Research ArticleChaotic Dynamics of Cage Behavior ina High-Speed Cylindrical Roller Bearing

Long Chen Xintao Xia Haotian Zheng and Ming Qiu

School of Mechatronics Engineering Henan University of Science and Technology Luoyang 471039 China

Correspondence should be addressed to Long Chen haustchenlong163com

Received 7 August 2015 Revised 14 December 2015 Accepted 15 December 2015

Academic Editor Jussi Sopanen

Copyright copy 2016 Long Chen et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

This paper presents a mathematical model to investigate the nonlinear dynamic behavior of cage in high-speed cylindrical bearingVariations of cage behavior due to varying cage eccentricity and cage guidance gap are observed Hydrodynamic behavior in cagecontacts is taken into consideration for a more realistic calculation of acting forces owing to high working speed Analysis of real-time cage dynamic behavior on radial plane is carried out using chaos theory based on the theoretical and mathematical modelestablished in the paper The analytical results of this paper provide a solid foundation for designing and manufacturing of high-speed cylindrical roller bearing

1 Introductions

Radial cylindrical roller bearings are designed to carry radialloads and be applied under high-speed conditions Usuallycylindrical roller bearings may be obtained as a unit whichincludes two steel rings each of which having a hardenedraceway on which hardened cylindrical rollers roll Therollers are usually held in an angularly spaced relationship bya cageThe cage ismade frommachined brass or pressed steelBrass cage is widely used in high-speed application

There are normally twomethods to distinguish the type ofcylindrical roller bearings One is classified by arrangementof the ribs Depending on the type of bearing either theinner or the outer ring has two roller guiding ribs The otheris classified by the types of cage guidance There are threetypes of cage guidance as briefly demonstrated in Figure 1They are outer-ring-rib guidance (Figure 1(a)) inner-ring-rib guidance (Figure 1(b)) and roller guidance (Figure 1(c))respectively The weight of the cage acts on rollers directlyof roller guidance bearing and it acts on innerouter ringrespectively of innerouter guidance bearing when it ismounted horizontally in most common application

As shown in Figure 1 four kinds of gaps between compo-nents can be found in the section They are radial clearance(119866119903) roller gap (Δ) cage guidance gap (119862

119903) and cage axial

gap (119862119886) respectively These gaps are defined as the max-

imum possible displacements between relative componentsin radialaxial direction There is another important variablenamed pocket clearance (119879) in rolling bearing design andit can be observed from Figure 2 Obviously pocket clear-ance can be defined by the difference between cage pocketdiameter and roller diameter The cage pocket diameter hasto be optimized to avoid faster wear of cage in terms of betterlubrication film forming and decreased roller-cage bridgeimpact forces

According to the general design guidelines of rollingbearings the value of cage axial gap (119862

119886) is larger than roller

axial gap (Δ) the value of pocket clearance (119879) is larger thancage guidance gap (119862

119903) and the value of cage guidance gap

(119862119903) is larger than bearing radial clearance (119866

119903) usuallyThese

rules ensure that the rollers contact with raceways directlyThe rollersrsquo skewing can be adjusted by ring ribs according tothe rules Then the cage is in a certain state of being ldquofreerdquo inthe bearing

When cylindrical bearings operate at a high speed theygenerate vibrations and noise The principal forces whichdrive these vibrations are time varying nonlinear contactforces which exist among the various components of thebearings rings rollers and cage In the last decades alot of efforts have been devoted to studying the stability

Hindawi Publishing CorporationShock and VibrationVolume 2016 Article ID 9120505 12 pageshttpdxdoiorg10115520169120505

2 Shock and Vibration

Outer ring

Inner ring

Cage Rollers

Δ2 Δ2

Ca2 Ca2

Gr

Cr

(a) Outer-ring-rib guidance

Δ2 Δ2

Ca2Ca2

Gr

Cr

(b) Inner-ring-rib guidance

Δ2 Δ2

Ca2 Ca2

Gr

Cr

(c) Roller guidance

Figure 1 Cage guidance in cylindrical roller bearings

and nonlinear dynamic behavior of flexible rotor bearingsObviously one of themost importantmechanical elements tobe taken into account is bearings due to their large influenceon the dynamic behavior of rotating machinery (Tiwari et al[1] Adam Jr [2])

Many researchers developed special technique ondynamic behavior considering details of geometricparameters and applicable conditions of rolling bearingsAkturk et al [3] and Upadhyay et al [4] presented theoreticalinvestigations of varying preload the influence of the numberof the balls and ball diameters on vibration characteristicsof a rotor bearing system Akturk [5] and Harsha et al [6]researched the effect of surface waviness on vibrations of ballbearings Sopanen and Mikkola [7] and Upadhyay et al [8]investigated dynamic behaviors of high-speed rotation withlocalized and distributed defects Harsha [9] and Villa et al[10] presented a nonlinear dynamic analysis of a flexibleunbalanced rotor supported by ball bearings

