the tooth root the location of a bending failure may be through the gear rim rather than from root fillet to root fillet
YB is not a factor used to convert calculated root stresses at the 30deg tangent to actual stresses in a thin rim tension fillet Actually the compression fillet can be more susceptible to fatigue
YB is a simplified empirical factor used to de-rate thin rim gears (external as well as internal) when no detailed calcula-tion of stresses in both tension and compression fillets are available
Figure 34 Examples on thin rims
YB is applicable in the range 175 lt sRmn lt 35
YB = 115 middot ln (8324 middot mnsR)
(for sRmn ge 35 YB = 1)
(for sRmn le 175 use 315)
σF as calculated in 321 is to be multiplied with YB when sRmn lt 35 Thus YB is used for both high and low cycle fatigue
Note This method is considered to be on the safe side for external gear rims However for internal gear rims without any flange or web stiffeners the method may not be on the safe side and it is advised to check with the method in 315316
315 Stresses in Thin Rims For rim thickness sR lt 35 mn the safety against rim cracking has to be checked
The following method may be used
3151 General The stresses in the standardised 30ordm tangent section tension side are slightly reduced due to decreasing stress correction factor with decreasing relative rim thickness sRmn On the other hand during the complete stress cycle of that fillet a certain amount of compression stresses are also introduced The complete stress range remains approximately constant Therefore the standardised calculation of stresses at the 30ordm tangent may be retained for thin rims as one of the necessary criteria
The maximum stress range for thin rims usually occurs at the 60ordm ndash 80ordm tangents both for laquotensionraquo and laquocompressionraquo side The following method assumes the 75ordm tangent to be the decisive Therefore in addition to the am criterion ap-plied at the 30ordm tangent it is necessary to evaluate the max and min stresses at the 75ordm tangent for both laquotensionraquo (loaded flank) fillet and laquocompressionraquo (back-flank) fillet For this purpose the whole stress cycle of each fillet should be considered but usually the following simplification is justified
Figure 35 Nomenclature of fillets
Index laquoTraquo means laquotensileraquo fillet laquoCraquo means laquocompressionraquo fillet
σFTmin and σFCmax are determined on basis of the nominal rim stresses times the stress concentration factor Y75
σFCmin and σFTmax are determined on basis of superposition of nominal rim stresses times Y75 plus the tooth bending stresses at 75ordm tangent
30 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
3152 Stress concentration factors at the 75ordm tangents The nominal rim stress consists of bending stresses due to local bending moments tangential stresses due to the tan-gential force Ft and radial shear stresses due to Fr
The major influence is given by the bending stresses The influence of the tangential stresses is minor and even though its stress concentration factor is slightly higher than for bending it is considered to be safe enough when the sum of these nominal stresses are combined with the stress concen-tration factor for bending The influence of the radial shear stress is neglected
The stress concentration factor relating nominal rim stresses to local fillet stresses at the 75ordm tangent may be calculated as
75n
R75
n
R
75
ρmsρ185
ms3
Y+
sdot=
where ρ75 is the root radius at the 75ordm tangent ref to mn Usually ρ75 is closed to the tool radius ρao and
ρ75 = ρao
is a safe approximation compensating for the am simplifi-cations to the laquounsaferaquo side
The tooth root stresses of the loaded tooth are decreasing with decreasing relative rim thickness approximately with the empirical factor
15
R
ncorr s
m311Y
minus=
3153 Nominal rim stresses The bending moment applied to the rim consists of a part of the tooth tilt moment Ft (hF + 05 sR) and the bending caused by the radial force Fr
The sectional modulus (first moment of area) which is used for determination of the nominal bending stresses is not nec-essarily the same for the 2 am bending moments If flanges webs etc outside the toothed section contribute to stiffening the rim against various deflections the influence of these stiffeners should be considered Eg an end flange will have an almost negligible influence on the effective sectional modulus for the stresses due to tooth tilt as the deflection caused by the tilt moment is rather small and would not much involve the flange On the other hand the radial forces as for instance from the meshes in an annulus would cause considerable radial deflections that the flange might restrict to a substantial amount When considering the stiff-ening of such flanges or webs on basis of simplified models it is advised to use an effective rim thickness sR = sR + 02 mn for the first moment of area of the rim (toothed part) cross section
For a high number of rim teeth it may be assumed that the rim bending moments in the fillets adjacent to the loaded tooth are of the same magnitude as right under the applied
force This assumption is reasonable for an annulus but rather much on the safe side for a hollow pinion
The influence of Ft on the nominal tangential stress is simpli-fied by half of it for compressive stresses (σ1) and the other half for tensile stresses (σ2) Applying these assumptions the nominal rim stresses adjacent to the loaded tooth are
( ) ( )A2
FW
f RFW
s05hF05 t
R
r
T
RFt1 minus
ϑminus
+minus=σ
( ) ( )A2
FW
f RFW
s05hF05 t
R
r
T
RFt2 +
ϑminus
+=σ
where σ1 σ2 see Fig 35
A = minimum area of cross section (usually bR sR)
WT = the sectional modulus of rim with respect to tooth tilt moment (usually 6sb 2
rR )
WR = the sectional modulus of the rim including the influence of stiffeners as flanges webs etc (WR ge WT)
bR = the width of the rim
R = the radius of the neutral axis in the rim ie from wheel centre to midpoint of rim
( )ϑf = a function for bending moment distribution around the rim For a rim (pinion) with one mesh only the ( )ϑf at the position of load application is 024 For an annulus with 3 or more meshes ( )ϑf at the position of each load application is approx 3 planets ( )ϑf = 019 4 planets ( )ϑf = 014 5 planets ( )ϑf = 011 6 planets ( )ϑf = 009
It must be checked if the max (tensile) stress in the compres-sion fillet really occurs when the fillet is adjacent to the loaded tooth In principle the stress variation through a complete rotation should be considered and the max value used The max value is usually never less than 0 For an annulus eg the tilt moment is zero in the mid position be-tween the planet meshes whilst the bending moment due to the radial forces is half of that at the mesh but with opposite sign
If these formulae are applied to idler gears as eg planets the influence of nominal tangential stresses must be cor-rected by deleting Ft(2 A) for σ1 and using FtA for σ2 Further the influence of Fr on the nominal bending stresses is usually negligible due to the planet bearing support
3154 Root fillet stresses Determination of min and max stresses in the laquotensionraquo fillet
Classification Notes- No 412 31 May 2003
DET NORSKE VERITAS
Minimum stress
751FTmin YσK σ =
Maximum stress
corrF752 Yσ03Yσ KσFTmax +=
where
03 is an empirical factor relating the tension stresses (σF) at the 30ordm tangent to the part of the tension stresses at the 75ordm tangent which add to the rim related stresses (03 also takes into account that full superposition of nominal stresses times stress concentration factors from both laquosides of the corner filletraquo would result in too high stresses)
αβγ sdotsdotsdotsdot= FFvA KKKKKK
Determination of min and max stresses in the laquocompres-sionraquo fillet
Minimum stress
corrF751FCmin Yσ036Yσ Kσ minus=
Maximum stress
752FCmax Yσ Kσ =
where
036 is an empirical factor relating the tension stresses (σF) at the 30ordm tangent to the part of the compression stresses at the 75ordm tangent which add to the rim related stresses
For gears with reversed loads as idler gears and planets there is no distinct laquotensionraquo or laquocompressionraquo fillet The mini-mum stress σFmin is the minimum of σFTmin and σFCmin (usually the latter is decisive) The maximum stress σFTmax is the maximum of σFTmax and σFCmax (usually the former is deci-sive)
316 Permissible Stresses in Thin Rims
3161 General The safety against fatigue fracture respectively overload fracture is to be at least at the same level as for solid gears The laquoordinaryraquo criteria at the 30ordm tangent apply as given in 31 through 313
Additionally the following criteria at the 75ordm tangent may apply
3162 For gt3middot106 cycles The permissible stresses for the laquotensionraquo fillets and for the laquocompressionraquo fillets are determined by means of a relevant fatigue diagram
If the actual tooth root stress (tensile or compressive) ex-ceeds the yield strength to the material the induced residual stresses are to be taken into account
For determination of permissible stresses the following is defined
R = stress ratio ie FTmax
FTminσσ respectively
FCmax
FCmin
σσ
∆σ = stress range ie σFTmax ndash σFTmin resp σFCmax ndash σFCmin
(For idler gears and planets Fmax
FminσσR =
and ∆σ = σFmax minus σFmin)
The permissible stress range ∆σρ for the laquotensionraquo respec-tively laquocompressionraquo fillets can be calculated as
For R gt minus1 FPp σ
R1R1031
13∆σ
minus+
+=
For minus infin lt R lt minus1 FPp σ
R1R10151
13∆σ
minus+
+=
where
σFP = see 32 determined for unidirectional stresses (YM = 1)
If the yield strength σy is exceeded in either tension or com-pression residual stresses are induced This may be consid-ered by correcting the stress ratio R for the respective fillets (tension or compression)
Eg if yFCmin σσ gt (ie exceeded in compression) the
difference yFCmin σσ∆ minus= affects the stress ratio as
∆σ
σ∆σ∆σR
FCmax
y
FCmax
FCmin
+
minus=
++
=
Similarly the stress ratio in the tension fillet may require cor-rection
If the yield strength is exceeded in tension σFTmax gt σy the difference ∆ = σFTmax ndash σy affects the stress ratio as
y
FTmin
FTmax
FTmin
σ∆σ
∆σ∆σR minus=
minusminus
=
Checking for possible exceeding of the yield strength has to be made with the highest torque and with the KAP if this ex-ceeds KA
3163 Forle 103 cycles The permissible stress range ∆σp is not to exceed
For R gt minus1 FPstpst σ
R1R1051
15∆σ
minus+
+=
For ndash infin lt R lt minus1 FPstpst σ
R1R10251
15∆σ
minus+
+=
For all values of R ∆σpst is limited by
not surface hardened F
y
Sσ225
32 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
surface hardened CF
YSHV5
Definition of ∆σ and R see 3162 with particular attention to possible correction of R if the yield strength is exceeded
σFPst see 32 determined for unidirectional stresses (YM = 1) and lt 103 cycles
3164 For 103 lt cycles lt 3106 ∆σp is to be determined by linear interpolation a log-log dia-gram
∆σp at NL load cycles is
6L 103p
exp
L
6
N p ∆σN103∆σ sdot
sdot=
6
3
103p
10p
∆σ
∆σlog28760exp
sdot
=
Classification Notes- No 412 33 May 2003
DET NORSKE VERITAS
4 Calculation of Scuffing Load Capacity
41 Introduction High surface temperatures due to high loads and sliding ve-locities can cause lubricant films to break down This sei-zure or welding together of areas of tooth surface is termed scuffing
In contrast to pitting and fatigue breakage which show a dis-tinct incubation period a single short overloading can lead to a scuffing failure In the ISO-TR13989 two criteria are mentioned The method used in this Classification Note is based on the principles of the flash temperature criterion
Note Bulk temperature in excess of 120ordmC for long periods may have an adverse effect on the surface durability see 211
42 General Criteria In no point along the path of contact the local contact tem-perature may exceed the permissible temperature ie
oilS
oilSB S
ϑ+ϑminusϑ
leϑ
50SB minusϑleϑ
where
Bϑ = max contact temperature along the path of contact
maxflaMBB ϑ+ϑ=ϑ
MBϑ = bulk temperature see 434
maxflaϑ = max flash temperature along the path of con-tact see 44
Sϑ = scuffing temperature see below
oilϑ = oil temperature before it reaches the mesh (max applicable for the actual load case to be used ie normally alarm temperature except for ice classes where a max expected tem-perature applies)
SS = required safety factor according to the Rules
The scuffing temperature Sϑ may be calculated as
L2
wrelT
002
40S XFZGX
ν100112085780
sdot
sdot++=ϑ
where
XwrelT = relative welding factor
XwrelT
Through hardened steel 10
Phosphated steel 125
Copper-plated steel 150
Nitrided steel 150
Less than 10 retained austenite
115
10 ndash 20 retained austenite 10
Casehardened steel
20 ndash 30 retained austenite 085
Austenitic steel 045
FZG = load stage according to FZG-Test A8390 (Note This is the load stage where scuffing oc-curs However due to scatter in test results cal-culations are to be made with one load stage less than the specification)
XL = lubricant factor
= 10 for mineral oils
= 08 for polyalfaolefins
= 07 for non-water-soluble polyglycols
= 06 for water-soluble polyglycols
= 15 for traction fluids
= 13 for phosphate esters
ν40 = kinematic oil viscosity at 40˚C (mm2s)
Application of other test methods such as the Ryder the FZG-Ryder R46574 and the FZG L-42 Test 141195110 may be specially considered
For high speed gears with very short time of contact Sϑ may be increased as follows provided use of EP-oils
Addition to the calculated scuffing temperature Sϑ
If micros 18tc ge no addition
If micros 18tc lt add ( )cwrelT t18X18 minussdotsdot
where
ct = contact time ( micros ) which is the time needed to cross the Hertzian contact width
( ) [ ]microsu1cosβn
uσ340tb1
Hc +sdotsdot
sdot=
σH as calculated in 221
For bevel gears use vu in stead of u
34 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
43 Influence Factors
431 Coefficient of friction The following coefficient of friction may apply
L025a
005oil
02
redC ΣC
Bt X R ηρv
w0048micro minus
=
where
wBt = specific tooth load (Nmm)
vΣC = sum of tangential velocities at pitch point At pitch line velocities gt 50 ms the limiting value of vΣC at v = 50 ms is to be used
ρredC = relative radius of curvature (transversal plane) at the pitch point
Cylindrical gears
HαHβvγAbt
Bt KKKKKb
Fw sdotsdotsdotsdotsdot= (see 1)
twΣC αsin v2v =
bCredC β cos ρρ =
Bevel gears
HαHβvγAtmb
Bt KKKKKb
Fw sdotsdotsdotsdotsdot= (see 1)
vtmtΣC αsin v2v =
bmvCredC β cos ρρ =
ηoil = dynamic viscosity (mPa s) at oilϑ calculated as
1000
ρ νη oiloil =
where ρ in kgm3 approximated as
( ) 70 15oil15 minusϑminusρ=ρ and νoil is kinematic viscosity at oilϑ and may be calculated by means of the following equation
( ) ( )++=+ 08νloglog08νloglog 100oil ( )
sdotminus
ϑ+minus313log373log
273log373log oil
( ) ( )( )08νloglog08νloglog 10040 +minus+
Ra = composite arithmetic mean roughness (micron) of pinion an wheel calculated as ( )2a1aa RR 05R +=This is defined as the roughness on the new flanks ie as manufactured
XL = see 42
432 Effective tip relief Ceff Ceff is the effective tip relief that amount of tip relief which compensates for the elastic deformation of the gear mesh ie
zero load at the tooth tip It is assumed (simplified) to be equal for both pinion and wheel
Cylindrical gears
for helical
γ
γ=cb
KKFC Abt
eff (see 1)
For spur
cbKKF
C Abteff
γ= (see 1)
Alternatively for spur and helical gears the non-linear ap-proach in 111 may be used (taking Ceff=δ)
Bevel gears
bcKFC Ambt
effγ
= (see 1)
where
γ
α
ε2ε44c
+sdot
=γ
433 Tip relief and extension Cylindrical gears
The extension of the tip relief is not to result in an effective contact ratio 1ltεα when the gear is unloaded (exceptions to this may only apply for applications where the gear is not to run at light loads) This means that the unrelieved part of the path of contact is to be minimum pbt It is further assumed that this unrelieved part is placed centrally on the path of contact
If root relief applies it has to be calculated as equivalent tip relief Ie pinion root relief (at mesh position A) is added to Ca2 and wheel root relief (at mesh position E) is added to Ca1 If no design tip relief or root relief on the mating gear is specified (ie if Ca1+Croot2 = 0 and visa versa) use the run-ning in amount see 112
Bevel gears
Bevel gears are to have heightwise crowning ie no distinct relievedunrelieved area This may be treated as tip and root relief For calculation purposes the root relief is combined with the tip relief of the mating member into an equivalent tip relief If no resulting tip relieves are specified the equivalent tip relives may be calculated as an approxima-tion based on the tool crowning Ca tool (per mille of tool module m0) as follows
2root1aeq1a CCC +=
1root2aeq2a CCC +=
where
Classification Notes- No 412 35 May 2003
DET NORSKE VERITAS
2
0
n0101atoolal m
)m2(mA1middotmCC
minus+minussdot=
( )
sdotminusminussdotsdotminus+= vtvb1
2vb1
21vavt1n1 αtan dddαsin 05x1mA
2
0
n020atool22a m
)mm(2A1middotmCC
minus+minussdot=
( )
sdotminusminussdotsdotminus+= vtvb2
2vb2
2va2vt2n2 αtan dddαsin 05x1mA
2
0
20atool1root1 m
A1mCC
minussdotsdot=
2
0
10atool2root2 m
A1mCC
minussdotsdot=
If the pressure angles of the cutter blades are modified (and verified) to balance the tip relieves the following may be assumed
( )eq2aeq1aeq2aeq1a C andC calculated of sum50CC sdot==
Throughout the following Ca1 and Ca2 mean the equivalent tip relieves Ca1eq and Ca2eq
434 Bulk temperature The bulk temperature may be calculated as
flaaveragempsoilMB X X 50 ϑ+ϑ=ϑ
where
Xs = lubrication factor
= 12 for spray lubrication
= 10 for dip lubrication (provided both pinion and wheel are dip lubricated and tip speed lt 5 ms)
= 10 for spray lubrication with additional cooling spray (spray on both pinion and wheel or spray on pinion and dip lubrication of wheel)
= 02 for meshes fully submerged in oil
Xmp = contact factor ( )pmp nX += 150
np = number of mesh contact on the pinion (for small gear ratii the number of wheel meshes should be used if higher)
flaaverageϑ
= average of the integrated flash temperature (see 44) along the path of contact
( )
AE
E
Ayyyfla
flaaverage
d
ΓminusΓ
ΓΓϑ=ϑint =
For high speed gears (v gt 50 ms) it may be necessary to assess the bulk temperature on the basis of a thermal rating of the entire gear transmission
44 The Flash Temperature flaϑ
441 Basic formula The local flash temperature flaϑ may be calculated as
( ) 41redy
y2ly
211
43Btcorrfla
unXwX3250
y ρ
ρminusρ
micro=ϑ Γ
(For bevel gears replace u with uv)
and is to be calculated stepwise along the path of contact from A to E
where
micro = coefficient of friction see 431
Xcorr = correction factor taking empirically into account the increased scuffing risk in the beginning of the approach path due to mesh starting without any previously built up oil film and possible shuffling away oil before meshing if insufficient tip relief
( )
3
AD
yaeffcorr 50
CC1X
ΓminusΓε
Γminus+=
α
Ca = tip relief of driven member Xcorr is only applicable in the approach path and if Ca lt Ceff otherwise 10
wBt = unit load see 431
yXΓ = load sharing factor see 443
n1 = pinion rpm
Γy ρly etc see 442
442 Geometrical relations The various radii of flank curvature (transversal plane) are
ρ1y = pinion flank radius at mesh point y
ρ2y = wheel flank radius at mesh point y
ρredy = equivalent radius of curvature at mesh point y
y2y1
y2y1redy
ρ+ρ
ρρ=ρ
Cylindrical gears
twy
1y αsin au1Γ1
ρ+
+=
twy
2y αsin au1Γu
ρ+
minus=
Note that for internal gears a and u are negative
36 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
Bevel gears
vtvv
y1y αsin a
u1Γ1
ρ+
+=
vtvv
yv2y αsin a
u1Γu
ρ+
minus=
Γ is the parameter on the path of contact and y is any point between A end E
At the respective ends Γ has the following values
Root piniontip wheel
Cylindrical gears
( )
minus
minusminus= 1
αtan 1dd
zzΓ
tw
2b2a2
1
2A
Bevel gears
( )
minus
minusminus= 1
αtan 1dd
uΓvt
2vb2va2
vA
Tip pinionroot wheel
Cylindrical gears
( )
1αtan
1ddΓ
tw
2b1a1
E minusminus
=
Bevel gears
( )
1αtan
1ddΓ
vt
2vb1va1
E minusminus
=
At inner point of single pair contact
Cylindrical gears
tw1
EB α tan zπ2ΓΓ minus=
Bevel gears
vtv1
EB α tan zπ2ΓΓ minus=
At outer point of single pair contact
Cylindrical gears
tw1AD α tan z
π2ΓΓ +=
Bevel gears
vt v1
AD αtan z π2ΓΓ +=
At pitch point ΓC = 0
The points F and G (only applicable to cylindrical gears) limiting the extension of tip relief (so as to maintain a mini-mum contact ratio of unity for unloaded gears) are at
2
BAF
Γ+Γ=Γ
2
EDG
Γ+Γ=Γ
443 Load sharing factor XΓ The load sharing factor XΓ accounts for the load sharing be-tween the various pairs of teeth in mesh along the path of contact
XΓ is to be calculated stepwise from A to E using the pa-rameter Γy
4431 Cylindrical gears with β = 0 and no tip relief
Figure 41
ByAAB
Ay for31
31X
yΓltΓleΓ
ΓminusΓ
ΓminusΓ+=Γ
DyB for1Xy
ΓltΓleΓ=Γ
EyDDE
yE for31
31X
yΓleΓltΓ
ΓminusΓ
ΓminusΓ+=Γ
4432 Cylindrical gears with β = 0 and tip relief Tip relief on the pinion reduces XΓ in the range G ndash E and increases correspondingly XΓ in the range F ndash B
Tip relief on the wheel reduces XΓ in the range A ndash F and increases correspondingly XΓ in the range D ndash G
Following remains generally
DyB for1Xy
ΓleΓleΓ=Γ
21XXGF== ΓΓ
In the following it must be distinguished between Ca lt Ceff respectively Ca gt Ceff This is shown by an example below where Ca1 lt Ceff and Ca2 gt Ceff
Classification Notes- No 412 37 May 2003
DET NORSKE VERITAS
Figure 42
Note When Ca gt Ceff the path of contact is shortened by A ndash A respectively E ndash E The single pair contact path is extended into B respectively D If this shift is significant it is necessary to consider the negative effect on surface durability (B) and bending stresses (B and D)
Range A - F
For effa2 CC le
minus=Γ
eff
2a
CC1
31X
A
FyAeff
2a
AF
Ay for C3
C61XX
AyΓleΓleΓ
sdot
+ΓminusΓ
ΓminusΓ+= ΓΓ
For effa2 CC ge
AyAfor0Xy
ΓleΓleΓ=Γ
with ( )21
CC
1CC
eff
2a
eff
2a
AFAAminus
minusΓminusΓ+Γ=Γ
FyAeff
2a
AF
Ay
eff
2a for 21
CC
CC1X
yΓleΓleΓ
minus
ΓminusΓ
ΓminusΓ+minus=Γ
Range F - B
For effa1 CC le
ByFeff
1a
FB
Fy forC3
C61
21X
yΓleΓleΓ
sdot
+ΓminusΓ
ΓminusΓ+=Γ
For effa1 CC ge
ByFeff
1a
FB
Fy for21
CC
21X
yΓleΓleΓ
minus
ΓminusΓ
ΓminusΓ+=Γ
ByB for1Xy
ΓleΓleΓ=Γ
with 1
CC2
eff
1a
FBFB
minus
ΓminusΓ+Γ=Γ
Range D ndash G
For effa2 CC le
GyDeff
2a
DG
Dy
eff
2a for C 3C
61
C 3C
32X
yΓleΓleΓ
+
ΓminusΓ
ΓminusΓminus+=Γ F
or effa2 CC ge
DyD for1Xy
ΓleΓleΓ=Γ
with ( )21
CC
1CC
eff
2a
eff
2a
DGDDminus
minusΓminusΓ+Γ=Γ
GyDeff
2a
DG
Dy
eff
2a for21
CC
CCX ΓleΓleΓ
minus
ΓminusΓ
ΓminusΓminus=Γ
Range G - E
For effa1 CC le
minus=Γ
eff
1a
CC1
31X
E
EyGeff
1a
GE
Gy forC3
C61
21X
yΓleΓleΓ
sdot
+ΓminusΓ
ΓminusΓminus=Γ
For effa1 CC gt
EyGeff
1a
GE
Gy for21
CC
21X
yΓleΓleΓ
minus
ΓminusΓ
ΓminusΓminus=Γ
EyE for0Xy
ΓleΓleΓ=Γ
with 1
CC2
eff
1a
GEGE
minus
ΓminusΓ+Γ=Γ
4433 Gears with β gt 0 buttressing Due to oblique contact lines over the flanks a certain but-tressing may occur near A and E
This applies to both cylindrical and bevel gears with tip relief lt Ceff The buttressing Xbutt is simplified as a linear function within the ranges A ndash H respectively I ndash E
Figure 43
1εwhen13X βbutt EAge=
1εwhenε031X ββbutt EAlt+=
38 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
Cylindrical gears
bIEAH βsin 02ΓΓΓΓ =minus=minus
Bevel gears
bmIEAH βsin 02ΓΓΓΓ =minus=minus
4434 Cylindrical gears with 2εγ le and no tip relief
yXΓ is obtained by multiplication of
yXΓ in 4431 with
Xbutt in 4433
4435 Gears with 2εγ gt and no tip relief
Applicable to both cylindrical and bevel gears
IyH for1Xy
ΓleΓleΓε
=α
Γ
IyHybutt and forX1Xy
ΓgtΓΓltΓε
=α
Γ
Figure 44
4436 Cylindrical gears with 2εγ le and tip relief
yXΓ is obtained by multiplication of
yXΓ in 4432 with Xbutt
in 4433
4437 Cylindrical gears with 2εγ gt and tip relief
Tip relief on the pinion (respectively wheel) reduces XГ in the range G ndash E (respectively A ndash F) and increases XГ in the range F ndash G
yXΓ is obtained by multiplication of
yXΓ as described below
with Xbutt in 4433
In the XГ example below the influence of tip relief is shown (without the influence of Xbutt) by means of
eff2aeff1a CC and CC ltgt
Tip relief gt Ceff causes new end points A respectively E of the path of contact
Figure 45
Range A ndash F
( ) ( )( ) eff
a2αa1α
AF
Ay
eff
2aeff
C 12C 13εC 1ε
CCCX
y +εε++minus
ΓminusΓ
ΓminusΓ+
εminus
=ααα
Γ
eff2aFyA CC if for leΓleΓleΓ
eff2aFyA CifCfor and geΓleΓleΓ
eff2aAyA CC iffor0Xy
gtΓleΓleΓ=Γ
( ) ( ) ( )( ) ( ) a2αa1α
αeffa2AFAA C 1ε 3C 1ε
1ε 2 CC with++minus+minus
ΓminusΓ+Γ=Γ
Range F ndash G
( )( )( ) GyF
eff
2a1a forC 1 2CC 11X
yΓleΓleΓ
+εε+minusε
+ε
=αα
α
αΓ
Range G ndash E
( ) ( )( ) eff
2a1a
GE
Gy
C 1 2C 1C 1 3XX
GFy +εεminusε++ε
ΓminusΓ
ΓminusΓminus=
αα
ααΓΓ minus
effa1EyG CC if for leΓleΓleΓ
effa1EyG CC if Γfor and geΓleΓle
eff1aEyE CC if for0Xy
geΓleΓleΓ=Γ
( ) ( ) ( )( ) ( ) 2a1a
eff1aGEEE C 1C 1 3
1 2 CC withminusε++ε+εminus
ΓminusΓminusΓ=Γαα
α
4438 Bevel gears with εγ more than approx 18 and heightwise crowning For Ca1 = Ca2 = Ceff the following applies
Figure 46
Classification Notes- No 412 39 May 2003
DET NORSKE VERITAS
( )AEM 50 Γ+Γ=Γ
( )( )2AD
3
2My651X
y ΓminusΓε
ΓminusΓminus
ε=
ααΓ
For tip relief lt Ceff yXΓ is found by linear interpolation be-
tween ( )effay CCX =Γ and ( )0CayX =Γ as in 4435
The interpolation is to be made stepwise from A to M with the influence of Ca2 and from M to E with the influence of Ca1 (For 2a1a CC ne there is a discontinuity at M)
Eg with Ca1 = 04 Ceff and Ca2 = 055 Ceff then
Range A ndash M
)CC()0C( eff2ay2ayyX 550X 450X =Γ=ΓΓ +=
Range M ndash E
)CC()0C( eff1ay1ayyX 40X 06X =Γ=ΓΓ +=
For tip relief gt Ceff the new end points A and E are found as
( )
minusΓminusΓ
ε+Γ=Γ α 1
CC
6 eff
2aADAA
( )
minusΓminusΓ
ε+Γ=Γ α 1
CC
6 eff
1aADEE
Range A ndash A
0Xy=Γ
Range A ndash M
( )( )
ΓminusΓ
ΓminusΓminus
minusε=
αΓ 2
MA
2My
eff
2a1
CC4
351Xy
Range M ndash E
( )( )
ΓminusΓ
ΓminusΓminus
minusε=
αΓ 2
ME
2My
eff
1a1
CC4
351Xy
Range E ndash E
0Xy=Γ
40 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
Appendix A Fatigue Damage Accumulation
The Palmgren-Miner cumulative damage calculation princi-ple is used The procedure may be applied as follows
A1 Stress Spectrum From the individual torque classes the torques (Ti) at the peak values of class intervals and the associated number of cycles (NLi) for both pinion and wheel are to be listed from the highest to the lowest torque
(In case of a cyclic torque variation within the torque classes it is advised to use the peak torque If the cyclic variation is such that the same teeth will repeatedly suffer the peak torque this is a must)
The stress spectra for tooth roots and flanks (σFi σHi) with all relevant factors (except KA) are to be calculated on the basis of the torque spectrum The load dependent K-factors are to be determined for each torque class
A2 σminusN-curve The stress versus load cycle curves for tooth roots and flanks (both pinion and wheel) are to be drawn on the basis of per-missible stresses (ie including the demanded minimum safety factors) as determined in 2 respectively 3 If different safety levels for high cycle fatigue and low cycle fatigue are desired this may be expressed by different demand safety factors applied at the endurance limit respectively at static strength
A3 Damage accumulation The individual damage ratio Di at ith stress level is defined as
Fi
Lii N
ND =
where
NLi = The number of applied cycles at ith stress
NFi = The number of cycles to failure at ith stress
Basically stresses σi below the permissible stress level for infinite life (if a constant ZN or YN is accepted) do not con-tribute to the damage sum However calculating the actual safety factor Sact as described below all the σi for which the product S σi is bigger than or equal to the permissible stress level for infinite life contribute to the damage sum and thus to the determination of Sact The final value of S is decisive
(NFi can be found mathematically by putting the permissible stress σpi equal the actual stress σi thereby finding the actual life factor This life factor can be solved with regard to load cycles ie NFi)
The damage sum ΣDi is not to exceed unity
If ΣDi ne 1 the safety against cumulative fatigue damage is different from the applied demand safety factor For deter-mination of this theoretical safety factor an iteration proce-dure is required as described in the following flowchart
S is correction factor with which the actual safety factor Sact can be found
Sact is the demand safety factor (used in determination of the permissible stresses in the σ ndash N ndash curve) times the correc-tion factor S
The full procedure is to be applied for pinion and wheel tooth roots and flanks
Note If alternating stresses occur in a spectrum of mainly pulsating stresses the alternating stresses may be replaced by equiva-lent pulsating stresses ie by means of division with the ac-tual mean stress influence factor YM
Classification Notes- No 412 41 May 2003
DET NORSKE VERITAS
Appendix B Application Factors for Diesel Driven Gears
For diesel driven gears the application factor KA depends on torsional vibrations Both normal operation and misfiring conditions have to be considered
Normally these two running conditions can be covered by only one calculation
B1 Definitions
Normal operation KAnorm = 0
normv0
TTT +
where
T0 = rated nominal torque
Tv norm = vibratory torque amplitude for normal opera-tion (see rules Pt4 Ch3 Sec1 G301 for defi-nition of ldquonormalrdquo irregularity)
Misfiring operation KA misf = 0
misfv
TTT +
where
T = remaining nominal torque when one cylinder out of action
Tv misf = vibratory torque amplitude in misfiring con-dition This refers to a permissible misfiring condition ie a condition that does not require automatic or immediate corrective actions as speed or pitch reduction
The normal operation is assumed to last for a very high num-ber of cycles such as 1010
The misfiring operation is assumed to last for a limited dura-tion such as 107 cycles
B2 Determination of decisive load Assuming life factor at 1010 cycles as YN = ZN = 092 which usually is relevant the calculation may be performed only once with the combination having the highest value of appli-cation factorlife factor
For bending stresses and scuffing the higher value of
092
K normA and 098
K misfA
For contact stresses the higher value of
092K normA and
113K misfA (but
097K misfA for nitrided gears)
B3 Simplified procedure Note that this is only a guidance and is not a binding con-vention
T = oTZ
1Zsdot
minus where Z = number of cylinders
( ) ideal Videal Vmisf Vnorm V TTT24ZT +minus=
where TV ideal = vibratory torque with all cylinders perfectly equal See also rules Pt4 Ch3 Sec1 G300
When using trends from torsional vibration analysis and measurements the following may be used
TV ideal To is close to zero for engines with few cylinders and using a suitable elastic coupling and increases with relative coupling stiffness and number of cylinders
TV misf To may be high for engines with few cylinders and decreases with number of cylinders
This can be indicated as
200Z
TT
ο
idealV asymp and 80Z04
TT
o
misfV minusasymp
Inserting this into the formulae for the two application fac-tors the following guidance can be given
118112K misfA minusasymp
115110K normA minusasymp
Since KA norm is to be combined with the lower life factors the decisive load condition will be the normal one and a KA of 115 will cover most relevant cases when a suitable elas-tic coupling is chosen
42 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
Appendix C Calculation of Pinion-Rack
Pinion-racks used for elevating of mobile offshore units are open gears that are subjected to wear and tear With normal specifications such as only a few hundred total operation cycles (site to site) the tooth bending stresses for static strength will be decisive for the lay out however with ex-ception of surface hardened pinions where case crushing has to be considered
In the following the use of part 1 and 3 for pinion-racks is shown including relevant simplifications
C1 Pinion tooth root stresses Since the load spectrum normally is dominated by high torques and few load cycles (in the range up to 1-2000) the static strength is decisive
The actual stress is calculated as
β1FSaFan1
tF1 KYY
mbFσ sdotsdotsdotsdot
=
b1 is limited to b2 + 2middotmn
YFa and YSa replace YF and YS because load application at tooth tip has to be assumed for such inaccurate gears
Pinions often use a non-involute profile in the dedendum part of the flank eg a constant radius equal the radius of curva-ture at reference circle For such pinions sFn and hFa are to be measured directly on a sectional drawing of the pinion tooth
Due to high loads and narrow facewidths it may be assumed that KFβ2 = KHβ = 10 However when b1 gt b2 then KFβ1 gt10
If no detailed documentation of KFβ1 is available the fol-lowing may be used
KFβ1 = 1 + 015middot(b1b2 ndash 1)
The permissible stress (not surface hardened) is calculated as
δrelTstF
Fst1FP1 Y
Sσσ sdot=
The mean stress influence due to leg lifting may be disre-garded
The actual and permissible stresses should be calculated for the relevant loads as given in the rules
C2 Rack tooth root stresses The actual stress is calculated as
SaFan2
tF2 YY
mbFσ sdotsdotsdot
=
See C1 for details
The permissible stress is calculated as
δrelTstF
Fst2FP2 Y
Sσσ sdot=
For alloyed steels (Ni Cr Mo) with high toughness and duc-tility the value of YδrelTst may be put equal to YSa
C3 Surface hardened pinions For surface hardened pinions the maximum load is not to cause crushing of the hardened layer of the flank
In principle the calculation described in 213 may be used but when the theoretical Hertzian stress exceeds the ap-proximately 18 times the yield strength of the rack material plastic deformation will occur This will limit the peak Hertzian stress but increases the contact width and thus the penetration of stresses into the depth
An approximation may be based on an assessment of contact width determined by means of equal areas under the theoreti-cal (elastic range) Hertzian contact and the elasto-plastic contact stress (limited to 18 middot σy) with the unknown width
10 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
1815 Kv in the supercritical range Cylindrical gears N ge 15
Bevel gears N ge 15
Special care should be taken as to influence of higher vibra-tion modes andor influence of coupled bending (ie lateral shaft vibrations) and torsional vibrations between pinion and wheel These influences are not covered by the following approach
Kv = Cv5 Bp + Cv6 Bf + Cv7
Cv5 accounts for the pitch error influence
Cv5 = 047
Cv6 accounts for the profile error influence
Cv6 = 047 for εγ le 2
Cv6 = 174ε
012
γ minus for εγ gt 2
Cv7 relates the maximum externally applied tooth loading to the maximum tooth loading of ideal accurate gears operating in the supercritical speed sector when the circumferential vibration becomes very soft
Cv7 = 075 for εγ le 15
Cv7 = [ ] 8750)2(sin 0125 +minusβεπ for 15 lt εγ le 25
Cv7 = 10 for εγ gt 25
1816 Kv in the intermediate range Cylindrical gears 115 lt N lt 15
Bevel gears 125 lt N lt 15
Comments raised in 1814 and 1815 should be observed
Kv is determined by linear interpolation between Kv for N = 115 respectively 125 and N = 15 as
Cylindrical gears
( ) ( ) ( )[ ]51Nv151Nv51Nvv KK350
N51KK === minussdot
minus
+=
Bevel gears
( ) ( ) ( )[ ]51Nv251Nv51Nvv KK250
N51KK === minussdot
minus
+=
182 Multi-resonance method For high speed gear (vgt40 ms) for multimesh medium speed gears for gears with significant lateral shaft flexibility etc it is advised to determine Kv on basis of relevant dy-namic analysis
Incorporating lateral shaft compliance requires transforma-tion of even a simple pinion-wheel system into a lumped multi-mass system It is advised to incorporate all relevant inertias and torsional shaft stiffnesses into an equivalent (to pinion speed) system Thereby the mesh stiffness appears as an equivalent torsional stiffness
cγ b (db12)2 (Nmrad)
The natural frequencies are found by solving the set of dif-ferential equations (one equation per inertia) Note that for a gear put on a laterally flexible shaft the coupling bending-torsionals is arranged by introducing the gear mass and the lateral stiffness with its relation to the torsional displacement and torque in that shaft
Only the natural frequency (ies) having high relative dis-placement and relative torque through the actual pinion-wheel flexible element need(s) to be considered as critical frequency (ies)
Kv may be determined by means of the method mentioned in 181 thereby using N as the least favourable ratio (in case of more than one pinion-wheel dominated natural frequency) Ie the N-ratio that results in the highest Kv has to be consid-ered
The level of the dynamic factor may also be determined on basis of simulation technique using numeric time integration with relevant tooth stiffness variation and pitchprofile er-rors
19 Face Load Factors KHβ and KFβ The face load factors KHβ for contact stresses and for scuff-ing KFβ for tooth root stresses account for non-uniform load distribution across the facewidth
KHβ is defined as the ratio between the maximum load per unit facewidth and the mean load per unit facewidth
KFβ is defined as the ratio between the maximum tooth root stress per unit facewidth and the mean tooth root stress per unit facewidth The mean tooth root stress relates to the con-sidered facewidth b1 respectively b2
Note that facewidth in this context is the design facewidth b even if the ends are unloaded as often applies to eg bevel gears
The plane of contact is considered
191 Relations between KHβ and KFβ
KFβ = expβHK 2(hb)hb1
1exp++
=
where hb is the ratio tooth heightfacewidth The maximum of h1b1 and h2b2 is to be used but not higher than 13 For double helical gears use only the facewidth of one helix
If the tooth root facewidth (b1 or b2) is considerably wider than b the value of KFβ(1or2) is to be specially considered as it may even exceed KHβ
Eg in pinion-rack lifting systems for jack up rigs where b = b2 asymp mn and b1 asymp 3 mn the typical KHβ asymp KFβ2 asymp 1 and KFβ1 asymp 13
192 Measurement of face load factors Primarily
KFβ may be determined by a number of strain gauges distrib-uted over the facewidth Such strain gauges must be put in
Classification Notes- No 412 11 May 2003
DET NORSKE VERITAS
exactly the same position relative to the root fillet Relations in 191 apply for conversion to KHβ
Secondarily
KHβ may be evaluated by observed contact patterns on vari-ous defined load levels It is imperative that the various test loads are well defined Usually it is also necessary to evalu-ate the elastic deflections Some teeth at each 90 degrees are to be painted with a suitable lacquer Always consider the poorest of the contact patterns
After having run the gear for a suitable time at test load 1 (the lowest) observe the contact pattern with respect to ex-tension over the facewidth Evaluate that KHβ by means of the methods mentioned in this section Proceed in the same way for the next higher test load etc until there is a full face contact pattern From these data the initial mesh misalign-ment (ie without elastic deflections) can be found by ex-trapolation and then also the KHβ at design load can be found by calculation and extrapolation See example
Figure 11 Example of experimental determination of KHβ
It must be considered that inaccurate gears may accumulate a larger observed contact pattern than the actual single mesh to mesh contact patterns This is particularly important for lapped bevel gears Ground or hard metal hobbed bevel gears are assumed to present an accumulated contact pattern that is practically equal the actual single mesh to mesh con-tact patterns As a rough guidance the (observed) accumu-lated contact pattern of lapped bevel gears may be reduced by 10 in order to assess the single mesh to mesh contact pattern which is used in 199
193 Theoretical determination of KHβ The methods described in 193 to 198 may be used for cy-lindrical gears The principles may to some extent also be used for bevel gears but a more practical approach is given in 199
General For gears where the tooth contact pattern cannot be verified during assembly or under load all assumptions are to be well on the safe side
KHβ is to be determined in the plane of contact
The influence parameters considered in this method are
bull mean mesh stiffness cγ (see 111) (if necessary also variable stiffness over b)
bull mean unit load Fmb = Fbt KA Kγ Kvb (for double helical gears see 17 for use of Kγ)
bull misalignment fsh due to elastic deflections of shafts and gear bodies (both pinion and wheel)
bull misalignment fdefl due to elastic deflections of and work-ing positions in bearings
bull misalignment fbe due to bearing clearance tolerances bull misalignment fma due to manufacturing tolerances bull helix modifications as crowning end relief helix correc-
tion bull running in amount yβ (see 112)
In practice several other parameters such as centrifugal ex-pansion thermal expansion housing deflection etc contrib-ute to KHβ However these parameters are not taken into account unless in special cases when being considered as particularly important
When all or most of the am parameters are to be considered the most practical way to determine KHβ is by means of a graphical approach described in 1931
If cγ can be considered constant over the facewidth and no helix modifications apply KHβ can be determined analyti-cally as described in 1932
1931 Graphical method The graphical method utilises the superposition principle and is as follows
bull Calculate the mean mesh deflection Mδ as a function of
γm c and bF see 111
bull Draw a base line with length b and draw up a rectangu-lar with height δM (The area δM b is proportional to the transmitted force)
bull Calculate the elastic deflection fsh in the plane of contact Balance this deflection curve around a zero line so that the areas above and below this zero line are equal
Figure 12 fsh balanced around zero line
bull Superimpose these ordinates of the fsh curve to the previ-ous load distribution curve (The area under this new load distribution curve is still δM b)
bull Calculate the bearing deflections andor working posi-tions in the bearings and evaluate the influence fdefl in the plane of contact This is a straight line and is bal-anced around a zero line as indicated in Fig 14 but with one distinct direction Superimpose these ordinates to the previous load distribution curve
12 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
bull The amount of crowning end relief or helix correction (defined in the plane of contact) is to be balanced around a zero line similarly to fsh
Figure 13 Crowning Cc balanced aound zero line
bull Superimpose these ordinates to the previous load distri-bution curve In case of high crowning etc as eg often applied to bevel gears the new load distribution curve may cross the base line (the real zero line) The result is areas with negative load that is not real as the load in those areas should be zero Thus corrective actions must be made but for practical reasons it may be postponed to after next operation
bull The amount of initial mesh misalignment bema ff + (defined in the plane of contact) is to be bal-
anced around a zero line If the direction of bema ff + is known (due to initial contact check) or if the direction of fbe is known due to design (eg overhang bevel pin-ion) this should be taken into account If direction un-known the influence of bema ff + in both directions as well as equal zero should be considered
Figure 14 fma+fbe in both directions balanced around zero line
Superimpose these ordinates to the previous load distri-bution curve This results in up to 3 different curves of which the one with the highest peak is to be chosen for further evaluation
bull If the chosen load distribution curve crosses the base line (ie mathematically negative load) the curve is to be corrected by adding the negative areas and dividing this with the active facewidth The (constant) ordinates of this rectangular correction area are to be subtracted from the positive part of the load distribution curve It is advisable to check that the area covered under this new load distribution curve is still equal δM b
bull If cγ cannot be considered as constant over b then cor-rect the ordinates of the load distribution curve with the local (on various positions over the facewidth) ratio between local mesh stiffness and average mesh stiffness cγ (average over the active facewidth only) Note that the result is to be a curve that covers the same area δM b as before
bull The influence of running in yβ is to be determined as in 112 whereby the value for Fβx is to be taken as twice the distance between the peak of the load distribution curve and δM
bull Determine
KHβ = M
β
δycurveofpeak minus
1932 Simplified analytical method for cylindrical gears The analytical approach is similar to 1931 but has a more limited application as cγ is assumed constant over the face-width and no helix modification applies
bull Calculate the elastic deflection fsh in the plane of contact Balance this deflection curve around a zero line so that the area above and below this zero line are equal see Fig 12 The max positive ordinate is frac12∆fsh
bull Calculate the initial mesh alignment as Fβx= deflbemash ffff plusmnplusmnplusmn∆ The negative signs may only be used if this is justified andor verified by a contact pattern test Otherwise al-ways use positive signs If a negative sign is justified the value of Fβx is not to be taken less than the largest of each of these elements
bull Calculate the effective mesh misalignment as Fβy = Fβx - yβ (yβ see 112)
bull Determine
2KforF2
bFc1K βH
m
βγγH le+=β
or
2KforF
bFc2K Hβ
m
βγγH gt=β
where cγ as used here is the effective mesh stiffness see 111
194 Determination of fsh fsh is the mesh misalignment due to elastic deflections Usu-ally it is sufficient to consider the combined mesh deflection of the pinion body and shaft and the wheel shaft The calcu-lation is to be made in the plane of contact (of the considered gear mesh) and to consider all forces (incl axial) acting on the shafts Forces from other meshes can be parted into components parallel respectively vertical to the considered plane of contact Forces vertical to this plane of contact have no influence on fsh
It is advised to use following diameters for toothed elements
d + 2 x mn for bending and shear deflection
d + 2 mn (x ndash ha0 + 02) for torsional deflection
Usually fsh is calculated on basis of an evenly distributed load If the analysis of KHβ shows a considerable maldis-
Classification Notes- No 412 13 May 2003
DET NORSKE VERITAS
tribution in term of hard end contact or if it is known by other reasons that there exists a hard end contact the load should be correspondingly distributed when calculating fsh In fact the whole KHβ procedure can be used iteratively 2-3 iterations will be enough even for almost triangular load distributions
195 Determination of fdefl fdefl is the mesh misalignment in the plane of contact due to bearing deflections and working positions (housing deflec-tion may be included if determined)
First the journal working positions in the bearings are to be determined The influence of external moments and forces must be considered This is of special importance for twin pinion single output gears with all 3 shafts in one plane
For rolling bearings fdefl is further determined on basis of the elastic deflection of the bearings An elastic bearing deflec-tion depends on the bearing load and size and number of rolling elements Note that the bearing clearance tolerances are not included here
For fluid film bearings fdefl is further determined on basis of the lift and angular shift of the shafts due to lubrication oil film thickness Note that fbe takes into account the influence of the bearing clearance tolerance
When working positions bearing deflections and oil film lift are combined for all bearings the angular misalignment as projected into the plane of the contact is to be determined fdefl is this angular misalignment (radians) times the face-width
196 Determination of fbe fbe is the mesh misalignment in the plane of contact due to tolerances in bearing clearances In principle fbe and fdefl could be combined But as fdefl can be determined by analy-sis and has a distinct direction and fbe is dependent on toler-ances and in most cases has no distinct direction (ie + toler-ance) it is practicable to separate these two influences
Due to different bearing clearance tolerances in both pinion and wheel shafts the two shaft axis will have an angular mis-alignment in the plane of contact that is superimposed to the working positions determined in 195 fbe is the facewidth times this angular misalignment Note that fbe may have a distinct direction or be given as a + tolerance or a combina-tion of both For combination of + tolerance it is adviced to use
fbe= 22be
21be ff ++plusmn
fbe is particularly important for overhang designs for gears with widely different kinds of bearings on each side and when the bearings have wide tolerances on clearances In general it shall be possible to replace standard bearings with-out causing the real load distribution to exceed the design premises For slow speed gears with journal bearings the expected wear should also be considered
197 Determination of fma fma is the mesh misalignment due to manufacturing toler-ances (helix slope deviation) of pinion fHβ1 wheel fHβ2 and housing bore
For gear without specifically approved requirements to as-sembly control the value of fma is to be determined as
fma= 22Hβ
21Hβ ff +
For gears with specially approved assembly control the value of fma will depend on those specific requirements
198 Comments to various gear types For double helical gears KHβ is to be determined for both helices Usually an even load share between the helices can be assumed If not the calculation is to be made as de-scribed in 171
For planetary gears the free floating sun pinion suffers only twist no bending It must be noted that the total twist is the sum of the twist due to each mesh If the value of 1K γ ne this must be taken into account when calculating the total sun pinion twist (ie twist calculated with the force per mesh without Kγ and multiplied with the number of planets)
When planets are mounted on spherical bearings the mesh misalignments sun-planet respectively planet-annulus will be balanced Ie the misalignment will be the average between the two theoretical individual misalignments The faceload distribution on the flanks of the planets can take full advan-tage of this However as the sun and annulus mesh with several planets with possibly different lead errors the sun and annulus cannot obtain the above mentioned advantage to the full extent
199 Determination of KHβ for bevel gears If a theoretical approach similar to 193-198 is not docu-mented the following may be used
testeff
H Kb
b851851K sdot
minussdot=β
beff b represents the relative active facewidth (regarding lapped gears see 192 last part)
Higher values than beff b = 090 are normally not to be used in the formula
For dual directional gears it may be difficult to obtain a high beff b in both directions In that case the smaller beff b is to be used
Ktest represents the influence of the bearing arrangement shaft stiffness bearing stiffness housing stiffness etc on the faceload distribution and the verification thereof Expected variations in length- and height-wise tooth profile is also accounted for to some extent
a) Ktest = 1 For ground or hard metal hobbed gears with the specified contact pattern verified at full rating or at full torque slow turning at a condition representative for the thermal expansion at normal operation
14 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
It also applies when the bearing arrangementsupport has insignificant elastic deflections and thermal axial expansion However each initial mesh contact must be verified to be within acceptance criteria that are cali-brated against a type test at full load Reproduction of the gear tooth length- and height-wise profile must also be verified This can be made through 3D measurements or by initial contact movements caused by defined axial offsets of the pinion (tolerances to be agreed upon)
b) Ktest = 1 + 04middot(beffbndash06) For designs with possible influence of thermal expansion in the axial direction of the pinion The initial mesh contact verified with low load or spin test where the acceptance criteria are calibrated against a type test at full load
c) Ktest = 12 if mesh is only checked by toolmakerrsquos blue or by spin test contact For gears in this category beffb gt 085 is not to be used in the calculation
110 Transversal Load Distribution Factors KHα and KFα The transverse load distribution factors KHα for contact stresses and for scuffing KFα for tooth root stresses account for the effects of pitch and profile errors on the transversal load distribution between 2 or more pairs of teeth in mesh
The following relations may be used
Cylindrical gears
( )
minus+==
tH
αptγγHαFα F
byfc0409
2ε
KK
valid for 2εγ le
( ) ( )tH
αptγ
γ
γHαFα F
byfcε
1ε20409KK
minusminus+==
valid for 2ε γ gt
where
FtH = Ft KA Kγ Kv KHβ
cγ = See 111
γα = See 112
fpt = Maximum single pitch deviation (microm) of pinion or wheel or maximum total profile form devia-tion Fα of pinion or wheel if this is larger than the maximum single pitch deviation
Note In case of adequate equivalent tip relief adapted to the load half of the above mentioned fpt can be introduced A tip relief is considered adequate when the average of Ca1 and Ca2 is within plusmn40 of the value of Ceff in 432
Limitations of KHα and KFα
If the calculated values for
KFα = KHα lt 1 use KFα = KHα = 10
If the calculated value of KHα gt 2εα
γ
Zε
ε use KHα = 2
εα
γ
Zε
ε
If the calculated value of KFα gt εα
γ
Yεε
use KFα = εα
γ
Yεε
where Yε = αnε
075025+ (for εαn see 331c)
Bevel gears
For ground or hard metal hobbed gears KFα = KHα = 1
For lapped gears KFα = KHα = 11
111 Tooth Stiffness Constants cacute and cγ The tooth stiffness is defined as the load which is necessary to deform one or several meshing gear teeth having 1 mm facewidth by an amount of 1 microm in the plane of contact cacute is the maximum stiffness of a single pair of teeth cγ is the mean value of the mesh stiffness in a transverse plane (brief term mesh stiffness)
Both valid for high unit load (Unit load = Ft middot KA middot Kγb)
Cylindrical gears The real stiffness is a combination of the progressive Hertzian contact stiffness and the linear tooth bending stiff-nesses For high unit loads the Hertzian stiffness has little importance and can be disregarded This approach is on the safe side for determination of KHβ and KHα However for moderate or low loads Kv may be underestimated due to determination of a too high resonance speed
The linear approach is described in A
An optional approach for inclusion of the non-linear stiffness is described in B
A The linear approach
BRCCq
βcos08acutec =
and
( )025ε075cc αγ +prime=
where
( )[ ]n02a01a
B α2000212
hh12051C minusminus
+minus+=
12n1n
x000635z
025791z
015551004723q minus++=
12
2n
22
1n
1 x005290z
x024188x000193z
x011654+minusminusminus
+ 000182 x22
Classification Notes- No 412 15 May 2003
DET NORSKE VERITAS
(for internal gears use zn2 equal infinite and x2 = 0) ha0 = hfp for all practical purposes CR considers the increased flexibility of the wheel teeth if the wheel is not a solid disc and may be calculated as
( )( )nR m5s
sR e5
bbln1C +=
where
bs = thickness of a central web
sR = average thickness of rim (net value from tooth root to inside of rim)
The formula is valid for bs b ge 02 and sRmn ge 1 Outside this range of validity and if the web is not centrally posi-tioned CR has to be specially considered
Note CR is the ratio between the average mesh stiffness over the facewidth and the mesh stiffness of a gear pair of solid discs The local mesh stiffness in way of the web corresponds to the mesh stiffness with CR = 1 The local mesh stiffness where there is no web support will be less than calculated with CR above Thus eg a centrally positioned web will have an effect corresponding to a longitudinal crowning of the teeth See also 1931 regarding KHβ
B The non-linear approach
In the following an example is given on how to consider the non-linearity
The relation between unit load Fb as a function of mesh deflection δ is assumed to be a progressive curve up to 500 Nmm and from there on a straight line This straight line when extended to the baseline is assumed to intersect at 10microm
With these assumptions the unit force Fb as a function of mesh deflection δ can be expressed as
( )10δKbF
minus= for 500bFgt
minus=
500Fb10δK
bF for 500
bFlt
with γAt KK
bF
bF
sdotsdot= etc (Nmm) ie unit load incorporat-
ing the relevant factors as
KA middot Kγ for determination of Kv
KA middot Kγ middot Kv for determination of KHβ
KA middot Kγ middot Kv middot KHβ for determination of KHα
δ = mesh deflection (microm)
K = applicable stiffness (c or cγ)
Use of stiffnesses for KV KHβ and KHα
For calculation of Kv and KHα the stiffness is calculated as follows
When Fb lt 500
the stiffness is determined as δ∆
∆ bF
where the increment is chosen as eg ∆ Fb = 10 and thus
50010Fb10
K10Fb∆δ +
++
=
When F b gt 500 the stiffness is c or cγ For calculation of KHβ the mesh deflection δ is used directly
or an equivalent stiffness determined as δsdotb
F
Bevel gears
In lack of more detailed relationship between stiffness and geometry the following may be used
b085
b13cacute eff=
b085b
16c effγ =
beff not to be used in excess of 085 b in these formulae
Bevel gears with heightwise and lengthwise crowning have progressive mesh stiffness The values mentioned above are only valid for high loads They should not be used for de-termination of Ceff (see 432) or KHβ (see 1931)
112 Running-in Allowances The running-in allowances account for the influence of run-ning-in wear on the various error elements
yα respectively yβ are the running-in amounts which reduce the influence of pitch and profile errors respectively influ-ence of localised faceload
Cay is defined as the running-in amount that compensates for lack of tip relief
The following relations may be used
For not surface hardened steel
ptHlim
α fσ160y =
yβ = βxlimH
f320σ
with the following upper limits
V lt 5 ms 5-10 ms gt 10 ms
yα max none
limH
12800σ limH
6400σ
yβ max none
limH
25600σ limH
12800σ
For surface hardened steel
yα = 0075 fpt but not more than 3 for any speed
yβ = 015 Fβx but not more than 6 for any speed
16 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
For all kinds of steel
5145189718
1C2
limHay +
minusσ
=
When pinion and wheel material differ the following ap-plies
bull Use the larger of fpt1 - yα1 and fpt2 - yα2 to replace fpt - yα in the calculation of KHα and Kv
bull Use ( )2β1ββ yy21y += in the calculation of KHβ
bull Use ( )2ay1aya CC21C += in the calculation of Kv
bull Use ( )2ay1ay2a1a CC21CC +== in the scuffing
calculation if no design tip relief is foreseen
Classification Notes- No 412 17 May 2003
DET NORSKE VERITAS
2 Calculation of Surface Durability
21 Scope and General Remarks Part 2 includes the calculations of flank surface durability as limited by pitting spalling case crushing and subsurface yielding Endurance and time limited flank surface fatigue is calculated by means of 22 ndash 212 In a way also tooth frac-tures starting from the flank due to subsurface fatigue is in-cluded through the criteria in 213
Pitting itself is not considered as a critical damage for slow speed gears However pits can create a severe notch effect that may result in tooth breakage This is particularly im-portant for surface hardened teeth but also for high strength through hardened teeth For high-speed gears pitting is not permitted
Spalling and case crushing are considered similar to pitting but may have a more severe effect on tooth breakage due to the larger material breakouts initiated below the surface Subsurface fatigue is considered in 213
For jacking gears (self-elevating offshore units) or similar slow speed gears designed for very limited life the max static (or very slow running) surface load for surface hard-ened flanks is limited by the subsurface yield strength
For case hardened gears operating with relatively thin lubri-cation oil films grey staining (micropitting) may be the lim-iting criterion for the gear rating Specific calculation meth-ods for this purpose are not given here but are under consid-eration for future revisions Thus depending on experience with similar gear designs limitations on surface durability rating other than those according to 22 - 213 may be ap-plied
22 Basic Equations Calculation of surface durability (pitting) for spur gears is based on the contact stress at the inner point of single pair contact or the contact at the pitch point whichever is greater
Calculation of surface durability for helical gears is based on the contact stress at the pitch point
For helical gears with 0 lt εβ lt 1 a linear interpolation be-tween the above mentioned applies
Calculation of surface durability for spiral bevel gears is based on the contact stress at the midpoint of the zone of contact
Alternatively for bevel gears the contact stress may be cal-culated with the program ldquoBECALrdquo In that case KA and Kv are to be included in the applied tooth force but not KHβ and KHα The calculated (real) Hertzian stresses are to be multi-plied with ZK in order to be comparable with the permissible contact stresses
The contact stresses calculated with the method in part 2 are based on the Hertzian theory but do not always represent the real Hertzian stresses
The corresponding permissible contact stresses σHP are to be calculated for both pinion and wheel
221 Contact stress Cylindrical gears
( )HαHβvγA
1
tβεEHDBH KKKKK
bud1uF
ZZZZZσ+
=
where
ZBD = Zone factor for inner point of single pair contact for pinion resp wheel (see 232)
ZH = Zone factor for pitch point (see 231)
ZE = Elasticity factor (see 24)
Zε = Contact ratio factor (see 25)
Zβ = Helix angle factor (see 26)
Ft KA Kγ Kv KHβ KHα see 15 ndash 110
d1 b u see 12 ndash 15
Bevel gears
( )HαHβvγA
v1v
vmtKEMH KKKKK
bud1uF
ZZZ105σ+
sdot=
where
105 is a correlation factor to reach real Hertzian stresses (when ZK = 1)
ZE KA etc see above
ZM = mid-zone factor see 233
ZK = bevel gear factor see 27
Fmt dv1 uv see 12 ndash 15
It is assumed that the heightwise crowning is chosen so as to result in the maximum contact stresses at or near the mid-point of the flanks
222 Permissible contact stress
XWRvLH
NHlimHP ZZZZZ
SZσ
σ =
where
σH lim = Endurance limit for contact stresses (see 28)
ZN = Life factor for contact stresses (see 29)
SH = Required safety factor according to the rules
ZLZvZR = Oil film influence factors (see 210)
ZW = Work hardening factor (see 211)
ZX = Size factor (see 212)
18 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
23 Zone Factors ZH ZBD and ZM
231 Zone factor ZH The zone factor ZH accounts for the influence on contact stresses of the tooth flank curvature at the pitch point and converts the tangential force at the reference cylinder to the normal force at the pitch cylinder
wtt2
wtbH sinααcos
cosαcosβ2Z =
232 Zone factors ZBD The zone factors ZBD account for the influence on contact stresses of the tooth flank curvature at the inner point of sin-gle pair contact in relation to ZH Index B refers to pinion D to wheel
For εβ ge 1 ZBD = 1
For internal gears ZD = 1
For εβ = 0 (spur gears)
( )
π
minusεminusminus
π
minusminus
α=
α2
2
2b
2a
1
2
1b
1a
wtB
z211
dd
z21
dd
tanZ
( )
π
minusεminusminus
π
minusminus
α=
α1
2
1b
1a
2
2
2b
2a
wtD
z211
dd
z21
dd
tanZ
If ZB lt 1 use ZB = 1
If ZD lt 1 use ZD = 1
For 0 lt εβ lt 1
ZBD = ZBD (for spur gears) ndash εβ (ZBD (for spur gears) ndash 1)
233 Zone factor ZM The mid-zone factor ZM accounts for the influence of the contact stress at the mid point of the flank and applies to spi-ral bevel gears
εminusminus
εminusminus
αβ=
αα btm2
2vb2
2vabtm2
1vb2val
2v1vvtbmM
pddpdd
ddtancos2Z
This factor is the product of ZH and ZM-B in ISO 10300 with the condition that the heightwise crowning is sufficient to move the peak load towards the midpoint
234 Inner contact point For cylindrical or bevel gears with very low number of teeth the inner contact point (A) may be close to the base circle In order to avoid a wear edge near A it is required to have suitable tip relief on the wheel
24 Elasticity Factor ZE The elasticity factor ZE accounts for the influence of the material properties as modulus of elasticity and Poissonrsquos ratio on the contact stresses
For steel against steel ZE = 1898
25 Contact Ratio Factor Zε The contact ratio factor Zε accounts for the influence of the transverse contact ratio εα and the overlap ratio εβ on the contact stresses
αε1Z =ε for 1εβ ge
( )α
ββ
αε ε
εε1
3ε4
Z +minusminus
= for εβ lt 1
26 Helix Angle Factor Zβ The helix angle factor Zβ accounts for the influence of helix angle (independent of its influence on Zε) on the surface du-rability
cosβZβ =
27 Bevel Gear Factor ZK The bevel gear factor accounts for the difference between the real Hertzian stresses in spiral bevel gears and the contact stresses assumed responsible for surface fatigue (pitting) ZK adjusts the contact stresses in such a way that the same per-missible stresses as for cylindrical gears may apply
The following may be used ZK = 080
28 Values of Endurance Limit σHlim and Static Strength 5H10σ 3H10σ
σHlim is the limit of contact stress that may be sustained for 5middot107 cycles without the occurrence of progressive pitting
For most materials 5middot107 cycles are considered to be the be-ginning of the endurance strength range or lower knee of the σ-N curve (See also Life Factor ZN) However for nitrided steels 2middot106 apply
For this purpose pitting is defined by
bull for not surface hardened gears pitted area ge 2 of total active flank area
bull for surface hardened gears pitted area ge 05 of total active flank area or ge 4 of one particular tooth flank area 510Hσ and 310Hσ are the contact stresses which the given
material can withstand for 105 respectively 103 cycles without subsurface yielding or flank damages as pitting spalling or case crushing when adequate case depth applies
The following listed values for σHlim 510Hσ and 310Hσ may only be used for materials subjected to a quality control as
Classification Notes- No 412 19 May 2003
DET NORSKE VERITAS
the one referred to in the rules Results of approved fatigue tests may also be used as the basis for establishing these values The defined survival probability is 99
σHlim σH105 σH10
3
Alloyed case hardened steels (surface hardness 58-63 HRC) - of specially approved high grade - of normal grade
1650 1500
2500 2400
3100 3100
Nitrided steel of approved grade gas nitrided (surface hardness 700-800 HV) 1250 13 σHlim 13 σHlim
Alloyed quenched and tempered steel bath or gas nitrided (surface hardness 500-700 HV)
1000
13 σHlim
13 σHlim
Alloyed flame or induction hardened steel (surface hardness 500-650 HV) 075 HV + 750 16 σHlim 45 HV
Alloyed quenched and tempered steel 14 HV + 350 16 σHlim 45 HV
Carbon steel 15 HV + 250 16 σHlim 16 σHlim
These values refer to forged or hot rolled steel For cast steel the values for σHlim are to be reduced by 15
29 Life Factor ZN The life factor ZN takes account of a higher permissible contact stress if only limited life (number of cycles NL) is demanded or lower permissible contact stress if very high number of cycles apply
If this is not documented by approved fatigue tests the fol-lowing method may be used
For all steels except nitrided
7L 105N sdotge ZN = 1 or
01570
L
7
N N105Z
sdot=
Ie ZN = 092 for 1010 cycles
The ZN = 1 from 5middot107 on may only be used when the mate-rial cleanliness is of approved high grade (see Rules Pt4 Ch2) and the lubrication is optimised by a specially approved filtering process
105 lt NL lt 5middot107 510N
logZ037
L
7
N N105Z
sdot=
NL = 105 WXRVLHlim
WstX10H1010NN ZZZZZσ
ZZσZZ
55
5=
103 lt NL lt 105 )Z(Zlog05
L
5
10NN
5N103N10
5N10ZZ
==
3L 10N le
WXRVLlimH
Wst10X10H10NN ZZZZZ
ZZZZ
33
3σ
σ==
(but not less than ZN105)
For nitrided steels
6L 102N sdotge ZN = 1 or
00980
L
6
N N102Z
sdot=
Ie ZN = 092 for 1010 cycles
The ZN = 1 from 2middot106 on may only be used when the mate-rial cleanliness is of approved high grade (see Rules Pt4 Ch2) and the lubrication is optimised by a specially approved filtering process
105 lt NL lt 2middot106 510N
Zlog76860
L
6
N N102Z
sdot=
5L 10N le
XWRVL
10XWst10NN ZZZZZ
ZZ13ZZ
55 ==
Note that when no index indicating number of cycles is used the factors are valid for 5middot107 (respectively 2middot106 for nitrid-ing) cycles
210 Influence Factors on Lubrication Film ZL ZV and ZR The lubricant factor ZL accounts for the influence of the type of lubricant and its viscosity the speed factor ZV ac-counts for the influence of the pitch line velocity and the roughness factor ZR accounts for influence of the surface roughness on the surface endurance capacity
The following methods may be applied in connection with the endurance limit
20 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
Surface hardened steels Not surface hardened steels
ZL ( )24013421
360910ν+
+
( )24013421
680830ν+
+
ZV ( )v3280
140930+
+ ( )v3280300850
++
ZR
080
ZrelR3
150
ZrelR3
where
ν40 = Kinematic oil viscosity at 40ordmC (mm2s) For case hardened steels the influence of a high bulk temperature (see 4 Scuffing) should be con-sidered Eg bulk temperatures in excess of 120ordmC for long periods may cause reduced flank surface endurance limits
For values of ν40 gt 500 use ν40 = 500
RZrel = The mean roughness between pinion and wheel (after running in) relative to an equivalent radius of curvature at the pitch point ρc = 10mm
RZrel
= ( ) 31
cZ2Z1 ρ
10RR50
+
RZ = Mean peak to valley roughness (microm) (DIN defini-tion) (roughly RZ = 6 Ra)
For 5
L 10N le ZL ZV ZR = 10
211 Work Hardening Factor ZW The work hardening factor ZW accounts for the increase of surface durability of a soft steel gear when meshing the soft steel gear with a surface hardened or substantially harder gear with a smooth surface
The following approximation may be used for the endurance limit
Surface hardened steel against not surface hardened steel 150
ZeqW R
31700
130HB21Z
minus
minus=
where HB = the Brinell hardness of the soft member For HB gt 470 use HB = 470 For HB lt 130 use HB = 130
RZeq = equivalent roughness
033
c40
066
ZS
ZHZHZEQ ρvν
15000RRRR
=
If RZeq gt 16 then use RZeq = 16
If RZeq lt 15 then use RZeq = 15
where
RZH = surface roughness of the hard member before run in
RZS = surface roughness of the soft member before run in
ν40 = see 210
If values of ZW lt 1 are evaluated ZW = 1 should be used for flank endurance However the low value for ZW may indi-cate a potential wear problem
Through hardened pinion against softer wheel
( )
minussdotminus+= 000829
HBHB0008981u1Z
2
1W
For 21HBHB
2
1 le use ZW = 1
For 71HBHB
2
1 gt use 17HBHB
2
1 =
For u gt 20 use u = 20
For static strength (lt 105 cycles)
Surface hardened against not surface hardened
ZWst = 105
Through hardened pinion against softer wheel
ZWst = 1
212 Size Factor ZX The size factor accounts for statistics indicating that the stress levels at which fatigue damage occurs decrease with an increase of component size as a consequence of the influ-ence on subsurface defects combined with small stress gradi-ents and of the influence of size on material quality
ZX may be taken unity provided that subsurface fatigue for surface hardened pinions and wheels is considered eg as in the following subsection 213
213 Subsurface Fatigue This is only applicable to surface hardened pinions and wheels The main objective is to have a subsurface safety against fatigue (endurance limit) or deformation (static strength) which is at least as high as the safety SH required for the surface The following method may be used as an approximation unless otherwise documented
The high cycle fatigue (gt3106 cycles) is assumed to mainly depend on the orthogonal shear stresses Static strength (lt103 cycles) is assumed to depend mainly on equivalent
Classification Notes- No 412 21 May 2003
DET NORSKE VERITAS
stresses (von Mises) Both are influenced by residual stresses but this is only considered roughly and empirically
The subsurface working stresses at depths inside the peak of the orthogonal shear stresses respectively the equivalent stresses are only dependent on the (real) Hertzian stresses Surface related conditions as expressed by ZL ZV and ZR are assumed to have a negligible influence
The real Hertzian stresses σHR are determined as
For helical gears with εβ gt 1
σHR = σH
For helical gears with εβ lt 1 and spur gears
ε
α
ββ ε
ε+εminus
sdotσ=σZ
1
HHR
For bevel gears
KHHR Z
1σσ sdot=
The necessary hardness HV is given as a function of the net depth tz (net = after grinding or hard metal hobbing and per-pendicular to the flank)
The coordinates tz and HV are to be compared with the de-sign specification such as
bull for flame and induction hardening tHVmin HVmin bull for nitriding t400min HV = 400 bull for case hardening t550min HV = 550 t400min HV = 400
and t300min HV = 300 (the latter only if the core hardness lt 300 If the core hardness gt 400 the t400 is to be re-placed by a fictive t400 = 16 t550)
In addition the specified surface hardness is not to be less than the max necessary hardness (at tz = 05aH) This applies to all hardening methods
For high cycle fatigue (gt3 106 cycles) the following applies
sdot+
minussdotsdotsdot= o90
05azt
05azt
cosSσ04HV
H
HHHR
applicable to 05a
zt
Hge
For 05at
H
z lt the value for 05at
H
z = applies
56300ρSσ12a cHHR
Hsdotsdot
sdot=
Where aH is half the hertzian contact width multiplied by an empirical factor of 12 that takes into account the possible influence of reduced compressive residual stresses (or even tensile residual stresses) on the local fatigue strength
If any of the specified hardness depths including the surface hardness is below the curve described by HV = f (tz) the actual safety factor against subsurface fatigue is determined as follows
reduce SH stepwise in the formula for HV and aH until all specified hardness depths and surface hardness balance with the corrected curve The safety factor obtained through this method is the safety against subsurface fatigue
For static strength (lt103 cycles) the following applies
sdot+
minussdotsdotsdot= o90
07at
06a
zt
cosSσ019HV
Hst
z
HstHHR
applicable to 06at
Hst
z ge
56300ρSσa cHHR
Hstsdotsdot
=
In the case of insufficient specified hardness depths the same procedure for determination of the actual safety factor as above applies
For limited life fatigue (103 lt cycles lt 3106 )
For this purpose it is necessary to extend the correction of safety factors to include also higher values than required Ie in the case of more than sufficient hardness and depths the safety factor in the formulae for both high cycle fatigue and static strength are to be increased until necessary and speci-fied values balance
The actual safety factor for a given number of cycles N be-tween 103 and 3106 is found by linear interpolation in a dou-ble logarithmic diagram
minussdotminus
= sdot logN3477
logSlogSlogS
310H6103HHN
36 10H103H Slog86281Slog86280 sdot+sdot sdot
22 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
3 Calculation of Tooth Strength
31 Scope and General Remarks Part 3 include the calculation of tooth root strength as limited by tooth root cracking (surface or subsurface initiated) and yielding
For rim thickness sR ge 35middotmn the strength is calculated by means of 32 ndash 313 For cylindrical gears the calculation is based on the assumption that the highest tooth root tensile stress arises by application of the force at the outer point of single tooth pair contact of the virtual spur gears The method has however a few limitations that are mentioned in 36
For bevel gears the calculation is based on force application at the tooth tip of the virtual cylindrical gear Subsequently the stress is converted to load application at the mid point of the flank due to the heightwise crowning
Bevel gears may also be calculated with the program BECAL In that case KA and Kv are to be included in the applied tooth force but not KFβ and KFα
In case of a thin annulus or a thin gear rim etc radial crack-ing can occur rather than tangential cracking (from root fillet to root fillet) Cracking can also start from the compression fillet rather than the tension fillet For rim thickness sR lt 35middotmn a special calculation procedure is given in 315 and 316 and a simplified procedure in 314
A tooth breakage is often the end of the life of a gear trans-mission Therefore a high safety SF against breakage is re-quired
It should be noted that this part 3 does not cover fractures caused by
bull oil holes in the tooth root space bull wear steps on the flank bull flank surface distress such as pits spalls or grey staining
Especially the latter is known to cause oblique fractures starting from the active flank predominately in spiral bevel gears but also sometimes in cylindrical gears
Specific calculation methods for these purposes are not given here but are under consideration for future revisions Thus depending on experience with similar gear designs limita-tions other than those outlined in part 3 may be applied
32 Tooth Root Stresses The local tooth root stress is defined as the max principal stress in the tooth root caused by application of the tooth force Ie the stress ratio R = 0 Other stress ratios such as for eg idler gears (R asymp -12) shrunk on gear rims (R gt 0) etc are considered by correcting the permissible stress level
321 Local tooth root stress The local tooth root stress for pinion and wheel may be as-sessed by strain gauge measurements or FE calculations or similar For both measurements and calculations all details are to be agreed in advance
Normally the stresses for pinion and wheel are calculated as
Cylindrical gears
FαFβvγAβSFn
tF K K K K K Y Y Y
m bFσ =
where
YF = Tooth form factor (see 33)
YS = Stress correction factor (see 34)
Yβ = Helix angle factor (see 36)
Ft KA Kγ Kv KFβ KFα see 15 ndash 110
b see 13
Bevel gears
FαFβvγASaFamn
mtF K K K K K Y Y Y
m bFσ ε=
where
YFa = Tooth form factor see 33
YSa = Stress correction factor see 34
Yε = Contact ratio factor see 35
Fmt KA etc see 15 ndash 110
b see 13
322 Permissible tooth root stress The permissible local tooth root stress for pinion respectively wheel for a given number of cycles N is
CXRrelTrelTF
NMFEFP YYYY
SYY
δσ
=σ
Note that all these factors YM etc are applicable to 3middot106 cycles when used in this formula for σFP The influence of other number of cycles on these factors is covered by the calculation of YN
where
σFE = Local tooth root bending endurance limit of reference test gear (see 37)
YM = Mean stress influence factor which accounts for other loads than constant load direction eg idler gears temporary change of load di-rection pre-stress due to shrinkage etc (see 38)
YN = Life factor for tooth root stresses related to reference test gear dimensions (see 39)
SF = Required safety factor according to the rules
YδrelT = Relative notch sensitivity factor of the gear to be determined related to the reference test gear (see 310)
YRrelT = Relative (root fillet) surface condition factor of the gear to be determined related to the reference test gear (see 311)
Classification Notes- No 412 23 May 2003
DET NORSKE VERITAS
YX = Size factor (see 312)
YC = Case depth factor (see 313)
33 Tooth Form Factors YF YFa The tooth form factors YF and YFa take into account the in-fluence of the tooth form on the nominal bending stress
YF applies to load application at the outer point of single tooth pair contact of the virtual spur gear pair and is used for cylindrical gears
YFa applies to load application at the tooth tip and is used for bevel gears
Both YF and YFa are based on the distance between the con-tact points of the 30˚-tangents at the root fillet of the tooth profile for external gears respectively 60˚ tangents for inter-nal gears
Figure 31 External tooth in normal section
Figure 32 Internal tooth in normal section
Definitions
n
2
n
Fn
enFn
Fe
F
α cosms
α cosmh6
Y
=
n
2
n
Fn
anFn
Fa
Fa
α cosms
α cosmh6
Y
=
In the case of helical gears YF and YFa are determined in the normal section ie for a virtual number of teeth
YFa differs from YF by the bending moment arm hFa and αFan and can be determined by the same procedure as YF with exception of hFe and αFan For hFa and αFan all indices e will change to a (tip)
The following formulae apply to cylindrical gears but may also be used for bevel gears when replacing
mn with mnm
zn with zvn
αt with αvt
β with βm
with undercut without undercut
Fig 33 Dimensions and basic rack profile of the teeth (finished profile)
Tool and basic rack data such as hfP ρfp and spr etc are re-ferred to mn ie dimensionless
331 Determination of parameters
( )n
n
prnfPnfP m
α cossαsin 1ρ
αtan h4πE
minusminusminusminus=
For external gears fPfP ρρ =
For internal gears ( )0z
195fPfP0
fPfP 10363156ρhxρρ
sdotminus+
+=
where
z0 = number of teeth of pinion cutter
x0 = addendum modification coefficient of pinion cutter
hfP = addendum of pinion cutter
ρfP = tip radius of pinion cutter
xhρG fpfp +minus=
24 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
τmE
2π
z2H
nnminus
minus=
with
3πτ = for external gears
6πτ = for internal gears
Htan zG2
nminusϑ=ϑ
(to be solved iteratively suitable start value6π
=ϑ for exter-
nal gears and 3π for internal gears)
a) Tooth root chord sFn For external gears
minus
ϑ+
ϑminus= ρ
cosG3
3πsinz
ms
fpnn
Fn
For bevel gears with a tooth thickness modification
xsm affects mainly sFn but also hFe and αFen The total influence of xsm on YFa Ysa can be approximated by only adding 2 xsm to sFn mn
For internal gears
minus
ϑ+
ϑminus= ρ
cosG
6πsinz
ms
fPnn
Fn
b) Root fillet radius ρF at 30ordm tangent
( )G2coszcosG2ρ
mρ
2n
2
fpn
F
minusϑϑ+=
c) Determination of bending moment arm hF dn = zn mn
b2α
αn βcosε
ε =
dan = dn + 2 ha
pbn = π mn cos αn
dbn = dn cos αn
( )4
d1εpzz
2dd
zz2d
2bn
2
αnbn
2bn
2an
en +
minusminus
minus=
en
bnen d
dcos arcα =
ennnn
e α invα invα x tan 22π
z1
minus+
+=γ
αFen = αen ndash γe
For external gears
( )
minus=
n
enFenee
n
Fe
mdαtan sin γ γcos
21
mh
]ρcos
G3πcosz fpn +
ϑminus
ϑminusminus
For internal gears
( ) minus
sdotsdotminus=
n
enFenee
n
Fe
mdα tanγ sinγ cos
21
mh
minus
ϑminus
ϑminussdot ρ
cosG3
6πcosz fPn
332 Gearing with εαn gt 2 For deep tooth form gearing ( )25ε2 αn lele produced with a verified grade of accuracy of 4 or better and with applied profile modification to obtain a trapezoidal load distribution along the path of contact the YF may be corrected by the factor YDT as
250ε205for 0666ε2366Y αnαnDT leleminus=
205εfor 10Y αnDT lt=
34 Stress Correction Factors YS YSa The stress correction factors YS and YSa take into account the conversion of the nominal bending stress to the local tooth root stress Thereby YS and YSa cover the stress increasing effect of the notch (fillet) and the fact that not only bending stresses arise at the root A part of the local stress is inde-pendent of the bending moment arm This part increases the more the decisive point of load application approaches the critical tooth root section
Therefore in addition to its dependence on the notch radius the stress correction is also dependent on the position of the load application ie the size of the bending moment arm
YS applies to the load application at the outer point of single tooth pair contact YSa to the load application at tooth tip
YS can be determined as follows
( )
+
+=L23121
1
sS qL01312Y
where Fe
Fn
hsL = and
F
Fns ρ2
sq = (see 33)
YSa can be calculated by replacing hFe with hFa in the above formulae
Note a) Range of validity 1 lt qs lt 8
In case of sharper root radii (ie produced with tools having too sharp tip radii) YS resp YSa must be specially considered
b) In case of grinding notches (due to insufficient protuberance of the hob) YS resp YSa can rise considerably and must be multiplied with
Classification Notes- No 412 25 May 2003
DET NORSKE VERITAS
g
g
ρt
0613
13
minus
where
tg = depth of the grinding notch
ρg = radius of the grinding notch
c) The formulae for YS resp YSa are only valid for αn = 20˚ However the same formulae can be used as a safe approximation for other pressure angles
35 Contact Ratio Factor Yε The contact ratio factor Yε covers the conversion from load application at the tooth tip to the load application at the mid point of the flank (heightwise) for bevel gears
The following may be used
Yε = 0625
36 Helix Angle Factor Yβ The helix angle factor Yβ takes into account the difference between the helical gear and the virtual spur gear in the nor-mal section on which the calculation is based in the first step In this way it is accounted for that the conditions for tooth root stresses are more favourable because the lines of contact are sloping over the flank
The following may be used (β input in degrees)
Yβ = 1 ndash εβ β120
When εβ gt 1 use εβ = 1 and when β gt 30deg use β = 30deg in the formula
However the above equation for Yβ may only be used for gears with β gt 25deg if adequate tip relief is applied to both pinion and wheel (adequate = at least 05 middot Ceff see 432)
37 Values of Endurance Limit σFE σFE is the local tooth root stress (max principal) which the material can endure permanently with 99 survival prob-ability 3106 load cycles is regarded as the beginning of the endurance limit or the lower knee of the σ ndash N curve σFE is defined as the unidirectional pulsating stress with a minimum stress of zero (disregarding residual stresses due to heat treatment) Other stress conditions such as alternating or pre-stressed etc are covered by the conversion factor YM
σFE can be found by pulsating tests or gear running tests for any material in any condition If the approval of the gear is to be based on the results of such tests all details on the testing conditions have to be approved by the Society Fur-ther the tests may have to be made under the Societys su-pervision
If no fatigue tests are available the following listed values for σFE may be used for materials subjected to a quality con-trol as the one referred to in the rules
σFE
Alloyed case hardened steels 1) (fillet surface hardness 58 ndash 63 HRC)
bull of specially approved high grade
1050
bull of normal grade
minus CrNiMo steels with approved process
1000
minus CrNi and CrNiMo steels generally 920 minus MnCr steels generally 850
Nitriding steel of approved grade quenched tempered and gas nitrided (surface hardness 700 ndash 800 HV)
840
Alloyed quenched and tempered steel bath or gas nitrided (surface hardness 500 ndash 700 HV)
720
Alloyed quenched and tempered steel flame or induction hardened 2) (incl entire root fillet) (fillet surface hardness 500 ndash 650 HV)
07 HV + 300
Alloyed quenched and tempered steel flame or induction hardened (excl entire root fillet) (σB = uts of base material)
025 σB + 125
Alloyed quenched and tempered steel 04 σB + 200
Carbon steel 025 σB + 250
Note All numbers given above are valid for separate forgings and for blanks cut from bars forged according to a qualified procedure see Pt 4 Ch 2 Sec 3 For rolled steel the values are to be reduced with 10 For blanks cut from forged bars that are not qualified as mentioned above the values are to be reduced with 20 For cast steel reduce with 40
1) These values are valid for a root radius
bull being unground If however any grinding is made in the root fillet area in such a way that the residual stresses may be affected σFE is to be reduced by 20 (If the grinding also leaves a notch see 34)
bull with fillet surface hardness 58 ndash 63 HRC In case of lower surface hardness than 58 HRC σFE is to be reduced with 20(58 ndash HRC) where HRC is the detected hardness (This may lead to a permissible tooth root stress that varies along the facewidth If so the actual tooth root stresses may also be considered along facewidth)
bull not being shot peened In case of approved shot peening σFE may be increased by 200 for gears where σFE is reduced by 20 due to root grinding Otherwise σFE may be increased by 100 for
6nm le and 100 ndash 5 (mn - 6) for mn gt 6 However the possible adverse influence on the flanks regarding grey staining should be considered and if necessary the flanks should be masked
2) The fillet is not to be ground after surface hardening Regarding possible root grinding see 1)
26 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
38 Mean stress influence Factor YM The mean stress influence factor YM takes into account the influence of other working stress conditions than pure pulsa-tions (R = 0) such as eg load reversals idler gears planets and shrink-fitted gears
YM (YMst) are defined as the ratio between the endurance (or static) strength with a stress ratio R ne 0 and the endurance (or static) strength with R = 0
YM and YMst apply only to a calculation method that assesses the positive (tensile) stresses and is therefore suitable for comparison between the calculated (positive) working stress σF and the permissible stress σFP calculated with YM or YMst
For thin rings (annulus) in epicyclic gears where the com-pression fillet may be decisive special considerations apply see 316
The following method may be used within a stress ratio ndash12 lt R lt 05
381 For idlers planets and PTO with ice class
M1M1R1
1Yor Y MstM
+minus
minus=
where
R = stress ratio = min stress divided by max stress
For designs with the same force applied on both forward- and back-flank R may be assumed to ndash 12
For designs with considerably different forces on forward- and back-flank such as eg a marine propulsion wheel with a power take off pinion R may be assessed as
branchmain theoffacewidth unit per forcepto offacewidt unit per force21minus
For a power take off (PTO) with ice class see 161 c
M considers the mean stress influence on the endurance (or static) strength amplitudes
M is defined as the reduction of the endurance strength am-plitude for a certain increase of the mean stress divided by that increase of the mean stress
Following M values may be used
Endurance limit
Static strength
Case hardened 08 ndash 015 Ys 1) 07
If shot peened 04 06
Nitrided 03 03
Induction or flame hardened
04 06
Not surface hardened steel 03 05
Cast steels 04 06 1)For bevel gears use Ys=2 for determination of M
The listed M values for the endurance limit are independent of the fillet shape (Ys) except for case hardening In princi-ple there is a dependency but wide variations usually only occur for case hardening eg smooth semicircular fillets versus grinding notches
382 For gears with periodical change of rotational direction For case hardened gears with full load applied periodically in both directions such as side thrusters the same formula for YM as for idlers (with R = ndash 12) may be used together with the M values for endurance limit This simplified approach is valid when the number of changes of direction exceeds 100 and the total number of load cycles exceeds 3middot106
For gears of other materials YM will normally be higher than for a pure idler provided the number of changes of direction is below 3middot106 A linear interpolation in a diagram with loga-rithmic number of changes of direction may be used ie from YM = 09 with one change to YM (idler) for 3middot106 changes This is applicable to YM for endurance limit For static strength use YM as for idlers
For gears with occasional full load in reversed direction such as the main wheel in a reversing gear box YM = 09 may be used
383 For gears with shrinkage stresses and unidirectional load For endurance strength
FE
fitM σ
σM1
M21Y+
minus=
σFE is the endurance limit for R = 0
For static strength YMst = 1 and σfit accounted for in 39b
σfit is the shrinkage stress in the fillet (30˚ tangent) and may be found by multiplying the nominal tangential (hoop) stress with a stress concentration factor
n
Ffit m
ρ215scf minus=
384 For shrink-fitted idlers and planets When combined conditions apply such as idlers with shrink-age stresses the design factor for endurance strength is
( ) ( ) FE
fitM σ
σR1M1
M2
M1M1R1
1Yminussdot+
minus
+minus
minus=
Symbols as above but note that the stress ratio R in this par-ticular connection should disregard the influence of σfit ie R normally equal ndash 12
For static strength
M1M1R1
1YMst
+minus
minus=
The effect of σfit is accounted for in 39 b
Classification Notes- No 412 27 May 2003
DET NORSKE VERITAS
385 Additional requirements for peak loads The total stress range (σmax ndash σmin) in a tooth root fillet is not to exceed
F
y
Sσ225
for not surface hardened fillets
FSHV5 for surface hardened fillets
39 Life Factor YN The life factor YN takes into account that in the case of limited life (number of cycles) a higher tooth root stress can be permitted and that lower stresses may apply for very high number of cycles
Decisive for the strength at limited life is the σ ndash N ndash curve of the respective material for given hardening module fillet radius roughness in the tooth root etc Ie the factors YδrelT YRelT YX and YM have an influence on YN
If no σ ndash N ndash curve for the actual material and hardening etc is available the following method may be used
Determination of the σ ndash N ndash curve
a) Calculate the permissible stress σFP for the beginning of the endurance limit (3middot106 cycles) including the influ-ence of all relevant factors as SF YδrelT YRelT YX YM and YC ie σFP = σFE middotYM middotYδrelT middotYRelT middotYX middotYC SF
b) Calculate the permissible laquostaticraquo stress (le 103 load cy-cles) including the influence of all relevant factors as SFst YδrelTst YMst and YCst ( )fitσCstYδrelTstYMstYFstσ
FstS1
FPstσ minussdotsdotsdot=
where σFst is the local tooth root stress which the material can resist without cracking (surface hardened materials) or unacceptable deformation (not surface hardened materials) with 99 survival probability
σFst
Alloyed case hardened steel 1) 2300
Nitriding steel quenched tempered and gas nitrided (surface hardness 700 ndash 800 HV)
1250
Alloyed quenched and tempered steel bath or gas nitrided (surface hardness 500 ndash 700 HV)
1050
Alloyed quenched and tempered steel flame or induction hardened (fillet surface hardness 500 ndash 650 HV)
18 HV + 800
Steel with not surface hardened fillets the smaller value of 2)
18 σB or 225 σy
1) This is valid for a fillet surface hardness of 58 ndash 63 HRC In case of lower fillet surface hardness than 58 HRC σFst is to be reduced with 30(58 ndash HRC) where HRC is the actual hardness Shot peening or grinding notches are not considered to have any significant influence on σFst 2) Actual stresses exceeding the yield point (σy or σ02) will alter the residual stresses locally in the ldquotensionrdquo fillet re-spectively ldquocompressionrdquo fillet This is only to be utilised for gears that are not later loaded with a high number of cycles at lower loads that could cause fatigue in the ldquocompressionrdquo fillet
c) Calculate YN as
NL gt 3middot106
YN = 1 or 001
L
6
N N103Y
sdot= ie Yn = 092 for 1010
The YN = 1 from 3middot106 on may only be used when special material cleanness applies see rules Pt4 Ch2
103ltNLlt3middot106exp
L
6
N N103Y
sdot=
cycles6103forσcycles310forσlog02876exp
FP
FPst
sdot=
NL lt 103 cycles6103forσcycles310forσ
NYFP
FPst
sdot=
or simply use σFPst as mentioned in b) directly
Guidance on number of load cycles NL for various applica-tions
bull For propulsion purpose normally NL = 1010 at full load (yachts etc may have lower values)
bull For auxiliary gears driving generators that normally operate with 70-90 of rated power NL = 108 with rated power may be applied
28 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
310 Relative Notch Sensitivity Factor YδrelT The dynamic (respectively static) relative notch sensitivity factor YδrelT (YδrelTst) indicate to which extent the theoreti-cally concentrated stress lies above the endurance limits (re-spectively static strengths) in the case of fatigue (respectively overload) breakage
YδrelT is a function of the material and the relative stress gra-dient It differs for static strength and endurance limit
The following method may be used
For endurance limit
for not surface hardened fillets
( )
024
s024
relT σ103133q21σ1012201351
Ysdotsdotminus
+sdotsdotminus+= minus
minus
δ
for all surface hardened fillets except nitrided
( )
106q21002451
Y srelT
++=δ
for nitrided fillets
( )
1347q2101421
Y srelT
++=δ
For static strength
for not surface hardened fillets1)
( )( )( )025
02
02502s
relTst σ3000821σ300 1Y0821Y
+
minus+=δ
for surface hardened fillets except nitrided
YδrelTst = 044 YS + 012
for nitrided fillets
YδrelTst = 06 + 02 YS 1) These values are only valid if the local stresses do not
exceed the yield point and thereby alter the residual stress level See also 39b footnote 2
311 Relative Surface Condition Factor YRrelT The relative surface condition factor YRrelT takes into ac-count the dependence of the tooth root strength on the sur-face condition in the tooth root fillet mainly the dependence on the peak to valley surface roughness
YRrelT differs for endurance limit and static strength
The following method may be used
For endurance limit
YRrelT = 1675 ndash 053 (Ry + 1)01 for surface hardened steels and alloyed quenched and tempered steels except nitrided
YRrelT = 53 ndash 42 (Ry + 1)001 for carbon steels
YδrelT = 43 ndash 326 (Ry + 1)0005
for nitrided steels
For static strength
YRrelTst = 1 for all Ry and all materials
For a fillet without any longitudinal machining trace Ry asymp Rz
312 Size Factor YX The size factor YX takes into account the decrease of the strength with increasing size YX differs for endurance limit and static strength
The following may be used
For endurance limit
YX = 1 for mn le 5 generally
YX = 103 ndash 0006 mn for 5 lt mn lt 30
YX = 085 for mn ge 30
for not surface hard-ened steels
YX = 105 ndash 001 mn for 5 lt mn ge 25
YX = 08 for mn ge 25
for surface hardened steels
For static strength
YXst = 1 for all mn and all materials
313 Case Depth Factor YC The case depth factor YC takes into account the influence of hardening depth on tooth root strength
YC applies only to surface hardened tooth roots and is dif-ferent for endurance limit and static strength
In case of insufficient hardening depth fatigue cracks can develop in the transition zone between the hardened layer and the core For static strength yielding shall not occur in the transition zone as this would alter the surface residual stresses and therewith also the fatigue strength
The major parameters are case depth stress gradient permis-sible surface respectively subsurface stresses and subsurface residual stresses
The following simplified method for YC may be used
YC consists of a ratio between permissible subsurface stress (incl influence of expected residual stresses) and permissible surface stress This ratio is multiplied with a bracket con-taining the influence of case depth and stress gradient (The empirical numbers in the bracket are based on a high number of teeth and are somewhat on the safe side for low number of teeth)
YC and YCst may be calculated as given below but calculated values above 10 are to be put equal 10
For endurance limit
+
+=nFFE
C m02ρt31
σconstY
Classification Notes- No 412 29 May 2003
DET NORSKE VERITAS
For static strength
+
+=nFFst
Cst m02ρt31
σconstY
where const and t are connected as
Hardening process
t = endurance limit const =
static strength const =
t550 640 1900
t400 500 1200 Case hard-ening
t300 380 800
Nitriding t400 500 1200
Induction- or flame hardening
tHVmin 11 HVmin 25 HVmin
For symbols see 213
In addition to these requirements to minimum case depths for endurance limit some upper limitations apply to case hard-ened gears
The max depth to 550 HV should not exceed
1) 13 of the top land thickness san unless adequate tip relief is applied (see 110)
2) 025 mn If this is exceeded the following applies addi-tionally in connection with endurance limit
minusminus= 025
mt
1Yn
max550C
314 Thin rim factor YB Where the rim thickness is not sufficient to provide full sup-port for the tooth root the location of a bending failure may be through the gear rim rather than from root fillet to root fillet
YB is not a factor used to convert calculated root stresses at the 30deg tangent to actual stresses in a thin rim tension fillet Actually the compression fillet can be more susceptible to fatigue
YB is a simplified empirical factor used to de-rate thin rim gears (external as well as internal) when no detailed calcula-tion of stresses in both tension and compression fillets are available
Figure 34 Examples on thin rims
YB is applicable in the range 175 lt sRmn lt 35
YB = 115 middot ln (8324 middot mnsR)
(for sRmn ge 35 YB = 1)
(for sRmn le 175 use 315)
σF as calculated in 321 is to be multiplied with YB when sRmn lt 35 Thus YB is used for both high and low cycle fatigue
Note This method is considered to be on the safe side for external gear rims However for internal gear rims without any flange or web stiffeners the method may not be on the safe side and it is advised to check with the method in 315316
315 Stresses in Thin Rims For rim thickness sR lt 35 mn the safety against rim cracking has to be checked
The following method may be used
3151 General The stresses in the standardised 30ordm tangent section tension side are slightly reduced due to decreasing stress correction factor with decreasing relative rim thickness sRmn On the other hand during the complete stress cycle of that fillet a certain amount of compression stresses are also introduced The complete stress range remains approximately constant Therefore the standardised calculation of stresses at the 30ordm tangent may be retained for thin rims as one of the necessary criteria
The maximum stress range for thin rims usually occurs at the 60ordm ndash 80ordm tangents both for laquotensionraquo and laquocompressionraquo side The following method assumes the 75ordm tangent to be the decisive Therefore in addition to the am criterion ap-plied at the 30ordm tangent it is necessary to evaluate the max and min stresses at the 75ordm tangent for both laquotensionraquo (loaded flank) fillet and laquocompressionraquo (back-flank) fillet For this purpose the whole stress cycle of each fillet should be considered but usually the following simplification is justified
Figure 35 Nomenclature of fillets
Index laquoTraquo means laquotensileraquo fillet laquoCraquo means laquocompressionraquo fillet
σFTmin and σFCmax are determined on basis of the nominal rim stresses times the stress concentration factor Y75
σFCmin and σFTmax are determined on basis of superposition of nominal rim stresses times Y75 plus the tooth bending stresses at 75ordm tangent
30 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
3152 Stress concentration factors at the 75ordm tangents The nominal rim stress consists of bending stresses due to local bending moments tangential stresses due to the tan-gential force Ft and radial shear stresses due to Fr
The major influence is given by the bending stresses The influence of the tangential stresses is minor and even though its stress concentration factor is slightly higher than for bending it is considered to be safe enough when the sum of these nominal stresses are combined with the stress concen-tration factor for bending The influence of the radial shear stress is neglected
The stress concentration factor relating nominal rim stresses to local fillet stresses at the 75ordm tangent may be calculated as
75n
R75
n
R
75
ρmsρ185
ms3
Y+
sdot=
where ρ75 is the root radius at the 75ordm tangent ref to mn Usually ρ75 is closed to the tool radius ρao and
ρ75 = ρao
is a safe approximation compensating for the am simplifi-cations to the laquounsaferaquo side
The tooth root stresses of the loaded tooth are decreasing with decreasing relative rim thickness approximately with the empirical factor
15
R
ncorr s
m311Y
minus=
3153 Nominal rim stresses The bending moment applied to the rim consists of a part of the tooth tilt moment Ft (hF + 05 sR) and the bending caused by the radial force Fr
The sectional modulus (first moment of area) which is used for determination of the nominal bending stresses is not nec-essarily the same for the 2 am bending moments If flanges webs etc outside the toothed section contribute to stiffening the rim against various deflections the influence of these stiffeners should be considered Eg an end flange will have an almost negligible influence on the effective sectional modulus for the stresses due to tooth tilt as the deflection caused by the tilt moment is rather small and would not much involve the flange On the other hand the radial forces as for instance from the meshes in an annulus would cause considerable radial deflections that the flange might restrict to a substantial amount When considering the stiff-ening of such flanges or webs on basis of simplified models it is advised to use an effective rim thickness sR = sR + 02 mn for the first moment of area of the rim (toothed part) cross section
For a high number of rim teeth it may be assumed that the rim bending moments in the fillets adjacent to the loaded tooth are of the same magnitude as right under the applied
force This assumption is reasonable for an annulus but rather much on the safe side for a hollow pinion
The influence of Ft on the nominal tangential stress is simpli-fied by half of it for compressive stresses (σ1) and the other half for tensile stresses (σ2) Applying these assumptions the nominal rim stresses adjacent to the loaded tooth are
( ) ( )A2
FW
f RFW
s05hF05 t
R
r
T
RFt1 minus
ϑminus
+minus=σ
( ) ( )A2
FW
f RFW
s05hF05 t
R
r
T
RFt2 +
ϑminus
+=σ
where σ1 σ2 see Fig 35
A = minimum area of cross section (usually bR sR)
WT = the sectional modulus of rim with respect to tooth tilt moment (usually 6sb 2
rR )
WR = the sectional modulus of the rim including the influence of stiffeners as flanges webs etc (WR ge WT)
bR = the width of the rim
R = the radius of the neutral axis in the rim ie from wheel centre to midpoint of rim
( )ϑf = a function for bending moment distribution around the rim For a rim (pinion) with one mesh only the ( )ϑf at the position of load application is 024 For an annulus with 3 or more meshes ( )ϑf at the position of each load application is approx 3 planets ( )ϑf = 019 4 planets ( )ϑf = 014 5 planets ( )ϑf = 011 6 planets ( )ϑf = 009
It must be checked if the max (tensile) stress in the compres-sion fillet really occurs when the fillet is adjacent to the loaded tooth In principle the stress variation through a complete rotation should be considered and the max value used The max value is usually never less than 0 For an annulus eg the tilt moment is zero in the mid position be-tween the planet meshes whilst the bending moment due to the radial forces is half of that at the mesh but with opposite sign
If these formulae are applied to idler gears as eg planets the influence of nominal tangential stresses must be cor-rected by deleting Ft(2 A) for σ1 and using FtA for σ2 Further the influence of Fr on the nominal bending stresses is usually negligible due to the planet bearing support
3154 Root fillet stresses Determination of min and max stresses in the laquotensionraquo fillet
Classification Notes- No 412 31 May 2003
DET NORSKE VERITAS
Minimum stress
751FTmin YσK σ =
Maximum stress
corrF752 Yσ03Yσ KσFTmax +=
where
03 is an empirical factor relating the tension stresses (σF) at the 30ordm tangent to the part of the tension stresses at the 75ordm tangent which add to the rim related stresses (03 also takes into account that full superposition of nominal stresses times stress concentration factors from both laquosides of the corner filletraquo would result in too high stresses)
αβγ sdotsdotsdotsdot= FFvA KKKKKK
Determination of min and max stresses in the laquocompres-sionraquo fillet
Minimum stress
corrF751FCmin Yσ036Yσ Kσ minus=
Maximum stress
752FCmax Yσ Kσ =
where
036 is an empirical factor relating the tension stresses (σF) at the 30ordm tangent to the part of the compression stresses at the 75ordm tangent which add to the rim related stresses
For gears with reversed loads as idler gears and planets there is no distinct laquotensionraquo or laquocompressionraquo fillet The mini-mum stress σFmin is the minimum of σFTmin and σFCmin (usually the latter is decisive) The maximum stress σFTmax is the maximum of σFTmax and σFCmax (usually the former is deci-sive)
316 Permissible Stresses in Thin Rims
3161 General The safety against fatigue fracture respectively overload fracture is to be at least at the same level as for solid gears The laquoordinaryraquo criteria at the 30ordm tangent apply as given in 31 through 313
Additionally the following criteria at the 75ordm tangent may apply
3162 For gt3middot106 cycles The permissible stresses for the laquotensionraquo fillets and for the laquocompressionraquo fillets are determined by means of a relevant fatigue diagram
If the actual tooth root stress (tensile or compressive) ex-ceeds the yield strength to the material the induced residual stresses are to be taken into account
For determination of permissible stresses the following is defined
R = stress ratio ie FTmax
FTminσσ respectively
FCmax
FCmin
σσ
∆σ = stress range ie σFTmax ndash σFTmin resp σFCmax ndash σFCmin
(For idler gears and planets Fmax
FminσσR =
and ∆σ = σFmax minus σFmin)
The permissible stress range ∆σρ for the laquotensionraquo respec-tively laquocompressionraquo fillets can be calculated as
For R gt minus1 FPp σ
R1R1031
13∆σ
minus+
+=
For minus infin lt R lt minus1 FPp σ
R1R10151
13∆σ
minus+
+=
where
σFP = see 32 determined for unidirectional stresses (YM = 1)
If the yield strength σy is exceeded in either tension or com-pression residual stresses are induced This may be consid-ered by correcting the stress ratio R for the respective fillets (tension or compression)
Eg if yFCmin σσ gt (ie exceeded in compression) the
difference yFCmin σσ∆ minus= affects the stress ratio as
∆σ
σ∆σ∆σR
FCmax
y
FCmax
FCmin
+
minus=
++
=
Similarly the stress ratio in the tension fillet may require cor-rection
If the yield strength is exceeded in tension σFTmax gt σy the difference ∆ = σFTmax ndash σy affects the stress ratio as
y
FTmin
FTmax
FTmin
σ∆σ
∆σ∆σR minus=
minusminus
=
Checking for possible exceeding of the yield strength has to be made with the highest torque and with the KAP if this ex-ceeds KA
3163 Forle 103 cycles The permissible stress range ∆σp is not to exceed
For R gt minus1 FPstpst σ
R1R1051
15∆σ
minus+
+=
For ndash infin lt R lt minus1 FPstpst σ
R1R10251
15∆σ
minus+
+=
For all values of R ∆σpst is limited by
not surface hardened F
y
Sσ225
32 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
surface hardened CF
YSHV5
Definition of ∆σ and R see 3162 with particular attention to possible correction of R if the yield strength is exceeded
σFPst see 32 determined for unidirectional stresses (YM = 1) and lt 103 cycles
3164 For 103 lt cycles lt 3106 ∆σp is to be determined by linear interpolation a log-log dia-gram
∆σp at NL load cycles is
6L 103p
exp
L
6
N p ∆σN103∆σ sdot
sdot=
6
3
103p
10p
∆σ
∆σlog28760exp
sdot
=
Classification Notes- No 412 33 May 2003
DET NORSKE VERITAS
4 Calculation of Scuffing Load Capacity
41 Introduction High surface temperatures due to high loads and sliding ve-locities can cause lubricant films to break down This sei-zure or welding together of areas of tooth surface is termed scuffing
In contrast to pitting and fatigue breakage which show a dis-tinct incubation period a single short overloading can lead to a scuffing failure In the ISO-TR13989 two criteria are mentioned The method used in this Classification Note is based on the principles of the flash temperature criterion
Note Bulk temperature in excess of 120ordmC for long periods may have an adverse effect on the surface durability see 211
42 General Criteria In no point along the path of contact the local contact tem-perature may exceed the permissible temperature ie
oilS
oilSB S
ϑ+ϑminusϑ
leϑ
50SB minusϑleϑ
where
Bϑ = max contact temperature along the path of contact
maxflaMBB ϑ+ϑ=ϑ
MBϑ = bulk temperature see 434
maxflaϑ = max flash temperature along the path of con-tact see 44
Sϑ = scuffing temperature see below
oilϑ = oil temperature before it reaches the mesh (max applicable for the actual load case to be used ie normally alarm temperature except for ice classes where a max expected tem-perature applies)
SS = required safety factor according to the Rules
The scuffing temperature Sϑ may be calculated as
L2
wrelT
002
40S XFZGX
ν100112085780
sdot
sdot++=ϑ
where
XwrelT = relative welding factor
XwrelT
Through hardened steel 10
Phosphated steel 125
Copper-plated steel 150
Nitrided steel 150
Less than 10 retained austenite
115
10 ndash 20 retained austenite 10
Casehardened steel
20 ndash 30 retained austenite 085
Austenitic steel 045
FZG = load stage according to FZG-Test A8390 (Note This is the load stage where scuffing oc-curs However due to scatter in test results cal-culations are to be made with one load stage less than the specification)
XL = lubricant factor
= 10 for mineral oils
= 08 for polyalfaolefins
= 07 for non-water-soluble polyglycols
= 06 for water-soluble polyglycols
= 15 for traction fluids
= 13 for phosphate esters
ν40 = kinematic oil viscosity at 40˚C (mm2s)
Application of other test methods such as the Ryder the FZG-Ryder R46574 and the FZG L-42 Test 141195110 may be specially considered
For high speed gears with very short time of contact Sϑ may be increased as follows provided use of EP-oils
Addition to the calculated scuffing temperature Sϑ
If micros 18tc ge no addition
If micros 18tc lt add ( )cwrelT t18X18 minussdotsdot
where
ct = contact time ( micros ) which is the time needed to cross the Hertzian contact width
( ) [ ]microsu1cosβn
uσ340tb1
Hc +sdotsdot
sdot=
σH as calculated in 221
For bevel gears use vu in stead of u
34 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
43 Influence Factors
431 Coefficient of friction The following coefficient of friction may apply
L025a
005oil
02
redC ΣC
Bt X R ηρv
w0048micro minus
=
where
wBt = specific tooth load (Nmm)
vΣC = sum of tangential velocities at pitch point At pitch line velocities gt 50 ms the limiting value of vΣC at v = 50 ms is to be used
ρredC = relative radius of curvature (transversal plane) at the pitch point
Cylindrical gears
HαHβvγAbt
Bt KKKKKb
Fw sdotsdotsdotsdotsdot= (see 1)
twΣC αsin v2v =
bCredC β cos ρρ =
Bevel gears
HαHβvγAtmb
Bt KKKKKb
Fw sdotsdotsdotsdotsdot= (see 1)
vtmtΣC αsin v2v =
bmvCredC β cos ρρ =
ηoil = dynamic viscosity (mPa s) at oilϑ calculated as
1000
ρ νη oiloil =
where ρ in kgm3 approximated as
( ) 70 15oil15 minusϑminusρ=ρ and νoil is kinematic viscosity at oilϑ and may be calculated by means of the following equation
( ) ( )++=+ 08νloglog08νloglog 100oil ( )
sdotminus
ϑ+minus313log373log
273log373log oil
( ) ( )( )08νloglog08νloglog 10040 +minus+
Ra = composite arithmetic mean roughness (micron) of pinion an wheel calculated as ( )2a1aa RR 05R +=This is defined as the roughness on the new flanks ie as manufactured
XL = see 42
432 Effective tip relief Ceff Ceff is the effective tip relief that amount of tip relief which compensates for the elastic deformation of the gear mesh ie
zero load at the tooth tip It is assumed (simplified) to be equal for both pinion and wheel
Cylindrical gears
for helical
γ
γ=cb
KKFC Abt
eff (see 1)
For spur
cbKKF
C Abteff
γ= (see 1)
Alternatively for spur and helical gears the non-linear ap-proach in 111 may be used (taking Ceff=δ)
Bevel gears
bcKFC Ambt
effγ
= (see 1)
where
γ
α
ε2ε44c
+sdot
=γ
433 Tip relief and extension Cylindrical gears
The extension of the tip relief is not to result in an effective contact ratio 1ltεα when the gear is unloaded (exceptions to this may only apply for applications where the gear is not to run at light loads) This means that the unrelieved part of the path of contact is to be minimum pbt It is further assumed that this unrelieved part is placed centrally on the path of contact
If root relief applies it has to be calculated as equivalent tip relief Ie pinion root relief (at mesh position A) is added to Ca2 and wheel root relief (at mesh position E) is added to Ca1 If no design tip relief or root relief on the mating gear is specified (ie if Ca1+Croot2 = 0 and visa versa) use the run-ning in amount see 112
Bevel gears
Bevel gears are to have heightwise crowning ie no distinct relievedunrelieved area This may be treated as tip and root relief For calculation purposes the root relief is combined with the tip relief of the mating member into an equivalent tip relief If no resulting tip relieves are specified the equivalent tip relives may be calculated as an approxima-tion based on the tool crowning Ca tool (per mille of tool module m0) as follows
2root1aeq1a CCC +=
1root2aeq2a CCC +=
where
Classification Notes- No 412 35 May 2003
DET NORSKE VERITAS
2
0
n0101atoolal m
)m2(mA1middotmCC
minus+minussdot=
( )
sdotminusminussdotsdotminus+= vtvb1
2vb1
21vavt1n1 αtan dddαsin 05x1mA
2
0
n020atool22a m
)mm(2A1middotmCC
minus+minussdot=
( )
sdotminusminussdotsdotminus+= vtvb2
2vb2
2va2vt2n2 αtan dddαsin 05x1mA
2
0
20atool1root1 m
A1mCC
minussdotsdot=
2
0
10atool2root2 m
A1mCC
minussdotsdot=
If the pressure angles of the cutter blades are modified (and verified) to balance the tip relieves the following may be assumed
( )eq2aeq1aeq2aeq1a C andC calculated of sum50CC sdot==
Throughout the following Ca1 and Ca2 mean the equivalent tip relieves Ca1eq and Ca2eq
434 Bulk temperature The bulk temperature may be calculated as
flaaveragempsoilMB X X 50 ϑ+ϑ=ϑ
where
Xs = lubrication factor
= 12 for spray lubrication
= 10 for dip lubrication (provided both pinion and wheel are dip lubricated and tip speed lt 5 ms)
= 10 for spray lubrication with additional cooling spray (spray on both pinion and wheel or spray on pinion and dip lubrication of wheel)
= 02 for meshes fully submerged in oil
Xmp = contact factor ( )pmp nX += 150
np = number of mesh contact on the pinion (for small gear ratii the number of wheel meshes should be used if higher)
flaaverageϑ
= average of the integrated flash temperature (see 44) along the path of contact
( )
AE
E
Ayyyfla
flaaverage
d
ΓminusΓ
ΓΓϑ=ϑint =
For high speed gears (v gt 50 ms) it may be necessary to assess the bulk temperature on the basis of a thermal rating of the entire gear transmission
44 The Flash Temperature flaϑ
441 Basic formula The local flash temperature flaϑ may be calculated as
( ) 41redy
y2ly
211
43Btcorrfla
unXwX3250
y ρ
ρminusρ
micro=ϑ Γ
(For bevel gears replace u with uv)
and is to be calculated stepwise along the path of contact from A to E
where
micro = coefficient of friction see 431
Xcorr = correction factor taking empirically into account the increased scuffing risk in the beginning of the approach path due to mesh starting without any previously built up oil film and possible shuffling away oil before meshing if insufficient tip relief
( )
3
AD
yaeffcorr 50
CC1X
ΓminusΓε
Γminus+=
α
Ca = tip relief of driven member Xcorr is only applicable in the approach path and if Ca lt Ceff otherwise 10
wBt = unit load see 431
yXΓ = load sharing factor see 443
n1 = pinion rpm
Γy ρly etc see 442
442 Geometrical relations The various radii of flank curvature (transversal plane) are
ρ1y = pinion flank radius at mesh point y
ρ2y = wheel flank radius at mesh point y
ρredy = equivalent radius of curvature at mesh point y
y2y1
y2y1redy
ρ+ρ
ρρ=ρ
Cylindrical gears
twy
1y αsin au1Γ1
ρ+
+=
twy
2y αsin au1Γu
ρ+
minus=
Note that for internal gears a and u are negative
36 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
Bevel gears
vtvv
y1y αsin a
u1Γ1
ρ+
+=
vtvv
yv2y αsin a
u1Γu
ρ+
minus=
Γ is the parameter on the path of contact and y is any point between A end E
At the respective ends Γ has the following values
Root piniontip wheel
Cylindrical gears
( )
minus
minusminus= 1
αtan 1dd
zzΓ
tw
2b2a2
1
2A
Bevel gears
( )
minus
minusminus= 1
αtan 1dd
uΓvt
2vb2va2
vA
Tip pinionroot wheel
Cylindrical gears
( )
1αtan
1ddΓ
tw
2b1a1
E minusminus
=
Bevel gears
( )
1αtan
1ddΓ
vt
2vb1va1
E minusminus
=
At inner point of single pair contact
Cylindrical gears
tw1
EB α tan zπ2ΓΓ minus=
Bevel gears
vtv1
EB α tan zπ2ΓΓ minus=
At outer point of single pair contact
Cylindrical gears
tw1AD α tan z
π2ΓΓ +=
Bevel gears
vt v1
AD αtan z π2ΓΓ +=
At pitch point ΓC = 0
The points F and G (only applicable to cylindrical gears) limiting the extension of tip relief (so as to maintain a mini-mum contact ratio of unity for unloaded gears) are at
2
BAF
Γ+Γ=Γ
2
EDG
Γ+Γ=Γ
443 Load sharing factor XΓ The load sharing factor XΓ accounts for the load sharing be-tween the various pairs of teeth in mesh along the path of contact
XΓ is to be calculated stepwise from A to E using the pa-rameter Γy
4431 Cylindrical gears with β = 0 and no tip relief
Figure 41
ByAAB
Ay for31
31X
yΓltΓleΓ
ΓminusΓ
ΓminusΓ+=Γ
DyB for1Xy
ΓltΓleΓ=Γ
EyDDE
yE for31
31X
yΓleΓltΓ
ΓminusΓ
ΓminusΓ+=Γ
4432 Cylindrical gears with β = 0 and tip relief Tip relief on the pinion reduces XΓ in the range G ndash E and increases correspondingly XΓ in the range F ndash B
Tip relief on the wheel reduces XΓ in the range A ndash F and increases correspondingly XΓ in the range D ndash G
Following remains generally
DyB for1Xy
ΓleΓleΓ=Γ
21XXGF== ΓΓ
In the following it must be distinguished between Ca lt Ceff respectively Ca gt Ceff This is shown by an example below where Ca1 lt Ceff and Ca2 gt Ceff
Classification Notes- No 412 37 May 2003
DET NORSKE VERITAS
Figure 42
Note When Ca gt Ceff the path of contact is shortened by A ndash A respectively E ndash E The single pair contact path is extended into B respectively D If this shift is significant it is necessary to consider the negative effect on surface durability (B) and bending stresses (B and D)
Range A - F
For effa2 CC le
minus=Γ
eff
2a
CC1
31X
A
FyAeff
2a
AF
Ay for C3
C61XX
AyΓleΓleΓ
sdot
+ΓminusΓ
ΓminusΓ+= ΓΓ
For effa2 CC ge
AyAfor0Xy
ΓleΓleΓ=Γ
with ( )21
CC
1CC
eff
2a
eff
2a
AFAAminus
minusΓminusΓ+Γ=Γ
FyAeff
2a
AF
Ay
eff
2a for 21
CC
CC1X
yΓleΓleΓ
minus
ΓminusΓ
ΓminusΓ+minus=Γ
Range F - B
For effa1 CC le
ByFeff
1a
FB
Fy forC3
C61
21X
yΓleΓleΓ
sdot
+ΓminusΓ
ΓminusΓ+=Γ
For effa1 CC ge
ByFeff
1a
FB
Fy for21
CC
21X
yΓleΓleΓ
minus
ΓminusΓ
ΓminusΓ+=Γ
ByB for1Xy
ΓleΓleΓ=Γ
with 1
CC2
eff
1a
FBFB
minus
ΓminusΓ+Γ=Γ
Range D ndash G
For effa2 CC le
GyDeff
2a
DG
Dy
eff
2a for C 3C
61
C 3C
32X
yΓleΓleΓ
+
ΓminusΓ
ΓminusΓminus+=Γ F
or effa2 CC ge
DyD for1Xy
ΓleΓleΓ=Γ
with ( )21
CC
1CC
eff
2a
eff
2a
DGDDminus
minusΓminusΓ+Γ=Γ
GyDeff
2a
DG
Dy
eff
2a for21
CC
CCX ΓleΓleΓ
minus
ΓminusΓ
ΓminusΓminus=Γ
Range G - E
For effa1 CC le
minus=Γ
eff
1a
CC1
31X
E
EyGeff
1a
GE
Gy forC3
C61
21X
yΓleΓleΓ
sdot
+ΓminusΓ
ΓminusΓminus=Γ
For effa1 CC gt
EyGeff
1a
GE
Gy for21
CC
21X
yΓleΓleΓ
minus
ΓminusΓ
ΓminusΓminus=Γ
EyE for0Xy
ΓleΓleΓ=Γ
with 1
CC2
eff
1a
GEGE
minus
ΓminusΓ+Γ=Γ
4433 Gears with β gt 0 buttressing Due to oblique contact lines over the flanks a certain but-tressing may occur near A and E
This applies to both cylindrical and bevel gears with tip relief lt Ceff The buttressing Xbutt is simplified as a linear function within the ranges A ndash H respectively I ndash E
Figure 43
1εwhen13X βbutt EAge=
1εwhenε031X ββbutt EAlt+=
38 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
Cylindrical gears
bIEAH βsin 02ΓΓΓΓ =minus=minus
Bevel gears
bmIEAH βsin 02ΓΓΓΓ =minus=minus
4434 Cylindrical gears with 2εγ le and no tip relief
yXΓ is obtained by multiplication of
yXΓ in 4431 with
Xbutt in 4433
4435 Gears with 2εγ gt and no tip relief
Applicable to both cylindrical and bevel gears
IyH for1Xy
ΓleΓleΓε
=α
Γ
IyHybutt and forX1Xy
ΓgtΓΓltΓε
=α
Γ
Figure 44
4436 Cylindrical gears with 2εγ le and tip relief
yXΓ is obtained by multiplication of
yXΓ in 4432 with Xbutt
in 4433
4437 Cylindrical gears with 2εγ gt and tip relief
Tip relief on the pinion (respectively wheel) reduces XГ in the range G ndash E (respectively A ndash F) and increases XГ in the range F ndash G
yXΓ is obtained by multiplication of
yXΓ as described below
with Xbutt in 4433
In the XГ example below the influence of tip relief is shown (without the influence of Xbutt) by means of
eff2aeff1a CC and CC ltgt
Tip relief gt Ceff causes new end points A respectively E of the path of contact
Figure 45
Range A ndash F
( ) ( )( ) eff
a2αa1α
AF
Ay
eff
2aeff
C 12C 13εC 1ε
CCCX
y +εε++minus
ΓminusΓ
ΓminusΓ+
εminus
=ααα
Γ
eff2aFyA CC if for leΓleΓleΓ
eff2aFyA CifCfor and geΓleΓleΓ
eff2aAyA CC iffor0Xy
gtΓleΓleΓ=Γ
( ) ( ) ( )( ) ( ) a2αa1α
αeffa2AFAA C 1ε 3C 1ε
1ε 2 CC with++minus+minus
ΓminusΓ+Γ=Γ
Range F ndash G
( )( )( ) GyF
eff
2a1a forC 1 2CC 11X
yΓleΓleΓ
+εε+minusε
+ε
=αα
α
αΓ
Range G ndash E
( ) ( )( ) eff
2a1a
GE
Gy
C 1 2C 1C 1 3XX
GFy +εεminusε++ε
ΓminusΓ
ΓminusΓminus=
αα
ααΓΓ minus
effa1EyG CC if for leΓleΓleΓ
effa1EyG CC if Γfor and geΓleΓle
eff1aEyE CC if for0Xy
geΓleΓleΓ=Γ
( ) ( ) ( )( ) ( ) 2a1a
eff1aGEEE C 1C 1 3
1 2 CC withminusε++ε+εminus
ΓminusΓminusΓ=Γαα
α
4438 Bevel gears with εγ more than approx 18 and heightwise crowning For Ca1 = Ca2 = Ceff the following applies
Figure 46
Classification Notes- No 412 39 May 2003
DET NORSKE VERITAS
( )AEM 50 Γ+Γ=Γ
( )( )2AD
3
2My651X
y ΓminusΓε
ΓminusΓminus
ε=
ααΓ
For tip relief lt Ceff yXΓ is found by linear interpolation be-
tween ( )effay CCX =Γ and ( )0CayX =Γ as in 4435
The interpolation is to be made stepwise from A to M with the influence of Ca2 and from M to E with the influence of Ca1 (For 2a1a CC ne there is a discontinuity at M)
Eg with Ca1 = 04 Ceff and Ca2 = 055 Ceff then
Range A ndash M
)CC()0C( eff2ay2ayyX 550X 450X =Γ=ΓΓ +=
Range M ndash E
)CC()0C( eff1ay1ayyX 40X 06X =Γ=ΓΓ +=
For tip relief gt Ceff the new end points A and E are found as
( )
minusΓminusΓ
ε+Γ=Γ α 1
CC
6 eff
2aADAA
( )
minusΓminusΓ
ε+Γ=Γ α 1
CC
6 eff
1aADEE
Range A ndash A
0Xy=Γ
Range A ndash M
( )( )
ΓminusΓ
ΓminusΓminus
minusε=
αΓ 2
MA
2My
eff
2a1
CC4
351Xy
Range M ndash E
( )( )
ΓminusΓ
ΓminusΓminus
minusε=
αΓ 2
ME
2My
eff
1a1
CC4
351Xy
Range E ndash E
0Xy=Γ
40 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
Appendix A Fatigue Damage Accumulation
The Palmgren-Miner cumulative damage calculation princi-ple is used The procedure may be applied as follows
A1 Stress Spectrum From the individual torque classes the torques (Ti) at the peak values of class intervals and the associated number of cycles (NLi) for both pinion and wheel are to be listed from the highest to the lowest torque
(In case of a cyclic torque variation within the torque classes it is advised to use the peak torque If the cyclic variation is such that the same teeth will repeatedly suffer the peak torque this is a must)
The stress spectra for tooth roots and flanks (σFi σHi) with all relevant factors (except KA) are to be calculated on the basis of the torque spectrum The load dependent K-factors are to be determined for each torque class
A2 σminusN-curve The stress versus load cycle curves for tooth roots and flanks (both pinion and wheel) are to be drawn on the basis of per-missible stresses (ie including the demanded minimum safety factors) as determined in 2 respectively 3 If different safety levels for high cycle fatigue and low cycle fatigue are desired this may be expressed by different demand safety factors applied at the endurance limit respectively at static strength
A3 Damage accumulation The individual damage ratio Di at ith stress level is defined as
Fi
Lii N
ND =
where
NLi = The number of applied cycles at ith stress
NFi = The number of cycles to failure at ith stress
Basically stresses σi below the permissible stress level for infinite life (if a constant ZN or YN is accepted) do not con-tribute to the damage sum However calculating the actual safety factor Sact as described below all the σi for which the product S σi is bigger than or equal to the permissible stress level for infinite life contribute to the damage sum and thus to the determination of Sact The final value of S is decisive
(NFi can be found mathematically by putting the permissible stress σpi equal the actual stress σi thereby finding the actual life factor This life factor can be solved with regard to load cycles ie NFi)
The damage sum ΣDi is not to exceed unity
If ΣDi ne 1 the safety against cumulative fatigue damage is different from the applied demand safety factor For deter-mination of this theoretical safety factor an iteration proce-dure is required as described in the following flowchart
S is correction factor with which the actual safety factor Sact can be found
Sact is the demand safety factor (used in determination of the permissible stresses in the σ ndash N ndash curve) times the correc-tion factor S
The full procedure is to be applied for pinion and wheel tooth roots and flanks
Note If alternating stresses occur in a spectrum of mainly pulsating stresses the alternating stresses may be replaced by equiva-lent pulsating stresses ie by means of division with the ac-tual mean stress influence factor YM
Classification Notes- No 412 41 May 2003
DET NORSKE VERITAS
Appendix B Application Factors for Diesel Driven Gears
For diesel driven gears the application factor KA depends on torsional vibrations Both normal operation and misfiring conditions have to be considered
Normally these two running conditions can be covered by only one calculation
B1 Definitions
Normal operation KAnorm = 0
normv0
TTT +
where
T0 = rated nominal torque
Tv norm = vibratory torque amplitude for normal opera-tion (see rules Pt4 Ch3 Sec1 G301 for defi-nition of ldquonormalrdquo irregularity)
Misfiring operation KA misf = 0
misfv
TTT +
where
T = remaining nominal torque when one cylinder out of action
Tv misf = vibratory torque amplitude in misfiring con-dition This refers to a permissible misfiring condition ie a condition that does not require automatic or immediate corrective actions as speed or pitch reduction
The normal operation is assumed to last for a very high num-ber of cycles such as 1010
The misfiring operation is assumed to last for a limited dura-tion such as 107 cycles
B2 Determination of decisive load Assuming life factor at 1010 cycles as YN = ZN = 092 which usually is relevant the calculation may be performed only once with the combination having the highest value of appli-cation factorlife factor
For bending stresses and scuffing the higher value of
092
K normA and 098
K misfA
For contact stresses the higher value of
092K normA and
113K misfA (but
097K misfA for nitrided gears)
B3 Simplified procedure Note that this is only a guidance and is not a binding con-vention
T = oTZ
1Zsdot
minus where Z = number of cylinders
( ) ideal Videal Vmisf Vnorm V TTT24ZT +minus=
where TV ideal = vibratory torque with all cylinders perfectly equal See also rules Pt4 Ch3 Sec1 G300
When using trends from torsional vibration analysis and measurements the following may be used
TV ideal To is close to zero for engines with few cylinders and using a suitable elastic coupling and increases with relative coupling stiffness and number of cylinders
TV misf To may be high for engines with few cylinders and decreases with number of cylinders
This can be indicated as
200Z
TT
ο
idealV asymp and 80Z04
TT
o
misfV minusasymp
Inserting this into the formulae for the two application fac-tors the following guidance can be given
118112K misfA minusasymp
115110K normA minusasymp
Since KA norm is to be combined with the lower life factors the decisive load condition will be the normal one and a KA of 115 will cover most relevant cases when a suitable elas-tic coupling is chosen
42 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
Appendix C Calculation of Pinion-Rack
Pinion-racks used for elevating of mobile offshore units are open gears that are subjected to wear and tear With normal specifications such as only a few hundred total operation cycles (site to site) the tooth bending stresses for static strength will be decisive for the lay out however with ex-ception of surface hardened pinions where case crushing has to be considered
In the following the use of part 1 and 3 for pinion-racks is shown including relevant simplifications
C1 Pinion tooth root stresses Since the load spectrum normally is dominated by high torques and few load cycles (in the range up to 1-2000) the static strength is decisive
The actual stress is calculated as
β1FSaFan1
tF1 KYY
mbFσ sdotsdotsdotsdot
=
b1 is limited to b2 + 2middotmn
YFa and YSa replace YF and YS because load application at tooth tip has to be assumed for such inaccurate gears
Pinions often use a non-involute profile in the dedendum part of the flank eg a constant radius equal the radius of curva-ture at reference circle For such pinions sFn and hFa are to be measured directly on a sectional drawing of the pinion tooth
Due to high loads and narrow facewidths it may be assumed that KFβ2 = KHβ = 10 However when b1 gt b2 then KFβ1 gt10
If no detailed documentation of KFβ1 is available the fol-lowing may be used
KFβ1 = 1 + 015middot(b1b2 ndash 1)
The permissible stress (not surface hardened) is calculated as
δrelTstF
Fst1FP1 Y
Sσσ sdot=
The mean stress influence due to leg lifting may be disre-garded
The actual and permissible stresses should be calculated for the relevant loads as given in the rules
C2 Rack tooth root stresses The actual stress is calculated as
SaFan2
tF2 YY
mbFσ sdotsdotsdot
=
See C1 for details
The permissible stress is calculated as
δrelTstF
Fst2FP2 Y
Sσσ sdot=
For alloyed steels (Ni Cr Mo) with high toughness and duc-tility the value of YδrelTst may be put equal to YSa
C3 Surface hardened pinions For surface hardened pinions the maximum load is not to cause crushing of the hardened layer of the flank
In principle the calculation described in 213 may be used but when the theoretical Hertzian stress exceeds the ap-proximately 18 times the yield strength of the rack material plastic deformation will occur This will limit the peak Hertzian stress but increases the contact width and thus the penetration of stresses into the depth
An approximation may be based on an assessment of contact width determined by means of equal areas under the theoreti-cal (elastic range) Hertzian contact and the elasto-plastic contact stress (limited to 18 middot σy) with the unknown width
Classification Notes- No 412 11 May 2003
DET NORSKE VERITAS
exactly the same position relative to the root fillet Relations in 191 apply for conversion to KHβ
Secondarily
KHβ may be evaluated by observed contact patterns on vari-ous defined load levels It is imperative that the various test loads are well defined Usually it is also necessary to evalu-ate the elastic deflections Some teeth at each 90 degrees are to be painted with a suitable lacquer Always consider the poorest of the contact patterns
After having run the gear for a suitable time at test load 1 (the lowest) observe the contact pattern with respect to ex-tension over the facewidth Evaluate that KHβ by means of the methods mentioned in this section Proceed in the same way for the next higher test load etc until there is a full face contact pattern From these data the initial mesh misalign-ment (ie without elastic deflections) can be found by ex-trapolation and then also the KHβ at design load can be found by calculation and extrapolation See example
Figure 11 Example of experimental determination of KHβ
It must be considered that inaccurate gears may accumulate a larger observed contact pattern than the actual single mesh to mesh contact patterns This is particularly important for lapped bevel gears Ground or hard metal hobbed bevel gears are assumed to present an accumulated contact pattern that is practically equal the actual single mesh to mesh con-tact patterns As a rough guidance the (observed) accumu-lated contact pattern of lapped bevel gears may be reduced by 10 in order to assess the single mesh to mesh contact pattern which is used in 199
193 Theoretical determination of KHβ The methods described in 193 to 198 may be used for cy-lindrical gears The principles may to some extent also be used for bevel gears but a more practical approach is given in 199
General For gears where the tooth contact pattern cannot be verified during assembly or under load all assumptions are to be well on the safe side
KHβ is to be determined in the plane of contact
The influence parameters considered in this method are
bull mean mesh stiffness cγ (see 111) (if necessary also variable stiffness over b)
bull mean unit load Fmb = Fbt KA Kγ Kvb (for double helical gears see 17 for use of Kγ)
bull misalignment fsh due to elastic deflections of shafts and gear bodies (both pinion and wheel)
bull misalignment fdefl due to elastic deflections of and work-ing positions in bearings
bull misalignment fbe due to bearing clearance tolerances bull misalignment fma due to manufacturing tolerances bull helix modifications as crowning end relief helix correc-
tion bull running in amount yβ (see 112)
In practice several other parameters such as centrifugal ex-pansion thermal expansion housing deflection etc contrib-ute to KHβ However these parameters are not taken into account unless in special cases when being considered as particularly important
When all or most of the am parameters are to be considered the most practical way to determine KHβ is by means of a graphical approach described in 1931
If cγ can be considered constant over the facewidth and no helix modifications apply KHβ can be determined analyti-cally as described in 1932
1931 Graphical method The graphical method utilises the superposition principle and is as follows
bull Calculate the mean mesh deflection Mδ as a function of
γm c and bF see 111
bull Draw a base line with length b and draw up a rectangu-lar with height δM (The area δM b is proportional to the transmitted force)
bull Calculate the elastic deflection fsh in the plane of contact Balance this deflection curve around a zero line so that the areas above and below this zero line are equal
Figure 12 fsh balanced around zero line
bull Superimpose these ordinates of the fsh curve to the previ-ous load distribution curve (The area under this new load distribution curve is still δM b)
bull Calculate the bearing deflections andor working posi-tions in the bearings and evaluate the influence fdefl in the plane of contact This is a straight line and is bal-anced around a zero line as indicated in Fig 14 but with one distinct direction Superimpose these ordinates to the previous load distribution curve
12 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
bull The amount of crowning end relief or helix correction (defined in the plane of contact) is to be balanced around a zero line similarly to fsh
Figure 13 Crowning Cc balanced aound zero line
bull Superimpose these ordinates to the previous load distri-bution curve In case of high crowning etc as eg often applied to bevel gears the new load distribution curve may cross the base line (the real zero line) The result is areas with negative load that is not real as the load in those areas should be zero Thus corrective actions must be made but for practical reasons it may be postponed to after next operation
bull The amount of initial mesh misalignment bema ff + (defined in the plane of contact) is to be bal-
anced around a zero line If the direction of bema ff + is known (due to initial contact check) or if the direction of fbe is known due to design (eg overhang bevel pin-ion) this should be taken into account If direction un-known the influence of bema ff + in both directions as well as equal zero should be considered
Figure 14 fma+fbe in both directions balanced around zero line
Superimpose these ordinates to the previous load distri-bution curve This results in up to 3 different curves of which the one with the highest peak is to be chosen for further evaluation
bull If the chosen load distribution curve crosses the base line (ie mathematically negative load) the curve is to be corrected by adding the negative areas and dividing this with the active facewidth The (constant) ordinates of this rectangular correction area are to be subtracted from the positive part of the load distribution curve It is advisable to check that the area covered under this new load distribution curve is still equal δM b
bull If cγ cannot be considered as constant over b then cor-rect the ordinates of the load distribution curve with the local (on various positions over the facewidth) ratio between local mesh stiffness and average mesh stiffness cγ (average over the active facewidth only) Note that the result is to be a curve that covers the same area δM b as before
bull The influence of running in yβ is to be determined as in 112 whereby the value for Fβx is to be taken as twice the distance between the peak of the load distribution curve and δM
bull Determine
KHβ = M
β
δycurveofpeak minus
1932 Simplified analytical method for cylindrical gears The analytical approach is similar to 1931 but has a more limited application as cγ is assumed constant over the face-width and no helix modification applies
bull Calculate the elastic deflection fsh in the plane of contact Balance this deflection curve around a zero line so that the area above and below this zero line are equal see Fig 12 The max positive ordinate is frac12∆fsh
bull Calculate the initial mesh alignment as Fβx= deflbemash ffff plusmnplusmnplusmn∆ The negative signs may only be used if this is justified andor verified by a contact pattern test Otherwise al-ways use positive signs If a negative sign is justified the value of Fβx is not to be taken less than the largest of each of these elements
bull Calculate the effective mesh misalignment as Fβy = Fβx - yβ (yβ see 112)
bull Determine
2KforF2
bFc1K βH
m
βγγH le+=β
or
2KforF
bFc2K Hβ
m
βγγH gt=β
where cγ as used here is the effective mesh stiffness see 111
194 Determination of fsh fsh is the mesh misalignment due to elastic deflections Usu-ally it is sufficient to consider the combined mesh deflection of the pinion body and shaft and the wheel shaft The calcu-lation is to be made in the plane of contact (of the considered gear mesh) and to consider all forces (incl axial) acting on the shafts Forces from other meshes can be parted into components parallel respectively vertical to the considered plane of contact Forces vertical to this plane of contact have no influence on fsh
It is advised to use following diameters for toothed elements
d + 2 x mn for bending and shear deflection
d + 2 mn (x ndash ha0 + 02) for torsional deflection
Usually fsh is calculated on basis of an evenly distributed load If the analysis of KHβ shows a considerable maldis-
Classification Notes- No 412 13 May 2003
DET NORSKE VERITAS
tribution in term of hard end contact or if it is known by other reasons that there exists a hard end contact the load should be correspondingly distributed when calculating fsh In fact the whole KHβ procedure can be used iteratively 2-3 iterations will be enough even for almost triangular load distributions
195 Determination of fdefl fdefl is the mesh misalignment in the plane of contact due to bearing deflections and working positions (housing deflec-tion may be included if determined)
First the journal working positions in the bearings are to be determined The influence of external moments and forces must be considered This is of special importance for twin pinion single output gears with all 3 shafts in one plane
For rolling bearings fdefl is further determined on basis of the elastic deflection of the bearings An elastic bearing deflec-tion depends on the bearing load and size and number of rolling elements Note that the bearing clearance tolerances are not included here
For fluid film bearings fdefl is further determined on basis of the lift and angular shift of the shafts due to lubrication oil film thickness Note that fbe takes into account the influence of the bearing clearance tolerance
When working positions bearing deflections and oil film lift are combined for all bearings the angular misalignment as projected into the plane of the contact is to be determined fdefl is this angular misalignment (radians) times the face-width
196 Determination of fbe fbe is the mesh misalignment in the plane of contact due to tolerances in bearing clearances In principle fbe and fdefl could be combined But as fdefl can be determined by analy-sis and has a distinct direction and fbe is dependent on toler-ances and in most cases has no distinct direction (ie + toler-ance) it is practicable to separate these two influences
Due to different bearing clearance tolerances in both pinion and wheel shafts the two shaft axis will have an angular mis-alignment in the plane of contact that is superimposed to the working positions determined in 195 fbe is the facewidth times this angular misalignment Note that fbe may have a distinct direction or be given as a + tolerance or a combina-tion of both For combination of + tolerance it is adviced to use
fbe= 22be
21be ff ++plusmn
fbe is particularly important for overhang designs for gears with widely different kinds of bearings on each side and when the bearings have wide tolerances on clearances In general it shall be possible to replace standard bearings with-out causing the real load distribution to exceed the design premises For slow speed gears with journal bearings the expected wear should also be considered
197 Determination of fma fma is the mesh misalignment due to manufacturing toler-ances (helix slope deviation) of pinion fHβ1 wheel fHβ2 and housing bore
For gear without specifically approved requirements to as-sembly control the value of fma is to be determined as
fma= 22Hβ
21Hβ ff +
For gears with specially approved assembly control the value of fma will depend on those specific requirements
198 Comments to various gear types For double helical gears KHβ is to be determined for both helices Usually an even load share between the helices can be assumed If not the calculation is to be made as de-scribed in 171
For planetary gears the free floating sun pinion suffers only twist no bending It must be noted that the total twist is the sum of the twist due to each mesh If the value of 1K γ ne this must be taken into account when calculating the total sun pinion twist (ie twist calculated with the force per mesh without Kγ and multiplied with the number of planets)
When planets are mounted on spherical bearings the mesh misalignments sun-planet respectively planet-annulus will be balanced Ie the misalignment will be the average between the two theoretical individual misalignments The faceload distribution on the flanks of the planets can take full advan-tage of this However as the sun and annulus mesh with several planets with possibly different lead errors the sun and annulus cannot obtain the above mentioned advantage to the full extent
199 Determination of KHβ for bevel gears If a theoretical approach similar to 193-198 is not docu-mented the following may be used
testeff
H Kb
b851851K sdot
minussdot=β
beff b represents the relative active facewidth (regarding lapped gears see 192 last part)
Higher values than beff b = 090 are normally not to be used in the formula
For dual directional gears it may be difficult to obtain a high beff b in both directions In that case the smaller beff b is to be used
Ktest represents the influence of the bearing arrangement shaft stiffness bearing stiffness housing stiffness etc on the faceload distribution and the verification thereof Expected variations in length- and height-wise tooth profile is also accounted for to some extent
a) Ktest = 1 For ground or hard metal hobbed gears with the specified contact pattern verified at full rating or at full torque slow turning at a condition representative for the thermal expansion at normal operation
14 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
It also applies when the bearing arrangementsupport has insignificant elastic deflections and thermal axial expansion However each initial mesh contact must be verified to be within acceptance criteria that are cali-brated against a type test at full load Reproduction of the gear tooth length- and height-wise profile must also be verified This can be made through 3D measurements or by initial contact movements caused by defined axial offsets of the pinion (tolerances to be agreed upon)
b) Ktest = 1 + 04middot(beffbndash06) For designs with possible influence of thermal expansion in the axial direction of the pinion The initial mesh contact verified with low load or spin test where the acceptance criteria are calibrated against a type test at full load
c) Ktest = 12 if mesh is only checked by toolmakerrsquos blue or by spin test contact For gears in this category beffb gt 085 is not to be used in the calculation
110 Transversal Load Distribution Factors KHα and KFα The transverse load distribution factors KHα for contact stresses and for scuffing KFα for tooth root stresses account for the effects of pitch and profile errors on the transversal load distribution between 2 or more pairs of teeth in mesh
The following relations may be used
Cylindrical gears
( )
minus+==
tH
αptγγHαFα F
byfc0409
2ε
KK
valid for 2εγ le
( ) ( )tH
αptγ
γ
γHαFα F
byfcε
1ε20409KK
minusminus+==
valid for 2ε γ gt
where
FtH = Ft KA Kγ Kv KHβ
cγ = See 111
γα = See 112
fpt = Maximum single pitch deviation (microm) of pinion or wheel or maximum total profile form devia-tion Fα of pinion or wheel if this is larger than the maximum single pitch deviation
Note In case of adequate equivalent tip relief adapted to the load half of the above mentioned fpt can be introduced A tip relief is considered adequate when the average of Ca1 and Ca2 is within plusmn40 of the value of Ceff in 432
Limitations of KHα and KFα
If the calculated values for
KFα = KHα lt 1 use KFα = KHα = 10
If the calculated value of KHα gt 2εα
γ
Zε
ε use KHα = 2
εα
γ
Zε
ε
If the calculated value of KFα gt εα
γ
Yεε
use KFα = εα
γ
Yεε
where Yε = αnε
075025+ (for εαn see 331c)
Bevel gears
For ground or hard metal hobbed gears KFα = KHα = 1
For lapped gears KFα = KHα = 11
111 Tooth Stiffness Constants cacute and cγ The tooth stiffness is defined as the load which is necessary to deform one or several meshing gear teeth having 1 mm facewidth by an amount of 1 microm in the plane of contact cacute is the maximum stiffness of a single pair of teeth cγ is the mean value of the mesh stiffness in a transverse plane (brief term mesh stiffness)
Both valid for high unit load (Unit load = Ft middot KA middot Kγb)
Cylindrical gears The real stiffness is a combination of the progressive Hertzian contact stiffness and the linear tooth bending stiff-nesses For high unit loads the Hertzian stiffness has little importance and can be disregarded This approach is on the safe side for determination of KHβ and KHα However for moderate or low loads Kv may be underestimated due to determination of a too high resonance speed
The linear approach is described in A
An optional approach for inclusion of the non-linear stiffness is described in B
A The linear approach
BRCCq
βcos08acutec =
and
( )025ε075cc αγ +prime=
where
( )[ ]n02a01a
B α2000212
hh12051C minusminus
+minus+=
12n1n
x000635z
025791z
015551004723q minus++=
12
2n
22
1n
1 x005290z
x024188x000193z
x011654+minusminusminus
+ 000182 x22
Classification Notes- No 412 15 May 2003
DET NORSKE VERITAS
(for internal gears use zn2 equal infinite and x2 = 0) ha0 = hfp for all practical purposes CR considers the increased flexibility of the wheel teeth if the wheel is not a solid disc and may be calculated as
( )( )nR m5s
sR e5
bbln1C +=
where
bs = thickness of a central web
sR = average thickness of rim (net value from tooth root to inside of rim)
The formula is valid for bs b ge 02 and sRmn ge 1 Outside this range of validity and if the web is not centrally posi-tioned CR has to be specially considered
Note CR is the ratio between the average mesh stiffness over the facewidth and the mesh stiffness of a gear pair of solid discs The local mesh stiffness in way of the web corresponds to the mesh stiffness with CR = 1 The local mesh stiffness where there is no web support will be less than calculated with CR above Thus eg a centrally positioned web will have an effect corresponding to a longitudinal crowning of the teeth See also 1931 regarding KHβ
B The non-linear approach
In the following an example is given on how to consider the non-linearity
The relation between unit load Fb as a function of mesh deflection δ is assumed to be a progressive curve up to 500 Nmm and from there on a straight line This straight line when extended to the baseline is assumed to intersect at 10microm
With these assumptions the unit force Fb as a function of mesh deflection δ can be expressed as
( )10δKbF
minus= for 500bFgt
minus=
500Fb10δK
bF for 500
bFlt
with γAt KK
bF
bF
sdotsdot= etc (Nmm) ie unit load incorporat-
ing the relevant factors as
KA middot Kγ for determination of Kv
KA middot Kγ middot Kv for determination of KHβ
KA middot Kγ middot Kv middot KHβ for determination of KHα
δ = mesh deflection (microm)
K = applicable stiffness (c or cγ)
Use of stiffnesses for KV KHβ and KHα
For calculation of Kv and KHα the stiffness is calculated as follows
When Fb lt 500
the stiffness is determined as δ∆
∆ bF
where the increment is chosen as eg ∆ Fb = 10 and thus
50010Fb10
K10Fb∆δ +
++
=
When F b gt 500 the stiffness is c or cγ For calculation of KHβ the mesh deflection δ is used directly
or an equivalent stiffness determined as δsdotb
F
Bevel gears
In lack of more detailed relationship between stiffness and geometry the following may be used
b085
b13cacute eff=
b085b
16c effγ =
beff not to be used in excess of 085 b in these formulae
Bevel gears with heightwise and lengthwise crowning have progressive mesh stiffness The values mentioned above are only valid for high loads They should not be used for de-termination of Ceff (see 432) or KHβ (see 1931)
112 Running-in Allowances The running-in allowances account for the influence of run-ning-in wear on the various error elements
yα respectively yβ are the running-in amounts which reduce the influence of pitch and profile errors respectively influ-ence of localised faceload
Cay is defined as the running-in amount that compensates for lack of tip relief
The following relations may be used
For not surface hardened steel
ptHlim
α fσ160y =
yβ = βxlimH
f320σ
with the following upper limits
V lt 5 ms 5-10 ms gt 10 ms
yα max none
limH
12800σ limH
6400σ
yβ max none
limH
25600σ limH
12800σ
For surface hardened steel
yα = 0075 fpt but not more than 3 for any speed
yβ = 015 Fβx but not more than 6 for any speed
16 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
For all kinds of steel
5145189718
1C2
limHay +
minusσ
=
When pinion and wheel material differ the following ap-plies
bull Use the larger of fpt1 - yα1 and fpt2 - yα2 to replace fpt - yα in the calculation of KHα and Kv
bull Use ( )2β1ββ yy21y += in the calculation of KHβ
bull Use ( )2ay1aya CC21C += in the calculation of Kv
bull Use ( )2ay1ay2a1a CC21CC +== in the scuffing
calculation if no design tip relief is foreseen
Classification Notes- No 412 17 May 2003
DET NORSKE VERITAS
2 Calculation of Surface Durability
21 Scope and General Remarks Part 2 includes the calculations of flank surface durability as limited by pitting spalling case crushing and subsurface yielding Endurance and time limited flank surface fatigue is calculated by means of 22 ndash 212 In a way also tooth frac-tures starting from the flank due to subsurface fatigue is in-cluded through the criteria in 213
Pitting itself is not considered as a critical damage for slow speed gears However pits can create a severe notch effect that may result in tooth breakage This is particularly im-portant for surface hardened teeth but also for high strength through hardened teeth For high-speed gears pitting is not permitted
Spalling and case crushing are considered similar to pitting but may have a more severe effect on tooth breakage due to the larger material breakouts initiated below the surface Subsurface fatigue is considered in 213
For jacking gears (self-elevating offshore units) or similar slow speed gears designed for very limited life the max static (or very slow running) surface load for surface hard-ened flanks is limited by the subsurface yield strength
For case hardened gears operating with relatively thin lubri-cation oil films grey staining (micropitting) may be the lim-iting criterion for the gear rating Specific calculation meth-ods for this purpose are not given here but are under consid-eration for future revisions Thus depending on experience with similar gear designs limitations on surface durability rating other than those according to 22 - 213 may be ap-plied
22 Basic Equations Calculation of surface durability (pitting) for spur gears is based on the contact stress at the inner point of single pair contact or the contact at the pitch point whichever is greater
Calculation of surface durability for helical gears is based on the contact stress at the pitch point
For helical gears with 0 lt εβ lt 1 a linear interpolation be-tween the above mentioned applies
Calculation of surface durability for spiral bevel gears is based on the contact stress at the midpoint of the zone of contact
Alternatively for bevel gears the contact stress may be cal-culated with the program ldquoBECALrdquo In that case KA and Kv are to be included in the applied tooth force but not KHβ and KHα The calculated (real) Hertzian stresses are to be multi-plied with ZK in order to be comparable with the permissible contact stresses
The contact stresses calculated with the method in part 2 are based on the Hertzian theory but do not always represent the real Hertzian stresses
The corresponding permissible contact stresses σHP are to be calculated for both pinion and wheel
221 Contact stress Cylindrical gears
( )HαHβvγA
1
tβεEHDBH KKKKK
bud1uF
ZZZZZσ+
=
where
ZBD = Zone factor for inner point of single pair contact for pinion resp wheel (see 232)
ZH = Zone factor for pitch point (see 231)
ZE = Elasticity factor (see 24)
Zε = Contact ratio factor (see 25)
Zβ = Helix angle factor (see 26)
Ft KA Kγ Kv KHβ KHα see 15 ndash 110
d1 b u see 12 ndash 15
Bevel gears
( )HαHβvγA
v1v
vmtKEMH KKKKK
bud1uF
ZZZ105σ+
sdot=
where
105 is a correlation factor to reach real Hertzian stresses (when ZK = 1)
ZE KA etc see above
ZM = mid-zone factor see 233
ZK = bevel gear factor see 27
Fmt dv1 uv see 12 ndash 15
It is assumed that the heightwise crowning is chosen so as to result in the maximum contact stresses at or near the mid-point of the flanks
222 Permissible contact stress
XWRvLH
NHlimHP ZZZZZ
SZσ
σ =
where
σH lim = Endurance limit for contact stresses (see 28)
ZN = Life factor for contact stresses (see 29)
SH = Required safety factor according to the rules
ZLZvZR = Oil film influence factors (see 210)
ZW = Work hardening factor (see 211)
ZX = Size factor (see 212)
18 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
23 Zone Factors ZH ZBD and ZM
231 Zone factor ZH The zone factor ZH accounts for the influence on contact stresses of the tooth flank curvature at the pitch point and converts the tangential force at the reference cylinder to the normal force at the pitch cylinder
wtt2
wtbH sinααcos
cosαcosβ2Z =
232 Zone factors ZBD The zone factors ZBD account for the influence on contact stresses of the tooth flank curvature at the inner point of sin-gle pair contact in relation to ZH Index B refers to pinion D to wheel
For εβ ge 1 ZBD = 1
For internal gears ZD = 1
For εβ = 0 (spur gears)
( )
π
minusεminusminus
π
minusminus
α=
α2
2
2b
2a
1
2
1b
1a
wtB
z211
dd
z21
dd
tanZ
( )
π
minusεminusminus
π
minusminus
α=
α1
2
1b
1a
2
2
2b
2a
wtD
z211
dd
z21
dd
tanZ
If ZB lt 1 use ZB = 1
If ZD lt 1 use ZD = 1
For 0 lt εβ lt 1
ZBD = ZBD (for spur gears) ndash εβ (ZBD (for spur gears) ndash 1)
233 Zone factor ZM The mid-zone factor ZM accounts for the influence of the contact stress at the mid point of the flank and applies to spi-ral bevel gears
εminusminus
εminusminus
αβ=
αα btm2
2vb2
2vabtm2
1vb2val
2v1vvtbmM
pddpdd
ddtancos2Z
This factor is the product of ZH and ZM-B in ISO 10300 with the condition that the heightwise crowning is sufficient to move the peak load towards the midpoint
234 Inner contact point For cylindrical or bevel gears with very low number of teeth the inner contact point (A) may be close to the base circle In order to avoid a wear edge near A it is required to have suitable tip relief on the wheel
24 Elasticity Factor ZE The elasticity factor ZE accounts for the influence of the material properties as modulus of elasticity and Poissonrsquos ratio on the contact stresses
For steel against steel ZE = 1898
25 Contact Ratio Factor Zε The contact ratio factor Zε accounts for the influence of the transverse contact ratio εα and the overlap ratio εβ on the contact stresses
αε1Z =ε for 1εβ ge
( )α
ββ
αε ε
εε1
3ε4
Z +minusminus
= for εβ lt 1
26 Helix Angle Factor Zβ The helix angle factor Zβ accounts for the influence of helix angle (independent of its influence on Zε) on the surface du-rability
cosβZβ =
27 Bevel Gear Factor ZK The bevel gear factor accounts for the difference between the real Hertzian stresses in spiral bevel gears and the contact stresses assumed responsible for surface fatigue (pitting) ZK adjusts the contact stresses in such a way that the same per-missible stresses as for cylindrical gears may apply
The following may be used ZK = 080
28 Values of Endurance Limit σHlim and Static Strength 5H10σ 3H10σ
σHlim is the limit of contact stress that may be sustained for 5middot107 cycles without the occurrence of progressive pitting
For most materials 5middot107 cycles are considered to be the be-ginning of the endurance strength range or lower knee of the σ-N curve (See also Life Factor ZN) However for nitrided steels 2middot106 apply
For this purpose pitting is defined by
bull for not surface hardened gears pitted area ge 2 of total active flank area
bull for surface hardened gears pitted area ge 05 of total active flank area or ge 4 of one particular tooth flank area 510Hσ and 310Hσ are the contact stresses which the given
material can withstand for 105 respectively 103 cycles without subsurface yielding or flank damages as pitting spalling or case crushing when adequate case depth applies
The following listed values for σHlim 510Hσ and 310Hσ may only be used for materials subjected to a quality control as
Classification Notes- No 412 19 May 2003
DET NORSKE VERITAS
the one referred to in the rules Results of approved fatigue tests may also be used as the basis for establishing these values The defined survival probability is 99
σHlim σH105 σH10
3
Alloyed case hardened steels (surface hardness 58-63 HRC) - of specially approved high grade - of normal grade
1650 1500
2500 2400
3100 3100
Nitrided steel of approved grade gas nitrided (surface hardness 700-800 HV) 1250 13 σHlim 13 σHlim
Alloyed quenched and tempered steel bath or gas nitrided (surface hardness 500-700 HV)
1000
13 σHlim
13 σHlim
Alloyed flame or induction hardened steel (surface hardness 500-650 HV) 075 HV + 750 16 σHlim 45 HV
Alloyed quenched and tempered steel 14 HV + 350 16 σHlim 45 HV
Carbon steel 15 HV + 250 16 σHlim 16 σHlim
These values refer to forged or hot rolled steel For cast steel the values for σHlim are to be reduced by 15
29 Life Factor ZN The life factor ZN takes account of a higher permissible contact stress if only limited life (number of cycles NL) is demanded or lower permissible contact stress if very high number of cycles apply
If this is not documented by approved fatigue tests the fol-lowing method may be used
For all steels except nitrided
7L 105N sdotge ZN = 1 or
01570
L
7
N N105Z
sdot=
Ie ZN = 092 for 1010 cycles
The ZN = 1 from 5middot107 on may only be used when the mate-rial cleanliness is of approved high grade (see Rules Pt4 Ch2) and the lubrication is optimised by a specially approved filtering process
105 lt NL lt 5middot107 510N
logZ037
L
7
N N105Z
sdot=
NL = 105 WXRVLHlim
WstX10H1010NN ZZZZZσ
ZZσZZ
55
5=
103 lt NL lt 105 )Z(Zlog05
L
5
10NN
5N103N10
5N10ZZ
==
3L 10N le
WXRVLlimH
Wst10X10H10NN ZZZZZ
ZZZZ
33
3σ
σ==
(but not less than ZN105)
For nitrided steels
6L 102N sdotge ZN = 1 or
00980
L
6
N N102Z
sdot=
Ie ZN = 092 for 1010 cycles
The ZN = 1 from 2middot106 on may only be used when the mate-rial cleanliness is of approved high grade (see Rules Pt4 Ch2) and the lubrication is optimised by a specially approved filtering process
105 lt NL lt 2middot106 510N
Zlog76860
L
6
N N102Z
sdot=
5L 10N le
XWRVL
10XWst10NN ZZZZZ
ZZ13ZZ
55 ==
Note that when no index indicating number of cycles is used the factors are valid for 5middot107 (respectively 2middot106 for nitrid-ing) cycles
210 Influence Factors on Lubrication Film ZL ZV and ZR The lubricant factor ZL accounts for the influence of the type of lubricant and its viscosity the speed factor ZV ac-counts for the influence of the pitch line velocity and the roughness factor ZR accounts for influence of the surface roughness on the surface endurance capacity
The following methods may be applied in connection with the endurance limit
20 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
Surface hardened steels Not surface hardened steels
ZL ( )24013421
360910ν+
+
( )24013421
680830ν+
+
ZV ( )v3280
140930+
+ ( )v3280300850
++
ZR
080
ZrelR3
150
ZrelR3
where
ν40 = Kinematic oil viscosity at 40ordmC (mm2s) For case hardened steels the influence of a high bulk temperature (see 4 Scuffing) should be con-sidered Eg bulk temperatures in excess of 120ordmC for long periods may cause reduced flank surface endurance limits
For values of ν40 gt 500 use ν40 = 500
RZrel = The mean roughness between pinion and wheel (after running in) relative to an equivalent radius of curvature at the pitch point ρc = 10mm
RZrel
= ( ) 31
cZ2Z1 ρ
10RR50
+
RZ = Mean peak to valley roughness (microm) (DIN defini-tion) (roughly RZ = 6 Ra)
For 5
L 10N le ZL ZV ZR = 10
211 Work Hardening Factor ZW The work hardening factor ZW accounts for the increase of surface durability of a soft steel gear when meshing the soft steel gear with a surface hardened or substantially harder gear with a smooth surface
The following approximation may be used for the endurance limit
Surface hardened steel against not surface hardened steel 150
ZeqW R
31700
130HB21Z
minus
minus=
where HB = the Brinell hardness of the soft member For HB gt 470 use HB = 470 For HB lt 130 use HB = 130
RZeq = equivalent roughness
033
c40
066
ZS
ZHZHZEQ ρvν
15000RRRR
=
If RZeq gt 16 then use RZeq = 16
If RZeq lt 15 then use RZeq = 15
where
RZH = surface roughness of the hard member before run in
RZS = surface roughness of the soft member before run in
ν40 = see 210
If values of ZW lt 1 are evaluated ZW = 1 should be used for flank endurance However the low value for ZW may indi-cate a potential wear problem
Through hardened pinion against softer wheel
( )
minussdotminus+= 000829
HBHB0008981u1Z
2
1W
For 21HBHB
2
1 le use ZW = 1
For 71HBHB
2
1 gt use 17HBHB
2
1 =
For u gt 20 use u = 20
For static strength (lt 105 cycles)
Surface hardened against not surface hardened
ZWst = 105
Through hardened pinion against softer wheel
ZWst = 1
212 Size Factor ZX The size factor accounts for statistics indicating that the stress levels at which fatigue damage occurs decrease with an increase of component size as a consequence of the influ-ence on subsurface defects combined with small stress gradi-ents and of the influence of size on material quality
ZX may be taken unity provided that subsurface fatigue for surface hardened pinions and wheels is considered eg as in the following subsection 213
213 Subsurface Fatigue This is only applicable to surface hardened pinions and wheels The main objective is to have a subsurface safety against fatigue (endurance limit) or deformation (static strength) which is at least as high as the safety SH required for the surface The following method may be used as an approximation unless otherwise documented
The high cycle fatigue (gt3106 cycles) is assumed to mainly depend on the orthogonal shear stresses Static strength (lt103 cycles) is assumed to depend mainly on equivalent
Classification Notes- No 412 21 May 2003
DET NORSKE VERITAS
stresses (von Mises) Both are influenced by residual stresses but this is only considered roughly and empirically
The subsurface working stresses at depths inside the peak of the orthogonal shear stresses respectively the equivalent stresses are only dependent on the (real) Hertzian stresses Surface related conditions as expressed by ZL ZV and ZR are assumed to have a negligible influence
The real Hertzian stresses σHR are determined as
For helical gears with εβ gt 1
σHR = σH
For helical gears with εβ lt 1 and spur gears
ε
α
ββ ε
ε+εminus
sdotσ=σZ
1
HHR
For bevel gears
KHHR Z
1σσ sdot=
The necessary hardness HV is given as a function of the net depth tz (net = after grinding or hard metal hobbing and per-pendicular to the flank)
The coordinates tz and HV are to be compared with the de-sign specification such as
bull for flame and induction hardening tHVmin HVmin bull for nitriding t400min HV = 400 bull for case hardening t550min HV = 550 t400min HV = 400
and t300min HV = 300 (the latter only if the core hardness lt 300 If the core hardness gt 400 the t400 is to be re-placed by a fictive t400 = 16 t550)
In addition the specified surface hardness is not to be less than the max necessary hardness (at tz = 05aH) This applies to all hardening methods
For high cycle fatigue (gt3 106 cycles) the following applies
sdot+
minussdotsdotsdot= o90
05azt
05azt
cosSσ04HV
H
HHHR
applicable to 05a
zt
Hge
For 05at
H
z lt the value for 05at
H
z = applies
56300ρSσ12a cHHR
Hsdotsdot
sdot=
Where aH is half the hertzian contact width multiplied by an empirical factor of 12 that takes into account the possible influence of reduced compressive residual stresses (or even tensile residual stresses) on the local fatigue strength
If any of the specified hardness depths including the surface hardness is below the curve described by HV = f (tz) the actual safety factor against subsurface fatigue is determined as follows
reduce SH stepwise in the formula for HV and aH until all specified hardness depths and surface hardness balance with the corrected curve The safety factor obtained through this method is the safety against subsurface fatigue
For static strength (lt103 cycles) the following applies
sdot+
minussdotsdotsdot= o90
07at
06a
zt
cosSσ019HV
Hst
z
HstHHR
applicable to 06at
Hst
z ge
56300ρSσa cHHR
Hstsdotsdot
=
In the case of insufficient specified hardness depths the same procedure for determination of the actual safety factor as above applies
For limited life fatigue (103 lt cycles lt 3106 )
For this purpose it is necessary to extend the correction of safety factors to include also higher values than required Ie in the case of more than sufficient hardness and depths the safety factor in the formulae for both high cycle fatigue and static strength are to be increased until necessary and speci-fied values balance
The actual safety factor for a given number of cycles N be-tween 103 and 3106 is found by linear interpolation in a dou-ble logarithmic diagram
minussdotminus
= sdot logN3477
logSlogSlogS
310H6103HHN
36 10H103H Slog86281Slog86280 sdot+sdot sdot
22 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
3 Calculation of Tooth Strength
31 Scope and General Remarks Part 3 include the calculation of tooth root strength as limited by tooth root cracking (surface or subsurface initiated) and yielding
For rim thickness sR ge 35middotmn the strength is calculated by means of 32 ndash 313 For cylindrical gears the calculation is based on the assumption that the highest tooth root tensile stress arises by application of the force at the outer point of single tooth pair contact of the virtual spur gears The method has however a few limitations that are mentioned in 36
For bevel gears the calculation is based on force application at the tooth tip of the virtual cylindrical gear Subsequently the stress is converted to load application at the mid point of the flank due to the heightwise crowning
Bevel gears may also be calculated with the program BECAL In that case KA and Kv are to be included in the applied tooth force but not KFβ and KFα
In case of a thin annulus or a thin gear rim etc radial crack-ing can occur rather than tangential cracking (from root fillet to root fillet) Cracking can also start from the compression fillet rather than the tension fillet For rim thickness sR lt 35middotmn a special calculation procedure is given in 315 and 316 and a simplified procedure in 314
A tooth breakage is often the end of the life of a gear trans-mission Therefore a high safety SF against breakage is re-quired
It should be noted that this part 3 does not cover fractures caused by
bull oil holes in the tooth root space bull wear steps on the flank bull flank surface distress such as pits spalls or grey staining
Especially the latter is known to cause oblique fractures starting from the active flank predominately in spiral bevel gears but also sometimes in cylindrical gears
Specific calculation methods for these purposes are not given here but are under consideration for future revisions Thus depending on experience with similar gear designs limita-tions other than those outlined in part 3 may be applied
32 Tooth Root Stresses The local tooth root stress is defined as the max principal stress in the tooth root caused by application of the tooth force Ie the stress ratio R = 0 Other stress ratios such as for eg idler gears (R asymp -12) shrunk on gear rims (R gt 0) etc are considered by correcting the permissible stress level
321 Local tooth root stress The local tooth root stress for pinion and wheel may be as-sessed by strain gauge measurements or FE calculations or similar For both measurements and calculations all details are to be agreed in advance
Normally the stresses for pinion and wheel are calculated as
Cylindrical gears
FαFβvγAβSFn
tF K K K K K Y Y Y
m bFσ =
where
YF = Tooth form factor (see 33)
YS = Stress correction factor (see 34)
Yβ = Helix angle factor (see 36)
Ft KA Kγ Kv KFβ KFα see 15 ndash 110
b see 13
Bevel gears
FαFβvγASaFamn
mtF K K K K K Y Y Y
m bFσ ε=
where
YFa = Tooth form factor see 33
YSa = Stress correction factor see 34
Yε = Contact ratio factor see 35
Fmt KA etc see 15 ndash 110
b see 13
322 Permissible tooth root stress The permissible local tooth root stress for pinion respectively wheel for a given number of cycles N is
CXRrelTrelTF
NMFEFP YYYY
SYY
δσ
=σ
Note that all these factors YM etc are applicable to 3middot106 cycles when used in this formula for σFP The influence of other number of cycles on these factors is covered by the calculation of YN
where
σFE = Local tooth root bending endurance limit of reference test gear (see 37)
YM = Mean stress influence factor which accounts for other loads than constant load direction eg idler gears temporary change of load di-rection pre-stress due to shrinkage etc (see 38)
YN = Life factor for tooth root stresses related to reference test gear dimensions (see 39)
SF = Required safety factor according to the rules
YδrelT = Relative notch sensitivity factor of the gear to be determined related to the reference test gear (see 310)
YRrelT = Relative (root fillet) surface condition factor of the gear to be determined related to the reference test gear (see 311)
Classification Notes- No 412 23 May 2003
DET NORSKE VERITAS
YX = Size factor (see 312)
YC = Case depth factor (see 313)
33 Tooth Form Factors YF YFa The tooth form factors YF and YFa take into account the in-fluence of the tooth form on the nominal bending stress
YF applies to load application at the outer point of single tooth pair contact of the virtual spur gear pair and is used for cylindrical gears
YFa applies to load application at the tooth tip and is used for bevel gears
Both YF and YFa are based on the distance between the con-tact points of the 30˚-tangents at the root fillet of the tooth profile for external gears respectively 60˚ tangents for inter-nal gears
Figure 31 External tooth in normal section
Figure 32 Internal tooth in normal section
Definitions
n
2
n
Fn
enFn
Fe
F
α cosms
α cosmh6
Y
=
n
2
n
Fn
anFn
Fa
Fa
α cosms
α cosmh6
Y
=
In the case of helical gears YF and YFa are determined in the normal section ie for a virtual number of teeth
YFa differs from YF by the bending moment arm hFa and αFan and can be determined by the same procedure as YF with exception of hFe and αFan For hFa and αFan all indices e will change to a (tip)
The following formulae apply to cylindrical gears but may also be used for bevel gears when replacing
mn with mnm
zn with zvn
αt with αvt
β with βm
with undercut without undercut
Fig 33 Dimensions and basic rack profile of the teeth (finished profile)
Tool and basic rack data such as hfP ρfp and spr etc are re-ferred to mn ie dimensionless
331 Determination of parameters
( )n
n
prnfPnfP m
α cossαsin 1ρ
αtan h4πE
minusminusminusminus=
For external gears fPfP ρρ =
For internal gears ( )0z
195fPfP0
fPfP 10363156ρhxρρ
sdotminus+
+=
where
z0 = number of teeth of pinion cutter
x0 = addendum modification coefficient of pinion cutter
hfP = addendum of pinion cutter
ρfP = tip radius of pinion cutter
xhρG fpfp +minus=
24 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
τmE
2π
z2H
nnminus
minus=
with
3πτ = for external gears
6πτ = for internal gears
Htan zG2
nminusϑ=ϑ
(to be solved iteratively suitable start value6π
=ϑ for exter-
nal gears and 3π for internal gears)
a) Tooth root chord sFn For external gears
minus
ϑ+
ϑminus= ρ
cosG3
3πsinz
ms
fpnn
Fn
For bevel gears with a tooth thickness modification
xsm affects mainly sFn but also hFe and αFen The total influence of xsm on YFa Ysa can be approximated by only adding 2 xsm to sFn mn
For internal gears
minus
ϑ+
ϑminus= ρ
cosG
6πsinz
ms
fPnn
Fn
b) Root fillet radius ρF at 30ordm tangent
( )G2coszcosG2ρ
mρ
2n
2
fpn
F
minusϑϑ+=
c) Determination of bending moment arm hF dn = zn mn
b2α
αn βcosε
ε =
dan = dn + 2 ha
pbn = π mn cos αn
dbn = dn cos αn
( )4
d1εpzz
2dd
zz2d
2bn
2
αnbn
2bn
2an
en +
minusminus
minus=
en
bnen d
dcos arcα =
ennnn
e α invα invα x tan 22π
z1
minus+
+=γ
αFen = αen ndash γe
For external gears
( )
minus=
n
enFenee
n
Fe
mdαtan sin γ γcos
21
mh
]ρcos
G3πcosz fpn +
ϑminus
ϑminusminus
For internal gears
( ) minus
sdotsdotminus=
n
enFenee
n
Fe
mdα tanγ sinγ cos
21
mh
minus
ϑminus
ϑminussdot ρ
cosG3
6πcosz fPn
332 Gearing with εαn gt 2 For deep tooth form gearing ( )25ε2 αn lele produced with a verified grade of accuracy of 4 or better and with applied profile modification to obtain a trapezoidal load distribution along the path of contact the YF may be corrected by the factor YDT as
250ε205for 0666ε2366Y αnαnDT leleminus=
205εfor 10Y αnDT lt=
34 Stress Correction Factors YS YSa The stress correction factors YS and YSa take into account the conversion of the nominal bending stress to the local tooth root stress Thereby YS and YSa cover the stress increasing effect of the notch (fillet) and the fact that not only bending stresses arise at the root A part of the local stress is inde-pendent of the bending moment arm This part increases the more the decisive point of load application approaches the critical tooth root section
Therefore in addition to its dependence on the notch radius the stress correction is also dependent on the position of the load application ie the size of the bending moment arm
YS applies to the load application at the outer point of single tooth pair contact YSa to the load application at tooth tip
YS can be determined as follows
( )
+
+=L23121
1
sS qL01312Y
where Fe
Fn
hsL = and
F
Fns ρ2
sq = (see 33)
YSa can be calculated by replacing hFe with hFa in the above formulae
Note a) Range of validity 1 lt qs lt 8
In case of sharper root radii (ie produced with tools having too sharp tip radii) YS resp YSa must be specially considered
b) In case of grinding notches (due to insufficient protuberance of the hob) YS resp YSa can rise considerably and must be multiplied with
Classification Notes- No 412 25 May 2003
DET NORSKE VERITAS
g
g
ρt
0613
13
minus
where
tg = depth of the grinding notch
ρg = radius of the grinding notch
c) The formulae for YS resp YSa are only valid for αn = 20˚ However the same formulae can be used as a safe approximation for other pressure angles
35 Contact Ratio Factor Yε The contact ratio factor Yε covers the conversion from load application at the tooth tip to the load application at the mid point of the flank (heightwise) for bevel gears
The following may be used
Yε = 0625
36 Helix Angle Factor Yβ The helix angle factor Yβ takes into account the difference between the helical gear and the virtual spur gear in the nor-mal section on which the calculation is based in the first step In this way it is accounted for that the conditions for tooth root stresses are more favourable because the lines of contact are sloping over the flank
The following may be used (β input in degrees)
Yβ = 1 ndash εβ β120
When εβ gt 1 use εβ = 1 and when β gt 30deg use β = 30deg in the formula
However the above equation for Yβ may only be used for gears with β gt 25deg if adequate tip relief is applied to both pinion and wheel (adequate = at least 05 middot Ceff see 432)
37 Values of Endurance Limit σFE σFE is the local tooth root stress (max principal) which the material can endure permanently with 99 survival prob-ability 3106 load cycles is regarded as the beginning of the endurance limit or the lower knee of the σ ndash N curve σFE is defined as the unidirectional pulsating stress with a minimum stress of zero (disregarding residual stresses due to heat treatment) Other stress conditions such as alternating or pre-stressed etc are covered by the conversion factor YM
σFE can be found by pulsating tests or gear running tests for any material in any condition If the approval of the gear is to be based on the results of such tests all details on the testing conditions have to be approved by the Society Fur-ther the tests may have to be made under the Societys su-pervision
If no fatigue tests are available the following listed values for σFE may be used for materials subjected to a quality con-trol as the one referred to in the rules
σFE
Alloyed case hardened steels 1) (fillet surface hardness 58 ndash 63 HRC)
bull of specially approved high grade
1050
bull of normal grade
minus CrNiMo steels with approved process
1000
minus CrNi and CrNiMo steels generally 920 minus MnCr steels generally 850
Nitriding steel of approved grade quenched tempered and gas nitrided (surface hardness 700 ndash 800 HV)
840
Alloyed quenched and tempered steel bath or gas nitrided (surface hardness 500 ndash 700 HV)
720
Alloyed quenched and tempered steel flame or induction hardened 2) (incl entire root fillet) (fillet surface hardness 500 ndash 650 HV)
07 HV + 300
Alloyed quenched and tempered steel flame or induction hardened (excl entire root fillet) (σB = uts of base material)
025 σB + 125
Alloyed quenched and tempered steel 04 σB + 200
Carbon steel 025 σB + 250
Note All numbers given above are valid for separate forgings and for blanks cut from bars forged according to a qualified procedure see Pt 4 Ch 2 Sec 3 For rolled steel the values are to be reduced with 10 For blanks cut from forged bars that are not qualified as mentioned above the values are to be reduced with 20 For cast steel reduce with 40
1) These values are valid for a root radius
bull being unground If however any grinding is made in the root fillet area in such a way that the residual stresses may be affected σFE is to be reduced by 20 (If the grinding also leaves a notch see 34)
bull with fillet surface hardness 58 ndash 63 HRC In case of lower surface hardness than 58 HRC σFE is to be reduced with 20(58 ndash HRC) where HRC is the detected hardness (This may lead to a permissible tooth root stress that varies along the facewidth If so the actual tooth root stresses may also be considered along facewidth)
bull not being shot peened In case of approved shot peening σFE may be increased by 200 for gears where σFE is reduced by 20 due to root grinding Otherwise σFE may be increased by 100 for
6nm le and 100 ndash 5 (mn - 6) for mn gt 6 However the possible adverse influence on the flanks regarding grey staining should be considered and if necessary the flanks should be masked
2) The fillet is not to be ground after surface hardening Regarding possible root grinding see 1)
26 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
38 Mean stress influence Factor YM The mean stress influence factor YM takes into account the influence of other working stress conditions than pure pulsa-tions (R = 0) such as eg load reversals idler gears planets and shrink-fitted gears
YM (YMst) are defined as the ratio between the endurance (or static) strength with a stress ratio R ne 0 and the endurance (or static) strength with R = 0
YM and YMst apply only to a calculation method that assesses the positive (tensile) stresses and is therefore suitable for comparison between the calculated (positive) working stress σF and the permissible stress σFP calculated with YM or YMst
For thin rings (annulus) in epicyclic gears where the com-pression fillet may be decisive special considerations apply see 316
The following method may be used within a stress ratio ndash12 lt R lt 05
381 For idlers planets and PTO with ice class
M1M1R1
1Yor Y MstM
+minus
minus=
where
R = stress ratio = min stress divided by max stress
For designs with the same force applied on both forward- and back-flank R may be assumed to ndash 12
For designs with considerably different forces on forward- and back-flank such as eg a marine propulsion wheel with a power take off pinion R may be assessed as
branchmain theoffacewidth unit per forcepto offacewidt unit per force21minus
For a power take off (PTO) with ice class see 161 c
M considers the mean stress influence on the endurance (or static) strength amplitudes
M is defined as the reduction of the endurance strength am-plitude for a certain increase of the mean stress divided by that increase of the mean stress
Following M values may be used
Endurance limit
Static strength
Case hardened 08 ndash 015 Ys 1) 07
If shot peened 04 06
Nitrided 03 03
Induction or flame hardened
04 06
Not surface hardened steel 03 05
Cast steels 04 06 1)For bevel gears use Ys=2 for determination of M
The listed M values for the endurance limit are independent of the fillet shape (Ys) except for case hardening In princi-ple there is a dependency but wide variations usually only occur for case hardening eg smooth semicircular fillets versus grinding notches
382 For gears with periodical change of rotational direction For case hardened gears with full load applied periodically in both directions such as side thrusters the same formula for YM as for idlers (with R = ndash 12) may be used together with the M values for endurance limit This simplified approach is valid when the number of changes of direction exceeds 100 and the total number of load cycles exceeds 3middot106
For gears of other materials YM will normally be higher than for a pure idler provided the number of changes of direction is below 3middot106 A linear interpolation in a diagram with loga-rithmic number of changes of direction may be used ie from YM = 09 with one change to YM (idler) for 3middot106 changes This is applicable to YM for endurance limit For static strength use YM as for idlers
For gears with occasional full load in reversed direction such as the main wheel in a reversing gear box YM = 09 may be used
383 For gears with shrinkage stresses and unidirectional load For endurance strength
FE
fitM σ
σM1
M21Y+
minus=
σFE is the endurance limit for R = 0
For static strength YMst = 1 and σfit accounted for in 39b
σfit is the shrinkage stress in the fillet (30˚ tangent) and may be found by multiplying the nominal tangential (hoop) stress with a stress concentration factor
n
Ffit m
ρ215scf minus=
384 For shrink-fitted idlers and planets When combined conditions apply such as idlers with shrink-age stresses the design factor for endurance strength is
( ) ( ) FE
fitM σ
σR1M1
M2
M1M1R1
1Yminussdot+
minus
+minus
minus=
Symbols as above but note that the stress ratio R in this par-ticular connection should disregard the influence of σfit ie R normally equal ndash 12
For static strength
M1M1R1
1YMst
+minus
minus=
The effect of σfit is accounted for in 39 b
Classification Notes- No 412 27 May 2003
DET NORSKE VERITAS
385 Additional requirements for peak loads The total stress range (σmax ndash σmin) in a tooth root fillet is not to exceed
F
y
Sσ225
for not surface hardened fillets
FSHV5 for surface hardened fillets
39 Life Factor YN The life factor YN takes into account that in the case of limited life (number of cycles) a higher tooth root stress can be permitted and that lower stresses may apply for very high number of cycles
Decisive for the strength at limited life is the σ ndash N ndash curve of the respective material for given hardening module fillet radius roughness in the tooth root etc Ie the factors YδrelT YRelT YX and YM have an influence on YN
If no σ ndash N ndash curve for the actual material and hardening etc is available the following method may be used
Determination of the σ ndash N ndash curve
a) Calculate the permissible stress σFP for the beginning of the endurance limit (3middot106 cycles) including the influ-ence of all relevant factors as SF YδrelT YRelT YX YM and YC ie σFP = σFE middotYM middotYδrelT middotYRelT middotYX middotYC SF
b) Calculate the permissible laquostaticraquo stress (le 103 load cy-cles) including the influence of all relevant factors as SFst YδrelTst YMst and YCst ( )fitσCstYδrelTstYMstYFstσ
FstS1
FPstσ minussdotsdotsdot=
where σFst is the local tooth root stress which the material can resist without cracking (surface hardened materials) or unacceptable deformation (not surface hardened materials) with 99 survival probability
σFst
Alloyed case hardened steel 1) 2300
Nitriding steel quenched tempered and gas nitrided (surface hardness 700 ndash 800 HV)
1250
Alloyed quenched and tempered steel bath or gas nitrided (surface hardness 500 ndash 700 HV)
1050
Alloyed quenched and tempered steel flame or induction hardened (fillet surface hardness 500 ndash 650 HV)
18 HV + 800
Steel with not surface hardened fillets the smaller value of 2)
18 σB or 225 σy
1) This is valid for a fillet surface hardness of 58 ndash 63 HRC In case of lower fillet surface hardness than 58 HRC σFst is to be reduced with 30(58 ndash HRC) where HRC is the actual hardness Shot peening or grinding notches are not considered to have any significant influence on σFst 2) Actual stresses exceeding the yield point (σy or σ02) will alter the residual stresses locally in the ldquotensionrdquo fillet re-spectively ldquocompressionrdquo fillet This is only to be utilised for gears that are not later loaded with a high number of cycles at lower loads that could cause fatigue in the ldquocompressionrdquo fillet
c) Calculate YN as
NL gt 3middot106
YN = 1 or 001
L
6
N N103Y
sdot= ie Yn = 092 for 1010
The YN = 1 from 3middot106 on may only be used when special material cleanness applies see rules Pt4 Ch2
103ltNLlt3middot106exp
L
6
N N103Y
sdot=
cycles6103forσcycles310forσlog02876exp
FP
FPst
sdot=
NL lt 103 cycles6103forσcycles310forσ
NYFP
FPst
sdot=
or simply use σFPst as mentioned in b) directly
Guidance on number of load cycles NL for various applica-tions
bull For propulsion purpose normally NL = 1010 at full load (yachts etc may have lower values)
bull For auxiliary gears driving generators that normally operate with 70-90 of rated power NL = 108 with rated power may be applied
28 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
310 Relative Notch Sensitivity Factor YδrelT The dynamic (respectively static) relative notch sensitivity factor YδrelT (YδrelTst) indicate to which extent the theoreti-cally concentrated stress lies above the endurance limits (re-spectively static strengths) in the case of fatigue (respectively overload) breakage
YδrelT is a function of the material and the relative stress gra-dient It differs for static strength and endurance limit
The following method may be used
For endurance limit
for not surface hardened fillets
( )
024
s024
relT σ103133q21σ1012201351
Ysdotsdotminus
+sdotsdotminus+= minus
minus
δ
for all surface hardened fillets except nitrided
( )
106q21002451
Y srelT
++=δ
for nitrided fillets
( )
1347q2101421
Y srelT
++=δ
For static strength
for not surface hardened fillets1)
( )( )( )025
02
02502s
relTst σ3000821σ300 1Y0821Y
+
minus+=δ
for surface hardened fillets except nitrided
YδrelTst = 044 YS + 012
for nitrided fillets
YδrelTst = 06 + 02 YS 1) These values are only valid if the local stresses do not
exceed the yield point and thereby alter the residual stress level See also 39b footnote 2
311 Relative Surface Condition Factor YRrelT The relative surface condition factor YRrelT takes into ac-count the dependence of the tooth root strength on the sur-face condition in the tooth root fillet mainly the dependence on the peak to valley surface roughness
YRrelT differs for endurance limit and static strength
The following method may be used
For endurance limit
YRrelT = 1675 ndash 053 (Ry + 1)01 for surface hardened steels and alloyed quenched and tempered steels except nitrided
YRrelT = 53 ndash 42 (Ry + 1)001 for carbon steels
YδrelT = 43 ndash 326 (Ry + 1)0005
for nitrided steels
For static strength
YRrelTst = 1 for all Ry and all materials
For a fillet without any longitudinal machining trace Ry asymp Rz
312 Size Factor YX The size factor YX takes into account the decrease of the strength with increasing size YX differs for endurance limit and static strength
The following may be used
For endurance limit
YX = 1 for mn le 5 generally
YX = 103 ndash 0006 mn for 5 lt mn lt 30
YX = 085 for mn ge 30
for not surface hard-ened steels
YX = 105 ndash 001 mn for 5 lt mn ge 25
YX = 08 for mn ge 25
for surface hardened steels
For static strength
YXst = 1 for all mn and all materials
313 Case Depth Factor YC The case depth factor YC takes into account the influence of hardening depth on tooth root strength
YC applies only to surface hardened tooth roots and is dif-ferent for endurance limit and static strength
In case of insufficient hardening depth fatigue cracks can develop in the transition zone between the hardened layer and the core For static strength yielding shall not occur in the transition zone as this would alter the surface residual stresses and therewith also the fatigue strength
The major parameters are case depth stress gradient permis-sible surface respectively subsurface stresses and subsurface residual stresses
The following simplified method for YC may be used
YC consists of a ratio between permissible subsurface stress (incl influence of expected residual stresses) and permissible surface stress This ratio is multiplied with a bracket con-taining the influence of case depth and stress gradient (The empirical numbers in the bracket are based on a high number of teeth and are somewhat on the safe side for low number of teeth)
YC and YCst may be calculated as given below but calculated values above 10 are to be put equal 10
For endurance limit
+
+=nFFE
C m02ρt31
σconstY
Classification Notes- No 412 29 May 2003
DET NORSKE VERITAS
For static strength
+
+=nFFst
Cst m02ρt31
σconstY
where const and t are connected as
Hardening process
t = endurance limit const =
static strength const =
t550 640 1900
t400 500 1200 Case hard-ening
t300 380 800
Nitriding t400 500 1200
Induction- or flame hardening
tHVmin 11 HVmin 25 HVmin
For symbols see 213
In addition to these requirements to minimum case depths for endurance limit some upper limitations apply to case hard-ened gears
The max depth to 550 HV should not exceed
1) 13 of the top land thickness san unless adequate tip relief is applied (see 110)
2) 025 mn If this is exceeded the following applies addi-tionally in connection with endurance limit
minusminus= 025
mt
1Yn
max550C
314 Thin rim factor YB Where the rim thickness is not sufficient to provide full sup-port for the tooth root the location of a bending failure may be through the gear rim rather than from root fillet to root fillet
YB is not a factor used to convert calculated root stresses at the 30deg tangent to actual stresses in a thin rim tension fillet Actually the compression fillet can be more susceptible to fatigue
YB is a simplified empirical factor used to de-rate thin rim gears (external as well as internal) when no detailed calcula-tion of stresses in both tension and compression fillets are available
