research article frictional effects on gear tooth contact...

Hindawi Publishing CorporationAdvances in TribologyVolume 2013, Article ID 181048, 8 pageshttp://dx.doi.org/10.1155/2013/181048

Research ArticleFrictional Effects on Gear Tooth Contact Analysis

Zheng Li and Ken Mao

The Gearbox Research Institute, Dalian Huarui Heavy Group Co., Ltd., School of Engineering, Warwick University,Coventry CV4 7AL, UK

Correspondence should be addressed to Ken Mao; [email protected]

Received 16 February 2013; Accepted 20 June 2013

Academic Editor: Patrick De Baets

Copyright © 2013 Z. Li and K. Mao. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The present paper concentrates on the investigations regarding the situations of frictional shear stress of gear teeth and the relevantfrictional effects on bending stresses and transmission error in gearmeshing. Sliding friction is one of themajor reasons causing gearfailure and vibration; the adequate consideration of frictional effects is essential for understanding gear contact behavior accurately.An analysis of tooth frictional effect on gear performance in spur gear is presented using finite element method. Nonlinear finiteelementmodel for gear tooth contact with rolling/sliding is then developed.The contact zones formultiple tooth pairs are identifiedand the associated integration situation is derived. The illustrated bending stress and transmission error results with static anddynamic boundary conditions indicate the significant effects due to the sliding friction between the surfaces of contacted gear teeth,and the friction effect can not be ignored. To understand the particular static and dynamic frictional effects on gear tooth contactanalysis, some significant phenomena of gained results will also be discussed. The potentially significant contribution of toothfrictional shear stress is presented, particularly in the case of gear tooth contact analysis with both static and dynamic boundaryconditions.

1. Introduction

Inmodern world, many reliable gearmanufacture techniqueshave existed and operated formodern industry; the advancedhigh performance gears products are indispensable com-ponents for lots of power-train transmission systems. Asa basic transmission part, gears have been used for threethousand years in history, and the investigation of gear designtechniques and product performance kept developing all thetime. In 1892, Wilfred Lewis first presented a formula forevaluating the bending stress of gear teeth in which the toothform entered into the equation.The formula still remains thebasis for most gear design today, and the parameter proposedas “Lewis stress” can indicate the bending stress of loadedcontacted gear teeth accurately. But in further investigationon the basis of the application of Lewis stress, the nonlinearresponses due to frictional effects in meshing process havebecome obvious gradually, and many scientists have beentried to investigate any reliable techniques to predict fric-tional effect on gear performance. In 1982, Barakat [1] pre-sented a technique for the determination of the instantaneouscoefficient of friction at gear tooth contact by considering

tooth fillet strains using finite element method. Ishibashi etal. [2] researched the friction situation of contacted teethat start of running in 1987, and they considered lubricantconditions in gear meshing and found that reduction inthe tooth surface roughness is very effective to decrease thecoefficient of friction and to prevent the cold scoring whichmight become the origins of severe failure in the lengthyrunning.The primary investigations of frictional effects werejust focused on how to determine the coefficient of frictionor the particular amount of friction in gear meshing, butthe significant frictional effects on gear static and dynamicperformance were still secrets.

In 1990s, some gear tooth contact analysis started toconsider frictional effects in the investigations; it really madevery significant progress of gear performance investigation,and the presented technologies regarding predicting thelinear and nonlinear responses to gear tooth contact weremore accurate and reliable during this period. In 1994,Vijayarangan and Ganesan [3] analyzed static contact stressof spur gear tooth with frictional effect using finite elementmethod and Lagrangianmultiplier technique.They discussedthe actual length of contact against the calculated length of

2 Advances in Tribology

contact and presented the variation of contact stress along thecontact surface in a direction normal to the mating surfaces.In 1996, Rebbechi et al. [4] did research whichwas focused ondynamic friction status; they measured the strain at the rootfillets of two successive teeth, and the results of their researchdemonstrated that the friction coefficient does not appear tobe significantly influenced by the sliding reversal at the pitchpoint and that the values of coefficient of friction found arein accord with those in general use.The coefficient of frictionwas found to increase at low sliding speed. The investigationconclusions of this period were much more reliable than lastdecade, but due to the lack of high precision tools (computersoftware and experiment equipment), the accuracy of resultswas still limited; the evaluation of new findings usually usedempirical results as measurement.