The importance of energy efficiency has been increasingand has become a quality criterion for bearing producers andusers in recent years Hencemore andmore researchers drewtheir attention on dynamic behaviors of the cage in rollingbearing Houpert [11] developed a simulation software tosimulate cage behavior and carried out relative experimentalvalidation Harsha [12] analyzed the nonlinear dynamics ofball bearings due to cage run-out and varying number ofballs He presented the results in the form of fast Fouriertransformations (FFT) and phase trajectories It is impliedfrom the obtained FFT that due to the nonuniform spacingthe ball passage frequency is modulated with the cagefrequency Bercea et al [13] and Sakaguchi and Harada [14]investigated cage behavior of tapered roller bearings andmade a comparison between numerical and experimentalresults

In this paper a theoretical investigation considering allpossible contacts of cage is conducted to observe its dynamicbehavior Owing to the investigation object applied in high-speed working condition hydrodynamics in the contacts istaken into consideration for a more realistic calculation ofacting forces by simulation model with the hypothesis of

enough lubrication to supply all the contacts Brass is chosenas the raw material of the cage in the paper consideringthe working speed of the bearing Brass cages are machinedby lathe and drilling in processing technique in effect Thiscauses uneven quality distribution and error of the cageroundnessThendifferent cage eccentricity and cage gap drawour attention and they are analyzed In addition lubricationdynamic viscosity exponent is another key factor in theapplication of high-speed bearings Influence of differentlubrication dynamic viscosity exponents is investigated aswell Chaos theory is applied for analyzing above-mentionedparameters Both Poincare map and bifurcation diagram areobtained and the key factor which influences the dynamicbehavior is found

2 Static Displacements of the Cage

The bearing under study has the outer ring fixed to a rigidsupport and the inner ring fixed rigidly to the shaft Aconstant vertical radial load acts on the bearing and it shiftsthe inner ring to the outer ring through rollers The staticradial displacement is influenced by many factors such asclearance load and number of the rollers

21 Displacements of Different Guidance Types As illustratedin the instructions different guidance types mean differentcage structure and different contact position Then differentdisplacement caused by different guidance type is describedfirst here

211 Outer-Ring-Rib-Guidance Bearing As can be seen inFigure 1(a) radial gap between cage outer diameter and innerdiameter of outer ring rib is the only moveable space forouter-ring-rib-guidance cylindrical roller bearing Then thedisplacement is confirmed by cage guidance gap (119862

119903) and it

can be given by

119884119888119900=119862119903

2 (1)

Shock and Vibration 3

F

Supporting zone

Outer ring

Inner ring

Cage Rollers

R-C contact region

XY

Z

120596o

120596w

120596c

120596i

OOOD

OC

120593

i c o

2120579

Gr

Cr

(a) Schematic diagram

TT

XY

Z

T

T

120596o 120596c120596i

(b) Pocket clearance

Figure 2 Schematic diagram and pocket clearance in a cylindrical roller bearing

212 Inner-Ring-Rib-Guidance Bearing Inner ring displace-ment of inner-ring-rib-guided bearing is caused by the clear-ance (119866

119903) cage guidance gap (119862

119903) and the elastic deformation

(120575119903) in the contact lines of the raceways and the rollers

The value of elastic deformation depends on the value ofthe load and the number of the rollers under load Thensupporting zone as can be seen in Figure 2(a) should beconfirmed firstlyThe angular extent of the supporting zone isdetermined by the radial clearance (119866

119903) of a cylindrical roller

bearing such as

2120579 = cosminus1119866119903

2120575119903

(2)

Elastic deformation between the raceway and roller showsa nonlinear relation which is obtained by using the Hertziantheory [15] The local Hertzian contact force and deflectionrelationship for a bearing may be written as

119876119894= 119870119894120575119894

109

119876119900= 119870119900120575119900

109

(3)

The local Hertzian contact forces in (3) are determinedby external load (119865) number of rollers in supporting zone(119873) and the radial clearance (119866

119903) The calculation method is

a very standardized procedure and will not be described hereThe total normal approach between two raceways under loadseparated by a roller is the sum of the approaches between theroller and each raceway Hence

120575119903= 120575119903119894+ 120575119903119900 (4)

Then the radial displacement of the cage can be obtainedfrom above equations as

119884119888119894= 120575119903+1

2119866119903+1

2119862119903 (5)

213 Roller Guidance The normal approach between tworaceways under load influences the cage displacementdirectly in this kind of guidance type The cage displacementin roller-guidance bearing can be given as

119884119888119908= 120575119903+1

2119866119903+ 119879 (6)

22 Cage Displacement Caused by Rotating Bearings areused to bear various kinds of loads while keeping a shaftrotating Rollers are subjected to dynamic loading due tospeed effects As a general case it will be initially assumedthat both inner and outer rings are rotating in a bearing asillustrated in Figure 2 Consequently