Figure 34 Examples on thin rims
YB is applicable in the range 175 lt sRmn lt 35
YB = 115 middot ln (8324 middot mnsR)
(for sRmn ge 35 YB = 1)
(for sRmn le 175 use 315)
σF as calculated in 321 is to be multiplied with YB when sRmn lt 35 Thus YB is used for both high and low cycle fatigue
Note This method is considered to be on the safe side for external gear rims However for internal gear rims without any flange or web stiffeners the method may not be on the safe side and it is advised to check with the method in 315316
315 Stresses in Thin Rims For rim thickness sR lt 35 mn the safety against rim cracking has to be checked
The following method may be used
3151 General The stresses in the standardised 30ordm tangent section tension side are slightly reduced due to decreasing stress correction factor with decreasing relative rim thickness sRmn On the other hand during the complete stress cycle of that fillet a certain amount of compression stresses are also introduced The complete stress range remains approximately constant Therefore the standardised calculation of stresses at the 30ordm tangent may be retained for thin rims as one of the necessary criteria
The maximum stress range for thin rims usually occurs at the 60ordm ndash 80ordm tangents both for laquotensionraquo and laquocompressionraquo side The following method assumes the 75ordm tangent to be the decisive Therefore in addition to the am criterion ap-plied at the 30ordm tangent it is necessary to evaluate the max and min stresses at the 75ordm tangent for both laquotensionraquo (loaded flank) fillet and laquocompressionraquo (back-flank) fillet For this purpose the whole stress cycle of each fillet should be considered but usually the following simplification is justified
Figure 35 Nomenclature of fillets
Index laquoTraquo means laquotensileraquo fillet laquoCraquo means laquocompressionraquo fillet
σFTmin and σFCmax are determined on basis of the nominal rim stresses times the stress concentration factor Y75
σFCmin and σFTmax are determined on basis of superposition of nominal rim stresses times Y75 plus the tooth bending stresses at 75ordm tangent
30 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
3152 Stress concentration factors at the 75ordm tangents The nominal rim stress consists of bending stresses due to local bending moments tangential stresses due to the tan-gential force Ft and radial shear stresses due to Fr
The major influence is given by the bending stresses The influence of the tangential stresses is minor and even though its stress concentration factor is slightly higher than for bending it is considered to be safe enough when the sum of these nominal stresses are combined with the stress concen-tration factor for bending The influence of the radial shear stress is neglected
The stress concentration factor relating nominal rim stresses to local fillet stresses at the 75ordm tangent may be calculated as
75n
R75
n
R
75
ρmsρ185
ms3
Y+
sdot=
where ρ75 is the root radius at the 75ordm tangent ref to mn Usually ρ75 is closed to the tool radius ρao and
ρ75 = ρao
is a safe approximation compensating for the am simplifi-cations to the laquounsaferaquo side
The tooth root stresses of the loaded tooth are decreasing with decreasing relative rim thickness approximately with the empirical factor
15
R
ncorr s
m311Y
minus=
3153 Nominal rim stresses The bending moment applied to the rim consists of a part of the tooth tilt moment Ft (hF + 05 sR) and the bending caused by the radial force Fr
The sectional modulus (first moment of area) which is used for determination of the nominal bending stresses is not nec-essarily the same for the 2 am bending moments If flanges webs etc outside the toothed section contribute to stiffening the rim against various deflections the influence of these stiffeners should be considered Eg an end flange will have an almost negligible influence on the effective sectional modulus for the stresses due to tooth tilt as the deflection caused by the tilt moment is rather small and would not much involve the flange On the other hand the radial forces as for instance from the meshes in an annulus would cause considerable radial deflections that the flange might restrict to a substantial amount When considering the stiff-ening of such flanges or webs on basis of simplified models it is advised to use an effective rim thickness sR = sR + 02 mn for the first moment of area of the rim (toothed part) cross section
For a high number of rim teeth it may be assumed that the rim bending moments in the fillets adjacent to the loaded tooth are of the same magnitude as right under the applied
force This assumption is reasonable for an annulus but rather much on the safe side for a hollow pinion
The influence of Ft on the nominal tangential stress is simpli-fied by half of it for compressive stresses (σ1) and the other half for tensile stresses (σ2) Applying these assumptions the nominal rim stresses adjacent to the loaded tooth are
( ) ( )A2
FW
f RFW
s05hF05 t
R
r
T
RFt1 minus
ϑminus
+minus=σ
( ) ( )A2
FW
f RFW
s05hF05 t
R
r
T
RFt2 +
ϑminus
+=σ
where σ1 σ2 see Fig 35
A = minimum area of cross section (usually bR sR)
WT = the sectional modulus of rim with respect to tooth tilt moment (usually 6sb 2
rR )
WR = the sectional modulus of the rim including the influence of stiffeners as flanges webs etc (WR ge WT)
bR = the width of the rim
R = the radius of the neutral axis in the rim ie from wheel centre to midpoint of rim
( )ϑf = a function for bending moment distribution around the rim For a rim (pinion) with one mesh only the ( )ϑf at the position of load application is 024 For an annulus with 3 or more meshes ( )ϑf at the position of each load application is approx 3 planets ( )ϑf = 019 4 planets ( )ϑf = 014 5 planets ( )ϑf = 011 6 planets ( )ϑf = 009
It must be checked if the max (tensile) stress in the compres-sion fillet really occurs when the fillet is adjacent to the loaded tooth In principle the stress variation through a complete rotation should be considered and the max value used The max value is usually never less than 0 For an annulus eg the tilt moment is zero in the mid position be-tween the planet meshes whilst the bending moment due to the radial forces is half of that at the mesh but with opposite sign
If these formulae are applied to idler gears as eg planets the influence of nominal tangential stresses must be cor-rected by deleting Ft(2 A) for σ1 and using FtA for σ2 Further the influence of Fr on the nominal bending stresses is usually negligible due to the planet bearing support
3154 Root fillet stresses Determination of min and max stresses in the laquotensionraquo fillet
Classification Notes- No 412 31 May 2003
DET NORSKE VERITAS
Minimum stress
751FTmin YσK σ =
Maximum stress
corrF752 Yσ03Yσ KσFTmax +=
where
03 is an empirical factor relating the tension stresses (σF) at the 30ordm tangent to the part of the tension stresses at the 75ordm tangent which add to the rim related stresses (03 also takes into account that full superposition of nominal stresses times stress concentration factors from both laquosides of the corner filletraquo would result in too high stresses)
αβγ sdotsdotsdotsdot= FFvA KKKKKK
Determination of min and max stresses in the laquocompres-sionraquo fillet
Minimum stress
corrF751FCmin Yσ036Yσ Kσ minus=
Maximum stress
752FCmax Yσ Kσ =
where
036 is an empirical factor relating the tension stresses (σF) at the 30ordm tangent to the part of the compression stresses at the 75ordm tangent which add to the rim related stresses
For gears with reversed loads as idler gears and planets there is no distinct laquotensionraquo or laquocompressionraquo fillet The mini-mum stress σFmin is the minimum of σFTmin and σFCmin (usually the latter is decisive) The maximum stress σFTmax is the maximum of σFTmax and σFCmax (usually the former is deci-sive)
316 Permissible Stresses in Thin Rims
3161 General The safety against fatigue fracture respectively overload fracture is to be at least at the same level as for solid gears The laquoordinaryraquo criteria at the 30ordm tangent apply as given in 31 through 313
Additionally the following criteria at the 75ordm tangent may apply
3162 For gt3middot106 cycles The permissible stresses for the laquotensionraquo fillets and for the laquocompressionraquo fillets are determined by means of a relevant fatigue diagram
If the actual tooth root stress (tensile or compressive) ex-ceeds the yield strength to the material the induced residual stresses are to be taken into account
For determination of permissible stresses the following is defined
R = stress ratio ie FTmax
FTminσσ respectively
FCmax
FCmin
σσ
∆σ = stress range ie σFTmax ndash σFTmin resp σFCmax ndash σFCmin
(For idler gears and planets Fmax
FminσσR =
and ∆σ = σFmax minus σFmin)
The permissible stress range ∆σρ for the laquotensionraquo respec-tively laquocompressionraquo fillets can be calculated as
For R gt minus1 FPp σ
R1R1031
13∆σ
minus+
+=
For minus infin lt R lt minus1 FPp σ
R1R10151
13∆σ
minus+
+=
where
σFP = see 32 determined for unidirectional stresses (YM = 1)
If the yield strength σy is exceeded in either tension or com-pression residual stresses are induced This may be consid-ered by correcting the stress ratio R for the respective fillets (tension or compression)
Eg if yFCmin σσ gt (ie exceeded in compression) the
difference yFCmin σσ∆ minus= affects the stress ratio as
∆σ
σ∆σ∆σR
FCmax
y
FCmax
FCmin
+
minus=
++
=
Similarly the stress ratio in the tension fillet may require cor-rection
If the yield strength is exceeded in tension σFTmax gt σy the difference ∆ = σFTmax ndash σy affects the stress ratio as
y
FTmin
FTmax
FTmin
σ∆σ
∆σ∆σR minus=
minusminus
=
Checking for possible exceeding of the yield strength has to be made with the highest torque and with the KAP if this ex-ceeds KA
3163 Forle 103 cycles The permissible stress range ∆σp is not to exceed
For R gt minus1 FPstpst σ
R1R1051
15∆σ
minus+
+=
For ndash infin lt R lt minus1 FPstpst σ
R1R10251
15∆σ
minus+
+=
For all values of R ∆σpst is limited by
not surface hardened F
y
Sσ225
32 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
surface hardened CF
YSHV5
Definition of ∆σ and R see 3162 with particular attention to possible correction of R if the yield strength is exceeded
σFPst see 32 determined for unidirectional stresses (YM = 1) and lt 103 cycles
3164 For 103 lt cycles lt 3106 ∆σp is to be determined by linear interpolation a log-log dia-gram
∆σp at NL load cycles is
6L 103p
exp
L
6
N p ∆σN103∆σ sdot
sdot=
6
3
103p
10p
∆σ
∆σlog28760exp
sdot
=
Classification Notes- No 412 33 May 2003
DET NORSKE VERITAS
4 Calculation of Scuffing Load Capacity
41 Introduction High surface temperatures due to high loads and sliding ve-locities can cause lubricant films to break down This sei-zure or welding together of areas of tooth surface is termed scuffing
In contrast to pitting and fatigue breakage which show a dis-tinct incubation period a single short overloading can lead to a scuffing failure In the ISO-TR13989 two criteria are mentioned The method used in this Classification Note is based on the principles of the flash temperature criterion
Note Bulk temperature in excess of 120ordmC for long periods may have an adverse effect on the surface durability see 211
42 General Criteria In no point along the path of contact the local contact tem-perature may exceed the permissible temperature ie
oilS
oilSB S
ϑ+ϑminusϑ
leϑ
50SB minusϑleϑ
where
Bϑ = max contact temperature along the path of contact
maxflaMBB ϑ+ϑ=ϑ
MBϑ = bulk temperature see 434
maxflaϑ = max flash temperature along the path of con-tact see 44
Sϑ = scuffing temperature see below
oilϑ = oil temperature before it reaches the mesh (max applicable for the actual load case to be used ie normally alarm temperature except for ice classes where a max expected tem-perature applies)
SS = required safety factor according to the Rules
The scuffing temperature Sϑ may be calculated as
L2
wrelT
002
40S XFZGX
ν100112085780
sdot
sdot++=ϑ
where
XwrelT = relative welding factor
XwrelT
Through hardened steel 10
Phosphated steel 125
Copper-plated steel 150
Nitrided steel 150
Less than 10 retained austenite
115
10 ndash 20 retained austenite 10
Casehardened steel
20 ndash 30 retained austenite 085
Austenitic steel 045
FZG = load stage according to FZG-Test A8390 (Note This is the load stage where scuffing oc-curs However due to scatter in test results cal-culations are to be made with one load stage less than the specification)
XL = lubricant factor
= 10 for mineral oils
= 08 for polyalfaolefins
= 07 for non-water-soluble polyglycols
= 06 for water-soluble polyglycols
= 15 for traction fluids
= 13 for phosphate esters
ν40 = kinematic oil viscosity at 40˚C (mm2s)
Application of other test methods such as the Ryder the FZG-Ryder R46574 and the FZG L-42 Test 141195110 may be specially considered
For high speed gears with very short time of contact Sϑ may be increased as follows provided use of EP-oils
Addition to the calculated scuffing temperature Sϑ
If micros 18tc ge no addition
If micros 18tc lt add ( )cwrelT t18X18 minussdotsdot
where
ct = contact time ( micros ) which is the time needed to cross the Hertzian contact width
( ) [ ]microsu1cosβn
uσ340tb1
Hc +sdotsdot
sdot=
σH as calculated in 221
For bevel gears use vu in stead of u
34 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
43 Influence Factors
431 Coefficient of friction The following coefficient of friction may apply
L025a
005oil
02
redC ΣC
Bt X R ηρv
w0048micro minus
=
where
wBt = specific tooth load (Nmm)
vΣC = sum of tangential velocities at pitch point At pitch line velocities gt 50 ms the limiting value of vΣC at v = 50 ms is to be used
ρredC = relative radius of curvature (transversal plane) at the pitch point
Cylindrical gears
HαHβvγAbt
Bt KKKKKb
Fw sdotsdotsdotsdotsdot= (see 1)
twΣC αsin v2v =
bCredC β cos ρρ =
Bevel gears
HαHβvγAtmb
Bt KKKKKb
Fw sdotsdotsdotsdotsdot= (see 1)
vtmtΣC αsin v2v =
bmvCredC β cos ρρ =
ηoil = dynamic viscosity (mPa s) at oilϑ calculated as
1000
ρ νη oiloil =
where ρ in kgm3 approximated as
( ) 70 15oil15 minusϑminusρ=ρ and νoil is kinematic viscosity at oilϑ and may be calculated by means of the following equation
( ) ( )++=+ 08νloglog08νloglog 100oil ( )
sdotminus
ϑ+minus313log373log
273log373log oil
( ) ( )( )08νloglog08νloglog 10040 +minus+
Ra = composite arithmetic mean roughness (micron) of pinion an wheel calculated as ( )2a1aa RR 05R +=This is defined as the roughness on the new flanks ie as manufactured
XL = see 42
432 Effective tip relief Ceff Ceff is the effective tip relief that amount of tip relief which compensates for the elastic deformation of the gear mesh ie
zero load at the tooth tip It is assumed (simplified) to be equal for both pinion and wheel
Cylindrical gears
for helical
γ
γ=cb
KKFC Abt
eff (see 1)
For spur
cbKKF
C Abteff
γ= (see 1)
Alternatively for spur and helical gears the non-linear ap-proach in 111 may be used (taking Ceff=δ)
Bevel gears
bcKFC Ambt
effγ
= (see 1)
where
γ
α
ε2ε44c
+sdot
=γ
433 Tip relief and extension Cylindrical gears
The extension of the tip relief is not to result in an effective contact ratio 1ltεα when the gear is unloaded (exceptions to this may only apply for applications where the gear is not to run at light loads) This means that the unrelieved part of the path of contact is to be minimum pbt It is further assumed that this unrelieved part is placed centrally on the path of contact
If root relief applies it has to be calculated as equivalent tip relief Ie pinion root relief (at mesh position A) is added to Ca2 and wheel root relief (at mesh position E) is added to Ca1 If no design tip relief or root relief on the mating gear is specified (ie if Ca1+Croot2 = 0 and visa versa) use the run-ning in amount see 112
Bevel gears
Bevel gears are to have heightwise crowning ie no distinct relievedunrelieved area This may be treated as tip and root relief For calculation purposes the root relief is combined with the tip relief of the mating member into an equivalent tip relief If no resulting tip relieves are specified the equivalent tip relives may be calculated as an approxima-tion based on the tool crowning Ca tool (per mille of tool module m0) as follows
2root1aeq1a CCC +=
1root2aeq2a CCC +=
where
Classification Notes- No 412 35 May 2003
DET NORSKE VERITAS
2
0
n0101atoolal m
)m2(mA1middotmCC
minus+minussdot=
( )
sdotminusminussdotsdotminus+= vtvb1
2vb1
21vavt1n1 αtan dddαsin 05x1mA
2
0
n020atool22a m
)mm(2A1middotmCC
minus+minussdot=
( )
sdotminusminussdotsdotminus+= vtvb2
2vb2
2va2vt2n2 αtan dddαsin 05x1mA
2
0
20atool1root1 m
A1mCC
minussdotsdot=
2
0
10atool2root2 m
A1mCC
minussdotsdot=
If the pressure angles of the cutter blades are modified (and verified) to balance the tip relieves the following may be assumed
( )eq2aeq1aeq2aeq1a C andC calculated of sum50CC sdot==
Throughout the following Ca1 and Ca2 mean the equivalent tip relieves Ca1eq and Ca2eq
434 Bulk temperature The bulk temperature may be calculated as
flaaveragempsoilMB X X 50 ϑ+ϑ=ϑ
where
Xs = lubrication factor
= 12 for spray lubrication
= 10 for dip lubrication (provided both pinion and wheel are dip lubricated and tip speed lt 5 ms)
= 10 for spray lubrication with additional cooling spray (spray on both pinion and wheel or spray on pinion and dip lubrication of wheel)
= 02 for meshes fully submerged in oil
Xmp = contact factor ( )pmp nX += 150
np = number of mesh contact on the pinion (for small gear ratii the number of wheel meshes should be used if higher)
flaaverageϑ
= average of the integrated flash temperature (see 44) along the path of contact
( )
AE
E
Ayyyfla
flaaverage
d
ΓminusΓ
ΓΓϑ=ϑint =
For high speed gears (v gt 50 ms) it may be necessary to assess the bulk temperature on the basis of a thermal rating of the entire gear transmission
44 The Flash Temperature flaϑ
441 Basic formula The local flash temperature flaϑ may be calculated as
( ) 41redy
y2ly
211
43Btcorrfla
unXwX3250
y ρ
ρminusρ
micro=ϑ Γ
(For bevel gears replace u with uv)
and is to be calculated stepwise along the path of contact from A to E
where
micro = coefficient of friction see 431
Xcorr = correction factor taking empirically into account the increased scuffing risk in the beginning of the approach path due to mesh starting without any previously built up oil film and possible shuffling away oil before meshing if insufficient tip relief
( )
3
AD
yaeffcorr 50
CC1X
ΓminusΓε
Γminus+=
α
Ca = tip relief of driven member Xcorr is only applicable in the approach path and if Ca lt Ceff otherwise 10
wBt = unit load see 431
yXΓ = load sharing factor see 443
n1 = pinion rpm
Γy ρly etc see 442
442 Geometrical relations The various radii of flank curvature (transversal plane) are
ρ1y = pinion flank radius at mesh point y
ρ2y = wheel flank radius at mesh point y
ρredy = equivalent radius of curvature at mesh point y
y2y1
y2y1redy
ρ+ρ
ρρ=ρ
Cylindrical gears
twy
1y αsin au1Γ1
ρ+
+=
twy
2y αsin au1Γu
ρ+
minus=
Note that for internal gears a and u are negative
36 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
Bevel gears
vtvv
y1y αsin a
u1Γ1
ρ+
+=
vtvv
yv2y αsin a
u1Γu
ρ+
minus=
Γ is the parameter on the path of contact and y is any point between A end E
At the respective ends Γ has the following values
Root piniontip wheel
Cylindrical gears
( )
minus
minusminus= 1
αtan 1dd
zzΓ
tw
2b2a2
1
2A
Bevel gears
( )
minus
minusminus= 1
αtan 1dd
uΓvt
2vb2va2
vA
Tip pinionroot wheel
Cylindrical gears
( )
1αtan
1ddΓ
tw
2b1a1
E minusminus
=
Bevel gears
( )
1αtan
1ddΓ
vt
2vb1va1
E minusminus
=
At inner point of single pair contact
Cylindrical gears
tw1
EB α tan zπ2ΓΓ minus=
Bevel gears
vtv1
EB α tan zπ2ΓΓ minus=
At outer point of single pair contact
Cylindrical gears
tw1AD α tan z
π2ΓΓ +=
Bevel gears
vt v1
AD αtan z π2ΓΓ +=
At pitch point ΓC = 0
The points F and G (only applicable to cylindrical gears) limiting the extension of tip relief (so as to maintain a mini-mum contact ratio of unity for unloaded gears) are at
2
BAF
Γ+Γ=Γ
2
EDG
Γ+Γ=Γ
443 Load sharing factor XΓ The load sharing factor XΓ accounts for the load sharing be-tween the various pairs of teeth in mesh along the path of contact
XΓ is to be calculated stepwise from A to E using the pa-rameter Γy
4431 Cylindrical gears with β = 0 and no tip relief
Figure 41
ByAAB
Ay for31
31X
yΓltΓleΓ
ΓminusΓ
ΓminusΓ+=Γ
DyB for1Xy
ΓltΓleΓ=Γ
EyDDE
yE for31
31X
yΓleΓltΓ
ΓminusΓ
ΓminusΓ+=Γ
4432 Cylindrical gears with β = 0 and tip relief Tip relief on the pinion reduces XΓ in the range G ndash E and increases correspondingly XΓ in the range F ndash B
Tip relief on the wheel reduces XΓ in the range A ndash F and increases correspondingly XΓ in the range D ndash G
Following remains generally
DyB for1Xy
ΓleΓleΓ=Γ
21XXGF== ΓΓ
In the following it must be distinguished between Ca lt Ceff respectively Ca gt Ceff This is shown by an example below where Ca1 lt Ceff and Ca2 gt Ceff
Classification Notes- No 412 37 May 2003
DET NORSKE VERITAS
Figure 42
Note When Ca gt Ceff the path of contact is shortened by A ndash A respectively E ndash E The single pair contact path is extended into B respectively D If this shift is significant it is necessary to consider the negative effect on surface durability (B) and bending stresses (B and D)
Range A - F
For effa2 CC le
minus=Γ
eff
2a
CC1
31X
A
FyAeff
2a
AF
Ay for C3
C61XX
AyΓleΓleΓ
sdot
+ΓminusΓ
ΓminusΓ+= ΓΓ
For effa2 CC ge
AyAfor0Xy
ΓleΓleΓ=Γ
with ( )21
CC
1CC
eff
2a
eff
2a
AFAAminus
minusΓminusΓ+Γ=Γ
FyAeff
2a
AF
Ay
eff
2a for 21
CC
CC1X
yΓleΓleΓ
minus
ΓminusΓ
ΓminusΓ+minus=Γ
Range F - B
For effa1 CC le
ByFeff
1a
FB
Fy forC3
C61
21X
yΓleΓleΓ
sdot
+ΓminusΓ
ΓminusΓ+=Γ
For effa1 CC ge
ByFeff
1a
FB
Fy for21
CC
21X
yΓleΓleΓ
minus
ΓminusΓ
ΓminusΓ+=Γ
ByB for1Xy
ΓleΓleΓ=Γ
with 1
CC2
eff
1a
FBFB
minus
ΓminusΓ+Γ=Γ
Range D ndash G
For effa2 CC le
GyDeff
2a
DG
Dy
eff
2a for C 3C
61
C 3C
32X
yΓleΓleΓ
+
ΓminusΓ
ΓminusΓminus+=Γ F
or effa2 CC ge
DyD for1Xy
ΓleΓleΓ=Γ
with ( )21
CC
1CC
eff
2a
eff
2a
DGDDminus
minusΓminusΓ+Γ=Γ
GyDeff
2a
DG
Dy
eff
2a for21
CC
CCX ΓleΓleΓ
minus
ΓminusΓ
ΓminusΓminus=Γ
Range G - E
For effa1 CC le
minus=Γ
eff
1a
CC1
31X
E
EyGeff
1a
GE
Gy forC3
C61
21X
yΓleΓleΓ
sdot
+ΓminusΓ
ΓminusΓminus=Γ
For effa1 CC gt
EyGeff
1a
GE
Gy for21
CC
21X
yΓleΓleΓ
minus
ΓminusΓ
ΓminusΓminus=Γ
EyE for0Xy
ΓleΓleΓ=Γ
with 1
CC2
eff
1a
GEGE
minus
ΓminusΓ+Γ=Γ
4433 Gears with β gt 0 buttressing Due to oblique contact lines over the flanks a certain but-tressing may occur near A and E
This applies to both cylindrical and bevel gears with tip relief lt Ceff The buttressing Xbutt is simplified as a linear function within the ranges A ndash H respectively I ndash E
Figure 43
1εwhen13X βbutt EAge=
1εwhenε031X ββbutt EAlt+=
38 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
Cylindrical gears
bIEAH βsin 02ΓΓΓΓ =minus=minus
Bevel gears
bmIEAH βsin 02ΓΓΓΓ =minus=minus
4434 Cylindrical gears with 2εγ le and no tip relief
yXΓ is obtained by multiplication of
yXΓ in 4431 with
Xbutt in 4433
4435 Gears with 2εγ gt and no tip relief
Applicable to both cylindrical and bevel gears
IyH for1Xy
ΓleΓleΓε
=α
Γ
IyHybutt and forX1Xy
ΓgtΓΓltΓε
=α
Γ
Figure 44
4436 Cylindrical gears with 2εγ le and tip relief
yXΓ is obtained by multiplication of
yXΓ in 4432 with Xbutt
in 4433
4437 Cylindrical gears with 2εγ gt and tip relief
Tip relief on the pinion (respectively wheel) reduces XГ in the range G ndash E (respectively A ndash F) and increases XГ in the range F ndash G
yXΓ is obtained by multiplication of
yXΓ as described below
with Xbutt in 4433
In the XГ example below the influence of tip relief is shown (without the influence of Xbutt) by means of
eff2aeff1a CC and CC ltgt
Tip relief gt Ceff causes new end points A respectively E of the path of contact
Figure 45
Range A ndash F
( ) ( )( ) eff
a2αa1α
AF
Ay
eff
2aeff
C 12C 13εC 1ε
CCCX
y +εε++minus
ΓminusΓ
ΓminusΓ+
εminus
=ααα
Γ
eff2aFyA CC if for leΓleΓleΓ
eff2aFyA CifCfor and geΓleΓleΓ
eff2aAyA CC iffor0Xy
gtΓleΓleΓ=Γ
( ) ( ) ( )( ) ( ) a2αa1α
αeffa2AFAA C 1ε 3C 1ε
1ε 2 CC with++minus+minus
ΓminusΓ+Γ=Γ
Range F ndash G
( )( )( ) GyF
eff
2a1a forC 1 2CC 11X
yΓleΓleΓ
+εε+minusε
+ε
=αα
α
αΓ
Range G ndash E
( ) ( )( ) eff
2a1a
GE
Gy
C 1 2C 1C 1 3XX
GFy +εεminusε++ε
ΓminusΓ
ΓminusΓminus=
αα
ααΓΓ minus
effa1EyG CC if for leΓleΓleΓ
effa1EyG CC if Γfor and geΓleΓle
eff1aEyE CC if for0Xy
geΓleΓleΓ=Γ
( ) ( ) ( )( ) ( ) 2a1a
eff1aGEEE C 1C 1 3
1 2 CC withminusε++ε+εminus
ΓminusΓminusΓ=Γαα
α
4438 Bevel gears with εγ more than approx 18 and heightwise crowning For Ca1 = Ca2 = Ceff the following applies
Figure 46
Classification Notes- No 412 39 May 2003
DET NORSKE VERITAS
( )AEM 50 Γ+Γ=Γ
( )( )2AD
3
2My651X
y ΓminusΓε
ΓminusΓminus
ε=
ααΓ
For tip relief lt Ceff yXΓ is found by linear interpolation be-
tween ( )effay CCX =Γ and ( )0CayX =Γ as in 4435
The interpolation is to be made stepwise from A to M with the influence of Ca2 and from M to E with the influence of Ca1 (For 2a1a CC ne there is a discontinuity at M)
Eg with Ca1 = 04 Ceff and Ca2 = 055 Ceff then
Range A ndash M
)CC()0C( eff2ay2ayyX 550X 450X =Γ=ΓΓ +=
Range M ndash E
)CC()0C( eff1ay1ayyX 40X 06X =Γ=ΓΓ +=
For tip relief gt Ceff the new end points A and E are found as
( )
minusΓminusΓ
ε+Γ=Γ α 1
CC
6 eff
2aADAA
( )
minusΓminusΓ
ε+Γ=Γ α 1
CC
6 eff
1aADEE
Range A ndash A
0Xy=Γ
Range A ndash M
( )( )
ΓminusΓ
ΓminusΓminus
minusε=
αΓ 2
MA
2My
eff
2a1
CC4
351Xy
Range M ndash E
( )( )
ΓminusΓ
ΓminusΓminus
minusε=
αΓ 2
ME
2My
eff
1a1
CC4
351Xy
Range E ndash E
0Xy=Γ
40 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
Appendix A Fatigue Damage Accumulation
The Palmgren-Miner cumulative damage calculation princi-ple is used The procedure may be applied as follows
A1 Stress Spectrum From the individual torque classes the torques (Ti) at the peak values of class intervals and the associated number of cycles (NLi) for both pinion and wheel are to be listed from the highest to the lowest torque
(In case of a cyclic torque variation within the torque classes it is advised to use the peak torque If the cyclic variation is such that the same teeth will repeatedly suffer the peak torque this is a must)
The stress spectra for tooth roots and flanks (σFi σHi) with all relevant factors (except KA) are to be calculated on the basis of the torque spectrum The load dependent K-factors are to be determined for each torque class
A2 σminusN-curve The stress versus load cycle curves for tooth roots and flanks (both pinion and wheel) are to be drawn on the basis of per-missible stresses (ie including the demanded minimum safety factors) as determined in 2 respectively 3 If different safety levels for high cycle fatigue and low cycle fatigue are desired this may be expressed by different demand safety factors applied at the endurance limit respectively at static strength
A3 Damage accumulation The individual damage ratio Di at ith stress level is defined as
Fi
Lii N
ND =
where
NLi = The number of applied cycles at ith stress
NFi = The number of cycles to failure at ith stress
Basically stresses σi below the permissible stress level for infinite life (if a constant ZN or YN is accepted) do not con-tribute to the damage sum However calculating the actual safety factor Sact as described below all the σi for which the product S σi is bigger than or equal to the permissible stress level for infinite life contribute to the damage sum and thus to the determination of Sact The final value of S is decisive
(NFi can be found mathematically by putting the permissible stress σpi equal the actual stress σi thereby finding the actual life factor This life factor can be solved with regard to load cycles ie NFi)
The damage sum ΣDi is not to exceed unity
If ΣDi ne 1 the safety against cumulative fatigue damage is different from the applied demand safety factor For deter-mination of this theoretical safety factor an iteration proce-dure is required as described in the following flowchart
S is correction factor with which the actual safety factor Sact can be found
Sact is the demand safety factor (used in determination of the permissible stresses in the σ ndash N ndash curve) times the correc-tion factor S
The full procedure is to be applied for pinion and wheel tooth roots and flanks
Note If alternating stresses occur in a spectrum of mainly pulsating stresses the alternating stresses may be replaced by equiva-lent pulsating stresses ie by means of division with the ac-tual mean stress influence factor YM
Classification Notes- No 412 41 May 2003
DET NORSKE VERITAS
Appendix B Application Factors for Diesel Driven Gears
For diesel driven gears the application factor KA depends on torsional vibrations Both normal operation and misfiring conditions have to be considered
Normally these two running conditions can be covered by only one calculation
B1 Definitions
Normal operation KAnorm = 0
normv0
TTT +
where
T0 = rated nominal torque
Tv norm = vibratory torque amplitude for normal opera-tion (see rules Pt4 Ch3 Sec1 G301 for defi-nition of ldquonormalrdquo irregularity)
Misfiring operation KA misf = 0
misfv
TTT +
where
T = remaining nominal torque when one cylinder out of action
Tv misf = vibratory torque amplitude in misfiring con-dition This refers to a permissible misfiring condition ie a condition that does not require automatic or immediate corrective actions as speed or pitch reduction
The normal operation is assumed to last for a very high num-ber of cycles such as 1010
The misfiring operation is assumed to last for a limited dura-tion such as 107 cycles
B2 Determination of decisive load Assuming life factor at 1010 cycles as YN = ZN = 092 which usually is relevant the calculation may be performed only once with the combination having the highest value of appli-cation factorlife factor
For bending stresses and scuffing the higher value of
092
K normA and 098
K misfA
For contact stresses the higher value of
092K normA and
113K misfA (but
097K misfA for nitrided gears)
B3 Simplified procedure Note that this is only a guidance and is not a binding con-vention
T = oTZ
1Zsdot
minus where Z = number of cylinders
( ) ideal Videal Vmisf Vnorm V TTT24ZT +minus=
where TV ideal = vibratory torque with all cylinders perfectly equal See also rules Pt4 Ch3 Sec1 G300
When using trends from torsional vibration analysis and measurements the following may be used
TV ideal To is close to zero for engines with few cylinders and using a suitable elastic coupling and increases with relative coupling stiffness and number of cylinders
TV misf To may be high for engines with few cylinders and decreases with number of cylinders
This can be indicated as
200Z
TT
ο
idealV asymp and 80Z04
TT
o
misfV minusasymp
Inserting this into the formulae for the two application fac-tors the following guidance can be given
118112K misfA minusasymp
115110K normA minusasymp
Since KA norm is to be combined with the lower life factors the decisive load condition will be the normal one and a KA of 115 will cover most relevant cases when a suitable elas-tic coupling is chosen
42 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
Appendix C Calculation of Pinion-Rack
Pinion-racks used for elevating of mobile offshore units are open gears that are subjected to wear and tear With normal specifications such as only a few hundred total operation cycles (site to site) the tooth bending stresses for static strength will be decisive for the lay out however with ex-ception of surface hardened pinions where case crushing has to be considered
In the following the use of part 1 and 3 for pinion-racks is shown including relevant simplifications
C1 Pinion tooth root stresses Since the load spectrum normally is dominated by high torques and few load cycles (in the range up to 1-2000) the static strength is decisive
The actual stress is calculated as
β1FSaFan1
tF1 KYY
mbFσ sdotsdotsdotsdot
=
b1 is limited to b2 + 2middotmn
YFa and YSa replace YF and YS because load application at tooth tip has to be assumed for such inaccurate gears
Pinions often use a non-involute profile in the dedendum part of the flank eg a constant radius equal the radius of curva-ture at reference circle For such pinions sFn and hFa are to be measured directly on a sectional drawing of the pinion tooth
Due to high loads and narrow facewidths it may be assumed that KFβ2 = KHβ = 10 However when b1 gt b2 then KFβ1 gt10
If no detailed documentation of KFβ1 is available the fol-lowing may be used
KFβ1 = 1 + 015middot(b1b2 ndash 1)
The permissible stress (not surface hardened) is calculated as
δrelTstF
Fst1FP1 Y
Sσσ sdot=
The mean stress influence due to leg lifting may be disre-garded
The actual and permissible stresses should be calculated for the relevant loads as given in the rules
C2 Rack tooth root stresses The actual stress is calculated as
SaFan2
tF2 YY
mbFσ sdotsdotsdot
=
See C1 for details
The permissible stress is calculated as
δrelTstF
Fst2FP2 Y
Sσσ sdot=
For alloyed steels (Ni Cr Mo) with high toughness and duc-tility the value of YδrelTst may be put equal to YSa
C3 Surface hardened pinions For surface hardened pinions the maximum load is not to cause crushing of the hardened layer of the flank
In principle the calculation described in 213 may be used but when the theoretical Hertzian stress exceeds the ap-proximately 18 times the yield strength of the rack material plastic deformation will occur This will limit the peak Hertzian stress but increases the contact width and thus the penetration of stresses into the depth
An approximation may be based on an assessment of contact width determined by means of equal areas under the theoreti-cal (elastic range) Hertzian contact and the elasto-plastic contact stress (limited to 18 middot σy) with the unknown width
12 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
bull The amount of crowning end relief or helix correction (defined in the plane of contact) is to be balanced around a zero line similarly to fsh
Figure 13 Crowning Cc balanced aound zero line
bull Superimpose these ordinates to the previous load distri-bution curve In case of high crowning etc as eg often applied to bevel gears the new load distribution curve may cross the base line (the real zero line) The result is areas with negative load that is not real as the load in those areas should be zero Thus corrective actions must be made but for practical reasons it may be postponed to after next operation
bull The amount of initial mesh misalignment bema ff + (defined in the plane of contact) is to be bal-
anced around a zero line If the direction of bema ff + is known (due to initial contact check) or if the direction of fbe is known due to design (eg overhang bevel pin-ion) this should be taken into account If direction un-known the influence of bema ff + in both directions as well as equal zero should be considered
Figure 14 fma+fbe in both directions balanced around zero line
Superimpose these ordinates to the previous load distri-bution curve This results in up to 3 different curves of which the one with the highest peak is to be chosen for further evaluation
bull If the chosen load distribution curve crosses the base line (ie mathematically negative load) the curve is to be corrected by adding the negative areas and dividing this with the active facewidth The (constant) ordinates of this rectangular correction area are to be subtracted from the positive part of the load distribution curve It is advisable to check that the area covered under this new load distribution curve is still equal δM b
bull If cγ cannot be considered as constant over b then cor-rect the ordinates of the load distribution curve with the local (on various positions over the facewidth) ratio between local mesh stiffness and average mesh stiffness cγ (average over the active facewidth only) Note that the result is to be a curve that covers the same area δM b as before
bull The influence of running in yβ is to be determined as in 112 whereby the value for Fβx is to be taken as twice the distance between the peak of the load distribution curve and δM
bull Determine
KHβ = M
β
δycurveofpeak minus
1932 Simplified analytical method for cylindrical gears The analytical approach is similar to 1931 but has a more limited application as cγ is assumed constant over the face-width and no helix modification applies
bull Calculate the elastic deflection fsh in the plane of contact Balance this deflection curve around a zero line so that the area above and below this zero line are equal see Fig 12 The max positive ordinate is frac12∆fsh
bull Calculate the initial mesh alignment as Fβx= deflbemash ffff plusmnplusmnplusmn∆ The negative signs may only be used if this is justified andor verified by a contact pattern test Otherwise al-ways use positive signs If a negative sign is justified the value of Fβx is not to be taken less than the largest of each of these elements
bull Calculate the effective mesh misalignment as Fβy = Fβx - yβ (yβ see 112)
bull Determine
2KforF2
bFc1K βH
m
βγγH le+=β
or
2KforF
bFc2K Hβ
m
βγγH gt=β
where cγ as used here is the effective mesh stiffness see 111
194 Determination of fsh fsh is the mesh misalignment due to elastic deflections Usu-ally it is sufficient to consider the combined mesh deflection of the pinion body and shaft and the wheel shaft The calcu-lation is to be made in the plane of contact (of the considered gear mesh) and to consider all forces (incl axial) acting on the shafts Forces from other meshes can be parted into components parallel respectively vertical to the considered plane of contact Forces vertical to this plane of contact have no influence on fsh
It is advised to use following diameters for toothed elements
d + 2 x mn for bending and shear deflection
d + 2 mn (x ndash ha0 + 02) for torsional deflection
Usually fsh is calculated on basis of an evenly distributed load If the analysis of KHβ shows a considerable maldis-
Classification Notes- No 412 13 May 2003
DET NORSKE VERITAS
tribution in term of hard end contact or if it is known by other reasons that there exists a hard end contact the load should be correspondingly distributed when calculating fsh In fact the whole KHβ procedure can be used iteratively 2-3 iterations will be enough even for almost triangular load distributions
195 Determination of fdefl fdefl is the mesh misalignment in the plane of contact due to bearing deflections and working positions (housing deflec-tion may be included if determined)
First the journal working positions in the bearings are to be determined The influence of external moments and forces must be considered This is of special importance for twin pinion single output gears with all 3 shafts in one plane
For rolling bearings fdefl is further determined on basis of the elastic deflection of the bearings An elastic bearing deflec-tion depends on the bearing load and size and number of rolling elements Note that the bearing clearance tolerances are not included here
For fluid film bearings fdefl is further determined on basis of the lift and angular shift of the shafts due to lubrication oil film thickness Note that fbe takes into account the influence of the bearing clearance tolerance
When working positions bearing deflections and oil film lift are combined for all bearings the angular misalignment as projected into the plane of the contact is to be determined fdefl is this angular misalignment (radians) times the face-width
196 Determination of fbe fbe is the mesh misalignment in the plane of contact due to tolerances in bearing clearances In principle fbe and fdefl could be combined But as fdefl can be determined by analy-sis and has a distinct direction and fbe is dependent on toler-ances and in most cases has no distinct direction (ie + toler-ance) it is practicable to separate these two influences
Due to different bearing clearance tolerances in both pinion and wheel shafts the two shaft axis will have an angular mis-alignment in the plane of contact that is superimposed to the working positions determined in 195 fbe is the facewidth times this angular misalignment Note that fbe may have a distinct direction or be given as a + tolerance or a combina-tion of both For combination of + tolerance it is adviced to use
fbe= 22be
21be ff ++plusmn
fbe is particularly important for overhang designs for gears with widely different kinds of bearings on each side and when the bearings have wide tolerances on clearances In general it shall be possible to replace standard bearings with-out causing the real load distribution to exceed the design premises For slow speed gears with journal bearings the expected wear should also be considered
197 Determination of fma fma is the mesh misalignment due to manufacturing toler-ances (helix slope deviation) of pinion fHβ1 wheel fHβ2 and housing bore
For gear without specifically approved requirements to as-sembly control the value of fma is to be determined as
fma= 22Hβ
21Hβ ff +
For gears with specially approved assembly control the value of fma will depend on those specific requirements
198 Comments to various gear types For double helical gears KHβ is to be determined for both helices Usually an even load share between the helices can be assumed If not the calculation is to be made as de-scribed in 171
For planetary gears the free floating sun pinion suffers only twist no bending It must be noted that the total twist is the sum of the twist due to each mesh If the value of 1K γ ne this must be taken into account when calculating the total sun pinion twist (ie twist calculated with the force per mesh without Kγ and multiplied with the number of planets)
When planets are mounted on spherical bearings the mesh misalignments sun-planet respectively planet-annulus will be balanced Ie the misalignment will be the average between the two theoretical individual misalignments The faceload distribution on the flanks of the planets can take full advan-tage of this However as the sun and annulus mesh with several planets with possibly different lead errors the sun and annulus cannot obtain the above mentioned advantage to the full extent
199 Determination of KHβ for bevel gears If a theoretical approach similar to 193-198 is not docu-mented the following may be used
testeff
H Kb
b851851K sdot
minussdot=β
beff b represents the relative active facewidth (regarding lapped gears see 192 last part)
Higher values than beff b = 090 are normally not to be used in the formula
For dual directional gears it may be difficult to obtain a high beff b in both directions In that case the smaller beff b is to be used
Ktest represents the influence of the bearing arrangement shaft stiffness bearing stiffness housing stiffness etc on the faceload distribution and the verification thereof Expected variations in length- and height-wise tooth profile is also accounted for to some extent
a) Ktest = 1 For ground or hard metal hobbed gears with the specified contact pattern verified at full rating or at full torque slow turning at a condition representative for the thermal expansion at normal operation
14 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
It also applies when the bearing arrangementsupport has insignificant elastic deflections and thermal axial expansion However each initial mesh contact must be verified to be within acceptance criteria that are cali-brated against a type test at full load Reproduction of the gear tooth length- and height-wise profile must also be verified This can be made through 3D measurements or by initial contact movements caused by defined axial offsets of the pinion (tolerances to be agreed upon)
b) Ktest = 1 + 04middot(beffbndash06) For designs with possible influence of thermal expansion in the axial direction of the pinion The initial mesh contact verified with low load or spin test where the acceptance criteria are calibrated against a type test at full load
c) Ktest = 12 if mesh is only checked by toolmakerrsquos blue or by spin test contact For gears in this category beffb gt 085 is not to be used in the calculation
110 Transversal Load Distribution Factors KHα and KFα The transverse load distribution factors KHα for contact stresses and for scuffing KFα for tooth root stresses account for the effects of pitch and profile errors on the transversal load distribution between 2 or more pairs of teeth in mesh
The following relations may be used
Cylindrical gears
( )
minus+==
tH
αptγγHαFα F
byfc0409
2ε
KK
valid for 2εγ le
( ) ( )tH
αptγ
γ
γHαFα F
byfcε
1ε20409KK
minusminus+==
valid for 2ε γ gt
where
FtH = Ft KA Kγ Kv KHβ
cγ = See 111
γα = See 112
fpt = Maximum single pitch deviation (microm) of pinion or wheel or maximum total profile form devia-tion Fα of pinion or wheel if this is larger than the maximum single pitch deviation
Note In case of adequate equivalent tip relief adapted to the load half of the above mentioned fpt can be introduced A tip relief is considered adequate when the average of Ca1 and Ca2 is within plusmn40 of the value of Ceff in 432
Limitations of KHα and KFα
If the calculated values for
KFα = KHα lt 1 use KFα = KHα = 10
If the calculated value of KHα gt 2εα
γ
Zε
ε use KHα = 2
εα
γ
Zε
ε
If the calculated value of KFα gt εα
γ
Yεε
use KFα = εα
γ
Yεε
where Yε = αnε
075025+ (for εαn see 331c)
Bevel gears
For ground or hard metal hobbed gears KFα = KHα = 1
For lapped gears KFα = KHα = 11
111 Tooth Stiffness Constants cacute and cγ The tooth stiffness is defined as the load which is necessary to deform one or several meshing gear teeth having 1 mm facewidth by an amount of 1 microm in the plane of contact cacute is the maximum stiffness of a single pair of teeth cγ is the mean value of the mesh stiffness in a transverse plane (brief term mesh stiffness)
Both valid for high unit load (Unit load = Ft middot KA middot Kγb)
Cylindrical gears The real stiffness is a combination of the progressive Hertzian contact stiffness and the linear tooth bending stiff-nesses For high unit loads the Hertzian stiffness has little importance and can be disregarded This approach is on the safe side for determination of KHβ and KHα However for moderate or low loads Kv may be underestimated due to determination of a too high resonance speed
The linear approach is described in A
An optional approach for inclusion of the non-linear stiffness is described in B
A The linear approach
BRCCq
βcos08acutec =
and
( )025ε075cc αγ +prime=
where
( )[ ]n02a01a
B α2000212
hh12051C minusminus
+minus+=
12n1n
x000635z
025791z
015551004723q minus++=
12
2n
22
1n
1 x005290z
x024188x000193z
x011654+minusminusminus
+ 000182 x22
Classification Notes- No 412 15 May 2003
DET NORSKE VERITAS
(for internal gears use zn2 equal infinite and x2 = 0) ha0 = hfp for all practical purposes CR considers the increased flexibility of the wheel teeth if the wheel is not a solid disc and may be calculated as
( )( )nR m5s
sR e5
bbln1C +=
where
bs = thickness of a central web
sR = average thickness of rim (net value from tooth root to inside of rim)
The formula is valid for bs b ge 02 and sRmn ge 1 Outside this range of validity and if the web is not centrally posi-tioned CR has to be specially considered
Note CR is the ratio between the average mesh stiffness over the facewidth and the mesh stiffness of a gear pair of solid discs The local mesh stiffness in way of the web corresponds to the mesh stiffness with CR = 1 The local mesh stiffness where there is no web support will be less than calculated with CR above Thus eg a centrally positioned web will have an effect corresponding to a longitudinal crowning of the teeth See also 1931 regarding KHβ
B The non-linear approach
In the following an example is given on how to consider the non-linearity
The relation between unit load Fb as a function of mesh deflection δ is assumed to be a progressive curve up to 500 Nmm and from there on a straight line This straight line when extended to the baseline is assumed to intersect at 10microm
With these assumptions the unit force Fb as a function of mesh deflection δ can be expressed as
( )10δKbF
minus= for 500bFgt
minus=
500Fb10δK
bF for 500
bFlt
with γAt KK
bF
bF
sdotsdot= etc (Nmm) ie unit load incorporat-
ing the relevant factors as
KA middot Kγ for determination of Kv
KA middot Kγ middot Kv for determination of KHβ
KA middot Kγ middot Kv middot KHβ for determination of KHα
δ = mesh deflection (microm)
K = applicable stiffness (c or cγ)
Use of stiffnesses for KV KHβ and KHα
For calculation of Kv and KHα the stiffness is calculated as follows
When Fb lt 500
the stiffness is determined as δ∆
∆ bF
where the increment is chosen as eg ∆ Fb = 10 and thus
50010Fb10
K10Fb∆δ +
++
=
When F b gt 500 the stiffness is c or cγ For calculation of KHβ the mesh deflection δ is used directly
or an equivalent stiffness determined as δsdotb
F
Bevel gears
In lack of more detailed relationship between stiffness and geometry the following may be used
b085
b13cacute eff=
b085b
16c effγ =
beff not to be used in excess of 085 b in these formulae
Bevel gears with heightwise and lengthwise crowning have progressive mesh stiffness The values mentioned above are only valid for high loads They should not be used for de-termination of Ceff (see 432) or KHβ (see 1931)
112 Running-in Allowances The running-in allowances account for the influence of run-ning-in wear on the various error elements
yα respectively yβ are the running-in amounts which reduce the influence of pitch and profile errors respectively influ-ence of localised faceload
Cay is defined as the running-in amount that compensates for lack of tip relief
The following relations may be used
For not surface hardened steel
ptHlim
α fσ160y =
yβ = βxlimH
f320σ
with the following upper limits
V lt 5 ms 5-10 ms gt 10 ms
yα max none
limH
12800σ limH
6400σ
yβ max none
limH
25600σ limH
12800σ
For surface hardened steel
yα = 0075 fpt but not more than 3 for any speed
yβ = 015 Fβx but not more than 6 for any speed
16 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
For all kinds of steel
5145189718
1C2
limHay +
minusσ
=
When pinion and wheel material differ the following ap-plies
bull Use the larger of fpt1 - yα1 and fpt2 - yα2 to replace fpt - yα in the calculation of KHα and Kv
bull Use ( )2β1ββ yy21y += in the calculation of KHβ
bull Use ( )2ay1aya CC21C += in the calculation of Kv
bull Use ( )2ay1ay2a1a CC21CC +== in the scuffing
calculation if no design tip relief is foreseen
Classification Notes- No 412 17 May 2003
DET NORSKE VERITAS
2 Calculation of Surface Durability
21 Scope and General Remarks Part 2 includes the calculations of flank surface durability as limited by pitting spalling case crushing and subsurface yielding Endurance and time limited flank surface fatigue is calculated by means of 22 ndash 212 In a way also tooth frac-tures starting from the flank due to subsurface fatigue is in-cluded through the criteria in 213
Pitting itself is not considered as a critical damage for slow speed gears However pits can create a severe notch effect that may result in tooth breakage This is particularly im-portant for surface hardened teeth but also for high strength through hardened teeth For high-speed gears pitting is not permitted
Spalling and case crushing are considered similar to pitting but may have a more severe effect on tooth breakage due to the larger material breakouts initiated below the surface Subsurface fatigue is considered in 213
For jacking gears (self-elevating offshore units) or similar slow speed gears designed for very limited life the max static (or very slow running) surface load for surface hard-ened flanks is limited by the subsurface yield strength
For case hardened gears operating with relatively thin lubri-cation oil films grey staining (micropitting) may be the lim-iting criterion for the gear rating Specific calculation meth-ods for this purpose are not given here but are under consid-eration for future revisions Thus depending on experience with similar gear designs limitations on surface durability rating other than those according to 22 - 213 may be ap-plied
22 Basic Equations Calculation of surface durability (pitting) for spur gears is based on the contact stress at the inner point of single pair contact or the contact at the pitch point whichever is greater
Calculation of surface durability for helical gears is based on the contact stress at the pitch point
For helical gears with 0 lt εβ lt 1 a linear interpolation be-tween the above mentioned applies
Calculation of surface durability for spiral bevel gears is based on the contact stress at the midpoint of the zone of contact
Alternatively for bevel gears the contact stress may be cal-culated with the program ldquoBECALrdquo In that case KA and Kv are to be included in the applied tooth force but not KHβ and KHα The calculated (real) Hertzian stresses are to be multi-plied with ZK in order to be comparable with the permissible contact stresses
The contact stresses calculated with the method in part 2 are based on the Hertzian theory but do not always represent the real Hertzian stresses
The corresponding permissible contact stresses σHP are to be calculated for both pinion and wheel
221 Contact stress Cylindrical gears
( )HαHβvγA
1
tβεEHDBH KKKKK
bud1uF
ZZZZZσ+
=
where
ZBD = Zone factor for inner point of single pair contact for pinion resp wheel (see 232)
ZH = Zone factor for pitch point (see 231)
ZE = Elasticity factor (see 24)
Zε = Contact ratio factor (see 25)
Zβ = Helix angle factor (see 26)
Ft KA Kγ Kv KHβ KHα see 15 ndash 110
d1 b u see 12 ndash 15
Bevel gears
( )HαHβvγA
v1v
vmtKEMH KKKKK
bud1uF
ZZZ105σ+
sdot=
where
105 is a correlation factor to reach real Hertzian stresses (when ZK = 1)
ZE KA etc see above
ZM = mid-zone factor see 233
ZK = bevel gear factor see 27
Fmt dv1 uv see 12 ndash 15
It is assumed that the heightwise crowning is chosen so as to result in the maximum contact stresses at or near the mid-point of the flanks
222 Permissible contact stress
XWRvLH
NHlimHP ZZZZZ
SZσ
σ =
where
σH lim = Endurance limit for contact stresses (see 28)
ZN = Life factor for contact stresses (see 29)
SH = Required safety factor according to the rules
ZLZvZR = Oil film influence factors (see 210)
ZW = Work hardening factor (see 211)
ZX = Size factor (see 212)
18 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
23 Zone Factors ZH ZBD and ZM
231 Zone factor ZH The zone factor ZH accounts for the influence on contact stresses of the tooth flank curvature at the pitch point and converts the tangential force at the reference cylinder to the normal force at the pitch cylinder
wtt2
wtbH sinααcos
cosαcosβ2Z =
232 Zone factors ZBD The zone factors ZBD account for the influence on contact stresses of the tooth flank curvature at the inner point of sin-gle pair contact in relation to ZH Index B refers to pinion D to wheel
For εβ ge 1 ZBD = 1
For internal gears ZD = 1
For εβ = 0 (spur gears)
( )
π
minusεminusminus
π
minusminus
α=
α2
2
2b
2a
1
2
1b
1a
wtB
z211
dd
z21
dd
tanZ
( )
π
minusεminusminus
π
minusminus
α=
α1
2
1b
1a
2
2
2b
2a
wtD
z211
dd
z21
dd
tanZ
If ZB lt 1 use ZB = 1
If ZD lt 1 use ZD = 1
For 0 lt εβ lt 1
ZBD = ZBD (for spur gears) ndash εβ (ZBD (for spur gears) ndash 1)
233 Zone factor ZM The mid-zone factor ZM accounts for the influence of the contact stress at the mid point of the flank and applies to spi-ral bevel gears
εminusminus
εminusminus
αβ=
αα btm2
2vb2
2vabtm2
1vb2val
2v1vvtbmM
pddpdd
ddtancos2Z
This factor is the product of ZH and ZM-B in ISO 10300 with the condition that the heightwise crowning is sufficient to move the peak load towards the midpoint
234 Inner contact point For cylindrical or bevel gears with very low number of teeth the inner contact point (A) may be close to the base circle In order to avoid a wear edge near A it is required to have suitable tip relief on the wheel
24 Elasticity Factor ZE The elasticity factor ZE accounts for the influence of the material properties as modulus of elasticity and Poissonrsquos ratio on the contact stresses
For steel against steel ZE = 1898
25 Contact Ratio Factor Zε The contact ratio factor Zε accounts for the influence of the transverse contact ratio εα and the overlap ratio εβ on the contact stresses
αε1Z =ε for 1εβ ge
( )α
ββ
αε ε
εε1
3ε4
Z +minusminus
= for εβ lt 1
26 Helix Angle Factor Zβ The helix angle factor Zβ accounts for the influence of helix angle (independent of its influence on Zε) on the surface du-rability
cosβZβ =
27 Bevel Gear Factor ZK The bevel gear factor accounts for the difference between the real Hertzian stresses in spiral bevel gears and the contact stresses assumed responsible for surface fatigue (pitting) ZK adjusts the contact stresses in such a way that the same per-missible stresses as for cylindrical gears may apply
The following may be used ZK = 080
28 Values of Endurance Limit σHlim and Static Strength 5H10σ 3H10σ
σHlim is the limit of contact stress that may be sustained for 5middot107 cycles without the occurrence of progressive pitting
For most materials 5middot107 cycles are considered to be the be-ginning of the endurance strength range or lower knee of the σ-N curve (See also Life Factor ZN) However for nitrided steels 2middot106 apply
For this purpose pitting is defined by
bull for not surface hardened gears pitted area ge 2 of total active flank area
bull for surface hardened gears pitted area ge 05 of total active flank area or ge 4 of one particular tooth flank area 510Hσ and 310Hσ are the contact stresses which the given
material can withstand for 105 respectively 103 cycles without subsurface yielding or flank damages as pitting spalling or case crushing when adequate case depth applies
The following listed values for σHlim 510Hσ and 310Hσ may only be used for materials subjected to a quality control as
Classification Notes- No 412 19 May 2003
DET NORSKE VERITAS
the one referred to in the rules Results of approved fatigue tests may also be used as the basis for establishing these values The defined survival probability is 99
σHlim σH105 σH10
3
Alloyed case hardened steels (surface hardness 58-63 HRC) - of specially approved high grade - of normal grade
1650 1500
2500 2400
3100 3100
Nitrided steel of approved grade gas nitrided (surface hardness 700-800 HV) 1250 13 σHlim 13 σHlim
Alloyed quenched and tempered steel bath or gas nitrided (surface hardness 500-700 HV)
1000
13 σHlim
13 σHlim
Alloyed flame or induction hardened steel (surface hardness 500-650 HV) 075 HV + 750 16 σHlim 45 HV
Alloyed quenched and tempered steel 14 HV + 350 16 σHlim 45 HV
Carbon steel 15 HV + 250 16 σHlim 16 σHlim
These values refer to forged or hot rolled steel For cast steel the values for σHlim are to be reduced by 15
29 Life Factor ZN The life factor ZN takes account of a higher permissible contact stress if only limited life (number of cycles NL) is demanded or lower permissible contact stress if very high number of cycles apply
If this is not documented by approved fatigue tests the fol-lowing method may be used
For all steels except nitrided
7L 105N sdotge ZN = 1 or
01570
L
7
N N105Z
sdot=
Ie ZN = 092 for 1010 cycles
The ZN = 1 from 5middot107 on may only be used when the mate-rial cleanliness is of approved high grade (see Rules Pt4 Ch2) and the lubrication is optimised by a specially approved filtering process
105 lt NL lt 5middot107 510N
logZ037
L
7
N N105Z
sdot=
NL = 105 WXRVLHlim
WstX10H1010NN ZZZZZσ
ZZσZZ
55
5=
103 lt NL lt 105 )Z(Zlog05
L
5
10NN
5N103N10
5N10ZZ
==
3L 10N le
WXRVLlimH
Wst10X10H10NN ZZZZZ
ZZZZ
33
3σ
σ==
(but not less than ZN105)
For nitrided steels
6L 102N sdotge ZN = 1 or
00980
L
6
N N102Z
sdot=
Ie ZN = 092 for 1010 cycles
The ZN = 1 from 2middot106 on may only be used when the mate-rial cleanliness is of approved high grade (see Rules Pt4 Ch2) and the lubrication is optimised by a specially approved filtering process
105 lt NL lt 2middot106 510N
Zlog76860
L
6
N N102Z
sdot=
5L 10N le
XWRVL
10XWst10NN ZZZZZ
ZZ13ZZ
55 ==
Note that when no index indicating number of cycles is used the factors are valid for 5middot107 (respectively 2middot106 for nitrid-ing) cycles
210 Influence Factors on Lubrication Film ZL ZV and ZR The lubricant factor ZL accounts for the influence of the type of lubricant and its viscosity the speed factor ZV ac-counts for the influence of the pitch line velocity and the roughness factor ZR accounts for influence of the surface roughness on the surface endurance capacity
The following methods may be applied in connection with the endurance limit
20 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
Surface hardened steels Not surface hardened steels
ZL ( )24013421
360910ν+
+
( )24013421
680830ν+
+
ZV ( )v3280
140930+
+ ( )v3280300850
++
ZR
080
ZrelR3
150
ZrelR3
where
ν40 = Kinematic oil viscosity at 40ordmC (mm2s) For case hardened steels the influence of a high bulk temperature (see 4 Scuffing) should be con-sidered Eg bulk temperatures in excess of 120ordmC for long periods may cause reduced flank surface endurance limits
For values of ν40 gt 500 use ν40 = 500
RZrel = The mean roughness between pinion and wheel (after running in) relative to an equivalent radius of curvature at the pitch point ρc = 10mm
RZrel
= ( ) 31
cZ2Z1 ρ
10RR50
+
RZ = Mean peak to valley roughness (microm) (DIN defini-tion) (roughly RZ = 6 Ra)
For 5
L 10N le ZL ZV ZR = 10
211 Work Hardening Factor ZW The work hardening factor ZW accounts for the increase of surface durability of a soft steel gear when meshing the soft steel gear with a surface hardened or substantially harder gear with a smooth surface
The following approximation may be used for the endurance limit
Surface hardened steel against not surface hardened steel 150
ZeqW R
31700
130HB21Z
minus
minus=
where HB = the Brinell hardness of the soft member For HB gt 470 use HB = 470 For HB lt 130 use HB = 130
RZeq = equivalent roughness
033
c40
066
ZS
ZHZHZEQ ρvν
15000RRRR
=
If RZeq gt 16 then use RZeq = 16
If RZeq lt 15 then use RZeq = 15
where
RZH = surface roughness of the hard member before run in
RZS = surface roughness of the soft member before run in
ν40 = see 210
If values of ZW lt 1 are evaluated ZW = 1 should be used for flank endurance However the low value for ZW may indi-cate a potential wear problem
Through hardened pinion against softer wheel
( )
minussdotminus+= 000829
HBHB0008981u1Z
2
1W
For 21HBHB
2
1 le use ZW = 1
For 71HBHB
2
1 gt use 17HBHB
2
1 =
For u gt 20 use u = 20
For static strength (lt 105 cycles)
Surface hardened against not surface hardened
ZWst = 105
Through hardened pinion against softer wheel
ZWst = 1
212 Size Factor ZX The size factor accounts for statistics indicating that the stress levels at which fatigue damage occurs decrease with an increase of component size as a consequence of the influ-ence on subsurface defects combined with small stress gradi-ents and of the influence of size on material quality
ZX may be taken unity provided that subsurface fatigue for surface hardened pinions and wheels is considered eg as in the following subsection 213
213 Subsurface Fatigue This is only applicable to surface hardened pinions and wheels The main objective is to have a subsurface safety against fatigue (endurance limit) or deformation (static strength) which is at least as high as the safety SH required for the surface The following method may be used as an approximation unless otherwise documented
The high cycle fatigue (gt3106 cycles) is assumed to mainly depend on the orthogonal shear stresses Static strength (lt103 cycles) is assumed to depend mainly on equivalent
Classification Notes- No 412 21 May 2003
DET NORSKE VERITAS
stresses (von Mises) Both are influenced by residual stresses but this is only considered roughly and empirically
The subsurface working stresses at depths inside the peak of the orthogonal shear stresses respectively the equivalent stresses are only dependent on the (real) Hertzian stresses Surface related conditions as expressed by ZL ZV and ZR are assumed to have a negligible influence
The real Hertzian stresses σHR are determined as
For helical gears with εβ gt 1
σHR = σH
For helical gears with εβ lt 1 and spur gears
ε
α
ββ ε
ε+εminus
sdotσ=σZ
1
HHR
For bevel gears
KHHR Z
1σσ sdot=
The necessary hardness HV is given as a function of the net depth tz (net = after grinding or hard metal hobbing and per-pendicular to the flank)
The coordinates tz and HV are to be compared with the de-sign specification such as
bull for flame and induction hardening tHVmin HVmin bull for nitriding t400min HV = 400 bull for case hardening t550min HV = 550 t400min HV = 400
and t300min HV = 300 (the latter only if the core hardness lt 300 If the core hardness gt 400 the t400 is to be re-placed by a fictive t400 = 16 t550)
In addition the specified surface hardness is not to be less than the max necessary hardness (at tz = 05aH) This applies to all hardening methods
For high cycle fatigue (gt3 106 cycles) the following applies
sdot+
minussdotsdotsdot= o90
05azt
05azt
cosSσ04HV
H
HHHR
applicable to 05a
zt
Hge
For 05at
H
z lt the value for 05at
H
z = applies
56300ρSσ12a cHHR
Hsdotsdot
sdot=
Where aH is half the hertzian contact width multiplied by an empirical factor of 12 that takes into account the possible influence of reduced compressive residual stresses (or even tensile residual stresses) on the local fatigue strength
If any of the specified hardness depths including the surface hardness is below the curve described by HV = f (tz) the actual safety factor against subsurface fatigue is determined as follows
reduce SH stepwise in the formula for HV and aH until all specified hardness depths and surface hardness balance with the corrected curve The safety factor obtained through this method is the safety against subsurface fatigue
For static strength (lt103 cycles) the following applies
sdot+
minussdotsdotsdot= o90
07at
06a
zt
cosSσ019HV
Hst
z
HstHHR
applicable to 06at
Hst
z ge
56300ρSσa cHHR
Hstsdotsdot
=
In the case of insufficient specified hardness depths the same procedure for determination of the actual safety factor as above applies
For limited life fatigue (103 lt cycles lt 3106 )
For this purpose it is necessary to extend the correction of safety factors to include also higher values than required Ie in the case of more than sufficient hardness and depths the safety factor in the formulae for both high cycle fatigue and static strength are to be increased until necessary and speci-fied values balance
The actual safety factor for a given number of cycles N be-tween 103 and 3106 is found by linear interpolation in a dou-ble logarithmic diagram
minussdotminus
= sdot logN3477
logSlogSlogS
310H6103HHN
36 10H103H Slog86281Slog86280 sdot+sdot sdot
22 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
3 Calculation of Tooth Strength
31 Scope and General Remarks Part 3 include the calculation of tooth root strength as limited by tooth root cracking (surface or subsurface initiated) and yielding
For rim thickness sR ge 35middotmn the strength is calculated by means of 32 ndash 313 For cylindrical gears the calculation is based on the assumption that the highest tooth root tensile stress arises by application of the force at the outer point of single tooth pair contact of the virtual spur gears The method has however a few limitations that are mentioned in 36
For bevel gears the calculation is based on force application at the tooth tip of the virtual cylindrical gear Subsequently the stress is converted to load application at the mid point of the flank due to the heightwise crowning
Bevel gears may also be calculated with the program BECAL In that case KA and Kv are to be included in the applied tooth force but not KFβ and KFα
In case of a thin annulus or a thin gear rim etc radial crack-ing can occur rather than tangential cracking (from root fillet to root fillet) Cracking can also start from the compression fillet rather than the tension fillet For rim thickness sR lt 35middotmn a special calculation procedure is given in 315 and 316 and a simplified procedure in 314
A tooth breakage is often the end of the life of a gear trans-mission Therefore a high safety SF against breakage is re-quired
It should be noted that this part 3 does not cover fractures caused by
bull oil holes in the tooth root space bull wear steps on the flank bull flank surface distress such as pits spalls or grey staining
Especially the latter is known to cause oblique fractures starting from the active flank predominately in spiral bevel gears but also sometimes in cylindrical gears
Specific calculation methods for these purposes are not given here but are under consideration for future revisions Thus depending on experience with similar gear designs limita-tions other than those outlined in part 3 may be applied
32 Tooth Root Stresses The local tooth root stress is defined as the max principal stress in the tooth root caused by application of the tooth force Ie the stress ratio R = 0 Other stress ratios such as for eg idler gears (R asymp -12) shrunk on gear rims (R gt 0) etc are considered by correcting the permissible stress level
321 Local tooth root stress The local tooth root stress for pinion and wheel may be as-sessed by strain gauge measurements or FE calculations or similar For both measurements and calculations all details are to be agreed in advance
Normally the stresses for pinion and wheel are calculated as
Cylindrical gears
FαFβvγAβSFn
tF K K K K K Y Y Y
m bFσ =
where
YF = Tooth form factor (see 33)
YS = Stress correction factor (see 34)
Yβ = Helix angle factor (see 36)
Ft KA Kγ Kv KFβ KFα see 15 ndash 110
b see 13
Bevel gears
FαFβvγASaFamn
mtF K K K K K Y Y Y
m bFσ ε=
where
YFa = Tooth form factor see 33
YSa = Stress correction factor see 34
Yε = Contact ratio factor see 35
Fmt KA etc see 15 ndash 110
b see 13
322 Permissible tooth root stress The permissible local tooth root stress for pinion respectively wheel for a given number of cycles N is
CXRrelTrelTF
NMFEFP YYYY
SYY
δσ
=σ
Note that all these factors YM etc are applicable to 3middot106 cycles when used in this formula for σFP The influence of other number of cycles on these factors is covered by the calculation of YN
where
σFE = Local tooth root bending endurance limit of reference test gear (see 37)
YM = Mean stress influence factor which accounts for other loads than constant load direction eg idler gears temporary change of load di-rection pre-stress due to shrinkage etc (see 38)
YN = Life factor for tooth root stresses related to reference test gear dimensions (see 39)
SF = Required safety factor according to the rules
YδrelT = Relative notch sensitivity factor of the gear to be determined related to the reference test gear (see 310)
YRrelT = Relative (root fillet) surface condition factor of the gear to be determined related to the reference test gear (see 311)
Classification Notes- No 412 23 May 2003
DET NORSKE VERITAS
YX = Size factor (see 312)
YC = Case depth factor (see 313)
33 Tooth Form Factors YF YFa The tooth form factors YF and YFa take into account the in-fluence of the tooth form on the nominal bending stress
YF applies to load application at the outer point of single tooth pair contact of the virtual spur gear pair and is used for cylindrical gears
YFa applies to load application at the tooth tip and is used for bevel gears
Both YF and YFa are based on the distance between the con-tact points of the 30˚-tangents at the root fillet of the tooth profile for external gears respectively 60˚ tangents for inter-nal gears
Figure 31 External tooth in normal section
Figure 32 Internal tooth in normal section
Definitions
n
2
n
Fn
enFn
Fe
F
α cosms
α cosmh6
Y
=
n
2
n
Fn
anFn
Fa
Fa
α cosms
α cosmh6
Y
=
In the case of helical gears YF and YFa are determined in the normal section ie for a virtual number of teeth
YFa differs from YF by the bending moment arm hFa and αFan and can be determined by the same procedure as YF with exception of hFe and αFan For hFa and αFan all indices e will change to a (tip)
The following formulae apply to cylindrical gears but may also be used for bevel gears when replacing
mn with mnm
zn with zvn
αt with αvt
β with βm
with undercut without undercut
Fig 33 Dimensions and basic rack profile of the teeth (finished profile)
Tool and basic rack data such as hfP ρfp and spr etc are re-ferred to mn ie dimensionless
331 Determination of parameters
( )n
n
prnfPnfP m
α cossαsin 1ρ
αtan h4πE
minusminusminusminus=
For external gears fPfP ρρ =
For internal gears ( )0z
195fPfP0
fPfP 10363156ρhxρρ
sdotminus+
+=
where
z0 = number of teeth of pinion cutter
x0 = addendum modification coefficient of pinion cutter
hfP = addendum of pinion cutter
ρfP = tip radius of pinion cutter
xhρG fpfp +minus=
24 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
τmE
2π
z2H
nnminus
minus=
with
3πτ = for external gears
6πτ = for internal gears
Htan zG2
nminusϑ=ϑ
(to be solved iteratively suitable start value6π
=ϑ for exter-
nal gears and 3π for internal gears)
a) Tooth root chord sFn For external gears
minus
ϑ+
ϑminus= ρ
cosG3
3πsinz
ms
fpnn
Fn
For bevel gears with a tooth thickness modification
xsm affects mainly sFn but also hFe and αFen The total influence of xsm on YFa Ysa can be approximated by only adding 2 xsm to sFn mn
For internal gears
minus
ϑ+
ϑminus= ρ
cosG
6πsinz
ms
fPnn
Fn
b) Root fillet radius ρF at 30ordm tangent
( )G2coszcosG2ρ
mρ
2n
2
fpn
F
minusϑϑ+=
c) Determination of bending moment arm hF dn = zn mn
b2α
αn βcosε
ε =
dan = dn + 2 ha
pbn = π mn cos αn
dbn = dn cos αn
( )4
d1εpzz
2dd
zz2d
2bn
2
αnbn
2bn
2an
en +
minusminus
minus=
en
bnen d
dcos arcα =
ennnn
e α invα invα x tan 22π
z1
minus+
+=γ
αFen = αen ndash γe
For external gears
( )
minus=
n
enFenee
n
Fe
mdαtan sin γ γcos
21
mh
]ρcos
G3πcosz fpn +
ϑminus
ϑminusminus
For internal gears
( ) minus
sdotsdotminus=
n
enFenee
n
Fe
mdα tanγ sinγ cos
21
mh
minus
ϑminus
ϑminussdot ρ
cosG3
6πcosz fPn
332 Gearing with εαn gt 2 For deep tooth form gearing ( )25ε2 αn lele produced with a verified grade of accuracy of 4 or better and with applied profile modification to obtain a trapezoidal load distribution along the path of contact the YF may be corrected by the factor YDT as
250ε205for 0666ε2366Y αnαnDT leleminus=
205εfor 10Y αnDT lt=
34 Stress Correction Factors YS YSa The stress correction factors YS and YSa take into account the conversion of the nominal bending stress to the local tooth root stress Thereby YS and YSa cover the stress increasing effect of the notch (fillet) and the fact that not only bending stresses arise at the root A part of the local stress is inde-pendent of the bending moment arm This part increases the more the decisive point of load application approaches the critical tooth root section
Therefore in addition to its dependence on the notch radius the stress correction is also dependent on the position of the load application ie the size of the bending moment arm
YS applies to the load application at the outer point of single tooth pair contact YSa to the load application at tooth tip
YS can be determined as follows
( )
+
+=L23121
1
sS qL01312Y
where Fe
Fn
hsL = and
F
Fns ρ2
sq = (see 33)
YSa can be calculated by replacing hFe with hFa in the above formulae
Note a) Range of validity 1 lt qs lt 8
In case of sharper root radii (ie produced with tools having too sharp tip radii) YS resp YSa must be specially considered
b) In case of grinding notches (due to insufficient protuberance of the hob) YS resp YSa can rise considerably and must be multiplied with
Classification Notes- No 412 25 May 2003
DET NORSKE VERITAS
g
g
ρt
0613
13
minus
where
tg = depth of the grinding notch
ρg = radius of the grinding notch
c) The formulae for YS resp YSa are only valid for αn = 20˚ However the same formulae can be used as a safe approximation for other pressure angles
35 Contact Ratio Factor Yε The contact ratio factor Yε covers the conversion from load application at the tooth tip to the load application at the mid point of the flank (heightwise) for bevel gears
The following may be used
Yε = 0625
36 Helix Angle Factor Yβ The helix angle factor Yβ takes into account the difference between the helical gear and the virtual spur gear in the nor-mal section on which the calculation is based in the first step In this way it is accounted for that the conditions for tooth root stresses are more favourable because the lines of contact are sloping over the flank
The following may be used (β input in degrees)
Yβ = 1 ndash εβ β120
When εβ gt 1 use εβ = 1 and when β gt 30deg use β = 30deg in the formula
However the above equation for Yβ may only be used for gears with β gt 25deg if adequate tip relief is applied to both pinion and wheel (adequate = at least 05 middot Ceff see 432)
37 Values of Endurance Limit σFE σFE is the local tooth root stress (max principal) which the material can endure permanently with 99 survival prob-ability 3106 load cycles is regarded as the beginning of the endurance limit or the lower knee of the σ ndash N curve σFE is defined as the unidirectional pulsating stress with a minimum stress of zero (disregarding residual stresses due to heat treatment) Other stress conditions such as alternating or pre-stressed etc are covered by the conversion factor YM
σFE can be found by pulsating tests or gear running tests for any material in any condition If the approval of the gear is to be based on the results of such tests all details on the testing conditions have to be approved by the Society Fur-ther the tests may have to be made under the Societys su-pervision
If no fatigue tests are available the following listed values for σFE may be used for materials subjected to a quality con-trol as the one referred to in the rules
σFE
Alloyed case hardened steels 1) (fillet surface hardness 58 ndash 63 HRC)
bull of specially approved high grade
1050
bull of normal grade
minus CrNiMo steels with approved process
1000
minus CrNi and CrNiMo steels generally 920 minus MnCr steels generally 850
Nitriding steel of approved grade quenched tempered and gas nitrided (surface hardness 700 ndash 800 HV)
840
Alloyed quenched and tempered steel bath or gas nitrided (surface hardness 500 ndash 700 HV)
720
Alloyed quenched and tempered steel flame or induction hardened 2) (incl entire root fillet) (fillet surface hardness 500 ndash 650 HV)
07 HV + 300
Alloyed quenched and tempered steel flame or induction hardened (excl entire root fillet) (σB = uts of base material)
025 σB + 125
Alloyed quenched and tempered steel 04 σB + 200
Carbon steel 025 σB + 250
Note All numbers given above are valid for separate forgings and for blanks cut from bars forged according to a qualified procedure see Pt 4 Ch 2 Sec 3 For rolled steel the values are to be reduced with 10 For blanks cut from forged bars that are not qualified as mentioned above the values are to be reduced with 20 For cast steel reduce with 40
1) These values are valid for a root radius
bull being unground If however any grinding is made in the root fillet area in such a way that the residual stresses may be affected σFE is to be reduced by 20 (If the grinding also leaves a notch see 34)
bull with fillet surface hardness 58 ndash 63 HRC In case of lower surface hardness than 58 HRC σFE is to be reduced with 20(58 ndash HRC) where HRC is the detected hardness (This may lead to a permissible tooth root stress that varies along the facewidth If so the actual tooth root stresses may also be considered along facewidth)
bull not being shot peened In case of approved shot peening σFE may be increased by 200 for gears where σFE is reduced by 20 due to root grinding Otherwise σFE may be increased by 100 for
6nm le and 100 ndash 5 (mn - 6) for mn gt 6 However the possible adverse influence on the flanks regarding grey staining should be considered and if necessary the flanks should be masked
2) The fillet is not to be ground after surface hardening Regarding possible root grinding see 1)
26 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
38 Mean stress influence Factor YM The mean stress influence factor YM takes into account the influence of other working stress conditions than pure pulsa-tions (R = 0) such as eg load reversals idler gears planets and shrink-fitted gears
YM (YMst) are defined as the ratio between the endurance (or static) strength with a stress ratio R ne 0 and the endurance (or static) strength with R = 0
YM and YMst apply only to a calculation method that assesses the positive (tensile) stresses and is therefore suitable for comparison between the calculated (positive) working stress σF and the permissible stress σFP calculated with YM or YMst
For thin rings (annulus) in epicyclic gears where the com-pression fillet may be decisive special considerations apply see 316
The following method may be used within a stress ratio ndash12 lt R lt 05
381 For idlers planets and PTO with ice class
M1M1R1
1Yor Y MstM
+minus
minus=
where
R = stress ratio = min stress divided by max stress
For designs with the same force applied on both forward- and back-flank R may be assumed to ndash 12
For designs with considerably different forces on forward- and back-flank such as eg a marine propulsion wheel with a power take off pinion R may be assessed as
branchmain theoffacewidth unit per forcepto offacewidt unit per force21minus
For a power take off (PTO) with ice class see 161 c
M considers the mean stress influence on the endurance (or static) strength amplitudes
M is defined as the reduction of the endurance strength am-plitude for a certain increase of the mean stress divided by that increase of the mean stress
Following M values may be used
Endurance limit
Static strength
Case hardened 08 ndash 015 Ys 1) 07
If shot peened 04 06
Nitrided 03 03
Induction or flame hardened
04 06
Not surface hardened steel 03 05
Cast steels 04 06 1)For bevel gears use Ys=2 for determination of M
The listed M values for the endurance limit are independent of the fillet shape (Ys) except for case hardening In princi-ple there is a dependency but wide variations usually only occur for case hardening eg smooth semicircular fillets versus grinding notches
382 For gears with periodical change of rotational direction For case hardened gears with full load applied periodically in both directions such as side thrusters the same formula for YM as for idlers (with R = ndash 12) may be used together with the M values for endurance limit This simplified approach is valid when the number of changes of direction exceeds 100 and the total number of load cycles exceeds 3middot106
For gears of other materials YM will normally be higher than for a pure idler provided the number of changes of direction is below 3middot106 A linear interpolation in a diagram with loga-rithmic number of changes of direction may be used ie from YM = 09 with one change to YM (idler) for 3middot106 changes This is applicable to YM for endurance limit For static strength use YM as for idlers
For gears with occasional full load in reversed direction such as the main wheel in a reversing gear box YM = 09 may be used
383 For gears with shrinkage stresses and unidirectional load For endurance strength
FE
fitM σ
σM1
M21Y+
minus=
σFE is the endurance limit for R = 0
For static strength YMst = 1 and σfit accounted for in 39b
σfit is the shrinkage stress in the fillet (30˚ tangent) and may be found by multiplying the nominal tangential (hoop) stress with a stress concentration factor
n
Ffit m
ρ215scf minus=
384 For shrink-fitted idlers and planets When combined conditions apply such as idlers with shrink-age stresses the design factor for endurance strength is
( ) ( ) FE
fitM σ
σR1M1
M2
M1M1R1
1Yminussdot+
minus
+minus
minus=
Symbols as above but note that the stress ratio R in this par-ticular connection should disregard the influence of σfit ie R normally equal ndash 12
For static strength
M1M1R1
1YMst
+minus
minus=
The effect of σfit is accounted for in 39 b
Classification Notes- No 412 27 May 2003
DET NORSKE VERITAS
385 Additional requirements for peak loads The total stress range (σmax ndash σmin) in a tooth root fillet is not to exceed
F
y
Sσ225
for not surface hardened fillets
FSHV5 for surface hardened fillets
39 Life Factor YN The life factor YN takes into account that in the case of limited life (number of cycles) a higher tooth root stress can be permitted and that lower stresses may apply for very high number of cycles
Decisive for the strength at limited life is the σ ndash N ndash curve of the respective material for given hardening module fillet radius roughness in the tooth root etc Ie the factors YδrelT YRelT YX and YM have an influence on YN
If no σ ndash N ndash curve for the actual material and hardening etc is available the following method may be used
Determination of the σ ndash N ndash curve
a) Calculate the permissible stress σFP for the beginning of the endurance limit (3middot106 cycles) including the influ-ence of all relevant factors as SF YδrelT YRelT YX YM and YC ie σFP = σFE middotYM middotYδrelT middotYRelT middotYX middotYC SF
b) Calculate the permissible laquostaticraquo stress (le 103 load cy-cles) including the influence of all relevant factors as SFst YδrelTst YMst and YCst ( )fitσCstYδrelTstYMstYFstσ
FstS1
FPstσ minussdotsdotsdot=
where σFst is the local tooth root stress which the material can resist without cracking (surface hardened materials) or unacceptable deformation (not surface hardened materials) with 99 survival probability
σFst
Alloyed case hardened steel 1) 2300
Nitriding steel quenched tempered and gas nitrided (surface hardness 700 ndash 800 HV)
1250
Alloyed quenched and tempered steel bath or gas nitrided (surface hardness 500 ndash 700 HV)
1050
Alloyed quenched and tempered steel flame or induction hardened (fillet surface hardness 500 ndash 650 HV)
18 HV + 800
Steel with not surface hardened fillets the smaller value of 2)
18 σB or 225 σy
1) This is valid for a fillet surface hardness of 58 ndash 63 HRC In case of lower fillet surface hardness than 58 HRC σFst is to be reduced with 30(58 ndash HRC) where HRC is the actual hardness Shot peening or grinding notches are not considered to have any significant influence on σFst 2) Actual stresses exceeding the yield point (σy or σ02) will alter the residual stresses locally in the ldquotensionrdquo fillet re-spectively ldquocompressionrdquo fillet This is only to be utilised for gears that are not later loaded with a high number of cycles at lower loads that could cause fatigue in the ldquocompressionrdquo fillet
c) Calculate YN as
NL gt 3middot106
YN = 1 or 001
L
6
N N103Y
sdot= ie Yn = 092 for 1010
The YN = 1 from 3middot106 on may only be used when special material cleanness applies see rules Pt4 Ch2
103ltNLlt3middot106exp
L
6
N N103Y
sdot=
cycles6103forσcycles310forσlog02876exp
FP
FPst
sdot=
NL lt 103 cycles6103forσcycles310forσ
NYFP
FPst
sdot=
or simply use σFPst as mentioned in b) directly
Guidance on number of load cycles NL for various applica-tions
bull For propulsion purpose normally NL = 1010 at full load (yachts etc may have lower values)
bull For auxiliary gears driving generators that normally operate with 70-90 of rated power NL = 108 with rated power may be applied
28 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
310 Relative Notch Sensitivity Factor YδrelT The dynamic (respectively static) relative notch sensitivity factor YδrelT (YδrelTst) indicate to which extent the theoreti-cally concentrated stress lies above the endurance limits (re-spectively static strengths) in the case of fatigue (respectively overload) breakage
YδrelT is a function of the material and the relative stress gra-dient It differs for static strength and endurance limit
The following method may be used
For endurance limit
for not surface hardened fillets
( )
024
s024
relT σ103133q21σ1012201351
Ysdotsdotminus
+sdotsdotminus+= minus
minus
δ
for all surface hardened fillets except nitrided
( )
106q21002451
Y srelT
++=δ
for nitrided fillets
( )
1347q2101421
Y srelT
++=δ
For static strength
for not surface hardened fillets1)
( )( )( )025
02
02502s
relTst σ3000821σ300 1Y0821Y
+
minus+=δ
for surface hardened fillets except nitrided
YδrelTst = 044 YS + 012
for nitrided fillets
YδrelTst = 06 + 02 YS 1) These values are only valid if the local stresses do not
exceed the yield point and thereby alter the residual stress level See also 39b footnote 2
311 Relative Surface Condition Factor YRrelT The relative surface condition factor YRrelT takes into ac-count the dependence of the tooth root strength on the sur-face condition in the tooth root fillet mainly the dependence on the peak to valley surface roughness
YRrelT differs for endurance limit and static strength
The following method may be used
For endurance limit
YRrelT = 1675 ndash 053 (Ry + 1)01 for surface hardened steels and alloyed quenched and tempered steels except nitrided
YRrelT = 53 ndash 42 (Ry + 1)001 for carbon steels
YδrelT = 43 ndash 326 (Ry + 1)0005
for nitrided steels
For static strength
YRrelTst = 1 for all Ry and all materials
For a fillet without any longitudinal machining trace Ry asymp Rz
312 Size Factor YX The size factor YX takes into account the decrease of the strength with increasing size YX differs for endurance limit and static strength
The following may be used
For endurance limit
YX = 1 for mn le 5 generally
YX = 103 ndash 0006 mn for 5 lt mn lt 30
YX = 085 for mn ge 30
for not surface hard-ened steels
YX = 105 ndash 001 mn for 5 lt mn ge 25
YX = 08 for mn ge 25
for surface hardened steels
For static strength
YXst = 1 for all mn and all materials
313 Case Depth Factor YC The case depth factor YC takes into account the influence of hardening depth on tooth root strength
YC applies only to surface hardened tooth roots and is dif-ferent for endurance limit and static strength
In case of insufficient hardening depth fatigue cracks can develop in the transition zone between the hardened layer and the core For static strength yielding shall not occur in the transition zone as this would alter the surface residual stresses and therewith also the fatigue strength
The major parameters are case depth stress gradient permis-sible surface respectively subsurface stresses and subsurface residual stresses
The following simplified method for YC may be used
YC consists of a ratio between permissible subsurface stress (incl influence of expected residual stresses) and permissible surface stress This ratio is multiplied with a bracket con-taining the influence of case depth and stress gradient (The empirical numbers in the bracket are based on a high number of teeth and are somewhat on the safe side for low number of teeth)
YC and YCst may be calculated as given below but calculated values above 10 are to be put equal 10
For endurance limit
+
+=nFFE
C m02ρt31
σconstY
Classification Notes- No 412 29 May 2003
DET NORSKE VERITAS
For static strength
+
+=nFFst
Cst m02ρt31
σconstY
where const and t are connected as
Hardening process
t = endurance limit const =
static strength const =
t550 640 1900
t400 500 1200 Case hard-ening
t300 380 800
Nitriding t400 500 1200
Induction- or flame hardening
tHVmin 11 HVmin 25 HVmin
For symbols see 213
In addition to these requirements to minimum case depths for endurance limit some upper limitations apply to case hard-ened gears
The max depth to 550 HV should not exceed
1) 13 of the top land thickness san unless adequate tip relief is applied (see 110)
2) 025 mn If this is exceeded the following applies addi-tionally in connection with endurance limit
minusminus= 025
mt
1Yn
max550C
314 Thin rim factor YB Where the rim thickness is not sufficient to provide full sup-port for the tooth root the location of a bending failure may be through the gear rim rather than from root fillet to root fillet
YB is not a factor used to convert calculated root stresses at the 30deg tangent to actual stresses in a thin rim tension fillet Actually the compression fillet can be more susceptible to fatigue
YB is a simplified empirical factor used to de-rate thin rim gears (external as well as internal) when no detailed calcula-tion of stresses in both tension and compression fillets are available
Figure 34 Examples on thin rims
YB is applicable in the range 175 lt sRmn lt 35
YB = 115 middot ln (8324 middot mnsR)
(for sRmn ge 35 YB = 1)
(for sRmn le 175 use 315)
σF as calculated in 321 is to be multiplied with YB when sRmn lt 35 Thus YB is used for both high and low cycle fatigue
Note This method is considered to be on the safe side for external gear rims However for internal gear rims without any flange or web stiffeners the method may not be on the safe side and it is advised to check with the method in 315316
315 Stresses in Thin Rims For rim thickness sR lt 35 mn the safety against rim cracking has to be checked
The following method may be used
3151 General The stresses in the standardised 30ordm tangent section tension side are slightly reduced due to decreasing stress correction factor with decreasing relative rim thickness sRmn On the other hand during the complete stress cycle of that fillet a certain amount of compression stresses are also introduced The complete stress range remains approximately constant Therefore the standardised calculation of stresses at the 30ordm tangent may be retained for thin rims as one of the necessary criteria
The maximum stress range for thin rims usually occurs at the 60ordm ndash 80ordm tangents both for laquotensionraquo and laquocompressionraquo side The following method assumes the 75ordm tangent to be the decisive Therefore in addition to the am criterion ap-plied at the 30ordm tangent it is necessary to evaluate the max and min stresses at the 75ordm tangent for both laquotensionraquo (loaded flank) fillet and laquocompressionraquo (back-flank) fillet For this purpose the whole stress cycle of each fillet should be considered but usually the following simplification is justified
Figure 35 Nomenclature of fillets
Index laquoTraquo means laquotensileraquo fillet laquoCraquo means laquocompressionraquo fillet
σFTmin and σFCmax are determined on basis of the nominal rim stresses times the stress concentration factor Y75
σFCmin and σFTmax are determined on basis of superposition of nominal rim stresses times Y75 plus the tooth bending stresses at 75ordm tangent
30 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
3152 Stress concentration factors at the 75ordm tangents The nominal rim stress consists of bending stresses due to local bending moments tangential stresses due to the tan-gential force Ft and radial shear stresses due to Fr
The major influence is given by the bending stresses The influence of the tangential stresses is minor and even though its stress concentration factor is slightly higher than for bending it is considered to be safe enough when the sum of these nominal stresses are combined with the stress concen-tration factor for bending The influence of the radial shear stress is neglected
The stress concentration factor relating nominal rim stresses to local fillet stresses at the 75ordm tangent may be calculated as
75n
R75
n
R
75
ρmsρ185
ms3
Y+
sdot=
where ρ75 is the root radius at the 75ordm tangent ref to mn Usually ρ75 is closed to the tool radius ρao and
ρ75 = ρao
is a safe approximation compensating for the am simplifi-cations to the laquounsaferaquo side
The tooth root stresses of the loaded tooth are decreasing with decreasing relative rim thickness approximately with the empirical factor
15
R
ncorr s
m311Y
minus=
3153 Nominal rim stresses The bending moment applied to the rim consists of a part of the tooth tilt moment Ft (hF + 05 sR) and the bending caused by the radial force Fr
The sectional modulus (first moment of area) which is used for determination of the nominal bending stresses is not nec-essarily the same for the 2 am bending moments If flanges webs etc outside the toothed section contribute to stiffening the rim against various deflections the influence of these stiffeners should be considered Eg an end flange will have an almost negligible influence on the effective sectional modulus for the stresses due to tooth tilt as the deflection caused by the tilt moment is rather small and would not much involve the flange On the other hand the radial forces as for instance from the meshes in an annulus would cause considerable radial deflections that the flange might restrict to a substantial amount When considering the stiff-ening of such flanges or webs on basis of simplified models it is advised to use an effective rim thickness sR = sR + 02 mn for the first moment of area of the rim (toothed part) cross section
For a high number of rim teeth it may be assumed that the rim bending moments in the fillets adjacent to the loaded tooth are of the same magnitude as right under the applied
force This assumption is reasonable for an annulus but rather much on the safe side for a hollow pinion
The influence of Ft on the nominal tangential stress is simpli-fied by half of it for compressive stresses (σ1) and the other half for tensile stresses (σ2) Applying these assumptions the nominal rim stresses adjacent to the loaded tooth are
( ) ( )A2
FW
f RFW
s05hF05 t
R
r
T
RFt1 minus
ϑminus
+minus=σ
( ) ( )A2
FW
f RFW
s05hF05 t
R
r
T
RFt2 +
ϑminus
+=σ
where σ1 σ2 see Fig 35
A = minimum area of cross section (usually bR sR)
WT = the sectional modulus of rim with respect to tooth tilt moment (usually 6sb 2
rR )
WR = the sectional modulus of the rim including the influence of stiffeners as flanges webs etc (WR ge WT)
bR = the width of the rim
R = the radius of the neutral axis in the rim ie from wheel centre to midpoint of rim
( )ϑf = a function for bending moment distribution around the rim For a rim (pinion) with one mesh only the ( )ϑf at the position of load application is 024 For an annulus with 3 or more meshes ( )ϑf at the position of each load application is approx 3 planets ( )ϑf = 019 4 planets ( )ϑf = 014 5 planets ( )ϑf = 011 6 planets ( )ϑf = 009
It must be checked if the max (tensile) stress in the compres-sion fillet really occurs when the fillet is adjacent to the loaded tooth In principle the stress variation through a complete rotation should be considered and the max value used The max value is usually never less than 0 For an annulus eg the tilt moment is zero in the mid position be-tween the planet meshes whilst the bending moment due to the radial forces is half of that at the mesh but with opposite sign
If these formulae are applied to idler gears as eg planets the influence of nominal tangential stresses must be cor-rected by deleting Ft(2 A) for σ1 and using FtA for σ2 Further the influence of Fr on the nominal bending stresses is usually negligible due to the planet bearing support
3154 Root fillet stresses Determination of min and max stresses in the laquotensionraquo fillet
Classification Notes- No 412 31 May 2003
DET NORSKE VERITAS
Minimum stress
751FTmin YσK σ =
Maximum stress
corrF752 Yσ03Yσ KσFTmax +=
where
03 is an empirical factor relating the tension stresses (σF) at the 30ordm tangent to the part of the tension stresses at the 75ordm tangent which add to the rim related stresses (03 also takes into account that full superposition of nominal stresses times stress concentration factors from both laquosides of the corner filletraquo would result in too high stresses)
αβγ sdotsdotsdotsdot= FFvA KKKKKK
Determination of min and max stresses in the laquocompres-sionraquo fillet
Minimum stress
corrF751FCmin Yσ036Yσ Kσ minus=
Maximum stress
752FCmax Yσ Kσ =
where
036 is an empirical factor relating the tension stresses (σF) at the 30ordm tangent to the part of the compression stresses at the 75ordm tangent which add to the rim related stresses
For gears with reversed loads as idler gears and planets there is no distinct laquotensionraquo or laquocompressionraquo fillet The mini-mum stress σFmin is the minimum of σFTmin and σFCmin (usually the latter is decisive) The maximum stress σFTmax is the maximum of σFTmax and σFCmax (usually the former is deci-sive)
316 Permissible Stresses in Thin Rims
3161 General The safety against fatigue fracture respectively overload fracture is to be at least at the same level as for solid gears The laquoordinaryraquo criteria at the 30ordm tangent apply as given in 31 through 313
Additionally the following criteria at the 75ordm tangent may apply
3162 For gt3middot106 cycles The permissible stresses for the laquotensionraquo fillets and for the laquocompressionraquo fillets are determined by means of a relevant fatigue diagram
If the actual tooth root stress (tensile or compressive) ex-ceeds the yield strength to the material the induced residual stresses are to be taken into account
For determination of permissible stresses the following is defined
R = stress ratio ie FTmax
FTminσσ respectively
FCmax
FCmin
σσ
∆σ = stress range ie σFTmax ndash σFTmin resp σFCmax ndash σFCmin
(For idler gears and planets Fmax
FminσσR =
and ∆σ = σFmax minus σFmin)
The permissible stress range ∆σρ for the laquotensionraquo respec-tively laquocompressionraquo fillets can be calculated as
For R gt minus1 FPp σ
R1R1031
13∆σ
minus+
+=
For minus infin lt R lt minus1 FPp σ
R1R10151
13∆σ
minus+
+=
where
σFP = see 32 determined for unidirectional stresses (YM = 1)
If the yield strength σy is exceeded in either tension or com-pression residual stresses are induced This may be consid-ered by correcting the stress ratio R for the respective fillets (tension or compression)
Eg if yFCmin σσ gt (ie exceeded in compression) the
difference yFCmin σσ∆ minus= affects the stress ratio as
∆σ
σ∆σ∆σR
FCmax
y
FCmax
FCmin
+
minus=
++
=
Similarly the stress ratio in the tension fillet may require cor-rection
If the yield strength is exceeded in tension σFTmax gt σy the difference ∆ = σFTmax ndash σy affects the stress ratio as
y
FTmin
FTmax
FTmin
σ∆σ
∆σ∆σR minus=
minusminus
=
Checking for possible exceeding of the yield strength has to be made with the highest torque and with the KAP if this ex-ceeds KA
3163 Forle 103 cycles The permissible stress range ∆σp is not to exceed
For R gt minus1 FPstpst σ
R1R1051
15∆σ
minus+
+=
For ndash infin lt R lt minus1 FPstpst σ
R1R10251
15∆σ
minus+
+=
For all values of R ∆σpst is limited by
not surface hardened F
y
Sσ225
32 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
surface hardened CF
YSHV5
Definition of ∆σ and R see 3162 with particular attention to possible correction of R if the yield strength is exceeded
σFPst see 32 determined for unidirectional stresses (YM = 1) and lt 103 cycles
3164 For 103 lt cycles lt 3106 ∆σp is to be determined by linear interpolation a log-log dia-gram
∆σp at NL load cycles is
6L 103p
exp
L
6
N p ∆σN103∆σ sdot
sdot=
6
3
103p
10p
∆σ
∆σlog28760exp
sdot
=
Classification Notes- No 412 33 May 2003
DET NORSKE VERITAS
4 Calculation of Scuffing Load Capacity
41 Introduction High surface temperatures due to high loads and sliding ve-locities can cause lubricant films to break down This sei-zure or welding together of areas of tooth surface is termed scuffing
In contrast to pitting and fatigue breakage which show a dis-tinct incubation period a single short overloading can lead to a scuffing failure In the ISO-TR13989 two criteria are mentioned The method used in this Classification Note is based on the principles of the flash temperature criterion
Note Bulk temperature in excess of 120ordmC for long periods may have an adverse effect on the surface durability see 211
42 General Criteria In no point along the path of contact the local contact tem-perature may exceed the permissible temperature ie
oilS
oilSB S
ϑ+ϑminusϑ
leϑ
50SB minusϑleϑ
where
Bϑ = max contact temperature along the path of contact
maxflaMBB ϑ+ϑ=ϑ
MBϑ = bulk temperature see 434
maxflaϑ = max flash temperature along the path of con-tact see 44
Sϑ = scuffing temperature see below
oilϑ = oil temperature before it reaches the mesh (max applicable for the actual load case to be used ie normally alarm temperature except for ice classes where a max expected tem-perature applies)
SS = required safety factor according to the Rules
The scuffing temperature Sϑ may be calculated as
L2
wrelT
002
40S XFZGX
ν100112085780
sdot
sdot++=ϑ
where
XwrelT = relative welding factor
XwrelT
Through hardened steel 10
Phosphated steel 125
Copper-plated steel 150
Nitrided steel 150
Less than 10 retained austenite
115
10 ndash 20 retained austenite 10
Casehardened steel
20 ndash 30 retained austenite 085
Austenitic steel 045
FZG = load stage according to FZG-Test A8390 (Note This is the load stage where scuffing oc-curs However due to scatter in test results cal-culations are to be made with one load stage less than the specification)
XL = lubricant factor
= 10 for mineral oils
= 08 for polyalfaolefins
= 07 for non-water-soluble polyglycols
= 06 for water-soluble polyglycols
= 15 for traction fluids
= 13 for phosphate esters
ν40 = kinematic oil viscosity at 40˚C (mm2s)
Application of other test methods such as the Ryder the FZG-Ryder R46574 and the FZG L-42 Test 141195110 may be specially considered
For high speed gears with very short time of contact Sϑ may be increased as follows provided use of EP-oils
Addition to the calculated scuffing temperature Sϑ
If micros 18tc ge no addition
If micros 18tc lt add ( )cwrelT t18X18 minussdotsdot
where
ct = contact time ( micros ) which is the time needed to cross the Hertzian contact width
( ) [ ]microsu1cosβn
uσ340tb1
Hc +sdotsdot
sdot=
σH as calculated in 221
For bevel gears use vu in stead of u
34 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
43 Influence Factors
431 Coefficient of friction The following coefficient of friction may apply
L025a
005oil
02
redC ΣC
Bt X R ηρv
w0048micro minus
=
where
wBt = specific tooth load (Nmm)
vΣC = sum of tangential velocities at pitch point At pitch line velocities gt 50 ms the limiting value of vΣC at v = 50 ms is to be used
ρredC = relative radius of curvature (transversal plane) at the pitch point
Cylindrical gears
HαHβvγAbt
Bt KKKKKb
Fw sdotsdotsdotsdotsdot= (see 1)
twΣC αsin v2v =
bCredC β cos ρρ =
Bevel gears
HαHβvγAtmb
Bt KKKKKb
Fw sdotsdotsdotsdotsdot= (see 1)
vtmtΣC αsin v2v =
bmvCredC β cos ρρ =
ηoil = dynamic viscosity (mPa s) at oilϑ calculated as
1000
ρ νη oiloil =
where ρ in kgm3 approximated as
( ) 70 15oil15 minusϑminusρ=ρ and νoil is kinematic viscosity at oilϑ and may be calculated by means of the following equation
( ) ( )++=+ 08νloglog08νloglog 100oil ( )
sdotminus
ϑ+minus313log373log
273log373log oil
( ) ( )( )08νloglog08νloglog 10040 +minus+
Ra = composite arithmetic mean roughness (micron) of pinion an wheel calculated as ( )2a1aa RR 05R +=This is defined as the roughness on the new flanks ie as manufactured
XL = see 42
432 Effective tip relief Ceff Ceff is the effective tip relief that amount of tip relief which compensates for the elastic deformation of the gear mesh ie
zero load at the tooth tip It is assumed (simplified) to be equal for both pinion and wheel
Cylindrical gears
for helical
γ
γ=cb
KKFC Abt
eff (see 1)
For spur
cbKKF
C Abteff
γ= (see 1)
Alternatively for spur and helical gears the non-linear ap-proach in 111 may be used (taking Ceff=δ)
Bevel gears
bcKFC Ambt
effγ
= (see 1)
where
γ
α
ε2ε44c
+sdot
=γ
433 Tip relief and extension Cylindrical gears
The extension of the tip relief is not to result in an effective contact ratio 1ltεα when the gear is unloaded (exceptions to this may only apply for applications where the gear is not to run at light loads) This means that the unrelieved part of the path of contact is to be minimum pbt It is further assumed that this unrelieved part is placed centrally on the path of contact
If root relief applies it has to be calculated as equivalent tip relief Ie pinion root relief (at mesh position A) is added to Ca2 and wheel root relief (at mesh position E) is added to Ca1 If no design tip relief or root relief on the mating gear is specified (ie if Ca1+Croot2 = 0 and visa versa) use the run-ning in amount see 112
Bevel gears
Bevel gears are to have heightwise crowning ie no distinct relievedunrelieved area This may be treated as tip and root relief For calculation purposes the root relief is combined with the tip relief of the mating member into an equivalent tip relief If no resulting tip relieves are specified the equivalent tip relives may be calculated as an approxima-tion based on the tool crowning Ca tool (per mille of tool module m0) as follows
2root1aeq1a CCC +=
1root2aeq2a CCC +=
where
Classification Notes- No 412 35 May 2003
DET NORSKE VERITAS
2
0
n0101atoolal m
)m2(mA1middotmCC
minus+minussdot=
( )
sdotminusminussdotsdotminus+= vtvb1
2vb1
21vavt1n1 αtan dddαsin 05x1mA
2
0
n020atool22a m
)mm(2A1middotmCC
minus+minussdot=
( )
sdotminusminussdotsdotminus+= vtvb2
2vb2
2va2vt2n2 αtan dddαsin 05x1mA
2
0
20atool1root1 m
A1mCC
minussdotsdot=
2
0
10atool2root2 m
A1mCC
minussdotsdot=
If the pressure angles of the cutter blades are modified (and verified) to balance the tip relieves the following may be assumed
( )eq2aeq1aeq2aeq1a C andC calculated of sum50CC sdot==
Throughout the following Ca1 and Ca2 mean the equivalent tip relieves Ca1eq and Ca2eq
434 Bulk temperature The bulk temperature may be calculated as
flaaveragempsoilMB X X 50 ϑ+ϑ=ϑ
where
Xs = lubrication factor
= 12 for spray lubrication
= 10 for dip lubrication (provided both pinion and wheel are dip lubricated and tip speed lt 5 ms)
= 10 for spray lubrication with additional cooling spray (spray on both pinion and wheel or spray on pinion and dip lubrication of wheel)
= 02 for meshes fully submerged in oil
Xmp = contact factor ( )pmp nX += 150
np = number of mesh contact on the pinion (for small gear ratii the number of wheel meshes should be used if higher)
flaaverageϑ
= average of the integrated flash temperature (see 44) along the path of contact
( )
AE
E
Ayyyfla
flaaverage
d
ΓminusΓ
ΓΓϑ=ϑint =
For high speed gears (v gt 50 ms) it may be necessary to assess the bulk temperature on the basis of a thermal rating of the entire gear transmission
44 The Flash Temperature flaϑ
441 Basic formula The local flash temperature flaϑ may be calculated as
( ) 41redy
y2ly
211
43Btcorrfla
unXwX3250
y ρ
ρminusρ
micro=ϑ Γ
(For bevel gears replace u with uv)
and is to be calculated stepwise along the path of contact from A to E
where
micro = coefficient of friction see 431
Xcorr = correction factor taking empirically into account the increased scuffing risk in the beginning of the approach path due to mesh starting without any previously built up oil film and possible shuffling away oil before meshing if insufficient tip relief
( )
3
AD
yaeffcorr 50
CC1X
ΓminusΓε
Γminus+=
α
Ca = tip relief of driven member Xcorr is only applicable in the approach path and if Ca lt Ceff otherwise 10
wBt = unit load see 431
yXΓ = load sharing factor see 443
n1 = pinion rpm
Γy ρly etc see 442
442 Geometrical relations The various radii of flank curvature (transversal plane) are
ρ1y = pinion flank radius at mesh point y
ρ2y = wheel flank radius at mesh point y
ρredy = equivalent radius of curvature at mesh point y
y2y1
y2y1redy
ρ+ρ
ρρ=ρ
Cylindrical gears
twy
1y αsin au1Γ1
ρ+
+=
twy
2y αsin au1Γu
ρ+
minus=
Note that for internal gears a and u are negative
36 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
Bevel gears
vtvv
y1y αsin a
u1Γ1
ρ+
+=
vtvv
yv2y αsin a
u1Γu
ρ+
minus=
Γ is the parameter on the path of contact and y is any point between A end E
At the respective ends Γ has the following values
Root piniontip wheel
Cylindrical gears
( )
minus
minusminus= 1
αtan 1dd
zzΓ
tw
2b2a2
1
2A
Bevel gears
( )
minus
minusminus= 1
αtan 1dd
uΓvt
2vb2va2
vA
Tip pinionroot wheel
Cylindrical gears
( )
1αtan
1ddΓ
tw
2b1a1
E minusminus
=
Bevel gears
( )
1αtan
1ddΓ
vt
2vb1va1
E minusminus
=
At inner point of single pair contact
Cylindrical gears
tw1
EB α tan zπ2ΓΓ minus=
Bevel gears
vtv1
EB α tan zπ2ΓΓ minus=
At outer point of single pair contact
Cylindrical gears
tw1AD α tan z
π2ΓΓ +=
Bevel gears
vt v1
AD αtan z π2ΓΓ +=
At pitch point ΓC = 0
The points F and G (only applicable to cylindrical gears) limiting the extension of tip relief (so as to maintain a mini-mum contact ratio of unity for unloaded gears) are at
2
BAF
Γ+Γ=Γ
2
EDG
Γ+Γ=Γ
443 Load sharing factor XΓ The load sharing factor XΓ accounts for the load sharing be-tween the various pairs of teeth in mesh along the path of contact
XΓ is to be calculated stepwise from A to E using the pa-rameter Γy
4431 Cylindrical gears with β = 0 and no tip relief
Figure 41
ByAAB
Ay for31
31X
yΓltΓleΓ
ΓminusΓ
ΓminusΓ+=Γ
DyB for1Xy
ΓltΓleΓ=Γ
EyDDE
yE for31
31X
yΓleΓltΓ
ΓminusΓ
ΓminusΓ+=Γ
4432 Cylindrical gears with β = 0 and tip relief Tip relief on the pinion reduces XΓ in the range G ndash E and increases correspondingly XΓ in the range F ndash B
Tip relief on the wheel reduces XΓ in the range A ndash F and increases correspondingly XΓ in the range D ndash G
Following remains generally
DyB for1Xy
ΓleΓleΓ=Γ
21XXGF== ΓΓ
In the following it must be distinguished between Ca lt Ceff respectively Ca gt Ceff This is shown by an example below where Ca1 lt Ceff and Ca2 gt Ceff
Classification Notes- No 412 37 May 2003
DET NORSKE VERITAS
Figure 42
Note When Ca gt Ceff the path of contact is shortened by A ndash A respectively E ndash E The single pair contact path is extended into B respectively D If this shift is significant it is necessary to consider the negative effect on surface durability (B) and bending stresses (B and D)
Range A - F
For effa2 CC le
minus=Γ
eff
2a
CC1
31X
A
FyAeff
2a
AF
Ay for C3
C61XX
AyΓleΓleΓ
sdot
+ΓminusΓ
ΓminusΓ+= ΓΓ
For effa2 CC ge
AyAfor0Xy
ΓleΓleΓ=Γ
with ( )21
CC
1CC
eff
2a
eff
2a
AFAAminus
minusΓminusΓ+Γ=Γ
FyAeff
2a
AF
Ay
eff
2a for 21
CC
CC1X
yΓleΓleΓ
minus
ΓminusΓ
ΓminusΓ+minus=Γ
Range F - B
For effa1 CC le
ByFeff
1a
FB
Fy forC3
C61
21X
yΓleΓleΓ
sdot
+ΓminusΓ
ΓminusΓ+=Γ
For effa1 CC ge
ByFeff
1a
FB
Fy for21
CC
21X
yΓleΓleΓ
minus
ΓminusΓ
ΓminusΓ+=Γ
ByB for1Xy
ΓleΓleΓ=Γ
with 1
CC2
eff
1a
FBFB
minus
ΓminusΓ+Γ=Γ
Range D ndash G
For effa2 CC le
GyDeff
2a
DG
Dy
eff
2a for C 3C
61
C 3C
32X
yΓleΓleΓ
+
ΓminusΓ
ΓminusΓminus+=Γ F
or effa2 CC ge
DyD for1Xy
ΓleΓleΓ=Γ
with ( )21
CC
1CC
eff
2a
eff
2a
DGDDminus
minusΓminusΓ+Γ=Γ
GyDeff
2a
DG
Dy
eff
2a for21
CC
CCX ΓleΓleΓ
minus
ΓminusΓ
ΓminusΓminus=Γ
Range G - E
For effa1 CC le
minus=Γ
eff
1a
CC1
31X
E
EyGeff
1a
GE
Gy forC3
C61
21X
yΓleΓleΓ
sdot
+ΓminusΓ
ΓminusΓminus=Γ
For effa1 CC gt
EyGeff
1a
GE
Gy for21
CC
21X
yΓleΓleΓ
minus
ΓminusΓ
ΓminusΓminus=Γ
EyE for0Xy
ΓleΓleΓ=Γ
with 1
CC2
eff
1a
GEGE
minus
ΓminusΓ+Γ=Γ
4433 Gears with β gt 0 buttressing Due to oblique contact lines over the flanks a certain but-tressing may occur near A and E
This applies to both cylindrical and bevel gears with tip relief lt Ceff The buttressing Xbutt is simplified as a linear function within the ranges A ndash H respectively I ndash E
Figure 43
1εwhen13X βbutt EAge=
1εwhenε031X ββbutt EAlt+=
38 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
Cylindrical gears
bIEAH βsin 02ΓΓΓΓ =minus=minus
Bevel gears
bmIEAH βsin 02ΓΓΓΓ =minus=minus
4434 Cylindrical gears with 2εγ le and no tip relief
yXΓ is obtained by multiplication of
yXΓ in 4431 with
Xbutt in 4433
4435 Gears with 2εγ gt and no tip relief
Applicable to both cylindrical and bevel gears
IyH for1Xy
ΓleΓleΓε
=α
Γ
IyHybutt and forX1Xy
ΓgtΓΓltΓε
=α
Γ
Figure 44
4436 Cylindrical gears with 2εγ le and tip relief
yXΓ is obtained by multiplication of
yXΓ in 4432 with Xbutt
in 4433
4437 Cylindrical gears with 2εγ gt and tip relief
Tip relief on the pinion (respectively wheel) reduces XГ in the range G ndash E (respectively A ndash F) and increases XГ in the range F ndash G
yXΓ is obtained by multiplication of
yXΓ as described below
with Xbutt in 4433
In the XГ example below the influence of tip relief is shown (without the influence of Xbutt) by means of
eff2aeff1a CC and CC ltgt
Tip relief gt Ceff causes new end points A respectively E of the path of contact
Figure 45
Range A ndash F
( ) ( )( ) eff
a2αa1α
AF
Ay
eff
2aeff
C 12C 13εC 1ε
CCCX
y +εε++minus
ΓminusΓ
ΓminusΓ+
εminus
=ααα
Γ
eff2aFyA CC if for leΓleΓleΓ
eff2aFyA CifCfor and geΓleΓleΓ
eff2aAyA CC iffor0Xy
gtΓleΓleΓ=Γ
( ) ( ) ( )( ) ( ) a2αa1α
αeffa2AFAA C 1ε 3C 1ε
1ε 2 CC with++minus+minus
ΓminusΓ+Γ=Γ
Range F ndash G
( )( )( ) GyF
eff
2a1a forC 1 2CC 11X
yΓleΓleΓ
+εε+minusε
+ε
=αα
α
αΓ
Range G ndash E
( ) ( )( ) eff
2a1a
GE
Gy
C 1 2C 1C 1 3XX
GFy +εεminusε++ε
ΓminusΓ
ΓminusΓminus=
αα
ααΓΓ minus
effa1EyG CC if for leΓleΓleΓ
effa1EyG CC if Γfor and geΓleΓle
eff1aEyE CC if for0Xy
geΓleΓleΓ=Γ
( ) ( ) ( )( ) ( ) 2a1a
eff1aGEEE C 1C 1 3
1 2 CC withminusε++ε+εminus
ΓminusΓminusΓ=Γαα
α
4438 Bevel gears with εγ more than approx 18 and heightwise crowning For Ca1 = Ca2 = Ceff the following applies
Figure 46
Classification Notes- No 412 39 May 2003
DET NORSKE VERITAS
( )AEM 50 Γ+Γ=Γ
( )( )2AD
3
2My651X
y ΓminusΓε
ΓminusΓminus
ε=
ααΓ
For tip relief lt Ceff yXΓ is found by linear interpolation be-
tween ( )effay CCX =Γ and ( )0CayX =Γ as in 4435
The interpolation is to be made stepwise from A to M with the influence of Ca2 and from M to E with the influence of Ca1 (For 2a1a CC ne there is a discontinuity at M)
Eg with Ca1 = 04 Ceff and Ca2 = 055 Ceff then
Range A ndash M
)CC()0C( eff2ay2ayyX 550X 450X =Γ=ΓΓ +=
Range M ndash E
)CC()0C( eff1ay1ayyX 40X 06X =Γ=ΓΓ +=
For tip relief gt Ceff the new end points A and E are found as
( )
minusΓminusΓ
ε+Γ=Γ α 1
CC
6 eff
2aADAA
( )
minusΓminusΓ
ε+Γ=Γ α 1
CC
6 eff
1aADEE
Range A ndash A
0Xy=Γ
Range A ndash M
( )( )
ΓminusΓ
ΓminusΓminus
minusε=
αΓ 2
MA
2My
eff
2a1
CC4
351Xy
Range M ndash E
( )( )
ΓminusΓ
ΓminusΓminus
minusε=
αΓ 2
ME
2My
eff
1a1
CC4
351Xy
Range E ndash E
0Xy=Γ
40 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
Appendix A Fatigue Damage Accumulation
The Palmgren-Miner cumulative damage calculation princi-ple is used The procedure may be applied as follows
A1 Stress Spectrum From the individual torque classes the torques (Ti) at the peak values of class intervals and the associated number of cycles (NLi) for both pinion and wheel are to be listed from the highest to the lowest torque
(In case of a cyclic torque variation within the torque classes it is advised to use the peak torque If the cyclic variation is such that the same teeth will repeatedly suffer the peak torque this is a must)
The stress spectra for tooth roots and flanks (σFi σHi) with all relevant factors (except KA) are to be calculated on the basis of the torque spectrum The load dependent K-factors are to be determined for each torque class
A2 σminusN-curve The stress versus load cycle curves for tooth roots and flanks (both pinion and wheel) are to be drawn on the basis of per-missible stresses (ie including the demanded minimum safety factors) as determined in 2 respectively 3 If different safety levels for high cycle fatigue and low cycle fatigue are desired this may be expressed by different demand safety factors applied at the endurance limit respectively at static strength
A3 Damage accumulation The individual damage ratio Di at ith stress level is defined as
Fi
Lii N
ND =
where
NLi = The number of applied cycles at ith stress
NFi = The number of cycles to failure at ith stress
Basically stresses σi below the permissible stress level for infinite life (if a constant ZN or YN is accepted) do not con-tribute to the damage sum However calculating the actual safety factor Sact as described below all the σi for which the product S σi is bigger than or equal to the permissible stress level for infinite life contribute to the damage sum and thus to the determination of Sact The final value of S is decisive
(NFi can be found mathematically by putting the permissible stress σpi equal the actual stress σi thereby finding the actual life factor This life factor can be solved with regard to load cycles ie NFi)
The damage sum ΣDi is not to exceed unity
If ΣDi ne 1 the safety against cumulative fatigue damage is different from the applied demand safety factor For deter-mination of this theoretical safety factor an iteration proce-dure is required as described in the following flowchart
S is correction factor with which the actual safety factor Sact can be found
Sact is the demand safety factor (used in determination of the permissible stresses in the σ ndash N ndash curve) times the correc-tion factor S
The full procedure is to be applied for pinion and wheel tooth roots and flanks
Note If alternating stresses occur in a spectrum of mainly pulsating stresses the alternating stresses may be replaced by equiva-lent pulsating stresses ie by means of division with the ac-tual mean stress influence factor YM
Classification Notes- No 412 41 May 2003
DET NORSKE VERITAS
Appendix B Application Factors for Diesel Driven Gears
For diesel driven gears the application factor KA depends on torsional vibrations Both normal operation and misfiring conditions have to be considered
Normally these two running conditions can be covered by only one calculation
B1 Definitions
Normal operation KAnorm = 0
normv0
TTT +
where
T0 = rated nominal torque
Tv norm = vibratory torque amplitude for normal opera-tion (see rules Pt4 Ch3 Sec1 G301 for defi-nition of ldquonormalrdquo irregularity)
Misfiring operation KA misf = 0
misfv
TTT +
where
T = remaining nominal torque when one cylinder out of action
Tv misf = vibratory torque amplitude in misfiring con-dition This refers to a permissible misfiring condition ie a condition that does not require automatic or immediate corrective actions as speed or pitch reduction
The normal operation is assumed to last for a very high num-ber of cycles such as 1010
The misfiring operation is assumed to last for a limited dura-tion such as 107 cycles
B2 Determination of decisive load Assuming life factor at 1010 cycles as YN = ZN = 092 which usually is relevant the calculation may be performed only once with the combination having the highest value of appli-cation factorlife factor
For bending stresses and scuffing the higher value of
092
K normA and 098
K misfA
For contact stresses the higher value of
092K normA and
113K misfA (but
097K misfA for nitrided gears)
B3 Simplified procedure Note that this is only a guidance and is not a binding con-vention
T = oTZ
1Zsdot
minus where Z = number of cylinders
( ) ideal Videal Vmisf Vnorm V TTT24ZT +minus=
where TV ideal = vibratory torque with all cylinders perfectly equal See also rules Pt4 Ch3 Sec1 G300
When using trends from torsional vibration analysis and measurements the following may be used
TV ideal To is close to zero for engines with few cylinders and using a suitable elastic coupling and increases with relative coupling stiffness and number of cylinders
TV misf To may be high for engines with few cylinders and decreases with number of cylinders
This can be indicated as
200Z
TT
ο
idealV asymp and 80Z04
TT
o
misfV minusasymp
Inserting this into the formulae for the two application fac-tors the following guidance can be given
118112K misfA minusasymp
115110K normA minusasymp
Since KA norm is to be combined with the lower life factors the decisive load condition will be the normal one and a KA of 115 will cover most relevant cases when a suitable elas-tic coupling is chosen
42 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
Appendix C Calculation of Pinion-Rack
Pinion-racks used for elevating of mobile offshore units are open gears that are subjected to wear and tear With normal specifications such as only a few hundred total operation cycles (site to site) the tooth bending stresses for static strength will be decisive for the lay out however with ex-ception of surface hardened pinions where case crushing has to be considered
In the following the use of part 1 and 3 for pinion-racks is shown including relevant simplifications
C1 Pinion tooth root stresses Since the load spectrum normally is dominated by high torques and few load cycles (in the range up to 1-2000) the static strength is decisive
The actual stress is calculated as
β1FSaFan1
tF1 KYY
mbFσ sdotsdotsdotsdot
=
b1 is limited to b2 + 2middotmn
YFa and YSa replace YF and YS because load application at tooth tip has to be assumed for such inaccurate gears
Pinions often use a non-involute profile in the dedendum part of the flank eg a constant radius equal the radius of curva-ture at reference circle For such pinions sFn and hFa are to be measured directly on a sectional drawing of the pinion tooth
Due to high loads and narrow facewidths it may be assumed that KFβ2 = KHβ = 10 However when b1 gt b2 then KFβ1 gt10
If no detailed documentation of KFβ1 is available the fol-lowing may be used
KFβ1 = 1 + 015middot(b1b2 ndash 1)
The permissible stress (not surface hardened) is calculated as
δrelTstF
Fst1FP1 Y
Sσσ sdot=
The mean stress influence due to leg lifting may be disre-garded
The actual and permissible stresses should be calculated for the relevant loads as given in the rules
C2 Rack tooth root stresses The actual stress is calculated as
SaFan2
tF2 YY
mbFσ sdotsdotsdot
=
See C1 for details
The permissible stress is calculated as
δrelTstF
Fst2FP2 Y
Sσσ sdot=
For alloyed steels (Ni Cr Mo) with high toughness and duc-tility the value of YδrelTst may be put equal to YSa
C3 Surface hardened pinions For surface hardened pinions the maximum load is not to cause crushing of the hardened layer of the flank
In principle the calculation described in 213 may be used but when the theoretical Hertzian stress exceeds the ap-proximately 18 times the yield strength of the rack material plastic deformation will occur This will limit the peak Hertzian stress but increases the contact width and thus the penetration of stresses into the depth
An approximation may be based on an assessment of contact width determined by means of equal areas under the theoreti-cal (elastic range) Hertzian contact and the elasto-plastic contact stress (limited to 18 middot σy) with the unknown width
Classification Notes- No 412 13 May 2003
DET NORSKE VERITAS
tribution in term of hard end contact or if it is known by other reasons that there exists a hard end contact the load should be correspondingly distributed when calculating fsh In fact the whole KHβ procedure can be used iteratively 2-3 iterations will be enough even for almost triangular load distributions
195 Determination of fdefl fdefl is the mesh misalignment in the plane of contact due to bearing deflections and working positions (housing deflec-tion may be included if determined)
First the journal working positions in the bearings are to be determined The influence of external moments and forces must be considered This is of special importance for twin pinion single output gears with all 3 shafts in one plane
For rolling bearings fdefl is further determined on basis of the elastic deflection of the bearings An elastic bearing deflec-tion depends on the bearing load and size and number of rolling elements Note that the bearing clearance tolerances are not included here
For fluid film bearings fdefl is further determined on basis of the lift and angular shift of the shafts due to lubrication oil film thickness Note that fbe takes into account the influence of the bearing clearance tolerance
When working positions bearing deflections and oil film lift are combined for all bearings the angular misalignment as projected into the plane of the contact is to be determined fdefl is this angular misalignment (radians) times the face-width
196 Determination of fbe fbe is the mesh misalignment in the plane of contact due to tolerances in bearing clearances In principle fbe and fdefl could be combined But as fdefl can be determined by analy-sis and has a distinct direction and fbe is dependent on toler-ances and in most cases has no distinct direction (ie + toler-ance) it is practicable to separate these two influences
Due to different bearing clearance tolerances in both pinion and wheel shafts the two shaft axis will have an angular mis-alignment in the plane of contact that is superimposed to the working positions determined in 195 fbe is the facewidth times this angular misalignment Note that fbe may have a distinct direction or be given as a + tolerance or a combina-tion of both For combination of + tolerance it is adviced to use
fbe= 22be
21be ff ++plusmn
fbe is particularly important for overhang designs for gears with widely different kinds of bearings on each side and when the bearings have wide tolerances on clearances In general it shall be possible to replace standard bearings with-out causing the real load distribution to exceed the design premises For slow speed gears with journal bearings the expected wear should also be considered
197 Determination of fma fma is the mesh misalignment due to manufacturing toler-ances (helix slope deviation) of pinion fHβ1 wheel fHβ2 and housing bore
For gear without specifically approved requirements to as-sembly control the value of fma is to be determined as
fma= 22Hβ
21Hβ ff +
For gears with specially approved assembly control the value of fma will depend on those specific requirements
198 Comments to various gear types For double helical gears KHβ is to be determined for both helices Usually an even load share between the helices can be assumed If not the calculation is to be made as de-scribed in 171
For planetary gears the free floating sun pinion suffers only twist no bending It must be noted that the total twist is the sum of the twist due to each mesh If the value of 1K γ ne this must be taken into account when calculating the total sun pinion twist (ie twist calculated with the force per mesh without Kγ and multiplied with the number of planets)
When planets are mounted on spherical bearings the mesh misalignments sun-planet respectively planet-annulus will be balanced Ie the misalignment will be the average between the two theoretical individual misalignments The faceload distribution on the flanks of the planets can take full advan-tage of this However as the sun and annulus mesh with several planets with possibly different lead errors the sun and annulus cannot obtain the above mentioned advantage to the full extent
199 Determination of KHβ for bevel gears If a theoretical approach similar to 193-198 is not docu-mented the following may be used
testeff
H Kb
b851851K sdot
minussdot=β
beff b represents the relative active facewidth (regarding lapped gears see 192 last part)
Higher values than beff b = 090 are normally not to be used in the formula
For dual directional gears it may be difficult to obtain a high beff b in both directions In that case the smaller beff b is to be used
Ktest represents the influence of the bearing arrangement shaft stiffness bearing stiffness housing stiffness etc on the faceload distribution and the verification thereof Expected variations in length- and height-wise tooth profile is also accounted for to some extent
a) Ktest = 1 For ground or hard metal hobbed gears with the specified contact pattern verified at full rating or at full torque slow turning at a condition representative for the thermal expansion at normal operation
14 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
It also applies when the bearing arrangementsupport has insignificant elastic deflections and thermal axial expansion However each initial mesh contact must be verified to be within acceptance criteria that are cali-brated against a type test at full load Reproduction of the gear tooth length- and height-wise profile must also be verified This can be made through 3D measurements or by initial contact movements caused by defined axial offsets of the pinion (tolerances to be agreed upon)
b) Ktest = 1 + 04middot(beffbndash06) For designs with possible influence of thermal expansion in the axial direction of the pinion The initial mesh contact verified with low load or spin test where the acceptance criteria are calibrated against a type test at full load
c) Ktest = 12 if mesh is only checked by toolmakerrsquos blue or by spin test contact For gears in this category beffb gt 085 is not to be used in the calculation
110 Transversal Load Distribution Factors KHα and KFα The transverse load distribution factors KHα for contact stresses and for scuffing KFα for tooth root stresses account for the effects of pitch and profile errors on the transversal load distribution between 2 or more pairs of teeth in mesh
The following relations may be used
Cylindrical gears
( )
minus+==
tH
αptγγHαFα F
byfc0409
2ε
KK
valid for 2εγ le
( ) ( )tH
αptγ
γ
γHαFα F
byfcε
1ε20409KK
minusminus+==
valid for 2ε γ gt
where
FtH = Ft KA Kγ Kv KHβ
cγ = See 111
γα = See 112
fpt = Maximum single pitch deviation (microm) of pinion or wheel or maximum total profile form devia-tion Fα of pinion or wheel if this is larger than the maximum single pitch deviation
Note In case of adequate equivalent tip relief adapted to the load half of the above mentioned fpt can be introduced A tip relief is considered adequate when the average of Ca1 and Ca2 is within plusmn40 of the value of Ceff in 432
Limitations of KHα and KFα
If the calculated values for
KFα = KHα lt 1 use KFα = KHα = 10
If the calculated value of KHα gt 2εα
γ
Zε
ε use KHα = 2
εα
γ
Zε
ε
If the calculated value of KFα gt εα
γ
Yεε
use KFα = εα
γ
Yεε
where Yε = αnε
075025+ (for εαn see 331c)
Bevel gears
For ground or hard metal hobbed gears KFα = KHα = 1
For lapped gears KFα = KHα = 11
111 Tooth Stiffness Constants cacute and cγ The tooth stiffness is defined as the load which is necessary to deform one or several meshing gear teeth having 1 mm facewidth by an amount of 1 microm in the plane of contact cacute is the maximum stiffness of a single pair of teeth cγ is the mean value of the mesh stiffness in a transverse plane (brief term mesh stiffness)
Both valid for high unit load (Unit load = Ft middot KA middot Kγb)
Cylindrical gears The real stiffness is a combination of the progressive Hertzian contact stiffness and the linear tooth bending stiff-nesses For high unit loads the Hertzian stiffness has little importance and can be disregarded This approach is on the safe side for determination of KHβ and KHα However for moderate or low loads Kv may be underestimated due to determination of a too high resonance speed
The linear approach is described in A
An optional approach for inclusion of the non-linear stiffness is described in B
A The linear approach
BRCCq
βcos08acutec =
and
( )025ε075cc αγ +prime=
where
( )[ ]n02a01a
B α2000212
hh12051C minusminus
+minus+=
12n1n
x000635z
025791z
015551004723q minus++=
12
2n
22
1n
1 x005290z
x024188x000193z
x011654+minusminusminus
+ 000182 x22
Classification Notes- No 412 15 May 2003
DET NORSKE VERITAS
(for internal gears use zn2 equal infinite and x2 = 0) ha0 = hfp for all practical purposes CR considers the increased flexibility of the wheel teeth if the wheel is not a solid disc and may be calculated as
( )( )nR m5s
sR e5
bbln1C +=
where
bs = thickness of a central web
sR = average thickness of rim (net value from tooth root to inside of rim)
The formula is valid for bs b ge 02 and sRmn ge 1 Outside this range of validity and if the web is not centrally posi-tioned CR has to be specially considered
Note CR is the ratio between the average mesh stiffness over the facewidth and the mesh stiffness of a gear pair of solid discs The local mesh stiffness in way of the web corresponds to the mesh stiffness with CR = 1 The local mesh stiffness where there is no web support will be less than calculated with CR above Thus eg a centrally positioned web will have an effect corresponding to a longitudinal crowning of the teeth See also 1931 regarding KHβ
B The non-linear approach
In the following an example is given on how to consider the non-linearity
The relation between unit load Fb as a function of mesh deflection δ is assumed to be a progressive curve up to 500 Nmm and from there on a straight line This straight line when extended to the baseline is assumed to intersect at 10microm
With these assumptions the unit force Fb as a function of mesh deflection δ can be expressed as
( )10δKbF
minus= for 500bFgt
minus=
500Fb10δK
bF for 500
bFlt
with γAt KK
bF
bF
sdotsdot= etc (Nmm) ie unit load incorporat-
ing the relevant factors as
KA middot Kγ for determination of Kv
KA middot Kγ middot Kv for determination of KHβ
KA middot Kγ middot Kv middot KHβ for determination of KHα
δ = mesh deflection (microm)
K = applicable stiffness (c or cγ)
Use of stiffnesses for KV KHβ and KHα
For calculation of Kv and KHα the stiffness is calculated as follows
When Fb lt 500
the stiffness is determined as δ∆
∆ bF
where the increment is chosen as eg ∆ Fb = 10 and thus
50010Fb10
K10Fb∆δ +
++
=
When F b gt 500 the stiffness is c or cγ For calculation of KHβ the mesh deflection δ is used directly
or an equivalent stiffness determined as δsdotb
F
Bevel gears
In lack of more detailed relationship between stiffness and geometry the following may be used
b085
b13cacute eff=
b085b
16c effγ =
beff not to be used in excess of 085 b in these formulae
Bevel gears with heightwise and lengthwise crowning have progressive mesh stiffness The values mentioned above are only valid for high loads They should not be used for de-termination of Ceff (see 432) or KHβ (see 1931)
112 Running-in Allowances The running-in allowances account for the influence of run-ning-in wear on the various error elements
yα respectively yβ are the running-in amounts which reduce the influence of pitch and profile errors respectively influ-ence of localised faceload
Cay is defined as the running-in amount that compensates for lack of tip relief
The following relations may be used
For not surface hardened steel
ptHlim
α fσ160y =
yβ = βxlimH
f320σ
with the following upper limits
V lt 5 ms 5-10 ms gt 10 ms
yα max none
limH
12800σ limH
6400σ
yβ max none
limH
25600σ limH
12800σ
For surface hardened steel
yα = 0075 fpt but not more than 3 for any speed
yβ = 015 Fβx but not more than 6 for any speed
16 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
For all kinds of steel
5145189718
1C2
limHay +
minusσ
=
When pinion and wheel material differ the following ap-plies
bull Use the larger of fpt1 - yα1 and fpt2 - yα2 to replace fpt - yα in the calculation of KHα and Kv
bull Use ( )2β1ββ yy21y += in the calculation of KHβ
bull Use ( )2ay1aya CC21C += in the calculation of Kv
bull Use ( )2ay1ay2a1a CC21CC +== in the scuffing
calculation if no design tip relief is foreseen
Classification Notes- No 412 17 May 2003
DET NORSKE VERITAS
2 Calculation of Surface Durability
21 Scope and General Remarks Part 2 includes the calculations of flank surface durability as limited by pitting spalling case crushing and subsurface yielding Endurance and time limited flank surface fatigue is calculated by means of 22 ndash 212 In a way also tooth frac-tures starting from the flank due to subsurface fatigue is in-cluded through the criteria in 213
Pitting itself is not considered as a critical damage for slow speed gears However pits can create a severe notch effect that may result in tooth breakage This is particularly im-portant for surface hardened teeth but also for high strength through hardened teeth For high-speed gears pitting is not permitted
Spalling and case crushing are considered similar to pitting but may have a more severe effect on tooth breakage due to the larger material breakouts initiated below the surface Subsurface fatigue is considered in 213
For jacking gears (self-elevating offshore units) or similar slow speed gears designed for very limited life the max static (or very slow running) surface load for surface hard-ened flanks is limited by the subsurface yield strength
For case hardened gears operating with relatively thin lubri-cation oil films grey staining (micropitting) may be the lim-iting criterion for the gear rating Specific calculation meth-ods for this purpose are not given here but are under consid-eration for future revisions Thus depending on experience with similar gear designs limitations on surface durability rating other than those according to 22 - 213 may be ap-plied
22 Basic Equations Calculation of surface durability (pitting) for spur gears is based on the contact stress at the inner point of single pair contact or the contact at the pitch point whichever is greater
Calculation of surface durability for helical gears is based on the contact stress at the pitch point
For helical gears with 0 lt εβ lt 1 a linear interpolation be-tween the above mentioned applies
Calculation of surface durability for spiral bevel gears is based on the contact stress at the midpoint of the zone of contact
Alternatively for bevel gears the contact stress may be cal-culated with the program ldquoBECALrdquo In that case KA and Kv are to be included in the applied tooth force but not KHβ and KHα The calculated (real) Hertzian stresses are to be multi-plied with ZK in order to be comparable with the permissible contact stresses
The contact stresses calculated with the method in part 2 are based on the Hertzian theory but do not always represent the real Hertzian stresses
The corresponding permissible contact stresses σHP are to be calculated for both pinion and wheel
221 Contact stress Cylindrical gears
( )HαHβvγA
1
tβεEHDBH KKKKK
bud1uF
ZZZZZσ+
=
where
ZBD = Zone factor for inner point of single pair contact for pinion resp wheel (see 232)
ZH = Zone factor for pitch point (see 231)
ZE = Elasticity factor (see 24)
Zε = Contact ratio factor (see 25)
Zβ = Helix angle factor (see 26)
Ft KA Kγ Kv KHβ KHα see 15 ndash 110
d1 b u see 12 ndash 15
Bevel gears
( )HαHβvγA
v1v
vmtKEMH KKKKK
bud1uF
ZZZ105σ+
sdot=
where
105 is a correlation factor to reach real Hertzian stresses (when ZK = 1)
ZE KA etc see above
ZM = mid-zone factor see 233
ZK = bevel gear factor see 27
Fmt dv1 uv see 12 ndash 15
It is assumed that the heightwise crowning is chosen so as to result in the maximum contact stresses at or near the mid-point of the flanks
222 Permissible contact stress
XWRvLH
NHlimHP ZZZZZ
SZσ
σ =
where
σH lim = Endurance limit for contact stresses (see 28)
ZN = Life factor for contact stresses (see 29)
SH = Required safety factor according to the rules
ZLZvZR = Oil film influence factors (see 210)
ZW = Work hardening factor (see 211)
ZX = Size factor (see 212)
18 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
23 Zone Factors ZH ZBD and ZM
231 Zone factor ZH The zone factor ZH accounts for the influence on contact stresses of the tooth flank curvature at the pitch point and converts the tangential force at the reference cylinder to the normal force at the pitch cylinder
wtt2
wtbH sinααcos
cosαcosβ2Z =
232 Zone factors ZBD The zone factors ZBD account for the influence on contact stresses of the tooth flank curvature at the inner point of sin-gle pair contact in relation to ZH Index B refers to pinion D to wheel
For εβ ge 1 ZBD = 1
For internal gears ZD = 1
For εβ = 0 (spur gears)
( )
π
minusεminusminus
π
minusminus
α=
α2
2
2b
2a
1
2
1b
1a
wtB
z211
dd
z21
dd
tanZ
( )
π
minusεminusminus
π
minusminus
α=
α1
2
1b
1a
2
2
2b
2a
wtD
z211
dd
z21
dd
tanZ
If ZB lt 1 use ZB = 1
If ZD lt 1 use ZD = 1
For 0 lt εβ lt 1
ZBD = ZBD (for spur gears) ndash εβ (ZBD (for spur gears) ndash 1)
233 Zone factor ZM The mid-zone factor ZM accounts for the influence of the contact stress at the mid point of the flank and applies to spi-ral bevel gears
εminusminus
εminusminus
αβ=
αα btm2
2vb2
2vabtm2
1vb2val
2v1vvtbmM
pddpdd
ddtancos2Z
This factor is the product of ZH and ZM-B in ISO 10300 with the condition that the heightwise crowning is sufficient to move the peak load towards the midpoint
234 Inner contact point For cylindrical or bevel gears with very low number of teeth the inner contact point (A) may be close to the base circle In order to avoid a wear edge near A it is required to have suitable tip relief on the wheel
24 Elasticity Factor ZE The elasticity factor ZE accounts for the influence of the material properties as modulus of elasticity and Poissonrsquos ratio on the contact stresses
For steel against steel ZE = 1898
25 Contact Ratio Factor Zε The contact ratio factor Zε accounts for the influence of the transverse contact ratio εα and the overlap ratio εβ on the contact stresses
αε1Z =ε for 1εβ ge
( )α
ββ
αε ε
εε1
3ε4
Z +minusminus
= for εβ lt 1
26 Helix Angle Factor Zβ The helix angle factor Zβ accounts for the influence of helix angle (independent of its influence on Zε) on the surface du-rability
cosβZβ =
27 Bevel Gear Factor ZK The bevel gear factor accounts for the difference between the real Hertzian stresses in spiral bevel gears and the contact stresses assumed responsible for surface fatigue (pitting) ZK adjusts the contact stresses in such a way that the same per-missible stresses as for cylindrical gears may apply
The following may be used ZK = 080
28 Values of Endurance Limit σHlim and Static Strength 5H10σ 3H10σ
σHlim is the limit of contact stress that may be sustained for 5middot107 cycles without the occurrence of progressive pitting
For most materials 5middot107 cycles are considered to be the be-ginning of the endurance strength range or lower knee of the σ-N curve (See also Life Factor ZN) However for nitrided steels 2middot106 apply
For this purpose pitting is defined by
bull for not surface hardened gears pitted area ge 2 of total active flank area
bull for surface hardened gears pitted area ge 05 of total active flank area or ge 4 of one particular tooth flank area 510Hσ and 310Hσ are the contact stresses which the given
material can withstand for 105 respectively 103 cycles without subsurface yielding or flank damages as pitting spalling or case crushing when adequate case depth applies
The following listed values for σHlim 510Hσ and 310Hσ may only be used for materials subjected to a quality control as
Classification Notes- No 412 19 May 2003
DET NORSKE VERITAS
the one referred to in the rules Results of approved fatigue tests may also be used as the basis for establishing these values The defined survival probability is 99
σHlim σH105 σH10
3
Alloyed case hardened steels (surface hardness 58-63 HRC) - of specially approved high grade - of normal grade
1650 1500
2500 2400
3100 3100
Nitrided steel of approved grade gas nitrided (surface hardness 700-800 HV) 1250 13 σHlim 13 σHlim
Alloyed quenched and tempered steel bath or gas nitrided (surface hardness 500-700 HV)
1000
13 σHlim
13 σHlim
Alloyed flame or induction hardened steel (surface hardness 500-650 HV) 075 HV + 750 16 σHlim 45 HV
Alloyed quenched and tempered steel 14 HV + 350 16 σHlim 45 HV
Carbon steel 15 HV + 250 16 σHlim 16 σHlim
These values refer to forged or hot rolled steel For cast steel the values for σHlim are to be reduced by 15
29 Life Factor ZN The life factor ZN takes account of a higher permissible contact stress if only limited life (number of cycles NL) is demanded or lower permissible contact stress if very high number of cycles apply
If this is not documented by approved fatigue tests the fol-lowing method may be used
For all steels except nitrided
7L 105N sdotge ZN = 1 or
01570
L
7
N N105Z
sdot=
Ie ZN = 092 for 1010 cycles
The ZN = 1 from 5middot107 on may only be used when the mate-rial cleanliness is of approved high grade (see Rules Pt4 Ch2) and the lubrication is optimised by a specially approved filtering process
105 lt NL lt 5middot107 510N
logZ037
L
7
N N105Z
sdot=
NL = 105 WXRVLHlim
WstX10H1010NN ZZZZZσ
ZZσZZ
55
5=
103 lt NL lt 105 )Z(Zlog05
L
5
10NN
5N103N10
5N10ZZ
==
3L 10N le
WXRVLlimH
Wst10X10H10NN ZZZZZ
ZZZZ
33
3σ
σ==
(but not less than ZN105)
For nitrided steels
6L 102N sdotge ZN = 1 or
00980
L
6
N N102Z
sdot=
Ie ZN = 092 for 1010 cycles
The ZN = 1 from 2middot106 on may only be used when the mate-rial cleanliness is of approved high grade (see Rules Pt4 Ch2) and the lubrication is optimised by a specially approved filtering process
105 lt NL lt 2middot106 510N
Zlog76860
L
6
N N102Z
sdot=
5L 10N le
XWRVL
10XWst10NN ZZZZZ
ZZ13ZZ
55 ==
Note that when no index indicating number of cycles is used the factors are valid for 5middot107 (respectively 2middot106 for nitrid-ing) cycles
210 Influence Factors on Lubrication Film ZL ZV and ZR The lubricant factor ZL accounts for the influence of the type of lubricant and its viscosity the speed factor ZV ac-counts for the influence of the pitch line velocity and the roughness factor ZR accounts for influence of the surface roughness on the surface endurance capacity
The following methods may be applied in connection with the endurance limit
20 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
Surface hardened steels Not surface hardened steels
ZL ( )24013421
360910ν+
+
( )24013421
680830ν+
+
ZV ( )v3280
140930+
+ ( )v3280300850
++
ZR
080
ZrelR3
150
ZrelR3
where
ν40 = Kinematic oil viscosity at 40ordmC (mm2s) For case hardened steels the influence of a high bulk temperature (see 4 Scuffing) should be con-sidered Eg bulk temperatures in excess of 120ordmC for long periods may cause reduced flank surface endurance limits
For values of ν40 gt 500 use ν40 = 500
RZrel = The mean roughness between pinion and wheel (after running in) relative to an equivalent radius of curvature at the pitch point ρc = 10mm
RZrel
= ( ) 31
cZ2Z1 ρ
10RR50
+
RZ = Mean peak to valley roughness (microm) (DIN defini-tion) (roughly RZ = 6 Ra)
For 5
L 10N le ZL ZV ZR = 10
211 Work Hardening Factor ZW The work hardening factor ZW accounts for the increase of surface durability of a soft steel gear when meshing the soft steel gear with a surface hardened or substantially harder gear with a smooth surface
The following approximation may be used for the endurance limit
Surface hardened steel against not surface hardened steel 150
ZeqW R
31700
130HB21Z
minus
minus=
where HB = the Brinell hardness of the soft member For HB gt 470 use HB = 470 For HB lt 130 use HB = 130
RZeq = equivalent roughness
033
c40
066
ZS
ZHZHZEQ ρvν
15000RRRR
=
If RZeq gt 16 then use RZeq = 16
If RZeq lt 15 then use RZeq = 15
where
RZH = surface roughness of the hard member before run in
RZS = surface roughness of the soft member before run in
ν40 = see 210
If values of ZW lt 1 are evaluated ZW = 1 should be used for flank endurance However the low value for ZW may indi-cate a potential wear problem
Through hardened pinion against softer wheel
( )
minussdotminus+= 000829
HBHB0008981u1Z
2
1W
For 21HBHB
2
1 le use ZW = 1
For 71HBHB
2
1 gt use 17HBHB
2
1 =
For u gt 20 use u = 20
For static strength (lt 105 cycles)
Surface hardened against not surface hardened
ZWst = 105
Through hardened pinion against softer wheel
ZWst = 1
212 Size Factor ZX The size factor accounts for statistics indicating that the stress levels at which fatigue damage occurs decrease with an increase of component size as a consequence of the influ-ence on subsurface defects combined with small stress gradi-ents and of the influence of size on material quality
ZX may be taken unity provided that subsurface fatigue for surface hardened pinions and wheels is considered eg as in the following subsection 213
213 Subsurface Fatigue This is only applicable to surface hardened pinions and wheels The main objective is to have a subsurface safety against fatigue (endurance limit) or deformation (static strength) which is at least as high as the safety SH required for the surface The following method may be used as an approximation unless otherwise documented
The high cycle fatigue (gt3106 cycles) is assumed to mainly depend on the orthogonal shear stresses Static strength (lt103 cycles) is assumed to depend mainly on equivalent
Classification Notes- No 412 21 May 2003
DET NORSKE VERITAS
stresses (von Mises) Both are influenced by residual stresses but this is only considered roughly and empirically
The subsurface working stresses at depths inside the peak of the orthogonal shear stresses respectively the equivalent stresses are only dependent on the (real) Hertzian stresses Surface related conditions as expressed by ZL ZV and ZR are assumed to have a negligible influence
The real Hertzian stresses σHR are determined as
For helical gears with εβ gt 1
σHR = σH
For helical gears with εβ lt 1 and spur gears
ε
α
ββ ε
ε+εminus
sdotσ=σZ
1
HHR
For bevel gears
KHHR Z
1σσ sdot=
The necessary hardness HV is given as a function of the net depth tz (net = after grinding or hard metal hobbing and per-pendicular to the flank)
The coordinates tz and HV are to be compared with the de-sign specification such as
bull for flame and induction hardening tHVmin HVmin bull for nitriding t400min HV = 400 bull for case hardening t550min HV = 550 t400min HV = 400
and t300min HV = 300 (the latter only if the core hardness lt 300 If the core hardness gt 400 the t400 is to be re-placed by a fictive t400 = 16 t550)
In addition the specified surface hardness is not to be less than the max necessary hardness (at tz = 05aH) This applies to all hardening methods
For high cycle fatigue (gt3 106 cycles) the following applies
sdot+
minussdotsdotsdot= o90
05azt
05azt
cosSσ04HV
H
HHHR
applicable to 05a
zt
Hge
For 05at
H
z lt the value for 05at
H
z = applies
56300ρSσ12a cHHR
Hsdotsdot
sdot=
Where aH is half the hertzian contact width multiplied by an empirical factor of 12 that takes into account the possible influence of reduced compressive residual stresses (or even tensile residual stresses) on the local fatigue strength
If any of the specified hardness depths including the surface hardness is below the curve described by HV = f (tz) the actual safety factor against subsurface fatigue is determined as follows
reduce SH stepwise in the formula for HV and aH until all specified hardness depths and surface hardness balance with the corrected curve The safety factor obtained through this method is the safety against subsurface fatigue
For static strength (lt103 cycles) the following applies
sdot+
minussdotsdotsdot= o90
07at
06a
zt
cosSσ019HV
Hst
z
HstHHR
applicable to 06at
Hst
z ge
56300ρSσa cHHR
Hstsdotsdot
=
In the case of insufficient specified hardness depths the same procedure for determination of the actual safety factor as above applies
For limited life fatigue (103 lt cycles lt 3106 )
For this purpose it is necessary to extend the correction of safety factors to include also higher values than required Ie in the case of more than sufficient hardness and depths the safety factor in the formulae for both high cycle fatigue and static strength are to be increased until necessary and speci-fied values balance
The actual safety factor for a given number of cycles N be-tween 103 and 3106 is found by linear interpolation in a dou-ble logarithmic diagram
minussdotminus
= sdot logN3477
logSlogSlogS
310H6103HHN
36 10H103H Slog86281Slog86280 sdot+sdot sdot
22 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
3 Calculation of Tooth Strength
31 Scope and General Remarks Part 3 include the calculation of tooth root strength as limited by tooth root cracking (surface or subsurface initiated) and yielding
For rim thickness sR ge 35middotmn the strength is calculated by means of 32 ndash 313 For cylindrical gears the calculation is based on the assumption that the highest tooth root tensile stress arises by application of the force at the outer point of single tooth pair contact of the virtual spur gears The method has however a few limitations that are mentioned in 36
For bevel gears the calculation is based on force application at the tooth tip of the virtual cylindrical gear Subsequently the stress is converted to load application at the mid point of the flank due to the heightwise crowning
Bevel gears may also be calculated with the program BECAL In that case KA and Kv are to be included in the applied tooth force but not KFβ and KFα
In case of a thin annulus or a thin gear rim etc radial crack-ing can occur rather than tangential cracking (from root fillet to root fillet) Cracking can also start from the compression fillet rather than the tension fillet For rim thickness sR lt 35middotmn a special calculation procedure is given in 315 and 316 and a simplified procedure in 314
A tooth breakage is often the end of the life of a gear trans-mission Therefore a high safety SF against breakage is re-quired
It should be noted that this part 3 does not cover fractures caused by
bull oil holes in the tooth root space bull wear steps on the flank bull flank surface distress such as pits spalls or grey staining
Especially the latter is known to cause oblique fractures starting from the active flank predominately in spiral bevel gears but also sometimes in cylindrical gears
Specific calculation methods for these purposes are not given here but are under consideration for future revisions Thus depending on experience with similar gear designs limita-tions other than those outlined in part 3 may be applied
32 Tooth Root Stresses The local tooth root stress is defined as the max principal stress in the tooth root caused by application of the tooth force Ie the stress ratio R = 0 Other stress ratios such as for eg idler gears (R asymp -12) shrunk on gear rims (R gt 0) etc are considered by correcting the permissible stress level
321 Local tooth root stress The local tooth root stress for pinion and wheel may be as-sessed by strain gauge measurements or FE calculations or similar For both measurements and calculations all details are to be agreed in advance
Normally the stresses for pinion and wheel are calculated as
Cylindrical gears
FαFβvγAβSFn
tF K K K K K Y Y Y
m bFσ =
where
YF = Tooth form factor (see 33)
YS = Stress correction factor (see 34)
Yβ = Helix angle factor (see 36)
Ft KA Kγ Kv KFβ KFα see 15 ndash 110
b see 13
Bevel gears
FαFβvγASaFamn
mtF K K K K K Y Y Y
m bFσ ε=
where
YFa = Tooth form factor see 33
YSa = Stress correction factor see 34
Yε = Contact ratio factor see 35
Fmt KA etc see 15 ndash 110
b see 13
322 Permissible tooth root stress The permissible local tooth root stress for pinion respectively wheel for a given number of cycles N is
CXRrelTrelTF
NMFEFP YYYY
SYY
δσ
=σ
Note that all these factors YM etc are applicable to 3middot106 cycles when used in this formula for σFP The influence of other number of cycles on these factors is covered by the calculation of YN
where
σFE = Local tooth root bending endurance limit of reference test gear (see 37)
YM = Mean stress influence factor which accounts for other loads than constant load direction eg idler gears temporary change of load di-rection pre-stress due to shrinkage etc (see 38)
YN = Life factor for tooth root stresses related to reference test gear dimensions (see 39)
SF = Required safety factor according to the rules
YδrelT = Relative notch sensitivity factor of the gear to be determined related to the reference test gear (see 310)
YRrelT = Relative (root fillet) surface condition factor of the gear to be determined related to the reference test gear (see 311)
Classification Notes- No 412 23 May 2003
DET NORSKE VERITAS
YX = Size factor (see 312)
YC = Case depth factor (see 313)
33 Tooth Form Factors YF YFa The tooth form factors YF and YFa take into account the in-fluence of the tooth form on the nominal bending stress
YF applies to load application at the outer point of single tooth pair contact of the virtual spur gear pair and is used for cylindrical gears
YFa applies to load application at the tooth tip and is used for bevel gears
Both YF and YFa are based on the distance between the con-tact points of the 30˚-tangents at the root fillet of the tooth profile for external gears respectively 60˚ tangents for inter-nal gears
Figure 31 External tooth in normal section
Figure 32 Internal tooth in normal section
Definitions
n
2
n
Fn
enFn
Fe
F
α cosms
α cosmh6
Y
=
n
2
n
Fn
anFn
Fa
Fa
α cosms
α cosmh6
Y
=
In the case of helical gears YF and YFa are determined in the normal section ie for a virtual number of teeth
YFa differs from YF by the bending moment arm hFa and αFan and can be determined by the same procedure as YF with exception of hFe and αFan For hFa and αFan all indices e will change to a (tip)
The following formulae apply to cylindrical gears but may also be used for bevel gears when replacing
mn with mnm
zn with zvn
αt with αvt
β with βm
with undercut without undercut
Fig 33 Dimensions and basic rack profile of the teeth (finished profile)
Tool and basic rack data such as hfP ρfp and spr etc are re-ferred to mn ie dimensionless
331 Determination of parameters
( )n
n
prnfPnfP m
α cossαsin 1ρ
αtan h4πE
minusminusminusminus=
For external gears fPfP ρρ =
For internal gears ( )0z
195fPfP0
fPfP 10363156ρhxρρ
sdotminus+
+=
where
z0 = number of teeth of pinion cutter
x0 = addendum modification coefficient of pinion cutter
hfP = addendum of pinion cutter
ρfP = tip radius of pinion cutter
xhρG fpfp +minus=
24 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
τmE
2π
z2H
nnminus
minus=
with
3πτ = for external gears
6πτ = for internal gears
Htan zG2
nminusϑ=ϑ
(to be solved iteratively suitable start value6π
=ϑ for exter-
nal gears and 3π for internal gears)
a) Tooth root chord sFn For external gears
minus
ϑ+
ϑminus= ρ
cosG3
3πsinz
ms
fpnn
Fn
For bevel gears with a tooth thickness modification
xsm affects mainly sFn but also hFe and αFen The total influence of xsm on YFa Ysa can be approximated by only adding 2 xsm to sFn mn
For internal gears
minus
ϑ+
ϑminus= ρ
cosG
6πsinz
ms
fPnn
Fn
b) Root fillet radius ρF at 30ordm tangent
( )G2coszcosG2ρ
mρ
2n
2
fpn
F
minusϑϑ+=
c) Determination of bending moment arm hF dn = zn mn
b2α
αn βcosε
ε =
dan = dn + 2 ha
pbn = π mn cos αn
dbn = dn cos αn
( )4
d1εpzz
2dd
zz2d
2bn
2
αnbn
2bn
2an
en +
minusminus
minus=
en
bnen d
dcos arcα =
ennnn
e α invα invα x tan 22π
z1
minus+
+=γ
αFen = αen ndash γe
For external gears
( )
minus=
n
enFenee
n
Fe
mdαtan sin γ γcos
21
mh
]ρcos
G3πcosz fpn +
ϑminus
ϑminusminus
For internal gears
( ) minus
sdotsdotminus=
n
enFenee
n
Fe
mdα tanγ sinγ cos
21
mh
minus
ϑminus
ϑminussdot ρ
cosG3
6πcosz fPn
332 Gearing with εαn gt 2 For deep tooth form gearing ( )25ε2 αn lele produced with a verified grade of accuracy of 4 or better and with applied profile modification to obtain a trapezoidal load distribution along the path of contact the YF may be corrected by the factor YDT as
250ε205for 0666ε2366Y αnαnDT leleminus=
205εfor 10Y αnDT lt=
34 Stress Correction Factors YS YSa The stress correction factors YS and YSa take into account the conversion of the nominal bending stress to the local tooth root stress Thereby YS and YSa cover the stress increasing effect of the notch (fillet) and the fact that not only bending stresses arise at the root A part of the local stress is inde-pendent of the bending moment arm This part increases the more the decisive point of load application approaches the critical tooth root section
Therefore in addition to its dependence on the notch radius the stress correction is also dependent on the position of the load application ie the size of the bending moment arm
YS applies to the load application at the outer point of single tooth pair contact YSa to the load application at tooth tip
YS can be determined as follows
( )
+
+=L23121
1
sS qL01312Y
where Fe
Fn
hsL = and
F
Fns ρ2
sq = (see 33)
YSa can be calculated by replacing hFe with hFa in the above formulae
Note a) Range of validity 1 lt qs lt 8
In case of sharper root radii (ie produced with tools having too sharp tip radii) YS resp YSa must be specially considered
b) In case of grinding notches (due to insufficient protuberance of the hob) YS resp YSa can rise considerably and must be multiplied with
Classification Notes- No 412 25 May 2003
DET NORSKE VERITAS
g
g
ρt
0613
13
minus
where
tg = depth of the grinding notch
ρg = radius of the grinding notch
c) The formulae for YS resp YSa are only valid for αn = 20˚ However the same formulae can be used as a safe approximation for other pressure angles
35 Contact Ratio Factor Yε The contact ratio factor Yε covers the conversion from load application at the tooth tip to the load application at the mid point of the flank (heightwise) for bevel gears
The following may be used
Yε = 0625
36 Helix Angle Factor Yβ The helix angle factor Yβ takes into account the difference between the helical gear and the virtual spur gear in the nor-mal section on which the calculation is based in the first step In this way it is accounted for that the conditions for tooth root stresses are more favourable because the lines of contact are sloping over the flank
The following may be used (β input in degrees)
Yβ = 1 ndash εβ β120
When εβ gt 1 use εβ = 1 and when β gt 30deg use β = 30deg in the formula
However the above equation for Yβ may only be used for gears with β gt 25deg if adequate tip relief is applied to both pinion and wheel (adequate = at least 05 middot Ceff see 432)
37 Values of Endurance Limit σFE σFE is the local tooth root stress (max principal) which the material can endure permanently with 99 survival prob-ability 3106 load cycles is regarded as the beginning of the endurance limit or the lower knee of the σ ndash N curve σFE is defined as the unidirectional pulsating stress with a minimum stress of zero (disregarding residual stresses due to heat treatment) Other stress conditions such as alternating or pre-stressed etc are covered by the conversion factor YM
σFE can be found by pulsating tests or gear running tests for any material in any condition If the approval of the gear is to be based on the results of such tests all details on the testing conditions have to be approved by the Society Fur-ther the tests may have to be made under the Societys su-pervision
If no fatigue tests are available the following listed values for σFE may be used for materials subjected to a quality con-trol as the one referred to in the rules
σFE
Alloyed case hardened steels 1) (fillet surface hardness 58 ndash 63 HRC)
bull of specially approved high grade
1050
bull of normal grade
minus CrNiMo steels with approved process
1000
minus CrNi and CrNiMo steels generally 920 minus MnCr steels generally 850
Nitriding steel of approved grade quenched tempered and gas nitrided (surface hardness 700 ndash 800 HV)
840
Alloyed quenched and tempered steel bath or gas nitrided (surface hardness 500 ndash 700 HV)
720
Alloyed quenched and tempered steel flame or induction hardened 2) (incl entire root fillet) (fillet surface hardness 500 ndash 650 HV)
07 HV + 300
Alloyed quenched and tempered steel flame or induction hardened (excl entire root fillet) (σB = uts of base material)
025 σB + 125
Alloyed quenched and tempered steel 04 σB + 200
Carbon steel 025 σB + 250
Note All numbers given above are valid for separate forgings and for blanks cut from bars forged according to a qualified procedure see Pt 4 Ch 2 Sec 3 For rolled steel the values are to be reduced with 10 For blanks cut from forged bars that are not qualified as mentioned above the values are to be reduced with 20 For cast steel reduce with 40
1) These values are valid for a root radius
bull being unground If however any grinding is made in the root fillet area in such a way that the residual stresses may be affected σFE is to be reduced by 20 (If the grinding also leaves a notch see 34)
bull with fillet surface hardness 58 ndash 63 HRC In case of lower surface hardness than 58 HRC σFE is to be reduced with 20(58 ndash HRC) where HRC is the detected hardness (This may lead to a permissible tooth root stress that varies along the facewidth If so the actual tooth root stresses may also be considered along facewidth)
bull not being shot peened In case of approved shot peening σFE may be increased by 200 for gears where σFE is reduced by 20 due to root grinding Otherwise σFE may be increased by 100 for
6nm le and 100 ndash 5 (mn - 6) for mn gt 6 However the possible adverse influence on the flanks regarding grey staining should be considered and if necessary the flanks should be masked
2) The fillet is not to be ground after surface hardening Regarding possible root grinding see 1)
26 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
38 Mean stress influence Factor YM The mean stress influence factor YM takes into account the influence of other working stress conditions than pure pulsa-tions (R = 0) such as eg load reversals idler gears planets and shrink-fitted gears
YM (YMst) are defined as the ratio between the endurance (or static) strength with a stress ratio R ne 0 and the endurance (or static) strength with R = 0
YM and YMst apply only to a calculation method that assesses the positive (tensile) stresses and is therefore suitable for comparison between the calculated (positive) working stress σF and the permissible stress σFP calculated with YM or YMst
For thin rings (annulus) in epicyclic gears where the com-pression fillet may be decisive special considerations apply see 316
The following method may be used within a stress ratio ndash12 lt R lt 05
381 For idlers planets and PTO with ice class
M1M1R1
1Yor Y MstM
+minus
minus=
where
R = stress ratio = min stress divided by max stress
For designs with the same force applied on both forward- and back-flank R may be assumed to ndash 12
For designs with considerably different forces on forward- and back-flank such as eg a marine propulsion wheel with a power take off pinion R may be assessed as
branchmain theoffacewidth unit per forcepto offacewidt unit per force21minus
For a power take off (PTO) with ice class see 161 c
M considers the mean stress influence on the endurance (or static) strength amplitudes
M is defined as the reduction of the endurance strength am-plitude for a certain increase of the mean stress divided by that increase of the mean stress
Following M values may be used
Endurance limit
Static strength
Case hardened 08 ndash 015 Ys 1) 07
If shot peened 04 06
Nitrided 03 03
Induction or flame hardened
04 06
Not surface hardened steel 03 05
Cast steels 04 06 1)For bevel gears use Ys=2 for determination of M
The listed M values for the endurance limit are independent of the fillet shape (Ys) except for case hardening In princi-ple there is a dependency but wide variations usually only occur for case hardening eg smooth semicircular fillets versus grinding notches
382 For gears with periodical change of rotational direction For case hardened gears with full load applied periodically in both directions such as side thrusters the same formula for YM as for idlers (with R = ndash 12) may be used together with the M values for endurance limit This simplified approach is valid when the number of changes of direction exceeds 100 and the total number of load cycles exceeds 3middot106
For gears of other materials YM will normally be higher than for a pure idler provided the number of changes of direction is below 3middot106 A linear interpolation in a diagram with loga-rithmic number of changes of direction may be used ie from YM = 09 with one change to YM (idler) for 3middot106 changes This is applicable to YM for endurance limit For static strength use YM as for idlers
For gears with occasional full load in reversed direction such as the main wheel in a reversing gear box YM = 09 may be used
383 For gears with shrinkage stresses and unidirectional load For endurance strength
FE
fitM σ
σM1
M21Y+
minus=
σFE is the endurance limit for R = 0
For static strength YMst = 1 and σfit accounted for in 39b
σfit is the shrinkage stress in the fillet (30˚ tangent) and may be found by multiplying the nominal tangential (hoop) stress with a stress concentration factor
n
Ffit m
ρ215scf minus=
384 For shrink-fitted idlers and planets When combined conditions apply such as idlers with shrink-age stresses the design factor for endurance strength is
( ) ( ) FE
fitM σ
σR1M1
M2
M1M1R1
1Yminussdot+
minus
+minus
minus=
Symbols as above but note that the stress ratio R in this par-ticular connection should disregard the influence of σfit ie R normally equal ndash 12
For static strength
M1M1R1
1YMst
+minus
minus=
The effect of σfit is accounted for in 39 b
Classification Notes- No 412 27 May 2003
DET NORSKE VERITAS
385 Additional requirements for peak loads The total stress range (σmax ndash σmin) in a tooth root fillet is not to exceed
F
y
Sσ225
for not surface hardened fillets
FSHV5 for surface hardened fillets
39 Life Factor YN The life factor YN takes into account that in the case of limited life (number of cycles) a higher tooth root stress can be permitted and that lower stresses may apply for very high number of cycles
Decisive for the strength at limited life is the σ ndash N ndash curve of the respective material for given hardening module fillet radius roughness in the tooth root etc Ie the factors YδrelT YRelT YX and YM have an influence on YN
If no σ ndash N ndash curve for the actual material and hardening etc is available the following method may be used
Determination of the σ ndash N ndash curve
a) Calculate the permissible stress σFP for the beginning of the endurance limit (3middot106 cycles) including the influ-ence of all relevant factors as SF YδrelT YRelT YX YM and YC ie σFP = σFE middotYM middotYδrelT middotYRelT middotYX middotYC SF
b) Calculate the permissible laquostaticraquo stress (le 103 load cy-cles) including the influence of all relevant factors as SFst YδrelTst YMst and YCst ( )fitσCstYδrelTstYMstYFstσ
FstS1
FPstσ minussdotsdotsdot=
where σFst is the local tooth root stress which the material can resist without cracking (surface hardened materials) or unacceptable deformation (not surface hardened materials) with 99 survival probability
σFst
Alloyed case hardened steel 1) 2300
Nitriding steel quenched tempered and gas nitrided (surface hardness 700 ndash 800 HV)
1250
Alloyed quenched and tempered steel bath or gas nitrided (surface hardness 500 ndash 700 HV)
1050
Alloyed quenched and tempered steel flame or induction hardened (fillet surface hardness 500 ndash 650 HV)
18 HV + 800
Steel with not surface hardened fillets the smaller value of 2)
18 σB or 225 σy
1) This is valid for a fillet surface hardness of 58 ndash 63 HRC In case of lower fillet surface hardness than 58 HRC σFst is to be reduced with 30(58 ndash HRC) where HRC is the actual hardness Shot peening or grinding notches are not considered to have any significant influence on σFst 2) Actual stresses exceeding the yield point (σy or σ02) will alter the residual stresses locally in the ldquotensionrdquo fillet re-spectively ldquocompressionrdquo fillet This is only to be utilised for gears that are not later loaded with a high number of cycles at lower loads that could cause fatigue in the ldquocompressionrdquo fillet
c) Calculate YN as
NL gt 3middot106
YN = 1 or 001
L
6
N N103Y
sdot= ie Yn = 092 for 1010
The YN = 1 from 3middot106 on may only be used when special material cleanness applies see rules Pt4 Ch2
103ltNLlt3middot106exp
L
6
N N103Y
sdot=
cycles6103forσcycles310forσlog02876exp
FP
FPst
sdot=
NL lt 103 cycles6103forσcycles310forσ
NYFP
FPst
sdot=
or simply use σFPst as mentioned in b) directly
Guidance on number of load cycles NL for various applica-tions
bull For propulsion purpose normally NL = 1010 at full load (yachts etc may have lower values)
bull For auxiliary gears driving generators that normally operate with 70-90 of rated power NL = 108 with rated power may be applied
28 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
310 Relative Notch Sensitivity Factor YδrelT The dynamic (respectively static) relative notch sensitivity factor YδrelT (YδrelTst) indicate to which extent the theoreti-cally concentrated stress lies above the endurance limits (re-spectively static strengths) in the case of fatigue (respectively overload) breakage
YδrelT is a function of the material and the relative stress gra-dient It differs for static strength and endurance limit
The following method may be used
For endurance limit
for not surface hardened fillets
( )
024
s024
relT σ103133q21σ1012201351
Ysdotsdotminus
+sdotsdotminus+= minus
minus
δ
for all surface hardened fillets except nitrided
( )
106q21002451
Y srelT
++=δ
for nitrided fillets
( )
1347q2101421
Y srelT
++=δ
For static strength
for not surface hardened fillets1)
( )( )( )025
02
02502s
relTst σ3000821σ300 1Y0821Y
+
minus+=δ
for surface hardened fillets except nitrided
YδrelTst = 044 YS + 012
for nitrided fillets
YδrelTst = 06 + 02 YS 1) These values are only valid if the local stresses do not
exceed the yield point and thereby alter the residual stress level See also 39b footnote 2
311 Relative Surface Condition Factor YRrelT The relative surface condition factor YRrelT takes into ac-count the dependence of the tooth root strength on the sur-face condition in the tooth root fillet mainly the dependence on the peak to valley surface roughness
YRrelT differs for endurance limit and static strength
The following method may be used
For endurance limit
YRrelT = 1675 ndash 053 (Ry + 1)01 for surface hardened steels and alloyed quenched and tempered steels except nitrided
YRrelT = 53 ndash 42 (Ry + 1)001 for carbon steels
YδrelT = 43 ndash 326 (Ry + 1)0005
for nitrided steels
For static strength
YRrelTst = 1 for all Ry and all materials
For a fillet without any longitudinal machining trace Ry asymp Rz
312 Size Factor YX The size factor YX takes into account the decrease of the strength with increasing size YX differs for endurance limit and static strength
The following may be used
For endurance limit
YX = 1 for mn le 5 generally
YX = 103 ndash 0006 mn for 5 lt mn lt 30
YX = 085 for mn ge 30
for not surface hard-ened steels
YX = 105 ndash 001 mn for 5 lt mn ge 25
YX = 08 for mn ge 25
for surface hardened steels
For static strength
YXst = 1 for all mn and all materials
313 Case Depth Factor YC The case depth factor YC takes into account the influence of hardening depth on tooth root strength
YC applies only to surface hardened tooth roots and is dif-ferent for endurance limit and static strength
In case of insufficient hardening depth fatigue cracks can develop in the transition zone between the hardened layer and the core For static strength yielding shall not occur in the transition zone as this would alter the surface residual stresses and therewith also the fatigue strength
The major parameters are case depth stress gradient permis-sible surface respectively subsurface stresses and subsurface residual stresses
The following simplified method for YC may be used
YC consists of a ratio between permissible subsurface stress (incl influence of expected residual stresses) and permissible surface stress This ratio is multiplied with a bracket con-taining the influence of case depth and stress gradient (The empirical numbers in the bracket are based on a high number of teeth and are somewhat on the safe side for low number of teeth)
YC and YCst may be calculated as given below but calculated values above 10 are to be put equal 10
For endurance limit
+
+=nFFE
C m02ρt31
σconstY
Classification Notes- No 412 29 May 2003
DET NORSKE VERITAS
For static strength
+
+=nFFst
Cst m02ρt31
σconstY
where const and t are connected as
Hardening process
t = endurance limit const =
static strength const =
t550 640 1900
t400 500 1200 Case hard-ening
t300 380 800
Nitriding t400 500 1200
Induction- or flame hardening
tHVmin 11 HVmin 25 HVmin
For symbols see 213
In addition to these requirements to minimum case depths for endurance limit some upper limitations apply to case hard-ened gears
The max depth to 550 HV should not exceed
1) 13 of the top land thickness san unless adequate tip relief is applied (see 110)
2) 025 mn If this is exceeded the following applies addi-tionally in connection with endurance limit
minusminus= 025
mt
1Yn
max550C
314 Thin rim factor YB Where the rim thickness is not sufficient to provide full sup-port for the tooth root the location of a bending failure may be through the gear rim rather than from root fillet to root fillet
YB is not a factor used to convert calculated root stresses at the 30deg tangent to actual stresses in a thin rim tension fillet Actually the compression fillet can be more susceptible to fatigue
YB is a simplified empirical factor used to de-rate thin rim gears (external as well as internal) when no detailed calcula-tion of stresses in both tension and compression fillets are available
Figure 34 Examples on thin rims
YB is applicable in the range 175 lt sRmn lt 35
YB = 115 middot ln (8324 middot mnsR)
(for sRmn ge 35 YB = 1)
(for sRmn le 175 use 315)
σF as calculated in 321 is to be multiplied with YB when sRmn lt 35 Thus YB is used for both high and low cycle fatigue
Note This method is considered to be on the safe side for external gear rims However for internal gear rims without any flange or web stiffeners the method may not be on the safe side and it is advised to check with the method in 315316
315 Stresses in Thin Rims For rim thickness sR lt 35 mn the safety against rim cracking has to be checked
The following method may be used
3151 General The stresses in the standardised 30ordm tangent section tension side are slightly reduced due to decreasing stress correction factor with decreasing relative rim thickness sRmn On the other hand during the complete stress cycle of that fillet a certain amount of compression stresses are also introduced The complete stress range remains approximately constant Therefore the standardised calculation of stresses at the 30ordm tangent may be retained for thin rims as one of the necessary criteria
The maximum stress range for thin rims usually occurs at the 60ordm ndash 80ordm tangents both for laquotensionraquo and laquocompressionraquo side The following method assumes the 75ordm tangent to be the decisive Therefore in addition to the am criterion ap-plied at the 30ordm tangent it is necessary to evaluate the max and min stresses at the 75ordm tangent for both laquotensionraquo (loaded flank) fillet and laquocompressionraquo (back-flank) fillet For this purpose the whole stress cycle of each fillet should be considered but usually the following simplification is justified
Figure 35 Nomenclature of fillets
Index laquoTraquo means laquotensileraquo fillet laquoCraquo means laquocompressionraquo fillet
σFTmin and σFCmax are determined on basis of the nominal rim stresses times the stress concentration factor Y75
σFCmin and σFTmax are determined on basis of superposition of nominal rim stresses times Y75 plus the tooth bending stresses at 75ordm tangent
30 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
3152 Stress concentration factors at the 75ordm tangents The nominal rim stress consists of bending stresses due to local bending moments tangential stresses due to the tan-gential force Ft and radial shear stresses due to Fr
The major influence is given by the bending stresses The influence of the tangential stresses is minor and even though its stress concentration factor is slightly higher than for bending it is considered to be safe enough when the sum of these nominal stresses are combined with the stress concen-tration factor for bending The influence of the radial shear stress is neglected
The stress concentration factor relating nominal rim stresses to local fillet stresses at the 75ordm tangent may be calculated as
75n
R75
n
R
75
ρmsρ185
ms3
Y+
sdot=
where ρ75 is the root radius at the 75ordm tangent ref to mn Usually ρ75 is closed to the tool radius ρao and
ρ75 = ρao
is a safe approximation compensating for the am simplifi-cations to the laquounsaferaquo side
The tooth root stresses of the loaded tooth are decreasing with decreasing relative rim thickness approximately with the empirical factor
15
R
ncorr s
m311Y
minus=
3153 Nominal rim stresses The bending moment applied to the rim consists of a part of the tooth tilt moment Ft (hF + 05 sR) and the bending caused by the radial force Fr
The sectional modulus (first moment of area) which is used for determination of the nominal bending stresses is not nec-essarily the same for the 2 am bending moments If flanges webs etc outside the toothed section contribute to stiffening the rim against various deflections the influence of these stiffeners should be considered Eg an end flange will have an almost negligible influence on the effective sectional modulus for the stresses due to tooth tilt as the deflection caused by the tilt moment is rather small and would not much involve the flange On the other hand the radial forces as for instance from the meshes in an annulus would cause considerable radial deflections that the flange might restrict to a substantial amount When considering the stiff-ening of such flanges or webs on basis of simplified models it is advised to use an effective rim thickness sR = sR + 02 mn for the first moment of area of the rim (toothed part) cross section
For a high number of rim teeth it may be assumed that the rim bending moments in the fillets adjacent to the loaded tooth are of the same magnitude as right under the applied
force This assumption is reasonable for an annulus but rather much on the safe side for a hollow pinion
The influence of Ft on the nominal tangential stress is simpli-fied by half of it for compressive stresses (σ1) and the other half for tensile stresses (σ2) Applying these assumptions the nominal rim stresses adjacent to the loaded tooth are
( ) ( )A2
FW
f RFW
s05hF05 t
R
r
T
RFt1 minus
ϑminus
+minus=σ
( ) ( )A2
FW
f RFW
s05hF05 t
R
r
T
RFt2 +
ϑminus
+=σ
where σ1 σ2 see Fig 35
A = minimum area of cross section (usually bR sR)
WT = the sectional modulus of rim with respect to tooth tilt moment (usually 6sb 2
rR )
WR = the sectional modulus of the rim including the influence of stiffeners as flanges webs etc (WR ge WT)
bR = the width of the rim
R = the radius of the neutral axis in the rim ie from wheel centre to midpoint of rim
( )ϑf = a function for bending moment distribution around the rim For a rim (pinion) with one mesh only the ( )ϑf at the position of load application is 024 For an annulus with 3 or more meshes ( )ϑf at the position of each load application is approx 3 planets ( )ϑf = 019 4 planets ( )ϑf = 014 5 planets ( )ϑf = 011 6 planets ( )ϑf = 009
It must be checked if the max (tensile) stress in the compres-sion fillet really occurs when the fillet is adjacent to the loaded tooth In principle the stress variation through a complete rotation should be considered and the max value used The max value is usually never less than 0 For an annulus eg the tilt moment is zero in the mid position be-tween the planet meshes whilst the bending moment due to the radial forces is half of that at the mesh but with opposite sign
If these formulae are applied to idler gears as eg planets the influence of nominal tangential stresses must be cor-rected by deleting Ft(2 A) for σ1 and using FtA for σ2 Further the influence of Fr on the nominal bending stresses is usually negligible due to the planet bearing support
3154 Root fillet stresses Determination of min and max stresses in the laquotensionraquo fillet
Classification Notes- No 412 31 May 2003
DET NORSKE VERITAS
Minimum stress
751FTmin YσK σ =
Maximum stress
corrF752 Yσ03Yσ KσFTmax +=
where
03 is an empirical factor relating the tension stresses (σF) at the 30ordm tangent to the part of the tension stresses at the 75ordm tangent which add to the rim related stresses (03 also takes into account that full superposition of nominal stresses times stress concentration factors from both laquosides of the corner filletraquo would result in too high stresses)
αβγ sdotsdotsdotsdot= FFvA KKKKKK
Determination of min and max stresses in the laquocompres-sionraquo fillet
Minimum stress
corrF751FCmin Yσ036Yσ Kσ minus=
Maximum stress
752FCmax Yσ Kσ =
where
036 is an empirical factor relating the tension stresses (σF) at the 30ordm tangent to the part of the compression stresses at the 75ordm tangent which add to the rim related stresses
For gears with reversed loads as idler gears and planets there is no distinct laquotensionraquo or laquocompressionraquo fillet The mini-mum stress σFmin is the minimum of σFTmin and σFCmin (usually the latter is decisive) The maximum stress σFTmax is the maximum of σFTmax and σFCmax (usually the former is deci-sive)
316 Permissible Stresses in Thin Rims
3161 General The safety against fatigue fracture respectively overload fracture is to be at least at the same level as for solid gears The laquoordinaryraquo criteria at the 30ordm tangent apply as given in 31 through 313
Additionally the following criteria at the 75ordm tangent may apply
3162 For gt3middot106 cycles The permissible stresses for the laquotensionraquo fillets and for the laquocompressionraquo fillets are determined by means of a relevant fatigue diagram
If the actual tooth root stress (tensile or compressive) ex-ceeds the yield strength to the material the induced residual stresses are to be taken into account
For determination of permissible stresses the following is defined
R = stress ratio ie FTmax
FTminσσ respectively
FCmax
FCmin
σσ
∆σ = stress range ie σFTmax ndash σFTmin resp σFCmax ndash σFCmin
(For idler gears and planets Fmax
FminσσR =
and ∆σ = σFmax minus σFmin)
The permissible stress range ∆σρ for the laquotensionraquo respec-tively laquocompressionraquo fillets can be calculated as
For R gt minus1 FPp σ
R1R1031
13∆σ
minus+
+=
For minus infin lt R lt minus1 FPp σ
R1R10151
13∆σ
minus+
+=
where
σFP = see 32 determined for unidirectional stresses (YM = 1)
If the yield strength σy is exceeded in either tension or com-pression residual stresses are induced This may be consid-ered by correcting the stress ratio R for the respective fillets (tension or compression)
Eg if yFCmin σσ gt (ie exceeded in compression) the
difference yFCmin σσ∆ minus= affects the stress ratio as
∆σ
σ∆σ∆σR
FCmax
y
FCmax
FCmin
+
minus=
++
=
Similarly the stress ratio in the tension fillet may require cor-rection
If the yield strength is exceeded in tension σFTmax gt σy the difference ∆ = σFTmax ndash σy affects the stress ratio as
y
FTmin
FTmax
FTmin
σ∆σ
∆σ∆σR minus=
minusminus
=
Checking for possible exceeding of the yield strength has to be made with the highest torque and with the KAP if this ex-ceeds KA
3163 Forle 103 cycles The permissible stress range ∆σp is not to exceed
For R gt minus1 FPstpst σ
R1R1051
15∆σ
minus+
+=
For ndash infin lt R lt minus1 FPstpst σ
R1R10251
15∆σ
minus+
+=
For all values of R ∆σpst is limited by
not surface hardened F
y
Sσ225
32 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
surface hardened CF
YSHV5
Definition of ∆σ and R see 3162 with particular attention to possible correction of R if the yield strength is exceeded
σFPst see 32 determined for unidirectional stresses (YM = 1) and lt 103 cycles
3164 For 103 lt cycles lt 3106 ∆σp is to be determined by linear interpolation a log-log dia-gram
∆σp at NL load cycles is
6L 103p
exp
L
6
N p ∆σN103∆σ sdot
sdot=
6
3
103p
10p
∆σ
∆σlog28760exp
sdot
=
Classification Notes- No 412 33 May 2003
DET NORSKE VERITAS
4 Calculation of Scuffing Load Capacity
41 Introduction High surface temperatures due to high loads and sliding ve-locities can cause lubricant films to break down This sei-zure or welding together of areas of tooth surface is termed scuffing
In contrast to pitting and fatigue breakage which show a dis-tinct incubation period a single short overloading can lead to a scuffing failure In the ISO-TR13989 two criteria are mentioned The method used in this Classification Note is based on the principles of the flash temperature criterion
Note Bulk temperature in excess of 120ordmC for long periods may have an adverse effect on the surface durability see 211
42 General Criteria In no point along the path of contact the local contact tem-perature may exceed the permissible temperature ie
oilS
oilSB S
ϑ+ϑminusϑ
leϑ
50SB minusϑleϑ
where
Bϑ = max contact temperature along the path of contact
maxflaMBB ϑ+ϑ=ϑ
MBϑ = bulk temperature see 434
maxflaϑ = max flash temperature along the path of con-tact see 44
Sϑ = scuffing temperature see below
oilϑ = oil temperature before it reaches the mesh (max applicable for the actual load case to be used ie normally alarm temperature except for ice classes where a max expected tem-perature applies)
SS = required safety factor according to the Rules
The scuffing temperature Sϑ may be calculated as
L2
wrelT
002
40S XFZGX
ν100112085780
sdot
sdot++=ϑ
where
XwrelT = relative welding factor
XwrelT
Through hardened steel 10
Phosphated steel 125
Copper-plated steel 150
Nitrided steel 150
Less than 10 retained austenite
115
10 ndash 20 retained austenite 10
Casehardened steel
20 ndash 30 retained austenite 085
Austenitic steel 045
FZG = load stage according to FZG-Test A8390 (Note This is the load stage where scuffing oc-curs However due to scatter in test results cal-culations are to be made with one load stage less than the specification)
XL = lubricant factor
= 10 for mineral oils
= 08 for polyalfaolefins
= 07 for non-water-soluble polyglycols
= 06 for water-soluble polyglycols
= 15 for traction fluids
= 13 for phosphate esters
ν40 = kinematic oil viscosity at 40˚C (mm2s)
Application of other test methods such as the Ryder the FZG-Ryder R46574 and the FZG L-42 Test 141195110 may be specially considered
For high speed gears with very short time of contact Sϑ may be increased as follows provided use of EP-oils
Addition to the calculated scuffing temperature Sϑ
If micros 18tc ge no addition
If micros 18tc lt add ( )cwrelT t18X18 minussdotsdot
where
ct = contact time ( micros ) which is the time needed to cross the Hertzian contact width
( ) [ ]microsu1cosβn
uσ340tb1
Hc +sdotsdot
sdot=
σH as calculated in 221
For bevel gears use vu in stead of u
34 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
43 Influence Factors
431 Coefficient of friction The following coefficient of friction may apply
L025a
005oil
02
redC ΣC
Bt X R ηρv
w0048micro minus
=
where
wBt = specific tooth load (Nmm)
vΣC = sum of tangential velocities at pitch point At pitch line velocities gt 50 ms the limiting value of vΣC at v = 50 ms is to be used
ρredC = relative radius of curvature (transversal plane) at the pitch point
Cylindrical gears
HαHβvγAbt
Bt KKKKKb
Fw sdotsdotsdotsdotsdot= (see 1)
twΣC αsin v2v =
bCredC β cos ρρ =
Bevel gears
HαHβvγAtmb
Bt KKKKKb
Fw sdotsdotsdotsdotsdot= (see 1)
vtmtΣC αsin v2v =
bmvCredC β cos ρρ =
ηoil = dynamic viscosity (mPa s) at oilϑ calculated as
1000
ρ νη oiloil =
where ρ in kgm3 approximated as
( ) 70 15oil15 minusϑminusρ=ρ and νoil is kinematic viscosity at oilϑ and may be calculated by means of the following equation
( ) ( )++=+ 08νloglog08νloglog 100oil ( )
sdotminus
ϑ+minus313log373log
273log373log oil
( ) ( )( )08νloglog08νloglog 10040 +minus+
Ra = composite arithmetic mean roughness (micron) of pinion an wheel calculated as ( )2a1aa RR 05R +=This is defined as the roughness on the new flanks ie as manufactured
XL = see 42
432 Effective tip relief Ceff Ceff is the effective tip relief that amount of tip relief which compensates for the elastic deformation of the gear mesh ie
zero load at the tooth tip It is assumed (simplified) to be equal for both pinion and wheel
Cylindrical gears
for helical
γ
γ=cb
KKFC Abt
eff (see 1)
For spur
cbKKF
C Abteff
γ= (see 1)
Alternatively for spur and helical gears the non-linear ap-proach in 111 may be used (taking Ceff=δ)
Bevel gears
bcKFC Ambt
effγ
= (see 1)
where
γ
α
ε2ε44c
+sdot
=γ
433 Tip relief and extension Cylindrical gears
The extension of the tip relief is not to result in an effective contact ratio 1ltεα when the gear is unloaded (exceptions to this may only apply for applications where the gear is not to run at light loads) This means that the unrelieved part of the path of contact is to be minimum pbt It is further assumed that this unrelieved part is placed centrally on the path of contact
If root relief applies it has to be calculated as equivalent tip relief Ie pinion root relief (at mesh position A) is added to Ca2 and wheel root relief (at mesh position E) is added to Ca1 If no design tip relief or root relief on the mating gear is specified (ie if Ca1+Croot2 = 0 and visa versa) use the run-ning in amount see 112
Bevel gears
Bevel gears are to have heightwise crowning ie no distinct relievedunrelieved area This may be treated as tip and root relief For calculation purposes the root relief is combined with the tip relief of the mating member into an equivalent tip relief If no resulting tip relieves are specified the equivalent tip relives may be calculated as an approxima-tion based on the tool crowning Ca tool (per mille of tool module m0) as follows
2root1aeq1a CCC +=
1root2aeq2a CCC +=
where
Classification Notes- No 412 35 May 2003
DET NORSKE VERITAS
2
0
n0101atoolal m
)m2(mA1middotmCC
minus+minussdot=
( )
sdotminusminussdotsdotminus+= vtvb1
2vb1
21vavt1n1 αtan dddαsin 05x1mA
2
0
n020atool22a m
)mm(2A1middotmCC
minus+minussdot=
( )
sdotminusminussdotsdotminus+= vtvb2
2vb2
2va2vt2n2 αtan dddαsin 05x1mA
2
0
20atool1root1 m
A1mCC
minussdotsdot=
2
0
10atool2root2 m
A1mCC
minussdotsdot=
If the pressure angles of the cutter blades are modified (and verified) to balance the tip relieves the following may be assumed
( )eq2aeq1aeq2aeq1a C andC calculated of sum50CC sdot==
Throughout the following Ca1 and Ca2 mean the equivalent tip relieves Ca1eq and Ca2eq
434 Bulk temperature The bulk temperature may be calculated as
flaaveragempsoilMB X X 50 ϑ+ϑ=ϑ
where
Xs = lubrication factor
= 12 for spray lubrication
= 10 for dip lubrication (provided both pinion and wheel are dip lubricated and tip speed lt 5 ms)