Nowadays, many software tools with perfect precisionhave been developed and kept updating; the researches ofgear performance and frictional effects have become moreconvenient and accurate with the help of various finite ele-ment simulation software tools. In 2002, Velex and Sainsot [5]presented an analytical analysis of tooth friction excitations inerrorless spur and helical gears base on Coulomb model. Justfor the most typical spur gear, Mihalidis et al. [6] calculatedthe coefficient of friction at several contact points alongthe path of contact based on a mixed lubrication modelin the same year; their production was very significant forfollowing deeper research of gear contact friction. In recentyears, some new conclusions have been presented, and thefrictional effects status and nonlinear performances causedby such slide friction have been paid more attention bygear designers. In 2007, Kim and Singh [7] presented a newgear surface roughness induced noise source model whiletaking into account the sliding contacts between meshinggear teeth, and they considered dynamic condition for pre-dicting transmission noise. He et al. [8] proposed a newdynamic contacted model which considered sliding friction;the model is highly accurate enough and significant for gearperformance analyzing. The nonlinear dynamic responsesand frictional effects are really complicated problems ofgear research, especially the relationship between individualdynamic response (noise and vibration) and frictional effects,and the conclusions will build significant basis for furtherrelevant investigations.

According to empirical conception foundations and theliterature, this paper will analyze the frictional effects on geartooth bending stress and transmission error inmeshing usingfinite elementmethod and attempt to explain the phenomena.For dynamic response, the transmission error is the mainreason of noise and vibration of system and can be affectedby sliding friction obviously. The reliable microgeometrymodification scheme is effective and necessary to decreaseharmful noise and vibration, but the frictional effect is stillavailable for transmission error regulation.

2. Tooth Profile Generation and FEM Model

The tooth profile of model was generated by mathematicalmethod; the principle of generation is calculating sufficient

X2

r2

O1

Y1

X1

M1

O X

Y

Gear blank rotation

Rack pitch line

Rack movement

O2

Y2

ba

∠𝛼1

∠𝜙2

Figure 1: Involute profile generation. 𝛼1: pressure angle, 𝜙

2: gear

rotation angle corresponding to rack movement of 𝑏1.

point coordinates which belong in tooth profile and thenconnecting all of the points by straight lines. The wholeprocesses can be divided into two parts: involute profilegeneration and root curve generation.

2.1. Involute Profile Generation. According to Figure 1, theequation for 𝑂

1𝑎 line is

𝑦1=

𝑥1

tan𝛼1

. (1)

If a point 𝑀1in 𝑂1𝑎 line will contact with the gear, the

rack needs to be moved a distance 𝑏1(𝑂1𝑂) to achieve the

normal of point𝑀1cross with point 𝑂 according to the gear

conjugating theory:

𝑏1=

𝑦1

tan𝛼1

+ 𝑥1. (2)

For coordinate transfer from 𝑂1𝑋1𝑌1to 𝑂𝑋𝑌,

𝑥 = 𝑥1− 𝑏1,

𝑦 = 𝑦1.

(3)

Then for coordinate transfer from 𝑂𝑋𝑌 to 𝑂2𝑋2𝑌2,

(i) transfer

𝑥= 𝑥,

𝑦= 𝑦 − 𝑟

2,

(4)

(ii) rotate

𝑥2= 𝑥 cos𝜙

2− 𝑦 sin𝜙

2,

𝑦2= 𝑥 sin𝜙

2+ 𝑦 cos𝜙

2,

⇒

𝑥2= 𝑥 cos𝜙

2− (𝑦 − 𝑟

2) sin𝜙

2,

𝑦2= 𝑥 sin𝜙

2+ (𝑦 − 𝑟

2) cos𝜙

2,

⇒

𝑥2= 𝑥 cos𝜙

2− 𝑦 sin𝜙

2+ 𝑟2sin𝜙2,

𝑦2= 𝑥 sin𝜙

2+ 𝑦 cos𝜙

2− 𝑟2cos𝜙2.