V119894=1

2120596119894119889119894

V119900=1

2120596119900119889119900

(7)

Then the cage speed or the speed of rotation of the set ofrollers around the origin is

120596119888=1

2[120596119894(1 minus

119889119908

119889119898

) + 120596119900(1 +

119889119908

119889119898

)] (8)

The centrifugal force caused by the 119895th roller is calculatedas

119865119895cen= 119898119895

1

2120596119888

2

119889119898 (9)

According to design guidelines of rolling bearings thevalue of 119862

119886is larger than that of Δ the value of 119879 is larger

than that of 119862119903 and the value of 119862

119903is larger than that of

119866119903generally Thus the rollersrsquo centrifugal forces are acting

4 Shock and Vibration

X

Y

p

e

120596o

120596c

OO

OD

OC

2120579

120572

120588

F(jminus2)cw

fy

fx

Rg

RcCr

Fjcen

Figure 3 Forces generated by motion on the cage

on outer raceway Subsequently the centrifugal forces of therollers cause larger contact deformation It can be added tothe original deformation on the outer raceway as

120575119903119900= (

119865cen + 119876119900119870119900

)

910

(10)

Substituting (10) into (4) equations (5) and (6) willchange along with (4) According to different expressions ofcage displacements both inner ring guidance and roller guid-ance bearings will generate complex dynamic displacementswhen they are operating at high speedsThenwe can drawourconclusion that both inner ring guidance and roller guidanceare not applicable under high-speed conditions Outer-ring-rib-guidance bearing is chosen as the research object in thefollowing analysis naturally

3 Physical Model

The cage has two kinds of contacts in a rotating outer-ring-guided cylindrical bearing One is the cage-rib contact andthe other is the roller-pocket contactThe forces generated bythe contacts are marked in Figure 3 Cage-rib contact statusis determined by the relative rotational speed (120596

119888119900) and the

cage guidance gap (119862119903) The cage has physical contact with

guiding ring in the starting phase owing to a low relativespeed However the cage and the rib are separated by oilfilm eventually with the increasing of relative speed Thencage-rib contact is considered as a fluid-structure interactionproblem Similarly roller-cage contact status is determined byrotational speed of the roller around its own axil (120596

119888119908) and the

pocket clearance (119879) The cage and the rollers are separatedby oil film with the increasing of speed too Note that thecage is driven by the rollers in supporting zone and the rollersare driven by cage in nonloaded zone The driving forces byrollers on the cage can be seen in Figure 3 as well Differentdrivingmode of the rollers in different force directions causesimpact loads on the cage The impact between the cage andthe roller in a very short time results in large impact forces

31 Instantaneous Motion Forces Different contacts causedifferent motion forces and there is no strong relationshipamong the motion forces then linear superposition methodcan be adopted for calculating motion forces on the cage

The relative rotational velocity of the cage to the rib isdefined as

120596119888119900= 120596119888minus 120596119900 (11)

And the cage guidance gap determined by cage diameterand rib can be expressed as

119862119903= 119889119892minus 119889119888 (12)

Brass cage is usually machined from centrifugally casttubing and then nonuniform distribution of the mass causesstatic imbalance (119890) of the cage Nondimensional eccentricityof the cage to cage guidance gap can be given as

120588 =119890

119862119903

(13)

Spiridon [16] established an elastic system with twodegrees of freedom (DOF) that considered both cage-roller stiffness (119870

119888119908) and cage structural stiffness (119870

119888) for

cageroller contact As described in Figure 4 the dynamicequilibrium condition written for each of the two massesprovides the following set of two differential equations

119870119888119908(120575119908minus 120575119888)109

+119872119908

1198892

120575119908

1198891199052= 0

minus119870119888119908(120575119908minus 120575119888)109

+ 119870119888120575119888+119872119888

1198892

120575119908

1198891199052= 0

(14)

119870119888119908

can be derived from the Hertzian contact theorywhereas119870

119888can be obtained analyticallyThe cage-roller stress

(119876119888119908) can be obtained from (3) and its variation trend for

the specified roller is shown in Figure 5 when the bearing isrotating

Impact force of cageroller contact can be given as

119891ℎ119909= 119876119888119908[sin120596

119888119900119905 + sin (120587 minus 120596

119888119900119905)] cos 120579

119891ℎ119910= 119876119888119908[sin120596

119888119900119905 minus sin (120587 minus 120596

119888119900119905)] sin 120579

(15)

As for the cage-rib dynamic model it can encompassradial-plane orbital cage motion with 2-DOF equationsas well According to Figure 4 motion equation can beexpressed as

119872119863119863= minus2119870

119904(119874119862minus 119874119874) minus 2119861

119904(119862minus 119874)

+ 2119872119863120588119892119890119910

minus 119865119904(119883119863 119863 119863 119884119863 119863 119863)