= 10 for spray lubrication with additional cooling spray (spray on both pinion and wheel or spray on pinion and dip lubrication of wheel)
= 02 for meshes fully submerged in oil
Xmp = contact factor ( )pmp nX += 150
np = number of mesh contact on the pinion (for small gear ratii the number of wheel meshes should be used if higher)
flaaverageϑ
= average of the integrated flash temperature (see 44) along the path of contact
( )
AE
E
Ayyyfla
flaaverage
d
ΓminusΓ
ΓΓϑ=ϑint =
For high speed gears (v gt 50 ms) it may be necessary to assess the bulk temperature on the basis of a thermal rating of the entire gear transmission
44 The Flash Temperature flaϑ
441 Basic formula The local flash temperature flaϑ may be calculated as
( ) 41redy
y2ly
211
43Btcorrfla
unXwX3250
y ρ
ρminusρ
micro=ϑ Γ
(For bevel gears replace u with uv)
and is to be calculated stepwise along the path of contact from A to E
where
micro = coefficient of friction see 431
Xcorr = correction factor taking empirically into account the increased scuffing risk in the beginning of the approach path due to mesh starting without any previously built up oil film and possible shuffling away oil before meshing if insufficient tip relief
( )
3
AD
yaeffcorr 50
CC1X
ΓminusΓε
Γminus+=
α
Ca = tip relief of driven member Xcorr is only applicable in the approach path and if Ca lt Ceff otherwise 10
wBt = unit load see 431
yXΓ = load sharing factor see 443
n1 = pinion rpm
Γy ρly etc see 442
442 Geometrical relations The various radii of flank curvature (transversal plane) are
ρ1y = pinion flank radius at mesh point y
ρ2y = wheel flank radius at mesh point y
ρredy = equivalent radius of curvature at mesh point y
y2y1
y2y1redy
ρ+ρ
ρρ=ρ
Cylindrical gears
twy
1y αsin au1Γ1
ρ+
+=
twy
2y αsin au1Γu
ρ+
minus=
Note that for internal gears a and u are negative
36 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
Bevel gears
vtvv
y1y αsin a
u1Γ1
ρ+
+=
vtvv
yv2y αsin a
u1Γu
ρ+
minus=
Γ is the parameter on the path of contact and y is any point between A end E
At the respective ends Γ has the following values
Root piniontip wheel
Cylindrical gears
( )
minus
minusminus= 1
αtan 1dd
zzΓ
tw
2b2a2
1
2A
Bevel gears
( )
minus
minusminus= 1
αtan 1dd
uΓvt
2vb2va2
vA
Tip pinionroot wheel
Cylindrical gears
( )
1αtan
1ddΓ
tw
2b1a1
E minusminus
=
Bevel gears
( )
1αtan
1ddΓ
vt
2vb1va1
E minusminus
=
At inner point of single pair contact
Cylindrical gears
tw1
EB α tan zπ2ΓΓ minus=
Bevel gears
vtv1
EB α tan zπ2ΓΓ minus=
At outer point of single pair contact
Cylindrical gears
tw1AD α tan z
π2ΓΓ +=
Bevel gears
vt v1
AD αtan z π2ΓΓ +=
At pitch point ΓC = 0
The points F and G (only applicable to cylindrical gears) limiting the extension of tip relief (so as to maintain a mini-mum contact ratio of unity for unloaded gears) are at
2
BAF
Γ+Γ=Γ
2
EDG
Γ+Γ=Γ
443 Load sharing factor XΓ The load sharing factor XΓ accounts for the load sharing be-tween the various pairs of teeth in mesh along the path of contact
XΓ is to be calculated stepwise from A to E using the pa-rameter Γy
4431 Cylindrical gears with β = 0 and no tip relief
Figure 41
ByAAB
Ay for31
31X
yΓltΓleΓ
ΓminusΓ
ΓminusΓ+=Γ
DyB for1Xy
ΓltΓleΓ=Γ
EyDDE
yE for31
31X
yΓleΓltΓ
ΓminusΓ
ΓminusΓ+=Γ
4432 Cylindrical gears with β = 0 and tip relief Tip relief on the pinion reduces XΓ in the range G ndash E and increases correspondingly XΓ in the range F ndash B
Tip relief on the wheel reduces XΓ in the range A ndash F and increases correspondingly XΓ in the range D ndash G
Following remains generally
DyB for1Xy
ΓleΓleΓ=Γ
21XXGF== ΓΓ
In the following it must be distinguished between Ca lt Ceff respectively Ca gt Ceff This is shown by an example below where Ca1 lt Ceff and Ca2 gt Ceff
Classification Notes- No 412 37 May 2003
DET NORSKE VERITAS
Figure 42
Note When Ca gt Ceff the path of contact is shortened by A ndash A respectively E ndash E The single pair contact path is extended into B respectively D If this shift is significant it is necessary to consider the negative effect on surface durability (B) and bending stresses (B and D)
Range A - F
For effa2 CC le
minus=Γ
eff
2a
CC1
31X
A
FyAeff
2a
AF
Ay for C3
C61XX
AyΓleΓleΓ
sdot
+ΓminusΓ
ΓminusΓ+= ΓΓ
For effa2 CC ge
AyAfor0Xy
ΓleΓleΓ=Γ
with ( )21
CC
1CC
eff
2a
eff
2a
AFAAminus
minusΓminusΓ+Γ=Γ
FyAeff
2a
AF
Ay
eff
2a for 21
CC
CC1X
yΓleΓleΓ
minus
ΓminusΓ
ΓminusΓ+minus=Γ
Range F - B
For effa1 CC le
ByFeff
1a
FB
Fy forC3
C61
21X
yΓleΓleΓ
sdot
+ΓminusΓ
ΓminusΓ+=Γ
For effa1 CC ge
ByFeff
1a
FB
Fy for21
CC
21X
yΓleΓleΓ
minus
ΓminusΓ
ΓminusΓ+=Γ
ByB for1Xy
ΓleΓleΓ=Γ
with 1
CC2
eff
1a
FBFB
minus
ΓminusΓ+Γ=Γ
Range D ndash G
For effa2 CC le
GyDeff
2a
DG
Dy
eff
2a for C 3C
61
C 3C
32X
yΓleΓleΓ
+
ΓminusΓ
ΓminusΓminus+=Γ F
or effa2 CC ge
DyD for1Xy
ΓleΓleΓ=Γ
with ( )21
CC
1CC
eff
2a
eff
2a
DGDDminus
minusΓminusΓ+Γ=Γ
GyDeff
2a
DG
Dy
eff
2a for21
CC
CCX ΓleΓleΓ
minus
ΓminusΓ
ΓminusΓminus=Γ
Range G - E
For effa1 CC le
minus=Γ
eff
1a
CC1
31X
E
EyGeff
1a
GE
Gy forC3
C61
21X
yΓleΓleΓ
sdot
+ΓminusΓ
ΓminusΓminus=Γ
For effa1 CC gt
EyGeff
1a
GE
Gy for21
CC
21X
yΓleΓleΓ
minus
ΓminusΓ
ΓminusΓminus=Γ
EyE for0Xy
ΓleΓleΓ=Γ
with 1
CC2
eff
1a
GEGE
minus
ΓminusΓ+Γ=Γ
4433 Gears with β gt 0 buttressing Due to oblique contact lines over the flanks a certain but-tressing may occur near A and E
This applies to both cylindrical and bevel gears with tip relief lt Ceff The buttressing Xbutt is simplified as a linear function within the ranges A ndash H respectively I ndash E
Figure 43
1εwhen13X βbutt EAge=
1εwhenε031X ββbutt EAlt+=
38 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
Cylindrical gears
bIEAH βsin 02ΓΓΓΓ =minus=minus
Bevel gears
bmIEAH βsin 02ΓΓΓΓ =minus=minus
4434 Cylindrical gears with 2εγ le and no tip relief
yXΓ is obtained by multiplication of
yXΓ in 4431 with
Xbutt in 4433
4435 Gears with 2εγ gt and no tip relief
Applicable to both cylindrical and bevel gears
IyH for1Xy
ΓleΓleΓε
=α
Γ
IyHybutt and forX1Xy
ΓgtΓΓltΓε
=α
Γ
Figure 44
4436 Cylindrical gears with 2εγ le and tip relief
yXΓ is obtained by multiplication of
yXΓ in 4432 with Xbutt
in 4433
4437 Cylindrical gears with 2εγ gt and tip relief
Tip relief on the pinion (respectively wheel) reduces XГ in the range G ndash E (respectively A ndash F) and increases XГ in the range F ndash G
yXΓ is obtained by multiplication of
yXΓ as described below
with Xbutt in 4433
In the XГ example below the influence of tip relief is shown (without the influence of Xbutt) by means of
eff2aeff1a CC and CC ltgt
Tip relief gt Ceff causes new end points A respectively E of the path of contact
Figure 45
Range A ndash F
( ) ( )( ) eff
a2αa1α
AF
Ay
eff
2aeff
C 12C 13εC 1ε
CCCX
y +εε++minus
ΓminusΓ
ΓminusΓ+
εminus
=ααα
Γ
eff2aFyA CC if for leΓleΓleΓ
eff2aFyA CifCfor and geΓleΓleΓ
eff2aAyA CC iffor0Xy
gtΓleΓleΓ=Γ
( ) ( ) ( )( ) ( ) a2αa1α
αeffa2AFAA C 1ε 3C 1ε
1ε 2 CC with++minus+minus
ΓminusΓ+Γ=Γ
Range F ndash G
( )( )( ) GyF
eff
2a1a forC 1 2CC 11X
yΓleΓleΓ
+εε+minusε
+ε
=αα
α
αΓ
Range G ndash E
( ) ( )( ) eff
2a1a
GE
Gy
C 1 2C 1C 1 3XX
GFy +εεminusε++ε
ΓminusΓ
ΓminusΓminus=
αα
ααΓΓ minus
effa1EyG CC if for leΓleΓleΓ
effa1EyG CC if Γfor and geΓleΓle
eff1aEyE CC if for0Xy
geΓleΓleΓ=Γ
( ) ( ) ( )( ) ( ) 2a1a
eff1aGEEE C 1C 1 3
1 2 CC withminusε++ε+εminus
ΓminusΓminusΓ=Γαα
α
4438 Bevel gears with εγ more than approx 18 and heightwise crowning For Ca1 = Ca2 = Ceff the following applies
Figure 46
Classification Notes- No 412 39 May 2003
DET NORSKE VERITAS
( )AEM 50 Γ+Γ=Γ
( )( )2AD
3
2My651X
y ΓminusΓε
ΓminusΓminus
ε=
ααΓ
For tip relief lt Ceff yXΓ is found by linear interpolation be-
tween ( )effay CCX =Γ and ( )0CayX =Γ as in 4435
The interpolation is to be made stepwise from A to M with the influence of Ca2 and from M to E with the influence of Ca1 (For 2a1a CC ne there is a discontinuity at M)
Eg with Ca1 = 04 Ceff and Ca2 = 055 Ceff then
Range A ndash M
)CC()0C( eff2ay2ayyX 550X 450X =Γ=ΓΓ +=
Range M ndash E
)CC()0C( eff1ay1ayyX 40X 06X =Γ=ΓΓ +=
For tip relief gt Ceff the new end points A and E are found as
( )
minusΓminusΓ
ε+Γ=Γ α 1
CC
6 eff
2aADAA
( )
minusΓminusΓ
ε+Γ=Γ α 1
CC
6 eff
1aADEE
Range A ndash A
0Xy=Γ
Range A ndash M
( )( )
ΓminusΓ
ΓminusΓminus
minusε=
αΓ 2
MA
2My
eff
2a1
CC4
351Xy
Range M ndash E
( )( )
ΓminusΓ
ΓminusΓminus
minusε=
αΓ 2
ME
2My
eff
1a1
CC4
351Xy
Range E ndash E
0Xy=Γ
40 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
Appendix A Fatigue Damage Accumulation
The Palmgren-Miner cumulative damage calculation princi-ple is used The procedure may be applied as follows
A1 Stress Spectrum From the individual torque classes the torques (Ti) at the peak values of class intervals and the associated number of cycles (NLi) for both pinion and wheel are to be listed from the highest to the lowest torque
(In case of a cyclic torque variation within the torque classes it is advised to use the peak torque If the cyclic variation is such that the same teeth will repeatedly suffer the peak torque this is a must)
The stress spectra for tooth roots and flanks (σFi σHi) with all relevant factors (except KA) are to be calculated on the basis of the torque spectrum The load dependent K-factors are to be determined for each torque class
A2 σminusN-curve The stress versus load cycle curves for tooth roots and flanks (both pinion and wheel) are to be drawn on the basis of per-missible stresses (ie including the demanded minimum safety factors) as determined in 2 respectively 3 If different safety levels for high cycle fatigue and low cycle fatigue are desired this may be expressed by different demand safety factors applied at the endurance limit respectively at static strength
A3 Damage accumulation The individual damage ratio Di at ith stress level is defined as
Fi
Lii N
ND =
where
NLi = The number of applied cycles at ith stress
NFi = The number of cycles to failure at ith stress
Basically stresses σi below the permissible stress level for infinite life (if a constant ZN or YN is accepted) do not con-tribute to the damage sum However calculating the actual safety factor Sact as described below all the σi for which the product S σi is bigger than or equal to the permissible stress level for infinite life contribute to the damage sum and thus to the determination of Sact The final value of S is decisive
(NFi can be found mathematically by putting the permissible stress σpi equal the actual stress σi thereby finding the actual life factor This life factor can be solved with regard to load cycles ie NFi)
The damage sum ΣDi is not to exceed unity
If ΣDi ne 1 the safety against cumulative fatigue damage is different from the applied demand safety factor For deter-mination of this theoretical safety factor an iteration proce-dure is required as described in the following flowchart
S is correction factor with which the actual safety factor Sact can be found
Sact is the demand safety factor (used in determination of the permissible stresses in the σ ndash N ndash curve) times the correc-tion factor S
The full procedure is to be applied for pinion and wheel tooth roots and flanks
Note If alternating stresses occur in a spectrum of mainly pulsating stresses the alternating stresses may be replaced by equiva-lent pulsating stresses ie by means of division with the ac-tual mean stress influence factor YM
Classification Notes- No 412 41 May 2003
DET NORSKE VERITAS
Appendix B Application Factors for Diesel Driven Gears
For diesel driven gears the application factor KA depends on torsional vibrations Both normal operation and misfiring conditions have to be considered
Normally these two running conditions can be covered by only one calculation
B1 Definitions
Normal operation KAnorm = 0
normv0
TTT +
where
T0 = rated nominal torque
Tv norm = vibratory torque amplitude for normal opera-tion (see rules Pt4 Ch3 Sec1 G301 for defi-nition of ldquonormalrdquo irregularity)
Misfiring operation KA misf = 0
misfv
TTT +
where
T = remaining nominal torque when one cylinder out of action
Tv misf = vibratory torque amplitude in misfiring con-dition This refers to a permissible misfiring condition ie a condition that does not require automatic or immediate corrective actions as speed or pitch reduction
The normal operation is assumed to last for a very high num-ber of cycles such as 1010
The misfiring operation is assumed to last for a limited dura-tion such as 107 cycles
B2 Determination of decisive load Assuming life factor at 1010 cycles as YN = ZN = 092 which usually is relevant the calculation may be performed only once with the combination having the highest value of appli-cation factorlife factor
For bending stresses and scuffing the higher value of
092
K normA and 098
K misfA
For contact stresses the higher value of
092K normA and
113K misfA (but
097K misfA for nitrided gears)
B3 Simplified procedure Note that this is only a guidance and is not a binding con-vention
T = oTZ
1Zsdot
minus where Z = number of cylinders
( ) ideal Videal Vmisf Vnorm V TTT24ZT +minus=
where TV ideal = vibratory torque with all cylinders perfectly equal See also rules Pt4 Ch3 Sec1 G300
When using trends from torsional vibration analysis and measurements the following may be used
TV ideal To is close to zero for engines with few cylinders and using a suitable elastic coupling and increases with relative coupling stiffness and number of cylinders
TV misf To may be high for engines with few cylinders and decreases with number of cylinders
This can be indicated as
200Z
TT
ο
idealV asymp and 80Z04
TT
o
misfV minusasymp
Inserting this into the formulae for the two application fac-tors the following guidance can be given
118112K misfA minusasymp
115110K normA minusasymp
Since KA norm is to be combined with the lower life factors the decisive load condition will be the normal one and a KA of 115 will cover most relevant cases when a suitable elas-tic coupling is chosen
42 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
Appendix C Calculation of Pinion-Rack
Pinion-racks used for elevating of mobile offshore units are open gears that are subjected to wear and tear With normal specifications such as only a few hundred total operation cycles (site to site) the tooth bending stresses for static strength will be decisive for the lay out however with ex-ception of surface hardened pinions where case crushing has to be considered
In the following the use of part 1 and 3 for pinion-racks is shown including relevant simplifications
C1 Pinion tooth root stresses Since the load spectrum normally is dominated by high torques and few load cycles (in the range up to 1-2000) the static strength is decisive
The actual stress is calculated as
β1FSaFan1
tF1 KYY
mbFσ sdotsdotsdotsdot
=
b1 is limited to b2 + 2middotmn
YFa and YSa replace YF and YS because load application at tooth tip has to be assumed for such inaccurate gears
Pinions often use a non-involute profile in the dedendum part of the flank eg a constant radius equal the radius of curva-ture at reference circle For such pinions sFn and hFa are to be measured directly on a sectional drawing of the pinion tooth
Due to high loads and narrow facewidths it may be assumed that KFβ2 = KHβ = 10 However when b1 gt b2 then KFβ1 gt10
If no detailed documentation of KFβ1 is available the fol-lowing may be used
KFβ1 = 1 + 015middot(b1b2 ndash 1)
The permissible stress (not surface hardened) is calculated as
δrelTstF
Fst1FP1 Y
Sσσ sdot=
The mean stress influence due to leg lifting may be disre-garded
The actual and permissible stresses should be calculated for the relevant loads as given in the rules
C2 Rack tooth root stresses The actual stress is calculated as
SaFan2
tF2 YY
mbFσ sdotsdotsdot
=
See C1 for details
The permissible stress is calculated as
δrelTstF
Fst2FP2 Y
Sσσ sdot=
For alloyed steels (Ni Cr Mo) with high toughness and duc-tility the value of YδrelTst may be put equal to YSa
C3 Surface hardened pinions For surface hardened pinions the maximum load is not to cause crushing of the hardened layer of the flank
In principle the calculation described in 213 may be used but when the theoretical Hertzian stress exceeds the ap-proximately 18 times the yield strength of the rack material plastic deformation will occur This will limit the peak Hertzian stress but increases the contact width and thus the penetration of stresses into the depth
An approximation may be based on an assessment of contact width determined by means of equal areas under the theoreti-cal (elastic range) Hertzian contact and the elasto-plastic contact stress (limited to 18 middot σy) with the unknown width
14 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
It also applies when the bearing arrangementsupport has insignificant elastic deflections and thermal axial expansion However each initial mesh contact must be verified to be within acceptance criteria that are cali-brated against a type test at full load Reproduction of the gear tooth length- and height-wise profile must also be verified This can be made through 3D measurements or by initial contact movements caused by defined axial offsets of the pinion (tolerances to be agreed upon)
b) Ktest = 1 + 04middot(beffbndash06) For designs with possible influence of thermal expansion in the axial direction of the pinion The initial mesh contact verified with low load or spin test where the acceptance criteria are calibrated against a type test at full load
c) Ktest = 12 if mesh is only checked by toolmakerrsquos blue or by spin test contact For gears in this category beffb gt 085 is not to be used in the calculation
110 Transversal Load Distribution Factors KHα and KFα The transverse load distribution factors KHα for contact stresses and for scuffing KFα for tooth root stresses account for the effects of pitch and profile errors on the transversal load distribution between 2 or more pairs of teeth in mesh
The following relations may be used
Cylindrical gears
( )
minus+==
tH
αptγγHαFα F
byfc0409
2ε
KK
valid for 2εγ le
( ) ( )tH
αptγ
γ
γHαFα F
byfcε
1ε20409KK
minusminus+==
valid for 2ε γ gt
where
FtH = Ft KA Kγ Kv KHβ
cγ = See 111
γα = See 112
fpt = Maximum single pitch deviation (microm) of pinion or wheel or maximum total profile form devia-tion Fα of pinion or wheel if this is larger than the maximum single pitch deviation
Note In case of adequate equivalent tip relief adapted to the load half of the above mentioned fpt can be introduced A tip relief is considered adequate when the average of Ca1 and Ca2 is within plusmn40 of the value of Ceff in 432
Limitations of KHα and KFα
If the calculated values for
KFα = KHα lt 1 use KFα = KHα = 10
If the calculated value of KHα gt 2εα
γ
Zε
ε use KHα = 2
εα
γ
Zε
ε
If the calculated value of KFα gt εα
γ
Yεε
use KFα = εα
γ
Yεε
where Yε = αnε
075025+ (for εαn see 331c)
Bevel gears
For ground or hard metal hobbed gears KFα = KHα = 1
For lapped gears KFα = KHα = 11
111 Tooth Stiffness Constants cacute and cγ The tooth stiffness is defined as the load which is necessary to deform one or several meshing gear teeth having 1 mm facewidth by an amount of 1 microm in the plane of contact cacute is the maximum stiffness of a single pair of teeth cγ is the mean value of the mesh stiffness in a transverse plane (brief term mesh stiffness)
Both valid for high unit load (Unit load = Ft middot KA middot Kγb)
Cylindrical gears The real stiffness is a combination of the progressive Hertzian contact stiffness and the linear tooth bending stiff-nesses For high unit loads the Hertzian stiffness has little importance and can be disregarded This approach is on the safe side for determination of KHβ and KHα However for moderate or low loads Kv may be underestimated due to determination of a too high resonance speed
The linear approach is described in A
An optional approach for inclusion of the non-linear stiffness is described in B
A The linear approach
BRCCq
βcos08acutec =
and
( )025ε075cc αγ +prime=
where
( )[ ]n02a01a
B α2000212
hh12051C minusminus
+minus+=
12n1n
x000635z
025791z
015551004723q minus++=
12
2n
22
1n
1 x005290z
x024188x000193z
x011654+minusminusminus
+ 000182 x22
Classification Notes- No 412 15 May 2003
DET NORSKE VERITAS
(for internal gears use zn2 equal infinite and x2 = 0) ha0 = hfp for all practical purposes CR considers the increased flexibility of the wheel teeth if the wheel is not a solid disc and may be calculated as
( )( )nR m5s
sR e5
bbln1C +=
where
bs = thickness of a central web
sR = average thickness of rim (net value from tooth root to inside of rim)
The formula is valid for bs b ge 02 and sRmn ge 1 Outside this range of validity and if the web is not centrally posi-tioned CR has to be specially considered
Note CR is the ratio between the average mesh stiffness over the facewidth and the mesh stiffness of a gear pair of solid discs The local mesh stiffness in way of the web corresponds to the mesh stiffness with CR = 1 The local mesh stiffness where there is no web support will be less than calculated with CR above Thus eg a centrally positioned web will have an effect corresponding to a longitudinal crowning of the teeth See also 1931 regarding KHβ
B The non-linear approach
In the following an example is given on how to consider the non-linearity
The relation between unit load Fb as a function of mesh deflection δ is assumed to be a progressive curve up to 500 Nmm and from there on a straight line This straight line when extended to the baseline is assumed to intersect at 10microm
With these assumptions the unit force Fb as a function of mesh deflection δ can be expressed as
( )10δKbF
minus= for 500bFgt
minus=
500Fb10δK
bF for 500
bFlt
with γAt KK
bF
bF
sdotsdot= etc (Nmm) ie unit load incorporat-
ing the relevant factors as
KA middot Kγ for determination of Kv
KA middot Kγ middot Kv for determination of KHβ
KA middot Kγ middot Kv middot KHβ for determination of KHα
δ = mesh deflection (microm)
K = applicable stiffness (c or cγ)
Use of stiffnesses for KV KHβ and KHα
For calculation of Kv and KHα the stiffness is calculated as follows
When Fb lt 500
the stiffness is determined as δ∆
∆ bF
where the increment is chosen as eg ∆ Fb = 10 and thus
50010Fb10
K10Fb∆δ +
++
=
When F b gt 500 the stiffness is c or cγ For calculation of KHβ the mesh deflection δ is used directly
or an equivalent stiffness determined as δsdotb
F
Bevel gears
In lack of more detailed relationship between stiffness and geometry the following may be used
b085
b13cacute eff=
b085b
16c effγ =
beff not to be used in excess of 085 b in these formulae
Bevel gears with heightwise and lengthwise crowning have progressive mesh stiffness The values mentioned above are only valid for high loads They should not be used for de-termination of Ceff (see 432) or KHβ (see 1931)
112 Running-in Allowances The running-in allowances account for the influence of run-ning-in wear on the various error elements
yα respectively yβ are the running-in amounts which reduce the influence of pitch and profile errors respectively influ-ence of localised faceload
Cay is defined as the running-in amount that compensates for lack of tip relief
The following relations may be used
For not surface hardened steel
ptHlim
α fσ160y =
yβ = βxlimH
f320σ
with the following upper limits
V lt 5 ms 5-10 ms gt 10 ms
yα max none
limH
12800σ limH
6400σ
yβ max none
limH
25600σ limH
12800σ
For surface hardened steel
yα = 0075 fpt but not more than 3 for any speed
yβ = 015 Fβx but not more than 6 for any speed
16 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
For all kinds of steel
5145189718
1C2
limHay +
minusσ
=
When pinion and wheel material differ the following ap-plies
bull Use the larger of fpt1 - yα1 and fpt2 - yα2 to replace fpt - yα in the calculation of KHα and Kv
bull Use ( )2β1ββ yy21y += in the calculation of KHβ
bull Use ( )2ay1aya CC21C += in the calculation of Kv
bull Use ( )2ay1ay2a1a CC21CC +== in the scuffing
calculation if no design tip relief is foreseen
Classification Notes- No 412 17 May 2003
DET NORSKE VERITAS
2 Calculation of Surface Durability
21 Scope and General Remarks Part 2 includes the calculations of flank surface durability as limited by pitting spalling case crushing and subsurface yielding Endurance and time limited flank surface fatigue is calculated by means of 22 ndash 212 In a way also tooth frac-tures starting from the flank due to subsurface fatigue is in-cluded through the criteria in 213
Pitting itself is not considered as a critical damage for slow speed gears However pits can create a severe notch effect that may result in tooth breakage This is particularly im-portant for surface hardened teeth but also for high strength through hardened teeth For high-speed gears pitting is not permitted
Spalling and case crushing are considered similar to pitting but may have a more severe effect on tooth breakage due to the larger material breakouts initiated below the surface Subsurface fatigue is considered in 213
For jacking gears (self-elevating offshore units) or similar slow speed gears designed for very limited life the max static (or very slow running) surface load for surface hard-ened flanks is limited by the subsurface yield strength
For case hardened gears operating with relatively thin lubri-cation oil films grey staining (micropitting) may be the lim-iting criterion for the gear rating Specific calculation meth-ods for this purpose are not given here but are under consid-eration for future revisions Thus depending on experience with similar gear designs limitations on surface durability rating other than those according to 22 - 213 may be ap-plied
22 Basic Equations Calculation of surface durability (pitting) for spur gears is based on the contact stress at the inner point of single pair contact or the contact at the pitch point whichever is greater
Calculation of surface durability for helical gears is based on the contact stress at the pitch point
For helical gears with 0 lt εβ lt 1 a linear interpolation be-tween the above mentioned applies
Calculation of surface durability for spiral bevel gears is based on the contact stress at the midpoint of the zone of contact
Alternatively for bevel gears the contact stress may be cal-culated with the program ldquoBECALrdquo In that case KA and Kv are to be included in the applied tooth force but not KHβ and KHα The calculated (real) Hertzian stresses are to be multi-plied with ZK in order to be comparable with the permissible contact stresses
The contact stresses calculated with the method in part 2 are based on the Hertzian theory but do not always represent the real Hertzian stresses
The corresponding permissible contact stresses σHP are to be calculated for both pinion and wheel
221 Contact stress Cylindrical gears
( )HαHβvγA
1
tβεEHDBH KKKKK
bud1uF
ZZZZZσ+
=
where
ZBD = Zone factor for inner point of single pair contact for pinion resp wheel (see 232)
ZH = Zone factor for pitch point (see 231)
ZE = Elasticity factor (see 24)
Zε = Contact ratio factor (see 25)
Zβ = Helix angle factor (see 26)
Ft KA Kγ Kv KHβ KHα see 15 ndash 110
d1 b u see 12 ndash 15
Bevel gears
( )HαHβvγA
v1v
vmtKEMH KKKKK
bud1uF
ZZZ105σ+
sdot=
where
105 is a correlation factor to reach real Hertzian stresses (when ZK = 1)
ZE KA etc see above
ZM = mid-zone factor see 233
ZK = bevel gear factor see 27
Fmt dv1 uv see 12 ndash 15
It is assumed that the heightwise crowning is chosen so as to result in the maximum contact stresses at or near the mid-point of the flanks
222 Permissible contact stress
XWRvLH
NHlimHP ZZZZZ
SZσ
σ =
where
σH lim = Endurance limit for contact stresses (see 28)
ZN = Life factor for contact stresses (see 29)
SH = Required safety factor according to the rules
ZLZvZR = Oil film influence factors (see 210)
ZW = Work hardening factor (see 211)
ZX = Size factor (see 212)
18 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
23 Zone Factors ZH ZBD and ZM
231 Zone factor ZH The zone factor ZH accounts for the influence on contact stresses of the tooth flank curvature at the pitch point and converts the tangential force at the reference cylinder to the normal force at the pitch cylinder
wtt2
wtbH sinααcos
cosαcosβ2Z =
232 Zone factors ZBD The zone factors ZBD account for the influence on contact stresses of the tooth flank curvature at the inner point of sin-gle pair contact in relation to ZH Index B refers to pinion D to wheel
For εβ ge 1 ZBD = 1
For internal gears ZD = 1
For εβ = 0 (spur gears)
( )
π
minusεminusminus
π
minusminus
α=
α2
2
2b
2a
1
2
1b
1a
wtB
z211
dd
z21
dd
tanZ
( )
π
minusεminusminus
π
minusminus
α=
α1
2
1b
1a
2
2
2b
2a
wtD
z211
dd
z21
dd
tanZ
If ZB lt 1 use ZB = 1
If ZD lt 1 use ZD = 1
For 0 lt εβ lt 1
ZBD = ZBD (for spur gears) ndash εβ (ZBD (for spur gears) ndash 1)
233 Zone factor ZM The mid-zone factor ZM accounts for the influence of the contact stress at the mid point of the flank and applies to spi-ral bevel gears
εminusminus
εminusminus
αβ=
αα btm2
2vb2
2vabtm2
1vb2val
2v1vvtbmM
pddpdd
ddtancos2Z
This factor is the product of ZH and ZM-B in ISO 10300 with the condition that the heightwise crowning is sufficient to move the peak load towards the midpoint
234 Inner contact point For cylindrical or bevel gears with very low number of teeth the inner contact point (A) may be close to the base circle In order to avoid a wear edge near A it is required to have suitable tip relief on the wheel
24 Elasticity Factor ZE The elasticity factor ZE accounts for the influence of the material properties as modulus of elasticity and Poissonrsquos ratio on the contact stresses
For steel against steel ZE = 1898
25 Contact Ratio Factor Zε The contact ratio factor Zε accounts for the influence of the transverse contact ratio εα and the overlap ratio εβ on the contact stresses
αε1Z =ε for 1εβ ge
( )α
ββ
αε ε
εε1
3ε4
Z +minusminus
= for εβ lt 1
26 Helix Angle Factor Zβ The helix angle factor Zβ accounts for the influence of helix angle (independent of its influence on Zε) on the surface du-rability
cosβZβ =
27 Bevel Gear Factor ZK The bevel gear factor accounts for the difference between the real Hertzian stresses in spiral bevel gears and the contact stresses assumed responsible for surface fatigue (pitting) ZK adjusts the contact stresses in such a way that the same per-missible stresses as for cylindrical gears may apply
The following may be used ZK = 080
28 Values of Endurance Limit σHlim and Static Strength 5H10σ 3H10σ
σHlim is the limit of contact stress that may be sustained for 5middot107 cycles without the occurrence of progressive pitting
For most materials 5middot107 cycles are considered to be the be-ginning of the endurance strength range or lower knee of the σ-N curve (See also Life Factor ZN) However for nitrided steels 2middot106 apply
For this purpose pitting is defined by
bull for not surface hardened gears pitted area ge 2 of total active flank area
bull for surface hardened gears pitted area ge 05 of total active flank area or ge 4 of one particular tooth flank area 510Hσ and 310Hσ are the contact stresses which the given
material can withstand for 105 respectively 103 cycles without subsurface yielding or flank damages as pitting spalling or case crushing when adequate case depth applies
The following listed values for σHlim 510Hσ and 310Hσ may only be used for materials subjected to a quality control as
Classification Notes- No 412 19 May 2003
DET NORSKE VERITAS
the one referred to in the rules Results of approved fatigue tests may also be used as the basis for establishing these values The defined survival probability is 99
σHlim σH105 σH10
3
Alloyed case hardened steels (surface hardness 58-63 HRC) - of specially approved high grade - of normal grade
1650 1500
2500 2400
3100 3100
Nitrided steel of approved grade gas nitrided (surface hardness 700-800 HV) 1250 13 σHlim 13 σHlim
Alloyed quenched and tempered steel bath or gas nitrided (surface hardness 500-700 HV)
1000
13 σHlim
13 σHlim
Alloyed flame or induction hardened steel (surface hardness 500-650 HV) 075 HV + 750 16 σHlim 45 HV
Alloyed quenched and tempered steel 14 HV + 350 16 σHlim 45 HV
Carbon steel 15 HV + 250 16 σHlim 16 σHlim
These values refer to forged or hot rolled steel For cast steel the values for σHlim are to be reduced by 15
29 Life Factor ZN The life factor ZN takes account of a higher permissible contact stress if only limited life (number of cycles NL) is demanded or lower permissible contact stress if very high number of cycles apply
If this is not documented by approved fatigue tests the fol-lowing method may be used
For all steels except nitrided
7L 105N sdotge ZN = 1 or
01570
L
7
N N105Z
sdot=
Ie ZN = 092 for 1010 cycles
The ZN = 1 from 5middot107 on may only be used when the mate-rial cleanliness is of approved high grade (see Rules Pt4 Ch2) and the lubrication is optimised by a specially approved filtering process
105 lt NL lt 5middot107 510N
logZ037
L
7
N N105Z
sdot=
NL = 105 WXRVLHlim
WstX10H1010NN ZZZZZσ
ZZσZZ
55
5=
103 lt NL lt 105 )Z(Zlog05
L
5
10NN
5N103N10
5N10ZZ
==
3L 10N le
WXRVLlimH
Wst10X10H10NN ZZZZZ
ZZZZ
33
3σ
σ==
(but not less than ZN105)
For nitrided steels
6L 102N sdotge ZN = 1 or
00980
L
6
N N102Z
sdot=
Ie ZN = 092 for 1010 cycles
The ZN = 1 from 2middot106 on may only be used when the mate-rial cleanliness is of approved high grade (see Rules Pt4 Ch2) and the lubrication is optimised by a specially approved filtering process
105 lt NL lt 2middot106 510N
Zlog76860
L
6
N N102Z
sdot=
5L 10N le
XWRVL
10XWst10NN ZZZZZ
ZZ13ZZ
55 ==
Note that when no index indicating number of cycles is used the factors are valid for 5middot107 (respectively 2middot106 for nitrid-ing) cycles
210 Influence Factors on Lubrication Film ZL ZV and ZR The lubricant factor ZL accounts for the influence of the type of lubricant and its viscosity the speed factor ZV ac-counts for the influence of the pitch line velocity and the roughness factor ZR accounts for influence of the surface roughness on the surface endurance capacity
The following methods may be applied in connection with the endurance limit
20 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
Surface hardened steels Not surface hardened steels
ZL ( )24013421
360910ν+
+
( )24013421
680830ν+
+
ZV ( )v3280
140930+
+ ( )v3280300850
++
ZR
080
ZrelR3
150
ZrelR3
where
ν40 = Kinematic oil viscosity at 40ordmC (mm2s) For case hardened steels the influence of a high bulk temperature (see 4 Scuffing) should be con-sidered Eg bulk temperatures in excess of 120ordmC for long periods may cause reduced flank surface endurance limits
For values of ν40 gt 500 use ν40 = 500
RZrel = The mean roughness between pinion and wheel (after running in) relative to an equivalent radius of curvature at the pitch point ρc = 10mm
RZrel
= ( ) 31
cZ2Z1 ρ
10RR50
+
RZ = Mean peak to valley roughness (microm) (DIN defini-tion) (roughly RZ = 6 Ra)
For 5
L 10N le ZL ZV ZR = 10
211 Work Hardening Factor ZW The work hardening factor ZW accounts for the increase of surface durability of a soft steel gear when meshing the soft steel gear with a surface hardened or substantially harder gear with a smooth surface
The following approximation may be used for the endurance limit
Surface hardened steel against not surface hardened steel 150
ZeqW R
31700
130HB21Z
minus
minus=
where HB = the Brinell hardness of the soft member For HB gt 470 use HB = 470 For HB lt 130 use HB = 130
RZeq = equivalent roughness
033
c40
066
ZS
ZHZHZEQ ρvν
15000RRRR
=
If RZeq gt 16 then use RZeq = 16
If RZeq lt 15 then use RZeq = 15
where
RZH = surface roughness of the hard member before run in
RZS = surface roughness of the soft member before run in
ν40 = see 210
If values of ZW lt 1 are evaluated ZW = 1 should be used for flank endurance However the low value for ZW may indi-cate a potential wear problem
Through hardened pinion against softer wheel
( )
minussdotminus+= 000829
HBHB0008981u1Z
2
1W
For 21HBHB
2
1 le use ZW = 1
For 71HBHB
2
1 gt use 17HBHB
2
1 =
For u gt 20 use u = 20
For static strength (lt 105 cycles)
Surface hardened against not surface hardened
ZWst = 105
Through hardened pinion against softer wheel
ZWst = 1
212 Size Factor ZX The size factor accounts for statistics indicating that the stress levels at which fatigue damage occurs decrease with an increase of component size as a consequence of the influ-ence on subsurface defects combined with small stress gradi-ents and of the influence of size on material quality
ZX may be taken unity provided that subsurface fatigue for surface hardened pinions and wheels is considered eg as in the following subsection 213
213 Subsurface Fatigue This is only applicable to surface hardened pinions and wheels The main objective is to have a subsurface safety against fatigue (endurance limit) or deformation (static strength) which is at least as high as the safety SH required for the surface The following method may be used as an approximation unless otherwise documented
The high cycle fatigue (gt3106 cycles) is assumed to mainly depend on the orthogonal shear stresses Static strength (lt103 cycles) is assumed to depend mainly on equivalent
Classification Notes- No 412 21 May 2003
DET NORSKE VERITAS
stresses (von Mises) Both are influenced by residual stresses but this is only considered roughly and empirically
The subsurface working stresses at depths inside the peak of the orthogonal shear stresses respectively the equivalent stresses are only dependent on the (real) Hertzian stresses Surface related conditions as expressed by ZL ZV and ZR are assumed to have a negligible influence
The real Hertzian stresses σHR are determined as
For helical gears with εβ gt 1
σHR = σH
For helical gears with εβ lt 1 and spur gears
ε
α
ββ ε
ε+εminus
sdotσ=σZ
1
HHR
For bevel gears
KHHR Z
1σσ sdot=
The necessary hardness HV is given as a function of the net depth tz (net = after grinding or hard metal hobbing and per-pendicular to the flank)
The coordinates tz and HV are to be compared with the de-sign specification such as
bull for flame and induction hardening tHVmin HVmin bull for nitriding t400min HV = 400 bull for case hardening t550min HV = 550 t400min HV = 400
and t300min HV = 300 (the latter only if the core hardness lt 300 If the core hardness gt 400 the t400 is to be re-placed by a fictive t400 = 16 t550)
In addition the specified surface hardness is not to be less than the max necessary hardness (at tz = 05aH) This applies to all hardening methods
For high cycle fatigue (gt3 106 cycles) the following applies
sdot+
minussdotsdotsdot= o90
05azt
05azt
cosSσ04HV
H
HHHR
applicable to 05a
zt
Hge
For 05at
H
z lt the value for 05at
H
z = applies
56300ρSσ12a cHHR
Hsdotsdot
sdot=
Where aH is half the hertzian contact width multiplied by an empirical factor of 12 that takes into account the possible influence of reduced compressive residual stresses (or even tensile residual stresses) on the local fatigue strength
If any of the specified hardness depths including the surface hardness is below the curve described by HV = f (tz) the actual safety factor against subsurface fatigue is determined as follows
reduce SH stepwise in the formula for HV and aH until all specified hardness depths and surface hardness balance with the corrected curve The safety factor obtained through this method is the safety against subsurface fatigue
For static strength (lt103 cycles) the following applies
sdot+
minussdotsdotsdot= o90
07at
06a
zt
cosSσ019HV
Hst
z
HstHHR
applicable to 06at
Hst
z ge
56300ρSσa cHHR
Hstsdotsdot
=
In the case of insufficient specified hardness depths the same procedure for determination of the actual safety factor as above applies
For limited life fatigue (103 lt cycles lt 3106 )
For this purpose it is necessary to extend the correction of safety factors to include also higher values than required Ie in the case of more than sufficient hardness and depths the safety factor in the formulae for both high cycle fatigue and static strength are to be increased until necessary and speci-fied values balance
The actual safety factor for a given number of cycles N be-tween 103 and 3106 is found by linear interpolation in a dou-ble logarithmic diagram
minussdotminus
= sdot logN3477
logSlogSlogS
310H6103HHN
36 10H103H Slog86281Slog86280 sdot+sdot sdot
22 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
3 Calculation of Tooth Strength
31 Scope and General Remarks Part 3 include the calculation of tooth root strength as limited by tooth root cracking (surface or subsurface initiated) and yielding
For rim thickness sR ge 35middotmn the strength is calculated by means of 32 ndash 313 For cylindrical gears the calculation is based on the assumption that the highest tooth root tensile stress arises by application of the force at the outer point of single tooth pair contact of the virtual spur gears The method has however a few limitations that are mentioned in 36
For bevel gears the calculation is based on force application at the tooth tip of the virtual cylindrical gear Subsequently the stress is converted to load application at the mid point of the flank due to the heightwise crowning
Bevel gears may also be calculated with the program BECAL In that case KA and Kv are to be included in the applied tooth force but not KFβ and KFα
In case of a thin annulus or a thin gear rim etc radial crack-ing can occur rather than tangential cracking (from root fillet to root fillet) Cracking can also start from the compression fillet rather than the tension fillet For rim thickness sR lt 35middotmn a special calculation procedure is given in 315 and 316 and a simplified procedure in 314
A tooth breakage is often the end of the life of a gear trans-mission Therefore a high safety SF against breakage is re-quired
It should be noted that this part 3 does not cover fractures caused by
bull oil holes in the tooth root space bull wear steps on the flank bull flank surface distress such as pits spalls or grey staining
Especially the latter is known to cause oblique fractures starting from the active flank predominately in spiral bevel gears but also sometimes in cylindrical gears
Specific calculation methods for these purposes are not given here but are under consideration for future revisions Thus depending on experience with similar gear designs limita-tions other than those outlined in part 3 may be applied
32 Tooth Root Stresses The local tooth root stress is defined as the max principal stress in the tooth root caused by application of the tooth force Ie the stress ratio R = 0 Other stress ratios such as for eg idler gears (R asymp -12) shrunk on gear rims (R gt 0) etc are considered by correcting the permissible stress level
321 Local tooth root stress The local tooth root stress for pinion and wheel may be as-sessed by strain gauge measurements or FE calculations or similar For both measurements and calculations all details are to be agreed in advance
Normally the stresses for pinion and wheel are calculated as
Cylindrical gears
FαFβvγAβSFn
tF K K K K K Y Y Y
m bFσ =
where
YF = Tooth form factor (see 33)
YS = Stress correction factor (see 34)
Yβ = Helix angle factor (see 36)
Ft KA Kγ Kv KFβ KFα see 15 ndash 110
b see 13
Bevel gears
FαFβvγASaFamn
mtF K K K K K Y Y Y
m bFσ ε=
where
YFa = Tooth form factor see 33
YSa = Stress correction factor see 34
Yε = Contact ratio factor see 35
Fmt KA etc see 15 ndash 110
b see 13
322 Permissible tooth root stress The permissible local tooth root stress for pinion respectively wheel for a given number of cycles N is
CXRrelTrelTF
NMFEFP YYYY
SYY
δσ
=σ
Note that all these factors YM etc are applicable to 3middot106 cycles when used in this formula for σFP The influence of other number of cycles on these factors is covered by the calculation of YN
where
σFE = Local tooth root bending endurance limit of reference test gear (see 37)
YM = Mean stress influence factor which accounts for other loads than constant load direction eg idler gears temporary change of load di-rection pre-stress due to shrinkage etc (see 38)
YN = Life factor for tooth root stresses related to reference test gear dimensions (see 39)
SF = Required safety factor according to the rules
YδrelT = Relative notch sensitivity factor of the gear to be determined related to the reference test gear (see 310)
YRrelT = Relative (root fillet) surface condition factor of the gear to be determined related to the reference test gear (see 311)
Classification Notes- No 412 23 May 2003
DET NORSKE VERITAS
YX = Size factor (see 312)
YC = Case depth factor (see 313)
33 Tooth Form Factors YF YFa The tooth form factors YF and YFa take into account the in-fluence of the tooth form on the nominal bending stress
YF applies to load application at the outer point of single tooth pair contact of the virtual spur gear pair and is used for cylindrical gears
YFa applies to load application at the tooth tip and is used for bevel gears
Both YF and YFa are based on the distance between the con-tact points of the 30˚-tangents at the root fillet of the tooth profile for external gears respectively 60˚ tangents for inter-nal gears
Figure 31 External tooth in normal section
Figure 32 Internal tooth in normal section
Definitions
n
2
n
Fn
enFn
Fe
F
α cosms
α cosmh6
Y
=
n
2
n
Fn
anFn
Fa
Fa
α cosms
α cosmh6
Y
=
In the case of helical gears YF and YFa are determined in the normal section ie for a virtual number of teeth
YFa differs from YF by the bending moment arm hFa and αFan and can be determined by the same procedure as YF with exception of hFe and αFan For hFa and αFan all indices e will change to a (tip)
The following formulae apply to cylindrical gears but may also be used for bevel gears when replacing
mn with mnm
zn with zvn
αt with αvt
β with βm
with undercut without undercut
Fig 33 Dimensions and basic rack profile of the teeth (finished profile)
Tool and basic rack data such as hfP ρfp and spr etc are re-ferred to mn ie dimensionless
331 Determination of parameters
( )n
n
prnfPnfP m
α cossαsin 1ρ
αtan h4πE
minusminusminusminus=
For external gears fPfP ρρ =
For internal gears ( )0z
195fPfP0
fPfP 10363156ρhxρρ
sdotminus+
+=
where
z0 = number of teeth of pinion cutter
x0 = addendum modification coefficient of pinion cutter
hfP = addendum of pinion cutter
ρfP = tip radius of pinion cutter
xhρG fpfp +minus=
24 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
τmE
2π
z2H
nnminus
minus=
with
3πτ = for external gears
6πτ = for internal gears
Htan zG2
nminusϑ=ϑ
(to be solved iteratively suitable start value6π
=ϑ for exter-
nal gears and 3π for internal gears)
a) Tooth root chord sFn For external gears
minus
ϑ+
ϑminus= ρ
cosG3
3πsinz
ms
fpnn
Fn
For bevel gears with a tooth thickness modification
xsm affects mainly sFn but also hFe and αFen The total influence of xsm on YFa Ysa can be approximated by only adding 2 xsm to sFn mn
For internal gears
minus
ϑ+
ϑminus= ρ
cosG
6πsinz
ms
fPnn
Fn
b) Root fillet radius ρF at 30ordm tangent
( )G2coszcosG2ρ
mρ
2n
2
fpn
F
minusϑϑ+=
c) Determination of bending moment arm hF dn = zn mn
b2α
αn βcosε
ε =
dan = dn + 2 ha
pbn = π mn cos αn
dbn = dn cos αn
( )4
d1εpzz
2dd
zz2d
2bn
2
αnbn
2bn
2an
en +
minusminus
minus=
en
bnen d
dcos arcα =
ennnn
e α invα invα x tan 22π
z1
minus+
+=γ
αFen = αen ndash γe
For external gears
( )
minus=
n
enFenee
n
Fe
mdαtan sin γ γcos
21
mh
]ρcos
G3πcosz fpn +
ϑminus
ϑminusminus
For internal gears
( ) minus
sdotsdotminus=
n
enFenee
n
Fe
mdα tanγ sinγ cos
21
mh
minus
ϑminus
ϑminussdot ρ
cosG3
6πcosz fPn
332 Gearing with εαn gt 2 For deep tooth form gearing ( )25ε2 αn lele produced with a verified grade of accuracy of 4 or better and with applied profile modification to obtain a trapezoidal load distribution along the path of contact the YF may be corrected by the factor YDT as
250ε205for 0666ε2366Y αnαnDT leleminus=
205εfor 10Y αnDT lt=
34 Stress Correction Factors YS YSa The stress correction factors YS and YSa take into account the conversion of the nominal bending stress to the local tooth root stress Thereby YS and YSa cover the stress increasing effect of the notch (fillet) and the fact that not only bending stresses arise at the root A part of the local stress is inde-pendent of the bending moment arm This part increases the more the decisive point of load application approaches the critical tooth root section
Therefore in addition to its dependence on the notch radius the stress correction is also dependent on the position of the load application ie the size of the bending moment arm
YS applies to the load application at the outer point of single tooth pair contact YSa to the load application at tooth tip
YS can be determined as follows
( )
+
+=L23121
1
sS qL01312Y
where Fe
Fn
hsL = and
F
Fns ρ2
sq = (see 33)
YSa can be calculated by replacing hFe with hFa in the above formulae
Note a) Range of validity 1 lt qs lt 8
In case of sharper root radii (ie produced with tools having too sharp tip radii) YS resp YSa must be specially considered
b) In case of grinding notches (due to insufficient protuberance of the hob) YS resp YSa can rise considerably and must be multiplied with
Classification Notes- No 412 25 May 2003
DET NORSKE VERITAS
g
g
ρt
0613
13
minus
where
tg = depth of the grinding notch
ρg = radius of the grinding notch
c) The formulae for YS resp YSa are only valid for αn = 20˚ However the same formulae can be used as a safe approximation for other pressure angles
35 Contact Ratio Factor Yε The contact ratio factor Yε covers the conversion from load application at the tooth tip to the load application at the mid point of the flank (heightwise) for bevel gears
The following may be used
Yε = 0625
36 Helix Angle Factor Yβ The helix angle factor Yβ takes into account the difference between the helical gear and the virtual spur gear in the nor-mal section on which the calculation is based in the first step In this way it is accounted for that the conditions for tooth root stresses are more favourable because the lines of contact are sloping over the flank
The following may be used (β input in degrees)
Yβ = 1 ndash εβ β120
When εβ gt 1 use εβ = 1 and when β gt 30deg use β = 30deg in the formula
However the above equation for Yβ may only be used for gears with β gt 25deg if adequate tip relief is applied to both pinion and wheel (adequate = at least 05 middot Ceff see 432)
37 Values of Endurance Limit σFE σFE is the local tooth root stress (max principal) which the material can endure permanently with 99 survival prob-ability 3106 load cycles is regarded as the beginning of the endurance limit or the lower knee of the σ ndash N curve σFE is defined as the unidirectional pulsating stress with a minimum stress of zero (disregarding residual stresses due to heat treatment) Other stress conditions such as alternating or pre-stressed etc are covered by the conversion factor YM
σFE can be found by pulsating tests or gear running tests for any material in any condition If the approval of the gear is to be based on the results of such tests all details on the testing conditions have to be approved by the Society Fur-ther the tests may have to be made under the Societys su-pervision
If no fatigue tests are available the following listed values for σFE may be used for materials subjected to a quality con-trol as the one referred to in the rules
σFE
Alloyed case hardened steels 1) (fillet surface hardness 58 ndash 63 HRC)
bull of specially approved high grade
1050
bull of normal grade
minus CrNiMo steels with approved process
1000
minus CrNi and CrNiMo steels generally 920 minus MnCr steels generally 850
Nitriding steel of approved grade quenched tempered and gas nitrided (surface hardness 700 ndash 800 HV)
840
Alloyed quenched and tempered steel bath or gas nitrided (surface hardness 500 ndash 700 HV)
720
Alloyed quenched and tempered steel flame or induction hardened 2) (incl entire root fillet) (fillet surface hardness 500 ndash 650 HV)
07 HV + 300
Alloyed quenched and tempered steel flame or induction hardened (excl entire root fillet) (σB = uts of base material)
025 σB + 125
Alloyed quenched and tempered steel 04 σB + 200
Carbon steel 025 σB + 250
Note All numbers given above are valid for separate forgings and for blanks cut from bars forged according to a qualified procedure see Pt 4 Ch 2 Sec 3 For rolled steel the values are to be reduced with 10 For blanks cut from forged bars that are not qualified as mentioned above the values are to be reduced with 20 For cast steel reduce with 40
1) These values are valid for a root radius
bull being unground If however any grinding is made in the root fillet area in such a way that the residual stresses may be affected σFE is to be reduced by 20 (If the grinding also leaves a notch see 34)
bull with fillet surface hardness 58 ndash 63 HRC In case of lower surface hardness than 58 HRC σFE is to be reduced with 20(58 ndash HRC) where HRC is the detected hardness (This may lead to a permissible tooth root stress that varies along the facewidth If so the actual tooth root stresses may also be considered along facewidth)
bull not being shot peened In case of approved shot peening σFE may be increased by 200 for gears where σFE is reduced by 20 due to root grinding Otherwise σFE may be increased by 100 for
6nm le and 100 ndash 5 (mn - 6) for mn gt 6 However the possible adverse influence on the flanks regarding grey staining should be considered and if necessary the flanks should be masked
2) The fillet is not to be ground after surface hardening Regarding possible root grinding see 1)
26 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
38 Mean stress influence Factor YM The mean stress influence factor YM takes into account the influence of other working stress conditions than pure pulsa-tions (R = 0) such as eg load reversals idler gears planets and shrink-fitted gears
YM (YMst) are defined as the ratio between the endurance (or static) strength with a stress ratio R ne 0 and the endurance (or static) strength with R = 0
YM and YMst apply only to a calculation method that assesses the positive (tensile) stresses and is therefore suitable for comparison between the calculated (positive) working stress σF and the permissible stress σFP calculated with YM or YMst
For thin rings (annulus) in epicyclic gears where the com-pression fillet may be decisive special considerations apply see 316
The following method may be used within a stress ratio ndash12 lt R lt 05
381 For idlers planets and PTO with ice class
M1M1R1
1Yor Y MstM
+minus
minus=
where
R = stress ratio = min stress divided by max stress
For designs with the same force applied on both forward- and back-flank R may be assumed to ndash 12
For designs with considerably different forces on forward- and back-flank such as eg a marine propulsion wheel with a power take off pinion R may be assessed as
branchmain theoffacewidth unit per forcepto offacewidt unit per force21minus
For a power take off (PTO) with ice class see 161 c
M considers the mean stress influence on the endurance (or static) strength amplitudes
M is defined as the reduction of the endurance strength am-plitude for a certain increase of the mean stress divided by that increase of the mean stress
Following M values may be used
Endurance limit
Static strength
Case hardened 08 ndash 015 Ys 1) 07
If shot peened 04 06
Nitrided 03 03
Induction or flame hardened
04 06
Not surface hardened steel 03 05
Cast steels 04 06 1)For bevel gears use Ys=2 for determination of M
The listed M values for the endurance limit are independent of the fillet shape (Ys) except for case hardening In princi-ple there is a dependency but wide variations usually only occur for case hardening eg smooth semicircular fillets versus grinding notches
382 For gears with periodical change of rotational direction For case hardened gears with full load applied periodically in both directions such as side thrusters the same formula for YM as for idlers (with R = ndash 12) may be used together with the M values for endurance limit This simplified approach is valid when the number of changes of direction exceeds 100 and the total number of load cycles exceeds 3middot106
For gears of other materials YM will normally be higher than for a pure idler provided the number of changes of direction is below 3middot106 A linear interpolation in a diagram with loga-rithmic number of changes of direction may be used ie from YM = 09 with one change to YM (idler) for 3middot106 changes This is applicable to YM for endurance limit For static strength use YM as for idlers
For gears with occasional full load in reversed direction such as the main wheel in a reversing gear box YM = 09 may be used
383 For gears with shrinkage stresses and unidirectional load For endurance strength
FE
fitM σ
σM1
M21Y+
minus=
σFE is the endurance limit for R = 0
For static strength YMst = 1 and σfit accounted for in 39b
σfit is the shrinkage stress in the fillet (30˚ tangent) and may be found by multiplying the nominal tangential (hoop) stress with a stress concentration factor
n
Ffit m
ρ215scf minus=
384 For shrink-fitted idlers and planets When combined conditions apply such as idlers with shrink-age stresses the design factor for endurance strength is
( ) ( ) FE
fitM σ
σR1M1
M2
M1M1R1
1Yminussdot+
minus
+minus
minus=
Symbols as above but note that the stress ratio R in this par-ticular connection should disregard the influence of σfit ie R normally equal ndash 12
For static strength
M1M1R1
1YMst
+minus
minus=
The effect of σfit is accounted for in 39 b
Classification Notes- No 412 27 May 2003
DET NORSKE VERITAS
385 Additional requirements for peak loads The total stress range (σmax ndash σmin) in a tooth root fillet is not to exceed
F
y
Sσ225
for not surface hardened fillets
FSHV5 for surface hardened fillets
39 Life Factor YN The life factor YN takes into account that in the case of limited life (number of cycles) a higher tooth root stress can be permitted and that lower stresses may apply for very high number of cycles
Decisive for the strength at limited life is the σ ndash N ndash curve of the respective material for given hardening module fillet radius roughness in the tooth root etc Ie the factors YδrelT YRelT YX and YM have an influence on YN
If no σ ndash N ndash curve for the actual material and hardening etc is available the following method may be used
Determination of the σ ndash N ndash curve
a) Calculate the permissible stress σFP for the beginning of the endurance limit (3middot106 cycles) including the influ-ence of all relevant factors as SF YδrelT YRelT YX YM and YC ie σFP = σFE middotYM middotYδrelT middotYRelT middotYX middotYC SF
b) Calculate the permissible laquostaticraquo stress (le 103 load cy-cles) including the influence of all relevant factors as SFst YδrelTst YMst and YCst ( )fitσCstYδrelTstYMstYFstσ
FstS1
FPstσ minussdotsdotsdot=
where σFst is the local tooth root stress which the material can resist without cracking (surface hardened materials) or unacceptable deformation (not surface hardened materials) with 99 survival probability
σFst
Alloyed case hardened steel 1) 2300
Nitriding steel quenched tempered and gas nitrided (surface hardness 700 ndash 800 HV)
1250
Alloyed quenched and tempered steel bath or gas nitrided (surface hardness 500 ndash 700 HV)
1050
Alloyed quenched and tempered steel flame or induction hardened (fillet surface hardness 500 ndash 650 HV)
18 HV + 800
Steel with not surface hardened fillets the smaller value of 2)
18 σB or 225 σy
1) This is valid for a fillet surface hardness of 58 ndash 63 HRC In case of lower fillet surface hardness than 58 HRC σFst is to be reduced with 30(58 ndash HRC) where HRC is the actual hardness Shot peening or grinding notches are not considered to have any significant influence on σFst 2) Actual stresses exceeding the yield point (σy or σ02) will alter the residual stresses locally in the ldquotensionrdquo fillet re-spectively ldquocompressionrdquo fillet This is only to be utilised for gears that are not later loaded with a high number of cycles at lower loads that could cause fatigue in the ldquocompressionrdquo fillet
c) Calculate YN as
NL gt 3middot106
YN = 1 or 001
L
6
N N103Y
sdot= ie Yn = 092 for 1010
The YN = 1 from 3middot106 on may only be used when special material cleanness applies see rules Pt4 Ch2
103ltNLlt3middot106exp
L
6
N N103Y
sdot=
cycles6103forσcycles310forσlog02876exp
FP
FPst
sdot=
NL lt 103 cycles6103forσcycles310forσ
NYFP
FPst
sdot=
or simply use σFPst as mentioned in b) directly
Guidance on number of load cycles NL for various applica-tions
bull For propulsion purpose normally NL = 1010 at full load (yachts etc may have lower values)
bull For auxiliary gears driving generators that normally operate with 70-90 of rated power NL = 108 with rated power may be applied
28 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
310 Relative Notch Sensitivity Factor YδrelT The dynamic (respectively static) relative notch sensitivity factor YδrelT (YδrelTst) indicate to which extent the theoreti-cally concentrated stress lies above the endurance limits (re-spectively static strengths) in the case of fatigue (respectively overload) breakage
YδrelT is a function of the material and the relative stress gra-dient It differs for static strength and endurance limit
The following method may be used
For endurance limit
for not surface hardened fillets
( )
024
s024
relT σ103133q21σ1012201351
Ysdotsdotminus
+sdotsdotminus+= minus
minus
δ
for all surface hardened fillets except nitrided
( )
106q21002451
Y srelT
++=δ
for nitrided fillets
( )
1347q2101421
Y srelT
++=δ
For static strength
for not surface hardened fillets1)
( )( )( )025
02
02502s
relTst σ3000821σ300 1Y0821Y
+
minus+=δ
for surface hardened fillets except nitrided
YδrelTst = 044 YS + 012
for nitrided fillets
YδrelTst = 06 + 02 YS 1) These values are only valid if the local stresses do not
exceed the yield point and thereby alter the residual stress level See also 39b footnote 2
311 Relative Surface Condition Factor YRrelT The relative surface condition factor YRrelT takes into ac-count the dependence of the tooth root strength on the sur-face condition in the tooth root fillet mainly the dependence on the peak to valley surface roughness
YRrelT differs for endurance limit and static strength
The following method may be used
For endurance limit
YRrelT = 1675 ndash 053 (Ry + 1)01 for surface hardened steels and alloyed quenched and tempered steels except nitrided
YRrelT = 53 ndash 42 (Ry + 1)001 for carbon steels
YδrelT = 43 ndash 326 (Ry + 1)0005
for nitrided steels
For static strength
YRrelTst = 1 for all Ry and all materials
For a fillet without any longitudinal machining trace Ry asymp Rz
312 Size Factor YX The size factor YX takes into account the decrease of the strength with increasing size YX differs for endurance limit and static strength
The following may be used
For endurance limit
YX = 1 for mn le 5 generally
YX = 103 ndash 0006 mn for 5 lt mn lt 30
YX = 085 for mn ge 30
for not surface hard-ened steels
YX = 105 ndash 001 mn for 5 lt mn ge 25
YX = 08 for mn ge 25
for surface hardened steels
For static strength
YXst = 1 for all mn and all materials
313 Case Depth Factor YC The case depth factor YC takes into account the influence of hardening depth on tooth root strength
YC applies only to surface hardened tooth roots and is dif-ferent for endurance limit and static strength
In case of insufficient hardening depth fatigue cracks can develop in the transition zone between the hardened layer and the core For static strength yielding shall not occur in the transition zone as this would alter the surface residual stresses and therewith also the fatigue strength
The major parameters are case depth stress gradient permis-sible surface respectively subsurface stresses and subsurface residual stresses
The following simplified method for YC may be used
YC consists of a ratio between permissible subsurface stress (incl influence of expected residual stresses) and permissible surface stress This ratio is multiplied with a bracket con-taining the influence of case depth and stress gradient (The empirical numbers in the bracket are based on a high number of teeth and are somewhat on the safe side for low number of teeth)
YC and YCst may be calculated as given below but calculated values above 10 are to be put equal 10
For endurance limit
+
+=nFFE
C m02ρt31
σconstY
Classification Notes- No 412 29 May 2003
DET NORSKE VERITAS
For static strength
+
+=nFFst
Cst m02ρt31
σconstY
where const and t are connected as
Hardening process
t = endurance limit const =
static strength const =
t550 640 1900
t400 500 1200 Case hard-ening
t300 380 800
Nitriding t400 500 1200
Induction- or flame hardening
tHVmin 11 HVmin 25 HVmin
For symbols see 213
In addition to these requirements to minimum case depths for endurance limit some upper limitations apply to case hard-ened gears
The max depth to 550 HV should not exceed
1) 13 of the top land thickness san unless adequate tip relief is applied (see 110)
2) 025 mn If this is exceeded the following applies addi-tionally in connection with endurance limit
minusminus= 025
mt
1Yn
max550C
314 Thin rim factor YB Where the rim thickness is not sufficient to provide full sup-port for the tooth root the location of a bending failure may be through the gear rim rather than from root fillet to root fillet
YB is not a factor used to convert calculated root stresses at the 30deg tangent to actual stresses in a thin rim tension fillet Actually the compression fillet can be more susceptible to fatigue
YB is a simplified empirical factor used to de-rate thin rim gears (external as well as internal) when no detailed calcula-tion of stresses in both tension and compression fillets are available
Figure 34 Examples on thin rims
YB is applicable in the range 175 lt sRmn lt 35
YB = 115 middot ln (8324 middot mnsR)
(for sRmn ge 35 YB = 1)
(for sRmn le 175 use 315)
σF as calculated in 321 is to be multiplied with YB when sRmn lt 35 Thus YB is used for both high and low cycle fatigue
Note This method is considered to be on the safe side for external gear rims However for internal gear rims without any flange or web stiffeners the method may not be on the safe side and it is advised to check with the method in 315316
315 Stresses in Thin Rims For rim thickness sR lt 35 mn the safety against rim cracking has to be checked
The following method may be used
3151 General The stresses in the standardised 30ordm tangent section tension side are slightly reduced due to decreasing stress correction factor with decreasing relative rim thickness sRmn On the other hand during the complete stress cycle of that fillet a certain amount of compression stresses are also introduced The complete stress range remains approximately constant Therefore the standardised calculation of stresses at the 30ordm tangent may be retained for thin rims as one of the necessary criteria
The maximum stress range for thin rims usually occurs at the 60ordm ndash 80ordm tangents both for laquotensionraquo and laquocompressionraquo side The following method assumes the 75ordm tangent to be the decisive Therefore in addition to the am criterion ap-plied at the 30ordm tangent it is necessary to evaluate the max and min stresses at the 75ordm tangent for both laquotensionraquo (loaded flank) fillet and laquocompressionraquo (back-flank) fillet For this purpose the whole stress cycle of each fillet should be considered but usually the following simplification is justified
Figure 35 Nomenclature of fillets
Index laquoTraquo means laquotensileraquo fillet laquoCraquo means laquocompressionraquo fillet
σFTmin and σFCmax are determined on basis of the nominal rim stresses times the stress concentration factor Y75
σFCmin and σFTmax are determined on basis of superposition of nominal rim stresses times Y75 plus the tooth bending stresses at 75ordm tangent
30 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
3152 Stress concentration factors at the 75ordm tangents The nominal rim stress consists of bending stresses due to local bending moments tangential stresses due to the tan-gential force Ft and radial shear stresses due to Fr
The major influence is given by the bending stresses The influence of the tangential stresses is minor and even though its stress concentration factor is slightly higher than for bending it is considered to be safe enough when the sum of these nominal stresses are combined with the stress concen-tration factor for bending The influence of the radial shear stress is neglected
The stress concentration factor relating nominal rim stresses to local fillet stresses at the 75ordm tangent may be calculated as
75n
R75
n
R
75
ρmsρ185
ms3
Y+
sdot=
where ρ75 is the root radius at the 75ordm tangent ref to mn Usually ρ75 is closed to the tool radius ρao and
ρ75 = ρao
is a safe approximation compensating for the am simplifi-cations to the laquounsaferaquo side
The tooth root stresses of the loaded tooth are decreasing with decreasing relative rim thickness approximately with the empirical factor
15
R
ncorr s
m311Y
minus=
3153 Nominal rim stresses The bending moment applied to the rim consists of a part of the tooth tilt moment Ft (hF + 05 sR) and the bending caused by the radial force Fr
The sectional modulus (first moment of area) which is used for determination of the nominal bending stresses is not nec-essarily the same for the 2 am bending moments If flanges webs etc outside the toothed section contribute to stiffening the rim against various deflections the influence of these stiffeners should be considered Eg an end flange will have an almost negligible influence on the effective sectional modulus for the stresses due to tooth tilt as the deflection caused by the tilt moment is rather small and would not much involve the flange On the other hand the radial forces as for instance from the meshes in an annulus would cause considerable radial deflections that the flange might restrict to a substantial amount When considering the stiff-ening of such flanges or webs on basis of simplified models it is advised to use an effective rim thickness sR = sR + 02 mn for the first moment of area of the rim (toothed part) cross section
For a high number of rim teeth it may be assumed that the rim bending moments in the fillets adjacent to the loaded tooth are of the same magnitude as right under the applied
force This assumption is reasonable for an annulus but rather much on the safe side for a hollow pinion
The influence of Ft on the nominal tangential stress is simpli-fied by half of it for compressive stresses (σ1) and the other half for tensile stresses (σ2) Applying these assumptions the nominal rim stresses adjacent to the loaded tooth are
( ) ( )A2
FW
f RFW
s05hF05 t
R
r
T
RFt1 minus
ϑminus
+minus=σ
( ) ( )A2
FW
f RFW
s05hF05 t
R
r
T
RFt2 +
ϑminus
+=σ
where σ1 σ2 see Fig 35
A = minimum area of cross section (usually bR sR)
WT = the sectional modulus of rim with respect to tooth tilt moment (usually 6sb 2
rR )
WR = the sectional modulus of the rim including the influence of stiffeners as flanges webs etc (WR ge WT)
bR = the width of the rim
R = the radius of the neutral axis in the rim ie from wheel centre to midpoint of rim
( )ϑf = a function for bending moment distribution around the rim For a rim (pinion) with one mesh only the ( )ϑf at the position of load application is 024 For an annulus with 3 or more meshes ( )ϑf at the position of each load application is approx 3 planets ( )ϑf = 019 4 planets ( )ϑf = 014 5 planets ( )ϑf = 011 6 planets ( )ϑf = 009
It must be checked if the max (tensile) stress in the compres-sion fillet really occurs when the fillet is adjacent to the loaded tooth In principle the stress variation through a complete rotation should be considered and the max value used The max value is usually never less than 0 For an annulus eg the tilt moment is zero in the mid position be-tween the planet meshes whilst the bending moment due to the radial forces is half of that at the mesh but with opposite sign
If these formulae are applied to idler gears as eg planets the influence of nominal tangential stresses must be cor-rected by deleting Ft(2 A) for σ1 and using FtA for σ2 Further the influence of Fr on the nominal bending stresses is usually negligible due to the planet bearing support
3154 Root fillet stresses Determination of min and max stresses in the laquotensionraquo fillet
Classification Notes- No 412 31 May 2003
DET NORSKE VERITAS
Minimum stress
751FTmin YσK σ =
Maximum stress
corrF752 Yσ03Yσ KσFTmax +=
where
03 is an empirical factor relating the tension stresses (σF) at the 30ordm tangent to the part of the tension stresses at the 75ordm tangent which add to the rim related stresses (03 also takes into account that full superposition of nominal stresses times stress concentration factors from both laquosides of the corner filletraquo would result in too high stresses)
αβγ sdotsdotsdotsdot= FFvA KKKKKK
Determination of min and max stresses in the laquocompres-sionraquo fillet
Minimum stress
corrF751FCmin Yσ036Yσ Kσ minus=
Maximum stress
752FCmax Yσ Kσ =
where
036 is an empirical factor relating the tension stresses (σF) at the 30ordm tangent to the part of the compression stresses at the 75ordm tangent which add to the rim related stresses
For gears with reversed loads as idler gears and planets there is no distinct laquotensionraquo or laquocompressionraquo fillet The mini-mum stress σFmin is the minimum of σFTmin and σFCmin (usually the latter is decisive) The maximum stress σFTmax is the maximum of σFTmax and σFCmax (usually the former is deci-sive)
316 Permissible Stresses in Thin Rims
3161 General The safety against fatigue fracture respectively overload fracture is to be at least at the same level as for solid gears The laquoordinaryraquo criteria at the 30ordm tangent apply as given in 31 through 313
Additionally the following criteria at the 75ordm tangent may apply
3162 For gt3middot106 cycles The permissible stresses for the laquotensionraquo fillets and for the laquocompressionraquo fillets are determined by means of a relevant fatigue diagram
If the actual tooth root stress (tensile or compressive) ex-ceeds the yield strength to the material the induced residual stresses are to be taken into account
For determination of permissible stresses the following is defined
R = stress ratio ie FTmax
FTminσσ respectively
FCmax
FCmin
σσ
∆σ = stress range ie σFTmax ndash σFTmin resp σFCmax ndash σFCmin
(For idler gears and planets Fmax
FminσσR =
and ∆σ = σFmax minus σFmin)
The permissible stress range ∆σρ for the laquotensionraquo respec-tively laquocompressionraquo fillets can be calculated as
For R gt minus1 FPp σ
R1R1031
13∆σ
minus+
+=
For minus infin lt R lt minus1 FPp σ
R1R10151
13∆σ
minus+
+=
where
σFP = see 32 determined for unidirectional stresses (YM = 1)
If the yield strength σy is exceeded in either tension or com-pression residual stresses are induced This may be consid-ered by correcting the stress ratio R for the respective fillets (tension or compression)
Eg if yFCmin σσ gt (ie exceeded in compression) the
difference yFCmin σσ∆ minus= affects the stress ratio as
∆σ
σ∆σ∆σR
FCmax
y
FCmax
FCmin
+
minus=
++
=
Similarly the stress ratio in the tension fillet may require cor-rection
If the yield strength is exceeded in tension σFTmax gt σy the difference ∆ = σFTmax ndash σy affects the stress ratio as
y
FTmin
FTmax
FTmin
σ∆σ
∆σ∆σR minus=
minusminus
=
Checking for possible exceeding of the yield strength has to be made with the highest torque and with the KAP if this ex-ceeds KA
3163 Forle 103 cycles The permissible stress range ∆σp is not to exceed
For R gt minus1 FPstpst σ
R1R1051
15∆σ
minus+
+=
For ndash infin lt R lt minus1 FPstpst σ
R1R10251
15∆σ
minus+
+=
For all values of R ∆σpst is limited by
not surface hardened F
y
Sσ225
32 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
surface hardened CF
YSHV5
Definition of ∆σ and R see 3162 with particular attention to possible correction of R if the yield strength is exceeded
σFPst see 32 determined for unidirectional stresses (YM = 1) and lt 103 cycles
3164 For 103 lt cycles lt 3106 ∆σp is to be determined by linear interpolation a log-log dia-gram
∆σp at NL load cycles is
6L 103p
exp
L
6
N p ∆σN103∆σ sdot
sdot=
6
3
103p
10p
∆σ
∆σlog28760exp
sdot
=
Classification Notes- No 412 33 May 2003
DET NORSKE VERITAS
4 Calculation of Scuffing Load Capacity
41 Introduction High surface temperatures due to high loads and sliding ve-locities can cause lubricant films to break down This sei-zure or welding together of areas of tooth surface is termed scuffing
In contrast to pitting and fatigue breakage which show a dis-tinct incubation period a single short overloading can lead to a scuffing failure In the ISO-TR13989 two criteria are mentioned The method used in this Classification Note is based on the principles of the flash temperature criterion
Note Bulk temperature in excess of 120ordmC for long periods may have an adverse effect on the surface durability see 211
42 General Criteria In no point along the path of contact the local contact tem-perature may exceed the permissible temperature ie
oilS
oilSB S
ϑ+ϑminusϑ
leϑ
50SB minusϑleϑ
where
Bϑ = max contact temperature along the path of contact
maxflaMBB ϑ+ϑ=ϑ
MBϑ = bulk temperature see 434
maxflaϑ = max flash temperature along the path of con-tact see 44
Sϑ = scuffing temperature see below
oilϑ = oil temperature before it reaches the mesh (max applicable for the actual load case to be used ie normally alarm temperature except for ice classes where a max expected tem-perature applies)
SS = required safety factor according to the Rules
The scuffing temperature Sϑ may be calculated as
L2
wrelT
002
40S XFZGX
ν100112085780
sdot
sdot++=ϑ
where
XwrelT = relative welding factor
XwrelT
Through hardened steel 10
Phosphated steel 125
Copper-plated steel 150
Nitrided steel 150
Less than 10 retained austenite
115
10 ndash 20 retained austenite 10
Casehardened steel
20 ndash 30 retained austenite 085
Austenitic steel 045
FZG = load stage according to FZG-Test A8390 (Note This is the load stage where scuffing oc-curs However due to scatter in test results cal-culations are to be made with one load stage less than the specification)
XL = lubricant factor
= 10 for mineral oils
= 08 for polyalfaolefins
= 07 for non-water-soluble polyglycols
= 06 for water-soluble polyglycols
= 15 for traction fluids
= 13 for phosphate esters
ν40 = kinematic oil viscosity at 40˚C (mm2s)
Application of other test methods such as the Ryder the FZG-Ryder R46574 and the FZG L-42 Test 141195110 may be specially considered
For high speed gears with very short time of contact Sϑ may be increased as follows provided use of EP-oils
Addition to the calculated scuffing temperature Sϑ
If micros 18tc ge no addition
If micros 18tc lt add ( )cwrelT t18X18 minussdotsdot
where
ct = contact time ( micros ) which is the time needed to cross the Hertzian contact width
( ) [ ]microsu1cosβn
uσ340tb1
Hc +sdotsdot
sdot=
σH as calculated in 221
For bevel gears use vu in stead of u
34 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
43 Influence Factors
431 Coefficient of friction The following coefficient of friction may apply
L025a
005oil
02
redC ΣC
Bt X R ηρv
w0048micro minus
=
where
wBt = specific tooth load (Nmm)
vΣC = sum of tangential velocities at pitch point At pitch line velocities gt 50 ms the limiting value of vΣC at v = 50 ms is to be used
ρredC = relative radius of curvature (transversal plane) at the pitch point
Cylindrical gears
HαHβvγAbt
Bt KKKKKb
Fw sdotsdotsdotsdotsdot= (see 1)
twΣC αsin v2v =
bCredC β cos ρρ =
Bevel gears
HαHβvγAtmb
Bt KKKKKb
Fw sdotsdotsdotsdotsdot= (see 1)
vtmtΣC αsin v2v =
bmvCredC β cos ρρ =
ηoil = dynamic viscosity (mPa s) at oilϑ calculated as
1000
ρ νη oiloil =
where ρ in kgm3 approximated as
( ) 70 15oil15 minusϑminusρ=ρ and νoil is kinematic viscosity at oilϑ and may be calculated by means of the following equation
( ) ( )++=+ 08νloglog08νloglog 100oil ( )
sdotminus
ϑ+minus313log373log
273log373log oil
( ) ( )( )08νloglog08νloglog 10040 +minus+
Ra = composite arithmetic mean roughness (micron) of pinion an wheel calculated as ( )2a1aa RR 05R +=This is defined as the roughness on the new flanks ie as manufactured
XL = see 42
432 Effective tip relief Ceff Ceff is the effective tip relief that amount of tip relief which compensates for the elastic deformation of the gear mesh ie
zero load at the tooth tip It is assumed (simplified) to be equal for both pinion and wheel
Cylindrical gears
for helical
γ
γ=cb
KKFC Abt
eff (see 1)
For spur
cbKKF
C Abteff
γ= (see 1)
Alternatively for spur and helical gears the non-linear ap-proach in 111 may be used (taking Ceff=δ)
Bevel gears
bcKFC Ambt
effγ
= (see 1)
where
γ
α
ε2ε44c
+sdot
=γ
433 Tip relief and extension Cylindrical gears
The extension of the tip relief is not to result in an effective contact ratio 1ltεα when the gear is unloaded (exceptions to this may only apply for applications where the gear is not to run at light loads) This means that the unrelieved part of the path of contact is to be minimum pbt It is further assumed that this unrelieved part is placed centrally on the path of contact
If root relief applies it has to be calculated as equivalent tip relief Ie pinion root relief (at mesh position A) is added to Ca2 and wheel root relief (at mesh position E) is added to Ca1 If no design tip relief or root relief on the mating gear is specified (ie if Ca1+Croot2 = 0 and visa versa) use the run-ning in amount see 112
Bevel gears
Bevel gears are to have heightwise crowning ie no distinct relievedunrelieved area This may be treated as tip and root relief For calculation purposes the root relief is combined with the tip relief of the mating member into an equivalent tip relief If no resulting tip relieves are specified the equivalent tip relives may be calculated as an approxima-tion based on the tool crowning Ca tool (per mille of tool module m0) as follows
2root1aeq1a CCC +=
1root2aeq2a CCC +=
where
Classification Notes- No 412 35 May 2003
DET NORSKE VERITAS
2
0
n0101atoolal m
)m2(mA1middotmCC
minus+minussdot=
( )
sdotminusminussdotsdotminus+= vtvb1
2vb1
21vavt1n1 αtan dddαsin 05x1mA
2
0
n020atool22a m
)mm(2A1middotmCC
minus+minussdot=
( )
sdotminusminussdotsdotminus+= vtvb2
2vb2
2va2vt2n2 αtan dddαsin 05x1mA
2
0
20atool1root1 m
A1mCC
minussdotsdot=
2
0
10atool2root2 m
A1mCC
minussdotsdot=
If the pressure angles of the cutter blades are modified (and verified) to balance the tip relieves the following may be assumed
( )eq2aeq1aeq2aeq1a C andC calculated of sum50CC sdot==
Throughout the following Ca1 and Ca2 mean the equivalent tip relieves Ca1eq and Ca2eq
434 Bulk temperature The bulk temperature may be calculated as
flaaveragempsoilMB X X 50 ϑ+ϑ=ϑ
where
Xs = lubrication factor
= 12 for spray lubrication
= 10 for dip lubrication (provided both pinion and wheel are dip lubricated and tip speed lt 5 ms)
= 10 for spray lubrication with additional cooling spray (spray on both pinion and wheel or spray on pinion and dip lubrication of wheel)
= 02 for meshes fully submerged in oil
Xmp = contact factor ( )pmp nX += 150
np = number of mesh contact on the pinion (for small gear ratii the number of wheel meshes should be used if higher)
flaaverageϑ
= average of the integrated flash temperature (see 44) along the path of contact
( )
AE
E
Ayyyfla
flaaverage
d
ΓminusΓ
ΓΓϑ=ϑint =
For high speed gears (v gt 50 ms) it may be necessary to assess the bulk temperature on the basis of a thermal rating of the entire gear transmission
44 The Flash Temperature flaϑ
441 Basic formula The local flash temperature flaϑ may be calculated as
( ) 41redy
y2ly
211
43Btcorrfla
unXwX3250
y ρ
ρminusρ
micro=ϑ Γ
(For bevel gears replace u with uv)
and is to be calculated stepwise along the path of contact from A to E
where
micro = coefficient of friction see 431
Xcorr = correction factor taking empirically into account the increased scuffing risk in the beginning of the approach path due to mesh starting without any previously built up oil film and possible shuffling away oil before meshing if insufficient tip relief
( )
3
AD
yaeffcorr 50
CC1X
ΓminusΓε
Γminus+=
α
Ca = tip relief of driven member Xcorr is only applicable in the approach path and if Ca lt Ceff otherwise 10
wBt = unit load see 431
yXΓ = load sharing factor see 443
n1 = pinion rpm
Γy ρly etc see 442
442 Geometrical relations The various radii of flank curvature (transversal plane) are
ρ1y = pinion flank radius at mesh point y
ρ2y = wheel flank radius at mesh point y
ρredy = equivalent radius of curvature at mesh point y
y2y1
y2y1redy
ρ+ρ
ρρ=ρ
Cylindrical gears
twy
1y αsin au1Γ1
ρ+
+=
twy
2y αsin au1Γu
ρ+
minus=
Note that for internal gears a and u are negative
36 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
Bevel gears
vtvv
y1y αsin a
u1Γ1
ρ+
+=
vtvv
yv2y αsin a
u1Γu
ρ+
minus=
Γ is the parameter on the path of contact and y is any point between A end E
At the respective ends Γ has the following values
Root piniontip wheel
Cylindrical gears
( )
minus
minusminus= 1
αtan 1dd
zzΓ
tw
2b2a2
1
2A
Bevel gears
( )
minus
minusminus= 1
αtan 1dd
uΓvt
2vb2va2
vA
Tip pinionroot wheel
Cylindrical gears
( )
1αtan
1ddΓ
tw
2b1a1
E minusminus
=
Bevel gears
( )
1αtan
1ddΓ
vt
2vb1va1
E minusminus
=
At inner point of single pair contact
Cylindrical gears
tw1
EB α tan zπ2ΓΓ minus=
Bevel gears
vtv1
EB α tan zπ2ΓΓ minus=
At outer point of single pair contact
Cylindrical gears
tw1AD α tan z
π2ΓΓ +=
Bevel gears
vt v1
AD αtan z π2ΓΓ +=
At pitch point ΓC = 0
The points F and G (only applicable to cylindrical gears) limiting the extension of tip relief (so as to maintain a mini-mum contact ratio of unity for unloaded gears) are at
2
BAF
Γ+Γ=Γ
2
EDG
Γ+Γ=Γ
443 Load sharing factor XΓ The load sharing factor XΓ accounts for the load sharing be-tween the various pairs of teeth in mesh along the path of contact
XΓ is to be calculated stepwise from A to E using the pa-rameter Γy
4431 Cylindrical gears with β = 0 and no tip relief
Figure 41
ByAAB
Ay for31
31X
yΓltΓleΓ
ΓminusΓ
ΓminusΓ+=Γ
DyB for1Xy
ΓltΓleΓ=Γ
EyDDE
yE for31
31X
yΓleΓltΓ
ΓminusΓ
ΓminusΓ+=Γ
4432 Cylindrical gears with β = 0 and tip relief Tip relief on the pinion reduces XΓ in the range G ndash E and increases correspondingly XΓ in the range F ndash B
Tip relief on the wheel reduces XΓ in the range A ndash F and increases correspondingly XΓ in the range D ndash G
Following remains generally
DyB for1Xy
ΓleΓleΓ=Γ
21XXGF== ΓΓ
In the following it must be distinguished between Ca lt Ceff respectively Ca gt Ceff This is shown by an example below where Ca1 lt Ceff and Ca2 gt Ceff
Classification Notes- No 412 37 May 2003
DET NORSKE VERITAS
Figure 42
Note When Ca gt Ceff the path of contact is shortened by A ndash A respectively E ndash E The single pair contact path is extended into B respectively D If this shift is significant it is necessary to consider the negative effect on surface durability (B) and bending stresses (B and D)
Range A - F
For effa2 CC le
minus=Γ
eff
2a
CC1
31X
A
FyAeff
2a
AF
Ay for C3
C61XX
AyΓleΓleΓ
sdot
+ΓminusΓ
ΓminusΓ+= ΓΓ
For effa2 CC ge
AyAfor0Xy
ΓleΓleΓ=Γ
with ( )21
CC
1CC
eff
2a
eff
2a
AFAAminus
minusΓminusΓ+Γ=Γ
FyAeff
2a
AF
Ay
eff
2a for 21
CC
CC1X
yΓleΓleΓ
minus
ΓminusΓ
ΓminusΓ+minus=Γ
Range F - B
For effa1 CC le
ByFeff
1a
FB
Fy forC3
C61
21X
yΓleΓleΓ
sdot
+ΓminusΓ
ΓminusΓ+=Γ
For effa1 CC ge
ByFeff
1a
FB
Fy for21
CC
21X
yΓleΓleΓ
minus
ΓminusΓ
ΓminusΓ+=Γ
ByB for1Xy
ΓleΓleΓ=Γ
with 1
CC2
eff
1a
FBFB
minus
ΓminusΓ+Γ=Γ
Range D ndash G
For effa2 CC le
GyDeff
2a
DG
Dy
eff
2a for C 3C
61
C 3C
32X
yΓleΓleΓ
+
ΓminusΓ
ΓminusΓminus+=Γ F
or effa2 CC ge
DyD for1Xy
ΓleΓleΓ=Γ
with ( )21
CC
1CC
eff
2a
eff
2a
DGDDminus
minusΓminusΓ+Γ=Γ
GyDeff
2a
DG
Dy
eff
2a for21
CC
CCX ΓleΓleΓ
minus
ΓminusΓ
ΓminusΓminus=Γ
Range G - E
For effa1 CC le
minus=Γ
eff
1a
CC1
31X
E
EyGeff
1a
GE
Gy forC3
C61
21X
yΓleΓleΓ
sdot
+ΓminusΓ
ΓminusΓminus=Γ
For effa1 CC gt
EyGeff
1a
GE
Gy for21
CC
21X
yΓleΓleΓ
minus
ΓminusΓ
ΓminusΓminus=Γ
EyE for0Xy
ΓleΓleΓ=Γ
with 1
CC2
eff
1a
GEGE
minus
ΓminusΓ+Γ=Γ
4433 Gears with β gt 0 buttressing Due to oblique contact lines over the flanks a certain but-tressing may occur near A and E
This applies to both cylindrical and bevel gears with tip relief lt Ceff The buttressing Xbutt is simplified as a linear function within the ranges A ndash H respectively I ndash E
Figure 43
1εwhen13X βbutt EAge=
1εwhenε031X ββbutt EAlt+=
38 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
Cylindrical gears
bIEAH βsin 02ΓΓΓΓ =minus=minus
Bevel gears
bmIEAH βsin 02ΓΓΓΓ =minus=minus
4434 Cylindrical gears with 2εγ le and no tip relief
yXΓ is obtained by multiplication of
yXΓ in 4431 with
Xbutt in 4433
4435 Gears with 2εγ gt and no tip relief
Applicable to both cylindrical and bevel gears
IyH for1Xy
ΓleΓleΓε
=α
Γ
IyHybutt and forX1Xy
ΓgtΓΓltΓε
=α
Γ
Figure 44
4436 Cylindrical gears with 2εγ le and tip relief
yXΓ is obtained by multiplication of
yXΓ in 4432 with Xbutt
in 4433
4437 Cylindrical gears with 2εγ gt and tip relief
Tip relief on the pinion (respectively wheel) reduces XГ in the range G ndash E (respectively A ndash F) and increases XГ in the range F ndash G
yXΓ is obtained by multiplication of
yXΓ as described below
with Xbutt in 4433
In the XГ example below the influence of tip relief is shown (without the influence of Xbutt) by means of
eff2aeff1a CC and CC ltgt
Tip relief gt Ceff causes new end points A respectively E of the path of contact
Figure 45
Range A ndash F
( ) ( )( ) eff
a2αa1α
AF
Ay
eff
2aeff
C 12C 13εC 1ε
CCCX
y +εε++minus
ΓminusΓ
ΓminusΓ+
εminus
=ααα
Γ
eff2aFyA CC if for leΓleΓleΓ
eff2aFyA CifCfor and geΓleΓleΓ
eff2aAyA CC iffor0Xy
gtΓleΓleΓ=Γ
( ) ( ) ( )( ) ( ) a2αa1α
αeffa2AFAA C 1ε 3C 1ε
1ε 2 CC with++minus+minus
ΓminusΓ+Γ=Γ
Range F ndash G
( )( )( ) GyF
eff
2a1a forC 1 2CC 11X
yΓleΓleΓ
+εε+minusε
+ε
=αα
α
αΓ
Range G ndash E
( ) ( )( ) eff
2a1a
GE
Gy
C 1 2C 1C 1 3XX
GFy +εεminusε++ε
ΓminusΓ
ΓminusΓminus=
αα
ααΓΓ minus
effa1EyG CC if for leΓleΓleΓ
effa1EyG CC if Γfor and geΓleΓle
eff1aEyE CC if for0Xy
geΓleΓleΓ=Γ
( ) ( ) ( )( ) ( ) 2a1a
eff1aGEEE C 1C 1 3
1 2 CC withminusε++ε+εminus
ΓminusΓminusΓ=Γαα
α
4438 Bevel gears with εγ more than approx 18 and heightwise crowning For Ca1 = Ca2 = Ceff the following applies
Figure 46
Classification Notes- No 412 39 May 2003
DET NORSKE VERITAS
( )AEM 50 Γ+Γ=Γ
( )( )2AD
3
2My651X
y ΓminusΓε
ΓminusΓminus
ε=
ααΓ
For tip relief lt Ceff yXΓ is found by linear interpolation be-
tween ( )effay CCX =Γ and ( )0CayX =Γ as in 4435
The interpolation is to be made stepwise from A to M with the influence of Ca2 and from M to E with the influence of Ca1 (For 2a1a CC ne there is a discontinuity at M)
Eg with Ca1 = 04 Ceff and Ca2 = 055 Ceff then
Range A ndash M
)CC()0C( eff2ay2ayyX 550X 450X =Γ=ΓΓ +=
Range M ndash E
)CC()0C( eff1ay1ayyX 40X 06X =Γ=ΓΓ +=
For tip relief gt Ceff the new end points A and E are found as
( )
minusΓminusΓ
ε+Γ=Γ α 1
CC
6 eff
2aADAA
( )
minusΓminusΓ
ε+Γ=Γ α 1
CC
6 eff
1aADEE
Range A ndash A
0Xy=Γ
Range A ndash M
( )( )
ΓminusΓ
ΓminusΓminus
minusε=
αΓ 2
MA
2My
eff
2a1
CC4
351Xy
Range M ndash E
( )( )
ΓminusΓ
ΓminusΓminus
minusε=
αΓ 2
ME
2My
eff
1a1
CC4
351Xy
Range E ndash E
0Xy=Γ
40 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
Appendix A Fatigue Damage Accumulation
The Palmgren-Miner cumulative damage calculation princi-ple is used The procedure may be applied as follows
A1 Stress Spectrum From the individual torque classes the torques (Ti) at the peak values of class intervals and the associated number of cycles (NLi) for both pinion and wheel are to be listed from the highest to the lowest torque
(In case of a cyclic torque variation within the torque classes it is advised to use the peak torque If the cyclic variation is such that the same teeth will repeatedly suffer the peak torque this is a must)
The stress spectra for tooth roots and flanks (σFi σHi) with all relevant factors (except KA) are to be calculated on the basis of the torque spectrum The load dependent K-factors are to be determined for each torque class
A2 σminusN-curve The stress versus load cycle curves for tooth roots and flanks (both pinion and wheel) are to be drawn on the basis of per-missible stresses (ie including the demanded minimum safety factors) as determined in 2 respectively 3 If different safety levels for high cycle fatigue and low cycle fatigue are desired this may be expressed by different demand safety factors applied at the endurance limit respectively at static strength
A3 Damage accumulation The individual damage ratio Di at ith stress level is defined as
Fi
Lii N
ND =
where
NLi = The number of applied cycles at ith stress
NFi = The number of cycles to failure at ith stress
Basically stresses σi below the permissible stress level for infinite life (if a constant ZN or YN is accepted) do not con-tribute to the damage sum However calculating the actual safety factor Sact as described below all the σi for which the product S σi is bigger than or equal to the permissible stress level for infinite life contribute to the damage sum and thus to the determination of Sact The final value of S is decisive
(NFi can be found mathematically by putting the permissible stress σpi equal the actual stress σi thereby finding the actual life factor This life factor can be solved with regard to load cycles ie NFi)
The damage sum ΣDi is not to exceed unity
If ΣDi ne 1 the safety against cumulative fatigue damage is different from the applied demand safety factor For deter-mination of this theoretical safety factor an iteration proce-dure is required as described in the following flowchart
S is correction factor with which the actual safety factor Sact can be found
Sact is the demand safety factor (used in determination of the permissible stresses in the σ ndash N ndash curve) times the correc-tion factor S
The full procedure is to be applied for pinion and wheel tooth roots and flanks
Note If alternating stresses occur in a spectrum of mainly pulsating stresses the alternating stresses may be replaced by equiva-lent pulsating stresses ie by means of division with the ac-tual mean stress influence factor YM
Classification Notes- No 412 41 May 2003
DET NORSKE VERITAS
Appendix B Application Factors for Diesel Driven Gears
For diesel driven gears the application factor KA depends on torsional vibrations Both normal operation and misfiring conditions have to be considered
Normally these two running conditions can be covered by only one calculation
B1 Definitions
Normal operation KAnorm = 0
normv0
TTT +
where
T0 = rated nominal torque
Tv norm = vibratory torque amplitude for normal opera-tion (see rules Pt4 Ch3 Sec1 G301 for defi-nition of ldquonormalrdquo irregularity)
Misfiring operation KA misf = 0
misfv
TTT +
where
T = remaining nominal torque when one cylinder out of action
Tv misf = vibratory torque amplitude in misfiring con-dition This refers to a permissible misfiring condition ie a condition that does not require automatic or immediate corrective actions as speed or pitch reduction
The normal operation is assumed to last for a very high num-ber of cycles such as 1010
The misfiring operation is assumed to last for a limited dura-tion such as 107 cycles
B2 Determination of decisive load Assuming life factor at 1010 cycles as YN = ZN = 092 which usually is relevant the calculation may be performed only once with the combination having the highest value of appli-cation factorlife factor
For bending stresses and scuffing the higher value of
092
K normA and 098
K misfA
For contact stresses the higher value of
092K normA and
113K misfA (but
097K misfA for nitrided gears)
B3 Simplified procedure Note that this is only a guidance and is not a binding con-vention
T = oTZ
1Zsdot
minus where Z = number of cylinders
( ) ideal Videal Vmisf Vnorm V TTT24ZT +minus=
where TV ideal = vibratory torque with all cylinders perfectly equal See also rules Pt4 Ch3 Sec1 G300
When using trends from torsional vibration analysis and measurements the following may be used
TV ideal To is close to zero for engines with few cylinders and using a suitable elastic coupling and increases with relative coupling stiffness and number of cylinders
TV misf To may be high for engines with few cylinders and decreases with number of cylinders
This can be indicated as
200Z
TT
ο
idealV asymp and 80Z04
TT
o
misfV minusasymp
Inserting this into the formulae for the two application fac-tors the following guidance can be given
118112K misfA minusasymp
115110K normA minusasymp
Since KA norm is to be combined with the lower life factors the decisive load condition will be the normal one and a KA of 115 will cover most relevant cases when a suitable elas-tic coupling is chosen
42 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
Appendix C Calculation of Pinion-Rack
Pinion-racks used for elevating of mobile offshore units are open gears that are subjected to wear and tear With normal specifications such as only a few hundred total operation cycles (site to site) the tooth bending stresses for static strength will be decisive for the lay out however with ex-ception of surface hardened pinions where case crushing has to be considered
In the following the use of part 1 and 3 for pinion-racks is shown including relevant simplifications
C1 Pinion tooth root stresses Since the load spectrum normally is dominated by high torques and few load cycles (in the range up to 1-2000) the static strength is decisive
The actual stress is calculated as
β1FSaFan1
tF1 KYY
mbFσ sdotsdotsdotsdot
=
b1 is limited to b2 + 2middotmn
YFa and YSa replace YF and YS because load application at tooth tip has to be assumed for such inaccurate gears
Pinions often use a non-involute profile in the dedendum part of the flank eg a constant radius equal the radius of curva-ture at reference circle For such pinions sFn and hFa are to be measured directly on a sectional drawing of the pinion tooth
Due to high loads and narrow facewidths it may be assumed that KFβ2 = KHβ = 10 However when b1 gt b2 then KFβ1 gt10
If no detailed documentation of KFβ1 is available the fol-lowing may be used
KFβ1 = 1 + 015middot(b1b2 ndash 1)
The permissible stress (not surface hardened) is calculated as
δrelTstF
Fst1FP1 Y
Sσσ sdot=
The mean stress influence due to leg lifting may be disre-garded
The actual and permissible stresses should be calculated for the relevant loads as given in the rules
C2 Rack tooth root stresses The actual stress is calculated as
SaFan2
tF2 YY
mbFσ sdotsdotsdot
=
See C1 for details
The permissible stress is calculated as
δrelTstF
Fst2FP2 Y
Sσσ sdot=
For alloyed steels (Ni Cr Mo) with high toughness and duc-tility the value of YδrelTst may be put equal to YSa
C3 Surface hardened pinions For surface hardened pinions the maximum load is not to cause crushing of the hardened layer of the flank
In principle the calculation described in 213 may be used but when the theoretical Hertzian stress exceeds the ap-proximately 18 times the yield strength of the rack material plastic deformation will occur This will limit the peak Hertzian stress but increases the contact width and thus the penetration of stresses into the depth
An approximation may be based on an assessment of contact width determined by means of equal areas under the theoreti-cal (elastic range) Hertzian contact and the elasto-plastic contact stress (limited to 18 middot σy) with the unknown width
Classification Notes- No 412 15 May 2003
DET NORSKE VERITAS
(for internal gears use zn2 equal infinite and x2 = 0) ha0 = hfp for all practical purposes CR considers the increased flexibility of the wheel teeth if the wheel is not a solid disc and may be calculated as
( )( )nR m5s
sR e5
bbln1C +=
where
bs = thickness of a central web
sR = average thickness of rim (net value from tooth root to inside of rim)
The formula is valid for bs b ge 02 and sRmn ge 1 Outside this range of validity and if the web is not centrally posi-tioned CR has to be specially considered
Note CR is the ratio between the average mesh stiffness over the facewidth and the mesh stiffness of a gear pair of solid discs The local mesh stiffness in way of the web corresponds to the mesh stiffness with CR = 1 The local mesh stiffness where there is no web support will be less than calculated with CR above Thus eg a centrally positioned web will have an effect corresponding to a longitudinal crowning of the teeth See also 1931 regarding KHβ
B The non-linear approach
In the following an example is given on how to consider the non-linearity
The relation between unit load Fb as a function of mesh deflection δ is assumed to be a progressive curve up to 500 Nmm and from there on a straight line This straight line when extended to the baseline is assumed to intersect at 10microm
With these assumptions the unit force Fb as a function of mesh deflection δ can be expressed as
( )10δKbF
minus= for 500bFgt
minus=
500Fb10δK
bF for 500
bFlt
with γAt KK
bF
bF
sdotsdot= etc (Nmm) ie unit load incorporat-
ing the relevant factors as
KA middot Kγ for determination of Kv
KA middot Kγ middot Kv for determination of KHβ
KA middot Kγ middot Kv middot KHβ for determination of KHα
δ = mesh deflection (microm)
K = applicable stiffness (c or cγ)
Use of stiffnesses for KV KHβ and KHα
For calculation of Kv and KHα the stiffness is calculated as follows
When Fb lt 500
the stiffness is determined as δ∆
∆ bF
where the increment is chosen as eg ∆ Fb = 10 and thus
50010Fb10
K10Fb∆δ +
++
=
When F b gt 500 the stiffness is c or cγ For calculation of KHβ the mesh deflection δ is used directly
or an equivalent stiffness determined as δsdotb
F
Bevel gears
In lack of more detailed relationship between stiffness and geometry the following may be used
b085
b13cacute eff=
b085b
16c effγ =
beff not to be used in excess of 085 b in these formulae
Bevel gears with heightwise and lengthwise crowning have progressive mesh stiffness The values mentioned above are only valid for high loads They should not be used for de-termination of Ceff (see 432) or KHβ (see 1931)
112 Running-in Allowances The running-in allowances account for the influence of run-ning-in wear on the various error elements
yα respectively yβ are the running-in amounts which reduce the influence of pitch and profile errors respectively influ-ence of localised faceload
Cay is defined as the running-in amount that compensates for lack of tip relief
The following relations may be used
For not surface hardened steel
ptHlim
α fσ160y =
yβ = βxlimH
f320σ
with the following upper limits
V lt 5 ms 5-10 ms gt 10 ms
yα max none
limH
12800σ limH
6400σ
yβ max none
limH
25600σ limH
12800σ
For surface hardened steel
yα = 0075 fpt but not more than 3 for any speed
yβ = 015 Fβx but not more than 6 for any speed
16 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
For all kinds of steel
5145189718
1C2
limHay +
minusσ
=
When pinion and wheel material differ the following ap-plies
bull Use the larger of fpt1 - yα1 and fpt2 - yα2 to replace fpt - yα in the calculation of KHα and Kv
bull Use ( )2β1ββ yy21y += in the calculation of KHβ
bull Use ( )2ay1aya CC21C += in the calculation of Kv
bull Use ( )2ay1ay2a1a CC21CC +== in the scuffing
calculation if no design tip relief is foreseen
Classification Notes- No 412 17 May 2003
DET NORSKE VERITAS
2 Calculation of Surface Durability
21 Scope and General Remarks Part 2 includes the calculations of flank surface durability as limited by pitting spalling case crushing and subsurface yielding Endurance and time limited flank surface fatigue is calculated by means of 22 ndash 212 In a way also tooth frac-tures starting from the flank due to subsurface fatigue is in-cluded through the criteria in 213
Pitting itself is not considered as a critical damage for slow speed gears However pits can create a severe notch effect that may result in tooth breakage This is particularly im-portant for surface hardened teeth but also for high strength through hardened teeth For high-speed gears pitting is not permitted
Spalling and case crushing are considered similar to pitting but may have a more severe effect on tooth breakage due to the larger material breakouts initiated below the surface Subsurface fatigue is considered in 213
For jacking gears (self-elevating offshore units) or similar slow speed gears designed for very limited life the max static (or very slow running) surface load for surface hard-ened flanks is limited by the subsurface yield strength
For case hardened gears operating with relatively thin lubri-cation oil films grey staining (micropitting) may be the lim-iting criterion for the gear rating Specific calculation meth-ods for this purpose are not given here but are under consid-eration for future revisions Thus depending on experience with similar gear designs limitations on surface durability rating other than those according to 22 - 213 may be ap-plied
22 Basic Equations Calculation of surface durability (pitting) for spur gears is based on the contact stress at the inner point of single pair contact or the contact at the pitch point whichever is greater
Calculation of surface durability for helical gears is based on the contact stress at the pitch point
For helical gears with 0 lt εβ lt 1 a linear interpolation be-tween the above mentioned applies
Calculation of surface durability for spiral bevel gears is based on the contact stress at the midpoint of the zone of contact
Alternatively for bevel gears the contact stress may be cal-culated with the program ldquoBECALrdquo In that case KA and Kv are to be included in the applied tooth force but not KHβ and KHα The calculated (real) Hertzian stresses are to be multi-plied with ZK in order to be comparable with the permissible contact stresses
The contact stresses calculated with the method in part 2 are based on the Hertzian theory but do not always represent the real Hertzian stresses
The corresponding permissible contact stresses σHP are to be calculated for both pinion and wheel
221 Contact stress Cylindrical gears
( )HαHβvγA
1
tβεEHDBH KKKKK
bud1uF
ZZZZZσ+
=
where
ZBD = Zone factor for inner point of single pair contact for pinion resp wheel (see 232)
ZH = Zone factor for pitch point (see 231)
ZE = Elasticity factor (see 24)
Zε = Contact ratio factor (see 25)
Zβ = Helix angle factor (see 26)
Ft KA Kγ Kv KHβ KHα see 15 ndash 110
d1 b u see 12 ndash 15
Bevel gears
( )HαHβvγA
v1v
vmtKEMH KKKKK
bud1uF
ZZZ105σ+
sdot=
where
105 is a correlation factor to reach real Hertzian stresses (when ZK = 1)
ZE KA etc see above
ZM = mid-zone factor see 233
ZK = bevel gear factor see 27
Fmt dv1 uv see 12 ndash 15
It is assumed that the heightwise crowning is chosen so as to result in the maximum contact stresses at or near the mid-point of the flanks
222 Permissible contact stress
XWRvLH
NHlimHP ZZZZZ
SZσ
σ =
where
σH lim = Endurance limit for contact stresses (see 28)
ZN = Life factor for contact stresses (see 29)
SH = Required safety factor according to the rules
ZLZvZR = Oil film influence factors (see 210)
ZW = Work hardening factor (see 211)
ZX = Size factor (see 212)
18 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
23 Zone Factors ZH ZBD and ZM
231 Zone factor ZH The zone factor ZH accounts for the influence on contact stresses of the tooth flank curvature at the pitch point and converts the tangential force at the reference cylinder to the normal force at the pitch cylinder
wtt2
wtbH sinααcos
cosαcosβ2Z =
232 Zone factors ZBD The zone factors ZBD account for the influence on contact stresses of the tooth flank curvature at the inner point of sin-gle pair contact in relation to ZH Index B refers to pinion D to wheel
For εβ ge 1 ZBD = 1
For internal gears ZD = 1
For εβ = 0 (spur gears)
( )
π
minusεminusminus
π
minusminus
α=
α2
2
2b
2a
1
2
1b
1a
wtB
z211
dd
z21
dd
tanZ
( )
π
minusεminusminus
π
minusminus
α=
α1
2
1b
1a
2
2
2b
2a
wtD
z211
dd
z21
dd
tanZ
If ZB lt 1 use ZB = 1
If ZD lt 1 use ZD = 1
For 0 lt εβ lt 1
ZBD = ZBD (for spur gears) ndash εβ (ZBD (for spur gears) ndash 1)
233 Zone factor ZM The mid-zone factor ZM accounts for the influence of the contact stress at the mid point of the flank and applies to spi-ral bevel gears
εminusminus
εminusminus
αβ=
αα btm2
2vb2
2vabtm2
1vb2val
2v1vvtbmM
pddpdd
ddtancos2Z
This factor is the product of ZH and ZM-B in ISO 10300 with the condition that the heightwise crowning is sufficient to move the peak load towards the midpoint
234 Inner contact point For cylindrical or bevel gears with very low number of teeth the inner contact point (A) may be close to the base circle In order to avoid a wear edge near A it is required to have suitable tip relief on the wheel
24 Elasticity Factor ZE The elasticity factor ZE accounts for the influence of the material properties as modulus of elasticity and Poissonrsquos ratio on the contact stresses
For steel against steel ZE = 1898
25 Contact Ratio Factor Zε The contact ratio factor Zε accounts for the influence of the transverse contact ratio εα and the overlap ratio εβ on the contact stresses
αε1Z =ε for 1εβ ge
( )α
ββ
αε ε
εε1
3ε4
Z +minusminus
= for εβ lt 1
26 Helix Angle Factor Zβ The helix angle factor Zβ accounts for the influence of helix angle (independent of its influence on Zε) on the surface du-rability
cosβZβ =
27 Bevel Gear Factor ZK The bevel gear factor accounts for the difference between the real Hertzian stresses in spiral bevel gears and the contact stresses assumed responsible for surface fatigue (pitting) ZK adjusts the contact stresses in such a way that the same per-missible stresses as for cylindrical gears may apply
The following may be used ZK = 080
28 Values of Endurance Limit σHlim and Static Strength 5H10σ 3H10σ
σHlim is the limit of contact stress that may be sustained for 5middot107 cycles without the occurrence of progressive pitting
For most materials 5middot107 cycles are considered to be the be-ginning of the endurance strength range or lower knee of the σ-N curve (See also Life Factor ZN) However for nitrided steels 2middot106 apply
For this purpose pitting is defined by
bull for not surface hardened gears pitted area ge 2 of total active flank area
bull for surface hardened gears pitted area ge 05 of total active flank area or ge 4 of one particular tooth flank area 510Hσ and 310Hσ are the contact stresses which the given
material can withstand for 105 respectively 103 cycles without subsurface yielding or flank damages as pitting spalling or case crushing when adequate case depth applies
The following listed values for σHlim 510Hσ and 310Hσ may only be used for materials subjected to a quality control as
Classification Notes- No 412 19 May 2003
DET NORSKE VERITAS
the one referred to in the rules Results of approved fatigue tests may also be used as the basis for establishing these values The defined survival probability is 99
σHlim σH105 σH10
3
Alloyed case hardened steels (surface hardness 58-63 HRC) - of specially approved high grade - of normal grade
1650 1500
2500 2400
3100 3100
Nitrided steel of approved grade gas nitrided (surface hardness 700-800 HV) 1250 13 σHlim 13 σHlim
Alloyed quenched and tempered steel bath or gas nitrided (surface hardness 500-700 HV)
1000
13 σHlim
13 σHlim
Alloyed flame or induction hardened steel (surface hardness 500-650 HV) 075 HV + 750 16 σHlim 45 HV
Alloyed quenched and tempered steel 14 HV + 350 16 σHlim 45 HV
Carbon steel 15 HV + 250 16 σHlim 16 σHlim
These values refer to forged or hot rolled steel For cast steel the values for σHlim are to be reduced by 15
29 Life Factor ZN The life factor ZN takes account of a higher permissible contact stress if only limited life (number of cycles NL) is demanded or lower permissible contact stress if very high number of cycles apply
If this is not documented by approved fatigue tests the fol-lowing method may be used
For all steels except nitrided
7L 105N sdotge ZN = 1 or
01570
L
7
N N105Z
sdot=
Ie ZN = 092 for 1010 cycles
The ZN = 1 from 5middot107 on may only be used when the mate-rial cleanliness is of approved high grade (see Rules Pt4 Ch2) and the lubrication is optimised by a specially approved filtering process
105 lt NL lt 5middot107 510N
logZ037
L
7
N N105Z
sdot=
NL = 105 WXRVLHlim
WstX10H1010NN ZZZZZσ
ZZσZZ
55
5=
103 lt NL lt 105 )Z(Zlog05
L
5
10NN
5N103N10
5N10ZZ
==
3L 10N le
WXRVLlimH
Wst10X10H10NN ZZZZZ
ZZZZ
33
3σ
σ==
(but not less than ZN105)
For nitrided steels
6L 102N sdotge ZN = 1 or
00980
L
6
N N102Z
sdot=
Ie ZN = 092 for 1010 cycles
The ZN = 1 from 2middot106 on may only be used when the mate-rial cleanliness is of approved high grade (see Rules Pt4 Ch2) and the lubrication is optimised by a specially approved filtering process
105 lt NL lt 2middot106 510N
Zlog76860
L
6
N N102Z
sdot=
5L 10N le
XWRVL
10XWst10NN ZZZZZ
ZZ13ZZ
55 ==
Note that when no index indicating number of cycles is used the factors are valid for 5middot107 (respectively 2middot106 for nitrid-ing) cycles
210 Influence Factors on Lubrication Film ZL ZV and ZR The lubricant factor ZL accounts for the influence of the type of lubricant and its viscosity the speed factor ZV ac-counts for the influence of the pitch line velocity and the roughness factor ZR accounts for influence of the surface roughness on the surface endurance capacity
The following methods may be applied in connection with the endurance limit
20 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
Surface hardened steels Not surface hardened steels
ZL ( )24013421
360910ν+
+
( )24013421
680830ν+
+
ZV ( )v3280
140930+
+ ( )v3280300850
++
ZR
080
ZrelR3
150
ZrelR3
where
ν40 = Kinematic oil viscosity at 40ordmC (mm2s) For case hardened steels the influence of a high bulk temperature (see 4 Scuffing) should be con-sidered Eg bulk temperatures in excess of 120ordmC for long periods may cause reduced flank surface endurance limits
For values of ν40 gt 500 use ν40 = 500
RZrel = The mean roughness between pinion and wheel (after running in) relative to an equivalent radius of curvature at the pitch point ρc = 10mm
RZrel
= ( ) 31
cZ2Z1 ρ
10RR50
+
RZ = Mean peak to valley roughness (microm) (DIN defini-tion) (roughly RZ = 6 Ra)
For 5
L 10N le ZL ZV ZR = 10
211 Work Hardening Factor ZW The work hardening factor ZW accounts for the increase of surface durability of a soft steel gear when meshing the soft steel gear with a surface hardened or substantially harder gear with a smooth surface
The following approximation may be used for the endurance limit
Surface hardened steel against not surface hardened steel 150
ZeqW R
31700
130HB21Z
minus
minus=
where HB = the Brinell hardness of the soft member For HB gt 470 use HB = 470 For HB lt 130 use HB = 130
RZeq = equivalent roughness
033
c40
066
ZS
ZHZHZEQ ρvν
15000RRRR
=
If RZeq gt 16 then use RZeq = 16
If RZeq lt 15 then use RZeq = 15
where
RZH = surface roughness of the hard member before run in
RZS = surface roughness of the soft member before run in
ν40 = see 210
If values of ZW lt 1 are evaluated ZW = 1 should be used for flank endurance However the low value for ZW may indi-cate a potential wear problem
Through hardened pinion against softer wheel
( )
minussdotminus+= 000829
HBHB0008981u1Z
2
1W
For 21HBHB
2
1 le use ZW = 1
For 71HBHB
2
1 gt use 17HBHB
2
1 =
For u gt 20 use u = 20
For static strength (lt 105 cycles)
Surface hardened against not surface hardened
ZWst = 105
Through hardened pinion against softer wheel
ZWst = 1
212 Size Factor ZX The size factor accounts for statistics indicating that the stress levels at which fatigue damage occurs decrease with an increase of component size as a consequence of the influ-ence on subsurface defects combined with small stress gradi-ents and of the influence of size on material quality
ZX may be taken unity provided that subsurface fatigue for surface hardened pinions and wheels is considered eg as in the following subsection 213
213 Subsurface Fatigue This is only applicable to surface hardened pinions and wheels The main objective is to have a subsurface safety against fatigue (endurance limit) or deformation (static strength) which is at least as high as the safety SH required for the surface The following method may be used as an approximation unless otherwise documented
The high cycle fatigue (gt3106 cycles) is assumed to mainly depend on the orthogonal shear stresses Static strength (lt103 cycles) is assumed to depend mainly on equivalent
Classification Notes- No 412 21 May 2003
DET NORSKE VERITAS
stresses (von Mises) Both are influenced by residual stresses but this is only considered roughly and empirically
The subsurface working stresses at depths inside the peak of the orthogonal shear stresses respectively the equivalent stresses are only dependent on the (real) Hertzian stresses Surface related conditions as expressed by ZL ZV and ZR are assumed to have a negligible influence
The real Hertzian stresses σHR are determined as
For helical gears with εβ gt 1
σHR = σH
For helical gears with εβ lt 1 and spur gears
ε
α
ββ ε
ε+εminus
sdotσ=σZ
1
HHR
For bevel gears
KHHR Z
1σσ sdot=
The necessary hardness HV is given as a function of the net depth tz (net = after grinding or hard metal hobbing and per-pendicular to the flank)
The coordinates tz and HV are to be compared with the de-sign specification such as
bull for flame and induction hardening tHVmin HVmin bull for nitriding t400min HV = 400 bull for case hardening t550min HV = 550 t400min HV = 400
and t300min HV = 300 (the latter only if the core hardness lt 300 If the core hardness gt 400 the t400 is to be re-placed by a fictive t400 = 16 t550)
In addition the specified surface hardness is not to be less than the max necessary hardness (at tz = 05aH) This applies to all hardening methods
For high cycle fatigue (gt3 106 cycles) the following applies
sdot+
minussdotsdotsdot= o90
05azt
05azt
cosSσ04HV
H
HHHR
applicable to 05a
zt
Hge
For 05at
H
z lt the value for 05at
H
z = applies
56300ρSσ12a cHHR
Hsdotsdot
sdot=
Where aH is half the hertzian contact width multiplied by an empirical factor of 12 that takes into account the possible influence of reduced compressive residual stresses (or even tensile residual stresses) on the local fatigue strength
If any of the specified hardness depths including the surface hardness is below the curve described by HV = f (tz) the actual safety factor against subsurface fatigue is determined as follows
reduce SH stepwise in the formula for HV and aH until all specified hardness depths and surface hardness balance with the corrected curve The safety factor obtained through this method is the safety against subsurface fatigue
For static strength (lt103 cycles) the following applies
sdot+
minussdotsdotsdot= o90
07at
06a
zt
cosSσ019HV
Hst
z
HstHHR
applicable to 06at
Hst
z ge
56300ρSσa cHHR
Hstsdotsdot
=
In the case of insufficient specified hardness depths the same procedure for determination of the actual safety factor as above applies
For limited life fatigue (103 lt cycles lt 3106 )
For this purpose it is necessary to extend the correction of safety factors to include also higher values than required Ie in the case of more than sufficient hardness and depths the safety factor in the formulae for both high cycle fatigue and static strength are to be increased until necessary and speci-fied values balance
The actual safety factor for a given number of cycles N be-tween 103 and 3106 is found by linear interpolation in a dou-ble logarithmic diagram
minussdotminus
= sdot logN3477
logSlogSlogS
310H6103HHN
36 10H103H Slog86281Slog86280 sdot+sdot sdot
22 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
3 Calculation of Tooth Strength
31 Scope and General Remarks Part 3 include the calculation of tooth root strength as limited by tooth root cracking (surface or subsurface initiated) and yielding
For rim thickness sR ge 35middotmn the strength is calculated by means of 32 ndash 313 For cylindrical gears the calculation is based on the assumption that the highest tooth root tensile stress arises by application of the force at the outer point of single tooth pair contact of the virtual spur gears The method has however a few limitations that are mentioned in 36
For bevel gears the calculation is based on force application at the tooth tip of the virtual cylindrical gear Subsequently the stress is converted to load application at the mid point of the flank due to the heightwise crowning
Bevel gears may also be calculated with the program BECAL In that case KA and Kv are to be included in the applied tooth force but not KFβ and KFα
In case of a thin annulus or a thin gear rim etc radial crack-ing can occur rather than tangential cracking (from root fillet to root fillet) Cracking can also start from the compression fillet rather than the tension fillet For rim thickness sR lt 35middotmn a special calculation procedure is given in 315 and 316 and a simplified procedure in 314
A tooth breakage is often the end of the life of a gear trans-mission Therefore a high safety SF against breakage is re-quired
It should be noted that this part 3 does not cover fractures caused by
bull oil holes in the tooth root space bull wear steps on the flank bull flank surface distress such as pits spalls or grey staining
Especially the latter is known to cause oblique fractures starting from the active flank predominately in spiral bevel gears but also sometimes in cylindrical gears
Specific calculation methods for these purposes are not given here but are under consideration for future revisions Thus depending on experience with similar gear designs limita-tions other than those outlined in part 3 may be applied
32 Tooth Root Stresses The local tooth root stress is defined as the max principal stress in the tooth root caused by application of the tooth force Ie the stress ratio R = 0 Other stress ratios such as for eg idler gears (R asymp -12) shrunk on gear rims (R gt 0) etc are considered by correcting the permissible stress level
321 Local tooth root stress The local tooth root stress for pinion and wheel may be as-sessed by strain gauge measurements or FE calculations or similar For both measurements and calculations all details are to be agreed in advance
Normally the stresses for pinion and wheel are calculated as
Cylindrical gears
FαFβvγAβSFn
tF K K K K K Y Y Y
m bFσ =
where
YF = Tooth form factor (see 33)
YS = Stress correction factor (see 34)
Yβ = Helix angle factor (see 36)
Ft KA Kγ Kv KFβ KFα see 15 ndash 110
b see 13
Bevel gears
FαFβvγASaFamn
mtF K K K K K Y Y Y
m bFσ ε=
where
YFa = Tooth form factor see 33
YSa = Stress correction factor see 34
Yε = Contact ratio factor see 35
Fmt KA etc see 15 ndash 110
b see 13
322 Permissible tooth root stress The permissible local tooth root stress for pinion respectively wheel for a given number of cycles N is
CXRrelTrelTF
NMFEFP YYYY
SYY
δσ
=σ
Note that all these factors YM etc are applicable to 3middot106 cycles when used in this formula for σFP The influence of other number of cycles on these factors is covered by the calculation of YN
where
σFE = Local tooth root bending endurance limit of reference test gear (see 37)
YM = Mean stress influence factor which accounts for other loads than constant load direction eg idler gears temporary change of load di-rection pre-stress due to shrinkage etc (see 38)
YN = Life factor for tooth root stresses related to reference test gear dimensions (see 39)
SF = Required safety factor according to the rules
YδrelT = Relative notch sensitivity factor of the gear to be determined related to the reference test gear (see 310)
YRrelT = Relative (root fillet) surface condition factor of the gear to be determined related to the reference test gear (see 311)
Classification Notes- No 412 23 May 2003
DET NORSKE VERITAS
YX = Size factor (see 312)
YC = Case depth factor (see 313)
33 Tooth Form Factors YF YFa The tooth form factors YF and YFa take into account the in-fluence of the tooth form on the nominal bending stress
YF applies to load application at the outer point of single tooth pair contact of the virtual spur gear pair and is used for cylindrical gears
YFa applies to load application at the tooth tip and is used for bevel gears
Both YF and YFa are based on the distance between the con-tact points of the 30˚-tangents at the root fillet of the tooth profile for external gears respectively 60˚ tangents for inter-nal gears
Figure 31 External tooth in normal section
Figure 32 Internal tooth in normal section
Definitions
n
2
n
Fn
enFn
Fe
F
α cosms
α cosmh6
Y
=
n
2
n
Fn
anFn
Fa
Fa
α cosms
α cosmh6
Y
=
In the case of helical gears YF and YFa are determined in the normal section ie for a virtual number of teeth
YFa differs from YF by the bending moment arm hFa and αFan and can be determined by the same procedure as YF with exception of hFe and αFan For hFa and αFan all indices e will change to a (tip)
The following formulae apply to cylindrical gears but may also be used for bevel gears when replacing
mn with mnm
zn with zvn
αt with αvt
β with βm
with undercut without undercut
Fig 33 Dimensions and basic rack profile of the teeth (finished profile)
Tool and basic rack data such as hfP ρfp and spr etc are re-ferred to mn ie dimensionless
331 Determination of parameters
( )n
n
prnfPnfP m
α cossαsin 1ρ
αtan h4πE
minusminusminusminus=
For external gears fPfP ρρ =
For internal gears ( )0z
195fPfP0
fPfP 10363156ρhxρρ
sdotminus+
+=
where
z0 = number of teeth of pinion cutter
x0 = addendum modification coefficient of pinion cutter
hfP = addendum of pinion cutter
ρfP = tip radius of pinion cutter
xhρG fpfp +minus=
24 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
τmE
2π
z2H
nnminus
minus=
with
3πτ = for external gears
6πτ = for internal gears
Htan zG2
nminusϑ=ϑ
(to be solved iteratively suitable start value6π
=ϑ for exter-
nal gears and 3π for internal gears)
a) Tooth root chord sFn For external gears
minus
ϑ+
ϑminus= ρ
cosG3
3πsinz
ms
fpnn
Fn
For bevel gears with a tooth thickness modification
xsm affects mainly sFn but also hFe and αFen The total influence of xsm on YFa Ysa can be approximated by only adding 2 xsm to sFn mn
For internal gears
minus
ϑ+
ϑminus= ρ
cosG
6πsinz
ms
fPnn
Fn
b) Root fillet radius ρF at 30ordm tangent
( )G2coszcosG2ρ
mρ
2n
2
fpn
F
minusϑϑ+=
c) Determination of bending moment arm hF dn = zn mn
b2α
αn βcosε
ε =
dan = dn + 2 ha
pbn = π mn cos αn
dbn = dn cos αn
( )4
d1εpzz
2dd
zz2d
2bn
2
αnbn
2bn
2an
en +
minusminus
minus=
en
bnen d
dcos arcα =
ennnn
e α invα invα x tan 22π
z1
minus+
+=γ
αFen = αen ndash γe
For external gears
( )
minus=
n
enFenee
n
Fe
mdαtan sin γ γcos
21
mh
]ρcos
G3πcosz fpn +
ϑminus
ϑminusminus
For internal gears
( ) minus
sdotsdotminus=
n
enFenee
n
Fe
mdα tanγ sinγ cos
21
mh
minus
ϑminus
ϑminussdot ρ
cosG3
6πcosz fPn
332 Gearing with εαn gt 2 For deep tooth form gearing ( )25ε2 αn lele produced with a verified grade of accuracy of 4 or better and with applied profile modification to obtain a trapezoidal load distribution along the path of contact the YF may be corrected by the factor YDT as
250ε205for 0666ε2366Y αnαnDT leleminus=
205εfor 10Y αnDT lt=
34 Stress Correction Factors YS YSa The stress correction factors YS and YSa take into account the conversion of the nominal bending stress to the local tooth root stress Thereby YS and YSa cover the stress increasing effect of the notch (fillet) and the fact that not only bending stresses arise at the root A part of the local stress is inde-pendent of the bending moment arm This part increases the more the decisive point of load application approaches the critical tooth root section
Therefore in addition to its dependence on the notch radius the stress correction is also dependent on the position of the load application ie the size of the bending moment arm
YS applies to the load application at the outer point of single tooth pair contact YSa to the load application at tooth tip
YS can be determined as follows
( )
+
+=L23121
1
sS qL01312Y
where Fe
Fn
hsL = and
F
Fns ρ2
sq = (see 33)
YSa can be calculated by replacing hFe with hFa in the above formulae
Note a) Range of validity 1 lt qs lt 8
In case of sharper root radii (ie produced with tools having too sharp tip radii) YS resp YSa must be specially considered
b) In case of grinding notches (due to insufficient protuberance of the hob) YS resp YSa can rise considerably and must be multiplied with
Classification Notes- No 412 25 May 2003
DET NORSKE VERITAS
g
g
ρt
0613
13
minus
where
tg = depth of the grinding notch
ρg = radius of the grinding notch
c) The formulae for YS resp YSa are only valid for αn = 20˚ However the same formulae can be used as a safe approximation for other pressure angles
35 Contact Ratio Factor Yε The contact ratio factor Yε covers the conversion from load application at the tooth tip to the load application at the mid point of the flank (heightwise) for bevel gears
The following may be used
Yε = 0625
36 Helix Angle Factor Yβ The helix angle factor Yβ takes into account the difference between the helical gear and the virtual spur gear in the nor-mal section on which the calculation is based in the first step In this way it is accounted for that the conditions for tooth root stresses are more favourable because the lines of contact are sloping over the flank
The following may be used (β input in degrees)
Yβ = 1 ndash εβ β120
When εβ gt 1 use εβ = 1 and when β gt 30deg use β = 30deg in the formula
However the above equation for Yβ may only be used for gears with β gt 25deg if adequate tip relief is applied to both pinion and wheel (adequate = at least 05 middot Ceff see 432)
37 Values of Endurance Limit σFE σFE is the local tooth root stress (max principal) which the material can endure permanently with 99 survival prob-ability 3106 load cycles is regarded as the beginning of the endurance limit or the lower knee of the σ ndash N curve σFE is defined as the unidirectional pulsating stress with a minimum stress of zero (disregarding residual stresses due to heat treatment) Other stress conditions such as alternating or pre-stressed etc are covered by the conversion factor YM
σFE can be found by pulsating tests or gear running tests for any material in any condition If the approval of the gear is to be based on the results of such tests all details on the testing conditions have to be approved by the Society Fur-ther the tests may have to be made under the Societys su-pervision
If no fatigue tests are available the following listed values for σFE may be used for materials subjected to a quality con-trol as the one referred to in the rules
σFE
Alloyed case hardened steels 1) (fillet surface hardness 58 ndash 63 HRC)
bull of specially approved high grade
1050
bull of normal grade
minus CrNiMo steels with approved process
1000
minus CrNi and CrNiMo steels generally 920 minus MnCr steels generally 850
Nitriding steel of approved grade quenched tempered and gas nitrided (surface hardness 700 ndash 800 HV)
840
Alloyed quenched and tempered steel bath or gas nitrided (surface hardness 500 ndash 700 HV)
720
Alloyed quenched and tempered steel flame or induction hardened 2) (incl entire root fillet) (fillet surface hardness 500 ndash 650 HV)
07 HV + 300
Alloyed quenched and tempered steel flame or induction hardened (excl entire root fillet) (σB = uts of base material)
025 σB + 125
Alloyed quenched and tempered steel 04 σB + 200
Carbon steel 025 σB + 250
Note All numbers given above are valid for separate forgings and for blanks cut from bars forged according to a qualified procedure see Pt 4 Ch 2 Sec 3 For rolled steel the values are to be reduced with 10 For blanks cut from forged bars that are not qualified as mentioned above the values are to be reduced with 20 For cast steel reduce with 40
1) These values are valid for a root radius
bull being unground If however any grinding is made in the root fillet area in such a way that the residual stresses may be affected σFE is to be reduced by 20 (If the grinding also leaves a notch see 34)
bull with fillet surface hardness 58 ndash 63 HRC In case of lower surface hardness than 58 HRC σFE is to be reduced with 20(58 ndash HRC) where HRC is the detected hardness (This may lead to a permissible tooth root stress that varies along the facewidth If so the actual tooth root stresses may also be considered along facewidth)
bull not being shot peened In case of approved shot peening σFE may be increased by 200 for gears where σFE is reduced by 20 due to root grinding Otherwise σFE may be increased by 100 for
6nm le and 100 ndash 5 (mn - 6) for mn gt 6 However the possible adverse influence on the flanks regarding grey staining should be considered and if necessary the flanks should be masked
2) The fillet is not to be ground after surface hardening Regarding possible root grinding see 1)
26 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
38 Mean stress influence Factor YM The mean stress influence factor YM takes into account the influence of other working stress conditions than pure pulsa-tions (R = 0) such as eg load reversals idler gears planets and shrink-fitted gears
YM (YMst) are defined as the ratio between the endurance (or static) strength with a stress ratio R ne 0 and the endurance (or static) strength with R = 0
YM and YMst apply only to a calculation method that assesses the positive (tensile) stresses and is therefore suitable for comparison between the calculated (positive) working stress σF and the permissible stress σFP calculated with YM or YMst
For thin rings (annulus) in epicyclic gears where the com-pression fillet may be decisive special considerations apply see 316
The following method may be used within a stress ratio ndash12 lt R lt 05
381 For idlers planets and PTO with ice class
M1M1R1
1Yor Y MstM
+minus
minus=
where
R = stress ratio = min stress divided by max stress
For designs with the same force applied on both forward- and back-flank R may be assumed to ndash 12
For designs with considerably different forces on forward- and back-flank such as eg a marine propulsion wheel with a power take off pinion R may be assessed as
branchmain theoffacewidth unit per forcepto offacewidt unit per force21minus
For a power take off (PTO) with ice class see 161 c
M considers the mean stress influence on the endurance (or static) strength amplitudes
M is defined as the reduction of the endurance strength am-plitude for a certain increase of the mean stress divided by that increase of the mean stress
Following M values may be used
Endurance limit
Static strength
Case hardened 08 ndash 015 Ys 1) 07
If shot peened 04 06
Nitrided 03 03
Induction or flame hardened
04 06
Not surface hardened steel 03 05
Cast steels 04 06 1)For bevel gears use Ys=2 for determination of M
The listed M values for the endurance limit are independent of the fillet shape (Ys) except for case hardening In princi-ple there is a dependency but wide variations usually only occur for case hardening eg smooth semicircular fillets versus grinding notches
382 For gears with periodical change of rotational direction For case hardened gears with full load applied periodically in both directions such as side thrusters the same formula for YM as for idlers (with R = ndash 12) may be used together with the M values for endurance limit This simplified approach is valid when the number of changes of direction exceeds 100 and the total number of load cycles exceeds 3middot106
For gears of other materials YM will normally be higher than for a pure idler provided the number of changes of direction is below 3middot106 A linear interpolation in a diagram with loga-rithmic number of changes of direction may be used ie from YM = 09 with one change to YM (idler) for 3middot106 changes This is applicable to YM for endurance limit For static strength use YM as for idlers
For gears with occasional full load in reversed direction such as the main wheel in a reversing gear box YM = 09 may be used
383 For gears with shrinkage stresses and unidirectional load For endurance strength
FE
fitM σ
σM1
M21Y+
minus=
σFE is the endurance limit for R = 0
For static strength YMst = 1 and σfit accounted for in 39b
σfit is the shrinkage stress in the fillet (30˚ tangent) and may be found by multiplying the nominal tangential (hoop) stress with a stress concentration factor
n
Ffit m
ρ215scf minus=
384 For shrink-fitted idlers and planets When combined conditions apply such as idlers with shrink-age stresses the design factor for endurance strength is
( ) ( ) FE
fitM σ
σR1M1
M2
M1M1R1
1Yminussdot+
minus
+minus
minus=
Symbols as above but note that the stress ratio R in this par-ticular connection should disregard the influence of σfit ie R normally equal ndash 12
For static strength
M1M1R1
1YMst
+minus
minus=
The effect of σfit is accounted for in 39 b
Classification Notes- No 412 27 May 2003
DET NORSKE VERITAS
385 Additional requirements for peak loads The total stress range (σmax ndash σmin) in a tooth root fillet is not to exceed
F
y
Sσ225
for not surface hardened fillets
FSHV5 for surface hardened fillets
39 Life Factor YN The life factor YN takes into account that in the case of limited life (number of cycles) a higher tooth root stress can be permitted and that lower stresses may apply for very high number of cycles
Decisive for the strength at limited life is the σ ndash N ndash curve of the respective material for given hardening module fillet radius roughness in the tooth root etc Ie the factors YδrelT YRelT YX and YM have an influence on YN
If no σ ndash N ndash curve for the actual material and hardening etc is available the following method may be used
Determination of the σ ndash N ndash curve
a) Calculate the permissible stress σFP for the beginning of the endurance limit (3middot106 cycles) including the influ-ence of all relevant factors as SF YδrelT YRelT YX YM and YC ie σFP = σFE middotYM middotYδrelT middotYRelT middotYX middotYC SF
b) Calculate the permissible laquostaticraquo stress (le 103 load cy-cles) including the influence of all relevant factors as SFst YδrelTst YMst and YCst ( )fitσCstYδrelTstYMstYFstσ
FstS1
FPstσ minussdotsdotsdot=
where σFst is the local tooth root stress which the material can resist without cracking (surface hardened materials) or unacceptable deformation (not surface hardened materials) with 99 survival probability
σFst
Alloyed case hardened steel 1) 2300
Nitriding steel quenched tempered and gas nitrided (surface hardness 700 ndash 800 HV)
1250
Alloyed quenched and tempered steel bath or gas nitrided (surface hardness 500 ndash 700 HV)
1050
Alloyed quenched and tempered steel flame or induction hardened (fillet surface hardness 500 ndash 650 HV)
18 HV + 800
Steel with not surface hardened fillets the smaller value of 2)
18 σB or 225 σy
1) This is valid for a fillet surface hardness of 58 ndash 63 HRC In case of lower fillet surface hardness than 58 HRC σFst is to be reduced with 30(58 ndash HRC) where HRC is the actual hardness Shot peening or grinding notches are not considered to have any significant influence on σFst 2) Actual stresses exceeding the yield point (σy or σ02) will alter the residual stresses locally in the ldquotensionrdquo fillet re-spectively ldquocompressionrdquo fillet This is only to be utilised for gears that are not later loaded with a high number of cycles at lower loads that could cause fatigue in the ldquocompressionrdquo fillet
c) Calculate YN as
NL gt 3middot106
YN = 1 or 001
L
6
N N103Y
sdot= ie Yn = 092 for 1010
The YN = 1 from 3middot106 on may only be used when special material cleanness applies see rules Pt4 Ch2
103ltNLlt3middot106exp
L
6
N N103Y
sdot=
cycles6103forσcycles310forσlog02876exp
FP
FPst
sdot=
NL lt 103 cycles6103forσcycles310forσ
NYFP
FPst
sdot=
or simply use σFPst as mentioned in b) directly
Guidance on number of load cycles NL for various applica-tions
bull For propulsion purpose normally NL = 1010 at full load (yachts etc may have lower values)
bull For auxiliary gears driving generators that normally operate with 70-90 of rated power NL = 108 with rated power may be applied
28 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
310 Relative Notch Sensitivity Factor YδrelT The dynamic (respectively static) relative notch sensitivity factor YδrelT (YδrelTst) indicate to which extent the theoreti-cally concentrated stress lies above the endurance limits (re-spectively static strengths) in the case of fatigue (respectively overload) breakage
YδrelT is a function of the material and the relative stress gra-dient It differs for static strength and endurance limit
The following method may be used
For endurance limit
for not surface hardened fillets
( )
024
s024
relT σ103133q21σ1012201351
Ysdotsdotminus
+sdotsdotminus+= minus
minus
δ
for all surface hardened fillets except nitrided
( )
106q21002451
Y srelT
++=δ
for nitrided fillets
( )
1347q2101421
Y srelT
++=δ
For static strength
for not surface hardened fillets1)
( )( )( )025
02
02502s
relTst σ3000821σ300 1Y0821Y
+
minus+=δ
for surface hardened fillets except nitrided
YδrelTst = 044 YS + 012
for nitrided fillets
YδrelTst = 06 + 02 YS 1) These values are only valid if the local stresses do not
exceed the yield point and thereby alter the residual stress level See also 39b footnote 2
311 Relative Surface Condition Factor YRrelT The relative surface condition factor YRrelT takes into ac-count the dependence of the tooth root strength on the sur-face condition in the tooth root fillet mainly the dependence on the peak to valley surface roughness
YRrelT differs for endurance limit and static strength
The following method may be used
For endurance limit
YRrelT = 1675 ndash 053 (Ry + 1)01 for surface hardened steels and alloyed quenched and tempered steels except nitrided
YRrelT = 53 ndash 42 (Ry + 1)001 for carbon steels
YδrelT = 43 ndash 326 (Ry + 1)0005
for nitrided steels
For static strength
YRrelTst = 1 for all Ry and all materials
For a fillet without any longitudinal machining trace Ry asymp Rz
312 Size Factor YX The size factor YX takes into account the decrease of the strength with increasing size YX differs for endurance limit and static strength
The following may be used
For endurance limit
YX = 1 for mn le 5 generally
YX = 103 ndash 0006 mn for 5 lt mn lt 30
YX = 085 for mn ge 30
for not surface hard-ened steels
YX = 105 ndash 001 mn for 5 lt mn ge 25
YX = 08 for mn ge 25
for surface hardened steels
For static strength
YXst = 1 for all mn and all materials
313 Case Depth Factor YC The case depth factor YC takes into account the influence of hardening depth on tooth root strength
YC applies only to surface hardened tooth roots and is dif-ferent for endurance limit and static strength
In case of insufficient hardening depth fatigue cracks can develop in the transition zone between the hardened layer and the core For static strength yielding shall not occur in the transition zone as this would alter the surface residual stresses and therewith also the fatigue strength
The major parameters are case depth stress gradient permis-sible surface respectively subsurface stresses and subsurface residual stresses
The following simplified method for YC may be used
YC consists of a ratio between permissible subsurface stress (incl influence of expected residual stresses) and permissible surface stress This ratio is multiplied with a bracket con-taining the influence of case depth and stress gradient (The empirical numbers in the bracket are based on a high number of teeth and are somewhat on the safe side for low number of teeth)
YC and YCst may be calculated as given below but calculated values above 10 are to be put equal 10
For endurance limit
+
+=nFFE
C m02ρt31
σconstY
Classification Notes- No 412 29 May 2003
DET NORSKE VERITAS
For static strength
+
+=nFFst
Cst m02ρt31
σconstY
where const and t are connected as
Hardening process
t = endurance limit const =
static strength const =
t550 640 1900
t400 500 1200 Case hard-ening
t300 380 800
Nitriding t400 500 1200
Induction- or flame hardening
tHVmin 11 HVmin 25 HVmin
For symbols see 213
In addition to these requirements to minimum case depths for endurance limit some upper limitations apply to case hard-ened gears
The max depth to 550 HV should not exceed
1) 13 of the top land thickness san unless adequate tip relief is applied (see 110)
2) 025 mn If this is exceeded the following applies addi-tionally in connection with endurance limit
minusminus= 025
mt
1Yn
max550C
314 Thin rim factor YB Where the rim thickness is not sufficient to provide full sup-port for the tooth root the location of a bending failure may be through the gear rim rather than from root fillet to root fillet
YB is not a factor used to convert calculated root stresses at the 30deg tangent to actual stresses in a thin rim tension fillet Actually the compression fillet can be more susceptible to fatigue
YB is a simplified empirical factor used to de-rate thin rim gears (external as well as internal) when no detailed calcula-tion of stresses in both tension and compression fillets are available
Figure 34 Examples on thin rims
YB is applicable in the range 175 lt sRmn lt 35
YB = 115 middot ln (8324 middot mnsR)
(for sRmn ge 35 YB = 1)
(for sRmn le 175 use 315)
σF as calculated in 321 is to be multiplied with YB when sRmn lt 35 Thus YB is used for both high and low cycle fatigue
Note This method is considered to be on the safe side for external gear rims However for internal gear rims without any flange or web stiffeners the method may not be on the safe side and it is advised to check with the method in 315316
315 Stresses in Thin Rims For rim thickness sR lt 35 mn the safety against rim cracking has to be checked
The following method may be used
3151 General The stresses in the standardised 30ordm tangent section tension side are slightly reduced due to decreasing stress correction factor with decreasing relative rim thickness sRmn On the other hand during the complete stress cycle of that fillet a certain amount of compression stresses are also introduced The complete stress range remains approximately constant Therefore the standardised calculation of stresses at the 30ordm tangent may be retained for thin rims as one of the necessary criteria
The maximum stress range for thin rims usually occurs at the 60ordm ndash 80ordm tangents both for laquotensionraquo and laquocompressionraquo side The following method assumes the 75ordm tangent to be the decisive Therefore in addition to the am criterion ap-plied at the 30ordm tangent it is necessary to evaluate the max and min stresses at the 75ordm tangent for both laquotensionraquo (loaded flank) fillet and laquocompressionraquo (back-flank) fillet For this purpose the whole stress cycle of each fillet should be considered but usually the following simplification is justified
Figure 35 Nomenclature of fillets
Index laquoTraquo means laquotensileraquo fillet laquoCraquo means laquocompressionraquo fillet
σFTmin and σFCmax are determined on basis of the nominal rim stresses times the stress concentration factor Y75
σFCmin and σFTmax are determined on basis of superposition of nominal rim stresses times Y75 plus the tooth bending stresses at 75ordm tangent
30 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
3152 Stress concentration factors at the 75ordm tangents The nominal rim stress consists of bending stresses due to local bending moments tangential stresses due to the tan-gential force Ft and radial shear stresses due to Fr
The major influence is given by the bending stresses The influence of the tangential stresses is minor and even though its stress concentration factor is slightly higher than for bending it is considered to be safe enough when the sum of these nominal stresses are combined with the stress concen-tration factor for bending The influence of the radial shear stress is neglected
The stress concentration factor relating nominal rim stresses to local fillet stresses at the 75ordm tangent may be calculated as
75n
R75
n
R
75
ρmsρ185
ms3
Y+
sdot=
where ρ75 is the root radius at the 75ordm tangent ref to mn Usually ρ75 is closed to the tool radius ρao and
ρ75 = ρao
is a safe approximation compensating for the am simplifi-cations to the laquounsaferaquo side
The tooth root stresses of the loaded tooth are decreasing with decreasing relative rim thickness approximately with the empirical factor
15
R
ncorr s
m311Y
minus=
3153 Nominal rim stresses The bending moment applied to the rim consists of a part of the tooth tilt moment Ft (hF + 05 sR) and the bending caused by the radial force Fr
The sectional modulus (first moment of area) which is used for determination of the nominal bending stresses is not nec-essarily the same for the 2 am bending moments If flanges webs etc outside the toothed section contribute to stiffening the rim against various deflections the influence of these stiffeners should be considered Eg an end flange will have an almost negligible influence on the effective sectional modulus for the stresses due to tooth tilt as the deflection caused by the tilt moment is rather small and would not much involve the flange On the other hand the radial forces as for instance from the meshes in an annulus would cause considerable radial deflections that the flange might restrict to a substantial amount When considering the stiff-ening of such flanges or webs on basis of simplified models it is advised to use an effective rim thickness sR = sR + 02 mn for the first moment of area of the rim (toothed part) cross section
For a high number of rim teeth it may be assumed that the rim bending moments in the fillets adjacent to the loaded tooth are of the same magnitude as right under the applied
force This assumption is reasonable for an annulus but rather much on the safe side for a hollow pinion
The influence of Ft on the nominal tangential stress is simpli-fied by half of it for compressive stresses (σ1) and the other half for tensile stresses (σ2) Applying these assumptions the nominal rim stresses adjacent to the loaded tooth are
( ) ( )A2
FW
f RFW
s05hF05 t
R
r
T
RFt1 minus
ϑminus
+minus=σ
( ) ( )A2
FW
f RFW
s05hF05 t
R
r
T
RFt2 +
ϑminus
+=σ
where σ1 σ2 see Fig 35
A = minimum area of cross section (usually bR sR)
WT = the sectional modulus of rim with respect to tooth tilt moment (usually 6sb 2
rR )
WR = the sectional modulus of the rim including the influence of stiffeners as flanges webs etc (WR ge WT)
bR = the width of the rim
R = the radius of the neutral axis in the rim ie from wheel centre to midpoint of rim
( )ϑf = a function for bending moment distribution around the rim For a rim (pinion) with one mesh only the ( )ϑf at the position of load application is 024 For an annulus with 3 or more meshes ( )ϑf at the position of each load application is approx 3 planets ( )ϑf = 019 4 planets ( )ϑf = 014 5 planets ( )ϑf = 011 6 planets ( )ϑf = 009
It must be checked if the max (tensile) stress in the compres-sion fillet really occurs when the fillet is adjacent to the loaded tooth In principle the stress variation through a complete rotation should be considered and the max value used The max value is usually never less than 0 For an annulus eg the tilt moment is zero in the mid position be-tween the planet meshes whilst the bending moment due to the radial forces is half of that at the mesh but with opposite sign
If these formulae are applied to idler gears as eg planets the influence of nominal tangential stresses must be cor-rected by deleting Ft(2 A) for σ1 and using FtA for σ2 Further the influence of Fr on the nominal bending stresses is usually negligible due to the planet bearing support
3154 Root fillet stresses Determination of min and max stresses in the laquotensionraquo fillet
Classification Notes- No 412 31 May 2003
DET NORSKE VERITAS
Minimum stress
751FTmin YσK σ =
Maximum stress
corrF752 Yσ03Yσ KσFTmax +=
where
03 is an empirical factor relating the tension stresses (σF) at the 30ordm tangent to the part of the tension stresses at the 75ordm tangent which add to the rim related stresses (03 also takes into account that full superposition of nominal stresses times stress concentration factors from both laquosides of the corner filletraquo would result in too high stresses)
αβγ sdotsdotsdotsdot= FFvA KKKKKK
Determination of min and max stresses in the laquocompres-sionraquo fillet
Minimum stress
corrF751FCmin Yσ036Yσ Kσ minus=
Maximum stress
752FCmax Yσ Kσ =
where
036 is an empirical factor relating the tension stresses (σF) at the 30ordm tangent to the part of the compression stresses at the 75ordm tangent which add to the rim related stresses
For gears with reversed loads as idler gears and planets there is no distinct laquotensionraquo or laquocompressionraquo fillet The mini-mum stress σFmin is the minimum of σFTmin and σFCmin (usually the latter is decisive) The maximum stress σFTmax is the maximum of σFTmax and σFCmax (usually the former is deci-sive)
316 Permissible Stresses in Thin Rims
3161 General The safety against fatigue fracture respectively overload fracture is to be at least at the same level as for solid gears The laquoordinaryraquo criteria at the 30ordm tangent apply as given in 31 through 313
Additionally the following criteria at the 75ordm tangent may apply
3162 For gt3middot106 cycles The permissible stresses for the laquotensionraquo fillets and for the laquocompressionraquo fillets are determined by means of a relevant fatigue diagram
If the actual tooth root stress (tensile or compressive) ex-ceeds the yield strength to the material the induced residual stresses are to be taken into account
For determination of permissible stresses the following is defined
R = stress ratio ie FTmax
FTminσσ respectively
FCmax
FCmin
σσ
∆σ = stress range ie σFTmax ndash σFTmin resp σFCmax ndash σFCmin
(For idler gears and planets Fmax
FminσσR =
and ∆σ = σFmax minus σFmin)
The permissible stress range ∆σρ for the laquotensionraquo respec-tively laquocompressionraquo fillets can be calculated as
For R gt minus1 FPp σ
R1R1031
13∆σ
minus+
+=
For minus infin lt R lt minus1 FPp σ
R1R10151
13∆σ
minus+
+=
where
σFP = see 32 determined for unidirectional stresses (YM = 1)
If the yield strength σy is exceeded in either tension or com-pression residual stresses are induced This may be consid-ered by correcting the stress ratio R for the respective fillets (tension or compression)
Eg if yFCmin σσ gt (ie exceeded in compression) the
difference yFCmin σσ∆ minus= affects the stress ratio as
∆σ
σ∆σ∆σR
FCmax
y
FCmax
FCmin
+
minus=
++
=
Similarly the stress ratio in the tension fillet may require cor-rection
If the yield strength is exceeded in tension σFTmax gt σy the difference ∆ = σFTmax ndash σy affects the stress ratio as
y
FTmin
FTmax
FTmin
σ∆σ
∆σ∆σR minus=
minusminus
=
Checking for possible exceeding of the yield strength has to be made with the highest torque and with the KAP if this ex-ceeds KA
3163 Forle 103 cycles The permissible stress range ∆σp is not to exceed
For R gt minus1 FPstpst σ
R1R1051
15∆σ
minus+
+=
For ndash infin lt R lt minus1 FPstpst σ
R1R10251
15∆σ
minus+
+=
For all values of R ∆σpst is limited by
not surface hardened F
y
Sσ225
32 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
surface hardened CF
YSHV5
Definition of ∆σ and R see 3162 with particular attention to possible correction of R if the yield strength is exceeded
σFPst see 32 determined for unidirectional stresses (YM = 1) and lt 103 cycles
3164 For 103 lt cycles lt 3106 ∆σp is to be determined by linear interpolation a log-log dia-gram
∆σp at NL load cycles is
6L 103p
exp
L
6
N p ∆σN103∆σ sdot
sdot=
6
3
103p
10p
∆σ
∆σlog28760exp
sdot
=
Classification Notes- No 412 33 May 2003
DET NORSKE VERITAS
4 Calculation of Scuffing Load Capacity
41 Introduction High surface temperatures due to high loads and sliding ve-locities can cause lubricant films to break down This sei-zure or welding together of areas of tooth surface is termed scuffing
In contrast to pitting and fatigue breakage which show a dis-tinct incubation period a single short overloading can lead to a scuffing failure In the ISO-TR13989 two criteria are mentioned The method used in this Classification Note is based on the principles of the flash temperature criterion
Note Bulk temperature in excess of 120ordmC for long periods may have an adverse effect on the surface durability see 211
42 General Criteria In no point along the path of contact the local contact tem-perature may exceed the permissible temperature ie
oilS
oilSB S
ϑ+ϑminusϑ
leϑ
50SB minusϑleϑ
where
Bϑ = max contact temperature along the path of contact
maxflaMBB ϑ+ϑ=ϑ
MBϑ = bulk temperature see 434
maxflaϑ = max flash temperature along the path of con-tact see 44
Sϑ = scuffing temperature see below
oilϑ = oil temperature before it reaches the mesh (max applicable for the actual load case to be used ie normally alarm temperature except for ice classes where a max expected tem-perature applies)
SS = required safety factor according to the Rules
The scuffing temperature Sϑ may be calculated as
L2
wrelT
002
40S XFZGX
ν100112085780
sdot
sdot++=ϑ
where
XwrelT = relative welding factor
XwrelT
Through hardened steel 10
Phosphated steel 125
Copper-plated steel 150
Nitrided steel 150
Less than 10 retained austenite
115
10 ndash 20 retained austenite 10
Casehardened steel
20 ndash 30 retained austenite 085
Austenitic steel 045
FZG = load stage according to FZG-Test A8390 (Note This is the load stage where scuffing oc-curs However due to scatter in test results cal-culations are to be made with one load stage less than the specification)
XL = lubricant factor
= 10 for mineral oils
= 08 for polyalfaolefins
= 07 for non-water-soluble polyglycols
= 06 for water-soluble polyglycols
= 15 for traction fluids
= 13 for phosphate esters
ν40 = kinematic oil viscosity at 40˚C (mm2s)
Application of other test methods such as the Ryder the FZG-Ryder R46574 and the FZG L-42 Test 141195110 may be specially considered
For high speed gears with very short time of contact Sϑ may be increased as follows provided use of EP-oils
Addition to the calculated scuffing temperature Sϑ
If micros 18tc ge no addition
If micros 18tc lt add ( )cwrelT t18X18 minussdotsdot
where
ct = contact time ( micros ) which is the time needed to cross the Hertzian contact width
( ) [ ]microsu1cosβn
uσ340tb1
Hc +sdotsdot
sdot=
σH as calculated in 221
For bevel gears use vu in stead of u
34 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
43 Influence Factors
431 Coefficient of friction The following coefficient of friction may apply
L025a
005oil
02
redC ΣC
Bt X R ηρv
w0048micro minus
=
where
wBt = specific tooth load (Nmm)
vΣC = sum of tangential velocities at pitch point At pitch line velocities gt 50 ms the limiting value of vΣC at v = 50 ms is to be used
ρredC = relative radius of curvature (transversal plane) at the pitch point
Cylindrical gears
HαHβvγAbt
Bt KKKKKb
Fw sdotsdotsdotsdotsdot= (see 1)
twΣC αsin v2v =
bCredC β cos ρρ =
Bevel gears
HαHβvγAtmb
Bt KKKKKb
Fw sdotsdotsdotsdotsdot= (see 1)
vtmtΣC αsin v2v =
bmvCredC β cos ρρ =
ηoil = dynamic viscosity (mPa s) at oilϑ calculated as
1000
ρ νη oiloil =
where ρ in kgm3 approximated as
( ) 70 15oil15 minusϑminusρ=ρ and νoil is kinematic viscosity at oilϑ and may be calculated by means of the following equation
( ) ( )++=+ 08νloglog08νloglog 100oil ( )
sdotminus
ϑ+minus313log373log
273log373log oil
( ) ( )( )08νloglog08νloglog 10040 +minus+
Ra = composite arithmetic mean roughness (micron) of pinion an wheel calculated as ( )2a1aa RR 05R +=This is defined as the roughness on the new flanks ie as manufactured
XL = see 42
432 Effective tip relief Ceff Ceff is the effective tip relief that amount of tip relief which compensates for the elastic deformation of the gear mesh ie
zero load at the tooth tip It is assumed (simplified) to be equal for both pinion and wheel
Cylindrical gears
for helical
γ
γ=cb
KKFC Abt
eff (see 1)
For spur
cbKKF
C Abteff
γ= (see 1)
Alternatively for spur and helical gears the non-linear ap-proach in 111 may be used (taking Ceff=δ)
Bevel gears
bcKFC Ambt
effγ
= (see 1)
where
γ
α
ε2ε44c
+sdot
=γ
433 Tip relief and extension Cylindrical gears
The extension of the tip relief is not to result in an effective contact ratio 1ltεα when the gear is unloaded (exceptions to this may only apply for applications where the gear is not to run at light loads) This means that the unrelieved part of the path of contact is to be minimum pbt It is further assumed that this unrelieved part is placed centrally on the path of contact
If root relief applies it has to be calculated as equivalent tip relief Ie pinion root relief (at mesh position A) is added to Ca2 and wheel root relief (at mesh position E) is added to Ca1 If no design tip relief or root relief on the mating gear is specified (ie if Ca1+Croot2 = 0 and visa versa) use the run-ning in amount see 112
Bevel gears
Bevel gears are to have heightwise crowning ie no distinct relievedunrelieved area This may be treated as tip and root relief For calculation purposes the root relief is combined with the tip relief of the mating member into an equivalent tip relief If no resulting tip relieves are specified the equivalent tip relives may be calculated as an approxima-tion based on the tool crowning Ca tool (per mille of tool module m0) as follows
2root1aeq1a CCC +=
1root2aeq2a CCC +=
where
Classification Notes- No 412 35 May 2003
DET NORSKE VERITAS
2
0
n0101atoolal m
)m2(mA1middotmCC
minus+minussdot=
( )
sdotminusminussdotsdotminus+= vtvb1
2vb1
21vavt1n1 αtan dddαsin 05x1mA
2
0
n020atool22a m
)mm(2A1middotmCC
minus+minussdot=
( )
sdotminusminussdotsdotminus+= vtvb2
2vb2
2va2vt2n2 αtan dddαsin 05x1mA
2
0
20atool1root1 m
A1mCC
minussdotsdot=
2
0
10atool2root2 m
A1mCC
minussdotsdot=
If the pressure angles of the cutter blades are modified (and verified) to balance the tip relieves the following may be assumed
( )eq2aeq1aeq2aeq1a C andC calculated of sum50CC sdot==
Throughout the following Ca1 and Ca2 mean the equivalent tip relieves Ca1eq and Ca2eq
434 Bulk temperature The bulk temperature may be calculated as
flaaveragempsoilMB X X 50 ϑ+ϑ=ϑ
where
Xs = lubrication factor
= 12 for spray lubrication
= 10 for dip lubrication (provided both pinion and wheel are dip lubricated and tip speed lt 5 ms)
= 10 for spray lubrication with additional cooling spray (spray on both pinion and wheel or spray on pinion and dip lubrication of wheel)
= 02 for meshes fully submerged in oil
Xmp = contact factor ( )pmp nX += 150
np = number of mesh contact on the pinion (for small gear ratii the number of wheel meshes should be used if higher)
flaaverageϑ
= average of the integrated flash temperature (see 44) along the path of contact
( )
AE
E
Ayyyfla
flaaverage
d
ΓminusΓ
ΓΓϑ=ϑint =
For high speed gears (v gt 50 ms) it may be necessary to assess the bulk temperature on the basis of a thermal rating of the entire gear transmission
44 The Flash Temperature flaϑ
441 Basic formula The local flash temperature flaϑ may be calculated as
( ) 41redy
y2ly
211
43Btcorrfla
unXwX3250
y ρ
ρminusρ
micro=ϑ Γ
(For bevel gears replace u with uv)
and is to be calculated stepwise along the path of contact from A to E
where
micro = coefficient of friction see 431
Xcorr = correction factor taking empirically into account the increased scuffing risk in the beginning of the approach path due to mesh starting without any previously built up oil film and possible shuffling away oil before meshing if insufficient tip relief
( )
3
AD
yaeffcorr 50
CC1X
ΓminusΓε
Γminus+=
α
Ca = tip relief of driven member Xcorr is only applicable in the approach path and if Ca lt Ceff otherwise 10
wBt = unit load see 431
yXΓ = load sharing factor see 443
n1 = pinion rpm
Γy ρly etc see 442
442 Geometrical relations The various radii of flank curvature (transversal plane) are
ρ1y = pinion flank radius at mesh point y
ρ2y = wheel flank radius at mesh point y
ρredy = equivalent radius of curvature at mesh point y
y2y1
y2y1redy
ρ+ρ
ρρ=ρ
Cylindrical gears
twy
1y αsin au1Γ1
ρ+
+=
twy
2y αsin au1Γu
ρ+
minus=
Note that for internal gears a and u are negative
36 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
Bevel gears
vtvv
y1y αsin a
u1Γ1
ρ+
+=
vtvv
yv2y αsin a
u1Γu
ρ+
minus=
Γ is the parameter on the path of contact and y is any point between A end E
At the respective ends Γ has the following values
Root piniontip wheel
Cylindrical gears
( )
minus
minusminus= 1
αtan 1dd
zzΓ
tw
2b2a2
1
2A
Bevel gears
( )
minus
minusminus= 1
αtan 1dd
uΓvt
2vb2va2
vA
Tip pinionroot wheel
Cylindrical gears
( )
1αtan
1ddΓ
tw
2b1a1
E minusminus
=
Bevel gears
( )
1αtan
1ddΓ
vt
2vb1va1
E minusminus
=
At inner point of single pair contact
Cylindrical gears
tw1
EB α tan zπ2ΓΓ minus=
Bevel gears
vtv1
EB α tan zπ2ΓΓ minus=
At outer point of single pair contact
Cylindrical gears
tw1AD α tan z
π2ΓΓ +=
Bevel gears
vt v1
AD αtan z π2ΓΓ +=
At pitch point ΓC = 0
The points F and G (only applicable to cylindrical gears) limiting the extension of tip relief (so as to maintain a mini-mum contact ratio of unity for unloaded gears) are at
2
BAF
Γ+Γ=Γ
2
EDG
Γ+Γ=Γ
443 Load sharing factor XΓ The load sharing factor XΓ accounts for the load sharing be-tween the various pairs of teeth in mesh along the path of contact
XΓ is to be calculated stepwise from A to E using the pa-rameter Γy
4431 Cylindrical gears with β = 0 and no tip relief
Figure 41
ByAAB
Ay for31
31X
yΓltΓleΓ
ΓminusΓ
ΓminusΓ+=Γ
DyB for1Xy
ΓltΓleΓ=Γ
EyDDE
yE for31
31X
yΓleΓltΓ
ΓminusΓ
ΓminusΓ+=Γ
4432 Cylindrical gears with β = 0 and tip relief Tip relief on the pinion reduces XΓ in the range G ndash E and increases correspondingly XΓ in the range F ndash B
Tip relief on the wheel reduces XΓ in the range A ndash F and increases correspondingly XΓ in the range D ndash G
Following remains generally
DyB for1Xy
ΓleΓleΓ=Γ
21XXGF== ΓΓ
In the following it must be distinguished between Ca lt Ceff respectively Ca gt Ceff This is shown by an example below where Ca1 lt Ceff and Ca2 gt Ceff
Classification Notes- No 412 37 May 2003
DET NORSKE VERITAS
Figure 42
Note When Ca gt Ceff the path of contact is shortened by A ndash A respectively E ndash E The single pair contact path is extended into B respectively D If this shift is significant it is necessary to consider the negative effect on surface durability (B) and bending stresses (B and D)
Range A - F
For effa2 CC le
minus=Γ
eff
2a
CC1
31X
A
FyAeff
2a
AF
Ay for C3
C61XX
AyΓleΓleΓ
sdot
+ΓminusΓ
ΓminusΓ+= ΓΓ
For effa2 CC ge
AyAfor0Xy
ΓleΓleΓ=Γ
with ( )21
CC
1CC
eff
2a
eff
2a
AFAAminus
minusΓminusΓ+Γ=Γ
FyAeff
2a
AF
Ay
eff
2a for 21
CC
CC1X
yΓleΓleΓ
minus
ΓminusΓ
ΓminusΓ+minus=Γ
Range F - B
For effa1 CC le
ByFeff
1a
FB
Fy forC3
C61
21X
yΓleΓleΓ
sdot
+ΓminusΓ
ΓminusΓ+=Γ
For effa1 CC ge
ByFeff
1a
FB
Fy for21
CC
21X
yΓleΓleΓ
minus
ΓminusΓ
ΓminusΓ+=Γ
ByB for1Xy
ΓleΓleΓ=Γ
with 1
CC2
eff
1a
FBFB
minus
ΓminusΓ+Γ=Γ
Range D ndash G
For effa2 CC le
GyDeff
2a
DG
Dy
eff
2a for C 3C
61
C 3C
32X
yΓleΓleΓ
+
ΓminusΓ
ΓminusΓminus+=Γ F
or effa2 CC ge
DyD for1Xy
ΓleΓleΓ=Γ
with ( )21
CC
1CC
eff
2a
eff
2a
DGDDminus
minusΓminusΓ+Γ=Γ
GyDeff
2a
DG
Dy
eff
2a for21
CC
CCX ΓleΓleΓ
minus
ΓminusΓ
ΓminusΓminus=Γ
Range G - E
For effa1 CC le
minus=Γ
eff
1a
CC1
31X
E
EyGeff
1a
GE
Gy forC3
C61
21X
yΓleΓleΓ
sdot
+ΓminusΓ
ΓminusΓminus=Γ
For effa1 CC gt
EyGeff
1a
GE
Gy for21
CC
21X
yΓleΓleΓ
minus
ΓminusΓ
ΓminusΓminus=Γ
EyE for0Xy
ΓleΓleΓ=Γ
with 1
CC2
eff
1a
GEGE
minus
ΓminusΓ+Γ=Γ
4433 Gears with β gt 0 buttressing Due to oblique contact lines over the flanks a certain but-tressing may occur near A and E
This applies to both cylindrical and bevel gears with tip relief lt Ceff The buttressing Xbutt is simplified as a linear function within the ranges A ndash H respectively I ndash E
Figure 43
1εwhen13X βbutt EAge=
1εwhenε031X ββbutt EAlt+=
38 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
Cylindrical gears
bIEAH βsin 02ΓΓΓΓ =minus=minus
Bevel gears
bmIEAH βsin 02ΓΓΓΓ =minus=minus
4434 Cylindrical gears with 2εγ le and no tip relief
yXΓ is obtained by multiplication of
yXΓ in 4431 with
Xbutt in 4433
4435 Gears with 2εγ gt and no tip relief
Applicable to both cylindrical and bevel gears
IyH for1Xy
ΓleΓleΓε
=α
Γ
IyHybutt and forX1Xy
ΓgtΓΓltΓε
=α
Γ
Figure 44
4436 Cylindrical gears with 2εγ le and tip relief
yXΓ is obtained by multiplication of
yXΓ in 4432 with Xbutt
in 4433
4437 Cylindrical gears with 2εγ gt and tip relief
Tip relief on the pinion (respectively wheel) reduces XГ in the range G ndash E (respectively A ndash F) and increases XГ in the range F ndash G
yXΓ is obtained by multiplication of
yXΓ as described below
with Xbutt in 4433
In the XГ example below the influence of tip relief is shown (without the influence of Xbutt) by means of
eff2aeff1a CC and CC ltgt
Tip relief gt Ceff causes new end points A respectively E of the path of contact
Figure 45
Range A ndash F
( ) ( )( ) eff
a2αa1α
AF
Ay
eff
2aeff
C 12C 13εC 1ε
CCCX
y +εε++minus
ΓminusΓ
ΓminusΓ+
εminus
=ααα
Γ
eff2aFyA CC if for leΓleΓleΓ
eff2aFyA CifCfor and geΓleΓleΓ
eff2aAyA CC iffor0Xy
gtΓleΓleΓ=Γ
( ) ( ) ( )( ) ( ) a2αa1α
αeffa2AFAA C 1ε 3C 1ε
1ε 2 CC with++minus+minus
ΓminusΓ+Γ=Γ
Range F ndash G
( )( )( ) GyF
eff
2a1a forC 1 2CC 11X
yΓleΓleΓ
+εε+minusε
+ε
=αα
α
αΓ
Range G ndash E
( ) ( )( ) eff
2a1a
GE
Gy
C 1 2C 1C 1 3XX
GFy +εεminusε++ε
ΓminusΓ
ΓminusΓminus=
αα
ααΓΓ minus
effa1EyG CC if for leΓleΓleΓ
effa1EyG CC if Γfor and geΓleΓle
eff1aEyE CC if for0Xy
geΓleΓleΓ=Γ
( ) ( ) ( )( ) ( ) 2a1a
eff1aGEEE C 1C 1 3
1 2 CC withminusε++ε+εminus
ΓminusΓminusΓ=Γαα
α
4438 Bevel gears with εγ more than approx 18 and heightwise crowning For Ca1 = Ca2 = Ceff the following applies
Figure 46
Classification Notes- No 412 39 May 2003
DET NORSKE VERITAS
( )AEM 50 Γ+Γ=Γ
( )( )2AD
3
2My651X
y ΓminusΓε
ΓminusΓminus
ε=
ααΓ
For tip relief lt Ceff yXΓ is found by linear interpolation be-
tween ( )effay CCX =Γ and ( )0CayX =Γ as in 4435
The interpolation is to be made stepwise from A to M with the influence of Ca2 and from M to E with the influence of Ca1 (For 2a1a CC ne there is a discontinuity at M)
Eg with Ca1 = 04 Ceff and Ca2 = 055 Ceff then
Range A ndash M
)CC()0C( eff2ay2ayyX 550X 450X =Γ=ΓΓ +=
Range M ndash E
)CC()0C( eff1ay1ayyX 40X 06X =Γ=ΓΓ +=
For tip relief gt Ceff the new end points A and E are found as
( )
minusΓminusΓ
ε+Γ=Γ α 1
CC
6 eff
2aADAA
( )
minusΓminusΓ
ε+Γ=Γ α 1
CC
6 eff
1aADEE
Range A ndash A
0Xy=Γ
Range A ndash M
( )( )
ΓminusΓ
ΓminusΓminus
minusε=
αΓ 2
MA
2My
eff
2a1
CC4
351Xy
Range M ndash E
( )( )
ΓminusΓ
ΓminusΓminus
minusε=
αΓ 2
ME
2My
eff
1a1
CC4
351Xy
Range E ndash E
0Xy=Γ
40 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
Appendix A Fatigue Damage Accumulation
The Palmgren-Miner cumulative damage calculation princi-ple is used The procedure may be applied as follows
A1 Stress Spectrum From the individual torque classes the torques (Ti) at the peak values of class intervals and the associated number of cycles (NLi) for both pinion and wheel are to be listed from the highest to the lowest torque
(In case of a cyclic torque variation within the torque classes it is advised to use the peak torque If the cyclic variation is such that the same teeth will repeatedly suffer the peak torque this is a must)
The stress spectra for tooth roots and flanks (σFi σHi) with all relevant factors (except KA) are to be calculated on the basis of the torque spectrum The load dependent K-factors are to be determined for each torque class
A2 σminusN-curve The stress versus load cycle curves for tooth roots and flanks (both pinion and wheel) are to be drawn on the basis of per-missible stresses (ie including the demanded minimum safety factors) as determined in 2 respectively 3 If different safety levels for high cycle fatigue and low cycle fatigue are desired this may be expressed by different demand safety factors applied at the endurance limit respectively at static strength
A3 Damage accumulation The individual damage ratio Di at ith stress level is defined as
Fi
Lii N
ND =
where
NLi = The number of applied cycles at ith stress
NFi = The number of cycles to failure at ith stress
Basically stresses σi below the permissible stress level for infinite life (if a constant ZN or YN is accepted) do not con-tribute to the damage sum However calculating the actual safety factor Sact as described below all the σi for which the product S σi is bigger than or equal to the permissible stress level for infinite life contribute to the damage sum and thus to the determination of Sact The final value of S is decisive
(NFi can be found mathematically by putting the permissible stress σpi equal the actual stress σi thereby finding the actual life factor This life factor can be solved with regard to load cycles ie NFi)
The damage sum ΣDi is not to exceed unity
If ΣDi ne 1 the safety against cumulative fatigue damage is different from the applied demand safety factor For deter-mination of this theoretical safety factor an iteration proce-dure is required as described in the following flowchart
S is correction factor with which the actual safety factor Sact can be found
Sact is the demand safety factor (used in determination of the permissible stresses in the σ ndash N ndash curve) times the correc-tion factor S
The full procedure is to be applied for pinion and wheel tooth roots and flanks
Note If alternating stresses occur in a spectrum of mainly pulsating stresses the alternating stresses may be replaced by equiva-lent pulsating stresses ie by means of division with the ac-tual mean stress influence factor YM
Classification Notes- No 412 41 May 2003
DET NORSKE VERITAS
Appendix B Application Factors for Diesel Driven Gears
For diesel driven gears the application factor KA depends on torsional vibrations Both normal operation and misfiring conditions have to be considered
Normally these two running conditions can be covered by only one calculation
B1 Definitions
Normal operation KAnorm = 0
normv0
TTT +
where
T0 = rated nominal torque
Tv norm = vibratory torque amplitude for normal opera-tion (see rules Pt4 Ch3 Sec1 G301 for defi-nition of ldquonormalrdquo irregularity)
Misfiring operation KA misf = 0
misfv
TTT +
where
T = remaining nominal torque when one cylinder out of action
Tv misf = vibratory torque amplitude in misfiring con-dition This refers to a permissible misfiring condition ie a condition that does not require automatic or immediate corrective actions as speed or pitch reduction
The normal operation is assumed to last for a very high num-ber of cycles such as 1010
The misfiring operation is assumed to last for a limited dura-tion such as 107 cycles
B2 Determination of decisive load Assuming life factor at 1010 cycles as YN = ZN = 092 which usually is relevant the calculation may be performed only once with the combination having the highest value of appli-cation factorlife factor
For bending stresses and scuffing the higher value of
092
K normA and 098
K misfA
For contact stresses the higher value of
092K normA and
113K misfA (but
097K misfA for nitrided gears)
B3 Simplified procedure Note that this is only a guidance and is not a binding con-vention
T = oTZ
1Zsdot
minus where Z = number of cylinders
( ) ideal Videal Vmisf Vnorm V TTT24ZT +minus=
where TV ideal = vibratory torque with all cylinders perfectly equal See also rules Pt4 Ch3 Sec1 G300
When using trends from torsional vibration analysis and measurements the following may be used
TV ideal To is close to zero for engines with few cylinders and using a suitable elastic coupling and increases with relative coupling stiffness and number of cylinders
TV misf To may be high for engines with few cylinders and decreases with number of cylinders
This can be indicated as
200Z
TT
ο
idealV asymp and 80Z04
TT
o
misfV minusasymp
Inserting this into the formulae for the two application fac-tors the following guidance can be given
118112K misfA minusasymp
115110K normA minusasymp
Since KA norm is to be combined with the lower life factors the decisive load condition will be the normal one and a KA of 115 will cover most relevant cases when a suitable elas-tic coupling is chosen
42 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
Appendix C Calculation of Pinion-Rack
Pinion-racks used for elevating of mobile offshore units are open gears that are subjected to wear and tear With normal specifications such as only a few hundred total operation cycles (site to site) the tooth bending stresses for static strength will be decisive for the lay out however with ex-ception of surface hardened pinions where case crushing has to be considered
In the following the use of part 1 and 3 for pinion-racks is shown including relevant simplifications
C1 Pinion tooth root stresses Since the load spectrum normally is dominated by high torques and few load cycles (in the range up to 1-2000) the static strength is decisive
The actual stress is calculated as
β1FSaFan1
tF1 KYY
mbFσ sdotsdotsdotsdot
=
b1 is limited to b2 + 2middotmn
YFa and YSa replace YF and YS because load application at tooth tip has to be assumed for such inaccurate gears
Pinions often use a non-involute profile in the dedendum part of the flank eg a constant radius equal the radius of curva-ture at reference circle For such pinions sFn and hFa are to be measured directly on a sectional drawing of the pinion tooth
Due to high loads and narrow facewidths it may be assumed that KFβ2 = KHβ = 10 However when b1 gt b2 then KFβ1 gt10
If no detailed documentation of KFβ1 is available the fol-lowing may be used
KFβ1 = 1 + 015middot(b1b2 ndash 1)
The permissible stress (not surface hardened) is calculated as
δrelTstF
Fst1FP1 Y
Sσσ sdot=
The mean stress influence due to leg lifting may be disre-garded
The actual and permissible stresses should be calculated for the relevant loads as given in the rules
C2 Rack tooth root stresses The actual stress is calculated as
SaFan2
tF2 YY
mbFσ sdotsdotsdot
=
See C1 for details
The permissible stress is calculated as
δrelTstF
Fst2FP2 Y
Sσσ sdot=
For alloyed steels (Ni Cr Mo) with high toughness and duc-tility the value of YδrelTst may be put equal to YSa
C3 Surface hardened pinions For surface hardened pinions the maximum load is not to cause crushing of the hardened layer of the flank
In principle the calculation described in 213 may be used but when the theoretical Hertzian stress exceeds the ap-proximately 18 times the yield strength of the rack material plastic deformation will occur This will limit the peak Hertzian stress but increases the contact width and thus the penetration of stresses into the depth
An approximation may be based on an assessment of contact width determined by means of equal areas under the theoreti-cal (elastic range) Hertzian contact and the elasto-plastic contact stress (limited to 18 middot σy) with the unknown width
16 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
For all kinds of steel
5145189718
1C2
limHay +
minusσ
=
When pinion and wheel material differ the following ap-plies
bull Use the larger of fpt1 - yα1 and fpt2 - yα2 to replace fpt - yα in the calculation of KHα and Kv
bull Use ( )2β1ββ yy21y += in the calculation of KHβ
bull Use ( )2ay1aya CC21C += in the calculation of Kv
bull Use ( )2ay1ay2a1a CC21CC +== in the scuffing
calculation if no design tip relief is foreseen
Classification Notes- No 412 17 May 2003
DET NORSKE VERITAS
2 Calculation of Surface Durability
21 Scope and General Remarks Part 2 includes the calculations of flank surface durability as limited by pitting spalling case crushing and subsurface yielding Endurance and time limited flank surface fatigue is calculated by means of 22 ndash 212 In a way also tooth frac-tures starting from the flank due to subsurface fatigue is in-cluded through the criteria in 213
Pitting itself is not considered as a critical damage for slow speed gears However pits can create a severe notch effect that may result in tooth breakage This is particularly im-portant for surface hardened teeth but also for high strength through hardened teeth For high-speed gears pitting is not permitted
Spalling and case crushing are considered similar to pitting but may have a more severe effect on tooth breakage due to the larger material breakouts initiated below the surface Subsurface fatigue is considered in 213
For jacking gears (self-elevating offshore units) or similar slow speed gears designed for very limited life the max static (or very slow running) surface load for surface hard-ened flanks is limited by the subsurface yield strength
For case hardened gears operating with relatively thin lubri-cation oil films grey staining (micropitting) may be the lim-iting criterion for the gear rating Specific calculation meth-ods for this purpose are not given here but are under consid-eration for future revisions Thus depending on experience with similar gear designs limitations on surface durability rating other than those according to 22 - 213 may be ap-plied
22 Basic Equations Calculation of surface durability (pitting) for spur gears is based on the contact stress at the inner point of single pair contact or the contact at the pitch point whichever is greater
Calculation of surface durability for helical gears is based on the contact stress at the pitch point
For helical gears with 0 lt εβ lt 1 a linear interpolation be-tween the above mentioned applies
Calculation of surface durability for spiral bevel gears is based on the contact stress at the midpoint of the zone of contact
Alternatively for bevel gears the contact stress may be cal-culated with the program ldquoBECALrdquo In that case KA and Kv are to be included in the applied tooth force but not KHβ and KHα The calculated (real) Hertzian stresses are to be multi-plied with ZK in order to be comparable with the permissible contact stresses
The contact stresses calculated with the method in part 2 are based on the Hertzian theory but do not always represent the real Hertzian stresses
The corresponding permissible contact stresses σHP are to be calculated for both pinion and wheel
221 Contact stress Cylindrical gears
( )HαHβvγA
1
tβεEHDBH KKKKK
bud1uF
ZZZZZσ+
=
where
ZBD = Zone factor for inner point of single pair contact for pinion resp wheel (see 232)
ZH = Zone factor for pitch point (see 231)
ZE = Elasticity factor (see 24)
Zε = Contact ratio factor (see 25)
Zβ = Helix angle factor (see 26)
Ft KA Kγ Kv KHβ KHα see 15 ndash 110
d1 b u see 12 ndash 15
Bevel gears
( )HαHβvγA
v1v
vmtKEMH KKKKK
bud1uF
ZZZ105σ+
sdot=
where
105 is a correlation factor to reach real Hertzian stresses (when ZK = 1)
ZE KA etc see above
ZM = mid-zone factor see 233
ZK = bevel gear factor see 27
Fmt dv1 uv see 12 ndash 15
It is assumed that the heightwise crowning is chosen so as to result in the maximum contact stresses at or near the mid-point of the flanks
222 Permissible contact stress
XWRvLH
NHlimHP ZZZZZ
SZσ
σ =
where
σH lim = Endurance limit for contact stresses (see 28)
ZN = Life factor for contact stresses (see 29)
SH = Required safety factor according to the rules
ZLZvZR = Oil film influence factors (see 210)
ZW = Work hardening factor (see 211)
ZX = Size factor (see 212)
18 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
23 Zone Factors ZH ZBD and ZM
231 Zone factor ZH The zone factor ZH accounts for the influence on contact stresses of the tooth flank curvature at the pitch point and converts the tangential force at the reference cylinder to the normal force at the pitch cylinder
wtt2
wtbH sinααcos
cosαcosβ2Z =
232 Zone factors ZBD The zone factors ZBD account for the influence on contact stresses of the tooth flank curvature at the inner point of sin-gle pair contact in relation to ZH Index B refers to pinion D to wheel
For εβ ge 1 ZBD = 1
For internal gears ZD = 1
For εβ = 0 (spur gears)
( )
π
minusεminusminus
π
minusminus
α=
α2
2
2b
2a
1
2
1b
1a
wtB
z211
dd
z21
dd
tanZ
( )
π
minusεminusminus
π
minusminus
α=
α1
2
1b
1a
2
2
2b
2a
wtD
z211
dd
z21
dd
tanZ
If ZB lt 1 use ZB = 1
If ZD lt 1 use ZD = 1
For 0 lt εβ lt 1
ZBD = ZBD (for spur gears) ndash εβ (ZBD (for spur gears) ndash 1)
233 Zone factor ZM The mid-zone factor ZM accounts for the influence of the contact stress at the mid point of the flank and applies to spi-ral bevel gears
εminusminus
εminusminus
αβ=
αα btm2
2vb2
2vabtm2
1vb2val
2v1vvtbmM
pddpdd
ddtancos2Z
This factor is the product of ZH and ZM-B in ISO 10300 with the condition that the heightwise crowning is sufficient to move the peak load towards the midpoint
234 Inner contact point For cylindrical or bevel gears with very low number of teeth the inner contact point (A) may be close to the base circle In order to avoid a wear edge near A it is required to have suitable tip relief on the wheel
24 Elasticity Factor ZE The elasticity factor ZE accounts for the influence of the material properties as modulus of elasticity and Poissonrsquos ratio on the contact stresses
For steel against steel ZE = 1898
25 Contact Ratio Factor Zε The contact ratio factor Zε accounts for the influence of the transverse contact ratio εα and the overlap ratio εβ on the contact stresses
αε1Z =ε for 1εβ ge
( )α
ββ
αε ε
εε1
3ε4
Z +minusminus
= for εβ lt 1
26 Helix Angle Factor Zβ The helix angle factor Zβ accounts for the influence of helix angle (independent of its influence on Zε) on the surface du-rability
cosβZβ =
27 Bevel Gear Factor ZK The bevel gear factor accounts for the difference between the real Hertzian stresses in spiral bevel gears and the contact stresses assumed responsible for surface fatigue (pitting) ZK adjusts the contact stresses in such a way that the same per-missible stresses as for cylindrical gears may apply
The following may be used ZK = 080
28 Values of Endurance Limit σHlim and Static Strength 5H10σ 3H10σ
σHlim is the limit of contact stress that may be sustained for 5middot107 cycles without the occurrence of progressive pitting
For most materials 5middot107 cycles are considered to be the be-ginning of the endurance strength range or lower knee of the σ-N curve (See also Life Factor ZN) However for nitrided steels 2middot106 apply
For this purpose pitting is defined by
bull for not surface hardened gears pitted area ge 2 of total active flank area
bull for surface hardened gears pitted area ge 05 of total active flank area or ge 4 of one particular tooth flank area 510Hσ and 310Hσ are the contact stresses which the given
material can withstand for 105 respectively 103 cycles without subsurface yielding or flank damages as pitting spalling or case crushing when adequate case depth applies
The following listed values for σHlim 510Hσ and 310Hσ may only be used for materials subjected to a quality control as
Classification Notes- No 412 19 May 2003
DET NORSKE VERITAS
the one referred to in the rules Results of approved fatigue tests may also be used as the basis for establishing these values The defined survival probability is 99
σHlim σH105 σH10
3
Alloyed case hardened steels (surface hardness 58-63 HRC) - of specially approved high grade - of normal grade
1650 1500
2500 2400
3100 3100
Nitrided steel of approved grade gas nitrided (surface hardness 700-800 HV) 1250 13 σHlim 13 σHlim
Alloyed quenched and tempered steel bath or gas nitrided (surface hardness 500-700 HV)
1000
13 σHlim
13 σHlim
Alloyed flame or induction hardened steel (surface hardness 500-650 HV) 075 HV + 750 16 σHlim 45 HV
Alloyed quenched and tempered steel 14 HV + 350 16 σHlim 45 HV
Carbon steel 15 HV + 250 16 σHlim 16 σHlim
These values refer to forged or hot rolled steel For cast steel the values for σHlim are to be reduced by 15
29 Life Factor ZN The life factor ZN takes account of a higher permissible contact stress if only limited life (number of cycles NL) is demanded or lower permissible contact stress if very high number of cycles apply
If this is not documented by approved fatigue tests the fol-lowing method may be used
For all steels except nitrided
7L 105N sdotge ZN = 1 or
01570
L
7
N N105Z
sdot=
Ie ZN = 092 for 1010 cycles
The ZN = 1 from 5middot107 on may only be used when the mate-rial cleanliness is of approved high grade (see Rules Pt4 Ch2) and the lubrication is optimised by a specially approved filtering process
105 lt NL lt 5middot107 510N
logZ037
L
7
N N105Z
sdot=
NL = 105 WXRVLHlim
WstX10H1010NN ZZZZZσ
ZZσZZ
55
5=
103 lt NL lt 105 )Z(Zlog05
L
5
10NN
5N103N10
5N10ZZ
==
3L 10N le
WXRVLlimH
Wst10X10H10NN ZZZZZ
ZZZZ
33
3σ
σ==
(but not less than ZN105)
For nitrided steels
6L 102N sdotge ZN = 1 or
00980
L
6
N N102Z
sdot=
Ie ZN = 092 for 1010 cycles
The ZN = 1 from 2middot106 on may only be used when the mate-rial cleanliness is of approved high grade (see Rules Pt4 Ch2) and the lubrication is optimised by a specially approved filtering process
105 lt NL lt 2middot106 510N
Zlog76860
L
6
N N102Z
sdot=
5L 10N le
XWRVL
10XWst10NN ZZZZZ
ZZ13ZZ
55 ==
Note that when no index indicating number of cycles is used the factors are valid for 5middot107 (respectively 2middot106 for nitrid-ing) cycles
210 Influence Factors on Lubrication Film ZL ZV and ZR The lubricant factor ZL accounts for the influence of the type of lubricant and its viscosity the speed factor ZV ac-counts for the influence of the pitch line velocity and the roughness factor ZR accounts for influence of the surface roughness on the surface endurance capacity
The following methods may be applied in connection with the endurance limit
20 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
Surface hardened steels Not surface hardened steels
ZL ( )24013421
360910ν+
+
( )24013421
680830ν+
+
ZV ( )v3280
140930+
+ ( )v3280300850
++
ZR
080
ZrelR3
150
ZrelR3
where
ν40 = Kinematic oil viscosity at 40ordmC (mm2s) For case hardened steels the influence of a high bulk temperature (see 4 Scuffing) should be con-sidered Eg bulk temperatures in excess of 120ordmC for long periods may cause reduced flank surface endurance limits
For values of ν40 gt 500 use ν40 = 500
RZrel = The mean roughness between pinion and wheel (after running in) relative to an equivalent radius of curvature at the pitch point ρc = 10mm
RZrel
= ( ) 31
cZ2Z1 ρ
10RR50
+
RZ = Mean peak to valley roughness (microm) (DIN defini-tion) (roughly RZ = 6 Ra)
For 5
L 10N le ZL ZV ZR = 10
211 Work Hardening Factor ZW The work hardening factor ZW accounts for the increase of surface durability of a soft steel gear when meshing the soft steel gear with a surface hardened or substantially harder gear with a smooth surface
The following approximation may be used for the endurance limit
Surface hardened steel against not surface hardened steel 150
ZeqW R
31700
130HB21Z
minus
minus=
where HB = the Brinell hardness of the soft member For HB gt 470 use HB = 470 For HB lt 130 use HB = 130
RZeq = equivalent roughness
033
c40
066
ZS
ZHZHZEQ ρvν
15000RRRR
=
If RZeq gt 16 then use RZeq = 16
If RZeq lt 15 then use RZeq = 15
where
RZH = surface roughness of the hard member before run in
RZS = surface roughness of the soft member before run in
ν40 = see 210
If values of ZW lt 1 are evaluated ZW = 1 should be used for flank endurance However the low value for ZW may indi-cate a potential wear problem
Through hardened pinion against softer wheel
( )
minussdotminus+= 000829
HBHB0008981u1Z
2
1W
For 21HBHB
2
1 le use ZW = 1
For 71HBHB
2
1 gt use 17HBHB
2
1 =
For u gt 20 use u = 20
For static strength (lt 105 cycles)
Surface hardened against not surface hardened
ZWst = 105
Through hardened pinion against softer wheel
ZWst = 1
212 Size Factor ZX The size factor accounts for statistics indicating that the stress levels at which fatigue damage occurs decrease with an increase of component size as a consequence of the influ-ence on subsurface defects combined with small stress gradi-ents and of the influence of size on material quality
ZX may be taken unity provided that subsurface fatigue for surface hardened pinions and wheels is considered eg as in the following subsection 213
213 Subsurface Fatigue This is only applicable to surface hardened pinions and wheels The main objective is to have a subsurface safety against fatigue (endurance limit) or deformation (static strength) which is at least as high as the safety SH required for the surface The following method may be used as an approximation unless otherwise documented
The high cycle fatigue (gt3106 cycles) is assumed to mainly depend on the orthogonal shear stresses Static strength (lt103 cycles) is assumed to depend mainly on equivalent
Classification Notes- No 412 21 May 2003
DET NORSKE VERITAS
stresses (von Mises) Both are influenced by residual stresses but this is only considered roughly and empirically
The subsurface working stresses at depths inside the peak of the orthogonal shear stresses respectively the equivalent stresses are only dependent on the (real) Hertzian stresses Surface related conditions as expressed by ZL ZV and ZR are assumed to have a negligible influence
The real Hertzian stresses σHR are determined as
For helical gears with εβ gt 1
σHR = σH
For helical gears with εβ lt 1 and spur gears
ε
α
ββ ε
ε+εminus
sdotσ=σZ
1
HHR
For bevel gears
KHHR Z
1σσ sdot=
The necessary hardness HV is given as a function of the net depth tz (net = after grinding or hard metal hobbing and per-pendicular to the flank)
The coordinates tz and HV are to be compared with the de-sign specification such as
bull for flame and induction hardening tHVmin HVmin bull for nitriding t400min HV = 400 bull for case hardening t550min HV = 550 t400min HV = 400
and t300min HV = 300 (the latter only if the core hardness lt 300 If the core hardness gt 400 the t400 is to be re-placed by a fictive t400 = 16 t550)
In addition the specified surface hardness is not to be less than the max necessary hardness (at tz = 05aH) This applies to all hardening methods
For high cycle fatigue (gt3 106 cycles) the following applies
sdot+
minussdotsdotsdot= o90
05azt
05azt
cosSσ04HV
H
HHHR
applicable to 05a
zt
Hge
For 05at
H
z lt the value for 05at
H
z = applies
56300ρSσ12a cHHR
Hsdotsdot
sdot=
Where aH is half the hertzian contact width multiplied by an empirical factor of 12 that takes into account the possible influence of reduced compressive residual stresses (or even tensile residual stresses) on the local fatigue strength
If any of the specified hardness depths including the surface hardness is below the curve described by HV = f (tz) the actual safety factor against subsurface fatigue is determined as follows
reduce SH stepwise in the formula for HV and aH until all specified hardness depths and surface hardness balance with the corrected curve The safety factor obtained through this method is the safety against subsurface fatigue
For static strength (lt103 cycles) the following applies
sdot+
minussdotsdotsdot= o90
07at
06a
zt
cosSσ019HV
Hst
z
HstHHR
applicable to 06at
Hst
z ge
56300ρSσa cHHR
Hstsdotsdot
=
In the case of insufficient specified hardness depths the same procedure for determination of the actual safety factor as above applies
For limited life fatigue (103 lt cycles lt 3106 )
For this purpose it is necessary to extend the correction of safety factors to include also higher values than required Ie in the case of more than sufficient hardness and depths the safety factor in the formulae for both high cycle fatigue and static strength are to be increased until necessary and speci-fied values balance
The actual safety factor for a given number of cycles N be-tween 103 and 3106 is found by linear interpolation in a dou-ble logarithmic diagram
minussdotminus
= sdot logN3477
logSlogSlogS
310H6103HHN
36 10H103H Slog86281Slog86280 sdot+sdot sdot
22 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
3 Calculation of Tooth Strength
31 Scope and General Remarks Part 3 include the calculation of tooth root strength as limited by tooth root cracking (surface or subsurface initiated) and yielding
For rim thickness sR ge 35middotmn the strength is calculated by means of 32 ndash 313 For cylindrical gears the calculation is based on the assumption that the highest tooth root tensile stress arises by application of the force at the outer point of single tooth pair contact of the virtual spur gears The method has however a few limitations that are mentioned in 36
For bevel gears the calculation is based on force application at the tooth tip of the virtual cylindrical gear Subsequently the stress is converted to load application at the mid point of the flank due to the heightwise crowning
Bevel gears may also be calculated with the program BECAL In that case KA and Kv are to be included in the applied tooth force but not KFβ and KFα
In case of a thin annulus or a thin gear rim etc radial crack-ing can occur rather than tangential cracking (from root fillet to root fillet) Cracking can also start from the compression fillet rather than the tension fillet For rim thickness sR lt 35middotmn a special calculation procedure is given in 315 and 316 and a simplified procedure in 314
A tooth breakage is often the end of the life of a gear trans-mission Therefore a high safety SF against breakage is re-quired
It should be noted that this part 3 does not cover fractures caused by
bull oil holes in the tooth root space bull wear steps on the flank bull flank surface distress such as pits spalls or grey staining
Especially the latter is known to cause oblique fractures starting from the active flank predominately in spiral bevel gears but also sometimes in cylindrical gears
Specific calculation methods for these purposes are not given here but are under consideration for future revisions Thus depending on experience with similar gear designs limita-tions other than those outlined in part 3 may be applied