(5)

Advances in Tribology 3

r

b

a

𝜆

𝜂

C𝜌𝜌

P

lr 𝜌

lc𝜌

C

𝜙

nn

X

Y

O

𝛼

rsin𝜙

Figure 2: Root profile generation.

b

a fm

cm

𝛼

C𝜌

r𝜌

Figure 3: The hob tooth profile.

Therefore, the point coordinates in system 𝑂2𝑋2𝑌2can be

written as following equations:

𝑥2= (𝑥1− 𝑏1) cos𝜙

2− (𝑦1− 𝑟2) sin𝜙

2,

𝑦2= (𝑥1− 𝑏1) sin𝜙

2+ (𝑦1− 𝑟2) cos𝜙

2,

(6)

where 𝑟2is the gear pitch circle radius, 𝜙

2= 𝑏1/𝑟2.

The gear profile generated by line 𝑂1𝑎 will be obtained

by solving the combined equations (1), (2), and (6); the finalsimultaneous equations are

𝑦1=

𝑥1

tan𝛼1

,

𝑏1=

𝑦1

tan𝛼1

+ 𝑥1,

𝑥2= (𝑥1− 𝑏1) cos𝜙

2− (𝑦1− 𝑟2) sin𝜙

2,

𝑦2= (𝑥1− 𝑏1) sin𝜙

2+ (𝑦1− 𝑟2) cos𝜙

2.

(7)

2.2. Root Curve Generation. Global coordinate system: 𝑂𝜂𝜆.The final simultaneous equations are expressed as following,and the meanings of necessary factors are indicated byFigures 2 and 3.

X

Y

Figure 4: Gear geometry and contact model.

The final simultaneous equations are

𝑎 = 𝑓𝑚 + 𝑐𝑚 − 𝑟𝜌,

𝑏 =

𝜋𝑚

2

,

𝑟𝜌=

𝑐𝑚

1 − sin𝛼,

𝜂 = 𝑟 sin𝜙 − ( 𝑎sin𝛼+ 𝑟𝜌) cos (𝛼 − 𝜙) ,

𝜆 = 𝑟 cos𝜙 − ( 𝑎sin𝛼+ 𝑟𝜌) sin (𝛼 − 𝜙) .

sin𝜙 = 1𝑟

(

𝑎

tan𝛼+ 𝑏) ,

(8)

where 𝑚 is module, 𝑓 is depth coefficient: whole depth/module, 𝑐 is Clearance coefficient, 𝛼: pressure angle, 𝛼 ispariable amount, the variable zone is 0∼90∘, and (𝜂, 𝜆) isrequired point coordinate.

In this research, a two-dimension finite element modelis built using ABAQUS to simulate standard involute spurgear pair. The gear will run at 100 rad/s with 30Nm load. Theprimary parameters of the gears are listed in Table 1. Figure 4shows the 2D gear pair model for the study.

3. Static Bending Stress andTransmission Error

Bending stress analysis is an important objective of staticperformance investigation because it is the primary reasonof various gear tooth failures. The empirical conceptions ofstatic gear stress analysis have already demonstrated that thegear tooth can be considered as cantilever beam, so the com-plicated contact question can be simplified to solve a simplecantilever beam strength calculation. The examinations ofbending stress will consult Lewis’ stress theories. Accordingto the common cantilever beams hypothesis for gear research,the Lewis stress formula can be obtained from the most basictheories regarding cantilever beam.


Table 1: Gear model parameters.

Type SpurModule 2mmNumber of teeth 24Pressure angle 20∘

Tooth width 3.142mmTip radius 26mmRoot radius 21.5mmModification factor 0Contact ratio 1.77Element size: 0.2mmMaterial

SteelDensity 7.8 g/cm3

Young’s modulus 210000MPaPoisson’s ratio 0.3

Transmission error is considered to be an importantexcitation mechanism for gear noise and vibration, andmany researches relate to gear noise and vibration arealways focused on transmission error situation to achieveparticular purposes. Transmission error has been definedas the difference between the actual position of the outputgear and the position it would occupy if the gear drive wereperfect. Actually, transmission error is an important kindof deformation for the gear set and it is the main sourceof noise and vibration because the deformation and frictionwill cause harmful impact and abrasion. In actual designand manufacture processes, the deformation and frictionmust exist, which means the transmission error cannot beeliminated. The frictional effect on gear meshing will alsoaffect transmission error obviously, and it will make thetransmission error anomalous to produce serious noise andvibration. The investigation of transmission error is verysignificant for optimizing dynamic performance by reducingnoise and vibration because the most effective method tocontrol harmful dynamic response is transmission errorregulation.