119872119862119862= minus119870119904(119874119862minus 119874119874) minus 119861119904(119862minus 119874) + 119891119890119909sdot 119890119909

+ 119891119890119910sdot 119890119910minus119872119874120588119892119890119910

(16)

Shock and Vibration 5

Roller

Cage120575c(t)

120575w(t)

(a) Cageroller contact

120575c(t) 120575w(t)

kckcw

Mc Mw

(b) Model of cageroller contact

Figure 4 A 2-DOF model of cageroller contact of the 119895th roller

Load

(N)

0 1 2 3 4 5 6minus40

minus20

0

20

40

60

80

100

120

t(s) times 10minus4

Figure 5 Demonstration on variation trend of cage-roller contactforce

The position vector velocity vector and accelerationvector of the unbalanced gravitational center of the cage canbe given as

119874119863= (119883119888+ 120588 cos120596119905) sdot 119890

119909+ (119884119888+ 120588 sin120596119905) sdot 119890

119910

119863= (119888+ 120588 cos120596119905) sdot 119890

119909+ (119888+ 120588 sin120596119905) sdot 119890

119910

119863= (119888+ 120588 cos120596119905) sdot 119890

119909+ (119888+ 120588 sin120596119905) sdot 119890

119910

(17)

In the same way the position velocity and accelerationvector of geometry center of the cage are

119874119888= 119883119888sdot 119890119909+ 119884119888sdot 119890119910

119888= 119888sdot 119890119909+ 119888sdot 119890119910

119888= 119888sdot 119890119909+ 119888sdot 119890119910

(18)

Similarly using above-mentioned exhibition the posi-tion velocity and acceleration vector of the center of the ribare

119874119900= 119883119900sdot 119890119909+ 119884119900sdot 119890119910

119900= 119883119900sdot 119890119909+ 119900sdot 119890119910

119900= 119883119900sdot 119890119909+ 119900sdot 119890119910

(19)

Substituting (15) (16) and (17) into (14) then the follow-ing two equations described in 119909-119910 plane can be obtained

119872119863119874

= minus2119870119904(119883119862minus 119883119874) minus 2119861

119904(119862minus 119874)

+ 2119872119863120588120596119888minus119900

2 cos120596119888minus119900119905 minus 119870119909119910119884119863minus 1198701015840

119909119910119884119863

3

119872119863119874

= minus2119870119904(119884119862minus 119884119874) minus 2119861

119904(119862minus 119874)

+ 2119872119863120588120596119888minus119900

2 sin120596119888minus119900119905 minus 119870119909119910119883119863minus 1198701015840

119909119910119883119863

3

minus 2119872119863119892

(20)

119872119862119862minus 119891119890119909minus 119870119904(119883119874minus 119883119862) minus 119861119904(119874minus 119862) = 0

119872119862119862minus 119891119890119910minus 119870119904(119884119874minus 119884119862) minus 119861119904(119862minus 119874)

= minus119872119862119892

(21)

Considering (20)-(21) the results can be obtained as

119891119890119909= 1205921

2

(119874minus 119862) minus 1205922(119874minus 119862) minus 1205923(119909119874minus 119909119862)

119891119890119910= 1205922

2

( 119910119874minus 119910119862) minus 1205922( 119910119874minus 119910119862) minus 1205923(119910119874minus 119910119862)

+ 119870119891119889119891119889

(22)

Here 119909119874 119910119874 119909119862 and 119910

119862are nondimensional form of

rib center (119883119874 119884119874) and cage center (119883

119862 119884119862) and they are

obtained by original value divided by 119862119903 The other factors in

(22) can be expressed as

1205921=2 (119872119863120596119888119900119861)2

119882

1205922= 2119870119904119861

1205923=2119870119904

32

119861

11987212

119863

120596119888119900

(23)

6 Shock and Vibration

When the linear superposition method is adopted tocontact forces of the cage in 119909-119910 plane the total force ondifferent directions can be given as

119891119909= 119891119890119909

119891119910= 119891119886119910+ 119891119890119910

(24)

32 Hydrodynamic Fluid-Film Force Cage-rib contact is afluid-structure interaction problem with bearings rotatingat a high speed The main supporting force generated bylubricant squeeze effects can be approximated to a short-width journal bearing theory as illustrated in Figure 3 Thendynamic-deviation interactive force between cage and rib canbe described as a continuous function of cage position andvelocity components Reynolds lubrication equation (RLE)provides the basis of lubrication theory [15] and the forcegiven on the cage by the film is expressed as

120597

120597120572[(1 + 120576 cos120572)3

120597119901

120597120572] + 1198772

119888

120597

120597119909[(1 + 120576 cos120572)2

120597119901

120597119910]

= minus6120583(119877119888

119862119903

)

2

[(120596119888119900minus 2) 120576 sin120572 minus 2120576 cos120572]