32 Tooth Root Stresses The local tooth root stress is defined as the max principal stress in the tooth root caused by application of the tooth force Ie the stress ratio R = 0 Other stress ratios such as for eg idler gears (R asymp -12) shrunk on gear rims (R gt 0) etc are considered by correcting the permissible stress level
321 Local tooth root stress The local tooth root stress for pinion and wheel may be as-sessed by strain gauge measurements or FE calculations or similar For both measurements and calculations all details are to be agreed in advance
Normally the stresses for pinion and wheel are calculated as
Cylindrical gears
FαFβvγAβSFn
tF K K K K K Y Y Y
m bFσ =
where
YF = Tooth form factor (see 33)
YS = Stress correction factor (see 34)
Yβ = Helix angle factor (see 36)
Ft KA Kγ Kv KFβ KFα see 15 ndash 110
b see 13
Bevel gears
FαFβvγASaFamn
mtF K K K K K Y Y Y
m bFσ ε=
where
YFa = Tooth form factor see 33
YSa = Stress correction factor see 34
Yε = Contact ratio factor see 35
Fmt KA etc see 15 ndash 110
b see 13
322 Permissible tooth root stress The permissible local tooth root stress for pinion respectively wheel for a given number of cycles N is
CXRrelTrelTF
NMFEFP YYYY
SYY
δσ
=σ
Note that all these factors YM etc are applicable to 3middot106 cycles when used in this formula for σFP The influence of other number of cycles on these factors is covered by the calculation of YN
where
σFE = Local tooth root bending endurance limit of reference test gear (see 37)
YM = Mean stress influence factor which accounts for other loads than constant load direction eg idler gears temporary change of load di-rection pre-stress due to shrinkage etc (see 38)
YN = Life factor for tooth root stresses related to reference test gear dimensions (see 39)
SF = Required safety factor according to the rules
YδrelT = Relative notch sensitivity factor of the gear to be determined related to the reference test gear (see 310)
YRrelT = Relative (root fillet) surface condition factor of the gear to be determined related to the reference test gear (see 311)
Classification Notes- No 412 23 May 2003
DET NORSKE VERITAS
YX = Size factor (see 312)
YC = Case depth factor (see 313)
33 Tooth Form Factors YF YFa The tooth form factors YF and YFa take into account the in-fluence of the tooth form on the nominal bending stress
YF applies to load application at the outer point of single tooth pair contact of the virtual spur gear pair and is used for cylindrical gears
YFa applies to load application at the tooth tip and is used for bevel gears
Both YF and YFa are based on the distance between the con-tact points of the 30˚-tangents at the root fillet of the tooth profile for external gears respectively 60˚ tangents for inter-nal gears
Figure 31 External tooth in normal section
Figure 32 Internal tooth in normal section
Definitions
n
2
n
Fn
enFn
Fe
F
α cosms
α cosmh6
Y
=
n
2
n
Fn
anFn
Fa
Fa
α cosms
α cosmh6
Y
=
In the case of helical gears YF and YFa are determined in the normal section ie for a virtual number of teeth
YFa differs from YF by the bending moment arm hFa and αFan and can be determined by the same procedure as YF with exception of hFe and αFan For hFa and αFan all indices e will change to a (tip)
The following formulae apply to cylindrical gears but may also be used for bevel gears when replacing
mn with mnm
zn with zvn
αt with αvt
β with βm
with undercut without undercut
Fig 33 Dimensions and basic rack profile of the teeth (finished profile)
Tool and basic rack data such as hfP ρfp and spr etc are re-ferred to mn ie dimensionless
331 Determination of parameters
( )n
n
prnfPnfP m
α cossαsin 1ρ
αtan h4πE
minusminusminusminus=
For external gears fPfP ρρ =
For internal gears ( )0z
195fPfP0
fPfP 10363156ρhxρρ
sdotminus+
+=
where
z0 = number of teeth of pinion cutter
x0 = addendum modification coefficient of pinion cutter
hfP = addendum of pinion cutter
ρfP = tip radius of pinion cutter
xhρG fpfp +minus=
24 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
τmE
2π
z2H
nnminus
minus=
with
3πτ = for external gears
6πτ = for internal gears
Htan zG2
nminusϑ=ϑ
(to be solved iteratively suitable start value6π
=ϑ for exter-
nal gears and 3π for internal gears)
a) Tooth root chord sFn For external gears
minus
ϑ+
ϑminus= ρ
cosG3
3πsinz
ms
fpnn
Fn
For bevel gears with a tooth thickness modification
xsm affects mainly sFn but also hFe and αFen The total influence of xsm on YFa Ysa can be approximated by only adding 2 xsm to sFn mn
For internal gears
minus
ϑ+
ϑminus= ρ
cosG
6πsinz
ms
fPnn
Fn
b) Root fillet radius ρF at 30ordm tangent
( )G2coszcosG2ρ
mρ
2n
2
fpn
F
minusϑϑ+=
c) Determination of bending moment arm hF dn = zn mn
b2α
αn βcosε
ε =
dan = dn + 2 ha
pbn = π mn cos αn
dbn = dn cos αn
( )4
d1εpzz
2dd
zz2d
2bn
2
αnbn
2bn
2an
en +
minusminus
minus=
en
bnen d
dcos arcα =
ennnn
e α invα invα x tan 22π
z1
minus+
+=γ
αFen = αen ndash γe
For external gears
( )
minus=
n
enFenee
n
Fe
mdαtan sin γ γcos
21
mh
]ρcos
G3πcosz fpn +
ϑminus
ϑminusminus
For internal gears
( ) minus
sdotsdotminus=
n
enFenee
n
Fe
mdα tanγ sinγ cos
21
mh
minus
ϑminus
ϑminussdot ρ
cosG3
6πcosz fPn
332 Gearing with εαn gt 2 For deep tooth form gearing ( )25ε2 αn lele produced with a verified grade of accuracy of 4 or better and with applied profile modification to obtain a trapezoidal load distribution along the path of contact the YF may be corrected by the factor YDT as
250ε205for 0666ε2366Y αnαnDT leleminus=
205εfor 10Y αnDT lt=
34 Stress Correction Factors YS YSa The stress correction factors YS and YSa take into account the conversion of the nominal bending stress to the local tooth root stress Thereby YS and YSa cover the stress increasing effect of the notch (fillet) and the fact that not only bending stresses arise at the root A part of the local stress is inde-pendent of the bending moment arm This part increases the more the decisive point of load application approaches the critical tooth root section
Therefore in addition to its dependence on the notch radius the stress correction is also dependent on the position of the load application ie the size of the bending moment arm
YS applies to the load application at the outer point of single tooth pair contact YSa to the load application at tooth tip
YS can be determined as follows
( )
+
+=L23121
1
sS qL01312Y
where Fe
Fn
hsL = and
F
Fns ρ2
sq = (see 33)
YSa can be calculated by replacing hFe with hFa in the above formulae
Note a) Range of validity 1 lt qs lt 8
In case of sharper root radii (ie produced with tools having too sharp tip radii) YS resp YSa must be specially considered
b) In case of grinding notches (due to insufficient protuberance of the hob) YS resp YSa can rise considerably and must be multiplied with
Classification Notes- No 412 25 May 2003
DET NORSKE VERITAS
g
g
ρt
0613
13
minus
where
tg = depth of the grinding notch
ρg = radius of the grinding notch
c) The formulae for YS resp YSa are only valid for αn = 20˚ However the same formulae can be used as a safe approximation for other pressure angles
35 Contact Ratio Factor Yε The contact ratio factor Yε covers the conversion from load application at the tooth tip to the load application at the mid point of the flank (heightwise) for bevel gears
The following may be used
Yε = 0625
36 Helix Angle Factor Yβ The helix angle factor Yβ takes into account the difference between the helical gear and the virtual spur gear in the nor-mal section on which the calculation is based in the first step In this way it is accounted for that the conditions for tooth root stresses are more favourable because the lines of contact are sloping over the flank
The following may be used (β input in degrees)
Yβ = 1 ndash εβ β120
When εβ gt 1 use εβ = 1 and when β gt 30deg use β = 30deg in the formula
However the above equation for Yβ may only be used for gears with β gt 25deg if adequate tip relief is applied to both pinion and wheel (adequate = at least 05 middot Ceff see 432)
37 Values of Endurance Limit σFE σFE is the local tooth root stress (max principal) which the material can endure permanently with 99 survival prob-ability 3106 load cycles is regarded as the beginning of the endurance limit or the lower knee of the σ ndash N curve σFE is defined as the unidirectional pulsating stress with a minimum stress of zero (disregarding residual stresses due to heat treatment) Other stress conditions such as alternating or pre-stressed etc are covered by the conversion factor YM
σFE can be found by pulsating tests or gear running tests for any material in any condition If the approval of the gear is to be based on the results of such tests all details on the testing conditions have to be approved by the Society Fur-ther the tests may have to be made under the Societys su-pervision
If no fatigue tests are available the following listed values for σFE may be used for materials subjected to a quality con-trol as the one referred to in the rules
σFE
Alloyed case hardened steels 1) (fillet surface hardness 58 ndash 63 HRC)
bull of specially approved high grade
1050
bull of normal grade
minus CrNiMo steels with approved process
1000
minus CrNi and CrNiMo steels generally 920 minus MnCr steels generally 850
Nitriding steel of approved grade quenched tempered and gas nitrided (surface hardness 700 ndash 800 HV)
840
Alloyed quenched and tempered steel bath or gas nitrided (surface hardness 500 ndash 700 HV)
720
Alloyed quenched and tempered steel flame or induction hardened 2) (incl entire root fillet) (fillet surface hardness 500 ndash 650 HV)
07 HV + 300
Alloyed quenched and tempered steel flame or induction hardened (excl entire root fillet) (σB = uts of base material)
025 σB + 125
Alloyed quenched and tempered steel 04 σB + 200
Carbon steel 025 σB + 250
Note All numbers given above are valid for separate forgings and for blanks cut from bars forged according to a qualified procedure see Pt 4 Ch 2 Sec 3 For rolled steel the values are to be reduced with 10 For blanks cut from forged bars that are not qualified as mentioned above the values are to be reduced with 20 For cast steel reduce with 40
1) These values are valid for a root radius
bull being unground If however any grinding is made in the root fillet area in such a way that the residual stresses may be affected σFE is to be reduced by 20 (If the grinding also leaves a notch see 34)
bull with fillet surface hardness 58 ndash 63 HRC In case of lower surface hardness than 58 HRC σFE is to be reduced with 20(58 ndash HRC) where HRC is the detected hardness (This may lead to a permissible tooth root stress that varies along the facewidth If so the actual tooth root stresses may also be considered along facewidth)
bull not being shot peened In case of approved shot peening σFE may be increased by 200 for gears where σFE is reduced by 20 due to root grinding Otherwise σFE may be increased by 100 for
6nm le and 100 ndash 5 (mn - 6) for mn gt 6 However the possible adverse influence on the flanks regarding grey staining should be considered and if necessary the flanks should be masked
2) The fillet is not to be ground after surface hardening Regarding possible root grinding see 1)
26 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
38 Mean stress influence Factor YM The mean stress influence factor YM takes into account the influence of other working stress conditions than pure pulsa-tions (R = 0) such as eg load reversals idler gears planets and shrink-fitted gears
YM (YMst) are defined as the ratio between the endurance (or static) strength with a stress ratio R ne 0 and the endurance (or static) strength with R = 0
YM and YMst apply only to a calculation method that assesses the positive (tensile) stresses and is therefore suitable for comparison between the calculated (positive) working stress σF and the permissible stress σFP calculated with YM or YMst
For thin rings (annulus) in epicyclic gears where the com-pression fillet may be decisive special considerations apply see 316
The following method may be used within a stress ratio ndash12 lt R lt 05
381 For idlers planets and PTO with ice class
M1M1R1
1Yor Y MstM
+minus
minus=
where
R = stress ratio = min stress divided by max stress
For designs with the same force applied on both forward- and back-flank R may be assumed to ndash 12
For designs with considerably different forces on forward- and back-flank such as eg a marine propulsion wheel with a power take off pinion R may be assessed as
branchmain theoffacewidth unit per forcepto offacewidt unit per force21minus
For a power take off (PTO) with ice class see 161 c
M considers the mean stress influence on the endurance (or static) strength amplitudes
M is defined as the reduction of the endurance strength am-plitude for a certain increase of the mean stress divided by that increase of the mean stress
Following M values may be used
Endurance limit
Static strength
Case hardened 08 ndash 015 Ys 1) 07
If shot peened 04 06
Nitrided 03 03
Induction or flame hardened
04 06
Not surface hardened steel 03 05
Cast steels 04 06 1)For bevel gears use Ys=2 for determination of M
The listed M values for the endurance limit are independent of the fillet shape (Ys) except for case hardening In princi-ple there is a dependency but wide variations usually only occur for case hardening eg smooth semicircular fillets versus grinding notches
382 For gears with periodical change of rotational direction For case hardened gears with full load applied periodically in both directions such as side thrusters the same formula for YM as for idlers (with R = ndash 12) may be used together with the M values for endurance limit This simplified approach is valid when the number of changes of direction exceeds 100 and the total number of load cycles exceeds 3middot106
For gears of other materials YM will normally be higher than for a pure idler provided the number of changes of direction is below 3middot106 A linear interpolation in a diagram with loga-rithmic number of changes of direction may be used ie from YM = 09 with one change to YM (idler) for 3middot106 changes This is applicable to YM for endurance limit For static strength use YM as for idlers
For gears with occasional full load in reversed direction such as the main wheel in a reversing gear box YM = 09 may be used
383 For gears with shrinkage stresses and unidirectional load For endurance strength
FE
fitM σ
σM1
M21Y+
minus=
σFE is the endurance limit for R = 0
For static strength YMst = 1 and σfit accounted for in 39b
σfit is the shrinkage stress in the fillet (30˚ tangent) and may be found by multiplying the nominal tangential (hoop) stress with a stress concentration factor
n
Ffit m
ρ215scf minus=
384 For shrink-fitted idlers and planets When combined conditions apply such as idlers with shrink-age stresses the design factor for endurance strength is
( ) ( ) FE
fitM σ
σR1M1
M2
M1M1R1
1Yminussdot+
minus
+minus
minus=
Symbols as above but note that the stress ratio R in this par-ticular connection should disregard the influence of σfit ie R normally equal ndash 12
For static strength
M1M1R1
1YMst
+minus
minus=
The effect of σfit is accounted for in 39 b
Classification Notes- No 412 27 May 2003
DET NORSKE VERITAS
385 Additional requirements for peak loads The total stress range (σmax ndash σmin) in a tooth root fillet is not to exceed
F
y
Sσ225
for not surface hardened fillets
FSHV5 for surface hardened fillets
39 Life Factor YN The life factor YN takes into account that in the case of limited life (number of cycles) a higher tooth root stress can be permitted and that lower stresses may apply for very high number of cycles
Decisive for the strength at limited life is the σ ndash N ndash curve of the respective material for given hardening module fillet radius roughness in the tooth root etc Ie the factors YδrelT YRelT YX and YM have an influence on YN
If no σ ndash N ndash curve for the actual material and hardening etc is available the following method may be used
Determination of the σ ndash N ndash curve
a) Calculate the permissible stress σFP for the beginning of the endurance limit (3middot106 cycles) including the influ-ence of all relevant factors as SF YδrelT YRelT YX YM and YC ie σFP = σFE middotYM middotYδrelT middotYRelT middotYX middotYC SF
b) Calculate the permissible laquostaticraquo stress (le 103 load cy-cles) including the influence of all relevant factors as SFst YδrelTst YMst and YCst ( )fitσCstYδrelTstYMstYFstσ
FstS1
FPstσ minussdotsdotsdot=
where σFst is the local tooth root stress which the material can resist without cracking (surface hardened materials) or unacceptable deformation (not surface hardened materials) with 99 survival probability
σFst
Alloyed case hardened steel 1) 2300
Nitriding steel quenched tempered and gas nitrided (surface hardness 700 ndash 800 HV)
1250
Alloyed quenched and tempered steel bath or gas nitrided (surface hardness 500 ndash 700 HV)
1050
Alloyed quenched and tempered steel flame or induction hardened (fillet surface hardness 500 ndash 650 HV)
18 HV + 800
Steel with not surface hardened fillets the smaller value of 2)
18 σB or 225 σy
1) This is valid for a fillet surface hardness of 58 ndash 63 HRC In case of lower fillet surface hardness than 58 HRC σFst is to be reduced with 30(58 ndash HRC) where HRC is the actual hardness Shot peening or grinding notches are not considered to have any significant influence on σFst 2) Actual stresses exceeding the yield point (σy or σ02) will alter the residual stresses locally in the ldquotensionrdquo fillet re-spectively ldquocompressionrdquo fillet This is only to be utilised for gears that are not later loaded with a high number of cycles at lower loads that could cause fatigue in the ldquocompressionrdquo fillet
c) Calculate YN as
NL gt 3middot106
YN = 1 or 001
L
6
N N103Y
sdot= ie Yn = 092 for 1010
The YN = 1 from 3middot106 on may only be used when special material cleanness applies see rules Pt4 Ch2
103ltNLlt3middot106exp
L
6
N N103Y
sdot=
cycles6103forσcycles310forσlog02876exp
FP
FPst
sdot=
NL lt 103 cycles6103forσcycles310forσ
NYFP
FPst
sdot=
or simply use σFPst as mentioned in b) directly
Guidance on number of load cycles NL for various applica-tions
bull For propulsion purpose normally NL = 1010 at full load (yachts etc may have lower values)
bull For auxiliary gears driving generators that normally operate with 70-90 of rated power NL = 108 with rated power may be applied
28 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
310 Relative Notch Sensitivity Factor YδrelT The dynamic (respectively static) relative notch sensitivity factor YδrelT (YδrelTst) indicate to which extent the theoreti-cally concentrated stress lies above the endurance limits (re-spectively static strengths) in the case of fatigue (respectively overload) breakage
YδrelT is a function of the material and the relative stress gra-dient It differs for static strength and endurance limit
The following method may be used
For endurance limit
for not surface hardened fillets
( )
024
s024
relT σ103133q21σ1012201351
Ysdotsdotminus
+sdotsdotminus+= minus
minus
δ
for all surface hardened fillets except nitrided
( )
106q21002451
Y srelT
++=δ
for nitrided fillets
( )
1347q2101421
Y srelT
++=δ
For static strength
for not surface hardened fillets1)
( )( )( )025
02
02502s
relTst σ3000821σ300 1Y0821Y
+
minus+=δ
for surface hardened fillets except nitrided
YδrelTst = 044 YS + 012
for nitrided fillets
YδrelTst = 06 + 02 YS 1) These values are only valid if the local stresses do not
exceed the yield point and thereby alter the residual stress level See also 39b footnote 2
311 Relative Surface Condition Factor YRrelT The relative surface condition factor YRrelT takes into ac-count the dependence of the tooth root strength on the sur-face condition in the tooth root fillet mainly the dependence on the peak to valley surface roughness
YRrelT differs for endurance limit and static strength
The following method may be used
For endurance limit
YRrelT = 1675 ndash 053 (Ry + 1)01 for surface hardened steels and alloyed quenched and tempered steels except nitrided
YRrelT = 53 ndash 42 (Ry + 1)001 for carbon steels
YδrelT = 43 ndash 326 (Ry + 1)0005
for nitrided steels
For static strength
YRrelTst = 1 for all Ry and all materials
For a fillet without any longitudinal machining trace Ry asymp Rz
312 Size Factor YX The size factor YX takes into account the decrease of the strength with increasing size YX differs for endurance limit and static strength
The following may be used
For endurance limit
YX = 1 for mn le 5 generally
YX = 103 ndash 0006 mn for 5 lt mn lt 30
YX = 085 for mn ge 30
for not surface hard-ened steels
YX = 105 ndash 001 mn for 5 lt mn ge 25
YX = 08 for mn ge 25
for surface hardened steels
For static strength
YXst = 1 for all mn and all materials
313 Case Depth Factor YC The case depth factor YC takes into account the influence of hardening depth on tooth root strength
YC applies only to surface hardened tooth roots and is dif-ferent for endurance limit and static strength
In case of insufficient hardening depth fatigue cracks can develop in the transition zone between the hardened layer and the core For static strength yielding shall not occur in the transition zone as this would alter the surface residual stresses and therewith also the fatigue strength
The major parameters are case depth stress gradient permis-sible surface respectively subsurface stresses and subsurface residual stresses
The following simplified method for YC may be used
YC consists of a ratio between permissible subsurface stress (incl influence of expected residual stresses) and permissible surface stress This ratio is multiplied with a bracket con-taining the influence of case depth and stress gradient (The empirical numbers in the bracket are based on a high number of teeth and are somewhat on the safe side for low number of teeth)
YC and YCst may be calculated as given below but calculated values above 10 are to be put equal 10
For endurance limit
+
+=nFFE
C m02ρt31
σconstY
Classification Notes- No 412 29 May 2003
DET NORSKE VERITAS
For static strength
+
+=nFFst
Cst m02ρt31
σconstY
where const and t are connected as
Hardening process
t = endurance limit const =
static strength const =
t550 640 1900
t400 500 1200 Case hard-ening
t300 380 800
Nitriding t400 500 1200
Induction- or flame hardening
tHVmin 11 HVmin 25 HVmin
For symbols see 213
In addition to these requirements to minimum case depths for endurance limit some upper limitations apply to case hard-ened gears
The max depth to 550 HV should not exceed
1) 13 of the top land thickness san unless adequate tip relief is applied (see 110)
2) 025 mn If this is exceeded the following applies addi-tionally in connection with endurance limit
minusminus= 025
mt
1Yn
max550C
314 Thin rim factor YB Where the rim thickness is not sufficient to provide full sup-port for the tooth root the location of a bending failure may be through the gear rim rather than from root fillet to root fillet
YB is not a factor used to convert calculated root stresses at the 30deg tangent to actual stresses in a thin rim tension fillet Actually the compression fillet can be more susceptible to fatigue
YB is a simplified empirical factor used to de-rate thin rim gears (external as well as internal) when no detailed calcula-tion of stresses in both tension and compression fillets are available
Figure 34 Examples on thin rims
YB is applicable in the range 175 lt sRmn lt 35
YB = 115 middot ln (8324 middot mnsR)
(for sRmn ge 35 YB = 1)
(for sRmn le 175 use 315)
σF as calculated in 321 is to be multiplied with YB when sRmn lt 35 Thus YB is used for both high and low cycle fatigue
Note This method is considered to be on the safe side for external gear rims However for internal gear rims without any flange or web stiffeners the method may not be on the safe side and it is advised to check with the method in 315316
315 Stresses in Thin Rims For rim thickness sR lt 35 mn the safety against rim cracking has to be checked
The following method may be used
3151 General The stresses in the standardised 30ordm tangent section tension side are slightly reduced due to decreasing stress correction factor with decreasing relative rim thickness sRmn On the other hand during the complete stress cycle of that fillet a certain amount of compression stresses are also introduced The complete stress range remains approximately constant Therefore the standardised calculation of stresses at the 30ordm tangent may be retained for thin rims as one of the necessary criteria
The maximum stress range for thin rims usually occurs at the 60ordm ndash 80ordm tangents both for laquotensionraquo and laquocompressionraquo side The following method assumes the 75ordm tangent to be the decisive Therefore in addition to the am criterion ap-plied at the 30ordm tangent it is necessary to evaluate the max and min stresses at the 75ordm tangent for both laquotensionraquo (loaded flank) fillet and laquocompressionraquo (back-flank) fillet For this purpose the whole stress cycle of each fillet should be considered but usually the following simplification is justified
Figure 35 Nomenclature of fillets
Index laquoTraquo means laquotensileraquo fillet laquoCraquo means laquocompressionraquo fillet
σFTmin and σFCmax are determined on basis of the nominal rim stresses times the stress concentration factor Y75
σFCmin and σFTmax are determined on basis of superposition of nominal rim stresses times Y75 plus the tooth bending stresses at 75ordm tangent
30 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
3152 Stress concentration factors at the 75ordm tangents The nominal rim stress consists of bending stresses due to local bending moments tangential stresses due to the tan-gential force Ft and radial shear stresses due to Fr
The major influence is given by the bending stresses The influence of the tangential stresses is minor and even though its stress concentration factor is slightly higher than for bending it is considered to be safe enough when the sum of these nominal stresses are combined with the stress concen-tration factor for bending The influence of the radial shear stress is neglected
The stress concentration factor relating nominal rim stresses to local fillet stresses at the 75ordm tangent may be calculated as
75n
R75
n
R
75
ρmsρ185
ms3
Y+
sdot=
where ρ75 is the root radius at the 75ordm tangent ref to mn Usually ρ75 is closed to the tool radius ρao and
ρ75 = ρao
is a safe approximation compensating for the am simplifi-cations to the laquounsaferaquo side
The tooth root stresses of the loaded tooth are decreasing with decreasing relative rim thickness approximately with the empirical factor
15
R
ncorr s
m311Y
minus=
3153 Nominal rim stresses The bending moment applied to the rim consists of a part of the tooth tilt moment Ft (hF + 05 sR) and the bending caused by the radial force Fr
The sectional modulus (first moment of area) which is used for determination of the nominal bending stresses is not nec-essarily the same for the 2 am bending moments If flanges webs etc outside the toothed section contribute to stiffening the rim against various deflections the influence of these stiffeners should be considered Eg an end flange will have an almost negligible influence on the effective sectional modulus for the stresses due to tooth tilt as the deflection caused by the tilt moment is rather small and would not much involve the flange On the other hand the radial forces as for instance from the meshes in an annulus would cause considerable radial deflections that the flange might restrict to a substantial amount When considering the stiff-ening of such flanges or webs on basis of simplified models it is advised to use an effective rim thickness sR = sR + 02 mn for the first moment of area of the rim (toothed part) cross section
For a high number of rim teeth it may be assumed that the rim bending moments in the fillets adjacent to the loaded tooth are of the same magnitude as right under the applied
force This assumption is reasonable for an annulus but rather much on the safe side for a hollow pinion
The influence of Ft on the nominal tangential stress is simpli-fied by half of it for compressive stresses (σ1) and the other half for tensile stresses (σ2) Applying these assumptions the nominal rim stresses adjacent to the loaded tooth are
( ) ( )A2
FW
f RFW
s05hF05 t
R
r
T
RFt1 minus
ϑminus
+minus=σ
( ) ( )A2
FW
f RFW
s05hF05 t
R
r
T
RFt2 +
ϑminus
+=σ
where σ1 σ2 see Fig 35
A = minimum area of cross section (usually bR sR)
WT = the sectional modulus of rim with respect to tooth tilt moment (usually 6sb 2
rR )
WR = the sectional modulus of the rim including the influence of stiffeners as flanges webs etc (WR ge WT)
bR = the width of the rim
R = the radius of the neutral axis in the rim ie from wheel centre to midpoint of rim
( )ϑf = a function for bending moment distribution around the rim For a rim (pinion) with one mesh only the ( )ϑf at the position of load application is 024 For an annulus with 3 or more meshes ( )ϑf at the position of each load application is approx 3 planets ( )ϑf = 019 4 planets ( )ϑf = 014 5 planets ( )ϑf = 011 6 planets ( )ϑf = 009
It must be checked if the max (tensile) stress in the compres-sion fillet really occurs when the fillet is adjacent to the loaded tooth In principle the stress variation through a complete rotation should be considered and the max value used The max value is usually never less than 0 For an annulus eg the tilt moment is zero in the mid position be-tween the planet meshes whilst the bending moment due to the radial forces is half of that at the mesh but with opposite sign
If these formulae are applied to idler gears as eg planets the influence of nominal tangential stresses must be cor-rected by deleting Ft(2 A) for σ1 and using FtA for σ2 Further the influence of Fr on the nominal bending stresses is usually negligible due to the planet bearing support
3154 Root fillet stresses Determination of min and max stresses in the laquotensionraquo fillet
Classification Notes- No 412 31 May 2003
DET NORSKE VERITAS
Minimum stress
751FTmin YσK σ =
Maximum stress
corrF752 Yσ03Yσ KσFTmax +=
where
03 is an empirical factor relating the tension stresses (σF) at the 30ordm tangent to the part of the tension stresses at the 75ordm tangent which add to the rim related stresses (03 also takes into account that full superposition of nominal stresses times stress concentration factors from both laquosides of the corner filletraquo would result in too high stresses)
αβγ sdotsdotsdotsdot= FFvA KKKKKK
Determination of min and max stresses in the laquocompres-sionraquo fillet
Minimum stress
corrF751FCmin Yσ036Yσ Kσ minus=
Maximum stress
752FCmax Yσ Kσ =
where
036 is an empirical factor relating the tension stresses (σF) at the 30ordm tangent to the part of the compression stresses at the 75ordm tangent which add to the rim related stresses
For gears with reversed loads as idler gears and planets there is no distinct laquotensionraquo or laquocompressionraquo fillet The mini-mum stress σFmin is the minimum of σFTmin and σFCmin (usually the latter is decisive) The maximum stress σFTmax is the maximum of σFTmax and σFCmax (usually the former is deci-sive)
316 Permissible Stresses in Thin Rims
3161 General The safety against fatigue fracture respectively overload fracture is to be at least at the same level as for solid gears The laquoordinaryraquo criteria at the 30ordm tangent apply as given in 31 through 313
Additionally the following criteria at the 75ordm tangent may apply
3162 For gt3middot106 cycles The permissible stresses for the laquotensionraquo fillets and for the laquocompressionraquo fillets are determined by means of a relevant fatigue diagram
If the actual tooth root stress (tensile or compressive) ex-ceeds the yield strength to the material the induced residual stresses are to be taken into account
For determination of permissible stresses the following is defined
R = stress ratio ie FTmax
FTminσσ respectively
FCmax
FCmin
σσ
∆σ = stress range ie σFTmax ndash σFTmin resp σFCmax ndash σFCmin
(For idler gears and planets Fmax
FminσσR =
and ∆σ = σFmax minus σFmin)
The permissible stress range ∆σρ for the laquotensionraquo respec-tively laquocompressionraquo fillets can be calculated as
For R gt minus1 FPp σ
R1R1031
13∆σ
minus+
+=
For minus infin lt R lt minus1 FPp σ
R1R10151
13∆σ
minus+
+=
where
σFP = see 32 determined for unidirectional stresses (YM = 1)
If the yield strength σy is exceeded in either tension or com-pression residual stresses are induced This may be consid-ered by correcting the stress ratio R for the respective fillets (tension or compression)
Eg if yFCmin σσ gt (ie exceeded in compression) the
difference yFCmin σσ∆ minus= affects the stress ratio as
∆σ
σ∆σ∆σR
FCmax
y
FCmax
FCmin
+
minus=
++
=
Similarly the stress ratio in the tension fillet may require cor-rection
If the yield strength is exceeded in tension σFTmax gt σy the difference ∆ = σFTmax ndash σy affects the stress ratio as
y
FTmin
FTmax
FTmin
σ∆σ
∆σ∆σR minus=
minusminus
=
Checking for possible exceeding of the yield strength has to be made with the highest torque and with the KAP if this ex-ceeds KA
3163 Forle 103 cycles The permissible stress range ∆σp is not to exceed
For R gt minus1 FPstpst σ
R1R1051
15∆σ
minus+
+=
For ndash infin lt R lt minus1 FPstpst σ
R1R10251
15∆σ
minus+
+=
For all values of R ∆σpst is limited by
not surface hardened F
y
Sσ225
32 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
surface hardened CF
YSHV5
Definition of ∆σ and R see 3162 with particular attention to possible correction of R if the yield strength is exceeded
σFPst see 32 determined for unidirectional stresses (YM = 1) and lt 103 cycles
3164 For 103 lt cycles lt 3106 ∆σp is to be determined by linear interpolation a log-log dia-gram
∆σp at NL load cycles is
6L 103p
exp
L
6
N p ∆σN103∆σ sdot
sdot=
6
3
103p
10p
∆σ
∆σlog28760exp
sdot
=
Classification Notes- No 412 33 May 2003
DET NORSKE VERITAS
4 Calculation of Scuffing Load Capacity
41 Introduction High surface temperatures due to high loads and sliding ve-locities can cause lubricant films to break down This sei-zure or welding together of areas of tooth surface is termed scuffing
In contrast to pitting and fatigue breakage which show a dis-tinct incubation period a single short overloading can lead to a scuffing failure In the ISO-TR13989 two criteria are mentioned The method used in this Classification Note is based on the principles of the flash temperature criterion
Note Bulk temperature in excess of 120ordmC for long periods may have an adverse effect on the surface durability see 211
42 General Criteria In no point along the path of contact the local contact tem-perature may exceed the permissible temperature ie
oilS
oilSB S
ϑ+ϑminusϑ
leϑ
50SB minusϑleϑ
where
Bϑ = max contact temperature along the path of contact
maxflaMBB ϑ+ϑ=ϑ
MBϑ = bulk temperature see 434
maxflaϑ = max flash temperature along the path of con-tact see 44
Sϑ = scuffing temperature see below
oilϑ = oil temperature before it reaches the mesh (max applicable for the actual load case to be used ie normally alarm temperature except for ice classes where a max expected tem-perature applies)
SS = required safety factor according to the Rules
The scuffing temperature Sϑ may be calculated as
L2
wrelT
002
40S XFZGX
ν100112085780
sdot
sdot++=ϑ
where
XwrelT = relative welding factor
XwrelT
Through hardened steel 10
Phosphated steel 125
Copper-plated steel 150
Nitrided steel 150
Less than 10 retained austenite
115
10 ndash 20 retained austenite 10
Casehardened steel
20 ndash 30 retained austenite 085
Austenitic steel 045
FZG = load stage according to FZG-Test A8390 (Note This is the load stage where scuffing oc-curs However due to scatter in test results cal-culations are to be made with one load stage less than the specification)
XL = lubricant factor
= 10 for mineral oils
= 08 for polyalfaolefins
= 07 for non-water-soluble polyglycols
= 06 for water-soluble polyglycols
= 15 for traction fluids
= 13 for phosphate esters
ν40 = kinematic oil viscosity at 40˚C (mm2s)
Application of other test methods such as the Ryder the FZG-Ryder R46574 and the FZG L-42 Test 141195110 may be specially considered
For high speed gears with very short time of contact Sϑ may be increased as follows provided use of EP-oils
Addition to the calculated scuffing temperature Sϑ
If micros 18tc ge no addition
If micros 18tc lt add ( )cwrelT t18X18 minussdotsdot
where
ct = contact time ( micros ) which is the time needed to cross the Hertzian contact width
( ) [ ]microsu1cosβn
uσ340tb1
Hc +sdotsdot
sdot=
σH as calculated in 221
For bevel gears use vu in stead of u
34 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
43 Influence Factors
431 Coefficient of friction The following coefficient of friction may apply
L025a
005oil
02
redC ΣC
Bt X R ηρv
w0048micro minus
=
where
wBt = specific tooth load (Nmm)
vΣC = sum of tangential velocities at pitch point At pitch line velocities gt 50 ms the limiting value of vΣC at v = 50 ms is to be used
ρredC = relative radius of curvature (transversal plane) at the pitch point
Cylindrical gears
HαHβvγAbt
Bt KKKKKb
Fw sdotsdotsdotsdotsdot= (see 1)
twΣC αsin v2v =
bCredC β cos ρρ =
Bevel gears
HαHβvγAtmb
Bt KKKKKb
Fw sdotsdotsdotsdotsdot= (see 1)
vtmtΣC αsin v2v =
bmvCredC β cos ρρ =
ηoil = dynamic viscosity (mPa s) at oilϑ calculated as
1000
ρ νη oiloil =
where ρ in kgm3 approximated as
( ) 70 15oil15 minusϑminusρ=ρ and νoil is kinematic viscosity at oilϑ and may be calculated by means of the following equation
( ) ( )++=+ 08νloglog08νloglog 100oil ( )
sdotminus
ϑ+minus313log373log
273log373log oil
( ) ( )( )08νloglog08νloglog 10040 +minus+
Ra = composite arithmetic mean roughness (micron) of pinion an wheel calculated as ( )2a1aa RR 05R +=This is defined as the roughness on the new flanks ie as manufactured
XL = see 42
432 Effective tip relief Ceff Ceff is the effective tip relief that amount of tip relief which compensates for the elastic deformation of the gear mesh ie
zero load at the tooth tip It is assumed (simplified) to be equal for both pinion and wheel
Cylindrical gears
for helical
γ
γ=cb
KKFC Abt
eff (see 1)
For spur
cbKKF
C Abteff
γ= (see 1)
Alternatively for spur and helical gears the non-linear ap-proach in 111 may be used (taking Ceff=δ)
Bevel gears
bcKFC Ambt
effγ
= (see 1)
where
γ
α
ε2ε44c
+sdot
=γ
433 Tip relief and extension Cylindrical gears
The extension of the tip relief is not to result in an effective contact ratio 1ltεα when the gear is unloaded (exceptions to this may only apply for applications where the gear is not to run at light loads) This means that the unrelieved part of the path of contact is to be minimum pbt It is further assumed that this unrelieved part is placed centrally on the path of contact
If root relief applies it has to be calculated as equivalent tip relief Ie pinion root relief (at mesh position A) is added to Ca2 and wheel root relief (at mesh position E) is added to Ca1 If no design tip relief or root relief on the mating gear is specified (ie if Ca1+Croot2 = 0 and visa versa) use the run-ning in amount see 112
Bevel gears
Bevel gears are to have heightwise crowning ie no distinct relievedunrelieved area This may be treated as tip and root relief For calculation purposes the root relief is combined with the tip relief of the mating member into an equivalent tip relief If no resulting tip relieves are specified the equivalent tip relives may be calculated as an approxima-tion based on the tool crowning Ca tool (per mille of tool module m0) as follows
2root1aeq1a CCC +=
1root2aeq2a CCC +=
where
Classification Notes- No 412 35 May 2003
DET NORSKE VERITAS
2
0
n0101atoolal m
)m2(mA1middotmCC
minus+minussdot=
( )
sdotminusminussdotsdotminus+= vtvb1
2vb1
21vavt1n1 αtan dddαsin 05x1mA
2
0
n020atool22a m
)mm(2A1middotmCC
minus+minussdot=
( )
sdotminusminussdotsdotminus+= vtvb2
2vb2
2va2vt2n2 αtan dddαsin 05x1mA
2
0
20atool1root1 m
A1mCC
minussdotsdot=
2
0
10atool2root2 m
A1mCC
minussdotsdot=
If the pressure angles of the cutter blades are modified (and verified) to balance the tip relieves the following may be assumed
( )eq2aeq1aeq2aeq1a C andC calculated of sum50CC sdot==
Throughout the following Ca1 and Ca2 mean the equivalent tip relieves Ca1eq and Ca2eq
434 Bulk temperature The bulk temperature may be calculated as
flaaveragempsoilMB X X 50 ϑ+ϑ=ϑ
where
Xs = lubrication factor
= 12 for spray lubrication
= 10 for dip lubrication (provided both pinion and wheel are dip lubricated and tip speed lt 5 ms)
= 10 for spray lubrication with additional cooling spray (spray on both pinion and wheel or spray on pinion and dip lubrication of wheel)
= 02 for meshes fully submerged in oil
Xmp = contact factor ( )pmp nX += 150
np = number of mesh contact on the pinion (for small gear ratii the number of wheel meshes should be used if higher)
flaaverageϑ
= average of the integrated flash temperature (see 44) along the path of contact
( )
AE
E
Ayyyfla
flaaverage
d
ΓminusΓ
ΓΓϑ=ϑint =
For high speed gears (v gt 50 ms) it may be necessary to assess the bulk temperature on the basis of a thermal rating of the entire gear transmission
44 The Flash Temperature flaϑ
441 Basic formula The local flash temperature flaϑ may be calculated as
( ) 41redy
y2ly
211
43Btcorrfla
unXwX3250
y ρ
ρminusρ
micro=ϑ Γ
(For bevel gears replace u with uv)
and is to be calculated stepwise along the path of contact from A to E
where
micro = coefficient of friction see 431
Xcorr = correction factor taking empirically into account the increased scuffing risk in the beginning of the approach path due to mesh starting without any previously built up oil film and possible shuffling away oil before meshing if insufficient tip relief
( )
3
AD
yaeffcorr 50
CC1X
ΓminusΓε
Γminus+=
α
Ca = tip relief of driven member Xcorr is only applicable in the approach path and if Ca lt Ceff otherwise 10
wBt = unit load see 431
yXΓ = load sharing factor see 443
n1 = pinion rpm
Γy ρly etc see 442
442 Geometrical relations The various radii of flank curvature (transversal plane) are
ρ1y = pinion flank radius at mesh point y
ρ2y = wheel flank radius at mesh point y
ρredy = equivalent radius of curvature at mesh point y
y2y1
y2y1redy
ρ+ρ
ρρ=ρ
Cylindrical gears
twy
1y αsin au1Γ1
ρ+
+=
twy
2y αsin au1Γu
ρ+
minus=
Note that for internal gears a and u are negative
36 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
Bevel gears
vtvv
y1y αsin a
u1Γ1
ρ+
+=
vtvv
yv2y αsin a
u1Γu
ρ+
minus=
Γ is the parameter on the path of contact and y is any point between A end E
At the respective ends Γ has the following values
Root piniontip wheel
Cylindrical gears
( )
minus
minusminus= 1
αtan 1dd
zzΓ
tw
2b2a2
1
2A
Bevel gears
( )
minus
minusminus= 1
αtan 1dd
uΓvt
2vb2va2
vA
Tip pinionroot wheel
Cylindrical gears
( )
1αtan
1ddΓ
tw
2b1a1
E minusminus
=
Bevel gears
( )
1αtan
1ddΓ
vt
2vb1va1
E minusminus
=
At inner point of single pair contact
Cylindrical gears
tw1
EB α tan zπ2ΓΓ minus=
Bevel gears
vtv1
EB α tan zπ2ΓΓ minus=
At outer point of single pair contact
Cylindrical gears
tw1AD α tan z
π2ΓΓ +=
Bevel gears
vt v1
AD αtan z π2ΓΓ +=
At pitch point ΓC = 0
The points F and G (only applicable to cylindrical gears) limiting the extension of tip relief (so as to maintain a mini-mum contact ratio of unity for unloaded gears) are at
2
BAF
Γ+Γ=Γ
2
EDG
Γ+Γ=Γ
443 Load sharing factor XΓ The load sharing factor XΓ accounts for the load sharing be-tween the various pairs of teeth in mesh along the path of contact
XΓ is to be calculated stepwise from A to E using the pa-rameter Γy
4431 Cylindrical gears with β = 0 and no tip relief
Figure 41
ByAAB
Ay for31
31X
yΓltΓleΓ
ΓminusΓ
ΓminusΓ+=Γ
DyB for1Xy
ΓltΓleΓ=Γ
EyDDE
yE for31
31X
yΓleΓltΓ
ΓminusΓ
ΓminusΓ+=Γ
4432 Cylindrical gears with β = 0 and tip relief Tip relief on the pinion reduces XΓ in the range G ndash E and increases correspondingly XΓ in the range F ndash B
Tip relief on the wheel reduces XΓ in the range A ndash F and increases correspondingly XΓ in the range D ndash G
Following remains generally
DyB for1Xy
ΓleΓleΓ=Γ
21XXGF== ΓΓ
In the following it must be distinguished between Ca lt Ceff respectively Ca gt Ceff This is shown by an example below where Ca1 lt Ceff and Ca2 gt Ceff
Classification Notes- No 412 37 May 2003
DET NORSKE VERITAS
Figure 42
Note When Ca gt Ceff the path of contact is shortened by A ndash A respectively E ndash E The single pair contact path is extended into B respectively D If this shift is significant it is necessary to consider the negative effect on surface durability (B) and bending stresses (B and D)
Range A - F
For effa2 CC le
minus=Γ
eff
2a
CC1
31X
A
FyAeff
2a
AF
Ay for C3
C61XX
AyΓleΓleΓ
sdot
+ΓminusΓ
ΓminusΓ+= ΓΓ
For effa2 CC ge
AyAfor0Xy
ΓleΓleΓ=Γ
with ( )21
CC
1CC
eff
2a
eff
2a
AFAAminus
minusΓminusΓ+Γ=Γ
FyAeff
2a
AF
Ay
eff
2a for 21
CC
CC1X
yΓleΓleΓ
minus
ΓminusΓ
ΓminusΓ+minus=Γ
Range F - B
For effa1 CC le
ByFeff
1a
FB
Fy forC3
C61
21X
yΓleΓleΓ
sdot
+ΓminusΓ
ΓminusΓ+=Γ
For effa1 CC ge
ByFeff
1a
FB
Fy for21
CC
21X
yΓleΓleΓ
minus
ΓminusΓ
ΓminusΓ+=Γ
ByB for1Xy
ΓleΓleΓ=Γ
with 1
CC2
eff
1a
FBFB
minus
ΓminusΓ+Γ=Γ
Range D ndash G
For effa2 CC le
GyDeff
2a
DG
Dy
eff
2a for C 3C
61
C 3C
32X
yΓleΓleΓ
+
ΓminusΓ
ΓminusΓminus+=Γ F
or effa2 CC ge
DyD for1Xy
ΓleΓleΓ=Γ
with ( )21
CC
1CC
eff
2a
eff
2a
DGDDminus
minusΓminusΓ+Γ=Γ
GyDeff
2a
DG
Dy
eff
2a for21
CC
CCX ΓleΓleΓ
minus
ΓminusΓ
ΓminusΓminus=Γ
Range G - E
For effa1 CC le
minus=Γ
eff
1a
CC1
31X
E
EyGeff
1a
GE
Gy forC3
C61
21X
yΓleΓleΓ
sdot
+ΓminusΓ
ΓminusΓminus=Γ
For effa1 CC gt
EyGeff
1a
GE
Gy for21
CC
21X
yΓleΓleΓ
minus
ΓminusΓ
ΓminusΓminus=Γ
EyE for0Xy
ΓleΓleΓ=Γ
with 1
CC2
eff
1a
GEGE
minus
ΓminusΓ+Γ=Γ
4433 Gears with β gt 0 buttressing Due to oblique contact lines over the flanks a certain but-tressing may occur near A and E
This applies to both cylindrical and bevel gears with tip relief lt Ceff The buttressing Xbutt is simplified as a linear function within the ranges A ndash H respectively I ndash E
Figure 43
1εwhen13X βbutt EAge=
1εwhenε031X ββbutt EAlt+=
38 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
Cylindrical gears
bIEAH βsin 02ΓΓΓΓ =minus=minus
Bevel gears
bmIEAH βsin 02ΓΓΓΓ =minus=minus
4434 Cylindrical gears with 2εγ le and no tip relief
yXΓ is obtained by multiplication of
yXΓ in 4431 with
Xbutt in 4433
4435 Gears with 2εγ gt and no tip relief
Applicable to both cylindrical and bevel gears
IyH for1Xy
ΓleΓleΓε
=α
Γ
IyHybutt and forX1Xy
ΓgtΓΓltΓε
=α
Γ
Figure 44
4436 Cylindrical gears with 2εγ le and tip relief
yXΓ is obtained by multiplication of
yXΓ in 4432 with Xbutt
in 4433
4437 Cylindrical gears with 2εγ gt and tip relief
Tip relief on the pinion (respectively wheel) reduces XГ in the range G ndash E (respectively A ndash F) and increases XГ in the range F ndash G
yXΓ is obtained by multiplication of
yXΓ as described below
with Xbutt in 4433
In the XГ example below the influence of tip relief is shown (without the influence of Xbutt) by means of
eff2aeff1a CC and CC ltgt
Tip relief gt Ceff causes new end points A respectively E of the path of contact
Figure 45
Range A ndash F
( ) ( )( ) eff
a2αa1α
AF
Ay
eff
2aeff
C 12C 13εC 1ε
CCCX
y +εε++minus
ΓminusΓ
ΓminusΓ+
εminus
=ααα
Γ
eff2aFyA CC if for leΓleΓleΓ
eff2aFyA CifCfor and geΓleΓleΓ
eff2aAyA CC iffor0Xy
gtΓleΓleΓ=Γ
( ) ( ) ( )( ) ( ) a2αa1α
αeffa2AFAA C 1ε 3C 1ε
1ε 2 CC with++minus+minus
ΓminusΓ+Γ=Γ
Range F ndash G
( )( )( ) GyF
eff
2a1a forC 1 2CC 11X
yΓleΓleΓ
+εε+minusε
+ε
=αα
α
αΓ
Range G ndash E
( ) ( )( ) eff
2a1a
GE
Gy
C 1 2C 1C 1 3XX
GFy +εεminusε++ε
ΓminusΓ
ΓminusΓminus=
αα
ααΓΓ minus
effa1EyG CC if for leΓleΓleΓ
effa1EyG CC if Γfor and geΓleΓle
eff1aEyE CC if for0Xy
geΓleΓleΓ=Γ
( ) ( ) ( )( ) ( ) 2a1a
eff1aGEEE C 1C 1 3
1 2 CC withminusε++ε+εminus
ΓminusΓminusΓ=Γαα
α
4438 Bevel gears with εγ more than approx 18 and heightwise crowning For Ca1 = Ca2 = Ceff the following applies
Figure 46
Classification Notes- No 412 39 May 2003
DET NORSKE VERITAS
( )AEM 50 Γ+Γ=Γ
( )( )2AD
3
2My651X
y ΓminusΓε
ΓminusΓminus
ε=
ααΓ
For tip relief lt Ceff yXΓ is found by linear interpolation be-
tween ( )effay CCX =Γ and ( )0CayX =Γ as in 4435
The interpolation is to be made stepwise from A to M with the influence of Ca2 and from M to E with the influence of Ca1 (For 2a1a CC ne there is a discontinuity at M)
Eg with Ca1 = 04 Ceff and Ca2 = 055 Ceff then
Range A ndash M
)CC()0C( eff2ay2ayyX 550X 450X =Γ=ΓΓ +=
Range M ndash E
)CC()0C( eff1ay1ayyX 40X 06X =Γ=ΓΓ +=
For tip relief gt Ceff the new end points A and E are found as
( )
minusΓminusΓ
ε+Γ=Γ α 1
CC
6 eff
2aADAA
( )
minusΓminusΓ
ε+Γ=Γ α 1
CC
6 eff
1aADEE
Range A ndash A
0Xy=Γ
Range A ndash M
( )( )
ΓminusΓ
ΓminusΓminus
minusε=
αΓ 2
MA
2My
eff
2a1
CC4
351Xy
Range M ndash E
( )( )
ΓminusΓ
ΓminusΓminus
minusε=
αΓ 2
ME
2My
eff
1a1
CC4
351Xy
Range E ndash E
0Xy=Γ
40 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
Appendix A Fatigue Damage Accumulation
The Palmgren-Miner cumulative damage calculation princi-ple is used The procedure may be applied as follows
A1 Stress Spectrum From the individual torque classes the torques (Ti) at the peak values of class intervals and the associated number of cycles (NLi) for both pinion and wheel are to be listed from the highest to the lowest torque
(In case of a cyclic torque variation within the torque classes it is advised to use the peak torque If the cyclic variation is such that the same teeth will repeatedly suffer the peak torque this is a must)
The stress spectra for tooth roots and flanks (σFi σHi) with all relevant factors (except KA) are to be calculated on the basis of the torque spectrum The load dependent K-factors are to be determined for each torque class
A2 σminusN-curve The stress versus load cycle curves for tooth roots and flanks (both pinion and wheel) are to be drawn on the basis of per-missible stresses (ie including the demanded minimum safety factors) as determined in 2 respectively 3 If different safety levels for high cycle fatigue and low cycle fatigue are desired this may be expressed by different demand safety factors applied at the endurance limit respectively at static strength
A3 Damage accumulation The individual damage ratio Di at ith stress level is defined as
Fi
Lii N
ND =
where
NLi = The number of applied cycles at ith stress
NFi = The number of cycles to failure at ith stress
Basically stresses σi below the permissible stress level for infinite life (if a constant ZN or YN is accepted) do not con-tribute to the damage sum However calculating the actual safety factor Sact as described below all the σi for which the product S σi is bigger than or equal to the permissible stress level for infinite life contribute to the damage sum and thus to the determination of Sact The final value of S is decisive
(NFi can be found mathematically by putting the permissible stress σpi equal the actual stress σi thereby finding the actual life factor This life factor can be solved with regard to load cycles ie NFi)
The damage sum ΣDi is not to exceed unity
If ΣDi ne 1 the safety against cumulative fatigue damage is different from the applied demand safety factor For deter-mination of this theoretical safety factor an iteration proce-dure is required as described in the following flowchart
S is correction factor with which the actual safety factor Sact can be found
Sact is the demand safety factor (used in determination of the permissible stresses in the σ ndash N ndash curve) times the correc-tion factor S
The full procedure is to be applied for pinion and wheel tooth roots and flanks
Note If alternating stresses occur in a spectrum of mainly pulsating stresses the alternating stresses may be replaced by equiva-lent pulsating stresses ie by means of division with the ac-tual mean stress influence factor YM
Classification Notes- No 412 41 May 2003
DET NORSKE VERITAS
Appendix B Application Factors for Diesel Driven Gears
For diesel driven gears the application factor KA depends on torsional vibrations Both normal operation and misfiring conditions have to be considered
Normally these two running conditions can be covered by only one calculation
B1 Definitions
Normal operation KAnorm = 0
normv0
TTT +
where
T0 = rated nominal torque
Tv norm = vibratory torque amplitude for normal opera-tion (see rules Pt4 Ch3 Sec1 G301 for defi-nition of ldquonormalrdquo irregularity)
Misfiring operation KA misf = 0
misfv
TTT +
where
T = remaining nominal torque when one cylinder out of action
Tv misf = vibratory torque amplitude in misfiring con-dition This refers to a permissible misfiring condition ie a condition that does not require automatic or immediate corrective actions as speed or pitch reduction
The normal operation is assumed to last for a very high num-ber of cycles such as 1010
The misfiring operation is assumed to last for a limited dura-tion such as 107 cycles
B2 Determination of decisive load Assuming life factor at 1010 cycles as YN = ZN = 092 which usually is relevant the calculation may be performed only once with the combination having the highest value of appli-cation factorlife factor
For bending stresses and scuffing the higher value of
092
K normA and 098
K misfA
For contact stresses the higher value of
092K normA and
113K misfA (but
097K misfA for nitrided gears)
B3 Simplified procedure Note that this is only a guidance and is not a binding con-vention
T = oTZ
1Zsdot
minus where Z = number of cylinders
( ) ideal Videal Vmisf Vnorm V TTT24ZT +minus=
where TV ideal = vibratory torque with all cylinders perfectly equal See also rules Pt4 Ch3 Sec1 G300
When using trends from torsional vibration analysis and measurements the following may be used
TV ideal To is close to zero for engines with few cylinders and using a suitable elastic coupling and increases with relative coupling stiffness and number of cylinders
TV misf To may be high for engines with few cylinders and decreases with number of cylinders
This can be indicated as
200Z
TT
ο
idealV asymp and 80Z04
TT
o
misfV minusasymp
Inserting this into the formulae for the two application fac-tors the following guidance can be given
118112K misfA minusasymp
115110K normA minusasymp
Since KA norm is to be combined with the lower life factors the decisive load condition will be the normal one and a KA of 115 will cover most relevant cases when a suitable elas-tic coupling is chosen
42 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
Appendix C Calculation of Pinion-Rack
Pinion-racks used for elevating of mobile offshore units are open gears that are subjected to wear and tear With normal specifications such as only a few hundred total operation cycles (site to site) the tooth bending stresses for static strength will be decisive for the lay out however with ex-ception of surface hardened pinions where case crushing has to be considered
In the following the use of part 1 and 3 for pinion-racks is shown including relevant simplifications
C1 Pinion tooth root stresses Since the load spectrum normally is dominated by high torques and few load cycles (in the range up to 1-2000) the static strength is decisive
The actual stress is calculated as
β1FSaFan1
tF1 KYY
mbFσ sdotsdotsdotsdot
=
b1 is limited to b2 + 2middotmn
YFa and YSa replace YF and YS because load application at tooth tip has to be assumed for such inaccurate gears
Pinions often use a non-involute profile in the dedendum part of the flank eg a constant radius equal the radius of curva-ture at reference circle For such pinions sFn and hFa are to be measured directly on a sectional drawing of the pinion tooth
Due to high loads and narrow facewidths it may be assumed that KFβ2 = KHβ = 10 However when b1 gt b2 then KFβ1 gt10
If no detailed documentation of KFβ1 is available the fol-lowing may be used
KFβ1 = 1 + 015middot(b1b2 ndash 1)
The permissible stress (not surface hardened) is calculated as
δrelTstF
Fst1FP1 Y
Sσσ sdot=
The mean stress influence due to leg lifting may be disre-garded
The actual and permissible stresses should be calculated for the relevant loads as given in the rules
C2 Rack tooth root stresses The actual stress is calculated as
SaFan2
tF2 YY
mbFσ sdotsdotsdot
=
See C1 for details
The permissible stress is calculated as
δrelTstF
Fst2FP2 Y
Sσσ sdot=
For alloyed steels (Ni Cr Mo) with high toughness and duc-tility the value of YδrelTst may be put equal to YSa
C3 Surface hardened pinions For surface hardened pinions the maximum load is not to cause crushing of the hardened layer of the flank
In principle the calculation described in 213 may be used but when the theoretical Hertzian stress exceeds the ap-proximately 18 times the yield strength of the rack material plastic deformation will occur This will limit the peak Hertzian stress but increases the contact width and thus the penetration of stresses into the depth
An approximation may be based on an assessment of contact width determined by means of equal areas under the theoreti-cal (elastic range) Hertzian contact and the elasto-plastic contact stress (limited to 18 middot σy) with the unknown width
Classification Notes- No 412 17 May 2003
DET NORSKE VERITAS
2 Calculation of Surface Durability
21 Scope and General Remarks Part 2 includes the calculations of flank surface durability as limited by pitting spalling case crushing and subsurface yielding Endurance and time limited flank surface fatigue is calculated by means of 22 ndash 212 In a way also tooth frac-tures starting from the flank due to subsurface fatigue is in-cluded through the criteria in 213
Pitting itself is not considered as a critical damage for slow speed gears However pits can create a severe notch effect that may result in tooth breakage This is particularly im-portant for surface hardened teeth but also for high strength through hardened teeth For high-speed gears pitting is not permitted
Spalling and case crushing are considered similar to pitting but may have a more severe effect on tooth breakage due to the larger material breakouts initiated below the surface Subsurface fatigue is considered in 213
For jacking gears (self-elevating offshore units) or similar slow speed gears designed for very limited life the max static (or very slow running) surface load for surface hard-ened flanks is limited by the subsurface yield strength
For case hardened gears operating with relatively thin lubri-cation oil films grey staining (micropitting) may be the lim-iting criterion for the gear rating Specific calculation meth-ods for this purpose are not given here but are under consid-eration for future revisions Thus depending on experience with similar gear designs limitations on surface durability rating other than those according to 22 - 213 may be ap-plied
22 Basic Equations Calculation of surface durability (pitting) for spur gears is based on the contact stress at the inner point of single pair contact or the contact at the pitch point whichever is greater
Calculation of surface durability for helical gears is based on the contact stress at the pitch point
For helical gears with 0 lt εβ lt 1 a linear interpolation be-tween the above mentioned applies
Calculation of surface durability for spiral bevel gears is based on the contact stress at the midpoint of the zone of contact
Alternatively for bevel gears the contact stress may be cal-culated with the program ldquoBECALrdquo In that case KA and Kv are to be included in the applied tooth force but not KHβ and KHα The calculated (real) Hertzian stresses are to be multi-plied with ZK in order to be comparable with the permissible contact stresses
The contact stresses calculated with the method in part 2 are based on the Hertzian theory but do not always represent the real Hertzian stresses
The corresponding permissible contact stresses σHP are to be calculated for both pinion and wheel
221 Contact stress Cylindrical gears
( )HαHβvγA
1
tβεEHDBH KKKKK
bud1uF
ZZZZZσ+
=
where
ZBD = Zone factor for inner point of single pair contact for pinion resp wheel (see 232)
ZH = Zone factor for pitch point (see 231)
ZE = Elasticity factor (see 24)
Zε = Contact ratio factor (see 25)
Zβ = Helix angle factor (see 26)
Ft KA Kγ Kv KHβ KHα see 15 ndash 110
d1 b u see 12 ndash 15
Bevel gears
( )HαHβvγA
v1v
vmtKEMH KKKKK
bud1uF
ZZZ105σ+
sdot=
where
105 is a correlation factor to reach real Hertzian stresses (when ZK = 1)
ZE KA etc see above
ZM = mid-zone factor see 233
ZK = bevel gear factor see 27
Fmt dv1 uv see 12 ndash 15
It is assumed that the heightwise crowning is chosen so as to result in the maximum contact stresses at or near the mid-point of the flanks
222 Permissible contact stress
XWRvLH
NHlimHP ZZZZZ
SZσ
σ =
where
σH lim = Endurance limit for contact stresses (see 28)
ZN = Life factor for contact stresses (see 29)
SH = Required safety factor according to the rules
ZLZvZR = Oil film influence factors (see 210)
ZW = Work hardening factor (see 211)
ZX = Size factor (see 212)
18 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
23 Zone Factors ZH ZBD and ZM
231 Zone factor ZH The zone factor ZH accounts for the influence on contact stresses of the tooth flank curvature at the pitch point and converts the tangential force at the reference cylinder to the normal force at the pitch cylinder
wtt2
wtbH sinααcos
cosαcosβ2Z =
232 Zone factors ZBD The zone factors ZBD account for the influence on contact stresses of the tooth flank curvature at the inner point of sin-gle pair contact in relation to ZH Index B refers to pinion D to wheel
For εβ ge 1 ZBD = 1
For internal gears ZD = 1
For εβ = 0 (spur gears)
( )
π
minusεminusminus
π
minusminus
α=
α2
2
2b
2a
1
2
1b
1a
wtB
z211
dd
z21
dd
tanZ
( )
π
minusεminusminus
π
minusminus
α=
α1
2
1b
1a
2
2
2b
2a
wtD
z211
dd
z21
dd
tanZ
If ZB lt 1 use ZB = 1
If ZD lt 1 use ZD = 1
For 0 lt εβ lt 1
ZBD = ZBD (for spur gears) ndash εβ (ZBD (for spur gears) ndash 1)
233 Zone factor ZM The mid-zone factor ZM accounts for the influence of the contact stress at the mid point of the flank and applies to spi-ral bevel gears
εminusminus
εminusminus
αβ=
αα btm2
2vb2
2vabtm2
1vb2val
2v1vvtbmM
pddpdd
ddtancos2Z
This factor is the product of ZH and ZM-B in ISO 10300 with the condition that the heightwise crowning is sufficient to move the peak load towards the midpoint
234 Inner contact point For cylindrical or bevel gears with very low number of teeth the inner contact point (A) may be close to the base circle In order to avoid a wear edge near A it is required to have suitable tip relief on the wheel
24 Elasticity Factor ZE The elasticity factor ZE accounts for the influence of the material properties as modulus of elasticity and Poissonrsquos ratio on the contact stresses
For steel against steel ZE = 1898
25 Contact Ratio Factor Zε The contact ratio factor Zε accounts for the influence of the transverse contact ratio εα and the overlap ratio εβ on the contact stresses
αε1Z =ε for 1εβ ge
( )α
ββ
αε ε
εε1
3ε4
Z +minusminus
= for εβ lt 1
26 Helix Angle Factor Zβ The helix angle factor Zβ accounts for the influence of helix angle (independent of its influence on Zε) on the surface du-rability
cosβZβ =
27 Bevel Gear Factor ZK The bevel gear factor accounts for the difference between the real Hertzian stresses in spiral bevel gears and the contact stresses assumed responsible for surface fatigue (pitting) ZK adjusts the contact stresses in such a way that the same per-missible stresses as for cylindrical gears may apply
The following may be used ZK = 080
28 Values of Endurance Limit σHlim and Static Strength 5H10σ 3H10σ
σHlim is the limit of contact stress that may be sustained for 5middot107 cycles without the occurrence of progressive pitting
For most materials 5middot107 cycles are considered to be the be-ginning of the endurance strength range or lower knee of the σ-N curve (See also Life Factor ZN) However for nitrided steels 2middot106 apply
For this purpose pitting is defined by
bull for not surface hardened gears pitted area ge 2 of total active flank area
bull for surface hardened gears pitted area ge 05 of total active flank area or ge 4 of one particular tooth flank area 510Hσ and 310Hσ are the contact stresses which the given
material can withstand for 105 respectively 103 cycles without subsurface yielding or flank damages as pitting spalling or case crushing when adequate case depth applies
The following listed values for σHlim 510Hσ and 310Hσ may only be used for materials subjected to a quality control as
Classification Notes- No 412 19 May 2003
DET NORSKE VERITAS
the one referred to in the rules Results of approved fatigue tests may also be used as the basis for establishing these values The defined survival probability is 99
σHlim σH105 σH10
3
Alloyed case hardened steels (surface hardness 58-63 HRC) - of specially approved high grade - of normal grade
1650 1500
2500 2400
3100 3100
Nitrided steel of approved grade gas nitrided (surface hardness 700-800 HV) 1250 13 σHlim 13 σHlim
Alloyed quenched and tempered steel bath or gas nitrided (surface hardness 500-700 HV)
1000
13 σHlim
13 σHlim
Alloyed flame or induction hardened steel (surface hardness 500-650 HV) 075 HV + 750 16 σHlim 45 HV
Alloyed quenched and tempered steel 14 HV + 350 16 σHlim 45 HV
Carbon steel 15 HV + 250 16 σHlim 16 σHlim
These values refer to forged or hot rolled steel For cast steel the values for σHlim are to be reduced by 15
29 Life Factor ZN The life factor ZN takes account of a higher permissible contact stress if only limited life (number of cycles NL) is demanded or lower permissible contact stress if very high number of cycles apply
If this is not documented by approved fatigue tests the fol-lowing method may be used
For all steels except nitrided
7L 105N sdotge ZN = 1 or
01570
L
7
N N105Z
sdot=
Ie ZN = 092 for 1010 cycles
The ZN = 1 from 5middot107 on may only be used when the mate-rial cleanliness is of approved high grade (see Rules Pt4 Ch2) and the lubrication is optimised by a specially approved filtering process
105 lt NL lt 5middot107 510N
logZ037
L
7
N N105Z
sdot=
NL = 105 WXRVLHlim
WstX10H1010NN ZZZZZσ
ZZσZZ
55
5=
103 lt NL lt 105 )Z(Zlog05
L
5
10NN
5N103N10
5N10ZZ
==
3L 10N le
WXRVLlimH
Wst10X10H10NN ZZZZZ
ZZZZ
33
3σ
σ==
(but not less than ZN105)
For nitrided steels
6L 102N sdotge ZN = 1 or
00980
L
6
N N102Z
sdot=
Ie ZN = 092 for 1010 cycles
The ZN = 1 from 2middot106 on may only be used when the mate-rial cleanliness is of approved high grade (see Rules Pt4 Ch2) and the lubrication is optimised by a specially approved filtering process
105 lt NL lt 2middot106 510N
Zlog76860
L
6
N N102Z
sdot=
5L 10N le
XWRVL
10XWst10NN ZZZZZ
ZZ13ZZ
55 ==
Note that when no index indicating number of cycles is used the factors are valid for 5middot107 (respectively 2middot106 for nitrid-ing) cycles
210 Influence Factors on Lubrication Film ZL ZV and ZR The lubricant factor ZL accounts for the influence of the type of lubricant and its viscosity the speed factor ZV ac-counts for the influence of the pitch line velocity and the roughness factor ZR accounts for influence of the surface roughness on the surface endurance capacity
The following methods may be applied in connection with the endurance limit
20 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
Surface hardened steels Not surface hardened steels
ZL ( )24013421
360910ν+
+
( )24013421
680830ν+
+
ZV ( )v3280
140930+
+ ( )v3280300850
++
ZR
080
ZrelR3
150
ZrelR3
where
ν40 = Kinematic oil viscosity at 40ordmC (mm2s) For case hardened steels the influence of a high bulk temperature (see 4 Scuffing) should be con-sidered Eg bulk temperatures in excess of 120ordmC for long periods may cause reduced flank surface endurance limits
For values of ν40 gt 500 use ν40 = 500
RZrel = The mean roughness between pinion and wheel (after running in) relative to an equivalent radius of curvature at the pitch point ρc = 10mm
RZrel
= ( ) 31
cZ2Z1 ρ
10RR50
+
RZ = Mean peak to valley roughness (microm) (DIN defini-tion) (roughly RZ = 6 Ra)
For 5
L 10N le ZL ZV ZR = 10
211 Work Hardening Factor ZW The work hardening factor ZW accounts for the increase of surface durability of a soft steel gear when meshing the soft steel gear with a surface hardened or substantially harder gear with a smooth surface
The following approximation may be used for the endurance limit
Surface hardened steel against not surface hardened steel 150
ZeqW R
31700
130HB21Z
minus
minus=
where HB = the Brinell hardness of the soft member For HB gt 470 use HB = 470 For HB lt 130 use HB = 130
RZeq = equivalent roughness
033
c40
066
ZS
ZHZHZEQ ρvν
15000RRRR
=
If RZeq gt 16 then use RZeq = 16
If RZeq lt 15 then use RZeq = 15
where
RZH = surface roughness of the hard member before run in
RZS = surface roughness of the soft member before run in
ν40 = see 210
If values of ZW lt 1 are evaluated ZW = 1 should be used for flank endurance However the low value for ZW may indi-cate a potential wear problem
Through hardened pinion against softer wheel
( )
minussdotminus+= 000829
HBHB0008981u1Z
2
1W
For 21HBHB
2
1 le use ZW = 1
For 71HBHB
2
1 gt use 17HBHB
2
1 =
For u gt 20 use u = 20
For static strength (lt 105 cycles)
Surface hardened against not surface hardened
ZWst = 105
Through hardened pinion against softer wheel
ZWst = 1
212 Size Factor ZX The size factor accounts for statistics indicating that the stress levels at which fatigue damage occurs decrease with an increase of component size as a consequence of the influ-ence on subsurface defects combined with small stress gradi-ents and of the influence of size on material quality
ZX may be taken unity provided that subsurface fatigue for surface hardened pinions and wheels is considered eg as in the following subsection 213
213 Subsurface Fatigue This is only applicable to surface hardened pinions and wheels The main objective is to have a subsurface safety against fatigue (endurance limit) or deformation (static strength) which is at least as high as the safety SH required for the surface The following method may be used as an approximation unless otherwise documented
The high cycle fatigue (gt3106 cycles) is assumed to mainly depend on the orthogonal shear stresses Static strength (lt103 cycles) is assumed to depend mainly on equivalent
Classification Notes- No 412 21 May 2003
DET NORSKE VERITAS
stresses (von Mises) Both are influenced by residual stresses but this is only considered roughly and empirically
The subsurface working stresses at depths inside the peak of the orthogonal shear stresses respectively the equivalent stresses are only dependent on the (real) Hertzian stresses Surface related conditions as expressed by ZL ZV and ZR are assumed to have a negligible influence
The real Hertzian stresses σHR are determined as
For helical gears with εβ gt 1
σHR = σH
For helical gears with εβ lt 1 and spur gears
ε
α
ββ ε
ε+εminus
sdotσ=σZ
1
HHR
For bevel gears
KHHR Z
1σσ sdot=
The necessary hardness HV is given as a function of the net depth tz (net = after grinding or hard metal hobbing and per-pendicular to the flank)
The coordinates tz and HV are to be compared with the de-sign specification such as
bull for flame and induction hardening tHVmin HVmin bull for nitriding t400min HV = 400 bull for case hardening t550min HV = 550 t400min HV = 400
and t300min HV = 300 (the latter only if the core hardness lt 300 If the core hardness gt 400 the t400 is to be re-placed by a fictive t400 = 16 t550)
In addition the specified surface hardness is not to be less than the max necessary hardness (at tz = 05aH) This applies to all hardening methods
For high cycle fatigue (gt3 106 cycles) the following applies
sdot+
minussdotsdotsdot= o90
05azt
05azt
cosSσ04HV
H
HHHR
applicable to 05a
zt
Hge
For 05at
H
z lt the value for 05at
H
z = applies
56300ρSσ12a cHHR
Hsdotsdot
sdot=
Where aH is half the hertzian contact width multiplied by an empirical factor of 12 that takes into account the possible influence of reduced compressive residual stresses (or even tensile residual stresses) on the local fatigue strength
If any of the specified hardness depths including the surface hardness is below the curve described by HV = f (tz) the actual safety factor against subsurface fatigue is determined as follows
reduce SH stepwise in the formula for HV and aH until all specified hardness depths and surface hardness balance with the corrected curve The safety factor obtained through this method is the safety against subsurface fatigue
For static strength (lt103 cycles) the following applies
sdot+
minussdotsdotsdot= o90
07at
06a
zt
cosSσ019HV
Hst
z
HstHHR
applicable to 06at
Hst
z ge
56300ρSσa cHHR
Hstsdotsdot
=
In the case of insufficient specified hardness depths the same procedure for determination of the actual safety factor as above applies
For limited life fatigue (103 lt cycles lt 3106 )
For this purpose it is necessary to extend the correction of safety factors to include also higher values than required Ie in the case of more than sufficient hardness and depths the safety factor in the formulae for both high cycle fatigue and static strength are to be increased until necessary and speci-fied values balance
The actual safety factor for a given number of cycles N be-tween 103 and 3106 is found by linear interpolation in a dou-ble logarithmic diagram
minussdotminus
= sdot logN3477
logSlogSlogS
310H6103HHN
36 10H103H Slog86281Slog86280 sdot+sdot sdot
22 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
3 Calculation of Tooth Strength
31 Scope and General Remarks Part 3 include the calculation of tooth root strength as limited by tooth root cracking (surface or subsurface initiated) and yielding
For rim thickness sR ge 35middotmn the strength is calculated by means of 32 ndash 313 For cylindrical gears the calculation is based on the assumption that the highest tooth root tensile stress arises by application of the force at the outer point of single tooth pair contact of the virtual spur gears The method has however a few limitations that are mentioned in 36
For bevel gears the calculation is based on force application at the tooth tip of the virtual cylindrical gear Subsequently the stress is converted to load application at the mid point of the flank due to the heightwise crowning
Bevel gears may also be calculated with the program BECAL In that case KA and Kv are to be included in the applied tooth force but not KFβ and KFα
In case of a thin annulus or a thin gear rim etc radial crack-ing can occur rather than tangential cracking (from root fillet to root fillet) Cracking can also start from the compression fillet rather than the tension fillet For rim thickness sR lt 35middotmn a special calculation procedure is given in 315 and 316 and a simplified procedure in 314
A tooth breakage is often the end of the life of a gear trans-mission Therefore a high safety SF against breakage is re-quired
It should be noted that this part 3 does not cover fractures caused by
bull oil holes in the tooth root space bull wear steps on the flank bull flank surface distress such as pits spalls or grey staining
Especially the latter is known to cause oblique fractures starting from the active flank predominately in spiral bevel gears but also sometimes in cylindrical gears
Specific calculation methods for these purposes are not given here but are under consideration for future revisions Thus depending on experience with similar gear designs limita-tions other than those outlined in part 3 may be applied
32 Tooth Root Stresses The local tooth root stress is defined as the max principal stress in the tooth root caused by application of the tooth force Ie the stress ratio R = 0 Other stress ratios such as for eg idler gears (R asymp -12) shrunk on gear rims (R gt 0) etc are considered by correcting the permissible stress level
321 Local tooth root stress The local tooth root stress for pinion and wheel may be as-sessed by strain gauge measurements or FE calculations or similar For both measurements and calculations all details are to be agreed in advance
Normally the stresses for pinion and wheel are calculated as
Cylindrical gears
FαFβvγAβSFn
tF K K K K K Y Y Y
m bFσ =
where
YF = Tooth form factor (see 33)
YS = Stress correction factor (see 34)
Yβ = Helix angle factor (see 36)
Ft KA Kγ Kv KFβ KFα see 15 ndash 110
b see 13
Bevel gears
FαFβvγASaFamn
mtF K K K K K Y Y Y
m bFσ ε=
where
YFa = Tooth form factor see 33
YSa = Stress correction factor see 34
Yε = Contact ratio factor see 35
Fmt KA etc see 15 ndash 110
b see 13
322 Permissible tooth root stress The permissible local tooth root stress for pinion respectively wheel for a given number of cycles N is
CXRrelTrelTF
NMFEFP YYYY
SYY
δσ
=σ
Note that all these factors YM etc are applicable to 3middot106 cycles when used in this formula for σFP The influence of other number of cycles on these factors is covered by the calculation of YN
where
σFE = Local tooth root bending endurance limit of reference test gear (see 37)
YM = Mean stress influence factor which accounts for other loads than constant load direction eg idler gears temporary change of load di-rection pre-stress due to shrinkage etc (see 38)
YN = Life factor for tooth root stresses related to reference test gear dimensions (see 39)
SF = Required safety factor according to the rules
YδrelT = Relative notch sensitivity factor of the gear to be determined related to the reference test gear (see 310)
YRrelT = Relative (root fillet) surface condition factor of the gear to be determined related to the reference test gear (see 311)
Classification Notes- No 412 23 May 2003
DET NORSKE VERITAS
YX = Size factor (see 312)
YC = Case depth factor (see 313)
33 Tooth Form Factors YF YFa The tooth form factors YF and YFa take into account the in-fluence of the tooth form on the nominal bending stress
YF applies to load application at the outer point of single tooth pair contact of the virtual spur gear pair and is used for cylindrical gears
YFa applies to load application at the tooth tip and is used for bevel gears
Both YF and YFa are based on the distance between the con-tact points of the 30˚-tangents at the root fillet of the tooth profile for external gears respectively 60˚ tangents for inter-nal gears
Figure 31 External tooth in normal section
Figure 32 Internal tooth in normal section
Definitions
n
2
n
Fn
enFn
Fe
F
α cosms
α cosmh6
Y
=
n
2
n
Fn
anFn
Fa
Fa
α cosms
α cosmh6
Y
=
In the case of helical gears YF and YFa are determined in the normal section ie for a virtual number of teeth
YFa differs from YF by the bending moment arm hFa and αFan and can be determined by the same procedure as YF with exception of hFe and αFan For hFa and αFan all indices e will change to a (tip)
The following formulae apply to cylindrical gears but may also be used for bevel gears when replacing
mn with mnm
zn with zvn
αt with αvt
β with βm
with undercut without undercut
Fig 33 Dimensions and basic rack profile of the teeth (finished profile)
Tool and basic rack data such as hfP ρfp and spr etc are re-ferred to mn ie dimensionless
331 Determination of parameters
( )n
n
prnfPnfP m
α cossαsin 1ρ
αtan h4πE
minusminusminusminus=
For external gears fPfP ρρ =
For internal gears ( )0z
195fPfP0
fPfP 10363156ρhxρρ
sdotminus+
+=
where
z0 = number of teeth of pinion cutter
x0 = addendum modification coefficient of pinion cutter
hfP = addendum of pinion cutter
ρfP = tip radius of pinion cutter
xhρG fpfp +minus=
24 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
τmE
2π
z2H
nnminus
minus=
with
3πτ = for external gears
6πτ = for internal gears
Htan zG2
nminusϑ=ϑ
(to be solved iteratively suitable start value6π
=ϑ for exter-
nal gears and 3π for internal gears)
a) Tooth root chord sFn For external gears
minus
ϑ+
ϑminus= ρ
cosG3
3πsinz
ms
fpnn
Fn
For bevel gears with a tooth thickness modification
xsm affects mainly sFn but also hFe and αFen The total influence of xsm on YFa Ysa can be approximated by only adding 2 xsm to sFn mn
For internal gears
minus
ϑ+
ϑminus= ρ
cosG
6πsinz
ms
fPnn
Fn
b) Root fillet radius ρF at 30ordm tangent
( )G2coszcosG2ρ
mρ
2n
2
fpn
F
minusϑϑ+=
c) Determination of bending moment arm hF dn = zn mn
b2α
αn βcosε
ε =
dan = dn + 2 ha
pbn = π mn cos αn
dbn = dn cos αn
( )4
d1εpzz
2dd
zz2d
2bn
2
αnbn
2bn
2an
en +
minusminus
minus=
en
bnen d
dcos arcα =
ennnn
e α invα invα x tan 22π
z1
minus+
+=γ
αFen = αen ndash γe
For external gears
( )
minus=
n
enFenee
n
Fe
mdαtan sin γ γcos
21
mh
]ρcos
G3πcosz fpn +
ϑminus
ϑminusminus
For internal gears
( ) minus
sdotsdotminus=
n
enFenee
n
Fe
mdα tanγ sinγ cos
21
mh
minus
ϑminus
ϑminussdot ρ
cosG3
6πcosz fPn
332 Gearing with εαn gt 2 For deep tooth form gearing ( )25ε2 αn lele produced with a verified grade of accuracy of 4 or better and with applied profile modification to obtain a trapezoidal load distribution along the path of contact the YF may be corrected by the factor YDT as
250ε205for 0666ε2366Y αnαnDT leleminus=
205εfor 10Y αnDT lt=
34 Stress Correction Factors YS YSa The stress correction factors YS and YSa take into account the conversion of the nominal bending stress to the local tooth root stress Thereby YS and YSa cover the stress increasing effect of the notch (fillet) and the fact that not only bending stresses arise at the root A part of the local stress is inde-pendent of the bending moment arm This part increases the more the decisive point of load application approaches the critical tooth root section
Therefore in addition to its dependence on the notch radius the stress correction is also dependent on the position of the load application ie the size of the bending moment arm
YS applies to the load application at the outer point of single tooth pair contact YSa to the load application at tooth tip
YS can be determined as follows
( )
+
+=L23121
1
sS qL01312Y
where Fe
Fn
hsL = and
F
Fns ρ2
sq = (see 33)
YSa can be calculated by replacing hFe with hFa in the above formulae
Note a) Range of validity 1 lt qs lt 8
In case of sharper root radii (ie produced with tools having too sharp tip radii) YS resp YSa must be specially considered
b) In case of grinding notches (due to insufficient protuberance of the hob) YS resp YSa can rise considerably and must be multiplied with
Classification Notes- No 412 25 May 2003
DET NORSKE VERITAS
g
g
ρt
0613
13
minus
where
tg = depth of the grinding notch
ρg = radius of the grinding notch
c) The formulae for YS resp YSa are only valid for αn = 20˚ However the same formulae can be used as a safe approximation for other pressure angles
35 Contact Ratio Factor Yε The contact ratio factor Yε covers the conversion from load application at the tooth tip to the load application at the mid point of the flank (heightwise) for bevel gears
The following may be used
Yε = 0625
36 Helix Angle Factor Yβ The helix angle factor Yβ takes into account the difference between the helical gear and the virtual spur gear in the nor-mal section on which the calculation is based in the first step In this way it is accounted for that the conditions for tooth root stresses are more favourable because the lines of contact are sloping over the flank
The following may be used (β input in degrees)
Yβ = 1 ndash εβ β120
When εβ gt 1 use εβ = 1 and when β gt 30deg use β = 30deg in the formula
However the above equation for Yβ may only be used for gears with β gt 25deg if adequate tip relief is applied to both pinion and wheel (adequate = at least 05 middot Ceff see 432)
37 Values of Endurance Limit σFE σFE is the local tooth root stress (max principal) which the material can endure permanently with 99 survival prob-ability 3106 load cycles is regarded as the beginning of the endurance limit or the lower knee of the σ ndash N curve σFE is defined as the unidirectional pulsating stress with a minimum stress of zero (disregarding residual stresses due to heat treatment) Other stress conditions such as alternating or pre-stressed etc are covered by the conversion factor YM
σFE can be found by pulsating tests or gear running tests for any material in any condition If the approval of the gear is to be based on the results of such tests all details on the testing conditions have to be approved by the Society Fur-ther the tests may have to be made under the Societys su-pervision
If no fatigue tests are available the following listed values for σFE may be used for materials subjected to a quality con-trol as the one referred to in the rules
σFE
Alloyed case hardened steels 1) (fillet surface hardness 58 ndash 63 HRC)
bull of specially approved high grade
1050
bull of normal grade
minus CrNiMo steels with approved process
1000
minus CrNi and CrNiMo steels generally 920 minus MnCr steels generally 850
Nitriding steel of approved grade quenched tempered and gas nitrided (surface hardness 700 ndash 800 HV)
840
Alloyed quenched and tempered steel bath or gas nitrided (surface hardness 500 ndash 700 HV)
720
Alloyed quenched and tempered steel flame or induction hardened 2) (incl entire root fillet) (fillet surface hardness 500 ndash 650 HV)
07 HV + 300
Alloyed quenched and tempered steel flame or induction hardened (excl entire root fillet) (σB = uts of base material)
025 σB + 125
Alloyed quenched and tempered steel 04 σB + 200
Carbon steel 025 σB + 250
Note All numbers given above are valid for separate forgings and for blanks cut from bars forged according to a qualified procedure see Pt 4 Ch 2 Sec 3 For rolled steel the values are to be reduced with 10 For blanks cut from forged bars that are not qualified as mentioned above the values are to be reduced with 20 For cast steel reduce with 40
1) These values are valid for a root radius
bull being unground If however any grinding is made in the root fillet area in such a way that the residual stresses may be affected σFE is to be reduced by 20 (If the grinding also leaves a notch see 34)
bull with fillet surface hardness 58 ndash 63 HRC In case of lower surface hardness than 58 HRC σFE is to be reduced with 20(58 ndash HRC) where HRC is the detected hardness (This may lead to a permissible tooth root stress that varies along the facewidth If so the actual tooth root stresses may also be considered along facewidth)
bull not being shot peened In case of approved shot peening σFE may be increased by 200 for gears where σFE is reduced by 20 due to root grinding Otherwise σFE may be increased by 100 for
6nm le and 100 ndash 5 (mn - 6) for mn gt 6 However the possible adverse influence on the flanks regarding grey staining should be considered and if necessary the flanks should be masked
2) The fillet is not to be ground after surface hardening Regarding possible root grinding see 1)
26 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
38 Mean stress influence Factor YM The mean stress influence factor YM takes into account the influence of other working stress conditions than pure pulsa-tions (R = 0) such as eg load reversals idler gears planets and shrink-fitted gears
YM (YMst) are defined as the ratio between the endurance (or static) strength with a stress ratio R ne 0 and the endurance (or static) strength with R = 0
YM and YMst apply only to a calculation method that assesses the positive (tensile) stresses and is therefore suitable for comparison between the calculated (positive) working stress σF and the permissible stress σFP calculated with YM or YMst
For thin rings (annulus) in epicyclic gears where the com-pression fillet may be decisive special considerations apply see 316
The following method may be used within a stress ratio ndash12 lt R lt 05
381 For idlers planets and PTO with ice class
M1M1R1
1Yor Y MstM
+minus
minus=
where
R = stress ratio = min stress divided by max stress
For designs with the same force applied on both forward- and back-flank R may be assumed to ndash 12
For designs with considerably different forces on forward- and back-flank such as eg a marine propulsion wheel with a power take off pinion R may be assessed as
branchmain theoffacewidth unit per forcepto offacewidt unit per force21minus
For a power take off (PTO) with ice class see 161 c
M considers the mean stress influence on the endurance (or static) strength amplitudes
M is defined as the reduction of the endurance strength am-plitude for a certain increase of the mean stress divided by that increase of the mean stress
Following M values may be used
Endurance limit
Static strength
Case hardened 08 ndash 015 Ys 1) 07
If shot peened 04 06
Nitrided 03 03
Induction or flame hardened
04 06
Not surface hardened steel 03 05
Cast steels 04 06 1)For bevel gears use Ys=2 for determination of M
The listed M values for the endurance limit are independent of the fillet shape (Ys) except for case hardening In princi-ple there is a dependency but wide variations usually only occur for case hardening eg smooth semicircular fillets versus grinding notches
382 For gears with periodical change of rotational direction For case hardened gears with full load applied periodically in both directions such as side thrusters the same formula for YM as for idlers (with R = ndash 12) may be used together with the M values for endurance limit This simplified approach is valid when the number of changes of direction exceeds 100 and the total number of load cycles exceeds 3middot106
For gears of other materials YM will normally be higher than for a pure idler provided the number of changes of direction is below 3middot106 A linear interpolation in a diagram with loga-rithmic number of changes of direction may be used ie from YM = 09 with one change to YM (idler) for 3middot106 changes This is applicable to YM for endurance limit For static strength use YM as for idlers
For gears with occasional full load in reversed direction such as the main wheel in a reversing gear box YM = 09 may be used
383 For gears with shrinkage stresses and unidirectional load For endurance strength
FE
fitM σ
σM1
M21Y+
minus=
σFE is the endurance limit for R = 0
For static strength YMst = 1 and σfit accounted for in 39b
σfit is the shrinkage stress in the fillet (30˚ tangent) and may be found by multiplying the nominal tangential (hoop) stress with a stress concentration factor
n
Ffit m
ρ215scf minus=
384 For shrink-fitted idlers and planets When combined conditions apply such as idlers with shrink-age stresses the design factor for endurance strength is
( ) ( ) FE
fitM σ
σR1M1
M2
M1M1R1
1Yminussdot+
minus
+minus
minus=
Symbols as above but note that the stress ratio R in this par-ticular connection should disregard the influence of σfit ie R normally equal ndash 12
For static strength
M1M1R1
1YMst
+minus
minus=
The effect of σfit is accounted for in 39 b
Classification Notes- No 412 27 May 2003
DET NORSKE VERITAS
385 Additional requirements for peak loads The total stress range (σmax ndash σmin) in a tooth root fillet is not to exceed
F
y
Sσ225
for not surface hardened fillets
FSHV5 for surface hardened fillets
39 Life Factor YN The life factor YN takes into account that in the case of limited life (number of cycles) a higher tooth root stress can be permitted and that lower stresses may apply for very high number of cycles
Decisive for the strength at limited life is the σ ndash N ndash curve of the respective material for given hardening module fillet radius roughness in the tooth root etc Ie the factors YδrelT YRelT YX and YM have an influence on YN
If no σ ndash N ndash curve for the actual material and hardening etc is available the following method may be used
Determination of the σ ndash N ndash curve
a) Calculate the permissible stress σFP for the beginning of the endurance limit (3middot106 cycles) including the influ-ence of all relevant factors as SF YδrelT YRelT YX YM and YC ie σFP = σFE middotYM middotYδrelT middotYRelT middotYX middotYC SF
b) Calculate the permissible laquostaticraquo stress (le 103 load cy-cles) including the influence of all relevant factors as SFst YδrelTst YMst and YCst ( )fitσCstYδrelTstYMstYFstσ
FstS1
FPstσ minussdotsdotsdot=
where σFst is the local tooth root stress which the material can resist without cracking (surface hardened materials) or unacceptable deformation (not surface hardened materials) with 99 survival probability
σFst
Alloyed case hardened steel 1) 2300
Nitriding steel quenched tempered and gas nitrided (surface hardness 700 ndash 800 HV)
1250
Alloyed quenched and tempered steel bath or gas nitrided (surface hardness 500 ndash 700 HV)
1050
Alloyed quenched and tempered steel flame or induction hardened (fillet surface hardness 500 ndash 650 HV)
18 HV + 800
Steel with not surface hardened fillets the smaller value of 2)
18 σB or 225 σy
1) This is valid for a fillet surface hardness of 58 ndash 63 HRC In case of lower fillet surface hardness than 58 HRC σFst is to be reduced with 30(58 ndash HRC) where HRC is the actual hardness Shot peening or grinding notches are not considered to have any significant influence on σFst 2) Actual stresses exceeding the yield point (σy or σ02) will alter the residual stresses locally in the ldquotensionrdquo fillet re-spectively ldquocompressionrdquo fillet This is only to be utilised for gears that are not later loaded with a high number of cycles at lower loads that could cause fatigue in the ldquocompressionrdquo fillet
c) Calculate YN as
NL gt 3middot106
YN = 1 or 001
L
6
N N103Y
sdot= ie Yn = 092 for 1010
The YN = 1 from 3middot106 on may only be used when special material cleanness applies see rules Pt4 Ch2
103ltNLlt3middot106exp
L
6
N N103Y
sdot=
cycles6103forσcycles310forσlog02876exp
FP
FPst
sdot=
NL lt 103 cycles6103forσcycles310forσ
NYFP
FPst
sdot=
or simply use σFPst as mentioned in b) directly
Guidance on number of load cycles NL for various applica-tions
bull For propulsion purpose normally NL = 1010 at full load (yachts etc may have lower values)
bull For auxiliary gears driving generators that normally operate with 70-90 of rated power NL = 108 with rated power may be applied
28 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
310 Relative Notch Sensitivity Factor YδrelT The dynamic (respectively static) relative notch sensitivity factor YδrelT (YδrelTst) indicate to which extent the theoreti-cally concentrated stress lies above the endurance limits (re-spectively static strengths) in the case of fatigue (respectively overload) breakage
YδrelT is a function of the material and the relative stress gra-dient It differs for static strength and endurance limit
The following method may be used
For endurance limit
for not surface hardened fillets
( )
024
s024
relT σ103133q21σ1012201351
Ysdotsdotminus
+sdotsdotminus+= minus
minus
δ
for all surface hardened fillets except nitrided
( )
106q21002451
Y srelT
++=δ
for nitrided fillets
( )
1347q2101421
Y srelT
++=δ
For static strength
for not surface hardened fillets1)
( )( )( )025
02
02502s
relTst σ3000821σ300 1Y0821Y
+
minus+=δ
for surface hardened fillets except nitrided
YδrelTst = 044 YS + 012
for nitrided fillets
YδrelTst = 06 + 02 YS 1) These values are only valid if the local stresses do not
exceed the yield point and thereby alter the residual stress level See also 39b footnote 2
311 Relative Surface Condition Factor YRrelT The relative surface condition factor YRrelT takes into ac-count the dependence of the tooth root strength on the sur-face condition in the tooth root fillet mainly the dependence on the peak to valley surface roughness
YRrelT differs for endurance limit and static strength
The following method may be used
For endurance limit
YRrelT = 1675 ndash 053 (Ry + 1)01 for surface hardened steels and alloyed quenched and tempered steels except nitrided
YRrelT = 53 ndash 42 (Ry + 1)001 for carbon steels
YδrelT = 43 ndash 326 (Ry + 1)0005
for nitrided steels
For static strength
YRrelTst = 1 for all Ry and all materials
For a fillet without any longitudinal machining trace Ry asymp Rz
312 Size Factor YX The size factor YX takes into account the decrease of the strength with increasing size YX differs for endurance limit and static strength
The following may be used
For endurance limit
YX = 1 for mn le 5 generally
YX = 103 ndash 0006 mn for 5 lt mn lt 30
YX = 085 for mn ge 30
for not surface hard-ened steels
YX = 105 ndash 001 mn for 5 lt mn ge 25
YX = 08 for mn ge 25
for surface hardened steels
For static strength
YXst = 1 for all mn and all materials
313 Case Depth Factor YC The case depth factor YC takes into account the influence of hardening depth on tooth root strength
YC applies only to surface hardened tooth roots and is dif-ferent for endurance limit and static strength
In case of insufficient hardening depth fatigue cracks can develop in the transition zone between the hardened layer and the core For static strength yielding shall not occur in the transition zone as this would alter the surface residual stresses and therewith also the fatigue strength
The major parameters are case depth stress gradient permis-sible surface respectively subsurface stresses and subsurface residual stresses
The following simplified method for YC may be used
YC consists of a ratio between permissible subsurface stress (incl influence of expected residual stresses) and permissible surface stress This ratio is multiplied with a bracket con-taining the influence of case depth and stress gradient (The empirical numbers in the bracket are based on a high number of teeth and are somewhat on the safe side for low number of teeth)
YC and YCst may be calculated as given below but calculated values above 10 are to be put equal 10
For endurance limit
+
+=nFFE
C m02ρt31
σconstY
Classification Notes- No 412 29 May 2003
DET NORSKE VERITAS
For static strength
+
+=nFFst
Cst m02ρt31
σconstY
where const and t are connected as
Hardening process
t = endurance limit const =
static strength const =
t550 640 1900
t400 500 1200 Case hard-ening
t300 380 800
Nitriding t400 500 1200
Induction- or flame hardening
tHVmin 11 HVmin 25 HVmin
For symbols see 213
In addition to these requirements to minimum case depths for endurance limit some upper limitations apply to case hard-ened gears
The max depth to 550 HV should not exceed
1) 13 of the top land thickness san unless adequate tip relief is applied (see 110)
2) 025 mn If this is exceeded the following applies addi-tionally in connection with endurance limit
minusminus= 025
mt
1Yn
max550C
314 Thin rim factor YB Where the rim thickness is not sufficient to provide full sup-port for the tooth root the location of a bending failure may be through the gear rim rather than from root fillet to root fillet
YB is not a factor used to convert calculated root stresses at the 30deg tangent to actual stresses in a thin rim tension fillet Actually the compression fillet can be more susceptible to fatigue
YB is a simplified empirical factor used to de-rate thin rim gears (external as well as internal) when no detailed calcula-tion of stresses in both tension and compression fillets are available
Figure 34 Examples on thin rims
YB is applicable in the range 175 lt sRmn lt 35
YB = 115 middot ln (8324 middot mnsR)
(for sRmn ge 35 YB = 1)
(for sRmn le 175 use 315)
σF as calculated in 321 is to be multiplied with YB when sRmn lt 35 Thus YB is used for both high and low cycle fatigue
Note This method is considered to be on the safe side for external gear rims However for internal gear rims without any flange or web stiffeners the method may not be on the safe side and it is advised to check with the method in 315316
315 Stresses in Thin Rims For rim thickness sR lt 35 mn the safety against rim cracking has to be checked
The following method may be used
3151 General The stresses in the standardised 30ordm tangent section tension side are slightly reduced due to decreasing stress correction factor with decreasing relative rim thickness sRmn On the other hand during the complete stress cycle of that fillet a certain amount of compression stresses are also introduced The complete stress range remains approximately constant Therefore the standardised calculation of stresses at the 30ordm tangent may be retained for thin rims as one of the necessary criteria
The maximum stress range for thin rims usually occurs at the 60ordm ndash 80ordm tangents both for laquotensionraquo and laquocompressionraquo side The following method assumes the 75ordm tangent to be the decisive Therefore in addition to the am criterion ap-plied at the 30ordm tangent it is necessary to evaluate the max and min stresses at the 75ordm tangent for both laquotensionraquo (loaded flank) fillet and laquocompressionraquo (back-flank) fillet For this purpose the whole stress cycle of each fillet should be considered but usually the following simplification is justified
Figure 35 Nomenclature of fillets
Index laquoTraquo means laquotensileraquo fillet laquoCraquo means laquocompressionraquo fillet
σFTmin and σFCmax are determined on basis of the nominal rim stresses times the stress concentration factor Y75
σFCmin and σFTmax are determined on basis of superposition of nominal rim stresses times Y75 plus the tooth bending stresses at 75ordm tangent
30 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
3152 Stress concentration factors at the 75ordm tangents The nominal rim stress consists of bending stresses due to local bending moments tangential stresses due to the tan-gential force Ft and radial shear stresses due to Fr
The major influence is given by the bending stresses The influence of the tangential stresses is minor and even though its stress concentration factor is slightly higher than for bending it is considered to be safe enough when the sum of these nominal stresses are combined with the stress concen-tration factor for bending The influence of the radial shear stress is neglected
The stress concentration factor relating nominal rim stresses to local fillet stresses at the 75ordm tangent may be calculated as
75n
R75
n
R
75
ρmsρ185
ms3
Y+
sdot=
where ρ75 is the root radius at the 75ordm tangent ref to mn Usually ρ75 is closed to the tool radius ρao and
ρ75 = ρao
is a safe approximation compensating for the am simplifi-cations to the laquounsaferaquo side
The tooth root stresses of the loaded tooth are decreasing with decreasing relative rim thickness approximately with the empirical factor
15
R
ncorr s
m311Y
minus=
3153 Nominal rim stresses The bending moment applied to the rim consists of a part of the tooth tilt moment Ft (hF + 05 sR) and the bending caused by the radial force Fr
The sectional modulus (first moment of area) which is used for determination of the nominal bending stresses is not nec-essarily the same for the 2 am bending moments If flanges webs etc outside the toothed section contribute to stiffening the rim against various deflections the influence of these stiffeners should be considered Eg an end flange will have an almost negligible influence on the effective sectional modulus for the stresses due to tooth tilt as the deflection caused by the tilt moment is rather small and would not much involve the flange On the other hand the radial forces as for instance from the meshes in an annulus would cause considerable radial deflections that the flange might restrict to a substantial amount When considering the stiff-ening of such flanges or webs on basis of simplified models it is advised to use an effective rim thickness sR = sR + 02 mn for the first moment of area of the rim (toothed part) cross section
For a high number of rim teeth it may be assumed that the rim bending moments in the fillets adjacent to the loaded tooth are of the same magnitude as right under the applied
force This assumption is reasonable for an annulus but rather much on the safe side for a hollow pinion
The influence of Ft on the nominal tangential stress is simpli-fied by half of it for compressive stresses (σ1) and the other half for tensile stresses (σ2) Applying these assumptions the nominal rim stresses adjacent to the loaded tooth are
( ) ( )A2
FW
f RFW
s05hF05 t
R
r
T
RFt1 minus
ϑminus
+minus=σ
( ) ( )A2
FW
f RFW
s05hF05 t
R
r
T
RFt2 +
ϑminus
+=σ
where σ1 σ2 see Fig 35
A = minimum area of cross section (usually bR sR)
WT = the sectional modulus of rim with respect to tooth tilt moment (usually 6sb 2
rR )
WR = the sectional modulus of the rim including the influence of stiffeners as flanges webs etc (WR ge WT)
bR = the width of the rim
R = the radius of the neutral axis in the rim ie from wheel centre to midpoint of rim
( )ϑf = a function for bending moment distribution around the rim For a rim (pinion) with one mesh only the ( )ϑf at the position of load application is 024 For an annulus with 3 or more meshes ( )ϑf at the position of each load application is approx 3 planets ( )ϑf = 019 4 planets ( )ϑf = 014 5 planets ( )ϑf = 011 6 planets ( )ϑf = 009
It must be checked if the max (tensile) stress in the compres-sion fillet really occurs when the fillet is adjacent to the loaded tooth In principle the stress variation through a complete rotation should be considered and the max value used The max value is usually never less than 0 For an annulus eg the tilt moment is zero in the mid position be-tween the planet meshes whilst the bending moment due to the radial forces is half of that at the mesh but with opposite sign
If these formulae are applied to idler gears as eg planets the influence of nominal tangential stresses must be cor-rected by deleting Ft(2 A) for σ1 and using FtA for σ2 Further the influence of Fr on the nominal bending stresses is usually negligible due to the planet bearing support
3154 Root fillet stresses Determination of min and max stresses in the laquotensionraquo fillet
Classification Notes- No 412 31 May 2003
DET NORSKE VERITAS
Minimum stress
751FTmin YσK σ =
Maximum stress
corrF752 Yσ03Yσ KσFTmax +=
where
03 is an empirical factor relating the tension stresses (σF) at the 30ordm tangent to the part of the tension stresses at the 75ordm tangent which add to the rim related stresses (03 also takes into account that full superposition of nominal stresses times stress concentration factors from both laquosides of the corner filletraquo would result in too high stresses)
αβγ sdotsdotsdotsdot= FFvA KKKKKK
Determination of min and max stresses in the laquocompres-sionraquo fillet
Minimum stress
corrF751FCmin Yσ036Yσ Kσ minus=
Maximum stress
752FCmax Yσ Kσ =
where
036 is an empirical factor relating the tension stresses (σF) at the 30ordm tangent to the part of the compression stresses at the 75ordm tangent which add to the rim related stresses
For gears with reversed loads as idler gears and planets there is no distinct laquotensionraquo or laquocompressionraquo fillet The mini-mum stress σFmin is the minimum of σFTmin and σFCmin (usually the latter is decisive) The maximum stress σFTmax is the maximum of σFTmax and σFCmax (usually the former is deci-sive)
316 Permissible Stresses in Thin Rims
3161 General The safety against fatigue fracture respectively overload fracture is to be at least at the same level as for solid gears The laquoordinaryraquo criteria at the 30ordm tangent apply as given in 31 through 313
Additionally the following criteria at the 75ordm tangent may apply
3162 For gt3middot106 cycles The permissible stresses for the laquotensionraquo fillets and for the laquocompressionraquo fillets are determined by means of a relevant fatigue diagram
If the actual tooth root stress (tensile or compressive) ex-ceeds the yield strength to the material the induced residual stresses are to be taken into account
For determination of permissible stresses the following is defined
R = stress ratio ie FTmax
FTminσσ respectively
FCmax
FCmin
σσ
∆σ = stress range ie σFTmax ndash σFTmin resp σFCmax ndash σFCmin
(For idler gears and planets Fmax
FminσσR =
and ∆σ = σFmax minus σFmin)
The permissible stress range ∆σρ for the laquotensionraquo respec-tively laquocompressionraquo fillets can be calculated as
For R gt minus1 FPp σ
R1R1031
13∆σ
minus+
+=
For minus infin lt R lt minus1 FPp σ
R1R10151
13∆σ
minus+
+=
where
σFP = see 32 determined for unidirectional stresses (YM = 1)
If the yield strength σy is exceeded in either tension or com-pression residual stresses are induced This may be consid-ered by correcting the stress ratio R for the respective fillets (tension or compression)
Eg if yFCmin σσ gt (ie exceeded in compression) the
difference yFCmin σσ∆ minus= affects the stress ratio as
∆σ
σ∆σ∆σR
FCmax
y
FCmax
FCmin
+
minus=
++
=
Similarly the stress ratio in the tension fillet may require cor-rection
If the yield strength is exceeded in tension σFTmax gt σy the difference ∆ = σFTmax ndash σy affects the stress ratio as
y
FTmin
FTmax
FTmin
σ∆σ
∆σ∆σR minus=
minusminus
=
Checking for possible exceeding of the yield strength has to be made with the highest torque and with the KAP if this ex-ceeds KA
3163 Forle 103 cycles The permissible stress range ∆σp is not to exceed
For R gt minus1 FPstpst σ
R1R1051
15∆σ
minus+
+=
For ndash infin lt R lt minus1 FPstpst σ
R1R10251
15∆σ
minus+
+=
For all values of R ∆σpst is limited by
not surface hardened F
y
Sσ225
32 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
surface hardened CF
YSHV5
Definition of ∆σ and R see 3162 with particular attention to possible correction of R if the yield strength is exceeded
σFPst see 32 determined for unidirectional stresses (YM = 1) and lt 103 cycles
3164 For 103 lt cycles lt 3106 ∆σp is to be determined by linear interpolation a log-log dia-gram
∆σp at NL load cycles is
6L 103p
exp
L
6
N p ∆σN103∆σ sdot
sdot=
6
3
103p
10p
∆σ
∆σlog28760exp
sdot
=
Classification Notes- No 412 33 May 2003
DET NORSKE VERITAS
4 Calculation of Scuffing Load Capacity
41 Introduction High surface temperatures due to high loads and sliding ve-locities can cause lubricant films to break down This sei-zure or welding together of areas of tooth surface is termed scuffing
In contrast to pitting and fatigue breakage which show a dis-tinct incubation period a single short overloading can lead to a scuffing failure In the ISO-TR13989 two criteria are mentioned The method used in this Classification Note is based on the principles of the flash temperature criterion
Note Bulk temperature in excess of 120ordmC for long periods may have an adverse effect on the surface durability see 211
42 General Criteria In no point along the path of contact the local contact tem-perature may exceed the permissible temperature ie
oilS
oilSB S
ϑ+ϑminusϑ
leϑ
50SB minusϑleϑ
where
Bϑ = max contact temperature along the path of contact
maxflaMBB ϑ+ϑ=ϑ
MBϑ = bulk temperature see 434
maxflaϑ = max flash temperature along the path of con-tact see 44
Sϑ = scuffing temperature see below
oilϑ = oil temperature before it reaches the mesh (max applicable for the actual load case to be used ie normally alarm temperature except for ice classes where a max expected tem-perature applies)
SS = required safety factor according to the Rules
The scuffing temperature Sϑ may be calculated as
L2
wrelT
002
40S XFZGX
ν100112085780
sdot
sdot++=ϑ
where
XwrelT = relative welding factor
XwrelT
Through hardened steel 10
Phosphated steel 125
Copper-plated steel 150
Nitrided steel 150
Less than 10 retained austenite
115
10 ndash 20 retained austenite 10
Casehardened steel
20 ndash 30 retained austenite 085
Austenitic steel 045
FZG = load stage according to FZG-Test A8390 (Note This is the load stage where scuffing oc-curs However due to scatter in test results cal-culations are to be made with one load stage less than the specification)
XL = lubricant factor
= 10 for mineral oils
= 08 for polyalfaolefins
= 07 for non-water-soluble polyglycols
= 06 for water-soluble polyglycols
= 15 for traction fluids
= 13 for phosphate esters
ν40 = kinematic oil viscosity at 40˚C (mm2s)
Application of other test methods such as the Ryder the FZG-Ryder R46574 and the FZG L-42 Test 141195110 may be specially considered
For high speed gears with very short time of contact Sϑ may be increased as follows provided use of EP-oils
Addition to the calculated scuffing temperature Sϑ
If micros 18tc ge no addition
If micros 18tc lt add ( )cwrelT t18X18 minussdotsdot
where
ct = contact time ( micros ) which is the time needed to cross the Hertzian contact width
( ) [ ]microsu1cosβn
uσ340tb1
Hc +sdotsdot
sdot=
σH as calculated in 221
For bevel gears use vu in stead of u
34 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
43 Influence Factors
431 Coefficient of friction The following coefficient of friction may apply
L025a
005oil
02
redC ΣC
Bt X R ηρv
w0048micro minus
=
where
wBt = specific tooth load (Nmm)
vΣC = sum of tangential velocities at pitch point At pitch line velocities gt 50 ms the limiting value of vΣC at v = 50 ms is to be used
ρredC = relative radius of curvature (transversal plane) at the pitch point
Cylindrical gears
HαHβvγAbt
Bt KKKKKb
Fw sdotsdotsdotsdotsdot= (see 1)
twΣC αsin v2v =
bCredC β cos ρρ =
Bevel gears
HαHβvγAtmb
Bt KKKKKb
Fw sdotsdotsdotsdotsdot= (see 1)
vtmtΣC αsin v2v =
bmvCredC β cos ρρ =
ηoil = dynamic viscosity (mPa s) at oilϑ calculated as
1000
ρ νη oiloil =
where ρ in kgm3 approximated as
( ) 70 15oil15 minusϑminusρ=ρ and νoil is kinematic viscosity at oilϑ and may be calculated by means of the following equation
( ) ( )++=+ 08νloglog08νloglog 100oil ( )
sdotminus
ϑ+minus313log373log
273log373log oil
( ) ( )( )08νloglog08νloglog 10040 +minus+
Ra = composite arithmetic mean roughness (micron) of pinion an wheel calculated as ( )2a1aa RR 05R +=This is defined as the roughness on the new flanks ie as manufactured
XL = see 42
432 Effective tip relief Ceff Ceff is the effective tip relief that amount of tip relief which compensates for the elastic deformation of the gear mesh ie
zero load at the tooth tip It is assumed (simplified) to be equal for both pinion and wheel
Cylindrical gears
for helical
γ
γ=cb
KKFC Abt
eff (see 1)
For spur
cbKKF
C Abteff
γ= (see 1)
Alternatively for spur and helical gears the non-linear ap-proach in 111 may be used (taking Ceff=δ)
Bevel gears
bcKFC Ambt
effγ
= (see 1)
where
γ
α
ε2ε44c
+sdot
=γ
433 Tip relief and extension Cylindrical gears
The extension of the tip relief is not to result in an effective contact ratio 1ltεα when the gear is unloaded (exceptions to this may only apply for applications where the gear is not to run at light loads) This means that the unrelieved part of the path of contact is to be minimum pbt It is further assumed that this unrelieved part is placed centrally on the path of contact
If root relief applies it has to be calculated as equivalent tip relief Ie pinion root relief (at mesh position A) is added to Ca2 and wheel root relief (at mesh position E) is added to Ca1 If no design tip relief or root relief on the mating gear is specified (ie if Ca1+Croot2 = 0 and visa versa) use the run-ning in amount see 112
Bevel gears
Bevel gears are to have heightwise crowning ie no distinct relievedunrelieved area This may be treated as tip and root relief For calculation purposes the root relief is combined with the tip relief of the mating member into an equivalent tip relief If no resulting tip relieves are specified the equivalent tip relives may be calculated as an approxima-tion based on the tool crowning Ca tool (per mille of tool module m0) as follows
2root1aeq1a CCC +=
1root2aeq2a CCC +=
where
Classification Notes- No 412 35 May 2003
DET NORSKE VERITAS
2
0
n0101atoolal m
)m2(mA1middotmCC
minus+minussdot=
( )
sdotminusminussdotsdotminus+= vtvb1
2vb1
21vavt1n1 αtan dddαsin 05x1mA
2
0
n020atool22a m
)mm(2A1middotmCC
minus+minussdot=
( )
sdotminusminussdotsdotminus+= vtvb2
2vb2
2va2vt2n2 αtan dddαsin 05x1mA
2
0
20atool1root1 m
A1mCC
minussdotsdot=
2
0
10atool2root2 m
A1mCC
minussdotsdot=
If the pressure angles of the cutter blades are modified (and verified) to balance the tip relieves the following may be assumed
( )eq2aeq1aeq2aeq1a C andC calculated of sum50CC sdot==
Throughout the following Ca1 and Ca2 mean the equivalent tip relieves Ca1eq and Ca2eq
434 Bulk temperature The bulk temperature may be calculated as
flaaveragempsoilMB X X 50 ϑ+ϑ=ϑ
where
Xs = lubrication factor
= 12 for spray lubrication
= 10 for dip lubrication (provided both pinion and wheel are dip lubricated and tip speed lt 5 ms)
= 10 for spray lubrication with additional cooling spray (spray on both pinion and wheel or spray on pinion and dip lubrication of wheel)
= 02 for meshes fully submerged in oil
Xmp = contact factor ( )pmp nX += 150
np = number of mesh contact on the pinion (for small gear ratii the number of wheel meshes should be used if higher)
flaaverageϑ
= average of the integrated flash temperature (see 44) along the path of contact
( )
AE
E
Ayyyfla
flaaverage
d
ΓminusΓ
ΓΓϑ=ϑint =
For high speed gears (v gt 50 ms) it may be necessary to assess the bulk temperature on the basis of a thermal rating of the entire gear transmission
44 The Flash Temperature flaϑ
441 Basic formula The local flash temperature flaϑ may be calculated as
( ) 41redy
y2ly
211
43Btcorrfla
unXwX3250
y ρ
ρminusρ
micro=ϑ Γ
(For bevel gears replace u with uv)
and is to be calculated stepwise along the path of contact from A to E
where
micro = coefficient of friction see 431
Xcorr = correction factor taking empirically into account the increased scuffing risk in the beginning of the approach path due to mesh starting without any previously built up oil film and possible shuffling away oil before meshing if insufficient tip relief
( )
3
AD
yaeffcorr 50
CC1X
ΓminusΓε
Γminus+=
α
Ca = tip relief of driven member Xcorr is only applicable in the approach path and if Ca lt Ceff otherwise 10
wBt = unit load see 431
yXΓ = load sharing factor see 443
n1 = pinion rpm
Γy ρly etc see 442
442 Geometrical relations The various radii of flank curvature (transversal plane) are
ρ1y = pinion flank radius at mesh point y
ρ2y = wheel flank radius at mesh point y
ρredy = equivalent radius of curvature at mesh point y
y2y1
y2y1redy
ρ+ρ
ρρ=ρ
Cylindrical gears
twy
1y αsin au1Γ1
ρ+
+=
twy
2y αsin au1Γu
ρ+
minus=
Note that for internal gears a and u are negative
36 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
Bevel gears
vtvv
y1y αsin a
u1Γ1
ρ+
+=
vtvv
yv2y αsin a
u1Γu
ρ+
minus=
Γ is the parameter on the path of contact and y is any point between A end E
At the respective ends Γ has the following values
Root piniontip wheel
Cylindrical gears
( )
minus
minusminus= 1
αtan 1dd
zzΓ
tw
2b2a2
1
2A
Bevel gears
( )
minus
minusminus= 1
αtan 1dd
uΓvt
2vb2va2
vA
Tip pinionroot wheel
Cylindrical gears
( )
1αtan
1ddΓ
tw
2b1a1
E minusminus
=
Bevel gears
( )
1αtan
1ddΓ
vt
2vb1va1
E minusminus
=
At inner point of single pair contact
Cylindrical gears
tw1
EB α tan zπ2ΓΓ minus=
Bevel gears
vtv1
EB α tan zπ2ΓΓ minus=
At outer point of single pair contact
Cylindrical gears
tw1AD α tan z
π2ΓΓ +=
Bevel gears
vt v1
AD αtan z π2ΓΓ +=
At pitch point ΓC = 0
The points F and G (only applicable to cylindrical gears) limiting the extension of tip relief (so as to maintain a mini-mum contact ratio of unity for unloaded gears) are at
2
BAF
Γ+Γ=Γ
2
EDG
Γ+Γ=Γ
443 Load sharing factor XΓ The load sharing factor XΓ accounts for the load sharing be-tween the various pairs of teeth in mesh along the path of contact
XΓ is to be calculated stepwise from A to E using the pa-rameter Γy
4431 Cylindrical gears with β = 0 and no tip relief
Figure 41
ByAAB
Ay for31
31X
yΓltΓleΓ
ΓminusΓ
ΓminusΓ+=Γ
DyB for1Xy
ΓltΓleΓ=Γ
EyDDE
yE for31
31X
yΓleΓltΓ
ΓminusΓ
ΓminusΓ+=Γ
4432 Cylindrical gears with β = 0 and tip relief Tip relief on the pinion reduces XΓ in the range G ndash E and increases correspondingly XΓ in the range F ndash B
Tip relief on the wheel reduces XΓ in the range A ndash F and increases correspondingly XΓ in the range D ndash G
Following remains generally
DyB for1Xy
ΓleΓleΓ=Γ
21XXGF== ΓΓ
In the following it must be distinguished between Ca lt Ceff respectively Ca gt Ceff This is shown by an example below where Ca1 lt Ceff and Ca2 gt Ceff
Classification Notes- No 412 37 May 2003
DET NORSKE VERITAS
Figure 42
Note When Ca gt Ceff the path of contact is shortened by A ndash A respectively E ndash E The single pair contact path is extended into B respectively D If this shift is significant it is necessary to consider the negative effect on surface durability (B) and bending stresses (B and D)
Range A - F
For effa2 CC le
minus=Γ
eff
2a
CC1
31X
A
FyAeff
2a
AF
Ay for C3
C61XX
AyΓleΓleΓ
sdot
+ΓminusΓ
ΓminusΓ+= ΓΓ
For effa2 CC ge
AyAfor0Xy
ΓleΓleΓ=Γ
with ( )21
CC
1CC
eff
2a
eff
2a
AFAAminus
minusΓminusΓ+Γ=Γ
FyAeff
2a
AF
Ay
eff
2a for 21
CC
CC1X
yΓleΓleΓ
minus
ΓminusΓ
ΓminusΓ+minus=Γ
Range F - B
For effa1 CC le
ByFeff
1a
FB
Fy forC3
C61
21X
yΓleΓleΓ
sdot
+ΓminusΓ
ΓminusΓ+=Γ
For effa1 CC ge
ByFeff
1a
FB
Fy for21
CC
21X
yΓleΓleΓ
minus
ΓminusΓ
ΓminusΓ+=Γ
ByB for1Xy
ΓleΓleΓ=Γ
with 1
CC2
eff
1a
FBFB
minus
ΓminusΓ+Γ=Γ
Range D ndash G
For effa2 CC le
GyDeff
2a
DG
Dy
eff
2a for C 3C
61
C 3C
32X
yΓleΓleΓ
+
ΓminusΓ
ΓminusΓminus+=Γ F
or effa2 CC ge
DyD for1Xy
ΓleΓleΓ=Γ
with ( )21
CC
1CC
eff
2a
eff
2a
DGDDminus
minusΓminusΓ+Γ=Γ
GyDeff
2a
DG
Dy
eff
2a for21
CC
CCX ΓleΓleΓ
minus
ΓminusΓ
ΓminusΓminus=Γ
Range G - E
For effa1 CC le
minus=Γ
eff
1a
CC1
31X
E
EyGeff
1a
GE
Gy forC3
C61
21X
yΓleΓleΓ
sdot
+ΓminusΓ
ΓminusΓminus=Γ
For effa1 CC gt
EyGeff
1a
GE
Gy for21
CC
21X
yΓleΓleΓ
minus
ΓminusΓ
ΓminusΓminus=Γ
EyE for0Xy
ΓleΓleΓ=Γ
with 1
CC2
eff
1a
GEGE
minus
ΓminusΓ+Γ=Γ
4433 Gears with β gt 0 buttressing Due to oblique contact lines over the flanks a certain but-tressing may occur near A and E
This applies to both cylindrical and bevel gears with tip relief lt Ceff The buttressing Xbutt is simplified as a linear function within the ranges A ndash H respectively I ndash E
Figure 43
1εwhen13X βbutt EAge=
1εwhenε031X ββbutt EAlt+=
38 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
Cylindrical gears
bIEAH βsin 02ΓΓΓΓ =minus=minus
Bevel gears
bmIEAH βsin 02ΓΓΓΓ =minus=minus
4434 Cylindrical gears with 2εγ le and no tip relief
yXΓ is obtained by multiplication of
yXΓ in 4431 with
Xbutt in 4433
4435 Gears with 2εγ gt and no tip relief
Applicable to both cylindrical and bevel gears
IyH for1Xy
ΓleΓleΓε
=α
Γ
IyHybutt and forX1Xy
ΓgtΓΓltΓε
=α
Γ
Figure 44
4436 Cylindrical gears with 2εγ le and tip relief
yXΓ is obtained by multiplication of
yXΓ in 4432 with Xbutt
in 4433
4437 Cylindrical gears with 2εγ gt and tip relief
Tip relief on the pinion (respectively wheel) reduces XГ in the range G ndash E (respectively A ndash F) and increases XГ in the range F ndash G
yXΓ is obtained by multiplication of
yXΓ as described below
with Xbutt in 4433
In the XГ example below the influence of tip relief is shown (without the influence of Xbutt) by means of
eff2aeff1a CC and CC ltgt
Tip relief gt Ceff causes new end points A respectively E of the path of contact
Figure 45
Range A ndash F
( ) ( )( ) eff
a2αa1α
AF
Ay
eff
2aeff
C 12C 13εC 1ε
CCCX
y +εε++minus
ΓminusΓ
ΓminusΓ+
εminus
=ααα
Γ
eff2aFyA CC if for leΓleΓleΓ
eff2aFyA CifCfor and geΓleΓleΓ
eff2aAyA CC iffor0Xy
gtΓleΓleΓ=Γ
( ) ( ) ( )( ) ( ) a2αa1α
αeffa2AFAA C 1ε 3C 1ε
1ε 2 CC with++minus+minus
ΓminusΓ+Γ=Γ
Range F ndash G
( )( )( ) GyF
eff
2a1a forC 1 2CC 11X
yΓleΓleΓ
+εε+minusε
+ε
=αα
α
αΓ
Range G ndash E
( ) ( )( ) eff
2a1a
GE
Gy
C 1 2C 1C 1 3XX
GFy +εεminusε++ε
ΓminusΓ
ΓminusΓminus=
αα
ααΓΓ minus
effa1EyG CC if for leΓleΓleΓ
effa1EyG CC if Γfor and geΓleΓle
eff1aEyE CC if for0Xy
geΓleΓleΓ=Γ
( ) ( ) ( )( ) ( ) 2a1a
eff1aGEEE C 1C 1 3
1 2 CC withminusε++ε+εminus
ΓminusΓminusΓ=Γαα
α
4438 Bevel gears with εγ more than approx 18 and heightwise crowning For Ca1 = Ca2 = Ceff the following applies
Figure 46
Classification Notes- No 412 39 May 2003
DET NORSKE VERITAS
( )AEM 50 Γ+Γ=Γ
( )( )2AD
3
2My651X
y ΓminusΓε
ΓminusΓminus
ε=
ααΓ
For tip relief lt Ceff yXΓ is found by linear interpolation be-
tween ( )effay CCX =Γ and ( )0CayX =Γ as in 4435
The interpolation is to be made stepwise from A to M with the influence of Ca2 and from M to E with the influence of Ca1 (For 2a1a CC ne there is a discontinuity at M)
Eg with Ca1 = 04 Ceff and Ca2 = 055 Ceff then
Range A ndash M
)CC()0C( eff2ay2ayyX 550X 450X =Γ=ΓΓ +=
Range M ndash E
)CC()0C( eff1ay1ayyX 40X 06X =Γ=ΓΓ +=
For tip relief gt Ceff the new end points A and E are found as
( )
minusΓminusΓ
ε+Γ=Γ α 1
CC
6 eff
2aADAA
( )
minusΓminusΓ
ε+Γ=Γ α 1
CC
6 eff
1aADEE
Range A ndash A
0Xy=Γ
Range A ndash M
( )( )
ΓminusΓ
ΓminusΓminus
minusε=
αΓ 2
MA
2My
eff
2a1
CC4
351Xy
Range M ndash E
( )( )
ΓminusΓ
ΓminusΓminus
minusε=
αΓ 2
ME
2My
eff
1a1
CC4
351Xy
Range E ndash E
0Xy=Γ
40 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
Appendix A Fatigue Damage Accumulation
The Palmgren-Miner cumulative damage calculation princi-ple is used The procedure may be applied as follows
A1 Stress Spectrum From the individual torque classes the torques (Ti) at the peak values of class intervals and the associated number of cycles (NLi) for both pinion and wheel are to be listed from the highest to the lowest torque
(In case of a cyclic torque variation within the torque classes it is advised to use the peak torque If the cyclic variation is such that the same teeth will repeatedly suffer the peak torque this is a must)
The stress spectra for tooth roots and flanks (σFi σHi) with all relevant factors (except KA) are to be calculated on the basis of the torque spectrum The load dependent K-factors are to be determined for each torque class
A2 σminusN-curve The stress versus load cycle curves for tooth roots and flanks (both pinion and wheel) are to be drawn on the basis of per-missible stresses (ie including the demanded minimum safety factors) as determined in 2 respectively 3 If different safety levels for high cycle fatigue and low cycle fatigue are desired this may be expressed by different demand safety factors applied at the endurance limit respectively at static strength
A3 Damage accumulation The individual damage ratio Di at ith stress level is defined as
Fi
Lii N
ND =
where
NLi = The number of applied cycles at ith stress
NFi = The number of cycles to failure at ith stress
Basically stresses σi below the permissible stress level for infinite life (if a constant ZN or YN is accepted) do not con-tribute to the damage sum However calculating the actual safety factor Sact as described below all the σi for which the product S σi is bigger than or equal to the permissible stress level for infinite life contribute to the damage sum and thus to the determination of Sact The final value of S is decisive
(NFi can be found mathematically by putting the permissible stress σpi equal the actual stress σi thereby finding the actual life factor This life factor can be solved with regard to load cycles ie NFi)
The damage sum ΣDi is not to exceed unity
If ΣDi ne 1 the safety against cumulative fatigue damage is different from the applied demand safety factor For deter-mination of this theoretical safety factor an iteration proce-dure is required as described in the following flowchart
S is correction factor with which the actual safety factor Sact can be found
Sact is the demand safety factor (used in determination of the permissible stresses in the σ ndash N ndash curve) times the correc-tion factor S
The full procedure is to be applied for pinion and wheel tooth roots and flanks
Note If alternating stresses occur in a spectrum of mainly pulsating stresses the alternating stresses may be replaced by equiva-lent pulsating stresses ie by means of division with the ac-tual mean stress influence factor YM
Classification Notes- No 412 41 May 2003
DET NORSKE VERITAS
Appendix B Application Factors for Diesel Driven Gears
For diesel driven gears the application factor KA depends on torsional vibrations Both normal operation and misfiring conditions have to be considered
Normally these two running conditions can be covered by only one calculation
B1 Definitions
Normal operation KAnorm = 0
normv0
TTT +
where
T0 = rated nominal torque
Tv norm = vibratory torque amplitude for normal opera-tion (see rules Pt4 Ch3 Sec1 G301 for defi-nition of ldquonormalrdquo irregularity)
Misfiring operation KA misf = 0
misfv
TTT +
where
T = remaining nominal torque when one cylinder out of action
Tv misf = vibratory torque amplitude in misfiring con-dition This refers to a permissible misfiring condition ie a condition that does not require automatic or immediate corrective actions as speed or pitch reduction
The normal operation is assumed to last for a very high num-ber of cycles such as 1010
The misfiring operation is assumed to last for a limited dura-tion such as 107 cycles
B2 Determination of decisive load Assuming life factor at 1010 cycles as YN = ZN = 092 which usually is relevant the calculation may be performed only once with the combination having the highest value of appli-cation factorlife factor
For bending stresses and scuffing the higher value of
092
K normA and 098
K misfA
For contact stresses the higher value of
092K normA and
113K misfA (but
097K misfA for nitrided gears)
B3 Simplified procedure Note that this is only a guidance and is not a binding con-vention
T = oTZ
1Zsdot
minus where Z = number of cylinders
( ) ideal Videal Vmisf Vnorm V TTT24ZT +minus=
where TV ideal = vibratory torque with all cylinders perfectly equal See also rules Pt4 Ch3 Sec1 G300
When using trends from torsional vibration analysis and measurements the following may be used
TV ideal To is close to zero for engines with few cylinders and using a suitable elastic coupling and increases with relative coupling stiffness and number of cylinders
TV misf To may be high for engines with few cylinders and decreases with number of cylinders
This can be indicated as
200Z
TT
ο
idealV asymp and 80Z04
TT
o
misfV minusasymp
Inserting this into the formulae for the two application fac-tors the following guidance can be given
118112K misfA minusasymp
115110K normA minusasymp
Since KA norm is to be combined with the lower life factors the decisive load condition will be the normal one and a KA of 115 will cover most relevant cases when a suitable elas-tic coupling is chosen
42 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
Appendix C Calculation of Pinion-Rack
Pinion-racks used for elevating of mobile offshore units are open gears that are subjected to wear and tear With normal specifications such as only a few hundred total operation cycles (site to site) the tooth bending stresses for static strength will be decisive for the lay out however with ex-ception of surface hardened pinions where case crushing has to be considered
In the following the use of part 1 and 3 for pinion-racks is shown including relevant simplifications
C1 Pinion tooth root stresses Since the load spectrum normally is dominated by high torques and few load cycles (in the range up to 1-2000) the static strength is decisive
The actual stress is calculated as
β1FSaFan1
tF1 KYY
mbFσ sdotsdotsdotsdot
=
b1 is limited to b2 + 2middotmn
YFa and YSa replace YF and YS because load application at tooth tip has to be assumed for such inaccurate gears
Pinions often use a non-involute profile in the dedendum part of the flank eg a constant radius equal the radius of curva-ture at reference circle For such pinions sFn and hFa are to be measured directly on a sectional drawing of the pinion tooth
Due to high loads and narrow facewidths it may be assumed that KFβ2 = KHβ = 10 However when b1 gt b2 then KFβ1 gt10
If no detailed documentation of KFβ1 is available the fol-lowing may be used
KFβ1 = 1 + 015middot(b1b2 ndash 1)
The permissible stress (not surface hardened) is calculated as
δrelTstF
Fst1FP1 Y
Sσσ sdot=
The mean stress influence due to leg lifting may be disre-garded
The actual and permissible stresses should be calculated for the relevant loads as given in the rules
C2 Rack tooth root stresses The actual stress is calculated as
SaFan2
tF2 YY
mbFσ sdotsdotsdot
=
See C1 for details
The permissible stress is calculated as
δrelTstF
Fst2FP2 Y
Sσσ sdot=
For alloyed steels (Ni Cr Mo) with high toughness and duc-tility the value of YδrelTst may be put equal to YSa
C3 Surface hardened pinions For surface hardened pinions the maximum load is not to cause crushing of the hardened layer of the flank
In principle the calculation described in 213 may be used but when the theoretical Hertzian stress exceeds the ap-proximately 18 times the yield strength of the rack material plastic deformation will occur This will limit the peak Hertzian stress but increases the contact width and thus the penetration of stresses into the depth
An approximation may be based on an assessment of contact width determined by means of equal areas under the theoreti-cal (elastic range) Hertzian contact and the elasto-plastic contact stress (limited to 18 middot σy) with the unknown width
18 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
23 Zone Factors ZH ZBD and ZM
231 Zone factor ZH The zone factor ZH accounts for the influence on contact stresses of the tooth flank curvature at the pitch point and converts the tangential force at the reference cylinder to the normal force at the pitch cylinder
wtt2
wtbH sinααcos
cosαcosβ2Z =
232 Zone factors ZBD The zone factors ZBD account for the influence on contact stresses of the tooth flank curvature at the inner point of sin-gle pair contact in relation to ZH Index B refers to pinion D to wheel
For εβ ge 1 ZBD = 1
For internal gears ZD = 1
For εβ = 0 (spur gears)
( )
π
minusεminusminus
π
minusminus
α=
α2
2
2b
2a
1
2
1b
1a
wtB
z211
dd
z21
dd
tanZ
( )
π
minusεminusminus
π
minusminus
α=
α1
2
1b
1a
2
2
2b
2a
wtD
z211
dd
z21
dd
tanZ
If ZB lt 1 use ZB = 1
If ZD lt 1 use ZD = 1
For 0 lt εβ lt 1
ZBD = ZBD (for spur gears) ndash εβ (ZBD (for spur gears) ndash 1)
233 Zone factor ZM The mid-zone factor ZM accounts for the influence of the contact stress at the mid point of the flank and applies to spi-ral bevel gears
εminusminus
εminusminus
αβ=
αα btm2
2vb2
2vabtm2
1vb2val
2v1vvtbmM
pddpdd
ddtancos2Z
This factor is the product of ZH and ZM-B in ISO 10300 with the condition that the heightwise crowning is sufficient to move the peak load towards the midpoint
234 Inner contact point For cylindrical or bevel gears with very low number of teeth the inner contact point (A) may be close to the base circle In order to avoid a wear edge near A it is required to have suitable tip relief on the wheel
24 Elasticity Factor ZE The elasticity factor ZE accounts for the influence of the material properties as modulus of elasticity and Poissonrsquos ratio on the contact stresses
For steel against steel ZE = 1898
25 Contact Ratio Factor Zε The contact ratio factor Zε accounts for the influence of the transverse contact ratio εα and the overlap ratio εβ on the contact stresses
αε1Z =ε for 1εβ ge
( )α
ββ
αε ε
εε1
3ε4
Z +minusminus
= for εβ lt 1
26 Helix Angle Factor Zβ The helix angle factor Zβ accounts for the influence of helix angle (independent of its influence on Zε) on the surface du-rability
cosβZβ =
27 Bevel Gear Factor ZK The bevel gear factor accounts for the difference between the real Hertzian stresses in spiral bevel gears and the contact stresses assumed responsible for surface fatigue (pitting) ZK adjusts the contact stresses in such a way that the same per-missible stresses as for cylindrical gears may apply
The following may be used ZK = 080
28 Values of Endurance Limit σHlim and Static Strength 5H10σ 3H10σ
σHlim is the limit of contact stress that may be sustained for 5middot107 cycles without the occurrence of progressive pitting
For most materials 5middot107 cycles are considered to be the be-ginning of the endurance strength range or lower knee of the σ-N curve (See also Life Factor ZN) However for nitrided steels 2middot106 apply
For this purpose pitting is defined by
bull for not surface hardened gears pitted area ge 2 of total active flank area
bull for surface hardened gears pitted area ge 05 of total active flank area or ge 4 of one particular tooth flank area 510Hσ and 310Hσ are the contact stresses which the given
material can withstand for 105 respectively 103 cycles without subsurface yielding or flank damages as pitting spalling or case crushing when adequate case depth applies
The following listed values for σHlim 510Hσ and 310Hσ may only be used for materials subjected to a quality control as
Classification Notes- No 412 19 May 2003
DET NORSKE VERITAS
the one referred to in the rules Results of approved fatigue tests may also be used as the basis for establishing these values The defined survival probability is 99
σHlim σH105 σH10
3
Alloyed case hardened steels (surface hardness 58-63 HRC) - of specially approved high grade - of normal grade
1650 1500
2500 2400
3100 3100
Nitrided steel of approved grade gas nitrided (surface hardness 700-800 HV) 1250 13 σHlim 13 σHlim
Alloyed quenched and tempered steel bath or gas nitrided (surface hardness 500-700 HV)
1000
13 σHlim
13 σHlim
Alloyed flame or induction hardened steel (surface hardness 500-650 HV) 075 HV + 750 16 σHlim 45 HV
Alloyed quenched and tempered steel 14 HV + 350 16 σHlim 45 HV
Carbon steel 15 HV + 250 16 σHlim 16 σHlim
These values refer to forged or hot rolled steel For cast steel the values for σHlim are to be reduced by 15
29 Life Factor ZN The life factor ZN takes account of a higher permissible contact stress if only limited life (number of cycles NL) is demanded or lower permissible contact stress if very high number of cycles apply
If this is not documented by approved fatigue tests the fol-lowing method may be used
For all steels except nitrided
7L 105N sdotge ZN = 1 or
01570
L
7
N N105Z
sdot=
Ie ZN = 092 for 1010 cycles
The ZN = 1 from 5middot107 on may only be used when the mate-rial cleanliness is of approved high grade (see Rules Pt4 Ch2) and the lubrication is optimised by a specially approved filtering process
105 lt NL lt 5middot107 510N
logZ037
L
7
N N105Z
sdot=
NL = 105 WXRVLHlim
WstX10H1010NN ZZZZZσ
ZZσZZ
55
5=
103 lt NL lt 105 )Z(Zlog05
L
5
10NN
5N103N10
5N10ZZ
==
3L 10N le
WXRVLlimH
Wst10X10H10NN ZZZZZ
ZZZZ
33
3σ
σ==
(but not less than ZN105)
For nitrided steels
6L 102N sdotge ZN = 1 or
00980
L
6
N N102Z
sdot=
Ie ZN = 092 for 1010 cycles
The ZN = 1 from 2middot106 on may only be used when the mate-rial cleanliness is of approved high grade (see Rules Pt4 Ch2) and the lubrication is optimised by a specially approved filtering process
105 lt NL lt 2middot106 510N
Zlog76860
L
6
N N102Z
sdot=
5L 10N le
XWRVL
10XWst10NN ZZZZZ
ZZ13ZZ
55 ==
Note that when no index indicating number of cycles is used the factors are valid for 5middot107 (respectively 2middot106 for nitrid-ing) cycles
210 Influence Factors on Lubrication Film ZL ZV and ZR The lubricant factor ZL accounts for the influence of the type of lubricant and its viscosity the speed factor ZV ac-counts for the influence of the pitch line velocity and the roughness factor ZR accounts for influence of the surface roughness on the surface endurance capacity
The following methods may be applied in connection with the endurance limit
20 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
Surface hardened steels Not surface hardened steels
ZL ( )24013421
360910ν+
+
( )24013421
680830ν+
+
ZV ( )v3280
140930+
+ ( )v3280300850
++
ZR
080
ZrelR3
150
ZrelR3
where
ν40 = Kinematic oil viscosity at 40ordmC (mm2s) For case hardened steels the influence of a high bulk temperature (see 4 Scuffing) should be con-sidered Eg bulk temperatures in excess of 120ordmC for long periods may cause reduced flank surface endurance limits
For values of ν40 gt 500 use ν40 = 500
RZrel = The mean roughness between pinion and wheel (after running in) relative to an equivalent radius of curvature at the pitch point ρc = 10mm
RZrel
= ( ) 31
cZ2Z1 ρ
10RR50
+
RZ = Mean peak to valley roughness (microm) (DIN defini-tion) (roughly RZ = 6 Ra)
For 5
L 10N le ZL ZV ZR = 10
211 Work Hardening Factor ZW The work hardening factor ZW accounts for the increase of surface durability of a soft steel gear when meshing the soft steel gear with a surface hardened or substantially harder gear with a smooth surface
The following approximation may be used for the endurance limit
Surface hardened steel against not surface hardened steel 150
ZeqW R
31700
130HB21Z
minus
minus=
where HB = the Brinell hardness of the soft member For HB gt 470 use HB = 470 For HB lt 130 use HB = 130
RZeq = equivalent roughness
033
c40
066
ZS
ZHZHZEQ ρvν
15000RRRR
=
If RZeq gt 16 then use RZeq = 16
If RZeq lt 15 then use RZeq = 15
where
RZH = surface roughness of the hard member before run in
RZS = surface roughness of the soft member before run in
ν40 = see 210
If values of ZW lt 1 are evaluated ZW = 1 should be used for flank endurance However the low value for ZW may indi-cate a potential wear problem
Through hardened pinion against softer wheel
( )
minussdotminus+= 000829
HBHB0008981u1Z
2
1W
For 21HBHB
2
1 le use ZW = 1
For 71HBHB
2
1 gt use 17HBHB
2
1 =
For u gt 20 use u = 20
For static strength (lt 105 cycles)
Surface hardened against not surface hardened
ZWst = 105
Through hardened pinion against softer wheel
ZWst = 1
212 Size Factor ZX The size factor accounts for statistics indicating that the stress levels at which fatigue damage occurs decrease with an increase of component size as a consequence of the influ-ence on subsurface defects combined with small stress gradi-ents and of the influence of size on material quality
ZX may be taken unity provided that subsurface fatigue for surface hardened pinions and wheels is considered eg as in the following subsection 213
213 Subsurface Fatigue This is only applicable to surface hardened pinions and wheels The main objective is to have a subsurface safety against fatigue (endurance limit) or deformation (static strength) which is at least as high as the safety SH required for the surface The following method may be used as an approximation unless otherwise documented
The high cycle fatigue (gt3106 cycles) is assumed to mainly depend on the orthogonal shear stresses Static strength (lt103 cycles) is assumed to depend mainly on equivalent
Classification Notes- No 412 21 May 2003
DET NORSKE VERITAS
stresses (von Mises) Both are influenced by residual stresses but this is only considered roughly and empirically
The subsurface working stresses at depths inside the peak of the orthogonal shear stresses respectively the equivalent stresses are only dependent on the (real) Hertzian stresses Surface related conditions as expressed by ZL ZV and ZR are assumed to have a negligible influence
The real Hertzian stresses σHR are determined as
For helical gears with εβ gt 1
σHR = σH
For helical gears with εβ lt 1 and spur gears
ε
α
ββ ε
ε+εminus
sdotσ=σZ
1
HHR
For bevel gears
KHHR Z
1σσ sdot=
The necessary hardness HV is given as a function of the net depth tz (net = after grinding or hard metal hobbing and per-pendicular to the flank)
The coordinates tz and HV are to be compared with the de-sign specification such as
bull for flame and induction hardening tHVmin HVmin bull for nitriding t400min HV = 400 bull for case hardening t550min HV = 550 t400min HV = 400
and t300min HV = 300 (the latter only if the core hardness lt 300 If the core hardness gt 400 the t400 is to be re-placed by a fictive t400 = 16 t550)
In addition the specified surface hardness is not to be less than the max necessary hardness (at tz = 05aH) This applies to all hardening methods
For high cycle fatigue (gt3 106 cycles) the following applies
sdot+
minussdotsdotsdot= o90
05azt
05azt
cosSσ04HV
H
HHHR
applicable to 05a
zt
Hge
For 05at
H
z lt the value for 05at
H
z = applies
56300ρSσ12a cHHR
Hsdotsdot
sdot=
Where aH is half the hertzian contact width multiplied by an empirical factor of 12 that takes into account the possible influence of reduced compressive residual stresses (or even tensile residual stresses) on the local fatigue strength
If any of the specified hardness depths including the surface hardness is below the curve described by HV = f (tz) the actual safety factor against subsurface fatigue is determined as follows
reduce SH stepwise in the formula for HV and aH until all specified hardness depths and surface hardness balance with the corrected curve The safety factor obtained through this method is the safety against subsurface fatigue
For static strength (lt103 cycles) the following applies
sdot+
minussdotsdotsdot= o90
07at
06a
zt
cosSσ019HV
Hst
z
HstHHR
applicable to 06at
Hst
z ge
56300ρSσa cHHR
Hstsdotsdot
=
In the case of insufficient specified hardness depths the same procedure for determination of the actual safety factor as above applies
For limited life fatigue (103 lt cycles lt 3106 )
For this purpose it is necessary to extend the correction of safety factors to include also higher values than required Ie in the case of more than sufficient hardness and depths the safety factor in the formulae for both high cycle fatigue and static strength are to be increased until necessary and speci-fied values balance
The actual safety factor for a given number of cycles N be-tween 103 and 3106 is found by linear interpolation in a dou-ble logarithmic diagram
minussdotminus
= sdot logN3477
logSlogSlogS
310H6103HHN
36 10H103H Slog86281Slog86280 sdot+sdot sdot
22 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
3 Calculation of Tooth Strength
31 Scope and General Remarks Part 3 include the calculation of tooth root strength as limited by tooth root cracking (surface or subsurface initiated) and yielding
For rim thickness sR ge 35middotmn the strength is calculated by means of 32 ndash 313 For cylindrical gears the calculation is based on the assumption that the highest tooth root tensile stress arises by application of the force at the outer point of single tooth pair contact of the virtual spur gears The method has however a few limitations that are mentioned in 36
For bevel gears the calculation is based on force application at the tooth tip of the virtual cylindrical gear Subsequently the stress is converted to load application at the mid point of the flank due to the heightwise crowning
Bevel gears may also be calculated with the program BECAL In that case KA and Kv are to be included in the applied tooth force but not KFβ and KFα
In case of a thin annulus or a thin gear rim etc radial crack-ing can occur rather than tangential cracking (from root fillet to root fillet) Cracking can also start from the compression fillet rather than the tension fillet For rim thickness sR lt 35middotmn a special calculation procedure is given in 315 and 316 and a simplified procedure in 314
A tooth breakage is often the end of the life of a gear trans-mission Therefore a high safety SF against breakage is re-quired
It should be noted that this part 3 does not cover fractures caused by
bull oil holes in the tooth root space bull wear steps on the flank bull flank surface distress such as pits spalls or grey staining
Especially the latter is known to cause oblique fractures starting from the active flank predominately in spiral bevel gears but also sometimes in cylindrical gears
Specific calculation methods for these purposes are not given here but are under consideration for future revisions Thus depending on experience with similar gear designs limita-tions other than those outlined in part 3 may be applied
32 Tooth Root Stresses The local tooth root stress is defined as the max principal stress in the tooth root caused by application of the tooth force Ie the stress ratio R = 0 Other stress ratios such as for eg idler gears (R asymp -12) shrunk on gear rims (R gt 0) etc are considered by correcting the permissible stress level
321 Local tooth root stress The local tooth root stress for pinion and wheel may be as-sessed by strain gauge measurements or FE calculations or similar For both measurements and calculations all details are to be agreed in advance
Normally the stresses for pinion and wheel are calculated as
Cylindrical gears
FαFβvγAβSFn
tF K K K K K Y Y Y
m bFσ =
where
YF = Tooth form factor (see 33)
YS = Stress correction factor (see 34)
Yβ = Helix angle factor (see 36)
Ft KA Kγ Kv KFβ KFα see 15 ndash 110
b see 13
Bevel gears
FαFβvγASaFamn
mtF K K K K K Y Y Y
m bFσ ε=
where
YFa = Tooth form factor see 33
YSa = Stress correction factor see 34
Yε = Contact ratio factor see 35
Fmt KA etc see 15 ndash 110
b see 13
322 Permissible tooth root stress The permissible local tooth root stress for pinion respectively wheel for a given number of cycles N is
CXRrelTrelTF
NMFEFP YYYY
SYY
δσ
=σ
Note that all these factors YM etc are applicable to 3middot106 cycles when used in this formula for σFP The influence of other number of cycles on these factors is covered by the calculation of YN
where
σFE = Local tooth root bending endurance limit of reference test gear (see 37)
YM = Mean stress influence factor which accounts for other loads than constant load direction eg idler gears temporary change of load di-rection pre-stress due to shrinkage etc (see 38)
YN = Life factor for tooth root stresses related to reference test gear dimensions (see 39)
SF = Required safety factor according to the rules
YδrelT = Relative notch sensitivity factor of the gear to be determined related to the reference test gear (see 310)
YRrelT = Relative (root fillet) surface condition factor of the gear to be determined related to the reference test gear (see 311)
Classification Notes- No 412 23 May 2003
DET NORSKE VERITAS
YX = Size factor (see 312)
YC = Case depth factor (see 313)
33 Tooth Form Factors YF YFa The tooth form factors YF and YFa take into account the in-fluence of the tooth form on the nominal bending stress
YF applies to load application at the outer point of single tooth pair contact of the virtual spur gear pair and is used for cylindrical gears
YFa applies to load application at the tooth tip and is used for bevel gears
Both YF and YFa are based on the distance between the con-tact points of the 30˚-tangents at the root fillet of the tooth profile for external gears respectively 60˚ tangents for inter-nal gears
Figure 31 External tooth in normal section
Figure 32 Internal tooth in normal section
Definitions
n
2
n
Fn
enFn
Fe
F
α cosms
α cosmh6
Y
=
n
2
n
Fn
anFn
Fa
Fa
α cosms
α cosmh6
Y
=
In the case of helical gears YF and YFa are determined in the normal section ie for a virtual number of teeth
YFa differs from YF by the bending moment arm hFa and αFan and can be determined by the same procedure as YF with exception of hFe and αFan For hFa and αFan all indices e will change to a (tip)
The following formulae apply to cylindrical gears but may also be used for bevel gears when replacing
mn with mnm
zn with zvn
αt with αvt
β with βm
with undercut without undercut
Fig 33 Dimensions and basic rack profile of the teeth (finished profile)
Tool and basic rack data such as hfP ρfp and spr etc are re-ferred to mn ie dimensionless
331 Determination of parameters
( )n
n
prnfPnfP m
α cossαsin 1ρ
αtan h4πE
minusminusminusminus=
For external gears fPfP ρρ =
For internal gears ( )0z
195fPfP0
fPfP 10363156ρhxρρ
sdotminus+
+=
where
z0 = number of teeth of pinion cutter
x0 = addendum modification coefficient of pinion cutter
hfP = addendum of pinion cutter
ρfP = tip radius of pinion cutter
xhρG fpfp +minus=
24 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
τmE
2π
z2H
nnminus
minus=
with
3πτ = for external gears
6πτ = for internal gears
Htan zG2
nminusϑ=ϑ
(to be solved iteratively suitable start value6π
=ϑ for exter-
nal gears and 3π for internal gears)
a) Tooth root chord sFn For external gears
minus
ϑ+
ϑminus= ρ
cosG3
3πsinz
ms
fpnn
Fn
For bevel gears with a tooth thickness modification
xsm affects mainly sFn but also hFe and αFen The total influence of xsm on YFa Ysa can be approximated by only adding 2 xsm to sFn mn
For internal gears
minus
ϑ+
ϑminus= ρ
cosG
6πsinz
ms
fPnn
Fn
b) Root fillet radius ρF at 30ordm tangent
( )G2coszcosG2ρ
mρ
2n
2
fpn
F
minusϑϑ+=
c) Determination of bending moment arm hF dn = zn mn
b2α
αn βcosε
ε =
dan = dn + 2 ha
pbn = π mn cos αn
dbn = dn cos αn
( )4
d1εpzz
2dd
zz2d
2bn
2
αnbn
2bn
2an
en +
minusminus
minus=
en
bnen d
dcos arcα =
ennnn
e α invα invα x tan 22π
z1
minus+
+=γ
αFen = αen ndash γe
For external gears
( )
minus=
n
enFenee
n
Fe
mdαtan sin γ γcos
21
mh
]ρcos
G3πcosz fpn +
ϑminus
ϑminusminus
For internal gears
( ) minus
sdotsdotminus=
n
enFenee
n
Fe
mdα tanγ sinγ cos
21
mh
minus
ϑminus
ϑminussdot ρ
cosG3
6πcosz fPn
332 Gearing with εαn gt 2 For deep tooth form gearing ( )25ε2 αn lele produced with a verified grade of accuracy of 4 or better and with applied profile modification to obtain a trapezoidal load distribution along the path of contact the YF may be corrected by the factor YDT as
250ε205for 0666ε2366Y αnαnDT leleminus=
205εfor 10Y αnDT lt=
34 Stress Correction Factors YS YSa The stress correction factors YS and YSa take into account the conversion of the nominal bending stress to the local tooth root stress Thereby YS and YSa cover the stress increasing effect of the notch (fillet) and the fact that not only bending stresses arise at the root A part of the local stress is inde-pendent of the bending moment arm This part increases the more the decisive point of load application approaches the critical tooth root section
Therefore in addition to its dependence on the notch radius the stress correction is also dependent on the position of the load application ie the size of the bending moment arm
YS applies to the load application at the outer point of single tooth pair contact YSa to the load application at tooth tip
YS can be determined as follows
( )
+
+=L23121
1
sS qL01312Y
where Fe
Fn
hsL = and
F
Fns ρ2
sq = (see 33)
YSa can be calculated by replacing hFe with hFa in the above formulae
Note a) Range of validity 1 lt qs lt 8
In case of sharper root radii (ie produced with tools having too sharp tip radii) YS resp YSa must be specially considered
b) In case of grinding notches (due to insufficient protuberance of the hob) YS resp YSa can rise considerably and must be multiplied with
Classification Notes- No 412 25 May 2003
DET NORSKE VERITAS
g
g
ρt
0613
13
minus
where
tg = depth of the grinding notch
ρg = radius of the grinding notch
c) The formulae for YS resp YSa are only valid for αn = 20˚ However the same formulae can be used as a safe approximation for other pressure angles
35 Contact Ratio Factor Yε The contact ratio factor Yε covers the conversion from load application at the tooth tip to the load application at the mid point of the flank (heightwise) for bevel gears
The following may be used
Yε = 0625
36 Helix Angle Factor Yβ The helix angle factor Yβ takes into account the difference between the helical gear and the virtual spur gear in the nor-mal section on which the calculation is based in the first step In this way it is accounted for that the conditions for tooth root stresses are more favourable because the lines of contact are sloping over the flank
The following may be used (β input in degrees)
Yβ = 1 ndash εβ β120
When εβ gt 1 use εβ = 1 and when β gt 30deg use β = 30deg in the formula
However the above equation for Yβ may only be used for gears with β gt 25deg if adequate tip relief is applied to both pinion and wheel (adequate = at least 05 middot Ceff see 432)
37 Values of Endurance Limit σFE σFE is the local tooth root stress (max principal) which the material can endure permanently with 99 survival prob-ability 3106 load cycles is regarded as the beginning of the endurance limit or the lower knee of the σ ndash N curve σFE is defined as the unidirectional pulsating stress with a minimum stress of zero (disregarding residual stresses due to heat treatment) Other stress conditions such as alternating or pre-stressed etc are covered by the conversion factor YM
σFE can be found by pulsating tests or gear running tests for any material in any condition If the approval of the gear is to be based on the results of such tests all details on the testing conditions have to be approved by the Society Fur-ther the tests may have to be made under the Societys su-pervision
If no fatigue tests are available the following listed values for σFE may be used for materials subjected to a quality con-trol as the one referred to in the rules
σFE
Alloyed case hardened steels 1) (fillet surface hardness 58 ndash 63 HRC)
bull of specially approved high grade
1050
bull of normal grade
minus CrNiMo steels with approved process
1000
minus CrNi and CrNiMo steels generally 920 minus MnCr steels generally 850
Nitriding steel of approved grade quenched tempered and gas nitrided (surface hardness 700 ndash 800 HV)
840
Alloyed quenched and tempered steel bath or gas nitrided (surface hardness 500 ndash 700 HV)
720
Alloyed quenched and tempered steel flame or induction hardened 2) (incl entire root fillet) (fillet surface hardness 500 ndash 650 HV)
07 HV + 300
Alloyed quenched and tempered steel flame or induction hardened (excl entire root fillet) (σB = uts of base material)
025 σB + 125
Alloyed quenched and tempered steel 04 σB + 200
Carbon steel 025 σB + 250
Note All numbers given above are valid for separate forgings and for blanks cut from bars forged according to a qualified procedure see Pt 4 Ch 2 Sec 3 For rolled steel the values are to be reduced with 10 For blanks cut from forged bars that are not qualified as mentioned above the values are to be reduced with 20 For cast steel reduce with 40
1) These values are valid for a root radius
bull being unground If however any grinding is made in the root fillet area in such a way that the residual stresses may be affected σFE is to be reduced by 20 (If the grinding also leaves a notch see 34)
bull with fillet surface hardness 58 ndash 63 HRC In case of lower surface hardness than 58 HRC σFE is to be reduced with 20(58 ndash HRC) where HRC is the detected hardness (This may lead to a permissible tooth root stress that varies along the facewidth If so the actual tooth root stresses may also be considered along facewidth)
bull not being shot peened In case of approved shot peening σFE may be increased by 200 for gears where σFE is reduced by 20 due to root grinding Otherwise σFE may be increased by 100 for
6nm le and 100 ndash 5 (mn - 6) for mn gt 6 However the possible adverse influence on the flanks regarding grey staining should be considered and if necessary the flanks should be masked
2) The fillet is not to be ground after surface hardening Regarding possible root grinding see 1)
26 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
38 Mean stress influence Factor YM The mean stress influence factor YM takes into account the influence of other working stress conditions than pure pulsa-tions (R = 0) such as eg load reversals idler gears planets and shrink-fitted gears
YM (YMst) are defined as the ratio between the endurance (or static) strength with a stress ratio R ne 0 and the endurance (or static) strength with R = 0
YM and YMst apply only to a calculation method that assesses the positive (tensile) stresses and is therefore suitable for comparison between the calculated (positive) working stress σF and the permissible stress σFP calculated with YM or YMst
For thin rings (annulus) in epicyclic gears where the com-pression fillet may be decisive special considerations apply see 316
The following method may be used within a stress ratio ndash12 lt R lt 05
381 For idlers planets and PTO with ice class
M1M1R1
1Yor Y MstM
+minus
minus=
where
R = stress ratio = min stress divided by max stress
For designs with the same force applied on both forward- and back-flank R may be assumed to ndash 12
For designs with considerably different forces on forward- and back-flank such as eg a marine propulsion wheel with a power take off pinion R may be assessed as
branchmain theoffacewidth unit per forcepto offacewidt unit per force21minus
For a power take off (PTO) with ice class see 161 c
M considers the mean stress influence on the endurance (or static) strength amplitudes
M is defined as the reduction of the endurance strength am-plitude for a certain increase of the mean stress divided by that increase of the mean stress
Following M values may be used
Endurance limit
Static strength
Case hardened 08 ndash 015 Ys 1) 07
If shot peened 04 06
Nitrided 03 03
Induction or flame hardened
04 06
Not surface hardened steel 03 05
Cast steels 04 06 1)For bevel gears use Ys=2 for determination of M
The listed M values for the endurance limit are independent of the fillet shape (Ys) except for case hardening In princi-ple there is a dependency but wide variations usually only occur for case hardening eg smooth semicircular fillets versus grinding notches
382 For gears with periodical change of rotational direction For case hardened gears with full load applied periodically in both directions such as side thrusters the same formula for YM as for idlers (with R = ndash 12) may be used together with the M values for endurance limit This simplified approach is valid when the number of changes of direction exceeds 100 and the total number of load cycles exceeds 3middot106
For gears of other materials YM will normally be higher than for a pure idler provided the number of changes of direction is below 3middot106 A linear interpolation in a diagram with loga-rithmic number of changes of direction may be used ie from YM = 09 with one change to YM (idler) for 3middot106 changes This is applicable to YM for endurance limit For static strength use YM as for idlers
For gears with occasional full load in reversed direction such as the main wheel in a reversing gear box YM = 09 may be used
383 For gears with shrinkage stresses and unidirectional load For endurance strength
FE
fitM σ
σM1
M21Y+
minus=
σFE is the endurance limit for R = 0
For static strength YMst = 1 and σfit accounted for in 39b
σfit is the shrinkage stress in the fillet (30˚ tangent) and may be found by multiplying the nominal tangential (hoop) stress with a stress concentration factor
n
Ffit m
ρ215scf minus=
384 For shrink-fitted idlers and planets When combined conditions apply such as idlers with shrink-age stresses the design factor for endurance strength is
( ) ( ) FE
fitM σ
σR1M1
M2
M1M1R1
1Yminussdot+
minus
+minus
minus=
Symbols as above but note that the stress ratio R in this par-ticular connection should disregard the influence of σfit ie R normally equal ndash 12
For static strength
M1M1R1
1YMst
+minus
minus=
The effect of σfit is accounted for in 39 b
Classification Notes- No 412 27 May 2003
DET NORSKE VERITAS
385 Additional requirements for peak loads The total stress range (σmax ndash σmin) in a tooth root fillet is not to exceed
F
y
Sσ225
for not surface hardened fillets
FSHV5 for surface hardened fillets
39 Life Factor YN The life factor YN takes into account that in the case of limited life (number of cycles) a higher tooth root stress can be permitted and that lower stresses may apply for very high number of cycles
Decisive for the strength at limited life is the σ ndash N ndash curve of the respective material for given hardening module fillet radius roughness in the tooth root etc Ie the factors YδrelT YRelT YX and YM have an influence on YN
If no σ ndash N ndash curve for the actual material and hardening etc is available the following method may be used
Determination of the σ ndash N ndash curve
a) Calculate the permissible stress σFP for the beginning of the endurance limit (3middot106 cycles) including the influ-ence of all relevant factors as SF YδrelT YRelT YX YM and YC ie σFP = σFE middotYM middotYδrelT middotYRelT middotYX middotYC SF
b) Calculate the permissible laquostaticraquo stress (le 103 load cy-cles) including the influence of all relevant factors as SFst YδrelTst YMst and YCst ( )fitσCstYδrelTstYMstYFstσ
FstS1
FPstσ minussdotsdotsdot=
where σFst is the local tooth root stress which the material can resist without cracking (surface hardened materials) or unacceptable deformation (not surface hardened materials) with 99 survival probability
σFst
Alloyed case hardened steel 1) 2300
Nitriding steel quenched tempered and gas nitrided (surface hardness 700 ndash 800 HV)
1250
Alloyed quenched and tempered steel bath or gas nitrided (surface hardness 500 ndash 700 HV)
1050
Alloyed quenched and tempered steel flame or induction hardened (fillet surface hardness 500 ndash 650 HV)
18 HV + 800
Steel with not surface hardened fillets the smaller value of 2)
18 σB or 225 σy
1) This is valid for a fillet surface hardness of 58 ndash 63 HRC In case of lower fillet surface hardness than 58 HRC σFst is to be reduced with 30(58 ndash HRC) where HRC is the actual hardness Shot peening or grinding notches are not considered to have any significant influence on σFst 2) Actual stresses exceeding the yield point (σy or σ02) will alter the residual stresses locally in the ldquotensionrdquo fillet re-spectively ldquocompressionrdquo fillet This is only to be utilised for gears that are not later loaded with a high number of cycles at lower loads that could cause fatigue in the ldquocompressionrdquo fillet
c) Calculate YN as
NL gt 3middot106
YN = 1 or 001
L
6
N N103Y
sdot= ie Yn = 092 for 1010
The YN = 1 from 3middot106 on may only be used when special material cleanness applies see rules Pt4 Ch2
103ltNLlt3middot106exp
L
6
N N103Y
sdot=
cycles6103forσcycles310forσlog02876exp
FP
FPst
sdot=
NL lt 103 cycles6103forσcycles310forσ
NYFP
FPst
sdot=
or simply use σFPst as mentioned in b) directly
Guidance on number of load cycles NL for various applica-tions
bull For propulsion purpose normally NL = 1010 at full load (yachts etc may have lower values)
bull For auxiliary gears driving generators that normally operate with 70-90 of rated power NL = 108 with rated power may be applied
28 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
310 Relative Notch Sensitivity Factor YδrelT The dynamic (respectively static) relative notch sensitivity factor YδrelT (YδrelTst) indicate to which extent the theoreti-cally concentrated stress lies above the endurance limits (re-spectively static strengths) in the case of fatigue (respectively overload) breakage
YδrelT is a function of the material and the relative stress gra-dient It differs for static strength and endurance limit
The following method may be used
For endurance limit
for not surface hardened fillets
( )
024
s024
relT σ103133q21σ1012201351
Ysdotsdotminus
+sdotsdotminus+= minus
minus
δ
for all surface hardened fillets except nitrided
( )
106q21002451
Y srelT
++=δ
for nitrided fillets
( )
1347q2101421
Y srelT
++=δ
For static strength
for not surface hardened fillets1)
( )( )( )025
02
02502s
relTst σ3000821σ300 1Y0821Y
+
minus+=δ
for surface hardened fillets except nitrided
YδrelTst = 044 YS + 012
for nitrided fillets
YδrelTst = 06 + 02 YS 1) These values are only valid if the local stresses do not
exceed the yield point and thereby alter the residual stress level See also 39b footnote 2
311 Relative Surface Condition Factor YRrelT The relative surface condition factor YRrelT takes into ac-count the dependence of the tooth root strength on the sur-face condition in the tooth root fillet mainly the dependence on the peak to valley surface roughness
YRrelT differs for endurance limit and static strength
The following method may be used
For endurance limit
YRrelT = 1675 ndash 053 (Ry + 1)01 for surface hardened steels and alloyed quenched and tempered steels except nitrided
YRrelT = 53 ndash 42 (Ry + 1)001 for carbon steels
YδrelT = 43 ndash 326 (Ry + 1)0005
for nitrided steels
For static strength
YRrelTst = 1 for all Ry and all materials
For a fillet without any longitudinal machining trace Ry asymp Rz
312 Size Factor YX The size factor YX takes into account the decrease of the strength with increasing size YX differs for endurance limit and static strength
The following may be used
For endurance limit
YX = 1 for mn le 5 generally
YX = 103 ndash 0006 mn for 5 lt mn lt 30
YX = 085 for mn ge 30
for not surface hard-ened steels
YX = 105 ndash 001 mn for 5 lt mn ge 25
YX = 08 for mn ge 25
for surface hardened steels
For static strength
YXst = 1 for all mn and all materials
313 Case Depth Factor YC The case depth factor YC takes into account the influence of hardening depth on tooth root strength
YC applies only to surface hardened tooth roots and is dif-ferent for endurance limit and static strength
In case of insufficient hardening depth fatigue cracks can develop in the transition zone between the hardened layer and the core For static strength yielding shall not occur in the transition zone as this would alter the surface residual stresses and therewith also the fatigue strength
The major parameters are case depth stress gradient permis-sible surface respectively subsurface stresses and subsurface residual stresses
The following simplified method for YC may be used
YC consists of a ratio between permissible subsurface stress (incl influence of expected residual stresses) and permissible surface stress This ratio is multiplied with a bracket con-taining the influence of case depth and stress gradient (The empirical numbers in the bracket are based on a high number of teeth and are somewhat on the safe side for low number of teeth)
YC and YCst may be calculated as given below but calculated values above 10 are to be put equal 10
For endurance limit
+
+=nFFE
C m02ρt31
σconstY
Classification Notes- No 412 29 May 2003
DET NORSKE VERITAS
For static strength
+
+=nFFst
Cst m02ρt31
σconstY
where const and t are connected as
Hardening process
t = endurance limit const =
static strength const =
t550 640 1900
t400 500 1200 Case hard-ening
t300 380 800
Nitriding t400 500 1200
Induction- or flame hardening
tHVmin 11 HVmin 25 HVmin
For symbols see 213
In addition to these requirements to minimum case depths for endurance limit some upper limitations apply to case hard-ened gears
The max depth to 550 HV should not exceed
1) 13 of the top land thickness san unless adequate tip relief is applied (see 110)
2) 025 mn If this is exceeded the following applies addi-tionally in connection with endurance limit
minusminus= 025
mt
1Yn
max550C
314 Thin rim factor YB Where the rim thickness is not sufficient to provide full sup-port for the tooth root the location of a bending failure may be through the gear rim rather than from root fillet to root fillet
YB is not a factor used to convert calculated root stresses at the 30deg tangent to actual stresses in a thin rim tension fillet Actually the compression fillet can be more susceptible to fatigue
YB is a simplified empirical factor used to de-rate thin rim gears (external as well as internal) when no detailed calcula-tion of stresses in both tension and compression fillets are available
Figure 34 Examples on thin rims
YB is applicable in the range 175 lt sRmn lt 35
YB = 115 middot ln (8324 middot mnsR)
(for sRmn ge 35 YB = 1)
(for sRmn le 175 use 315)
σF as calculated in 321 is to be multiplied with YB when sRmn lt 35 Thus YB is used for both high and low cycle fatigue
Note This method is considered to be on the safe side for external gear rims However for internal gear rims without any flange or web stiffeners the method may not be on the safe side and it is advised to check with the method in 315316
315 Stresses in Thin Rims For rim thickness sR lt 35 mn the safety against rim cracking has to be checked
The following method may be used
3151 General The stresses in the standardised 30ordm tangent section tension side are slightly reduced due to decreasing stress correction factor with decreasing relative rim thickness sRmn On the other hand during the complete stress cycle of that fillet a certain amount of compression stresses are also introduced The complete stress range remains approximately constant Therefore the standardised calculation of stresses at the 30ordm tangent may be retained for thin rims as one of the necessary criteria
The maximum stress range for thin rims usually occurs at the 60ordm ndash 80ordm tangents both for laquotensionraquo and laquocompressionraquo side The following method assumes the 75ordm tangent to be the decisive Therefore in addition to the am criterion ap-plied at the 30ordm tangent it is necessary to evaluate the max and min stresses at the 75ordm tangent for both laquotensionraquo (loaded flank) fillet and laquocompressionraquo (back-flank) fillet For this purpose the whole stress cycle of each fillet should be considered but usually the following simplification is justified
Figure 35 Nomenclature of fillets
Index laquoTraquo means laquotensileraquo fillet laquoCraquo means laquocompressionraquo fillet
σFTmin and σFCmax are determined on basis of the nominal rim stresses times the stress concentration factor Y75
σFCmin and σFTmax are determined on basis of superposition of nominal rim stresses times Y75 plus the tooth bending stresses at 75ordm tangent
30 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
3152 Stress concentration factors at the 75ordm tangents The nominal rim stress consists of bending stresses due to local bending moments tangential stresses due to the tan-gential force Ft and radial shear stresses due to Fr
The major influence is given by the bending stresses The influence of the tangential stresses is minor and even though its stress concentration factor is slightly higher than for bending it is considered to be safe enough when the sum of these nominal stresses are combined with the stress concen-tration factor for bending The influence of the radial shear stress is neglected
The stress concentration factor relating nominal rim stresses to local fillet stresses at the 75ordm tangent may be calculated as
75n
R75
n
R
75
ρmsρ185
ms3
Y+
sdot=
where ρ75 is the root radius at the 75ordm tangent ref to mn Usually ρ75 is closed to the tool radius ρao and
ρ75 = ρao
is a safe approximation compensating for the am simplifi-cations to the laquounsaferaquo side
The tooth root stresses of the loaded tooth are decreasing with decreasing relative rim thickness approximately with the empirical factor
15
R
ncorr s
m311Y
minus=
3153 Nominal rim stresses The bending moment applied to the rim consists of a part of the tooth tilt moment Ft (hF + 05 sR) and the bending caused by the radial force Fr
The sectional modulus (first moment of area) which is used for determination of the nominal bending stresses is not nec-essarily the same for the 2 am bending moments If flanges webs etc outside the toothed section contribute to stiffening the rim against various deflections the influence of these stiffeners should be considered Eg an end flange will have an almost negligible influence on the effective sectional modulus for the stresses due to tooth tilt as the deflection caused by the tilt moment is rather small and would not much involve the flange On the other hand the radial forces as for instance from the meshes in an annulus would cause considerable radial deflections that the flange might restrict to a substantial amount When considering the stiff-ening of such flanges or webs on basis of simplified models it is advised to use an effective rim thickness sR = sR + 02 mn for the first moment of area of the rim (toothed part) cross section
For a high number of rim teeth it may be assumed that the rim bending moments in the fillets adjacent to the loaded tooth are of the same magnitude as right under the applied
force This assumption is reasonable for an annulus but rather much on the safe side for a hollow pinion
The influence of Ft on the nominal tangential stress is simpli-fied by half of it for compressive stresses (σ1) and the other half for tensile stresses (σ2) Applying these assumptions the nominal rim stresses adjacent to the loaded tooth are
( ) ( )A2
FW
f RFW
s05hF05 t
R
r
T
RFt1 minus
ϑminus
+minus=σ
( ) ( )A2
FW
f RFW
s05hF05 t
R
r
T
RFt2 +
ϑminus
+=σ
where σ1 σ2 see Fig 35
A = minimum area of cross section (usually bR sR)
WT = the sectional modulus of rim with respect to tooth tilt moment (usually 6sb 2
rR )
WR = the sectional modulus of the rim including the influence of stiffeners as flanges webs etc (WR ge WT)
bR = the width of the rim
R = the radius of the neutral axis in the rim ie from wheel centre to midpoint of rim
( )ϑf = a function for bending moment distribution around the rim For a rim (pinion) with one mesh only the ( )ϑf at the position of load application is 024 For an annulus with 3 or more meshes ( )ϑf at the position of each load application is approx 3 planets ( )ϑf = 019 4 planets ( )ϑf = 014 5 planets ( )ϑf = 011 6 planets ( )ϑf = 009
It must be checked if the max (tensile) stress in the compres-sion fillet really occurs when the fillet is adjacent to the loaded tooth In principle the stress variation through a complete rotation should be considered and the max value used The max value is usually never less than 0 For an annulus eg the tilt moment is zero in the mid position be-tween the planet meshes whilst the bending moment due to the radial forces is half of that at the mesh but with opposite sign
If these formulae are applied to idler gears as eg planets the influence of nominal tangential stresses must be cor-rected by deleting Ft(2 A) for σ1 and using FtA for σ2 Further the influence of Fr on the nominal bending stresses is usually negligible due to the planet bearing support
3154 Root fillet stresses Determination of min and max stresses in the laquotensionraquo fillet
Classification Notes- No 412 31 May 2003
DET NORSKE VERITAS
Minimum stress
751FTmin YσK σ =
Maximum stress
corrF752 Yσ03Yσ KσFTmax +=
where
03 is an empirical factor relating the tension stresses (σF) at the 30ordm tangent to the part of the tension stresses at the 75ordm tangent which add to the rim related stresses (03 also takes into account that full superposition of nominal stresses times stress concentration factors from both laquosides of the corner filletraquo would result in too high stresses)
αβγ sdotsdotsdotsdot= FFvA KKKKKK
Determination of min and max stresses in the laquocompres-sionraquo fillet
Minimum stress
corrF751FCmin Yσ036Yσ Kσ minus=
Maximum stress
752FCmax Yσ Kσ =
where
036 is an empirical factor relating the tension stresses (σF) at the 30ordm tangent to the part of the compression stresses at the 75ordm tangent which add to the rim related stresses
For gears with reversed loads as idler gears and planets there is no distinct laquotensionraquo or laquocompressionraquo fillet The mini-mum stress σFmin is the minimum of σFTmin and σFCmin (usually the latter is decisive) The maximum stress σFTmax is the maximum of σFTmax and σFCmax (usually the former is deci-sive)
316 Permissible Stresses in Thin Rims
3161 General The safety against fatigue fracture respectively overload fracture is to be at least at the same level as for solid gears The laquoordinaryraquo criteria at the 30ordm tangent apply as given in 31 through 313
Additionally the following criteria at the 75ordm tangent may apply
3162 For gt3middot106 cycles The permissible stresses for the laquotensionraquo fillets and for the laquocompressionraquo fillets are determined by means of a relevant fatigue diagram
If the actual tooth root stress (tensile or compressive) ex-ceeds the yield strength to the material the induced residual stresses are to be taken into account
For determination of permissible stresses the following is defined
R = stress ratio ie FTmax
FTminσσ respectively
FCmax
FCmin
σσ
∆σ = stress range ie σFTmax ndash σFTmin resp σFCmax ndash σFCmin
(For idler gears and planets Fmax
FminσσR =
and ∆σ = σFmax minus σFmin)
The permissible stress range ∆σρ for the laquotensionraquo respec-tively laquocompressionraquo fillets can be calculated as
For R gt minus1 FPp σ
R1R1031
13∆σ
minus+
+=
For minus infin lt R lt minus1 FPp σ
R1R10151
13∆σ
minus+
+=
where
σFP = see 32 determined for unidirectional stresses (YM = 1)
If the yield strength σy is exceeded in either tension or com-pression residual stresses are induced This may be consid-ered by correcting the stress ratio R for the respective fillets (tension or compression)
Eg if yFCmin σσ gt (ie exceeded in compression) the
difference yFCmin σσ∆ minus= affects the stress ratio as
∆σ
σ∆σ∆σR
FCmax
y
FCmax
FCmin
+
minus=
++
=
Similarly the stress ratio in the tension fillet may require cor-rection
If the yield strength is exceeded in tension σFTmax gt σy the difference ∆ = σFTmax ndash σy affects the stress ratio as
y
FTmin
FTmax
FTmin
σ∆σ
∆σ∆σR minus=
minusminus
=
Checking for possible exceeding of the yield strength has to be made with the highest torque and with the KAP if this ex-ceeds KA
3163 Forle 103 cycles The permissible stress range ∆σp is not to exceed
For R gt minus1 FPstpst σ
R1R1051
15∆σ
minus+
+=
For ndash infin lt R lt minus1 FPstpst σ
R1R10251
15∆σ
minus+
+=
For all values of R ∆σpst is limited by
not surface hardened F
y
Sσ225
32 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
surface hardened CF
YSHV5
Definition of ∆σ and R see 3162 with particular attention to possible correction of R if the yield strength is exceeded
σFPst see 32 determined for unidirectional stresses (YM = 1) and lt 103 cycles
3164 For 103 lt cycles lt 3106 ∆σp is to be determined by linear interpolation a log-log dia-gram
∆σp at NL load cycles is
6L 103p
exp
L
6
N p ∆σN103∆σ sdot
sdot=
6
3
103p
10p
∆σ
∆σlog28760exp
sdot
=
Classification Notes- No 412 33 May 2003
DET NORSKE VERITAS
4 Calculation of Scuffing Load Capacity
41 Introduction High surface temperatures due to high loads and sliding ve-locities can cause lubricant films to break down This sei-zure or welding together of areas of tooth surface is termed scuffing
In contrast to pitting and fatigue breakage which show a dis-tinct incubation period a single short overloading can lead to a scuffing failure In the ISO-TR13989 two criteria are mentioned The method used in this Classification Note is based on the principles of the flash temperature criterion
Note Bulk temperature in excess of 120ordmC for long periods may have an adverse effect on the surface durability see 211
42 General Criteria In no point along the path of contact the local contact tem-perature may exceed the permissible temperature ie
oilS
oilSB S
ϑ+ϑminusϑ
leϑ
50SB minusϑleϑ
where
Bϑ = max contact temperature along the path of contact
maxflaMBB ϑ+ϑ=ϑ
MBϑ = bulk temperature see 434
maxflaϑ = max flash temperature along the path of con-tact see 44
Sϑ = scuffing temperature see below
oilϑ = oil temperature before it reaches the mesh (max applicable for the actual load case to be used ie normally alarm temperature except for ice classes where a max expected tem-perature applies)
SS = required safety factor according to the Rules
The scuffing temperature Sϑ may be calculated as
L2
wrelT
002
40S XFZGX
ν100112085780
sdot
sdot++=ϑ
where
XwrelT = relative welding factor
XwrelT
Through hardened steel 10
Phosphated steel 125
Copper-plated steel 150
Nitrided steel 150
Less than 10 retained austenite
115
10 ndash 20 retained austenite 10
Casehardened steel
20 ndash 30 retained austenite 085
Austenitic steel 045
FZG = load stage according to FZG-Test A8390 (Note This is the load stage where scuffing oc-curs However due to scatter in test results cal-culations are to be made with one load stage less than the specification)
XL = lubricant factor
= 10 for mineral oils
= 08 for polyalfaolefins
= 07 for non-water-soluble polyglycols
= 06 for water-soluble polyglycols
= 15 for traction fluids
= 13 for phosphate esters
ν40 = kinematic oil viscosity at 40˚C (mm2s)
Application of other test methods such as the Ryder the FZG-Ryder R46574 and the FZG L-42 Test 141195110 may be specially considered
For high speed gears with very short time of contact Sϑ may be increased as follows provided use of EP-oils
Addition to the calculated scuffing temperature Sϑ
If micros 18tc ge no addition
If micros 18tc lt add ( )cwrelT t18X18 minussdotsdot
where
ct = contact time ( micros ) which is the time needed to cross the Hertzian contact width
( ) [ ]microsu1cosβn
uσ340tb1
Hc +sdotsdot
sdot=
σH as calculated in 221
For bevel gears use vu in stead of u
34 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
43 Influence Factors
431 Coefficient of friction The following coefficient of friction may apply
L025a
005oil
02
redC ΣC
Bt X R ηρv
w0048micro minus
=
where
wBt = specific tooth load (Nmm)
vΣC = sum of tangential velocities at pitch point At pitch line velocities gt 50 ms the limiting value of vΣC at v = 50 ms is to be used
ρredC = relative radius of curvature (transversal plane) at the pitch point
Cylindrical gears
HαHβvγAbt
Bt KKKKKb
Fw sdotsdotsdotsdotsdot= (see 1)
twΣC αsin v2v =
bCredC β cos ρρ =
Bevel gears
HαHβvγAtmb
Bt KKKKKb
Fw sdotsdotsdotsdotsdot= (see 1)
vtmtΣC αsin v2v =
bmvCredC β cos ρρ =
ηoil = dynamic viscosity (mPa s) at oilϑ calculated as
1000
ρ νη oiloil =
where ρ in kgm3 approximated as
( ) 70 15oil15 minusϑminusρ=ρ and νoil is kinematic viscosity at oilϑ and may be calculated by means of the following equation
( ) ( )++=+ 08νloglog08νloglog 100oil ( )
sdotminus
ϑ+minus313log373log
273log373log oil
( ) ( )( )08νloglog08νloglog 10040 +minus+
Ra = composite arithmetic mean roughness (micron) of pinion an wheel calculated as ( )2a1aa RR 05R +=This is defined as the roughness on the new flanks ie as manufactured
XL = see 42
432 Effective tip relief Ceff Ceff is the effective tip relief that amount of tip relief which compensates for the elastic deformation of the gear mesh ie
zero load at the tooth tip It is assumed (simplified) to be equal for both pinion and wheel
Cylindrical gears
for helical
γ
γ=cb
KKFC Abt
eff (see 1)
For spur
cbKKF
C Abteff
γ= (see 1)
Alternatively for spur and helical gears the non-linear ap-proach in 111 may be used (taking Ceff=δ)
Bevel gears
bcKFC Ambt
effγ
= (see 1)
where
γ
α
ε2ε44c
+sdot
=γ
433 Tip relief and extension Cylindrical gears
The extension of the tip relief is not to result in an effective contact ratio 1ltεα when the gear is unloaded (exceptions to this may only apply for applications where the gear is not to run at light loads) This means that the unrelieved part of the path of contact is to be minimum pbt It is further assumed that this unrelieved part is placed centrally on the path of contact
If root relief applies it has to be calculated as equivalent tip relief Ie pinion root relief (at mesh position A) is added to Ca2 and wheel root relief (at mesh position E) is added to Ca1 If no design tip relief or root relief on the mating gear is specified (ie if Ca1+Croot2 = 0 and visa versa) use the run-ning in amount see 112
Bevel gears
Bevel gears are to have heightwise crowning ie no distinct relievedunrelieved area This may be treated as tip and root relief For calculation purposes the root relief is combined with the tip relief of the mating member into an equivalent tip relief If no resulting tip relieves are specified the equivalent tip relives may be calculated as an approxima-tion based on the tool crowning Ca tool (per mille of tool module m0) as follows
2root1aeq1a CCC +=
1root2aeq2a CCC +=
where
Classification Notes- No 412 35 May 2003
DET NORSKE VERITAS
2
0
n0101atoolal m
)m2(mA1middotmCC
minus+minussdot=
( )
sdotminusminussdotsdotminus+= vtvb1
2vb1
21vavt1n1 αtan dddαsin 05x1mA
2
0
n020atool22a m
)mm(2A1middotmCC
minus+minussdot=
( )
sdotminusminussdotsdotminus+= vtvb2
2vb2
2va2vt2n2 αtan dddαsin 05x1mA
2
0
20atool1root1 m
A1mCC
minussdotsdot=
2
0
10atool2root2 m
A1mCC
minussdotsdot=
If the pressure angles of the cutter blades are modified (and verified) to balance the tip relieves the following may be assumed
( )eq2aeq1aeq2aeq1a C andC calculated of sum50CC sdot==
Throughout the following Ca1 and Ca2 mean the equivalent tip relieves Ca1eq and Ca2eq
434 Bulk temperature The bulk temperature may be calculated as
flaaveragempsoilMB X X 50 ϑ+ϑ=ϑ
where
Xs = lubrication factor
= 12 for spray lubrication
= 10 for dip lubrication (provided both pinion and wheel are dip lubricated and tip speed lt 5 ms)
= 10 for spray lubrication with additional cooling spray (spray on both pinion and wheel or spray on pinion and dip lubrication of wheel)
= 02 for meshes fully submerged in oil
Xmp = contact factor ( )pmp nX += 150
np = number of mesh contact on the pinion (for small gear ratii the number of wheel meshes should be used if higher)
flaaverageϑ
= average of the integrated flash temperature (see 44) along the path of contact
( )
AE
E
Ayyyfla
flaaverage
d
ΓminusΓ
ΓΓϑ=ϑint =
For high speed gears (v gt 50 ms) it may be necessary to assess the bulk temperature on the basis of a thermal rating of the entire gear transmission
44 The Flash Temperature flaϑ
441 Basic formula The local flash temperature flaϑ may be calculated as
( ) 41redy
y2ly
211
43Btcorrfla
unXwX3250
y ρ
ρminusρ
micro=ϑ Γ
(For bevel gears replace u with uv)
and is to be calculated stepwise along the path of contact from A to E
where
micro = coefficient of friction see 431
Xcorr = correction factor taking empirically into account the increased scuffing risk in the beginning of the approach path due to mesh starting without any previously built up oil film and possible shuffling away oil before meshing if insufficient tip relief
( )
3
AD
yaeffcorr 50
CC1X
ΓminusΓε
Γminus+=
α
Ca = tip relief of driven member Xcorr is only applicable in the approach path and if Ca lt Ceff otherwise 10
wBt = unit load see 431
yXΓ = load sharing factor see 443
n1 = pinion rpm
Γy ρly etc see 442
442 Geometrical relations The various radii of flank curvature (transversal plane) are
ρ1y = pinion flank radius at mesh point y
ρ2y = wheel flank radius at mesh point y
ρredy = equivalent radius of curvature at mesh point y
y2y1
y2y1redy
ρ+ρ
ρρ=ρ
Cylindrical gears
twy
1y αsin au1Γ1
ρ+
+=
twy
2y αsin au1Γu
ρ+
minus=
Note that for internal gears a and u are negative
36 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
Bevel gears
vtvv
y1y αsin a
u1Γ1
ρ+
+=
vtvv
yv2y αsin a
u1Γu
ρ+
minus=
Γ is the parameter on the path of contact and y is any point between A end E
At the respective ends Γ has the following values
Root piniontip wheel
Cylindrical gears
( )
minus
minusminus= 1
αtan 1dd
zzΓ
tw
2b2a2
1
2A
Bevel gears
( )
minus
minusminus= 1
αtan 1dd
uΓvt
2vb2va2
vA
Tip pinionroot wheel
Cylindrical gears
( )
1αtan
1ddΓ
tw
2b1a1
E minusminus
=
Bevel gears
( )
1αtan
1ddΓ
vt
2vb1va1
E minusminus
=
At inner point of single pair contact
Cylindrical gears
tw1
EB α tan zπ2ΓΓ minus=
Bevel gears
vtv1
EB α tan zπ2ΓΓ minus=
At outer point of single pair contact
Cylindrical gears
tw1AD α tan z
π2ΓΓ +=
Bevel gears
vt v1
AD αtan z π2ΓΓ +=
At pitch point ΓC = 0
The points F and G (only applicable to cylindrical gears) limiting the extension of tip relief (so as to maintain a mini-mum contact ratio of unity for unloaded gears) are at
2
BAF
Γ+Γ=Γ
2
EDG
Γ+Γ=Γ
443 Load sharing factor XΓ The load sharing factor XΓ accounts for the load sharing be-tween the various pairs of teeth in mesh along the path of contact
XΓ is to be calculated stepwise from A to E using the pa-rameter Γy
4431 Cylindrical gears with β = 0 and no tip relief
Figure 41
ByAAB
Ay for31
31X
yΓltΓleΓ
ΓminusΓ
ΓminusΓ+=Γ
DyB for1Xy
ΓltΓleΓ=Γ
EyDDE
yE for31
31X
yΓleΓltΓ
ΓminusΓ
ΓminusΓ+=Γ
4432 Cylindrical gears with β = 0 and tip relief Tip relief on the pinion reduces XΓ in the range G ndash E and increases correspondingly XΓ in the range F ndash B
Tip relief on the wheel reduces XΓ in the range A ndash F and increases correspondingly XΓ in the range D ndash G
Following remains generally
DyB for1Xy
ΓleΓleΓ=Γ
21XXGF== ΓΓ
In the following it must be distinguished between Ca lt Ceff respectively Ca gt Ceff This is shown by an example below where Ca1 lt Ceff and Ca2 gt Ceff
Classification Notes- No 412 37 May 2003
DET NORSKE VERITAS
Figure 42
Note When Ca gt Ceff the path of contact is shortened by A ndash A respectively E ndash E The single pair contact path is extended into B respectively D If this shift is significant it is necessary to consider the negative effect on surface durability (B) and bending stresses (B and D)
Range A - F
For effa2 CC le
minus=Γ
eff
2a
CC1
31X
A
FyAeff
2a
AF
Ay for C3
C61XX
AyΓleΓleΓ
sdot
+ΓminusΓ
ΓminusΓ+= ΓΓ
For effa2 CC ge
AyAfor0Xy
ΓleΓleΓ=Γ
with ( )21
CC
1CC
eff
2a
eff
2a
AFAAminus
minusΓminusΓ+Γ=Γ
FyAeff
2a
AF
Ay
eff
2a for 21
CC
CC1X
yΓleΓleΓ
minus
ΓminusΓ
ΓminusΓ+minus=Γ
Range F - B
For effa1 CC le
ByFeff
1a
FB
Fy forC3
C61
21X
yΓleΓleΓ
sdot
+ΓminusΓ
ΓminusΓ+=Γ
For effa1 CC ge
ByFeff
1a
FB
Fy for21
CC
21X
yΓleΓleΓ
minus
ΓminusΓ
ΓminusΓ+=Γ
ByB for1Xy
ΓleΓleΓ=Γ
with 1
CC2
eff
1a
FBFB
minus
ΓminusΓ+Γ=Γ
Range D ndash G
For effa2 CC le
GyDeff
2a
DG
Dy
eff
2a for C 3C
61
C 3C
32X
yΓleΓleΓ
+
ΓminusΓ
ΓminusΓminus+=Γ F
or effa2 CC ge
DyD for1Xy
ΓleΓleΓ=Γ
with ( )21
CC
1CC
eff
2a
eff
2a
DGDDminus
minusΓminusΓ+Γ=Γ
GyDeff
2a
DG
Dy
eff
2a for21
CC
CCX ΓleΓleΓ
minus
ΓminusΓ
ΓminusΓminus=Γ
Range G - E
For effa1 CC le
minus=Γ
eff
1a
CC1
31X
E
EyGeff
1a
GE
Gy forC3
C61
21X
yΓleΓleΓ
sdot
+ΓminusΓ
ΓminusΓminus=Γ
For effa1 CC gt
EyGeff
1a
GE
Gy for21
CC
21X
yΓleΓleΓ
minus
ΓminusΓ
ΓminusΓminus=Γ
EyE for0Xy
ΓleΓleΓ=Γ
with 1
CC2
eff
1a
GEGE
minus
ΓminusΓ+Γ=Γ
4433 Gears with β gt 0 buttressing Due to oblique contact lines over the flanks a certain but-tressing may occur near A and E
This applies to both cylindrical and bevel gears with tip relief lt Ceff The buttressing Xbutt is simplified as a linear function within the ranges A ndash H respectively I ndash E
Figure 43
1εwhen13X βbutt EAge=
1εwhenε031X ββbutt EAlt+=
38 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
Cylindrical gears
bIEAH βsin 02ΓΓΓΓ =minus=minus
Bevel gears
bmIEAH βsin 02ΓΓΓΓ =minus=minus
4434 Cylindrical gears with 2εγ le and no tip relief
yXΓ is obtained by multiplication of
yXΓ in 4431 with
Xbutt in 4433
4435 Gears with 2εγ gt and no tip relief
Applicable to both cylindrical and bevel gears
IyH for1Xy
ΓleΓleΓε
=α
Γ
IyHybutt and forX1Xy
ΓgtΓΓltΓε
=α
Γ
Figure 44
4436 Cylindrical gears with 2εγ le and tip relief
yXΓ is obtained by multiplication of
yXΓ in 4432 with Xbutt
in 4433
4437 Cylindrical gears with 2εγ gt and tip relief
Tip relief on the pinion (respectively wheel) reduces XГ in the range G ndash E (respectively A ndash F) and increases XГ in the range F ndash G
yXΓ is obtained by multiplication of
yXΓ as described below
with Xbutt in 4433
In the XГ example below the influence of tip relief is shown (without the influence of Xbutt) by means of
eff2aeff1a CC and CC ltgt
Tip relief gt Ceff causes new end points A respectively E of the path of contact
Figure 45
Range A ndash F
( ) ( )( ) eff
a2αa1α
AF
Ay
eff
2aeff
C 12C 13εC 1ε
CCCX
y +εε++minus
ΓminusΓ
ΓminusΓ+
εminus
=ααα
Γ
eff2aFyA CC if for leΓleΓleΓ
eff2aFyA CifCfor and geΓleΓleΓ
eff2aAyA CC iffor0Xy
gtΓleΓleΓ=Γ
( ) ( ) ( )( ) ( ) a2αa1α
αeffa2AFAA C 1ε 3C 1ε
1ε 2 CC with++minus+minus
ΓminusΓ+Γ=Γ
Range F ndash G
( )( )( ) GyF
eff
2a1a forC 1 2CC 11X
yΓleΓleΓ
+εε+minusε
+ε
=αα
α
αΓ
Range G ndash E
( ) ( )( ) eff
2a1a
GE
Gy
C 1 2C 1C 1 3XX
GFy +εεminusε++ε
ΓminusΓ
ΓminusΓminus=
αα
ααΓΓ minus
effa1EyG CC if for leΓleΓleΓ
effa1EyG CC if Γfor and geΓleΓle
eff1aEyE CC if for0Xy
geΓleΓleΓ=Γ
( ) ( ) ( )( ) ( ) 2a1a
eff1aGEEE C 1C 1 3
1 2 CC withminusε++ε+εminus
ΓminusΓminusΓ=Γαα
α
4438 Bevel gears with εγ more than approx 18 and heightwise crowning For Ca1 = Ca2 = Ceff the following applies
Figure 46
Classification Notes- No 412 39 May 2003
DET NORSKE VERITAS
( )AEM 50 Γ+Γ=Γ
( )( )2AD
3
2My651X
y ΓminusΓε
ΓminusΓminus
ε=
ααΓ
For tip relief lt Ceff yXΓ is found by linear interpolation be-
tween ( )effay CCX =Γ and ( )0CayX =Γ as in 4435
The interpolation is to be made stepwise from A to M with the influence of Ca2 and from M to E with the influence of Ca1 (For 2a1a CC ne there is a discontinuity at M)
Eg with Ca1 = 04 Ceff and Ca2 = 055 Ceff then
Range A ndash M
)CC()0C( eff2ay2ayyX 550X 450X =Γ=ΓΓ +=
Range M ndash E
)CC()0C( eff1ay1ayyX 40X 06X =Γ=ΓΓ +=
For tip relief gt Ceff the new end points A and E are found as
( )
minusΓminusΓ
ε+Γ=Γ α 1
CC
6 eff
2aADAA
( )
minusΓminusΓ
ε+Γ=Γ α 1
CC
6 eff
1aADEE
Range A ndash A
0Xy=Γ
Range A ndash M
( )( )
ΓminusΓ
ΓminusΓminus
minusε=
αΓ 2
MA
2My
eff
2a1
CC4
351Xy
Range M ndash E
( )( )
ΓminusΓ
ΓminusΓminus
minusε=
αΓ 2
ME
2My
eff
1a1
CC4
351Xy
Range E ndash E
0Xy=Γ
40 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
Appendix A Fatigue Damage Accumulation
The Palmgren-Miner cumulative damage calculation princi-ple is used The procedure may be applied as follows
A1 Stress Spectrum From the individual torque classes the torques (Ti) at the peak values of class intervals and the associated number of cycles (NLi) for both pinion and wheel are to be listed from the highest to the lowest torque
(In case of a cyclic torque variation within the torque classes it is advised to use the peak torque If the cyclic variation is such that the same teeth will repeatedly suffer the peak torque this is a must)
The stress spectra for tooth roots and flanks (σFi σHi) with all relevant factors (except KA) are to be calculated on the basis of the torque spectrum The load dependent K-factors are to be determined for each torque class
A2 σminusN-curve The stress versus load cycle curves for tooth roots and flanks (both pinion and wheel) are to be drawn on the basis of per-missible stresses (ie including the demanded minimum safety factors) as determined in 2 respectively 3 If different safety levels for high cycle fatigue and low cycle fatigue are desired this may be expressed by different demand safety factors applied at the endurance limit respectively at static strength