In this research, the particular static bending tensilestress and transmission error results will be illustrated. Theinvestigation analyzes two coefficients of friction, one withvalue 0 and the other with value 0.05 to determine thesignificant frictional effects on static bending tensile stressand transmission error.

Figure 5 is the result of gear static bending tensile stresswith static boundary conditions group.The interaction pointsA and B are two special points which implicate certainsignificant contact events in gear meshing process.

The result lines indicate that the general trends of staticbending tensile stresses with different coefficient of frictionare very similar. Point A means that the gear tooth which thecritical point belongs to is starting to contact at the moment;the interaction point B of the two lines indicates that thereis no friction shear stress at the moment no matter howhigh the coefficient of friction of gear tooth surface is, and itimplicates that the meshing gear teeth are mainly contacting

−2000

200400600800

10001200

0 1 2Base pitch

Static bending tensile stress

𝜇 = 0

𝜇 = 0.05

B

A

Bend

ing

tens

ile st

ress

(MPa

)

Figure 5: Static bending tensile stress results.

at pitch point in nature. In fact, a complete gear teeth contactprocess can be divided into two different contact phases: startpoint-pitch point and pitch point-finish point; the integratedbending stress status is different during these two contactphases due to the friction. It is also the most significantfrictional effect on static bending stress.

According to both of the result lines in Figure 5, it canbe seen that the relative positions of the two result lines aredifferent.The position of line 𝜇= 0 is higher than line 𝜇= 0.05during the segment between points A and B. It indicates thatthe bending stress with low coefficient of friction is higher inthe first contact phase from start point to pitch point, but thestatus will be opposite in the second contact phase from pitchpoint to finish point. The implication of this phenomenonis that the friction can decrease bending tensile stress in thefirst contact phase but increase bending tensile stress in thesecond contact phase.The reason of this phenomenon can beexplained by simple analysis of forces in gearmeshing processas in the following discussions.

Whilst a pair of teeth starts to mesh, the input forcewill act on contact point with vertical direction to contactsurface. The component of input force in vertical direction(relative to horizontal surface) is the bending stress which isthe inspection object. If the friction between contact surfaceshas been ignored, the force analysis of gear tooth should beshown in Figure 6 where 𝐹

𝑛is input force, 𝐹

𝑡and 𝐹

𝑟are

vertical and horizontal components of 𝐹𝑛, and 𝑂 is contact

point.The force analysis bases on non-frictional situation of

gear contact, but the friction must exist between contactsurfaces due to relative sliding, and the bending stress willbe affected by frictional effect. In frictional situation, the geartooth contact process can be divided into two phases: startpoint-pitch point and pitch point-end point. It is necessary tonotice that the pitch point is a special node; there is no frictionforce in the moment when the contact point overlaps pitchpoint, and the relative sliding direction will change from thatmoment. That means the friction force will start to reversewhen the gear teeth are contacting at pitch point, and thebending stress status is different whilst the contact point staysbefore and after pitch point. According to the studies, theforce analysis with frictional effect can be shown as in Figures6(b) and 6(c).


O

Fr

FtFn

(a) Frictionless

O

Fr

FtFn

f

fr

ft

(b) The approach process

O

Fr

Ft

Fn

f

frft

(c) The recess process

Figure 6: Gear contact force.

Figure 6 indicates the relation of the input force andfriction force. The input force vertical component 𝐹V andhorizontal component 𝐹

ℎwhich cause bending stress of non-

frictional model should be

𝐹V = 𝐹𝑡,

𝐹ℎ= 𝐹𝑟.