(25)

And

119901 (120572) = 1198750+120596119888119900

119862119903

3[1198972

119888

4minus 1199102

]120576 sin120572

1 + 120576cos3120572 (26)

Solutions to the RLE are a nonlinear function of displace-ment and angle at cage center Hydrodynamic fluid-film forcecan be written as follows

119891119890= 120583119877119888119897119888(119897119888

119862119903

)

2

[(120596119888119900minus 2)

1205762

(1 minus 1205762)2

+120587

2

(1 + 21205762

) 120576

(1 minus 1205762)52

]

119891120572= 120583119877119888119897119888(119877119888

119862119903

)

2

sdot [(120596119888119900minus 2120572)

120587120576

4 (1 minus 1205762)32

+2120576 120576

(1 minus 1205762)2]

+ 21198771198881198971198881199010

(27)

The fluid-film force can also be represented as two com-ponent forces in horizontal and vertical force equilibrium

119891119909= 119891119890sdot cos120572 + 119891

120572sdot sin120572

119891119910= 119891119890sdot sin120572 minus 119891

120572sdot cos120572

(28)

33 Equilibrium of the Equations In hydrodynamic fluid-film bearing the fluid supported pressure is generated nor-mally by motion of the cage and depends on the dynamic

viscosity exponent of the lubricating fluid For the cagestructure and coordination please refer to Figure 6 Thegenerated forces can be expressed as

119891119890sdot cos120572 + 119891

120572sdot sin120572 = 120592

1

2

119862minus 1205922(119874minus 119862)

minus 1205923(119909119874minus 119909119862)

119891119890sdot sin120572 minus 119891

120572sdot cos120572 = 120592

1

2

119910119862minus 1205922( 119910119874minus 119910119862)

minus 1205923(119910119874minus 119910119862) + 119861119891119889119891119889

+ 119891119894119910

(29)

Displacement relations between the centers of cage andrib are

119909119888= 119909119900+ 120576 cos120593

119910119888= 119910119900+ 120576 sin120593

(30)

Similarly for the velocity

119888= 119900+ 120576 cos120593 minus 120576 sin120593 sdot

119910119888= 119910119900+ 120576 sin120593 minus 120576 cos120593 sdot

(31)

Substituting (27) (30) and (31) into (29) integration ofequation can be obtained

120576 (1 minus 1205762

)3

+ 120576 [1205922(1 minus 120576

2

)3

minus 1205921015840

1

120587

2(1 + 2120576

2

) (1 minus 1205762

)12

] minus 1205922120576 sin 2120572 sdot (1

minus 1205762

)3

+ 1205923120576 (1 minus 120576

2

)3

minus 120583 cos 2120572 minus 12059210158401(120596119888119900minus 2)

sdot 1205762

(1 minus 1205762

) = 0

[120576 (1 minus 1205762

)2

+ 120576 (1 minus 1205762

)2

2120583 sin 2120572 sdot (1 minus 1205762)2

]

+ [2 120576 (1 minus 1205762

)2

+ 1205922120576 (1 minus 120576

2

)2

]

+ 1205921015840

1[(120596119888119900minus 2120572) 120587120576 (1 minus 120576

2

)12

+ 2120576 120576]

+ 21198771198881198971198881198750 = 0

(32)

Equation (32) can be solved by modified Newmark-120573method to obtain the displacement and velocity of the cage

4 Chaos Analyses

Chaos Theory is a new way to analyze complexity whichHenri Poincare studied as the possibility of forecastingA small variation in the initial conditions can generate adifferent set of resultssolutions [17]

A bearing typed as NU2310 is chosen to analyze chaoticresponse Tomoya and Kaoru [18] tested the dynamic cagedisplacements of this bearing and the maximum displace-ments can be used to verify the validity of theoretical analysis

Shock and Vibration 7

e

x

y

X

Y

OO

OD

OC

Rg

Rc

120576

(a) Cage-rib contact

Mc

MD

xc(t)Kx

Kxy

Ky K998400xyBy

Bx

yc(t)

(b) Cage-rib contact model

Figure 6 2-DOF model of cagerib contact

The structural specifications of the bearing are listed inTable 1

The cage of the bearing is made of machined brass DINCuZn40Pb2F43 (GB ZCuZn38Mn2Pb2) and the rings of thebearing are made of through-harden rolling bearing steelDIN 100Cr6 (SAE 52100GB GCr15) The cage is guided bythe outer-ring rib Both static load rating and dynamic loadrating are 186KN According to the recommendation of themanufacturer reference speed rating is 6700 rpm and limitedspeed rating is 12000 rpm Then the weight of the cage anddamping and stiffness parameters can be obtained accordingto the relative parameters in Table 1 The displacement in119909 direction is zero and the displacement in 119910 directionis determined by the cage guidance gap according to (1)Bifurcation diagram of 119883