A3 Damage accumulation The individual damage ratio Di at ith stress level is defined as
Fi
Lii N
ND =
where
NLi = The number of applied cycles at ith stress
NFi = The number of cycles to failure at ith stress
Basically stresses σi below the permissible stress level for infinite life (if a constant ZN or YN is accepted) do not con-tribute to the damage sum However calculating the actual safety factor Sact as described below all the σi for which the product S σi is bigger than or equal to the permissible stress level for infinite life contribute to the damage sum and thus to the determination of Sact The final value of S is decisive
(NFi can be found mathematically by putting the permissible stress σpi equal the actual stress σi thereby finding the actual life factor This life factor can be solved with regard to load cycles ie NFi)
The damage sum ΣDi is not to exceed unity
If ΣDi ne 1 the safety against cumulative fatigue damage is different from the applied demand safety factor For deter-mination of this theoretical safety factor an iteration proce-dure is required as described in the following flowchart
S is correction factor with which the actual safety factor Sact can be found
Sact is the demand safety factor (used in determination of the permissible stresses in the σ ndash N ndash curve) times the correc-tion factor S
The full procedure is to be applied for pinion and wheel tooth roots and flanks
Note If alternating stresses occur in a spectrum of mainly pulsating stresses the alternating stresses may be replaced by equiva-lent pulsating stresses ie by means of division with the ac-tual mean stress influence factor YM
Classification Notes- No 412 41 May 2003
DET NORSKE VERITAS
Appendix B Application Factors for Diesel Driven Gears
For diesel driven gears the application factor KA depends on torsional vibrations Both normal operation and misfiring conditions have to be considered
Normally these two running conditions can be covered by only one calculation
B1 Definitions
Normal operation KAnorm = 0
normv0
TTT +
where
T0 = rated nominal torque
Tv norm = vibratory torque amplitude for normal opera-tion (see rules Pt4 Ch3 Sec1 G301 for defi-nition of ldquonormalrdquo irregularity)
Misfiring operation KA misf = 0
misfv
TTT +
where
T = remaining nominal torque when one cylinder out of action
Tv misf = vibratory torque amplitude in misfiring con-dition This refers to a permissible misfiring condition ie a condition that does not require automatic or immediate corrective actions as speed or pitch reduction
The normal operation is assumed to last for a very high num-ber of cycles such as 1010
The misfiring operation is assumed to last for a limited dura-tion such as 107 cycles
B2 Determination of decisive load Assuming life factor at 1010 cycles as YN = ZN = 092 which usually is relevant the calculation may be performed only once with the combination having the highest value of appli-cation factorlife factor
For bending stresses and scuffing the higher value of
092
K normA and 098
K misfA
For contact stresses the higher value of
092K normA and
113K misfA (but
097K misfA for nitrided gears)
B3 Simplified procedure Note that this is only a guidance and is not a binding con-vention
T = oTZ
1Zsdot
minus where Z = number of cylinders
( ) ideal Videal Vmisf Vnorm V TTT24ZT +minus=
where TV ideal = vibratory torque with all cylinders perfectly equal See also rules Pt4 Ch3 Sec1 G300
When using trends from torsional vibration analysis and measurements the following may be used
TV ideal To is close to zero for engines with few cylinders and using a suitable elastic coupling and increases with relative coupling stiffness and number of cylinders
TV misf To may be high for engines with few cylinders and decreases with number of cylinders
This can be indicated as
200Z
TT
ο
idealV asymp and 80Z04
TT
o
misfV minusasymp
Inserting this into the formulae for the two application fac-tors the following guidance can be given
118112K misfA minusasymp
115110K normA minusasymp
Since KA norm is to be combined with the lower life factors the decisive load condition will be the normal one and a KA of 115 will cover most relevant cases when a suitable elas-tic coupling is chosen
42 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
Appendix C Calculation of Pinion-Rack
Pinion-racks used for elevating of mobile offshore units are open gears that are subjected to wear and tear With normal specifications such as only a few hundred total operation cycles (site to site) the tooth bending stresses for static strength will be decisive for the lay out however with ex-ception of surface hardened pinions where case crushing has to be considered
In the following the use of part 1 and 3 for pinion-racks is shown including relevant simplifications
C1 Pinion tooth root stresses Since the load spectrum normally is dominated by high torques and few load cycles (in the range up to 1-2000) the static strength is decisive
The actual stress is calculated as
β1FSaFan1
tF1 KYY
mbFσ sdotsdotsdotsdot
=
b1 is limited to b2 + 2middotmn
YFa and YSa replace YF and YS because load application at tooth tip has to be assumed for such inaccurate gears
Pinions often use a non-involute profile in the dedendum part of the flank eg a constant radius equal the radius of curva-ture at reference circle For such pinions sFn and hFa are to be measured directly on a sectional drawing of the pinion tooth
Due to high loads and narrow facewidths it may be assumed that KFβ2 = KHβ = 10 However when b1 gt b2 then KFβ1 gt10
If no detailed documentation of KFβ1 is available the fol-lowing may be used
KFβ1 = 1 + 015middot(b1b2 ndash 1)
The permissible stress (not surface hardened) is calculated as
δrelTstF
Fst1FP1 Y
Sσσ sdot=
The mean stress influence due to leg lifting may be disre-garded
The actual and permissible stresses should be calculated for the relevant loads as given in the rules
C2 Rack tooth root stresses The actual stress is calculated as
SaFan2
tF2 YY
mbFσ sdotsdotsdot
=
See C1 for details
The permissible stress is calculated as
δrelTstF
Fst2FP2 Y
Sσσ sdot=
For alloyed steels (Ni Cr Mo) with high toughness and duc-tility the value of YδrelTst may be put equal to YSa
C3 Surface hardened pinions For surface hardened pinions the maximum load is not to cause crushing of the hardened layer of the flank
In principle the calculation described in 213 may be used but when the theoretical Hertzian stress exceeds the ap-proximately 18 times the yield strength of the rack material plastic deformation will occur This will limit the peak Hertzian stress but increases the contact width and thus the penetration of stresses into the depth
An approximation may be based on an assessment of contact width determined by means of equal areas under the theoreti-cal (elastic range) Hertzian contact and the elasto-plastic contact stress (limited to 18 middot σy) with the unknown width
Classification Notes- No 412 19 May 2003
DET NORSKE VERITAS
the one referred to in the rules Results of approved fatigue tests may also be used as the basis for establishing these values The defined survival probability is 99
σHlim σH105 σH10
3
Alloyed case hardened steels (surface hardness 58-63 HRC) - of specially approved high grade - of normal grade
1650 1500
2500 2400
3100 3100
Nitrided steel of approved grade gas nitrided (surface hardness 700-800 HV) 1250 13 σHlim 13 σHlim
Alloyed quenched and tempered steel bath or gas nitrided (surface hardness 500-700 HV)
1000
13 σHlim
13 σHlim
Alloyed flame or induction hardened steel (surface hardness 500-650 HV) 075 HV + 750 16 σHlim 45 HV
Alloyed quenched and tempered steel 14 HV + 350 16 σHlim 45 HV
Carbon steel 15 HV + 250 16 σHlim 16 σHlim
These values refer to forged or hot rolled steel For cast steel the values for σHlim are to be reduced by 15
29 Life Factor ZN The life factor ZN takes account of a higher permissible contact stress if only limited life (number of cycles NL) is demanded or lower permissible contact stress if very high number of cycles apply
If this is not documented by approved fatigue tests the fol-lowing method may be used
For all steels except nitrided
7L 105N sdotge ZN = 1 or
01570
L
7
N N105Z
sdot=
Ie ZN = 092 for 1010 cycles
The ZN = 1 from 5middot107 on may only be used when the mate-rial cleanliness is of approved high grade (see Rules Pt4 Ch2) and the lubrication is optimised by a specially approved filtering process
105 lt NL lt 5middot107 510N
logZ037
L
7
N N105Z
sdot=
NL = 105 WXRVLHlim
WstX10H1010NN ZZZZZσ
ZZσZZ
55
5=
103 lt NL lt 105 )Z(Zlog05
L
5
10NN
5N103N10
5N10ZZ
==
3L 10N le
WXRVLlimH
Wst10X10H10NN ZZZZZ
ZZZZ
33
3σ
σ==
(but not less than ZN105)
For nitrided steels
6L 102N sdotge ZN = 1 or
00980
L
6
N N102Z
sdot=
Ie ZN = 092 for 1010 cycles
The ZN = 1 from 2middot106 on may only be used when the mate-rial cleanliness is of approved high grade (see Rules Pt4 Ch2) and the lubrication is optimised by a specially approved filtering process
105 lt NL lt 2middot106 510N
Zlog76860
L
6
N N102Z
sdot=
5L 10N le
XWRVL
10XWst10NN ZZZZZ
ZZ13ZZ
55 ==
Note that when no index indicating number of cycles is used the factors are valid for 5middot107 (respectively 2middot106 for nitrid-ing) cycles
210 Influence Factors on Lubrication Film ZL ZV and ZR The lubricant factor ZL accounts for the influence of the type of lubricant and its viscosity the speed factor ZV ac-counts for the influence of the pitch line velocity and the roughness factor ZR accounts for influence of the surface roughness on the surface endurance capacity
The following methods may be applied in connection with the endurance limit
20 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
Surface hardened steels Not surface hardened steels
ZL ( )24013421
360910ν+
+
( )24013421
680830ν+
+
ZV ( )v3280
140930+
+ ( )v3280300850
++
ZR
080
ZrelR3
150
ZrelR3
where
ν40 = Kinematic oil viscosity at 40ordmC (mm2s) For case hardened steels the influence of a high bulk temperature (see 4 Scuffing) should be con-sidered Eg bulk temperatures in excess of 120ordmC for long periods may cause reduced flank surface endurance limits
For values of ν40 gt 500 use ν40 = 500
RZrel = The mean roughness between pinion and wheel (after running in) relative to an equivalent radius of curvature at the pitch point ρc = 10mm
RZrel
= ( ) 31
cZ2Z1 ρ
10RR50
+
RZ = Mean peak to valley roughness (microm) (DIN defini-tion) (roughly RZ = 6 Ra)
For 5
L 10N le ZL ZV ZR = 10
211 Work Hardening Factor ZW The work hardening factor ZW accounts for the increase of surface durability of a soft steel gear when meshing the soft steel gear with a surface hardened or substantially harder gear with a smooth surface
The following approximation may be used for the endurance limit
Surface hardened steel against not surface hardened steel 150
ZeqW R
31700
130HB21Z
minus
minus=
where HB = the Brinell hardness of the soft member For HB gt 470 use HB = 470 For HB lt 130 use HB = 130
RZeq = equivalent roughness
033
c40
066
ZS
ZHZHZEQ ρvν
15000RRRR
=
If RZeq gt 16 then use RZeq = 16
If RZeq lt 15 then use RZeq = 15
where
RZH = surface roughness of the hard member before run in
RZS = surface roughness of the soft member before run in
ν40 = see 210
If values of ZW lt 1 are evaluated ZW = 1 should be used for flank endurance However the low value for ZW may indi-cate a potential wear problem
Through hardened pinion against softer wheel
( )
minussdotminus+= 000829
HBHB0008981u1Z
2
1W
For 21HBHB
2
1 le use ZW = 1
For 71HBHB
2
1 gt use 17HBHB
2
1 =
For u gt 20 use u = 20
For static strength (lt 105 cycles)
Surface hardened against not surface hardened
ZWst = 105
Through hardened pinion against softer wheel
ZWst = 1
212 Size Factor ZX The size factor accounts for statistics indicating that the stress levels at which fatigue damage occurs decrease with an increase of component size as a consequence of the influ-ence on subsurface defects combined with small stress gradi-ents and of the influence of size on material quality
ZX may be taken unity provided that subsurface fatigue for surface hardened pinions and wheels is considered eg as in the following subsection 213
213 Subsurface Fatigue This is only applicable to surface hardened pinions and wheels The main objective is to have a subsurface safety against fatigue (endurance limit) or deformation (static strength) which is at least as high as the safety SH required for the surface The following method may be used as an approximation unless otherwise documented
The high cycle fatigue (gt3106 cycles) is assumed to mainly depend on the orthogonal shear stresses Static strength (lt103 cycles) is assumed to depend mainly on equivalent
Classification Notes- No 412 21 May 2003
DET NORSKE VERITAS
stresses (von Mises) Both are influenced by residual stresses but this is only considered roughly and empirically
The subsurface working stresses at depths inside the peak of the orthogonal shear stresses respectively the equivalent stresses are only dependent on the (real) Hertzian stresses Surface related conditions as expressed by ZL ZV and ZR are assumed to have a negligible influence
The real Hertzian stresses σHR are determined as
For helical gears with εβ gt 1
σHR = σH
For helical gears with εβ lt 1 and spur gears
ε
α
ββ ε
ε+εminus
sdotσ=σZ
1
HHR
For bevel gears
KHHR Z
1σσ sdot=
The necessary hardness HV is given as a function of the net depth tz (net = after grinding or hard metal hobbing and per-pendicular to the flank)
The coordinates tz and HV are to be compared with the de-sign specification such as
bull for flame and induction hardening tHVmin HVmin bull for nitriding t400min HV = 400 bull for case hardening t550min HV = 550 t400min HV = 400
and t300min HV = 300 (the latter only if the core hardness lt 300 If the core hardness gt 400 the t400 is to be re-placed by a fictive t400 = 16 t550)
In addition the specified surface hardness is not to be less than the max necessary hardness (at tz = 05aH) This applies to all hardening methods
For high cycle fatigue (gt3 106 cycles) the following applies
sdot+
minussdotsdotsdot= o90
05azt
05azt
cosSσ04HV
H
HHHR
applicable to 05a
zt
Hge
For 05at
H
z lt the value for 05at
H
z = applies
56300ρSσ12a cHHR
Hsdotsdot
sdot=
Where aH is half the hertzian contact width multiplied by an empirical factor of 12 that takes into account the possible influence of reduced compressive residual stresses (or even tensile residual stresses) on the local fatigue strength
If any of the specified hardness depths including the surface hardness is below the curve described by HV = f (tz) the actual safety factor against subsurface fatigue is determined as follows
reduce SH stepwise in the formula for HV and aH until all specified hardness depths and surface hardness balance with the corrected curve The safety factor obtained through this method is the safety against subsurface fatigue
For static strength (lt103 cycles) the following applies
sdot+
minussdotsdotsdot= o90
07at
06a
zt
cosSσ019HV
Hst
z
HstHHR
applicable to 06at
Hst
z ge
56300ρSσa cHHR
Hstsdotsdot
=
In the case of insufficient specified hardness depths the same procedure for determination of the actual safety factor as above applies
For limited life fatigue (103 lt cycles lt 3106 )
For this purpose it is necessary to extend the correction of safety factors to include also higher values than required Ie in the case of more than sufficient hardness and depths the safety factor in the formulae for both high cycle fatigue and static strength are to be increased until necessary and speci-fied values balance
The actual safety factor for a given number of cycles N be-tween 103 and 3106 is found by linear interpolation in a dou-ble logarithmic diagram
minussdotminus
= sdot logN3477
logSlogSlogS
310H6103HHN
36 10H103H Slog86281Slog86280 sdot+sdot sdot
22 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
3 Calculation of Tooth Strength
31 Scope and General Remarks Part 3 include the calculation of tooth root strength as limited by tooth root cracking (surface or subsurface initiated) and yielding
For rim thickness sR ge 35middotmn the strength is calculated by means of 32 ndash 313 For cylindrical gears the calculation is based on the assumption that the highest tooth root tensile stress arises by application of the force at the outer point of single tooth pair contact of the virtual spur gears The method has however a few limitations that are mentioned in 36
For bevel gears the calculation is based on force application at the tooth tip of the virtual cylindrical gear Subsequently the stress is converted to load application at the mid point of the flank due to the heightwise crowning
Bevel gears may also be calculated with the program BECAL In that case KA and Kv are to be included in the applied tooth force but not KFβ and KFα
In case of a thin annulus or a thin gear rim etc radial crack-ing can occur rather than tangential cracking (from root fillet to root fillet) Cracking can also start from the compression fillet rather than the tension fillet For rim thickness sR lt 35middotmn a special calculation procedure is given in 315 and 316 and a simplified procedure in 314
A tooth breakage is often the end of the life of a gear trans-mission Therefore a high safety SF against breakage is re-quired
It should be noted that this part 3 does not cover fractures caused by
bull oil holes in the tooth root space bull wear steps on the flank bull flank surface distress such as pits spalls or grey staining
Especially the latter is known to cause oblique fractures starting from the active flank predominately in spiral bevel gears but also sometimes in cylindrical gears
Specific calculation methods for these purposes are not given here but are under consideration for future revisions Thus depending on experience with similar gear designs limita-tions other than those outlined in part 3 may be applied
32 Tooth Root Stresses The local tooth root stress is defined as the max principal stress in the tooth root caused by application of the tooth force Ie the stress ratio R = 0 Other stress ratios such as for eg idler gears (R asymp -12) shrunk on gear rims (R gt 0) etc are considered by correcting the permissible stress level
321 Local tooth root stress The local tooth root stress for pinion and wheel may be as-sessed by strain gauge measurements or FE calculations or similar For both measurements and calculations all details are to be agreed in advance
Normally the stresses for pinion and wheel are calculated as
Cylindrical gears
FαFβvγAβSFn
tF K K K K K Y Y Y
m bFσ =
where
YF = Tooth form factor (see 33)
YS = Stress correction factor (see 34)
Yβ = Helix angle factor (see 36)
Ft KA Kγ Kv KFβ KFα see 15 ndash 110
b see 13
Bevel gears
FαFβvγASaFamn
mtF K K K K K Y Y Y
m bFσ ε=
where
YFa = Tooth form factor see 33
YSa = Stress correction factor see 34
Yε = Contact ratio factor see 35
Fmt KA etc see 15 ndash 110
b see 13
322 Permissible tooth root stress The permissible local tooth root stress for pinion respectively wheel for a given number of cycles N is
CXRrelTrelTF
NMFEFP YYYY
SYY
δσ
=σ
Note that all these factors YM etc are applicable to 3middot106 cycles when used in this formula for σFP The influence of other number of cycles on these factors is covered by the calculation of YN
where
σFE = Local tooth root bending endurance limit of reference test gear (see 37)
YM = Mean stress influence factor which accounts for other loads than constant load direction eg idler gears temporary change of load di-rection pre-stress due to shrinkage etc (see 38)
YN = Life factor for tooth root stresses related to reference test gear dimensions (see 39)
SF = Required safety factor according to the rules
YδrelT = Relative notch sensitivity factor of the gear to be determined related to the reference test gear (see 310)
YRrelT = Relative (root fillet) surface condition factor of the gear to be determined related to the reference test gear (see 311)
Classification Notes- No 412 23 May 2003
DET NORSKE VERITAS
YX = Size factor (see 312)
YC = Case depth factor (see 313)
33 Tooth Form Factors YF YFa The tooth form factors YF and YFa take into account the in-fluence of the tooth form on the nominal bending stress
YF applies to load application at the outer point of single tooth pair contact of the virtual spur gear pair and is used for cylindrical gears
YFa applies to load application at the tooth tip and is used for bevel gears
Both YF and YFa are based on the distance between the con-tact points of the 30˚-tangents at the root fillet of the tooth profile for external gears respectively 60˚ tangents for inter-nal gears
Figure 31 External tooth in normal section
Figure 32 Internal tooth in normal section
Definitions
n
2
n
Fn
enFn
Fe
F
α cosms
α cosmh6
Y
=
n
2
n
Fn
anFn
Fa
Fa
α cosms
α cosmh6
Y
=
In the case of helical gears YF and YFa are determined in the normal section ie for a virtual number of teeth
YFa differs from YF by the bending moment arm hFa and αFan and can be determined by the same procedure as YF with exception of hFe and αFan For hFa and αFan all indices e will change to a (tip)
The following formulae apply to cylindrical gears but may also be used for bevel gears when replacing
mn with mnm
zn with zvn
αt with αvt
β with βm
with undercut without undercut
Fig 33 Dimensions and basic rack profile of the teeth (finished profile)
Tool and basic rack data such as hfP ρfp and spr etc are re-ferred to mn ie dimensionless
331 Determination of parameters
( )n
n
prnfPnfP m
α cossαsin 1ρ
αtan h4πE
minusminusminusminus=
For external gears fPfP ρρ =
For internal gears ( )0z
195fPfP0
fPfP 10363156ρhxρρ
sdotminus+
+=
where
z0 = number of teeth of pinion cutter
x0 = addendum modification coefficient of pinion cutter
hfP = addendum of pinion cutter
ρfP = tip radius of pinion cutter
xhρG fpfp +minus=
24 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
τmE
2π
z2H
nnminus
minus=
with
3πτ = for external gears
6πτ = for internal gears
Htan zG2
nminusϑ=ϑ
(to be solved iteratively suitable start value6π
=ϑ for exter-
nal gears and 3π for internal gears)
a) Tooth root chord sFn For external gears
minus
ϑ+
ϑminus= ρ
cosG3
3πsinz
ms
fpnn
Fn
For bevel gears with a tooth thickness modification
xsm affects mainly sFn but also hFe and αFen The total influence of xsm on YFa Ysa can be approximated by only adding 2 xsm to sFn mn
For internal gears
minus
ϑ+
ϑminus= ρ
cosG
6πsinz
ms
fPnn
Fn
b) Root fillet radius ρF at 30ordm tangent
( )G2coszcosG2ρ
mρ
2n
2
fpn
F
minusϑϑ+=
c) Determination of bending moment arm hF dn = zn mn
b2α
αn βcosε
ε =
dan = dn + 2 ha
pbn = π mn cos αn
dbn = dn cos αn
( )4
d1εpzz
2dd
zz2d
2bn
2
αnbn
2bn
2an
en +
minusminus
minus=
en
bnen d
dcos arcα =
ennnn
e α invα invα x tan 22π
z1
minus+
+=γ
αFen = αen ndash γe
For external gears
( )
minus=
n
enFenee
n
Fe
mdαtan sin γ γcos
21
mh
]ρcos
G3πcosz fpn +
ϑminus
ϑminusminus
For internal gears
( ) minus
sdotsdotminus=
n
enFenee
n
Fe
mdα tanγ sinγ cos
21
mh
minus
ϑminus
ϑminussdot ρ
cosG3
6πcosz fPn
332 Gearing with εαn gt 2 For deep tooth form gearing ( )25ε2 αn lele produced with a verified grade of accuracy of 4 or better and with applied profile modification to obtain a trapezoidal load distribution along the path of contact the YF may be corrected by the factor YDT as
250ε205for 0666ε2366Y αnαnDT leleminus=
205εfor 10Y αnDT lt=
34 Stress Correction Factors YS YSa The stress correction factors YS and YSa take into account the conversion of the nominal bending stress to the local tooth root stress Thereby YS and YSa cover the stress increasing effect of the notch (fillet) and the fact that not only bending stresses arise at the root A part of the local stress is inde-pendent of the bending moment arm This part increases the more the decisive point of load application approaches the critical tooth root section
Therefore in addition to its dependence on the notch radius the stress correction is also dependent on the position of the load application ie the size of the bending moment arm
YS applies to the load application at the outer point of single tooth pair contact YSa to the load application at tooth tip
YS can be determined as follows
( )
+
+=L23121
1
sS qL01312Y
where Fe
Fn
hsL = and
F
Fns ρ2
sq = (see 33)
YSa can be calculated by replacing hFe with hFa in the above formulae
Note a) Range of validity 1 lt qs lt 8
In case of sharper root radii (ie produced with tools having too sharp tip radii) YS resp YSa must be specially considered
b) In case of grinding notches (due to insufficient protuberance of the hob) YS resp YSa can rise considerably and must be multiplied with
Classification Notes- No 412 25 May 2003
DET NORSKE VERITAS
g
g
ρt
0613
13
minus
where
tg = depth of the grinding notch
ρg = radius of the grinding notch
c) The formulae for YS resp YSa are only valid for αn = 20˚ However the same formulae can be used as a safe approximation for other pressure angles
35 Contact Ratio Factor Yε The contact ratio factor Yε covers the conversion from load application at the tooth tip to the load application at the mid point of the flank (heightwise) for bevel gears
The following may be used
Yε = 0625
36 Helix Angle Factor Yβ The helix angle factor Yβ takes into account the difference between the helical gear and the virtual spur gear in the nor-mal section on which the calculation is based in the first step In this way it is accounted for that the conditions for tooth root stresses are more favourable because the lines of contact are sloping over the flank
The following may be used (β input in degrees)
Yβ = 1 ndash εβ β120
When εβ gt 1 use εβ = 1 and when β gt 30deg use β = 30deg in the formula
However the above equation for Yβ may only be used for gears with β gt 25deg if adequate tip relief is applied to both pinion and wheel (adequate = at least 05 middot Ceff see 432)
37 Values of Endurance Limit σFE σFE is the local tooth root stress (max principal) which the material can endure permanently with 99 survival prob-ability 3106 load cycles is regarded as the beginning of the endurance limit or the lower knee of the σ ndash N curve σFE is defined as the unidirectional pulsating stress with a minimum stress of zero (disregarding residual stresses due to heat treatment) Other stress conditions such as alternating or pre-stressed etc are covered by the conversion factor YM
σFE can be found by pulsating tests or gear running tests for any material in any condition If the approval of the gear is to be based on the results of such tests all details on the testing conditions have to be approved by the Society Fur-ther the tests may have to be made under the Societys su-pervision
If no fatigue tests are available the following listed values for σFE may be used for materials subjected to a quality con-trol as the one referred to in the rules
σFE
Alloyed case hardened steels 1) (fillet surface hardness 58 ndash 63 HRC)
bull of specially approved high grade
1050
bull of normal grade
minus CrNiMo steels with approved process
1000
minus CrNi and CrNiMo steels generally 920 minus MnCr steels generally 850
Nitriding steel of approved grade quenched tempered and gas nitrided (surface hardness 700 ndash 800 HV)
840
Alloyed quenched and tempered steel bath or gas nitrided (surface hardness 500 ndash 700 HV)
720
Alloyed quenched and tempered steel flame or induction hardened 2) (incl entire root fillet) (fillet surface hardness 500 ndash 650 HV)
07 HV + 300
Alloyed quenched and tempered steel flame or induction hardened (excl entire root fillet) (σB = uts of base material)
025 σB + 125
Alloyed quenched and tempered steel 04 σB + 200
Carbon steel 025 σB + 250
Note All numbers given above are valid for separate forgings and for blanks cut from bars forged according to a qualified procedure see Pt 4 Ch 2 Sec 3 For rolled steel the values are to be reduced with 10 For blanks cut from forged bars that are not qualified as mentioned above the values are to be reduced with 20 For cast steel reduce with 40
1) These values are valid for a root radius
bull being unground If however any grinding is made in the root fillet area in such a way that the residual stresses may be affected σFE is to be reduced by 20 (If the grinding also leaves a notch see 34)
bull with fillet surface hardness 58 ndash 63 HRC In case of lower surface hardness than 58 HRC σFE is to be reduced with 20(58 ndash HRC) where HRC is the detected hardness (This may lead to a permissible tooth root stress that varies along the facewidth If so the actual tooth root stresses may also be considered along facewidth)
bull not being shot peened In case of approved shot peening σFE may be increased by 200 for gears where σFE is reduced by 20 due to root grinding Otherwise σFE may be increased by 100 for
6nm le and 100 ndash 5 (mn - 6) for mn gt 6 However the possible adverse influence on the flanks regarding grey staining should be considered and if necessary the flanks should be masked
2) The fillet is not to be ground after surface hardening Regarding possible root grinding see 1)
26 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
38 Mean stress influence Factor YM The mean stress influence factor YM takes into account the influence of other working stress conditions than pure pulsa-tions (R = 0) such as eg load reversals idler gears planets and shrink-fitted gears
YM (YMst) are defined as the ratio between the endurance (or static) strength with a stress ratio R ne 0 and the endurance (or static) strength with R = 0
YM and YMst apply only to a calculation method that assesses the positive (tensile) stresses and is therefore suitable for comparison between the calculated (positive) working stress σF and the permissible stress σFP calculated with YM or YMst
For thin rings (annulus) in epicyclic gears where the com-pression fillet may be decisive special considerations apply see 316
The following method may be used within a stress ratio ndash12 lt R lt 05
381 For idlers planets and PTO with ice class
M1M1R1
1Yor Y MstM
+minus
minus=
where
R = stress ratio = min stress divided by max stress
For designs with the same force applied on both forward- and back-flank R may be assumed to ndash 12
For designs with considerably different forces on forward- and back-flank such as eg a marine propulsion wheel with a power take off pinion R may be assessed as
branchmain theoffacewidth unit per forcepto offacewidt unit per force21minus
For a power take off (PTO) with ice class see 161 c
M considers the mean stress influence on the endurance (or static) strength amplitudes
M is defined as the reduction of the endurance strength am-plitude for a certain increase of the mean stress divided by that increase of the mean stress
Following M values may be used
Endurance limit
Static strength
Case hardened 08 ndash 015 Ys 1) 07
If shot peened 04 06
Nitrided 03 03
Induction or flame hardened
04 06
Not surface hardened steel 03 05
Cast steels 04 06 1)For bevel gears use Ys=2 for determination of M
The listed M values for the endurance limit are independent of the fillet shape (Ys) except for case hardening In princi-ple there is a dependency but wide variations usually only occur for case hardening eg smooth semicircular fillets versus grinding notches
382 For gears with periodical change of rotational direction For case hardened gears with full load applied periodically in both directions such as side thrusters the same formula for YM as for idlers (with R = ndash 12) may be used together with the M values for endurance limit This simplified approach is valid when the number of changes of direction exceeds 100 and the total number of load cycles exceeds 3middot106
For gears of other materials YM will normally be higher than for a pure idler provided the number of changes of direction is below 3middot106 A linear interpolation in a diagram with loga-rithmic number of changes of direction may be used ie from YM = 09 with one change to YM (idler) for 3middot106 changes This is applicable to YM for endurance limit For static strength use YM as for idlers
For gears with occasional full load in reversed direction such as the main wheel in a reversing gear box YM = 09 may be used
383 For gears with shrinkage stresses and unidirectional load For endurance strength
FE
fitM σ
σM1
M21Y+
minus=
σFE is the endurance limit for R = 0
For static strength YMst = 1 and σfit accounted for in 39b
σfit is the shrinkage stress in the fillet (30˚ tangent) and may be found by multiplying the nominal tangential (hoop) stress with a stress concentration factor
n
Ffit m
ρ215scf minus=
384 For shrink-fitted idlers and planets When combined conditions apply such as idlers with shrink-age stresses the design factor for endurance strength is
( ) ( ) FE
fitM σ
σR1M1
M2
M1M1R1
1Yminussdot+
minus
+minus
minus=
Symbols as above but note that the stress ratio R in this par-ticular connection should disregard the influence of σfit ie R normally equal ndash 12
For static strength
M1M1R1
1YMst
+minus
minus=
The effect of σfit is accounted for in 39 b
Classification Notes- No 412 27 May 2003
DET NORSKE VERITAS
385 Additional requirements for peak loads The total stress range (σmax ndash σmin) in a tooth root fillet is not to exceed
F
y
Sσ225
for not surface hardened fillets
FSHV5 for surface hardened fillets
39 Life Factor YN The life factor YN takes into account that in the case of limited life (number of cycles) a higher tooth root stress can be permitted and that lower stresses may apply for very high number of cycles
Decisive for the strength at limited life is the σ ndash N ndash curve of the respective material for given hardening module fillet radius roughness in the tooth root etc Ie the factors YδrelT YRelT YX and YM have an influence on YN
If no σ ndash N ndash curve for the actual material and hardening etc is available the following method may be used
Determination of the σ ndash N ndash curve
a) Calculate the permissible stress σFP for the beginning of the endurance limit (3middot106 cycles) including the influ-ence of all relevant factors as SF YδrelT YRelT YX YM and YC ie σFP = σFE middotYM middotYδrelT middotYRelT middotYX middotYC SF
b) Calculate the permissible laquostaticraquo stress (le 103 load cy-cles) including the influence of all relevant factors as SFst YδrelTst YMst and YCst ( )fitσCstYδrelTstYMstYFstσ
FstS1
FPstσ minussdotsdotsdot=
where σFst is the local tooth root stress which the material can resist without cracking (surface hardened materials) or unacceptable deformation (not surface hardened materials) with 99 survival probability
σFst
Alloyed case hardened steel 1) 2300
Nitriding steel quenched tempered and gas nitrided (surface hardness 700 ndash 800 HV)
1250
Alloyed quenched and tempered steel bath or gas nitrided (surface hardness 500 ndash 700 HV)
1050
Alloyed quenched and tempered steel flame or induction hardened (fillet surface hardness 500 ndash 650 HV)
18 HV + 800
Steel with not surface hardened fillets the smaller value of 2)
18 σB or 225 σy
1) This is valid for a fillet surface hardness of 58 ndash 63 HRC In case of lower fillet surface hardness than 58 HRC σFst is to be reduced with 30(58 ndash HRC) where HRC is the actual hardness Shot peening or grinding notches are not considered to have any significant influence on σFst 2) Actual stresses exceeding the yield point (σy or σ02) will alter the residual stresses locally in the ldquotensionrdquo fillet re-spectively ldquocompressionrdquo fillet This is only to be utilised for gears that are not later loaded with a high number of cycles at lower loads that could cause fatigue in the ldquocompressionrdquo fillet
c) Calculate YN as
NL gt 3middot106
YN = 1 or 001
L
6
N N103Y
sdot= ie Yn = 092 for 1010
The YN = 1 from 3middot106 on may only be used when special material cleanness applies see rules Pt4 Ch2
103ltNLlt3middot106exp
L
6
N N103Y
sdot=
cycles6103forσcycles310forσlog02876exp
FP
FPst
sdot=
NL lt 103 cycles6103forσcycles310forσ
NYFP
FPst
sdot=
or simply use σFPst as mentioned in b) directly
Guidance on number of load cycles NL for various applica-tions
bull For propulsion purpose normally NL = 1010 at full load (yachts etc may have lower values)
bull For auxiliary gears driving generators that normally operate with 70-90 of rated power NL = 108 with rated power may be applied
28 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
310 Relative Notch Sensitivity Factor YδrelT The dynamic (respectively static) relative notch sensitivity factor YδrelT (YδrelTst) indicate to which extent the theoreti-cally concentrated stress lies above the endurance limits (re-spectively static strengths) in the case of fatigue (respectively overload) breakage
YδrelT is a function of the material and the relative stress gra-dient It differs for static strength and endurance limit
The following method may be used
For endurance limit
for not surface hardened fillets
( )
024
s024
relT σ103133q21σ1012201351
Ysdotsdotminus
+sdotsdotminus+= minus
minus
δ
for all surface hardened fillets except nitrided
( )
106q21002451
Y srelT
++=δ
for nitrided fillets
( )
1347q2101421
Y srelT
++=δ
For static strength
for not surface hardened fillets1)
( )( )( )025
02
02502s
relTst σ3000821σ300 1Y0821Y
+
minus+=δ
for surface hardened fillets except nitrided
YδrelTst = 044 YS + 012
for nitrided fillets
YδrelTst = 06 + 02 YS 1) These values are only valid if the local stresses do not
exceed the yield point and thereby alter the residual stress level See also 39b footnote 2
311 Relative Surface Condition Factor YRrelT The relative surface condition factor YRrelT takes into ac-count the dependence of the tooth root strength on the sur-face condition in the tooth root fillet mainly the dependence on the peak to valley surface roughness
YRrelT differs for endurance limit and static strength
The following method may be used
For endurance limit
YRrelT = 1675 ndash 053 (Ry + 1)01 for surface hardened steels and alloyed quenched and tempered steels except nitrided
YRrelT = 53 ndash 42 (Ry + 1)001 for carbon steels
YδrelT = 43 ndash 326 (Ry + 1)0005
for nitrided steels
For static strength
YRrelTst = 1 for all Ry and all materials
For a fillet without any longitudinal machining trace Ry asymp Rz
312 Size Factor YX The size factor YX takes into account the decrease of the strength with increasing size YX differs for endurance limit and static strength
The following may be used
For endurance limit
YX = 1 for mn le 5 generally
YX = 103 ndash 0006 mn for 5 lt mn lt 30
YX = 085 for mn ge 30
for not surface hard-ened steels
YX = 105 ndash 001 mn for 5 lt mn ge 25
YX = 08 for mn ge 25
for surface hardened steels
For static strength
YXst = 1 for all mn and all materials
313 Case Depth Factor YC The case depth factor YC takes into account the influence of hardening depth on tooth root strength
YC applies only to surface hardened tooth roots and is dif-ferent for endurance limit and static strength
In case of insufficient hardening depth fatigue cracks can develop in the transition zone between the hardened layer and the core For static strength yielding shall not occur in the transition zone as this would alter the surface residual stresses and therewith also the fatigue strength
The major parameters are case depth stress gradient permis-sible surface respectively subsurface stresses and subsurface residual stresses
The following simplified method for YC may be used
YC consists of a ratio between permissible subsurface stress (incl influence of expected residual stresses) and permissible surface stress This ratio is multiplied with a bracket con-taining the influence of case depth and stress gradient (The empirical numbers in the bracket are based on a high number of teeth and are somewhat on the safe side for low number of teeth)
YC and YCst may be calculated as given below but calculated values above 10 are to be put equal 10
For endurance limit
+
+=nFFE
C m02ρt31
σconstY
Classification Notes- No 412 29 May 2003
DET NORSKE VERITAS
For static strength
+
+=nFFst
Cst m02ρt31
σconstY
where const and t are connected as
Hardening process
t = endurance limit const =
static strength const =
t550 640 1900
t400 500 1200 Case hard-ening
t300 380 800
Nitriding t400 500 1200
Induction- or flame hardening
tHVmin 11 HVmin 25 HVmin
For symbols see 213
In addition to these requirements to minimum case depths for endurance limit some upper limitations apply to case hard-ened gears
The max depth to 550 HV should not exceed
1) 13 of the top land thickness san unless adequate tip relief is applied (see 110)
2) 025 mn If this is exceeded the following applies addi-tionally in connection with endurance limit
minusminus= 025
mt
1Yn
max550C
314 Thin rim factor YB Where the rim thickness is not sufficient to provide full sup-port for the tooth root the location of a bending failure may be through the gear rim rather than from root fillet to root fillet
YB is not a factor used to convert calculated root stresses at the 30deg tangent to actual stresses in a thin rim tension fillet Actually the compression fillet can be more susceptible to fatigue
YB is a simplified empirical factor used to de-rate thin rim gears (external as well as internal) when no detailed calcula-tion of stresses in both tension and compression fillets are available
Figure 34 Examples on thin rims
YB is applicable in the range 175 lt sRmn lt 35
YB = 115 middot ln (8324 middot mnsR)
(for sRmn ge 35 YB = 1)
(for sRmn le 175 use 315)
σF as calculated in 321 is to be multiplied with YB when sRmn lt 35 Thus YB is used for both high and low cycle fatigue
Note This method is considered to be on the safe side for external gear rims However for internal gear rims without any flange or web stiffeners the method may not be on the safe side and it is advised to check with the method in 315316
315 Stresses in Thin Rims For rim thickness sR lt 35 mn the safety against rim cracking has to be checked
The following method may be used
3151 General The stresses in the standardised 30ordm tangent section tension side are slightly reduced due to decreasing stress correction factor with decreasing relative rim thickness sRmn On the other hand during the complete stress cycle of that fillet a certain amount of compression stresses are also introduced The complete stress range remains approximately constant Therefore the standardised calculation of stresses at the 30ordm tangent may be retained for thin rims as one of the necessary criteria
The maximum stress range for thin rims usually occurs at the 60ordm ndash 80ordm tangents both for laquotensionraquo and laquocompressionraquo side The following method assumes the 75ordm tangent to be the decisive Therefore in addition to the am criterion ap-plied at the 30ordm tangent it is necessary to evaluate the max and min stresses at the 75ordm tangent for both laquotensionraquo (loaded flank) fillet and laquocompressionraquo (back-flank) fillet For this purpose the whole stress cycle of each fillet should be considered but usually the following simplification is justified
Figure 35 Nomenclature of fillets
Index laquoTraquo means laquotensileraquo fillet laquoCraquo means laquocompressionraquo fillet
σFTmin and σFCmax are determined on basis of the nominal rim stresses times the stress concentration factor Y75
σFCmin and σFTmax are determined on basis of superposition of nominal rim stresses times Y75 plus the tooth bending stresses at 75ordm tangent
30 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
3152 Stress concentration factors at the 75ordm tangents The nominal rim stress consists of bending stresses due to local bending moments tangential stresses due to the tan-gential force Ft and radial shear stresses due to Fr
The major influence is given by the bending stresses The influence of the tangential stresses is minor and even though its stress concentration factor is slightly higher than for bending it is considered to be safe enough when the sum of these nominal stresses are combined with the stress concen-tration factor for bending The influence of the radial shear stress is neglected
The stress concentration factor relating nominal rim stresses to local fillet stresses at the 75ordm tangent may be calculated as
75n
R75
n
R
75
ρmsρ185
ms3
Y+
sdot=
where ρ75 is the root radius at the 75ordm tangent ref to mn Usually ρ75 is closed to the tool radius ρao and
ρ75 = ρao
is a safe approximation compensating for the am simplifi-cations to the laquounsaferaquo side
The tooth root stresses of the loaded tooth are decreasing with decreasing relative rim thickness approximately with the empirical factor
15
R
ncorr s
m311Y
minus=
3153 Nominal rim stresses The bending moment applied to the rim consists of a part of the tooth tilt moment Ft (hF + 05 sR) and the bending caused by the radial force Fr
The sectional modulus (first moment of area) which is used for determination of the nominal bending stresses is not nec-essarily the same for the 2 am bending moments If flanges webs etc outside the toothed section contribute to stiffening the rim against various deflections the influence of these stiffeners should be considered Eg an end flange will have an almost negligible influence on the effective sectional modulus for the stresses due to tooth tilt as the deflection caused by the tilt moment is rather small and would not much involve the flange On the other hand the radial forces as for instance from the meshes in an annulus would cause considerable radial deflections that the flange might restrict to a substantial amount When considering the stiff-ening of such flanges or webs on basis of simplified models it is advised to use an effective rim thickness sR = sR + 02 mn for the first moment of area of the rim (toothed part) cross section
For a high number of rim teeth it may be assumed that the rim bending moments in the fillets adjacent to the loaded tooth are of the same magnitude as right under the applied
force This assumption is reasonable for an annulus but rather much on the safe side for a hollow pinion
The influence of Ft on the nominal tangential stress is simpli-fied by half of it for compressive stresses (σ1) and the other half for tensile stresses (σ2) Applying these assumptions the nominal rim stresses adjacent to the loaded tooth are
( ) ( )A2
FW
f RFW
s05hF05 t
R
r
T
RFt1 minus
ϑminus
+minus=σ
( ) ( )A2
FW
f RFW
s05hF05 t
R
r
T
RFt2 +
ϑminus
+=σ
where σ1 σ2 see Fig 35
A = minimum area of cross section (usually bR sR)
WT = the sectional modulus of rim with respect to tooth tilt moment (usually 6sb 2
rR )
WR = the sectional modulus of the rim including the influence of stiffeners as flanges webs etc (WR ge WT)
bR = the width of the rim
R = the radius of the neutral axis in the rim ie from wheel centre to midpoint of rim
( )ϑf = a function for bending moment distribution around the rim For a rim (pinion) with one mesh only the ( )ϑf at the position of load application is 024 For an annulus with 3 or more meshes ( )ϑf at the position of each load application is approx 3 planets ( )ϑf = 019 4 planets ( )ϑf = 014 5 planets ( )ϑf = 011 6 planets ( )ϑf = 009
It must be checked if the max (tensile) stress in the compres-sion fillet really occurs when the fillet is adjacent to the loaded tooth In principle the stress variation through a complete rotation should be considered and the max value used The max value is usually never less than 0 For an annulus eg the tilt moment is zero in the mid position be-tween the planet meshes whilst the bending moment due to the radial forces is half of that at the mesh but with opposite sign
If these formulae are applied to idler gears as eg planets the influence of nominal tangential stresses must be cor-rected by deleting Ft(2 A) for σ1 and using FtA for σ2 Further the influence of Fr on the nominal bending stresses is usually negligible due to the planet bearing support
3154 Root fillet stresses Determination of min and max stresses in the laquotensionraquo fillet
Classification Notes- No 412 31 May 2003
DET NORSKE VERITAS
Minimum stress
751FTmin YσK σ =
Maximum stress
corrF752 Yσ03Yσ KσFTmax +=
where
03 is an empirical factor relating the tension stresses (σF) at the 30ordm tangent to the part of the tension stresses at the 75ordm tangent which add to the rim related stresses (03 also takes into account that full superposition of nominal stresses times stress concentration factors from both laquosides of the corner filletraquo would result in too high stresses)
αβγ sdotsdotsdotsdot= FFvA KKKKKK
Determination of min and max stresses in the laquocompres-sionraquo fillet
Minimum stress
corrF751FCmin Yσ036Yσ Kσ minus=
Maximum stress
752FCmax Yσ Kσ =
where
036 is an empirical factor relating the tension stresses (σF) at the 30ordm tangent to the part of the compression stresses at the 75ordm tangent which add to the rim related stresses
For gears with reversed loads as idler gears and planets there is no distinct laquotensionraquo or laquocompressionraquo fillet The mini-mum stress σFmin is the minimum of σFTmin and σFCmin (usually the latter is decisive) The maximum stress σFTmax is the maximum of σFTmax and σFCmax (usually the former is deci-sive)
316 Permissible Stresses in Thin Rims
3161 General The safety against fatigue fracture respectively overload fracture is to be at least at the same level as for solid gears The laquoordinaryraquo criteria at the 30ordm tangent apply as given in 31 through 313
Additionally the following criteria at the 75ordm tangent may apply
3162 For gt3middot106 cycles The permissible stresses for the laquotensionraquo fillets and for the laquocompressionraquo fillets are determined by means of a relevant fatigue diagram
If the actual tooth root stress (tensile or compressive) ex-ceeds the yield strength to the material the induced residual stresses are to be taken into account
For determination of permissible stresses the following is defined
R = stress ratio ie FTmax
FTminσσ respectively
FCmax
FCmin
σσ
∆σ = stress range ie σFTmax ndash σFTmin resp σFCmax ndash σFCmin
(For idler gears and planets Fmax
FminσσR =
and ∆σ = σFmax minus σFmin)
The permissible stress range ∆σρ for the laquotensionraquo respec-tively laquocompressionraquo fillets can be calculated as
For R gt minus1 FPp σ
R1R1031
13∆σ
minus+
+=
For minus infin lt R lt minus1 FPp σ
R1R10151
13∆σ
minus+
+=
where
σFP = see 32 determined for unidirectional stresses (YM = 1)
If the yield strength σy is exceeded in either tension or com-pression residual stresses are induced This may be consid-ered by correcting the stress ratio R for the respective fillets (tension or compression)
Eg if yFCmin σσ gt (ie exceeded in compression) the
difference yFCmin σσ∆ minus= affects the stress ratio as
∆σ
σ∆σ∆σR
FCmax
y
FCmax
FCmin
+
minus=
++
=
Similarly the stress ratio in the tension fillet may require cor-rection
If the yield strength is exceeded in tension σFTmax gt σy the difference ∆ = σFTmax ndash σy affects the stress ratio as
y
FTmin
FTmax
FTmin
σ∆σ
∆σ∆σR minus=
minusminus
=
Checking for possible exceeding of the yield strength has to be made with the highest torque and with the KAP if this ex-ceeds KA
3163 Forle 103 cycles The permissible stress range ∆σp is not to exceed
For R gt minus1 FPstpst σ
R1R1051
15∆σ
minus+
+=
For ndash infin lt R lt minus1 FPstpst σ
R1R10251
15∆σ
minus+
+=
For all values of R ∆σpst is limited by
not surface hardened F
y
Sσ225
32 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
surface hardened CF
YSHV5
Definition of ∆σ and R see 3162 with particular attention to possible correction of R if the yield strength is exceeded
σFPst see 32 determined for unidirectional stresses (YM = 1) and lt 103 cycles
3164 For 103 lt cycles lt 3106 ∆σp is to be determined by linear interpolation a log-log dia-gram
∆σp at NL load cycles is
6L 103p
exp
L
6
N p ∆σN103∆σ sdot
sdot=
6
3
103p
10p
∆σ
∆σlog28760exp
sdot
=
Classification Notes- No 412 33 May 2003
DET NORSKE VERITAS
4 Calculation of Scuffing Load Capacity
41 Introduction High surface temperatures due to high loads and sliding ve-locities can cause lubricant films to break down This sei-zure or welding together of areas of tooth surface is termed scuffing
In contrast to pitting and fatigue breakage which show a dis-tinct incubation period a single short overloading can lead to a scuffing failure In the ISO-TR13989 two criteria are mentioned The method used in this Classification Note is based on the principles of the flash temperature criterion
Note Bulk temperature in excess of 120ordmC for long periods may have an adverse effect on the surface durability see 211
42 General Criteria In no point along the path of contact the local contact tem-perature may exceed the permissible temperature ie
oilS
oilSB S
ϑ+ϑminusϑ
leϑ
50SB minusϑleϑ
where
Bϑ = max contact temperature along the path of contact
maxflaMBB ϑ+ϑ=ϑ
MBϑ = bulk temperature see 434
maxflaϑ = max flash temperature along the path of con-tact see 44
Sϑ = scuffing temperature see below
oilϑ = oil temperature before it reaches the mesh (max applicable for the actual load case to be used ie normally alarm temperature except for ice classes where a max expected tem-perature applies)
SS = required safety factor according to the Rules
The scuffing temperature Sϑ may be calculated as
L2
wrelT
002
40S XFZGX
ν100112085780
sdot
sdot++=ϑ
where
XwrelT = relative welding factor
XwrelT
Through hardened steel 10
Phosphated steel 125
Copper-plated steel 150
Nitrided steel 150
Less than 10 retained austenite
115
10 ndash 20 retained austenite 10
Casehardened steel
20 ndash 30 retained austenite 085
Austenitic steel 045
FZG = load stage according to FZG-Test A8390 (Note This is the load stage where scuffing oc-curs However due to scatter in test results cal-culations are to be made with one load stage less than the specification)
XL = lubricant factor
= 10 for mineral oils
= 08 for polyalfaolefins
= 07 for non-water-soluble polyglycols
= 06 for water-soluble polyglycols
= 15 for traction fluids
= 13 for phosphate esters
ν40 = kinematic oil viscosity at 40˚C (mm2s)
Application of other test methods such as the Ryder the FZG-Ryder R46574 and the FZG L-42 Test 141195110 may be specially considered
For high speed gears with very short time of contact Sϑ may be increased as follows provided use of EP-oils
Addition to the calculated scuffing temperature Sϑ
If micros 18tc ge no addition
If micros 18tc lt add ( )cwrelT t18X18 minussdotsdot
where
ct = contact time ( micros ) which is the time needed to cross the Hertzian contact width
( ) [ ]microsu1cosβn
uσ340tb1
Hc +sdotsdot
sdot=
σH as calculated in 221
For bevel gears use vu in stead of u
34 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
43 Influence Factors
431 Coefficient of friction The following coefficient of friction may apply
L025a
005oil
02
redC ΣC
Bt X R ηρv
w0048micro minus
=
where
wBt = specific tooth load (Nmm)
vΣC = sum of tangential velocities at pitch point At pitch line velocities gt 50 ms the limiting value of vΣC at v = 50 ms is to be used
ρredC = relative radius of curvature (transversal plane) at the pitch point
Cylindrical gears
HαHβvγAbt
Bt KKKKKb
Fw sdotsdotsdotsdotsdot= (see 1)
twΣC αsin v2v =
bCredC β cos ρρ =
Bevel gears
HαHβvγAtmb
Bt KKKKKb
Fw sdotsdotsdotsdotsdot= (see 1)
vtmtΣC αsin v2v =
bmvCredC β cos ρρ =
ηoil = dynamic viscosity (mPa s) at oilϑ calculated as
1000
ρ νη oiloil =
where ρ in kgm3 approximated as
( ) 70 15oil15 minusϑminusρ=ρ and νoil is kinematic viscosity at oilϑ and may be calculated by means of the following equation
( ) ( )++=+ 08νloglog08νloglog 100oil ( )
sdotminus
ϑ+minus313log373log
273log373log oil
( ) ( )( )08νloglog08νloglog 10040 +minus+
Ra = composite arithmetic mean roughness (micron) of pinion an wheel calculated as ( )2a1aa RR 05R +=This is defined as the roughness on the new flanks ie as manufactured
XL = see 42
432 Effective tip relief Ceff Ceff is the effective tip relief that amount of tip relief which compensates for the elastic deformation of the gear mesh ie
zero load at the tooth tip It is assumed (simplified) to be equal for both pinion and wheel
Cylindrical gears
for helical
γ
γ=cb
KKFC Abt
eff (see 1)
For spur
cbKKF
C Abteff
γ= (see 1)
Alternatively for spur and helical gears the non-linear ap-proach in 111 may be used (taking Ceff=δ)
Bevel gears
bcKFC Ambt
effγ
= (see 1)
where
γ
α
ε2ε44c
+sdot
=γ
433 Tip relief and extension Cylindrical gears
The extension of the tip relief is not to result in an effective contact ratio 1ltεα when the gear is unloaded (exceptions to this may only apply for applications where the gear is not to run at light loads) This means that the unrelieved part of the path of contact is to be minimum pbt It is further assumed that this unrelieved part is placed centrally on the path of contact
If root relief applies it has to be calculated as equivalent tip relief Ie pinion root relief (at mesh position A) is added to Ca2 and wheel root relief (at mesh position E) is added to Ca1 If no design tip relief or root relief on the mating gear is specified (ie if Ca1+Croot2 = 0 and visa versa) use the run-ning in amount see 112
Bevel gears
Bevel gears are to have heightwise crowning ie no distinct relievedunrelieved area This may be treated as tip and root relief For calculation purposes the root relief is combined with the tip relief of the mating member into an equivalent tip relief If no resulting tip relieves are specified the equivalent tip relives may be calculated as an approxima-tion based on the tool crowning Ca tool (per mille of tool module m0) as follows
2root1aeq1a CCC +=
1root2aeq2a CCC +=
where
Classification Notes- No 412 35 May 2003
DET NORSKE VERITAS
2
0
n0101atoolal m
)m2(mA1middotmCC
minus+minussdot=
( )
sdotminusminussdotsdotminus+= vtvb1
2vb1
21vavt1n1 αtan dddαsin 05x1mA
2
0
n020atool22a m
)mm(2A1middotmCC
minus+minussdot=
( )
sdotminusminussdotsdotminus+= vtvb2
2vb2
2va2vt2n2 αtan dddαsin 05x1mA
2
0
20atool1root1 m
A1mCC
minussdotsdot=
2
0
10atool2root2 m
A1mCC
minussdotsdot=
If the pressure angles of the cutter blades are modified (and verified) to balance the tip relieves the following may be assumed
( )eq2aeq1aeq2aeq1a C andC calculated of sum50CC sdot==
Throughout the following Ca1 and Ca2 mean the equivalent tip relieves Ca1eq and Ca2eq
434 Bulk temperature The bulk temperature may be calculated as
flaaveragempsoilMB X X 50 ϑ+ϑ=ϑ
where
Xs = lubrication factor
= 12 for spray lubrication
= 10 for dip lubrication (provided both pinion and wheel are dip lubricated and tip speed lt 5 ms)
= 10 for spray lubrication with additional cooling spray (spray on both pinion and wheel or spray on pinion and dip lubrication of wheel)
= 02 for meshes fully submerged in oil
Xmp = contact factor ( )pmp nX += 150
np = number of mesh contact on the pinion (for small gear ratii the number of wheel meshes should be used if higher)
flaaverageϑ
= average of the integrated flash temperature (see 44) along the path of contact
( )
AE
E
Ayyyfla
flaaverage
d
ΓminusΓ
ΓΓϑ=ϑint =
For high speed gears (v gt 50 ms) it may be necessary to assess the bulk temperature on the basis of a thermal rating of the entire gear transmission
44 The Flash Temperature flaϑ
441 Basic formula The local flash temperature flaϑ may be calculated as
( ) 41redy
y2ly
211
43Btcorrfla
unXwX3250
y ρ
ρminusρ
micro=ϑ Γ
(For bevel gears replace u with uv)
and is to be calculated stepwise along the path of contact from A to E
where
micro = coefficient of friction see 431
Xcorr = correction factor taking empirically into account the increased scuffing risk in the beginning of the approach path due to mesh starting without any previously built up oil film and possible shuffling away oil before meshing if insufficient tip relief
( )
3
AD
yaeffcorr 50
CC1X
ΓminusΓε
Γminus+=
α
Ca = tip relief of driven member Xcorr is only applicable in the approach path and if Ca lt Ceff otherwise 10
wBt = unit load see 431
yXΓ = load sharing factor see 443
n1 = pinion rpm
Γy ρly etc see 442
442 Geometrical relations The various radii of flank curvature (transversal plane) are
ρ1y = pinion flank radius at mesh point y
ρ2y = wheel flank radius at mesh point y
ρredy = equivalent radius of curvature at mesh point y
y2y1
y2y1redy
ρ+ρ
ρρ=ρ
Cylindrical gears
twy
1y αsin au1Γ1
ρ+
+=
twy
2y αsin au1Γu
ρ+
minus=
Note that for internal gears a and u are negative
36 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
Bevel gears
vtvv
y1y αsin a
u1Γ1
ρ+
+=
vtvv
yv2y αsin a
u1Γu
ρ+
minus=
Γ is the parameter on the path of contact and y is any point between A end E
At the respective ends Γ has the following values
Root piniontip wheel
Cylindrical gears
( )
minus
minusminus= 1
αtan 1dd
zzΓ
tw
2b2a2
1
2A
Bevel gears
( )
minus
minusminus= 1
αtan 1dd
uΓvt
2vb2va2
vA
Tip pinionroot wheel
Cylindrical gears
( )
1αtan
1ddΓ
tw
2b1a1
E minusminus
=
Bevel gears
( )
1αtan
1ddΓ
vt
2vb1va1
E minusminus
=
At inner point of single pair contact
Cylindrical gears
tw1
EB α tan zπ2ΓΓ minus=
Bevel gears
vtv1
EB α tan zπ2ΓΓ minus=
At outer point of single pair contact
Cylindrical gears
tw1AD α tan z
π2ΓΓ +=
Bevel gears
vt v1
AD αtan z π2ΓΓ +=
At pitch point ΓC = 0
The points F and G (only applicable to cylindrical gears) limiting the extension of tip relief (so as to maintain a mini-mum contact ratio of unity for unloaded gears) are at
2
BAF
Γ+Γ=Γ
2
EDG
Γ+Γ=Γ
443 Load sharing factor XΓ The load sharing factor XΓ accounts for the load sharing be-tween the various pairs of teeth in mesh along the path of contact
XΓ is to be calculated stepwise from A to E using the pa-rameter Γy
4431 Cylindrical gears with β = 0 and no tip relief
Figure 41
ByAAB
Ay for31
31X
yΓltΓleΓ
ΓminusΓ
ΓminusΓ+=Γ
DyB for1Xy
ΓltΓleΓ=Γ
EyDDE
yE for31
31X
yΓleΓltΓ
ΓminusΓ
ΓminusΓ+=Γ
4432 Cylindrical gears with β = 0 and tip relief Tip relief on the pinion reduces XΓ in the range G ndash E and increases correspondingly XΓ in the range F ndash B
Tip relief on the wheel reduces XΓ in the range A ndash F and increases correspondingly XΓ in the range D ndash G
Following remains generally
DyB for1Xy
ΓleΓleΓ=Γ
21XXGF== ΓΓ
In the following it must be distinguished between Ca lt Ceff respectively Ca gt Ceff This is shown by an example below where Ca1 lt Ceff and Ca2 gt Ceff
Classification Notes- No 412 37 May 2003
DET NORSKE VERITAS
Figure 42
Note When Ca gt Ceff the path of contact is shortened by A ndash A respectively E ndash E The single pair contact path is extended into B respectively D If this shift is significant it is necessary to consider the negative effect on surface durability (B) and bending stresses (B and D)
Range A - F
For effa2 CC le
minus=Γ
eff
2a
CC1
31X
A
FyAeff
2a
AF
Ay for C3
C61XX
AyΓleΓleΓ
sdot
+ΓminusΓ
ΓminusΓ+= ΓΓ
For effa2 CC ge
AyAfor0Xy
ΓleΓleΓ=Γ
with ( )21
CC
1CC
eff
2a
eff
2a
AFAAminus
minusΓminusΓ+Γ=Γ
FyAeff
2a
AF
Ay
eff
2a for 21
CC
CC1X
yΓleΓleΓ
minus
ΓminusΓ
ΓminusΓ+minus=Γ
Range F - B
For effa1 CC le
ByFeff
1a
FB
Fy forC3
C61
21X
yΓleΓleΓ
sdot
+ΓminusΓ
ΓminusΓ+=Γ
For effa1 CC ge
ByFeff
1a
FB
Fy for21
CC
21X
yΓleΓleΓ
minus
ΓminusΓ
ΓminusΓ+=Γ
ByB for1Xy
ΓleΓleΓ=Γ
with 1
CC2
eff
1a
FBFB
minus
ΓminusΓ+Γ=Γ
Range D ndash G
For effa2 CC le
GyDeff
2a
DG
Dy
eff
2a for C 3C
61
C 3C
32X
yΓleΓleΓ
+
ΓminusΓ
ΓminusΓminus+=Γ F
or effa2 CC ge
DyD for1Xy
ΓleΓleΓ=Γ
with ( )21
CC
1CC
eff
2a
eff
2a
DGDDminus
minusΓminusΓ+Γ=Γ
GyDeff
2a
DG
Dy
eff
2a for21
CC
CCX ΓleΓleΓ
minus
ΓminusΓ
ΓminusΓminus=Γ
Range G - E
For effa1 CC le
minus=Γ
eff
1a
CC1
31X
E
EyGeff
1a
GE
Gy forC3
C61
21X
yΓleΓleΓ
sdot
+ΓminusΓ
ΓminusΓminus=Γ
For effa1 CC gt
EyGeff
1a
GE
Gy for21
CC
21X
yΓleΓleΓ
minus
ΓminusΓ
ΓminusΓminus=Γ
EyE for0Xy
ΓleΓleΓ=Γ
with 1
CC2
eff
1a
GEGE
minus
ΓminusΓ+Γ=Γ
4433 Gears with β gt 0 buttressing Due to oblique contact lines over the flanks a certain but-tressing may occur near A and E
This applies to both cylindrical and bevel gears with tip relief lt Ceff The buttressing Xbutt is simplified as a linear function within the ranges A ndash H respectively I ndash E
Figure 43
1εwhen13X βbutt EAge=
1εwhenε031X ββbutt EAlt+=
38 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
Cylindrical gears
bIEAH βsin 02ΓΓΓΓ =minus=minus
Bevel gears
bmIEAH βsin 02ΓΓΓΓ =minus=minus
4434 Cylindrical gears with 2εγ le and no tip relief
yXΓ is obtained by multiplication of
yXΓ in 4431 with
Xbutt in 4433
4435 Gears with 2εγ gt and no tip relief
Applicable to both cylindrical and bevel gears
IyH for1Xy
ΓleΓleΓε
=α
Γ
IyHybutt and forX1Xy
ΓgtΓΓltΓε
=α
Γ
Figure 44
4436 Cylindrical gears with 2εγ le and tip relief
yXΓ is obtained by multiplication of
yXΓ in 4432 with Xbutt
in 4433
4437 Cylindrical gears with 2εγ gt and tip relief
Tip relief on the pinion (respectively wheel) reduces XГ in the range G ndash E (respectively A ndash F) and increases XГ in the range F ndash G
yXΓ is obtained by multiplication of
yXΓ as described below
with Xbutt in 4433
In the XГ example below the influence of tip relief is shown (without the influence of Xbutt) by means of
eff2aeff1a CC and CC ltgt
Tip relief gt Ceff causes new end points A respectively E of the path of contact
Figure 45
Range A ndash F
( ) ( )( ) eff
a2αa1α
AF
Ay
eff
2aeff
C 12C 13εC 1ε
CCCX
y +εε++minus
ΓminusΓ
ΓminusΓ+
εminus
=ααα
Γ
eff2aFyA CC if for leΓleΓleΓ
eff2aFyA CifCfor and geΓleΓleΓ
eff2aAyA CC iffor0Xy
gtΓleΓleΓ=Γ
( ) ( ) ( )( ) ( ) a2αa1α
αeffa2AFAA C 1ε 3C 1ε
1ε 2 CC with++minus+minus
ΓminusΓ+Γ=Γ
Range F ndash G
( )( )( ) GyF
eff
2a1a forC 1 2CC 11X
yΓleΓleΓ
+εε+minusε
+ε
=αα
α
αΓ
Range G ndash E
( ) ( )( ) eff
2a1a
GE
Gy
C 1 2C 1C 1 3XX
GFy +εεminusε++ε
ΓminusΓ
ΓminusΓminus=
αα
ααΓΓ minus
effa1EyG CC if for leΓleΓleΓ
effa1EyG CC if Γfor and geΓleΓle
eff1aEyE CC if for0Xy
geΓleΓleΓ=Γ
( ) ( ) ( )( ) ( ) 2a1a
eff1aGEEE C 1C 1 3
1 2 CC withminusε++ε+εminus
ΓminusΓminusΓ=Γαα
α
4438 Bevel gears with εγ more than approx 18 and heightwise crowning For Ca1 = Ca2 = Ceff the following applies
Figure 46
Classification Notes- No 412 39 May 2003
DET NORSKE VERITAS
( )AEM 50 Γ+Γ=Γ
( )( )2AD
3
2My651X
y ΓminusΓε
ΓminusΓminus
ε=
ααΓ
For tip relief lt Ceff yXΓ is found by linear interpolation be-
tween ( )effay CCX =Γ and ( )0CayX =Γ as in 4435
The interpolation is to be made stepwise from A to M with the influence of Ca2 and from M to E with the influence of Ca1 (For 2a1a CC ne there is a discontinuity at M)
Eg with Ca1 = 04 Ceff and Ca2 = 055 Ceff then
Range A ndash M
)CC()0C( eff2ay2ayyX 550X 450X =Γ=ΓΓ +=
Range M ndash E
)CC()0C( eff1ay1ayyX 40X 06X =Γ=ΓΓ +=
For tip relief gt Ceff the new end points A and E are found as
( )
minusΓminusΓ
ε+Γ=Γ α 1
CC
6 eff
2aADAA
( )
minusΓminusΓ
ε+Γ=Γ α 1
CC
6 eff
1aADEE
Range A ndash A
0Xy=Γ
Range A ndash M
( )( )
ΓminusΓ
ΓminusΓminus
minusε=
αΓ 2
MA
2My
eff
2a1
CC4
351Xy
Range M ndash E
( )( )
ΓminusΓ
ΓminusΓminus
minusε=
αΓ 2
ME
2My
eff
1a1
CC4
351Xy
Range E ndash E
0Xy=Γ
40 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
Appendix A Fatigue Damage Accumulation
The Palmgren-Miner cumulative damage calculation princi-ple is used The procedure may be applied as follows
A1 Stress Spectrum From the individual torque classes the torques (Ti) at the peak values of class intervals and the associated number of cycles (NLi) for both pinion and wheel are to be listed from the highest to the lowest torque
(In case of a cyclic torque variation within the torque classes it is advised to use the peak torque If the cyclic variation is such that the same teeth will repeatedly suffer the peak torque this is a must)
The stress spectra for tooth roots and flanks (σFi σHi) with all relevant factors (except KA) are to be calculated on the basis of the torque spectrum The load dependent K-factors are to be determined for each torque class
A2 σminusN-curve The stress versus load cycle curves for tooth roots and flanks (both pinion and wheel) are to be drawn on the basis of per-missible stresses (ie including the demanded minimum safety factors) as determined in 2 respectively 3 If different safety levels for high cycle fatigue and low cycle fatigue are desired this may be expressed by different demand safety factors applied at the endurance limit respectively at static strength
A3 Damage accumulation The individual damage ratio Di at ith stress level is defined as
Fi
Lii N
ND =
where
NLi = The number of applied cycles at ith stress
NFi = The number of cycles to failure at ith stress
Basically stresses σi below the permissible stress level for infinite life (if a constant ZN or YN is accepted) do not con-tribute to the damage sum However calculating the actual safety factor Sact as described below all the σi for which the product S σi is bigger than or equal to the permissible stress level for infinite life contribute to the damage sum and thus to the determination of Sact The final value of S is decisive
(NFi can be found mathematically by putting the permissible stress σpi equal the actual stress σi thereby finding the actual life factor This life factor can be solved with regard to load cycles ie NFi)
The damage sum ΣDi is not to exceed unity
If ΣDi ne 1 the safety against cumulative fatigue damage is different from the applied demand safety factor For deter-mination of this theoretical safety factor an iteration proce-dure is required as described in the following flowchart
S is correction factor with which the actual safety factor Sact can be found
Sact is the demand safety factor (used in determination of the permissible stresses in the σ ndash N ndash curve) times the correc-tion factor S
The full procedure is to be applied for pinion and wheel tooth roots and flanks
Note If alternating stresses occur in a spectrum of mainly pulsating stresses the alternating stresses may be replaced by equiva-lent pulsating stresses ie by means of division with the ac-tual mean stress influence factor YM
Classification Notes- No 412 41 May 2003
DET NORSKE VERITAS
Appendix B Application Factors for Diesel Driven Gears
For diesel driven gears the application factor KA depends on torsional vibrations Both normal operation and misfiring conditions have to be considered
Normally these two running conditions can be covered by only one calculation
B1 Definitions
Normal operation KAnorm = 0
normv0
TTT +
where
T0 = rated nominal torque
Tv norm = vibratory torque amplitude for normal opera-tion (see rules Pt4 Ch3 Sec1 G301 for defi-nition of ldquonormalrdquo irregularity)
Misfiring operation KA misf = 0
misfv
TTT +
where
T = remaining nominal torque when one cylinder out of action
Tv misf = vibratory torque amplitude in misfiring con-dition This refers to a permissible misfiring condition ie a condition that does not require automatic or immediate corrective actions as speed or pitch reduction
The normal operation is assumed to last for a very high num-ber of cycles such as 1010
The misfiring operation is assumed to last for a limited dura-tion such as 107 cycles
B2 Determination of decisive load Assuming life factor at 1010 cycles as YN = ZN = 092 which usually is relevant the calculation may be performed only once with the combination having the highest value of appli-cation factorlife factor
For bending stresses and scuffing the higher value of
092
K normA and 098
K misfA
For contact stresses the higher value of
092K normA and
113K misfA (but
097K misfA for nitrided gears)
B3 Simplified procedure Note that this is only a guidance and is not a binding con-vention
T = oTZ
1Zsdot
minus where Z = number of cylinders
( ) ideal Videal Vmisf Vnorm V TTT24ZT +minus=
where TV ideal = vibratory torque with all cylinders perfectly equal See also rules Pt4 Ch3 Sec1 G300
When using trends from torsional vibration analysis and measurements the following may be used
TV ideal To is close to zero for engines with few cylinders and using a suitable elastic coupling and increases with relative coupling stiffness and number of cylinders
TV misf To may be high for engines with few cylinders and decreases with number of cylinders
This can be indicated as
200Z
TT
ο
idealV asymp and 80Z04
TT
o
misfV minusasymp
Inserting this into the formulae for the two application fac-tors the following guidance can be given
118112K misfA minusasymp
115110K normA minusasymp
Since KA norm is to be combined with the lower life factors the decisive load condition will be the normal one and a KA of 115 will cover most relevant cases when a suitable elas-tic coupling is chosen
42 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
Appendix C Calculation of Pinion-Rack
Pinion-racks used for elevating of mobile offshore units are open gears that are subjected to wear and tear With normal specifications such as only a few hundred total operation cycles (site to site) the tooth bending stresses for static strength will be decisive for the lay out however with ex-ception of surface hardened pinions where case crushing has to be considered
In the following the use of part 1 and 3 for pinion-racks is shown including relevant simplifications
C1 Pinion tooth root stresses Since the load spectrum normally is dominated by high torques and few load cycles (in the range up to 1-2000) the static strength is decisive
The actual stress is calculated as
β1FSaFan1
tF1 KYY
mbFσ sdotsdotsdotsdot
=
b1 is limited to b2 + 2middotmn
YFa and YSa replace YF and YS because load application at tooth tip has to be assumed for such inaccurate gears
Pinions often use a non-involute profile in the dedendum part of the flank eg a constant radius equal the radius of curva-ture at reference circle For such pinions sFn and hFa are to be measured directly on a sectional drawing of the pinion tooth
Due to high loads and narrow facewidths it may be assumed that KFβ2 = KHβ = 10 However when b1 gt b2 then KFβ1 gt10
If no detailed documentation of KFβ1 is available the fol-lowing may be used
KFβ1 = 1 + 015middot(b1b2 ndash 1)
The permissible stress (not surface hardened) is calculated as
δrelTstF
Fst1FP1 Y
Sσσ sdot=
The mean stress influence due to leg lifting may be disre-garded
The actual and permissible stresses should be calculated for the relevant loads as given in the rules
C2 Rack tooth root stresses The actual stress is calculated as
SaFan2
tF2 YY
mbFσ sdotsdotsdot
=
See C1 for details
The permissible stress is calculated as
δrelTstF
Fst2FP2 Y
Sσσ sdot=
For alloyed steels (Ni Cr Mo) with high toughness and duc-tility the value of YδrelTst may be put equal to YSa
C3 Surface hardened pinions For surface hardened pinions the maximum load is not to cause crushing of the hardened layer of the flank
In principle the calculation described in 213 may be used but when the theoretical Hertzian stress exceeds the ap-proximately 18 times the yield strength of the rack material plastic deformation will occur This will limit the peak Hertzian stress but increases the contact width and thus the penetration of stresses into the depth
An approximation may be based on an assessment of contact width determined by means of equal areas under the theoreti-cal (elastic range) Hertzian contact and the elasto-plastic contact stress (limited to 18 middot σy) with the unknown width
20 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
Surface hardened steels Not surface hardened steels
ZL ( )24013421
360910ν+
+
( )24013421
680830ν+
+
ZV ( )v3280
140930+
+ ( )v3280300850
++
ZR
080
ZrelR3
150
ZrelR3
where
ν40 = Kinematic oil viscosity at 40ordmC (mm2s) For case hardened steels the influence of a high bulk temperature (see 4 Scuffing) should be con-sidered Eg bulk temperatures in excess of 120ordmC for long periods may cause reduced flank surface endurance limits
For values of ν40 gt 500 use ν40 = 500
RZrel = The mean roughness between pinion and wheel (after running in) relative to an equivalent radius of curvature at the pitch point ρc = 10mm
RZrel
= ( ) 31
cZ2Z1 ρ
10RR50
+
RZ = Mean peak to valley roughness (microm) (DIN defini-tion) (roughly RZ = 6 Ra)
For 5
L 10N le ZL ZV ZR = 10
211 Work Hardening Factor ZW The work hardening factor ZW accounts for the increase of surface durability of a soft steel gear when meshing the soft steel gear with a surface hardened or substantially harder gear with a smooth surface
The following approximation may be used for the endurance limit
Surface hardened steel against not surface hardened steel 150
ZeqW R
31700
130HB21Z
minus
minus=
where HB = the Brinell hardness of the soft member For HB gt 470 use HB = 470 For HB lt 130 use HB = 130
RZeq = equivalent roughness
033
c40
066
ZS
ZHZHZEQ ρvν
15000RRRR
=
If RZeq gt 16 then use RZeq = 16
If RZeq lt 15 then use RZeq = 15
where
RZH = surface roughness of the hard member before run in
RZS = surface roughness of the soft member before run in
ν40 = see 210
If values of ZW lt 1 are evaluated ZW = 1 should be used for flank endurance However the low value for ZW may indi-cate a potential wear problem
Through hardened pinion against softer wheel
( )
minussdotminus+= 000829
HBHB0008981u1Z
2
1W
For 21HBHB
2
1 le use ZW = 1
For 71HBHB
2
1 gt use 17HBHB
2
1 =
For u gt 20 use u = 20
For static strength (lt 105 cycles)
Surface hardened against not surface hardened
ZWst = 105
Through hardened pinion against softer wheel
ZWst = 1
212 Size Factor ZX The size factor accounts for statistics indicating that the stress levels at which fatigue damage occurs decrease with an increase of component size as a consequence of the influ-ence on subsurface defects combined with small stress gradi-ents and of the influence of size on material quality
ZX may be taken unity provided that subsurface fatigue for surface hardened pinions and wheels is considered eg as in the following subsection 213
213 Subsurface Fatigue This is only applicable to surface hardened pinions and wheels The main objective is to have a subsurface safety against fatigue (endurance limit) or deformation (static strength) which is at least as high as the safety SH required for the surface The following method may be used as an approximation unless otherwise documented
The high cycle fatigue (gt3106 cycles) is assumed to mainly depend on the orthogonal shear stresses Static strength (lt103 cycles) is assumed to depend mainly on equivalent
Classification Notes- No 412 21 May 2003
DET NORSKE VERITAS
stresses (von Mises) Both are influenced by residual stresses but this is only considered roughly and empirically
The subsurface working stresses at depths inside the peak of the orthogonal shear stresses respectively the equivalent stresses are only dependent on the (real) Hertzian stresses Surface related conditions as expressed by ZL ZV and ZR are assumed to have a negligible influence
The real Hertzian stresses σHR are determined as
For helical gears with εβ gt 1
σHR = σH
For helical gears with εβ lt 1 and spur gears
ε
α
ββ ε
ε+εminus
sdotσ=σZ
1
HHR
For bevel gears
KHHR Z
1σσ sdot=
The necessary hardness HV is given as a function of the net depth tz (net = after grinding or hard metal hobbing and per-pendicular to the flank)
The coordinates tz and HV are to be compared with the de-sign specification such as
bull for flame and induction hardening tHVmin HVmin bull for nitriding t400min HV = 400 bull for case hardening t550min HV = 550 t400min HV = 400
and t300min HV = 300 (the latter only if the core hardness lt 300 If the core hardness gt 400 the t400 is to be re-placed by a fictive t400 = 16 t550)
In addition the specified surface hardness is not to be less than the max necessary hardness (at tz = 05aH) This applies to all hardening methods
For high cycle fatigue (gt3 106 cycles) the following applies
sdot+
minussdotsdotsdot= o90
05azt
05azt
cosSσ04HV
H
HHHR
applicable to 05a
zt
Hge
For 05at
H
z lt the value for 05at
H
z = applies
56300ρSσ12a cHHR
Hsdotsdot
sdot=
Where aH is half the hertzian contact width multiplied by an empirical factor of 12 that takes into account the possible influence of reduced compressive residual stresses (or even tensile residual stresses) on the local fatigue strength
If any of the specified hardness depths including the surface hardness is below the curve described by HV = f (tz) the actual safety factor against subsurface fatigue is determined as follows
reduce SH stepwise in the formula for HV and aH until all specified hardness depths and surface hardness balance with the corrected curve The safety factor obtained through this method is the safety against subsurface fatigue
For static strength (lt103 cycles) the following applies
sdot+
minussdotsdotsdot= o90
07at
06a
zt
cosSσ019HV
Hst
z
HstHHR
applicable to 06at
Hst
z ge
56300ρSσa cHHR
Hstsdotsdot
=
In the case of insufficient specified hardness depths the same procedure for determination of the actual safety factor as above applies
For limited life fatigue (103 lt cycles lt 3106 )
For this purpose it is necessary to extend the correction of safety factors to include also higher values than required Ie in the case of more than sufficient hardness and depths the safety factor in the formulae for both high cycle fatigue and static strength are to be increased until necessary and speci-fied values balance
The actual safety factor for a given number of cycles N be-tween 103 and 3106 is found by linear interpolation in a dou-ble logarithmic diagram
minussdotminus
= sdot logN3477
logSlogSlogS
310H6103HHN
36 10H103H Slog86281Slog86280 sdot+sdot sdot
22 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
3 Calculation of Tooth Strength
31 Scope and General Remarks Part 3 include the calculation of tooth root strength as limited by tooth root cracking (surface or subsurface initiated) and yielding
For rim thickness sR ge 35middotmn the strength is calculated by means of 32 ndash 313 For cylindrical gears the calculation is based on the assumption that the highest tooth root tensile stress arises by application of the force at the outer point of single tooth pair contact of the virtual spur gears The method has however a few limitations that are mentioned in 36
For bevel gears the calculation is based on force application at the tooth tip of the virtual cylindrical gear Subsequently the stress is converted to load application at the mid point of the flank due to the heightwise crowning
Bevel gears may also be calculated with the program BECAL In that case KA and Kv are to be included in the applied tooth force but not KFβ and KFα
In case of a thin annulus or a thin gear rim etc radial crack-ing can occur rather than tangential cracking (from root fillet to root fillet) Cracking can also start from the compression fillet rather than the tension fillet For rim thickness sR lt 35middotmn a special calculation procedure is given in 315 and 316 and a simplified procedure in 314
A tooth breakage is often the end of the life of a gear trans-mission Therefore a high safety SF against breakage is re-quired
It should be noted that this part 3 does not cover fractures caused by
bull oil holes in the tooth root space bull wear steps on the flank bull flank surface distress such as pits spalls or grey staining
Especially the latter is known to cause oblique fractures starting from the active flank predominately in spiral bevel gears but also sometimes in cylindrical gears
Specific calculation methods for these purposes are not given here but are under consideration for future revisions Thus depending on experience with similar gear designs limita-tions other than those outlined in part 3 may be applied
32 Tooth Root Stresses The local tooth root stress is defined as the max principal stress in the tooth root caused by application of the tooth force Ie the stress ratio R = 0 Other stress ratios such as for eg idler gears (R asymp -12) shrunk on gear rims (R gt 0) etc are considered by correcting the permissible stress level
321 Local tooth root stress The local tooth root stress for pinion and wheel may be as-sessed by strain gauge measurements or FE calculations or similar For both measurements and calculations all details are to be agreed in advance
Normally the stresses for pinion and wheel are calculated as
Cylindrical gears
FαFβvγAβSFn
tF K K K K K Y Y Y
m bFσ =
where
YF = Tooth form factor (see 33)
YS = Stress correction factor (see 34)
Yβ = Helix angle factor (see 36)
Ft KA Kγ Kv KFβ KFα see 15 ndash 110
b see 13
Bevel gears
FαFβvγASaFamn
mtF K K K K K Y Y Y
m bFσ ε=
where
YFa = Tooth form factor see 33
YSa = Stress correction factor see 34
Yε = Contact ratio factor see 35
Fmt KA etc see 15 ndash 110
b see 13
322 Permissible tooth root stress The permissible local tooth root stress for pinion respectively wheel for a given number of cycles N is
CXRrelTrelTF
NMFEFP YYYY
SYY
δσ
=σ
Note that all these factors YM etc are applicable to 3middot106 cycles when used in this formula for σFP The influence of other number of cycles on these factors is covered by the calculation of YN
where
σFE = Local tooth root bending endurance limit of reference test gear (see 37)
YM = Mean stress influence factor which accounts for other loads than constant load direction eg idler gears temporary change of load di-rection pre-stress due to shrinkage etc (see 38)
YN = Life factor for tooth root stresses related to reference test gear dimensions (see 39)
SF = Required safety factor according to the rules
YδrelT = Relative notch sensitivity factor of the gear to be determined related to the reference test gear (see 310)
YRrelT = Relative (root fillet) surface condition factor of the gear to be determined related to the reference test gear (see 311)
Classification Notes- No 412 23 May 2003
DET NORSKE VERITAS
YX = Size factor (see 312)
YC = Case depth factor (see 313)
33 Tooth Form Factors YF YFa The tooth form factors YF and YFa take into account the in-fluence of the tooth form on the nominal bending stress
YF applies to load application at the outer point of single tooth pair contact of the virtual spur gear pair and is used for cylindrical gears
YFa applies to load application at the tooth tip and is used for bevel gears
Both YF and YFa are based on the distance between the con-tact points of the 30˚-tangents at the root fillet of the tooth profile for external gears respectively 60˚ tangents for inter-nal gears
Figure 31 External tooth in normal section
Figure 32 Internal tooth in normal section
Definitions
n
2
n
Fn
enFn
Fe
F
α cosms
α cosmh6
Y
=
n
2
n
Fn
anFn
Fa
Fa
α cosms
α cosmh6
Y
=
In the case of helical gears YF and YFa are determined in the normal section ie for a virtual number of teeth
YFa differs from YF by the bending moment arm hFa and αFan and can be determined by the same procedure as YF with exception of hFe and αFan For hFa and αFan all indices e will change to a (tip)
The following formulae apply to cylindrical gears but may also be used for bevel gears when replacing
mn with mnm
zn with zvn
αt with αvt
β with βm
with undercut without undercut
Fig 33 Dimensions and basic rack profile of the teeth (finished profile)
Tool and basic rack data such as hfP ρfp and spr etc are re-ferred to mn ie dimensionless
331 Determination of parameters
( )n
n
prnfPnfP m
α cossαsin 1ρ
αtan h4πE
minusminusminusminus=
For external gears fPfP ρρ =
For internal gears ( )0z
195fPfP0
fPfP 10363156ρhxρρ
sdotminus+
+=
where
z0 = number of teeth of pinion cutter
x0 = addendum modification coefficient of pinion cutter
hfP = addendum of pinion cutter
ρfP = tip radius of pinion cutter
xhρG fpfp +minus=
24 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
τmE
2π
z2H
nnminus
minus=
with
3πτ = for external gears
6πτ = for internal gears
Htan zG2
nminusϑ=ϑ
(to be solved iteratively suitable start value6π
=ϑ for exter-
nal gears and 3π for internal gears)
a) Tooth root chord sFn For external gears
minus
ϑ+
ϑminus= ρ
cosG3
3πsinz
ms
fpnn
Fn
For bevel gears with a tooth thickness modification
xsm affects mainly sFn but also hFe and αFen The total influence of xsm on YFa Ysa can be approximated by only adding 2 xsm to sFn mn
For internal gears
minus
ϑ+
ϑminus= ρ
cosG
6πsinz
ms
fPnn
Fn
b) Root fillet radius ρF at 30ordm tangent
( )G2coszcosG2ρ
mρ
2n
2
fpn
F
minusϑϑ+=
c) Determination of bending moment arm hF dn = zn mn
b2α
αn βcosε
ε =
dan = dn + 2 ha
pbn = π mn cos αn
dbn = dn cos αn
( )4
d1εpzz
2dd
zz2d
2bn
2
αnbn
2bn
2an
en +
minusminus
minus=
en
bnen d
dcos arcα =
ennnn
e α invα invα x tan 22π
z1
minus+
+=γ
αFen = αen ndash γe
For external gears
( )
minus=
n
enFenee
n
Fe
mdαtan sin γ γcos
21
mh
]ρcos
G3πcosz fpn +
ϑminus
ϑminusminus
For internal gears
( ) minus
sdotsdotminus=
n
enFenee
n
Fe
mdα tanγ sinγ cos
21
mh
minus
ϑminus
ϑminussdot ρ
cosG3
6πcosz fPn
332 Gearing with εαn gt 2 For deep tooth form gearing ( )25ε2 αn lele produced with a verified grade of accuracy of 4 or better and with applied profile modification to obtain a trapezoidal load distribution along the path of contact the YF may be corrected by the factor YDT as
250ε205for 0666ε2366Y αnαnDT leleminus=
205εfor 10Y αnDT lt=
34 Stress Correction Factors YS YSa The stress correction factors YS and YSa take into account the conversion of the nominal bending stress to the local tooth root stress Thereby YS and YSa cover the stress increasing effect of the notch (fillet) and the fact that not only bending stresses arise at the root A part of the local stress is inde-pendent of the bending moment arm This part increases the more the decisive point of load application approaches the critical tooth root section
Therefore in addition to its dependence on the notch radius the stress correction is also dependent on the position of the load application ie the size of the bending moment arm
YS applies to the load application at the outer point of single tooth pair contact YSa to the load application at tooth tip
YS can be determined as follows
( )
+
+=L23121
1
sS qL01312Y
where Fe
Fn
hsL = and
F
Fns ρ2
sq = (see 33)
YSa can be calculated by replacing hFe with hFa in the above formulae
Note a) Range of validity 1 lt qs lt 8
In case of sharper root radii (ie produced with tools having too sharp tip radii) YS resp YSa must be specially considered
b) In case of grinding notches (due to insufficient protuberance of the hob) YS resp YSa can rise considerably and must be multiplied with
Classification Notes- No 412 25 May 2003
DET NORSKE VERITAS
g
g
ρt
0613
13
minus
where
tg = depth of the grinding notch
ρg = radius of the grinding notch
c) The formulae for YS resp YSa are only valid for αn = 20˚ However the same formulae can be used as a safe approximation for other pressure angles
35 Contact Ratio Factor Yε The contact ratio factor Yε covers the conversion from load application at the tooth tip to the load application at the mid point of the flank (heightwise) for bevel gears
The following may be used
Yε = 0625
36 Helix Angle Factor Yβ The helix angle factor Yβ takes into account the difference between the helical gear and the virtual spur gear in the nor-mal section on which the calculation is based in the first step In this way it is accounted for that the conditions for tooth root stresses are more favourable because the lines of contact are sloping over the flank
The following may be used (β input in degrees)
Yβ = 1 ndash εβ β120
When εβ gt 1 use εβ = 1 and when β gt 30deg use β = 30deg in the formula
However the above equation for Yβ may only be used for gears with β gt 25deg if adequate tip relief is applied to both pinion and wheel (adequate = at least 05 middot Ceff see 432)
37 Values of Endurance Limit σFE σFE is the local tooth root stress (max principal) which the material can endure permanently with 99 survival prob-ability 3106 load cycles is regarded as the beginning of the endurance limit or the lower knee of the σ ndash N curve σFE is defined as the unidirectional pulsating stress with a minimum stress of zero (disregarding residual stresses due to heat treatment) Other stress conditions such as alternating or pre-stressed etc are covered by the conversion factor YM
σFE can be found by pulsating tests or gear running tests for any material in any condition If the approval of the gear is to be based on the results of such tests all details on the testing conditions have to be approved by the Society Fur-ther the tests may have to be made under the Societys su-pervision
If no fatigue tests are available the following listed values for σFE may be used for materials subjected to a quality con-trol as the one referred to in the rules
σFE
Alloyed case hardened steels 1) (fillet surface hardness 58 ndash 63 HRC)
bull of specially approved high grade
1050
bull of normal grade
minus CrNiMo steels with approved process
1000
minus CrNi and CrNiMo steels generally 920 minus MnCr steels generally 850
Nitriding steel of approved grade quenched tempered and gas nitrided (surface hardness 700 ndash 800 HV)
840
Alloyed quenched and tempered steel bath or gas nitrided (surface hardness 500 ndash 700 HV)
720
Alloyed quenched and tempered steel flame or induction hardened 2) (incl entire root fillet) (fillet surface hardness 500 ndash 650 HV)
07 HV + 300
Alloyed quenched and tempered steel flame or induction hardened (excl entire root fillet) (σB = uts of base material)
025 σB + 125
Alloyed quenched and tempered steel 04 σB + 200
Carbon steel 025 σB + 250
Note All numbers given above are valid for separate forgings and for blanks cut from bars forged according to a qualified procedure see Pt 4 Ch 2 Sec 3 For rolled steel the values are to be reduced with 10 For blanks cut from forged bars that are not qualified as mentioned above the values are to be reduced with 20 For cast steel reduce with 40
1) These values are valid for a root radius
bull being unground If however any grinding is made in the root fillet area in such a way that the residual stresses may be affected σFE is to be reduced by 20 (If the grinding also leaves a notch see 34)
bull with fillet surface hardness 58 ndash 63 HRC In case of lower surface hardness than 58 HRC σFE is to be reduced with 20(58 ndash HRC) where HRC is the detected hardness (This may lead to a permissible tooth root stress that varies along the facewidth If so the actual tooth root stresses may also be considered along facewidth)
bull not being shot peened In case of approved shot peening σFE may be increased by 200 for gears where σFE is reduced by 20 due to root grinding Otherwise σFE may be increased by 100 for
6nm le and 100 ndash 5 (mn - 6) for mn gt 6 However the possible adverse influence on the flanks regarding grey staining should be considered and if necessary the flanks should be masked
2) The fillet is not to be ground after surface hardening Regarding possible root grinding see 1)
26 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
38 Mean stress influence Factor YM The mean stress influence factor YM takes into account the influence of other working stress conditions than pure pulsa-tions (R = 0) such as eg load reversals idler gears planets and shrink-fitted gears
YM (YMst) are defined as the ratio between the endurance (or static) strength with a stress ratio R ne 0 and the endurance (or static) strength with R = 0
YM and YMst apply only to a calculation method that assesses the positive (tensile) stresses and is therefore suitable for comparison between the calculated (positive) working stress σF and the permissible stress σFP calculated with YM or YMst
For thin rings (annulus) in epicyclic gears where the com-pression fillet may be decisive special considerations apply see 316
The following method may be used within a stress ratio ndash12 lt R lt 05
381 For idlers planets and PTO with ice class
M1M1R1
1Yor Y MstM
+minus
minus=
where
R = stress ratio = min stress divided by max stress
For designs with the same force applied on both forward- and back-flank R may be assumed to ndash 12
For designs with considerably different forces on forward- and back-flank such as eg a marine propulsion wheel with a power take off pinion R may be assessed as
branchmain theoffacewidth unit per forcepto offacewidt unit per force21minus
For a power take off (PTO) with ice class see 161 c
M considers the mean stress influence on the endurance (or static) strength amplitudes
M is defined as the reduction of the endurance strength am-plitude for a certain increase of the mean stress divided by that increase of the mean stress
Following M values may be used
Endurance limit
Static strength
Case hardened 08 ndash 015 Ys 1) 07
If shot peened 04 06
Nitrided 03 03
Induction or flame hardened
04 06
Not surface hardened steel 03 05
Cast steels 04 06 1)For bevel gears use Ys=2 for determination of M
The listed M values for the endurance limit are independent of the fillet shape (Ys) except for case hardening In princi-ple there is a dependency but wide variations usually only occur for case hardening eg smooth semicircular fillets versus grinding notches
382 For gears with periodical change of rotational direction For case hardened gears with full load applied periodically in both directions such as side thrusters the same formula for YM as for idlers (with R = ndash 12) may be used together with the M values for endurance limit This simplified approach is valid when the number of changes of direction exceeds 100 and the total number of load cycles exceeds 3middot106
For gears of other materials YM will normally be higher than for a pure idler provided the number of changes of direction is below 3middot106 A linear interpolation in a diagram with loga-rithmic number of changes of direction may be used ie from YM = 09 with one change to YM (idler) for 3middot106 changes This is applicable to YM for endurance limit For static strength use YM as for idlers
For gears with occasional full load in reversed direction such as the main wheel in a reversing gear box YM = 09 may be used
383 For gears with shrinkage stresses and unidirectional load For endurance strength
FE
fitM σ
σM1
M21Y+
minus=
σFE is the endurance limit for R = 0
For static strength YMst = 1 and σfit accounted for in 39b
σfit is the shrinkage stress in the fillet (30˚ tangent) and may be found by multiplying the nominal tangential (hoop) stress with a stress concentration factor
n
Ffit m
ρ215scf minus=
384 For shrink-fitted idlers and planets When combined conditions apply such as idlers with shrink-age stresses the design factor for endurance strength is
( ) ( ) FE
fitM σ
σR1M1
M2
M1M1R1
1Yminussdot+
minus
+minus
minus=
Symbols as above but note that the stress ratio R in this par-ticular connection should disregard the influence of σfit ie R normally equal ndash 12
For static strength
M1M1R1
1YMst
+minus
minus=
The effect of σfit is accounted for in 39 b
Classification Notes- No 412 27 May 2003
DET NORSKE VERITAS
385 Additional requirements for peak loads The total stress range (σmax ndash σmin) in a tooth root fillet is not to exceed
F
y
Sσ225
for not surface hardened fillets
FSHV5 for surface hardened fillets
39 Life Factor YN The life factor YN takes into account that in the case of limited life (number of cycles) a higher tooth root stress can be permitted and that lower stresses may apply for very high number of cycles
Decisive for the strength at limited life is the σ ndash N ndash curve of the respective material for given hardening module fillet radius roughness in the tooth root etc Ie the factors YδrelT YRelT YX and YM have an influence on YN
If no σ ndash N ndash curve for the actual material and hardening etc is available the following method may be used
Determination of the σ ndash N ndash curve
a) Calculate the permissible stress σFP for the beginning of the endurance limit (3middot106 cycles) including the influ-ence of all relevant factors as SF YδrelT YRelT YX YM and YC ie σFP = σFE middotYM middotYδrelT middotYRelT middotYX middotYC SF
b) Calculate the permissible laquostaticraquo stress (le 103 load cy-cles) including the influence of all relevant factors as SFst YδrelTst YMst and YCst ( )fitσCstYδrelTstYMstYFstσ
FstS1
FPstσ minussdotsdotsdot=
where σFst is the local tooth root stress which the material can resist without cracking (surface hardened materials) or unacceptable deformation (not surface hardened materials) with 99 survival probability
σFst
Alloyed case hardened steel 1) 2300
Nitriding steel quenched tempered and gas nitrided (surface hardness 700 ndash 800 HV)
1250
Alloyed quenched and tempered steel bath or gas nitrided (surface hardness 500 ndash 700 HV)
1050
Alloyed quenched and tempered steel flame or induction hardened (fillet surface hardness 500 ndash 650 HV)
18 HV + 800
Steel with not surface hardened fillets the smaller value of 2)
18 σB or 225 σy
1) This is valid for a fillet surface hardness of 58 ndash 63 HRC In case of lower fillet surface hardness than 58 HRC σFst is to be reduced with 30(58 ndash HRC) where HRC is the actual hardness Shot peening or grinding notches are not considered to have any significant influence on σFst 2) Actual stresses exceeding the yield point (σy or σ02) will alter the residual stresses locally in the ldquotensionrdquo fillet re-spectively ldquocompressionrdquo fillet This is only to be utilised for gears that are not later loaded with a high number of cycles at lower loads that could cause fatigue in the ldquocompressionrdquo fillet
c) Calculate YN as
NL gt 3middot106
YN = 1 or 001
L
6
N N103Y
sdot= ie Yn = 092 for 1010
The YN = 1 from 3middot106 on may only be used when special material cleanness applies see rules Pt4 Ch2
103ltNLlt3middot106exp
L
6
N N103Y
sdot=
cycles6103forσcycles310forσlog02876exp
FP
FPst
sdot=
NL lt 103 cycles6103forσcycles310forσ
NYFP
FPst
sdot=
or simply use σFPst as mentioned in b) directly
Guidance on number of load cycles NL for various applica-tions
bull For propulsion purpose normally NL = 1010 at full load (yachts etc may have lower values)
bull For auxiliary gears driving generators that normally operate with 70-90 of rated power NL = 108 with rated power may be applied
28 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
310 Relative Notch Sensitivity Factor YδrelT The dynamic (respectively static) relative notch sensitivity factor YδrelT (YδrelTst) indicate to which extent the theoreti-cally concentrated stress lies above the endurance limits (re-spectively static strengths) in the case of fatigue (respectively overload) breakage
YδrelT is a function of the material and the relative stress gra-dient It differs for static strength and endurance limit
The following method may be used
For endurance limit
for not surface hardened fillets
( )
024
s024
relT σ103133q21σ1012201351
Ysdotsdotminus
+sdotsdotminus+= minus
minus
δ
for all surface hardened fillets except nitrided
( )
106q21002451
Y srelT
++=δ
for nitrided fillets
( )
1347q2101421
Y srelT
++=δ
For static strength
for not surface hardened fillets1)
( )( )( )025
02
02502s
relTst σ3000821σ300 1Y0821Y
+
minus+=δ
for surface hardened fillets except nitrided
YδrelTst = 044 YS + 012
for nitrided fillets
YδrelTst = 06 + 02 YS 1) These values are only valid if the local stresses do not
exceed the yield point and thereby alter the residual stress level See also 39b footnote 2
311 Relative Surface Condition Factor YRrelT The relative surface condition factor YRrelT takes into ac-count the dependence of the tooth root strength on the sur-face condition in the tooth root fillet mainly the dependence on the peak to valley surface roughness
YRrelT differs for endurance limit and static strength
The following method may be used
For endurance limit
YRrelT = 1675 ndash 053 (Ry + 1)01 for surface hardened steels and alloyed quenched and tempered steels except nitrided
YRrelT = 53 ndash 42 (Ry + 1)001 for carbon steels
YδrelT = 43 ndash 326 (Ry + 1)0005
for nitrided steels
For static strength
YRrelTst = 1 for all Ry and all materials
For a fillet without any longitudinal machining trace Ry asymp Rz
312 Size Factor YX The size factor YX takes into account the decrease of the strength with increasing size YX differs for endurance limit and static strength
The following may be used
For endurance limit
YX = 1 for mn le 5 generally
YX = 103 ndash 0006 mn for 5 lt mn lt 30
YX = 085 for mn ge 30
for not surface hard-ened steels
YX = 105 ndash 001 mn for 5 lt mn ge 25
YX = 08 for mn ge 25
for surface hardened steels
For static strength
YXst = 1 for all mn and all materials
313 Case Depth Factor YC The case depth factor YC takes into account the influence of hardening depth on tooth root strength
YC applies only to surface hardened tooth roots and is dif-ferent for endurance limit and static strength
In case of insufficient hardening depth fatigue cracks can develop in the transition zone between the hardened layer and the core For static strength yielding shall not occur in the transition zone as this would alter the surface residual stresses and therewith also the fatigue strength
The major parameters are case depth stress gradient permis-sible surface respectively subsurface stresses and subsurface residual stresses
The following simplified method for YC may be used
YC consists of a ratio between permissible subsurface stress (incl influence of expected residual stresses) and permissible surface stress This ratio is multiplied with a bracket con-taining the influence of case depth and stress gradient (The empirical numbers in the bracket are based on a high number of teeth and are somewhat on the safe side for low number of teeth)
YC and YCst may be calculated as given below but calculated values above 10 are to be put equal 10
For endurance limit
+
+=nFFE
C m02ρt31
σconstY
Classification Notes- No 412 29 May 2003
DET NORSKE VERITAS
For static strength
+
+=nFFst
Cst m02ρt31
σconstY
where const and t are connected as
Hardening process
t = endurance limit const =
static strength const =
t550 640 1900
t400 500 1200 Case hard-ening
t300 380 800
Nitriding t400 500 1200
Induction- or flame hardening
tHVmin 11 HVmin 25 HVmin
For symbols see 213
In addition to these requirements to minimum case depths for endurance limit some upper limitations apply to case hard-ened gears
The max depth to 550 HV should not exceed
1) 13 of the top land thickness san unless adequate tip relief is applied (see 110)
2) 025 mn If this is exceeded the following applies addi-tionally in connection with endurance limit
minusminus= 025
mt
1Yn
max550C
314 Thin rim factor YB Where the rim thickness is not sufficient to provide full sup-port for the tooth root the location of a bending failure may be through the gear rim rather than from root fillet to root fillet
YB is not a factor used to convert calculated root stresses at the 30deg tangent to actual stresses in a thin rim tension fillet Actually the compression fillet can be more susceptible to fatigue
YB is a simplified empirical factor used to de-rate thin rim gears (external as well as internal) when no detailed calcula-tion of stresses in both tension and compression fillets are available
Figure 34 Examples on thin rims
YB is applicable in the range 175 lt sRmn lt 35
YB = 115 middot ln (8324 middot mnsR)
(for sRmn ge 35 YB = 1)
(for sRmn le 175 use 315)
σF as calculated in 321 is to be multiplied with YB when sRmn lt 35 Thus YB is used for both high and low cycle fatigue
Note This method is considered to be on the safe side for external gear rims However for internal gear rims without any flange or web stiffeners the method may not be on the safe side and it is advised to check with the method in 315316
315 Stresses in Thin Rims For rim thickness sR lt 35 mn the safety against rim cracking has to be checked
The following method may be used
3151 General The stresses in the standardised 30ordm tangent section tension side are slightly reduced due to decreasing stress correction factor with decreasing relative rim thickness sRmn On the other hand during the complete stress cycle of that fillet a certain amount of compression stresses are also introduced The complete stress range remains approximately constant Therefore the standardised calculation of stresses at the 30ordm tangent may be retained for thin rims as one of the necessary criteria
The maximum stress range for thin rims usually occurs at the 60ordm ndash 80ordm tangents both for laquotensionraquo and laquocompressionraquo side The following method assumes the 75ordm tangent to be the decisive Therefore in addition to the am criterion ap-plied at the 30ordm tangent it is necessary to evaluate the max and min stresses at the 75ordm tangent for both laquotensionraquo (loaded flank) fillet and laquocompressionraquo (back-flank) fillet For this purpose the whole stress cycle of each fillet should be considered but usually the following simplification is justified
Figure 35 Nomenclature of fillets
Index laquoTraquo means laquotensileraquo fillet laquoCraquo means laquocompressionraquo fillet
σFTmin and σFCmax are determined on basis of the nominal rim stresses times the stress concentration factor Y75
σFCmin and σFTmax are determined on basis of superposition of nominal rim stresses times Y75 plus the tooth bending stresses at 75ordm tangent
30 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
3152 Stress concentration factors at the 75ordm tangents The nominal rim stress consists of bending stresses due to local bending moments tangential stresses due to the tan-gential force Ft and radial shear stresses due to Fr
The major influence is given by the bending stresses The influence of the tangential stresses is minor and even though its stress concentration factor is slightly higher than for bending it is considered to be safe enough when the sum of these nominal stresses are combined with the stress concen-tration factor for bending The influence of the radial shear stress is neglected
The stress concentration factor relating nominal rim stresses to local fillet stresses at the 75ordm tangent may be calculated as
75n
R75
n
R
75
ρmsρ185
ms3
Y+
sdot=
where ρ75 is the root radius at the 75ordm tangent ref to mn Usually ρ75 is closed to the tool radius ρao and
ρ75 = ρao
is a safe approximation compensating for the am simplifi-cations to the laquounsaferaquo side
The tooth root stresses of the loaded tooth are decreasing with decreasing relative rim thickness approximately with the empirical factor
15
R
ncorr s
m311Y
minus=
3153 Nominal rim stresses The bending moment applied to the rim consists of a part of the tooth tilt moment Ft (hF + 05 sR) and the bending caused by the radial force Fr
The sectional modulus (first moment of area) which is used for determination of the nominal bending stresses is not nec-essarily the same for the 2 am bending moments If flanges webs etc outside the toothed section contribute to stiffening the rim against various deflections the influence of these stiffeners should be considered Eg an end flange will have an almost negligible influence on the effective sectional modulus for the stresses due to tooth tilt as the deflection caused by the tilt moment is rather small and would not much involve the flange On the other hand the radial forces as for instance from the meshes in an annulus would cause considerable radial deflections that the flange might restrict to a substantial amount When considering the stiff-ening of such flanges or webs on basis of simplified models it is advised to use an effective rim thickness sR = sR + 02 mn for the first moment of area of the rim (toothed part) cross section
For a high number of rim teeth it may be assumed that the rim bending moments in the fillets adjacent to the loaded tooth are of the same magnitude as right under the applied
force This assumption is reasonable for an annulus but rather much on the safe side for a hollow pinion
The influence of Ft on the nominal tangential stress is simpli-fied by half of it for compressive stresses (σ1) and the other half for tensile stresses (σ2) Applying these assumptions the nominal rim stresses adjacent to the loaded tooth are
( ) ( )A2
FW
f RFW
s05hF05 t
R
r
T
RFt1 minus
ϑminus
+minus=σ
( ) ( )A2
FW
f RFW
s05hF05 t
R
r
T
RFt2 +
ϑminus
+=σ
where σ1 σ2 see Fig 35
A = minimum area of cross section (usually bR sR)
WT = the sectional modulus of rim with respect to tooth tilt moment (usually 6sb 2
rR )
WR = the sectional modulus of the rim including the influence of stiffeners as flanges webs etc (WR ge WT)
bR = the width of the rim
R = the radius of the neutral axis in the rim ie from wheel centre to midpoint of rim
( )ϑf = a function for bending moment distribution around the rim For a rim (pinion) with one mesh only the ( )ϑf at the position of load application is 024 For an annulus with 3 or more meshes ( )ϑf at the position of each load application is approx 3 planets ( )ϑf = 019 4 planets ( )ϑf = 014 5 planets ( )ϑf = 011 6 planets ( )ϑf = 009
It must be checked if the max (tensile) stress in the compres-sion fillet really occurs when the fillet is adjacent to the loaded tooth In principle the stress variation through a complete rotation should be considered and the max value used The max value is usually never less than 0 For an annulus eg the tilt moment is zero in the mid position be-tween the planet meshes whilst the bending moment due to the radial forces is half of that at the mesh but with opposite sign
If these formulae are applied to idler gears as eg planets the influence of nominal tangential stresses must be cor-rected by deleting Ft(2 A) for σ1 and using FtA for σ2 Further the influence of Fr on the nominal bending stresses is usually negligible due to the planet bearing support
3154 Root fillet stresses Determination of min and max stresses in the laquotensionraquo fillet
Classification Notes- No 412 31 May 2003
DET NORSKE VERITAS
Minimum stress
751FTmin YσK σ =
Maximum stress
corrF752 Yσ03Yσ KσFTmax +=
where
03 is an empirical factor relating the tension stresses (σF) at the 30ordm tangent to the part of the tension stresses at the 75ordm tangent which add to the rim related stresses (03 also takes into account that full superposition of nominal stresses times stress concentration factors from both laquosides of the corner filletraquo would result in too high stresses)
αβγ sdotsdotsdotsdot= FFvA KKKKKK
Determination of min and max stresses in the laquocompres-sionraquo fillet
Minimum stress
corrF751FCmin Yσ036Yσ Kσ minus=
Maximum stress
752FCmax Yσ Kσ =
where
036 is an empirical factor relating the tension stresses (σF) at the 30ordm tangent to the part of the compression stresses at the 75ordm tangent which add to the rim related stresses
For gears with reversed loads as idler gears and planets there is no distinct laquotensionraquo or laquocompressionraquo fillet The mini-mum stress σFmin is the minimum of σFTmin and σFCmin (usually the latter is decisive) The maximum stress σFTmax is the maximum of σFTmax and σFCmax (usually the former is deci-sive)
316 Permissible Stresses in Thin Rims
3161 General The safety against fatigue fracture respectively overload fracture is to be at least at the same level as for solid gears The laquoordinaryraquo criteria at the 30ordm tangent apply as given in 31 through 313
Additionally the following criteria at the 75ordm tangent may apply
3162 For gt3middot106 cycles The permissible stresses for the laquotensionraquo fillets and for the laquocompressionraquo fillets are determined by means of a relevant fatigue diagram
If the actual tooth root stress (tensile or compressive) ex-ceeds the yield strength to the material the induced residual stresses are to be taken into account
For determination of permissible stresses the following is defined
R = stress ratio ie FTmax
FTminσσ respectively
FCmax
FCmin
σσ
∆σ = stress range ie σFTmax ndash σFTmin resp σFCmax ndash σFCmin
(For idler gears and planets Fmax
FminσσR =
and ∆σ = σFmax minus σFmin)
The permissible stress range ∆σρ for the laquotensionraquo respec-tively laquocompressionraquo fillets can be calculated as
For R gt minus1 FPp σ
R1R1031
13∆σ
minus+
+=
For minus infin lt R lt minus1 FPp σ
R1R10151
13∆σ
minus+
+=
where
σFP = see 32 determined for unidirectional stresses (YM = 1)
If the yield strength σy is exceeded in either tension or com-pression residual stresses are induced This may be consid-ered by correcting the stress ratio R for the respective fillets (tension or compression)
Eg if yFCmin σσ gt (ie exceeded in compression) the
difference yFCmin σσ∆ minus= affects the stress ratio as
∆σ
σ∆σ∆σR
FCmax
y
FCmax
FCmin
+
minus=
++
=
Similarly the stress ratio in the tension fillet may require cor-rection
If the yield strength is exceeded in tension σFTmax gt σy the difference ∆ = σFTmax ndash σy affects the stress ratio as
y
FTmin
FTmax
FTmin
σ∆σ
∆σ∆σR minus=
minusminus
=
Checking for possible exceeding of the yield strength has to be made with the highest torque and with the KAP if this ex-ceeds KA
3163 Forle 103 cycles The permissible stress range ∆σp is not to exceed
For R gt minus1 FPstpst σ
R1R1051
15∆σ
minus+
+=
For ndash infin lt R lt minus1 FPstpst σ
R1R10251
15∆σ
minus+
+=
For all values of R ∆σpst is limited by
not surface hardened F
y
Sσ225
32 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
surface hardened CF
YSHV5
Definition of ∆σ and R see 3162 with particular attention to possible correction of R if the yield strength is exceeded
σFPst see 32 determined for unidirectional stresses (YM = 1) and lt 103 cycles
3164 For 103 lt cycles lt 3106 ∆σp is to be determined by linear interpolation a log-log dia-gram
∆σp at NL load cycles is
6L 103p
exp
L
6
N p ∆σN103∆σ sdot
sdot=
6
3
103p
10p
∆σ
∆σlog28760exp
sdot
=
Classification Notes- No 412 33 May 2003
DET NORSKE VERITAS
4 Calculation of Scuffing Load Capacity
41 Introduction High surface temperatures due to high loads and sliding ve-locities can cause lubricant films to break down This sei-zure or welding together of areas of tooth surface is termed scuffing
In contrast to pitting and fatigue breakage which show a dis-tinct incubation period a single short overloading can lead to a scuffing failure In the ISO-TR13989 two criteria are mentioned The method used in this Classification Note is based on the principles of the flash temperature criterion
Note Bulk temperature in excess of 120ordmC for long periods may have an adverse effect on the surface durability see 211
42 General Criteria In no point along the path of contact the local contact tem-perature may exceed the permissible temperature ie
oilS
oilSB S
ϑ+ϑminusϑ
leϑ
50SB minusϑleϑ
where
Bϑ = max contact temperature along the path of contact
maxflaMBB ϑ+ϑ=ϑ
MBϑ = bulk temperature see 434
maxflaϑ = max flash temperature along the path of con-tact see 44
Sϑ = scuffing temperature see below
oilϑ = oil temperature before it reaches the mesh (max applicable for the actual load case to be used ie normally alarm temperature except for ice classes where a max expected tem-perature applies)
SS = required safety factor according to the Rules
The scuffing temperature Sϑ may be calculated as
L2
wrelT
002
40S XFZGX
ν100112085780
sdot
sdot++=ϑ
where
XwrelT = relative welding factor
XwrelT
Through hardened steel 10
Phosphated steel 125
Copper-plated steel 150
Nitrided steel 150
Less than 10 retained austenite
115
10 ndash 20 retained austenite 10
Casehardened steel
20 ndash 30 retained austenite 085
Austenitic steel 045
FZG = load stage according to FZG-Test A8390 (Note This is the load stage where scuffing oc-curs However due to scatter in test results cal-culations are to be made with one load stage less than the specification)
XL = lubricant factor
= 10 for mineral oils
= 08 for polyalfaolefins
= 07 for non-water-soluble polyglycols
= 06 for water-soluble polyglycols
= 15 for traction fluids
= 13 for phosphate esters
ν40 = kinematic oil viscosity at 40˚C (mm2s)
Application of other test methods such as the Ryder the FZG-Ryder R46574 and the FZG L-42 Test 141195110 may be specially considered
For high speed gears with very short time of contact Sϑ may be increased as follows provided use of EP-oils
Addition to the calculated scuffing temperature Sϑ
If micros 18tc ge no addition
If micros 18tc lt add ( )cwrelT t18X18 minussdotsdot
where
ct = contact time ( micros ) which is the time needed to cross the Hertzian contact width
( ) [ ]microsu1cosβn
uσ340tb1
Hc +sdotsdot
sdot=
σH as calculated in 221
For bevel gears use vu in stead of u
34 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
43 Influence Factors
431 Coefficient of friction The following coefficient of friction may apply
L025a
005oil
02
redC ΣC
Bt X R ηρv
w0048micro minus
=
where
wBt = specific tooth load (Nmm)
vΣC = sum of tangential velocities at pitch point At pitch line velocities gt 50 ms the limiting value of vΣC at v = 50 ms is to be used
ρredC = relative radius of curvature (transversal plane) at the pitch point
Cylindrical gears
HαHβvγAbt
Bt KKKKKb
Fw sdotsdotsdotsdotsdot= (see 1)
twΣC αsin v2v =
bCredC β cos ρρ =
Bevel gears
HαHβvγAtmb
Bt KKKKKb
Fw sdotsdotsdotsdotsdot= (see 1)
vtmtΣC αsin v2v =
bmvCredC β cos ρρ =
ηoil = dynamic viscosity (mPa s) at oilϑ calculated as
1000
ρ νη oiloil =
where ρ in kgm3 approximated as
( ) 70 15oil15 minusϑminusρ=ρ and νoil is kinematic viscosity at oilϑ and may be calculated by means of the following equation
( ) ( )++=+ 08νloglog08νloglog 100oil ( )
sdotminus
ϑ+minus313log373log
273log373log oil
( ) ( )( )08νloglog08νloglog 10040 +minus+
Ra = composite arithmetic mean roughness (micron) of pinion an wheel calculated as ( )2a1aa RR 05R +=This is defined as the roughness on the new flanks ie as manufactured
XL = see 42
432 Effective tip relief Ceff Ceff is the effective tip relief that amount of tip relief which compensates for the elastic deformation of the gear mesh ie
zero load at the tooth tip It is assumed (simplified) to be equal for both pinion and wheel
Cylindrical gears
for helical
γ
γ=cb
KKFC Abt
eff (see 1)
For spur
cbKKF
C Abteff
γ= (see 1)
Alternatively for spur and helical gears the non-linear ap-proach in 111 may be used (taking Ceff=δ)
Bevel gears
bcKFC Ambt
effγ
= (see 1)
where
γ
α
ε2ε44c
+sdot
=γ
433 Tip relief and extension Cylindrical gears
The extension of the tip relief is not to result in an effective contact ratio 1ltεα when the gear is unloaded (exceptions to this may only apply for applications where the gear is not to run at light loads) This means that the unrelieved part of the path of contact is to be minimum pbt It is further assumed that this unrelieved part is placed centrally on the path of contact
If root relief applies it has to be calculated as equivalent tip relief Ie pinion root relief (at mesh position A) is added to Ca2 and wheel root relief (at mesh position E) is added to Ca1 If no design tip relief or root relief on the mating gear is specified (ie if Ca1+Croot2 = 0 and visa versa) use the run-ning in amount see 112
Bevel gears
Bevel gears are to have heightwise crowning ie no distinct relievedunrelieved area This may be treated as tip and root relief For calculation purposes the root relief is combined with the tip relief of the mating member into an equivalent tip relief If no resulting tip relieves are specified the equivalent tip relives may be calculated as an approxima-tion based on the tool crowning Ca tool (per mille of tool module m0) as follows
2root1aeq1a CCC +=
1root2aeq2a CCC +=
where
Classification Notes- No 412 35 May 2003
DET NORSKE VERITAS
2
0
n0101atoolal m
)m2(mA1middotmCC
minus+minussdot=
( )
sdotminusminussdotsdotminus+= vtvb1
2vb1
21vavt1n1 αtan dddαsin 05x1mA
2
0
n020atool22a m
)mm(2A1middotmCC
minus+minussdot=
( )
sdotminusminussdotsdotminus+= vtvb2
2vb2
2va2vt2n2 αtan dddαsin 05x1mA
2
0
20atool1root1 m
A1mCC
minussdotsdot=
2
0
10atool2root2 m
A1mCC
minussdotsdot=
If the pressure angles of the cutter blades are modified (and verified) to balance the tip relieves the following may be assumed
( )eq2aeq1aeq2aeq1a C andC calculated of sum50CC sdot==
Throughout the following Ca1 and Ca2 mean the equivalent tip relieves Ca1eq and Ca2eq
434 Bulk temperature The bulk temperature may be calculated as
flaaveragempsoilMB X X 50 ϑ+ϑ=ϑ
where
Xs = lubrication factor
= 12 for spray lubrication
= 10 for dip lubrication (provided both pinion and wheel are dip lubricated and tip speed lt 5 ms)
= 10 for spray lubrication with additional cooling spray (spray on both pinion and wheel or spray on pinion and dip lubrication of wheel)
= 02 for meshes fully submerged in oil
Xmp = contact factor ( )pmp nX += 150
np = number of mesh contact on the pinion (for small gear ratii the number of wheel meshes should be used if higher)
flaaverageϑ
= average of the integrated flash temperature (see 44) along the path of contact
( )
AE
E
Ayyyfla
flaaverage
d
ΓminusΓ
ΓΓϑ=ϑint =
For high speed gears (v gt 50 ms) it may be necessary to assess the bulk temperature on the basis of a thermal rating of the entire gear transmission
44 The Flash Temperature flaϑ
441 Basic formula The local flash temperature flaϑ may be calculated as
( ) 41redy
y2ly
211
43Btcorrfla
unXwX3250
y ρ
ρminusρ
micro=ϑ Γ
(For bevel gears replace u with uv)
and is to be calculated stepwise along the path of contact from A to E
where
micro = coefficient of friction see 431
Xcorr = correction factor taking empirically into account the increased scuffing risk in the beginning of the approach path due to mesh starting without any previously built up oil film and possible shuffling away oil before meshing if insufficient tip relief
( )
3
AD
yaeffcorr 50
CC1X
ΓminusΓε
Γminus+=
α
Ca = tip relief of driven member Xcorr is only applicable in the approach path and if Ca lt Ceff otherwise 10
wBt = unit load see 431
yXΓ = load sharing factor see 443
n1 = pinion rpm
Γy ρly etc see 442
442 Geometrical relations The various radii of flank curvature (transversal plane) are
ρ1y = pinion flank radius at mesh point y
ρ2y = wheel flank radius at mesh point y
ρredy = equivalent radius of curvature at mesh point y
y2y1
y2y1redy
ρ+ρ
ρρ=ρ
Cylindrical gears
twy
1y αsin au1Γ1
ρ+
+=
twy
2y αsin au1Γu
ρ+
minus=
Note that for internal gears a and u are negative
36 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
Bevel gears
vtvv
y1y αsin a
u1Γ1
ρ+
+=
vtvv
yv2y αsin a
u1Γu
ρ+
minus=
Γ is the parameter on the path of contact and y is any point between A end E
At the respective ends Γ has the following values
Root piniontip wheel
Cylindrical gears
( )
minus
minusminus= 1
αtan 1dd
zzΓ
tw
2b2a2
1
2A
Bevel gears
( )
minus
minusminus= 1
αtan 1dd
uΓvt
2vb2va2
vA
Tip pinionroot wheel
Cylindrical gears
( )
1αtan
1ddΓ
tw
2b1a1
E minusminus
=
Bevel gears
( )
1αtan
1ddΓ
vt
2vb1va1
E minusminus
=
At inner point of single pair contact
Cylindrical gears
tw1
EB α tan zπ2ΓΓ minus=
Bevel gears
vtv1
EB α tan zπ2ΓΓ minus=
At outer point of single pair contact
Cylindrical gears
tw1AD α tan z
π2ΓΓ +=
Bevel gears
vt v1
AD αtan z π2ΓΓ +=
At pitch point ΓC = 0
The points F and G (only applicable to cylindrical gears) limiting the extension of tip relief (so as to maintain a mini-mum contact ratio of unity for unloaded gears) are at
2
BAF
Γ+Γ=Γ
2
EDG
Γ+Γ=Γ
443 Load sharing factor XΓ The load sharing factor XΓ accounts for the load sharing be-tween the various pairs of teeth in mesh along the path of contact
XΓ is to be calculated stepwise from A to E using the pa-rameter Γy
4431 Cylindrical gears with β = 0 and no tip relief
Figure 41
ByAAB
Ay for31
31X
yΓltΓleΓ
ΓminusΓ
ΓminusΓ+=Γ
DyB for1Xy
ΓltΓleΓ=Γ
EyDDE
yE for31
31X
yΓleΓltΓ
ΓminusΓ
ΓminusΓ+=Γ
4432 Cylindrical gears with β = 0 and tip relief Tip relief on the pinion reduces XΓ in the range G ndash E and increases correspondingly XΓ in the range F ndash B
Tip relief on the wheel reduces XΓ in the range A ndash F and increases correspondingly XΓ in the range D ndash G
Following remains generally
DyB for1Xy
ΓleΓleΓ=Γ
21XXGF== ΓΓ
In the following it must be distinguished between Ca lt Ceff respectively Ca gt Ceff This is shown by an example below where Ca1 lt Ceff and Ca2 gt Ceff
Classification Notes- No 412 37 May 2003
DET NORSKE VERITAS
Figure 42
Note When Ca gt Ceff the path of contact is shortened by A ndash A respectively E ndash E The single pair contact path is extended into B respectively D If this shift is significant it is necessary to consider the negative effect on surface durability (B) and bending stresses (B and D)
Range A - F
For effa2 CC le
minus=Γ
eff
2a
CC1
31X
A
FyAeff
2a
AF
Ay for C3
C61XX
AyΓleΓleΓ
sdot
+ΓminusΓ
ΓminusΓ+= ΓΓ
For effa2 CC ge
AyAfor0Xy
ΓleΓleΓ=Γ
with ( )21
CC
1CC
eff
2a
eff
2a
AFAAminus
minusΓminusΓ+Γ=Γ
FyAeff
2a
AF
Ay
eff
2a for 21
CC
CC1X
yΓleΓleΓ
minus
ΓminusΓ
ΓminusΓ+minus=Γ
Range F - B
For effa1 CC le
ByFeff
1a
FB
Fy forC3
C61
21X
yΓleΓleΓ
sdot
+ΓminusΓ
ΓminusΓ+=Γ
For effa1 CC ge
ByFeff
1a
FB
Fy for21
CC
21X
yΓleΓleΓ
minus
ΓminusΓ
ΓminusΓ+=Γ
ByB for1Xy
ΓleΓleΓ=Γ
with 1
CC2
eff
1a
FBFB
minus
ΓminusΓ+Γ=Γ
Range D ndash G
For effa2 CC le
GyDeff
2a
DG
Dy
eff
2a for C 3C
61
C 3C
32X
yΓleΓleΓ
+
ΓminusΓ
ΓminusΓminus+=Γ F
or effa2 CC ge
DyD for1Xy
ΓleΓleΓ=Γ
with ( )21
CC
1CC
eff
2a
eff
2a
DGDDminus
minusΓminusΓ+Γ=Γ
GyDeff
2a
DG
Dy
eff
2a for21
CC
CCX ΓleΓleΓ
minus
ΓminusΓ
ΓminusΓminus=Γ
Range G - E
For effa1 CC le
minus=Γ
eff
1a
CC1
31X
E
EyGeff
1a
GE
Gy forC3
C61
21X
yΓleΓleΓ
sdot
+ΓminusΓ
ΓminusΓminus=Γ
For effa1 CC gt
EyGeff
1a
GE
Gy for21
CC
21X
yΓleΓleΓ
minus
ΓminusΓ
ΓminusΓminus=Γ
EyE for0Xy
ΓleΓleΓ=Γ
with 1
CC2
eff
1a
GEGE
minus
ΓminusΓ+Γ=Γ
4433 Gears with β gt 0 buttressing Due to oblique contact lines over the flanks a certain but-tressing may occur near A and E
This applies to both cylindrical and bevel gears with tip relief lt Ceff The buttressing Xbutt is simplified as a linear function within the ranges A ndash H respectively I ndash E
Figure 43
1εwhen13X βbutt EAge=
1εwhenε031X ββbutt EAlt+=
38 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
Cylindrical gears
bIEAH βsin 02ΓΓΓΓ =minus=minus
Bevel gears
bmIEAH βsin 02ΓΓΓΓ =minus=minus
4434 Cylindrical gears with 2εγ le and no tip relief
yXΓ is obtained by multiplication of
yXΓ in 4431 with
Xbutt in 4433
4435 Gears with 2εγ gt and no tip relief
Applicable to both cylindrical and bevel gears
IyH for1Xy
ΓleΓleΓε
=α
Γ
IyHybutt and forX1Xy
ΓgtΓΓltΓε
=α
Γ
Figure 44
4436 Cylindrical gears with 2εγ le and tip relief
yXΓ is obtained by multiplication of
yXΓ in 4432 with Xbutt
in 4433
4437 Cylindrical gears with 2εγ gt and tip relief
Tip relief on the pinion (respectively wheel) reduces XГ in the range G ndash E (respectively A ndash F) and increases XГ in the range F ndash G
yXΓ is obtained by multiplication of
yXΓ as described below
with Xbutt in 4433
In the XГ example below the influence of tip relief is shown (without the influence of Xbutt) by means of
eff2aeff1a CC and CC ltgt
Tip relief gt Ceff causes new end points A respectively E of the path of contact
Figure 45
Range A ndash F
( ) ( )( ) eff
a2αa1α
AF
Ay
eff
2aeff
C 12C 13εC 1ε
CCCX
y +εε++minus
ΓminusΓ
ΓminusΓ+
εminus
=ααα
Γ
eff2aFyA CC if for leΓleΓleΓ
eff2aFyA CifCfor and geΓleΓleΓ
eff2aAyA CC iffor0Xy
gtΓleΓleΓ=Γ
( ) ( ) ( )( ) ( ) a2αa1α
αeffa2AFAA C 1ε 3C 1ε
1ε 2 CC with++minus+minus
ΓminusΓ+Γ=Γ
Range F ndash G
( )( )( ) GyF
eff
2a1a forC 1 2CC 11X
yΓleΓleΓ
+εε+minusε
+ε
=αα
α
αΓ
Range G ndash E
( ) ( )( ) eff
2a1a
GE
Gy
C 1 2C 1C 1 3XX
GFy +εεminusε++ε
ΓminusΓ
ΓminusΓminus=
αα
ααΓΓ minus
effa1EyG CC if for leΓleΓleΓ
effa1EyG CC if Γfor and geΓleΓle
eff1aEyE CC if for0Xy
geΓleΓleΓ=Γ
( ) ( ) ( )( ) ( ) 2a1a
eff1aGEEE C 1C 1 3
1 2 CC withminusε++ε+εminus
ΓminusΓminusΓ=Γαα
α
4438 Bevel gears with εγ more than approx 18 and heightwise crowning For Ca1 = Ca2 = Ceff the following applies
Figure 46
Classification Notes- No 412 39 May 2003
DET NORSKE VERITAS
( )AEM 50 Γ+Γ=Γ
( )( )2AD
3
2My651X
y ΓminusΓε
ΓminusΓminus
ε=
ααΓ
For tip relief lt Ceff yXΓ is found by linear interpolation be-
tween ( )effay CCX =Γ and ( )0CayX =Γ as in 4435
The interpolation is to be made stepwise from A to M with the influence of Ca2 and from M to E with the influence of Ca1 (For 2a1a CC ne there is a discontinuity at M)
Eg with Ca1 = 04 Ceff and Ca2 = 055 Ceff then
Range A ndash M
)CC()0C( eff2ay2ayyX 550X 450X =Γ=ΓΓ +=
Range M ndash E
)CC()0C( eff1ay1ayyX 40X 06X =Γ=ΓΓ +=
For tip relief gt Ceff the new end points A and E are found as
( )
minusΓminusΓ
ε+Γ=Γ α 1
CC
6 eff
2aADAA
( )
minusΓminusΓ
ε+Γ=Γ α 1
CC
6 eff
1aADEE
Range A ndash A
0Xy=Γ
Range A ndash M
( )( )
ΓminusΓ
ΓminusΓminus
minusε=
αΓ 2
MA
2My
eff
2a1
CC4
351Xy
Range M ndash E
( )( )
ΓminusΓ
ΓminusΓminus
minusε=
αΓ 2
ME
2My
eff
1a1
CC4
351Xy
Range E ndash E
0Xy=Γ
40 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
Appendix A Fatigue Damage Accumulation
The Palmgren-Miner cumulative damage calculation princi-ple is used The procedure may be applied as follows
A1 Stress Spectrum From the individual torque classes the torques (Ti) at the peak values of class intervals and the associated number of cycles (NLi) for both pinion and wheel are to be listed from the highest to the lowest torque
(In case of a cyclic torque variation within the torque classes it is advised to use the peak torque If the cyclic variation is such that the same teeth will repeatedly suffer the peak torque this is a must)
The stress spectra for tooth roots and flanks (σFi σHi) with all relevant factors (except KA) are to be calculated on the basis of the torque spectrum The load dependent K-factors are to be determined for each torque class
A2 σminusN-curve The stress versus load cycle curves for tooth roots and flanks (both pinion and wheel) are to be drawn on the basis of per-missible stresses (ie including the demanded minimum safety factors) as determined in 2 respectively 3 If different safety levels for high cycle fatigue and low cycle fatigue are desired this may be expressed by different demand safety factors applied at the endurance limit respectively at static strength
A3 Damage accumulation The individual damage ratio Di at ith stress level is defined as
Fi
Lii N
ND =
where
NLi = The number of applied cycles at ith stress
NFi = The number of cycles to failure at ith stress
Basically stresses σi below the permissible stress level for infinite life (if a constant ZN or YN is accepted) do not con-tribute to the damage sum However calculating the actual safety factor Sact as described below all the σi for which the product S σi is bigger than or equal to the permissible stress level for infinite life contribute to the damage sum and thus to the determination of Sact The final value of S is decisive
(NFi can be found mathematically by putting the permissible stress σpi equal the actual stress σi thereby finding the actual life factor This life factor can be solved with regard to load cycles ie NFi)
The damage sum ΣDi is not to exceed unity
If ΣDi ne 1 the safety against cumulative fatigue damage is different from the applied demand safety factor For deter-mination of this theoretical safety factor an iteration proce-dure is required as described in the following flowchart
S is correction factor with which the actual safety factor Sact can be found
Sact is the demand safety factor (used in determination of the permissible stresses in the σ ndash N ndash curve) times the correc-tion factor S
The full procedure is to be applied for pinion and wheel tooth roots and flanks
Note If alternating stresses occur in a spectrum of mainly pulsating stresses the alternating stresses may be replaced by equiva-lent pulsating stresses ie by means of division with the ac-tual mean stress influence factor YM
Classification Notes- No 412 41 May 2003
DET NORSKE VERITAS
Appendix B Application Factors for Diesel Driven Gears
For diesel driven gears the application factor KA depends on torsional vibrations Both normal operation and misfiring conditions have to be considered
Normally these two running conditions can be covered by only one calculation
B1 Definitions
Normal operation KAnorm = 0
normv0
TTT +
where
T0 = rated nominal torque
Tv norm = vibratory torque amplitude for normal opera-tion (see rules Pt4 Ch3 Sec1 G301 for defi-nition of ldquonormalrdquo irregularity)
Misfiring operation KA misf = 0
misfv
TTT +
where
T = remaining nominal torque when one cylinder out of action
Tv misf = vibratory torque amplitude in misfiring con-dition This refers to a permissible misfiring condition ie a condition that does not require automatic or immediate corrective actions as speed or pitch reduction
The normal operation is assumed to last for a very high num-ber of cycles such as 1010
The misfiring operation is assumed to last for a limited dura-tion such as 107 cycles
B2 Determination of decisive load Assuming life factor at 1010 cycles as YN = ZN = 092 which usually is relevant the calculation may be performed only once with the combination having the highest value of appli-cation factorlife factor
For bending stresses and scuffing the higher value of
092
K normA and 098
K misfA
For contact stresses the higher value of
092K normA and
113K misfA (but
097K misfA for nitrided gears)
B3 Simplified procedure Note that this is only a guidance and is not a binding con-vention
T = oTZ
1Zsdot
minus where Z = number of cylinders
( ) ideal Videal Vmisf Vnorm V TTT24ZT +minus=
where TV ideal = vibratory torque with all cylinders perfectly equal See also rules Pt4 Ch3 Sec1 G300
When using trends from torsional vibration analysis and measurements the following may be used
TV ideal To is close to zero for engines with few cylinders and using a suitable elastic coupling and increases with relative coupling stiffness and number of cylinders
TV misf To may be high for engines with few cylinders and decreases with number of cylinders
This can be indicated as
200Z
TT
ο
idealV asymp and 80Z04
TT
o
misfV minusasymp
Inserting this into the formulae for the two application fac-tors the following guidance can be given
118112K misfA minusasymp
115110K normA minusasymp
Since KA norm is to be combined with the lower life factors the decisive load condition will be the normal one and a KA of 115 will cover most relevant cases when a suitable elas-tic coupling is chosen
42 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
Appendix C Calculation of Pinion-Rack
Pinion-racks used for elevating of mobile offshore units are open gears that are subjected to wear and tear With normal specifications such as only a few hundred total operation cycles (site to site) the tooth bending stresses for static strength will be decisive for the lay out however with ex-ception of surface hardened pinions where case crushing has to be considered
In the following the use of part 1 and 3 for pinion-racks is shown including relevant simplifications
C1 Pinion tooth root stresses Since the load spectrum normally is dominated by high torques and few load cycles (in the range up to 1-2000) the static strength is decisive
The actual stress is calculated as
β1FSaFan1
tF1 KYY
mbFσ sdotsdotsdotsdot
=
b1 is limited to b2 + 2middotmn
YFa and YSa replace YF and YS because load application at tooth tip has to be assumed for such inaccurate gears
Pinions often use a non-involute profile in the dedendum part of the flank eg a constant radius equal the radius of curva-ture at reference circle For such pinions sFn and hFa are to be measured directly on a sectional drawing of the pinion tooth
Due to high loads and narrow facewidths it may be assumed that KFβ2 = KHβ = 10 However when b1 gt b2 then KFβ1 gt10
If no detailed documentation of KFβ1 is available the fol-lowing may be used
KFβ1 = 1 + 015middot(b1b2 ndash 1)
The permissible stress (not surface hardened) is calculated as
δrelTstF
Fst1FP1 Y
Sσσ sdot=
The mean stress influence due to leg lifting may be disre-garded
The actual and permissible stresses should be calculated for the relevant loads as given in the rules
C2 Rack tooth root stresses The actual stress is calculated as
SaFan2
tF2 YY
mbFσ sdotsdotsdot
=
See C1 for details
The permissible stress is calculated as
δrelTstF
Fst2FP2 Y
Sσσ sdot=
For alloyed steels (Ni Cr Mo) with high toughness and duc-tility the value of YδrelTst may be put equal to YSa
C3 Surface hardened pinions For surface hardened pinions the maximum load is not to cause crushing of the hardened layer of the flank
In principle the calculation described in 213 may be used but when the theoretical Hertzian stress exceeds the ap-proximately 18 times the yield strength of the rack material plastic deformation will occur This will limit the peak Hertzian stress but increases the contact width and thus the penetration of stresses into the depth
An approximation may be based on an assessment of contact width determined by means of equal areas under the theoreti-cal (elastic range) Hertzian contact and the elasto-plastic contact stress (limited to 18 middot σy) with the unknown width
Classification Notes- No 412 21 May 2003
DET NORSKE VERITAS
stresses (von Mises) Both are influenced by residual stresses but this is only considered roughly and empirically
The subsurface working stresses at depths inside the peak of the orthogonal shear stresses respectively the equivalent stresses are only dependent on the (real) Hertzian stresses Surface related conditions as expressed by ZL ZV and ZR are assumed to have a negligible influence
The real Hertzian stresses σHR are determined as
For helical gears with εβ gt 1
σHR = σH
For helical gears with εβ lt 1 and spur gears
ε
α
ββ ε
ε+εminus
sdotσ=σZ
1
HHR
For bevel gears
KHHR Z
1σσ sdot=
The necessary hardness HV is given as a function of the net depth tz (net = after grinding or hard metal hobbing and per-pendicular to the flank)
The coordinates tz and HV are to be compared with the de-sign specification such as
bull for flame and induction hardening tHVmin HVmin bull for nitriding t400min HV = 400 bull for case hardening t550min HV = 550 t400min HV = 400
and t300min HV = 300 (the latter only if the core hardness lt 300 If the core hardness gt 400 the t400 is to be re-placed by a fictive t400 = 16 t550)
In addition the specified surface hardness is not to be less than the max necessary hardness (at tz = 05aH) This applies to all hardening methods
For high cycle fatigue (gt3 106 cycles) the following applies
sdot+
minussdotsdotsdot= o90
05azt
05azt
cosSσ04HV
H
HHHR
applicable to 05a
zt
Hge
For 05at
H
z lt the value for 05at
H
z = applies
56300ρSσ12a cHHR
Hsdotsdot
sdot=
Where aH is half the hertzian contact width multiplied by an empirical factor of 12 that takes into account the possible influence of reduced compressive residual stresses (or even tensile residual stresses) on the local fatigue strength
If any of the specified hardness depths including the surface hardness is below the curve described by HV = f (tz) the actual safety factor against subsurface fatigue is determined as follows
reduce SH stepwise in the formula for HV and aH until all specified hardness depths and surface hardness balance with the corrected curve The safety factor obtained through this method is the safety against subsurface fatigue
For static strength (lt103 cycles) the following applies
sdot+
minussdotsdotsdot= o90
07at
06a
zt
cosSσ019HV
Hst
z
HstHHR
applicable to 06at
Hst
z ge
56300ρSσa cHHR
Hstsdotsdot
=
In the case of insufficient specified hardness depths the same procedure for determination of the actual safety factor as above applies
For limited life fatigue (103 lt cycles lt 3106 )
For this purpose it is necessary to extend the correction of safety factors to include also higher values than required Ie in the case of more than sufficient hardness and depths the safety factor in the formulae for both high cycle fatigue and static strength are to be increased until necessary and speci-fied values balance
The actual safety factor for a given number of cycles N be-tween 103 and 3106 is found by linear interpolation in a dou-ble logarithmic diagram
minussdotminus
= sdot logN3477
logSlogSlogS
310H6103HHN
36 10H103H Slog86281Slog86280 sdot+sdot sdot
22 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
3 Calculation of Tooth Strength
31 Scope and General Remarks Part 3 include the calculation of tooth root strength as limited by tooth root cracking (surface or subsurface initiated) and yielding
For rim thickness sR ge 35middotmn the strength is calculated by means of 32 ndash 313 For cylindrical gears the calculation is based on the assumption that the highest tooth root tensile stress arises by application of the force at the outer point of single tooth pair contact of the virtual spur gears The method has however a few limitations that are mentioned in 36
For bevel gears the calculation is based on force application at the tooth tip of the virtual cylindrical gear Subsequently the stress is converted to load application at the mid point of the flank due to the heightwise crowning
Bevel gears may also be calculated with the program BECAL In that case KA and Kv are to be included in the applied tooth force but not KFβ and KFα
In case of a thin annulus or a thin gear rim etc radial crack-ing can occur rather than tangential cracking (from root fillet to root fillet) Cracking can also start from the compression fillet rather than the tension fillet For rim thickness sR lt 35middotmn a special calculation procedure is given in 315 and 316 and a simplified procedure in 314
A tooth breakage is often the end of the life of a gear trans-mission Therefore a high safety SF against breakage is re-quired
It should be noted that this part 3 does not cover fractures caused by
bull oil holes in the tooth root space bull wear steps on the flank bull flank surface distress such as pits spalls or grey staining
Especially the latter is known to cause oblique fractures starting from the active flank predominately in spiral bevel gears but also sometimes in cylindrical gears
Specific calculation methods for these purposes are not given here but are under consideration for future revisions Thus depending on experience with similar gear designs limita-tions other than those outlined in part 3 may be applied
32 Tooth Root Stresses The local tooth root stress is defined as the max principal stress in the tooth root caused by application of the tooth force Ie the stress ratio R = 0 Other stress ratios such as for eg idler gears (R asymp -12) shrunk on gear rims (R gt 0) etc are considered by correcting the permissible stress level
321 Local tooth root stress The local tooth root stress for pinion and wheel may be as-sessed by strain gauge measurements or FE calculations or similar For both measurements and calculations all details are to be agreed in advance
Normally the stresses for pinion and wheel are calculated as
Cylindrical gears
FαFβvγAβSFn
tF K K K K K Y Y Y
m bFσ =
where
YF = Tooth form factor (see 33)
YS = Stress correction factor (see 34)
Yβ = Helix angle factor (see 36)
Ft KA Kγ Kv KFβ KFα see 15 ndash 110
b see 13
Bevel gears
FαFβvγASaFamn
mtF K K K K K Y Y Y
m bFσ ε=
where
YFa = Tooth form factor see 33
YSa = Stress correction factor see 34
Yε = Contact ratio factor see 35
Fmt KA etc see 15 ndash 110
b see 13
322 Permissible tooth root stress The permissible local tooth root stress for pinion respectively wheel for a given number of cycles N is
CXRrelTrelTF
NMFEFP YYYY
SYY
δσ
=σ
Note that all these factors YM etc are applicable to 3middot106 cycles when used in this formula for σFP The influence of other number of cycles on these factors is covered by the calculation of YN
where
σFE = Local tooth root bending endurance limit of reference test gear (see 37)
YM = Mean stress influence factor which accounts for other loads than constant load direction eg idler gears temporary change of load di-rection pre-stress due to shrinkage etc (see 38)
YN = Life factor for tooth root stresses related to reference test gear dimensions (see 39)
SF = Required safety factor according to the rules
YδrelT = Relative notch sensitivity factor of the gear to be determined related to the reference test gear (see 310)
YRrelT = Relative (root fillet) surface condition factor of the gear to be determined related to the reference test gear (see 311)
Classification Notes- No 412 23 May 2003
DET NORSKE VERITAS
YX = Size factor (see 312)
YC = Case depth factor (see 313)
33 Tooth Form Factors YF YFa The tooth form factors YF and YFa take into account the in-fluence of the tooth form on the nominal bending stress
YF applies to load application at the outer point of single tooth pair contact of the virtual spur gear pair and is used for cylindrical gears
YFa applies to load application at the tooth tip and is used for bevel gears
Both YF and YFa are based on the distance between the con-tact points of the 30˚-tangents at the root fillet of the tooth profile for external gears respectively 60˚ tangents for inter-nal gears
Figure 31 External tooth in normal section
Figure 32 Internal tooth in normal section
Definitions
n
2
n
Fn
enFn
Fe
F
α cosms
α cosmh6
Y
=
n
2
n
Fn
anFn
Fa
Fa
α cosms
α cosmh6
Y
=
In the case of helical gears YF and YFa are determined in the normal section ie for a virtual number of teeth
YFa differs from YF by the bending moment arm hFa and αFan and can be determined by the same procedure as YF with exception of hFe and αFan For hFa and αFan all indices e will change to a (tip)
The following formulae apply to cylindrical gears but may also be used for bevel gears when replacing
mn with mnm
zn with zvn
αt with αvt
β with βm
with undercut without undercut
Fig 33 Dimensions and basic rack profile of the teeth (finished profile)
Tool and basic rack data such as hfP ρfp and spr etc are re-ferred to mn ie dimensionless
331 Determination of parameters
( )n
n
prnfPnfP m
α cossαsin 1ρ
αtan h4πE
minusminusminusminus=
For external gears fPfP ρρ =
For internal gears ( )0z
195fPfP0
fPfP 10363156ρhxρρ
sdotminus+
+=
where
z0 = number of teeth of pinion cutter
x0 = addendum modification coefficient of pinion cutter
hfP = addendum of pinion cutter
ρfP = tip radius of pinion cutter
xhρG fpfp +minus=
24 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
τmE
2π
z2H
nnminus
minus=
with
3πτ = for external gears
6πτ = for internal gears
Htan zG2
nminusϑ=ϑ
(to be solved iteratively suitable start value6π
=ϑ for exter-
nal gears and 3π for internal gears)
a) Tooth root chord sFn For external gears
minus
ϑ+
ϑminus= ρ
cosG3
3πsinz
ms
fpnn
Fn
For bevel gears with a tooth thickness modification
xsm affects mainly sFn but also hFe and αFen The total influence of xsm on YFa Ysa can be approximated by only adding 2 xsm to sFn mn
For internal gears
minus
ϑ+
ϑminus= ρ
cosG
6πsinz
ms
fPnn
Fn
b) Root fillet radius ρF at 30ordm tangent
( )G2coszcosG2ρ
mρ
2n
2
fpn
F
minusϑϑ+=
c) Determination of bending moment arm hF dn = zn mn
b2α
αn βcosε
ε =
dan = dn + 2 ha
pbn = π mn cos αn
dbn = dn cos αn
( )4
d1εpzz
2dd
zz2d
2bn
2
αnbn
2bn
2an
en +
minusminus
minus=
en
bnen d
dcos arcα =
ennnn
e α invα invα x tan 22π
z1
minus+
+=γ
αFen = αen ndash γe
For external gears
( )
minus=
n
enFenee
n
Fe
mdαtan sin γ γcos
21
mh
]ρcos
G3πcosz fpn +
ϑminus
ϑminusminus
For internal gears
( ) minus
sdotsdotminus=
n
enFenee
n
Fe
mdα tanγ sinγ cos
21
mh
minus
ϑminus
ϑminussdot ρ
cosG3
6πcosz fPn
332 Gearing with εαn gt 2 For deep tooth form gearing ( )25ε2 αn lele produced with a verified grade of accuracy of 4 or better and with applied profile modification to obtain a trapezoidal load distribution along the path of contact the YF may be corrected by the factor YDT as
250ε205for 0666ε2366Y αnαnDT leleminus=
205εfor 10Y αnDT lt=
34 Stress Correction Factors YS YSa The stress correction factors YS and YSa take into account the conversion of the nominal bending stress to the local tooth root stress Thereby YS and YSa cover the stress increasing effect of the notch (fillet) and the fact that not only bending stresses arise at the root A part of the local stress is inde-pendent of the bending moment arm This part increases the more the decisive point of load application approaches the critical tooth root section
Therefore in addition to its dependence on the notch radius the stress correction is also dependent on the position of the load application ie the size of the bending moment arm
YS applies to the load application at the outer point of single tooth pair contact YSa to the load application at tooth tip
YS can be determined as follows
( )
+
+=L23121
1
sS qL01312Y
where Fe
Fn
hsL = and
F
Fns ρ2
sq = (see 33)
YSa can be calculated by replacing hFe with hFa in the above formulae
Note a) Range of validity 1 lt qs lt 8
In case of sharper root radii (ie produced with tools having too sharp tip radii) YS resp YSa must be specially considered
b) In case of grinding notches (due to insufficient protuberance of the hob) YS resp YSa can rise considerably and must be multiplied with
Classification Notes- No 412 25 May 2003
DET NORSKE VERITAS
g
g
ρt
0613
13
minus
where
tg = depth of the grinding notch
ρg = radius of the grinding notch
c) The formulae for YS resp YSa are only valid for αn = 20˚ However the same formulae can be used as a safe approximation for other pressure angles
35 Contact Ratio Factor Yε The contact ratio factor Yε covers the conversion from load application at the tooth tip to the load application at the mid point of the flank (heightwise) for bevel gears
The following may be used
Yε = 0625
36 Helix Angle Factor Yβ The helix angle factor Yβ takes into account the difference between the helical gear and the virtual spur gear in the nor-mal section on which the calculation is based in the first step In this way it is accounted for that the conditions for tooth root stresses are more favourable because the lines of contact are sloping over the flank
The following may be used (β input in degrees)
Yβ = 1 ndash εβ β120
When εβ gt 1 use εβ = 1 and when β gt 30deg use β = 30deg in the formula
However the above equation for Yβ may only be used for gears with β gt 25deg if adequate tip relief is applied to both pinion and wheel (adequate = at least 05 middot Ceff see 432)
37 Values of Endurance Limit σFE σFE is the local tooth root stress (max principal) which the material can endure permanently with 99 survival prob-ability 3106 load cycles is regarded as the beginning of the endurance limit or the lower knee of the σ ndash N curve σFE is defined as the unidirectional pulsating stress with a minimum stress of zero (disregarding residual stresses due to heat treatment) Other stress conditions such as alternating or pre-stressed etc are covered by the conversion factor YM
σFE can be found by pulsating tests or gear running tests for any material in any condition If the approval of the gear is to be based on the results of such tests all details on the testing conditions have to be approved by the Society Fur-ther the tests may have to be made under the Societys su-pervision
If no fatigue tests are available the following listed values for σFE may be used for materials subjected to a quality con-trol as the one referred to in the rules
σFE
Alloyed case hardened steels 1) (fillet surface hardness 58 ndash 63 HRC)
bull of specially approved high grade
1050
bull of normal grade
minus CrNiMo steels with approved process
1000
minus CrNi and CrNiMo steels generally 920 minus MnCr steels generally 850
Nitriding steel of approved grade quenched tempered and gas nitrided (surface hardness 700 ndash 800 HV)
840
Alloyed quenched and tempered steel bath or gas nitrided (surface hardness 500 ndash 700 HV)
720
Alloyed quenched and tempered steel flame or induction hardened 2) (incl entire root fillet) (fillet surface hardness 500 ndash 650 HV)
07 HV + 300
Alloyed quenched and tempered steel flame or induction hardened (excl entire root fillet) (σB = uts of base material)
025 σB + 125
Alloyed quenched and tempered steel 04 σB + 200
Carbon steel 025 σB + 250
Note All numbers given above are valid for separate forgings and for blanks cut from bars forged according to a qualified procedure see Pt 4 Ch 2 Sec 3 For rolled steel the values are to be reduced with 10 For blanks cut from forged bars that are not qualified as mentioned above the values are to be reduced with 20 For cast steel reduce with 40
1) These values are valid for a root radius
bull being unground If however any grinding is made in the root fillet area in such a way that the residual stresses may be affected σFE is to be reduced by 20 (If the grinding also leaves a notch see 34)
bull with fillet surface hardness 58 ndash 63 HRC In case of lower surface hardness than 58 HRC σFE is to be reduced with 20(58 ndash HRC) where HRC is the detected hardness (This may lead to a permissible tooth root stress that varies along the facewidth If so the actual tooth root stresses may also be considered along facewidth)
bull not being shot peened In case of approved shot peening σFE may be increased by 200 for gears where σFE is reduced by 20 due to root grinding Otherwise σFE may be increased by 100 for
6nm le and 100 ndash 5 (mn - 6) for mn gt 6 However the possible adverse influence on the flanks regarding grey staining should be considered and if necessary the flanks should be masked
2) The fillet is not to be ground after surface hardening Regarding possible root grinding see 1)
26 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
38 Mean stress influence Factor YM The mean stress influence factor YM takes into account the influence of other working stress conditions than pure pulsa-tions (R = 0) such as eg load reversals idler gears planets and shrink-fitted gears
YM (YMst) are defined as the ratio between the endurance (or static) strength with a stress ratio R ne 0 and the endurance (or static) strength with R = 0
YM and YMst apply only to a calculation method that assesses the positive (tensile) stresses and is therefore suitable for comparison between the calculated (positive) working stress σF and the permissible stress σFP calculated with YM or YMst
For thin rings (annulus) in epicyclic gears where the com-pression fillet may be decisive special considerations apply see 316
The following method may be used within a stress ratio ndash12 lt R lt 05
381 For idlers planets and PTO with ice class
M1M1R1
1Yor Y MstM
+minus
minus=
where
R = stress ratio = min stress divided by max stress
For designs with the same force applied on both forward- and back-flank R may be assumed to ndash 12
For designs with considerably different forces on forward- and back-flank such as eg a marine propulsion wheel with a power take off pinion R may be assessed as
branchmain theoffacewidth unit per forcepto offacewidt unit per force21minus
For a power take off (PTO) with ice class see 161 c
M considers the mean stress influence on the endurance (or static) strength amplitudes
M is defined as the reduction of the endurance strength am-plitude for a certain increase of the mean stress divided by that increase of the mean stress
Following M values may be used
Endurance limit
Static strength
Case hardened 08 ndash 015 Ys 1) 07
If shot peened 04 06
Nitrided 03 03
Induction or flame hardened
04 06
Not surface hardened steel 03 05
Cast steels 04 06 1)For bevel gears use Ys=2 for determination of M
The listed M values for the endurance limit are independent of the fillet shape (Ys) except for case hardening In princi-ple there is a dependency but wide variations usually only occur for case hardening eg smooth semicircular fillets versus grinding notches
382 For gears with periodical change of rotational direction For case hardened gears with full load applied periodically in both directions such as side thrusters the same formula for YM as for idlers (with R = ndash 12) may be used together with the M values for endurance limit This simplified approach is valid when the number of changes of direction exceeds 100 and the total number of load cycles exceeds 3middot106
For gears of other materials YM will normally be higher than for a pure idler provided the number of changes of direction is below 3middot106 A linear interpolation in a diagram with loga-rithmic number of changes of direction may be used ie from YM = 09 with one change to YM (idler) for 3middot106 changes This is applicable to YM for endurance limit For static strength use YM as for idlers
For gears with occasional full load in reversed direction such as the main wheel in a reversing gear box YM = 09 may be used
383 For gears with shrinkage stresses and unidirectional load For endurance strength
FE
fitM σ
σM1
M21Y+
minus=
σFE is the endurance limit for R = 0
For static strength YMst = 1 and σfit accounted for in 39b
σfit is the shrinkage stress in the fillet (30˚ tangent) and may be found by multiplying the nominal tangential (hoop) stress with a stress concentration factor
n
Ffit m
ρ215scf minus=
384 For shrink-fitted idlers and planets When combined conditions apply such as idlers with shrink-age stresses the design factor for endurance strength is
( ) ( ) FE
fitM σ
σR1M1
M2
M1M1R1
1Yminussdot+
minus
+minus
minus=
Symbols as above but note that the stress ratio R in this par-ticular connection should disregard the influence of σfit ie R normally equal ndash 12
For static strength
M1M1R1
1YMst
+minus
minus=
The effect of σfit is accounted for in 39 b
Classification Notes- No 412 27 May 2003
DET NORSKE VERITAS
385 Additional requirements for peak loads The total stress range (σmax ndash σmin) in a tooth root fillet is not to exceed
F
y
Sσ225
for not surface hardened fillets
FSHV5 for surface hardened fillets
39 Life Factor YN The life factor YN takes into account that in the case of limited life (number of cycles) a higher tooth root stress can be permitted and that lower stresses may apply for very high number of cycles
Decisive for the strength at limited life is the σ ndash N ndash curve of the respective material for given hardening module fillet radius roughness in the tooth root etc Ie the factors YδrelT YRelT YX and YM have an influence on YN
If no σ ndash N ndash curve for the actual material and hardening etc is available the following method may be used
Determination of the σ ndash N ndash curve
a) Calculate the permissible stress σFP for the beginning of the endurance limit (3middot106 cycles) including the influ-ence of all relevant factors as SF YδrelT YRelT YX YM and YC ie σFP = σFE middotYM middotYδrelT middotYRelT middotYX middotYC SF
b) Calculate the permissible laquostaticraquo stress (le 103 load cy-cles) including the influence of all relevant factors as SFst YδrelTst YMst and YCst ( )fitσCstYδrelTstYMstYFstσ
FstS1
FPstσ minussdotsdotsdot=
where σFst is the local tooth root stress which the material can resist without cracking (surface hardened materials) or unacceptable deformation (not surface hardened materials) with 99 survival probability
σFst
Alloyed case hardened steel 1) 2300
Nitriding steel quenched tempered and gas nitrided (surface hardness 700 ndash 800 HV)
1250
Alloyed quenched and tempered steel bath or gas nitrided (surface hardness 500 ndash 700 HV)
1050
Alloyed quenched and tempered steel flame or induction hardened (fillet surface hardness 500 ndash 650 HV)
18 HV + 800
Steel with not surface hardened fillets the smaller value of 2)
18 σB or 225 σy
1) This is valid for a fillet surface hardness of 58 ndash 63 HRC In case of lower fillet surface hardness than 58 HRC σFst is to be reduced with 30(58 ndash HRC) where HRC is the actual hardness Shot peening or grinding notches are not considered to have any significant influence on σFst 2) Actual stresses exceeding the yield point (σy or σ02) will alter the residual stresses locally in the ldquotensionrdquo fillet re-spectively ldquocompressionrdquo fillet This is only to be utilised for gears that are not later loaded with a high number of cycles at lower loads that could cause fatigue in the ldquocompressionrdquo fillet
c) Calculate YN as
NL gt 3middot106
YN = 1 or 001
L
6
N N103Y
sdot= ie Yn = 092 for 1010
The YN = 1 from 3middot106 on may only be used when special material cleanness applies see rules Pt4 Ch2
103ltNLlt3middot106exp
L
6
N N103Y
sdot=
cycles6103forσcycles310forσlog02876exp
FP
FPst
sdot=
NL lt 103 cycles6103forσcycles310forσ
NYFP
FPst
sdot=
or simply use σFPst as mentioned in b) directly
Guidance on number of load cycles NL for various applica-tions
bull For propulsion purpose normally NL = 1010 at full load (yachts etc may have lower values)
bull For auxiliary gears driving generators that normally operate with 70-90 of rated power NL = 108 with rated power may be applied
28 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
310 Relative Notch Sensitivity Factor YδrelT The dynamic (respectively static) relative notch sensitivity factor YδrelT (YδrelTst) indicate to which extent the theoreti-cally concentrated stress lies above the endurance limits (re-spectively static strengths) in the case of fatigue (respectively overload) breakage
YδrelT is a function of the material and the relative stress gra-dient It differs for static strength and endurance limit
The following method may be used
For endurance limit
for not surface hardened fillets
( )
024
s024
relT σ103133q21σ1012201351
Ysdotsdotminus
+sdotsdotminus+= minus
minus
δ
for all surface hardened fillets except nitrided
( )
106q21002451
Y srelT
++=δ
for nitrided fillets
( )
1347q2101421
Y srelT
++=δ
For static strength
for not surface hardened fillets1)
( )( )( )025
02
02502s
relTst σ3000821σ300 1Y0821Y
+
minus+=δ
for surface hardened fillets except nitrided
YδrelTst = 044 YS + 012
for nitrided fillets
YδrelTst = 06 + 02 YS 1) These values are only valid if the local stresses do not
exceed the yield point and thereby alter the residual stress level See also 39b footnote 2
311 Relative Surface Condition Factor YRrelT The relative surface condition factor YRrelT takes into ac-count the dependence of the tooth root strength on the sur-face condition in the tooth root fillet mainly the dependence on the peak to valley surface roughness
YRrelT differs for endurance limit and static strength
The following method may be used
For endurance limit
YRrelT = 1675 ndash 053 (Ry + 1)01 for surface hardened steels and alloyed quenched and tempered steels except nitrided
YRrelT = 53 ndash 42 (Ry + 1)001 for carbon steels
YδrelT = 43 ndash 326 (Ry + 1)0005
for nitrided steels
For static strength
YRrelTst = 1 for all Ry and all materials
For a fillet without any longitudinal machining trace Ry asymp Rz
312 Size Factor YX The size factor YX takes into account the decrease of the strength with increasing size YX differs for endurance limit and static strength
The following may be used
For endurance limit
YX = 1 for mn le 5 generally
YX = 103 ndash 0006 mn for 5 lt mn lt 30
YX = 085 for mn ge 30
for not surface hard-ened steels
YX = 105 ndash 001 mn for 5 lt mn ge 25
YX = 08 for mn ge 25
for surface hardened steels
For static strength
YXst = 1 for all mn and all materials
313 Case Depth Factor YC The case depth factor YC takes into account the influence of hardening depth on tooth root strength
YC applies only to surface hardened tooth roots and is dif-ferent for endurance limit and static strength
In case of insufficient hardening depth fatigue cracks can develop in the transition zone between the hardened layer and the core For static strength yielding shall not occur in the transition zone as this would alter the surface residual stresses and therewith also the fatigue strength
The major parameters are case depth stress gradient permis-sible surface respectively subsurface stresses and subsurface residual stresses
The following simplified method for YC may be used
YC consists of a ratio between permissible subsurface stress (incl influence of expected residual stresses) and permissible surface stress This ratio is multiplied with a bracket con-taining the influence of case depth and stress gradient (The empirical numbers in the bracket are based on a high number of teeth and are somewhat on the safe side for low number of teeth)
YC and YCst may be calculated as given below but calculated values above 10 are to be put equal 10
For endurance limit
+
+=nFFE
C m02ρt31
σconstY
Classification Notes- No 412 29 May 2003
DET NORSKE VERITAS
For static strength
+
+=nFFst
Cst m02ρt31
σconstY
where const and t are connected as
Hardening process
t = endurance limit const =
static strength const =
t550 640 1900
t400 500 1200 Case hard-ening
t300 380 800
Nitriding t400 500 1200
Induction- or flame hardening
tHVmin 11 HVmin 25 HVmin
For symbols see 213
In addition to these requirements to minimum case depths for endurance limit some upper limitations apply to case hard-ened gears
The max depth to 550 HV should not exceed
1) 13 of the top land thickness san unless adequate tip relief is applied (see 110)
2) 025 mn If this is exceeded the following applies addi-tionally in connection with endurance limit
minusminus= 025
mt
1Yn
max550C
314 Thin rim factor YB Where the rim thickness is not sufficient to provide full sup-port for the tooth root the location of a bending failure may be through the gear rim rather than from root fillet to root fillet
YB is not a factor used to convert calculated root stresses at the 30deg tangent to actual stresses in a thin rim tension fillet Actually the compression fillet can be more susceptible to fatigue
YB is a simplified empirical factor used to de-rate thin rim gears (external as well as internal) when no detailed calcula-tion of stresses in both tension and compression fillets are available
Figure 34 Examples on thin rims
YB is applicable in the range 175 lt sRmn lt 35
YB = 115 middot ln (8324 middot mnsR)
(for sRmn ge 35 YB = 1)
(for sRmn le 175 use 315)
σF as calculated in 321 is to be multiplied with YB when sRmn lt 35 Thus YB is used for both high and low cycle fatigue
Note This method is considered to be on the safe side for external gear rims However for internal gear rims without any flange or web stiffeners the method may not be on the safe side and it is advised to check with the method in 315316
315 Stresses in Thin Rims For rim thickness sR lt 35 mn the safety against rim cracking has to be checked
The following method may be used
3151 General The stresses in the standardised 30ordm tangent section tension side are slightly reduced due to decreasing stress correction factor with decreasing relative rim thickness sRmn On the other hand during the complete stress cycle of that fillet a certain amount of compression stresses are also introduced The complete stress range remains approximately constant Therefore the standardised calculation of stresses at the 30ordm tangent may be retained for thin rims as one of the necessary criteria
The maximum stress range for thin rims usually occurs at the 60ordm ndash 80ordm tangents both for laquotensionraquo and laquocompressionraquo side The following method assumes the 75ordm tangent to be the decisive Therefore in addition to the am criterion ap-plied at the 30ordm tangent it is necessary to evaluate the max and min stresses at the 75ordm tangent for both laquotensionraquo (loaded flank) fillet and laquocompressionraquo (back-flank) fillet For this purpose the whole stress cycle of each fillet should be considered but usually the following simplification is justified
Figure 35 Nomenclature of fillets
Index laquoTraquo means laquotensileraquo fillet laquoCraquo means laquocompressionraquo fillet
σFTmin and σFCmax are determined on basis of the nominal rim stresses times the stress concentration factor Y75
σFCmin and σFTmax are determined on basis of superposition of nominal rim stresses times Y75 plus the tooth bending stresses at 75ordm tangent
30 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
3152 Stress concentration factors at the 75ordm tangents The nominal rim stress consists of bending stresses due to local bending moments tangential stresses due to the tan-gential force Ft and radial shear stresses due to Fr
The major influence is given by the bending stresses The influence of the tangential stresses is minor and even though its stress concentration factor is slightly higher than for bending it is considered to be safe enough when the sum of these nominal stresses are combined with the stress concen-tration factor for bending The influence of the radial shear stress is neglected
The stress concentration factor relating nominal rim stresses to local fillet stresses at the 75ordm tangent may be calculated as
75n
R75
n
R
75
ρmsρ185
ms3
Y+
sdot=
where ρ75 is the root radius at the 75ordm tangent ref to mn Usually ρ75 is closed to the tool radius ρao and
ρ75 = ρao
is a safe approximation compensating for the am simplifi-cations to the laquounsaferaquo side
The tooth root stresses of the loaded tooth are decreasing with decreasing relative rim thickness approximately with the empirical factor
15
R
ncorr s
m311Y
minus=
3153 Nominal rim stresses The bending moment applied to the rim consists of a part of the tooth tilt moment Ft (hF + 05 sR) and the bending caused by the radial force Fr
The sectional modulus (first moment of area) which is used for determination of the nominal bending stresses is not nec-essarily the same for the 2 am bending moments If flanges webs etc outside the toothed section contribute to stiffening the rim against various deflections the influence of these stiffeners should be considered Eg an end flange will have an almost negligible influence on the effective sectional modulus for the stresses due to tooth tilt as the deflection caused by the tilt moment is rather small and would not much involve the flange On the other hand the radial forces as for instance from the meshes in an annulus would cause considerable radial deflections that the flange might restrict to a substantial amount When considering the stiff-ening of such flanges or webs on basis of simplified models it is advised to use an effective rim thickness sR = sR + 02 mn for the first moment of area of the rim (toothed part) cross section
For a high number of rim teeth it may be assumed that the rim bending moments in the fillets adjacent to the loaded tooth are of the same magnitude as right under the applied
force This assumption is reasonable for an annulus but rather much on the safe side for a hollow pinion
The influence of Ft on the nominal tangential stress is simpli-fied by half of it for compressive stresses (σ1) and the other half for tensile stresses (σ2) Applying these assumptions the nominal rim stresses adjacent to the loaded tooth are
( ) ( )A2
FW
f RFW
s05hF05 t
R
r
T
RFt1 minus
ϑminus
+minus=σ
( ) ( )A2
FW
f RFW
s05hF05 t
R
r
T
RFt2 +
ϑminus
+=σ
where σ1 σ2 see Fig 35
A = minimum area of cross section (usually bR sR)
WT = the sectional modulus of rim with respect to tooth tilt moment (usually 6sb 2
rR )
WR = the sectional modulus of the rim including the influence of stiffeners as flanges webs etc (WR ge WT)
bR = the width of the rim
R = the radius of the neutral axis in the rim ie from wheel centre to midpoint of rim
( )ϑf = a function for bending moment distribution around the rim For a rim (pinion) with one mesh only the ( )ϑf at the position of load application is 024 For an annulus with 3 or more meshes ( )ϑf at the position of each load application is approx 3 planets ( )ϑf = 019 4 planets ( )ϑf = 014 5 planets ( )ϑf = 011 6 planets ( )ϑf = 009
It must be checked if the max (tensile) stress in the compres-sion fillet really occurs when the fillet is adjacent to the loaded tooth In principle the stress variation through a complete rotation should be considered and the max value used The max value is usually never less than 0 For an annulus eg the tilt moment is zero in the mid position be-tween the planet meshes whilst the bending moment due to the radial forces is half of that at the mesh but with opposite sign
If these formulae are applied to idler gears as eg planets the influence of nominal tangential stresses must be cor-rected by deleting Ft(2 A) for σ1 and using FtA for σ2 Further the influence of Fr on the nominal bending stresses is usually negligible due to the planet bearing support
3154 Root fillet stresses Determination of min and max stresses in the laquotensionraquo fillet
Classification Notes- No 412 31 May 2003
DET NORSKE VERITAS
Minimum stress
751FTmin YσK σ =
Maximum stress
corrF752 Yσ03Yσ KσFTmax +=
where
03 is an empirical factor relating the tension stresses (σF) at the 30ordm tangent to the part of the tension stresses at the 75ordm tangent which add to the rim related stresses (03 also takes into account that full superposition of nominal stresses times stress concentration factors from both laquosides of the corner filletraquo would result in too high stresses)
αβγ sdotsdotsdotsdot= FFvA KKKKKK
Determination of min and max stresses in the laquocompres-sionraquo fillet
Minimum stress
corrF751FCmin Yσ036Yσ Kσ minus=
Maximum stress
752FCmax Yσ Kσ =
where
036 is an empirical factor relating the tension stresses (σF) at the 30ordm tangent to the part of the compression stresses at the 75ordm tangent which add to the rim related stresses
For gears with reversed loads as idler gears and planets there is no distinct laquotensionraquo or laquocompressionraquo fillet The mini-mum stress σFmin is the minimum of σFTmin and σFCmin (usually the latter is decisive) The maximum stress σFTmax is the maximum of σFTmax and σFCmax (usually the former is deci-sive)
316 Permissible Stresses in Thin Rims
3161 General The safety against fatigue fracture respectively overload fracture is to be at least at the same level as for solid gears The laquoordinaryraquo criteria at the 30ordm tangent apply as given in 31 through 313
Additionally the following criteria at the 75ordm tangent may apply
3162 For gt3middot106 cycles The permissible stresses for the laquotensionraquo fillets and for the laquocompressionraquo fillets are determined by means of a relevant fatigue diagram
If the actual tooth root stress (tensile or compressive) ex-ceeds the yield strength to the material the induced residual stresses are to be taken into account
For determination of permissible stresses the following is defined
R = stress ratio ie FTmax
FTminσσ respectively
FCmax
FCmin
σσ
∆σ = stress range ie σFTmax ndash σFTmin resp σFCmax ndash σFCmin
(For idler gears and planets Fmax
FminσσR =
and ∆σ = σFmax minus σFmin)
The permissible stress range ∆σρ for the laquotensionraquo respec-tively laquocompressionraquo fillets can be calculated as
For R gt minus1 FPp σ
R1R1031
13∆σ
minus+
+=
For minus infin lt R lt minus1 FPp σ
R1R10151
13∆σ
minus+
+=
where
σFP = see 32 determined for unidirectional stresses (YM = 1)
If the yield strength σy is exceeded in either tension or com-pression residual stresses are induced This may be consid-ered by correcting the stress ratio R for the respective fillets (tension or compression)
Eg if yFCmin σσ gt (ie exceeded in compression) the
difference yFCmin σσ∆ minus= affects the stress ratio as
∆σ
σ∆σ∆σR
FCmax
y
FCmax
FCmin
+
minus=
++
=
Similarly the stress ratio in the tension fillet may require cor-rection
If the yield strength is exceeded in tension σFTmax gt σy the difference ∆ = σFTmax ndash σy affects the stress ratio as
y
FTmin
FTmax
FTmin
σ∆σ
∆σ∆σR minus=
minusminus
=
Checking for possible exceeding of the yield strength has to be made with the highest torque and with the KAP if this ex-ceeds KA
3163 Forle 103 cycles The permissible stress range ∆σp is not to exceed
For R gt minus1 FPstpst σ
R1R1051
15∆σ
minus+
+=
For ndash infin lt R lt minus1 FPstpst σ
R1R10251
15∆σ
minus+
+=
For all values of R ∆σpst is limited by
not surface hardened F
y
Sσ225
32 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
surface hardened CF
YSHV5
Definition of ∆σ and R see 3162 with particular attention to possible correction of R if the yield strength is exceeded
σFPst see 32 determined for unidirectional stresses (YM = 1) and lt 103 cycles
3164 For 103 lt cycles lt 3106 ∆σp is to be determined by linear interpolation a log-log dia-gram
∆σp at NL load cycles is
6L 103p
exp
L
6
N p ∆σN103∆σ sdot
sdot=
6
3
103p
10p
∆σ
∆σlog28760exp
sdot
=
Classification Notes- No 412 33 May 2003
DET NORSKE VERITAS
4 Calculation of Scuffing Load Capacity
41 Introduction High surface temperatures due to high loads and sliding ve-locities can cause lubricant films to break down This sei-zure or welding together of areas of tooth surface is termed scuffing
In contrast to pitting and fatigue breakage which show a dis-tinct incubation period a single short overloading can lead to a scuffing failure In the ISO-TR13989 two criteria are mentioned The method used in this Classification Note is based on the principles of the flash temperature criterion
Note Bulk temperature in excess of 120ordmC for long periods may have an adverse effect on the surface durability see 211
42 General Criteria In no point along the path of contact the local contact tem-perature may exceed the permissible temperature ie
oilS
oilSB S
ϑ+ϑminusϑ
leϑ
50SB minusϑleϑ
where
Bϑ = max contact temperature along the path of contact
maxflaMBB ϑ+ϑ=ϑ
MBϑ = bulk temperature see 434
maxflaϑ = max flash temperature along the path of con-tact see 44
Sϑ = scuffing temperature see below
oilϑ = oil temperature before it reaches the mesh (max applicable for the actual load case to be used ie normally alarm temperature except for ice classes where a max expected tem-perature applies)
SS = required safety factor according to the Rules
The scuffing temperature Sϑ may be calculated as
L2
wrelT
002
40S XFZGX
ν100112085780
sdot
sdot++=ϑ
where
XwrelT = relative welding factor
XwrelT
Through hardened steel 10
Phosphated steel 125
Copper-plated steel 150
Nitrided steel 150
Less than 10 retained austenite
115
10 ndash 20 retained austenite 10
Casehardened steel
20 ndash 30 retained austenite 085
Austenitic steel 045
FZG = load stage according to FZG-Test A8390 (Note This is the load stage where scuffing oc-curs However due to scatter in test results cal-culations are to be made with one load stage less than the specification)
XL = lubricant factor
= 10 for mineral oils
= 08 for polyalfaolefins
= 07 for non-water-soluble polyglycols
= 06 for water-soluble polyglycols
= 15 for traction fluids
= 13 for phosphate esters
ν40 = kinematic oil viscosity at 40˚C (mm2s)
Application of other test methods such as the Ryder the FZG-Ryder R46574 and the FZG L-42 Test 141195110 may be specially considered
For high speed gears with very short time of contact Sϑ may be increased as follows provided use of EP-oils
Addition to the calculated scuffing temperature Sϑ
If micros 18tc ge no addition
If micros 18tc lt add ( )cwrelT t18X18 minussdotsdot
where
ct = contact time ( micros ) which is the time needed to cross the Hertzian contact width
( ) [ ]microsu1cosβn
uσ340tb1
Hc +sdotsdot
sdot=
σH as calculated in 221
For bevel gears use vu in stead of u
34 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
43 Influence Factors
431 Coefficient of friction The following coefficient of friction may apply
L025a
005oil
02
redC ΣC
Bt X R ηρv
w0048micro minus
=
where
wBt = specific tooth load (Nmm)
vΣC = sum of tangential velocities at pitch point At pitch line velocities gt 50 ms the limiting value of vΣC at v = 50 ms is to be used
ρredC = relative radius of curvature (transversal plane) at the pitch point
Cylindrical gears
HαHβvγAbt
Bt KKKKKb
Fw sdotsdotsdotsdotsdot= (see 1)
twΣC αsin v2v =
bCredC β cos ρρ =
Bevel gears
HαHβvγAtmb
Bt KKKKKb
Fw sdotsdotsdotsdotsdot= (see 1)
vtmtΣC αsin v2v =
bmvCredC β cos ρρ =
ηoil = dynamic viscosity (mPa s) at oilϑ calculated as
1000
ρ νη oiloil =
where ρ in kgm3 approximated as
( ) 70 15oil15 minusϑminusρ=ρ and νoil is kinematic viscosity at oilϑ and may be calculated by means of the following equation
( ) ( )++=+ 08νloglog08νloglog 100oil ( )
sdotminus
ϑ+minus313log373log
273log373log oil
( ) ( )( )08νloglog08νloglog 10040 +minus+
Ra = composite arithmetic mean roughness (micron) of pinion an wheel calculated as ( )2a1aa RR 05R +=This is defined as the roughness on the new flanks ie as manufactured
XL = see 42
432 Effective tip relief Ceff Ceff is the effective tip relief that amount of tip relief which compensates for the elastic deformation of the gear mesh ie
zero load at the tooth tip It is assumed (simplified) to be equal for both pinion and wheel
Cylindrical gears
for helical
γ
γ=cb
KKFC Abt
eff (see 1)
For spur
cbKKF
C Abteff
γ= (see 1)
Alternatively for spur and helical gears the non-linear ap-proach in 111 may be used (taking Ceff=δ)
Bevel gears
bcKFC Ambt
effγ
= (see 1)
where
γ
α
ε2ε44c
+sdot
=γ
433 Tip relief and extension Cylindrical gears
The extension of the tip relief is not to result in an effective contact ratio 1ltεα when the gear is unloaded (exceptions to this may only apply for applications where the gear is not to run at light loads) This means that the unrelieved part of the path of contact is to be minimum pbt It is further assumed that this unrelieved part is placed centrally on the path of contact
If root relief applies it has to be calculated as equivalent tip relief Ie pinion root relief (at mesh position A) is added to Ca2 and wheel root relief (at mesh position E) is added to Ca1 If no design tip relief or root relief on the mating gear is specified (ie if Ca1+Croot2 = 0 and visa versa) use the run-ning in amount see 112
Bevel gears
Bevel gears are to have heightwise crowning ie no distinct relievedunrelieved area This may be treated as tip and root relief For calculation purposes the root relief is combined with the tip relief of the mating member into an equivalent tip relief If no resulting tip relieves are specified the equivalent tip relives may be calculated as an approxima-tion based on the tool crowning Ca tool (per mille of tool module m0) as follows
2root1aeq1a CCC +=
1root2aeq2a CCC +=
where
Classification Notes- No 412 35 May 2003
DET NORSKE VERITAS
2
0
n0101atoolal m
)m2(mA1middotmCC
minus+minussdot=
( )
sdotminusminussdotsdotminus+= vtvb1
2vb1
21vavt1n1 αtan dddαsin 05x1mA
2
0
n020atool22a m
)mm(2A1middotmCC
minus+minussdot=
( )
sdotminusminussdotsdotminus+= vtvb2
2vb2
2va2vt2n2 αtan dddαsin 05x1mA
2
0
20atool1root1 m
A1mCC
minussdotsdot=
2
0
10atool2root2 m
A1mCC
minussdotsdot=
If the pressure angles of the cutter blades are modified (and verified) to balance the tip relieves the following may be assumed
( )eq2aeq1aeq2aeq1a C andC calculated of sum50CC sdot==
Throughout the following Ca1 and Ca2 mean the equivalent tip relieves Ca1eq and Ca2eq
434 Bulk temperature The bulk temperature may be calculated as
flaaveragempsoilMB X X 50 ϑ+ϑ=ϑ
where
Xs = lubrication factor
= 12 for spray lubrication
= 10 for dip lubrication (provided both pinion and wheel are dip lubricated and tip speed lt 5 ms)
= 10 for spray lubrication with additional cooling spray (spray on both pinion and wheel or spray on pinion and dip lubrication of wheel)
= 02 for meshes fully submerged in oil
Xmp = contact factor ( )pmp nX += 150
np = number of mesh contact on the pinion (for small gear ratii the number of wheel meshes should be used if higher)
flaaverageϑ
= average of the integrated flash temperature (see 44) along the path of contact
( )
AE
E
Ayyyfla
flaaverage
d
ΓminusΓ
ΓΓϑ=ϑint =
For high speed gears (v gt 50 ms) it may be necessary to assess the bulk temperature on the basis of a thermal rating of the entire gear transmission
44 The Flash Temperature flaϑ
441 Basic formula The local flash temperature flaϑ may be calculated as
( ) 41redy
y2ly
211
43Btcorrfla
unXwX3250
y ρ
ρminusρ
micro=ϑ Γ
(For bevel gears replace u with uv)
and is to be calculated stepwise along the path of contact from A to E
where
micro = coefficient of friction see 431
Xcorr = correction factor taking empirically into account the increased scuffing risk in the beginning of the approach path due to mesh starting without any previously built up oil film and possible shuffling away oil before meshing if insufficient tip relief
( )
3
AD
yaeffcorr 50
CC1X
ΓminusΓε
Γminus+=
α
Ca = tip relief of driven member Xcorr is only applicable in the approach path and if Ca lt Ceff otherwise 10
wBt = unit load see 431
yXΓ = load sharing factor see 443
n1 = pinion rpm
Γy ρly etc see 442
442 Geometrical relations The various radii of flank curvature (transversal plane) are
ρ1y = pinion flank radius at mesh point y
ρ2y = wheel flank radius at mesh point y
ρredy = equivalent radius of curvature at mesh point y
y2y1
y2y1redy
ρ+ρ
ρρ=ρ
Cylindrical gears
twy
1y αsin au1Γ1
ρ+
+=
twy
2y αsin au1Γu
ρ+
minus=
Note that for internal gears a and u are negative
36 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
Bevel gears
vtvv
y1y αsin a
u1Γ1
ρ+
+=
vtvv
yv2y αsin a
u1Γu
ρ+
minus=
Γ is the parameter on the path of contact and y is any point between A end E
At the respective ends Γ has the following values
Root piniontip wheel
Cylindrical gears
( )
minus
minusminus= 1
αtan 1dd
zzΓ
tw
2b2a2
1
2A
Bevel gears
( )
minus
minusminus= 1
αtan 1dd
uΓvt
2vb2va2
vA
Tip pinionroot wheel
Cylindrical gears
( )
1αtan
1ddΓ
tw
2b1a1
E minusminus
=
Bevel gears
( )
1αtan
1ddΓ
vt
2vb1va1
E minusminus
=
At inner point of single pair contact
Cylindrical gears
tw1
EB α tan zπ2ΓΓ minus=
Bevel gears
vtv1
EB α tan zπ2ΓΓ minus=
At outer point of single pair contact
Cylindrical gears
tw1AD α tan z
π2ΓΓ +=
Bevel gears
vt v1
AD αtan z π2ΓΓ +=
At pitch point ΓC = 0
The points F and G (only applicable to cylindrical gears) limiting the extension of tip relief (so as to maintain a mini-mum contact ratio of unity for unloaded gears) are at
2
BAF
Γ+Γ=Γ
2
EDG
Γ+Γ=Γ
443 Load sharing factor XΓ The load sharing factor XΓ accounts for the load sharing be-tween the various pairs of teeth in mesh along the path of contact
XΓ is to be calculated stepwise from A to E using the pa-rameter Γy
4431 Cylindrical gears with β = 0 and no tip relief
Figure 41
ByAAB
Ay for31
31X
yΓltΓleΓ
ΓminusΓ
ΓminusΓ+=Γ
DyB for1Xy
ΓltΓleΓ=Γ
EyDDE
yE for31
31X
yΓleΓltΓ
ΓminusΓ
ΓminusΓ+=Γ
4432 Cylindrical gears with β = 0 and tip relief Tip relief on the pinion reduces XΓ in the range G ndash E and increases correspondingly XΓ in the range F ndash B
Tip relief on the wheel reduces XΓ in the range A ndash F and increases correspondingly XΓ in the range D ndash G
Following remains generally
DyB for1Xy
ΓleΓleΓ=Γ
21XXGF== ΓΓ
In the following it must be distinguished between Ca lt Ceff respectively Ca gt Ceff This is shown by an example below where Ca1 lt Ceff and Ca2 gt Ceff
Classification Notes- No 412 37 May 2003
DET NORSKE VERITAS
Figure 42
Note When Ca gt Ceff the path of contact is shortened by A ndash A respectively E ndash E The single pair contact path is extended into B respectively D If this shift is significant it is necessary to consider the negative effect on surface durability (B) and bending stresses (B and D)
Range A - F
For effa2 CC le
minus=Γ
eff
2a
CC1
31X
A
FyAeff
2a
AF
Ay for C3
C61XX
AyΓleΓleΓ
sdot
+ΓminusΓ
ΓminusΓ+= ΓΓ
For effa2 CC ge
AyAfor0Xy
ΓleΓleΓ=Γ
with ( )21
CC
1CC
eff
2a
eff
2a
AFAAminus
minusΓminusΓ+Γ=Γ
FyAeff
2a
AF
Ay
eff
2a for 21
CC
CC1X
yΓleΓleΓ
minus
ΓminusΓ
ΓminusΓ+minus=Γ
Range F - B
For effa1 CC le
ByFeff
1a
FB
Fy forC3
C61
21X
yΓleΓleΓ
sdot
+ΓminusΓ
ΓminusΓ+=Γ
For effa1 CC ge
ByFeff
1a
FB
Fy for21
CC
21X
yΓleΓleΓ
minus
ΓminusΓ
ΓminusΓ+=Γ
ByB for1Xy
ΓleΓleΓ=Γ
with 1
CC2
eff
1a
FBFB
minus
ΓminusΓ+Γ=Γ
Range D ndash G
For effa2 CC le
GyDeff
2a
DG
Dy
eff
2a for C 3C
61
C 3C
32X
yΓleΓleΓ
+
ΓminusΓ
ΓminusΓminus+=Γ F
or effa2 CC ge
DyD for1Xy
ΓleΓleΓ=Γ
with ( )21
CC
1CC
eff
2a
eff
2a
DGDDminus
minusΓminusΓ+Γ=Γ
GyDeff
2a
DG
Dy
eff
2a for21
CC
CCX ΓleΓleΓ
minus
ΓminusΓ
ΓminusΓminus=Γ
Range G - E
For effa1 CC le
minus=Γ
eff
1a
CC1
31X
E
EyGeff
1a
GE
Gy forC3
C61
21X
yΓleΓleΓ
sdot
+ΓminusΓ
ΓminusΓminus=Γ
For effa1 CC gt
EyGeff
1a
GE
Gy for21
CC
21X
yΓleΓleΓ
minus
ΓminusΓ
ΓminusΓminus=Γ
EyE for0Xy
ΓleΓleΓ=Γ
with 1
CC2
eff
1a
GEGE
minus
ΓminusΓ+Γ=Γ
4433 Gears with β gt 0 buttressing Due to oblique contact lines over the flanks a certain but-tressing may occur near A and E
This applies to both cylindrical and bevel gears with tip relief lt Ceff The buttressing Xbutt is simplified as a linear function within the ranges A ndash H respectively I ndash E
Figure 43
1εwhen13X βbutt EAge=
1εwhenε031X ββbutt EAlt+=
38 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
Cylindrical gears
bIEAH βsin 02ΓΓΓΓ =minus=minus
Bevel gears
bmIEAH βsin 02ΓΓΓΓ =minus=minus
4434 Cylindrical gears with 2εγ le and no tip relief
yXΓ is obtained by multiplication of
yXΓ in 4431 with
Xbutt in 4433
4435 Gears with 2εγ gt and no tip relief
Applicable to both cylindrical and bevel gears
IyH for1Xy
ΓleΓleΓε
=α
Γ
IyHybutt and forX1Xy
ΓgtΓΓltΓε
=α
Γ
Figure 44
4436 Cylindrical gears with 2εγ le and tip relief
yXΓ is obtained by multiplication of
yXΓ in 4432 with Xbutt
in 4433
4437 Cylindrical gears with 2εγ gt and tip relief
Tip relief on the pinion (respectively wheel) reduces XГ in the range G ndash E (respectively A ndash F) and increases XГ in the range F ndash G
yXΓ is obtained by multiplication of
yXΓ as described below
with Xbutt in 4433
In the XГ example below the influence of tip relief is shown (without the influence of Xbutt) by means of
eff2aeff1a CC and CC ltgt
Tip relief gt Ceff causes new end points A respectively E of the path of contact
Figure 45
Range A ndash F
( ) ( )( ) eff
a2αa1α
AF
Ay
eff
2aeff
C 12C 13εC 1ε
CCCX
y +εε++minus
ΓminusΓ
ΓminusΓ+
εminus
=ααα
Γ
eff2aFyA CC if for leΓleΓleΓ
eff2aFyA CifCfor and geΓleΓleΓ
eff2aAyA CC iffor0Xy
gtΓleΓleΓ=Γ
( ) ( ) ( )( ) ( ) a2αa1α
αeffa2AFAA C 1ε 3C 1ε
1ε 2 CC with++minus+minus
ΓminusΓ+Γ=Γ
Range F ndash G
( )( )( ) GyF
eff
2a1a forC 1 2CC 11X
yΓleΓleΓ
+εε+minusε
+ε
=αα
α
αΓ
Range G ndash E
( ) ( )( ) eff
2a1a
GE
Gy
C 1 2C 1C 1 3XX
GFy +εεminusε++ε
ΓminusΓ
ΓminusΓminus=
αα
ααΓΓ minus
effa1EyG CC if for leΓleΓleΓ
effa1EyG CC if Γfor and geΓleΓle
eff1aEyE CC if for0Xy
geΓleΓleΓ=Γ
( ) ( ) ( )( ) ( ) 2a1a
eff1aGEEE C 1C 1 3
1 2 CC withminusε++ε+εminus
ΓminusΓminusΓ=Γαα
α
4438 Bevel gears with εγ more than approx 18 and heightwise crowning For Ca1 = Ca2 = Ceff the following applies
Figure 46
Classification Notes- No 412 39 May 2003
DET NORSKE VERITAS
( )AEM 50 Γ+Γ=Γ
( )( )2AD
3
2My651X
y ΓminusΓε
ΓminusΓminus
ε=
ααΓ
For tip relief lt Ceff yXΓ is found by linear interpolation be-
tween ( )effay CCX =Γ and ( )0CayX =Γ as in 4435
The interpolation is to be made stepwise from A to M with the influence of Ca2 and from M to E with the influence of Ca1 (For 2a1a CC ne there is a discontinuity at M)
Eg with Ca1 = 04 Ceff and Ca2 = 055 Ceff then
Range A ndash M
)CC()0C( eff2ay2ayyX 550X 450X =Γ=ΓΓ +=
Range M ndash E
)CC()0C( eff1ay1ayyX 40X 06X =Γ=ΓΓ +=
For tip relief gt Ceff the new end points A and E are found as
( )
minusΓminusΓ
ε+Γ=Γ α 1
CC
6 eff
2aADAA
( )
minusΓminusΓ
ε+Γ=Γ α 1
CC
6 eff
1aADEE
Range A ndash A
0Xy=Γ
Range A ndash M
( )( )
ΓminusΓ
ΓminusΓminus
minusε=
αΓ 2
MA
2My
eff
2a1
CC4
351Xy
Range M ndash E
( )( )
ΓminusΓ
ΓminusΓminus
minusε=
αΓ 2
ME
2My
eff
1a1
CC4
351Xy
Range E ndash E
0Xy=Γ
40 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
Appendix A Fatigue Damage Accumulation
The Palmgren-Miner cumulative damage calculation princi-ple is used The procedure may be applied as follows
A1 Stress Spectrum From the individual torque classes the torques (Ti) at the peak values of class intervals and the associated number of cycles (NLi) for both pinion and wheel are to be listed from the highest to the lowest torque
(In case of a cyclic torque variation within the torque classes it is advised to use the peak torque If the cyclic variation is such that the same teeth will repeatedly suffer the peak torque this is a must)
The stress spectra for tooth roots and flanks (σFi σHi) with all relevant factors (except KA) are to be calculated on the basis of the torque spectrum The load dependent K-factors are to be determined for each torque class
A2 σminusN-curve The stress versus load cycle curves for tooth roots and flanks (both pinion and wheel) are to be drawn on the basis of per-missible stresses (ie including the demanded minimum safety factors) as determined in 2 respectively 3 If different safety levels for high cycle fatigue and low cycle fatigue are desired this may be expressed by different demand safety factors applied at the endurance limit respectively at static strength
A3 Damage accumulation The individual damage ratio Di at ith stress level is defined as
Fi
Lii N
ND =
where
NLi = The number of applied cycles at ith stress
NFi = The number of cycles to failure at ith stress
Basically stresses σi below the permissible stress level for infinite life (if a constant ZN or YN is accepted) do not con-tribute to the damage sum However calculating the actual safety factor Sact as described below all the σi for which the product S σi is bigger than or equal to the permissible stress level for infinite life contribute to the damage sum and thus to the determination of Sact The final value of S is decisive
(NFi can be found mathematically by putting the permissible stress σpi equal the actual stress σi thereby finding the actual life factor This life factor can be solved with regard to load cycles ie NFi)
The damage sum ΣDi is not to exceed unity
If ΣDi ne 1 the safety against cumulative fatigue damage is different from the applied demand safety factor For deter-mination of this theoretical safety factor an iteration proce-dure is required as described in the following flowchart
S is correction factor with which the actual safety factor Sact can be found
Sact is the demand safety factor (used in determination of the permissible stresses in the σ ndash N ndash curve) times the correc-tion factor S
The full procedure is to be applied for pinion and wheel tooth roots and flanks