(9)

In the first contact phase, the friction force components havebeen shown as in Figure 6(b). The current external contactforce (including input force and friction force) componentswhich cause bending stress should be

𝑓 = 𝐹𝑛× 𝜇,

𝐹V = 𝐹𝑡 − 𝑓𝑡,

𝐹ℎ= 𝐹𝑟+ 𝑓𝑟,

(10)

where 𝜇 is the coefficient of friction.In the second contact phase, the friction status is shown

as in Figure 6(c), friction force direction is opposite to thefirst contact phase, and the current external contact forcecomponents which cause bending stress should be

𝑓 = 𝐹𝑛× 𝜇,

𝐹V = 𝐹𝑡 + 𝑓𝑡,

𝐹ℎ= 𝐹𝑟− 𝑓𝑟,

(11)

where 𝜇 is coefficient of friction.The formulas explain significant phenomena owing to

sliding/rolling frictional effect. The friction force componentcounteracts the input force component in vertical, and thedirections of horizontal contact force component and theelastic deformation are different, so the high horizontal con-tact force componentswill decrease bending stress.Therefore,the bending tensile stress should be smaller than non-frictional model in the first phase. In the second contactphase, the friction force direction has become opposite, thefriction component and input force component in vertical arein the same direction to amplify resultant contact force, andthe horizontal input force component has also been weaken

00.0005

0.0010.0015

0.0020.0025

0.0030.0035

0.0040.0045

0 1 2Base pitch

Static transmission error

𝜇 = 0

𝜇 = 0.05

Tran

smiss

ion

erro

r(ra

d)

Figure 7: Static transmission error results.

due to friction force component. Consequently, it shouldbe larger than the non-frictional model. In addition, thesimilar phenomena have been displayed by relevant furtherinvestigations regarding static bending shearing stress. Infact, the perfect work conditions are impossible for actualmeshing process, the friction must exist between contactedsurfaces, and so a conclusion regarding frictional effects onbending stress base on static analysis can be summarizedlike this: the bending stress of gear tooth will be amplifiedby frictional effect in the phase in which the contact pointmoves from start point to pitch point and will be weaken byfrictional effect in the phase which the contact point movesfrom pitch point to end point.

Figure 7 is the result of gear static transmission error withstatic boundary conditions group. It reflects some significantphenomenon derived from frictional effect.

Some conclusions can be summarized according to theresult figure. First, the segments of individual result line ineach base pitch are the same. Second, generally speaking,transmission error is always considered as the main sourceof noise and vibration in gear meshing. The reason is thatthe transmission error leads to serious impact and extrusionbetween contacted gear teeth in meshing. The transmissionerror undulations in the result figure implicate that thegear must produce vibration in meshing due to the rotationdisplacement differences between pinion and wheel, and thevibration is the reason of referred impact and extrusion.


Third, the result indicates that the frictional effect on statictransmission error is very limited. There are no remarkabledifferences caused by friction. But in fact, the transmissionerror status can be influenced by lots of boundary conditions.The results of static analysis can only display the most typicalsituations of transmission error and the accuracy of statictransmission error result is not very satisfied and reliable.Therelevant investigations of dynamic analysis should be moreaccurate to reflect the actual situations.

To summarize, the frictional effect on static transmissionerror assuredly exists, but it is not very obvious and can beignored completely.

4. Dynamic Bending Stress andTransmission Error

The introduction regarding dynamic research schemes alsobases on the same contents as static analysis. In fact, thedynamic analysis is much more complicated than staticanalysis, and some significant phenomena which cannot beindicated in static analysis will be discussed carefully indynamic analysis. Many of dynamic boundary conditionswill lead obvious excitations to affect the characteristics ofgear contact significantly. A basic standpoint of this thesisis that the transmission error is a main source of harmfulnoise and vibration, but strictly speaking, it is not the onlyreason of those nonlinear responses.The empirical researcheshave indicated that the vibrations and noise are all derivedfrom various excitations of certain characteristic parametersof gear contact, and the transmission error is only oneparticular element of those excitations. It should considerthat the vibration and noise in gear meshing are particularresponses which are caused by various dynamic excitations,and the excitations can also lead some nonlinear responsesto make the dynamic performance of meshing gears morecomplicated. So the significant objectives of dynamic analysisare to investigate some typical responses to dynamic exci-tations and to decrease harmful influences as effectively aspossible.