119862versus spin speed increases with

different cage guide gaps cage eccentricities and dynamicviscosity exponent of lubrication oil The change curvesare described in Figure 7 respectively All three cases aresimulated and variation parameters of the cases are listed inTable 2 An interesting phenomenon that all the cases havetwo bifurcation zones and two steady zones is found in allcurves In relative low rotation speed transient responses canbe found The dynamic response of cage center goes intothe first ordered stage after the end of the resonance Thenthe first bifurcation response occurs The second orderedstatus arises after the first bifurcation ends and then thecurve goes into the second bifurcation Figure 7(a) comparesthe bifurcation diagrams of cage with different guidancegaps The first chaotic response zone of the bearing withsmaller guidance gap (case 1a) is from 120596

119888119900= 9100 rpm

to 120596119888119900= 12000 rpm and it is from 120596

119888119900= 7100 rpm to

120596119888119900= 22000 rpm for the bearing with larger guidance gap

(case 1b) The second chaotic response zone of the bearingwith smaller guidance gap is from 120596

119888119900= 25500 rpm and

it is from 120596119888119900

= 26000 rpm for the bearing with largerguidance gap Chaotic response zone is wider when the gap islarger Figure 7(b) shows bifurcation diagrams of cage with

different eccentricities The first chaotic zone of case 2a isfrom 120596

119888119900= 7600 rpm to 120596

119888119900= 14000 rpm and it is from

120596119888119900= 7800 rpm to 120596

119888119900= 13000 rpm of case 2b The second

chaotic zone of case 2a is from 120596119888119900= 22500 rpm and it is

from 120596119888119900= 19800 rpm of case 2b Chaotic response zone

increaseswith eccentricity In addition transient oscillation ismore drastic when comparing case 1a and case 2a Figure 7(c)depicts bifurcation diagrams of cage with different dynamicviscosity exponents The first chaotic zones of case 3a andcase 3b are from 120596

119888119900= 7650 rpm to 120596

119888119900= 13400 rpm The

second chaotic zones of case 3a and case 3b are from 120596119888119900=

21000 rpmThe change trend of case 3a curve is approachingto that of case 3b But the variation curve of cage with VG78is lower than that with VG56 due to higher dynamic viscosityexponent

The first bifurcation point 119884119862versus spin speed zooming

is shown in Figure 8 Fluctuation can be found even in steadyzone owing to the impact forces between roller and cagepockets In low-speed zone the wave frequency is low and itincreases with rotational speedThe difference of steady zonerange is shown more clearly in the enlarged view Bifurcationdiagrams of each case are different obviously and this denotesthat the position of119883

119862varies timely as well

Cage response behaviors of chaos which are subjected tothe cases in Table 2 at120596 = 8500RPM Poincare section pointsof 119884119862displacement in vertical direction and time history for

4th revolution to 4000th revolution are shown in Figure 9

5 Conclusion and Discussion

Rolling bearing life is typically calculated on the basis of itsload ratings relative to the applied loads and the requirementsregarding bearing life and reliability Both dynamic loadrating and rating life equations neglect cage design Actuallythe cage design has evolved to bear higher and higher load byincorporating more rollers and to lower costs by adopting

8 Shock and Vibration

2500 7500 12500 17500 22500 27500

Rotation speed (rpm)

Case 1b

Case 1a

03

02

01

0

minus01

minus02

minus03

Vert

ical

disp

lace

men

t of

cage

cent

er (m

m)

(a) Different cage guidance gaps

2500 7500 12500 17500 22500 27500

Rotation speed (rpm)

Case 2b

Case 2a

03

02

01

0

minus01

minus02

minus03

Vert

ical

disp

lace

men

t of

cage

cent

er (m

m)

(b) Different eccentricities

2500 7500 12500 17500 22500 27500

Rotation speed (rpm)

Case 3b

Case 3a

03

02

01

0

minus01

minus02

minus03

Vert

ical

disp

lace

men

t of

cage

cent

er (m

m)

(c) Different dynamic viscosity exponents

Figure 7 Bifurcation diagram points119883119862versus spin speed

Table 1 Specifications of the bearing structure and size

Bearing typeBoundary size Roller size Rib diameter Clearance

Boremm

Outsidemm

Widthmm Number Diameter

mmLengthmm mm Radial

120583mPocketmm

NU2310 Φ50 Φ110 40 13 Φ16 27 Φ921 40 03

Table 2 Case study

CaseVariation parameters

Guidance gap Eccentricity Dynamic viscosity exponent(mm) (mm) (Pasdots)