Note If alternating stresses occur in a spectrum of mainly pulsating stresses the alternating stresses may be replaced by equiva-lent pulsating stresses ie by means of division with the ac-tual mean stress influence factor YM
Classification Notes- No 412 41 May 2003
DET NORSKE VERITAS
Appendix B Application Factors for Diesel Driven Gears
For diesel driven gears the application factor KA depends on torsional vibrations Both normal operation and misfiring conditions have to be considered
Normally these two running conditions can be covered by only one calculation
B1 Definitions
Normal operation KAnorm = 0
normv0
TTT +
where
T0 = rated nominal torque
Tv norm = vibratory torque amplitude for normal opera-tion (see rules Pt4 Ch3 Sec1 G301 for defi-nition of ldquonormalrdquo irregularity)
Misfiring operation KA misf = 0
misfv
TTT +
where
T = remaining nominal torque when one cylinder out of action
Tv misf = vibratory torque amplitude in misfiring con-dition This refers to a permissible misfiring condition ie a condition that does not require automatic or immediate corrective actions as speed or pitch reduction
The normal operation is assumed to last for a very high num-ber of cycles such as 1010
The misfiring operation is assumed to last for a limited dura-tion such as 107 cycles
B2 Determination of decisive load Assuming life factor at 1010 cycles as YN = ZN = 092 which usually is relevant the calculation may be performed only once with the combination having the highest value of appli-cation factorlife factor
For bending stresses and scuffing the higher value of
092
K normA and 098
K misfA
For contact stresses the higher value of
092K normA and
113K misfA (but
097K misfA for nitrided gears)
B3 Simplified procedure Note that this is only a guidance and is not a binding con-vention
T = oTZ
1Zsdot
minus where Z = number of cylinders
( ) ideal Videal Vmisf Vnorm V TTT24ZT +minus=
where TV ideal = vibratory torque with all cylinders perfectly equal See also rules Pt4 Ch3 Sec1 G300
When using trends from torsional vibration analysis and measurements the following may be used
TV ideal To is close to zero for engines with few cylinders and using a suitable elastic coupling and increases with relative coupling stiffness and number of cylinders
TV misf To may be high for engines with few cylinders and decreases with number of cylinders
This can be indicated as
200Z
TT
ο
idealV asymp and 80Z04
TT
o
misfV minusasymp
Inserting this into the formulae for the two application fac-tors the following guidance can be given
118112K misfA minusasymp
115110K normA minusasymp
Since KA norm is to be combined with the lower life factors the decisive load condition will be the normal one and a KA of 115 will cover most relevant cases when a suitable elas-tic coupling is chosen
42 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
Appendix C Calculation of Pinion-Rack
Pinion-racks used for elevating of mobile offshore units are open gears that are subjected to wear and tear With normal specifications such as only a few hundred total operation cycles (site to site) the tooth bending stresses for static strength will be decisive for the lay out however with ex-ception of surface hardened pinions where case crushing has to be considered
In the following the use of part 1 and 3 for pinion-racks is shown including relevant simplifications
C1 Pinion tooth root stresses Since the load spectrum normally is dominated by high torques and few load cycles (in the range up to 1-2000) the static strength is decisive
The actual stress is calculated as
β1FSaFan1
tF1 KYY
mbFσ sdotsdotsdotsdot
=
b1 is limited to b2 + 2middotmn
YFa and YSa replace YF and YS because load application at tooth tip has to be assumed for such inaccurate gears
Pinions often use a non-involute profile in the dedendum part of the flank eg a constant radius equal the radius of curva-ture at reference circle For such pinions sFn and hFa are to be measured directly on a sectional drawing of the pinion tooth
Due to high loads and narrow facewidths it may be assumed that KFβ2 = KHβ = 10 However when b1 gt b2 then KFβ1 gt10
If no detailed documentation of KFβ1 is available the fol-lowing may be used
KFβ1 = 1 + 015middot(b1b2 ndash 1)
The permissible stress (not surface hardened) is calculated as
δrelTstF
Fst1FP1 Y
Sσσ sdot=
The mean stress influence due to leg lifting may be disre-garded
The actual and permissible stresses should be calculated for the relevant loads as given in the rules
C2 Rack tooth root stresses The actual stress is calculated as
SaFan2
tF2 YY
mbFσ sdotsdotsdot
=
See C1 for details
The permissible stress is calculated as
δrelTstF
Fst2FP2 Y
Sσσ sdot=
For alloyed steels (Ni Cr Mo) with high toughness and duc-tility the value of YδrelTst may be put equal to YSa
C3 Surface hardened pinions For surface hardened pinions the maximum load is not to cause crushing of the hardened layer of the flank
In principle the calculation described in 213 may be used but when the theoretical Hertzian stress exceeds the ap-proximately 18 times the yield strength of the rack material plastic deformation will occur This will limit the peak Hertzian stress but increases the contact width and thus the penetration of stresses into the depth
An approximation may be based on an assessment of contact width determined by means of equal areas under the theoreti-cal (elastic range) Hertzian contact and the elasto-plastic contact stress (limited to 18 middot σy) with the unknown width
22 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
3 Calculation of Tooth Strength
31 Scope and General Remarks Part 3 include the calculation of tooth root strength as limited by tooth root cracking (surface or subsurface initiated) and yielding
For rim thickness sR ge 35middotmn the strength is calculated by means of 32 ndash 313 For cylindrical gears the calculation is based on the assumption that the highest tooth root tensile stress arises by application of the force at the outer point of single tooth pair contact of the virtual spur gears The method has however a few limitations that are mentioned in 36
For bevel gears the calculation is based on force application at the tooth tip of the virtual cylindrical gear Subsequently the stress is converted to load application at the mid point of the flank due to the heightwise crowning
Bevel gears may also be calculated with the program BECAL In that case KA and Kv are to be included in the applied tooth force but not KFβ and KFα
In case of a thin annulus or a thin gear rim etc radial crack-ing can occur rather than tangential cracking (from root fillet to root fillet) Cracking can also start from the compression fillet rather than the tension fillet For rim thickness sR lt 35middotmn a special calculation procedure is given in 315 and 316 and a simplified procedure in 314
A tooth breakage is often the end of the life of a gear trans-mission Therefore a high safety SF against breakage is re-quired
It should be noted that this part 3 does not cover fractures caused by
bull oil holes in the tooth root space bull wear steps on the flank bull flank surface distress such as pits spalls or grey staining
Especially the latter is known to cause oblique fractures starting from the active flank predominately in spiral bevel gears but also sometimes in cylindrical gears
Specific calculation methods for these purposes are not given here but are under consideration for future revisions Thus depending on experience with similar gear designs limita-tions other than those outlined in part 3 may be applied
32 Tooth Root Stresses The local tooth root stress is defined as the max principal stress in the tooth root caused by application of the tooth force Ie the stress ratio R = 0 Other stress ratios such as for eg idler gears (R asymp -12) shrunk on gear rims (R gt 0) etc are considered by correcting the permissible stress level
321 Local tooth root stress The local tooth root stress for pinion and wheel may be as-sessed by strain gauge measurements or FE calculations or similar For both measurements and calculations all details are to be agreed in advance
Normally the stresses for pinion and wheel are calculated as
Cylindrical gears
FαFβvγAβSFn
tF K K K K K Y Y Y
m bFσ =
where
YF = Tooth form factor (see 33)
YS = Stress correction factor (see 34)
Yβ = Helix angle factor (see 36)
Ft KA Kγ Kv KFβ KFα see 15 ndash 110
b see 13
Bevel gears
FαFβvγASaFamn
mtF K K K K K Y Y Y
m bFσ ε=
where
YFa = Tooth form factor see 33
YSa = Stress correction factor see 34
Yε = Contact ratio factor see 35
Fmt KA etc see 15 ndash 110
b see 13
322 Permissible tooth root stress The permissible local tooth root stress for pinion respectively wheel for a given number of cycles N is
CXRrelTrelTF
NMFEFP YYYY
SYY
δσ
=σ
Note that all these factors YM etc are applicable to 3middot106 cycles when used in this formula for σFP The influence of other number of cycles on these factors is covered by the calculation of YN
where
σFE = Local tooth root bending endurance limit of reference test gear (see 37)
YM = Mean stress influence factor which accounts for other loads than constant load direction eg idler gears temporary change of load di-rection pre-stress due to shrinkage etc (see 38)
YN = Life factor for tooth root stresses related to reference test gear dimensions (see 39)
SF = Required safety factor according to the rules
YδrelT = Relative notch sensitivity factor of the gear to be determined related to the reference test gear (see 310)
YRrelT = Relative (root fillet) surface condition factor of the gear to be determined related to the reference test gear (see 311)
Classification Notes- No 412 23 May 2003
DET NORSKE VERITAS
YX = Size factor (see 312)
YC = Case depth factor (see 313)
33 Tooth Form Factors YF YFa The tooth form factors YF and YFa take into account the in-fluence of the tooth form on the nominal bending stress
YF applies to load application at the outer point of single tooth pair contact of the virtual spur gear pair and is used for cylindrical gears
YFa applies to load application at the tooth tip and is used for bevel gears
Both YF and YFa are based on the distance between the con-tact points of the 30˚-tangents at the root fillet of the tooth profile for external gears respectively 60˚ tangents for inter-nal gears
Figure 31 External tooth in normal section
Figure 32 Internal tooth in normal section
Definitions
n
2
n
Fn
enFn
Fe
F
α cosms
α cosmh6
Y
=
n
2
n
Fn
anFn
Fa
Fa
α cosms
α cosmh6
Y
=
In the case of helical gears YF and YFa are determined in the normal section ie for a virtual number of teeth
YFa differs from YF by the bending moment arm hFa and αFan and can be determined by the same procedure as YF with exception of hFe and αFan For hFa and αFan all indices e will change to a (tip)
The following formulae apply to cylindrical gears but may also be used for bevel gears when replacing
mn with mnm
zn with zvn
αt with αvt
β with βm
with undercut without undercut
Fig 33 Dimensions and basic rack profile of the teeth (finished profile)
Tool and basic rack data such as hfP ρfp and spr etc are re-ferred to mn ie dimensionless
331 Determination of parameters
( )n
n
prnfPnfP m
α cossαsin 1ρ
αtan h4πE
minusminusminusminus=
For external gears fPfP ρρ =
For internal gears ( )0z
195fPfP0
fPfP 10363156ρhxρρ
sdotminus+
+=
where
z0 = number of teeth of pinion cutter
x0 = addendum modification coefficient of pinion cutter
hfP = addendum of pinion cutter
ρfP = tip radius of pinion cutter
xhρG fpfp +minus=
24 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
τmE
2π
z2H
nnminus
minus=
with
3πτ = for external gears
6πτ = for internal gears
Htan zG2
nminusϑ=ϑ
(to be solved iteratively suitable start value6π
=ϑ for exter-
nal gears and 3π for internal gears)
a) Tooth root chord sFn For external gears
minus
ϑ+
ϑminus= ρ
cosG3
3πsinz
ms
fpnn
Fn
For bevel gears with a tooth thickness modification
xsm affects mainly sFn but also hFe and αFen The total influence of xsm on YFa Ysa can be approximated by only adding 2 xsm to sFn mn
For internal gears
minus
ϑ+
ϑminus= ρ
cosG
6πsinz
ms
fPnn
Fn
b) Root fillet radius ρF at 30ordm tangent
( )G2coszcosG2ρ
mρ
2n
2
fpn
F
minusϑϑ+=
c) Determination of bending moment arm hF dn = zn mn
b2α
αn βcosε
ε =
dan = dn + 2 ha
pbn = π mn cos αn
dbn = dn cos αn
( )4
d1εpzz
2dd
zz2d
2bn
2
αnbn
2bn
2an
en +
minusminus
minus=
en
bnen d
dcos arcα =
ennnn
e α invα invα x tan 22π
z1
minus+
+=γ
αFen = αen ndash γe
For external gears
( )
minus=
n
enFenee
n
Fe
mdαtan sin γ γcos
21
mh
]ρcos
G3πcosz fpn +
ϑminus
ϑminusminus
For internal gears
( ) minus
sdotsdotminus=
n
enFenee
n
Fe
mdα tanγ sinγ cos
21
mh
minus
ϑminus
ϑminussdot ρ
cosG3
6πcosz fPn
332 Gearing with εαn gt 2 For deep tooth form gearing ( )25ε2 αn lele produced with a verified grade of accuracy of 4 or better and with applied profile modification to obtain a trapezoidal load distribution along the path of contact the YF may be corrected by the factor YDT as
250ε205for 0666ε2366Y αnαnDT leleminus=
205εfor 10Y αnDT lt=
34 Stress Correction Factors YS YSa The stress correction factors YS and YSa take into account the conversion of the nominal bending stress to the local tooth root stress Thereby YS and YSa cover the stress increasing effect of the notch (fillet) and the fact that not only bending stresses arise at the root A part of the local stress is inde-pendent of the bending moment arm This part increases the more the decisive point of load application approaches the critical tooth root section
Therefore in addition to its dependence on the notch radius the stress correction is also dependent on the position of the load application ie the size of the bending moment arm
YS applies to the load application at the outer point of single tooth pair contact YSa to the load application at tooth tip
YS can be determined as follows
( )
+
+=L23121
1
sS qL01312Y
where Fe
Fn
hsL = and
F
Fns ρ2
sq = (see 33)
YSa can be calculated by replacing hFe with hFa in the above formulae
Note a) Range of validity 1 lt qs lt 8
In case of sharper root radii (ie produced with tools having too sharp tip radii) YS resp YSa must be specially considered
b) In case of grinding notches (due to insufficient protuberance of the hob) YS resp YSa can rise considerably and must be multiplied with
Classification Notes- No 412 25 May 2003
DET NORSKE VERITAS
g
g
ρt
0613
13
minus
where
tg = depth of the grinding notch
ρg = radius of the grinding notch
c) The formulae for YS resp YSa are only valid for αn = 20˚ However the same formulae can be used as a safe approximation for other pressure angles
35 Contact Ratio Factor Yε The contact ratio factor Yε covers the conversion from load application at the tooth tip to the load application at the mid point of the flank (heightwise) for bevel gears
The following may be used
Yε = 0625
36 Helix Angle Factor Yβ The helix angle factor Yβ takes into account the difference between the helical gear and the virtual spur gear in the nor-mal section on which the calculation is based in the first step In this way it is accounted for that the conditions for tooth root stresses are more favourable because the lines of contact are sloping over the flank
The following may be used (β input in degrees)
Yβ = 1 ndash εβ β120
When εβ gt 1 use εβ = 1 and when β gt 30deg use β = 30deg in the formula
However the above equation for Yβ may only be used for gears with β gt 25deg if adequate tip relief is applied to both pinion and wheel (adequate = at least 05 middot Ceff see 432)
37 Values of Endurance Limit σFE σFE is the local tooth root stress (max principal) which the material can endure permanently with 99 survival prob-ability 3106 load cycles is regarded as the beginning of the endurance limit or the lower knee of the σ ndash N curve σFE is defined as the unidirectional pulsating stress with a minimum stress of zero (disregarding residual stresses due to heat treatment) Other stress conditions such as alternating or pre-stressed etc are covered by the conversion factor YM
σFE can be found by pulsating tests or gear running tests for any material in any condition If the approval of the gear is to be based on the results of such tests all details on the testing conditions have to be approved by the Society Fur-ther the tests may have to be made under the Societys su-pervision
If no fatigue tests are available the following listed values for σFE may be used for materials subjected to a quality con-trol as the one referred to in the rules
σFE
Alloyed case hardened steels 1) (fillet surface hardness 58 ndash 63 HRC)
bull of specially approved high grade
1050
bull of normal grade
minus CrNiMo steels with approved process
1000
minus CrNi and CrNiMo steels generally 920 minus MnCr steels generally 850
Nitriding steel of approved grade quenched tempered and gas nitrided (surface hardness 700 ndash 800 HV)
840
Alloyed quenched and tempered steel bath or gas nitrided (surface hardness 500 ndash 700 HV)
720
Alloyed quenched and tempered steel flame or induction hardened 2) (incl entire root fillet) (fillet surface hardness 500 ndash 650 HV)
07 HV + 300
Alloyed quenched and tempered steel flame or induction hardened (excl entire root fillet) (σB = uts of base material)
025 σB + 125
Alloyed quenched and tempered steel 04 σB + 200
Carbon steel 025 σB + 250
Note All numbers given above are valid for separate forgings and for blanks cut from bars forged according to a qualified procedure see Pt 4 Ch 2 Sec 3 For rolled steel the values are to be reduced with 10 For blanks cut from forged bars that are not qualified as mentioned above the values are to be reduced with 20 For cast steel reduce with 40
1) These values are valid for a root radius
bull being unground If however any grinding is made in the root fillet area in such a way that the residual stresses may be affected σFE is to be reduced by 20 (If the grinding also leaves a notch see 34)
bull with fillet surface hardness 58 ndash 63 HRC In case of lower surface hardness than 58 HRC σFE is to be reduced with 20(58 ndash HRC) where HRC is the detected hardness (This may lead to a permissible tooth root stress that varies along the facewidth If so the actual tooth root stresses may also be considered along facewidth)
bull not being shot peened In case of approved shot peening σFE may be increased by 200 for gears where σFE is reduced by 20 due to root grinding Otherwise σFE may be increased by 100 for
6nm le and 100 ndash 5 (mn - 6) for mn gt 6 However the possible adverse influence on the flanks regarding grey staining should be considered and if necessary the flanks should be masked
2) The fillet is not to be ground after surface hardening Regarding possible root grinding see 1)
26 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
38 Mean stress influence Factor YM The mean stress influence factor YM takes into account the influence of other working stress conditions than pure pulsa-tions (R = 0) such as eg load reversals idler gears planets and shrink-fitted gears
YM (YMst) are defined as the ratio between the endurance (or static) strength with a stress ratio R ne 0 and the endurance (or static) strength with R = 0
YM and YMst apply only to a calculation method that assesses the positive (tensile) stresses and is therefore suitable for comparison between the calculated (positive) working stress σF and the permissible stress σFP calculated with YM or YMst
For thin rings (annulus) in epicyclic gears where the com-pression fillet may be decisive special considerations apply see 316
The following method may be used within a stress ratio ndash12 lt R lt 05
381 For idlers planets and PTO with ice class
M1M1R1
1Yor Y MstM
+minus
minus=
where
R = stress ratio = min stress divided by max stress
For designs with the same force applied on both forward- and back-flank R may be assumed to ndash 12
For designs with considerably different forces on forward- and back-flank such as eg a marine propulsion wheel with a power take off pinion R may be assessed as
branchmain theoffacewidth unit per forcepto offacewidt unit per force21minus
For a power take off (PTO) with ice class see 161 c
M considers the mean stress influence on the endurance (or static) strength amplitudes
M is defined as the reduction of the endurance strength am-plitude for a certain increase of the mean stress divided by that increase of the mean stress
Following M values may be used
Endurance limit
Static strength
Case hardened 08 ndash 015 Ys 1) 07
If shot peened 04 06
Nitrided 03 03
Induction or flame hardened
04 06
Not surface hardened steel 03 05
Cast steels 04 06 1)For bevel gears use Ys=2 for determination of M
The listed M values for the endurance limit are independent of the fillet shape (Ys) except for case hardening In princi-ple there is a dependency but wide variations usually only occur for case hardening eg smooth semicircular fillets versus grinding notches
382 For gears with periodical change of rotational direction For case hardened gears with full load applied periodically in both directions such as side thrusters the same formula for YM as for idlers (with R = ndash 12) may be used together with the M values for endurance limit This simplified approach is valid when the number of changes of direction exceeds 100 and the total number of load cycles exceeds 3middot106
For gears of other materials YM will normally be higher than for a pure idler provided the number of changes of direction is below 3middot106 A linear interpolation in a diagram with loga-rithmic number of changes of direction may be used ie from YM = 09 with one change to YM (idler) for 3middot106 changes This is applicable to YM for endurance limit For static strength use YM as for idlers
For gears with occasional full load in reversed direction such as the main wheel in a reversing gear box YM = 09 may be used
383 For gears with shrinkage stresses and unidirectional load For endurance strength
FE
fitM σ
σM1
M21Y+
minus=
σFE is the endurance limit for R = 0
For static strength YMst = 1 and σfit accounted for in 39b
σfit is the shrinkage stress in the fillet (30˚ tangent) and may be found by multiplying the nominal tangential (hoop) stress with a stress concentration factor
n
Ffit m
ρ215scf minus=
384 For shrink-fitted idlers and planets When combined conditions apply such as idlers with shrink-age stresses the design factor for endurance strength is
( ) ( ) FE
fitM σ
σR1M1
M2
M1M1R1
1Yminussdot+
minus
+minus
minus=
Symbols as above but note that the stress ratio R in this par-ticular connection should disregard the influence of σfit ie R normally equal ndash 12
For static strength
M1M1R1
1YMst
+minus
minus=
The effect of σfit is accounted for in 39 b
Classification Notes- No 412 27 May 2003
DET NORSKE VERITAS
385 Additional requirements for peak loads The total stress range (σmax ndash σmin) in a tooth root fillet is not to exceed
F
y
Sσ225
for not surface hardened fillets
FSHV5 for surface hardened fillets
39 Life Factor YN The life factor YN takes into account that in the case of limited life (number of cycles) a higher tooth root stress can be permitted and that lower stresses may apply for very high number of cycles
Decisive for the strength at limited life is the σ ndash N ndash curve of the respective material for given hardening module fillet radius roughness in the tooth root etc Ie the factors YδrelT YRelT YX and YM have an influence on YN
If no σ ndash N ndash curve for the actual material and hardening etc is available the following method may be used
Determination of the σ ndash N ndash curve
a) Calculate the permissible stress σFP for the beginning of the endurance limit (3middot106 cycles) including the influ-ence of all relevant factors as SF YδrelT YRelT YX YM and YC ie σFP = σFE middotYM middotYδrelT middotYRelT middotYX middotYC SF
b) Calculate the permissible laquostaticraquo stress (le 103 load cy-cles) including the influence of all relevant factors as SFst YδrelTst YMst and YCst ( )fitσCstYδrelTstYMstYFstσ
FstS1
FPstσ minussdotsdotsdot=
where σFst is the local tooth root stress which the material can resist without cracking (surface hardened materials) or unacceptable deformation (not surface hardened materials) with 99 survival probability
σFst
Alloyed case hardened steel 1) 2300
Nitriding steel quenched tempered and gas nitrided (surface hardness 700 ndash 800 HV)
1250
Alloyed quenched and tempered steel bath or gas nitrided (surface hardness 500 ndash 700 HV)
1050
Alloyed quenched and tempered steel flame or induction hardened (fillet surface hardness 500 ndash 650 HV)
18 HV + 800
Steel with not surface hardened fillets the smaller value of 2)
18 σB or 225 σy
1) This is valid for a fillet surface hardness of 58 ndash 63 HRC In case of lower fillet surface hardness than 58 HRC σFst is to be reduced with 30(58 ndash HRC) where HRC is the actual hardness Shot peening or grinding notches are not considered to have any significant influence on σFst 2) Actual stresses exceeding the yield point (σy or σ02) will alter the residual stresses locally in the ldquotensionrdquo fillet re-spectively ldquocompressionrdquo fillet This is only to be utilised for gears that are not later loaded with a high number of cycles at lower loads that could cause fatigue in the ldquocompressionrdquo fillet
c) Calculate YN as
NL gt 3middot106
YN = 1 or 001
L
6
N N103Y
sdot= ie Yn = 092 for 1010
The YN = 1 from 3middot106 on may only be used when special material cleanness applies see rules Pt4 Ch2
103ltNLlt3middot106exp
L
6
N N103Y
sdot=
cycles6103forσcycles310forσlog02876exp
FP
FPst
sdot=
NL lt 103 cycles6103forσcycles310forσ
NYFP
FPst
sdot=
or simply use σFPst as mentioned in b) directly
Guidance on number of load cycles NL for various applica-tions
bull For propulsion purpose normally NL = 1010 at full load (yachts etc may have lower values)
bull For auxiliary gears driving generators that normally operate with 70-90 of rated power NL = 108 with rated power may be applied
28 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
310 Relative Notch Sensitivity Factor YδrelT The dynamic (respectively static) relative notch sensitivity factor YδrelT (YδrelTst) indicate to which extent the theoreti-cally concentrated stress lies above the endurance limits (re-spectively static strengths) in the case of fatigue (respectively overload) breakage
YδrelT is a function of the material and the relative stress gra-dient It differs for static strength and endurance limit
The following method may be used
For endurance limit
for not surface hardened fillets
( )
024
s024
relT σ103133q21σ1012201351
Ysdotsdotminus
+sdotsdotminus+= minus
minus
δ
for all surface hardened fillets except nitrided
( )
106q21002451
Y srelT
++=δ
for nitrided fillets
( )
1347q2101421
Y srelT
++=δ
For static strength
for not surface hardened fillets1)
( )( )( )025
02
02502s
relTst σ3000821σ300 1Y0821Y
+
minus+=δ
for surface hardened fillets except nitrided
YδrelTst = 044 YS + 012
for nitrided fillets
YδrelTst = 06 + 02 YS 1) These values are only valid if the local stresses do not
exceed the yield point and thereby alter the residual stress level See also 39b footnote 2
311 Relative Surface Condition Factor YRrelT The relative surface condition factor YRrelT takes into ac-count the dependence of the tooth root strength on the sur-face condition in the tooth root fillet mainly the dependence on the peak to valley surface roughness
YRrelT differs for endurance limit and static strength
The following method may be used
For endurance limit
YRrelT = 1675 ndash 053 (Ry + 1)01 for surface hardened steels and alloyed quenched and tempered steels except nitrided
YRrelT = 53 ndash 42 (Ry + 1)001 for carbon steels
YδrelT = 43 ndash 326 (Ry + 1)0005
for nitrided steels
For static strength
YRrelTst = 1 for all Ry and all materials
For a fillet without any longitudinal machining trace Ry asymp Rz
312 Size Factor YX The size factor YX takes into account the decrease of the strength with increasing size YX differs for endurance limit and static strength
The following may be used
For endurance limit
YX = 1 for mn le 5 generally
YX = 103 ndash 0006 mn for 5 lt mn lt 30
YX = 085 for mn ge 30
for not surface hard-ened steels
YX = 105 ndash 001 mn for 5 lt mn ge 25
YX = 08 for mn ge 25
for surface hardened steels
For static strength
YXst = 1 for all mn and all materials
313 Case Depth Factor YC The case depth factor YC takes into account the influence of hardening depth on tooth root strength
YC applies only to surface hardened tooth roots and is dif-ferent for endurance limit and static strength
In case of insufficient hardening depth fatigue cracks can develop in the transition zone between the hardened layer and the core For static strength yielding shall not occur in the transition zone as this would alter the surface residual stresses and therewith also the fatigue strength
The major parameters are case depth stress gradient permis-sible surface respectively subsurface stresses and subsurface residual stresses
The following simplified method for YC may be used
YC consists of a ratio between permissible subsurface stress (incl influence of expected residual stresses) and permissible surface stress This ratio is multiplied with a bracket con-taining the influence of case depth and stress gradient (The empirical numbers in the bracket are based on a high number of teeth and are somewhat on the safe side for low number of teeth)
YC and YCst may be calculated as given below but calculated values above 10 are to be put equal 10
For endurance limit
+
+=nFFE
C m02ρt31
σconstY
Classification Notes- No 412 29 May 2003
DET NORSKE VERITAS
For static strength
+
+=nFFst
Cst m02ρt31
σconstY
where const and t are connected as
Hardening process
t = endurance limit const =
static strength const =
t550 640 1900
t400 500 1200 Case hard-ening
t300 380 800
Nitriding t400 500 1200
Induction- or flame hardening
tHVmin 11 HVmin 25 HVmin
For symbols see 213
In addition to these requirements to minimum case depths for endurance limit some upper limitations apply to case hard-ened gears
The max depth to 550 HV should not exceed
1) 13 of the top land thickness san unless adequate tip relief is applied (see 110)
2) 025 mn If this is exceeded the following applies addi-tionally in connection with endurance limit
minusminus= 025
mt
1Yn
max550C
314 Thin rim factor YB Where the rim thickness is not sufficient to provide full sup-port for the tooth root the location of a bending failure may be through the gear rim rather than from root fillet to root fillet
YB is not a factor used to convert calculated root stresses at the 30deg tangent to actual stresses in a thin rim tension fillet Actually the compression fillet can be more susceptible to fatigue
YB is a simplified empirical factor used to de-rate thin rim gears (external as well as internal) when no detailed calcula-tion of stresses in both tension and compression fillets are available
Figure 34 Examples on thin rims
YB is applicable in the range 175 lt sRmn lt 35
YB = 115 middot ln (8324 middot mnsR)
(for sRmn ge 35 YB = 1)
(for sRmn le 175 use 315)
σF as calculated in 321 is to be multiplied with YB when sRmn lt 35 Thus YB is used for both high and low cycle fatigue
Note This method is considered to be on the safe side for external gear rims However for internal gear rims without any flange or web stiffeners the method may not be on the safe side and it is advised to check with the method in 315316
315 Stresses in Thin Rims For rim thickness sR lt 35 mn the safety against rim cracking has to be checked
The following method may be used
3151 General The stresses in the standardised 30ordm tangent section tension side are slightly reduced due to decreasing stress correction factor with decreasing relative rim thickness sRmn On the other hand during the complete stress cycle of that fillet a certain amount of compression stresses are also introduced The complete stress range remains approximately constant Therefore the standardised calculation of stresses at the 30ordm tangent may be retained for thin rims as one of the necessary criteria
The maximum stress range for thin rims usually occurs at the 60ordm ndash 80ordm tangents both for laquotensionraquo and laquocompressionraquo side The following method assumes the 75ordm tangent to be the decisive Therefore in addition to the am criterion ap-plied at the 30ordm tangent it is necessary to evaluate the max and min stresses at the 75ordm tangent for both laquotensionraquo (loaded flank) fillet and laquocompressionraquo (back-flank) fillet For this purpose the whole stress cycle of each fillet should be considered but usually the following simplification is justified
Figure 35 Nomenclature of fillets
Index laquoTraquo means laquotensileraquo fillet laquoCraquo means laquocompressionraquo fillet
σFTmin and σFCmax are determined on basis of the nominal rim stresses times the stress concentration factor Y75
σFCmin and σFTmax are determined on basis of superposition of nominal rim stresses times Y75 plus the tooth bending stresses at 75ordm tangent
30 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
3152 Stress concentration factors at the 75ordm tangents The nominal rim stress consists of bending stresses due to local bending moments tangential stresses due to the tan-gential force Ft and radial shear stresses due to Fr
The major influence is given by the bending stresses The influence of the tangential stresses is minor and even though its stress concentration factor is slightly higher than for bending it is considered to be safe enough when the sum of these nominal stresses are combined with the stress concen-tration factor for bending The influence of the radial shear stress is neglected
The stress concentration factor relating nominal rim stresses to local fillet stresses at the 75ordm tangent may be calculated as
75n
R75
n
R
75
ρmsρ185
ms3
Y+
sdot=
where ρ75 is the root radius at the 75ordm tangent ref to mn Usually ρ75 is closed to the tool radius ρao and
ρ75 = ρao
is a safe approximation compensating for the am simplifi-cations to the laquounsaferaquo side
The tooth root stresses of the loaded tooth are decreasing with decreasing relative rim thickness approximately with the empirical factor
15
R
ncorr s
m311Y
minus=
3153 Nominal rim stresses The bending moment applied to the rim consists of a part of the tooth tilt moment Ft (hF + 05 sR) and the bending caused by the radial force Fr
The sectional modulus (first moment of area) which is used for determination of the nominal bending stresses is not nec-essarily the same for the 2 am bending moments If flanges webs etc outside the toothed section contribute to stiffening the rim against various deflections the influence of these stiffeners should be considered Eg an end flange will have an almost negligible influence on the effective sectional modulus for the stresses due to tooth tilt as the deflection caused by the tilt moment is rather small and would not much involve the flange On the other hand the radial forces as for instance from the meshes in an annulus would cause considerable radial deflections that the flange might restrict to a substantial amount When considering the stiff-ening of such flanges or webs on basis of simplified models it is advised to use an effective rim thickness sR = sR + 02 mn for the first moment of area of the rim (toothed part) cross section
For a high number of rim teeth it may be assumed that the rim bending moments in the fillets adjacent to the loaded tooth are of the same magnitude as right under the applied
force This assumption is reasonable for an annulus but rather much on the safe side for a hollow pinion
The influence of Ft on the nominal tangential stress is simpli-fied by half of it for compressive stresses (σ1) and the other half for tensile stresses (σ2) Applying these assumptions the nominal rim stresses adjacent to the loaded tooth are
( ) ( )A2
FW
f RFW
s05hF05 t
R
r
T
RFt1 minus
ϑminus
+minus=σ
( ) ( )A2
FW
f RFW
s05hF05 t
R
r
T
RFt2 +
ϑminus
+=σ
where σ1 σ2 see Fig 35
A = minimum area of cross section (usually bR sR)
WT = the sectional modulus of rim with respect to tooth tilt moment (usually 6sb 2
rR )
WR = the sectional modulus of the rim including the influence of stiffeners as flanges webs etc (WR ge WT)
bR = the width of the rim
R = the radius of the neutral axis in the rim ie from wheel centre to midpoint of rim
( )ϑf = a function for bending moment distribution around the rim For a rim (pinion) with one mesh only the ( )ϑf at the position of load application is 024 For an annulus with 3 or more meshes ( )ϑf at the position of each load application is approx 3 planets ( )ϑf = 019 4 planets ( )ϑf = 014 5 planets ( )ϑf = 011 6 planets ( )ϑf = 009
It must be checked if the max (tensile) stress in the compres-sion fillet really occurs when the fillet is adjacent to the loaded tooth In principle the stress variation through a complete rotation should be considered and the max value used The max value is usually never less than 0 For an annulus eg the tilt moment is zero in the mid position be-tween the planet meshes whilst the bending moment due to the radial forces is half of that at the mesh but with opposite sign
If these formulae are applied to idler gears as eg planets the influence of nominal tangential stresses must be cor-rected by deleting Ft(2 A) for σ1 and using FtA for σ2 Further the influence of Fr on the nominal bending stresses is usually negligible due to the planet bearing support
3154 Root fillet stresses Determination of min and max stresses in the laquotensionraquo fillet
Classification Notes- No 412 31 May 2003
DET NORSKE VERITAS
Minimum stress
751FTmin YσK σ =
Maximum stress
corrF752 Yσ03Yσ KσFTmax +=
where
03 is an empirical factor relating the tension stresses (σF) at the 30ordm tangent to the part of the tension stresses at the 75ordm tangent which add to the rim related stresses (03 also takes into account that full superposition of nominal stresses times stress concentration factors from both laquosides of the corner filletraquo would result in too high stresses)
αβγ sdotsdotsdotsdot= FFvA KKKKKK
Determination of min and max stresses in the laquocompres-sionraquo fillet
Minimum stress
corrF751FCmin Yσ036Yσ Kσ minus=
Maximum stress
752FCmax Yσ Kσ =
where
036 is an empirical factor relating the tension stresses (σF) at the 30ordm tangent to the part of the compression stresses at the 75ordm tangent which add to the rim related stresses
For gears with reversed loads as idler gears and planets there is no distinct laquotensionraquo or laquocompressionraquo fillet The mini-mum stress σFmin is the minimum of σFTmin and σFCmin (usually the latter is decisive) The maximum stress σFTmax is the maximum of σFTmax and σFCmax (usually the former is deci-sive)
316 Permissible Stresses in Thin Rims
3161 General The safety against fatigue fracture respectively overload fracture is to be at least at the same level as for solid gears The laquoordinaryraquo criteria at the 30ordm tangent apply as given in 31 through 313
Additionally the following criteria at the 75ordm tangent may apply
3162 For gt3middot106 cycles The permissible stresses for the laquotensionraquo fillets and for the laquocompressionraquo fillets are determined by means of a relevant fatigue diagram
If the actual tooth root stress (tensile or compressive) ex-ceeds the yield strength to the material the induced residual stresses are to be taken into account
For determination of permissible stresses the following is defined
R = stress ratio ie FTmax
FTminσσ respectively
FCmax
FCmin
σσ
∆σ = stress range ie σFTmax ndash σFTmin resp σFCmax ndash σFCmin
(For idler gears and planets Fmax
FminσσR =
and ∆σ = σFmax minus σFmin)
The permissible stress range ∆σρ for the laquotensionraquo respec-tively laquocompressionraquo fillets can be calculated as
For R gt minus1 FPp σ
R1R1031
13∆σ
minus+
+=
For minus infin lt R lt minus1 FPp σ
R1R10151
13∆σ
minus+
+=
where
σFP = see 32 determined for unidirectional stresses (YM = 1)
If the yield strength σy is exceeded in either tension or com-pression residual stresses are induced This may be consid-ered by correcting the stress ratio R for the respective fillets (tension or compression)
Eg if yFCmin σσ gt (ie exceeded in compression) the
difference yFCmin σσ∆ minus= affects the stress ratio as
∆σ
σ∆σ∆σR
FCmax
y
FCmax
FCmin
+
minus=
++
=
Similarly the stress ratio in the tension fillet may require cor-rection
If the yield strength is exceeded in tension σFTmax gt σy the difference ∆ = σFTmax ndash σy affects the stress ratio as
y
FTmin
FTmax
FTmin
σ∆σ
∆σ∆σR minus=
minusminus
=
Checking for possible exceeding of the yield strength has to be made with the highest torque and with the KAP if this ex-ceeds KA
3163 Forle 103 cycles The permissible stress range ∆σp is not to exceed
For R gt minus1 FPstpst σ
R1R1051
15∆σ
minus+
+=
For ndash infin lt R lt minus1 FPstpst σ
R1R10251
15∆σ
minus+
+=
For all values of R ∆σpst is limited by
not surface hardened F
y
Sσ225
32 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
surface hardened CF
YSHV5
Definition of ∆σ and R see 3162 with particular attention to possible correction of R if the yield strength is exceeded
σFPst see 32 determined for unidirectional stresses (YM = 1) and lt 103 cycles
3164 For 103 lt cycles lt 3106 ∆σp is to be determined by linear interpolation a log-log dia-gram
∆σp at NL load cycles is
6L 103p
exp
L
6
N p ∆σN103∆σ sdot
sdot=
6
3
103p
10p
∆σ
∆σlog28760exp
sdot
=
Classification Notes- No 412 33 May 2003
DET NORSKE VERITAS
4 Calculation of Scuffing Load Capacity
41 Introduction High surface temperatures due to high loads and sliding ve-locities can cause lubricant films to break down This sei-zure or welding together of areas of tooth surface is termed scuffing
In contrast to pitting and fatigue breakage which show a dis-tinct incubation period a single short overloading can lead to a scuffing failure In the ISO-TR13989 two criteria are mentioned The method used in this Classification Note is based on the principles of the flash temperature criterion
Note Bulk temperature in excess of 120ordmC for long periods may have an adverse effect on the surface durability see 211
42 General Criteria In no point along the path of contact the local contact tem-perature may exceed the permissible temperature ie
oilS
oilSB S
ϑ+ϑminusϑ
leϑ
50SB minusϑleϑ
where
Bϑ = max contact temperature along the path of contact
maxflaMBB ϑ+ϑ=ϑ
MBϑ = bulk temperature see 434
maxflaϑ = max flash temperature along the path of con-tact see 44
Sϑ = scuffing temperature see below
oilϑ = oil temperature before it reaches the mesh (max applicable for the actual load case to be used ie normally alarm temperature except for ice classes where a max expected tem-perature applies)
SS = required safety factor according to the Rules
The scuffing temperature Sϑ may be calculated as
L2
wrelT
002
40S XFZGX
ν100112085780
sdot
sdot++=ϑ
where
XwrelT = relative welding factor
XwrelT
Through hardened steel 10
Phosphated steel 125
Copper-plated steel 150
Nitrided steel 150
Less than 10 retained austenite
115
10 ndash 20 retained austenite 10
Casehardened steel
20 ndash 30 retained austenite 085
Austenitic steel 045
FZG = load stage according to FZG-Test A8390 (Note This is the load stage where scuffing oc-curs However due to scatter in test results cal-culations are to be made with one load stage less than the specification)
XL = lubricant factor
= 10 for mineral oils
= 08 for polyalfaolefins
= 07 for non-water-soluble polyglycols
= 06 for water-soluble polyglycols
= 15 for traction fluids
= 13 for phosphate esters
ν40 = kinematic oil viscosity at 40˚C (mm2s)
Application of other test methods such as the Ryder the FZG-Ryder R46574 and the FZG L-42 Test 141195110 may be specially considered
For high speed gears with very short time of contact Sϑ may be increased as follows provided use of EP-oils
Addition to the calculated scuffing temperature Sϑ
If micros 18tc ge no addition
If micros 18tc lt add ( )cwrelT t18X18 minussdotsdot
where
ct = contact time ( micros ) which is the time needed to cross the Hertzian contact width
( ) [ ]microsu1cosβn
uσ340tb1
Hc +sdotsdot
sdot=
σH as calculated in 221
For bevel gears use vu in stead of u
34 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
43 Influence Factors
431 Coefficient of friction The following coefficient of friction may apply
L025a
005oil
02
redC ΣC
Bt X R ηρv
w0048micro minus
=
where
wBt = specific tooth load (Nmm)
vΣC = sum of tangential velocities at pitch point At pitch line velocities gt 50 ms the limiting value of vΣC at v = 50 ms is to be used
ρredC = relative radius of curvature (transversal plane) at the pitch point
Cylindrical gears
HαHβvγAbt
Bt KKKKKb
Fw sdotsdotsdotsdotsdot= (see 1)
twΣC αsin v2v =
bCredC β cos ρρ =
Bevel gears
HαHβvγAtmb
Bt KKKKKb
Fw sdotsdotsdotsdotsdot= (see 1)
vtmtΣC αsin v2v =
bmvCredC β cos ρρ =
ηoil = dynamic viscosity (mPa s) at oilϑ calculated as
1000
ρ νη oiloil =
where ρ in kgm3 approximated as
( ) 70 15oil15 minusϑminusρ=ρ and νoil is kinematic viscosity at oilϑ and may be calculated by means of the following equation
( ) ( )++=+ 08νloglog08νloglog 100oil ( )
sdotminus
ϑ+minus313log373log
273log373log oil
( ) ( )( )08νloglog08νloglog 10040 +minus+
Ra = composite arithmetic mean roughness (micron) of pinion an wheel calculated as ( )2a1aa RR 05R +=This is defined as the roughness on the new flanks ie as manufactured
XL = see 42
432 Effective tip relief Ceff Ceff is the effective tip relief that amount of tip relief which compensates for the elastic deformation of the gear mesh ie
zero load at the tooth tip It is assumed (simplified) to be equal for both pinion and wheel
Cylindrical gears
for helical
γ
γ=cb
KKFC Abt
eff (see 1)
For spur
cbKKF
C Abteff
γ= (see 1)
Alternatively for spur and helical gears the non-linear ap-proach in 111 may be used (taking Ceff=δ)
Bevel gears
bcKFC Ambt
effγ
= (see 1)
where
γ
α
ε2ε44c
+sdot
=γ
433 Tip relief and extension Cylindrical gears
The extension of the tip relief is not to result in an effective contact ratio 1ltεα when the gear is unloaded (exceptions to this may only apply for applications where the gear is not to run at light loads) This means that the unrelieved part of the path of contact is to be minimum pbt It is further assumed that this unrelieved part is placed centrally on the path of contact
If root relief applies it has to be calculated as equivalent tip relief Ie pinion root relief (at mesh position A) is added to Ca2 and wheel root relief (at mesh position E) is added to Ca1 If no design tip relief or root relief on the mating gear is specified (ie if Ca1+Croot2 = 0 and visa versa) use the run-ning in amount see 112
Bevel gears
Bevel gears are to have heightwise crowning ie no distinct relievedunrelieved area This may be treated as tip and root relief For calculation purposes the root relief is combined with the tip relief of the mating member into an equivalent tip relief If no resulting tip relieves are specified the equivalent tip relives may be calculated as an approxima-tion based on the tool crowning Ca tool (per mille of tool module m0) as follows
2root1aeq1a CCC +=
1root2aeq2a CCC +=
where
Classification Notes- No 412 35 May 2003
DET NORSKE VERITAS
2
0
n0101atoolal m
)m2(mA1middotmCC
minus+minussdot=
( )
sdotminusminussdotsdotminus+= vtvb1
2vb1
21vavt1n1 αtan dddαsin 05x1mA
2
0
n020atool22a m
)mm(2A1middotmCC
minus+minussdot=
( )
sdotminusminussdotsdotminus+= vtvb2
2vb2
2va2vt2n2 αtan dddαsin 05x1mA
2
0
20atool1root1 m
A1mCC
minussdotsdot=
2
0
10atool2root2 m
A1mCC
minussdotsdot=
If the pressure angles of the cutter blades are modified (and verified) to balance the tip relieves the following may be assumed
( )eq2aeq1aeq2aeq1a C andC calculated of sum50CC sdot==
Throughout the following Ca1 and Ca2 mean the equivalent tip relieves Ca1eq and Ca2eq
434 Bulk temperature The bulk temperature may be calculated as
flaaveragempsoilMB X X 50 ϑ+ϑ=ϑ
where
Xs = lubrication factor
= 12 for spray lubrication
= 10 for dip lubrication (provided both pinion and wheel are dip lubricated and tip speed lt 5 ms)
= 10 for spray lubrication with additional cooling spray (spray on both pinion and wheel or spray on pinion and dip lubrication of wheel)
= 02 for meshes fully submerged in oil
Xmp = contact factor ( )pmp nX += 150
np = number of mesh contact on the pinion (for small gear ratii the number of wheel meshes should be used if higher)
flaaverageϑ
= average of the integrated flash temperature (see 44) along the path of contact
( )
AE
E
Ayyyfla
flaaverage
d
ΓminusΓ
ΓΓϑ=ϑint =
For high speed gears (v gt 50 ms) it may be necessary to assess the bulk temperature on the basis of a thermal rating of the entire gear transmission
44 The Flash Temperature flaϑ
441 Basic formula The local flash temperature flaϑ may be calculated as
( ) 41redy
y2ly
211
43Btcorrfla
unXwX3250
y ρ
ρminusρ
micro=ϑ Γ
(For bevel gears replace u with uv)
and is to be calculated stepwise along the path of contact from A to E
where
micro = coefficient of friction see 431
Xcorr = correction factor taking empirically into account the increased scuffing risk in the beginning of the approach path due to mesh starting without any previously built up oil film and possible shuffling away oil before meshing if insufficient tip relief
( )
3
AD
yaeffcorr 50
CC1X
ΓminusΓε
Γminus+=
α
Ca = tip relief of driven member Xcorr is only applicable in the approach path and if Ca lt Ceff otherwise 10
wBt = unit load see 431
yXΓ = load sharing factor see 443
n1 = pinion rpm
Γy ρly etc see 442
442 Geometrical relations The various radii of flank curvature (transversal plane) are
ρ1y = pinion flank radius at mesh point y
ρ2y = wheel flank radius at mesh point y
ρredy = equivalent radius of curvature at mesh point y
y2y1
y2y1redy
ρ+ρ
ρρ=ρ
Cylindrical gears
twy
1y αsin au1Γ1
ρ+
+=
twy
2y αsin au1Γu
ρ+
minus=
Note that for internal gears a and u are negative
36 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
Bevel gears
vtvv
y1y αsin a
u1Γ1
ρ+
+=
vtvv
yv2y αsin a
u1Γu
ρ+
minus=
Γ is the parameter on the path of contact and y is any point between A end E
At the respective ends Γ has the following values
Root piniontip wheel
Cylindrical gears
( )
minus
minusminus= 1
αtan 1dd
zzΓ
tw
2b2a2
1
2A
Bevel gears
( )
minus
minusminus= 1
αtan 1dd
uΓvt
2vb2va2
vA
Tip pinionroot wheel
Cylindrical gears
( )
1αtan
1ddΓ
tw
2b1a1
E minusminus
=
Bevel gears
( )
1αtan
1ddΓ
vt
2vb1va1
E minusminus
=
At inner point of single pair contact
Cylindrical gears
tw1
EB α tan zπ2ΓΓ minus=
Bevel gears
vtv1
EB α tan zπ2ΓΓ minus=
At outer point of single pair contact
Cylindrical gears
tw1AD α tan z
π2ΓΓ +=
Bevel gears
vt v1
AD αtan z π2ΓΓ +=
At pitch point ΓC = 0
The points F and G (only applicable to cylindrical gears) limiting the extension of tip relief (so as to maintain a mini-mum contact ratio of unity for unloaded gears) are at
2
BAF
Γ+Γ=Γ
2
EDG
Γ+Γ=Γ
443 Load sharing factor XΓ The load sharing factor XΓ accounts for the load sharing be-tween the various pairs of teeth in mesh along the path of contact
XΓ is to be calculated stepwise from A to E using the pa-rameter Γy
4431 Cylindrical gears with β = 0 and no tip relief
Figure 41
ByAAB
Ay for31
31X
yΓltΓleΓ
ΓminusΓ
ΓminusΓ+=Γ
DyB for1Xy
ΓltΓleΓ=Γ
EyDDE
yE for31
31X
yΓleΓltΓ
ΓminusΓ
ΓminusΓ+=Γ
4432 Cylindrical gears with β = 0 and tip relief Tip relief on the pinion reduces XΓ in the range G ndash E and increases correspondingly XΓ in the range F ndash B
Tip relief on the wheel reduces XΓ in the range A ndash F and increases correspondingly XΓ in the range D ndash G
Following remains generally
DyB for1Xy
ΓleΓleΓ=Γ
21XXGF== ΓΓ
In the following it must be distinguished between Ca lt Ceff respectively Ca gt Ceff This is shown by an example below where Ca1 lt Ceff and Ca2 gt Ceff
Classification Notes- No 412 37 May 2003
DET NORSKE VERITAS
Figure 42
Note When Ca gt Ceff the path of contact is shortened by A ndash A respectively E ndash E The single pair contact path is extended into B respectively D If this shift is significant it is necessary to consider the negative effect on surface durability (B) and bending stresses (B and D)
Range A - F
For effa2 CC le
minus=Γ
eff
2a
CC1
31X
A
FyAeff
2a
AF
Ay for C3
C61XX
AyΓleΓleΓ
sdot
+ΓminusΓ
ΓminusΓ+= ΓΓ
For effa2 CC ge
AyAfor0Xy
ΓleΓleΓ=Γ
with ( )21
CC
1CC
eff
2a
eff
2a
AFAAminus
minusΓminusΓ+Γ=Γ
FyAeff
2a
AF
Ay
eff
2a for 21
CC
CC1X
yΓleΓleΓ
minus
ΓminusΓ
ΓminusΓ+minus=Γ
Range F - B
For effa1 CC le
ByFeff
1a
FB
Fy forC3
C61
21X
yΓleΓleΓ
sdot
+ΓminusΓ
ΓminusΓ+=Γ
For effa1 CC ge
ByFeff
1a
FB
Fy for21
CC
21X
yΓleΓleΓ
minus
ΓminusΓ
ΓminusΓ+=Γ
ByB for1Xy
ΓleΓleΓ=Γ
with 1
CC2
eff
1a
FBFB
minus
ΓminusΓ+Γ=Γ
Range D ndash G
For effa2 CC le
GyDeff
2a
DG
Dy
eff
2a for C 3C
61
C 3C
32X
yΓleΓleΓ
+
ΓminusΓ
ΓminusΓminus+=Γ F
or effa2 CC ge
DyD for1Xy
ΓleΓleΓ=Γ
with ( )21
CC
1CC
eff
2a
eff
2a
DGDDminus
minusΓminusΓ+Γ=Γ
GyDeff
2a
DG
Dy
eff
2a for21
CC
CCX ΓleΓleΓ
minus
ΓminusΓ
ΓminusΓminus=Γ
Range G - E
For effa1 CC le
minus=Γ
eff
1a
CC1
31X
E
EyGeff
1a
GE
Gy forC3
C61
21X
yΓleΓleΓ
sdot
+ΓminusΓ
ΓminusΓminus=Γ
For effa1 CC gt
EyGeff
1a
GE
Gy for21
CC
21X
yΓleΓleΓ
minus
ΓminusΓ
ΓminusΓminus=Γ
EyE for0Xy
ΓleΓleΓ=Γ
with 1
CC2
eff
1a
GEGE
minus
ΓminusΓ+Γ=Γ
4433 Gears with β gt 0 buttressing Due to oblique contact lines over the flanks a certain but-tressing may occur near A and E
This applies to both cylindrical and bevel gears with tip relief lt Ceff The buttressing Xbutt is simplified as a linear function within the ranges A ndash H respectively I ndash E
Figure 43
1εwhen13X βbutt EAge=
1εwhenε031X ββbutt EAlt+=
38 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
Cylindrical gears
bIEAH βsin 02ΓΓΓΓ =minus=minus
Bevel gears
bmIEAH βsin 02ΓΓΓΓ =minus=minus
4434 Cylindrical gears with 2εγ le and no tip relief
yXΓ is obtained by multiplication of
yXΓ in 4431 with
Xbutt in 4433
4435 Gears with 2εγ gt and no tip relief
Applicable to both cylindrical and bevel gears
IyH for1Xy
ΓleΓleΓε
=α
Γ
IyHybutt and forX1Xy
ΓgtΓΓltΓε
=α
Γ
Figure 44
4436 Cylindrical gears with 2εγ le and tip relief
yXΓ is obtained by multiplication of
yXΓ in 4432 with Xbutt
in 4433
4437 Cylindrical gears with 2εγ gt and tip relief
Tip relief on the pinion (respectively wheel) reduces XГ in the range G ndash E (respectively A ndash F) and increases XГ in the range F ndash G
yXΓ is obtained by multiplication of
yXΓ as described below
with Xbutt in 4433
In the XГ example below the influence of tip relief is shown (without the influence of Xbutt) by means of
eff2aeff1a CC and CC ltgt
Tip relief gt Ceff causes new end points A respectively E of the path of contact
Figure 45
Range A ndash F
( ) ( )( ) eff
a2αa1α
AF
Ay
eff
2aeff
C 12C 13εC 1ε
CCCX
y +εε++minus
ΓminusΓ
ΓminusΓ+
εminus
=ααα
Γ
eff2aFyA CC if for leΓleΓleΓ
eff2aFyA CifCfor and geΓleΓleΓ
eff2aAyA CC iffor0Xy
gtΓleΓleΓ=Γ
( ) ( ) ( )( ) ( ) a2αa1α
αeffa2AFAA C 1ε 3C 1ε
1ε 2 CC with++minus+minus
ΓminusΓ+Γ=Γ
Range F ndash G
( )( )( ) GyF
eff
2a1a forC 1 2CC 11X
yΓleΓleΓ
+εε+minusε
+ε
=αα
α
αΓ
Range G ndash E
( ) ( )( ) eff
2a1a
GE
Gy
C 1 2C 1C 1 3XX
GFy +εεminusε++ε
ΓminusΓ
ΓminusΓminus=
αα
ααΓΓ minus
effa1EyG CC if for leΓleΓleΓ
effa1EyG CC if Γfor and geΓleΓle
eff1aEyE CC if for0Xy
geΓleΓleΓ=Γ
( ) ( ) ( )( ) ( ) 2a1a
eff1aGEEE C 1C 1 3
1 2 CC withminusε++ε+εminus
ΓminusΓminusΓ=Γαα
α
4438 Bevel gears with εγ more than approx 18 and heightwise crowning For Ca1 = Ca2 = Ceff the following applies
Figure 46
Classification Notes- No 412 39 May 2003
DET NORSKE VERITAS
( )AEM 50 Γ+Γ=Γ
( )( )2AD
3
2My651X
y ΓminusΓε
ΓminusΓminus
ε=
ααΓ
For tip relief lt Ceff yXΓ is found by linear interpolation be-
tween ( )effay CCX =Γ and ( )0CayX =Γ as in 4435
The interpolation is to be made stepwise from A to M with the influence of Ca2 and from M to E with the influence of Ca1 (For 2a1a CC ne there is a discontinuity at M)
Eg with Ca1 = 04 Ceff and Ca2 = 055 Ceff then
Range A ndash M
)CC()0C( eff2ay2ayyX 550X 450X =Γ=ΓΓ +=
Range M ndash E
)CC()0C( eff1ay1ayyX 40X 06X =Γ=ΓΓ +=
For tip relief gt Ceff the new end points A and E are found as
( )
minusΓminusΓ
ε+Γ=Γ α 1
CC
6 eff
2aADAA
( )
minusΓminusΓ
ε+Γ=Γ α 1
CC
6 eff
1aADEE
Range A ndash A
0Xy=Γ
Range A ndash M
( )( )
ΓminusΓ
ΓminusΓminus
minusε=
αΓ 2
MA
2My
eff
2a1
CC4
351Xy
Range M ndash E
( )( )
ΓminusΓ
ΓminusΓminus
minusε=
αΓ 2
ME
2My
eff
1a1
CC4
351Xy
Range E ndash E
0Xy=Γ
40 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
Appendix A Fatigue Damage Accumulation
The Palmgren-Miner cumulative damage calculation princi-ple is used The procedure may be applied as follows
A1 Stress Spectrum From the individual torque classes the torques (Ti) at the peak values of class intervals and the associated number of cycles (NLi) for both pinion and wheel are to be listed from the highest to the lowest torque
(In case of a cyclic torque variation within the torque classes it is advised to use the peak torque If the cyclic variation is such that the same teeth will repeatedly suffer the peak torque this is a must)
The stress spectra for tooth roots and flanks (σFi σHi) with all relevant factors (except KA) are to be calculated on the basis of the torque spectrum The load dependent K-factors are to be determined for each torque class
A2 σminusN-curve The stress versus load cycle curves for tooth roots and flanks (both pinion and wheel) are to be drawn on the basis of per-missible stresses (ie including the demanded minimum safety factors) as determined in 2 respectively 3 If different safety levels for high cycle fatigue and low cycle fatigue are desired this may be expressed by different demand safety factors applied at the endurance limit respectively at static strength
A3 Damage accumulation The individual damage ratio Di at ith stress level is defined as
Fi
Lii N
ND =
where
NLi = The number of applied cycles at ith stress
NFi = The number of cycles to failure at ith stress
Basically stresses σi below the permissible stress level for infinite life (if a constant ZN or YN is accepted) do not con-tribute to the damage sum However calculating the actual safety factor Sact as described below all the σi for which the product S σi is bigger than or equal to the permissible stress level for infinite life contribute to the damage sum and thus to the determination of Sact The final value of S is decisive
(NFi can be found mathematically by putting the permissible stress σpi equal the actual stress σi thereby finding the actual life factor This life factor can be solved with regard to load cycles ie NFi)
The damage sum ΣDi is not to exceed unity
If ΣDi ne 1 the safety against cumulative fatigue damage is different from the applied demand safety factor For deter-mination of this theoretical safety factor an iteration proce-dure is required as described in the following flowchart
S is correction factor with which the actual safety factor Sact can be found
Sact is the demand safety factor (used in determination of the permissible stresses in the σ ndash N ndash curve) times the correc-tion factor S
The full procedure is to be applied for pinion and wheel tooth roots and flanks
Note If alternating stresses occur in a spectrum of mainly pulsating stresses the alternating stresses may be replaced by equiva-lent pulsating stresses ie by means of division with the ac-tual mean stress influence factor YM
Classification Notes- No 412 41 May 2003
DET NORSKE VERITAS
Appendix B Application Factors for Diesel Driven Gears
For diesel driven gears the application factor KA depends on torsional vibrations Both normal operation and misfiring conditions have to be considered
Normally these two running conditions can be covered by only one calculation
B1 Definitions
Normal operation KAnorm = 0
normv0
TTT +
where
T0 = rated nominal torque
Tv norm = vibratory torque amplitude for normal opera-tion (see rules Pt4 Ch3 Sec1 G301 for defi-nition of ldquonormalrdquo irregularity)
Misfiring operation KA misf = 0
misfv
TTT +
where
T = remaining nominal torque when one cylinder out of action
Tv misf = vibratory torque amplitude in misfiring con-dition This refers to a permissible misfiring condition ie a condition that does not require automatic or immediate corrective actions as speed or pitch reduction
The normal operation is assumed to last for a very high num-ber of cycles such as 1010
The misfiring operation is assumed to last for a limited dura-tion such as 107 cycles
B2 Determination of decisive load Assuming life factor at 1010 cycles as YN = ZN = 092 which usually is relevant the calculation may be performed only once with the combination having the highest value of appli-cation factorlife factor
For bending stresses and scuffing the higher value of
092
K normA and 098
K misfA
For contact stresses the higher value of
092K normA and
113K misfA (but
097K misfA for nitrided gears)
B3 Simplified procedure Note that this is only a guidance and is not a binding con-vention
T = oTZ
1Zsdot
minus where Z = number of cylinders
( ) ideal Videal Vmisf Vnorm V TTT24ZT +minus=
where TV ideal = vibratory torque with all cylinders perfectly equal See also rules Pt4 Ch3 Sec1 G300
When using trends from torsional vibration analysis and measurements the following may be used
TV ideal To is close to zero for engines with few cylinders and using a suitable elastic coupling and increases with relative coupling stiffness and number of cylinders
TV misf To may be high for engines with few cylinders and decreases with number of cylinders
This can be indicated as
200Z
TT
ο
idealV asymp and 80Z04
TT
o
misfV minusasymp
Inserting this into the formulae for the two application fac-tors the following guidance can be given
118112K misfA minusasymp
115110K normA minusasymp
Since KA norm is to be combined with the lower life factors the decisive load condition will be the normal one and a KA of 115 will cover most relevant cases when a suitable elas-tic coupling is chosen
42 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
Appendix C Calculation of Pinion-Rack
Pinion-racks used for elevating of mobile offshore units are open gears that are subjected to wear and tear With normal specifications such as only a few hundred total operation cycles (site to site) the tooth bending stresses for static strength will be decisive for the lay out however with ex-ception of surface hardened pinions where case crushing has to be considered
In the following the use of part 1 and 3 for pinion-racks is shown including relevant simplifications
C1 Pinion tooth root stresses Since the load spectrum normally is dominated by high torques and few load cycles (in the range up to 1-2000) the static strength is decisive
The actual stress is calculated as
β1FSaFan1
tF1 KYY
mbFσ sdotsdotsdotsdot
=
b1 is limited to b2 + 2middotmn
YFa and YSa replace YF and YS because load application at tooth tip has to be assumed for such inaccurate gears
Pinions often use a non-involute profile in the dedendum part of the flank eg a constant radius equal the radius of curva-ture at reference circle For such pinions sFn and hFa are to be measured directly on a sectional drawing of the pinion tooth
Due to high loads and narrow facewidths it may be assumed that KFβ2 = KHβ = 10 However when b1 gt b2 then KFβ1 gt10
If no detailed documentation of KFβ1 is available the fol-lowing may be used
KFβ1 = 1 + 015middot(b1b2 ndash 1)
The permissible stress (not surface hardened) is calculated as
δrelTstF
Fst1FP1 Y
Sσσ sdot=
The mean stress influence due to leg lifting may be disre-garded
The actual and permissible stresses should be calculated for the relevant loads as given in the rules
C2 Rack tooth root stresses The actual stress is calculated as
SaFan2
tF2 YY
mbFσ sdotsdotsdot
=
See C1 for details
The permissible stress is calculated as
δrelTstF
Fst2FP2 Y
Sσσ sdot=
For alloyed steels (Ni Cr Mo) with high toughness and duc-tility the value of YδrelTst may be put equal to YSa
C3 Surface hardened pinions For surface hardened pinions the maximum load is not to cause crushing of the hardened layer of the flank
In principle the calculation described in 213 may be used but when the theoretical Hertzian stress exceeds the ap-proximately 18 times the yield strength of the rack material plastic deformation will occur This will limit the peak Hertzian stress but increases the contact width and thus the penetration of stresses into the depth
An approximation may be based on an assessment of contact width determined by means of equal areas under the theoreti-cal (elastic range) Hertzian contact and the elasto-plastic contact stress (limited to 18 middot σy) with the unknown width
Classification Notes- No 412 23 May 2003
DET NORSKE VERITAS
YX = Size factor (see 312)
YC = Case depth factor (see 313)
33 Tooth Form Factors YF YFa The tooth form factors YF and YFa take into account the in-fluence of the tooth form on the nominal bending stress
YF applies to load application at the outer point of single tooth pair contact of the virtual spur gear pair and is used for cylindrical gears
YFa applies to load application at the tooth tip and is used for bevel gears
Both YF and YFa are based on the distance between the con-tact points of the 30˚-tangents at the root fillet of the tooth profile for external gears respectively 60˚ tangents for inter-nal gears
Figure 31 External tooth in normal section
Figure 32 Internal tooth in normal section
Definitions
n
2
n
Fn
enFn
Fe
F
α cosms
α cosmh6
Y
=
n
2
n
Fn
anFn
Fa
Fa
α cosms
α cosmh6
Y
=
In the case of helical gears YF and YFa are determined in the normal section ie for a virtual number of teeth
YFa differs from YF by the bending moment arm hFa and αFan and can be determined by the same procedure as YF with exception of hFe and αFan For hFa and αFan all indices e will change to a (tip)
The following formulae apply to cylindrical gears but may also be used for bevel gears when replacing
mn with mnm
zn with zvn
αt with αvt
β with βm
with undercut without undercut
Fig 33 Dimensions and basic rack profile of the teeth (finished profile)
Tool and basic rack data such as hfP ρfp and spr etc are re-ferred to mn ie dimensionless
331 Determination of parameters
( )n
n
prnfPnfP m
α cossαsin 1ρ
αtan h4πE
minusminusminusminus=
For external gears fPfP ρρ =
For internal gears ( )0z
195fPfP0
fPfP 10363156ρhxρρ
sdotminus+
+=
where
z0 = number of teeth of pinion cutter
x0 = addendum modification coefficient of pinion cutter
hfP = addendum of pinion cutter
ρfP = tip radius of pinion cutter
xhρG fpfp +minus=
24 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
τmE
2π
z2H
nnminus
minus=
with
3πτ = for external gears
6πτ = for internal gears
Htan zG2
nminusϑ=ϑ
(to be solved iteratively suitable start value6π
=ϑ for exter-
nal gears and 3π for internal gears)
a) Tooth root chord sFn For external gears
minus
ϑ+
ϑminus= ρ
cosG3
3πsinz
ms
fpnn
Fn
For bevel gears with a tooth thickness modification
xsm affects mainly sFn but also hFe and αFen The total influence of xsm on YFa Ysa can be approximated by only adding 2 xsm to sFn mn
For internal gears
minus
ϑ+
ϑminus= ρ
cosG
6πsinz
ms
fPnn
Fn
b) Root fillet radius ρF at 30ordm tangent
( )G2coszcosG2ρ
mρ
2n
2
fpn
F
minusϑϑ+=
c) Determination of bending moment arm hF dn = zn mn
b2α
αn βcosε
ε =
dan = dn + 2 ha
pbn = π mn cos αn
dbn = dn cos αn
( )4
d1εpzz
2dd
zz2d
2bn
2
αnbn
2bn
2an
en +
minusminus
minus=
en
bnen d
dcos arcα =
ennnn
e α invα invα x tan 22π
z1
minus+
+=γ
αFen = αen ndash γe
For external gears
( )
minus=
n
enFenee
n
Fe
mdαtan sin γ γcos
21
mh
]ρcos
G3πcosz fpn +
ϑminus
ϑminusminus
For internal gears
( ) minus
sdotsdotminus=
n
enFenee
n
Fe
mdα tanγ sinγ cos
21
mh
minus
ϑminus
ϑminussdot ρ
cosG3
6πcosz fPn
332 Gearing with εαn gt 2 For deep tooth form gearing ( )25ε2 αn lele produced with a verified grade of accuracy of 4 or better and with applied profile modification to obtain a trapezoidal load distribution along the path of contact the YF may be corrected by the factor YDT as
250ε205for 0666ε2366Y αnαnDT leleminus=
205εfor 10Y αnDT lt=
34 Stress Correction Factors YS YSa The stress correction factors YS and YSa take into account the conversion of the nominal bending stress to the local tooth root stress Thereby YS and YSa cover the stress increasing effect of the notch (fillet) and the fact that not only bending stresses arise at the root A part of the local stress is inde-pendent of the bending moment arm This part increases the more the decisive point of load application approaches the critical tooth root section
Therefore in addition to its dependence on the notch radius the stress correction is also dependent on the position of the load application ie the size of the bending moment arm
YS applies to the load application at the outer point of single tooth pair contact YSa to the load application at tooth tip
YS can be determined as follows
( )
+
+=L23121
1
sS qL01312Y
where Fe
Fn
hsL = and
F
Fns ρ2
sq = (see 33)
YSa can be calculated by replacing hFe with hFa in the above formulae
Note a) Range of validity 1 lt qs lt 8
In case of sharper root radii (ie produced with tools having too sharp tip radii) YS resp YSa must be specially considered
b) In case of grinding notches (due to insufficient protuberance of the hob) YS resp YSa can rise considerably and must be multiplied with
Classification Notes- No 412 25 May 2003
DET NORSKE VERITAS
g
g
ρt
0613
13
minus
where
tg = depth of the grinding notch
ρg = radius of the grinding notch
c) The formulae for YS resp YSa are only valid for αn = 20˚ However the same formulae can be used as a safe approximation for other pressure angles
35 Contact Ratio Factor Yε The contact ratio factor Yε covers the conversion from load application at the tooth tip to the load application at the mid point of the flank (heightwise) for bevel gears
The following may be used
Yε = 0625
36 Helix Angle Factor Yβ The helix angle factor Yβ takes into account the difference between the helical gear and the virtual spur gear in the nor-mal section on which the calculation is based in the first step In this way it is accounted for that the conditions for tooth root stresses are more favourable because the lines of contact are sloping over the flank
The following may be used (β input in degrees)
Yβ = 1 ndash εβ β120
When εβ gt 1 use εβ = 1 and when β gt 30deg use β = 30deg in the formula
However the above equation for Yβ may only be used for gears with β gt 25deg if adequate tip relief is applied to both pinion and wheel (adequate = at least 05 middot Ceff see 432)
37 Values of Endurance Limit σFE σFE is the local tooth root stress (max principal) which the material can endure permanently with 99 survival prob-ability 3106 load cycles is regarded as the beginning of the endurance limit or the lower knee of the σ ndash N curve σFE is defined as the unidirectional pulsating stress with a minimum stress of zero (disregarding residual stresses due to heat treatment) Other stress conditions such as alternating or pre-stressed etc are covered by the conversion factor YM
σFE can be found by pulsating tests or gear running tests for any material in any condition If the approval of the gear is to be based on the results of such tests all details on the testing conditions have to be approved by the Society Fur-ther the tests may have to be made under the Societys su-pervision
If no fatigue tests are available the following listed values for σFE may be used for materials subjected to a quality con-trol as the one referred to in the rules
σFE
Alloyed case hardened steels 1) (fillet surface hardness 58 ndash 63 HRC)
bull of specially approved high grade
1050
bull of normal grade
minus CrNiMo steels with approved process
1000
minus CrNi and CrNiMo steels generally 920 minus MnCr steels generally 850
Nitriding steel of approved grade quenched tempered and gas nitrided (surface hardness 700 ndash 800 HV)
840
Alloyed quenched and tempered steel bath or gas nitrided (surface hardness 500 ndash 700 HV)
720
Alloyed quenched and tempered steel flame or induction hardened 2) (incl entire root fillet) (fillet surface hardness 500 ndash 650 HV)
07 HV + 300
Alloyed quenched and tempered steel flame or induction hardened (excl entire root fillet) (σB = uts of base material)
025 σB + 125
Alloyed quenched and tempered steel 04 σB + 200
Carbon steel 025 σB + 250
Note All numbers given above are valid for separate forgings and for blanks cut from bars forged according to a qualified procedure see Pt 4 Ch 2 Sec 3 For rolled steel the values are to be reduced with 10 For blanks cut from forged bars that are not qualified as mentioned above the values are to be reduced with 20 For cast steel reduce with 40
1) These values are valid for a root radius
bull being unground If however any grinding is made in the root fillet area in such a way that the residual stresses may be affected σFE is to be reduced by 20 (If the grinding also leaves a notch see 34)
bull with fillet surface hardness 58 ndash 63 HRC In case of lower surface hardness than 58 HRC σFE is to be reduced with 20(58 ndash HRC) where HRC is the detected hardness (This may lead to a permissible tooth root stress that varies along the facewidth If so the actual tooth root stresses may also be considered along facewidth)
bull not being shot peened In case of approved shot peening σFE may be increased by 200 for gears where σFE is reduced by 20 due to root grinding Otherwise σFE may be increased by 100 for
6nm le and 100 ndash 5 (mn - 6) for mn gt 6 However the possible adverse influence on the flanks regarding grey staining should be considered and if necessary the flanks should be masked
2) The fillet is not to be ground after surface hardening Regarding possible root grinding see 1)
26 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
38 Mean stress influence Factor YM The mean stress influence factor YM takes into account the influence of other working stress conditions than pure pulsa-tions (R = 0) such as eg load reversals idler gears planets and shrink-fitted gears
YM (YMst) are defined as the ratio between the endurance (or static) strength with a stress ratio R ne 0 and the endurance (or static) strength with R = 0
YM and YMst apply only to a calculation method that assesses the positive (tensile) stresses and is therefore suitable for comparison between the calculated (positive) working stress σF and the permissible stress σFP calculated with YM or YMst
For thin rings (annulus) in epicyclic gears where the com-pression fillet may be decisive special considerations apply see 316
The following method may be used within a stress ratio ndash12 lt R lt 05
381 For idlers planets and PTO with ice class
M1M1R1
1Yor Y MstM
+minus
minus=
where
R = stress ratio = min stress divided by max stress
For designs with the same force applied on both forward- and back-flank R may be assumed to ndash 12
For designs with considerably different forces on forward- and back-flank such as eg a marine propulsion wheel with a power take off pinion R may be assessed as
branchmain theoffacewidth unit per forcepto offacewidt unit per force21minus
For a power take off (PTO) with ice class see 161 c
M considers the mean stress influence on the endurance (or static) strength amplitudes
M is defined as the reduction of the endurance strength am-plitude for a certain increase of the mean stress divided by that increase of the mean stress
Following M values may be used
Endurance limit
Static strength
Case hardened 08 ndash 015 Ys 1) 07
If shot peened 04 06
Nitrided 03 03
Induction or flame hardened
04 06
Not surface hardened steel 03 05
Cast steels 04 06 1)For bevel gears use Ys=2 for determination of M
The listed M values for the endurance limit are independent of the fillet shape (Ys) except for case hardening In princi-ple there is a dependency but wide variations usually only occur for case hardening eg smooth semicircular fillets versus grinding notches
382 For gears with periodical change of rotational direction For case hardened gears with full load applied periodically in both directions such as side thrusters the same formula for YM as for idlers (with R = ndash 12) may be used together with the M values for endurance limit This simplified approach is valid when the number of changes of direction exceeds 100 and the total number of load cycles exceeds 3middot106
For gears of other materials YM will normally be higher than for a pure idler provided the number of changes of direction is below 3middot106 A linear interpolation in a diagram with loga-rithmic number of changes of direction may be used ie from YM = 09 with one change to YM (idler) for 3middot106 changes This is applicable to YM for endurance limit For static strength use YM as for idlers
For gears with occasional full load in reversed direction such as the main wheel in a reversing gear box YM = 09 may be used
383 For gears with shrinkage stresses and unidirectional load For endurance strength
FE
fitM σ
σM1
M21Y+
minus=
σFE is the endurance limit for R = 0
For static strength YMst = 1 and σfit accounted for in 39b
σfit is the shrinkage stress in the fillet (30˚ tangent) and may be found by multiplying the nominal tangential (hoop) stress with a stress concentration factor
n
Ffit m
ρ215scf minus=
384 For shrink-fitted idlers and planets When combined conditions apply such as idlers with shrink-age stresses the design factor for endurance strength is
( ) ( ) FE
fitM σ
σR1M1
M2
M1M1R1
1Yminussdot+
minus
+minus
minus=
Symbols as above but note that the stress ratio R in this par-ticular connection should disregard the influence of σfit ie R normally equal ndash 12
For static strength
M1M1R1
1YMst
+minus
minus=
The effect of σfit is accounted for in 39 b
Classification Notes- No 412 27 May 2003
DET NORSKE VERITAS
385 Additional requirements for peak loads The total stress range (σmax ndash σmin) in a tooth root fillet is not to exceed
F
y
Sσ225
for not surface hardened fillets
FSHV5 for surface hardened fillets
39 Life Factor YN The life factor YN takes into account that in the case of limited life (number of cycles) a higher tooth root stress can be permitted and that lower stresses may apply for very high number of cycles
Decisive for the strength at limited life is the σ ndash N ndash curve of the respective material for given hardening module fillet radius roughness in the tooth root etc Ie the factors YδrelT YRelT YX and YM have an influence on YN
If no σ ndash N ndash curve for the actual material and hardening etc is available the following method may be used
Determination of the σ ndash N ndash curve
a) Calculate the permissible stress σFP for the beginning of the endurance limit (3middot106 cycles) including the influ-ence of all relevant factors as SF YδrelT YRelT YX YM and YC ie σFP = σFE middotYM middotYδrelT middotYRelT middotYX middotYC SF
b) Calculate the permissible laquostaticraquo stress (le 103 load cy-cles) including the influence of all relevant factors as SFst YδrelTst YMst and YCst ( )fitσCstYδrelTstYMstYFstσ
FstS1
FPstσ minussdotsdotsdot=
where σFst is the local tooth root stress which the material can resist without cracking (surface hardened materials) or unacceptable deformation (not surface hardened materials) with 99 survival probability
σFst
Alloyed case hardened steel 1) 2300
Nitriding steel quenched tempered and gas nitrided (surface hardness 700 ndash 800 HV)
1250
Alloyed quenched and tempered steel bath or gas nitrided (surface hardness 500 ndash 700 HV)
1050
Alloyed quenched and tempered steel flame or induction hardened (fillet surface hardness 500 ndash 650 HV)
18 HV + 800
Steel with not surface hardened fillets the smaller value of 2)
18 σB or 225 σy
1) This is valid for a fillet surface hardness of 58 ndash 63 HRC In case of lower fillet surface hardness than 58 HRC σFst is to be reduced with 30(58 ndash HRC) where HRC is the actual hardness Shot peening or grinding notches are not considered to have any significant influence on σFst 2) Actual stresses exceeding the yield point (σy or σ02) will alter the residual stresses locally in the ldquotensionrdquo fillet re-spectively ldquocompressionrdquo fillet This is only to be utilised for gears that are not later loaded with a high number of cycles at lower loads that could cause fatigue in the ldquocompressionrdquo fillet
c) Calculate YN as
NL gt 3middot106
YN = 1 or 001
L
6
N N103Y
sdot= ie Yn = 092 for 1010
The YN = 1 from 3middot106 on may only be used when special material cleanness applies see rules Pt4 Ch2
103ltNLlt3middot106exp
L
6
N N103Y
sdot=
cycles6103forσcycles310forσlog02876exp
FP
FPst
sdot=
NL lt 103 cycles6103forσcycles310forσ
NYFP
FPst
sdot=
or simply use σFPst as mentioned in b) directly
Guidance on number of load cycles NL for various applica-tions
bull For propulsion purpose normally NL = 1010 at full load (yachts etc may have lower values)
bull For auxiliary gears driving generators that normally operate with 70-90 of rated power NL = 108 with rated power may be applied
28 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
310 Relative Notch Sensitivity Factor YδrelT The dynamic (respectively static) relative notch sensitivity factor YδrelT (YδrelTst) indicate to which extent the theoreti-cally concentrated stress lies above the endurance limits (re-spectively static strengths) in the case of fatigue (respectively overload) breakage
YδrelT is a function of the material and the relative stress gra-dient It differs for static strength and endurance limit
The following method may be used
For endurance limit
for not surface hardened fillets
( )
024
s024
relT σ103133q21σ1012201351
Ysdotsdotminus
+sdotsdotminus+= minus
minus
δ
for all surface hardened fillets except nitrided
( )
106q21002451
Y srelT
++=δ
for nitrided fillets
( )
1347q2101421
Y srelT
++=δ
For static strength
for not surface hardened fillets1)
( )( )( )025
02
02502s
relTst σ3000821σ300 1Y0821Y
+
minus+=δ
for surface hardened fillets except nitrided
YδrelTst = 044 YS + 012
for nitrided fillets
YδrelTst = 06 + 02 YS 1) These values are only valid if the local stresses do not
exceed the yield point and thereby alter the residual stress level See also 39b footnote 2
311 Relative Surface Condition Factor YRrelT The relative surface condition factor YRrelT takes into ac-count the dependence of the tooth root strength on the sur-face condition in the tooth root fillet mainly the dependence on the peak to valley surface roughness
YRrelT differs for endurance limit and static strength
The following method may be used
For endurance limit
YRrelT = 1675 ndash 053 (Ry + 1)01 for surface hardened steels and alloyed quenched and tempered steels except nitrided
YRrelT = 53 ndash 42 (Ry + 1)001 for carbon steels
YδrelT = 43 ndash 326 (Ry + 1)0005
for nitrided steels
For static strength
YRrelTst = 1 for all Ry and all materials
For a fillet without any longitudinal machining trace Ry asymp Rz
312 Size Factor YX The size factor YX takes into account the decrease of the strength with increasing size YX differs for endurance limit and static strength
The following may be used
For endurance limit
YX = 1 for mn le 5 generally
YX = 103 ndash 0006 mn for 5 lt mn lt 30
YX = 085 for mn ge 30
for not surface hard-ened steels
YX = 105 ndash 001 mn for 5 lt mn ge 25
YX = 08 for mn ge 25
for surface hardened steels
For static strength
YXst = 1 for all mn and all materials
313 Case Depth Factor YC The case depth factor YC takes into account the influence of hardening depth on tooth root strength
YC applies only to surface hardened tooth roots and is dif-ferent for endurance limit and static strength
In case of insufficient hardening depth fatigue cracks can develop in the transition zone between the hardened layer and the core For static strength yielding shall not occur in the transition zone as this would alter the surface residual stresses and therewith also the fatigue strength
The major parameters are case depth stress gradient permis-sible surface respectively subsurface stresses and subsurface residual stresses
The following simplified method for YC may be used
YC consists of a ratio between permissible subsurface stress (incl influence of expected residual stresses) and permissible surface stress This ratio is multiplied with a bracket con-taining the influence of case depth and stress gradient (The empirical numbers in the bracket are based on a high number of teeth and are somewhat on the safe side for low number of teeth)
YC and YCst may be calculated as given below but calculated values above 10 are to be put equal 10
For endurance limit
+
+=nFFE
C m02ρt31
σconstY
Classification Notes- No 412 29 May 2003
DET NORSKE VERITAS
For static strength
+
+=nFFst
Cst m02ρt31
σconstY
where const and t are connected as
Hardening process
t = endurance limit const =
static strength const =
t550 640 1900
t400 500 1200 Case hard-ening
t300 380 800
Nitriding t400 500 1200
Induction- or flame hardening
tHVmin 11 HVmin 25 HVmin
For symbols see 213
In addition to these requirements to minimum case depths for endurance limit some upper limitations apply to case hard-ened gears
The max depth to 550 HV should not exceed
1) 13 of the top land thickness san unless adequate tip relief is applied (see 110)
2) 025 mn If this is exceeded the following applies addi-tionally in connection with endurance limit
minusminus= 025
mt
1Yn
max550C
314 Thin rim factor YB Where the rim thickness is not sufficient to provide full sup-port for the tooth root the location of a bending failure may be through the gear rim rather than from root fillet to root fillet
YB is not a factor used to convert calculated root stresses at the 30deg tangent to actual stresses in a thin rim tension fillet Actually the compression fillet can be more susceptible to fatigue
YB is a simplified empirical factor used to de-rate thin rim gears (external as well as internal) when no detailed calcula-tion of stresses in both tension and compression fillets are available
Figure 34 Examples on thin rims
YB is applicable in the range 175 lt sRmn lt 35
YB = 115 middot ln (8324 middot mnsR)
(for sRmn ge 35 YB = 1)
(for sRmn le 175 use 315)
σF as calculated in 321 is to be multiplied with YB when sRmn lt 35 Thus YB is used for both high and low cycle fatigue
Note This method is considered to be on the safe side for external gear rims However for internal gear rims without any flange or web stiffeners the method may not be on the safe side and it is advised to check with the method in 315316
315 Stresses in Thin Rims For rim thickness sR lt 35 mn the safety against rim cracking has to be checked
The following method may be used
3151 General The stresses in the standardised 30ordm tangent section tension side are slightly reduced due to decreasing stress correction factor with decreasing relative rim thickness sRmn On the other hand during the complete stress cycle of that fillet a certain amount of compression stresses are also introduced The complete stress range remains approximately constant Therefore the standardised calculation of stresses at the 30ordm tangent may be retained for thin rims as one of the necessary criteria
The maximum stress range for thin rims usually occurs at the 60ordm ndash 80ordm tangents both for laquotensionraquo and laquocompressionraquo side The following method assumes the 75ordm tangent to be the decisive Therefore in addition to the am criterion ap-plied at the 30ordm tangent it is necessary to evaluate the max and min stresses at the 75ordm tangent for both laquotensionraquo (loaded flank) fillet and laquocompressionraquo (back-flank) fillet For this purpose the whole stress cycle of each fillet should be considered but usually the following simplification is justified
Figure 35 Nomenclature of fillets
Index laquoTraquo means laquotensileraquo fillet laquoCraquo means laquocompressionraquo fillet
σFTmin and σFCmax are determined on basis of the nominal rim stresses times the stress concentration factor Y75
σFCmin and σFTmax are determined on basis of superposition of nominal rim stresses times Y75 plus the tooth bending stresses at 75ordm tangent
30 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
3152 Stress concentration factors at the 75ordm tangents The nominal rim stress consists of bending stresses due to local bending moments tangential stresses due to the tan-gential force Ft and radial shear stresses due to Fr
The major influence is given by the bending stresses The influence of the tangential stresses is minor and even though its stress concentration factor is slightly higher than for bending it is considered to be safe enough when the sum of these nominal stresses are combined with the stress concen-tration factor for bending The influence of the radial shear stress is neglected
The stress concentration factor relating nominal rim stresses to local fillet stresses at the 75ordm tangent may be calculated as
75n
R75
n
R
75
ρmsρ185
ms3
Y+
sdot=
where ρ75 is the root radius at the 75ordm tangent ref to mn Usually ρ75 is closed to the tool radius ρao and
ρ75 = ρao
is a safe approximation compensating for the am simplifi-cations to the laquounsaferaquo side
The tooth root stresses of the loaded tooth are decreasing with decreasing relative rim thickness approximately with the empirical factor
15
R
ncorr s
m311Y
minus=
3153 Nominal rim stresses The bending moment applied to the rim consists of a part of the tooth tilt moment Ft (hF + 05 sR) and the bending caused by the radial force Fr
The sectional modulus (first moment of area) which is used for determination of the nominal bending stresses is not nec-essarily the same for the 2 am bending moments If flanges webs etc outside the toothed section contribute to stiffening the rim against various deflections the influence of these stiffeners should be considered Eg an end flange will have an almost negligible influence on the effective sectional modulus for the stresses due to tooth tilt as the deflection caused by the tilt moment is rather small and would not much involve the flange On the other hand the radial forces as for instance from the meshes in an annulus would cause considerable radial deflections that the flange might restrict to a substantial amount When considering the stiff-ening of such flanges or webs on basis of simplified models it is advised to use an effective rim thickness sR = sR + 02 mn for the first moment of area of the rim (toothed part) cross section
For a high number of rim teeth it may be assumed that the rim bending moments in the fillets adjacent to the loaded tooth are of the same magnitude as right under the applied
force This assumption is reasonable for an annulus but rather much on the safe side for a hollow pinion
The influence of Ft on the nominal tangential stress is simpli-fied by half of it for compressive stresses (σ1) and the other half for tensile stresses (σ2) Applying these assumptions the nominal rim stresses adjacent to the loaded tooth are
( ) ( )A2
FW
f RFW
s05hF05 t
R
r
T
RFt1 minus
ϑminus
+minus=σ
( ) ( )A2
FW
f RFW
s05hF05 t
R
r
T
RFt2 +
ϑminus
+=σ
where σ1 σ2 see Fig 35
A = minimum area of cross section (usually bR sR)
WT = the sectional modulus of rim with respect to tooth tilt moment (usually 6sb 2
rR )
WR = the sectional modulus of the rim including the influence of stiffeners as flanges webs etc (WR ge WT)
bR = the width of the rim
R = the radius of the neutral axis in the rim ie from wheel centre to midpoint of rim
( )ϑf = a function for bending moment distribution around the rim For a rim (pinion) with one mesh only the ( )ϑf at the position of load application is 024 For an annulus with 3 or more meshes ( )ϑf at the position of each load application is approx 3 planets ( )ϑf = 019 4 planets ( )ϑf = 014 5 planets ( )ϑf = 011 6 planets ( )ϑf = 009
It must be checked if the max (tensile) stress in the compres-sion fillet really occurs when the fillet is adjacent to the loaded tooth In principle the stress variation through a complete rotation should be considered and the max value used The max value is usually never less than 0 For an annulus eg the tilt moment is zero in the mid position be-tween the planet meshes whilst the bending moment due to the radial forces is half of that at the mesh but with opposite sign
If these formulae are applied to idler gears as eg planets the influence of nominal tangential stresses must be cor-rected by deleting Ft(2 A) for σ1 and using FtA for σ2 Further the influence of Fr on the nominal bending stresses is usually negligible due to the planet bearing support
3154 Root fillet stresses Determination of min and max stresses in the laquotensionraquo fillet
Classification Notes- No 412 31 May 2003
DET NORSKE VERITAS
Minimum stress
751FTmin YσK σ =
Maximum stress
corrF752 Yσ03Yσ KσFTmax +=
where
03 is an empirical factor relating the tension stresses (σF) at the 30ordm tangent to the part of the tension stresses at the 75ordm tangent which add to the rim related stresses (03 also takes into account that full superposition of nominal stresses times stress concentration factors from both laquosides of the corner filletraquo would result in too high stresses)
αβγ sdotsdotsdotsdot= FFvA KKKKKK
Determination of min and max stresses in the laquocompres-sionraquo fillet
Minimum stress
corrF751FCmin Yσ036Yσ Kσ minus=
Maximum stress
752FCmax Yσ Kσ =
where
036 is an empirical factor relating the tension stresses (σF) at the 30ordm tangent to the part of the compression stresses at the 75ordm tangent which add to the rim related stresses
For gears with reversed loads as idler gears and planets there is no distinct laquotensionraquo or laquocompressionraquo fillet The mini-mum stress σFmin is the minimum of σFTmin and σFCmin (usually the latter is decisive) The maximum stress σFTmax is the maximum of σFTmax and σFCmax (usually the former is deci-sive)
316 Permissible Stresses in Thin Rims
3161 General The safety against fatigue fracture respectively overload fracture is to be at least at the same level as for solid gears The laquoordinaryraquo criteria at the 30ordm tangent apply as given in 31 through 313
Additionally the following criteria at the 75ordm tangent may apply
3162 For gt3middot106 cycles The permissible stresses for the laquotensionraquo fillets and for the laquocompressionraquo fillets are determined by means of a relevant fatigue diagram
If the actual tooth root stress (tensile or compressive) ex-ceeds the yield strength to the material the induced residual stresses are to be taken into account
For determination of permissible stresses the following is defined
R = stress ratio ie FTmax
FTminσσ respectively
FCmax
FCmin
σσ
∆σ = stress range ie σFTmax ndash σFTmin resp σFCmax ndash σFCmin
(For idler gears and planets Fmax
FminσσR =
and ∆σ = σFmax minus σFmin)
The permissible stress range ∆σρ for the laquotensionraquo respec-tively laquocompressionraquo fillets can be calculated as
For R gt minus1 FPp σ
R1R1031
13∆σ
minus+
+=
For minus infin lt R lt minus1 FPp σ
R1R10151
13∆σ
minus+
+=
where
σFP = see 32 determined for unidirectional stresses (YM = 1)
If the yield strength σy is exceeded in either tension or com-pression residual stresses are induced This may be consid-ered by correcting the stress ratio R for the respective fillets (tension or compression)
Eg if yFCmin σσ gt (ie exceeded in compression) the
difference yFCmin σσ∆ minus= affects the stress ratio as
∆σ
σ∆σ∆σR
FCmax
y
FCmax
FCmin
+
minus=
++
=
Similarly the stress ratio in the tension fillet may require cor-rection
If the yield strength is exceeded in tension σFTmax gt σy the difference ∆ = σFTmax ndash σy affects the stress ratio as
y
FTmin
FTmax
FTmin
σ∆σ
∆σ∆σR minus=
minusminus
=
Checking for possible exceeding of the yield strength has to be made with the highest torque and with the KAP if this ex-ceeds KA
3163 Forle 103 cycles The permissible stress range ∆σp is not to exceed
For R gt minus1 FPstpst σ
R1R1051
15∆σ
minus+
+=
For ndash infin lt R lt minus1 FPstpst σ
R1R10251
15∆σ
minus+
+=
For all values of R ∆σpst is limited by
not surface hardened F
y
Sσ225
32 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
surface hardened CF
YSHV5
Definition of ∆σ and R see 3162 with particular attention to possible correction of R if the yield strength is exceeded
σFPst see 32 determined for unidirectional stresses (YM = 1) and lt 103 cycles
3164 For 103 lt cycles lt 3106 ∆σp is to be determined by linear interpolation a log-log dia-gram
∆σp at NL load cycles is
6L 103p
exp
L
6
N p ∆σN103∆σ sdot
sdot=
6
3
103p
10p
∆σ
∆σlog28760exp
sdot
=
Classification Notes- No 412 33 May 2003
DET NORSKE VERITAS
4 Calculation of Scuffing Load Capacity
41 Introduction High surface temperatures due to high loads and sliding ve-locities can cause lubricant films to break down This sei-zure or welding together of areas of tooth surface is termed scuffing
In contrast to pitting and fatigue breakage which show a dis-tinct incubation period a single short overloading can lead to a scuffing failure In the ISO-TR13989 two criteria are mentioned The method used in this Classification Note is based on the principles of the flash temperature criterion
Note Bulk temperature in excess of 120ordmC for long periods may have an adverse effect on the surface durability see 211
42 General Criteria In no point along the path of contact the local contact tem-perature may exceed the permissible temperature ie
oilS
oilSB S
ϑ+ϑminusϑ
leϑ
50SB minusϑleϑ
where
Bϑ = max contact temperature along the path of contact
maxflaMBB ϑ+ϑ=ϑ
MBϑ = bulk temperature see 434
maxflaϑ = max flash temperature along the path of con-tact see 44
Sϑ = scuffing temperature see below
oilϑ = oil temperature before it reaches the mesh (max applicable for the actual load case to be used ie normally alarm temperature except for ice classes where a max expected tem-perature applies)
SS = required safety factor according to the Rules
The scuffing temperature Sϑ may be calculated as
L2
wrelT
002
40S XFZGX
ν100112085780
sdot
sdot++=ϑ
where
XwrelT = relative welding factor
XwrelT
Through hardened steel 10
Phosphated steel 125
Copper-plated steel 150
Nitrided steel 150
Less than 10 retained austenite
115
10 ndash 20 retained austenite 10
Casehardened steel
20 ndash 30 retained austenite 085
Austenitic steel 045
FZG = load stage according to FZG-Test A8390 (Note This is the load stage where scuffing oc-curs However due to scatter in test results cal-culations are to be made with one load stage less than the specification)
XL = lubricant factor
= 10 for mineral oils
= 08 for polyalfaolefins
= 07 for non-water-soluble polyglycols
= 06 for water-soluble polyglycols
= 15 for traction fluids
= 13 for phosphate esters
ν40 = kinematic oil viscosity at 40˚C (mm2s)
Application of other test methods such as the Ryder the FZG-Ryder R46574 and the FZG L-42 Test 141195110 may be specially considered
For high speed gears with very short time of contact Sϑ may be increased as follows provided use of EP-oils
Addition to the calculated scuffing temperature Sϑ
If micros 18tc ge no addition
If micros 18tc lt add ( )cwrelT t18X18 minussdotsdot
where
ct = contact time ( micros ) which is the time needed to cross the Hertzian contact width
( ) [ ]microsu1cosβn
uσ340tb1
Hc +sdotsdot
sdot=
σH as calculated in 221
For bevel gears use vu in stead of u
34 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
43 Influence Factors
431 Coefficient of friction The following coefficient of friction may apply
L025a
005oil
02
redC ΣC
Bt X R ηρv
w0048micro minus
=
where
wBt = specific tooth load (Nmm)
vΣC = sum of tangential velocities at pitch point At pitch line velocities gt 50 ms the limiting value of vΣC at v = 50 ms is to be used
ρredC = relative radius of curvature (transversal plane) at the pitch point
Cylindrical gears
HαHβvγAbt
Bt KKKKKb
Fw sdotsdotsdotsdotsdot= (see 1)
twΣC αsin v2v =
bCredC β cos ρρ =
Bevel gears
HαHβvγAtmb
Bt KKKKKb
Fw sdotsdotsdotsdotsdot= (see 1)
vtmtΣC αsin v2v =
bmvCredC β cos ρρ =
ηoil = dynamic viscosity (mPa s) at oilϑ calculated as
1000
ρ νη oiloil =
where ρ in kgm3 approximated as
( ) 70 15oil15 minusϑminusρ=ρ and νoil is kinematic viscosity at oilϑ and may be calculated by means of the following equation
( ) ( )++=+ 08νloglog08νloglog 100oil ( )
sdotminus
ϑ+minus313log373log
273log373log oil
( ) ( )( )08νloglog08νloglog 10040 +minus+
Ra = composite arithmetic mean roughness (micron) of pinion an wheel calculated as ( )2a1aa RR 05R +=This is defined as the roughness on the new flanks ie as manufactured
XL = see 42
432 Effective tip relief Ceff Ceff is the effective tip relief that amount of tip relief which compensates for the elastic deformation of the gear mesh ie
zero load at the tooth tip It is assumed (simplified) to be equal for both pinion and wheel
Cylindrical gears
for helical
γ
γ=cb
KKFC Abt
eff (see 1)
For spur
cbKKF
C Abteff
γ= (see 1)
Alternatively for spur and helical gears the non-linear ap-proach in 111 may be used (taking Ceff=δ)
Bevel gears
bcKFC Ambt
effγ
= (see 1)
where
γ
α
ε2ε44c
+sdot
=γ
433 Tip relief and extension Cylindrical gears
The extension of the tip relief is not to result in an effective contact ratio 1ltεα when the gear is unloaded (exceptions to this may only apply for applications where the gear is not to run at light loads) This means that the unrelieved part of the path of contact is to be minimum pbt It is further assumed that this unrelieved part is placed centrally on the path of contact
If root relief applies it has to be calculated as equivalent tip relief Ie pinion root relief (at mesh position A) is added to Ca2 and wheel root relief (at mesh position E) is added to Ca1 If no design tip relief or root relief on the mating gear is specified (ie if Ca1+Croot2 = 0 and visa versa) use the run-ning in amount see 112
Bevel gears
Bevel gears are to have heightwise crowning ie no distinct relievedunrelieved area This may be treated as tip and root relief For calculation purposes the root relief is combined with the tip relief of the mating member into an equivalent tip relief If no resulting tip relieves are specified the equivalent tip relives may be calculated as an approxima-tion based on the tool crowning Ca tool (per mille of tool module m0) as follows
2root1aeq1a CCC +=
1root2aeq2a CCC +=
where
Classification Notes- No 412 35 May 2003
DET NORSKE VERITAS
2
0
n0101atoolal m
)m2(mA1middotmCC
minus+minussdot=
( )
sdotminusminussdotsdotminus+= vtvb1
2vb1
21vavt1n1 αtan dddαsin 05x1mA
2
0
n020atool22a m
)mm(2A1middotmCC
minus+minussdot=
( )
sdotminusminussdotsdotminus+= vtvb2
2vb2
2va2vt2n2 αtan dddαsin 05x1mA
2
0
20atool1root1 m
A1mCC
minussdotsdot=
2
0
10atool2root2 m
A1mCC
minussdotsdot=
If the pressure angles of the cutter blades are modified (and verified) to balance the tip relieves the following may be assumed
( )eq2aeq1aeq2aeq1a C andC calculated of sum50CC sdot==
Throughout the following Ca1 and Ca2 mean the equivalent tip relieves Ca1eq and Ca2eq
434 Bulk temperature The bulk temperature may be calculated as
flaaveragempsoilMB X X 50 ϑ+ϑ=ϑ
where
Xs = lubrication factor
= 12 for spray lubrication
= 10 for dip lubrication (provided both pinion and wheel are dip lubricated and tip speed lt 5 ms)
= 10 for spray lubrication with additional cooling spray (spray on both pinion and wheel or spray on pinion and dip lubrication of wheel)
= 02 for meshes fully submerged in oil
Xmp = contact factor ( )pmp nX += 150
np = number of mesh contact on the pinion (for small gear ratii the number of wheel meshes should be used if higher)
flaaverageϑ
= average of the integrated flash temperature (see 44) along the path of contact
( )
AE
E
Ayyyfla
flaaverage
d
ΓminusΓ
ΓΓϑ=ϑint =
For high speed gears (v gt 50 ms) it may be necessary to assess the bulk temperature on the basis of a thermal rating of the entire gear transmission
44 The Flash Temperature flaϑ
441 Basic formula The local flash temperature flaϑ may be calculated as
( ) 41redy
y2ly
211
43Btcorrfla
unXwX3250
y ρ
ρminusρ
micro=ϑ Γ
(For bevel gears replace u with uv)
and is to be calculated stepwise along the path of contact from A to E
where
micro = coefficient of friction see 431
Xcorr = correction factor taking empirically into account the increased scuffing risk in the beginning of the approach path due to mesh starting without any previously built up oil film and possible shuffling away oil before meshing if insufficient tip relief
( )
3
AD
yaeffcorr 50
CC1X
ΓminusΓε
Γminus+=
α
Ca = tip relief of driven member Xcorr is only applicable in the approach path and if Ca lt Ceff otherwise 10
wBt = unit load see 431
yXΓ = load sharing factor see 443
n1 = pinion rpm
Γy ρly etc see 442
442 Geometrical relations The various radii of flank curvature (transversal plane) are
ρ1y = pinion flank radius at mesh point y
ρ2y = wheel flank radius at mesh point y
ρredy = equivalent radius of curvature at mesh point y
y2y1
y2y1redy
ρ+ρ
ρρ=ρ
Cylindrical gears
twy
1y αsin au1Γ1
ρ+
+=
twy
2y αsin au1Γu
ρ+
minus=
Note that for internal gears a and u are negative
36 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
Bevel gears
vtvv
y1y αsin a
u1Γ1
ρ+
+=
vtvv
yv2y αsin a
u1Γu
ρ+
minus=
Γ is the parameter on the path of contact and y is any point between A end E
At the respective ends Γ has the following values
Root piniontip wheel
Cylindrical gears
( )
minus
minusminus= 1
αtan 1dd
zzΓ
tw
2b2a2
1
2A
Bevel gears
( )
minus
minusminus= 1
αtan 1dd
uΓvt
2vb2va2
vA
Tip pinionroot wheel
Cylindrical gears
( )
1αtan
1ddΓ
tw
2b1a1
E minusminus
=
Bevel gears
( )
1αtan
1ddΓ
vt
2vb1va1
E minusminus
=
At inner point of single pair contact
Cylindrical gears
tw1
EB α tan zπ2ΓΓ minus=
Bevel gears
vtv1
EB α tan zπ2ΓΓ minus=
At outer point of single pair contact
Cylindrical gears
tw1AD α tan z
π2ΓΓ +=
Bevel gears
vt v1
AD αtan z π2ΓΓ +=
At pitch point ΓC = 0
The points F and G (only applicable to cylindrical gears) limiting the extension of tip relief (so as to maintain a mini-mum contact ratio of unity for unloaded gears) are at
2
BAF
Γ+Γ=Γ
2
EDG
Γ+Γ=Γ
443 Load sharing factor XΓ The load sharing factor XΓ accounts for the load sharing be-tween the various pairs of teeth in mesh along the path of contact
XΓ is to be calculated stepwise from A to E using the pa-rameter Γy
4431 Cylindrical gears with β = 0 and no tip relief
Figure 41
ByAAB
Ay for31
31X
yΓltΓleΓ
ΓminusΓ
ΓminusΓ+=Γ
DyB for1Xy
ΓltΓleΓ=Γ
EyDDE
yE for31
31X
yΓleΓltΓ
ΓminusΓ
ΓminusΓ+=Γ
4432 Cylindrical gears with β = 0 and tip relief Tip relief on the pinion reduces XΓ in the range G ndash E and increases correspondingly XΓ in the range F ndash B
Tip relief on the wheel reduces XΓ in the range A ndash F and increases correspondingly XΓ in the range D ndash G
Following remains generally
DyB for1Xy
ΓleΓleΓ=Γ
21XXGF== ΓΓ
In the following it must be distinguished between Ca lt Ceff respectively Ca gt Ceff This is shown by an example below where Ca1 lt Ceff and Ca2 gt Ceff
Classification Notes- No 412 37 May 2003
DET NORSKE VERITAS
Figure 42
Note When Ca gt Ceff the path of contact is shortened by A ndash A respectively E ndash E The single pair contact path is extended into B respectively D If this shift is significant it is necessary to consider the negative effect on surface durability (B) and bending stresses (B and D)
Range A - F
For effa2 CC le
minus=Γ
eff
2a
CC1
31X
A
FyAeff
2a
AF
Ay for C3
C61XX
AyΓleΓleΓ
sdot
+ΓminusΓ
ΓminusΓ+= ΓΓ
For effa2 CC ge
AyAfor0Xy
ΓleΓleΓ=Γ
with ( )21
CC
1CC
eff
2a
eff
2a
AFAAminus
minusΓminusΓ+Γ=Γ
FyAeff
2a
AF
Ay
eff
2a for 21
CC
CC1X
yΓleΓleΓ
minus
ΓminusΓ
ΓminusΓ+minus=Γ
Range F - B
For effa1 CC le
ByFeff
1a
FB
Fy forC3
C61
21X
yΓleΓleΓ
sdot
+ΓminusΓ
ΓminusΓ+=Γ
For effa1 CC ge
ByFeff
1a
FB
Fy for21
CC
21X
yΓleΓleΓ
minus
ΓminusΓ
ΓminusΓ+=Γ
ByB for1Xy
ΓleΓleΓ=Γ
with 1
CC2
eff
1a
FBFB
minus
ΓminusΓ+Γ=Γ
Range D ndash G
For effa2 CC le
GyDeff
2a
DG
Dy
eff
2a for C 3C
61
C 3C
32X
yΓleΓleΓ
+
ΓminusΓ
ΓminusΓminus+=Γ F
or effa2 CC ge
DyD for1Xy
ΓleΓleΓ=Γ
with ( )21
CC
1CC
eff
2a
eff
2a
DGDDminus
minusΓminusΓ+Γ=Γ
GyDeff
2a
DG
Dy
eff
2a for21
CC
CCX ΓleΓleΓ
minus
ΓminusΓ
ΓminusΓminus=Γ
Range G - E
For effa1 CC le
minus=Γ
eff
1a
CC1
31X
E
EyGeff
1a
GE
Gy forC3
C61
21X
yΓleΓleΓ
sdot
+ΓminusΓ
ΓminusΓminus=Γ
For effa1 CC gt
EyGeff
1a
GE
Gy for21
CC
21X
yΓleΓleΓ
minus
ΓminusΓ
ΓminusΓminus=Γ
EyE for0Xy
ΓleΓleΓ=Γ
with 1
CC2
eff
1a
GEGE
minus
ΓminusΓ+Γ=Γ
4433 Gears with β gt 0 buttressing Due to oblique contact lines over the flanks a certain but-tressing may occur near A and E
This applies to both cylindrical and bevel gears with tip relief lt Ceff The buttressing Xbutt is simplified as a linear function within the ranges A ndash H respectively I ndash E
Figure 43
1εwhen13X βbutt EAge=
1εwhenε031X ββbutt EAlt+=
38 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
Cylindrical gears
bIEAH βsin 02ΓΓΓΓ =minus=minus
Bevel gears
bmIEAH βsin 02ΓΓΓΓ =minus=minus
4434 Cylindrical gears with 2εγ le and no tip relief
yXΓ is obtained by multiplication of
yXΓ in 4431 with
Xbutt in 4433
4435 Gears with 2εγ gt and no tip relief
Applicable to both cylindrical and bevel gears
IyH for1Xy
ΓleΓleΓε
=α
Γ
IyHybutt and forX1Xy
ΓgtΓΓltΓε
=α
Γ
Figure 44
4436 Cylindrical gears with 2εγ le and tip relief
yXΓ is obtained by multiplication of
yXΓ in 4432 with Xbutt
in 4433
4437 Cylindrical gears with 2εγ gt and tip relief
Tip relief on the pinion (respectively wheel) reduces XГ in the range G ndash E (respectively A ndash F) and increases XГ in the range F ndash G
yXΓ is obtained by multiplication of
yXΓ as described below
with Xbutt in 4433
In the XГ example below the influence of tip relief is shown (without the influence of Xbutt) by means of
eff2aeff1a CC and CC ltgt
Tip relief gt Ceff causes new end points A respectively E of the path of contact
Figure 45
Range A ndash F
( ) ( )( ) eff
a2αa1α
AF
Ay
eff
2aeff
C 12C 13εC 1ε
CCCX
y +εε++minus
ΓminusΓ
ΓminusΓ+
εminus
=ααα
Γ
eff2aFyA CC if for leΓleΓleΓ
eff2aFyA CifCfor and geΓleΓleΓ
eff2aAyA CC iffor0Xy
gtΓleΓleΓ=Γ
( ) ( ) ( )( ) ( ) a2αa1α
αeffa2AFAA C 1ε 3C 1ε
1ε 2 CC with++minus+minus
ΓminusΓ+Γ=Γ
Range F ndash G
( )( )( ) GyF
eff
2a1a forC 1 2CC 11X
yΓleΓleΓ
+εε+minusε
+ε
=αα
α
αΓ
Range G ndash E
( ) ( )( ) eff
2a1a
GE
Gy
C 1 2C 1C 1 3XX
GFy +εεminusε++ε
ΓminusΓ
ΓminusΓminus=
αα
ααΓΓ minus
effa1EyG CC if for leΓleΓleΓ
effa1EyG CC if Γfor and geΓleΓle
eff1aEyE CC if for0Xy
geΓleΓleΓ=Γ
( ) ( ) ( )( ) ( ) 2a1a
eff1aGEEE C 1C 1 3
1 2 CC withminusε++ε+εminus
ΓminusΓminusΓ=Γαα
α
4438 Bevel gears with εγ more than approx 18 and heightwise crowning For Ca1 = Ca2 = Ceff the following applies
Figure 46
Classification Notes- No 412 39 May 2003
DET NORSKE VERITAS
( )AEM 50 Γ+Γ=Γ
( )( )2AD
3
2My651X
y ΓminusΓε
ΓminusΓminus
ε=
ααΓ
For tip relief lt Ceff yXΓ is found by linear interpolation be-
tween ( )effay CCX =Γ and ( )0CayX =Γ as in 4435
The interpolation is to be made stepwise from A to M with the influence of Ca2 and from M to E with the influence of Ca1 (For 2a1a CC ne there is a discontinuity at M)
Eg with Ca1 = 04 Ceff and Ca2 = 055 Ceff then
Range A ndash M
)CC()0C( eff2ay2ayyX 550X 450X =Γ=ΓΓ +=
Range M ndash E
)CC()0C( eff1ay1ayyX 40X 06X =Γ=ΓΓ +=
For tip relief gt Ceff the new end points A and E are found as
( )
minusΓminusΓ
ε+Γ=Γ α 1
CC
6 eff
2aADAA
( )
minusΓminusΓ
ε+Γ=Γ α 1
CC
6 eff
1aADEE
Range A ndash A
0Xy=Γ
Range A ndash M
( )( )
ΓminusΓ
ΓminusΓminus
minusε=
αΓ 2
MA
2My
eff
2a1
CC4
351Xy
Range M ndash E
( )( )
ΓminusΓ
ΓminusΓminus
minusε=
αΓ 2
ME
2My
eff
1a1
CC4
351Xy
Range E ndash E
0Xy=Γ
40 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
Appendix A Fatigue Damage Accumulation
The Palmgren-Miner cumulative damage calculation princi-ple is used The procedure may be applied as follows
A1 Stress Spectrum From the individual torque classes the torques (Ti) at the peak values of class intervals and the associated number of cycles (NLi) for both pinion and wheel are to be listed from the highest to the lowest torque
(In case of a cyclic torque variation within the torque classes it is advised to use the peak torque If the cyclic variation is such that the same teeth will repeatedly suffer the peak torque this is a must)
The stress spectra for tooth roots and flanks (σFi σHi) with all relevant factors (except KA) are to be calculated on the basis of the torque spectrum The load dependent K-factors are to be determined for each torque class
A2 σminusN-curve The stress versus load cycle curves for tooth roots and flanks (both pinion and wheel) are to be drawn on the basis of per-missible stresses (ie including the demanded minimum safety factors) as determined in 2 respectively 3 If different safety levels for high cycle fatigue and low cycle fatigue are desired this may be expressed by different demand safety factors applied at the endurance limit respectively at static strength
A3 Damage accumulation The individual damage ratio Di at ith stress level is defined as
Fi
Lii N
ND =
where
NLi = The number of applied cycles at ith stress
NFi = The number of cycles to failure at ith stress
Basically stresses σi below the permissible stress level for infinite life (if a constant ZN or YN is accepted) do not con-tribute to the damage sum However calculating the actual safety factor Sact as described below all the σi for which the product S σi is bigger than or equal to the permissible stress level for infinite life contribute to the damage sum and thus to the determination of Sact The final value of S is decisive
(NFi can be found mathematically by putting the permissible stress σpi equal the actual stress σi thereby finding the actual life factor This life factor can be solved with regard to load cycles ie NFi)
The damage sum ΣDi is not to exceed unity
If ΣDi ne 1 the safety against cumulative fatigue damage is different from the applied demand safety factor For deter-mination of this theoretical safety factor an iteration proce-dure is required as described in the following flowchart
S is correction factor with which the actual safety factor Sact can be found
Sact is the demand safety factor (used in determination of the permissible stresses in the σ ndash N ndash curve) times the correc-tion factor S
The full procedure is to be applied for pinion and wheel tooth roots and flanks
Note If alternating stresses occur in a spectrum of mainly pulsating stresses the alternating stresses may be replaced by equiva-lent pulsating stresses ie by means of division with the ac-tual mean stress influence factor YM
Classification Notes- No 412 41 May 2003
DET NORSKE VERITAS
Appendix B Application Factors for Diesel Driven Gears
For diesel driven gears the application factor KA depends on torsional vibrations Both normal operation and misfiring conditions have to be considered
Normally these two running conditions can be covered by only one calculation
B1 Definitions
Normal operation KAnorm = 0
normv0
TTT +
where
T0 = rated nominal torque
Tv norm = vibratory torque amplitude for normal opera-tion (see rules Pt4 Ch3 Sec1 G301 for defi-nition of ldquonormalrdquo irregularity)
Misfiring operation KA misf = 0
misfv
TTT +
where
T = remaining nominal torque when one cylinder out of action
Tv misf = vibratory torque amplitude in misfiring con-dition This refers to a permissible misfiring condition ie a condition that does not require automatic or immediate corrective actions as speed or pitch reduction
The normal operation is assumed to last for a very high num-ber of cycles such as 1010
The misfiring operation is assumed to last for a limited dura-tion such as 107 cycles
B2 Determination of decisive load Assuming life factor at 1010 cycles as YN = ZN = 092 which usually is relevant the calculation may be performed only once with the combination having the highest value of appli-cation factorlife factor
For bending stresses and scuffing the higher value of
092
K normA and 098
K misfA
For contact stresses the higher value of
092K normA and
113K misfA (but
097K misfA for nitrided gears)
B3 Simplified procedure Note that this is only a guidance and is not a binding con-vention
T = oTZ
1Zsdot
minus where Z = number of cylinders
( ) ideal Videal Vmisf Vnorm V TTT24ZT +minus=
where TV ideal = vibratory torque with all cylinders perfectly equal See also rules Pt4 Ch3 Sec1 G300
When using trends from torsional vibration analysis and measurements the following may be used
TV ideal To is close to zero for engines with few cylinders and using a suitable elastic coupling and increases with relative coupling stiffness and number of cylinders
TV misf To may be high for engines with few cylinders and decreases with number of cylinders
This can be indicated as
200Z
TT
ο
idealV asymp and 80Z04
TT
o
misfV minusasymp
Inserting this into the formulae for the two application fac-tors the following guidance can be given
118112K misfA minusasymp
115110K normA minusasymp
Since KA norm is to be combined with the lower life factors the decisive load condition will be the normal one and a KA of 115 will cover most relevant cases when a suitable elas-tic coupling is chosen
42 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
Appendix C Calculation of Pinion-Rack
Pinion-racks used for elevating of mobile offshore units are open gears that are subjected to wear and tear With normal specifications such as only a few hundred total operation cycles (site to site) the tooth bending stresses for static strength will be decisive for the lay out however with ex-ception of surface hardened pinions where case crushing has to be considered
In the following the use of part 1 and 3 for pinion-racks is shown including relevant simplifications
C1 Pinion tooth root stresses Since the load spectrum normally is dominated by high torques and few load cycles (in the range up to 1-2000) the static strength is decisive
The actual stress is calculated as
β1FSaFan1
tF1 KYY
mbFσ sdotsdotsdotsdot
=
b1 is limited to b2 + 2middotmn
YFa and YSa replace YF and YS because load application at tooth tip has to be assumed for such inaccurate gears
Pinions often use a non-involute profile in the dedendum part of the flank eg a constant radius equal the radius of curva-ture at reference circle For such pinions sFn and hFa are to be measured directly on a sectional drawing of the pinion tooth
Due to high loads and narrow facewidths it may be assumed that KFβ2 = KHβ = 10 However when b1 gt b2 then KFβ1 gt10
If no detailed documentation of KFβ1 is available the fol-lowing may be used
KFβ1 = 1 + 015middot(b1b2 ndash 1)
The permissible stress (not surface hardened) is calculated as
δrelTstF
Fst1FP1 Y
Sσσ sdot=
The mean stress influence due to leg lifting may be disre-garded
The actual and permissible stresses should be calculated for the relevant loads as given in the rules
C2 Rack tooth root stresses The actual stress is calculated as
SaFan2
tF2 YY
mbFσ sdotsdotsdot
=
See C1 for details
The permissible stress is calculated as
δrelTstF
Fst2FP2 Y
Sσσ sdot=
For alloyed steels (Ni Cr Mo) with high toughness and duc-tility the value of YδrelTst may be put equal to YSa
C3 Surface hardened pinions For surface hardened pinions the maximum load is not to cause crushing of the hardened layer of the flank
In principle the calculation described in 213 may be used but when the theoretical Hertzian stress exceeds the ap-proximately 18 times the yield strength of the rack material plastic deformation will occur This will limit the peak Hertzian stress but increases the contact width and thus the penetration of stresses into the depth
An approximation may be based on an assessment of contact width determined by means of equal areas under the theoreti-cal (elastic range) Hertzian contact and the elasto-plastic contact stress (limited to 18 middot σy) with the unknown width
24 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
τmE
2π
z2H
nnminus
minus=
with
3πτ = for external gears
6πτ = for internal gears
Htan zG2
nminusϑ=ϑ
(to be solved iteratively suitable start value6π
=ϑ for exter-
nal gears and 3π for internal gears)
a) Tooth root chord sFn For external gears
minus
ϑ+
ϑminus= ρ
cosG3
3πsinz
ms
fpnn
Fn
For bevel gears with a tooth thickness modification
xsm affects mainly sFn but also hFe and αFen The total influence of xsm on YFa Ysa can be approximated by only adding 2 xsm to sFn mn
For internal gears
minus
ϑ+
ϑminus= ρ
cosG
6πsinz
ms
fPnn
Fn
b) Root fillet radius ρF at 30ordm tangent
( )G2coszcosG2ρ
mρ
2n
2
fpn
F
minusϑϑ+=
c) Determination of bending moment arm hF dn = zn mn
b2α
αn βcosε
ε =
dan = dn + 2 ha
pbn = π mn cos αn
dbn = dn cos αn
( )4
d1εpzz
2dd
zz2d
2bn
2
αnbn
2bn
2an
en +
minusminus
minus=
en
bnen d
dcos arcα =
ennnn
e α invα invα x tan 22π
z1
minus+
+=γ
αFen = αen ndash γe
For external gears
( )
minus=
n
enFenee
n
Fe
mdαtan sin γ γcos
21
mh
]ρcos
G3πcosz fpn +
ϑminus
ϑminusminus
For internal gears
( ) minus
sdotsdotminus=
n
enFenee
n
Fe
mdα tanγ sinγ cos
21
mh
minus
ϑminus
ϑminussdot ρ
cosG3
6πcosz fPn
332 Gearing with εαn gt 2 For deep tooth form gearing ( )25ε2 αn lele produced with a verified grade of accuracy of 4 or better and with applied profile modification to obtain a trapezoidal load distribution along the path of contact the YF may be corrected by the factor YDT as
250ε205for 0666ε2366Y αnαnDT leleminus=
205εfor 10Y αnDT lt=
34 Stress Correction Factors YS YSa The stress correction factors YS and YSa take into account the conversion of the nominal bending stress to the local tooth root stress Thereby YS and YSa cover the stress increasing effect of the notch (fillet) and the fact that not only bending stresses arise at the root A part of the local stress is inde-pendent of the bending moment arm This part increases the more the decisive point of load application approaches the critical tooth root section
Therefore in addition to its dependence on the notch radius the stress correction is also dependent on the position of the load application ie the size of the bending moment arm
YS applies to the load application at the outer point of single tooth pair contact YSa to the load application at tooth tip
YS can be determined as follows
( )
+
+=L23121
1
sS qL01312Y
where Fe
Fn
hsL = and
F
Fns ρ2
sq = (see 33)
YSa can be calculated by replacing hFe with hFa in the above formulae
Note a) Range of validity 1 lt qs lt 8
In case of sharper root radii (ie produced with tools having too sharp tip radii) YS resp YSa must be specially considered
b) In case of grinding notches (due to insufficient protuberance of the hob) YS resp YSa can rise considerably and must be multiplied with
Classification Notes- No 412 25 May 2003
DET NORSKE VERITAS
g
g
ρt
0613
13
minus
where
tg = depth of the grinding notch
ρg = radius of the grinding notch
c) The formulae for YS resp YSa are only valid for αn = 20˚ However the same formulae can be used as a safe approximation for other pressure angles
35 Contact Ratio Factor Yε The contact ratio factor Yε covers the conversion from load application at the tooth tip to the load application at the mid point of the flank (heightwise) for bevel gears
The following may be used
Yε = 0625
36 Helix Angle Factor Yβ The helix angle factor Yβ takes into account the difference between the helical gear and the virtual spur gear in the nor-mal section on which the calculation is based in the first step In this way it is accounted for that the conditions for tooth root stresses are more favourable because the lines of contact are sloping over the flank
The following may be used (β input in degrees)
Yβ = 1 ndash εβ β120
When εβ gt 1 use εβ = 1 and when β gt 30deg use β = 30deg in the formula
However the above equation for Yβ may only be used for gears with β gt 25deg if adequate tip relief is applied to both pinion and wheel (adequate = at least 05 middot Ceff see 432)
37 Values of Endurance Limit σFE σFE is the local tooth root stress (max principal) which the material can endure permanently with 99 survival prob-ability 3106 load cycles is regarded as the beginning of the endurance limit or the lower knee of the σ ndash N curve σFE is defined as the unidirectional pulsating stress with a minimum stress of zero (disregarding residual stresses due to heat treatment) Other stress conditions such as alternating or pre-stressed etc are covered by the conversion factor YM
σFE can be found by pulsating tests or gear running tests for any material in any condition If the approval of the gear is to be based on the results of such tests all details on the testing conditions have to be approved by the Society Fur-ther the tests may have to be made under the Societys su-pervision
If no fatigue tests are available the following listed values for σFE may be used for materials subjected to a quality con-trol as the one referred to in the rules
σFE
Alloyed case hardened steels 1) (fillet surface hardness 58 ndash 63 HRC)
bull of specially approved high grade
1050
bull of normal grade
minus CrNiMo steels with approved process
1000
minus CrNi and CrNiMo steels generally 920 minus MnCr steels generally 850
Nitriding steel of approved grade quenched tempered and gas nitrided (surface hardness 700 ndash 800 HV)
840
Alloyed quenched and tempered steel bath or gas nitrided (surface hardness 500 ndash 700 HV)
720
Alloyed quenched and tempered steel flame or induction hardened 2) (incl entire root fillet) (fillet surface hardness 500 ndash 650 HV)
07 HV + 300
Alloyed quenched and tempered steel flame or induction hardened (excl entire root fillet) (σB = uts of base material)
025 σB + 125
Alloyed quenched and tempered steel 04 σB + 200
Carbon steel 025 σB + 250
Note All numbers given above are valid for separate forgings and for blanks cut from bars forged according to a qualified procedure see Pt 4 Ch 2 Sec 3 For rolled steel the values are to be reduced with 10 For blanks cut from forged bars that are not qualified as mentioned above the values are to be reduced with 20 For cast steel reduce with 40
1) These values are valid for a root radius
bull being unground If however any grinding is made in the root fillet area in such a way that the residual stresses may be affected σFE is to be reduced by 20 (If the grinding also leaves a notch see 34)
bull with fillet surface hardness 58 ndash 63 HRC In case of lower surface hardness than 58 HRC σFE is to be reduced with 20(58 ndash HRC) where HRC is the detected hardness (This may lead to a permissible tooth root stress that varies along the facewidth If so the actual tooth root stresses may also be considered along facewidth)
bull not being shot peened In case of approved shot peening σFE may be increased by 200 for gears where σFE is reduced by 20 due to root grinding Otherwise σFE may be increased by 100 for
6nm le and 100 ndash 5 (mn - 6) for mn gt 6 However the possible adverse influence on the flanks regarding grey staining should be considered and if necessary the flanks should be masked
2) The fillet is not to be ground after surface hardening Regarding possible root grinding see 1)
26 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
38 Mean stress influence Factor YM The mean stress influence factor YM takes into account the influence of other working stress conditions than pure pulsa-tions (R = 0) such as eg load reversals idler gears planets and shrink-fitted gears
YM (YMst) are defined as the ratio between the endurance (or static) strength with a stress ratio R ne 0 and the endurance (or static) strength with R = 0
YM and YMst apply only to a calculation method that assesses the positive (tensile) stresses and is therefore suitable for comparison between the calculated (positive) working stress σF and the permissible stress σFP calculated with YM or YMst
For thin rings (annulus) in epicyclic gears where the com-pression fillet may be decisive special considerations apply see 316
The following method may be used within a stress ratio ndash12 lt R lt 05
381 For idlers planets and PTO with ice class
M1M1R1
1Yor Y MstM
+minus
minus=
where
R = stress ratio = min stress divided by max stress
For designs with the same force applied on both forward- and back-flank R may be assumed to ndash 12
For designs with considerably different forces on forward- and back-flank such as eg a marine propulsion wheel with a power take off pinion R may be assessed as
branchmain theoffacewidth unit per forcepto offacewidt unit per force21minus
For a power take off (PTO) with ice class see 161 c
M considers the mean stress influence on the endurance (or static) strength amplitudes
M is defined as the reduction of the endurance strength am-plitude for a certain increase of the mean stress divided by that increase of the mean stress
Following M values may be used
Endurance limit
Static strength
Case hardened 08 ndash 015 Ys 1) 07
If shot peened 04 06
Nitrided 03 03
Induction or flame hardened
04 06
Not surface hardened steel 03 05
Cast steels 04 06 1)For bevel gears use Ys=2 for determination of M
The listed M values for the endurance limit are independent of the fillet shape (Ys) except for case hardening In princi-ple there is a dependency but wide variations usually only occur for case hardening eg smooth semicircular fillets versus grinding notches
382 For gears with periodical change of rotational direction For case hardened gears with full load applied periodically in both directions such as side thrusters the same formula for YM as for idlers (with R = ndash 12) may be used together with the M values for endurance limit This simplified approach is valid when the number of changes of direction exceeds 100 and the total number of load cycles exceeds 3middot106
For gears of other materials YM will normally be higher than for a pure idler provided the number of changes of direction is below 3middot106 A linear interpolation in a diagram with loga-rithmic number of changes of direction may be used ie from YM = 09 with one change to YM (idler) for 3middot106 changes This is applicable to YM for endurance limit For static strength use YM as for idlers
For gears with occasional full load in reversed direction such as the main wheel in a reversing gear box YM = 09 may be used
383 For gears with shrinkage stresses and unidirectional load For endurance strength
FE
fitM σ
σM1
M21Y+
minus=
σFE is the endurance limit for R = 0
For static strength YMst = 1 and σfit accounted for in 39b
σfit is the shrinkage stress in the fillet (30˚ tangent) and may be found by multiplying the nominal tangential (hoop) stress with a stress concentration factor
n
Ffit m
ρ215scf minus=
384 For shrink-fitted idlers and planets When combined conditions apply such as idlers with shrink-age stresses the design factor for endurance strength is
( ) ( ) FE
fitM σ
σR1M1
M2
M1M1R1
1Yminussdot+
minus
+minus
minus=
Symbols as above but note that the stress ratio R in this par-ticular connection should disregard the influence of σfit ie R normally equal ndash 12
For static strength
M1M1R1
1YMst
+minus
minus=
The effect of σfit is accounted for in 39 b
Classification Notes- No 412 27 May 2003
DET NORSKE VERITAS
385 Additional requirements for peak loads The total stress range (σmax ndash σmin) in a tooth root fillet is not to exceed
F
y
Sσ225
for not surface hardened fillets
FSHV5 for surface hardened fillets
39 Life Factor YN The life factor YN takes into account that in the case of limited life (number of cycles) a higher tooth root stress can be permitted and that lower stresses may apply for very high number of cycles
Decisive for the strength at limited life is the σ ndash N ndash curve of the respective material for given hardening module fillet radius roughness in the tooth root etc Ie the factors YδrelT YRelT YX and YM have an influence on YN
If no σ ndash N ndash curve for the actual material and hardening etc is available the following method may be used
Determination of the σ ndash N ndash curve
a) Calculate the permissible stress σFP for the beginning of the endurance limit (3middot106 cycles) including the influ-ence of all relevant factors as SF YδrelT YRelT YX YM and YC ie σFP = σFE middotYM middotYδrelT middotYRelT middotYX middotYC SF
b) Calculate the permissible laquostaticraquo stress (le 103 load cy-cles) including the influence of all relevant factors as SFst YδrelTst YMst and YCst ( )fitσCstYδrelTstYMstYFstσ
FstS1
FPstσ minussdotsdotsdot=
where σFst is the local tooth root stress which the material can resist without cracking (surface hardened materials) or unacceptable deformation (not surface hardened materials) with 99 survival probability
σFst
Alloyed case hardened steel 1) 2300
Nitriding steel quenched tempered and gas nitrided (surface hardness 700 ndash 800 HV)
1250
Alloyed quenched and tempered steel bath or gas nitrided (surface hardness 500 ndash 700 HV)
1050
Alloyed quenched and tempered steel flame or induction hardened (fillet surface hardness 500 ndash 650 HV)
18 HV + 800
Steel with not surface hardened fillets the smaller value of 2)
18 σB or 225 σy
1) This is valid for a fillet surface hardness of 58 ndash 63 HRC In case of lower fillet surface hardness than 58 HRC σFst is to be reduced with 30(58 ndash HRC) where HRC is the actual hardness Shot peening or grinding notches are not considered to have any significant influence on σFst 2) Actual stresses exceeding the yield point (σy or σ02) will alter the residual stresses locally in the ldquotensionrdquo fillet re-spectively ldquocompressionrdquo fillet This is only to be utilised for gears that are not later loaded with a high number of cycles at lower loads that could cause fatigue in the ldquocompressionrdquo fillet
c) Calculate YN as
NL gt 3middot106
YN = 1 or 001
L
6
N N103Y
sdot= ie Yn = 092 for 1010
The YN = 1 from 3middot106 on may only be used when special material cleanness applies see rules Pt4 Ch2
103ltNLlt3middot106exp
L
6
N N103Y
sdot=
cycles6103forσcycles310forσlog02876exp
FP
FPst
sdot=
NL lt 103 cycles6103forσcycles310forσ
NYFP
FPst
sdot=
or simply use σFPst as mentioned in b) directly
Guidance on number of load cycles NL for various applica-tions
bull For propulsion purpose normally NL = 1010 at full load (yachts etc may have lower values)
bull For auxiliary gears driving generators that normally operate with 70-90 of rated power NL = 108 with rated power may be applied
28 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
310 Relative Notch Sensitivity Factor YδrelT The dynamic (respectively static) relative notch sensitivity factor YδrelT (YδrelTst) indicate to which extent the theoreti-cally concentrated stress lies above the endurance limits (re-spectively static strengths) in the case of fatigue (respectively overload) breakage
YδrelT is a function of the material and the relative stress gra-dient It differs for static strength and endurance limit
The following method may be used
For endurance limit
for not surface hardened fillets
( )
024
s024
relT σ103133q21σ1012201351
Ysdotsdotminus
+sdotsdotminus+= minus
minus
δ
for all surface hardened fillets except nitrided
( )
106q21002451
Y srelT
++=δ
for nitrided fillets
( )
1347q2101421
Y srelT
++=δ
For static strength
for not surface hardened fillets1)
( )( )( )025
02
02502s
relTst σ3000821σ300 1Y0821Y
+
minus+=δ
for surface hardened fillets except nitrided
YδrelTst = 044 YS + 012
for nitrided fillets
YδrelTst = 06 + 02 YS 1) These values are only valid if the local stresses do not
exceed the yield point and thereby alter the residual stress level See also 39b footnote 2
311 Relative Surface Condition Factor YRrelT The relative surface condition factor YRrelT takes into ac-count the dependence of the tooth root strength on the sur-face condition in the tooth root fillet mainly the dependence on the peak to valley surface roughness
YRrelT differs for endurance limit and static strength
The following method may be used
For endurance limit
YRrelT = 1675 ndash 053 (Ry + 1)01 for surface hardened steels and alloyed quenched and tempered steels except nitrided
YRrelT = 53 ndash 42 (Ry + 1)001 for carbon steels
YδrelT = 43 ndash 326 (Ry + 1)0005
for nitrided steels
For static strength
YRrelTst = 1 for all Ry and all materials
For a fillet without any longitudinal machining trace Ry asymp Rz
312 Size Factor YX The size factor YX takes into account the decrease of the strength with increasing size YX differs for endurance limit and static strength
The following may be used
For endurance limit
YX = 1 for mn le 5 generally
YX = 103 ndash 0006 mn for 5 lt mn lt 30
YX = 085 for mn ge 30
for not surface hard-ened steels
YX = 105 ndash 001 mn for 5 lt mn ge 25
YX = 08 for mn ge 25
for surface hardened steels
For static strength
YXst = 1 for all mn and all materials
313 Case Depth Factor YC The case depth factor YC takes into account the influence of hardening depth on tooth root strength
YC applies only to surface hardened tooth roots and is dif-ferent for endurance limit and static strength
In case of insufficient hardening depth fatigue cracks can develop in the transition zone between the hardened layer and the core For static strength yielding shall not occur in the transition zone as this would alter the surface residual stresses and therewith also the fatigue strength
The major parameters are case depth stress gradient permis-sible surface respectively subsurface stresses and subsurface residual stresses
The following simplified method for YC may be used
YC consists of a ratio between permissible subsurface stress (incl influence of expected residual stresses) and permissible surface stress This ratio is multiplied with a bracket con-taining the influence of case depth and stress gradient (The empirical numbers in the bracket are based on a high number of teeth and are somewhat on the safe side for low number of teeth)
YC and YCst may be calculated as given below but calculated values above 10 are to be put equal 10
For endurance limit
+
+=nFFE
C m02ρt31
σconstY
Classification Notes- No 412 29 May 2003
DET NORSKE VERITAS
For static strength
+
+=nFFst
Cst m02ρt31
σconstY
where const and t are connected as
Hardening process
t = endurance limit const =
static strength const =
t550 640 1900
t400 500 1200 Case hard-ening
t300 380 800
Nitriding t400 500 1200
Induction- or flame hardening
tHVmin 11 HVmin 25 HVmin
For symbols see 213
In addition to these requirements to minimum case depths for endurance limit some upper limitations apply to case hard-ened gears
The max depth to 550 HV should not exceed
1) 13 of the top land thickness san unless adequate tip relief is applied (see 110)
2) 025 mn If this is exceeded the following applies addi-tionally in connection with endurance limit
minusminus= 025
mt
1Yn
max550C
314 Thin rim factor YB Where the rim thickness is not sufficient to provide full sup-port for the tooth root the location of a bending failure may be through the gear rim rather than from root fillet to root fillet
YB is not a factor used to convert calculated root stresses at the 30deg tangent to actual stresses in a thin rim tension fillet Actually the compression fillet can be more susceptible to fatigue
YB is a simplified empirical factor used to de-rate thin rim gears (external as well as internal) when no detailed calcula-tion of stresses in both tension and compression fillets are available
Figure 34 Examples on thin rims
YB is applicable in the range 175 lt sRmn lt 35
YB = 115 middot ln (8324 middot mnsR)
(for sRmn ge 35 YB = 1)
(for sRmn le 175 use 315)
σF as calculated in 321 is to be multiplied with YB when sRmn lt 35 Thus YB is used for both high and low cycle fatigue
Note This method is considered to be on the safe side for external gear rims However for internal gear rims without any flange or web stiffeners the method may not be on the safe side and it is advised to check with the method in 315316
315 Stresses in Thin Rims For rim thickness sR lt 35 mn the safety against rim cracking has to be checked
The following method may be used
3151 General The stresses in the standardised 30ordm tangent section tension side are slightly reduced due to decreasing stress correction factor with decreasing relative rim thickness sRmn On the other hand during the complete stress cycle of that fillet a certain amount of compression stresses are also introduced The complete stress range remains approximately constant Therefore the standardised calculation of stresses at the 30ordm tangent may be retained for thin rims as one of the necessary criteria
The maximum stress range for thin rims usually occurs at the 60ordm ndash 80ordm tangents both for laquotensionraquo and laquocompressionraquo side The following method assumes the 75ordm tangent to be the decisive Therefore in addition to the am criterion ap-plied at the 30ordm tangent it is necessary to evaluate the max and min stresses at the 75ordm tangent for both laquotensionraquo (loaded flank) fillet and laquocompressionraquo (back-flank) fillet For this purpose the whole stress cycle of each fillet should be considered but usually the following simplification is justified
Figure 35 Nomenclature of fillets
Index laquoTraquo means laquotensileraquo fillet laquoCraquo means laquocompressionraquo fillet
σFTmin and σFCmax are determined on basis of the nominal rim stresses times the stress concentration factor Y75
σFCmin and σFTmax are determined on basis of superposition of nominal rim stresses times Y75 plus the tooth bending stresses at 75ordm tangent
30 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
3152 Stress concentration factors at the 75ordm tangents The nominal rim stress consists of bending stresses due to local bending moments tangential stresses due to the tan-gential force Ft and radial shear stresses due to Fr
The major influence is given by the bending stresses The influence of the tangential stresses is minor and even though its stress concentration factor is slightly higher than for bending it is considered to be safe enough when the sum of these nominal stresses are combined with the stress concen-tration factor for bending The influence of the radial shear stress is neglected
The stress concentration factor relating nominal rim stresses to local fillet stresses at the 75ordm tangent may be calculated as
75n
R75
n
R
75
ρmsρ185
ms3
Y+
sdot=
where ρ75 is the root radius at the 75ordm tangent ref to mn Usually ρ75 is closed to the tool radius ρao and
ρ75 = ρao
is a safe approximation compensating for the am simplifi-cations to the laquounsaferaquo side
The tooth root stresses of the loaded tooth are decreasing with decreasing relative rim thickness approximately with the empirical factor
15
R
ncorr s
m311Y
minus=
3153 Nominal rim stresses The bending moment applied to the rim consists of a part of the tooth tilt moment Ft (hF + 05 sR) and the bending caused by the radial force Fr
The sectional modulus (first moment of area) which is used for determination of the nominal bending stresses is not nec-essarily the same for the 2 am bending moments If flanges webs etc outside the toothed section contribute to stiffening the rim against various deflections the influence of these stiffeners should be considered Eg an end flange will have an almost negligible influence on the effective sectional modulus for the stresses due to tooth tilt as the deflection caused by the tilt moment is rather small and would not much involve the flange On the other hand the radial forces as for instance from the meshes in an annulus would cause considerable radial deflections that the flange might restrict to a substantial amount When considering the stiff-ening of such flanges or webs on basis of simplified models it is advised to use an effective rim thickness sR = sR + 02 mn for the first moment of area of the rim (toothed part) cross section
For a high number of rim teeth it may be assumed that the rim bending moments in the fillets adjacent to the loaded tooth are of the same magnitude as right under the applied
force This assumption is reasonable for an annulus but rather much on the safe side for a hollow pinion
The influence of Ft on the nominal tangential stress is simpli-fied by half of it for compressive stresses (σ1) and the other half for tensile stresses (σ2) Applying these assumptions the nominal rim stresses adjacent to the loaded tooth are
( ) ( )A2
FW
f RFW
s05hF05 t
R
r
T
RFt1 minus
ϑminus
+minus=σ
( ) ( )A2
FW
f RFW
s05hF05 t
R
r
T
RFt2 +
ϑminus
+=σ
where σ1 σ2 see Fig 35
A = minimum area of cross section (usually bR sR)
WT = the sectional modulus of rim with respect to tooth tilt moment (usually 6sb 2
rR )
WR = the sectional modulus of the rim including the influence of stiffeners as flanges webs etc (WR ge WT)
bR = the width of the rim
R = the radius of the neutral axis in the rim ie from wheel centre to midpoint of rim
( )ϑf = a function for bending moment distribution around the rim For a rim (pinion) with one mesh only the ( )ϑf at the position of load application is 024 For an annulus with 3 or more meshes ( )ϑf at the position of each load application is approx 3 planets ( )ϑf = 019 4 planets ( )ϑf = 014 5 planets ( )ϑf = 011 6 planets ( )ϑf = 009
It must be checked if the max (tensile) stress in the compres-sion fillet really occurs when the fillet is adjacent to the loaded tooth In principle the stress variation through a complete rotation should be considered and the max value used The max value is usually never less than 0 For an annulus eg the tilt moment is zero in the mid position be-tween the planet meshes whilst the bending moment due to the radial forces is half of that at the mesh but with opposite sign
If these formulae are applied to idler gears as eg planets the influence of nominal tangential stresses must be cor-rected by deleting Ft(2 A) for σ1 and using FtA for σ2 Further the influence of Fr on the nominal bending stresses is usually negligible due to the planet bearing support
3154 Root fillet stresses Determination of min and max stresses in the laquotensionraquo fillet
Classification Notes- No 412 31 May 2003
DET NORSKE VERITAS
Minimum stress
751FTmin YσK σ =
Maximum stress
corrF752 Yσ03Yσ KσFTmax +=
where
03 is an empirical factor relating the tension stresses (σF) at the 30ordm tangent to the part of the tension stresses at the 75ordm tangent which add to the rim related stresses (03 also takes into account that full superposition of nominal stresses times stress concentration factors from both laquosides of the corner filletraquo would result in too high stresses)
αβγ sdotsdotsdotsdot= FFvA KKKKKK
Determination of min and max stresses in the laquocompres-sionraquo fillet
Minimum stress
corrF751FCmin Yσ036Yσ Kσ minus=
Maximum stress
752FCmax Yσ Kσ =
where
036 is an empirical factor relating the tension stresses (σF) at the 30ordm tangent to the part of the compression stresses at the 75ordm tangent which add to the rim related stresses
For gears with reversed loads as idler gears and planets there is no distinct laquotensionraquo or laquocompressionraquo fillet The mini-mum stress σFmin is the minimum of σFTmin and σFCmin (usually the latter is decisive) The maximum stress σFTmax is the maximum of σFTmax and σFCmax (usually the former is deci-sive)
316 Permissible Stresses in Thin Rims
3161 General The safety against fatigue fracture respectively overload fracture is to be at least at the same level as for solid gears The laquoordinaryraquo criteria at the 30ordm tangent apply as given in 31 through 313
Additionally the following criteria at the 75ordm tangent may apply
3162 For gt3middot106 cycles The permissible stresses for the laquotensionraquo fillets and for the laquocompressionraquo fillets are determined by means of a relevant fatigue diagram
If the actual tooth root stress (tensile or compressive) ex-ceeds the yield strength to the material the induced residual stresses are to be taken into account
For determination of permissible stresses the following is defined
R = stress ratio ie FTmax
FTminσσ respectively
FCmax
FCmin
σσ
∆σ = stress range ie σFTmax ndash σFTmin resp σFCmax ndash σFCmin
(For idler gears and planets Fmax
FminσσR =
and ∆σ = σFmax minus σFmin)
The permissible stress range ∆σρ for the laquotensionraquo respec-tively laquocompressionraquo fillets can be calculated as
For R gt minus1 FPp σ
R1R1031
13∆σ
minus+
+=
For minus infin lt R lt minus1 FPp σ
R1R10151
13∆σ
minus+
+=
where
σFP = see 32 determined for unidirectional stresses (YM = 1)
If the yield strength σy is exceeded in either tension or com-pression residual stresses are induced This may be consid-ered by correcting the stress ratio R for the respective fillets (tension or compression)
Eg if yFCmin σσ gt (ie exceeded in compression) the
difference yFCmin σσ∆ minus= affects the stress ratio as
∆σ
σ∆σ∆σR
FCmax
y
FCmax
FCmin
+
minus=
++
=
Similarly the stress ratio in the tension fillet may require cor-rection
If the yield strength is exceeded in tension σFTmax gt σy the difference ∆ = σFTmax ndash σy affects the stress ratio as
y
FTmin
FTmax
FTmin
σ∆σ
∆σ∆σR minus=
minusminus
=
Checking for possible exceeding of the yield strength has to be made with the highest torque and with the KAP if this ex-ceeds KA
3163 Forle 103 cycles The permissible stress range ∆σp is not to exceed
For R gt minus1 FPstpst σ
R1R1051
15∆σ
minus+
+=
For ndash infin lt R lt minus1 FPstpst σ
R1R10251
15∆σ
minus+
+=
For all values of R ∆σpst is limited by
not surface hardened F
y
Sσ225
32 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
surface hardened CF
YSHV5
Definition of ∆σ and R see 3162 with particular attention to possible correction of R if the yield strength is exceeded
σFPst see 32 determined for unidirectional stresses (YM = 1) and lt 103 cycles
3164 For 103 lt cycles lt 3106 ∆σp is to be determined by linear interpolation a log-log dia-gram
∆σp at NL load cycles is
6L 103p
exp
L
6
N p ∆σN103∆σ sdot
sdot=
6
3
103p
10p
∆σ
∆σlog28760exp
sdot
=
Classification Notes- No 412 33 May 2003
DET NORSKE VERITAS
4 Calculation of Scuffing Load Capacity
41 Introduction High surface temperatures due to high loads and sliding ve-locities can cause lubricant films to break down This sei-zure or welding together of areas of tooth surface is termed scuffing
In contrast to pitting and fatigue breakage which show a dis-tinct incubation period a single short overloading can lead to a scuffing failure In the ISO-TR13989 two criteria are mentioned The method used in this Classification Note is based on the principles of the flash temperature criterion
Note Bulk temperature in excess of 120ordmC for long periods may have an adverse effect on the surface durability see 211
42 General Criteria In no point along the path of contact the local contact tem-perature may exceed the permissible temperature ie
oilS
oilSB S
ϑ+ϑminusϑ
leϑ
50SB minusϑleϑ
where
Bϑ = max contact temperature along the path of contact
maxflaMBB ϑ+ϑ=ϑ
MBϑ = bulk temperature see 434
maxflaϑ = max flash temperature along the path of con-tact see 44
Sϑ = scuffing temperature see below
oilϑ = oil temperature before it reaches the mesh (max applicable for the actual load case to be used ie normally alarm temperature except for ice classes where a max expected tem-perature applies)
SS = required safety factor according to the Rules
The scuffing temperature Sϑ may be calculated as
L2
wrelT
002
40S XFZGX
ν100112085780
sdot
sdot++=ϑ
where
XwrelT = relative welding factor
XwrelT
Through hardened steel 10
Phosphated steel 125
Copper-plated steel 150
Nitrided steel 150
Less than 10 retained austenite
115
10 ndash 20 retained austenite 10
Casehardened steel
20 ndash 30 retained austenite 085
Austenitic steel 045
FZG = load stage according to FZG-Test A8390 (Note This is the load stage where scuffing oc-curs However due to scatter in test results cal-culations are to be made with one load stage less than the specification)
XL = lubricant factor
= 10 for mineral oils
= 08 for polyalfaolefins
= 07 for non-water-soluble polyglycols
= 06 for water-soluble polyglycols
= 15 for traction fluids
= 13 for phosphate esters
ν40 = kinematic oil viscosity at 40˚C (mm2s)
Application of other test methods such as the Ryder the FZG-Ryder R46574 and the FZG L-42 Test 141195110 may be specially considered
For high speed gears with very short time of contact Sϑ may be increased as follows provided use of EP-oils
Addition to the calculated scuffing temperature Sϑ
If micros 18tc ge no addition
If micros 18tc lt add ( )cwrelT t18X18 minussdotsdot
where
ct = contact time ( micros ) which is the time needed to cross the Hertzian contact width
( ) [ ]microsu1cosβn
uσ340tb1
Hc +sdotsdot
sdot=
σH as calculated in 221
For bevel gears use vu in stead of u
34 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
43 Influence Factors
431 Coefficient of friction The following coefficient of friction may apply
L025a
005oil
02
redC ΣC
Bt X R ηρv
w0048micro minus
=
where
wBt = specific tooth load (Nmm)
vΣC = sum of tangential velocities at pitch point At pitch line velocities gt 50 ms the limiting value of vΣC at v = 50 ms is to be used
ρredC = relative radius of curvature (transversal plane) at the pitch point
Cylindrical gears
HαHβvγAbt
Bt KKKKKb
Fw sdotsdotsdotsdotsdot= (see 1)
twΣC αsin v2v =
bCredC β cos ρρ =
Bevel gears
HαHβvγAtmb
Bt KKKKKb
Fw sdotsdotsdotsdotsdot= (see 1)
vtmtΣC αsin v2v =
bmvCredC β cos ρρ =
ηoil = dynamic viscosity (mPa s) at oilϑ calculated as
1000
ρ νη oiloil =
where ρ in kgm3 approximated as
( ) 70 15oil15 minusϑminusρ=ρ and νoil is kinematic viscosity at oilϑ and may be calculated by means of the following equation
( ) ( )++=+ 08νloglog08νloglog 100oil ( )
sdotminus
ϑ+minus313log373log
273log373log oil
( ) ( )( )08νloglog08νloglog 10040 +minus+
Ra = composite arithmetic mean roughness (micron) of pinion an wheel calculated as ( )2a1aa RR 05R +=This is defined as the roughness on the new flanks ie as manufactured
XL = see 42
432 Effective tip relief Ceff Ceff is the effective tip relief that amount of tip relief which compensates for the elastic deformation of the gear mesh ie
zero load at the tooth tip It is assumed (simplified) to be equal for both pinion and wheel
Cylindrical gears
for helical
γ
γ=cb
KKFC Abt
eff (see 1)
For spur
cbKKF
C Abteff
γ= (see 1)
Alternatively for spur and helical gears the non-linear ap-proach in 111 may be used (taking Ceff=δ)
Bevel gears
bcKFC Ambt
effγ
= (see 1)
where
γ
α
ε2ε44c
+sdot
=γ
433 Tip relief and extension Cylindrical gears
The extension of the tip relief is not to result in an effective contact ratio 1ltεα when the gear is unloaded (exceptions to this may only apply for applications where the gear is not to run at light loads) This means that the unrelieved part of the path of contact is to be minimum pbt It is further assumed that this unrelieved part is placed centrally on the path of contact
If root relief applies it has to be calculated as equivalent tip relief Ie pinion root relief (at mesh position A) is added to Ca2 and wheel root relief (at mesh position E) is added to Ca1 If no design tip relief or root relief on the mating gear is specified (ie if Ca1+Croot2 = 0 and visa versa) use the run-ning in amount see 112
Bevel gears
Bevel gears are to have heightwise crowning ie no distinct relievedunrelieved area This may be treated as tip and root relief For calculation purposes the root relief is combined with the tip relief of the mating member into an equivalent tip relief If no resulting tip relieves are specified the equivalent tip relives may be calculated as an approxima-tion based on the tool crowning Ca tool (per mille of tool module m0) as follows
2root1aeq1a CCC +=
1root2aeq2a CCC +=
where
Classification Notes- No 412 35 May 2003
DET NORSKE VERITAS
2
0
n0101atoolal m
)m2(mA1middotmCC
minus+minussdot=
( )
sdotminusminussdotsdotminus+= vtvb1
2vb1
21vavt1n1 αtan dddαsin 05x1mA
2
0
n020atool22a m
)mm(2A1middotmCC
minus+minussdot=
( )
sdotminusminussdotsdotminus+= vtvb2
2vb2
2va2vt2n2 αtan dddαsin 05x1mA
2
0
20atool1root1 m
A1mCC
minussdotsdot=
2
0
10atool2root2 m
A1mCC
minussdotsdot=
If the pressure angles of the cutter blades are modified (and verified) to balance the tip relieves the following may be assumed
( )eq2aeq1aeq2aeq1a C andC calculated of sum50CC sdot==
Throughout the following Ca1 and Ca2 mean the equivalent tip relieves Ca1eq and Ca2eq
434 Bulk temperature The bulk temperature may be calculated as
flaaveragempsoilMB X X 50 ϑ+ϑ=ϑ
where
Xs = lubrication factor
= 12 for spray lubrication
= 10 for dip lubrication (provided both pinion and wheel are dip lubricated and tip speed lt 5 ms)
= 10 for spray lubrication with additional cooling spray (spray on both pinion and wheel or spray on pinion and dip lubrication of wheel)
= 02 for meshes fully submerged in oil
Xmp = contact factor ( )pmp nX += 150
np = number of mesh contact on the pinion (for small gear ratii the number of wheel meshes should be used if higher)
flaaverageϑ
= average of the integrated flash temperature (see 44) along the path of contact
( )
AE
E
Ayyyfla
flaaverage
d
ΓminusΓ
ΓΓϑ=ϑint =
For high speed gears (v gt 50 ms) it may be necessary to assess the bulk temperature on the basis of a thermal rating of the entire gear transmission
44 The Flash Temperature flaϑ
441 Basic formula The local flash temperature flaϑ may be calculated as
( ) 41redy
y2ly
211
43Btcorrfla
unXwX3250
y ρ
ρminusρ
micro=ϑ Γ
(For bevel gears replace u with uv)
and is to be calculated stepwise along the path of contact from A to E
where
micro = coefficient of friction see 431
Xcorr = correction factor taking empirically into account the increased scuffing risk in the beginning of the approach path due to mesh starting without any previously built up oil film and possible shuffling away oil before meshing if insufficient tip relief
( )
3
AD
yaeffcorr 50
CC1X
ΓminusΓε
Γminus+=
α
Ca = tip relief of driven member Xcorr is only applicable in the approach path and if Ca lt Ceff otherwise 10
wBt = unit load see 431
yXΓ = load sharing factor see 443
n1 = pinion rpm
Γy ρly etc see 442
442 Geometrical relations The various radii of flank curvature (transversal plane) are
ρ1y = pinion flank radius at mesh point y
ρ2y = wheel flank radius at mesh point y
ρredy = equivalent radius of curvature at mesh point y
y2y1
y2y1redy
ρ+ρ
ρρ=ρ
Cylindrical gears
twy
1y αsin au1Γ1
ρ+
+=
twy
2y αsin au1Γu
ρ+
minus=
Note that for internal gears a and u are negative
36 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
Bevel gears
vtvv
y1y αsin a
u1Γ1
ρ+
+=
vtvv
yv2y αsin a
u1Γu
ρ+
minus=
Γ is the parameter on the path of contact and y is any point between A end E
At the respective ends Γ has the following values
Root piniontip wheel
Cylindrical gears
( )
minus
minusminus= 1
αtan 1dd
zzΓ
tw
2b2a2
1
2A
Bevel gears
( )
minus
minusminus= 1
αtan 1dd
uΓvt
2vb2va2
vA
Tip pinionroot wheel
Cylindrical gears
( )
1αtan
1ddΓ
tw
2b1a1
E minusminus
=
Bevel gears
( )
1αtan
1ddΓ
vt
2vb1va1
E minusminus
=
At inner point of single pair contact
Cylindrical gears
tw1
EB α tan zπ2ΓΓ minus=
Bevel gears
vtv1
EB α tan zπ2ΓΓ minus=
At outer point of single pair contact
Cylindrical gears
tw1AD α tan z
π2ΓΓ +=
Bevel gears
vt v1
AD αtan z π2ΓΓ +=
At pitch point ΓC = 0
The points F and G (only applicable to cylindrical gears) limiting the extension of tip relief (so as to maintain a mini-mum contact ratio of unity for unloaded gears) are at
2
BAF
Γ+Γ=Γ
2
EDG
Γ+Γ=Γ
443 Load sharing factor XΓ The load sharing factor XΓ accounts for the load sharing be-tween the various pairs of teeth in mesh along the path of contact
XΓ is to be calculated stepwise from A to E using the pa-rameter Γy
4431 Cylindrical gears with β = 0 and no tip relief
Figure 41
ByAAB
Ay for31
31X
yΓltΓleΓ
ΓminusΓ
ΓminusΓ+=Γ
DyB for1Xy
ΓltΓleΓ=Γ
EyDDE
yE for31
31X
yΓleΓltΓ
ΓminusΓ
ΓminusΓ+=Γ
4432 Cylindrical gears with β = 0 and tip relief Tip relief on the pinion reduces XΓ in the range G ndash E and increases correspondingly XΓ in the range F ndash B
Tip relief on the wheel reduces XΓ in the range A ndash F and increases correspondingly XΓ in the range D ndash G
Following remains generally
DyB for1Xy
ΓleΓleΓ=Γ
21XXGF== ΓΓ
In the following it must be distinguished between Ca lt Ceff respectively Ca gt Ceff This is shown by an example below where Ca1 lt Ceff and Ca2 gt Ceff
Classification Notes- No 412 37 May 2003
DET NORSKE VERITAS
Figure 42
Note When Ca gt Ceff the path of contact is shortened by A ndash A respectively E ndash E The single pair contact path is extended into B respectively D If this shift is significant it is necessary to consider the negative effect on surface durability (B) and bending stresses (B and D)
Range A - F
For effa2 CC le
minus=Γ
eff
2a
CC1
31X
A
FyAeff
2a
AF
Ay for C3
C61XX
AyΓleΓleΓ
sdot
+ΓminusΓ
ΓminusΓ+= ΓΓ
For effa2 CC ge
AyAfor0Xy
ΓleΓleΓ=Γ
with ( )21
CC
1CC
eff
2a
eff
2a
AFAAminus
minusΓminusΓ+Γ=Γ
FyAeff
2a
AF
Ay
eff
2a for 21
CC
CC1X
yΓleΓleΓ
minus
ΓminusΓ
ΓminusΓ+minus=Γ
Range F - B
For effa1 CC le
ByFeff
1a
FB
Fy forC3
C61
21X
yΓleΓleΓ
sdot
+ΓminusΓ
ΓminusΓ+=Γ
For effa1 CC ge
ByFeff
1a
FB
Fy for21
CC
21X
yΓleΓleΓ
minus
ΓminusΓ
ΓminusΓ+=Γ
ByB for1Xy
ΓleΓleΓ=Γ
with 1
CC2
eff
1a
FBFB
minus
ΓminusΓ+Γ=Γ
Range D ndash G
For effa2 CC le
GyDeff
2a
DG
Dy
eff
2a for C 3C
61
C 3C
32X
yΓleΓleΓ
+
ΓminusΓ
ΓminusΓminus+=Γ F
or effa2 CC ge
DyD for1Xy
ΓleΓleΓ=Γ
with ( )21
CC
1CC
eff
2a
eff
2a
DGDDminus
minusΓminusΓ+Γ=Γ
GyDeff
2a
DG
Dy
eff
2a for21
CC
CCX ΓleΓleΓ
minus
ΓminusΓ
ΓminusΓminus=Γ
Range G - E
For effa1 CC le
minus=Γ
eff
1a
CC1
31X
E
EyGeff
1a
GE
Gy forC3
C61
21X
yΓleΓleΓ
sdot
+ΓminusΓ
ΓminusΓminus=Γ
For effa1 CC gt
EyGeff
1a
GE
Gy for21
CC
21X
yΓleΓleΓ
minus
ΓminusΓ
ΓminusΓminus=Γ
EyE for0Xy
ΓleΓleΓ=Γ
with 1
CC2
eff
1a
GEGE
minus
ΓminusΓ+Γ=Γ
4433 Gears with β gt 0 buttressing Due to oblique contact lines over the flanks a certain but-tressing may occur near A and E
This applies to both cylindrical and bevel gears with tip relief lt Ceff The buttressing Xbutt is simplified as a linear function within the ranges A ndash H respectively I ndash E
Figure 43
1εwhen13X βbutt EAge=
1εwhenε031X ββbutt EAlt+=
38 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
Cylindrical gears
bIEAH βsin 02ΓΓΓΓ =minus=minus
Bevel gears
bmIEAH βsin 02ΓΓΓΓ =minus=minus
4434 Cylindrical gears with 2εγ le and no tip relief
yXΓ is obtained by multiplication of
yXΓ in 4431 with
Xbutt in 4433
4435 Gears with 2εγ gt and no tip relief
Applicable to both cylindrical and bevel gears
IyH for1Xy
ΓleΓleΓε
=α
Γ
IyHybutt and forX1Xy
ΓgtΓΓltΓε
=α
Γ
Figure 44
4436 Cylindrical gears with 2εγ le and tip relief
yXΓ is obtained by multiplication of
yXΓ in 4432 with Xbutt
in 4433
4437 Cylindrical gears with 2εγ gt and tip relief
Tip relief on the pinion (respectively wheel) reduces XГ in the range G ndash E (respectively A ndash F) and increases XГ in the range F ndash G
yXΓ is obtained by multiplication of
yXΓ as described below
with Xbutt in 4433
In the XГ example below the influence of tip relief is shown (without the influence of Xbutt) by means of
eff2aeff1a CC and CC ltgt
Tip relief gt Ceff causes new end points A respectively E of the path of contact
Figure 45
Range A ndash F
( ) ( )( ) eff
a2αa1α
AF
Ay
eff
2aeff
C 12C 13εC 1ε
CCCX
y +εε++minus
ΓminusΓ
ΓminusΓ+
εminus
=ααα
Γ
eff2aFyA CC if for leΓleΓleΓ
eff2aFyA CifCfor and geΓleΓleΓ
eff2aAyA CC iffor0Xy
gtΓleΓleΓ=Γ
( ) ( ) ( )( ) ( ) a2αa1α
αeffa2AFAA C 1ε 3C 1ε
1ε 2 CC with++minus+minus
ΓminusΓ+Γ=Γ
Range F ndash G
( )( )( ) GyF
eff
2a1a forC 1 2CC 11X
yΓleΓleΓ
+εε+minusε
+ε
=αα
α
αΓ
Range G ndash E
( ) ( )( ) eff
2a1a
GE
Gy
C 1 2C 1C 1 3XX
GFy +εεminusε++ε
ΓminusΓ
ΓminusΓminus=
αα
ααΓΓ minus
effa1EyG CC if for leΓleΓleΓ
effa1EyG CC if Γfor and geΓleΓle
eff1aEyE CC if for0Xy
geΓleΓleΓ=Γ
( ) ( ) ( )( ) ( ) 2a1a
eff1aGEEE C 1C 1 3
1 2 CC withminusε++ε+εminus
ΓminusΓminusΓ=Γαα
α
4438 Bevel gears with εγ more than approx 18 and heightwise crowning For Ca1 = Ca2 = Ceff the following applies
Figure 46
Classification Notes- No 412 39 May 2003
DET NORSKE VERITAS
( )AEM 50 Γ+Γ=Γ
( )( )2AD
3
2My651X
y ΓminusΓε
ΓminusΓminus
ε=
ααΓ
For tip relief lt Ceff yXΓ is found by linear interpolation be-
tween ( )effay CCX =Γ and ( )0CayX =Γ as in 4435
The interpolation is to be made stepwise from A to M with the influence of Ca2 and from M to E with the influence of Ca1 (For 2a1a CC ne there is a discontinuity at M)
Eg with Ca1 = 04 Ceff and Ca2 = 055 Ceff then
Range A ndash M
)CC()0C( eff2ay2ayyX 550X 450X =Γ=ΓΓ +=
Range M ndash E
)CC()0C( eff1ay1ayyX 40X 06X =Γ=ΓΓ +=
For tip relief gt Ceff the new end points A and E are found as
( )
minusΓminusΓ
ε+Γ=Γ α 1
CC
6 eff
2aADAA
( )
minusΓminusΓ
ε+Γ=Γ α 1
CC
6 eff
1aADEE
Range A ndash A
0Xy=Γ
Range A ndash M
( )( )
ΓminusΓ
ΓminusΓminus
minusε=
αΓ 2
MA
2My
eff
2a1
CC4
351Xy
Range M ndash E
( )( )
ΓminusΓ
ΓminusΓminus
minusε=
αΓ 2
ME
2My
eff
1a1
CC4
351Xy
Range E ndash E
0Xy=Γ
40 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
Appendix A Fatigue Damage Accumulation
The Palmgren-Miner cumulative damage calculation princi-ple is used The procedure may be applied as follows
A1 Stress Spectrum From the individual torque classes the torques (Ti) at the peak values of class intervals and the associated number of cycles (NLi) for both pinion and wheel are to be listed from the highest to the lowest torque
(In case of a cyclic torque variation within the torque classes it is advised to use the peak torque If the cyclic variation is such that the same teeth will repeatedly suffer the peak torque this is a must)
The stress spectra for tooth roots and flanks (σFi σHi) with all relevant factors (except KA) are to be calculated on the basis of the torque spectrum The load dependent K-factors are to be determined for each torque class
A2 σminusN-curve The stress versus load cycle curves for tooth roots and flanks (both pinion and wheel) are to be drawn on the basis of per-missible stresses (ie including the demanded minimum safety factors) as determined in 2 respectively 3 If different safety levels for high cycle fatigue and low cycle fatigue are desired this may be expressed by different demand safety factors applied at the endurance limit respectively at static strength
A3 Damage accumulation The individual damage ratio Di at ith stress level is defined as
Fi
Lii N
ND =
where
NLi = The number of applied cycles at ith stress
NFi = The number of cycles to failure at ith stress
Basically stresses σi below the permissible stress level for infinite life (if a constant ZN or YN is accepted) do not con-tribute to the damage sum However calculating the actual safety factor Sact as described below all the σi for which the product S σi is bigger than or equal to the permissible stress level for infinite life contribute to the damage sum and thus to the determination of Sact The final value of S is decisive
(NFi can be found mathematically by putting the permissible stress σpi equal the actual stress σi thereby finding the actual life factor This life factor can be solved with regard to load cycles ie NFi)
The damage sum ΣDi is not to exceed unity
If ΣDi ne 1 the safety against cumulative fatigue damage is different from the applied demand safety factor For deter-mination of this theoretical safety factor an iteration proce-dure is required as described in the following flowchart
S is correction factor with which the actual safety factor Sact can be found
Sact is the demand safety factor (used in determination of the permissible stresses in the σ ndash N ndash curve) times the correc-tion factor S
The full procedure is to be applied for pinion and wheel tooth roots and flanks
Note If alternating stresses occur in a spectrum of mainly pulsating stresses the alternating stresses may be replaced by equiva-lent pulsating stresses ie by means of division with the ac-tual mean stress influence factor YM
Classification Notes- No 412 41 May 2003
DET NORSKE VERITAS
Appendix B Application Factors for Diesel Driven Gears
For diesel driven gears the application factor KA depends on torsional vibrations Both normal operation and misfiring conditions have to be considered
Normally these two running conditions can be covered by only one calculation
B1 Definitions
Normal operation KAnorm = 0
normv0
TTT +
where
T0 = rated nominal torque
Tv norm = vibratory torque amplitude for normal opera-tion (see rules Pt4 Ch3 Sec1 G301 for defi-nition of ldquonormalrdquo irregularity)
Misfiring operation KA misf = 0
misfv
TTT +
where
T = remaining nominal torque when one cylinder out of action
Tv misf = vibratory torque amplitude in misfiring con-dition This refers to a permissible misfiring condition ie a condition that does not require automatic or immediate corrective actions as speed or pitch reduction
The normal operation is assumed to last for a very high num-ber of cycles such as 1010
The misfiring operation is assumed to last for a limited dura-tion such as 107 cycles
B2 Determination of decisive load Assuming life factor at 1010 cycles as YN = ZN = 092 which usually is relevant the calculation may be performed only once with the combination having the highest value of appli-cation factorlife factor
For bending stresses and scuffing the higher value of
092
K normA and 098
K misfA
For contact stresses the higher value of
092K normA and
113K misfA (but
097K misfA for nitrided gears)
B3 Simplified procedure Note that this is only a guidance and is not a binding con-vention
T = oTZ
1Zsdot
minus where Z = number of cylinders
( ) ideal Videal Vmisf Vnorm V TTT24ZT +minus=
where TV ideal = vibratory torque with all cylinders perfectly equal See also rules Pt4 Ch3 Sec1 G300
When using trends from torsional vibration analysis and measurements the following may be used
TV ideal To is close to zero for engines with few cylinders and using a suitable elastic coupling and increases with relative coupling stiffness and number of cylinders
TV misf To may be high for engines with few cylinders and decreases with number of cylinders
This can be indicated as
200Z
TT
ο
idealV asymp and 80Z04
TT
o
misfV minusasymp
Inserting this into the formulae for the two application fac-tors the following guidance can be given
118112K misfA minusasymp
115110K normA minusasymp
Since KA norm is to be combined with the lower life factors the decisive load condition will be the normal one and a KA of 115 will cover most relevant cases when a suitable elas-tic coupling is chosen
42 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
Appendix C Calculation of Pinion-Rack
Pinion-racks used for elevating of mobile offshore units are open gears that are subjected to wear and tear With normal specifications such as only a few hundred total operation cycles (site to site) the tooth bending stresses for static strength will be decisive for the lay out however with ex-ception of surface hardened pinions where case crushing has to be considered
In the following the use of part 1 and 3 for pinion-racks is shown including relevant simplifications
C1 Pinion tooth root stresses Since the load spectrum normally is dominated by high torques and few load cycles (in the range up to 1-2000) the static strength is decisive
The actual stress is calculated as
β1FSaFan1
tF1 KYY
mbFσ sdotsdotsdotsdot
=
b1 is limited to b2 + 2middotmn
YFa and YSa replace YF and YS because load application at tooth tip has to be assumed for such inaccurate gears
Pinions often use a non-involute profile in the dedendum part of the flank eg a constant radius equal the radius of curva-ture at reference circle For such pinions sFn and hFa are to be measured directly on a sectional drawing of the pinion tooth
Due to high loads and narrow facewidths it may be assumed that KFβ2 = KHβ = 10 However when b1 gt b2 then KFβ1 gt10
If no detailed documentation of KFβ1 is available the fol-lowing may be used
KFβ1 = 1 + 015middot(b1b2 ndash 1)
The permissible stress (not surface hardened) is calculated as
δrelTstF
Fst1FP1 Y
Sσσ sdot=
The mean stress influence due to leg lifting may be disre-garded
The actual and permissible stresses should be calculated for the relevant loads as given in the rules
C2 Rack tooth root stresses The actual stress is calculated as
SaFan2
tF2 YY
mbFσ sdotsdotsdot
=
See C1 for details
The permissible stress is calculated as
δrelTstF
Fst2FP2 Y
Sσσ sdot=
For alloyed steels (Ni Cr Mo) with high toughness and duc-tility the value of YδrelTst may be put equal to YSa
C3 Surface hardened pinions For surface hardened pinions the maximum load is not to cause crushing of the hardened layer of the flank
In principle the calculation described in 213 may be used but when the theoretical Hertzian stress exceeds the ap-proximately 18 times the yield strength of the rack material plastic deformation will occur This will limit the peak Hertzian stress but increases the contact width and thus the penetration of stresses into the depth
An approximation may be based on an assessment of contact width determined by means of equal areas under the theoreti-cal (elastic range) Hertzian contact and the elasto-plastic contact stress (limited to 18 middot σy) with the unknown width
Classification Notes- No 412 25 May 2003
DET NORSKE VERITAS
g
g
ρt
0613
13
minus
where
tg = depth of the grinding notch
ρg = radius of the grinding notch
c) The formulae for YS resp YSa are only valid for αn = 20˚ However the same formulae can be used as a safe approximation for other pressure angles
35 Contact Ratio Factor Yε The contact ratio factor Yε covers the conversion from load application at the tooth tip to the load application at the mid point of the flank (heightwise) for bevel gears
The following may be used
Yε = 0625
36 Helix Angle Factor Yβ The helix angle factor Yβ takes into account the difference between the helical gear and the virtual spur gear in the nor-mal section on which the calculation is based in the first step In this way it is accounted for that the conditions for tooth root stresses are more favourable because the lines of contact are sloping over the flank
The following may be used (β input in degrees)
Yβ = 1 ndash εβ β120
When εβ gt 1 use εβ = 1 and when β gt 30deg use β = 30deg in the formula
However the above equation for Yβ may only be used for gears with β gt 25deg if adequate tip relief is applied to both pinion and wheel (adequate = at least 05 middot Ceff see 432)
37 Values of Endurance Limit σFE σFE is the local tooth root stress (max principal) which the material can endure permanently with 99 survival prob-ability 3106 load cycles is regarded as the beginning of the endurance limit or the lower knee of the σ ndash N curve σFE is defined as the unidirectional pulsating stress with a minimum stress of zero (disregarding residual stresses due to heat treatment) Other stress conditions such as alternating or pre-stressed etc are covered by the conversion factor YM
σFE can be found by pulsating tests or gear running tests for any material in any condition If the approval of the gear is to be based on the results of such tests all details on the testing conditions have to be approved by the Society Fur-ther the tests may have to be made under the Societys su-pervision
If no fatigue tests are available the following listed values for σFE may be used for materials subjected to a quality con-trol as the one referred to in the rules
σFE
Alloyed case hardened steels 1) (fillet surface hardness 58 ndash 63 HRC)
bull of specially approved high grade
1050
bull of normal grade
minus CrNiMo steels with approved process
1000
minus CrNi and CrNiMo steels generally 920 minus MnCr steels generally 850
Nitriding steel of approved grade quenched tempered and gas nitrided (surface hardness 700 ndash 800 HV)
840
Alloyed quenched and tempered steel bath or gas nitrided (surface hardness 500 ndash 700 HV)
720
Alloyed quenched and tempered steel flame or induction hardened 2) (incl entire root fillet) (fillet surface hardness 500 ndash 650 HV)
07 HV + 300
Alloyed quenched and tempered steel flame or induction hardened (excl entire root fillet) (σB = uts of base material)
025 σB + 125
Alloyed quenched and tempered steel 04 σB + 200
Carbon steel 025 σB + 250
Note All numbers given above are valid for separate forgings and for blanks cut from bars forged according to a qualified procedure see Pt 4 Ch 2 Sec 3 For rolled steel the values are to be reduced with 10 For blanks cut from forged bars that are not qualified as mentioned above the values are to be reduced with 20 For cast steel reduce with 40
1) These values are valid for a root radius
bull being unground If however any grinding is made in the root fillet area in such a way that the residual stresses may be affected σFE is to be reduced by 20 (If the grinding also leaves a notch see 34)
bull with fillet surface hardness 58 ndash 63 HRC In case of lower surface hardness than 58 HRC σFE is to be reduced with 20(58 ndash HRC) where HRC is the detected hardness (This may lead to a permissible tooth root stress that varies along the facewidth If so the actual tooth root stresses may also be considered along facewidth)
bull not being shot peened In case of approved shot peening σFE may be increased by 200 for gears where σFE is reduced by 20 due to root grinding Otherwise σFE may be increased by 100 for
6nm le and 100 ndash 5 (mn - 6) for mn gt 6 However the possible adverse influence on the flanks regarding grey staining should be considered and if necessary the flanks should be masked
2) The fillet is not to be ground after surface hardening Regarding possible root grinding see 1)
26 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
38 Mean stress influence Factor YM The mean stress influence factor YM takes into account the influence of other working stress conditions than pure pulsa-tions (R = 0) such as eg load reversals idler gears planets and shrink-fitted gears
YM (YMst) are defined as the ratio between the endurance (or static) strength with a stress ratio R ne 0 and the endurance (or static) strength with R = 0
YM and YMst apply only to a calculation method that assesses the positive (tensile) stresses and is therefore suitable for comparison between the calculated (positive) working stress σF and the permissible stress σFP calculated with YM or YMst
For thin rings (annulus) in epicyclic gears where the com-pression fillet may be decisive special considerations apply see 316
The following method may be used within a stress ratio ndash12 lt R lt 05
381 For idlers planets and PTO with ice class
M1M1R1
1Yor Y MstM
+minus
minus=
where
R = stress ratio = min stress divided by max stress
For designs with the same force applied on both forward- and back-flank R may be assumed to ndash 12
For designs with considerably different forces on forward- and back-flank such as eg a marine propulsion wheel with a power take off pinion R may be assessed as
branchmain theoffacewidth unit per forcepto offacewidt unit per force21minus
For a power take off (PTO) with ice class see 161 c
M considers the mean stress influence on the endurance (or static) strength amplitudes
M is defined as the reduction of the endurance strength am-plitude for a certain increase of the mean stress divided by that increase of the mean stress
Following M values may be used
Endurance limit
Static strength
Case hardened 08 ndash 015 Ys 1) 07
If shot peened 04 06
Nitrided 03 03
Induction or flame hardened
04 06
Not surface hardened steel 03 05
Cast steels 04 06 1)For bevel gears use Ys=2 for determination of M
The listed M values for the endurance limit are independent of the fillet shape (Ys) except for case hardening In princi-ple there is a dependency but wide variations usually only occur for case hardening eg smooth semicircular fillets versus grinding notches
382 For gears with periodical change of rotational direction For case hardened gears with full load applied periodically in both directions such as side thrusters the same formula for YM as for idlers (with R = ndash 12) may be used together with the M values for endurance limit This simplified approach is valid when the number of changes of direction exceeds 100 and the total number of load cycles exceeds 3middot106
For gears of other materials YM will normally be higher than for a pure idler provided the number of changes of direction is below 3middot106 A linear interpolation in a diagram with loga-rithmic number of changes of direction may be used ie from YM = 09 with one change to YM (idler) for 3middot106 changes This is applicable to YM for endurance limit For static strength use YM as for idlers
For gears with occasional full load in reversed direction such as the main wheel in a reversing gear box YM = 09 may be used
383 For gears with shrinkage stresses and unidirectional load For endurance strength
FE
fitM σ
σM1
M21Y+
minus=
σFE is the endurance limit for R = 0
For static strength YMst = 1 and σfit accounted for in 39b
σfit is the shrinkage stress in the fillet (30˚ tangent) and may be found by multiplying the nominal tangential (hoop) stress with a stress concentration factor
n
Ffit m
ρ215scf minus=
384 For shrink-fitted idlers and planets When combined conditions apply such as idlers with shrink-age stresses the design factor for endurance strength is
( ) ( ) FE
fitM σ
σR1M1
M2
M1M1R1
1Yminussdot+
minus
+minus
minus=
Symbols as above but note that the stress ratio R in this par-ticular connection should disregard the influence of σfit ie R normally equal ndash 12
For static strength
M1M1R1
1YMst
+minus
minus=
The effect of σfit is accounted for in 39 b
Classification Notes- No 412 27 May 2003
DET NORSKE VERITAS
385 Additional requirements for peak loads The total stress range (σmax ndash σmin) in a tooth root fillet is not to exceed
F
y
Sσ225
for not surface hardened fillets
FSHV5 for surface hardened fillets
39 Life Factor YN The life factor YN takes into account that in the case of limited life (number of cycles) a higher tooth root stress can be permitted and that lower stresses may apply for very high number of cycles
Decisive for the strength at limited life is the σ ndash N ndash curve of the respective material for given hardening module fillet radius roughness in the tooth root etc Ie the factors YδrelT YRelT YX and YM have an influence on YN
If no σ ndash N ndash curve for the actual material and hardening etc is available the following method may be used
Determination of the σ ndash N ndash curve
a) Calculate the permissible stress σFP for the beginning of the endurance limit (3middot106 cycles) including the influ-ence of all relevant factors as SF YδrelT YRelT YX YM and YC ie σFP = σFE middotYM middotYδrelT middotYRelT middotYX middotYC SF
b) Calculate the permissible laquostaticraquo stress (le 103 load cy-cles) including the influence of all relevant factors as SFst YδrelTst YMst and YCst ( )fitσCstYδrelTstYMstYFstσ
FstS1
FPstσ minussdotsdotsdot=
where σFst is the local tooth root stress which the material can resist without cracking (surface hardened materials) or unacceptable deformation (not surface hardened materials) with 99 survival probability
σFst
Alloyed case hardened steel 1) 2300
Nitriding steel quenched tempered and gas nitrided (surface hardness 700 ndash 800 HV)
1250
Alloyed quenched and tempered steel bath or gas nitrided (surface hardness 500 ndash 700 HV)
1050
Alloyed quenched and tempered steel flame or induction hardened (fillet surface hardness 500 ndash 650 HV)
18 HV + 800
Steel with not surface hardened fillets the smaller value of 2)
18 σB or 225 σy
1) This is valid for a fillet surface hardness of 58 ndash 63 HRC In case of lower fillet surface hardness than 58 HRC σFst is to be reduced with 30(58 ndash HRC) where HRC is the actual hardness Shot peening or grinding notches are not considered to have any significant influence on σFst 2) Actual stresses exceeding the yield point (σy or σ02) will alter the residual stresses locally in the ldquotensionrdquo fillet re-spectively ldquocompressionrdquo fillet This is only to be utilised for gears that are not later loaded with a high number of cycles at lower loads that could cause fatigue in the ldquocompressionrdquo fillet
c) Calculate YN as
NL gt 3middot106
YN = 1 or 001
L
6
N N103Y
sdot= ie Yn = 092 for 1010
The YN = 1 from 3middot106 on may only be used when special material cleanness applies see rules Pt4 Ch2
103ltNLlt3middot106exp
L
6
N N103Y
sdot=
cycles6103forσcycles310forσlog02876exp
FP
FPst
sdot=
NL lt 103 cycles6103forσcycles310forσ
NYFP
FPst
sdot=
or simply use σFPst as mentioned in b) directly
Guidance on number of load cycles NL for various applica-tions
bull For propulsion purpose normally NL = 1010 at full load (yachts etc may have lower values)
bull For auxiliary gears driving generators that normally operate with 70-90 of rated power NL = 108 with rated power may be applied
28 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
310 Relative Notch Sensitivity Factor YδrelT The dynamic (respectively static) relative notch sensitivity factor YδrelT (YδrelTst) indicate to which extent the theoreti-cally concentrated stress lies above the endurance limits (re-spectively static strengths) in the case of fatigue (respectively overload) breakage
YδrelT is a function of the material and the relative stress gra-dient It differs for static strength and endurance limit
The following method may be used
For endurance limit
for not surface hardened fillets
( )
024
s024
relT σ103133q21σ1012201351
Ysdotsdotminus
+sdotsdotminus+= minus
minus
δ
for all surface hardened fillets except nitrided
( )
106q21002451
Y srelT
++=δ
for nitrided fillets
( )
1347q2101421
Y srelT
++=δ
For static strength
for not surface hardened fillets1)
( )( )( )025
02
02502s
relTst σ3000821σ300 1Y0821Y
+
minus+=δ
for surface hardened fillets except nitrided
YδrelTst = 044 YS + 012
for nitrided fillets
YδrelTst = 06 + 02 YS 1) These values are only valid if the local stresses do not
exceed the yield point and thereby alter the residual stress level See also 39b footnote 2
311 Relative Surface Condition Factor YRrelT The relative surface condition factor YRrelT takes into ac-count the dependence of the tooth root strength on the sur-face condition in the tooth root fillet mainly the dependence on the peak to valley surface roughness
YRrelT differs for endurance limit and static strength
The following method may be used
For endurance limit
YRrelT = 1675 ndash 053 (Ry + 1)01 for surface hardened steels and alloyed quenched and tempered steels except nitrided
YRrelT = 53 ndash 42 (Ry + 1)001 for carbon steels
YδrelT = 43 ndash 326 (Ry + 1)0005
for nitrided steels
For static strength
YRrelTst = 1 for all Ry and all materials
For a fillet without any longitudinal machining trace Ry asymp Rz
312 Size Factor YX The size factor YX takes into account the decrease of the strength with increasing size YX differs for endurance limit and static strength
The following may be used
For endurance limit
YX = 1 for mn le 5 generally
YX = 103 ndash 0006 mn for 5 lt mn lt 30
YX = 085 for mn ge 30
for not surface hard-ened steels
YX = 105 ndash 001 mn for 5 lt mn ge 25
YX = 08 for mn ge 25
for surface hardened steels
For static strength
YXst = 1 for all mn and all materials
313 Case Depth Factor YC The case depth factor YC takes into account the influence of hardening depth on tooth root strength
YC applies only to surface hardened tooth roots and is dif-ferent for endurance limit and static strength
In case of insufficient hardening depth fatigue cracks can develop in the transition zone between the hardened layer and the core For static strength yielding shall not occur in the transition zone as this would alter the surface residual stresses and therewith also the fatigue strength
The major parameters are case depth stress gradient permis-sible surface respectively subsurface stresses and subsurface residual stresses
The following simplified method for YC may be used
YC consists of a ratio between permissible subsurface stress (incl influence of expected residual stresses) and permissible surface stress This ratio is multiplied with a bracket con-taining the influence of case depth and stress gradient (The empirical numbers in the bracket are based on a high number of teeth and are somewhat on the safe side for low number of teeth)
YC and YCst may be calculated as given below but calculated values above 10 are to be put equal 10
For endurance limit
+
+=nFFE
C m02ρt31
σconstY
Classification Notes- No 412 29 May 2003
DET NORSKE VERITAS
For static strength
+
+=nFFst
Cst m02ρt31
σconstY
where const and t are connected as
Hardening process
t = endurance limit const =
static strength const =
t550 640 1900
t400 500 1200 Case hard-ening
t300 380 800
Nitriding t400 500 1200
Induction- or flame hardening
tHVmin 11 HVmin 25 HVmin
For symbols see 213
In addition to these requirements to minimum case depths for endurance limit some upper limitations apply to case hard-ened gears
The max depth to 550 HV should not exceed
1) 13 of the top land thickness san unless adequate tip relief is applied (see 110)
2) 025 mn If this is exceeded the following applies addi-tionally in connection with endurance limit
minusminus= 025
mt
1Yn
max550C
314 Thin rim factor YB Where the rim thickness is not sufficient to provide full sup-port for the tooth root the location of a bending failure may be through the gear rim rather than from root fillet to root fillet
YB is not a factor used to convert calculated root stresses at the 30deg tangent to actual stresses in a thin rim tension fillet Actually the compression fillet can be more susceptible to fatigue
YB is a simplified empirical factor used to de-rate thin rim gears (external as well as internal) when no detailed calcula-tion of stresses in both tension and compression fillets are available
Figure 34 Examples on thin rims
YB is applicable in the range 175 lt sRmn lt 35
YB = 115 middot ln (8324 middot mnsR)
(for sRmn ge 35 YB = 1)
(for sRmn le 175 use 315)
σF as calculated in 321 is to be multiplied with YB when sRmn lt 35 Thus YB is used for both high and low cycle fatigue
Note This method is considered to be on the safe side for external gear rims However for internal gear rims without any flange or web stiffeners the method may not be on the safe side and it is advised to check with the method in 315316
315 Stresses in Thin Rims For rim thickness sR lt 35 mn the safety against rim cracking has to be checked
The following method may be used
3151 General The stresses in the standardised 30ordm tangent section tension side are slightly reduced due to decreasing stress correction factor with decreasing relative rim thickness sRmn On the other hand during the complete stress cycle of that fillet a certain amount of compression stresses are also introduced The complete stress range remains approximately constant Therefore the standardised calculation of stresses at the 30ordm tangent may be retained for thin rims as one of the necessary criteria
The maximum stress range for thin rims usually occurs at the 60ordm ndash 80ordm tangents both for laquotensionraquo and laquocompressionraquo side The following method assumes the 75ordm tangent to be the decisive Therefore in addition to the am criterion ap-plied at the 30ordm tangent it is necessary to evaluate the max and min stresses at the 75ordm tangent for both laquotensionraquo (loaded flank) fillet and laquocompressionraquo (back-flank) fillet For this purpose the whole stress cycle of each fillet should be considered but usually the following simplification is justified
Figure 35 Nomenclature of fillets
Index laquoTraquo means laquotensileraquo fillet laquoCraquo means laquocompressionraquo fillet
σFTmin and σFCmax are determined on basis of the nominal rim stresses times the stress concentration factor Y75
σFCmin and σFTmax are determined on basis of superposition of nominal rim stresses times Y75 plus the tooth bending stresses at 75ordm tangent
30 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
3152 Stress concentration factors at the 75ordm tangents The nominal rim stress consists of bending stresses due to local bending moments tangential stresses due to the tan-gential force Ft and radial shear stresses due to Fr
The major influence is given by the bending stresses The influence of the tangential stresses is minor and even though its stress concentration factor is slightly higher than for bending it is considered to be safe enough when the sum of these nominal stresses are combined with the stress concen-tration factor for bending The influence of the radial shear stress is neglected
The stress concentration factor relating nominal rim stresses to local fillet stresses at the 75ordm tangent may be calculated as
75n
R75
n
R
75
ρmsρ185
ms3
Y+
sdot=
where ρ75 is the root radius at the 75ordm tangent ref to mn Usually ρ75 is closed to the tool radius ρao and
ρ75 = ρao
is a safe approximation compensating for the am simplifi-cations to the laquounsaferaquo side
The tooth root stresses of the loaded tooth are decreasing with decreasing relative rim thickness approximately with the empirical factor
15
R
ncorr s
m311Y
minus=
3153 Nominal rim stresses The bending moment applied to the rim consists of a part of the tooth tilt moment Ft (hF + 05 sR) and the bending caused by the radial force Fr
The sectional modulus (first moment of area) which is used for determination of the nominal bending stresses is not nec-essarily the same for the 2 am bending moments If flanges webs etc outside the toothed section contribute to stiffening the rim against various deflections the influence of these stiffeners should be considered Eg an end flange will have an almost negligible influence on the effective sectional modulus for the stresses due to tooth tilt as the deflection caused by the tilt moment is rather small and would not much involve the flange On the other hand the radial forces as for instance from the meshes in an annulus would cause considerable radial deflections that the flange might restrict to a substantial amount When considering the stiff-ening of such flanges or webs on basis of simplified models it is advised to use an effective rim thickness sR = sR + 02 mn for the first moment of area of the rim (toothed part) cross section
For a high number of rim teeth it may be assumed that the rim bending moments in the fillets adjacent to the loaded tooth are of the same magnitude as right under the applied
force This assumption is reasonable for an annulus but rather much on the safe side for a hollow pinion
The influence of Ft on the nominal tangential stress is simpli-fied by half of it for compressive stresses (σ1) and the other half for tensile stresses (σ2) Applying these assumptions the nominal rim stresses adjacent to the loaded tooth are
( ) ( )A2
FW
f RFW
s05hF05 t
R
r
T
RFt1 minus
ϑminus
+minus=σ
( ) ( )A2
FW
f RFW
s05hF05 t
R
r
T
RFt2 +
ϑminus
+=σ
where σ1 σ2 see Fig 35
A = minimum area of cross section (usually bR sR)
WT = the sectional modulus of rim with respect to tooth tilt moment (usually 6sb 2
rR )
WR = the sectional modulus of the rim including the influence of stiffeners as flanges webs etc (WR ge WT)
bR = the width of the rim
R = the radius of the neutral axis in the rim ie from wheel centre to midpoint of rim
( )ϑf = a function for bending moment distribution around the rim For a rim (pinion) with one mesh only the ( )ϑf at the position of load application is 024 For an annulus with 3 or more meshes ( )ϑf at the position of each load application is approx 3 planets ( )ϑf = 019 4 planets ( )ϑf = 014 5 planets ( )ϑf = 011 6 planets ( )ϑf = 009
It must be checked if the max (tensile) stress in the compres-sion fillet really occurs when the fillet is adjacent to the loaded tooth In principle the stress variation through a complete rotation should be considered and the max value used The max value is usually never less than 0 For an annulus eg the tilt moment is zero in the mid position be-tween the planet meshes whilst the bending moment due to the radial forces is half of that at the mesh but with opposite sign
If these formulae are applied to idler gears as eg planets the influence of nominal tangential stresses must be cor-rected by deleting Ft(2 A) for σ1 and using FtA for σ2 Further the influence of Fr on the nominal bending stresses is usually negligible due to the planet bearing support
3154 Root fillet stresses Determination of min and max stresses in the laquotensionraquo fillet
Classification Notes- No 412 31 May 2003
DET NORSKE VERITAS
Minimum stress
751FTmin YσK σ =
Maximum stress
corrF752 Yσ03Yσ KσFTmax +=
where
03 is an empirical factor relating the tension stresses (σF) at the 30ordm tangent to the part of the tension stresses at the 75ordm tangent which add to the rim related stresses (03 also takes into account that full superposition of nominal stresses times stress concentration factors from both laquosides of the corner filletraquo would result in too high stresses)
αβγ sdotsdotsdotsdot= FFvA KKKKKK
Determination of min and max stresses in the laquocompres-sionraquo fillet
Minimum stress
corrF751FCmin Yσ036Yσ Kσ minus=
Maximum stress
752FCmax Yσ Kσ =
where
036 is an empirical factor relating the tension stresses (σF) at the 30ordm tangent to the part of the compression stresses at the 75ordm tangent which add to the rim related stresses
For gears with reversed loads as idler gears and planets there is no distinct laquotensionraquo or laquocompressionraquo fillet The mini-mum stress σFmin is the minimum of σFTmin and σFCmin (usually the latter is decisive) The maximum stress σFTmax is the maximum of σFTmax and σFCmax (usually the former is deci-sive)