The excitations include two categories mainly whichare “external” and “internal” excitations, and the internalexcitations are significant dynamic excitations of meshinggears. The external excitations are always caused by someexternal boundary conditions such as load, input momentand power, and engine speed; the particular responses aresimilar to other transmission systems. Internal excitationsof gear set are dynamic excitations in gear meshing pro-cess, and it includes three basic types: stiffness excitation,transmission error excitation, andmeshing impact excitation.However, all of the internal and external excitations mustexist in actual meshing process and affect gear performanceseriously. So the excitations make the actual situations ofgear meshing more complicated and lead many problemsin dynamic analysis. Although the finite element methodcan simulate the gear meshing process better and produceless error than other mathematical analytical methods, it isstill hard to gain very accurate results and achieve perfectobjectives.

−2000

200400600800

1000120014001600

0 1 2Base pitch

Dynamic bending tensile stress

𝜇 = 0

𝜇 = 0.05

Bend

ing

tens

ile st

ress

(MPa

)

Figure 8: Dynamic bending tensile stress results (𝜔 = 100 rad/s).

It is also necessary to mention that the dynamic investi-gations in this research are only on the basis of some empir-ical conclusions and simple linear and nonlinear vibrationtheories, so the conclusions and discussions are not veryaccurate in nature. Furthermore, the research strategy can notcover all complicated dynamic responses caused by variousexternal and internal excitations in gear meshing completelyit is only hoped to display and discuss some basically typicalphenomena which are derived from dynamic responses.Figure 8 displays the particular situation of dynamic bendingtensile stress with the angular velocity being 100 rad/s. It is fardifferent from the results of static analysis.

There are serious fluctuations of each result line andthe phenomenon implicates the internal excitations in gearmeshing. But based on the results of static analysis as areference, the general varying trend of dynamic bendingstress is still similar to static bending stress, and it is easy toexplain that the undulations of dynamic bending stress mustderive from the internal excitations. The undulation is a kindof typical dynamic transient response, and according to theresult figure, the internal excitations will not affect the steadyresponses very seriously.

The static bending stress results have indicated the dif-ference between the relative positions of two tensile stressresult lines in different contact phases due to frictional effect,and the interaction point of the two lines means the gearteeth are contacting at pitch point. It is an important andsignificant conclusion regarding frictional effect on staticbending stress. In dynamic analysis, the situation is similar,but the general situation of relative positions of two resultlines is not very obvious due to the undulations, and thesignificant interaction point is hard to display to implicate themoment when the teeth are contacting at pitch point. Theseparticular phenomena are the most significant differencesbetween the results of static and dynamic analysis.

Figure 9 is the dynamic transmission error result during 2base pitches. To compare with the result of static analysis, thegeneral varying trends of dynamic and static TE result lines𝜇 = 0.05 are similar; it implicates that the steady responseof dynamic transmission error also accords with the staticrules. The undulations of dynamic transmission error linesderive from various dynamic excitations, and in particular,the vibrations of gear set and the impacts between contactinggear teeth are main sources. In addition, the steady response


00.0010.0020.0030.0040.0050.0060.007

0 1 2Base pitch

Dynamic transmission error

𝜇 = 0

𝜇 = 0.05

Tran

smiss

ion

erro

r(r

ad)

Figure 9: Dynamic transmission error results (𝜔 = 100 rad/s).

of dynamic TE result lines 𝜇 = 0.05 line in the first basepitch does not obviously accord with the basic static rules. Itis believed that the reason of this situation is the gear’s inertiaduring the wheel gear run from quiet to unit rotary speed, sothe steady response during this period is not obvious due tothe wheel gear’s inertia.

According to the result lines in Figure 9, the frictionaleffects on dynamic transmission error can be indicated. Instatic analysis, the friction will not affect transmission errorsignificantly, but the frictional effect on dynamic transmis-sion error has been amplified seriously and could not beignored any more. Figure 9 shows the line 𝜇 = 0.05 islittle more steady and regular than the line 𝜇 = 0. Thisphenomenon implicates that the friction has affected both thesteady and transient responses by decreasing the amplitude ofundulations. Generally speaking, the difference between themaximum andminimumpeak values of transmission error isthemain reference to estimate the noise intensity roughly; thesituation of transmission error results in Figure 9 implicatesthat the frictional effect can decrease gear noise obviously dueto dynamic excitations.