1a 045 23 VG561b 060 23 VG562a 045 35 VG562b 045 57 VG563a 045 42 VG783b 045 42 VG32

different material and improve high-speed performance byusing optimized geometry structure This paper pays closeattention to influence from cage on speed characteristics ofcylindrical roller bearings The most suitable cage guidancetype is confirmed by mathematical derivation firstly Twomajor factors originating from manufacturing process andone involved in applying process are considered to establishdynamical mathematical model of cage with appropriate cageguidance type Analysis of real-time dynamic behavior of

a cage on radial plane was carried out using chaos theorybased on the theoretical mathematical model The chaoticanalysis is limited by the initial values of the system Hencethe initial values are determined firstly in analytical processPreliminary conclusions are listed as follows

(1) There are two bifurcation zones of cage dynamiccurves and the bearing should be away from these twozones in practiceThe range of the 1st bifurcation zoneismainly influenced by the value of cage guidance gapand eccentricity especially the cage guidance gapThelubrication dynamic viscosity exponent has relativelylittle influence on cage dynamic

(2) Numerical simulations show that the roller tends toacceleratedecelerate in switching process of support-ing zoneloaded zone which leads to single or mul-tiple roller-cage bridge impacts Impact force causesperiodic small oscillation on the curves Oscillationamplitude is determined by the pocket gap

(3) Eccentricity of the cage causes dramatic transientresponses in low rotational speed zone The transientresponses are easily tending to cause knocking and

Shock and Vibration 9

7000 10000 13000 16000 19000 22000

Rotation speed (rpm)

Case 1b

Case 1a

002

001

0

minus001

minus002

minus003

minus004

minus005

minus006

minus007

minus008

Vert

ical

disp

lace

men

t of

cage

cent

er (m

m)

(a) Different cage guidance gaps

Rotation speed (rpm)

Case 2b

Case 2a

0

minus001

minus002

minus003

006

005

004

003

002

001

7500 8800 10100 11400 12700 14000

Vert

ical

disp

lace

men

t of

cage

cent

er (m

m)

(b) Different eccentricities

Case 3b

Case 3a

Rotation speed (rpm)7500 8800 10100 11400 12700 14000

0

minus001

minus002

minus003

minus004

005

004

003

002

001

Vert

ical

disp

lace

men

t of

cage

cent

er (m

m)

(c) Different dynamic viscosity exponents

Figure 8 The first bifurcation zone from Figure 7 in enlarged scale

faster wear of cage pocket in terms of worse lubrica-tion film forming The steady zone is narrower wheneccentricity increases In addition divergent trendcan be found in the second bifurcation zonewith lagereccentricity

(4) According to the zooming of the first bifurcationzone variations of any parameters cause totally dif-ferent bifurcations We can draw our conclusion thatmovement of cage is a chaotic motion

(5) The Poincare maps show that position and rotationalspeed of the occurrence bifurcation zones are verydifferent with varied parameters

(6) The change of fluid dynamic viscosity exponent willnot affect the region of bifurcation due to the weaknonlinear of fluid film force

Considering cage dynamics of high-speed cylindricalroller bearing the values of cage guidance gap and eccen-tricity are crucial factors From the point of view of themanufacturers the very small cage guidance gap is difficult toobtain owing to the material (brass) of the cage used in high-speed bearingThe authors strongly suggest that the designershould minimize the gap within the confine of processingcapacity As for the cage eccentricity dynamic balance ofcage is widely used in ultra-high-speed bearings Analysesfrom the paper show that cage eccentricity brings violent

oscillation in low-speed zone and oscillation amplitude inchaotic zone is higher with the increase of cage eccentricityHence dynamic balance is another important factor inmanufacturing process With respect to lubrication dynamicviscosity exponent this study indicates that influence fromdynamic viscosity exponent on cage dynamics is not signifi-cant as the prediction

As a matter of fact researches in the paper are merelyrestricted to cage dynamic response What is to be pointedout is that dynamic response of a cylindrical roller bearingis a complex and interrelated problem among componentsHence there are many imperfect points in the paper whichshould be noted here Preliminary discussions are listed asfollows

(1) The displacements of the cage discussed in the paperare only in 119909-119910 plane During the rotation of thebearing rollers will inevitably have a certain slip-page towards the rings due to insufficient tractionfriction outside the load-zone churning momentfrom lubricant the friction loss between roller-pocketcontact and roller-rib contacts in the actual runningprocess Rollerraceways contact has not been takeninto account in the paper for the sake of simplifyingthe calculation

(2) Damping coefficient and contact stiffness involvedin the calculation are obtained according to material

10 Shock and Vibration

minus03 minus02 minus01 0 01 02minus400

minus200

0

200

400

Vertical displacement of cage center (mm)

Vert

ical

velo

city

(mm

s)

(a) Smaller cage guidance

minus03 minus02 minus01 0 01 02Vertical displacement of cage center (mm)

minus400

minus200

0

200

400

Vert

ical

velo

city

(mm

s)

(b) Bigger cage guidance

minus02 minus01 0 01Vertical displacement of cage center (mm)

minus300

minus200

0

200

300

Vert

ical

velo

city

(mm

s)