316 Permissible Stresses in Thin Rims
3161 General The safety against fatigue fracture respectively overload fracture is to be at least at the same level as for solid gears The laquoordinaryraquo criteria at the 30ordm tangent apply as given in 31 through 313
Additionally the following criteria at the 75ordm tangent may apply
3162 For gt3middot106 cycles The permissible stresses for the laquotensionraquo fillets and for the laquocompressionraquo fillets are determined by means of a relevant fatigue diagram
If the actual tooth root stress (tensile or compressive) ex-ceeds the yield strength to the material the induced residual stresses are to be taken into account
For determination of permissible stresses the following is defined
R = stress ratio ie FTmax
FTminσσ respectively
FCmax
FCmin
σσ
∆σ = stress range ie σFTmax ndash σFTmin resp σFCmax ndash σFCmin
(For idler gears and planets Fmax
FminσσR =
and ∆σ = σFmax minus σFmin)
The permissible stress range ∆σρ for the laquotensionraquo respec-tively laquocompressionraquo fillets can be calculated as
For R gt minus1 FPp σ
R1R1031
13∆σ
minus+
+=
For minus infin lt R lt minus1 FPp σ
R1R10151
13∆σ
minus+
+=
where
σFP = see 32 determined for unidirectional stresses (YM = 1)
If the yield strength σy is exceeded in either tension or com-pression residual stresses are induced This may be consid-ered by correcting the stress ratio R for the respective fillets (tension or compression)
Eg if yFCmin σσ gt (ie exceeded in compression) the
difference yFCmin σσ∆ minus= affects the stress ratio as
∆σ
σ∆σ∆σR
FCmax
y
FCmax
FCmin
+
minus=
++
=
Similarly the stress ratio in the tension fillet may require cor-rection
If the yield strength is exceeded in tension σFTmax gt σy the difference ∆ = σFTmax ndash σy affects the stress ratio as
y
FTmin
FTmax
FTmin
σ∆σ
∆σ∆σR minus=
minusminus
=
Checking for possible exceeding of the yield strength has to be made with the highest torque and with the KAP if this ex-ceeds KA
3163 Forle 103 cycles The permissible stress range ∆σp is not to exceed
For R gt minus1 FPstpst σ
R1R1051
15∆σ
minus+
+=
For ndash infin lt R lt minus1 FPstpst σ
R1R10251
15∆σ
minus+
+=
For all values of R ∆σpst is limited by
not surface hardened F
y
Sσ225
32 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
surface hardened CF
YSHV5
Definition of ∆σ and R see 3162 with particular attention to possible correction of R if the yield strength is exceeded
σFPst see 32 determined for unidirectional stresses (YM = 1) and lt 103 cycles
3164 For 103 lt cycles lt 3106 ∆σp is to be determined by linear interpolation a log-log dia-gram
∆σp at NL load cycles is
6L 103p
exp
L
6
N p ∆σN103∆σ sdot
sdot=
6
3
103p
10p
∆σ
∆σlog28760exp
sdot
=
Classification Notes- No 412 33 May 2003
DET NORSKE VERITAS
4 Calculation of Scuffing Load Capacity
41 Introduction High surface temperatures due to high loads and sliding ve-locities can cause lubricant films to break down This sei-zure or welding together of areas of tooth surface is termed scuffing
In contrast to pitting and fatigue breakage which show a dis-tinct incubation period a single short overloading can lead to a scuffing failure In the ISO-TR13989 two criteria are mentioned The method used in this Classification Note is based on the principles of the flash temperature criterion
Note Bulk temperature in excess of 120ordmC for long periods may have an adverse effect on the surface durability see 211
42 General Criteria In no point along the path of contact the local contact tem-perature may exceed the permissible temperature ie
oilS
oilSB S
ϑ+ϑminusϑ
leϑ
50SB minusϑleϑ
where
Bϑ = max contact temperature along the path of contact
maxflaMBB ϑ+ϑ=ϑ
MBϑ = bulk temperature see 434
maxflaϑ = max flash temperature along the path of con-tact see 44
Sϑ = scuffing temperature see below
oilϑ = oil temperature before it reaches the mesh (max applicable for the actual load case to be used ie normally alarm temperature except for ice classes where a max expected tem-perature applies)
SS = required safety factor according to the Rules
The scuffing temperature Sϑ may be calculated as
L2
wrelT
002
40S XFZGX
ν100112085780
sdot
sdot++=ϑ
where
XwrelT = relative welding factor
XwrelT
Through hardened steel 10
Phosphated steel 125
Copper-plated steel 150
Nitrided steel 150
Less than 10 retained austenite
115
10 ndash 20 retained austenite 10
Casehardened steel
20 ndash 30 retained austenite 085
Austenitic steel 045
FZG = load stage according to FZG-Test A8390 (Note This is the load stage where scuffing oc-curs However due to scatter in test results cal-culations are to be made with one load stage less than the specification)
XL = lubricant factor
= 10 for mineral oils
= 08 for polyalfaolefins
= 07 for non-water-soluble polyglycols
= 06 for water-soluble polyglycols
= 15 for traction fluids
= 13 for phosphate esters
ν40 = kinematic oil viscosity at 40˚C (mm2s)
Application of other test methods such as the Ryder the FZG-Ryder R46574 and the FZG L-42 Test 141195110 may be specially considered
For high speed gears with very short time of contact Sϑ may be increased as follows provided use of EP-oils
Addition to the calculated scuffing temperature Sϑ
If micros 18tc ge no addition
If micros 18tc lt add ( )cwrelT t18X18 minussdotsdot
where
ct = contact time ( micros ) which is the time needed to cross the Hertzian contact width
( ) [ ]microsu1cosβn
uσ340tb1
Hc +sdotsdot
sdot=
σH as calculated in 221
For bevel gears use vu in stead of u
34 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
43 Influence Factors
431 Coefficient of friction The following coefficient of friction may apply
L025a
005oil
02
redC ΣC
Bt X R ηρv
w0048micro minus
=
where
wBt = specific tooth load (Nmm)
vΣC = sum of tangential velocities at pitch point At pitch line velocities gt 50 ms the limiting value of vΣC at v = 50 ms is to be used
ρredC = relative radius of curvature (transversal plane) at the pitch point
Cylindrical gears
HαHβvγAbt
Bt KKKKKb
Fw sdotsdotsdotsdotsdot= (see 1)
twΣC αsin v2v =
bCredC β cos ρρ =
Bevel gears
HαHβvγAtmb
Bt KKKKKb
Fw sdotsdotsdotsdotsdot= (see 1)
vtmtΣC αsin v2v =
bmvCredC β cos ρρ =
ηoil = dynamic viscosity (mPa s) at oilϑ calculated as
1000
ρ νη oiloil =
where ρ in kgm3 approximated as
( ) 70 15oil15 minusϑminusρ=ρ and νoil is kinematic viscosity at oilϑ and may be calculated by means of the following equation
( ) ( )++=+ 08νloglog08νloglog 100oil ( )
sdotminus
ϑ+minus313log373log
273log373log oil
( ) ( )( )08νloglog08νloglog 10040 +minus+
Ra = composite arithmetic mean roughness (micron) of pinion an wheel calculated as ( )2a1aa RR 05R +=This is defined as the roughness on the new flanks ie as manufactured
XL = see 42
432 Effective tip relief Ceff Ceff is the effective tip relief that amount of tip relief which compensates for the elastic deformation of the gear mesh ie
zero load at the tooth tip It is assumed (simplified) to be equal for both pinion and wheel
Cylindrical gears
for helical
γ
γ=cb
KKFC Abt
eff (see 1)
For spur
cbKKF
C Abteff
γ= (see 1)
Alternatively for spur and helical gears the non-linear ap-proach in 111 may be used (taking Ceff=δ)
Bevel gears
bcKFC Ambt
effγ
= (see 1)
where
γ
α
ε2ε44c
+sdot
=γ
433 Tip relief and extension Cylindrical gears
The extension of the tip relief is not to result in an effective contact ratio 1ltεα when the gear is unloaded (exceptions to this may only apply for applications where the gear is not to run at light loads) This means that the unrelieved part of the path of contact is to be minimum pbt It is further assumed that this unrelieved part is placed centrally on the path of contact
If root relief applies it has to be calculated as equivalent tip relief Ie pinion root relief (at mesh position A) is added to Ca2 and wheel root relief (at mesh position E) is added to Ca1 If no design tip relief or root relief on the mating gear is specified (ie if Ca1+Croot2 = 0 and visa versa) use the run-ning in amount see 112
Bevel gears
Bevel gears are to have heightwise crowning ie no distinct relievedunrelieved area This may be treated as tip and root relief For calculation purposes the root relief is combined with the tip relief of the mating member into an equivalent tip relief If no resulting tip relieves are specified the equivalent tip relives may be calculated as an approxima-tion based on the tool crowning Ca tool (per mille of tool module m0) as follows
2root1aeq1a CCC +=
1root2aeq2a CCC +=
where
Classification Notes- No 412 35 May 2003
DET NORSKE VERITAS
2
0
n0101atoolal m
)m2(mA1middotmCC
minus+minussdot=
( )
sdotminusminussdotsdotminus+= vtvb1
2vb1
21vavt1n1 αtan dddαsin 05x1mA
2
0
n020atool22a m
)mm(2A1middotmCC
minus+minussdot=
( )
sdotminusminussdotsdotminus+= vtvb2
2vb2
2va2vt2n2 αtan dddαsin 05x1mA
2
0
20atool1root1 m
A1mCC
minussdotsdot=
2
0
10atool2root2 m
A1mCC
minussdotsdot=
If the pressure angles of the cutter blades are modified (and verified) to balance the tip relieves the following may be assumed
( )eq2aeq1aeq2aeq1a C andC calculated of sum50CC sdot==
Throughout the following Ca1 and Ca2 mean the equivalent tip relieves Ca1eq and Ca2eq
434 Bulk temperature The bulk temperature may be calculated as
flaaveragempsoilMB X X 50 ϑ+ϑ=ϑ
where
Xs = lubrication factor
= 12 for spray lubrication
= 10 for dip lubrication (provided both pinion and wheel are dip lubricated and tip speed lt 5 ms)
= 10 for spray lubrication with additional cooling spray (spray on both pinion and wheel or spray on pinion and dip lubrication of wheel)
= 02 for meshes fully submerged in oil
Xmp = contact factor ( )pmp nX += 150
np = number of mesh contact on the pinion (for small gear ratii the number of wheel meshes should be used if higher)
flaaverageϑ
= average of the integrated flash temperature (see 44) along the path of contact
( )
AE
E
Ayyyfla
flaaverage
d
ΓminusΓ
ΓΓϑ=ϑint =
For high speed gears (v gt 50 ms) it may be necessary to assess the bulk temperature on the basis of a thermal rating of the entire gear transmission
44 The Flash Temperature flaϑ
441 Basic formula The local flash temperature flaϑ may be calculated as
( ) 41redy
y2ly
211
43Btcorrfla
unXwX3250
y ρ
ρminusρ
micro=ϑ Γ
(For bevel gears replace u with uv)
and is to be calculated stepwise along the path of contact from A to E
where
micro = coefficient of friction see 431
Xcorr = correction factor taking empirically into account the increased scuffing risk in the beginning of the approach path due to mesh starting without any previously built up oil film and possible shuffling away oil before meshing if insufficient tip relief
( )
3
AD
yaeffcorr 50
CC1X
ΓminusΓε
Γminus+=
α
Ca = tip relief of driven member Xcorr is only applicable in the approach path and if Ca lt Ceff otherwise 10
wBt = unit load see 431
yXΓ = load sharing factor see 443
n1 = pinion rpm
Γy ρly etc see 442
442 Geometrical relations The various radii of flank curvature (transversal plane) are
ρ1y = pinion flank radius at mesh point y
ρ2y = wheel flank radius at mesh point y
ρredy = equivalent radius of curvature at mesh point y
y2y1
y2y1redy
ρ+ρ
ρρ=ρ
Cylindrical gears
twy
1y αsin au1Γ1
ρ+
+=
twy
2y αsin au1Γu
ρ+
minus=
Note that for internal gears a and u are negative
36 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
Bevel gears
vtvv
y1y αsin a
u1Γ1
ρ+
+=
vtvv
yv2y αsin a
u1Γu
ρ+
minus=
Γ is the parameter on the path of contact and y is any point between A end E
At the respective ends Γ has the following values
Root piniontip wheel
Cylindrical gears
( )
minus
minusminus= 1
αtan 1dd
zzΓ
tw
2b2a2
1
2A
Bevel gears
( )
minus
minusminus= 1
αtan 1dd
uΓvt
2vb2va2
vA
Tip pinionroot wheel
Cylindrical gears
( )
1αtan
1ddΓ
tw
2b1a1
E minusminus
=
Bevel gears
( )
1αtan
1ddΓ
vt
2vb1va1
E minusminus
=
At inner point of single pair contact
Cylindrical gears
tw1
EB α tan zπ2ΓΓ minus=
Bevel gears
vtv1
EB α tan zπ2ΓΓ minus=
At outer point of single pair contact
Cylindrical gears
tw1AD α tan z
π2ΓΓ +=
Bevel gears
vt v1
AD αtan z π2ΓΓ +=
At pitch point ΓC = 0
The points F and G (only applicable to cylindrical gears) limiting the extension of tip relief (so as to maintain a mini-mum contact ratio of unity for unloaded gears) are at
2
BAF
Γ+Γ=Γ
2
EDG
Γ+Γ=Γ
443 Load sharing factor XΓ The load sharing factor XΓ accounts for the load sharing be-tween the various pairs of teeth in mesh along the path of contact
XΓ is to be calculated stepwise from A to E using the pa-rameter Γy
4431 Cylindrical gears with β = 0 and no tip relief
Figure 41
ByAAB
Ay for31
31X
yΓltΓleΓ
ΓminusΓ
ΓminusΓ+=Γ
DyB for1Xy
ΓltΓleΓ=Γ
EyDDE
yE for31
31X
yΓleΓltΓ
ΓminusΓ
ΓminusΓ+=Γ
4432 Cylindrical gears with β = 0 and tip relief Tip relief on the pinion reduces XΓ in the range G ndash E and increases correspondingly XΓ in the range F ndash B
Tip relief on the wheel reduces XΓ in the range A ndash F and increases correspondingly XΓ in the range D ndash G
Following remains generally
DyB for1Xy
ΓleΓleΓ=Γ
21XXGF== ΓΓ
In the following it must be distinguished between Ca lt Ceff respectively Ca gt Ceff This is shown by an example below where Ca1 lt Ceff and Ca2 gt Ceff
Classification Notes- No 412 37 May 2003
DET NORSKE VERITAS
Figure 42
Note When Ca gt Ceff the path of contact is shortened by A ndash A respectively E ndash E The single pair contact path is extended into B respectively D If this shift is significant it is necessary to consider the negative effect on surface durability (B) and bending stresses (B and D)
Range A - F
For effa2 CC le
minus=Γ
eff
2a
CC1
31X
A
FyAeff
2a
AF
Ay for C3
C61XX
AyΓleΓleΓ
sdot
+ΓminusΓ
ΓminusΓ+= ΓΓ
For effa2 CC ge
AyAfor0Xy
ΓleΓleΓ=Γ
with ( )21
CC
1CC
eff
2a
eff
2a
AFAAminus
minusΓminusΓ+Γ=Γ
FyAeff
2a
AF
Ay
eff
2a for 21
CC
CC1X
yΓleΓleΓ
minus
ΓminusΓ
ΓminusΓ+minus=Γ
Range F - B
For effa1 CC le
ByFeff
1a
FB
Fy forC3
C61
21X
yΓleΓleΓ
sdot
+ΓminusΓ
ΓminusΓ+=Γ
For effa1 CC ge
ByFeff
1a
FB
Fy for21
CC
21X
yΓleΓleΓ
minus
ΓminusΓ
ΓminusΓ+=Γ
ByB for1Xy
ΓleΓleΓ=Γ
with 1
CC2
eff
1a
FBFB
minus
ΓminusΓ+Γ=Γ
Range D ndash G
For effa2 CC le
GyDeff
2a
DG
Dy
eff
2a for C 3C
61
C 3C
32X
yΓleΓleΓ
+
ΓminusΓ
ΓminusΓminus+=Γ F
or effa2 CC ge
DyD for1Xy
ΓleΓleΓ=Γ
with ( )21
CC
1CC
eff
2a
eff
2a
DGDDminus
minusΓminusΓ+Γ=Γ
GyDeff
2a
DG
Dy
eff
2a for21
CC
CCX ΓleΓleΓ
minus
ΓminusΓ
ΓminusΓminus=Γ
Range G - E
For effa1 CC le
minus=Γ
eff
1a
CC1
31X
E
EyGeff
1a
GE
Gy forC3
C61
21X
yΓleΓleΓ
sdot
+ΓminusΓ
ΓminusΓminus=Γ
For effa1 CC gt
EyGeff
1a
GE
Gy for21
CC
21X
yΓleΓleΓ
minus
ΓminusΓ
ΓminusΓminus=Γ
EyE for0Xy
ΓleΓleΓ=Γ
with 1
CC2
eff
1a
GEGE
minus
ΓminusΓ+Γ=Γ
4433 Gears with β gt 0 buttressing Due to oblique contact lines over the flanks a certain but-tressing may occur near A and E
This applies to both cylindrical and bevel gears with tip relief lt Ceff The buttressing Xbutt is simplified as a linear function within the ranges A ndash H respectively I ndash E
Figure 43
1εwhen13X βbutt EAge=
1εwhenε031X ββbutt EAlt+=
38 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
Cylindrical gears
bIEAH βsin 02ΓΓΓΓ =minus=minus
Bevel gears
bmIEAH βsin 02ΓΓΓΓ =minus=minus
4434 Cylindrical gears with 2εγ le and no tip relief
yXΓ is obtained by multiplication of
yXΓ in 4431 with
Xbutt in 4433
4435 Gears with 2εγ gt and no tip relief
Applicable to both cylindrical and bevel gears
IyH for1Xy
ΓleΓleΓε
=α
Γ
IyHybutt and forX1Xy
ΓgtΓΓltΓε
=α
Γ
Figure 44
4436 Cylindrical gears with 2εγ le and tip relief
yXΓ is obtained by multiplication of
yXΓ in 4432 with Xbutt
in 4433
4437 Cylindrical gears with 2εγ gt and tip relief
Tip relief on the pinion (respectively wheel) reduces XГ in the range G ndash E (respectively A ndash F) and increases XГ in the range F ndash G
yXΓ is obtained by multiplication of
yXΓ as described below
with Xbutt in 4433
In the XГ example below the influence of tip relief is shown (without the influence of Xbutt) by means of
eff2aeff1a CC and CC ltgt
Tip relief gt Ceff causes new end points A respectively E of the path of contact
Figure 45
Range A ndash F
( ) ( )( ) eff
a2αa1α
AF
Ay
eff
2aeff
C 12C 13εC 1ε
CCCX
y +εε++minus
ΓminusΓ
ΓminusΓ+
εminus
=ααα
Γ
eff2aFyA CC if for leΓleΓleΓ
eff2aFyA CifCfor and geΓleΓleΓ
eff2aAyA CC iffor0Xy
gtΓleΓleΓ=Γ
( ) ( ) ( )( ) ( ) a2αa1α
αeffa2AFAA C 1ε 3C 1ε
1ε 2 CC with++minus+minus
ΓminusΓ+Γ=Γ
Range F ndash G
( )( )( ) GyF
eff
2a1a forC 1 2CC 11X
yΓleΓleΓ
+εε+minusε
+ε
=αα
α
αΓ
Range G ndash E
( ) ( )( ) eff
2a1a
GE
Gy
C 1 2C 1C 1 3XX
GFy +εεminusε++ε
ΓminusΓ
ΓminusΓminus=
αα
ααΓΓ minus
effa1EyG CC if for leΓleΓleΓ
effa1EyG CC if Γfor and geΓleΓle
eff1aEyE CC if for0Xy
geΓleΓleΓ=Γ
( ) ( ) ( )( ) ( ) 2a1a
eff1aGEEE C 1C 1 3
1 2 CC withminusε++ε+εminus
ΓminusΓminusΓ=Γαα
α
4438 Bevel gears with εγ more than approx 18 and heightwise crowning For Ca1 = Ca2 = Ceff the following applies
Figure 46
Classification Notes- No 412 39 May 2003
DET NORSKE VERITAS
( )AEM 50 Γ+Γ=Γ
( )( )2AD
3
2My651X
y ΓminusΓε
ΓminusΓminus
ε=
ααΓ
For tip relief lt Ceff yXΓ is found by linear interpolation be-
tween ( )effay CCX =Γ and ( )0CayX =Γ as in 4435
The interpolation is to be made stepwise from A to M with the influence of Ca2 and from M to E with the influence of Ca1 (For 2a1a CC ne there is a discontinuity at M)
Eg with Ca1 = 04 Ceff and Ca2 = 055 Ceff then
Range A ndash M
)CC()0C( eff2ay2ayyX 550X 450X =Γ=ΓΓ +=
Range M ndash E
)CC()0C( eff1ay1ayyX 40X 06X =Γ=ΓΓ +=
For tip relief gt Ceff the new end points A and E are found as
( )
minusΓminusΓ
ε+Γ=Γ α 1
CC
6 eff
2aADAA
( )
minusΓminusΓ
ε+Γ=Γ α 1
CC
6 eff
1aADEE
Range A ndash A
0Xy=Γ
Range A ndash M
( )( )
ΓminusΓ
ΓminusΓminus
minusε=
αΓ 2
MA
2My
eff
2a1
CC4
351Xy
Range M ndash E
( )( )
ΓminusΓ
ΓminusΓminus
minusε=
αΓ 2
ME
2My
eff
1a1
CC4
351Xy
Range E ndash E
0Xy=Γ
40 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
Appendix A Fatigue Damage Accumulation
The Palmgren-Miner cumulative damage calculation princi-ple is used The procedure may be applied as follows
A1 Stress Spectrum From the individual torque classes the torques (Ti) at the peak values of class intervals and the associated number of cycles (NLi) for both pinion and wheel are to be listed from the highest to the lowest torque
(In case of a cyclic torque variation within the torque classes it is advised to use the peak torque If the cyclic variation is such that the same teeth will repeatedly suffer the peak torque this is a must)
The stress spectra for tooth roots and flanks (σFi σHi) with all relevant factors (except KA) are to be calculated on the basis of the torque spectrum The load dependent K-factors are to be determined for each torque class
A2 σminusN-curve The stress versus load cycle curves for tooth roots and flanks (both pinion and wheel) are to be drawn on the basis of per-missible stresses (ie including the demanded minimum safety factors) as determined in 2 respectively 3 If different safety levels for high cycle fatigue and low cycle fatigue are desired this may be expressed by different demand safety factors applied at the endurance limit respectively at static strength
A3 Damage accumulation The individual damage ratio Di at ith stress level is defined as
Fi
Lii N
ND =
where
NLi = The number of applied cycles at ith stress
NFi = The number of cycles to failure at ith stress
Basically stresses σi below the permissible stress level for infinite life (if a constant ZN or YN is accepted) do not con-tribute to the damage sum However calculating the actual safety factor Sact as described below all the σi for which the product S σi is bigger than or equal to the permissible stress level for infinite life contribute to the damage sum and thus to the determination of Sact The final value of S is decisive
(NFi can be found mathematically by putting the permissible stress σpi equal the actual stress σi thereby finding the actual life factor This life factor can be solved with regard to load cycles ie NFi)
The damage sum ΣDi is not to exceed unity
If ΣDi ne 1 the safety against cumulative fatigue damage is different from the applied demand safety factor For deter-mination of this theoretical safety factor an iteration proce-dure is required as described in the following flowchart
S is correction factor with which the actual safety factor Sact can be found
Sact is the demand safety factor (used in determination of the permissible stresses in the σ ndash N ndash curve) times the correc-tion factor S
The full procedure is to be applied for pinion and wheel tooth roots and flanks
Note If alternating stresses occur in a spectrum of mainly pulsating stresses the alternating stresses may be replaced by equiva-lent pulsating stresses ie by means of division with the ac-tual mean stress influence factor YM
Classification Notes- No 412 41 May 2003
DET NORSKE VERITAS
Appendix B Application Factors for Diesel Driven Gears
For diesel driven gears the application factor KA depends on torsional vibrations Both normal operation and misfiring conditions have to be considered
Normally these two running conditions can be covered by only one calculation
B1 Definitions
Normal operation KAnorm = 0
normv0
TTT +
where
T0 = rated nominal torque
Tv norm = vibratory torque amplitude for normal opera-tion (see rules Pt4 Ch3 Sec1 G301 for defi-nition of ldquonormalrdquo irregularity)
Misfiring operation KA misf = 0
misfv
TTT +
where
T = remaining nominal torque when one cylinder out of action
Tv misf = vibratory torque amplitude in misfiring con-dition This refers to a permissible misfiring condition ie a condition that does not require automatic or immediate corrective actions as speed or pitch reduction
The normal operation is assumed to last for a very high num-ber of cycles such as 1010
The misfiring operation is assumed to last for a limited dura-tion such as 107 cycles
B2 Determination of decisive load Assuming life factor at 1010 cycles as YN = ZN = 092 which usually is relevant the calculation may be performed only once with the combination having the highest value of appli-cation factorlife factor
For bending stresses and scuffing the higher value of
092
K normA and 098
K misfA
For contact stresses the higher value of
092K normA and
113K misfA (but
097K misfA for nitrided gears)
B3 Simplified procedure Note that this is only a guidance and is not a binding con-vention
T = oTZ
1Zsdot
minus where Z = number of cylinders
( ) ideal Videal Vmisf Vnorm V TTT24ZT +minus=
where TV ideal = vibratory torque with all cylinders perfectly equal See also rules Pt4 Ch3 Sec1 G300
When using trends from torsional vibration analysis and measurements the following may be used
TV ideal To is close to zero for engines with few cylinders and using a suitable elastic coupling and increases with relative coupling stiffness and number of cylinders
TV misf To may be high for engines with few cylinders and decreases with number of cylinders
This can be indicated as
200Z
TT
ο
idealV asymp and 80Z04
TT
o
misfV minusasymp
Inserting this into the formulae for the two application fac-tors the following guidance can be given
118112K misfA minusasymp
115110K normA minusasymp
Since KA norm is to be combined with the lower life factors the decisive load condition will be the normal one and a KA of 115 will cover most relevant cases when a suitable elas-tic coupling is chosen
42 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
Appendix C Calculation of Pinion-Rack
Pinion-racks used for elevating of mobile offshore units are open gears that are subjected to wear and tear With normal specifications such as only a few hundred total operation cycles (site to site) the tooth bending stresses for static strength will be decisive for the lay out however with ex-ception of surface hardened pinions where case crushing has to be considered
In the following the use of part 1 and 3 for pinion-racks is shown including relevant simplifications
C1 Pinion tooth root stresses Since the load spectrum normally is dominated by high torques and few load cycles (in the range up to 1-2000) the static strength is decisive
The actual stress is calculated as
β1FSaFan1
tF1 KYY
mbFσ sdotsdotsdotsdot
=
b1 is limited to b2 + 2middotmn
YFa and YSa replace YF and YS because load application at tooth tip has to be assumed for such inaccurate gears
Pinions often use a non-involute profile in the dedendum part of the flank eg a constant radius equal the radius of curva-ture at reference circle For such pinions sFn and hFa are to be measured directly on a sectional drawing of the pinion tooth
Due to high loads and narrow facewidths it may be assumed that KFβ2 = KHβ = 10 However when b1 gt b2 then KFβ1 gt10
If no detailed documentation of KFβ1 is available the fol-lowing may be used
KFβ1 = 1 + 015middot(b1b2 ndash 1)
The permissible stress (not surface hardened) is calculated as
δrelTstF
Fst1FP1 Y
Sσσ sdot=
The mean stress influence due to leg lifting may be disre-garded
The actual and permissible stresses should be calculated for the relevant loads as given in the rules
C2 Rack tooth root stresses The actual stress is calculated as
SaFan2
tF2 YY
mbFσ sdotsdotsdot
=
See C1 for details
The permissible stress is calculated as
δrelTstF
Fst2FP2 Y
Sσσ sdot=
For alloyed steels (Ni Cr Mo) with high toughness and duc-tility the value of YδrelTst may be put equal to YSa
C3 Surface hardened pinions For surface hardened pinions the maximum load is not to cause crushing of the hardened layer of the flank
In principle the calculation described in 213 may be used but when the theoretical Hertzian stress exceeds the ap-proximately 18 times the yield strength of the rack material plastic deformation will occur This will limit the peak Hertzian stress but increases the contact width and thus the penetration of stresses into the depth
An approximation may be based on an assessment of contact width determined by means of equal areas under the theoreti-cal (elastic range) Hertzian contact and the elasto-plastic contact stress (limited to 18 middot σy) with the unknown width
26 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
38 Mean stress influence Factor YM The mean stress influence factor YM takes into account the influence of other working stress conditions than pure pulsa-tions (R = 0) such as eg load reversals idler gears planets and shrink-fitted gears
YM (YMst) are defined as the ratio between the endurance (or static) strength with a stress ratio R ne 0 and the endurance (or static) strength with R = 0
YM and YMst apply only to a calculation method that assesses the positive (tensile) stresses and is therefore suitable for comparison between the calculated (positive) working stress σF and the permissible stress σFP calculated with YM or YMst
For thin rings (annulus) in epicyclic gears where the com-pression fillet may be decisive special considerations apply see 316
The following method may be used within a stress ratio ndash12 lt R lt 05
381 For idlers planets and PTO with ice class
M1M1R1
1Yor Y MstM
+minus
minus=
where
R = stress ratio = min stress divided by max stress
For designs with the same force applied on both forward- and back-flank R may be assumed to ndash 12
For designs with considerably different forces on forward- and back-flank such as eg a marine propulsion wheel with a power take off pinion R may be assessed as
branchmain theoffacewidth unit per forcepto offacewidt unit per force21minus
For a power take off (PTO) with ice class see 161 c
M considers the mean stress influence on the endurance (or static) strength amplitudes
M is defined as the reduction of the endurance strength am-plitude for a certain increase of the mean stress divided by that increase of the mean stress
Following M values may be used
Endurance limit
Static strength
Case hardened 08 ndash 015 Ys 1) 07
If shot peened 04 06
Nitrided 03 03
Induction or flame hardened
04 06
Not surface hardened steel 03 05
Cast steels 04 06 1)For bevel gears use Ys=2 for determination of M
The listed M values for the endurance limit are independent of the fillet shape (Ys) except for case hardening In princi-ple there is a dependency but wide variations usually only occur for case hardening eg smooth semicircular fillets versus grinding notches
382 For gears with periodical change of rotational direction For case hardened gears with full load applied periodically in both directions such as side thrusters the same formula for YM as for idlers (with R = ndash 12) may be used together with the M values for endurance limit This simplified approach is valid when the number of changes of direction exceeds 100 and the total number of load cycles exceeds 3middot106
For gears of other materials YM will normally be higher than for a pure idler provided the number of changes of direction is below 3middot106 A linear interpolation in a diagram with loga-rithmic number of changes of direction may be used ie from YM = 09 with one change to YM (idler) for 3middot106 changes This is applicable to YM for endurance limit For static strength use YM as for idlers
For gears with occasional full load in reversed direction such as the main wheel in a reversing gear box YM = 09 may be used
383 For gears with shrinkage stresses and unidirectional load For endurance strength
FE
fitM σ
σM1
M21Y+
minus=
σFE is the endurance limit for R = 0
For static strength YMst = 1 and σfit accounted for in 39b
σfit is the shrinkage stress in the fillet (30˚ tangent) and may be found by multiplying the nominal tangential (hoop) stress with a stress concentration factor
n
Ffit m
ρ215scf minus=
384 For shrink-fitted idlers and planets When combined conditions apply such as idlers with shrink-age stresses the design factor for endurance strength is
( ) ( ) FE
fitM σ
σR1M1
M2
M1M1R1
1Yminussdot+
minus
+minus
minus=
Symbols as above but note that the stress ratio R in this par-ticular connection should disregard the influence of σfit ie R normally equal ndash 12
For static strength
M1M1R1
1YMst
+minus
minus=
The effect of σfit is accounted for in 39 b
Classification Notes- No 412 27 May 2003
DET NORSKE VERITAS
385 Additional requirements for peak loads The total stress range (σmax ndash σmin) in a tooth root fillet is not to exceed
F
y
Sσ225
for not surface hardened fillets
FSHV5 for surface hardened fillets
39 Life Factor YN The life factor YN takes into account that in the case of limited life (number of cycles) a higher tooth root stress can be permitted and that lower stresses may apply for very high number of cycles
Decisive for the strength at limited life is the σ ndash N ndash curve of the respective material for given hardening module fillet radius roughness in the tooth root etc Ie the factors YδrelT YRelT YX and YM have an influence on YN
If no σ ndash N ndash curve for the actual material and hardening etc is available the following method may be used
Determination of the σ ndash N ndash curve
a) Calculate the permissible stress σFP for the beginning of the endurance limit (3middot106 cycles) including the influ-ence of all relevant factors as SF YδrelT YRelT YX YM and YC ie σFP = σFE middotYM middotYδrelT middotYRelT middotYX middotYC SF
b) Calculate the permissible laquostaticraquo stress (le 103 load cy-cles) including the influence of all relevant factors as SFst YδrelTst YMst and YCst ( )fitσCstYδrelTstYMstYFstσ
FstS1
FPstσ minussdotsdotsdot=
where σFst is the local tooth root stress which the material can resist without cracking (surface hardened materials) or unacceptable deformation (not surface hardened materials) with 99 survival probability
σFst
Alloyed case hardened steel 1) 2300
Nitriding steel quenched tempered and gas nitrided (surface hardness 700 ndash 800 HV)
1250
Alloyed quenched and tempered steel bath or gas nitrided (surface hardness 500 ndash 700 HV)
1050
Alloyed quenched and tempered steel flame or induction hardened (fillet surface hardness 500 ndash 650 HV)
18 HV + 800
Steel with not surface hardened fillets the smaller value of 2)
18 σB or 225 σy
1) This is valid for a fillet surface hardness of 58 ndash 63 HRC In case of lower fillet surface hardness than 58 HRC σFst is to be reduced with 30(58 ndash HRC) where HRC is the actual hardness Shot peening or grinding notches are not considered to have any significant influence on σFst 2) Actual stresses exceeding the yield point (σy or σ02) will alter the residual stresses locally in the ldquotensionrdquo fillet re-spectively ldquocompressionrdquo fillet This is only to be utilised for gears that are not later loaded with a high number of cycles at lower loads that could cause fatigue in the ldquocompressionrdquo fillet
c) Calculate YN as
NL gt 3middot106
YN = 1 or 001
L
6
N N103Y
sdot= ie Yn = 092 for 1010
The YN = 1 from 3middot106 on may only be used when special material cleanness applies see rules Pt4 Ch2
103ltNLlt3middot106exp
L
6
N N103Y
sdot=
cycles6103forσcycles310forσlog02876exp
FP
FPst
sdot=
NL lt 103 cycles6103forσcycles310forσ
NYFP
FPst
sdot=
or simply use σFPst as mentioned in b) directly
Guidance on number of load cycles NL for various applica-tions
bull For propulsion purpose normally NL = 1010 at full load (yachts etc may have lower values)
bull For auxiliary gears driving generators that normally operate with 70-90 of rated power NL = 108 with rated power may be applied
28 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
310 Relative Notch Sensitivity Factor YδrelT The dynamic (respectively static) relative notch sensitivity factor YδrelT (YδrelTst) indicate to which extent the theoreti-cally concentrated stress lies above the endurance limits (re-spectively static strengths) in the case of fatigue (respectively overload) breakage
YδrelT is a function of the material and the relative stress gra-dient It differs for static strength and endurance limit
The following method may be used
For endurance limit
for not surface hardened fillets
( )
024
s024
relT σ103133q21σ1012201351
Ysdotsdotminus
+sdotsdotminus+= minus
minus
δ
for all surface hardened fillets except nitrided
( )
106q21002451
Y srelT
++=δ
for nitrided fillets
( )
1347q2101421
Y srelT
++=δ
For static strength
for not surface hardened fillets1)
( )( )( )025
02
02502s
relTst σ3000821σ300 1Y0821Y
+
minus+=δ
for surface hardened fillets except nitrided
YδrelTst = 044 YS + 012
for nitrided fillets
YδrelTst = 06 + 02 YS 1) These values are only valid if the local stresses do not
exceed the yield point and thereby alter the residual stress level See also 39b footnote 2
311 Relative Surface Condition Factor YRrelT The relative surface condition factor YRrelT takes into ac-count the dependence of the tooth root strength on the sur-face condition in the tooth root fillet mainly the dependence on the peak to valley surface roughness
YRrelT differs for endurance limit and static strength
The following method may be used
For endurance limit
YRrelT = 1675 ndash 053 (Ry + 1)01 for surface hardened steels and alloyed quenched and tempered steels except nitrided
YRrelT = 53 ndash 42 (Ry + 1)001 for carbon steels
YδrelT = 43 ndash 326 (Ry + 1)0005
for nitrided steels
For static strength
YRrelTst = 1 for all Ry and all materials
For a fillet without any longitudinal machining trace Ry asymp Rz
312 Size Factor YX The size factor YX takes into account the decrease of the strength with increasing size YX differs for endurance limit and static strength
The following may be used
For endurance limit
YX = 1 for mn le 5 generally
YX = 103 ndash 0006 mn for 5 lt mn lt 30
YX = 085 for mn ge 30
for not surface hard-ened steels
YX = 105 ndash 001 mn for 5 lt mn ge 25
YX = 08 for mn ge 25
for surface hardened steels
For static strength
YXst = 1 for all mn and all materials
313 Case Depth Factor YC The case depth factor YC takes into account the influence of hardening depth on tooth root strength
YC applies only to surface hardened tooth roots and is dif-ferent for endurance limit and static strength
In case of insufficient hardening depth fatigue cracks can develop in the transition zone between the hardened layer and the core For static strength yielding shall not occur in the transition zone as this would alter the surface residual stresses and therewith also the fatigue strength
The major parameters are case depth stress gradient permis-sible surface respectively subsurface stresses and subsurface residual stresses
The following simplified method for YC may be used
YC consists of a ratio between permissible subsurface stress (incl influence of expected residual stresses) and permissible surface stress This ratio is multiplied with a bracket con-taining the influence of case depth and stress gradient (The empirical numbers in the bracket are based on a high number of teeth and are somewhat on the safe side for low number of teeth)
YC and YCst may be calculated as given below but calculated values above 10 are to be put equal 10
For endurance limit
+
+=nFFE
C m02ρt31
σconstY
Classification Notes- No 412 29 May 2003
DET NORSKE VERITAS
For static strength
+
+=nFFst
Cst m02ρt31
σconstY
where const and t are connected as
Hardening process
t = endurance limit const =
static strength const =
t550 640 1900
t400 500 1200 Case hard-ening
t300 380 800
Nitriding t400 500 1200
Induction- or flame hardening
tHVmin 11 HVmin 25 HVmin
For symbols see 213
In addition to these requirements to minimum case depths for endurance limit some upper limitations apply to case hard-ened gears
The max depth to 550 HV should not exceed
1) 13 of the top land thickness san unless adequate tip relief is applied (see 110)
2) 025 mn If this is exceeded the following applies addi-tionally in connection with endurance limit
minusminus= 025
mt
1Yn
max550C
314 Thin rim factor YB Where the rim thickness is not sufficient to provide full sup-port for the tooth root the location of a bending failure may be through the gear rim rather than from root fillet to root fillet
YB is not a factor used to convert calculated root stresses at the 30deg tangent to actual stresses in a thin rim tension fillet Actually the compression fillet can be more susceptible to fatigue
YB is a simplified empirical factor used to de-rate thin rim gears (external as well as internal) when no detailed calcula-tion of stresses in both tension and compression fillets are available
Figure 34 Examples on thin rims
YB is applicable in the range 175 lt sRmn lt 35
YB = 115 middot ln (8324 middot mnsR)
(for sRmn ge 35 YB = 1)
(for sRmn le 175 use 315)
σF as calculated in 321 is to be multiplied with YB when sRmn lt 35 Thus YB is used for both high and low cycle fatigue
Note This method is considered to be on the safe side for external gear rims However for internal gear rims without any flange or web stiffeners the method may not be on the safe side and it is advised to check with the method in 315316
315 Stresses in Thin Rims For rim thickness sR lt 35 mn the safety against rim cracking has to be checked
The following method may be used
3151 General The stresses in the standardised 30ordm tangent section tension side are slightly reduced due to decreasing stress correction factor with decreasing relative rim thickness sRmn On the other hand during the complete stress cycle of that fillet a certain amount of compression stresses are also introduced The complete stress range remains approximately constant Therefore the standardised calculation of stresses at the 30ordm tangent may be retained for thin rims as one of the necessary criteria
The maximum stress range for thin rims usually occurs at the 60ordm ndash 80ordm tangents both for laquotensionraquo and laquocompressionraquo side The following method assumes the 75ordm tangent to be the decisive Therefore in addition to the am criterion ap-plied at the 30ordm tangent it is necessary to evaluate the max and min stresses at the 75ordm tangent for both laquotensionraquo (loaded flank) fillet and laquocompressionraquo (back-flank) fillet For this purpose the whole stress cycle of each fillet should be considered but usually the following simplification is justified
Figure 35 Nomenclature of fillets
Index laquoTraquo means laquotensileraquo fillet laquoCraquo means laquocompressionraquo fillet
σFTmin and σFCmax are determined on basis of the nominal rim stresses times the stress concentration factor Y75
σFCmin and σFTmax are determined on basis of superposition of nominal rim stresses times Y75 plus the tooth bending stresses at 75ordm tangent
30 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
3152 Stress concentration factors at the 75ordm tangents The nominal rim stress consists of bending stresses due to local bending moments tangential stresses due to the tan-gential force Ft and radial shear stresses due to Fr
The major influence is given by the bending stresses The influence of the tangential stresses is minor and even though its stress concentration factor is slightly higher than for bending it is considered to be safe enough when the sum of these nominal stresses are combined with the stress concen-tration factor for bending The influence of the radial shear stress is neglected
The stress concentration factor relating nominal rim stresses to local fillet stresses at the 75ordm tangent may be calculated as
75n
R75
n
R
75
ρmsρ185
ms3
Y+
sdot=
where ρ75 is the root radius at the 75ordm tangent ref to mn Usually ρ75 is closed to the tool radius ρao and
ρ75 = ρao
is a safe approximation compensating for the am simplifi-cations to the laquounsaferaquo side
The tooth root stresses of the loaded tooth are decreasing with decreasing relative rim thickness approximately with the empirical factor
15
R
ncorr s
m311Y
minus=
3153 Nominal rim stresses The bending moment applied to the rim consists of a part of the tooth tilt moment Ft (hF + 05 sR) and the bending caused by the radial force Fr
The sectional modulus (first moment of area) which is used for determination of the nominal bending stresses is not nec-essarily the same for the 2 am bending moments If flanges webs etc outside the toothed section contribute to stiffening the rim against various deflections the influence of these stiffeners should be considered Eg an end flange will have an almost negligible influence on the effective sectional modulus for the stresses due to tooth tilt as the deflection caused by the tilt moment is rather small and would not much involve the flange On the other hand the radial forces as for instance from the meshes in an annulus would cause considerable radial deflections that the flange might restrict to a substantial amount When considering the stiff-ening of such flanges or webs on basis of simplified models it is advised to use an effective rim thickness sR = sR + 02 mn for the first moment of area of the rim (toothed part) cross section
For a high number of rim teeth it may be assumed that the rim bending moments in the fillets adjacent to the loaded tooth are of the same magnitude as right under the applied
force This assumption is reasonable for an annulus but rather much on the safe side for a hollow pinion
The influence of Ft on the nominal tangential stress is simpli-fied by half of it for compressive stresses (σ1) and the other half for tensile stresses (σ2) Applying these assumptions the nominal rim stresses adjacent to the loaded tooth are
( ) ( )A2
FW
f RFW
s05hF05 t
R
r
T
RFt1 minus
ϑminus
+minus=σ
( ) ( )A2
FW
f RFW
s05hF05 t
R
r
T
RFt2 +
ϑminus
+=σ
where σ1 σ2 see Fig 35
A = minimum area of cross section (usually bR sR)
WT = the sectional modulus of rim with respect to tooth tilt moment (usually 6sb 2
rR )
WR = the sectional modulus of the rim including the influence of stiffeners as flanges webs etc (WR ge WT)
bR = the width of the rim
R = the radius of the neutral axis in the rim ie from wheel centre to midpoint of rim
( )ϑf = a function for bending moment distribution around the rim For a rim (pinion) with one mesh only the ( )ϑf at the position of load application is 024 For an annulus with 3 or more meshes ( )ϑf at the position of each load application is approx 3 planets ( )ϑf = 019 4 planets ( )ϑf = 014 5 planets ( )ϑf = 011 6 planets ( )ϑf = 009
It must be checked if the max (tensile) stress in the compres-sion fillet really occurs when the fillet is adjacent to the loaded tooth In principle the stress variation through a complete rotation should be considered and the max value used The max value is usually never less than 0 For an annulus eg the tilt moment is zero in the mid position be-tween the planet meshes whilst the bending moment due to the radial forces is half of that at the mesh but with opposite sign
If these formulae are applied to idler gears as eg planets the influence of nominal tangential stresses must be cor-rected by deleting Ft(2 A) for σ1 and using FtA for σ2 Further the influence of Fr on the nominal bending stresses is usually negligible due to the planet bearing support
3154 Root fillet stresses Determination of min and max stresses in the laquotensionraquo fillet
Classification Notes- No 412 31 May 2003
DET NORSKE VERITAS
Minimum stress
751FTmin YσK σ =
Maximum stress
corrF752 Yσ03Yσ KσFTmax +=
where
03 is an empirical factor relating the tension stresses (σF) at the 30ordm tangent to the part of the tension stresses at the 75ordm tangent which add to the rim related stresses (03 also takes into account that full superposition of nominal stresses times stress concentration factors from both laquosides of the corner filletraquo would result in too high stresses)
αβγ sdotsdotsdotsdot= FFvA KKKKKK
Determination of min and max stresses in the laquocompres-sionraquo fillet
Minimum stress
corrF751FCmin Yσ036Yσ Kσ minus=
Maximum stress
752FCmax Yσ Kσ =
where
036 is an empirical factor relating the tension stresses (σF) at the 30ordm tangent to the part of the compression stresses at the 75ordm tangent which add to the rim related stresses
For gears with reversed loads as idler gears and planets there is no distinct laquotensionraquo or laquocompressionraquo fillet The mini-mum stress σFmin is the minimum of σFTmin and σFCmin (usually the latter is decisive) The maximum stress σFTmax is the maximum of σFTmax and σFCmax (usually the former is deci-sive)
316 Permissible Stresses in Thin Rims
3161 General The safety against fatigue fracture respectively overload fracture is to be at least at the same level as for solid gears The laquoordinaryraquo criteria at the 30ordm tangent apply as given in 31 through 313
Additionally the following criteria at the 75ordm tangent may apply
3162 For gt3middot106 cycles The permissible stresses for the laquotensionraquo fillets and for the laquocompressionraquo fillets are determined by means of a relevant fatigue diagram
If the actual tooth root stress (tensile or compressive) ex-ceeds the yield strength to the material the induced residual stresses are to be taken into account
For determination of permissible stresses the following is defined
R = stress ratio ie FTmax
FTminσσ respectively
FCmax
FCmin
σσ
∆σ = stress range ie σFTmax ndash σFTmin resp σFCmax ndash σFCmin
(For idler gears and planets Fmax
FminσσR =
and ∆σ = σFmax minus σFmin)
The permissible stress range ∆σρ for the laquotensionraquo respec-tively laquocompressionraquo fillets can be calculated as
For R gt minus1 FPp σ
R1R1031
13∆σ
minus+
+=
For minus infin lt R lt minus1 FPp σ
R1R10151
13∆σ
minus+
+=
where
σFP = see 32 determined for unidirectional stresses (YM = 1)
If the yield strength σy is exceeded in either tension or com-pression residual stresses are induced This may be consid-ered by correcting the stress ratio R for the respective fillets (tension or compression)
Eg if yFCmin σσ gt (ie exceeded in compression) the
difference yFCmin σσ∆ minus= affects the stress ratio as
∆σ
σ∆σ∆σR
FCmax
y
FCmax
FCmin
+
minus=
++
=
Similarly the stress ratio in the tension fillet may require cor-rection
If the yield strength is exceeded in tension σFTmax gt σy the difference ∆ = σFTmax ndash σy affects the stress ratio as
y
FTmin
FTmax
FTmin
σ∆σ
∆σ∆σR minus=
minusminus
=
Checking for possible exceeding of the yield strength has to be made with the highest torque and with the KAP if this ex-ceeds KA
3163 Forle 103 cycles The permissible stress range ∆σp is not to exceed
For R gt minus1 FPstpst σ
R1R1051
15∆σ
minus+
+=
For ndash infin lt R lt minus1 FPstpst σ
R1R10251
15∆σ
minus+
+=
For all values of R ∆σpst is limited by
not surface hardened F
y
Sσ225
32 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
surface hardened CF
YSHV5
Definition of ∆σ and R see 3162 with particular attention to possible correction of R if the yield strength is exceeded
σFPst see 32 determined for unidirectional stresses (YM = 1) and lt 103 cycles
3164 For 103 lt cycles lt 3106 ∆σp is to be determined by linear interpolation a log-log dia-gram
∆σp at NL load cycles is
6L 103p
exp
L
6
N p ∆σN103∆σ sdot
sdot=
6
3
103p
10p
∆σ
∆σlog28760exp
sdot
=
Classification Notes- No 412 33 May 2003
DET NORSKE VERITAS
4 Calculation of Scuffing Load Capacity
41 Introduction High surface temperatures due to high loads and sliding ve-locities can cause lubricant films to break down This sei-zure or welding together of areas of tooth surface is termed scuffing
In contrast to pitting and fatigue breakage which show a dis-tinct incubation period a single short overloading can lead to a scuffing failure In the ISO-TR13989 two criteria are mentioned The method used in this Classification Note is based on the principles of the flash temperature criterion
Note Bulk temperature in excess of 120ordmC for long periods may have an adverse effect on the surface durability see 211
42 General Criteria In no point along the path of contact the local contact tem-perature may exceed the permissible temperature ie
oilS
oilSB S
ϑ+ϑminusϑ
leϑ
50SB minusϑleϑ
where
Bϑ = max contact temperature along the path of contact
maxflaMBB ϑ+ϑ=ϑ
MBϑ = bulk temperature see 434
maxflaϑ = max flash temperature along the path of con-tact see 44
Sϑ = scuffing temperature see below
oilϑ = oil temperature before it reaches the mesh (max applicable for the actual load case to be used ie normally alarm temperature except for ice classes where a max expected tem-perature applies)
SS = required safety factor according to the Rules
The scuffing temperature Sϑ may be calculated as
L2
wrelT
002
40S XFZGX
ν100112085780
sdot
sdot++=ϑ
where
XwrelT = relative welding factor
XwrelT
Through hardened steel 10
Phosphated steel 125
Copper-plated steel 150
Nitrided steel 150
Less than 10 retained austenite
115
10 ndash 20 retained austenite 10
Casehardened steel
20 ndash 30 retained austenite 085
Austenitic steel 045
FZG = load stage according to FZG-Test A8390 (Note This is the load stage where scuffing oc-curs However due to scatter in test results cal-culations are to be made with one load stage less than the specification)
XL = lubricant factor
= 10 for mineral oils
= 08 for polyalfaolefins
= 07 for non-water-soluble polyglycols
= 06 for water-soluble polyglycols
= 15 for traction fluids
= 13 for phosphate esters
ν40 = kinematic oil viscosity at 40˚C (mm2s)
Application of other test methods such as the Ryder the FZG-Ryder R46574 and the FZG L-42 Test 141195110 may be specially considered
For high speed gears with very short time of contact Sϑ may be increased as follows provided use of EP-oils
Addition to the calculated scuffing temperature Sϑ
If micros 18tc ge no addition
If micros 18tc lt add ( )cwrelT t18X18 minussdotsdot
where
ct = contact time ( micros ) which is the time needed to cross the Hertzian contact width
( ) [ ]microsu1cosβn
uσ340tb1
Hc +sdotsdot
sdot=
σH as calculated in 221
For bevel gears use vu in stead of u
34 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
43 Influence Factors
431 Coefficient of friction The following coefficient of friction may apply
L025a
005oil
02
redC ΣC
Bt X R ηρv
w0048micro minus
=
where
wBt = specific tooth load (Nmm)
vΣC = sum of tangential velocities at pitch point At pitch line velocities gt 50 ms the limiting value of vΣC at v = 50 ms is to be used
ρredC = relative radius of curvature (transversal plane) at the pitch point
Cylindrical gears
HαHβvγAbt
Bt KKKKKb
Fw sdotsdotsdotsdotsdot= (see 1)
twΣC αsin v2v =
bCredC β cos ρρ =
Bevel gears
HαHβvγAtmb
Bt KKKKKb
Fw sdotsdotsdotsdotsdot= (see 1)
vtmtΣC αsin v2v =
bmvCredC β cos ρρ =
ηoil = dynamic viscosity (mPa s) at oilϑ calculated as
1000
ρ νη oiloil =
where ρ in kgm3 approximated as
( ) 70 15oil15 minusϑminusρ=ρ and νoil is kinematic viscosity at oilϑ and may be calculated by means of the following equation
( ) ( )++=+ 08νloglog08νloglog 100oil ( )
sdotminus
ϑ+minus313log373log
273log373log oil
( ) ( )( )08νloglog08νloglog 10040 +minus+
Ra = composite arithmetic mean roughness (micron) of pinion an wheel calculated as ( )2a1aa RR 05R +=This is defined as the roughness on the new flanks ie as manufactured
XL = see 42
432 Effective tip relief Ceff Ceff is the effective tip relief that amount of tip relief which compensates for the elastic deformation of the gear mesh ie
zero load at the tooth tip It is assumed (simplified) to be equal for both pinion and wheel
Cylindrical gears
for helical
γ
γ=cb
KKFC Abt
eff (see 1)
For spur
cbKKF
C Abteff
γ= (see 1)
Alternatively for spur and helical gears the non-linear ap-proach in 111 may be used (taking Ceff=δ)
Bevel gears
bcKFC Ambt
effγ
= (see 1)
where
γ
α
ε2ε44c
+sdot
=γ
433 Tip relief and extension Cylindrical gears
The extension of the tip relief is not to result in an effective contact ratio 1ltεα when the gear is unloaded (exceptions to this may only apply for applications where the gear is not to run at light loads) This means that the unrelieved part of the path of contact is to be minimum pbt It is further assumed that this unrelieved part is placed centrally on the path of contact
If root relief applies it has to be calculated as equivalent tip relief Ie pinion root relief (at mesh position A) is added to Ca2 and wheel root relief (at mesh position E) is added to Ca1 If no design tip relief or root relief on the mating gear is specified (ie if Ca1+Croot2 = 0 and visa versa) use the run-ning in amount see 112
Bevel gears
Bevel gears are to have heightwise crowning ie no distinct relievedunrelieved area This may be treated as tip and root relief For calculation purposes the root relief is combined with the tip relief of the mating member into an equivalent tip relief If no resulting tip relieves are specified the equivalent tip relives may be calculated as an approxima-tion based on the tool crowning Ca tool (per mille of tool module m0) as follows
2root1aeq1a CCC +=
1root2aeq2a CCC +=
where
Classification Notes- No 412 35 May 2003
DET NORSKE VERITAS
2
0
n0101atoolal m
)m2(mA1middotmCC
minus+minussdot=
( )
sdotminusminussdotsdotminus+= vtvb1
2vb1
21vavt1n1 αtan dddαsin 05x1mA
2
0
n020atool22a m
)mm(2A1middotmCC
minus+minussdot=
( )
sdotminusminussdotsdotminus+= vtvb2
2vb2
2va2vt2n2 αtan dddαsin 05x1mA
2
0
20atool1root1 m
A1mCC
minussdotsdot=
2
0
10atool2root2 m
A1mCC
minussdotsdot=
If the pressure angles of the cutter blades are modified (and verified) to balance the tip relieves the following may be assumed
( )eq2aeq1aeq2aeq1a C andC calculated of sum50CC sdot==
Throughout the following Ca1 and Ca2 mean the equivalent tip relieves Ca1eq and Ca2eq
434 Bulk temperature The bulk temperature may be calculated as
flaaveragempsoilMB X X 50 ϑ+ϑ=ϑ
where
Xs = lubrication factor
= 12 for spray lubrication
= 10 for dip lubrication (provided both pinion and wheel are dip lubricated and tip speed lt 5 ms)
= 10 for spray lubrication with additional cooling spray (spray on both pinion and wheel or spray on pinion and dip lubrication of wheel)
= 02 for meshes fully submerged in oil
Xmp = contact factor ( )pmp nX += 150
np = number of mesh contact on the pinion (for small gear ratii the number of wheel meshes should be used if higher)
flaaverageϑ
= average of the integrated flash temperature (see 44) along the path of contact
( )
AE
E
Ayyyfla
flaaverage
d
ΓminusΓ
ΓΓϑ=ϑint =
For high speed gears (v gt 50 ms) it may be necessary to assess the bulk temperature on the basis of a thermal rating of the entire gear transmission
44 The Flash Temperature flaϑ
441 Basic formula The local flash temperature flaϑ may be calculated as
( ) 41redy
y2ly
211
43Btcorrfla
unXwX3250
y ρ
ρminusρ
micro=ϑ Γ
(For bevel gears replace u with uv)
and is to be calculated stepwise along the path of contact from A to E
where
micro = coefficient of friction see 431
Xcorr = correction factor taking empirically into account the increased scuffing risk in the beginning of the approach path due to mesh starting without any previously built up oil film and possible shuffling away oil before meshing if insufficient tip relief
( )
3
AD
yaeffcorr 50
CC1X
ΓminusΓε
Γminus+=
α
Ca = tip relief of driven member Xcorr is only applicable in the approach path and if Ca lt Ceff otherwise 10
wBt = unit load see 431
yXΓ = load sharing factor see 443
n1 = pinion rpm
Γy ρly etc see 442
442 Geometrical relations The various radii of flank curvature (transversal plane) are
ρ1y = pinion flank radius at mesh point y
ρ2y = wheel flank radius at mesh point y
ρredy = equivalent radius of curvature at mesh point y
y2y1
y2y1redy
ρ+ρ
ρρ=ρ
Cylindrical gears
twy
1y αsin au1Γ1
ρ+
+=
twy
2y αsin au1Γu
ρ+
minus=
Note that for internal gears a and u are negative
36 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
Bevel gears
vtvv
y1y αsin a
u1Γ1
ρ+
+=
vtvv
yv2y αsin a
u1Γu
ρ+
minus=
Γ is the parameter on the path of contact and y is any point between A end E
At the respective ends Γ has the following values
Root piniontip wheel
Cylindrical gears
( )
minus
minusminus= 1
αtan 1dd
zzΓ
tw
2b2a2
1
2A
Bevel gears
( )
minus
minusminus= 1
αtan 1dd
uΓvt
2vb2va2
vA
Tip pinionroot wheel
Cylindrical gears
( )
1αtan
1ddΓ
tw
2b1a1
E minusminus
=
Bevel gears
( )
1αtan
1ddΓ
vt
2vb1va1
E minusminus
=
At inner point of single pair contact
Cylindrical gears
tw1
EB α tan zπ2ΓΓ minus=
Bevel gears
vtv1
EB α tan zπ2ΓΓ minus=
At outer point of single pair contact
Cylindrical gears
tw1AD α tan z
π2ΓΓ +=
Bevel gears
vt v1
AD αtan z π2ΓΓ +=
At pitch point ΓC = 0
The points F and G (only applicable to cylindrical gears) limiting the extension of tip relief (so as to maintain a mini-mum contact ratio of unity for unloaded gears) are at
2
BAF
Γ+Γ=Γ
2
EDG
Γ+Γ=Γ
443 Load sharing factor XΓ The load sharing factor XΓ accounts for the load sharing be-tween the various pairs of teeth in mesh along the path of contact
XΓ is to be calculated stepwise from A to E using the pa-rameter Γy
4431 Cylindrical gears with β = 0 and no tip relief
Figure 41
ByAAB
Ay for31
31X
yΓltΓleΓ
ΓminusΓ
ΓminusΓ+=Γ
DyB for1Xy
ΓltΓleΓ=Γ
EyDDE
yE for31
31X
yΓleΓltΓ
ΓminusΓ
ΓminusΓ+=Γ
4432 Cylindrical gears with β = 0 and tip relief Tip relief on the pinion reduces XΓ in the range G ndash E and increases correspondingly XΓ in the range F ndash B
Tip relief on the wheel reduces XΓ in the range A ndash F and increases correspondingly XΓ in the range D ndash G
Following remains generally
DyB for1Xy
ΓleΓleΓ=Γ
21XXGF== ΓΓ
In the following it must be distinguished between Ca lt Ceff respectively Ca gt Ceff This is shown by an example below where Ca1 lt Ceff and Ca2 gt Ceff
Classification Notes- No 412 37 May 2003
DET NORSKE VERITAS
Figure 42
Note When Ca gt Ceff the path of contact is shortened by A ndash A respectively E ndash E The single pair contact path is extended into B respectively D If this shift is significant it is necessary to consider the negative effect on surface durability (B) and bending stresses (B and D)
Range A - F
For effa2 CC le
minus=Γ
eff
2a
CC1
31X
A
FyAeff
2a
AF
Ay for C3
C61XX
AyΓleΓleΓ
sdot
+ΓminusΓ
ΓminusΓ+= ΓΓ
For effa2 CC ge
AyAfor0Xy
ΓleΓleΓ=Γ
with ( )21
CC
1CC
eff
2a
eff
2a
AFAAminus
minusΓminusΓ+Γ=Γ
FyAeff
2a
AF
Ay
eff
2a for 21
CC
CC1X
yΓleΓleΓ
minus
ΓminusΓ
ΓminusΓ+minus=Γ
Range F - B
For effa1 CC le
ByFeff
1a
FB
Fy forC3
C61
21X
yΓleΓleΓ
sdot
+ΓminusΓ
ΓminusΓ+=Γ
For effa1 CC ge
ByFeff
1a
FB
Fy for21
CC
21X
yΓleΓleΓ
minus
ΓminusΓ
ΓminusΓ+=Γ
ByB for1Xy
ΓleΓleΓ=Γ
with 1
CC2
eff
1a
FBFB
minus
ΓminusΓ+Γ=Γ
Range D ndash G
For effa2 CC le
GyDeff
2a
DG
Dy
eff
2a for C 3C
61
C 3C
32X
yΓleΓleΓ
+
ΓminusΓ
ΓminusΓminus+=Γ F
or effa2 CC ge
DyD for1Xy
ΓleΓleΓ=Γ
with ( )21
CC
1CC
eff
2a
eff
2a
DGDDminus
minusΓminusΓ+Γ=Γ
GyDeff
2a
DG
Dy
eff
2a for21
CC
CCX ΓleΓleΓ
minus
ΓminusΓ
ΓminusΓminus=Γ
Range G - E
For effa1 CC le
minus=Γ
eff
1a
CC1
31X
E
EyGeff
1a
GE
Gy forC3
C61
21X
yΓleΓleΓ
sdot
+ΓminusΓ
ΓminusΓminus=Γ
For effa1 CC gt
EyGeff
1a
GE
Gy for21
CC
21X
yΓleΓleΓ
minus
ΓminusΓ
ΓminusΓminus=Γ
EyE for0Xy
ΓleΓleΓ=Γ
with 1
CC2
eff
1a
GEGE
minus
ΓminusΓ+Γ=Γ
4433 Gears with β gt 0 buttressing Due to oblique contact lines over the flanks a certain but-tressing may occur near A and E
This applies to both cylindrical and bevel gears with tip relief lt Ceff The buttressing Xbutt is simplified as a linear function within the ranges A ndash H respectively I ndash E
Figure 43
1εwhen13X βbutt EAge=
1εwhenε031X ββbutt EAlt+=
38 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
Cylindrical gears
bIEAH βsin 02ΓΓΓΓ =minus=minus
Bevel gears
bmIEAH βsin 02ΓΓΓΓ =minus=minus
4434 Cylindrical gears with 2εγ le and no tip relief
yXΓ is obtained by multiplication of
yXΓ in 4431 with
Xbutt in 4433
4435 Gears with 2εγ gt and no tip relief
Applicable to both cylindrical and bevel gears
IyH for1Xy
ΓleΓleΓε
=α
Γ
IyHybutt and forX1Xy
ΓgtΓΓltΓε
=α
Γ
Figure 44
4436 Cylindrical gears with 2εγ le and tip relief
yXΓ is obtained by multiplication of
yXΓ in 4432 with Xbutt
in 4433
4437 Cylindrical gears with 2εγ gt and tip relief
Tip relief on the pinion (respectively wheel) reduces XГ in the range G ndash E (respectively A ndash F) and increases XГ in the range F ndash G
yXΓ is obtained by multiplication of
yXΓ as described below
with Xbutt in 4433
In the XГ example below the influence of tip relief is shown (without the influence of Xbutt) by means of
eff2aeff1a CC and CC ltgt
Tip relief gt Ceff causes new end points A respectively E of the path of contact
Figure 45
Range A ndash F
( ) ( )( ) eff
a2αa1α
AF
Ay
eff
2aeff
C 12C 13εC 1ε
CCCX
y +εε++minus
ΓminusΓ
ΓminusΓ+
εminus
=ααα
Γ
eff2aFyA CC if for leΓleΓleΓ
eff2aFyA CifCfor and geΓleΓleΓ
eff2aAyA CC iffor0Xy
gtΓleΓleΓ=Γ
( ) ( ) ( )( ) ( ) a2αa1α
αeffa2AFAA C 1ε 3C 1ε
1ε 2 CC with++minus+minus
ΓminusΓ+Γ=Γ
Range F ndash G
( )( )( ) GyF
eff
2a1a forC 1 2CC 11X
yΓleΓleΓ
+εε+minusε
+ε
=αα
α
αΓ
Range G ndash E
( ) ( )( ) eff
2a1a
GE
Gy
C 1 2C 1C 1 3XX
GFy +εεminusε++ε
ΓminusΓ
ΓminusΓminus=
αα
ααΓΓ minus
effa1EyG CC if for leΓleΓleΓ
effa1EyG CC if Γfor and geΓleΓle
eff1aEyE CC if for0Xy
geΓleΓleΓ=Γ
( ) ( ) ( )( ) ( ) 2a1a
eff1aGEEE C 1C 1 3
1 2 CC withminusε++ε+εminus
ΓminusΓminusΓ=Γαα
α
4438 Bevel gears with εγ more than approx 18 and heightwise crowning For Ca1 = Ca2 = Ceff the following applies
Figure 46
Classification Notes- No 412 39 May 2003
DET NORSKE VERITAS
( )AEM 50 Γ+Γ=Γ
( )( )2AD
3
2My651X
y ΓminusΓε
ΓminusΓminus
ε=
ααΓ
For tip relief lt Ceff yXΓ is found by linear interpolation be-
tween ( )effay CCX =Γ and ( )0CayX =Γ as in 4435
The interpolation is to be made stepwise from A to M with the influence of Ca2 and from M to E with the influence of Ca1 (For 2a1a CC ne there is a discontinuity at M)
Eg with Ca1 = 04 Ceff and Ca2 = 055 Ceff then
Range A ndash M
)CC()0C( eff2ay2ayyX 550X 450X =Γ=ΓΓ +=
Range M ndash E
)CC()0C( eff1ay1ayyX 40X 06X =Γ=ΓΓ +=
For tip relief gt Ceff the new end points A and E are found as
( )
minusΓminusΓ
ε+Γ=Γ α 1
CC
6 eff
2aADAA
( )
minusΓminusΓ
ε+Γ=Γ α 1
CC
6 eff
1aADEE
Range A ndash A
0Xy=Γ
Range A ndash M
( )( )
ΓminusΓ
ΓminusΓminus
minusε=
αΓ 2
MA
2My
eff
2a1
CC4
351Xy
Range M ndash E
( )( )
ΓminusΓ
ΓminusΓminus
minusε=
αΓ 2
ME
2My
eff
1a1
CC4
351Xy
Range E ndash E
0Xy=Γ
40 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
Appendix A Fatigue Damage Accumulation
The Palmgren-Miner cumulative damage calculation princi-ple is used The procedure may be applied as follows
A1 Stress Spectrum From the individual torque classes the torques (Ti) at the peak values of class intervals and the associated number of cycles (NLi) for both pinion and wheel are to be listed from the highest to the lowest torque
(In case of a cyclic torque variation within the torque classes it is advised to use the peak torque If the cyclic variation is such that the same teeth will repeatedly suffer the peak torque this is a must)
The stress spectra for tooth roots and flanks (σFi σHi) with all relevant factors (except KA) are to be calculated on the basis of the torque spectrum The load dependent K-factors are to be determined for each torque class
A2 σminusN-curve The stress versus load cycle curves for tooth roots and flanks (both pinion and wheel) are to be drawn on the basis of per-missible stresses (ie including the demanded minimum safety factors) as determined in 2 respectively 3 If different safety levels for high cycle fatigue and low cycle fatigue are desired this may be expressed by different demand safety factors applied at the endurance limit respectively at static strength
A3 Damage accumulation The individual damage ratio Di at ith stress level is defined as
Fi
Lii N
ND =
where
NLi = The number of applied cycles at ith stress
NFi = The number of cycles to failure at ith stress
Basically stresses σi below the permissible stress level for infinite life (if a constant ZN or YN is accepted) do not con-tribute to the damage sum However calculating the actual safety factor Sact as described below all the σi for which the product S σi is bigger than or equal to the permissible stress level for infinite life contribute to the damage sum and thus to the determination of Sact The final value of S is decisive
(NFi can be found mathematically by putting the permissible stress σpi equal the actual stress σi thereby finding the actual life factor This life factor can be solved with regard to load cycles ie NFi)
The damage sum ΣDi is not to exceed unity
If ΣDi ne 1 the safety against cumulative fatigue damage is different from the applied demand safety factor For deter-mination of this theoretical safety factor an iteration proce-dure is required as described in the following flowchart
S is correction factor with which the actual safety factor Sact can be found
Sact is the demand safety factor (used in determination of the permissible stresses in the σ ndash N ndash curve) times the correc-tion factor S
The full procedure is to be applied for pinion and wheel tooth roots and flanks
Note If alternating stresses occur in a spectrum of mainly pulsating stresses the alternating stresses may be replaced by equiva-lent pulsating stresses ie by means of division with the ac-tual mean stress influence factor YM
Classification Notes- No 412 41 May 2003
DET NORSKE VERITAS
Appendix B Application Factors for Diesel Driven Gears
For diesel driven gears the application factor KA depends on torsional vibrations Both normal operation and misfiring conditions have to be considered
Normally these two running conditions can be covered by only one calculation
B1 Definitions
Normal operation KAnorm = 0
normv0
TTT +
where
T0 = rated nominal torque
Tv norm = vibratory torque amplitude for normal opera-tion (see rules Pt4 Ch3 Sec1 G301 for defi-nition of ldquonormalrdquo irregularity)
Misfiring operation KA misf = 0
misfv
TTT +
where
T = remaining nominal torque when one cylinder out of action
Tv misf = vibratory torque amplitude in misfiring con-dition This refers to a permissible misfiring condition ie a condition that does not require automatic or immediate corrective actions as speed or pitch reduction
The normal operation is assumed to last for a very high num-ber of cycles such as 1010
The misfiring operation is assumed to last for a limited dura-tion such as 107 cycles
B2 Determination of decisive load Assuming life factor at 1010 cycles as YN = ZN = 092 which usually is relevant the calculation may be performed only once with the combination having the highest value of appli-cation factorlife factor
For bending stresses and scuffing the higher value of
092
K normA and 098
K misfA
For contact stresses the higher value of
092K normA and
113K misfA (but
097K misfA for nitrided gears)
B3 Simplified procedure Note that this is only a guidance and is not a binding con-vention
T = oTZ
1Zsdot
minus where Z = number of cylinders
( ) ideal Videal Vmisf Vnorm V TTT24ZT +minus=
where TV ideal = vibratory torque with all cylinders perfectly equal See also rules Pt4 Ch3 Sec1 G300
When using trends from torsional vibration analysis and measurements the following may be used
TV ideal To is close to zero for engines with few cylinders and using a suitable elastic coupling and increases with relative coupling stiffness and number of cylinders
TV misf To may be high for engines with few cylinders and decreases with number of cylinders
This can be indicated as
200Z
TT
ο
idealV asymp and 80Z04
TT
o
misfV minusasymp
Inserting this into the formulae for the two application fac-tors the following guidance can be given
118112K misfA minusasymp
115110K normA minusasymp
Since KA norm is to be combined with the lower life factors the decisive load condition will be the normal one and a KA of 115 will cover most relevant cases when a suitable elas-tic coupling is chosen
42 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
Appendix C Calculation of Pinion-Rack
Pinion-racks used for elevating of mobile offshore units are open gears that are subjected to wear and tear With normal specifications such as only a few hundred total operation cycles (site to site) the tooth bending stresses for static strength will be decisive for the lay out however with ex-ception of surface hardened pinions where case crushing has to be considered
In the following the use of part 1 and 3 for pinion-racks is shown including relevant simplifications
C1 Pinion tooth root stresses Since the load spectrum normally is dominated by high torques and few load cycles (in the range up to 1-2000) the static strength is decisive
The actual stress is calculated as
β1FSaFan1
tF1 KYY
mbFσ sdotsdotsdotsdot
=
b1 is limited to b2 + 2middotmn
YFa and YSa replace YF and YS because load application at tooth tip has to be assumed for such inaccurate gears
Pinions often use a non-involute profile in the dedendum part of the flank eg a constant radius equal the radius of curva-ture at reference circle For such pinions sFn and hFa are to be measured directly on a sectional drawing of the pinion tooth
Due to high loads and narrow facewidths it may be assumed that KFβ2 = KHβ = 10 However when b1 gt b2 then KFβ1 gt10
If no detailed documentation of KFβ1 is available the fol-lowing may be used
KFβ1 = 1 + 015middot(b1b2 ndash 1)
The permissible stress (not surface hardened) is calculated as
δrelTstF
Fst1FP1 Y
Sσσ sdot=
The mean stress influence due to leg lifting may be disre-garded
The actual and permissible stresses should be calculated for the relevant loads as given in the rules
C2 Rack tooth root stresses The actual stress is calculated as
SaFan2
tF2 YY
mbFσ sdotsdotsdot
=
See C1 for details
The permissible stress is calculated as
δrelTstF
Fst2FP2 Y
Sσσ sdot=
For alloyed steels (Ni Cr Mo) with high toughness and duc-tility the value of YδrelTst may be put equal to YSa
C3 Surface hardened pinions For surface hardened pinions the maximum load is not to cause crushing of the hardened layer of the flank
In principle the calculation described in 213 may be used but when the theoretical Hertzian stress exceeds the ap-proximately 18 times the yield strength of the rack material plastic deformation will occur This will limit the peak Hertzian stress but increases the contact width and thus the penetration of stresses into the depth
An approximation may be based on an assessment of contact width determined by means of equal areas under the theoreti-cal (elastic range) Hertzian contact and the elasto-plastic contact stress (limited to 18 middot σy) with the unknown width
Classification Notes- No 412 27 May 2003
DET NORSKE VERITAS
385 Additional requirements for peak loads The total stress range (σmax ndash σmin) in a tooth root fillet is not to exceed
F
y
Sσ225
for not surface hardened fillets
FSHV5 for surface hardened fillets
39 Life Factor YN The life factor YN takes into account that in the case of limited life (number of cycles) a higher tooth root stress can be permitted and that lower stresses may apply for very high number of cycles
Decisive for the strength at limited life is the σ ndash N ndash curve of the respective material for given hardening module fillet radius roughness in the tooth root etc Ie the factors YδrelT YRelT YX and YM have an influence on YN
If no σ ndash N ndash curve for the actual material and hardening etc is available the following method may be used
Determination of the σ ndash N ndash curve
a) Calculate the permissible stress σFP for the beginning of the endurance limit (3middot106 cycles) including the influ-ence of all relevant factors as SF YδrelT YRelT YX YM and YC ie σFP = σFE middotYM middotYδrelT middotYRelT middotYX middotYC SF
b) Calculate the permissible laquostaticraquo stress (le 103 load cy-cles) including the influence of all relevant factors as SFst YδrelTst YMst and YCst ( )fitσCstYδrelTstYMstYFstσ
FstS1
FPstσ minussdotsdotsdot=
where σFst is the local tooth root stress which the material can resist without cracking (surface hardened materials) or unacceptable deformation (not surface hardened materials) with 99 survival probability
σFst
Alloyed case hardened steel 1) 2300
Nitriding steel quenched tempered and gas nitrided (surface hardness 700 ndash 800 HV)
1250
Alloyed quenched and tempered steel bath or gas nitrided (surface hardness 500 ndash 700 HV)
1050
Alloyed quenched and tempered steel flame or induction hardened (fillet surface hardness 500 ndash 650 HV)
18 HV + 800
Steel with not surface hardened fillets the smaller value of 2)
18 σB or 225 σy
1) This is valid for a fillet surface hardness of 58 ndash 63 HRC In case of lower fillet surface hardness than 58 HRC σFst is to be reduced with 30(58 ndash HRC) where HRC is the actual hardness Shot peening or grinding notches are not considered to have any significant influence on σFst 2) Actual stresses exceeding the yield point (σy or σ02) will alter the residual stresses locally in the ldquotensionrdquo fillet re-spectively ldquocompressionrdquo fillet This is only to be utilised for gears that are not later loaded with a high number of cycles at lower loads that could cause fatigue in the ldquocompressionrdquo fillet
c) Calculate YN as
NL gt 3middot106
YN = 1 or 001
L
6
N N103Y
sdot= ie Yn = 092 for 1010
The YN = 1 from 3middot106 on may only be used when special material cleanness applies see rules Pt4 Ch2
103ltNLlt3middot106exp
L
6
N N103Y
sdot=
cycles6103forσcycles310forσlog02876exp
FP
FPst
sdot=
NL lt 103 cycles6103forσcycles310forσ
NYFP
FPst
sdot=
or simply use σFPst as mentioned in b) directly
Guidance on number of load cycles NL for various applica-tions
bull For propulsion purpose normally NL = 1010 at full load (yachts etc may have lower values)
bull For auxiliary gears driving generators that normally operate with 70-90 of rated power NL = 108 with rated power may be applied
28 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
310 Relative Notch Sensitivity Factor YδrelT The dynamic (respectively static) relative notch sensitivity factor YδrelT (YδrelTst) indicate to which extent the theoreti-cally concentrated stress lies above the endurance limits (re-spectively static strengths) in the case of fatigue (respectively overload) breakage
YδrelT is a function of the material and the relative stress gra-dient It differs for static strength and endurance limit
The following method may be used
For endurance limit
for not surface hardened fillets
( )
024
s024
relT σ103133q21σ1012201351
Ysdotsdotminus
+sdotsdotminus+= minus
minus
δ
for all surface hardened fillets except nitrided
( )
106q21002451
Y srelT
++=δ
for nitrided fillets
( )
1347q2101421
Y srelT
++=δ
For static strength
for not surface hardened fillets1)
( )( )( )025
02
02502s
relTst σ3000821σ300 1Y0821Y
+
minus+=δ
for surface hardened fillets except nitrided
YδrelTst = 044 YS + 012
for nitrided fillets
YδrelTst = 06 + 02 YS 1) These values are only valid if the local stresses do not
exceed the yield point and thereby alter the residual stress level See also 39b footnote 2
311 Relative Surface Condition Factor YRrelT The relative surface condition factor YRrelT takes into ac-count the dependence of the tooth root strength on the sur-face condition in the tooth root fillet mainly the dependence on the peak to valley surface roughness
YRrelT differs for endurance limit and static strength
The following method may be used
For endurance limit
YRrelT = 1675 ndash 053 (Ry + 1)01 for surface hardened steels and alloyed quenched and tempered steels except nitrided
YRrelT = 53 ndash 42 (Ry + 1)001 for carbon steels
YδrelT = 43 ndash 326 (Ry + 1)0005
for nitrided steels
For static strength
YRrelTst = 1 for all Ry and all materials
For a fillet without any longitudinal machining trace Ry asymp Rz
312 Size Factor YX The size factor YX takes into account the decrease of the strength with increasing size YX differs for endurance limit and static strength
The following may be used
For endurance limit
YX = 1 for mn le 5 generally
YX = 103 ndash 0006 mn for 5 lt mn lt 30
YX = 085 for mn ge 30
for not surface hard-ened steels
YX = 105 ndash 001 mn for 5 lt mn ge 25
YX = 08 for mn ge 25
for surface hardened steels
For static strength
YXst = 1 for all mn and all materials
313 Case Depth Factor YC The case depth factor YC takes into account the influence of hardening depth on tooth root strength
YC applies only to surface hardened tooth roots and is dif-ferent for endurance limit and static strength
In case of insufficient hardening depth fatigue cracks can develop in the transition zone between the hardened layer and the core For static strength yielding shall not occur in the transition zone as this would alter the surface residual stresses and therewith also the fatigue strength
The major parameters are case depth stress gradient permis-sible surface respectively subsurface stresses and subsurface residual stresses
The following simplified method for YC may be used
YC consists of a ratio between permissible subsurface stress (incl influence of expected residual stresses) and permissible surface stress This ratio is multiplied with a bracket con-taining the influence of case depth and stress gradient (The empirical numbers in the bracket are based on a high number of teeth and are somewhat on the safe side for low number of teeth)
YC and YCst may be calculated as given below but calculated values above 10 are to be put equal 10
For endurance limit
+
+=nFFE
C m02ρt31
σconstY
Classification Notes- No 412 29 May 2003
DET NORSKE VERITAS
For static strength
+
+=nFFst
Cst m02ρt31
σconstY
where const and t are connected as
Hardening process
t = endurance limit const =
static strength const =
t550 640 1900
t400 500 1200 Case hard-ening
t300 380 800
Nitriding t400 500 1200
Induction- or flame hardening
tHVmin 11 HVmin 25 HVmin
For symbols see 213
In addition to these requirements to minimum case depths for endurance limit some upper limitations apply to case hard-ened gears
The max depth to 550 HV should not exceed
1) 13 of the top land thickness san unless adequate tip relief is applied (see 110)
2) 025 mn If this is exceeded the following applies addi-tionally in connection with endurance limit
minusminus= 025
mt
1Yn
max550C
314 Thin rim factor YB Where the rim thickness is not sufficient to provide full sup-port for the tooth root the location of a bending failure may be through the gear rim rather than from root fillet to root fillet
YB is not a factor used to convert calculated root stresses at the 30deg tangent to actual stresses in a thin rim tension fillet Actually the compression fillet can be more susceptible to fatigue
YB is a simplified empirical factor used to de-rate thin rim gears (external as well as internal) when no detailed calcula-tion of stresses in both tension and compression fillets are available
Figure 34 Examples on thin rims
YB is applicable in the range 175 lt sRmn lt 35
YB = 115 middot ln (8324 middot mnsR)
(for sRmn ge 35 YB = 1)
(for sRmn le 175 use 315)
σF as calculated in 321 is to be multiplied with YB when sRmn lt 35 Thus YB is used for both high and low cycle fatigue
Note This method is considered to be on the safe side for external gear rims However for internal gear rims without any flange or web stiffeners the method may not be on the safe side and it is advised to check with the method in 315316
315 Stresses in Thin Rims For rim thickness sR lt 35 mn the safety against rim cracking has to be checked
The following method may be used
3151 General The stresses in the standardised 30ordm tangent section tension side are slightly reduced due to decreasing stress correction factor with decreasing relative rim thickness sRmn On the other hand during the complete stress cycle of that fillet a certain amount of compression stresses are also introduced The complete stress range remains approximately constant Therefore the standardised calculation of stresses at the 30ordm tangent may be retained for thin rims as one of the necessary criteria
The maximum stress range for thin rims usually occurs at the 60ordm ndash 80ordm tangents both for laquotensionraquo and laquocompressionraquo side The following method assumes the 75ordm tangent to be the decisive Therefore in addition to the am criterion ap-plied at the 30ordm tangent it is necessary to evaluate the max and min stresses at the 75ordm tangent for both laquotensionraquo (loaded flank) fillet and laquocompressionraquo (back-flank) fillet For this purpose the whole stress cycle of each fillet should be considered but usually the following simplification is justified
Figure 35 Nomenclature of fillets
Index laquoTraquo means laquotensileraquo fillet laquoCraquo means laquocompressionraquo fillet
σFTmin and σFCmax are determined on basis of the nominal rim stresses times the stress concentration factor Y75
σFCmin and σFTmax are determined on basis of superposition of nominal rim stresses times Y75 plus the tooth bending stresses at 75ordm tangent
30 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
3152 Stress concentration factors at the 75ordm tangents The nominal rim stress consists of bending stresses due to local bending moments tangential stresses due to the tan-gential force Ft and radial shear stresses due to Fr
The major influence is given by the bending stresses The influence of the tangential stresses is minor and even though its stress concentration factor is slightly higher than for bending it is considered to be safe enough when the sum of these nominal stresses are combined with the stress concen-tration factor for bending The influence of the radial shear stress is neglected
The stress concentration factor relating nominal rim stresses to local fillet stresses at the 75ordm tangent may be calculated as
75n
R75
n
R
75
ρmsρ185
ms3
Y+
sdot=
where ρ75 is the root radius at the 75ordm tangent ref to mn Usually ρ75 is closed to the tool radius ρao and
ρ75 = ρao
is a safe approximation compensating for the am simplifi-cations to the laquounsaferaquo side
The tooth root stresses of the loaded tooth are decreasing with decreasing relative rim thickness approximately with the empirical factor
15
R
ncorr s
m311Y
minus=
3153 Nominal rim stresses The bending moment applied to the rim consists of a part of the tooth tilt moment Ft (hF + 05 sR) and the bending caused by the radial force Fr
The sectional modulus (first moment of area) which is used for determination of the nominal bending stresses is not nec-essarily the same for the 2 am bending moments If flanges webs etc outside the toothed section contribute to stiffening the rim against various deflections the influence of these stiffeners should be considered Eg an end flange will have an almost negligible influence on the effective sectional modulus for the stresses due to tooth tilt as the deflection caused by the tilt moment is rather small and would not much involve the flange On the other hand the radial forces as for instance from the meshes in an annulus would cause considerable radial deflections that the flange might restrict to a substantial amount When considering the stiff-ening of such flanges or webs on basis of simplified models it is advised to use an effective rim thickness sR = sR + 02 mn for the first moment of area of the rim (toothed part) cross section
For a high number of rim teeth it may be assumed that the rim bending moments in the fillets adjacent to the loaded tooth are of the same magnitude as right under the applied
force This assumption is reasonable for an annulus but rather much on the safe side for a hollow pinion
The influence of Ft on the nominal tangential stress is simpli-fied by half of it for compressive stresses (σ1) and the other half for tensile stresses (σ2) Applying these assumptions the nominal rim stresses adjacent to the loaded tooth are
( ) ( )A2
FW
f RFW
s05hF05 t
R
r
T
RFt1 minus
ϑminus
+minus=σ
( ) ( )A2
FW
f RFW
s05hF05 t
R
r
T
RFt2 +
ϑminus
+=σ
where σ1 σ2 see Fig 35
A = minimum area of cross section (usually bR sR)
WT = the sectional modulus of rim with respect to tooth tilt moment (usually 6sb 2
rR )
WR = the sectional modulus of the rim including the influence of stiffeners as flanges webs etc (WR ge WT)
bR = the width of the rim
R = the radius of the neutral axis in the rim ie from wheel centre to midpoint of rim
( )ϑf = a function for bending moment distribution around the rim For a rim (pinion) with one mesh only the ( )ϑf at the position of load application is 024 For an annulus with 3 or more meshes ( )ϑf at the position of each load application is approx 3 planets ( )ϑf = 019 4 planets ( )ϑf = 014 5 planets ( )ϑf = 011 6 planets ( )ϑf = 009
It must be checked if the max (tensile) stress in the compres-sion fillet really occurs when the fillet is adjacent to the loaded tooth In principle the stress variation through a complete rotation should be considered and the max value used The max value is usually never less than 0 For an annulus eg the tilt moment is zero in the mid position be-tween the planet meshes whilst the bending moment due to the radial forces is half of that at the mesh but with opposite sign
If these formulae are applied to idler gears as eg planets the influence of nominal tangential stresses must be cor-rected by deleting Ft(2 A) for σ1 and using FtA for σ2 Further the influence of Fr on the nominal bending stresses is usually negligible due to the planet bearing support
3154 Root fillet stresses Determination of min and max stresses in the laquotensionraquo fillet
Classification Notes- No 412 31 May 2003
DET NORSKE VERITAS
Minimum stress
751FTmin YσK σ =
Maximum stress
corrF752 Yσ03Yσ KσFTmax +=
where
03 is an empirical factor relating the tension stresses (σF) at the 30ordm tangent to the part of the tension stresses at the 75ordm tangent which add to the rim related stresses (03 also takes into account that full superposition of nominal stresses times stress concentration factors from both laquosides of the corner filletraquo would result in too high stresses)
αβγ sdotsdotsdotsdot= FFvA KKKKKK
Determination of min and max stresses in the laquocompres-sionraquo fillet
Minimum stress
corrF751FCmin Yσ036Yσ Kσ minus=
Maximum stress
752FCmax Yσ Kσ =
where
036 is an empirical factor relating the tension stresses (σF) at the 30ordm tangent to the part of the compression stresses at the 75ordm tangent which add to the rim related stresses
For gears with reversed loads as idler gears and planets there is no distinct laquotensionraquo or laquocompressionraquo fillet The mini-mum stress σFmin is the minimum of σFTmin and σFCmin (usually the latter is decisive) The maximum stress σFTmax is the maximum of σFTmax and σFCmax (usually the former is deci-sive)
316 Permissible Stresses in Thin Rims
3161 General The safety against fatigue fracture respectively overload fracture is to be at least at the same level as for solid gears The laquoordinaryraquo criteria at the 30ordm tangent apply as given in 31 through 313
Additionally the following criteria at the 75ordm tangent may apply
3162 For gt3middot106 cycles The permissible stresses for the laquotensionraquo fillets and for the laquocompressionraquo fillets are determined by means of a relevant fatigue diagram
If the actual tooth root stress (tensile or compressive) ex-ceeds the yield strength to the material the induced residual stresses are to be taken into account
For determination of permissible stresses the following is defined
R = stress ratio ie FTmax
FTminσσ respectively
FCmax
FCmin
σσ
∆σ = stress range ie σFTmax ndash σFTmin resp σFCmax ndash σFCmin
(For idler gears and planets Fmax
FminσσR =
and ∆σ = σFmax minus σFmin)
The permissible stress range ∆σρ for the laquotensionraquo respec-tively laquocompressionraquo fillets can be calculated as
For R gt minus1 FPp σ
R1R1031
13∆σ
minus+
+=
For minus infin lt R lt minus1 FPp σ
R1R10151
13∆σ
minus+
+=
where
σFP = see 32 determined for unidirectional stresses (YM = 1)
If the yield strength σy is exceeded in either tension or com-pression residual stresses are induced This may be consid-ered by correcting the stress ratio R for the respective fillets (tension or compression)
Eg if yFCmin σσ gt (ie exceeded in compression) the
difference yFCmin σσ∆ minus= affects the stress ratio as
∆σ
σ∆σ∆σR
FCmax
y
FCmax
FCmin
+
minus=
++
=
Similarly the stress ratio in the tension fillet may require cor-rection
If the yield strength is exceeded in tension σFTmax gt σy the difference ∆ = σFTmax ndash σy affects the stress ratio as
y
FTmin
FTmax
FTmin
σ∆σ
∆σ∆σR minus=
minusminus
=
Checking for possible exceeding of the yield strength has to be made with the highest torque and with the KAP if this ex-ceeds KA
3163 Forle 103 cycles The permissible stress range ∆σp is not to exceed
For R gt minus1 FPstpst σ
R1R1051
15∆σ
minus+
+=
For ndash infin lt R lt minus1 FPstpst σ
R1R10251
15∆σ
minus+
+=
For all values of R ∆σpst is limited by
not surface hardened F
y
Sσ225
32 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
surface hardened CF
YSHV5
Definition of ∆σ and R see 3162 with particular attention to possible correction of R if the yield strength is exceeded
σFPst see 32 determined for unidirectional stresses (YM = 1) and lt 103 cycles
3164 For 103 lt cycles lt 3106 ∆σp is to be determined by linear interpolation a log-log dia-gram
∆σp at NL load cycles is
6L 103p
exp
L
6
N p ∆σN103∆σ sdot
sdot=
6
3
103p
10p
∆σ
∆σlog28760exp
sdot
=
Classification Notes- No 412 33 May 2003
DET NORSKE VERITAS
4 Calculation of Scuffing Load Capacity
41 Introduction High surface temperatures due to high loads and sliding ve-locities can cause lubricant films to break down This sei-zure or welding together of areas of tooth surface is termed scuffing
In contrast to pitting and fatigue breakage which show a dis-tinct incubation period a single short overloading can lead to a scuffing failure In the ISO-TR13989 two criteria are mentioned The method used in this Classification Note is based on the principles of the flash temperature criterion
Note Bulk temperature in excess of 120ordmC for long periods may have an adverse effect on the surface durability see 211
42 General Criteria In no point along the path of contact the local contact tem-perature may exceed the permissible temperature ie
oilS
oilSB S
ϑ+ϑminusϑ
leϑ
50SB minusϑleϑ
where
Bϑ = max contact temperature along the path of contact
maxflaMBB ϑ+ϑ=ϑ
MBϑ = bulk temperature see 434
maxflaϑ = max flash temperature along the path of con-tact see 44
Sϑ = scuffing temperature see below
oilϑ = oil temperature before it reaches the mesh (max applicable for the actual load case to be used ie normally alarm temperature except for ice classes where a max expected tem-perature applies)
SS = required safety factor according to the Rules
The scuffing temperature Sϑ may be calculated as
L2
wrelT
002
40S XFZGX
ν100112085780
sdot
sdot++=ϑ
where
XwrelT = relative welding factor
XwrelT
Through hardened steel 10
Phosphated steel 125
Copper-plated steel 150
Nitrided steel 150
Less than 10 retained austenite
115
10 ndash 20 retained austenite 10
Casehardened steel
20 ndash 30 retained austenite 085
Austenitic steel 045
FZG = load stage according to FZG-Test A8390 (Note This is the load stage where scuffing oc-curs However due to scatter in test results cal-culations are to be made with one load stage less than the specification)
XL = lubricant factor
= 10 for mineral oils
= 08 for polyalfaolefins
= 07 for non-water-soluble polyglycols
= 06 for water-soluble polyglycols
= 15 for traction fluids
= 13 for phosphate esters
ν40 = kinematic oil viscosity at 40˚C (mm2s)
Application of other test methods such as the Ryder the FZG-Ryder R46574 and the FZG L-42 Test 141195110 may be specially considered
For high speed gears with very short time of contact Sϑ may be increased as follows provided use of EP-oils
Addition to the calculated scuffing temperature Sϑ
If micros 18tc ge no addition
If micros 18tc lt add ( )cwrelT t18X18 minussdotsdot
where
ct = contact time ( micros ) which is the time needed to cross the Hertzian contact width
( ) [ ]microsu1cosβn
uσ340tb1
Hc +sdotsdot
sdot=
σH as calculated in 221
For bevel gears use vu in stead of u
34 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
43 Influence Factors
431 Coefficient of friction The following coefficient of friction may apply
L025a
005oil
02
redC ΣC
Bt X R ηρv
w0048micro minus
=
where
wBt = specific tooth load (Nmm)
vΣC = sum of tangential velocities at pitch point At pitch line velocities gt 50 ms the limiting value of vΣC at v = 50 ms is to be used
ρredC = relative radius of curvature (transversal plane) at the pitch point
Cylindrical gears
HαHβvγAbt
Bt KKKKKb
Fw sdotsdotsdotsdotsdot= (see 1)
twΣC αsin v2v =
bCredC β cos ρρ =
Bevel gears
HαHβvγAtmb
Bt KKKKKb
Fw sdotsdotsdotsdotsdot= (see 1)
vtmtΣC αsin v2v =
bmvCredC β cos ρρ =
ηoil = dynamic viscosity (mPa s) at oilϑ calculated as
1000
ρ νη oiloil =
where ρ in kgm3 approximated as
( ) 70 15oil15 minusϑminusρ=ρ and νoil is kinematic viscosity at oilϑ and may be calculated by means of the following equation
( ) ( )++=+ 08νloglog08νloglog 100oil ( )
sdotminus
ϑ+minus313log373log
273log373log oil
( ) ( )( )08νloglog08νloglog 10040 +minus+
Ra = composite arithmetic mean roughness (micron) of pinion an wheel calculated as ( )2a1aa RR 05R +=This is defined as the roughness on the new flanks ie as manufactured
XL = see 42
432 Effective tip relief Ceff Ceff is the effective tip relief that amount of tip relief which compensates for the elastic deformation of the gear mesh ie
zero load at the tooth tip It is assumed (simplified) to be equal for both pinion and wheel
Cylindrical gears
for helical
γ
γ=cb
KKFC Abt
eff (see 1)
For spur
cbKKF
C Abteff
γ= (see 1)
Alternatively for spur and helical gears the non-linear ap-proach in 111 may be used (taking Ceff=δ)
Bevel gears
bcKFC Ambt
effγ
= (see 1)
where
γ
α
ε2ε44c
+sdot
=γ
433 Tip relief and extension Cylindrical gears
The extension of the tip relief is not to result in an effective contact ratio 1ltεα when the gear is unloaded (exceptions to this may only apply for applications where the gear is not to run at light loads) This means that the unrelieved part of the path of contact is to be minimum pbt It is further assumed that this unrelieved part is placed centrally on the path of contact
If root relief applies it has to be calculated as equivalent tip relief Ie pinion root relief (at mesh position A) is added to Ca2 and wheel root relief (at mesh position E) is added to Ca1 If no design tip relief or root relief on the mating gear is specified (ie if Ca1+Croot2 = 0 and visa versa) use the run-ning in amount see 112
Bevel gears
Bevel gears are to have heightwise crowning ie no distinct relievedunrelieved area This may be treated as tip and root relief For calculation purposes the root relief is combined with the tip relief of the mating member into an equivalent tip relief If no resulting tip relieves are specified the equivalent tip relives may be calculated as an approxima-tion based on the tool crowning Ca tool (per mille of tool module m0) as follows
2root1aeq1a CCC +=
1root2aeq2a CCC +=
where
Classification Notes- No 412 35 May 2003
DET NORSKE VERITAS
2
0
n0101atoolal m
)m2(mA1middotmCC
minus+minussdot=
( )
sdotminusminussdotsdotminus+= vtvb1
2vb1
21vavt1n1 αtan dddαsin 05x1mA
2
0
n020atool22a m
)mm(2A1middotmCC
minus+minussdot=
( )
sdotminusminussdotsdotminus+= vtvb2
2vb2
2va2vt2n2 αtan dddαsin 05x1mA
2
0
20atool1root1 m
A1mCC
minussdotsdot=
2
0
10atool2root2 m
A1mCC
minussdotsdot=
If the pressure angles of the cutter blades are modified (and verified) to balance the tip relieves the following may be assumed
( )eq2aeq1aeq2aeq1a C andC calculated of sum50CC sdot==
Throughout the following Ca1 and Ca2 mean the equivalent tip relieves Ca1eq and Ca2eq
434 Bulk temperature The bulk temperature may be calculated as
flaaveragempsoilMB X X 50 ϑ+ϑ=ϑ
where
Xs = lubrication factor
= 12 for spray lubrication
= 10 for dip lubrication (provided both pinion and wheel are dip lubricated and tip speed lt 5 ms)
= 10 for spray lubrication with additional cooling spray (spray on both pinion and wheel or spray on pinion and dip lubrication of wheel)
= 02 for meshes fully submerged in oil
Xmp = contact factor ( )pmp nX += 150
np = number of mesh contact on the pinion (for small gear ratii the number of wheel meshes should be used if higher)
flaaverageϑ
= average of the integrated flash temperature (see 44) along the path of contact
( )
AE
E
Ayyyfla
flaaverage
d
ΓminusΓ
ΓΓϑ=ϑint =
For high speed gears (v gt 50 ms) it may be necessary to assess the bulk temperature on the basis of a thermal rating of the entire gear transmission
44 The Flash Temperature flaϑ
441 Basic formula The local flash temperature flaϑ may be calculated as
( ) 41redy
y2ly
211
43Btcorrfla
unXwX3250
y ρ
ρminusρ
micro=ϑ Γ
(For bevel gears replace u with uv)
and is to be calculated stepwise along the path of contact from A to E
where
micro = coefficient of friction see 431
Xcorr = correction factor taking empirically into account the increased scuffing risk in the beginning of the approach path due to mesh starting without any previously built up oil film and possible shuffling away oil before meshing if insufficient tip relief
( )
3
AD
yaeffcorr 50
CC1X
ΓminusΓε
Γminus+=
α
Ca = tip relief of driven member Xcorr is only applicable in the approach path and if Ca lt Ceff otherwise 10
wBt = unit load see 431
yXΓ = load sharing factor see 443
n1 = pinion rpm
Γy ρly etc see 442
442 Geometrical relations The various radii of flank curvature (transversal plane) are
ρ1y = pinion flank radius at mesh point y
ρ2y = wheel flank radius at mesh point y
ρredy = equivalent radius of curvature at mesh point y
y2y1
y2y1redy
ρ+ρ
ρρ=ρ
Cylindrical gears
twy
1y αsin au1Γ1
ρ+
+=
twy
2y αsin au1Γu
ρ+
minus=
Note that for internal gears a and u are negative
36 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
Bevel gears
vtvv
y1y αsin a
u1Γ1
ρ+
+=
vtvv
yv2y αsin a
u1Γu
ρ+
minus=
Γ is the parameter on the path of contact and y is any point between A end E
At the respective ends Γ has the following values
Root piniontip wheel
Cylindrical gears
( )
minus
minusminus= 1
αtan 1dd
zzΓ
tw
2b2a2
1
2A
Bevel gears
( )
minus
minusminus= 1
αtan 1dd
uΓvt
2vb2va2
vA
Tip pinionroot wheel
Cylindrical gears
( )
1αtan
1ddΓ
tw
2b1a1
E minusminus
=
Bevel gears
( )
1αtan
1ddΓ
vt
2vb1va1
E minusminus
=
At inner point of single pair contact
Cylindrical gears
tw1
EB α tan zπ2ΓΓ minus=
Bevel gears
vtv1
EB α tan zπ2ΓΓ minus=
At outer point of single pair contact
Cylindrical gears
tw1AD α tan z
π2ΓΓ +=
Bevel gears
vt v1
AD αtan z π2ΓΓ +=
At pitch point ΓC = 0
The points F and G (only applicable to cylindrical gears) limiting the extension of tip relief (so as to maintain a mini-mum contact ratio of unity for unloaded gears) are at
2
BAF
Γ+Γ=Γ
2
EDG
Γ+Γ=Γ
443 Load sharing factor XΓ The load sharing factor XΓ accounts for the load sharing be-tween the various pairs of teeth in mesh along the path of contact
XΓ is to be calculated stepwise from A to E using the pa-rameter Γy
4431 Cylindrical gears with β = 0 and no tip relief
Figure 41
ByAAB
Ay for31
31X
yΓltΓleΓ
ΓminusΓ
ΓminusΓ+=Γ
DyB for1Xy
ΓltΓleΓ=Γ
EyDDE
yE for31
31X
yΓleΓltΓ
ΓminusΓ
ΓminusΓ+=Γ
4432 Cylindrical gears with β = 0 and tip relief Tip relief on the pinion reduces XΓ in the range G ndash E and increases correspondingly XΓ in the range F ndash B
Tip relief on the wheel reduces XΓ in the range A ndash F and increases correspondingly XΓ in the range D ndash G
Following remains generally
DyB for1Xy
ΓleΓleΓ=Γ
21XXGF== ΓΓ
In the following it must be distinguished between Ca lt Ceff respectively Ca gt Ceff This is shown by an example below where Ca1 lt Ceff and Ca2 gt Ceff
Classification Notes- No 412 37 May 2003
DET NORSKE VERITAS
Figure 42
Note When Ca gt Ceff the path of contact is shortened by A ndash A respectively E ndash E The single pair contact path is extended into B respectively D If this shift is significant it is necessary to consider the negative effect on surface durability (B) and bending stresses (B and D)
Range A - F
For effa2 CC le
minus=Γ
eff
2a
CC1
31X
A
FyAeff
2a
AF
Ay for C3
C61XX
AyΓleΓleΓ
sdot
+ΓminusΓ
ΓminusΓ+= ΓΓ
For effa2 CC ge
AyAfor0Xy
ΓleΓleΓ=Γ
with ( )21
CC
1CC
eff
2a
eff
2a
AFAAminus
minusΓminusΓ+Γ=Γ
FyAeff
2a
AF
Ay
eff
2a for 21
CC
CC1X
yΓleΓleΓ
minus
ΓminusΓ
ΓminusΓ+minus=Γ
Range F - B
For effa1 CC le
ByFeff
1a
FB
Fy forC3
C61
21X
yΓleΓleΓ
sdot
+ΓminusΓ
ΓminusΓ+=Γ
For effa1 CC ge
ByFeff
1a
FB
Fy for21
CC
21X
yΓleΓleΓ
minus
ΓminusΓ
ΓminusΓ+=Γ
ByB for1Xy
ΓleΓleΓ=Γ
with 1
CC2
eff
1a
FBFB
minus
ΓminusΓ+Γ=Γ
Range D ndash G
For effa2 CC le
GyDeff
2a
DG
Dy
eff
2a for C 3C
61
C 3C
32X
yΓleΓleΓ
+
ΓminusΓ
ΓminusΓminus+=Γ F
or effa2 CC ge
DyD for1Xy
ΓleΓleΓ=Γ
with ( )21
CC
1CC
eff
2a
eff
2a
DGDDminus
minusΓminusΓ+Γ=Γ
GyDeff
2a
DG
Dy
eff
2a for21
CC
CCX ΓleΓleΓ
minus
ΓminusΓ
ΓminusΓminus=Γ
Range G - E
For effa1 CC le
minus=Γ
eff
1a
CC1
31X
E
EyGeff
1a
GE
Gy forC3
C61
21X
yΓleΓleΓ
sdot
+ΓminusΓ
ΓminusΓminus=Γ
For effa1 CC gt
EyGeff
1a
GE
Gy for21
CC
21X
yΓleΓleΓ
minus
ΓminusΓ
ΓminusΓminus=Γ
EyE for0Xy
ΓleΓleΓ=Γ
with 1
CC2
eff
1a
GEGE
minus
ΓminusΓ+Γ=Γ
4433 Gears with β gt 0 buttressing Due to oblique contact lines over the flanks a certain but-tressing may occur near A and E
This applies to both cylindrical and bevel gears with tip relief lt Ceff The buttressing Xbutt is simplified as a linear function within the ranges A ndash H respectively I ndash E
Figure 43
1εwhen13X βbutt EAge=
1εwhenε031X ββbutt EAlt+=
38 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
Cylindrical gears
bIEAH βsin 02ΓΓΓΓ =minus=minus
Bevel gears
bmIEAH βsin 02ΓΓΓΓ =minus=minus
4434 Cylindrical gears with 2εγ le and no tip relief
yXΓ is obtained by multiplication of
yXΓ in 4431 with
Xbutt in 4433
4435 Gears with 2εγ gt and no tip relief
Applicable to both cylindrical and bevel gears
IyH for1Xy
ΓleΓleΓε
=α
Γ
IyHybutt and forX1Xy
ΓgtΓΓltΓε
=α
Γ
Figure 44
4436 Cylindrical gears with 2εγ le and tip relief
yXΓ is obtained by multiplication of
yXΓ in 4432 with Xbutt
in 4433
4437 Cylindrical gears with 2εγ gt and tip relief
Tip relief on the pinion (respectively wheel) reduces XГ in the range G ndash E (respectively A ndash F) and increases XГ in the range F ndash G
yXΓ is obtained by multiplication of
yXΓ as described below
with Xbutt in 4433
In the XГ example below the influence of tip relief is shown (without the influence of Xbutt) by means of
eff2aeff1a CC and CC ltgt
Tip relief gt Ceff causes new end points A respectively E of the path of contact
Figure 45
Range A ndash F
( ) ( )( ) eff
a2αa1α
AF
Ay
eff
2aeff
C 12C 13εC 1ε
CCCX
y +εε++minus
ΓminusΓ
ΓminusΓ+
εminus
=ααα
Γ
eff2aFyA CC if for leΓleΓleΓ
eff2aFyA CifCfor and geΓleΓleΓ
eff2aAyA CC iffor0Xy
gtΓleΓleΓ=Γ
( ) ( ) ( )( ) ( ) a2αa1α
αeffa2AFAA C 1ε 3C 1ε
1ε 2 CC with++minus+minus
ΓminusΓ+Γ=Γ
Range F ndash G
( )( )( ) GyF
eff
2a1a forC 1 2CC 11X
yΓleΓleΓ
+εε+minusε
+ε
=αα
α
αΓ
Range G ndash E
( ) ( )( ) eff
2a1a
GE
Gy
C 1 2C 1C 1 3XX
GFy +εεminusε++ε
ΓminusΓ
ΓminusΓminus=
αα
ααΓΓ minus
effa1EyG CC if for leΓleΓleΓ
effa1EyG CC if Γfor and geΓleΓle
eff1aEyE CC if for0Xy
geΓleΓleΓ=Γ
( ) ( ) ( )( ) ( ) 2a1a
eff1aGEEE C 1C 1 3
1 2 CC withminusε++ε+εminus
ΓminusΓminusΓ=Γαα
α
4438 Bevel gears with εγ more than approx 18 and heightwise crowning For Ca1 = Ca2 = Ceff the following applies
Figure 46
Classification Notes- No 412 39 May 2003
DET NORSKE VERITAS
( )AEM 50 Γ+Γ=Γ
( )( )2AD
3
2My651X
y ΓminusΓε
ΓminusΓminus
ε=
ααΓ
For tip relief lt Ceff yXΓ is found by linear interpolation be-
tween ( )effay CCX =Γ and ( )0CayX =Γ as in 4435
The interpolation is to be made stepwise from A to M with the influence of Ca2 and from M to E with the influence of Ca1 (For 2a1a CC ne there is a discontinuity at M)
Eg with Ca1 = 04 Ceff and Ca2 = 055 Ceff then
Range A ndash M
)CC()0C( eff2ay2ayyX 550X 450X =Γ=ΓΓ +=
Range M ndash E
)CC()0C( eff1ay1ayyX 40X 06X =Γ=ΓΓ +=
For tip relief gt Ceff the new end points A and E are found as
( )
minusΓminusΓ
ε+Γ=Γ α 1
CC
6 eff
2aADAA
( )
minusΓminusΓ
ε+Γ=Γ α 1
CC
6 eff
1aADEE
Range A ndash A
0Xy=Γ
Range A ndash M
( )( )
ΓminusΓ
ΓminusΓminus
minusε=
αΓ 2
MA
2My
eff
2a1
CC4
351Xy
Range M ndash E
( )( )
ΓminusΓ
ΓminusΓminus
minusε=
αΓ 2
ME
2My
eff
1a1
CC4
351Xy
Range E ndash E
0Xy=Γ
40 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
Appendix A Fatigue Damage Accumulation
The Palmgren-Miner cumulative damage calculation princi-ple is used The procedure may be applied as follows
A1 Stress Spectrum From the individual torque classes the torques (Ti) at the peak values of class intervals and the associated number of cycles (NLi) for both pinion and wheel are to be listed from the highest to the lowest torque
(In case of a cyclic torque variation within the torque classes it is advised to use the peak torque If the cyclic variation is such that the same teeth will repeatedly suffer the peak torque this is a must)
The stress spectra for tooth roots and flanks (σFi σHi) with all relevant factors (except KA) are to be calculated on the basis of the torque spectrum The load dependent K-factors are to be determined for each torque class
A2 σminusN-curve The stress versus load cycle curves for tooth roots and flanks (both pinion and wheel) are to be drawn on the basis of per-missible stresses (ie including the demanded minimum safety factors) as determined in 2 respectively 3 If different safety levels for high cycle fatigue and low cycle fatigue are desired this may be expressed by different demand safety factors applied at the endurance limit respectively at static strength
A3 Damage accumulation The individual damage ratio Di at ith stress level is defined as
Fi
Lii N
ND =
where
NLi = The number of applied cycles at ith stress
NFi = The number of cycles to failure at ith stress
Basically stresses σi below the permissible stress level for infinite life (if a constant ZN or YN is accepted) do not con-tribute to the damage sum However calculating the actual safety factor Sact as described below all the σi for which the product S σi is bigger than or equal to the permissible stress level for infinite life contribute to the damage sum and thus to the determination of Sact The final value of S is decisive
(NFi can be found mathematically by putting the permissible stress σpi equal the actual stress σi thereby finding the actual life factor This life factor can be solved with regard to load cycles ie NFi)
The damage sum ΣDi is not to exceed unity
If ΣDi ne 1 the safety against cumulative fatigue damage is different from the applied demand safety factor For deter-mination of this theoretical safety factor an iteration proce-dure is required as described in the following flowchart
S is correction factor with which the actual safety factor Sact can be found
Sact is the demand safety factor (used in determination of the permissible stresses in the σ ndash N ndash curve) times the correc-tion factor S
The full procedure is to be applied for pinion and wheel tooth roots and flanks
Note If alternating stresses occur in a spectrum of mainly pulsating stresses the alternating stresses may be replaced by equiva-lent pulsating stresses ie by means of division with the ac-tual mean stress influence factor YM
Classification Notes- No 412 41 May 2003
DET NORSKE VERITAS
Appendix B Application Factors for Diesel Driven Gears
For diesel driven gears the application factor KA depends on torsional vibrations Both normal operation and misfiring conditions have to be considered
Normally these two running conditions can be covered by only one calculation
B1 Definitions
Normal operation KAnorm = 0
normv0
TTT +
where
T0 = rated nominal torque
Tv norm = vibratory torque amplitude for normal opera-tion (see rules Pt4 Ch3 Sec1 G301 for defi-nition of ldquonormalrdquo irregularity)
Misfiring operation KA misf = 0
misfv
TTT +
where
T = remaining nominal torque when one cylinder out of action
Tv misf = vibratory torque amplitude in misfiring con-dition This refers to a permissible misfiring condition ie a condition that does not require automatic or immediate corrective actions as speed or pitch reduction
The normal operation is assumed to last for a very high num-ber of cycles such as 1010
The misfiring operation is assumed to last for a limited dura-tion such as 107 cycles
B2 Determination of decisive load Assuming life factor at 1010 cycles as YN = ZN = 092 which usually is relevant the calculation may be performed only once with the combination having the highest value of appli-cation factorlife factor
For bending stresses and scuffing the higher value of
092
K normA and 098
K misfA
For contact stresses the higher value of
092K normA and
113K misfA (but
097K misfA for nitrided gears)
B3 Simplified procedure Note that this is only a guidance and is not a binding con-vention
T = oTZ
1Zsdot
minus where Z = number of cylinders
( ) ideal Videal Vmisf Vnorm V TTT24ZT +minus=
where TV ideal = vibratory torque with all cylinders perfectly equal See also rules Pt4 Ch3 Sec1 G300
When using trends from torsional vibration analysis and measurements the following may be used
TV ideal To is close to zero for engines with few cylinders and using a suitable elastic coupling and increases with relative coupling stiffness and number of cylinders
TV misf To may be high for engines with few cylinders and decreases with number of cylinders
This can be indicated as
200Z
TT
ο
idealV asymp and 80Z04
TT
o
misfV minusasymp
Inserting this into the formulae for the two application fac-tors the following guidance can be given
118112K misfA minusasymp
115110K normA minusasymp
Since KA norm is to be combined with the lower life factors the decisive load condition will be the normal one and a KA of 115 will cover most relevant cases when a suitable elas-tic coupling is chosen
42 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
Appendix C Calculation of Pinion-Rack
Pinion-racks used for elevating of mobile offshore units are open gears that are subjected to wear and tear With normal specifications such as only a few hundred total operation cycles (site to site) the tooth bending stresses for static strength will be decisive for the lay out however with ex-ception of surface hardened pinions where case crushing has to be considered
In the following the use of part 1 and 3 for pinion-racks is shown including relevant simplifications
C1 Pinion tooth root stresses Since the load spectrum normally is dominated by high torques and few load cycles (in the range up to 1-2000) the static strength is decisive
The actual stress is calculated as
β1FSaFan1
tF1 KYY
mbFσ sdotsdotsdotsdot
=
b1 is limited to b2 + 2middotmn
YFa and YSa replace YF and YS because load application at tooth tip has to be assumed for such inaccurate gears
Pinions often use a non-involute profile in the dedendum part of the flank eg a constant radius equal the radius of curva-ture at reference circle For such pinions sFn and hFa are to be measured directly on a sectional drawing of the pinion tooth
Due to high loads and narrow facewidths it may be assumed that KFβ2 = KHβ = 10 However when b1 gt b2 then KFβ1 gt10
If no detailed documentation of KFβ1 is available the fol-lowing may be used
KFβ1 = 1 + 015middot(b1b2 ndash 1)
The permissible stress (not surface hardened) is calculated as
δrelTstF
Fst1FP1 Y
Sσσ sdot=
The mean stress influence due to leg lifting may be disre-garded
The actual and permissible stresses should be calculated for the relevant loads as given in the rules
C2 Rack tooth root stresses The actual stress is calculated as
SaFan2
tF2 YY
mbFσ sdotsdotsdot
=
See C1 for details
The permissible stress is calculated as
δrelTstF
Fst2FP2 Y
Sσσ sdot=
For alloyed steels (Ni Cr Mo) with high toughness and duc-tility the value of YδrelTst may be put equal to YSa
C3 Surface hardened pinions For surface hardened pinions the maximum load is not to cause crushing of the hardened layer of the flank
In principle the calculation described in 213 may be used but when the theoretical Hertzian stress exceeds the ap-proximately 18 times the yield strength of the rack material plastic deformation will occur This will limit the peak Hertzian stress but increases the contact width and thus the penetration of stresses into the depth
An approximation may be based on an assessment of contact width determined by means of equal areas under the theoreti-cal (elastic range) Hertzian contact and the elasto-plastic contact stress (limited to 18 middot σy) with the unknown width
28 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
310 Relative Notch Sensitivity Factor YδrelT The dynamic (respectively static) relative notch sensitivity factor YδrelT (YδrelTst) indicate to which extent the theoreti-cally concentrated stress lies above the endurance limits (re-spectively static strengths) in the case of fatigue (respectively overload) breakage
YδrelT is a function of the material and the relative stress gra-dient It differs for static strength and endurance limit
The following method may be used
For endurance limit
for not surface hardened fillets
( )
024
s024
relT σ103133q21σ1012201351
Ysdotsdotminus
+sdotsdotminus+= minus
minus
δ
for all surface hardened fillets except nitrided
( )
106q21002451
Y srelT
++=δ
for nitrided fillets
( )
1347q2101421
Y srelT
++=δ
For static strength
for not surface hardened fillets1)
( )( )( )025
02
02502s
relTst σ3000821σ300 1Y0821Y
+
minus+=δ
for surface hardened fillets except nitrided
YδrelTst = 044 YS + 012
for nitrided fillets
YδrelTst = 06 + 02 YS 1) These values are only valid if the local stresses do not
exceed the yield point and thereby alter the residual stress level See also 39b footnote 2
311 Relative Surface Condition Factor YRrelT The relative surface condition factor YRrelT takes into ac-count the dependence of the tooth root strength on the sur-face condition in the tooth root fillet mainly the dependence on the peak to valley surface roughness
YRrelT differs for endurance limit and static strength
The following method may be used
For endurance limit
YRrelT = 1675 ndash 053 (Ry + 1)01 for surface hardened steels and alloyed quenched and tempered steels except nitrided
YRrelT = 53 ndash 42 (Ry + 1)001 for carbon steels
YδrelT = 43 ndash 326 (Ry + 1)0005
for nitrided steels
For static strength
YRrelTst = 1 for all Ry and all materials
For a fillet without any longitudinal machining trace Ry asymp Rz
312 Size Factor YX The size factor YX takes into account the decrease of the strength with increasing size YX differs for endurance limit and static strength
The following may be used
For endurance limit
YX = 1 for mn le 5 generally
YX = 103 ndash 0006 mn for 5 lt mn lt 30
YX = 085 for mn ge 30
for not surface hard-ened steels
YX = 105 ndash 001 mn for 5 lt mn ge 25
YX = 08 for mn ge 25
for surface hardened steels
For static strength
YXst = 1 for all mn and all materials
313 Case Depth Factor YC The case depth factor YC takes into account the influence of hardening depth on tooth root strength
YC applies only to surface hardened tooth roots and is dif-ferent for endurance limit and static strength
In case of insufficient hardening depth fatigue cracks can develop in the transition zone between the hardened layer and the core For static strength yielding shall not occur in the transition zone as this would alter the surface residual stresses and therewith also the fatigue strength
The major parameters are case depth stress gradient permis-sible surface respectively subsurface stresses and subsurface residual stresses
The following simplified method for YC may be used
YC consists of a ratio between permissible subsurface stress (incl influence of expected residual stresses) and permissible surface stress This ratio is multiplied with a bracket con-taining the influence of case depth and stress gradient (The empirical numbers in the bracket are based on a high number of teeth and are somewhat on the safe side for low number of teeth)
YC and YCst may be calculated as given below but calculated values above 10 are to be put equal 10
For endurance limit
+
+=nFFE
C m02ρt31
σconstY
Classification Notes- No 412 29 May 2003
DET NORSKE VERITAS
For static strength
+
+=nFFst
Cst m02ρt31
σconstY
where const and t are connected as
Hardening process
t = endurance limit const =
static strength const =
t550 640 1900
t400 500 1200 Case hard-ening
t300 380 800
Nitriding t400 500 1200
Induction- or flame hardening
tHVmin 11 HVmin 25 HVmin
For symbols see 213
In addition to these requirements to minimum case depths for endurance limit some upper limitations apply to case hard-ened gears
The max depth to 550 HV should not exceed
1) 13 of the top land thickness san unless adequate tip relief is applied (see 110)
2) 025 mn If this is exceeded the following applies addi-tionally in connection with endurance limit
minusminus= 025
mt
1Yn
max550C
314 Thin rim factor YB Where the rim thickness is not sufficient to provide full sup-port for the tooth root the location of a bending failure may be through the gear rim rather than from root fillet to root fillet
YB is not a factor used to convert calculated root stresses at the 30deg tangent to actual stresses in a thin rim tension fillet Actually the compression fillet can be more susceptible to fatigue
YB is a simplified empirical factor used to de-rate thin rim gears (external as well as internal) when no detailed calcula-tion of stresses in both tension and compression fillets are available
Figure 34 Examples on thin rims
YB is applicable in the range 175 lt sRmn lt 35
YB = 115 middot ln (8324 middot mnsR)
(for sRmn ge 35 YB = 1)
(for sRmn le 175 use 315)
σF as calculated in 321 is to be multiplied with YB when sRmn lt 35 Thus YB is used for both high and low cycle fatigue
Note This method is considered to be on the safe side for external gear rims However for internal gear rims without any flange or web stiffeners the method may not be on the safe side and it is advised to check with the method in 315316
315 Stresses in Thin Rims For rim thickness sR lt 35 mn the safety against rim cracking has to be checked
The following method may be used
3151 General The stresses in the standardised 30ordm tangent section tension side are slightly reduced due to decreasing stress correction factor with decreasing relative rim thickness sRmn On the other hand during the complete stress cycle of that fillet a certain amount of compression stresses are also introduced The complete stress range remains approximately constant Therefore the standardised calculation of stresses at the 30ordm tangent may be retained for thin rims as one of the necessary criteria
The maximum stress range for thin rims usually occurs at the 60ordm ndash 80ordm tangents both for laquotensionraquo and laquocompressionraquo side The following method assumes the 75ordm tangent to be the decisive Therefore in addition to the am criterion ap-plied at the 30ordm tangent it is necessary to evaluate the max and min stresses at the 75ordm tangent for both laquotensionraquo (loaded flank) fillet and laquocompressionraquo (back-flank) fillet For this purpose the whole stress cycle of each fillet should be considered but usually the following simplification is justified
Figure 35 Nomenclature of fillets
Index laquoTraquo means laquotensileraquo fillet laquoCraquo means laquocompressionraquo fillet
σFTmin and σFCmax are determined on basis of the nominal rim stresses times the stress concentration factor Y75
σFCmin and σFTmax are determined on basis of superposition of nominal rim stresses times Y75 plus the tooth bending stresses at 75ordm tangent
30 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
3152 Stress concentration factors at the 75ordm tangents The nominal rim stress consists of bending stresses due to local bending moments tangential stresses due to the tan-gential force Ft and radial shear stresses due to Fr
The major influence is given by the bending stresses The influence of the tangential stresses is minor and even though its stress concentration factor is slightly higher than for bending it is considered to be safe enough when the sum of these nominal stresses are combined with the stress concen-tration factor for bending The influence of the radial shear stress is neglected
The stress concentration factor relating nominal rim stresses to local fillet stresses at the 75ordm tangent may be calculated as
75n
R75
n
R
75
ρmsρ185
ms3
Y+
sdot=
where ρ75 is the root radius at the 75ordm tangent ref to mn Usually ρ75 is closed to the tool radius ρao and
ρ75 = ρao
is a safe approximation compensating for the am simplifi-cations to the laquounsaferaquo side
The tooth root stresses of the loaded tooth are decreasing with decreasing relative rim thickness approximately with the empirical factor
15
R
ncorr s
m311Y
minus=
3153 Nominal rim stresses The bending moment applied to the rim consists of a part of the tooth tilt moment Ft (hF + 05 sR) and the bending caused by the radial force Fr
The sectional modulus (first moment of area) which is used for determination of the nominal bending stresses is not nec-essarily the same for the 2 am bending moments If flanges webs etc outside the toothed section contribute to stiffening the rim against various deflections the influence of these stiffeners should be considered Eg an end flange will have an almost negligible influence on the effective sectional modulus for the stresses due to tooth tilt as the deflection caused by the tilt moment is rather small and would not much involve the flange On the other hand the radial forces as for instance from the meshes in an annulus would cause considerable radial deflections that the flange might restrict to a substantial amount When considering the stiff-ening of such flanges or webs on basis of simplified models it is advised to use an effective rim thickness sR = sR + 02 mn for the first moment of area of the rim (toothed part) cross section
For a high number of rim teeth it may be assumed that the rim bending moments in the fillets adjacent to the loaded tooth are of the same magnitude as right under the applied
force This assumption is reasonable for an annulus but rather much on the safe side for a hollow pinion
The influence of Ft on the nominal tangential stress is simpli-fied by half of it for compressive stresses (σ1) and the other half for tensile stresses (σ2) Applying these assumptions the nominal rim stresses adjacent to the loaded tooth are
( ) ( )A2
FW
f RFW
s05hF05 t
R
r
T
RFt1 minus
ϑminus
+minus=σ
( ) ( )A2
FW
f RFW
s05hF05 t
R
r
T
RFt2 +
ϑminus
+=σ
where σ1 σ2 see Fig 35
A = minimum area of cross section (usually bR sR)
WT = the sectional modulus of rim with respect to tooth tilt moment (usually 6sb 2
rR )
WR = the sectional modulus of the rim including the influence of stiffeners as flanges webs etc (WR ge WT)
bR = the width of the rim
R = the radius of the neutral axis in the rim ie from wheel centre to midpoint of rim
( )ϑf = a function for bending moment distribution around the rim For a rim (pinion) with one mesh only the ( )ϑf at the position of load application is 024 For an annulus with 3 or more meshes ( )ϑf at the position of each load application is approx 3 planets ( )ϑf = 019 4 planets ( )ϑf = 014 5 planets ( )ϑf = 011 6 planets ( )ϑf = 009
It must be checked if the max (tensile) stress in the compres-sion fillet really occurs when the fillet is adjacent to the loaded tooth In principle the stress variation through a complete rotation should be considered and the max value used The max value is usually never less than 0 For an annulus eg the tilt moment is zero in the mid position be-tween the planet meshes whilst the bending moment due to the radial forces is half of that at the mesh but with opposite sign
If these formulae are applied to idler gears as eg planets the influence of nominal tangential stresses must be cor-rected by deleting Ft(2 A) for σ1 and using FtA for σ2 Further the influence of Fr on the nominal bending stresses is usually negligible due to the planet bearing support
3154 Root fillet stresses Determination of min and max stresses in the laquotensionraquo fillet
Classification Notes- No 412 31 May 2003
DET NORSKE VERITAS
Minimum stress
751FTmin YσK σ =
Maximum stress
corrF752 Yσ03Yσ KσFTmax +=
where
03 is an empirical factor relating the tension stresses (σF) at the 30ordm tangent to the part of the tension stresses at the 75ordm tangent which add to the rim related stresses (03 also takes into account that full superposition of nominal stresses times stress concentration factors from both laquosides of the corner filletraquo would result in too high stresses)
αβγ sdotsdotsdotsdot= FFvA KKKKKK
Determination of min and max stresses in the laquocompres-sionraquo fillet
Minimum stress
corrF751FCmin Yσ036Yσ Kσ minus=
Maximum stress
752FCmax Yσ Kσ =
where
036 is an empirical factor relating the tension stresses (σF) at the 30ordm tangent to the part of the compression stresses at the 75ordm tangent which add to the rim related stresses
For gears with reversed loads as idler gears and planets there is no distinct laquotensionraquo or laquocompressionraquo fillet The mini-mum stress σFmin is the minimum of σFTmin and σFCmin (usually the latter is decisive) The maximum stress σFTmax is the maximum of σFTmax and σFCmax (usually the former is deci-sive)
316 Permissible Stresses in Thin Rims
3161 General The safety against fatigue fracture respectively overload fracture is to be at least at the same level as for solid gears The laquoordinaryraquo criteria at the 30ordm tangent apply as given in 31 through 313
Additionally the following criteria at the 75ordm tangent may apply
3162 For gt3middot106 cycles The permissible stresses for the laquotensionraquo fillets and for the laquocompressionraquo fillets are determined by means of a relevant fatigue diagram
If the actual tooth root stress (tensile or compressive) ex-ceeds the yield strength to the material the induced residual stresses are to be taken into account
For determination of permissible stresses the following is defined
R = stress ratio ie FTmax
FTminσσ respectively
FCmax
FCmin
σσ
∆σ = stress range ie σFTmax ndash σFTmin resp σFCmax ndash σFCmin
(For idler gears and planets Fmax
FminσσR =
and ∆σ = σFmax minus σFmin)
The permissible stress range ∆σρ for the laquotensionraquo respec-tively laquocompressionraquo fillets can be calculated as
For R gt minus1 FPp σ
R1R1031
13∆σ
minus+
+=
For minus infin lt R lt minus1 FPp σ
R1R10151
13∆σ
minus+
+=
where
σFP = see 32 determined for unidirectional stresses (YM = 1)
If the yield strength σy is exceeded in either tension or com-pression residual stresses are induced This may be consid-ered by correcting the stress ratio R for the respective fillets (tension or compression)
Eg if yFCmin σσ gt (ie exceeded in compression) the
difference yFCmin σσ∆ minus= affects the stress ratio as
∆σ
σ∆σ∆σR
FCmax
y
FCmax
FCmin
+
minus=
++
=
Similarly the stress ratio in the tension fillet may require cor-rection
If the yield strength is exceeded in tension σFTmax gt σy the difference ∆ = σFTmax ndash σy affects the stress ratio as
y
FTmin
FTmax
FTmin
σ∆σ
∆σ∆σR minus=
minusminus
=
Checking for possible exceeding of the yield strength has to be made with the highest torque and with the KAP if this ex-ceeds KA
3163 Forle 103 cycles The permissible stress range ∆σp is not to exceed
For R gt minus1 FPstpst σ
R1R1051
15∆σ
minus+
+=
For ndash infin lt R lt minus1 FPstpst σ
R1R10251
15∆σ
minus+
+=
For all values of R ∆σpst is limited by
not surface hardened F
y
Sσ225
32 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
surface hardened CF
YSHV5
Definition of ∆σ and R see 3162 with particular attention to possible correction of R if the yield strength is exceeded
σFPst see 32 determined for unidirectional stresses (YM = 1) and lt 103 cycles
3164 For 103 lt cycles lt 3106 ∆σp is to be determined by linear interpolation a log-log dia-gram
∆σp at NL load cycles is
6L 103p
exp
L
6
N p ∆σN103∆σ sdot
sdot=
6
3
103p
10p
∆σ
∆σlog28760exp
sdot
=
Classification Notes- No 412 33 May 2003
DET NORSKE VERITAS
4 Calculation of Scuffing Load Capacity
41 Introduction High surface temperatures due to high loads and sliding ve-locities can cause lubricant films to break down This sei-zure or welding together of areas of tooth surface is termed scuffing
In contrast to pitting and fatigue breakage which show a dis-tinct incubation period a single short overloading can lead to a scuffing failure In the ISO-TR13989 two criteria are mentioned The method used in this Classification Note is based on the principles of the flash temperature criterion
Note Bulk temperature in excess of 120ordmC for long periods may have an adverse effect on the surface durability see 211
42 General Criteria In no point along the path of contact the local contact tem-perature may exceed the permissible temperature ie
oilS
oilSB S
ϑ+ϑminusϑ
leϑ
50SB minusϑleϑ
where
Bϑ = max contact temperature along the path of contact
maxflaMBB ϑ+ϑ=ϑ
MBϑ = bulk temperature see 434
maxflaϑ = max flash temperature along the path of con-tact see 44
Sϑ = scuffing temperature see below
oilϑ = oil temperature before it reaches the mesh (max applicable for the actual load case to be used ie normally alarm temperature except for ice classes where a max expected tem-perature applies)
SS = required safety factor according to the Rules
The scuffing temperature Sϑ may be calculated as
L2
wrelT
002
40S XFZGX
ν100112085780
sdot
sdot++=ϑ
where
XwrelT = relative welding factor
XwrelT
Through hardened steel 10
Phosphated steel 125
Copper-plated steel 150
Nitrided steel 150
Less than 10 retained austenite
115
10 ndash 20 retained austenite 10
Casehardened steel
20 ndash 30 retained austenite 085
Austenitic steel 045
FZG = load stage according to FZG-Test A8390 (Note This is the load stage where scuffing oc-curs However due to scatter in test results cal-culations are to be made with one load stage less than the specification)
XL = lubricant factor
= 10 for mineral oils
= 08 for polyalfaolefins
= 07 for non-water-soluble polyglycols
= 06 for water-soluble polyglycols
= 15 for traction fluids
= 13 for phosphate esters
ν40 = kinematic oil viscosity at 40˚C (mm2s)
Application of other test methods such as the Ryder the FZG-Ryder R46574 and the FZG L-42 Test 141195110 may be specially considered
For high speed gears with very short time of contact Sϑ may be increased as follows provided use of EP-oils
Addition to the calculated scuffing temperature Sϑ
If micros 18tc ge no addition
If micros 18tc lt add ( )cwrelT t18X18 minussdotsdot
where
ct = contact time ( micros ) which is the time needed to cross the Hertzian contact width
( ) [ ]microsu1cosβn
uσ340tb1
Hc +sdotsdot
sdot=
σH as calculated in 221
For bevel gears use vu in stead of u
34 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
43 Influence Factors
431 Coefficient of friction The following coefficient of friction may apply
L025a
005oil
02
redC ΣC
Bt X R ηρv
w0048micro minus
=
where
wBt = specific tooth load (Nmm)
vΣC = sum of tangential velocities at pitch point At pitch line velocities gt 50 ms the limiting value of vΣC at v = 50 ms is to be used
ρredC = relative radius of curvature (transversal plane) at the pitch point
Cylindrical gears
HαHβvγAbt
Bt KKKKKb
Fw sdotsdotsdotsdotsdot= (see 1)
twΣC αsin v2v =
bCredC β cos ρρ =
Bevel gears
HαHβvγAtmb
Bt KKKKKb
Fw sdotsdotsdotsdotsdot= (see 1)
vtmtΣC αsin v2v =
bmvCredC β cos ρρ =
ηoil = dynamic viscosity (mPa s) at oilϑ calculated as
1000
ρ νη oiloil =
where ρ in kgm3 approximated as
( ) 70 15oil15 minusϑminusρ=ρ and νoil is kinematic viscosity at oilϑ and may be calculated by means of the following equation
( ) ( )++=+ 08νloglog08νloglog 100oil ( )
sdotminus
ϑ+minus313log373log
273log373log oil
( ) ( )( )08νloglog08νloglog 10040 +minus+
Ra = composite arithmetic mean roughness (micron) of pinion an wheel calculated as ( )2a1aa RR 05R +=This is defined as the roughness on the new flanks ie as manufactured
XL = see 42
432 Effective tip relief Ceff Ceff is the effective tip relief that amount of tip relief which compensates for the elastic deformation of the gear mesh ie
zero load at the tooth tip It is assumed (simplified) to be equal for both pinion and wheel
Cylindrical gears
for helical
γ
γ=cb
KKFC Abt
eff (see 1)
For spur
cbKKF
C Abteff
γ= (see 1)
Alternatively for spur and helical gears the non-linear ap-proach in 111 may be used (taking Ceff=δ)
Bevel gears
bcKFC Ambt
effγ
= (see 1)
where
γ
α
ε2ε44c
+sdot
=γ
433 Tip relief and extension Cylindrical gears
The extension of the tip relief is not to result in an effective contact ratio 1ltεα when the gear is unloaded (exceptions to this may only apply for applications where the gear is not to run at light loads) This means that the unrelieved part of the path of contact is to be minimum pbt It is further assumed that this unrelieved part is placed centrally on the path of contact
If root relief applies it has to be calculated as equivalent tip relief Ie pinion root relief (at mesh position A) is added to Ca2 and wheel root relief (at mesh position E) is added to Ca1 If no design tip relief or root relief on the mating gear is specified (ie if Ca1+Croot2 = 0 and visa versa) use the run-ning in amount see 112
Bevel gears
Bevel gears are to have heightwise crowning ie no distinct relievedunrelieved area This may be treated as tip and root relief For calculation purposes the root relief is combined with the tip relief of the mating member into an equivalent tip relief If no resulting tip relieves are specified the equivalent tip relives may be calculated as an approxima-tion based on the tool crowning Ca tool (per mille of tool module m0) as follows
2root1aeq1a CCC +=
1root2aeq2a CCC +=
where
Classification Notes- No 412 35 May 2003
DET NORSKE VERITAS
2
0
n0101atoolal m
)m2(mA1middotmCC
minus+minussdot=
( )
sdotminusminussdotsdotminus+= vtvb1
2vb1
21vavt1n1 αtan dddαsin 05x1mA
2
0
n020atool22a m
)mm(2A1middotmCC
minus+minussdot=
( )
sdotminusminussdotsdotminus+= vtvb2
2vb2
2va2vt2n2 αtan dddαsin 05x1mA
2
0
20atool1root1 m
A1mCC
minussdotsdot=
2
0
10atool2root2 m
A1mCC
minussdotsdot=
If the pressure angles of the cutter blades are modified (and verified) to balance the tip relieves the following may be assumed
( )eq2aeq1aeq2aeq1a C andC calculated of sum50CC sdot==
Throughout the following Ca1 and Ca2 mean the equivalent tip relieves Ca1eq and Ca2eq
434 Bulk temperature The bulk temperature may be calculated as
flaaveragempsoilMB X X 50 ϑ+ϑ=ϑ
where
Xs = lubrication factor
= 12 for spray lubrication
= 10 for dip lubrication (provided both pinion and wheel are dip lubricated and tip speed lt 5 ms)
= 10 for spray lubrication with additional cooling spray (spray on both pinion and wheel or spray on pinion and dip lubrication of wheel)
= 02 for meshes fully submerged in oil
Xmp = contact factor ( )pmp nX += 150
np = number of mesh contact on the pinion (for small gear ratii the number of wheel meshes should be used if higher)
flaaverageϑ
= average of the integrated flash temperature (see 44) along the path of contact
( )
AE
E
Ayyyfla
flaaverage
d
ΓminusΓ
ΓΓϑ=ϑint =
For high speed gears (v gt 50 ms) it may be necessary to assess the bulk temperature on the basis of a thermal rating of the entire gear transmission
44 The Flash Temperature flaϑ
441 Basic formula The local flash temperature flaϑ may be calculated as
( ) 41redy
y2ly
211
43Btcorrfla
unXwX3250
y ρ
ρminusρ
micro=ϑ Γ
(For bevel gears replace u with uv)
and is to be calculated stepwise along the path of contact from A to E
where
micro = coefficient of friction see 431
Xcorr = correction factor taking empirically into account the increased scuffing risk in the beginning of the approach path due to mesh starting without any previously built up oil film and possible shuffling away oil before meshing if insufficient tip relief
( )
3
AD
yaeffcorr 50
CC1X
ΓminusΓε
Γminus+=
α
Ca = tip relief of driven member Xcorr is only applicable in the approach path and if Ca lt Ceff otherwise 10
wBt = unit load see 431
yXΓ = load sharing factor see 443
n1 = pinion rpm
Γy ρly etc see 442
442 Geometrical relations The various radii of flank curvature (transversal plane) are
ρ1y = pinion flank radius at mesh point y
ρ2y = wheel flank radius at mesh point y
ρredy = equivalent radius of curvature at mesh point y
y2y1
y2y1redy
ρ+ρ
ρρ=ρ
Cylindrical gears
twy
1y αsin au1Γ1
ρ+
+=
twy
2y αsin au1Γu
ρ+
minus=
Note that for internal gears a and u are negative
36 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
Bevel gears
vtvv
y1y αsin a
u1Γ1
ρ+
+=
vtvv
yv2y αsin a
u1Γu
ρ+
minus=
Γ is the parameter on the path of contact and y is any point between A end E
At the respective ends Γ has the following values
Root piniontip wheel
Cylindrical gears
( )
minus
minusminus= 1
αtan 1dd
zzΓ
tw
2b2a2
1
2A
Bevel gears
( )
minus
minusminus= 1
αtan 1dd
uΓvt
2vb2va2
vA
Tip pinionroot wheel
Cylindrical gears
( )
1αtan
1ddΓ
tw
2b1a1
E minusminus
=
Bevel gears
( )
1αtan
1ddΓ
vt
2vb1va1
E minusminus
=
At inner point of single pair contact
Cylindrical gears
tw1
EB α tan zπ2ΓΓ minus=
Bevel gears
vtv1
EB α tan zπ2ΓΓ minus=
At outer point of single pair contact
Cylindrical gears
tw1AD α tan z
π2ΓΓ +=
Bevel gears
vt v1
AD αtan z π2ΓΓ +=
At pitch point ΓC = 0
The points F and G (only applicable to cylindrical gears) limiting the extension of tip relief (so as to maintain a mini-mum contact ratio of unity for unloaded gears) are at
2
BAF
Γ+Γ=Γ
2
EDG
Γ+Γ=Γ
443 Load sharing factor XΓ The load sharing factor XΓ accounts for the load sharing be-tween the various pairs of teeth in mesh along the path of contact
XΓ is to be calculated stepwise from A to E using the pa-rameter Γy
4431 Cylindrical gears with β = 0 and no tip relief
Figure 41
ByAAB
Ay for31
31X
yΓltΓleΓ
ΓminusΓ
ΓminusΓ+=Γ
DyB for1Xy
ΓltΓleΓ=Γ
EyDDE
yE for31
31X
yΓleΓltΓ
ΓminusΓ
ΓminusΓ+=Γ
4432 Cylindrical gears with β = 0 and tip relief Tip relief on the pinion reduces XΓ in the range G ndash E and increases correspondingly XΓ in the range F ndash B
Tip relief on the wheel reduces XΓ in the range A ndash F and increases correspondingly XΓ in the range D ndash G
Following remains generally
DyB for1Xy
ΓleΓleΓ=Γ
21XXGF== ΓΓ
In the following it must be distinguished between Ca lt Ceff respectively Ca gt Ceff This is shown by an example below where Ca1 lt Ceff and Ca2 gt Ceff
Classification Notes- No 412 37 May 2003
DET NORSKE VERITAS
Figure 42
Note When Ca gt Ceff the path of contact is shortened by A ndash A respectively E ndash E The single pair contact path is extended into B respectively D If this shift is significant it is necessary to consider the negative effect on surface durability (B) and bending stresses (B and D)
Range A - F
For effa2 CC le
minus=Γ
eff
2a
CC1
31X
A
FyAeff
2a
AF
Ay for C3
C61XX
AyΓleΓleΓ
sdot
+ΓminusΓ
ΓminusΓ+= ΓΓ
For effa2 CC ge
AyAfor0Xy
ΓleΓleΓ=Γ
with ( )21
CC
1CC
eff
2a
eff
2a
AFAAminus
minusΓminusΓ+Γ=Γ
FyAeff
2a
AF
Ay
eff
2a for 21
CC
CC1X
yΓleΓleΓ
minus
ΓminusΓ
ΓminusΓ+minus=Γ
Range F - B
For effa1 CC le
ByFeff
1a
FB
Fy forC3
C61
21X
yΓleΓleΓ
sdot
+ΓminusΓ
ΓminusΓ+=Γ
For effa1 CC ge
ByFeff
1a
FB
Fy for21
CC
21X
yΓleΓleΓ
minus
ΓminusΓ
ΓminusΓ+=Γ
ByB for1Xy
ΓleΓleΓ=Γ
with 1
CC2
eff
1a
FBFB
minus
ΓminusΓ+Γ=Γ
Range D ndash G
For effa2 CC le
GyDeff
2a
DG
Dy
eff
2a for C 3C
61
C 3C
32X
yΓleΓleΓ
+
ΓminusΓ
ΓminusΓminus+=Γ F
or effa2 CC ge
DyD for1Xy
ΓleΓleΓ=Γ
with ( )21
CC
1CC
eff
2a
eff
2a
DGDDminus
minusΓminusΓ+Γ=Γ
GyDeff
2a
DG
Dy
eff
2a for21
CC
CCX ΓleΓleΓ
minus
ΓminusΓ
ΓminusΓminus=Γ
Range G - E
For effa1 CC le
minus=Γ
eff
1a
CC1
31X
E
EyGeff
1a
GE
Gy forC3
C61
21X
yΓleΓleΓ
sdot
+ΓminusΓ
ΓminusΓminus=Γ
For effa1 CC gt
EyGeff
1a
GE
Gy for21
CC
21X
yΓleΓleΓ
minus
ΓminusΓ
ΓminusΓminus=Γ
EyE for0Xy
ΓleΓleΓ=Γ
with 1
CC2
eff
1a
GEGE
minus
ΓminusΓ+Γ=Γ
4433 Gears with β gt 0 buttressing Due to oblique contact lines over the flanks a certain but-tressing may occur near A and E
This applies to both cylindrical and bevel gears with tip relief lt Ceff The buttressing Xbutt is simplified as a linear function within the ranges A ndash H respectively I ndash E
Figure 43
1εwhen13X βbutt EAge=
1εwhenε031X ββbutt EAlt+=
38 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
Cylindrical gears
bIEAH βsin 02ΓΓΓΓ =minus=minus
Bevel gears
bmIEAH βsin 02ΓΓΓΓ =minus=minus
4434 Cylindrical gears with 2εγ le and no tip relief
yXΓ is obtained by multiplication of
yXΓ in 4431 with
Xbutt in 4433
4435 Gears with 2εγ gt and no tip relief
Applicable to both cylindrical and bevel gears
IyH for1Xy
ΓleΓleΓε
=α
Γ
IyHybutt and forX1Xy
ΓgtΓΓltΓε
=α
Γ
Figure 44
4436 Cylindrical gears with 2εγ le and tip relief
yXΓ is obtained by multiplication of
yXΓ in 4432 with Xbutt
in 4433
4437 Cylindrical gears with 2εγ gt and tip relief
Tip relief on the pinion (respectively wheel) reduces XГ in the range G ndash E (respectively A ndash F) and increases XГ in the range F ndash G
yXΓ is obtained by multiplication of
yXΓ as described below
with Xbutt in 4433
In the XГ example below the influence of tip relief is shown (without the influence of Xbutt) by means of
eff2aeff1a CC and CC ltgt
Tip relief gt Ceff causes new end points A respectively E of the path of contact
Figure 45
Range A ndash F
( ) ( )( ) eff
a2αa1α
AF
Ay
eff
2aeff
C 12C 13εC 1ε
CCCX
y +εε++minus
ΓminusΓ
ΓminusΓ+
εminus
=ααα
Γ
eff2aFyA CC if for leΓleΓleΓ
eff2aFyA CifCfor and geΓleΓleΓ
eff2aAyA CC iffor0Xy
gtΓleΓleΓ=Γ
( ) ( ) ( )( ) ( ) a2αa1α
αeffa2AFAA C 1ε 3C 1ε
1ε 2 CC with++minus+minus
ΓminusΓ+Γ=Γ
Range F ndash G
( )( )( ) GyF
eff
2a1a forC 1 2CC 11X
yΓleΓleΓ
+εε+minusε
+ε
=αα
α
αΓ
Range G ndash E
( ) ( )( ) eff
2a1a
GE
Gy
C 1 2C 1C 1 3XX
GFy +εεminusε++ε
ΓminusΓ
ΓminusΓminus=
αα
ααΓΓ minus
effa1EyG CC if for leΓleΓleΓ
effa1EyG CC if Γfor and geΓleΓle
eff1aEyE CC if for0Xy
geΓleΓleΓ=Γ
( ) ( ) ( )( ) ( ) 2a1a
eff1aGEEE C 1C 1 3
1 2 CC withminusε++ε+εminus
ΓminusΓminusΓ=Γαα
α
4438 Bevel gears with εγ more than approx 18 and heightwise crowning For Ca1 = Ca2 = Ceff the following applies
Figure 46
Classification Notes- No 412 39 May 2003
DET NORSKE VERITAS
( )AEM 50 Γ+Γ=Γ
( )( )2AD
3
2My651X
y ΓminusΓε
ΓminusΓminus
ε=
ααΓ
For tip relief lt Ceff yXΓ is found by linear interpolation be-
tween ( )effay CCX =Γ and ( )0CayX =Γ as in 4435
The interpolation is to be made stepwise from A to M with the influence of Ca2 and from M to E with the influence of Ca1 (For 2a1a CC ne there is a discontinuity at M)
Eg with Ca1 = 04 Ceff and Ca2 = 055 Ceff then
Range A ndash M
)CC()0C( eff2ay2ayyX 550X 450X =Γ=ΓΓ +=
Range M ndash E
)CC()0C( eff1ay1ayyX 40X 06X =Γ=ΓΓ +=
For tip relief gt Ceff the new end points A and E are found as
( )
minusΓminusΓ
ε+Γ=Γ α 1
CC
6 eff
2aADAA
( )
minusΓminusΓ
ε+Γ=Γ α 1
CC
6 eff
1aADEE
Range A ndash A
0Xy=Γ
Range A ndash M
( )( )
ΓminusΓ
ΓminusΓminus
minusε=
αΓ 2
MA
2My
eff
2a1
CC4
351Xy
Range M ndash E
( )( )
ΓminusΓ
ΓminusΓminus
minusε=
αΓ 2
ME
2My
eff
1a1
CC4
351Xy
Range E ndash E
0Xy=Γ
40 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
Appendix A Fatigue Damage Accumulation
The Palmgren-Miner cumulative damage calculation princi-ple is used The procedure may be applied as follows
A1 Stress Spectrum From the individual torque classes the torques (Ti) at the peak values of class intervals and the associated number of cycles (NLi) for both pinion and wheel are to be listed from the highest to the lowest torque
(In case of a cyclic torque variation within the torque classes it is advised to use the peak torque If the cyclic variation is such that the same teeth will repeatedly suffer the peak torque this is a must)
The stress spectra for tooth roots and flanks (σFi σHi) with all relevant factors (except KA) are to be calculated on the basis of the torque spectrum The load dependent K-factors are to be determined for each torque class
A2 σminusN-curve The stress versus load cycle curves for tooth roots and flanks (both pinion and wheel) are to be drawn on the basis of per-missible stresses (ie including the demanded minimum safety factors) as determined in 2 respectively 3 If different safety levels for high cycle fatigue and low cycle fatigue are desired this may be expressed by different demand safety factors applied at the endurance limit respectively at static strength
A3 Damage accumulation The individual damage ratio Di at ith stress level is defined as
Fi
Lii N
ND =
where
NLi = The number of applied cycles at ith stress
NFi = The number of cycles to failure at ith stress
Basically stresses σi below the permissible stress level for infinite life (if a constant ZN or YN is accepted) do not con-tribute to the damage sum However calculating the actual safety factor Sact as described below all the σi for which the product S σi is bigger than or equal to the permissible stress level for infinite life contribute to the damage sum and thus to the determination of Sact The final value of S is decisive
(NFi can be found mathematically by putting the permissible stress σpi equal the actual stress σi thereby finding the actual life factor This life factor can be solved with regard to load cycles ie NFi)
The damage sum ΣDi is not to exceed unity
If ΣDi ne 1 the safety against cumulative fatigue damage is different from the applied demand safety factor For deter-mination of this theoretical safety factor an iteration proce-dure is required as described in the following flowchart
S is correction factor with which the actual safety factor Sact can be found
Sact is the demand safety factor (used in determination of the permissible stresses in the σ ndash N ndash curve) times the correc-tion factor S
The full procedure is to be applied for pinion and wheel tooth roots and flanks
Note If alternating stresses occur in a spectrum of mainly pulsating stresses the alternating stresses may be replaced by equiva-lent pulsating stresses ie by means of division with the ac-tual mean stress influence factor YM
Classification Notes- No 412 41 May 2003
DET NORSKE VERITAS
Appendix B Application Factors for Diesel Driven Gears
For diesel driven gears the application factor KA depends on torsional vibrations Both normal operation and misfiring conditions have to be considered
Normally these two running conditions can be covered by only one calculation
B1 Definitions
Normal operation KAnorm = 0
normv0
TTT +
where
T0 = rated nominal torque
Tv norm = vibratory torque amplitude for normal opera-tion (see rules Pt4 Ch3 Sec1 G301 for defi-nition of ldquonormalrdquo irregularity)
Misfiring operation KA misf = 0
misfv
TTT +
where
T = remaining nominal torque when one cylinder out of action
Tv misf = vibratory torque amplitude in misfiring con-dition This refers to a permissible misfiring condition ie a condition that does not require automatic or immediate corrective actions as speed or pitch reduction
The normal operation is assumed to last for a very high num-ber of cycles such as 1010
The misfiring operation is assumed to last for a limited dura-tion such as 107 cycles
B2 Determination of decisive load Assuming life factor at 1010 cycles as YN = ZN = 092 which usually is relevant the calculation may be performed only once with the combination having the highest value of appli-cation factorlife factor
For bending stresses and scuffing the higher value of
092
K normA and 098
K misfA
For contact stresses the higher value of
092K normA and
113K misfA (but
097K misfA for nitrided gears)
B3 Simplified procedure Note that this is only a guidance and is not a binding con-vention
T = oTZ
1Zsdot
minus where Z = number of cylinders
( ) ideal Videal Vmisf Vnorm V TTT24ZT +minus=
where TV ideal = vibratory torque with all cylinders perfectly equal See also rules Pt4 Ch3 Sec1 G300
When using trends from torsional vibration analysis and measurements the following may be used
TV ideal To is close to zero for engines with few cylinders and using a suitable elastic coupling and increases with relative coupling stiffness and number of cylinders
TV misf To may be high for engines with few cylinders and decreases with number of cylinders
This can be indicated as
200Z
TT
ο
idealV asymp and 80Z04
TT
o
misfV minusasymp
Inserting this into the formulae for the two application fac-tors the following guidance can be given
118112K misfA minusasymp
115110K normA minusasymp
Since KA norm is to be combined with the lower life factors the decisive load condition will be the normal one and a KA of 115 will cover most relevant cases when a suitable elas-tic coupling is chosen
42 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
Appendix C Calculation of Pinion-Rack
Pinion-racks used for elevating of mobile offshore units are open gears that are subjected to wear and tear With normal specifications such as only a few hundred total operation cycles (site to site) the tooth bending stresses for static strength will be decisive for the lay out however with ex-ception of surface hardened pinions where case crushing has to be considered
In the following the use of part 1 and 3 for pinion-racks is shown including relevant simplifications
C1 Pinion tooth root stresses Since the load spectrum normally is dominated by high torques and few load cycles (in the range up to 1-2000) the static strength is decisive
The actual stress is calculated as
β1FSaFan1
tF1 KYY
mbFσ sdotsdotsdotsdot
=
b1 is limited to b2 + 2middotmn
YFa and YSa replace YF and YS because load application at tooth tip has to be assumed for such inaccurate gears
Pinions often use a non-involute profile in the dedendum part of the flank eg a constant radius equal the radius of curva-ture at reference circle For such pinions sFn and hFa are to be measured directly on a sectional drawing of the pinion tooth
Due to high loads and narrow facewidths it may be assumed that KFβ2 = KHβ = 10 However when b1 gt b2 then KFβ1 gt10
If no detailed documentation of KFβ1 is available the fol-lowing may be used
KFβ1 = 1 + 015middot(b1b2 ndash 1)
The permissible stress (not surface hardened) is calculated as
δrelTstF
Fst1FP1 Y
Sσσ sdot=
The mean stress influence due to leg lifting may be disre-garded
The actual and permissible stresses should be calculated for the relevant loads as given in the rules
C2 Rack tooth root stresses The actual stress is calculated as
SaFan2
tF2 YY
mbFσ sdotsdotsdot
=
See C1 for details
The permissible stress is calculated as
δrelTstF
Fst2FP2 Y
Sσσ sdot=
For alloyed steels (Ni Cr Mo) with high toughness and duc-tility the value of YδrelTst may be put equal to YSa
C3 Surface hardened pinions For surface hardened pinions the maximum load is not to cause crushing of the hardened layer of the flank
In principle the calculation described in 213 may be used but when the theoretical Hertzian stress exceeds the ap-proximately 18 times the yield strength of the rack material plastic deformation will occur This will limit the peak Hertzian stress but increases the contact width and thus the penetration of stresses into the depth
An approximation may be based on an assessment of contact width determined by means of equal areas under the theoreti-cal (elastic range) Hertzian contact and the elasto-plastic contact stress (limited to 18 middot σy) with the unknown width
Classification Notes- No 412 29 May 2003
DET NORSKE VERITAS
For static strength
+
+=nFFst
Cst m02ρt31
σconstY
where const and t are connected as
Hardening process
t = endurance limit const =
static strength const =
t550 640 1900
t400 500 1200 Case hard-ening
t300 380 800
Nitriding t400 500 1200
Induction- or flame hardening
tHVmin 11 HVmin 25 HVmin
For symbols see 213
In addition to these requirements to minimum case depths for endurance limit some upper limitations apply to case hard-ened gears
The max depth to 550 HV should not exceed
1) 13 of the top land thickness san unless adequate tip relief is applied (see 110)
2) 025 mn If this is exceeded the following applies addi-tionally in connection with endurance limit
minusminus= 025
mt
1Yn
max550C
314 Thin rim factor YB Where the rim thickness is not sufficient to provide full sup-port for the tooth root the location of a bending failure may be through the gear rim rather than from root fillet to root fillet
YB is not a factor used to convert calculated root stresses at the 30deg tangent to actual stresses in a thin rim tension fillet Actually the compression fillet can be more susceptible to fatigue
YB is a simplified empirical factor used to de-rate thin rim gears (external as well as internal) when no detailed calcula-tion of stresses in both tension and compression fillets are available
Figure 34 Examples on thin rims
YB is applicable in the range 175 lt sRmn lt 35
YB = 115 middot ln (8324 middot mnsR)
(for sRmn ge 35 YB = 1)
(for sRmn le 175 use 315)
σF as calculated in 321 is to be multiplied with YB when sRmn lt 35 Thus YB is used for both high and low cycle fatigue
Note This method is considered to be on the safe side for external gear rims However for internal gear rims without any flange or web stiffeners the method may not be on the safe side and it is advised to check with the method in 315316
315 Stresses in Thin Rims For rim thickness sR lt 35 mn the safety against rim cracking has to be checked
The following method may be used
3151 General The stresses in the standardised 30ordm tangent section tension side are slightly reduced due to decreasing stress correction factor with decreasing relative rim thickness sRmn On the other hand during the complete stress cycle of that fillet a certain amount of compression stresses are also introduced The complete stress range remains approximately constant Therefore the standardised calculation of stresses at the 30ordm tangent may be retained for thin rims as one of the necessary criteria
The maximum stress range for thin rims usually occurs at the 60ordm ndash 80ordm tangents both for laquotensionraquo and laquocompressionraquo side The following method assumes the 75ordm tangent to be the decisive Therefore in addition to the am criterion ap-plied at the 30ordm tangent it is necessary to evaluate the max and min stresses at the 75ordm tangent for both laquotensionraquo (loaded flank) fillet and laquocompressionraquo (back-flank) fillet For this purpose the whole stress cycle of each fillet should be considered but usually the following simplification is justified
Figure 35 Nomenclature of fillets
Index laquoTraquo means laquotensileraquo fillet laquoCraquo means laquocompressionraquo fillet
σFTmin and σFCmax are determined on basis of the nominal rim stresses times the stress concentration factor Y75
σFCmin and σFTmax are determined on basis of superposition of nominal rim stresses times Y75 plus the tooth bending stresses at 75ordm tangent
30 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
3152 Stress concentration factors at the 75ordm tangents The nominal rim stress consists of bending stresses due to local bending moments tangential stresses due to the tan-gential force Ft and radial shear stresses due to Fr
The major influence is given by the bending stresses The influence of the tangential stresses is minor and even though its stress concentration factor is slightly higher than for bending it is considered to be safe enough when the sum of these nominal stresses are combined with the stress concen-tration factor for bending The influence of the radial shear stress is neglected
The stress concentration factor relating nominal rim stresses to local fillet stresses at the 75ordm tangent may be calculated as
75n
R75
n
R
75
ρmsρ185
ms3
Y+
sdot=
where ρ75 is the root radius at the 75ordm tangent ref to mn Usually ρ75 is closed to the tool radius ρao and
ρ75 = ρao
is a safe approximation compensating for the am simplifi-cations to the laquounsaferaquo side
The tooth root stresses of the loaded tooth are decreasing with decreasing relative rim thickness approximately with the empirical factor
15
R
ncorr s
m311Y
minus=
3153 Nominal rim stresses The bending moment applied to the rim consists of a part of the tooth tilt moment Ft (hF + 05 sR) and the bending caused by the radial force Fr
The sectional modulus (first moment of area) which is used for determination of the nominal bending stresses is not nec-essarily the same for the 2 am bending moments If flanges webs etc outside the toothed section contribute to stiffening the rim against various deflections the influence of these stiffeners should be considered Eg an end flange will have an almost negligible influence on the effective sectional modulus for the stresses due to tooth tilt as the deflection caused by the tilt moment is rather small and would not much involve the flange On the other hand the radial forces as for instance from the meshes in an annulus would cause considerable radial deflections that the flange might restrict to a substantial amount When considering the stiff-ening of such flanges or webs on basis of simplified models it is advised to use an effective rim thickness sR = sR + 02 mn for the first moment of area of the rim (toothed part) cross section
For a high number of rim teeth it may be assumed that the rim bending moments in the fillets adjacent to the loaded tooth are of the same magnitude as right under the applied
force This assumption is reasonable for an annulus but rather much on the safe side for a hollow pinion
The influence of Ft on the nominal tangential stress is simpli-fied by half of it for compressive stresses (σ1) and the other half for tensile stresses (σ2) Applying these assumptions the nominal rim stresses adjacent to the loaded tooth are
( ) ( )A2
FW
f RFW
s05hF05 t
R
r
T
RFt1 minus
ϑminus
+minus=σ
( ) ( )A2
FW
f RFW
s05hF05 t
R
r
T
RFt2 +
ϑminus
+=σ
where σ1 σ2 see Fig 35
A = minimum area of cross section (usually bR sR)
WT = the sectional modulus of rim with respect to tooth tilt moment (usually 6sb 2
rR )
WR = the sectional modulus of the rim including the influence of stiffeners as flanges webs etc (WR ge WT)
bR = the width of the rim
R = the radius of the neutral axis in the rim ie from wheel centre to midpoint of rim
( )ϑf = a function for bending moment distribution around the rim For a rim (pinion) with one mesh only the ( )ϑf at the position of load application is 024 For an annulus with 3 or more meshes ( )ϑf at the position of each load application is approx 3 planets ( )ϑf = 019 4 planets ( )ϑf = 014 5 planets ( )ϑf = 011 6 planets ( )ϑf = 009
It must be checked if the max (tensile) stress in the compres-sion fillet really occurs when the fillet is adjacent to the loaded tooth In principle the stress variation through a complete rotation should be considered and the max value used The max value is usually never less than 0 For an annulus eg the tilt moment is zero in the mid position be-tween the planet meshes whilst the bending moment due to the radial forces is half of that at the mesh but with opposite sign
If these formulae are applied to idler gears as eg planets the influence of nominal tangential stresses must be cor-rected by deleting Ft(2 A) for σ1 and using FtA for σ2 Further the influence of Fr on the nominal bending stresses is usually negligible due to the planet bearing support
3154 Root fillet stresses Determination of min and max stresses in the laquotensionraquo fillet
Classification Notes- No 412 31 May 2003
DET NORSKE VERITAS
Minimum stress
751FTmin YσK σ =
Maximum stress
corrF752 Yσ03Yσ KσFTmax +=
where
03 is an empirical factor relating the tension stresses (σF) at the 30ordm tangent to the part of the tension stresses at the 75ordm tangent which add to the rim related stresses (03 also takes into account that full superposition of nominal stresses times stress concentration factors from both laquosides of the corner filletraquo would result in too high stresses)
αβγ sdotsdotsdotsdot= FFvA KKKKKK
Determination of min and max stresses in the laquocompres-sionraquo fillet
Minimum stress
corrF751FCmin Yσ036Yσ Kσ minus=
Maximum stress
752FCmax Yσ Kσ =
where
036 is an empirical factor relating the tension stresses (σF) at the 30ordm tangent to the part of the compression stresses at the 75ordm tangent which add to the rim related stresses
For gears with reversed loads as idler gears and planets there is no distinct laquotensionraquo or laquocompressionraquo fillet The mini-mum stress σFmin is the minimum of σFTmin and σFCmin (usually the latter is decisive) The maximum stress σFTmax is the maximum of σFTmax and σFCmax (usually the former is deci-sive)
316 Permissible Stresses in Thin Rims
3161 General The safety against fatigue fracture respectively overload fracture is to be at least at the same level as for solid gears The laquoordinaryraquo criteria at the 30ordm tangent apply as given in 31 through 313
Additionally the following criteria at the 75ordm tangent may apply
3162 For gt3middot106 cycles The permissible stresses for the laquotensionraquo fillets and for the laquocompressionraquo fillets are determined by means of a relevant fatigue diagram
If the actual tooth root stress (tensile or compressive) ex-ceeds the yield strength to the material the induced residual stresses are to be taken into account
For determination of permissible stresses the following is defined
R = stress ratio ie FTmax
FTminσσ respectively
FCmax
FCmin
σσ
∆σ = stress range ie σFTmax ndash σFTmin resp σFCmax ndash σFCmin
(For idler gears and planets Fmax
FminσσR =
and ∆σ = σFmax minus σFmin)
The permissible stress range ∆σρ for the laquotensionraquo respec-tively laquocompressionraquo fillets can be calculated as
For R gt minus1 FPp σ
R1R1031
13∆σ
minus+
+=
For minus infin lt R lt minus1 FPp σ
R1R10151
13∆σ
minus+
+=
where
σFP = see 32 determined for unidirectional stresses (YM = 1)
If the yield strength σy is exceeded in either tension or com-pression residual stresses are induced This may be consid-ered by correcting the stress ratio R for the respective fillets (tension or compression)
Eg if yFCmin σσ gt (ie exceeded in compression) the
difference yFCmin σσ∆ minus= affects the stress ratio as
∆σ
σ∆σ∆σR
FCmax
y
FCmax
FCmin
+
minus=
++
=
Similarly the stress ratio in the tension fillet may require cor-rection
If the yield strength is exceeded in tension σFTmax gt σy the difference ∆ = σFTmax ndash σy affects the stress ratio as
y
FTmin
FTmax
FTmin
σ∆σ
∆σ∆σR minus=
minusminus
=
Checking for possible exceeding of the yield strength has to be made with the highest torque and with the KAP if this ex-ceeds KA
3163 Forle 103 cycles The permissible stress range ∆σp is not to exceed
For R gt minus1 FPstpst σ
R1R1051
15∆σ
minus+
+=
For ndash infin lt R lt minus1 FPstpst σ
R1R10251
15∆σ
minus+
+=
For all values of R ∆σpst is limited by
not surface hardened F
y
Sσ225
32 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
surface hardened CF
YSHV5
Definition of ∆σ and R see 3162 with particular attention to possible correction of R if the yield strength is exceeded
σFPst see 32 determined for unidirectional stresses (YM = 1) and lt 103 cycles
3164 For 103 lt cycles lt 3106 ∆σp is to be determined by linear interpolation a log-log dia-gram
∆σp at NL load cycles is
6L 103p
exp
L
6
N p ∆σN103∆σ sdot
sdot=
6
3
103p
10p
∆σ
∆σlog28760exp
sdot
=
Classification Notes- No 412 33 May 2003
DET NORSKE VERITAS
4 Calculation of Scuffing Load Capacity
41 Introduction High surface temperatures due to high loads and sliding ve-locities can cause lubricant films to break down This sei-zure or welding together of areas of tooth surface is termed scuffing
In contrast to pitting and fatigue breakage which show a dis-tinct incubation period a single short overloading can lead to a scuffing failure In the ISO-TR13989 two criteria are mentioned The method used in this Classification Note is based on the principles of the flash temperature criterion
Note Bulk temperature in excess of 120ordmC for long periods may have an adverse effect on the surface durability see 211
42 General Criteria In no point along the path of contact the local contact tem-perature may exceed the permissible temperature ie
oilS
oilSB S
ϑ+ϑminusϑ
leϑ
50SB minusϑleϑ
where
Bϑ = max contact temperature along the path of contact
maxflaMBB ϑ+ϑ=ϑ
MBϑ = bulk temperature see 434
maxflaϑ = max flash temperature along the path of con-tact see 44
Sϑ = scuffing temperature see below
oilϑ = oil temperature before it reaches the mesh (max applicable for the actual load case to be used ie normally alarm temperature except for ice classes where a max expected tem-perature applies)
SS = required safety factor according to the Rules
The scuffing temperature Sϑ may be calculated as
L2
wrelT
002
40S XFZGX
ν100112085780
sdot
sdot++=ϑ
where
XwrelT = relative welding factor
XwrelT
Through hardened steel 10
Phosphated steel 125
Copper-plated steel 150
Nitrided steel 150
Less than 10 retained austenite
115
10 ndash 20 retained austenite 10
Casehardened steel
20 ndash 30 retained austenite 085
Austenitic steel 045
FZG = load stage according to FZG-Test A8390 (Note This is the load stage where scuffing oc-curs However due to scatter in test results cal-culations are to be made with one load stage less than the specification)
XL = lubricant factor
= 10 for mineral oils
= 08 for polyalfaolefins
= 07 for non-water-soluble polyglycols
= 06 for water-soluble polyglycols
= 15 for traction fluids
= 13 for phosphate esters
ν40 = kinematic oil viscosity at 40˚C (mm2s)
Application of other test methods such as the Ryder the FZG-Ryder R46574 and the FZG L-42 Test 141195110 may be specially considered
For high speed gears with very short time of contact Sϑ may be increased as follows provided use of EP-oils
Addition to the calculated scuffing temperature Sϑ
If micros 18tc ge no addition
If micros 18tc lt add ( )cwrelT t18X18 minussdotsdot
where
ct = contact time ( micros ) which is the time needed to cross the Hertzian contact width
( ) [ ]microsu1cosβn
uσ340tb1
Hc +sdotsdot
sdot=
σH as calculated in 221
For bevel gears use vu in stead of u
34 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
43 Influence Factors
431 Coefficient of friction The following coefficient of friction may apply
L025a
005oil
02
redC ΣC
Bt X R ηρv
w0048micro minus
=
where
wBt = specific tooth load (Nmm)
vΣC = sum of tangential velocities at pitch point At pitch line velocities gt 50 ms the limiting value of vΣC at v = 50 ms is to be used
ρredC = relative radius of curvature (transversal plane) at the pitch point
Cylindrical gears
HαHβvγAbt
Bt KKKKKb
Fw sdotsdotsdotsdotsdot= (see 1)
twΣC αsin v2v =
bCredC β cos ρρ =
Bevel gears
HαHβvγAtmb
Bt KKKKKb
Fw sdotsdotsdotsdotsdot= (see 1)
vtmtΣC αsin v2v =
bmvCredC β cos ρρ =
ηoil = dynamic viscosity (mPa s) at oilϑ calculated as
1000
ρ νη oiloil =
where ρ in kgm3 approximated as
( ) 70 15oil15 minusϑminusρ=ρ and νoil is kinematic viscosity at oilϑ and may be calculated by means of the following equation
( ) ( )++=+ 08νloglog08νloglog 100oil ( )
sdotminus
ϑ+minus313log373log
273log373log oil
( ) ( )( )08νloglog08νloglog 10040 +minus+
Ra = composite arithmetic mean roughness (micron) of pinion an wheel calculated as ( )2a1aa RR 05R +=This is defined as the roughness on the new flanks ie as manufactured
XL = see 42
432 Effective tip relief Ceff Ceff is the effective tip relief that amount of tip relief which compensates for the elastic deformation of the gear mesh ie
zero load at the tooth tip It is assumed (simplified) to be equal for both pinion and wheel
Cylindrical gears
for helical
γ
γ=cb
KKFC Abt
eff (see 1)
For spur
cbKKF
C Abteff
γ= (see 1)
Alternatively for spur and helical gears the non-linear ap-proach in 111 may be used (taking Ceff=δ)
Bevel gears
bcKFC Ambt
effγ
= (see 1)
where
γ
α
ε2ε44c
+sdot
=γ
433 Tip relief and extension Cylindrical gears
The extension of the tip relief is not to result in an effective contact ratio 1ltεα when the gear is unloaded (exceptions to this may only apply for applications where the gear is not to run at light loads) This means that the unrelieved part of the path of contact is to be minimum pbt It is further assumed that this unrelieved part is placed centrally on the path of contact
If root relief applies it has to be calculated as equivalent tip relief Ie pinion root relief (at mesh position A) is added to Ca2 and wheel root relief (at mesh position E) is added to Ca1 If no design tip relief or root relief on the mating gear is specified (ie if Ca1+Croot2 = 0 and visa versa) use the run-ning in amount see 112
Bevel gears
Bevel gears are to have heightwise crowning ie no distinct relievedunrelieved area This may be treated as tip and root relief For calculation purposes the root relief is combined with the tip relief of the mating member into an equivalent tip relief If no resulting tip relieves are specified the equivalent tip relives may be calculated as an approxima-tion based on the tool crowning Ca tool (per mille of tool module m0) as follows
2root1aeq1a CCC +=
1root2aeq2a CCC +=
where
Classification Notes- No 412 35 May 2003
DET NORSKE VERITAS
2
0
n0101atoolal m
)m2(mA1middotmCC
minus+minussdot=
( )
sdotminusminussdotsdotminus+= vtvb1
2vb1
21vavt1n1 αtan dddαsin 05x1mA
2
0
n020atool22a m
)mm(2A1middotmCC
minus+minussdot=
( )
sdotminusminussdotsdotminus+= vtvb2
2vb2
2va2vt2n2 αtan dddαsin 05x1mA
2
0
20atool1root1 m
A1mCC
minussdotsdot=
2
0
10atool2root2 m
A1mCC
minussdotsdot=
If the pressure angles of the cutter blades are modified (and verified) to balance the tip relieves the following may be assumed
( )eq2aeq1aeq2aeq1a C andC calculated of sum50CC sdot==
Throughout the following Ca1 and Ca2 mean the equivalent tip relieves Ca1eq and Ca2eq
434 Bulk temperature The bulk temperature may be calculated as
flaaveragempsoilMB X X 50 ϑ+ϑ=ϑ
where
Xs = lubrication factor
= 12 for spray lubrication
= 10 for dip lubrication (provided both pinion and wheel are dip lubricated and tip speed lt 5 ms)
= 10 for spray lubrication with additional cooling spray (spray on both pinion and wheel or spray on pinion and dip lubrication of wheel)
= 02 for meshes fully submerged in oil
Xmp = contact factor ( )pmp nX += 150
np = number of mesh contact on the pinion (for small gear ratii the number of wheel meshes should be used if higher)
flaaverageϑ
= average of the integrated flash temperature (see 44) along the path of contact
( )
AE
E
Ayyyfla
flaaverage
d
ΓminusΓ
ΓΓϑ=ϑint =
For high speed gears (v gt 50 ms) it may be necessary to assess the bulk temperature on the basis of a thermal rating of the entire gear transmission
44 The Flash Temperature flaϑ
441 Basic formula The local flash temperature flaϑ may be calculated as
( ) 41redy
y2ly
211
43Btcorrfla
unXwX3250
y ρ
ρminusρ
micro=ϑ Γ
(For bevel gears replace u with uv)
and is to be calculated stepwise along the path of contact from A to E
where
micro = coefficient of friction see 431
Xcorr = correction factor taking empirically into account the increased scuffing risk in the beginning of the approach path due to mesh starting without any previously built up oil film and possible shuffling away oil before meshing if insufficient tip relief
( )
3
AD
yaeffcorr 50
CC1X
ΓminusΓε
Γminus+=
α
Ca = tip relief of driven member Xcorr is only applicable in the approach path and if Ca lt Ceff otherwise 10
wBt = unit load see 431
yXΓ = load sharing factor see 443
n1 = pinion rpm
Γy ρly etc see 442
442 Geometrical relations The various radii of flank curvature (transversal plane) are
ρ1y = pinion flank radius at mesh point y
ρ2y = wheel flank radius at mesh point y
ρredy = equivalent radius of curvature at mesh point y
y2y1
y2y1redy
ρ+ρ
ρρ=ρ
Cylindrical gears
twy
1y αsin au1Γ1
ρ+
+=
twy
2y αsin au1Γu
ρ+
minus=
Note that for internal gears a and u are negative
36 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
Bevel gears
vtvv
y1y αsin a
u1Γ1
ρ+
+=
vtvv
yv2y αsin a
u1Γu
ρ+
minus=
Γ is the parameter on the path of contact and y is any point between A end E
At the respective ends Γ has the following values
Root piniontip wheel
Cylindrical gears
( )
minus
minusminus= 1
αtan 1dd
zzΓ
tw
2b2a2
1
2A
Bevel gears
( )
minus
minusminus= 1
αtan 1dd
uΓvt
2vb2va2
vA
Tip pinionroot wheel
Cylindrical gears
( )
1αtan
1ddΓ
tw
2b1a1
E minusminus
=
Bevel gears
( )
1αtan
1ddΓ
vt
2vb1va1
E minusminus
=
At inner point of single pair contact
Cylindrical gears
tw1
EB α tan zπ2ΓΓ minus=
Bevel gears
vtv1
EB α tan zπ2ΓΓ minus=
At outer point of single pair contact
Cylindrical gears
tw1AD α tan z
π2ΓΓ +=
Bevel gears
vt v1
AD αtan z π2ΓΓ +=
At pitch point ΓC = 0
The points F and G (only applicable to cylindrical gears) limiting the extension of tip relief (so as to maintain a mini-mum contact ratio of unity for unloaded gears) are at
2
BAF
Γ+Γ=Γ
2
EDG
Γ+Γ=Γ
443 Load sharing factor XΓ The load sharing factor XΓ accounts for the load sharing be-tween the various pairs of teeth in mesh along the path of contact
XΓ is to be calculated stepwise from A to E using the pa-rameter Γy
4431 Cylindrical gears with β = 0 and no tip relief
Figure 41
ByAAB
Ay for31
31X
yΓltΓleΓ
ΓminusΓ
ΓminusΓ+=Γ
DyB for1Xy
ΓltΓleΓ=Γ
EyDDE
yE for31
31X
yΓleΓltΓ
ΓminusΓ
ΓminusΓ+=Γ
4432 Cylindrical gears with β = 0 and tip relief Tip relief on the pinion reduces XΓ in the range G ndash E and increases correspondingly XΓ in the range F ndash B
Tip relief on the wheel reduces XΓ in the range A ndash F and increases correspondingly XΓ in the range D ndash G
Following remains generally
DyB for1Xy
ΓleΓleΓ=Γ
21XXGF== ΓΓ
In the following it must be distinguished between Ca lt Ceff respectively Ca gt Ceff This is shown by an example below where Ca1 lt Ceff and Ca2 gt Ceff
Classification Notes- No 412 37 May 2003
DET NORSKE VERITAS
Figure 42
Note When Ca gt Ceff the path of contact is shortened by A ndash A respectively E ndash E The single pair contact path is extended into B respectively D If this shift is significant it is necessary to consider the negative effect on surface durability (B) and bending stresses (B and D)
Range A - F
For effa2 CC le
minus=Γ
eff
2a
CC1
31X
A
FyAeff
2a
AF
Ay for C3
C61XX
AyΓleΓleΓ
sdot
+ΓminusΓ
ΓminusΓ+= ΓΓ
For effa2 CC ge
AyAfor0Xy
ΓleΓleΓ=Γ
with ( )21
CC
1CC
eff
2a
eff
2a
AFAAminus
minusΓminusΓ+Γ=Γ
FyAeff
2a
AF
Ay
eff
2a for 21
CC
CC1X
yΓleΓleΓ
minus
ΓminusΓ
ΓminusΓ+minus=Γ
Range F - B
For effa1 CC le
ByFeff
1a
FB
Fy forC3
C61
21X
yΓleΓleΓ
sdot
+ΓminusΓ
ΓminusΓ+=Γ
For effa1 CC ge
ByFeff
1a
FB
Fy for21
CC
21X
yΓleΓleΓ
minus
ΓminusΓ
ΓminusΓ+=Γ
ByB for1Xy
ΓleΓleΓ=Γ
with 1
CC2
eff
1a
FBFB
minus
ΓminusΓ+Γ=Γ
Range D ndash G
For effa2 CC le
GyDeff
2a
DG
Dy
eff
2a for C 3C
61
C 3C
32X
yΓleΓleΓ
+
ΓminusΓ
ΓminusΓminus+=Γ F
or effa2 CC ge
DyD for1Xy
ΓleΓleΓ=Γ
with ( )21
CC
1CC
eff
2a
eff
2a
DGDDminus
minusΓminusΓ+Γ=Γ
GyDeff
2a
DG
Dy
eff
2a for21
CC
CCX ΓleΓleΓ
minus
ΓminusΓ
ΓminusΓminus=Γ
Range G - E
For effa1 CC le
minus=Γ
eff
1a
CC1
31X
E
EyGeff
1a
GE
Gy forC3
C61
21X
yΓleΓleΓ
sdot
+ΓminusΓ
ΓminusΓminus=Γ
For effa1 CC gt
EyGeff
1a
GE
Gy for21
CC
21X
yΓleΓleΓ
minus
ΓminusΓ
ΓminusΓminus=Γ
EyE for0Xy
ΓleΓleΓ=Γ
with 1
CC2
eff
1a
GEGE
minus
ΓminusΓ+Γ=Γ
4433 Gears with β gt 0 buttressing Due to oblique contact lines over the flanks a certain but-tressing may occur near A and E
This applies to both cylindrical and bevel gears with tip relief lt Ceff The buttressing Xbutt is simplified as a linear function within the ranges A ndash H respectively I ndash E
Figure 43
1εwhen13X βbutt EAge=
1εwhenε031X ββbutt EAlt+=
38 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
Cylindrical gears
bIEAH βsin 02ΓΓΓΓ =minus=minus
Bevel gears
bmIEAH βsin 02ΓΓΓΓ =minus=minus
4434 Cylindrical gears with 2εγ le and no tip relief
yXΓ is obtained by multiplication of
yXΓ in 4431 with
Xbutt in 4433
4435 Gears with 2εγ gt and no tip relief
Applicable to both cylindrical and bevel gears
IyH for1Xy
ΓleΓleΓε
=α
Γ
IyHybutt and forX1Xy
ΓgtΓΓltΓε
=α
Γ
Figure 44
4436 Cylindrical gears with 2εγ le and tip relief
yXΓ is obtained by multiplication of
yXΓ in 4432 with Xbutt
in 4433
4437 Cylindrical gears with 2εγ gt and tip relief
Tip relief on the pinion (respectively wheel) reduces XГ in the range G ndash E (respectively A ndash F) and increases XГ in the range F ndash G
yXΓ is obtained by multiplication of
yXΓ as described below
with Xbutt in 4433
In the XГ example below the influence of tip relief is shown (without the influence of Xbutt) by means of
eff2aeff1a CC and CC ltgt
Tip relief gt Ceff causes new end points A respectively E of the path of contact
Figure 45
Range A ndash F
( ) ( )( ) eff
a2αa1α
AF
Ay
eff
2aeff
C 12C 13εC 1ε
CCCX
y +εε++minus
ΓminusΓ
ΓminusΓ+
εminus
=ααα
Γ
eff2aFyA CC if for leΓleΓleΓ
eff2aFyA CifCfor and geΓleΓleΓ
eff2aAyA CC iffor0Xy
gtΓleΓleΓ=Γ
( ) ( ) ( )( ) ( ) a2αa1α
αeffa2AFAA C 1ε 3C 1ε
1ε 2 CC with++minus+minus
ΓminusΓ+Γ=Γ
Range F ndash G
( )( )( ) GyF
eff
2a1a forC 1 2CC 11X
yΓleΓleΓ
+εε+minusε
+ε
=αα
α
αΓ
Range G ndash E
( ) ( )( ) eff
2a1a
GE
Gy
C 1 2C 1C 1 3XX
GFy +εεminusε++ε
ΓminusΓ
ΓminusΓminus=
αα
ααΓΓ minus
effa1EyG CC if for leΓleΓleΓ
effa1EyG CC if Γfor and geΓleΓle
eff1aEyE CC if for0Xy
geΓleΓleΓ=Γ
( ) ( ) ( )( ) ( ) 2a1a
eff1aGEEE C 1C 1 3
1 2 CC withminusε++ε+εminus
ΓminusΓminusΓ=Γαα
α
4438 Bevel gears with εγ more than approx 18 and heightwise crowning For Ca1 = Ca2 = Ceff the following applies
Figure 46
Classification Notes- No 412 39 May 2003
DET NORSKE VERITAS
( )AEM 50 Γ+Γ=Γ
( )( )2AD
3
2My651X
y ΓminusΓε
ΓminusΓminus
ε=
ααΓ
For tip relief lt Ceff yXΓ is found by linear interpolation be-
tween ( )effay CCX =Γ and ( )0CayX =Γ as in 4435
The interpolation is to be made stepwise from A to M with the influence of Ca2 and from M to E with the influence of Ca1 (For 2a1a CC ne there is a discontinuity at M)
Eg with Ca1 = 04 Ceff and Ca2 = 055 Ceff then
Range A ndash M
)CC()0C( eff2ay2ayyX 550X 450X =Γ=ΓΓ +=
Range M ndash E
)CC()0C( eff1ay1ayyX 40X 06X =Γ=ΓΓ +=
For tip relief gt Ceff the new end points A and E are found as
( )
minusΓminusΓ
ε+Γ=Γ α 1
CC
6 eff
2aADAA
( )
minusΓminusΓ
ε+Γ=Γ α 1
CC
6 eff
1aADEE
Range A ndash A
0Xy=Γ
Range A ndash M
( )( )
ΓminusΓ
ΓminusΓminus
minusε=
αΓ 2
MA
2My
eff
2a1
CC4
351Xy
Range M ndash E
( )( )
ΓminusΓ
ΓminusΓminus
minusε=
αΓ 2
ME
2My
eff
1a1
CC4
351Xy
Range E ndash E
0Xy=Γ
40 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
Appendix A Fatigue Damage Accumulation
The Palmgren-Miner cumulative damage calculation princi-ple is used The procedure may be applied as follows
A1 Stress Spectrum From the individual torque classes the torques (Ti) at the peak values of class intervals and the associated number of cycles (NLi) for both pinion and wheel are to be listed from the highest to the lowest torque
(In case of a cyclic torque variation within the torque classes it is advised to use the peak torque If the cyclic variation is such that the same teeth will repeatedly suffer the peak torque this is a must)
The stress spectra for tooth roots and flanks (σFi σHi) with all relevant factors (except KA) are to be calculated on the basis of the torque spectrum The load dependent K-factors are to be determined for each torque class
A2 σminusN-curve The stress versus load cycle curves for tooth roots and flanks (both pinion and wheel) are to be drawn on the basis of per-missible stresses (ie including the demanded minimum safety factors) as determined in 2 respectively 3 If different safety levels for high cycle fatigue and low cycle fatigue are desired this may be expressed by different demand safety factors applied at the endurance limit respectively at static strength
A3 Damage accumulation The individual damage ratio Di at ith stress level is defined as
Fi
Lii N
ND =
where
NLi = The number of applied cycles at ith stress
NFi = The number of cycles to failure at ith stress
Basically stresses σi below the permissible stress level for infinite life (if a constant ZN or YN is accepted) do not con-tribute to the damage sum However calculating the actual safety factor Sact as described below all the σi for which the product S σi is bigger than or equal to the permissible stress level for infinite life contribute to the damage sum and thus to the determination of Sact The final value of S is decisive
(NFi can be found mathematically by putting the permissible stress σpi equal the actual stress σi thereby finding the actual life factor This life factor can be solved with regard to load cycles ie NFi)
The damage sum ΣDi is not to exceed unity
If ΣDi ne 1 the safety against cumulative fatigue damage is different from the applied demand safety factor For deter-mination of this theoretical safety factor an iteration proce-dure is required as described in the following flowchart
S is correction factor with which the actual safety factor Sact can be found
Sact is the demand safety factor (used in determination of the permissible stresses in the σ ndash N ndash curve) times the correc-tion factor S
The full procedure is to be applied for pinion and wheel tooth roots and flanks
Note If alternating stresses occur in a spectrum of mainly pulsating stresses the alternating stresses may be replaced by equiva-lent pulsating stresses ie by means of division with the ac-tual mean stress influence factor YM
Classification Notes- No 412 41 May 2003
DET NORSKE VERITAS
Appendix B Application Factors for Diesel Driven Gears
For diesel driven gears the application factor KA depends on torsional vibrations Both normal operation and misfiring conditions have to be considered
Normally these two running conditions can be covered by only one calculation
B1 Definitions
Normal operation KAnorm = 0
normv0
TTT +
where
T0 = rated nominal torque
Tv norm = vibratory torque amplitude for normal opera-tion (see rules Pt4 Ch3 Sec1 G301 for defi-nition of ldquonormalrdquo irregularity)
Misfiring operation KA misf = 0
misfv
TTT +
where
T = remaining nominal torque when one cylinder out of action
Tv misf = vibratory torque amplitude in misfiring con-dition This refers to a permissible misfiring condition ie a condition that does not require automatic or immediate corrective actions as speed or pitch reduction
The normal operation is assumed to last for a very high num-ber of cycles such as 1010
The misfiring operation is assumed to last for a limited dura-tion such as 107 cycles
B2 Determination of decisive load Assuming life factor at 1010 cycles as YN = ZN = 092 which usually is relevant the calculation may be performed only once with the combination having the highest value of appli-cation factorlife factor
For bending stresses and scuffing the higher value of
092
K normA and 098
K misfA
For contact stresses the higher value of
092K normA and
113K misfA (but
097K misfA for nitrided gears)
B3 Simplified procedure Note that this is only a guidance and is not a binding con-vention
T = oTZ
1Zsdot
minus where Z = number of cylinders
( ) ideal Videal Vmisf Vnorm V TTT24ZT +minus=
where TV ideal = vibratory torque with all cylinders perfectly equal See also rules Pt4 Ch3 Sec1 G300
When using trends from torsional vibration analysis and measurements the following may be used
TV ideal To is close to zero for engines with few cylinders and using a suitable elastic coupling and increases with relative coupling stiffness and number of cylinders
TV misf To may be high for engines with few cylinders and decreases with number of cylinders
This can be indicated as
200Z
TT
ο
idealV asymp and 80Z04
TT
o
misfV minusasymp
Inserting this into the formulae for the two application fac-tors the following guidance can be given
118112K misfA minusasymp
115110K normA minusasymp
Since KA norm is to be combined with the lower life factors the decisive load condition will be the normal one and a KA of 115 will cover most relevant cases when a suitable elas-tic coupling is chosen
42 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
Appendix C Calculation of Pinion-Rack
Pinion-racks used for elevating of mobile offshore units are open gears that are subjected to wear and tear With normal specifications such as only a few hundred total operation cycles (site to site) the tooth bending stresses for static strength will be decisive for the lay out however with ex-ception of surface hardened pinions where case crushing has to be considered
In the following the use of part 1 and 3 for pinion-racks is shown including relevant simplifications
C1 Pinion tooth root stresses Since the load spectrum normally is dominated by high torques and few load cycles (in the range up to 1-2000) the static strength is decisive
The actual stress is calculated as
β1FSaFan1
tF1 KYY
mbFσ sdotsdotsdotsdot
=
b1 is limited to b2 + 2middotmn
YFa and YSa replace YF and YS because load application at tooth tip has to be assumed for such inaccurate gears
Pinions often use a non-involute profile in the dedendum part of the flank eg a constant radius equal the radius of curva-ture at reference circle For such pinions sFn and hFa are to be measured directly on a sectional drawing of the pinion tooth
Due to high loads and narrow facewidths it may be assumed that KFβ2 = KHβ = 10 However when b1 gt b2 then KFβ1 gt10
If no detailed documentation of KFβ1 is available the fol-lowing may be used
KFβ1 = 1 + 015middot(b1b2 ndash 1)
The permissible stress (not surface hardened) is calculated as
δrelTstF
Fst1FP1 Y
Sσσ sdot=
The mean stress influence due to leg lifting may be disre-garded
The actual and permissible stresses should be calculated for the relevant loads as given in the rules
C2 Rack tooth root stresses The actual stress is calculated as
SaFan2
tF2 YY
mbFσ sdotsdotsdot
=
See C1 for details
The permissible stress is calculated as
δrelTstF
Fst2FP2 Y
Sσσ sdot=
For alloyed steels (Ni Cr Mo) with high toughness and duc-tility the value of YδrelTst may be put equal to YSa
C3 Surface hardened pinions For surface hardened pinions the maximum load is not to cause crushing of the hardened layer of the flank
In principle the calculation described in 213 may be used but when the theoretical Hertzian stress exceeds the ap-proximately 18 times the yield strength of the rack material plastic deformation will occur This will limit the peak Hertzian stress but increases the contact width and thus the penetration of stresses into the depth
An approximation may be based on an assessment of contact width determined by means of equal areas under the theoreti-cal (elastic range) Hertzian contact and the elasto-plastic contact stress (limited to 18 middot σy) with the unknown width
30 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
3152 Stress concentration factors at the 75ordm tangents The nominal rim stress consists of bending stresses due to local bending moments tangential stresses due to the tan-gential force Ft and radial shear stresses due to Fr
The major influence is given by the bending stresses The influence of the tangential stresses is minor and even though its stress concentration factor is slightly higher than for bending it is considered to be safe enough when the sum of these nominal stresses are combined with the stress concen-tration factor for bending The influence of the radial shear stress is neglected
The stress concentration factor relating nominal rim stresses to local fillet stresses at the 75ordm tangent may be calculated as
75n
R75
n
R
75
ρmsρ185
ms3
Y+
sdot=
where ρ75 is the root radius at the 75ordm tangent ref to mn Usually ρ75 is closed to the tool radius ρao and
ρ75 = ρao
is a safe approximation compensating for the am simplifi-cations to the laquounsaferaquo side
The tooth root stresses of the loaded tooth are decreasing with decreasing relative rim thickness approximately with the empirical factor
15
R
ncorr s
m311Y
minus=
3153 Nominal rim stresses The bending moment applied to the rim consists of a part of the tooth tilt moment Ft (hF + 05 sR) and the bending caused by the radial force Fr
The sectional modulus (first moment of area) which is used for determination of the nominal bending stresses is not nec-essarily the same for the 2 am bending moments If flanges webs etc outside the toothed section contribute to stiffening the rim against various deflections the influence of these stiffeners should be considered Eg an end flange will have an almost negligible influence on the effective sectional modulus for the stresses due to tooth tilt as the deflection caused by the tilt moment is rather small and would not much involve the flange On the other hand the radial forces as for instance from the meshes in an annulus would cause considerable radial deflections that the flange might restrict to a substantial amount When considering the stiff-ening of such flanges or webs on basis of simplified models it is advised to use an effective rim thickness sR = sR + 02 mn for the first moment of area of the rim (toothed part) cross section
For a high number of rim teeth it may be assumed that the rim bending moments in the fillets adjacent to the loaded tooth are of the same magnitude as right under the applied
force This assumption is reasonable for an annulus but rather much on the safe side for a hollow pinion
The influence of Ft on the nominal tangential stress is simpli-fied by half of it for compressive stresses (σ1) and the other half for tensile stresses (σ2) Applying these assumptions the nominal rim stresses adjacent to the loaded tooth are
( ) ( )A2
FW
f RFW
s05hF05 t
R
r
T
RFt1 minus
ϑminus
+minus=σ
( ) ( )A2
FW
f RFW
s05hF05 t
R
r
T
RFt2 +
ϑminus
+=σ
where σ1 σ2 see Fig 35
A = minimum area of cross section (usually bR sR)
WT = the sectional modulus of rim with respect to tooth tilt moment (usually 6sb 2
rR )
WR = the sectional modulus of the rim including the influence of stiffeners as flanges webs etc (WR ge WT)
bR = the width of the rim
R = the radius of the neutral axis in the rim ie from wheel centre to midpoint of rim
( )ϑf = a function for bending moment distribution around the rim For a rim (pinion) with one mesh only the ( )ϑf at the position of load application is 024 For an annulus with 3 or more meshes ( )ϑf at the position of each load application is approx 3 planets ( )ϑf = 019 4 planets ( )ϑf = 014 5 planets ( )ϑf = 011 6 planets ( )ϑf = 009
It must be checked if the max (tensile) stress in the compres-sion fillet really occurs when the fillet is adjacent to the loaded tooth In principle the stress variation through a complete rotation should be considered and the max value used The max value is usually never less than 0 For an annulus eg the tilt moment is zero in the mid position be-tween the planet meshes whilst the bending moment due to the radial forces is half of that at the mesh but with opposite sign
If these formulae are applied to idler gears as eg planets the influence of nominal tangential stresses must be cor-rected by deleting Ft(2 A) for σ1 and using FtA for σ2 Further the influence of Fr on the nominal bending stresses is usually negligible due to the planet bearing support
3154 Root fillet stresses Determination of min and max stresses in the laquotensionraquo fillet
Classification Notes- No 412 31 May 2003
DET NORSKE VERITAS
Minimum stress
751FTmin YσK σ =
Maximum stress
corrF752 Yσ03Yσ KσFTmax +=
where
03 is an empirical factor relating the tension stresses (σF) at the 30ordm tangent to the part of the tension stresses at the 75ordm tangent which add to the rim related stresses (03 also takes into account that full superposition of nominal stresses times stress concentration factors from both laquosides of the corner filletraquo would result in too high stresses)
αβγ sdotsdotsdotsdot= FFvA KKKKKK
Determination of min and max stresses in the laquocompres-sionraquo fillet
Minimum stress
corrF751FCmin Yσ036Yσ Kσ minus=
Maximum stress
752FCmax Yσ Kσ =
where
036 is an empirical factor relating the tension stresses (σF) at the 30ordm tangent to the part of the compression stresses at the 75ordm tangent which add to the rim related stresses
For gears with reversed loads as idler gears and planets there is no distinct laquotensionraquo or laquocompressionraquo fillet The mini-mum stress σFmin is the minimum of σFTmin and σFCmin (usually the latter is decisive) The maximum stress σFTmax is the maximum of σFTmax and σFCmax (usually the former is deci-sive)
316 Permissible Stresses in Thin Rims
3161 General The safety against fatigue fracture respectively overload fracture is to be at least at the same level as for solid gears The laquoordinaryraquo criteria at the 30ordm tangent apply as given in 31 through 313
Additionally the following criteria at the 75ordm tangent may apply
3162 For gt3middot106 cycles The permissible stresses for the laquotensionraquo fillets and for the laquocompressionraquo fillets are determined by means of a relevant fatigue diagram
If the actual tooth root stress (tensile or compressive) ex-ceeds the yield strength to the material the induced residual stresses are to be taken into account
For determination of permissible stresses the following is defined
R = stress ratio ie FTmax
FTminσσ respectively
FCmax
FCmin
σσ
∆σ = stress range ie σFTmax ndash σFTmin resp σFCmax ndash σFCmin
(For idler gears and planets Fmax
FminσσR =
and ∆σ = σFmax minus σFmin)
The permissible stress range ∆σρ for the laquotensionraquo respec-tively laquocompressionraquo fillets can be calculated as
For R gt minus1 FPp σ
R1R1031
13∆σ
minus+
+=
For minus infin lt R lt minus1 FPp σ
R1R10151
13∆σ
minus+
+=
where
σFP = see 32 determined for unidirectional stresses (YM = 1)
If the yield strength σy is exceeded in either tension or com-pression residual stresses are induced This may be consid-ered by correcting the stress ratio R for the respective fillets (tension or compression)
Eg if yFCmin σσ gt (ie exceeded in compression) the
difference yFCmin σσ∆ minus= affects the stress ratio as
∆σ
σ∆σ∆σR
FCmax
y
FCmax
FCmin
+
minus=
++
=
Similarly the stress ratio in the tension fillet may require cor-rection
If the yield strength is exceeded in tension σFTmax gt σy the difference ∆ = σFTmax ndash σy affects the stress ratio as
y
FTmin
FTmax
FTmin
σ∆σ
∆σ∆σR minus=
minusminus
=
Checking for possible exceeding of the yield strength has to be made with the highest torque and with the KAP if this ex-ceeds KA
3163 Forle 103 cycles The permissible stress range ∆σp is not to exceed
For R gt minus1 FPstpst σ
R1R1051
15∆σ
minus+
+=
For ndash infin lt R lt minus1 FPstpst σ
R1R10251
15∆σ
minus+
+=
For all values of R ∆σpst is limited by
not surface hardened F
y
Sσ225
32 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
surface hardened CF
YSHV5
Definition of ∆σ and R see 3162 with particular attention to possible correction of R if the yield strength is exceeded
σFPst see 32 determined for unidirectional stresses (YM = 1) and lt 103 cycles
3164 For 103 lt cycles lt 3106 ∆σp is to be determined by linear interpolation a log-log dia-gram
∆σp at NL load cycles is
6L 103p
exp
L
6
N p ∆σN103∆σ sdot
sdot=
6
3
103p
10p
∆σ
∆σlog28760exp
sdot
=
Classification Notes- No 412 33 May 2003
DET NORSKE VERITAS
4 Calculation of Scuffing Load Capacity
41 Introduction High surface temperatures due to high loads and sliding ve-locities can cause lubricant films to break down This sei-zure or welding together of areas of tooth surface is termed scuffing
In contrast to pitting and fatigue breakage which show a dis-tinct incubation period a single short overloading can lead to a scuffing failure In the ISO-TR13989 two criteria are mentioned The method used in this Classification Note is based on the principles of the flash temperature criterion
Note Bulk temperature in excess of 120ordmC for long periods may have an adverse effect on the surface durability see 211
42 General Criteria In no point along the path of contact the local contact tem-perature may exceed the permissible temperature ie
oilS
oilSB S
ϑ+ϑminusϑ
leϑ
50SB minusϑleϑ
where
Bϑ = max contact temperature along the path of contact
maxflaMBB ϑ+ϑ=ϑ
MBϑ = bulk temperature see 434
maxflaϑ = max flash temperature along the path of con-tact see 44
Sϑ = scuffing temperature see below