5. Summary

The paper analyzed the frictional effects on bending tensilestress and transmission error in static and dynamic boundaryconditions. Base on the discussions and explanations in thispaper, the integrated frictional effects can be summarized.The frictional effects on bending tensile stress are listed asfollows.

(1) The friction can decrease bending stress during thefirst contact phase, but the bending stress will beincreased due to friction during the second contactphase.

(2) The moment when the frictional shear stress disap-pears implicates that the gear teeth are contactingat pitch point, and the frictional effects will becomeopposite from now on.

(3) The degree of frictional effects on bending stressdepends on the coefficient of friction of contactsurface and input torque.

(4) Dynamic frictional effects on bending stress are sim-ilar to static frictional effects. But the dynamic fric-tional effects are not very obvious, and the momentwhen the friction shear stress disappears is hard to beindicated due to the various dynamic excitations.

(5) Friction can weaken the bending stress undulations(typical transient response) due to various dynamicexcitations.

The frictional effects on transmission error should beconcluded as follows.

(1) In static analysis, the friction can increase resultvalues, but the noise will be decreased slightly due toreliable friction.

(2) In dynamic analysis, the friction can decrease harm-ful responses to dynamic excitations. The transientresponseswill also beweakened, and the transmissionerror result line will become more steady and regular.

(3) Friction can also affect noise in dynamic analysis, butit is believed that too serious friction can make thetransmission error result line more complicated.

(4) Dynamic frictional effects on transmission error arenot steady, and it will be affected and changed gradu-ally during the gear meshing.

(5) The steady responses will not be changed significantlydue to friction.

The results of research indicate that the frictional effectis an important boundary condition in gear design. For gearstatic performance, friction will affect the bending stressobviously; it should be considered carefully in gear toothstrength analysis. For dynamic performance, although thetooth profile modification can regulate complicated trans-mission error, the frictional effect will affect modificationscheme in high load work conditions. To sum up, frictioneffect should not be ignored; it will not only affect staticstress and transmission but also produce other boundaryconditions which may affect gear performance; for example,the sliding friction will produce thermal in meshing, andthere is a little difference of the tooth profile curve by thermaleffect, and the profile modification scheme should considerit to regulate. In actual manufacture processes, the frictionaleffect will affect production performance seriously; designersshould pay more attention to avoid or use frictional effect foroptimizing gear performance.

References

[1] M. A. Barakat, “Determination of instantaneous coefficient ofgear tooth friction,” Journal of Engineering Sciences, vol. 8, no. 1,pp. 9–19, 1982.

[2] A. Ishibashi, S. Ezoe, and S. Tanaka, “Friction coefficientsbetween contacting teeth at start of running,” pp. 845–850,Institution ofMechanical Engineers, Proceedings of the IMechEConference.

[3] S. Vijayarangan and N. Ganesan, “Static contact stress analysisof a spur gear tooth using the finite element method, including


frictional effects,” Computers and Structures, vol. 51, no. 6, pp.765–770, 1994.

[4] B. Rebbechi, F. B. Oswald, and D. P. Townsend, Measurementof Gear Tooth Dynamic Friction, vol. 88, American Society ofMechanical Engineers, Design Engineering Division, 1996.

[5] P. Velex and P. Sainsot, “An analytical study of tooth frictionexcitations in errorless spur and helical gears,” Mechanism andMachine Theory, vol. 37, no. 7, pp. 641–658, 2002.

[6] A. Mihalidis, V. Bakolas, K. Panagiotidis, and N. Drivakos,“Prediction of the friction coefficient of spur gear pairs,” VDIBerichte, vol. 2, no. 1665, pp. 705–719, 2002.

[7] S. Kim and R. Singh, “Gear surface roughness induced noiseprediction based on a linear time-varying model with slidingfriction,” JVC/Journal of Vibration and Control, vol. 13, no. 7, pp.1045–1063, 2007.

[8] S. He, R. Gunda, and R. Singh, “Inclusion of sliding friction incontact dynamicsmodel for helical gears,” Journal ofMechanicalDesign, vol. 129, no. 1, pp. 48–57, 2007.

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