oilϑ = oil temperature before it reaches the mesh (max applicable for the actual load case to be used ie normally alarm temperature except for ice classes where a max expected tem-perature applies)
SS = required safety factor according to the Rules
The scuffing temperature Sϑ may be calculated as
L2
wrelT
002
40S XFZGX
ν100112085780
sdot
sdot++=ϑ
where
XwrelT = relative welding factor
XwrelT
Through hardened steel 10
Phosphated steel 125
Copper-plated steel 150
Nitrided steel 150
Less than 10 retained austenite
115
10 ndash 20 retained austenite 10
Casehardened steel
20 ndash 30 retained austenite 085
Austenitic steel 045
FZG = load stage according to FZG-Test A8390 (Note This is the load stage where scuffing oc-curs However due to scatter in test results cal-culations are to be made with one load stage less than the specification)
XL = lubricant factor
= 10 for mineral oils
= 08 for polyalfaolefins
= 07 for non-water-soluble polyglycols
= 06 for water-soluble polyglycols
= 15 for traction fluids
= 13 for phosphate esters
ν40 = kinematic oil viscosity at 40˚C (mm2s)
Application of other test methods such as the Ryder the FZG-Ryder R46574 and the FZG L-42 Test 141195110 may be specially considered
For high speed gears with very short time of contact Sϑ may be increased as follows provided use of EP-oils
Addition to the calculated scuffing temperature Sϑ
If micros 18tc ge no addition
If micros 18tc lt add ( )cwrelT t18X18 minussdotsdot
where
ct = contact time ( micros ) which is the time needed to cross the Hertzian contact width
( ) [ ]microsu1cosβn
uσ340tb1
Hc +sdotsdot
sdot=
σH as calculated in 221
For bevel gears use vu in stead of u
34 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
43 Influence Factors
431 Coefficient of friction The following coefficient of friction may apply
L025a
005oil
02
redC ΣC
Bt X R ηρv
w0048micro minus
=
where
wBt = specific tooth load (Nmm)
vΣC = sum of tangential velocities at pitch point At pitch line velocities gt 50 ms the limiting value of vΣC at v = 50 ms is to be used
ρredC = relative radius of curvature (transversal plane) at the pitch point
Cylindrical gears
HαHβvγAbt
Bt KKKKKb
Fw sdotsdotsdotsdotsdot= (see 1)
twΣC αsin v2v =
bCredC β cos ρρ =
Bevel gears
HαHβvγAtmb
Bt KKKKKb
Fw sdotsdotsdotsdotsdot= (see 1)
vtmtΣC αsin v2v =
bmvCredC β cos ρρ =
ηoil = dynamic viscosity (mPa s) at oilϑ calculated as
1000
ρ νη oiloil =
where ρ in kgm3 approximated as
( ) 70 15oil15 minusϑminusρ=ρ and νoil is kinematic viscosity at oilϑ and may be calculated by means of the following equation
( ) ( )++=+ 08νloglog08νloglog 100oil ( )
sdotminus
ϑ+minus313log373log
273log373log oil
( ) ( )( )08νloglog08νloglog 10040 +minus+
Ra = composite arithmetic mean roughness (micron) of pinion an wheel calculated as ( )2a1aa RR 05R +=This is defined as the roughness on the new flanks ie as manufactured
XL = see 42
432 Effective tip relief Ceff Ceff is the effective tip relief that amount of tip relief which compensates for the elastic deformation of the gear mesh ie
zero load at the tooth tip It is assumed (simplified) to be equal for both pinion and wheel
Cylindrical gears
for helical
γ
γ=cb
KKFC Abt
eff (see 1)
For spur
cbKKF
C Abteff
γ= (see 1)
Alternatively for spur and helical gears the non-linear ap-proach in 111 may be used (taking Ceff=δ)
Bevel gears
bcKFC Ambt
effγ
= (see 1)
where
γ
α
ε2ε44c
+sdot
=γ
433 Tip relief and extension Cylindrical gears
The extension of the tip relief is not to result in an effective contact ratio 1ltεα when the gear is unloaded (exceptions to this may only apply for applications where the gear is not to run at light loads) This means that the unrelieved part of the path of contact is to be minimum pbt It is further assumed that this unrelieved part is placed centrally on the path of contact
If root relief applies it has to be calculated as equivalent tip relief Ie pinion root relief (at mesh position A) is added to Ca2 and wheel root relief (at mesh position E) is added to Ca1 If no design tip relief or root relief on the mating gear is specified (ie if Ca1+Croot2 = 0 and visa versa) use the run-ning in amount see 112
Bevel gears
Bevel gears are to have heightwise crowning ie no distinct relievedunrelieved area This may be treated as tip and root relief For calculation purposes the root relief is combined with the tip relief of the mating member into an equivalent tip relief If no resulting tip relieves are specified the equivalent tip relives may be calculated as an approxima-tion based on the tool crowning Ca tool (per mille of tool module m0) as follows
2root1aeq1a CCC +=
1root2aeq2a CCC +=
where
Classification Notes- No 412 35 May 2003
DET NORSKE VERITAS
2
0
n0101atoolal m
)m2(mA1middotmCC
minus+minussdot=
( )
sdotminusminussdotsdotminus+= vtvb1
2vb1
21vavt1n1 αtan dddαsin 05x1mA
2
0
n020atool22a m
)mm(2A1middotmCC
minus+minussdot=
( )
sdotminusminussdotsdotminus+= vtvb2
2vb2
2va2vt2n2 αtan dddαsin 05x1mA
2
0
20atool1root1 m
A1mCC
minussdotsdot=
2
0
10atool2root2 m
A1mCC
minussdotsdot=
If the pressure angles of the cutter blades are modified (and verified) to balance the tip relieves the following may be assumed
( )eq2aeq1aeq2aeq1a C andC calculated of sum50CC sdot==
Throughout the following Ca1 and Ca2 mean the equivalent tip relieves Ca1eq and Ca2eq
434 Bulk temperature The bulk temperature may be calculated as
flaaveragempsoilMB X X 50 ϑ+ϑ=ϑ
where
Xs = lubrication factor
= 12 for spray lubrication
= 10 for dip lubrication (provided both pinion and wheel are dip lubricated and tip speed lt 5 ms)
= 10 for spray lubrication with additional cooling spray (spray on both pinion and wheel or spray on pinion and dip lubrication of wheel)
= 02 for meshes fully submerged in oil
Xmp = contact factor ( )pmp nX += 150
np = number of mesh contact on the pinion (for small gear ratii the number of wheel meshes should be used if higher)
flaaverageϑ
= average of the integrated flash temperature (see 44) along the path of contact
( )
AE
E
Ayyyfla
flaaverage
d
ΓminusΓ
ΓΓϑ=ϑint =
For high speed gears (v gt 50 ms) it may be necessary to assess the bulk temperature on the basis of a thermal rating of the entire gear transmission
44 The Flash Temperature flaϑ
441 Basic formula The local flash temperature flaϑ may be calculated as
( ) 41redy
y2ly
211
43Btcorrfla
unXwX3250
y ρ
ρminusρ
micro=ϑ Γ
(For bevel gears replace u with uv)
and is to be calculated stepwise along the path of contact from A to E
where
micro = coefficient of friction see 431
Xcorr = correction factor taking empirically into account the increased scuffing risk in the beginning of the approach path due to mesh starting without any previously built up oil film and possible shuffling away oil before meshing if insufficient tip relief
( )
3
AD
yaeffcorr 50
CC1X
ΓminusΓε
Γminus+=
α
Ca = tip relief of driven member Xcorr is only applicable in the approach path and if Ca lt Ceff otherwise 10
wBt = unit load see 431
yXΓ = load sharing factor see 443
n1 = pinion rpm
Γy ρly etc see 442
442 Geometrical relations The various radii of flank curvature (transversal plane) are
ρ1y = pinion flank radius at mesh point y
ρ2y = wheel flank radius at mesh point y
ρredy = equivalent radius of curvature at mesh point y
y2y1
y2y1redy
ρ+ρ
ρρ=ρ
Cylindrical gears
twy
1y αsin au1Γ1
ρ+
+=
twy
2y αsin au1Γu
ρ+
minus=
Note that for internal gears a and u are negative
36 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
Bevel gears
vtvv
y1y αsin a
u1Γ1
ρ+
+=
vtvv
yv2y αsin a
u1Γu
ρ+
minus=
Γ is the parameter on the path of contact and y is any point between A end E
At the respective ends Γ has the following values
Root piniontip wheel
Cylindrical gears
( )
minus
minusminus= 1
αtan 1dd
zzΓ
tw
2b2a2
1
2A
Bevel gears
( )
minus
minusminus= 1
αtan 1dd
uΓvt
2vb2va2
vA
Tip pinionroot wheel
Cylindrical gears
( )
1αtan
1ddΓ
tw
2b1a1
E minusminus
=
Bevel gears
( )
1αtan
1ddΓ
vt
2vb1va1
E minusminus
=
At inner point of single pair contact
Cylindrical gears
tw1
EB α tan zπ2ΓΓ minus=
Bevel gears
vtv1
EB α tan zπ2ΓΓ minus=
At outer point of single pair contact
Cylindrical gears
tw1AD α tan z
π2ΓΓ +=
Bevel gears
vt v1
AD αtan z π2ΓΓ +=
At pitch point ΓC = 0
The points F and G (only applicable to cylindrical gears) limiting the extension of tip relief (so as to maintain a mini-mum contact ratio of unity for unloaded gears) are at
2
BAF
Γ+Γ=Γ
2
EDG
Γ+Γ=Γ
443 Load sharing factor XΓ The load sharing factor XΓ accounts for the load sharing be-tween the various pairs of teeth in mesh along the path of contact
XΓ is to be calculated stepwise from A to E using the pa-rameter Γy
4431 Cylindrical gears with β = 0 and no tip relief
Figure 41
ByAAB
Ay for31
31X
yΓltΓleΓ
ΓminusΓ
ΓminusΓ+=Γ
DyB for1Xy
ΓltΓleΓ=Γ
EyDDE
yE for31
31X
yΓleΓltΓ
ΓminusΓ
ΓminusΓ+=Γ
4432 Cylindrical gears with β = 0 and tip relief Tip relief on the pinion reduces XΓ in the range G ndash E and increases correspondingly XΓ in the range F ndash B
Tip relief on the wheel reduces XΓ in the range A ndash F and increases correspondingly XΓ in the range D ndash G
Following remains generally
DyB for1Xy
ΓleΓleΓ=Γ
21XXGF== ΓΓ
In the following it must be distinguished between Ca lt Ceff respectively Ca gt Ceff This is shown by an example below where Ca1 lt Ceff and Ca2 gt Ceff
Classification Notes- No 412 37 May 2003
DET NORSKE VERITAS
Figure 42
Note When Ca gt Ceff the path of contact is shortened by A ndash A respectively E ndash E The single pair contact path is extended into B respectively D If this shift is significant it is necessary to consider the negative effect on surface durability (B) and bending stresses (B and D)
Range A - F
For effa2 CC le
minus=Γ
eff
2a
CC1
31X
A
FyAeff
2a
AF
Ay for C3
C61XX
AyΓleΓleΓ
sdot
+ΓminusΓ
ΓminusΓ+= ΓΓ
For effa2 CC ge
AyAfor0Xy
ΓleΓleΓ=Γ
with ( )21
CC
1CC
eff
2a
eff
2a
AFAAminus
minusΓminusΓ+Γ=Γ
FyAeff
2a
AF
Ay
eff
2a for 21
CC
CC1X
yΓleΓleΓ
minus
ΓminusΓ
ΓminusΓ+minus=Γ
Range F - B
For effa1 CC le
ByFeff
1a
FB
Fy forC3
C61
21X
yΓleΓleΓ
sdot
+ΓminusΓ
ΓminusΓ+=Γ
For effa1 CC ge
ByFeff
1a
FB
Fy for21
CC
21X
yΓleΓleΓ
minus
ΓminusΓ
ΓminusΓ+=Γ
ByB for1Xy
ΓleΓleΓ=Γ
with 1
CC2
eff
1a
FBFB
minus
ΓminusΓ+Γ=Γ
Range D ndash G
For effa2 CC le
GyDeff
2a
DG
Dy
eff
2a for C 3C
61
C 3C
32X
yΓleΓleΓ
+
ΓminusΓ
ΓminusΓminus+=Γ F
or effa2 CC ge
DyD for1Xy
ΓleΓleΓ=Γ
with ( )21
CC
1CC
eff
2a
eff
2a
DGDDminus
minusΓminusΓ+Γ=Γ
GyDeff
2a
DG
Dy
eff
2a for21
CC
CCX ΓleΓleΓ
minus
ΓminusΓ
ΓminusΓminus=Γ
Range G - E
For effa1 CC le
minus=Γ
eff
1a
CC1
31X
E
EyGeff
1a
GE
Gy forC3
C61
21X
yΓleΓleΓ
sdot
+ΓminusΓ
ΓminusΓminus=Γ
For effa1 CC gt
EyGeff
1a
GE
Gy for21
CC
21X
yΓleΓleΓ
minus
ΓminusΓ
ΓminusΓminus=Γ
EyE for0Xy
ΓleΓleΓ=Γ
with 1
CC2
eff
1a
GEGE
minus
ΓminusΓ+Γ=Γ
4433 Gears with β gt 0 buttressing Due to oblique contact lines over the flanks a certain but-tressing may occur near A and E
This applies to both cylindrical and bevel gears with tip relief lt Ceff The buttressing Xbutt is simplified as a linear function within the ranges A ndash H respectively I ndash E
Figure 43
1εwhen13X βbutt EAge=
1εwhenε031X ββbutt EAlt+=
38 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
Cylindrical gears
bIEAH βsin 02ΓΓΓΓ =minus=minus
Bevel gears
bmIEAH βsin 02ΓΓΓΓ =minus=minus
4434 Cylindrical gears with 2εγ le and no tip relief
yXΓ is obtained by multiplication of
yXΓ in 4431 with
Xbutt in 4433
4435 Gears with 2εγ gt and no tip relief
Applicable to both cylindrical and bevel gears
IyH for1Xy
ΓleΓleΓε
=α
Γ
IyHybutt and forX1Xy
ΓgtΓΓltΓε
=α
Γ
Figure 44
4436 Cylindrical gears with 2εγ le and tip relief
yXΓ is obtained by multiplication of
yXΓ in 4432 with Xbutt
in 4433
4437 Cylindrical gears with 2εγ gt and tip relief
Tip relief on the pinion (respectively wheel) reduces XГ in the range G ndash E (respectively A ndash F) and increases XГ in the range F ndash G
yXΓ is obtained by multiplication of
yXΓ as described below
with Xbutt in 4433
In the XГ example below the influence of tip relief is shown (without the influence of Xbutt) by means of
eff2aeff1a CC and CC ltgt
Tip relief gt Ceff causes new end points A respectively E of the path of contact
Figure 45
Range A ndash F
( ) ( )( ) eff
a2αa1α
AF
Ay
eff
2aeff
C 12C 13εC 1ε
CCCX
y +εε++minus
ΓminusΓ
ΓminusΓ+
εminus
=ααα
Γ
eff2aFyA CC if for leΓleΓleΓ
eff2aFyA CifCfor and geΓleΓleΓ
eff2aAyA CC iffor0Xy
gtΓleΓleΓ=Γ
( ) ( ) ( )( ) ( ) a2αa1α
αeffa2AFAA C 1ε 3C 1ε
1ε 2 CC with++minus+minus
ΓminusΓ+Γ=Γ
Range F ndash G
( )( )( ) GyF
eff
2a1a forC 1 2CC 11X
yΓleΓleΓ
+εε+minusε
+ε
=αα
α
αΓ
Range G ndash E
( ) ( )( ) eff
2a1a
GE
Gy
C 1 2C 1C 1 3XX
GFy +εεminusε++ε
ΓminusΓ
ΓminusΓminus=
αα
ααΓΓ minus
effa1EyG CC if for leΓleΓleΓ
effa1EyG CC if Γfor and geΓleΓle
eff1aEyE CC if for0Xy
geΓleΓleΓ=Γ
( ) ( ) ( )( ) ( ) 2a1a
eff1aGEEE C 1C 1 3
1 2 CC withminusε++ε+εminus
ΓminusΓminusΓ=Γαα
α
4438 Bevel gears with εγ more than approx 18 and heightwise crowning For Ca1 = Ca2 = Ceff the following applies
Figure 46
Classification Notes- No 412 39 May 2003
DET NORSKE VERITAS
( )AEM 50 Γ+Γ=Γ
( )( )2AD
3
2My651X
y ΓminusΓε
ΓminusΓminus
ε=
ααΓ
For tip relief lt Ceff yXΓ is found by linear interpolation be-
tween ( )effay CCX =Γ and ( )0CayX =Γ as in 4435
The interpolation is to be made stepwise from A to M with the influence of Ca2 and from M to E with the influence of Ca1 (For 2a1a CC ne there is a discontinuity at M)
Eg with Ca1 = 04 Ceff and Ca2 = 055 Ceff then
Range A ndash M
)CC()0C( eff2ay2ayyX 550X 450X =Γ=ΓΓ +=
Range M ndash E
)CC()0C( eff1ay1ayyX 40X 06X =Γ=ΓΓ +=
For tip relief gt Ceff the new end points A and E are found as
( )
minusΓminusΓ
ε+Γ=Γ α 1
CC
6 eff
2aADAA
( )
minusΓminusΓ
ε+Γ=Γ α 1
CC
6 eff
1aADEE
Range A ndash A
0Xy=Γ
Range A ndash M
( )( )
ΓminusΓ
ΓminusΓminus
minusε=
αΓ 2
MA
2My
eff
2a1
CC4
351Xy
Range M ndash E
( )( )
ΓminusΓ
ΓminusΓminus
minusε=
αΓ 2
ME
2My
eff
1a1
CC4
351Xy
Range E ndash E
0Xy=Γ
40 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
Appendix A Fatigue Damage Accumulation
The Palmgren-Miner cumulative damage calculation princi-ple is used The procedure may be applied as follows
A1 Stress Spectrum From the individual torque classes the torques (Ti) at the peak values of class intervals and the associated number of cycles (NLi) for both pinion and wheel are to be listed from the highest to the lowest torque
(In case of a cyclic torque variation within the torque classes it is advised to use the peak torque If the cyclic variation is such that the same teeth will repeatedly suffer the peak torque this is a must)
The stress spectra for tooth roots and flanks (σFi σHi) with all relevant factors (except KA) are to be calculated on the basis of the torque spectrum The load dependent K-factors are to be determined for each torque class
A2 σminusN-curve The stress versus load cycle curves for tooth roots and flanks (both pinion and wheel) are to be drawn on the basis of per-missible stresses (ie including the demanded minimum safety factors) as determined in 2 respectively 3 If different safety levels for high cycle fatigue and low cycle fatigue are desired this may be expressed by different demand safety factors applied at the endurance limit respectively at static strength
A3 Damage accumulation The individual damage ratio Di at ith stress level is defined as
Fi
Lii N
ND =
where
NLi = The number of applied cycles at ith stress
NFi = The number of cycles to failure at ith stress
Basically stresses σi below the permissible stress level for infinite life (if a constant ZN or YN is accepted) do not con-tribute to the damage sum However calculating the actual safety factor Sact as described below all the σi for which the product S σi is bigger than or equal to the permissible stress level for infinite life contribute to the damage sum and thus to the determination of Sact The final value of S is decisive
(NFi can be found mathematically by putting the permissible stress σpi equal the actual stress σi thereby finding the actual life factor This life factor can be solved with regard to load cycles ie NFi)
The damage sum ΣDi is not to exceed unity
If ΣDi ne 1 the safety against cumulative fatigue damage is different from the applied demand safety factor For deter-mination of this theoretical safety factor an iteration proce-dure is required as described in the following flowchart
S is correction factor with which the actual safety factor Sact can be found
Sact is the demand safety factor (used in determination of the permissible stresses in the σ ndash N ndash curve) times the correc-tion factor S
The full procedure is to be applied for pinion and wheel tooth roots and flanks
Note If alternating stresses occur in a spectrum of mainly pulsating stresses the alternating stresses may be replaced by equiva-lent pulsating stresses ie by means of division with the ac-tual mean stress influence factor YM
Classification Notes- No 412 41 May 2003
DET NORSKE VERITAS
Appendix B Application Factors for Diesel Driven Gears
For diesel driven gears the application factor KA depends on torsional vibrations Both normal operation and misfiring conditions have to be considered
Normally these two running conditions can be covered by only one calculation
B1 Definitions
Normal operation KAnorm = 0
normv0
TTT +
where
T0 = rated nominal torque
Tv norm = vibratory torque amplitude for normal opera-tion (see rules Pt4 Ch3 Sec1 G301 for defi-nition of ldquonormalrdquo irregularity)
Misfiring operation KA misf = 0
misfv
TTT +
where
T = remaining nominal torque when one cylinder out of action
Tv misf = vibratory torque amplitude in misfiring con-dition This refers to a permissible misfiring condition ie a condition that does not require automatic or immediate corrective actions as speed or pitch reduction
The normal operation is assumed to last for a very high num-ber of cycles such as 1010
The misfiring operation is assumed to last for a limited dura-tion such as 107 cycles
B2 Determination of decisive load Assuming life factor at 1010 cycles as YN = ZN = 092 which usually is relevant the calculation may be performed only once with the combination having the highest value of appli-cation factorlife factor
For bending stresses and scuffing the higher value of
092
K normA and 098
K misfA
For contact stresses the higher value of
092K normA and
113K misfA (but
097K misfA for nitrided gears)
B3 Simplified procedure Note that this is only a guidance and is not a binding con-vention
T = oTZ
1Zsdot
minus where Z = number of cylinders
( ) ideal Videal Vmisf Vnorm V TTT24ZT +minus=
where TV ideal = vibratory torque with all cylinders perfectly equal See also rules Pt4 Ch3 Sec1 G300
When using trends from torsional vibration analysis and measurements the following may be used
TV ideal To is close to zero for engines with few cylinders and using a suitable elastic coupling and increases with relative coupling stiffness and number of cylinders
TV misf To may be high for engines with few cylinders and decreases with number of cylinders
This can be indicated as
200Z
TT
ο
idealV asymp and 80Z04
TT
o
misfV minusasymp
Inserting this into the formulae for the two application fac-tors the following guidance can be given
118112K misfA minusasymp
115110K normA minusasymp
Since KA norm is to be combined with the lower life factors the decisive load condition will be the normal one and a KA of 115 will cover most relevant cases when a suitable elas-tic coupling is chosen
42 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
Appendix C Calculation of Pinion-Rack
Pinion-racks used for elevating of mobile offshore units are open gears that are subjected to wear and tear With normal specifications such as only a few hundred total operation cycles (site to site) the tooth bending stresses for static strength will be decisive for the lay out however with ex-ception of surface hardened pinions where case crushing has to be considered
In the following the use of part 1 and 3 for pinion-racks is shown including relevant simplifications
C1 Pinion tooth root stresses Since the load spectrum normally is dominated by high torques and few load cycles (in the range up to 1-2000) the static strength is decisive
The actual stress is calculated as
β1FSaFan1
tF1 KYY
mbFσ sdotsdotsdotsdot
=
b1 is limited to b2 + 2middotmn
YFa and YSa replace YF and YS because load application at tooth tip has to be assumed for such inaccurate gears
Pinions often use a non-involute profile in the dedendum part of the flank eg a constant radius equal the radius of curva-ture at reference circle For such pinions sFn and hFa are to be measured directly on a sectional drawing of the pinion tooth
Due to high loads and narrow facewidths it may be assumed that KFβ2 = KHβ = 10 However when b1 gt b2 then KFβ1 gt10
If no detailed documentation of KFβ1 is available the fol-lowing may be used
KFβ1 = 1 + 015middot(b1b2 ndash 1)
The permissible stress (not surface hardened) is calculated as
δrelTstF
Fst1FP1 Y
Sσσ sdot=
The mean stress influence due to leg lifting may be disre-garded
The actual and permissible stresses should be calculated for the relevant loads as given in the rules
C2 Rack tooth root stresses The actual stress is calculated as
SaFan2
tF2 YY
mbFσ sdotsdotsdot
=
See C1 for details
The permissible stress is calculated as
δrelTstF
Fst2FP2 Y
Sσσ sdot=
For alloyed steels (Ni Cr Mo) with high toughness and duc-tility the value of YδrelTst may be put equal to YSa
C3 Surface hardened pinions For surface hardened pinions the maximum load is not to cause crushing of the hardened layer of the flank
In principle the calculation described in 213 may be used but when the theoretical Hertzian stress exceeds the ap-proximately 18 times the yield strength of the rack material plastic deformation will occur This will limit the peak Hertzian stress but increases the contact width and thus the penetration of stresses into the depth
An approximation may be based on an assessment of contact width determined by means of equal areas under the theoreti-cal (elastic range) Hertzian contact and the elasto-plastic contact stress (limited to 18 middot σy) with the unknown width
Classification Notes- No 412 31 May 2003
DET NORSKE VERITAS
Minimum stress
751FTmin YσK σ =
Maximum stress
corrF752 Yσ03Yσ KσFTmax +=
where
03 is an empirical factor relating the tension stresses (σF) at the 30ordm tangent to the part of the tension stresses at the 75ordm tangent which add to the rim related stresses (03 also takes into account that full superposition of nominal stresses times stress concentration factors from both laquosides of the corner filletraquo would result in too high stresses)
αβγ sdotsdotsdotsdot= FFvA KKKKKK
Determination of min and max stresses in the laquocompres-sionraquo fillet
Minimum stress
corrF751FCmin Yσ036Yσ Kσ minus=
Maximum stress
752FCmax Yσ Kσ =
where
036 is an empirical factor relating the tension stresses (σF) at the 30ordm tangent to the part of the compression stresses at the 75ordm tangent which add to the rim related stresses
For gears with reversed loads as idler gears and planets there is no distinct laquotensionraquo or laquocompressionraquo fillet The mini-mum stress σFmin is the minimum of σFTmin and σFCmin (usually the latter is decisive) The maximum stress σFTmax is the maximum of σFTmax and σFCmax (usually the former is deci-sive)
316 Permissible Stresses in Thin Rims
3161 General The safety against fatigue fracture respectively overload fracture is to be at least at the same level as for solid gears The laquoordinaryraquo criteria at the 30ordm tangent apply as given in 31 through 313
Additionally the following criteria at the 75ordm tangent may apply
3162 For gt3middot106 cycles The permissible stresses for the laquotensionraquo fillets and for the laquocompressionraquo fillets are determined by means of a relevant fatigue diagram
If the actual tooth root stress (tensile or compressive) ex-ceeds the yield strength to the material the induced residual stresses are to be taken into account
For determination of permissible stresses the following is defined
R = stress ratio ie FTmax
FTminσσ respectively
FCmax
FCmin
σσ
∆σ = stress range ie σFTmax ndash σFTmin resp σFCmax ndash σFCmin
(For idler gears and planets Fmax
FminσσR =
and ∆σ = σFmax minus σFmin)
The permissible stress range ∆σρ for the laquotensionraquo respec-tively laquocompressionraquo fillets can be calculated as
For R gt minus1 FPp σ
R1R1031
13∆σ
minus+
+=
For minus infin lt R lt minus1 FPp σ
R1R10151
13∆σ
minus+
+=
where
σFP = see 32 determined for unidirectional stresses (YM = 1)
If the yield strength σy is exceeded in either tension or com-pression residual stresses are induced This may be consid-ered by correcting the stress ratio R for the respective fillets (tension or compression)
Eg if yFCmin σσ gt (ie exceeded in compression) the
difference yFCmin σσ∆ minus= affects the stress ratio as
∆σ
σ∆σ∆σR
FCmax
y
FCmax
FCmin
+
minus=
++
=
Similarly the stress ratio in the tension fillet may require cor-rection
If the yield strength is exceeded in tension σFTmax gt σy the difference ∆ = σFTmax ndash σy affects the stress ratio as
y
FTmin
FTmax
FTmin
σ∆σ
∆σ∆σR minus=
minusminus
=
Checking for possible exceeding of the yield strength has to be made with the highest torque and with the KAP if this ex-ceeds KA
3163 Forle 103 cycles The permissible stress range ∆σp is not to exceed
For R gt minus1 FPstpst σ
R1R1051
15∆σ
minus+
+=
For ndash infin lt R lt minus1 FPstpst σ
R1R10251
15∆σ
minus+
+=
For all values of R ∆σpst is limited by
not surface hardened F
y
Sσ225
32 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
surface hardened CF
YSHV5
Definition of ∆σ and R see 3162 with particular attention to possible correction of R if the yield strength is exceeded
σFPst see 32 determined for unidirectional stresses (YM = 1) and lt 103 cycles
3164 For 103 lt cycles lt 3106 ∆σp is to be determined by linear interpolation a log-log dia-gram
∆σp at NL load cycles is
6L 103p
exp
L
6
N p ∆σN103∆σ sdot
sdot=
6
3
103p
10p
∆σ
∆σlog28760exp
sdot
=
Classification Notes- No 412 33 May 2003
DET NORSKE VERITAS
4 Calculation of Scuffing Load Capacity
41 Introduction High surface temperatures due to high loads and sliding ve-locities can cause lubricant films to break down This sei-zure or welding together of areas of tooth surface is termed scuffing
In contrast to pitting and fatigue breakage which show a dis-tinct incubation period a single short overloading can lead to a scuffing failure In the ISO-TR13989 two criteria are mentioned The method used in this Classification Note is based on the principles of the flash temperature criterion
Note Bulk temperature in excess of 120ordmC for long periods may have an adverse effect on the surface durability see 211
42 General Criteria In no point along the path of contact the local contact tem-perature may exceed the permissible temperature ie
oilS
oilSB S
ϑ+ϑminusϑ
leϑ
50SB minusϑleϑ
where
Bϑ = max contact temperature along the path of contact
maxflaMBB ϑ+ϑ=ϑ
MBϑ = bulk temperature see 434
maxflaϑ = max flash temperature along the path of con-tact see 44
Sϑ = scuffing temperature see below
oilϑ = oil temperature before it reaches the mesh (max applicable for the actual load case to be used ie normally alarm temperature except for ice classes where a max expected tem-perature applies)
SS = required safety factor according to the Rules
The scuffing temperature Sϑ may be calculated as
L2
wrelT
002
40S XFZGX
ν100112085780
sdot
sdot++=ϑ
where
XwrelT = relative welding factor
XwrelT
Through hardened steel 10
Phosphated steel 125
Copper-plated steel 150
Nitrided steel 150
Less than 10 retained austenite
115
10 ndash 20 retained austenite 10
Casehardened steel
20 ndash 30 retained austenite 085
Austenitic steel 045
FZG = load stage according to FZG-Test A8390 (Note This is the load stage where scuffing oc-curs However due to scatter in test results cal-culations are to be made with one load stage less than the specification)
XL = lubricant factor
= 10 for mineral oils
= 08 for polyalfaolefins
= 07 for non-water-soluble polyglycols
= 06 for water-soluble polyglycols
= 15 for traction fluids
= 13 for phosphate esters
ν40 = kinematic oil viscosity at 40˚C (mm2s)
Application of other test methods such as the Ryder the FZG-Ryder R46574 and the FZG L-42 Test 141195110 may be specially considered
For high speed gears with very short time of contact Sϑ may be increased as follows provided use of EP-oils
Addition to the calculated scuffing temperature Sϑ
If micros 18tc ge no addition
If micros 18tc lt add ( )cwrelT t18X18 minussdotsdot
where
ct = contact time ( micros ) which is the time needed to cross the Hertzian contact width
( ) [ ]microsu1cosβn
uσ340tb1
Hc +sdotsdot
sdot=
σH as calculated in 221
For bevel gears use vu in stead of u
34 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
43 Influence Factors
431 Coefficient of friction The following coefficient of friction may apply
L025a
005oil
02
redC ΣC
Bt X R ηρv
w0048micro minus
=
where
wBt = specific tooth load (Nmm)
vΣC = sum of tangential velocities at pitch point At pitch line velocities gt 50 ms the limiting value of vΣC at v = 50 ms is to be used
ρredC = relative radius of curvature (transversal plane) at the pitch point
Cylindrical gears
HαHβvγAbt
Bt KKKKKb
Fw sdotsdotsdotsdotsdot= (see 1)
twΣC αsin v2v =
bCredC β cos ρρ =
Bevel gears
HαHβvγAtmb
Bt KKKKKb
Fw sdotsdotsdotsdotsdot= (see 1)
vtmtΣC αsin v2v =
bmvCredC β cos ρρ =
ηoil = dynamic viscosity (mPa s) at oilϑ calculated as
1000
ρ νη oiloil =
where ρ in kgm3 approximated as
( ) 70 15oil15 minusϑminusρ=ρ and νoil is kinematic viscosity at oilϑ and may be calculated by means of the following equation
( ) ( )++=+ 08νloglog08νloglog 100oil ( )
sdotminus
ϑ+minus313log373log
273log373log oil
( ) ( )( )08νloglog08νloglog 10040 +minus+
Ra = composite arithmetic mean roughness (micron) of pinion an wheel calculated as ( )2a1aa RR 05R +=This is defined as the roughness on the new flanks ie as manufactured
XL = see 42
432 Effective tip relief Ceff Ceff is the effective tip relief that amount of tip relief which compensates for the elastic deformation of the gear mesh ie
zero load at the tooth tip It is assumed (simplified) to be equal for both pinion and wheel
Cylindrical gears
for helical
γ
γ=cb
KKFC Abt
eff (see 1)
For spur
cbKKF
C Abteff
γ= (see 1)
Alternatively for spur and helical gears the non-linear ap-proach in 111 may be used (taking Ceff=δ)
Bevel gears
bcKFC Ambt
effγ
= (see 1)
where
γ
α
ε2ε44c
+sdot
=γ
433 Tip relief and extension Cylindrical gears
The extension of the tip relief is not to result in an effective contact ratio 1ltεα when the gear is unloaded (exceptions to this may only apply for applications where the gear is not to run at light loads) This means that the unrelieved part of the path of contact is to be minimum pbt It is further assumed that this unrelieved part is placed centrally on the path of contact
If root relief applies it has to be calculated as equivalent tip relief Ie pinion root relief (at mesh position A) is added to Ca2 and wheel root relief (at mesh position E) is added to Ca1 If no design tip relief or root relief on the mating gear is specified (ie if Ca1+Croot2 = 0 and visa versa) use the run-ning in amount see 112
Bevel gears
Bevel gears are to have heightwise crowning ie no distinct relievedunrelieved area This may be treated as tip and root relief For calculation purposes the root relief is combined with the tip relief of the mating member into an equivalent tip relief If no resulting tip relieves are specified the equivalent tip relives may be calculated as an approxima-tion based on the tool crowning Ca tool (per mille of tool module m0) as follows
2root1aeq1a CCC +=
1root2aeq2a CCC +=
where
Classification Notes- No 412 35 May 2003
DET NORSKE VERITAS
2
0
n0101atoolal m
)m2(mA1middotmCC
minus+minussdot=
( )
sdotminusminussdotsdotminus+= vtvb1
2vb1
21vavt1n1 αtan dddαsin 05x1mA
2
0
n020atool22a m
)mm(2A1middotmCC
minus+minussdot=
( )
sdotminusminussdotsdotminus+= vtvb2
2vb2
2va2vt2n2 αtan dddαsin 05x1mA
2
0
20atool1root1 m
A1mCC
minussdotsdot=
2
0
10atool2root2 m
A1mCC
minussdotsdot=
If the pressure angles of the cutter blades are modified (and verified) to balance the tip relieves the following may be assumed
( )eq2aeq1aeq2aeq1a C andC calculated of sum50CC sdot==
Throughout the following Ca1 and Ca2 mean the equivalent tip relieves Ca1eq and Ca2eq
434 Bulk temperature The bulk temperature may be calculated as
flaaveragempsoilMB X X 50 ϑ+ϑ=ϑ
where
Xs = lubrication factor
= 12 for spray lubrication
= 10 for dip lubrication (provided both pinion and wheel are dip lubricated and tip speed lt 5 ms)
= 10 for spray lubrication with additional cooling spray (spray on both pinion and wheel or spray on pinion and dip lubrication of wheel)
= 02 for meshes fully submerged in oil
Xmp = contact factor ( )pmp nX += 150
np = number of mesh contact on the pinion (for small gear ratii the number of wheel meshes should be used if higher)
flaaverageϑ
= average of the integrated flash temperature (see 44) along the path of contact
( )
AE
E
Ayyyfla
flaaverage
d
ΓminusΓ
ΓΓϑ=ϑint =
For high speed gears (v gt 50 ms) it may be necessary to assess the bulk temperature on the basis of a thermal rating of the entire gear transmission
44 The Flash Temperature flaϑ
441 Basic formula The local flash temperature flaϑ may be calculated as
( ) 41redy
y2ly
211
43Btcorrfla
unXwX3250
y ρ
ρminusρ
micro=ϑ Γ
(For bevel gears replace u with uv)
and is to be calculated stepwise along the path of contact from A to E
where
micro = coefficient of friction see 431
Xcorr = correction factor taking empirically into account the increased scuffing risk in the beginning of the approach path due to mesh starting without any previously built up oil film and possible shuffling away oil before meshing if insufficient tip relief
( )
3
AD
yaeffcorr 50
CC1X
ΓminusΓε
Γminus+=
α
Ca = tip relief of driven member Xcorr is only applicable in the approach path and if Ca lt Ceff otherwise 10
wBt = unit load see 431
yXΓ = load sharing factor see 443
n1 = pinion rpm
Γy ρly etc see 442
442 Geometrical relations The various radii of flank curvature (transversal plane) are
ρ1y = pinion flank radius at mesh point y
ρ2y = wheel flank radius at mesh point y
ρredy = equivalent radius of curvature at mesh point y
y2y1
y2y1redy
ρ+ρ
ρρ=ρ
Cylindrical gears
twy
1y αsin au1Γ1
ρ+
+=
twy
2y αsin au1Γu
ρ+
minus=
Note that for internal gears a and u are negative
36 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
Bevel gears
vtvv
y1y αsin a
u1Γ1
ρ+
+=
vtvv
yv2y αsin a
u1Γu
ρ+
minus=
Γ is the parameter on the path of contact and y is any point between A end E
At the respective ends Γ has the following values
Root piniontip wheel
Cylindrical gears
( )
minus
minusminus= 1
αtan 1dd
zzΓ
tw
2b2a2
1
2A
Bevel gears
( )
minus
minusminus= 1
αtan 1dd
uΓvt
2vb2va2
vA
Tip pinionroot wheel
Cylindrical gears
( )
1αtan
1ddΓ
tw
2b1a1
E minusminus
=
Bevel gears
( )
1αtan
1ddΓ
vt
2vb1va1
E minusminus
=
At inner point of single pair contact
Cylindrical gears
tw1
EB α tan zπ2ΓΓ minus=
Bevel gears
vtv1
EB α tan zπ2ΓΓ minus=
At outer point of single pair contact
Cylindrical gears
tw1AD α tan z
π2ΓΓ +=
Bevel gears
vt v1
AD αtan z π2ΓΓ +=
At pitch point ΓC = 0
The points F and G (only applicable to cylindrical gears) limiting the extension of tip relief (so as to maintain a mini-mum contact ratio of unity for unloaded gears) are at
2
BAF
Γ+Γ=Γ
2
EDG
Γ+Γ=Γ
443 Load sharing factor XΓ The load sharing factor XΓ accounts for the load sharing be-tween the various pairs of teeth in mesh along the path of contact
XΓ is to be calculated stepwise from A to E using the pa-rameter Γy
4431 Cylindrical gears with β = 0 and no tip relief
Figure 41
ByAAB
Ay for31
31X
yΓltΓleΓ
ΓminusΓ
ΓminusΓ+=Γ
DyB for1Xy
ΓltΓleΓ=Γ
EyDDE
yE for31
31X
yΓleΓltΓ
ΓminusΓ
ΓminusΓ+=Γ
4432 Cylindrical gears with β = 0 and tip relief Tip relief on the pinion reduces XΓ in the range G ndash E and increases correspondingly XΓ in the range F ndash B
Tip relief on the wheel reduces XΓ in the range A ndash F and increases correspondingly XΓ in the range D ndash G
Following remains generally
DyB for1Xy
ΓleΓleΓ=Γ
21XXGF== ΓΓ
In the following it must be distinguished between Ca lt Ceff respectively Ca gt Ceff This is shown by an example below where Ca1 lt Ceff and Ca2 gt Ceff
Classification Notes- No 412 37 May 2003
DET NORSKE VERITAS
Figure 42
Note When Ca gt Ceff the path of contact is shortened by A ndash A respectively E ndash E The single pair contact path is extended into B respectively D If this shift is significant it is necessary to consider the negative effect on surface durability (B) and bending stresses (B and D)
Range A - F
For effa2 CC le
minus=Γ
eff
2a
CC1
31X
A
FyAeff
2a
AF
Ay for C3
C61XX
AyΓleΓleΓ
sdot
+ΓminusΓ
ΓminusΓ+= ΓΓ
For effa2 CC ge
AyAfor0Xy
ΓleΓleΓ=Γ
with ( )21
CC
1CC
eff
2a
eff
2a
AFAAminus
minusΓminusΓ+Γ=Γ
FyAeff
2a
AF
Ay
eff
2a for 21
CC
CC1X
yΓleΓleΓ
minus
ΓminusΓ
ΓminusΓ+minus=Γ
Range F - B
For effa1 CC le
ByFeff
1a
FB
Fy forC3
C61
21X
yΓleΓleΓ
sdot
+ΓminusΓ
ΓminusΓ+=Γ
For effa1 CC ge
ByFeff
1a
FB
Fy for21
CC
21X
yΓleΓleΓ
minus
ΓminusΓ
ΓminusΓ+=Γ
ByB for1Xy
ΓleΓleΓ=Γ
with 1
CC2
eff
1a
FBFB
minus
ΓminusΓ+Γ=Γ
Range D ndash G
For effa2 CC le
GyDeff
2a
DG
Dy
eff
2a for C 3C
61
C 3C
32X
yΓleΓleΓ
+
ΓminusΓ
ΓminusΓminus+=Γ F
or effa2 CC ge
DyD for1Xy
ΓleΓleΓ=Γ
with ( )21
CC
1CC
eff
2a
eff
2a
DGDDminus
minusΓminusΓ+Γ=Γ
GyDeff
2a
DG
Dy
eff
2a for21
CC
CCX ΓleΓleΓ
minus
ΓminusΓ
ΓminusΓminus=Γ
Range G - E
For effa1 CC le
minus=Γ
eff
1a
CC1
31X
E
EyGeff
1a
GE
Gy forC3
C61
21X
yΓleΓleΓ
sdot
+ΓminusΓ
ΓminusΓminus=Γ
For effa1 CC gt
EyGeff
1a
GE
Gy for21
CC
21X
yΓleΓleΓ
minus
ΓminusΓ
ΓminusΓminus=Γ
EyE for0Xy
ΓleΓleΓ=Γ
with 1
CC2
eff
1a
GEGE
minus
ΓminusΓ+Γ=Γ
4433 Gears with β gt 0 buttressing Due to oblique contact lines over the flanks a certain but-tressing may occur near A and E
This applies to both cylindrical and bevel gears with tip relief lt Ceff The buttressing Xbutt is simplified as a linear function within the ranges A ndash H respectively I ndash E
Figure 43
1εwhen13X βbutt EAge=
1εwhenε031X ββbutt EAlt+=
38 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
Cylindrical gears
bIEAH βsin 02ΓΓΓΓ =minus=minus
Bevel gears
bmIEAH βsin 02ΓΓΓΓ =minus=minus
4434 Cylindrical gears with 2εγ le and no tip relief
yXΓ is obtained by multiplication of
yXΓ in 4431 with
Xbutt in 4433
4435 Gears with 2εγ gt and no tip relief
Applicable to both cylindrical and bevel gears
IyH for1Xy
ΓleΓleΓε
=α
Γ
IyHybutt and forX1Xy
ΓgtΓΓltΓε
=α
Γ
Figure 44
4436 Cylindrical gears with 2εγ le and tip relief
yXΓ is obtained by multiplication of
yXΓ in 4432 with Xbutt
in 4433
4437 Cylindrical gears with 2εγ gt and tip relief
Tip relief on the pinion (respectively wheel) reduces XГ in the range G ndash E (respectively A ndash F) and increases XГ in the range F ndash G
yXΓ is obtained by multiplication of
yXΓ as described below
with Xbutt in 4433
In the XГ example below the influence of tip relief is shown (without the influence of Xbutt) by means of
eff2aeff1a CC and CC ltgt
Tip relief gt Ceff causes new end points A respectively E of the path of contact
Figure 45
Range A ndash F
( ) ( )( ) eff
a2αa1α
AF
Ay
eff
2aeff
C 12C 13εC 1ε
CCCX
y +εε++minus
ΓminusΓ
ΓminusΓ+
εminus
=ααα
Γ
eff2aFyA CC if for leΓleΓleΓ
eff2aFyA CifCfor and geΓleΓleΓ
eff2aAyA CC iffor0Xy
gtΓleΓleΓ=Γ
( ) ( ) ( )( ) ( ) a2αa1α
αeffa2AFAA C 1ε 3C 1ε
1ε 2 CC with++minus+minus
ΓminusΓ+Γ=Γ
Range F ndash G
( )( )( ) GyF
eff
2a1a forC 1 2CC 11X
yΓleΓleΓ
+εε+minusε
+ε
=αα
α
αΓ
Range G ndash E
( ) ( )( ) eff
2a1a
GE
Gy
C 1 2C 1C 1 3XX
GFy +εεminusε++ε
ΓminusΓ
ΓminusΓminus=
αα
ααΓΓ minus
effa1EyG CC if for leΓleΓleΓ
effa1EyG CC if Γfor and geΓleΓle
eff1aEyE CC if for0Xy
geΓleΓleΓ=Γ
( ) ( ) ( )( ) ( ) 2a1a
eff1aGEEE C 1C 1 3
1 2 CC withminusε++ε+εminus
ΓminusΓminusΓ=Γαα
α
4438 Bevel gears with εγ more than approx 18 and heightwise crowning For Ca1 = Ca2 = Ceff the following applies
Figure 46
Classification Notes- No 412 39 May 2003
DET NORSKE VERITAS
( )AEM 50 Γ+Γ=Γ
( )( )2AD
3
2My651X
y ΓminusΓε
ΓminusΓminus
ε=
ααΓ
For tip relief lt Ceff yXΓ is found by linear interpolation be-
tween ( )effay CCX =Γ and ( )0CayX =Γ as in 4435
The interpolation is to be made stepwise from A to M with the influence of Ca2 and from M to E with the influence of Ca1 (For 2a1a CC ne there is a discontinuity at M)
Eg with Ca1 = 04 Ceff and Ca2 = 055 Ceff then
Range A ndash M
)CC()0C( eff2ay2ayyX 550X 450X =Γ=ΓΓ +=
Range M ndash E
)CC()0C( eff1ay1ayyX 40X 06X =Γ=ΓΓ +=
For tip relief gt Ceff the new end points A and E are found as
( )
minusΓminusΓ
ε+Γ=Γ α 1
CC
6 eff
2aADAA
( )
minusΓminusΓ
ε+Γ=Γ α 1
CC
6 eff
1aADEE
Range A ndash A
0Xy=Γ
Range A ndash M
( )( )
ΓminusΓ
ΓminusΓminus
minusε=
αΓ 2
MA
2My
eff
2a1
CC4
351Xy
Range M ndash E
( )( )
ΓminusΓ
ΓminusΓminus
minusε=
αΓ 2
ME
2My
eff
1a1
CC4
351Xy
Range E ndash E
0Xy=Γ
40 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
Appendix A Fatigue Damage Accumulation
The Palmgren-Miner cumulative damage calculation princi-ple is used The procedure may be applied as follows
A1 Stress Spectrum From the individual torque classes the torques (Ti) at the peak values of class intervals and the associated number of cycles (NLi) for both pinion and wheel are to be listed from the highest to the lowest torque
(In case of a cyclic torque variation within the torque classes it is advised to use the peak torque If the cyclic variation is such that the same teeth will repeatedly suffer the peak torque this is a must)
The stress spectra for tooth roots and flanks (σFi σHi) with all relevant factors (except KA) are to be calculated on the basis of the torque spectrum The load dependent K-factors are to be determined for each torque class
A2 σminusN-curve The stress versus load cycle curves for tooth roots and flanks (both pinion and wheel) are to be drawn on the basis of per-missible stresses (ie including the demanded minimum safety factors) as determined in 2 respectively 3 If different safety levels for high cycle fatigue and low cycle fatigue are desired this may be expressed by different demand safety factors applied at the endurance limit respectively at static strength
A3 Damage accumulation The individual damage ratio Di at ith stress level is defined as
Fi
Lii N
ND =
where
NLi = The number of applied cycles at ith stress
NFi = The number of cycles to failure at ith stress
Basically stresses σi below the permissible stress level for infinite life (if a constant ZN or YN is accepted) do not con-tribute to the damage sum However calculating the actual safety factor Sact as described below all the σi for which the product S σi is bigger than or equal to the permissible stress level for infinite life contribute to the damage sum and thus to the determination of Sact The final value of S is decisive
(NFi can be found mathematically by putting the permissible stress σpi equal the actual stress σi thereby finding the actual life factor This life factor can be solved with regard to load cycles ie NFi)
The damage sum ΣDi is not to exceed unity
If ΣDi ne 1 the safety against cumulative fatigue damage is different from the applied demand safety factor For deter-mination of this theoretical safety factor an iteration proce-dure is required as described in the following flowchart
S is correction factor with which the actual safety factor Sact can be found
Sact is the demand safety factor (used in determination of the permissible stresses in the σ ndash N ndash curve) times the correc-tion factor S
The full procedure is to be applied for pinion and wheel tooth roots and flanks
Note If alternating stresses occur in a spectrum of mainly pulsating stresses the alternating stresses may be replaced by equiva-lent pulsating stresses ie by means of division with the ac-tual mean stress influence factor YM
Classification Notes- No 412 41 May 2003
DET NORSKE VERITAS
Appendix B Application Factors for Diesel Driven Gears
For diesel driven gears the application factor KA depends on torsional vibrations Both normal operation and misfiring conditions have to be considered
Normally these two running conditions can be covered by only one calculation
B1 Definitions
Normal operation KAnorm = 0
normv0
TTT +
where
T0 = rated nominal torque
Tv norm = vibratory torque amplitude for normal opera-tion (see rules Pt4 Ch3 Sec1 G301 for defi-nition of ldquonormalrdquo irregularity)
Misfiring operation KA misf = 0
misfv
TTT +
where
T = remaining nominal torque when one cylinder out of action
Tv misf = vibratory torque amplitude in misfiring con-dition This refers to a permissible misfiring condition ie a condition that does not require automatic or immediate corrective actions as speed or pitch reduction
The normal operation is assumed to last for a very high num-ber of cycles such as 1010
The misfiring operation is assumed to last for a limited dura-tion such as 107 cycles
B2 Determination of decisive load Assuming life factor at 1010 cycles as YN = ZN = 092 which usually is relevant the calculation may be performed only once with the combination having the highest value of appli-cation factorlife factor
For bending stresses and scuffing the higher value of
092
K normA and 098
K misfA
For contact stresses the higher value of
092K normA and
113K misfA (but
097K misfA for nitrided gears)
B3 Simplified procedure Note that this is only a guidance and is not a binding con-vention
T = oTZ
1Zsdot
minus where Z = number of cylinders
( ) ideal Videal Vmisf Vnorm V TTT24ZT +minus=
where TV ideal = vibratory torque with all cylinders perfectly equal See also rules Pt4 Ch3 Sec1 G300
When using trends from torsional vibration analysis and measurements the following may be used
TV ideal To is close to zero for engines with few cylinders and using a suitable elastic coupling and increases with relative coupling stiffness and number of cylinders
TV misf To may be high for engines with few cylinders and decreases with number of cylinders
This can be indicated as
200Z
TT
ο
idealV asymp and 80Z04
TT
o
misfV minusasymp
Inserting this into the formulae for the two application fac-tors the following guidance can be given
118112K misfA minusasymp
115110K normA minusasymp
Since KA norm is to be combined with the lower life factors the decisive load condition will be the normal one and a KA of 115 will cover most relevant cases when a suitable elas-tic coupling is chosen
42 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
Appendix C Calculation of Pinion-Rack
Pinion-racks used for elevating of mobile offshore units are open gears that are subjected to wear and tear With normal specifications such as only a few hundred total operation cycles (site to site) the tooth bending stresses for static strength will be decisive for the lay out however with ex-ception of surface hardened pinions where case crushing has to be considered
In the following the use of part 1 and 3 for pinion-racks is shown including relevant simplifications
C1 Pinion tooth root stresses Since the load spectrum normally is dominated by high torques and few load cycles (in the range up to 1-2000) the static strength is decisive
The actual stress is calculated as
β1FSaFan1
tF1 KYY
mbFσ sdotsdotsdotsdot
=
b1 is limited to b2 + 2middotmn
YFa and YSa replace YF and YS because load application at tooth tip has to be assumed for such inaccurate gears
Pinions often use a non-involute profile in the dedendum part of the flank eg a constant radius equal the radius of curva-ture at reference circle For such pinions sFn and hFa are to be measured directly on a sectional drawing of the pinion tooth
Due to high loads and narrow facewidths it may be assumed that KFβ2 = KHβ = 10 However when b1 gt b2 then KFβ1 gt10
If no detailed documentation of KFβ1 is available the fol-lowing may be used
KFβ1 = 1 + 015middot(b1b2 ndash 1)
The permissible stress (not surface hardened) is calculated as
δrelTstF
Fst1FP1 Y
Sσσ sdot=
The mean stress influence due to leg lifting may be disre-garded
The actual and permissible stresses should be calculated for the relevant loads as given in the rules
C2 Rack tooth root stresses The actual stress is calculated as
SaFan2
tF2 YY
mbFσ sdotsdotsdot
=
See C1 for details
The permissible stress is calculated as
δrelTstF
Fst2FP2 Y
Sσσ sdot=
For alloyed steels (Ni Cr Mo) with high toughness and duc-tility the value of YδrelTst may be put equal to YSa
C3 Surface hardened pinions For surface hardened pinions the maximum load is not to cause crushing of the hardened layer of the flank
In principle the calculation described in 213 may be used but when the theoretical Hertzian stress exceeds the ap-proximately 18 times the yield strength of the rack material plastic deformation will occur This will limit the peak Hertzian stress but increases the contact width and thus the penetration of stresses into the depth
An approximation may be based on an assessment of contact width determined by means of equal areas under the theoreti-cal (elastic range) Hertzian contact and the elasto-plastic contact stress (limited to 18 middot σy) with the unknown width
32 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
surface hardened CF
YSHV5
Definition of ∆σ and R see 3162 with particular attention to possible correction of R if the yield strength is exceeded
σFPst see 32 determined for unidirectional stresses (YM = 1) and lt 103 cycles
3164 For 103 lt cycles lt 3106 ∆σp is to be determined by linear interpolation a log-log dia-gram
∆σp at NL load cycles is
6L 103p
exp
L
6
N p ∆σN103∆σ sdot
sdot=
6
3
103p
10p
∆σ
∆σlog28760exp
sdot
=
Classification Notes- No 412 33 May 2003
DET NORSKE VERITAS
4 Calculation of Scuffing Load Capacity
41 Introduction High surface temperatures due to high loads and sliding ve-locities can cause lubricant films to break down This sei-zure or welding together of areas of tooth surface is termed scuffing
In contrast to pitting and fatigue breakage which show a dis-tinct incubation period a single short overloading can lead to a scuffing failure In the ISO-TR13989 two criteria are mentioned The method used in this Classification Note is based on the principles of the flash temperature criterion
Note Bulk temperature in excess of 120ordmC for long periods may have an adverse effect on the surface durability see 211
42 General Criteria In no point along the path of contact the local contact tem-perature may exceed the permissible temperature ie
oilS
oilSB S
ϑ+ϑminusϑ
leϑ
50SB minusϑleϑ
where
Bϑ = max contact temperature along the path of contact
maxflaMBB ϑ+ϑ=ϑ
MBϑ = bulk temperature see 434
maxflaϑ = max flash temperature along the path of con-tact see 44
Sϑ = scuffing temperature see below
oilϑ = oil temperature before it reaches the mesh (max applicable for the actual load case to be used ie normally alarm temperature except for ice classes where a max expected tem-perature applies)
SS = required safety factor according to the Rules
The scuffing temperature Sϑ may be calculated as
L2
wrelT
002
40S XFZGX
ν100112085780
sdot
sdot++=ϑ
where
XwrelT = relative welding factor
XwrelT
Through hardened steel 10
Phosphated steel 125
Copper-plated steel 150
Nitrided steel 150
Less than 10 retained austenite
115
10 ndash 20 retained austenite 10
Casehardened steel
20 ndash 30 retained austenite 085
Austenitic steel 045
FZG = load stage according to FZG-Test A8390 (Note This is the load stage where scuffing oc-curs However due to scatter in test results cal-culations are to be made with one load stage less than the specification)
XL = lubricant factor
= 10 for mineral oils
= 08 for polyalfaolefins
= 07 for non-water-soluble polyglycols
= 06 for water-soluble polyglycols
= 15 for traction fluids
= 13 for phosphate esters
ν40 = kinematic oil viscosity at 40˚C (mm2s)
Application of other test methods such as the Ryder the FZG-Ryder R46574 and the FZG L-42 Test 141195110 may be specially considered
For high speed gears with very short time of contact Sϑ may be increased as follows provided use of EP-oils
Addition to the calculated scuffing temperature Sϑ
If micros 18tc ge no addition
If micros 18tc lt add ( )cwrelT t18X18 minussdotsdot
where
ct = contact time ( micros ) which is the time needed to cross the Hertzian contact width
( ) [ ]microsu1cosβn
uσ340tb1
Hc +sdotsdot
sdot=
σH as calculated in 221
For bevel gears use vu in stead of u
34 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
43 Influence Factors
431 Coefficient of friction The following coefficient of friction may apply
L025a
005oil
02
redC ΣC
Bt X R ηρv
w0048micro minus
=
where
wBt = specific tooth load (Nmm)
vΣC = sum of tangential velocities at pitch point At pitch line velocities gt 50 ms the limiting value of vΣC at v = 50 ms is to be used
ρredC = relative radius of curvature (transversal plane) at the pitch point
Cylindrical gears
HαHβvγAbt
Bt KKKKKb
Fw sdotsdotsdotsdotsdot= (see 1)
twΣC αsin v2v =
bCredC β cos ρρ =
Bevel gears
HαHβvγAtmb
Bt KKKKKb
Fw sdotsdotsdotsdotsdot= (see 1)
vtmtΣC αsin v2v =
bmvCredC β cos ρρ =
ηoil = dynamic viscosity (mPa s) at oilϑ calculated as
1000
ρ νη oiloil =
where ρ in kgm3 approximated as
( ) 70 15oil15 minusϑminusρ=ρ and νoil is kinematic viscosity at oilϑ and may be calculated by means of the following equation
( ) ( )++=+ 08νloglog08νloglog 100oil ( )
sdotminus
ϑ+minus313log373log
273log373log oil
( ) ( )( )08νloglog08νloglog 10040 +minus+
Ra = composite arithmetic mean roughness (micron) of pinion an wheel calculated as ( )2a1aa RR 05R +=This is defined as the roughness on the new flanks ie as manufactured
XL = see 42
432 Effective tip relief Ceff Ceff is the effective tip relief that amount of tip relief which compensates for the elastic deformation of the gear mesh ie
zero load at the tooth tip It is assumed (simplified) to be equal for both pinion and wheel
Cylindrical gears
for helical
γ
γ=cb
KKFC Abt
eff (see 1)
For spur
cbKKF
C Abteff
γ= (see 1)
Alternatively for spur and helical gears the non-linear ap-proach in 111 may be used (taking Ceff=δ)
Bevel gears
bcKFC Ambt
effγ
= (see 1)
where
γ
α
ε2ε44c
+sdot
=γ
433 Tip relief and extension Cylindrical gears
The extension of the tip relief is not to result in an effective contact ratio 1ltεα when the gear is unloaded (exceptions to this may only apply for applications where the gear is not to run at light loads) This means that the unrelieved part of the path of contact is to be minimum pbt It is further assumed that this unrelieved part is placed centrally on the path of contact
If root relief applies it has to be calculated as equivalent tip relief Ie pinion root relief (at mesh position A) is added to Ca2 and wheel root relief (at mesh position E) is added to Ca1 If no design tip relief or root relief on the mating gear is specified (ie if Ca1+Croot2 = 0 and visa versa) use the run-ning in amount see 112
Bevel gears
Bevel gears are to have heightwise crowning ie no distinct relievedunrelieved area This may be treated as tip and root relief For calculation purposes the root relief is combined with the tip relief of the mating member into an equivalent tip relief If no resulting tip relieves are specified the equivalent tip relives may be calculated as an approxima-tion based on the tool crowning Ca tool (per mille of tool module m0) as follows
2root1aeq1a CCC +=
1root2aeq2a CCC +=
where
Classification Notes- No 412 35 May 2003
DET NORSKE VERITAS
2
0
n0101atoolal m
)m2(mA1middotmCC
minus+minussdot=
( )
sdotminusminussdotsdotminus+= vtvb1
2vb1
21vavt1n1 αtan dddαsin 05x1mA
2
0
n020atool22a m
)mm(2A1middotmCC
minus+minussdot=
( )
sdotminusminussdotsdotminus+= vtvb2
2vb2
2va2vt2n2 αtan dddαsin 05x1mA
2
0
20atool1root1 m
A1mCC
minussdotsdot=
2
0
10atool2root2 m
A1mCC
minussdotsdot=
If the pressure angles of the cutter blades are modified (and verified) to balance the tip relieves the following may be assumed
( )eq2aeq1aeq2aeq1a C andC calculated of sum50CC sdot==
Throughout the following Ca1 and Ca2 mean the equivalent tip relieves Ca1eq and Ca2eq
434 Bulk temperature The bulk temperature may be calculated as
flaaveragempsoilMB X X 50 ϑ+ϑ=ϑ
where
Xs = lubrication factor
= 12 for spray lubrication
= 10 for dip lubrication (provided both pinion and wheel are dip lubricated and tip speed lt 5 ms)
= 10 for spray lubrication with additional cooling spray (spray on both pinion and wheel or spray on pinion and dip lubrication of wheel)
= 02 for meshes fully submerged in oil
Xmp = contact factor ( )pmp nX += 150
np = number of mesh contact on the pinion (for small gear ratii the number of wheel meshes should be used if higher)
flaaverageϑ
= average of the integrated flash temperature (see 44) along the path of contact
( )
AE
E
Ayyyfla
flaaverage
d
ΓminusΓ
ΓΓϑ=ϑint =
For high speed gears (v gt 50 ms) it may be necessary to assess the bulk temperature on the basis of a thermal rating of the entire gear transmission
44 The Flash Temperature flaϑ
441 Basic formula The local flash temperature flaϑ may be calculated as
( ) 41redy
y2ly
211
43Btcorrfla
unXwX3250
y ρ
ρminusρ
micro=ϑ Γ
(For bevel gears replace u with uv)
and is to be calculated stepwise along the path of contact from A to E
where
micro = coefficient of friction see 431
Xcorr = correction factor taking empirically into account the increased scuffing risk in the beginning of the approach path due to mesh starting without any previously built up oil film and possible shuffling away oil before meshing if insufficient tip relief
( )
3
AD
yaeffcorr 50
CC1X
ΓminusΓε
Γminus+=
α
Ca = tip relief of driven member Xcorr is only applicable in the approach path and if Ca lt Ceff otherwise 10
wBt = unit load see 431
yXΓ = load sharing factor see 443
n1 = pinion rpm
Γy ρly etc see 442
442 Geometrical relations The various radii of flank curvature (transversal plane) are
ρ1y = pinion flank radius at mesh point y
ρ2y = wheel flank radius at mesh point y
ρredy = equivalent radius of curvature at mesh point y
y2y1
y2y1redy
ρ+ρ
ρρ=ρ
Cylindrical gears
twy
1y αsin au1Γ1
ρ+
+=
twy
2y αsin au1Γu
ρ+
minus=
Note that for internal gears a and u are negative
36 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
Bevel gears
vtvv
y1y αsin a
u1Γ1
ρ+
+=
vtvv
yv2y αsin a
u1Γu
ρ+
minus=
Γ is the parameter on the path of contact and y is any point between A end E
At the respective ends Γ has the following values
Root piniontip wheel
Cylindrical gears
( )
minus
minusminus= 1
αtan 1dd
zzΓ
tw
2b2a2
1
2A
Bevel gears
( )
minus
minusminus= 1
αtan 1dd
uΓvt
2vb2va2
vA
Tip pinionroot wheel
Cylindrical gears
( )
1αtan
1ddΓ
tw
2b1a1
E minusminus
=
Bevel gears
( )
1αtan
1ddΓ
vt
2vb1va1
E minusminus
=
At inner point of single pair contact
Cylindrical gears
tw1
EB α tan zπ2ΓΓ minus=
Bevel gears
vtv1
EB α tan zπ2ΓΓ minus=
At outer point of single pair contact
Cylindrical gears
tw1AD α tan z
π2ΓΓ +=
Bevel gears
vt v1
AD αtan z π2ΓΓ +=
At pitch point ΓC = 0
The points F and G (only applicable to cylindrical gears) limiting the extension of tip relief (so as to maintain a mini-mum contact ratio of unity for unloaded gears) are at
2
BAF
Γ+Γ=Γ
2
EDG
Γ+Γ=Γ
443 Load sharing factor XΓ The load sharing factor XΓ accounts for the load sharing be-tween the various pairs of teeth in mesh along the path of contact
XΓ is to be calculated stepwise from A to E using the pa-rameter Γy
4431 Cylindrical gears with β = 0 and no tip relief
Figure 41
ByAAB
Ay for31
31X
yΓltΓleΓ
ΓminusΓ
ΓminusΓ+=Γ
DyB for1Xy
ΓltΓleΓ=Γ
EyDDE
yE for31
31X
yΓleΓltΓ
ΓminusΓ
ΓminusΓ+=Γ
4432 Cylindrical gears with β = 0 and tip relief Tip relief on the pinion reduces XΓ in the range G ndash E and increases correspondingly XΓ in the range F ndash B
Tip relief on the wheel reduces XΓ in the range A ndash F and increases correspondingly XΓ in the range D ndash G
Following remains generally
DyB for1Xy
ΓleΓleΓ=Γ
21XXGF== ΓΓ
In the following it must be distinguished between Ca lt Ceff respectively Ca gt Ceff This is shown by an example below where Ca1 lt Ceff and Ca2 gt Ceff
Classification Notes- No 412 37 May 2003
DET NORSKE VERITAS
Figure 42
Note When Ca gt Ceff the path of contact is shortened by A ndash A respectively E ndash E The single pair contact path is extended into B respectively D If this shift is significant it is necessary to consider the negative effect on surface durability (B) and bending stresses (B and D)
Range A - F
For effa2 CC le
minus=Γ
eff
2a
CC1
31X
A
FyAeff
2a
AF
Ay for C3
C61XX
AyΓleΓleΓ
sdot
+ΓminusΓ
ΓminusΓ+= ΓΓ
For effa2 CC ge
AyAfor0Xy
ΓleΓleΓ=Γ
with ( )21
CC
1CC
eff
2a
eff
2a
AFAAminus
minusΓminusΓ+Γ=Γ
FyAeff
2a
AF
Ay
eff
2a for 21
CC
CC1X
yΓleΓleΓ
minus
ΓminusΓ
ΓminusΓ+minus=Γ
Range F - B
For effa1 CC le
ByFeff
1a
FB
Fy forC3
C61
21X
yΓleΓleΓ
sdot
+ΓminusΓ
ΓminusΓ+=Γ
For effa1 CC ge
ByFeff
1a
FB
Fy for21
CC
21X
yΓleΓleΓ
minus
ΓminusΓ
ΓminusΓ+=Γ
ByB for1Xy
ΓleΓleΓ=Γ
with 1
CC2
eff
1a
FBFB
minus
ΓminusΓ+Γ=Γ
Range D ndash G
For effa2 CC le
GyDeff
2a
DG
Dy
eff
2a for C 3C
61
C 3C
32X
yΓleΓleΓ
+
ΓminusΓ
ΓminusΓminus+=Γ F
or effa2 CC ge
DyD for1Xy
ΓleΓleΓ=Γ
with ( )21
CC
1CC
eff
2a
eff
2a
DGDDminus
minusΓminusΓ+Γ=Γ
GyDeff
2a
DG
Dy
eff
2a for21
CC
CCX ΓleΓleΓ
minus
ΓminusΓ
ΓminusΓminus=Γ
Range G - E
For effa1 CC le
minus=Γ
eff
1a
CC1
31X
E
EyGeff
1a
GE
Gy forC3
C61
21X
yΓleΓleΓ
sdot
+ΓminusΓ
ΓminusΓminus=Γ
For effa1 CC gt
EyGeff
1a
GE
Gy for21
CC
21X
yΓleΓleΓ
minus
ΓminusΓ
ΓminusΓminus=Γ
EyE for0Xy
ΓleΓleΓ=Γ
with 1
CC2
eff
1a
GEGE
minus
ΓminusΓ+Γ=Γ
4433 Gears with β gt 0 buttressing Due to oblique contact lines over the flanks a certain but-tressing may occur near A and E
This applies to both cylindrical and bevel gears with tip relief lt Ceff The buttressing Xbutt is simplified as a linear function within the ranges A ndash H respectively I ndash E
Figure 43
1εwhen13X βbutt EAge=
1εwhenε031X ββbutt EAlt+=
38 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
Cylindrical gears
bIEAH βsin 02ΓΓΓΓ =minus=minus
Bevel gears
bmIEAH βsin 02ΓΓΓΓ =minus=minus
4434 Cylindrical gears with 2εγ le and no tip relief
yXΓ is obtained by multiplication of
yXΓ in 4431 with
Xbutt in 4433
4435 Gears with 2εγ gt and no tip relief
Applicable to both cylindrical and bevel gears
IyH for1Xy
ΓleΓleΓε
=α
Γ
IyHybutt and forX1Xy
ΓgtΓΓltΓε
=α
Γ
Figure 44
4436 Cylindrical gears with 2εγ le and tip relief
yXΓ is obtained by multiplication of
yXΓ in 4432 with Xbutt
in 4433
4437 Cylindrical gears with 2εγ gt and tip relief
Tip relief on the pinion (respectively wheel) reduces XГ in the range G ndash E (respectively A ndash F) and increases XГ in the range F ndash G
yXΓ is obtained by multiplication of
yXΓ as described below
with Xbutt in 4433
In the XГ example below the influence of tip relief is shown (without the influence of Xbutt) by means of
eff2aeff1a CC and CC ltgt
Tip relief gt Ceff causes new end points A respectively E of the path of contact
Figure 45
Range A ndash F
( ) ( )( ) eff
a2αa1α
AF
Ay
eff
2aeff
C 12C 13εC 1ε
CCCX
y +εε++minus
ΓminusΓ
ΓminusΓ+
εminus
=ααα
Γ
eff2aFyA CC if for leΓleΓleΓ
eff2aFyA CifCfor and geΓleΓleΓ
eff2aAyA CC iffor0Xy
gtΓleΓleΓ=Γ
( ) ( ) ( )( ) ( ) a2αa1α
αeffa2AFAA C 1ε 3C 1ε
1ε 2 CC with++minus+minus
ΓminusΓ+Γ=Γ
Range F ndash G
( )( )( ) GyF
eff
2a1a forC 1 2CC 11X
yΓleΓleΓ
+εε+minusε
+ε
=αα
α
αΓ
Range G ndash E
( ) ( )( ) eff
2a1a
GE
Gy
C 1 2C 1C 1 3XX
GFy +εεminusε++ε
ΓminusΓ
ΓminusΓminus=
αα
ααΓΓ minus
effa1EyG CC if for leΓleΓleΓ
effa1EyG CC if Γfor and geΓleΓle
eff1aEyE CC if for0Xy
geΓleΓleΓ=Γ
( ) ( ) ( )( ) ( ) 2a1a
eff1aGEEE C 1C 1 3
1 2 CC withminusε++ε+εminus
ΓminusΓminusΓ=Γαα
α
4438 Bevel gears with εγ more than approx 18 and heightwise crowning For Ca1 = Ca2 = Ceff the following applies
Figure 46
Classification Notes- No 412 39 May 2003
DET NORSKE VERITAS
( )AEM 50 Γ+Γ=Γ
( )( )2AD
3
2My651X
y ΓminusΓε
ΓminusΓminus
ε=
ααΓ
For tip relief lt Ceff yXΓ is found by linear interpolation be-
tween ( )effay CCX =Γ and ( )0CayX =Γ as in 4435
The interpolation is to be made stepwise from A to M with the influence of Ca2 and from M to E with the influence of Ca1 (For 2a1a CC ne there is a discontinuity at M)
Eg with Ca1 = 04 Ceff and Ca2 = 055 Ceff then
Range A ndash M
)CC()0C( eff2ay2ayyX 550X 450X =Γ=ΓΓ +=
Range M ndash E
)CC()0C( eff1ay1ayyX 40X 06X =Γ=ΓΓ +=
For tip relief gt Ceff the new end points A and E are found as
( )
minusΓminusΓ
ε+Γ=Γ α 1
CC
6 eff
2aADAA
( )
minusΓminusΓ
ε+Γ=Γ α 1
CC
6 eff
1aADEE
Range A ndash A
0Xy=Γ
Range A ndash M
( )( )
ΓminusΓ
ΓminusΓminus
minusε=
αΓ 2
MA
2My
eff
2a1
CC4
351Xy
Range M ndash E
( )( )
ΓminusΓ
ΓminusΓminus
minusε=
αΓ 2
ME
2My
eff
1a1
CC4
351Xy
Range E ndash E
0Xy=Γ
40 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
Appendix A Fatigue Damage Accumulation
The Palmgren-Miner cumulative damage calculation princi-ple is used The procedure may be applied as follows
A1 Stress Spectrum From the individual torque classes the torques (Ti) at the peak values of class intervals and the associated number of cycles (NLi) for both pinion and wheel are to be listed from the highest to the lowest torque
(In case of a cyclic torque variation within the torque classes it is advised to use the peak torque If the cyclic variation is such that the same teeth will repeatedly suffer the peak torque this is a must)
The stress spectra for tooth roots and flanks (σFi σHi) with all relevant factors (except KA) are to be calculated on the basis of the torque spectrum The load dependent K-factors are to be determined for each torque class
A2 σminusN-curve The stress versus load cycle curves for tooth roots and flanks (both pinion and wheel) are to be drawn on the basis of per-missible stresses (ie including the demanded minimum safety factors) as determined in 2 respectively 3 If different safety levels for high cycle fatigue and low cycle fatigue are desired this may be expressed by different demand safety factors applied at the endurance limit respectively at static strength
A3 Damage accumulation The individual damage ratio Di at ith stress level is defined as
Fi
Lii N
ND =
where
NLi = The number of applied cycles at ith stress
NFi = The number of cycles to failure at ith stress
Basically stresses σi below the permissible stress level for infinite life (if a constant ZN or YN is accepted) do not con-tribute to the damage sum However calculating the actual safety factor Sact as described below all the σi for which the product S σi is bigger than or equal to the permissible stress level for infinite life contribute to the damage sum and thus to the determination of Sact The final value of S is decisive
(NFi can be found mathematically by putting the permissible stress σpi equal the actual stress σi thereby finding the actual life factor This life factor can be solved with regard to load cycles ie NFi)
The damage sum ΣDi is not to exceed unity
If ΣDi ne 1 the safety against cumulative fatigue damage is different from the applied demand safety factor For deter-mination of this theoretical safety factor an iteration proce-dure is required as described in the following flowchart
S is correction factor with which the actual safety factor Sact can be found
Sact is the demand safety factor (used in determination of the permissible stresses in the σ ndash N ndash curve) times the correc-tion factor S
The full procedure is to be applied for pinion and wheel tooth roots and flanks
Note If alternating stresses occur in a spectrum of mainly pulsating stresses the alternating stresses may be replaced by equiva-lent pulsating stresses ie by means of division with the ac-tual mean stress influence factor YM
Classification Notes- No 412 41 May 2003
DET NORSKE VERITAS
Appendix B Application Factors for Diesel Driven Gears
For diesel driven gears the application factor KA depends on torsional vibrations Both normal operation and misfiring conditions have to be considered
Normally these two running conditions can be covered by only one calculation
B1 Definitions
Normal operation KAnorm = 0
normv0
TTT +
where
T0 = rated nominal torque
Tv norm = vibratory torque amplitude for normal opera-tion (see rules Pt4 Ch3 Sec1 G301 for defi-nition of ldquonormalrdquo irregularity)
Misfiring operation KA misf = 0
misfv
TTT +
where
T = remaining nominal torque when one cylinder out of action
Tv misf = vibratory torque amplitude in misfiring con-dition This refers to a permissible misfiring condition ie a condition that does not require automatic or immediate corrective actions as speed or pitch reduction
The normal operation is assumed to last for a very high num-ber of cycles such as 1010
The misfiring operation is assumed to last for a limited dura-tion such as 107 cycles
B2 Determination of decisive load Assuming life factor at 1010 cycles as YN = ZN = 092 which usually is relevant the calculation may be performed only once with the combination having the highest value of appli-cation factorlife factor
For bending stresses and scuffing the higher value of
092
K normA and 098
K misfA
For contact stresses the higher value of
092K normA and
113K misfA (but
097K misfA for nitrided gears)
B3 Simplified procedure Note that this is only a guidance and is not a binding con-vention
T = oTZ
1Zsdot
minus where Z = number of cylinders
( ) ideal Videal Vmisf Vnorm V TTT24ZT +minus=
where TV ideal = vibratory torque with all cylinders perfectly equal See also rules Pt4 Ch3 Sec1 G300
When using trends from torsional vibration analysis and measurements the following may be used
TV ideal To is close to zero for engines with few cylinders and using a suitable elastic coupling and increases with relative coupling stiffness and number of cylinders
TV misf To may be high for engines with few cylinders and decreases with number of cylinders
This can be indicated as
200Z
TT
ο
idealV asymp and 80Z04
TT
o
misfV minusasymp
Inserting this into the formulae for the two application fac-tors the following guidance can be given
118112K misfA minusasymp
115110K normA minusasymp
Since KA norm is to be combined with the lower life factors the decisive load condition will be the normal one and a KA of 115 will cover most relevant cases when a suitable elas-tic coupling is chosen
42 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
Appendix C Calculation of Pinion-Rack
Pinion-racks used for elevating of mobile offshore units are open gears that are subjected to wear and tear With normal specifications such as only a few hundred total operation cycles (site to site) the tooth bending stresses for static strength will be decisive for the lay out however with ex-ception of surface hardened pinions where case crushing has to be considered
In the following the use of part 1 and 3 for pinion-racks is shown including relevant simplifications
C1 Pinion tooth root stresses Since the load spectrum normally is dominated by high torques and few load cycles (in the range up to 1-2000) the static strength is decisive
The actual stress is calculated as
β1FSaFan1
tF1 KYY
mbFσ sdotsdotsdotsdot
=
b1 is limited to b2 + 2middotmn
YFa and YSa replace YF and YS because load application at tooth tip has to be assumed for such inaccurate gears
Pinions often use a non-involute profile in the dedendum part of the flank eg a constant radius equal the radius of curva-ture at reference circle For such pinions sFn and hFa are to be measured directly on a sectional drawing of the pinion tooth
Due to high loads and narrow facewidths it may be assumed that KFβ2 = KHβ = 10 However when b1 gt b2 then KFβ1 gt10
If no detailed documentation of KFβ1 is available the fol-lowing may be used
KFβ1 = 1 + 015middot(b1b2 ndash 1)
The permissible stress (not surface hardened) is calculated as
δrelTstF
Fst1FP1 Y
Sσσ sdot=
The mean stress influence due to leg lifting may be disre-garded
The actual and permissible stresses should be calculated for the relevant loads as given in the rules
C2 Rack tooth root stresses The actual stress is calculated as
SaFan2
tF2 YY
mbFσ sdotsdotsdot
=
See C1 for details
The permissible stress is calculated as
δrelTstF
Fst2FP2 Y
Sσσ sdot=
For alloyed steels (Ni Cr Mo) with high toughness and duc-tility the value of YδrelTst may be put equal to YSa
C3 Surface hardened pinions For surface hardened pinions the maximum load is not to cause crushing of the hardened layer of the flank
In principle the calculation described in 213 may be used but when the theoretical Hertzian stress exceeds the ap-proximately 18 times the yield strength of the rack material plastic deformation will occur This will limit the peak Hertzian stress but increases the contact width and thus the penetration of stresses into the depth
An approximation may be based on an assessment of contact width determined by means of equal areas under the theoreti-cal (elastic range) Hertzian contact and the elasto-plastic contact stress (limited to 18 middot σy) with the unknown width
Classification Notes- No 412 33 May 2003
DET NORSKE VERITAS
4 Calculation of Scuffing Load Capacity
41 Introduction High surface temperatures due to high loads and sliding ve-locities can cause lubricant films to break down This sei-zure or welding together of areas of tooth surface is termed scuffing
In contrast to pitting and fatigue breakage which show a dis-tinct incubation period a single short overloading can lead to a scuffing failure In the ISO-TR13989 two criteria are mentioned The method used in this Classification Note is based on the principles of the flash temperature criterion
Note Bulk temperature in excess of 120ordmC for long periods may have an adverse effect on the surface durability see 211
42 General Criteria In no point along the path of contact the local contact tem-perature may exceed the permissible temperature ie
oilS
oilSB S
ϑ+ϑminusϑ
leϑ
50SB minusϑleϑ
where
Bϑ = max contact temperature along the path of contact
maxflaMBB ϑ+ϑ=ϑ
MBϑ = bulk temperature see 434
maxflaϑ = max flash temperature along the path of con-tact see 44
Sϑ = scuffing temperature see below
oilϑ = oil temperature before it reaches the mesh (max applicable for the actual load case to be used ie normally alarm temperature except for ice classes where a max expected tem-perature applies)
SS = required safety factor according to the Rules
The scuffing temperature Sϑ may be calculated as
L2
wrelT
002
40S XFZGX
ν100112085780
sdot
sdot++=ϑ
where
XwrelT = relative welding factor
XwrelT
Through hardened steel 10
Phosphated steel 125
Copper-plated steel 150
Nitrided steel 150
Less than 10 retained austenite
115
10 ndash 20 retained austenite 10
Casehardened steel
20 ndash 30 retained austenite 085
Austenitic steel 045
FZG = load stage according to FZG-Test A8390 (Note This is the load stage where scuffing oc-curs However due to scatter in test results cal-culations are to be made with one load stage less than the specification)
XL = lubricant factor
= 10 for mineral oils
= 08 for polyalfaolefins
= 07 for non-water-soluble polyglycols
= 06 for water-soluble polyglycols
= 15 for traction fluids
= 13 for phosphate esters
ν40 = kinematic oil viscosity at 40˚C (mm2s)
Application of other test methods such as the Ryder the FZG-Ryder R46574 and the FZG L-42 Test 141195110 may be specially considered
For high speed gears with very short time of contact Sϑ may be increased as follows provided use of EP-oils
Addition to the calculated scuffing temperature Sϑ
If micros 18tc ge no addition
If micros 18tc lt add ( )cwrelT t18X18 minussdotsdot
where
ct = contact time ( micros ) which is the time needed to cross the Hertzian contact width
( ) [ ]microsu1cosβn
uσ340tb1
Hc +sdotsdot
sdot=
σH as calculated in 221
For bevel gears use vu in stead of u
34 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
43 Influence Factors
431 Coefficient of friction The following coefficient of friction may apply
L025a
005oil
02
redC ΣC
Bt X R ηρv
w0048micro minus
=
where
wBt = specific tooth load (Nmm)
vΣC = sum of tangential velocities at pitch point At pitch line velocities gt 50 ms the limiting value of vΣC at v = 50 ms is to be used
ρredC = relative radius of curvature (transversal plane) at the pitch point
Cylindrical gears
HαHβvγAbt
Bt KKKKKb
Fw sdotsdotsdotsdotsdot= (see 1)
twΣC αsin v2v =
bCredC β cos ρρ =
Bevel gears
HαHβvγAtmb
Bt KKKKKb
Fw sdotsdotsdotsdotsdot= (see 1)
vtmtΣC αsin v2v =
bmvCredC β cos ρρ =
ηoil = dynamic viscosity (mPa s) at oilϑ calculated as
1000
ρ νη oiloil =
where ρ in kgm3 approximated as
( ) 70 15oil15 minusϑminusρ=ρ and νoil is kinematic viscosity at oilϑ and may be calculated by means of the following equation
( ) ( )++=+ 08νloglog08νloglog 100oil ( )
sdotminus
ϑ+minus313log373log
273log373log oil
( ) ( )( )08νloglog08νloglog 10040 +minus+
Ra = composite arithmetic mean roughness (micron) of pinion an wheel calculated as ( )2a1aa RR 05R +=This is defined as the roughness on the new flanks ie as manufactured
XL = see 42
432 Effective tip relief Ceff Ceff is the effective tip relief that amount of tip relief which compensates for the elastic deformation of the gear mesh ie
zero load at the tooth tip It is assumed (simplified) to be equal for both pinion and wheel
Cylindrical gears
for helical
γ
γ=cb
KKFC Abt
eff (see 1)
For spur
cbKKF
C Abteff
γ= (see 1)
Alternatively for spur and helical gears the non-linear ap-proach in 111 may be used (taking Ceff=δ)
Bevel gears
bcKFC Ambt
effγ
= (see 1)
where
γ
α
ε2ε44c
+sdot
=γ
433 Tip relief and extension Cylindrical gears
The extension of the tip relief is not to result in an effective contact ratio 1ltεα when the gear is unloaded (exceptions to this may only apply for applications where the gear is not to run at light loads) This means that the unrelieved part of the path of contact is to be minimum pbt It is further assumed that this unrelieved part is placed centrally on the path of contact
If root relief applies it has to be calculated as equivalent tip relief Ie pinion root relief (at mesh position A) is added to Ca2 and wheel root relief (at mesh position E) is added to Ca1 If no design tip relief or root relief on the mating gear is specified (ie if Ca1+Croot2 = 0 and visa versa) use the run-ning in amount see 112
Bevel gears
Bevel gears are to have heightwise crowning ie no distinct relievedunrelieved area This may be treated as tip and root relief For calculation purposes the root relief is combined with the tip relief of the mating member into an equivalent tip relief If no resulting tip relieves are specified the equivalent tip relives may be calculated as an approxima-tion based on the tool crowning Ca tool (per mille of tool module m0) as follows
2root1aeq1a CCC +=
1root2aeq2a CCC +=
where
Classification Notes- No 412 35 May 2003
DET NORSKE VERITAS
2
0
n0101atoolal m
)m2(mA1middotmCC
minus+minussdot=
( )
sdotminusminussdotsdotminus+= vtvb1
2vb1
21vavt1n1 αtan dddαsin 05x1mA
2
0
n020atool22a m
)mm(2A1middotmCC
minus+minussdot=
( )
sdotminusminussdotsdotminus+= vtvb2
2vb2
2va2vt2n2 αtan dddαsin 05x1mA
2
0
20atool1root1 m
A1mCC
minussdotsdot=
2
0
10atool2root2 m
A1mCC
minussdotsdot=
If the pressure angles of the cutter blades are modified (and verified) to balance the tip relieves the following may be assumed
( )eq2aeq1aeq2aeq1a C andC calculated of sum50CC sdot==
Throughout the following Ca1 and Ca2 mean the equivalent tip relieves Ca1eq and Ca2eq
434 Bulk temperature The bulk temperature may be calculated as
flaaveragempsoilMB X X 50 ϑ+ϑ=ϑ
where
Xs = lubrication factor
= 12 for spray lubrication
= 10 for dip lubrication (provided both pinion and wheel are dip lubricated and tip speed lt 5 ms)
= 10 for spray lubrication with additional cooling spray (spray on both pinion and wheel or spray on pinion and dip lubrication of wheel)
= 02 for meshes fully submerged in oil
Xmp = contact factor ( )pmp nX += 150
np = number of mesh contact on the pinion (for small gear ratii the number of wheel meshes should be used if higher)
flaaverageϑ
= average of the integrated flash temperature (see 44) along the path of contact
( )
AE
E
Ayyyfla
flaaverage
d
ΓminusΓ
ΓΓϑ=ϑint =
For high speed gears (v gt 50 ms) it may be necessary to assess the bulk temperature on the basis of a thermal rating of the entire gear transmission
44 The Flash Temperature flaϑ
441 Basic formula The local flash temperature flaϑ may be calculated as
( ) 41redy
y2ly
211
43Btcorrfla
unXwX3250
y ρ
ρminusρ
micro=ϑ Γ
(For bevel gears replace u with uv)
and is to be calculated stepwise along the path of contact from A to E
where
micro = coefficient of friction see 431
Xcorr = correction factor taking empirically into account the increased scuffing risk in the beginning of the approach path due to mesh starting without any previously built up oil film and possible shuffling away oil before meshing if insufficient tip relief
( )
3
AD
yaeffcorr 50
CC1X
ΓminusΓε
Γminus+=
α
Ca = tip relief of driven member Xcorr is only applicable in the approach path and if Ca lt Ceff otherwise 10
wBt = unit load see 431
yXΓ = load sharing factor see 443
n1 = pinion rpm
Γy ρly etc see 442
442 Geometrical relations The various radii of flank curvature (transversal plane) are
ρ1y = pinion flank radius at mesh point y
ρ2y = wheel flank radius at mesh point y
ρredy = equivalent radius of curvature at mesh point y
y2y1
y2y1redy
ρ+ρ
ρρ=ρ
Cylindrical gears
twy
1y αsin au1Γ1
ρ+
+=
twy
2y αsin au1Γu
ρ+
minus=
Note that for internal gears a and u are negative
36 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
Bevel gears
vtvv
y1y αsin a
u1Γ1
ρ+
+=
vtvv
yv2y αsin a
u1Γu
ρ+
minus=
Γ is the parameter on the path of contact and y is any point between A end E
At the respective ends Γ has the following values
Root piniontip wheel
Cylindrical gears
( )
minus
minusminus= 1
αtan 1dd
zzΓ
tw
2b2a2
1
2A
Bevel gears
( )
minus
minusminus= 1
αtan 1dd
uΓvt
2vb2va2
vA
Tip pinionroot wheel
Cylindrical gears
( )
1αtan
1ddΓ
tw
2b1a1
E minusminus
=
Bevel gears
( )
1αtan
1ddΓ
vt
2vb1va1
E minusminus
=
At inner point of single pair contact
Cylindrical gears
tw1
EB α tan zπ2ΓΓ minus=
Bevel gears
vtv1
EB α tan zπ2ΓΓ minus=
At outer point of single pair contact
Cylindrical gears
tw1AD α tan z
π2ΓΓ +=
Bevel gears
vt v1
AD αtan z π2ΓΓ +=
At pitch point ΓC = 0
The points F and G (only applicable to cylindrical gears) limiting the extension of tip relief (so as to maintain a mini-mum contact ratio of unity for unloaded gears) are at
2
BAF
Γ+Γ=Γ
2
EDG
Γ+Γ=Γ
443 Load sharing factor XΓ The load sharing factor XΓ accounts for the load sharing be-tween the various pairs of teeth in mesh along the path of contact
XΓ is to be calculated stepwise from A to E using the pa-rameter Γy
4431 Cylindrical gears with β = 0 and no tip relief
Figure 41
ByAAB
Ay for31
31X
yΓltΓleΓ
ΓminusΓ
ΓminusΓ+=Γ
DyB for1Xy
ΓltΓleΓ=Γ
EyDDE
yE for31
31X
yΓleΓltΓ
ΓminusΓ
ΓminusΓ+=Γ
4432 Cylindrical gears with β = 0 and tip relief Tip relief on the pinion reduces XΓ in the range G ndash E and increases correspondingly XΓ in the range F ndash B
Tip relief on the wheel reduces XΓ in the range A ndash F and increases correspondingly XΓ in the range D ndash G
Following remains generally
DyB for1Xy
ΓleΓleΓ=Γ
21XXGF== ΓΓ
In the following it must be distinguished between Ca lt Ceff respectively Ca gt Ceff This is shown by an example below where Ca1 lt Ceff and Ca2 gt Ceff
Classification Notes- No 412 37 May 2003
DET NORSKE VERITAS
Figure 42
Note When Ca gt Ceff the path of contact is shortened by A ndash A respectively E ndash E The single pair contact path is extended into B respectively D If this shift is significant it is necessary to consider the negative effect on surface durability (B) and bending stresses (B and D)
Range A - F
For effa2 CC le
minus=Γ
eff
2a
CC1
31X
A
FyAeff
2a
AF
Ay for C3
C61XX
AyΓleΓleΓ
sdot
+ΓminusΓ
ΓminusΓ+= ΓΓ
For effa2 CC ge
AyAfor0Xy
ΓleΓleΓ=Γ
with ( )21
CC
1CC
eff
2a
eff
2a
AFAAminus
minusΓminusΓ+Γ=Γ
FyAeff
2a
AF
Ay
eff
2a for 21
CC
CC1X
yΓleΓleΓ
minus
ΓminusΓ
ΓminusΓ+minus=Γ
Range F - B
For effa1 CC le
ByFeff
1a
FB
Fy forC3
C61
21X
yΓleΓleΓ
sdot
+ΓminusΓ
ΓminusΓ+=Γ
For effa1 CC ge
ByFeff
1a
FB
Fy for21
CC
21X
yΓleΓleΓ
minus
ΓminusΓ
ΓminusΓ+=Γ
ByB for1Xy
ΓleΓleΓ=Γ
with 1
CC2
eff
1a
FBFB
minus
ΓminusΓ+Γ=Γ
Range D ndash G
For effa2 CC le
GyDeff
2a
DG
Dy
eff
2a for C 3C
61
C 3C
32X
yΓleΓleΓ
+
ΓminusΓ
ΓminusΓminus+=Γ F
or effa2 CC ge
DyD for1Xy
ΓleΓleΓ=Γ
with ( )21
CC
1CC
eff
2a
eff
2a
DGDDminus
minusΓminusΓ+Γ=Γ
GyDeff
2a
DG
Dy
eff
2a for21
CC
CCX ΓleΓleΓ
minus
ΓminusΓ
ΓminusΓminus=Γ
Range G - E
For effa1 CC le
minus=Γ
eff
1a
CC1
31X
E
EyGeff
1a
GE
Gy forC3
C61
21X
yΓleΓleΓ
sdot
+ΓminusΓ
ΓminusΓminus=Γ
For effa1 CC gt
EyGeff
1a
GE
Gy for21
CC
21X
yΓleΓleΓ
minus
ΓminusΓ
ΓminusΓminus=Γ
EyE for0Xy
ΓleΓleΓ=Γ
with 1
CC2
eff
1a
GEGE
minus
ΓminusΓ+Γ=Γ
4433 Gears with β gt 0 buttressing Due to oblique contact lines over the flanks a certain but-tressing may occur near A and E
This applies to both cylindrical and bevel gears with tip relief lt Ceff The buttressing Xbutt is simplified as a linear function within the ranges A ndash H respectively I ndash E
Figure 43
1εwhen13X βbutt EAge=
1εwhenε031X ββbutt EAlt+=
38 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
Cylindrical gears
bIEAH βsin 02ΓΓΓΓ =minus=minus
Bevel gears
bmIEAH βsin 02ΓΓΓΓ =minus=minus
4434 Cylindrical gears with 2εγ le and no tip relief
yXΓ is obtained by multiplication of
yXΓ in 4431 with
Xbutt in 4433
4435 Gears with 2εγ gt and no tip relief
Applicable to both cylindrical and bevel gears
IyH for1Xy
ΓleΓleΓε
=α
Γ
IyHybutt and forX1Xy
ΓgtΓΓltΓε
=α
Γ
Figure 44
4436 Cylindrical gears with 2εγ le and tip relief
yXΓ is obtained by multiplication of
yXΓ in 4432 with Xbutt
in 4433
4437 Cylindrical gears with 2εγ gt and tip relief
Tip relief on the pinion (respectively wheel) reduces XГ in the range G ndash E (respectively A ndash F) and increases XГ in the range F ndash G
yXΓ is obtained by multiplication of
yXΓ as described below
with Xbutt in 4433
In the XГ example below the influence of tip relief is shown (without the influence of Xbutt) by means of
eff2aeff1a CC and CC ltgt
Tip relief gt Ceff causes new end points A respectively E of the path of contact
Figure 45
Range A ndash F
( ) ( )( ) eff
a2αa1α
AF
Ay
eff
2aeff
C 12C 13εC 1ε
CCCX
y +εε++minus
ΓminusΓ
ΓminusΓ+
εminus
=ααα
Γ
eff2aFyA CC if for leΓleΓleΓ
eff2aFyA CifCfor and geΓleΓleΓ
eff2aAyA CC iffor0Xy
gtΓleΓleΓ=Γ
( ) ( ) ( )( ) ( ) a2αa1α
αeffa2AFAA C 1ε 3C 1ε
1ε 2 CC with++minus+minus
ΓminusΓ+Γ=Γ
Range F ndash G
( )( )( ) GyF
eff
2a1a forC 1 2CC 11X
yΓleΓleΓ
+εε+minusε
+ε
=αα
α
αΓ
Range G ndash E
( ) ( )( ) eff
2a1a
GE
Gy
C 1 2C 1C 1 3XX
GFy +εεminusε++ε
ΓminusΓ
ΓminusΓminus=
αα
ααΓΓ minus
effa1EyG CC if for leΓleΓleΓ
effa1EyG CC if Γfor and geΓleΓle
eff1aEyE CC if for0Xy
geΓleΓleΓ=Γ
( ) ( ) ( )( ) ( ) 2a1a
eff1aGEEE C 1C 1 3
1 2 CC withminusε++ε+εminus
ΓminusΓminusΓ=Γαα
α
4438 Bevel gears with εγ more than approx 18 and heightwise crowning For Ca1 = Ca2 = Ceff the following applies
Figure 46
Classification Notes- No 412 39 May 2003
DET NORSKE VERITAS
( )AEM 50 Γ+Γ=Γ
( )( )2AD
3
2My651X
y ΓminusΓε
ΓminusΓminus
ε=
ααΓ
For tip relief lt Ceff yXΓ is found by linear interpolation be-
tween ( )effay CCX =Γ and ( )0CayX =Γ as in 4435
The interpolation is to be made stepwise from A to M with the influence of Ca2 and from M to E with the influence of Ca1 (For 2a1a CC ne there is a discontinuity at M)
Eg with Ca1 = 04 Ceff and Ca2 = 055 Ceff then
Range A ndash M
)CC()0C( eff2ay2ayyX 550X 450X =Γ=ΓΓ +=
Range M ndash E
)CC()0C( eff1ay1ayyX 40X 06X =Γ=ΓΓ +=
For tip relief gt Ceff the new end points A and E are found as
( )
minusΓminusΓ
ε+Γ=Γ α 1
CC
6 eff
2aADAA
( )
minusΓminusΓ
ε+Γ=Γ α 1
CC
6 eff
1aADEE
Range A ndash A
0Xy=Γ
Range A ndash M
( )( )
ΓminusΓ
ΓminusΓminus
minusε=
αΓ 2
MA
2My
eff
2a1
CC4
351Xy
Range M ndash E
( )( )
ΓminusΓ
ΓminusΓminus
minusε=
αΓ 2
ME
2My
eff
1a1
CC4
351Xy
Range E ndash E
0Xy=Γ
40 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
Appendix A Fatigue Damage Accumulation
The Palmgren-Miner cumulative damage calculation princi-ple is used The procedure may be applied as follows
A1 Stress Spectrum From the individual torque classes the torques (Ti) at the peak values of class intervals and the associated number of cycles (NLi) for both pinion and wheel are to be listed from the highest to the lowest torque
(In case of a cyclic torque variation within the torque classes it is advised to use the peak torque If the cyclic variation is such that the same teeth will repeatedly suffer the peak torque this is a must)
The stress spectra for tooth roots and flanks (σFi σHi) with all relevant factors (except KA) are to be calculated on the basis of the torque spectrum The load dependent K-factors are to be determined for each torque class
A2 σminusN-curve The stress versus load cycle curves for tooth roots and flanks (both pinion and wheel) are to be drawn on the basis of per-missible stresses (ie including the demanded minimum safety factors) as determined in 2 respectively 3 If different safety levels for high cycle fatigue and low cycle fatigue are desired this may be expressed by different demand safety factors applied at the endurance limit respectively at static strength
A3 Damage accumulation The individual damage ratio Di at ith stress level is defined as
Fi
Lii N
ND =
where
NLi = The number of applied cycles at ith stress
NFi = The number of cycles to failure at ith stress
Basically stresses σi below the permissible stress level for infinite life (if a constant ZN or YN is accepted) do not con-tribute to the damage sum However calculating the actual safety factor Sact as described below all the σi for which the product S σi is bigger than or equal to the permissible stress level for infinite life contribute to the damage sum and thus to the determination of Sact The final value of S is decisive
(NFi can be found mathematically by putting the permissible stress σpi equal the actual stress σi thereby finding the actual life factor This life factor can be solved with regard to load cycles ie NFi)
The damage sum ΣDi is not to exceed unity
If ΣDi ne 1 the safety against cumulative fatigue damage is different from the applied demand safety factor For deter-mination of this theoretical safety factor an iteration proce-dure is required as described in the following flowchart
S is correction factor with which the actual safety factor Sact can be found
Sact is the demand safety factor (used in determination of the permissible stresses in the σ ndash N ndash curve) times the correc-tion factor S
The full procedure is to be applied for pinion and wheel tooth roots and flanks
Note If alternating stresses occur in a spectrum of mainly pulsating stresses the alternating stresses may be replaced by equiva-lent pulsating stresses ie by means of division with the ac-tual mean stress influence factor YM
Classification Notes- No 412 41 May 2003
DET NORSKE VERITAS
Appendix B Application Factors for Diesel Driven Gears
For diesel driven gears the application factor KA depends on torsional vibrations Both normal operation and misfiring conditions have to be considered
Normally these two running conditions can be covered by only one calculation
B1 Definitions
Normal operation KAnorm = 0
normv0
TTT +
where
T0 = rated nominal torque
Tv norm = vibratory torque amplitude for normal opera-tion (see rules Pt4 Ch3 Sec1 G301 for defi-nition of ldquonormalrdquo irregularity)
Misfiring operation KA misf = 0
misfv
TTT +
where
T = remaining nominal torque when one cylinder out of action
Tv misf = vibratory torque amplitude in misfiring con-dition This refers to a permissible misfiring condition ie a condition that does not require automatic or immediate corrective actions as speed or pitch reduction
The normal operation is assumed to last for a very high num-ber of cycles such as 1010
The misfiring operation is assumed to last for a limited dura-tion such as 107 cycles
B2 Determination of decisive load Assuming life factor at 1010 cycles as YN = ZN = 092 which usually is relevant the calculation may be performed only once with the combination having the highest value of appli-cation factorlife factor
For bending stresses and scuffing the higher value of
092
K normA and 098
K misfA
For contact stresses the higher value of
092K normA and
113K misfA (but
097K misfA for nitrided gears)
B3 Simplified procedure Note that this is only a guidance and is not a binding con-vention
T = oTZ
1Zsdot
minus where Z = number of cylinders
( ) ideal Videal Vmisf Vnorm V TTT24ZT +minus=
where TV ideal = vibratory torque with all cylinders perfectly equal See also rules Pt4 Ch3 Sec1 G300
When using trends from torsional vibration analysis and measurements the following may be used
TV ideal To is close to zero for engines with few cylinders and using a suitable elastic coupling and increases with relative coupling stiffness and number of cylinders
TV misf To may be high for engines with few cylinders and decreases with number of cylinders
This can be indicated as
200Z
TT
ο
idealV asymp and 80Z04
TT
o
misfV minusasymp
Inserting this into the formulae for the two application fac-tors the following guidance can be given
118112K misfA minusasymp
115110K normA minusasymp
Since KA norm is to be combined with the lower life factors the decisive load condition will be the normal one and a KA of 115 will cover most relevant cases when a suitable elas-tic coupling is chosen
42 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
Appendix C Calculation of Pinion-Rack
Pinion-racks used for elevating of mobile offshore units are open gears that are subjected to wear and tear With normal specifications such as only a few hundred total operation cycles (site to site) the tooth bending stresses for static strength will be decisive for the lay out however with ex-ception of surface hardened pinions where case crushing has to be considered
In the following the use of part 1 and 3 for pinion-racks is shown including relevant simplifications
C1 Pinion tooth root stresses Since the load spectrum normally is dominated by high torques and few load cycles (in the range up to 1-2000) the static strength is decisive
The actual stress is calculated as
β1FSaFan1
tF1 KYY
mbFσ sdotsdotsdotsdot
=
b1 is limited to b2 + 2middotmn
YFa and YSa replace YF and YS because load application at tooth tip has to be assumed for such inaccurate gears
Pinions often use a non-involute profile in the dedendum part of the flank eg a constant radius equal the radius of curva-ture at reference circle For such pinions sFn and hFa are to be measured directly on a sectional drawing of the pinion tooth
Due to high loads and narrow facewidths it may be assumed that KFβ2 = KHβ = 10 However when b1 gt b2 then KFβ1 gt10
If no detailed documentation of KFβ1 is available the fol-lowing may be used
KFβ1 = 1 + 015middot(b1b2 ndash 1)
The permissible stress (not surface hardened) is calculated as
δrelTstF
Fst1FP1 Y
Sσσ sdot=
The mean stress influence due to leg lifting may be disre-garded
The actual and permissible stresses should be calculated for the relevant loads as given in the rules
C2 Rack tooth root stresses The actual stress is calculated as
SaFan2
tF2 YY
mbFσ sdotsdotsdot
=
See C1 for details
The permissible stress is calculated as
δrelTstF
Fst2FP2 Y
Sσσ sdot=
For alloyed steels (Ni Cr Mo) with high toughness and duc-tility the value of YδrelTst may be put equal to YSa
C3 Surface hardened pinions For surface hardened pinions the maximum load is not to cause crushing of the hardened layer of the flank
In principle the calculation described in 213 may be used but when the theoretical Hertzian stress exceeds the ap-proximately 18 times the yield strength of the rack material plastic deformation will occur This will limit the peak Hertzian stress but increases the contact width and thus the penetration of stresses into the depth
An approximation may be based on an assessment of contact width determined by means of equal areas under the theoreti-cal (elastic range) Hertzian contact and the elasto-plastic contact stress (limited to 18 middot σy) with the unknown width
34 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
43 Influence Factors
431 Coefficient of friction The following coefficient of friction may apply
L025a
005oil
02
redC ΣC
Bt X R ηρv
w0048micro minus
=
where
wBt = specific tooth load (Nmm)
vΣC = sum of tangential velocities at pitch point At pitch line velocities gt 50 ms the limiting value of vΣC at v = 50 ms is to be used
ρredC = relative radius of curvature (transversal plane) at the pitch point
Cylindrical gears
HαHβvγAbt
Bt KKKKKb
Fw sdotsdotsdotsdotsdot= (see 1)
twΣC αsin v2v =
bCredC β cos ρρ =
Bevel gears
HαHβvγAtmb
Bt KKKKKb
Fw sdotsdotsdotsdotsdot= (see 1)
vtmtΣC αsin v2v =
bmvCredC β cos ρρ =
ηoil = dynamic viscosity (mPa s) at oilϑ calculated as
1000
ρ νη oiloil =
where ρ in kgm3 approximated as
( ) 70 15oil15 minusϑminusρ=ρ and νoil is kinematic viscosity at oilϑ and may be calculated by means of the following equation
( ) ( )++=+ 08νloglog08νloglog 100oil ( )
sdotminus
ϑ+minus313log373log
273log373log oil
( ) ( )( )08νloglog08νloglog 10040 +minus+
Ra = composite arithmetic mean roughness (micron) of pinion an wheel calculated as ( )2a1aa RR 05R +=This is defined as the roughness on the new flanks ie as manufactured
XL = see 42
432 Effective tip relief Ceff Ceff is the effective tip relief that amount of tip relief which compensates for the elastic deformation of the gear mesh ie
zero load at the tooth tip It is assumed (simplified) to be equal for both pinion and wheel
Cylindrical gears
for helical
γ
γ=cb
KKFC Abt
eff (see 1)
For spur
cbKKF
C Abteff
γ= (see 1)
Alternatively for spur and helical gears the non-linear ap-proach in 111 may be used (taking Ceff=δ)
Bevel gears
bcKFC Ambt
effγ
= (see 1)
where
γ
α
ε2ε44c
+sdot
=γ
433 Tip relief and extension Cylindrical gears
The extension of the tip relief is not to result in an effective contact ratio 1ltεα when the gear is unloaded (exceptions to this may only apply for applications where the gear is not to run at light loads) This means that the unrelieved part of the path of contact is to be minimum pbt It is further assumed that this unrelieved part is placed centrally on the path of contact
If root relief applies it has to be calculated as equivalent tip relief Ie pinion root relief (at mesh position A) is added to Ca2 and wheel root relief (at mesh position E) is added to Ca1 If no design tip relief or root relief on the mating gear is specified (ie if Ca1+Croot2 = 0 and visa versa) use the run-ning in amount see 112
Bevel gears
Bevel gears are to have heightwise crowning ie no distinct relievedunrelieved area This may be treated as tip and root relief For calculation purposes the root relief is combined with the tip relief of the mating member into an equivalent tip relief If no resulting tip relieves are specified the equivalent tip relives may be calculated as an approxima-tion based on the tool crowning Ca tool (per mille of tool module m0) as follows
2root1aeq1a CCC +=
1root2aeq2a CCC +=
where
Classification Notes- No 412 35 May 2003
DET NORSKE VERITAS
2
0
n0101atoolal m
)m2(mA1middotmCC
minus+minussdot=
( )
sdotminusminussdotsdotminus+= vtvb1
2vb1
21vavt1n1 αtan dddαsin 05x1mA
2
0
n020atool22a m
)mm(2A1middotmCC
minus+minussdot=
( )
sdotminusminussdotsdotminus+= vtvb2
2vb2
2va2vt2n2 αtan dddαsin 05x1mA
2
0
20atool1root1 m
A1mCC
minussdotsdot=
2
0
10atool2root2 m
A1mCC
minussdotsdot=
If the pressure angles of the cutter blades are modified (and verified) to balance the tip relieves the following may be assumed
( )eq2aeq1aeq2aeq1a C andC calculated of sum50CC sdot==
Throughout the following Ca1 and Ca2 mean the equivalent tip relieves Ca1eq and Ca2eq
434 Bulk temperature The bulk temperature may be calculated as
flaaveragempsoilMB X X 50 ϑ+ϑ=ϑ
where
Xs = lubrication factor
= 12 for spray lubrication
= 10 for dip lubrication (provided both pinion and wheel are dip lubricated and tip speed lt 5 ms)
= 10 for spray lubrication with additional cooling spray (spray on both pinion and wheel or spray on pinion and dip lubrication of wheel)
= 02 for meshes fully submerged in oil
Xmp = contact factor ( )pmp nX += 150
np = number of mesh contact on the pinion (for small gear ratii the number of wheel meshes should be used if higher)
flaaverageϑ
= average of the integrated flash temperature (see 44) along the path of contact
( )
AE
E
Ayyyfla
flaaverage
d
ΓminusΓ
ΓΓϑ=ϑint =
For high speed gears (v gt 50 ms) it may be necessary to assess the bulk temperature on the basis of a thermal rating of the entire gear transmission
44 The Flash Temperature flaϑ
441 Basic formula The local flash temperature flaϑ may be calculated as
( ) 41redy
y2ly
211
43Btcorrfla
unXwX3250
y ρ
ρminusρ
micro=ϑ Γ
(For bevel gears replace u with uv)
and is to be calculated stepwise along the path of contact from A to E
where
micro = coefficient of friction see 431
Xcorr = correction factor taking empirically into account the increased scuffing risk in the beginning of the approach path due to mesh starting without any previously built up oil film and possible shuffling away oil before meshing if insufficient tip relief
( )
3
AD
yaeffcorr 50
CC1X
ΓminusΓε
Γminus+=
α
Ca = tip relief of driven member Xcorr is only applicable in the approach path and if Ca lt Ceff otherwise 10
wBt = unit load see 431
yXΓ = load sharing factor see 443
n1 = pinion rpm
Γy ρly etc see 442
442 Geometrical relations The various radii of flank curvature (transversal plane) are
ρ1y = pinion flank radius at mesh point y
ρ2y = wheel flank radius at mesh point y
ρredy = equivalent radius of curvature at mesh point y
y2y1
y2y1redy
ρ+ρ
ρρ=ρ
Cylindrical gears
twy
1y αsin au1Γ1
ρ+
+=
twy
2y αsin au1Γu
ρ+
minus=
Note that for internal gears a and u are negative
36 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
Bevel gears
vtvv
y1y αsin a
u1Γ1
ρ+
+=
vtvv
yv2y αsin a
u1Γu
ρ+
minus=
Γ is the parameter on the path of contact and y is any point between A end E
At the respective ends Γ has the following values
Root piniontip wheel
Cylindrical gears
( )
minus
minusminus= 1
αtan 1dd
zzΓ
tw
2b2a2
1
2A
Bevel gears
( )
minus
minusminus= 1
αtan 1dd
uΓvt
2vb2va2
vA
Tip pinionroot wheel
Cylindrical gears
( )
1αtan
1ddΓ
tw
2b1a1
E minusminus
=
Bevel gears
( )
1αtan
1ddΓ
vt
2vb1va1
E minusminus
=
At inner point of single pair contact
Cylindrical gears
tw1
EB α tan zπ2ΓΓ minus=
Bevel gears
vtv1
EB α tan zπ2ΓΓ minus=
At outer point of single pair contact
Cylindrical gears
tw1AD α tan z
π2ΓΓ +=
Bevel gears
vt v1
AD αtan z π2ΓΓ +=
At pitch point ΓC = 0
The points F and G (only applicable to cylindrical gears) limiting the extension of tip relief (so as to maintain a mini-mum contact ratio of unity for unloaded gears) are at
2
BAF
Γ+Γ=Γ
2
EDG
Γ+Γ=Γ
443 Load sharing factor XΓ The load sharing factor XΓ accounts for the load sharing be-tween the various pairs of teeth in mesh along the path of contact
XΓ is to be calculated stepwise from A to E using the pa-rameter Γy
4431 Cylindrical gears with β = 0 and no tip relief
Figure 41
ByAAB
Ay for31
31X
yΓltΓleΓ
ΓminusΓ
ΓminusΓ+=Γ
DyB for1Xy
ΓltΓleΓ=Γ
EyDDE
yE for31
31X
yΓleΓltΓ
ΓminusΓ
ΓminusΓ+=Γ
4432 Cylindrical gears with β = 0 and tip relief Tip relief on the pinion reduces XΓ in the range G ndash E and increases correspondingly XΓ in the range F ndash B
Tip relief on the wheel reduces XΓ in the range A ndash F and increases correspondingly XΓ in the range D ndash G
Following remains generally
DyB for1Xy
ΓleΓleΓ=Γ
21XXGF== ΓΓ
In the following it must be distinguished between Ca lt Ceff respectively Ca gt Ceff This is shown by an example below where Ca1 lt Ceff and Ca2 gt Ceff
Classification Notes- No 412 37 May 2003
DET NORSKE VERITAS
Figure 42
Note When Ca gt Ceff the path of contact is shortened by A ndash A respectively E ndash E The single pair contact path is extended into B respectively D If this shift is significant it is necessary to consider the negative effect on surface durability (B) and bending stresses (B and D)
Range A - F
For effa2 CC le
minus=Γ
eff
2a
CC1
31X
A
FyAeff
2a
AF
Ay for C3
C61XX
AyΓleΓleΓ
sdot
+ΓminusΓ
ΓminusΓ+= ΓΓ
For effa2 CC ge
AyAfor0Xy
ΓleΓleΓ=Γ
with ( )21
CC
1CC
eff
2a
eff
2a
AFAAminus
minusΓminusΓ+Γ=Γ
FyAeff
2a
AF
Ay
eff
2a for 21
CC
CC1X
yΓleΓleΓ
minus
ΓminusΓ
ΓminusΓ+minus=Γ
Range F - B
For effa1 CC le
ByFeff
1a
FB
Fy forC3
C61
21X
yΓleΓleΓ
sdot
+ΓminusΓ
ΓminusΓ+=Γ
For effa1 CC ge
ByFeff
1a
FB
Fy for21
CC
21X
yΓleΓleΓ
minus
ΓminusΓ
ΓminusΓ+=Γ
ByB for1Xy
ΓleΓleΓ=Γ
with 1
CC2
eff
1a
FBFB
minus
ΓminusΓ+Γ=Γ
Range D ndash G
For effa2 CC le
GyDeff
2a
DG
Dy
eff
2a for C 3C
61
C 3C
32X
yΓleΓleΓ
+
ΓminusΓ
ΓminusΓminus+=Γ F
or effa2 CC ge
DyD for1Xy
ΓleΓleΓ=Γ
with ( )21
CC
1CC
eff
2a
eff
2a
DGDDminus
minusΓminusΓ+Γ=Γ
GyDeff
2a
DG
Dy
eff
2a for21
CC
CCX ΓleΓleΓ
minus
ΓminusΓ
ΓminusΓminus=Γ
Range G - E
For effa1 CC le
minus=Γ
eff
1a
CC1
31X
E
EyGeff
1a
GE
Gy forC3
C61
21X
yΓleΓleΓ
sdot
+ΓminusΓ
ΓminusΓminus=Γ
For effa1 CC gt
EyGeff
1a
GE
Gy for21
CC
21X
yΓleΓleΓ
minus
ΓminusΓ
ΓminusΓminus=Γ
EyE for0Xy
ΓleΓleΓ=Γ
with 1
CC2
eff
1a
GEGE
minus
ΓminusΓ+Γ=Γ
4433 Gears with β gt 0 buttressing Due to oblique contact lines over the flanks a certain but-tressing may occur near A and E
This applies to both cylindrical and bevel gears with tip relief lt Ceff The buttressing Xbutt is simplified as a linear function within the ranges A ndash H respectively I ndash E
Figure 43
1εwhen13X βbutt EAge=
1εwhenε031X ββbutt EAlt+=
38 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
Cylindrical gears
bIEAH βsin 02ΓΓΓΓ =minus=minus
Bevel gears
bmIEAH βsin 02ΓΓΓΓ =minus=minus
4434 Cylindrical gears with 2εγ le and no tip relief
yXΓ is obtained by multiplication of
yXΓ in 4431 with
Xbutt in 4433
4435 Gears with 2εγ gt and no tip relief
Applicable to both cylindrical and bevel gears
IyH for1Xy
ΓleΓleΓε
=α
Γ
IyHybutt and forX1Xy
ΓgtΓΓltΓε
=α
Γ
Figure 44
4436 Cylindrical gears with 2εγ le and tip relief
yXΓ is obtained by multiplication of
yXΓ in 4432 with Xbutt
in 4433
4437 Cylindrical gears with 2εγ gt and tip relief
Tip relief on the pinion (respectively wheel) reduces XГ in the range G ndash E (respectively A ndash F) and increases XГ in the range F ndash G
yXΓ is obtained by multiplication of
yXΓ as described below
with Xbutt in 4433
In the XГ example below the influence of tip relief is shown (without the influence of Xbutt) by means of
eff2aeff1a CC and CC ltgt
Tip relief gt Ceff causes new end points A respectively E of the path of contact
Figure 45
Range A ndash F
( ) ( )( ) eff
a2αa1α
AF
Ay
eff
2aeff
C 12C 13εC 1ε
CCCX
y +εε++minus
ΓminusΓ
ΓminusΓ+
εminus
=ααα
Γ
eff2aFyA CC if for leΓleΓleΓ
eff2aFyA CifCfor and geΓleΓleΓ
eff2aAyA CC iffor0Xy
gtΓleΓleΓ=Γ
( ) ( ) ( )( ) ( ) a2αa1α
αeffa2AFAA C 1ε 3C 1ε
1ε 2 CC with++minus+minus
ΓminusΓ+Γ=Γ
Range F ndash G
( )( )( ) GyF
eff
2a1a forC 1 2CC 11X
yΓleΓleΓ
+εε+minusε
+ε
=αα
α
αΓ
Range G ndash E
( ) ( )( ) eff
2a1a
GE
Gy
C 1 2C 1C 1 3XX
GFy +εεminusε++ε
ΓminusΓ
ΓminusΓminus=
αα
ααΓΓ minus
effa1EyG CC if for leΓleΓleΓ
effa1EyG CC if Γfor and geΓleΓle
eff1aEyE CC if for0Xy
geΓleΓleΓ=Γ
( ) ( ) ( )( ) ( ) 2a1a
eff1aGEEE C 1C 1 3
1 2 CC withminusε++ε+εminus
ΓminusΓminusΓ=Γαα
α
4438 Bevel gears with εγ more than approx 18 and heightwise crowning For Ca1 = Ca2 = Ceff the following applies
Figure 46
Classification Notes- No 412 39 May 2003
DET NORSKE VERITAS
( )AEM 50 Γ+Γ=Γ
( )( )2AD
3
2My651X
y ΓminusΓε
ΓminusΓminus
ε=
ααΓ
For tip relief lt Ceff yXΓ is found by linear interpolation be-
tween ( )effay CCX =Γ and ( )0CayX =Γ as in 4435
The interpolation is to be made stepwise from A to M with the influence of Ca2 and from M to E with the influence of Ca1 (For 2a1a CC ne there is a discontinuity at M)
Eg with Ca1 = 04 Ceff and Ca2 = 055 Ceff then
Range A ndash M
)CC()0C( eff2ay2ayyX 550X 450X =Γ=ΓΓ +=
Range M ndash E
)CC()0C( eff1ay1ayyX 40X 06X =Γ=ΓΓ +=
For tip relief gt Ceff the new end points A and E are found as
( )
minusΓminusΓ
ε+Γ=Γ α 1
CC
6 eff
2aADAA
( )
minusΓminusΓ
ε+Γ=Γ α 1
CC
6 eff
1aADEE
Range A ndash A
0Xy=Γ
Range A ndash M
( )( )
ΓminusΓ
ΓminusΓminus
minusε=
αΓ 2
MA
2My
eff
2a1
CC4
351Xy
Range M ndash E
( )( )
ΓminusΓ
ΓminusΓminus
minusε=
αΓ 2
ME
2My
eff
1a1
CC4
351Xy
Range E ndash E
0Xy=Γ
40 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
Appendix A Fatigue Damage Accumulation
The Palmgren-Miner cumulative damage calculation princi-ple is used The procedure may be applied as follows
A1 Stress Spectrum From the individual torque classes the torques (Ti) at the peak values of class intervals and the associated number of cycles (NLi) for both pinion and wheel are to be listed from the highest to the lowest torque
(In case of a cyclic torque variation within the torque classes it is advised to use the peak torque If the cyclic variation is such that the same teeth will repeatedly suffer the peak torque this is a must)
The stress spectra for tooth roots and flanks (σFi σHi) with all relevant factors (except KA) are to be calculated on the basis of the torque spectrum The load dependent K-factors are to be determined for each torque class
A2 σminusN-curve The stress versus load cycle curves for tooth roots and flanks (both pinion and wheel) are to be drawn on the basis of per-missible stresses (ie including the demanded minimum safety factors) as determined in 2 respectively 3 If different safety levels for high cycle fatigue and low cycle fatigue are desired this may be expressed by different demand safety factors applied at the endurance limit respectively at static strength
A3 Damage accumulation The individual damage ratio Di at ith stress level is defined as
Fi
Lii N
ND =
where
NLi = The number of applied cycles at ith stress
NFi = The number of cycles to failure at ith stress
Basically stresses σi below the permissible stress level for infinite life (if a constant ZN or YN is accepted) do not con-tribute to the damage sum However calculating the actual safety factor Sact as described below all the σi for which the product S σi is bigger than or equal to the permissible stress level for infinite life contribute to the damage sum and thus to the determination of Sact The final value of S is decisive
(NFi can be found mathematically by putting the permissible stress σpi equal the actual stress σi thereby finding the actual life factor This life factor can be solved with regard to load cycles ie NFi)
The damage sum ΣDi is not to exceed unity
If ΣDi ne 1 the safety against cumulative fatigue damage is different from the applied demand safety factor For deter-mination of this theoretical safety factor an iteration proce-dure is required as described in the following flowchart
S is correction factor with which the actual safety factor Sact can be found
Sact is the demand safety factor (used in determination of the permissible stresses in the σ ndash N ndash curve) times the correc-tion factor S
The full procedure is to be applied for pinion and wheel tooth roots and flanks
Note If alternating stresses occur in a spectrum of mainly pulsating stresses the alternating stresses may be replaced by equiva-lent pulsating stresses ie by means of division with the ac-tual mean stress influence factor YM
Classification Notes- No 412 41 May 2003
DET NORSKE VERITAS
Appendix B Application Factors for Diesel Driven Gears
For diesel driven gears the application factor KA depends on torsional vibrations Both normal operation and misfiring conditions have to be considered
Normally these two running conditions can be covered by only one calculation
B1 Definitions
Normal operation KAnorm = 0
normv0
TTT +
where
T0 = rated nominal torque
Tv norm = vibratory torque amplitude for normal opera-tion (see rules Pt4 Ch3 Sec1 G301 for defi-nition of ldquonormalrdquo irregularity)
Misfiring operation KA misf = 0
misfv
TTT +
where
T = remaining nominal torque when one cylinder out of action
Tv misf = vibratory torque amplitude in misfiring con-dition This refers to a permissible misfiring condition ie a condition that does not require automatic or immediate corrective actions as speed or pitch reduction
The normal operation is assumed to last for a very high num-ber of cycles such as 1010
The misfiring operation is assumed to last for a limited dura-tion such as 107 cycles
B2 Determination of decisive load Assuming life factor at 1010 cycles as YN = ZN = 092 which usually is relevant the calculation may be performed only once with the combination having the highest value of appli-cation factorlife factor
For bending stresses and scuffing the higher value of
092
K normA and 098
K misfA
For contact stresses the higher value of
092K normA and
113K misfA (but
097K misfA for nitrided gears)
B3 Simplified procedure Note that this is only a guidance and is not a binding con-vention
T = oTZ
1Zsdot
minus where Z = number of cylinders
( ) ideal Videal Vmisf Vnorm V TTT24ZT +minus=
where TV ideal = vibratory torque with all cylinders perfectly equal See also rules Pt4 Ch3 Sec1 G300
When using trends from torsional vibration analysis and measurements the following may be used
TV ideal To is close to zero for engines with few cylinders and using a suitable elastic coupling and increases with relative coupling stiffness and number of cylinders
TV misf To may be high for engines with few cylinders and decreases with number of cylinders
This can be indicated as
200Z
TT
ο
idealV asymp and 80Z04
TT
o
misfV minusasymp
Inserting this into the formulae for the two application fac-tors the following guidance can be given
118112K misfA minusasymp
115110K normA minusasymp
Since KA norm is to be combined with the lower life factors the decisive load condition will be the normal one and a KA of 115 will cover most relevant cases when a suitable elas-tic coupling is chosen
42 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
Appendix C Calculation of Pinion-Rack
Pinion-racks used for elevating of mobile offshore units are open gears that are subjected to wear and tear With normal specifications such as only a few hundred total operation cycles (site to site) the tooth bending stresses for static strength will be decisive for the lay out however with ex-ception of surface hardened pinions where case crushing has to be considered
In the following the use of part 1 and 3 for pinion-racks is shown including relevant simplifications
C1 Pinion tooth root stresses Since the load spectrum normally is dominated by high torques and few load cycles (in the range up to 1-2000) the static strength is decisive
The actual stress is calculated as
β1FSaFan1
tF1 KYY
mbFσ sdotsdotsdotsdot
=
b1 is limited to b2 + 2middotmn
YFa and YSa replace YF and YS because load application at tooth tip has to be assumed for such inaccurate gears
Pinions often use a non-involute profile in the dedendum part of the flank eg a constant radius equal the radius of curva-ture at reference circle For such pinions sFn and hFa are to be measured directly on a sectional drawing of the pinion tooth
Due to high loads and narrow facewidths it may be assumed that KFβ2 = KHβ = 10 However when b1 gt b2 then KFβ1 gt10
If no detailed documentation of KFβ1 is available the fol-lowing may be used
KFβ1 = 1 + 015middot(b1b2 ndash 1)
The permissible stress (not surface hardened) is calculated as
δrelTstF
Fst1FP1 Y
Sσσ sdot=
The mean stress influence due to leg lifting may be disre-garded
The actual and permissible stresses should be calculated for the relevant loads as given in the rules
C2 Rack tooth root stresses The actual stress is calculated as
SaFan2
tF2 YY
mbFσ sdotsdotsdot
=
See C1 for details
The permissible stress is calculated as
δrelTstF
Fst2FP2 Y
Sσσ sdot=
For alloyed steels (Ni Cr Mo) with high toughness and duc-tility the value of YδrelTst may be put equal to YSa
C3 Surface hardened pinions For surface hardened pinions the maximum load is not to cause crushing of the hardened layer of the flank
In principle the calculation described in 213 may be used but when the theoretical Hertzian stress exceeds the ap-proximately 18 times the yield strength of the rack material plastic deformation will occur This will limit the peak Hertzian stress but increases the contact width and thus the penetration of stresses into the depth
An approximation may be based on an assessment of contact width determined by means of equal areas under the theoreti-cal (elastic range) Hertzian contact and the elasto-plastic contact stress (limited to 18 middot σy) with the unknown width
Classification Notes- No 412 35 May 2003
DET NORSKE VERITAS
2
0
n0101atoolal m
)m2(mA1middotmCC
minus+minussdot=
( )
sdotminusminussdotsdotminus+= vtvb1
2vb1
21vavt1n1 αtan dddαsin 05x1mA
2
0
n020atool22a m
)mm(2A1middotmCC
minus+minussdot=
( )
sdotminusminussdotsdotminus+= vtvb2
2vb2
2va2vt2n2 αtan dddαsin 05x1mA
2
0
20atool1root1 m
A1mCC
minussdotsdot=
2
0
10atool2root2 m
A1mCC
minussdotsdot=
If the pressure angles of the cutter blades are modified (and verified) to balance the tip relieves the following may be assumed
( )eq2aeq1aeq2aeq1a C andC calculated of sum50CC sdot==
Throughout the following Ca1 and Ca2 mean the equivalent tip relieves Ca1eq and Ca2eq
434 Bulk temperature The bulk temperature may be calculated as
flaaveragempsoilMB X X 50 ϑ+ϑ=ϑ
where
Xs = lubrication factor
= 12 for spray lubrication
= 10 for dip lubrication (provided both pinion and wheel are dip lubricated and tip speed lt 5 ms)
= 10 for spray lubrication with additional cooling spray (spray on both pinion and wheel or spray on pinion and dip lubrication of wheel)
= 02 for meshes fully submerged in oil
Xmp = contact factor ( )pmp nX += 150
np = number of mesh contact on the pinion (for small gear ratii the number of wheel meshes should be used if higher)
flaaverageϑ
= average of the integrated flash temperature (see 44) along the path of contact
( )
AE
E
Ayyyfla
flaaverage
d
ΓminusΓ
ΓΓϑ=ϑint =
For high speed gears (v gt 50 ms) it may be necessary to assess the bulk temperature on the basis of a thermal rating of the entire gear transmission
44 The Flash Temperature flaϑ
441 Basic formula The local flash temperature flaϑ may be calculated as
( ) 41redy
y2ly
211
43Btcorrfla
unXwX3250
y ρ
ρminusρ
micro=ϑ Γ
(For bevel gears replace u with uv)
and is to be calculated stepwise along the path of contact from A to E
where
micro = coefficient of friction see 431
Xcorr = correction factor taking empirically into account the increased scuffing risk in the beginning of the approach path due to mesh starting without any previously built up oil film and possible shuffling away oil before meshing if insufficient tip relief
( )
3
AD
yaeffcorr 50
CC1X
ΓminusΓε
Γminus+=
α
Ca = tip relief of driven member Xcorr is only applicable in the approach path and if Ca lt Ceff otherwise 10
wBt = unit load see 431
yXΓ = load sharing factor see 443
n1 = pinion rpm
Γy ρly etc see 442
442 Geometrical relations The various radii of flank curvature (transversal plane) are
ρ1y = pinion flank radius at mesh point y
ρ2y = wheel flank radius at mesh point y
ρredy = equivalent radius of curvature at mesh point y
y2y1
y2y1redy
ρ+ρ
ρρ=ρ
Cylindrical gears
twy
1y αsin au1Γ1
ρ+
+=
twy
2y αsin au1Γu
ρ+
minus=
Note that for internal gears a and u are negative
36 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
Bevel gears
vtvv
y1y αsin a
u1Γ1
ρ+
+=
vtvv
yv2y αsin a
u1Γu
ρ+
minus=
Γ is the parameter on the path of contact and y is any point between A end E
At the respective ends Γ has the following values
Root piniontip wheel
Cylindrical gears
( )
minus
minusminus= 1
αtan 1dd
zzΓ
tw
2b2a2
1
2A
Bevel gears
( )
minus
minusminus= 1
αtan 1dd
uΓvt
2vb2va2
vA
Tip pinionroot wheel
Cylindrical gears
( )
1αtan
1ddΓ
tw
2b1a1
E minusminus
=
Bevel gears
( )
1αtan
1ddΓ
vt
2vb1va1
E minusminus
=
At inner point of single pair contact
Cylindrical gears
tw1
EB α tan zπ2ΓΓ minus=
Bevel gears
vtv1
EB α tan zπ2ΓΓ minus=
At outer point of single pair contact
Cylindrical gears
tw1AD α tan z
π2ΓΓ +=
Bevel gears
vt v1
AD αtan z π2ΓΓ +=
At pitch point ΓC = 0
The points F and G (only applicable to cylindrical gears) limiting the extension of tip relief (so as to maintain a mini-mum contact ratio of unity for unloaded gears) are at
2
BAF
Γ+Γ=Γ
2
EDG
Γ+Γ=Γ
443 Load sharing factor XΓ The load sharing factor XΓ accounts for the load sharing be-tween the various pairs of teeth in mesh along the path of contact
XΓ is to be calculated stepwise from A to E using the pa-rameter Γy
4431 Cylindrical gears with β = 0 and no tip relief
Figure 41
ByAAB
Ay for31
31X
yΓltΓleΓ
ΓminusΓ
ΓminusΓ+=Γ
DyB for1Xy
ΓltΓleΓ=Γ
EyDDE
yE for31
31X
yΓleΓltΓ
ΓminusΓ
ΓminusΓ+=Γ
4432 Cylindrical gears with β = 0 and tip relief Tip relief on the pinion reduces XΓ in the range G ndash E and increases correspondingly XΓ in the range F ndash B
Tip relief on the wheel reduces XΓ in the range A ndash F and increases correspondingly XΓ in the range D ndash G
Following remains generally
DyB for1Xy
ΓleΓleΓ=Γ
21XXGF== ΓΓ
In the following it must be distinguished between Ca lt Ceff respectively Ca gt Ceff This is shown by an example below where Ca1 lt Ceff and Ca2 gt Ceff
Classification Notes- No 412 37 May 2003
DET NORSKE VERITAS
Figure 42
Note When Ca gt Ceff the path of contact is shortened by A ndash A respectively E ndash E The single pair contact path is extended into B respectively D If this shift is significant it is necessary to consider the negative effect on surface durability (B) and bending stresses (B and D)
Range A - F
For effa2 CC le
minus=Γ
eff
2a
CC1
31X
A
FyAeff
2a
AF
Ay for C3
C61XX
AyΓleΓleΓ
sdot
+ΓminusΓ
ΓminusΓ+= ΓΓ
For effa2 CC ge
AyAfor0Xy
ΓleΓleΓ=Γ
with ( )21
CC
1CC
eff
2a
eff
2a
AFAAminus
minusΓminusΓ+Γ=Γ
FyAeff
2a
AF
Ay
eff
2a for 21
CC
CC1X
yΓleΓleΓ
minus
ΓminusΓ
ΓminusΓ+minus=Γ
Range F - B
For effa1 CC le
ByFeff
1a
FB
Fy forC3
C61
21X
yΓleΓleΓ
sdot
+ΓminusΓ
ΓminusΓ+=Γ
For effa1 CC ge
ByFeff
1a
FB
Fy for21
CC
21X
yΓleΓleΓ
minus
ΓminusΓ
ΓminusΓ+=Γ
ByB for1Xy
ΓleΓleΓ=Γ
with 1
CC2
eff
1a
FBFB
minus
ΓminusΓ+Γ=Γ
Range D ndash G
For effa2 CC le
GyDeff
2a
DG
Dy
eff
2a for C 3C
61
C 3C
32X
yΓleΓleΓ
+
ΓminusΓ
ΓminusΓminus+=Γ F
or effa2 CC ge
DyD for1Xy
ΓleΓleΓ=Γ
with ( )21
CC
1CC
eff
2a
eff
2a
DGDDminus
minusΓminusΓ+Γ=Γ
GyDeff
2a
DG
Dy
eff
2a for21
CC
CCX ΓleΓleΓ
minus
ΓminusΓ
ΓminusΓminus=Γ
Range G - E
For effa1 CC le
minus=Γ
eff
1a
CC1
31X
E
EyGeff
1a
GE
Gy forC3
C61
21X
yΓleΓleΓ
sdot
+ΓminusΓ
ΓminusΓminus=Γ
For effa1 CC gt
EyGeff
1a
GE
Gy for21
CC
21X
yΓleΓleΓ
minus
ΓminusΓ
ΓminusΓminus=Γ
EyE for0Xy
ΓleΓleΓ=Γ
with 1
CC2
eff
1a
GEGE
minus
ΓminusΓ+Γ=Γ
4433 Gears with β gt 0 buttressing Due to oblique contact lines over the flanks a certain but-tressing may occur near A and E
This applies to both cylindrical and bevel gears with tip relief lt Ceff The buttressing Xbutt is simplified as a linear function within the ranges A ndash H respectively I ndash E
Figure 43
1εwhen13X βbutt EAge=
1εwhenε031X ββbutt EAlt+=
38 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
Cylindrical gears
bIEAH βsin 02ΓΓΓΓ =minus=minus
Bevel gears
bmIEAH βsin 02ΓΓΓΓ =minus=minus
4434 Cylindrical gears with 2εγ le and no tip relief
yXΓ is obtained by multiplication of
yXΓ in 4431 with
Xbutt in 4433
4435 Gears with 2εγ gt and no tip relief
Applicable to both cylindrical and bevel gears
IyH for1Xy
ΓleΓleΓε
=α
Γ
IyHybutt and forX1Xy
ΓgtΓΓltΓε
=α
Γ
Figure 44
4436 Cylindrical gears with 2εγ le and tip relief
yXΓ is obtained by multiplication of
yXΓ in 4432 with Xbutt
in 4433
4437 Cylindrical gears with 2εγ gt and tip relief
Tip relief on the pinion (respectively wheel) reduces XГ in the range G ndash E (respectively A ndash F) and increases XГ in the range F ndash G
yXΓ is obtained by multiplication of
yXΓ as described below
with Xbutt in 4433
In the XГ example below the influence of tip relief is shown (without the influence of Xbutt) by means of
eff2aeff1a CC and CC ltgt
Tip relief gt Ceff causes new end points A respectively E of the path of contact
Figure 45
Range A ndash F
( ) ( )( ) eff
a2αa1α
AF
Ay
eff
2aeff
C 12C 13εC 1ε
CCCX
y +εε++minus
ΓminusΓ
ΓminusΓ+
εminus
=ααα
Γ
eff2aFyA CC if for leΓleΓleΓ
eff2aFyA CifCfor and geΓleΓleΓ
eff2aAyA CC iffor0Xy
gtΓleΓleΓ=Γ
( ) ( ) ( )( ) ( ) a2αa1α
αeffa2AFAA C 1ε 3C 1ε
1ε 2 CC with++minus+minus
ΓminusΓ+Γ=Γ
Range F ndash G
( )( )( ) GyF
eff
2a1a forC 1 2CC 11X
yΓleΓleΓ
+εε+minusε
+ε
=αα
α
αΓ
Range G ndash E
( ) ( )( ) eff
2a1a
GE
Gy
C 1 2C 1C 1 3XX
GFy +εεminusε++ε
ΓminusΓ
ΓminusΓminus=
αα
ααΓΓ minus
effa1EyG CC if for leΓleΓleΓ
effa1EyG CC if Γfor and geΓleΓle
eff1aEyE CC if for0Xy
geΓleΓleΓ=Γ
( ) ( ) ( )( ) ( ) 2a1a
eff1aGEEE C 1C 1 3
1 2 CC withminusε++ε+εminus
ΓminusΓminusΓ=Γαα
α
4438 Bevel gears with εγ more than approx 18 and heightwise crowning For Ca1 = Ca2 = Ceff the following applies
Figure 46
Classification Notes- No 412 39 May 2003
DET NORSKE VERITAS
( )AEM 50 Γ+Γ=Γ
( )( )2AD
3
2My651X
y ΓminusΓε
ΓminusΓminus
ε=
ααΓ
For tip relief lt Ceff yXΓ is found by linear interpolation be-
tween ( )effay CCX =Γ and ( )0CayX =Γ as in 4435
The interpolation is to be made stepwise from A to M with the influence of Ca2 and from M to E with the influence of Ca1 (For 2a1a CC ne there is a discontinuity at M)
Eg with Ca1 = 04 Ceff and Ca2 = 055 Ceff then
Range A ndash M
)CC()0C( eff2ay2ayyX 550X 450X =Γ=ΓΓ +=
Range M ndash E
)CC()0C( eff1ay1ayyX 40X 06X =Γ=ΓΓ +=
For tip relief gt Ceff the new end points A and E are found as
( )
minusΓminusΓ
ε+Γ=Γ α 1
CC
6 eff
2aADAA
( )
minusΓminusΓ
ε+Γ=Γ α 1
CC
6 eff
1aADEE
Range A ndash A
0Xy=Γ
Range A ndash M
( )( )
ΓminusΓ
ΓminusΓminus
minusε=
αΓ 2
MA
2My
eff
2a1
CC4
351Xy
Range M ndash E
( )( )
ΓminusΓ
ΓminusΓminus
minusε=
αΓ 2
ME
2My
eff
1a1
CC4
351Xy
Range E ndash E
0Xy=Γ
40 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
Appendix A Fatigue Damage Accumulation
The Palmgren-Miner cumulative damage calculation princi-ple is used The procedure may be applied as follows
A1 Stress Spectrum From the individual torque classes the torques (Ti) at the peak values of class intervals and the associated number of cycles (NLi) for both pinion and wheel are to be listed from the highest to the lowest torque
(In case of a cyclic torque variation within the torque classes it is advised to use the peak torque If the cyclic variation is such that the same teeth will repeatedly suffer the peak torque this is a must)
The stress spectra for tooth roots and flanks (σFi σHi) with all relevant factors (except KA) are to be calculated on the basis of the torque spectrum The load dependent K-factors are to be determined for each torque class
A2 σminusN-curve The stress versus load cycle curves for tooth roots and flanks (both pinion and wheel) are to be drawn on the basis of per-missible stresses (ie including the demanded minimum safety factors) as determined in 2 respectively 3 If different safety levels for high cycle fatigue and low cycle fatigue are desired this may be expressed by different demand safety factors applied at the endurance limit respectively at static strength
A3 Damage accumulation The individual damage ratio Di at ith stress level is defined as
Fi
Lii N
ND =
where
NLi = The number of applied cycles at ith stress
NFi = The number of cycles to failure at ith stress
Basically stresses σi below the permissible stress level for infinite life (if a constant ZN or YN is accepted) do not con-tribute to the damage sum However calculating the actual safety factor Sact as described below all the σi for which the product S σi is bigger than or equal to the permissible stress level for infinite life contribute to the damage sum and thus to the determination of Sact The final value of S is decisive
(NFi can be found mathematically by putting the permissible stress σpi equal the actual stress σi thereby finding the actual life factor This life factor can be solved with regard to load cycles ie NFi)
The damage sum ΣDi is not to exceed unity
If ΣDi ne 1 the safety against cumulative fatigue damage is different from the applied demand safety factor For deter-mination of this theoretical safety factor an iteration proce-dure is required as described in the following flowchart
S is correction factor with which the actual safety factor Sact can be found
Sact is the demand safety factor (used in determination of the permissible stresses in the σ ndash N ndash curve) times the correc-tion factor S
The full procedure is to be applied for pinion and wheel tooth roots and flanks
Note If alternating stresses occur in a spectrum of mainly pulsating stresses the alternating stresses may be replaced by equiva-lent pulsating stresses ie by means of division with the ac-tual mean stress influence factor YM
Classification Notes- No 412 41 May 2003
DET NORSKE VERITAS
Appendix B Application Factors for Diesel Driven Gears
For diesel driven gears the application factor KA depends on torsional vibrations Both normal operation and misfiring conditions have to be considered
Normally these two running conditions can be covered by only one calculation
B1 Definitions
Normal operation KAnorm = 0
normv0
TTT +
where
T0 = rated nominal torque
Tv norm = vibratory torque amplitude for normal opera-tion (see rules Pt4 Ch3 Sec1 G301 for defi-nition of ldquonormalrdquo irregularity)
Misfiring operation KA misf = 0
misfv
TTT +
where
T = remaining nominal torque when one cylinder out of action
Tv misf = vibratory torque amplitude in misfiring con-dition This refers to a permissible misfiring condition ie a condition that does not require automatic or immediate corrective actions as speed or pitch reduction
The normal operation is assumed to last for a very high num-ber of cycles such as 1010
The misfiring operation is assumed to last for a limited dura-tion such as 107 cycles
B2 Determination of decisive load Assuming life factor at 1010 cycles as YN = ZN = 092 which usually is relevant the calculation may be performed only once with the combination having the highest value of appli-cation factorlife factor
For bending stresses and scuffing the higher value of
092
K normA and 098
K misfA
For contact stresses the higher value of
092K normA and
113K misfA (but
097K misfA for nitrided gears)
B3 Simplified procedure Note that this is only a guidance and is not a binding con-vention
T = oTZ
1Zsdot
minus where Z = number of cylinders
( ) ideal Videal Vmisf Vnorm V TTT24ZT +minus=
where TV ideal = vibratory torque with all cylinders perfectly equal See also rules Pt4 Ch3 Sec1 G300
When using trends from torsional vibration analysis and measurements the following may be used
TV ideal To is close to zero for engines with few cylinders and using a suitable elastic coupling and increases with relative coupling stiffness and number of cylinders
TV misf To may be high for engines with few cylinders and decreases with number of cylinders
This can be indicated as
200Z
TT
ο
idealV asymp and 80Z04
TT
o
misfV minusasymp
Inserting this into the formulae for the two application fac-tors the following guidance can be given
118112K misfA minusasymp
115110K normA minusasymp
Since KA norm is to be combined with the lower life factors the decisive load condition will be the normal one and a KA of 115 will cover most relevant cases when a suitable elas-tic coupling is chosen
42 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
Appendix C Calculation of Pinion-Rack
Pinion-racks used for elevating of mobile offshore units are open gears that are subjected to wear and tear With normal specifications such as only a few hundred total operation cycles (site to site) the tooth bending stresses for static strength will be decisive for the lay out however with ex-ception of surface hardened pinions where case crushing has to be considered
In the following the use of part 1 and 3 for pinion-racks is shown including relevant simplifications
C1 Pinion tooth root stresses Since the load spectrum normally is dominated by high torques and few load cycles (in the range up to 1-2000) the static strength is decisive
The actual stress is calculated as
β1FSaFan1
tF1 KYY
mbFσ sdotsdotsdotsdot
=
b1 is limited to b2 + 2middotmn
YFa and YSa replace YF and YS because load application at tooth tip has to be assumed for such inaccurate gears
Pinions often use a non-involute profile in the dedendum part of the flank eg a constant radius equal the radius of curva-ture at reference circle For such pinions sFn and hFa are to be measured directly on a sectional drawing of the pinion tooth
Due to high loads and narrow facewidths it may be assumed that KFβ2 = KHβ = 10 However when b1 gt b2 then KFβ1 gt10
If no detailed documentation of KFβ1 is available the fol-lowing may be used
KFβ1 = 1 + 015middot(b1b2 ndash 1)
The permissible stress (not surface hardened) is calculated as
δrelTstF
Fst1FP1 Y
Sσσ sdot=
The mean stress influence due to leg lifting may be disre-garded
The actual and permissible stresses should be calculated for the relevant loads as given in the rules
C2 Rack tooth root stresses The actual stress is calculated as
SaFan2
tF2 YY
mbFσ sdotsdotsdot
=
See C1 for details
The permissible stress is calculated as
δrelTstF
Fst2FP2 Y
Sσσ sdot=
For alloyed steels (Ni Cr Mo) with high toughness and duc-tility the value of YδrelTst may be put equal to YSa
C3 Surface hardened pinions For surface hardened pinions the maximum load is not to cause crushing of the hardened layer of the flank
In principle the calculation described in 213 may be used but when the theoretical Hertzian stress exceeds the ap-proximately 18 times the yield strength of the rack material plastic deformation will occur This will limit the peak Hertzian stress but increases the contact width and thus the penetration of stresses into the depth
An approximation may be based on an assessment of contact width determined by means of equal areas under the theoreti-cal (elastic range) Hertzian contact and the elasto-plastic contact stress (limited to 18 middot σy) with the unknown width
36 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
Bevel gears
vtvv
y1y αsin a
u1Γ1
ρ+
+=
vtvv
yv2y αsin a
u1Γu
ρ+
minus=
Γ is the parameter on the path of contact and y is any point between A end E
At the respective ends Γ has the following values
Root piniontip wheel
Cylindrical gears
( )
minus
minusminus= 1
αtan 1dd
zzΓ
tw
2b2a2
1
2A
Bevel gears
( )
minus
minusminus= 1
αtan 1dd
uΓvt
2vb2va2
vA
Tip pinionroot wheel
Cylindrical gears
( )
1αtan
1ddΓ
tw
2b1a1
E minusminus
=
Bevel gears
( )
1αtan
1ddΓ
vt
2vb1va1
E minusminus
=
At inner point of single pair contact
Cylindrical gears
tw1
EB α tan zπ2ΓΓ minus=
Bevel gears
vtv1
EB α tan zπ2ΓΓ minus=
At outer point of single pair contact
Cylindrical gears
tw1AD α tan z
π2ΓΓ +=
Bevel gears
vt v1
AD αtan z π2ΓΓ +=
At pitch point ΓC = 0
The points F and G (only applicable to cylindrical gears) limiting the extension of tip relief (so as to maintain a mini-mum contact ratio of unity for unloaded gears) are at
2
BAF
Γ+Γ=Γ
2
EDG
Γ+Γ=Γ
443 Load sharing factor XΓ The load sharing factor XΓ accounts for the load sharing be-tween the various pairs of teeth in mesh along the path of contact
XΓ is to be calculated stepwise from A to E using the pa-rameter Γy
4431 Cylindrical gears with β = 0 and no tip relief
Figure 41
ByAAB
Ay for31
31X
yΓltΓleΓ
ΓminusΓ
ΓminusΓ+=Γ
DyB for1Xy
ΓltΓleΓ=Γ
EyDDE
yE for31
31X
yΓleΓltΓ
ΓminusΓ
ΓminusΓ+=Γ
4432 Cylindrical gears with β = 0 and tip relief Tip relief on the pinion reduces XΓ in the range G ndash E and increases correspondingly XΓ in the range F ndash B
Tip relief on the wheel reduces XΓ in the range A ndash F and increases correspondingly XΓ in the range D ndash G
Following remains generally
DyB for1Xy
ΓleΓleΓ=Γ
21XXGF== ΓΓ
In the following it must be distinguished between Ca lt Ceff respectively Ca gt Ceff This is shown by an example below where Ca1 lt Ceff and Ca2 gt Ceff
Classification Notes- No 412 37 May 2003
DET NORSKE VERITAS
Figure 42
Note When Ca gt Ceff the path of contact is shortened by A ndash A respectively E ndash E The single pair contact path is extended into B respectively D If this shift is significant it is necessary to consider the negative effect on surface durability (B) and bending stresses (B and D)
Range A - F
For effa2 CC le
minus=Γ
eff
2a
CC1
31X
A
FyAeff
2a
AF
Ay for C3
C61XX
AyΓleΓleΓ
sdot
+ΓminusΓ
ΓminusΓ+= ΓΓ
For effa2 CC ge
AyAfor0Xy
ΓleΓleΓ=Γ
with ( )21
CC
1CC
eff
2a
eff
2a
AFAAminus
minusΓminusΓ+Γ=Γ
FyAeff
2a
AF
Ay
eff
2a for 21
CC
CC1X
yΓleΓleΓ
minus
ΓminusΓ
ΓminusΓ+minus=Γ
Range F - B
For effa1 CC le
ByFeff
1a
FB
Fy forC3
C61
21X
yΓleΓleΓ
sdot
+ΓminusΓ
ΓminusΓ+=Γ
For effa1 CC ge
ByFeff
1a
FB
Fy for21
CC
21X
yΓleΓleΓ
minus
ΓminusΓ
ΓminusΓ+=Γ
ByB for1Xy
ΓleΓleΓ=Γ
with 1
CC2
eff
1a
FBFB
minus
ΓminusΓ+Γ=Γ
Range D ndash G
For effa2 CC le
GyDeff
2a
DG
Dy
eff
2a for C 3C
61
C 3C
32X
yΓleΓleΓ
+
ΓminusΓ
ΓminusΓminus+=Γ F
or effa2 CC ge
DyD for1Xy
ΓleΓleΓ=Γ
with ( )21
CC
1CC
eff
2a
eff
2a
DGDDminus
minusΓminusΓ+Γ=Γ
GyDeff
2a
DG
Dy
eff
2a for21
CC
CCX ΓleΓleΓ
minus
ΓminusΓ
ΓminusΓminus=Γ
Range G - E
For effa1 CC le
minus=Γ
eff
1a
CC1
31X
E
EyGeff
1a
GE
Gy forC3
C61
21X
yΓleΓleΓ
sdot
+ΓminusΓ
ΓminusΓminus=Γ
For effa1 CC gt
EyGeff
1a
GE
Gy for21
CC
21X
yΓleΓleΓ
minus
ΓminusΓ
ΓminusΓminus=Γ
EyE for0Xy
ΓleΓleΓ=Γ
with 1
CC2
eff
1a
GEGE
minus
ΓminusΓ+Γ=Γ
4433 Gears with β gt 0 buttressing Due to oblique contact lines over the flanks a certain but-tressing may occur near A and E
This applies to both cylindrical and bevel gears with tip relief lt Ceff The buttressing Xbutt is simplified as a linear function within the ranges A ndash H respectively I ndash E
Figure 43
1εwhen13X βbutt EAge=
1εwhenε031X ββbutt EAlt+=
38 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
Cylindrical gears
bIEAH βsin 02ΓΓΓΓ =minus=minus
Bevel gears
bmIEAH βsin 02ΓΓΓΓ =minus=minus
4434 Cylindrical gears with 2εγ le and no tip relief
yXΓ is obtained by multiplication of
yXΓ in 4431 with
Xbutt in 4433
4435 Gears with 2εγ gt and no tip relief
Applicable to both cylindrical and bevel gears
IyH for1Xy
ΓleΓleΓε
=α
Γ
IyHybutt and forX1Xy
ΓgtΓΓltΓε
=α
Γ
Figure 44
4436 Cylindrical gears with 2εγ le and tip relief
yXΓ is obtained by multiplication of
yXΓ in 4432 with Xbutt
in 4433
4437 Cylindrical gears with 2εγ gt and tip relief
Tip relief on the pinion (respectively wheel) reduces XГ in the range G ndash E (respectively A ndash F) and increases XГ in the range F ndash G
yXΓ is obtained by multiplication of
yXΓ as described below
with Xbutt in 4433
In the XГ example below the influence of tip relief is shown (without the influence of Xbutt) by means of
eff2aeff1a CC and CC ltgt
Tip relief gt Ceff causes new end points A respectively E of the path of contact
Figure 45
Range A ndash F
( ) ( )( ) eff
a2αa1α
AF
Ay
eff
2aeff
C 12C 13εC 1ε
CCCX
y +εε++minus
ΓminusΓ
ΓminusΓ+
εminus
=ααα
Γ
eff2aFyA CC if for leΓleΓleΓ
eff2aFyA CifCfor and geΓleΓleΓ
eff2aAyA CC iffor0Xy
gtΓleΓleΓ=Γ
( ) ( ) ( )( ) ( ) a2αa1α
αeffa2AFAA C 1ε 3C 1ε
1ε 2 CC with++minus+minus
ΓminusΓ+Γ=Γ
Range F ndash G
( )( )( ) GyF
eff
2a1a forC 1 2CC 11X
yΓleΓleΓ
+εε+minusε
+ε
=αα
α
αΓ
Range G ndash E
( ) ( )( ) eff
2a1a
GE
Gy
C 1 2C 1C 1 3XX
GFy +εεminusε++ε
ΓminusΓ
ΓminusΓminus=
αα
ααΓΓ minus
effa1EyG CC if for leΓleΓleΓ
effa1EyG CC if Γfor and geΓleΓle
eff1aEyE CC if for0Xy
geΓleΓleΓ=Γ
( ) ( ) ( )( ) ( ) 2a1a
eff1aGEEE C 1C 1 3
1 2 CC withminusε++ε+εminus
ΓminusΓminusΓ=Γαα
α
4438 Bevel gears with εγ more than approx 18 and heightwise crowning For Ca1 = Ca2 = Ceff the following applies
Figure 46
Classification Notes- No 412 39 May 2003
DET NORSKE VERITAS
( )AEM 50 Γ+Γ=Γ
( )( )2AD
3
2My651X
y ΓminusΓε
ΓminusΓminus
ε=
ααΓ
For tip relief lt Ceff yXΓ is found by linear interpolation be-
tween ( )effay CCX =Γ and ( )0CayX =Γ as in 4435
The interpolation is to be made stepwise from A to M with the influence of Ca2 and from M to E with the influence of Ca1 (For 2a1a CC ne there is a discontinuity at M)
Eg with Ca1 = 04 Ceff and Ca2 = 055 Ceff then
Range A ndash M
)CC()0C( eff2ay2ayyX 550X 450X =Γ=ΓΓ +=
Range M ndash E
)CC()0C( eff1ay1ayyX 40X 06X =Γ=ΓΓ +=
For tip relief gt Ceff the new end points A and E are found as
( )
minusΓminusΓ
ε+Γ=Γ α 1
CC
6 eff
2aADAA
( )
minusΓminusΓ
ε+Γ=Γ α 1
CC
6 eff
1aADEE
Range A ndash A
0Xy=Γ
Range A ndash M
( )( )
ΓminusΓ
ΓminusΓminus
minusε=
αΓ 2
MA
2My
eff
2a1
CC4
351Xy
Range M ndash E
( )( )
ΓminusΓ
ΓminusΓminus
minusε=
αΓ 2
ME
2My
eff
1a1
CC4
351Xy
Range E ndash E
0Xy=Γ
40 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
Appendix A Fatigue Damage Accumulation
The Palmgren-Miner cumulative damage calculation princi-ple is used The procedure may be applied as follows
A1 Stress Spectrum From the individual torque classes the torques (Ti) at the peak values of class intervals and the associated number of cycles (NLi) for both pinion and wheel are to be listed from the highest to the lowest torque
(In case of a cyclic torque variation within the torque classes it is advised to use the peak torque If the cyclic variation is such that the same teeth will repeatedly suffer the peak torque this is a must)
The stress spectra for tooth roots and flanks (σFi σHi) with all relevant factors (except KA) are to be calculated on the basis of the torque spectrum The load dependent K-factors are to be determined for each torque class
A2 σminusN-curve The stress versus load cycle curves for tooth roots and flanks (both pinion and wheel) are to be drawn on the basis of per-missible stresses (ie including the demanded minimum safety factors) as determined in 2 respectively 3 If different safety levels for high cycle fatigue and low cycle fatigue are desired this may be expressed by different demand safety factors applied at the endurance limit respectively at static strength
A3 Damage accumulation The individual damage ratio Di at ith stress level is defined as
Fi
Lii N
ND =
where
NLi = The number of applied cycles at ith stress
NFi = The number of cycles to failure at ith stress
Basically stresses σi below the permissible stress level for infinite life (if a constant ZN or YN is accepted) do not con-tribute to the damage sum However calculating the actual safety factor Sact as described below all the σi for which the product S σi is bigger than or equal to the permissible stress level for infinite life contribute to the damage sum and thus to the determination of Sact The final value of S is decisive
(NFi can be found mathematically by putting the permissible stress σpi equal the actual stress σi thereby finding the actual life factor This life factor can be solved with regard to load cycles ie NFi)
The damage sum ΣDi is not to exceed unity
If ΣDi ne 1 the safety against cumulative fatigue damage is different from the applied demand safety factor For deter-mination of this theoretical safety factor an iteration proce-dure is required as described in the following flowchart
S is correction factor with which the actual safety factor Sact can be found
Sact is the demand safety factor (used in determination of the permissible stresses in the σ ndash N ndash curve) times the correc-tion factor S
The full procedure is to be applied for pinion and wheel tooth roots and flanks
Note If alternating stresses occur in a spectrum of mainly pulsating stresses the alternating stresses may be replaced by equiva-lent pulsating stresses ie by means of division with the ac-tual mean stress influence factor YM
Classification Notes- No 412 41 May 2003
DET NORSKE VERITAS
Appendix B Application Factors for Diesel Driven Gears
For diesel driven gears the application factor KA depends on torsional vibrations Both normal operation and misfiring conditions have to be considered
Normally these two running conditions can be covered by only one calculation
B1 Definitions
Normal operation KAnorm = 0
normv0
TTT +
where
T0 = rated nominal torque
Tv norm = vibratory torque amplitude for normal opera-tion (see rules Pt4 Ch3 Sec1 G301 for defi-nition of ldquonormalrdquo irregularity)
Misfiring operation KA misf = 0
misfv
TTT +
where
T = remaining nominal torque when one cylinder out of action
Tv misf = vibratory torque amplitude in misfiring con-dition This refers to a permissible misfiring condition ie a condition that does not require automatic or immediate corrective actions as speed or pitch reduction
The normal operation is assumed to last for a very high num-ber of cycles such as 1010
The misfiring operation is assumed to last for a limited dura-tion such as 107 cycles
B2 Determination of decisive load Assuming life factor at 1010 cycles as YN = ZN = 092 which usually is relevant the calculation may be performed only once with the combination having the highest value of appli-cation factorlife factor
For bending stresses and scuffing the higher value of
092
K normA and 098
K misfA
For contact stresses the higher value of
092K normA and
113K misfA (but
097K misfA for nitrided gears)
B3 Simplified procedure Note that this is only a guidance and is not a binding con-vention
T = oTZ
1Zsdot
minus where Z = number of cylinders
( ) ideal Videal Vmisf Vnorm V TTT24ZT +minus=
where TV ideal = vibratory torque with all cylinders perfectly equal See also rules Pt4 Ch3 Sec1 G300
When using trends from torsional vibration analysis and measurements the following may be used
TV ideal To is close to zero for engines with few cylinders and using a suitable elastic coupling and increases with relative coupling stiffness and number of cylinders
TV misf To may be high for engines with few cylinders and decreases with number of cylinders
This can be indicated as
200Z
TT
ο
idealV asymp and 80Z04
TT
o
misfV minusasymp
Inserting this into the formulae for the two application fac-tors the following guidance can be given
118112K misfA minusasymp
115110K normA minusasymp
Since KA norm is to be combined with the lower life factors the decisive load condition will be the normal one and a KA of 115 will cover most relevant cases when a suitable elas-tic coupling is chosen
42 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
Appendix C Calculation of Pinion-Rack
Pinion-racks used for elevating of mobile offshore units are open gears that are subjected to wear and tear With normal specifications such as only a few hundred total operation cycles (site to site) the tooth bending stresses for static strength will be decisive for the lay out however with ex-ception of surface hardened pinions where case crushing has to be considered
In the following the use of part 1 and 3 for pinion-racks is shown including relevant simplifications
C1 Pinion tooth root stresses Since the load spectrum normally is dominated by high torques and few load cycles (in the range up to 1-2000) the static strength is decisive
The actual stress is calculated as
β1FSaFan1
tF1 KYY
mbFσ sdotsdotsdotsdot
=
b1 is limited to b2 + 2middotmn
YFa and YSa replace YF and YS because load application at tooth tip has to be assumed for such inaccurate gears
Pinions often use a non-involute profile in the dedendum part of the flank eg a constant radius equal the radius of curva-ture at reference circle For such pinions sFn and hFa are to be measured directly on a sectional drawing of the pinion tooth
Due to high loads and narrow facewidths it may be assumed that KFβ2 = KHβ = 10 However when b1 gt b2 then KFβ1 gt10
If no detailed documentation of KFβ1 is available the fol-lowing may be used
KFβ1 = 1 + 015middot(b1b2 ndash 1)
The permissible stress (not surface hardened) is calculated as
δrelTstF
Fst1FP1 Y
Sσσ sdot=
The mean stress influence due to leg lifting may be disre-garded
The actual and permissible stresses should be calculated for the relevant loads as given in the rules
C2 Rack tooth root stresses The actual stress is calculated as
SaFan2
tF2 YY
mbFσ sdotsdotsdot
=
See C1 for details
The permissible stress is calculated as
δrelTstF
Fst2FP2 Y
Sσσ sdot=
For alloyed steels (Ni Cr Mo) with high toughness and duc-tility the value of YδrelTst may be put equal to YSa
C3 Surface hardened pinions For surface hardened pinions the maximum load is not to cause crushing of the hardened layer of the flank
In principle the calculation described in 213 may be used but when the theoretical Hertzian stress exceeds the ap-proximately 18 times the yield strength of the rack material plastic deformation will occur This will limit the peak Hertzian stress but increases the contact width and thus the penetration of stresses into the depth
An approximation may be based on an assessment of contact width determined by means of equal areas under the theoreti-cal (elastic range) Hertzian contact and the elasto-plastic contact stress (limited to 18 middot σy) with the unknown width
Classification Notes- No 412 37 May 2003
DET NORSKE VERITAS
Figure 42
Note When Ca gt Ceff the path of contact is shortened by A ndash A respectively E ndash E The single pair contact path is extended into B respectively D If this shift is significant it is necessary to consider the negative effect on surface durability (B) and bending stresses (B and D)
Range A - F
For effa2 CC le
minus=Γ
eff
2a
CC1
31X
A
FyAeff
2a
AF
Ay for C3
C61XX
AyΓleΓleΓ
sdot
+ΓminusΓ
ΓminusΓ+= ΓΓ
For effa2 CC ge
AyAfor0Xy
ΓleΓleΓ=Γ
with ( )21
CC
1CC
eff
2a
eff
2a
AFAAminus
minusΓminusΓ+Γ=Γ
FyAeff
2a
AF
Ay
eff
2a for 21
CC
CC1X
yΓleΓleΓ
minus
ΓminusΓ
ΓminusΓ+minus=Γ
Range F - B
For effa1 CC le
ByFeff
1a
FB
Fy forC3
C61
21X
yΓleΓleΓ
sdot
+ΓminusΓ
ΓminusΓ+=Γ
For effa1 CC ge
ByFeff
1a
FB
Fy for21
CC
21X
yΓleΓleΓ
minus
ΓminusΓ
ΓminusΓ+=Γ
ByB for1Xy
ΓleΓleΓ=Γ
with 1
CC2
eff
1a
FBFB
minus
ΓminusΓ+Γ=Γ
Range D ndash G
For effa2 CC le
GyDeff
2a
DG
Dy
eff
2a for C 3C
61
C 3C
32X
yΓleΓleΓ
+
ΓminusΓ
ΓminusΓminus+=Γ F
or effa2 CC ge
DyD for1Xy
ΓleΓleΓ=Γ
with ( )21
CC
1CC
eff
2a
eff
2a
DGDDminus
minusΓminusΓ+Γ=Γ
GyDeff
2a
DG
Dy
eff
2a for21
CC
CCX ΓleΓleΓ
minus
ΓminusΓ
ΓminusΓminus=Γ
Range G - E
For effa1 CC le
minus=Γ
eff
1a
CC1
31X
E
EyGeff
1a
GE
Gy forC3
C61
21X
yΓleΓleΓ
sdot
+ΓminusΓ
ΓminusΓminus=Γ
For effa1 CC gt
EyGeff
1a
GE
Gy for21
CC
21X
yΓleΓleΓ
minus
ΓminusΓ
ΓminusΓminus=Γ
EyE for0Xy
ΓleΓleΓ=Γ
with 1
CC2
eff
1a
GEGE
minus
ΓminusΓ+Γ=Γ
4433 Gears with β gt 0 buttressing Due to oblique contact lines over the flanks a certain but-tressing may occur near A and E
This applies to both cylindrical and bevel gears with tip relief lt Ceff The buttressing Xbutt is simplified as a linear function within the ranges A ndash H respectively I ndash E
Figure 43
1εwhen13X βbutt EAge=
1εwhenε031X ββbutt EAlt+=
38 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
Cylindrical gears
bIEAH βsin 02ΓΓΓΓ =minus=minus
Bevel gears
bmIEAH βsin 02ΓΓΓΓ =minus=minus
4434 Cylindrical gears with 2εγ le and no tip relief
yXΓ is obtained by multiplication of
yXΓ in 4431 with
Xbutt in 4433
4435 Gears with 2εγ gt and no tip relief
Applicable to both cylindrical and bevel gears
IyH for1Xy
ΓleΓleΓε
=α
Γ
IyHybutt and forX1Xy
ΓgtΓΓltΓε
=α
Γ
Figure 44
4436 Cylindrical gears with 2εγ le and tip relief
yXΓ is obtained by multiplication of
yXΓ in 4432 with Xbutt
in 4433
4437 Cylindrical gears with 2εγ gt and tip relief
Tip relief on the pinion (respectively wheel) reduces XГ in the range G ndash E (respectively A ndash F) and increases XГ in the range F ndash G
yXΓ is obtained by multiplication of
yXΓ as described below
with Xbutt in 4433
In the XГ example below the influence of tip relief is shown (without the influence of Xbutt) by means of
eff2aeff1a CC and CC ltgt
Tip relief gt Ceff causes new end points A respectively E of the path of contact
Figure 45
Range A ndash F
( ) ( )( ) eff
a2αa1α
AF
Ay
eff
2aeff
C 12C 13εC 1ε
CCCX
y +εε++minus
ΓminusΓ
ΓminusΓ+
εminus
=ααα
Γ
eff2aFyA CC if for leΓleΓleΓ
eff2aFyA CifCfor and geΓleΓleΓ
eff2aAyA CC iffor0Xy
gtΓleΓleΓ=Γ
( ) ( ) ( )( ) ( ) a2αa1α
αeffa2AFAA C 1ε 3C 1ε
1ε 2 CC with++minus+minus
ΓminusΓ+Γ=Γ
Range F ndash G
( )( )( ) GyF
eff
2a1a forC 1 2CC 11X
yΓleΓleΓ
+εε+minusε
+ε
=αα
α
αΓ
Range G ndash E
( ) ( )( ) eff
2a1a
GE
Gy
C 1 2C 1C 1 3XX
GFy +εεminusε++ε
ΓminusΓ
ΓminusΓminus=
αα
ααΓΓ minus
effa1EyG CC if for leΓleΓleΓ
effa1EyG CC if Γfor and geΓleΓle
eff1aEyE CC if for0Xy
geΓleΓleΓ=Γ
( ) ( ) ( )( ) ( ) 2a1a
eff1aGEEE C 1C 1 3
1 2 CC withminusε++ε+εminus
ΓminusΓminusΓ=Γαα
α
4438 Bevel gears with εγ more than approx 18 and heightwise crowning For Ca1 = Ca2 = Ceff the following applies
Figure 46
Classification Notes- No 412 39 May 2003
DET NORSKE VERITAS
( )AEM 50 Γ+Γ=Γ
( )( )2AD
3
2My651X
y ΓminusΓε
ΓminusΓminus
ε=
ααΓ
For tip relief lt Ceff yXΓ is found by linear interpolation be-
tween ( )effay CCX =Γ and ( )0CayX =Γ as in 4435
The interpolation is to be made stepwise from A to M with the influence of Ca2 and from M to E with the influence of Ca1 (For 2a1a CC ne there is a discontinuity at M)
Eg with Ca1 = 04 Ceff and Ca2 = 055 Ceff then
Range A ndash M
)CC()0C( eff2ay2ayyX 550X 450X =Γ=ΓΓ +=
Range M ndash E
)CC()0C( eff1ay1ayyX 40X 06X =Γ=ΓΓ +=
For tip relief gt Ceff the new end points A and E are found as
( )
minusΓminusΓ
ε+Γ=Γ α 1
CC
6 eff
2aADAA
( )
minusΓminusΓ
ε+Γ=Γ α 1
CC
6 eff
1aADEE
Range A ndash A
0Xy=Γ
Range A ndash M
( )( )
ΓminusΓ
ΓminusΓminus
minusε=
αΓ 2
MA
2My
eff
2a1
CC4
351Xy
Range M ndash E
( )( )
ΓminusΓ
ΓminusΓminus
minusε=
αΓ 2
ME
2My
eff
1a1
CC4
351Xy
Range E ndash E
0Xy=Γ
40 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
Appendix A Fatigue Damage Accumulation
The Palmgren-Miner cumulative damage calculation princi-ple is used The procedure may be applied as follows
A1 Stress Spectrum From the individual torque classes the torques (Ti) at the peak values of class intervals and the associated number of cycles (NLi) for both pinion and wheel are to be listed from the highest to the lowest torque
(In case of a cyclic torque variation within the torque classes it is advised to use the peak torque If the cyclic variation is such that the same teeth will repeatedly suffer the peak torque this is a must)
The stress spectra for tooth roots and flanks (σFi σHi) with all relevant factors (except KA) are to be calculated on the basis of the torque spectrum The load dependent K-factors are to be determined for each torque class
A2 σminusN-curve The stress versus load cycle curves for tooth roots and flanks (both pinion and wheel) are to be drawn on the basis of per-missible stresses (ie including the demanded minimum safety factors) as determined in 2 respectively 3 If different safety levels for high cycle fatigue and low cycle fatigue are desired this may be expressed by different demand safety factors applied at the endurance limit respectively at static strength
A3 Damage accumulation The individual damage ratio Di at ith stress level is defined as
Fi
Lii N
ND =
where
NLi = The number of applied cycles at ith stress
NFi = The number of cycles to failure at ith stress
Basically stresses σi below the permissible stress level for infinite life (if a constant ZN or YN is accepted) do not con-tribute to the damage sum However calculating the actual safety factor Sact as described below all the σi for which the product S σi is bigger than or equal to the permissible stress level for infinite life contribute to the damage sum and thus to the determination of Sact The final value of S is decisive
(NFi can be found mathematically by putting the permissible stress σpi equal the actual stress σi thereby finding the actual life factor This life factor can be solved with regard to load cycles ie NFi)
The damage sum ΣDi is not to exceed unity
If ΣDi ne 1 the safety against cumulative fatigue damage is different from the applied demand safety factor For deter-mination of this theoretical safety factor an iteration proce-dure is required as described in the following flowchart
S is correction factor with which the actual safety factor Sact can be found
Sact is the demand safety factor (used in determination of the permissible stresses in the σ ndash N ndash curve) times the correc-tion factor S
The full procedure is to be applied for pinion and wheel tooth roots and flanks
Note If alternating stresses occur in a spectrum of mainly pulsating stresses the alternating stresses may be replaced by equiva-lent pulsating stresses ie by means of division with the ac-tual mean stress influence factor YM
Classification Notes- No 412 41 May 2003
DET NORSKE VERITAS
Appendix B Application Factors for Diesel Driven Gears
For diesel driven gears the application factor KA depends on torsional vibrations Both normal operation and misfiring conditions have to be considered
Normally these two running conditions can be covered by only one calculation
B1 Definitions
Normal operation KAnorm = 0
normv0
TTT +
where
T0 = rated nominal torque
Tv norm = vibratory torque amplitude for normal opera-tion (see rules Pt4 Ch3 Sec1 G301 for defi-nition of ldquonormalrdquo irregularity)
Misfiring operation KA misf = 0
misfv
TTT +
where
T = remaining nominal torque when one cylinder out of action
Tv misf = vibratory torque amplitude in misfiring con-dition This refers to a permissible misfiring condition ie a condition that does not require automatic or immediate corrective actions as speed or pitch reduction
The normal operation is assumed to last for a very high num-ber of cycles such as 1010
The misfiring operation is assumed to last for a limited dura-tion such as 107 cycles
B2 Determination of decisive load Assuming life factor at 1010 cycles as YN = ZN = 092 which usually is relevant the calculation may be performed only once with the combination having the highest value of appli-cation factorlife factor
For bending stresses and scuffing the higher value of
092
K normA and 098
K misfA
For contact stresses the higher value of
092K normA and
113K misfA (but
097K misfA for nitrided gears)
B3 Simplified procedure Note that this is only a guidance and is not a binding con-vention
T = oTZ
1Zsdot
minus where Z = number of cylinders
( ) ideal Videal Vmisf Vnorm V TTT24ZT +minus=
where TV ideal = vibratory torque with all cylinders perfectly equal See also rules Pt4 Ch3 Sec1 G300
When using trends from torsional vibration analysis and measurements the following may be used
TV ideal To is close to zero for engines with few cylinders and using a suitable elastic coupling and increases with relative coupling stiffness and number of cylinders
TV misf To may be high for engines with few cylinders and decreases with number of cylinders
This can be indicated as
200Z
TT
ο
idealV asymp and 80Z04
TT
o
misfV minusasymp
Inserting this into the formulae for the two application fac-tors the following guidance can be given
118112K misfA minusasymp
115110K normA minusasymp
Since KA norm is to be combined with the lower life factors the decisive load condition will be the normal one and a KA of 115 will cover most relevant cases when a suitable elas-tic coupling is chosen
42 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
Appendix C Calculation of Pinion-Rack
Pinion-racks used for elevating of mobile offshore units are open gears that are subjected to wear and tear With normal specifications such as only a few hundred total operation cycles (site to site) the tooth bending stresses for static strength will be decisive for the lay out however with ex-ception of surface hardened pinions where case crushing has to be considered
In the following the use of part 1 and 3 for pinion-racks is shown including relevant simplifications
C1 Pinion tooth root stresses Since the load spectrum normally is dominated by high torques and few load cycles (in the range up to 1-2000) the static strength is decisive
The actual stress is calculated as
β1FSaFan1
tF1 KYY
mbFσ sdotsdotsdotsdot
=
b1 is limited to b2 + 2middotmn
YFa and YSa replace YF and YS because load application at tooth tip has to be assumed for such inaccurate gears
Pinions often use a non-involute profile in the dedendum part of the flank eg a constant radius equal the radius of curva-ture at reference circle For such pinions sFn and hFa are to be measured directly on a sectional drawing of the pinion tooth
Due to high loads and narrow facewidths it may be assumed that KFβ2 = KHβ = 10 However when b1 gt b2 then KFβ1 gt10
If no detailed documentation of KFβ1 is available the fol-lowing may be used
KFβ1 = 1 + 015middot(b1b2 ndash 1)
The permissible stress (not surface hardened) is calculated as
δrelTstF
Fst1FP1 Y
Sσσ sdot=
The mean stress influence due to leg lifting may be disre-garded
The actual and permissible stresses should be calculated for the relevant loads as given in the rules
C2 Rack tooth root stresses The actual stress is calculated as
SaFan2
tF2 YY
mbFσ sdotsdotsdot
=
See C1 for details
The permissible stress is calculated as
δrelTstF
Fst2FP2 Y
Sσσ sdot=
For alloyed steels (Ni Cr Mo) with high toughness and duc-tility the value of YδrelTst may be put equal to YSa
C3 Surface hardened pinions For surface hardened pinions the maximum load is not to cause crushing of the hardened layer of the flank
In principle the calculation described in 213 may be used but when the theoretical Hertzian stress exceeds the ap-proximately 18 times the yield strength of the rack material plastic deformation will occur This will limit the peak Hertzian stress but increases the contact width and thus the penetration of stresses into the depth
An approximation may be based on an assessment of contact width determined by means of equal areas under the theoreti-cal (elastic range) Hertzian contact and the elasto-plastic contact stress (limited to 18 middot σy) with the unknown width
38 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
Cylindrical gears
bIEAH βsin 02ΓΓΓΓ =minus=minus
Bevel gears
bmIEAH βsin 02ΓΓΓΓ =minus=minus
4434 Cylindrical gears with 2εγ le and no tip relief
yXΓ is obtained by multiplication of
yXΓ in 4431 with
Xbutt in 4433
4435 Gears with 2εγ gt and no tip relief
Applicable to both cylindrical and bevel gears
IyH for1Xy
ΓleΓleΓε
=α
Γ
IyHybutt and forX1Xy
ΓgtΓΓltΓε
=α
Γ
Figure 44
4436 Cylindrical gears with 2εγ le and tip relief
yXΓ is obtained by multiplication of
yXΓ in 4432 with Xbutt
in 4433
4437 Cylindrical gears with 2εγ gt and tip relief
Tip relief on the pinion (respectively wheel) reduces XГ in the range G ndash E (respectively A ndash F) and increases XГ in the range F ndash G
yXΓ is obtained by multiplication of
yXΓ as described below
with Xbutt in 4433
In the XГ example below the influence of tip relief is shown (without the influence of Xbutt) by means of
eff2aeff1a CC and CC ltgt
Tip relief gt Ceff causes new end points A respectively E of the path of contact
Figure 45
Range A ndash F
( ) ( )( ) eff
a2αa1α
AF
Ay
eff
2aeff
C 12C 13εC 1ε
CCCX
y +εε++minus
ΓminusΓ
ΓminusΓ+
εminus
=ααα
Γ
eff2aFyA CC if for leΓleΓleΓ
eff2aFyA CifCfor and geΓleΓleΓ
eff2aAyA CC iffor0Xy
gtΓleΓleΓ=Γ
( ) ( ) ( )( ) ( ) a2αa1α
αeffa2AFAA C 1ε 3C 1ε
1ε 2 CC with++minus+minus
ΓminusΓ+Γ=Γ
Range F ndash G
( )( )( ) GyF
eff
2a1a forC 1 2CC 11X
yΓleΓleΓ
+εε+minusε
+ε
=αα
α
αΓ
Range G ndash E
( ) ( )( ) eff
2a1a
GE
Gy
C 1 2C 1C 1 3XX
GFy +εεminusε++ε
ΓminusΓ
ΓminusΓminus=
αα
ααΓΓ minus
effa1EyG CC if for leΓleΓleΓ
effa1EyG CC if Γfor and geΓleΓle
eff1aEyE CC if for0Xy
geΓleΓleΓ=Γ
( ) ( ) ( )( ) ( ) 2a1a
eff1aGEEE C 1C 1 3
1 2 CC withminusε++ε+εminus
ΓminusΓminusΓ=Γαα
α
4438 Bevel gears with εγ more than approx 18 and heightwise crowning For Ca1 = Ca2 = Ceff the following applies
Figure 46
Classification Notes- No 412 39 May 2003
DET NORSKE VERITAS
( )AEM 50 Γ+Γ=Γ
( )( )2AD
3
2My651X
y ΓminusΓε
ΓminusΓminus
ε=
ααΓ
For tip relief lt Ceff yXΓ is found by linear interpolation be-
tween ( )effay CCX =Γ and ( )0CayX =Γ as in 4435
The interpolation is to be made stepwise from A to M with the influence of Ca2 and from M to E with the influence of Ca1 (For 2a1a CC ne there is a discontinuity at M)
Eg with Ca1 = 04 Ceff and Ca2 = 055 Ceff then
Range A ndash M
)CC()0C( eff2ay2ayyX 550X 450X =Γ=ΓΓ +=
Range M ndash E
)CC()0C( eff1ay1ayyX 40X 06X =Γ=ΓΓ +=
For tip relief gt Ceff the new end points A and E are found as
( )
minusΓminusΓ
ε+Γ=Γ α 1
CC
6 eff
2aADAA
( )
minusΓminusΓ
ε+Γ=Γ α 1
CC
6 eff
1aADEE
Range A ndash A
0Xy=Γ
Range A ndash M
( )( )
ΓminusΓ
ΓminusΓminus
minusε=
αΓ 2
MA
2My
eff
2a1
CC4
351Xy
Range M ndash E
( )( )
ΓminusΓ
ΓminusΓminus
minusε=
αΓ 2
ME
2My
eff
1a1
CC4
351Xy
Range E ndash E
0Xy=Γ
40 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
Appendix A Fatigue Damage Accumulation
The Palmgren-Miner cumulative damage calculation princi-ple is used The procedure may be applied as follows
A1 Stress Spectrum From the individual torque classes the torques (Ti) at the peak values of class intervals and the associated number of cycles (NLi) for both pinion and wheel are to be listed from the highest to the lowest torque
(In case of a cyclic torque variation within the torque classes it is advised to use the peak torque If the cyclic variation is such that the same teeth will repeatedly suffer the peak torque this is a must)
The stress spectra for tooth roots and flanks (σFi σHi) with all relevant factors (except KA) are to be calculated on the basis of the torque spectrum The load dependent K-factors are to be determined for each torque class
A2 σminusN-curve The stress versus load cycle curves for tooth roots and flanks (both pinion and wheel) are to be drawn on the basis of per-missible stresses (ie including the demanded minimum safety factors) as determined in 2 respectively 3 If different safety levels for high cycle fatigue and low cycle fatigue are desired this may be expressed by different demand safety factors applied at the endurance limit respectively at static strength
A3 Damage accumulation The individual damage ratio Di at ith stress level is defined as
Fi
Lii N
ND =
where
NLi = The number of applied cycles at ith stress
NFi = The number of cycles to failure at ith stress
Basically stresses σi below the permissible stress level for infinite life (if a constant ZN or YN is accepted) do not con-tribute to the damage sum However calculating the actual safety factor Sact as described below all the σi for which the product S σi is bigger than or equal to the permissible stress level for infinite life contribute to the damage sum and thus to the determination of Sact The final value of S is decisive
(NFi can be found mathematically by putting the permissible stress σpi equal the actual stress σi thereby finding the actual life factor This life factor can be solved with regard to load cycles ie NFi)
The damage sum ΣDi is not to exceed unity
If ΣDi ne 1 the safety against cumulative fatigue damage is different from the applied demand safety factor For deter-mination of this theoretical safety factor an iteration proce-dure is required as described in the following flowchart
S is correction factor with which the actual safety factor Sact can be found
Sact is the demand safety factor (used in determination of the permissible stresses in the σ ndash N ndash curve) times the correc-tion factor S
The full procedure is to be applied for pinion and wheel tooth roots and flanks
Note If alternating stresses occur in a spectrum of mainly pulsating stresses the alternating stresses may be replaced by equiva-lent pulsating stresses ie by means of division with the ac-tual mean stress influence factor YM
Classification Notes- No 412 41 May 2003
DET NORSKE VERITAS
Appendix B Application Factors for Diesel Driven Gears
For diesel driven gears the application factor KA depends on torsional vibrations Both normal operation and misfiring conditions have to be considered
Normally these two running conditions can be covered by only one calculation
B1 Definitions
Normal operation KAnorm = 0
normv0
TTT +
where
T0 = rated nominal torque
Tv norm = vibratory torque amplitude for normal opera-tion (see rules Pt4 Ch3 Sec1 G301 for defi-nition of ldquonormalrdquo irregularity)
Misfiring operation KA misf = 0
misfv
TTT +
where
T = remaining nominal torque when one cylinder out of action
Tv misf = vibratory torque amplitude in misfiring con-dition This refers to a permissible misfiring condition ie a condition that does not require automatic or immediate corrective actions as speed or pitch reduction
The normal operation is assumed to last for a very high num-ber of cycles such as 1010
The misfiring operation is assumed to last for a limited dura-tion such as 107 cycles
B2 Determination of decisive load Assuming life factor at 1010 cycles as YN = ZN = 092 which usually is relevant the calculation may be performed only once with the combination having the highest value of appli-cation factorlife factor
For bending stresses and scuffing the higher value of
092
K normA and 098
K misfA
For contact stresses the higher value of
092K normA and
113K misfA (but
097K misfA for nitrided gears)
B3 Simplified procedure Note that this is only a guidance and is not a binding con-vention
T = oTZ
1Zsdot
minus where Z = number of cylinders
( ) ideal Videal Vmisf Vnorm V TTT24ZT +minus=
where TV ideal = vibratory torque with all cylinders perfectly equal See also rules Pt4 Ch3 Sec1 G300
When using trends from torsional vibration analysis and measurements the following may be used
TV ideal To is close to zero for engines with few cylinders and using a suitable elastic coupling and increases with relative coupling stiffness and number of cylinders
TV misf To may be high for engines with few cylinders and decreases with number of cylinders
This can be indicated as
200Z
TT
ο
idealV asymp and 80Z04
TT
o
misfV minusasymp
Inserting this into the formulae for the two application fac-tors the following guidance can be given
118112K misfA minusasymp
115110K normA minusasymp
Since KA norm is to be combined with the lower life factors the decisive load condition will be the normal one and a KA of 115 will cover most relevant cases when a suitable elas-tic coupling is chosen
42 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
Appendix C Calculation of Pinion-Rack
Pinion-racks used for elevating of mobile offshore units are open gears that are subjected to wear and tear With normal specifications such as only a few hundred total operation cycles (site to site) the tooth bending stresses for static strength will be decisive for the lay out however with ex-ception of surface hardened pinions where case crushing has to be considered
In the following the use of part 1 and 3 for pinion-racks is shown including relevant simplifications
C1 Pinion tooth root stresses Since the load spectrum normally is dominated by high torques and few load cycles (in the range up to 1-2000) the static strength is decisive
The actual stress is calculated as
β1FSaFan1
tF1 KYY
mbFσ sdotsdotsdotsdot
=
b1 is limited to b2 + 2middotmn
YFa and YSa replace YF and YS because load application at tooth tip has to be assumed for such inaccurate gears
Pinions often use a non-involute profile in the dedendum part of the flank eg a constant radius equal the radius of curva-ture at reference circle For such pinions sFn and hFa are to be measured directly on a sectional drawing of the pinion tooth
Due to high loads and narrow facewidths it may be assumed that KFβ2 = KHβ = 10 However when b1 gt b2 then KFβ1 gt10
If no detailed documentation of KFβ1 is available the fol-lowing may be used
KFβ1 = 1 + 015middot(b1b2 ndash 1)
The permissible stress (not surface hardened) is calculated as
δrelTstF
Fst1FP1 Y
Sσσ sdot=
The mean stress influence due to leg lifting may be disre-garded
The actual and permissible stresses should be calculated for the relevant loads as given in the rules
C2 Rack tooth root stresses The actual stress is calculated as
SaFan2
tF2 YY
mbFσ sdotsdotsdot
=
See C1 for details
The permissible stress is calculated as
δrelTstF
Fst2FP2 Y
Sσσ sdot=
For alloyed steels (Ni Cr Mo) with high toughness and duc-tility the value of YδrelTst may be put equal to YSa
C3 Surface hardened pinions For surface hardened pinions the maximum load is not to cause crushing of the hardened layer of the flank
In principle the calculation described in 213 may be used but when the theoretical Hertzian stress exceeds the ap-proximately 18 times the yield strength of the rack material plastic deformation will occur This will limit the peak Hertzian stress but increases the contact width and thus the penetration of stresses into the depth
An approximation may be based on an assessment of contact width determined by means of equal areas under the theoreti-cal (elastic range) Hertzian contact and the elasto-plastic contact stress (limited to 18 middot σy) with the unknown width
Classification Notes- No 412 39 May 2003
DET NORSKE VERITAS
( )AEM 50 Γ+Γ=Γ
( )( )2AD
3
2My651X
y ΓminusΓε
ΓminusΓminus
ε=
ααΓ
For tip relief lt Ceff yXΓ is found by linear interpolation be-
tween ( )effay CCX =Γ and ( )0CayX =Γ as in 4435
The interpolation is to be made stepwise from A to M with the influence of Ca2 and from M to E with the influence of Ca1 (For 2a1a CC ne there is a discontinuity at M)
Eg with Ca1 = 04 Ceff and Ca2 = 055 Ceff then
Range A ndash M
)CC()0C( eff2ay2ayyX 550X 450X =Γ=ΓΓ +=
Range M ndash E
)CC()0C( eff1ay1ayyX 40X 06X =Γ=ΓΓ +=
For tip relief gt Ceff the new end points A and E are found as
( )
minusΓminusΓ
ε+Γ=Γ α 1
CC
6 eff
2aADAA
( )
minusΓminusΓ
ε+Γ=Γ α 1
CC
6 eff
1aADEE
Range A ndash A
0Xy=Γ
Range A ndash M
( )( )
ΓminusΓ
ΓminusΓminus
minusε=
αΓ 2
MA
2My
eff
2a1
CC4
351Xy
Range M ndash E
( )( )
ΓminusΓ
ΓminusΓminus
minusε=
αΓ 2
ME
2My
eff
1a1
CC4
351Xy
Range E ndash E
0Xy=Γ
40 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
Appendix A Fatigue Damage Accumulation
The Palmgren-Miner cumulative damage calculation princi-ple is used The procedure may be applied as follows
A1 Stress Spectrum From the individual torque classes the torques (Ti) at the peak values of class intervals and the associated number of cycles (NLi) for both pinion and wheel are to be listed from the highest to the lowest torque
(In case of a cyclic torque variation within the torque classes it is advised to use the peak torque If the cyclic variation is such that the same teeth will repeatedly suffer the peak torque this is a must)
The stress spectra for tooth roots and flanks (σFi σHi) with all relevant factors (except KA) are to be calculated on the basis of the torque spectrum The load dependent K-factors are to be determined for each torque class
A2 σminusN-curve The stress versus load cycle curves for tooth roots and flanks (both pinion and wheel) are to be drawn on the basis of per-missible stresses (ie including the demanded minimum safety factors) as determined in 2 respectively 3 If different safety levels for high cycle fatigue and low cycle fatigue are desired this may be expressed by different demand safety factors applied at the endurance limit respectively at static strength
A3 Damage accumulation The individual damage ratio Di at ith stress level is defined as
Fi
Lii N
ND =
where
NLi = The number of applied cycles at ith stress
NFi = The number of cycles to failure at ith stress
Basically stresses σi below the permissible stress level for infinite life (if a constant ZN or YN is accepted) do not con-tribute to the damage sum However calculating the actual safety factor Sact as described below all the σi for which the product S σi is bigger than or equal to the permissible stress level for infinite life contribute to the damage sum and thus to the determination of Sact The final value of S is decisive
(NFi can be found mathematically by putting the permissible stress σpi equal the actual stress σi thereby finding the actual life factor This life factor can be solved with regard to load cycles ie NFi)
The damage sum ΣDi is not to exceed unity
If ΣDi ne 1 the safety against cumulative fatigue damage is different from the applied demand safety factor For deter-mination of this theoretical safety factor an iteration proce-dure is required as described in the following flowchart
S is correction factor with which the actual safety factor Sact can be found
Sact is the demand safety factor (used in determination of the permissible stresses in the σ ndash N ndash curve) times the correc-tion factor S
The full procedure is to be applied for pinion and wheel tooth roots and flanks
Note If alternating stresses occur in a spectrum of mainly pulsating stresses the alternating stresses may be replaced by equiva-lent pulsating stresses ie by means of division with the ac-tual mean stress influence factor YM
Classification Notes- No 412 41 May 2003
DET NORSKE VERITAS
Appendix B Application Factors for Diesel Driven Gears
For diesel driven gears the application factor KA depends on torsional vibrations Both normal operation and misfiring conditions have to be considered
Normally these two running conditions can be covered by only one calculation
B1 Definitions
Normal operation KAnorm = 0
normv0
TTT +
where
T0 = rated nominal torque
Tv norm = vibratory torque amplitude for normal opera-tion (see rules Pt4 Ch3 Sec1 G301 for defi-nition of ldquonormalrdquo irregularity)
Misfiring operation KA misf = 0
misfv
TTT +
where
T = remaining nominal torque when one cylinder out of action
Tv misf = vibratory torque amplitude in misfiring con-dition This refers to a permissible misfiring condition ie a condition that does not require automatic or immediate corrective actions as speed or pitch reduction
The normal operation is assumed to last for a very high num-ber of cycles such as 1010
The misfiring operation is assumed to last for a limited dura-tion such as 107 cycles
B2 Determination of decisive load Assuming life factor at 1010 cycles as YN = ZN = 092 which usually is relevant the calculation may be performed only once with the combination having the highest value of appli-cation factorlife factor
For bending stresses and scuffing the higher value of
092
K normA and 098
K misfA
For contact stresses the higher value of
092K normA and
113K misfA (but
097K misfA for nitrided gears)
B3 Simplified procedure Note that this is only a guidance and is not a binding con-vention
T = oTZ
1Zsdot
minus where Z = number of cylinders
( ) ideal Videal Vmisf Vnorm V TTT24ZT +minus=
where TV ideal = vibratory torque with all cylinders perfectly equal See also rules Pt4 Ch3 Sec1 G300
When using trends from torsional vibration analysis and measurements the following may be used
TV ideal To is close to zero for engines with few cylinders and using a suitable elastic coupling and increases with relative coupling stiffness and number of cylinders
TV misf To may be high for engines with few cylinders and decreases with number of cylinders
This can be indicated as
200Z
TT
ο
idealV asymp and 80Z04
TT
o
misfV minusasymp
Inserting this into the formulae for the two application fac-tors the following guidance can be given
118112K misfA minusasymp
115110K normA minusasymp
Since KA norm is to be combined with the lower life factors the decisive load condition will be the normal one and a KA of 115 will cover most relevant cases when a suitable elas-tic coupling is chosen
42 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
Appendix C Calculation of Pinion-Rack
Pinion-racks used for elevating of mobile offshore units are open gears that are subjected to wear and tear With normal specifications such as only a few hundred total operation cycles (site to site) the tooth bending stresses for static strength will be decisive for the lay out however with ex-ception of surface hardened pinions where case crushing has to be considered
In the following the use of part 1 and 3 for pinion-racks is shown including relevant simplifications
C1 Pinion tooth root stresses Since the load spectrum normally is dominated by high torques and few load cycles (in the range up to 1-2000) the static strength is decisive
The actual stress is calculated as
β1FSaFan1
tF1 KYY
mbFσ sdotsdotsdotsdot
=
b1 is limited to b2 + 2middotmn
YFa and YSa replace YF and YS because load application at tooth tip has to be assumed for such inaccurate gears
Pinions often use a non-involute profile in the dedendum part of the flank eg a constant radius equal the radius of curva-ture at reference circle For such pinions sFn and hFa are to be measured directly on a sectional drawing of the pinion tooth
Due to high loads and narrow facewidths it may be assumed that KFβ2 = KHβ = 10 However when b1 gt b2 then KFβ1 gt10
If no detailed documentation of KFβ1 is available the fol-lowing may be used
KFβ1 = 1 + 015middot(b1b2 ndash 1)
The permissible stress (not surface hardened) is calculated as
δrelTstF
Fst1FP1 Y
Sσσ sdot=
The mean stress influence due to leg lifting may be disre-garded
The actual and permissible stresses should be calculated for the relevant loads as given in the rules
C2 Rack tooth root stresses The actual stress is calculated as
SaFan2
tF2 YY
mbFσ sdotsdotsdot
=
See C1 for details
The permissible stress is calculated as
δrelTstF
Fst2FP2 Y
Sσσ sdot=
For alloyed steels (Ni Cr Mo) with high toughness and duc-tility the value of YδrelTst may be put equal to YSa
C3 Surface hardened pinions For surface hardened pinions the maximum load is not to cause crushing of the hardened layer of the flank
In principle the calculation described in 213 may be used but when the theoretical Hertzian stress exceeds the ap-proximately 18 times the yield strength of the rack material plastic deformation will occur This will limit the peak Hertzian stress but increases the contact width and thus the penetration of stresses into the depth
An approximation may be based on an assessment of contact width determined by means of equal areas under the theoreti-cal (elastic range) Hertzian contact and the elasto-plastic contact stress (limited to 18 middot σy) with the unknown width
40 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
Appendix A Fatigue Damage Accumulation
The Palmgren-Miner cumulative damage calculation princi-ple is used The procedure may be applied as follows
A1 Stress Spectrum From the individual torque classes the torques (Ti) at the peak values of class intervals and the associated number of cycles (NLi) for both pinion and wheel are to be listed from the highest to the lowest torque
(In case of a cyclic torque variation within the torque classes it is advised to use the peak torque If the cyclic variation is such that the same teeth will repeatedly suffer the peak torque this is a must)
The stress spectra for tooth roots and flanks (σFi σHi) with all relevant factors (except KA) are to be calculated on the basis of the torque spectrum The load dependent K-factors are to be determined for each torque class
A2 σminusN-curve The stress versus load cycle curves for tooth roots and flanks (both pinion and wheel) are to be drawn on the basis of per-missible stresses (ie including the demanded minimum safety factors) as determined in 2 respectively 3 If different safety levels for high cycle fatigue and low cycle fatigue are desired this may be expressed by different demand safety factors applied at the endurance limit respectively at static strength
A3 Damage accumulation The individual damage ratio Di at ith stress level is defined as
Fi
Lii N
ND =
where
NLi = The number of applied cycles at ith stress
NFi = The number of cycles to failure at ith stress
Basically stresses σi below the permissible stress level for infinite life (if a constant ZN or YN is accepted) do not con-tribute to the damage sum However calculating the actual safety factor Sact as described below all the σi for which the product S σi is bigger than or equal to the permissible stress level for infinite life contribute to the damage sum and thus to the determination of Sact The final value of S is decisive
(NFi can be found mathematically by putting the permissible stress σpi equal the actual stress σi thereby finding the actual life factor This life factor can be solved with regard to load cycles ie NFi)
The damage sum ΣDi is not to exceed unity
If ΣDi ne 1 the safety against cumulative fatigue damage is different from the applied demand safety factor For deter-mination of this theoretical safety factor an iteration proce-dure is required as described in the following flowchart
S is correction factor with which the actual safety factor Sact can be found
Sact is the demand safety factor (used in determination of the permissible stresses in the σ ndash N ndash curve) times the correc-tion factor S
The full procedure is to be applied for pinion and wheel tooth roots and flanks
Note If alternating stresses occur in a spectrum of mainly pulsating stresses the alternating stresses may be replaced by equiva-lent pulsating stresses ie by means of division with the ac-tual mean stress influence factor YM
Classification Notes- No 412 41 May 2003
DET NORSKE VERITAS
Appendix B Application Factors for Diesel Driven Gears
For diesel driven gears the application factor KA depends on torsional vibrations Both normal operation and misfiring conditions have to be considered
Normally these two running conditions can be covered by only one calculation
B1 Definitions
Normal operation KAnorm = 0
normv0
TTT +
where
T0 = rated nominal torque
Tv norm = vibratory torque amplitude for normal opera-tion (see rules Pt4 Ch3 Sec1 G301 for defi-nition of ldquonormalrdquo irregularity)
Misfiring operation KA misf = 0
misfv
TTT +
where
T = remaining nominal torque when one cylinder out of action
Tv misf = vibratory torque amplitude in misfiring con-dition This refers to a permissible misfiring condition ie a condition that does not require automatic or immediate corrective actions as speed or pitch reduction
The normal operation is assumed to last for a very high num-ber of cycles such as 1010
The misfiring operation is assumed to last for a limited dura-tion such as 107 cycles
B2 Determination of decisive load Assuming life factor at 1010 cycles as YN = ZN = 092 which usually is relevant the calculation may be performed only once with the combination having the highest value of appli-cation factorlife factor
For bending stresses and scuffing the higher value of
092
K normA and 098
K misfA
For contact stresses the higher value of
092K normA and
113K misfA (but
097K misfA for nitrided gears)
B3 Simplified procedure Note that this is only a guidance and is not a binding con-vention
T = oTZ
1Zsdot
minus where Z = number of cylinders
( ) ideal Videal Vmisf Vnorm V TTT24ZT +minus=
where TV ideal = vibratory torque with all cylinders perfectly equal See also rules Pt4 Ch3 Sec1 G300
When using trends from torsional vibration analysis and measurements the following may be used
TV ideal To is close to zero for engines with few cylinders and using a suitable elastic coupling and increases with relative coupling stiffness and number of cylinders
TV misf To may be high for engines with few cylinders and decreases with number of cylinders
This can be indicated as
200Z
TT
ο
idealV asymp and 80Z04
TT
o
misfV minusasymp
Inserting this into the formulae for the two application fac-tors the following guidance can be given
118112K misfA minusasymp
115110K normA minusasymp
Since KA norm is to be combined with the lower life factors the decisive load condition will be the normal one and a KA of 115 will cover most relevant cases when a suitable elas-tic coupling is chosen
42 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
Appendix C Calculation of Pinion-Rack
Pinion-racks used for elevating of mobile offshore units are open gears that are subjected to wear and tear With normal specifications such as only a few hundred total operation cycles (site to site) the tooth bending stresses for static strength will be decisive for the lay out however with ex-ception of surface hardened pinions where case crushing has to be considered
In the following the use of part 1 and 3 for pinion-racks is shown including relevant simplifications
C1 Pinion tooth root stresses Since the load spectrum normally is dominated by high torques and few load cycles (in the range up to 1-2000) the static strength is decisive
The actual stress is calculated as
β1FSaFan1
tF1 KYY
mbFσ sdotsdotsdotsdot
=
b1 is limited to b2 + 2middotmn
YFa and YSa replace YF and YS because load application at tooth tip has to be assumed for such inaccurate gears
Pinions often use a non-involute profile in the dedendum part of the flank eg a constant radius equal the radius of curva-ture at reference circle For such pinions sFn and hFa are to be measured directly on a sectional drawing of the pinion tooth
Due to high loads and narrow facewidths it may be assumed that KFβ2 = KHβ = 10 However when b1 gt b2 then KFβ1 gt10
If no detailed documentation of KFβ1 is available the fol-lowing may be used
KFβ1 = 1 + 015middot(b1b2 ndash 1)
The permissible stress (not surface hardened) is calculated as
δrelTstF
Fst1FP1 Y
Sσσ sdot=
The mean stress influence due to leg lifting may be disre-garded
The actual and permissible stresses should be calculated for the relevant loads as given in the rules
C2 Rack tooth root stresses The actual stress is calculated as
SaFan2
tF2 YY
mbFσ sdotsdotsdot
=
See C1 for details
The permissible stress is calculated as
δrelTstF
Fst2FP2 Y
Sσσ sdot=
For alloyed steels (Ni Cr Mo) with high toughness and duc-tility the value of YδrelTst may be put equal to YSa
C3 Surface hardened pinions For surface hardened pinions the maximum load is not to cause crushing of the hardened layer of the flank
In principle the calculation described in 213 may be used but when the theoretical Hertzian stress exceeds the ap-proximately 18 times the yield strength of the rack material plastic deformation will occur This will limit the peak Hertzian stress but increases the contact width and thus the penetration of stresses into the depth
An approximation may be based on an assessment of contact width determined by means of equal areas under the theoreti-cal (elastic range) Hertzian contact and the elasto-plastic contact stress (limited to 18 middot σy) with the unknown width
Classification Notes- No 412 41 May 2003
DET NORSKE VERITAS
Appendix B Application Factors for Diesel Driven Gears
For diesel driven gears the application factor KA depends on torsional vibrations Both normal operation and misfiring conditions have to be considered
Normally these two running conditions can be covered by only one calculation
B1 Definitions
Normal operation KAnorm = 0
normv0
TTT +
where
T0 = rated nominal torque
Tv norm = vibratory torque amplitude for normal opera-tion (see rules Pt4 Ch3 Sec1 G301 for defi-nition of ldquonormalrdquo irregularity)
Misfiring operation KA misf = 0
misfv
TTT +
where
T = remaining nominal torque when one cylinder out of action
Tv misf = vibratory torque amplitude in misfiring con-dition This refers to a permissible misfiring condition ie a condition that does not require automatic or immediate corrective actions as speed or pitch reduction
The normal operation is assumed to last for a very high num-ber of cycles such as 1010
The misfiring operation is assumed to last for a limited dura-tion such as 107 cycles
B2 Determination of decisive load Assuming life factor at 1010 cycles as YN = ZN = 092 which usually is relevant the calculation may be performed only once with the combination having the highest value of appli-cation factorlife factor
For bending stresses and scuffing the higher value of
092
K normA and 098
K misfA
For contact stresses the higher value of
092K normA and
113K misfA (but
097K misfA for nitrided gears)
B3 Simplified procedure Note that this is only a guidance and is not a binding con-vention
T = oTZ
1Zsdot
minus where Z = number of cylinders
( ) ideal Videal Vmisf Vnorm V TTT24ZT +minus=
where TV ideal = vibratory torque with all cylinders perfectly equal See also rules Pt4 Ch3 Sec1 G300
When using trends from torsional vibration analysis and measurements the following may be used
TV ideal To is close to zero for engines with few cylinders and using a suitable elastic coupling and increases with relative coupling stiffness and number of cylinders
TV misf To may be high for engines with few cylinders and decreases with number of cylinders
This can be indicated as
200Z
TT
ο
idealV asymp and 80Z04
TT
o
misfV minusasymp
Inserting this into the formulae for the two application fac-tors the following guidance can be given
118112K misfA minusasymp
115110K normA minusasymp
Since KA norm is to be combined with the lower life factors the decisive load condition will be the normal one and a KA of 115 will cover most relevant cases when a suitable elas-tic coupling is chosen
42 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
Appendix C Calculation of Pinion-Rack
Pinion-racks used for elevating of mobile offshore units are open gears that are subjected to wear and tear With normal specifications such as only a few hundred total operation cycles (site to site) the tooth bending stresses for static strength will be decisive for the lay out however with ex-ception of surface hardened pinions where case crushing has to be considered
In the following the use of part 1 and 3 for pinion-racks is shown including relevant simplifications
C1 Pinion tooth root stresses Since the load spectrum normally is dominated by high torques and few load cycles (in the range up to 1-2000) the static strength is decisive
The actual stress is calculated as
β1FSaFan1
tF1 KYY
mbFσ sdotsdotsdotsdot
=
b1 is limited to b2 + 2middotmn
YFa and YSa replace YF and YS because load application at tooth tip has to be assumed for such inaccurate gears
Pinions often use a non-involute profile in the dedendum part of the flank eg a constant radius equal the radius of curva-ture at reference circle For such pinions sFn and hFa are to be measured directly on a sectional drawing of the pinion tooth
Due to high loads and narrow facewidths it may be assumed that KFβ2 = KHβ = 10 However when b1 gt b2 then KFβ1 gt10
If no detailed documentation of KFβ1 is available the fol-lowing may be used
KFβ1 = 1 + 015middot(b1b2 ndash 1)
The permissible stress (not surface hardened) is calculated as
δrelTstF
Fst1FP1 Y
Sσσ sdot=
The mean stress influence due to leg lifting may be disre-garded
The actual and permissible stresses should be calculated for the relevant loads as given in the rules
C2 Rack tooth root stresses The actual stress is calculated as
SaFan2
tF2 YY
mbFσ sdotsdotsdot
=
See C1 for details
The permissible stress is calculated as
δrelTstF
Fst2FP2 Y
Sσσ sdot=
For alloyed steels (Ni Cr Mo) with high toughness and duc-tility the value of YδrelTst may be put equal to YSa
C3 Surface hardened pinions For surface hardened pinions the maximum load is not to cause crushing of the hardened layer of the flank
In principle the calculation described in 213 may be used but when the theoretical Hertzian stress exceeds the ap-proximately 18 times the yield strength of the rack material plastic deformation will occur This will limit the peak Hertzian stress but increases the contact width and thus the penetration of stresses into the depth
An approximation may be based on an assessment of contact width determined by means of equal areas under the theoreti-cal (elastic range) Hertzian contact and the elasto-plastic contact stress (limited to 18 middot σy) with the unknown width
42 Classification Notes- No 412
May 2003
DET NORSKE VERITAS
Appendix C Calculation of Pinion-Rack
Pinion-racks used for elevating of mobile offshore units are open gears that are subjected to wear and tear With normal specifications such as only a few hundred total operation cycles (site to site) the tooth bending stresses for static strength will be decisive for the lay out however with ex-ception of surface hardened pinions where case crushing has to be considered
In the following the use of part 1 and 3 for pinion-racks is shown including relevant simplifications
C1 Pinion tooth root stresses Since the load spectrum normally is dominated by high torques and few load cycles (in the range up to 1-2000) the static strength is decisive
The actual stress is calculated as
β1FSaFan1
tF1 KYY
mbFσ sdotsdotsdotsdot
=
b1 is limited to b2 + 2middotmn
YFa and YSa replace YF and YS because load application at tooth tip has to be assumed for such inaccurate gears
Pinions often use a non-involute profile in the dedendum part of the flank eg a constant radius equal the radius of curva-ture at reference circle For such pinions sFn and hFa are to be measured directly on a sectional drawing of the pinion tooth
Due to high loads and narrow facewidths it may be assumed that KFβ2 = KHβ = 10 However when b1 gt b2 then KFβ1 gt10
If no detailed documentation of KFβ1 is available the fol-lowing may be used
KFβ1 = 1 + 015middot(b1b2 ndash 1)
The permissible stress (not surface hardened) is calculated as
δrelTstF
Fst1FP1 Y
Sσσ sdot=
The mean stress influence due to leg lifting may be disre-garded
The actual and permissible stresses should be calculated for the relevant loads as given in the rules
C2 Rack tooth root stresses The actual stress is calculated as
SaFan2
tF2 YY
mbFσ sdotsdotsdot
=
See C1 for details
The permissible stress is calculated as
δrelTstF
Fst2FP2 Y
Sσσ sdot=
For alloyed steels (Ni Cr Mo) with high toughness and duc-tility the value of YδrelTst may be put equal to YSa
C3 Surface hardened pinions For surface hardened pinions the maximum load is not to cause crushing of the hardened layer of the flank
In principle the calculation described in 213 may be used but when the theoretical Hertzian stress exceeds the ap-proximately 18 times the yield strength of the rack material plastic deformation will occur This will limit the peak Hertzian stress but increases the contact width and thus the penetration of stresses into the depth
An approximation may be based on an assessment of contact width determined by means of equal areas under the theoreti-cal (elastic range) Hertzian contact and the elasto-plastic contact stress (limited to 18 middot σy) with the unknown width