(c) Smaller eccentricity

minus03 minus02 minus01 0 01 02Vertical displacement of cage center (mm)

minus400

minus200

0

200

400

Vert

ical

velo

city

(mm

s)

(d) Bigger eccentricity

minus03 minus02 minus01 0 01 02Vertical displacement of cage center (mm)

minus400

minus200

0

200

400

Vert

ical

velo

city

(mm

s)

(e) Higher dynamic viscosity exponent

minus03 minus02 minus01 0 01 02Vertical displacement of cage center (mm)

minus400

minus200

0

200

400

Vert

ical

velo

city

(mm

s)

(f) Lower dynamic viscosity exponent

Figure 9 Poincare section points of 119884119862displacement in vertical direction

property and geometry characteristic parametersActually the damping coefficient and contact stiff-ness can be described more accurately by immediatemovement and velocity Namely contact stiffness isthe first-order derivative of immediate movementand damping coefficient is the first-order derivative ofimmediate velocity For the same sake of simplifyingcalculation the accurate expressions are not substi-tuted into the equations

(3) The bearing under study has the outer ring fixed to arigid support and the inner ring fixed rigidly to theshaft Bearing house and the shaft are truly nonlinearDOF systems The influence of bearing house andshaft on cage dynamic is relatively small so it can besimplified in the calculation

(4) As illustrated in Section 3 cage is driven by the rollersin supporting zone and the rollers are driven by cage

Shock and Vibration 11

in nonloaded zone Driving forces on the total cageare balanced and they are ignored except the impactforces in switching zones of driven modes

Symbols

120572 Angle at cage center in rad Angular velocity in tangential direction in rads Angular acceleration in tangential direction in

rads2119861 Damping coefficient119862 Cage gap in mm119889 Diameter in mmsΔ Roller gap in mm120575 Deformation in mm119890 Cage eccentricity in mm120576 Displacement in normal direction in mm120576 Velocity in normal direction in mms120576 Acceleration in normal direction in mms2119865 Load in N119891 Load caused by motion in N119866 Clearance in mm120579 Angle in ∘119870 Hertzian stiffness119897 Length in mm119872 Mass in kg120583 Fluid dynamic viscosity in Pasdots119873 Number of the rollers120592 Dimensionless parameter of impact factor119874 Position of the center in mm Velocity vector of the center in mms Acceleration vector of the center in mms2119901 Hydrodynamic pressure in Pa1199010 Initial hydrodynamic pressure in Pa

119876 Hertz stress in Nmm2119877 Radius in mm120588 Dimensionless parameter of cage eccentricity to cage

gap119905 Time in sec119879 Pocket clearance in mmV Surface velocity in mms119883 Coordinate119883 direction distance in mm119909 Position in coordinate direction in mm Velocity in 119909 direction in mms Acceleration in 119909 direction in mms2119884 Coordinate 119884 direction distance in mm119910 Dimensionless parameter in coordinate direction119910 Velocity in 119910 direction in mms119910 Acceleration in 119910 direction in mms2120596 Rotation speed in rads

Subscripts

119886 Axial direction120572 Tangential direction119862 Cage center119888 Cagecen Centrifugal

119888119894 Relative relation between cage and innerring

119888119900 Relative relation between cage and outerring

119888119908 Relative relation between cage and roller119863 Unbalanced gravitational center119890 Normal direction119891119889 Added fluid

119892 Ribℎ Impact119894 Inner raceway119895 The 119895th roller119895cen Centrifugal force of the 119895th roller119898 Mean value119874 Rib center119900 Outer raceway119903 Radial direction119903119894 Inner raceway on radial direction119903119900 Outer raceway on radial direction119904 Polar coordinate119908 Roller119909 119909 direction119910 119910 direction

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The project is supported by National Natural Science Foun-dation of China (Grant no 51475144) and the Foundation ofInnovation and Research Team of Science and Technology inUniversities in Henan Province (Grant no 13IRTSTHN025)

References

[1] M Tiwari K Gupta and O Prakash ldquoDynamic response of anunbalanced rotor supported on ball bearingsrdquo Journal of Soundand Vibration vol 238 no 5 pp 757ndash779 2000

[2] M L Adam Jr Rotating Machinery VibrationmdashFrom Analysisto Trouble Shooting Marcel Dekker New York NY USA 2001

[3] N Akturk M Uneeb and R Gohar ldquoThe effects of number ofballs and preload on vibrations associated with ball bearingsrdquoJournal of Tribology vol 119 no 4 pp 747ndash753 1997

[4] S H Upadhyay S C Jain and S P Harsha ldquoNon-linearvibration signature analysis of a high-speed rotating shaft due toball size variations and varying number of ballsrdquo Proceedings ofthe Institution of Mechanical Engineers Part K Journal of Multi-body Dynamics vol 223 no 2 pp 83ndash105 2009

[5] N Akturk ldquoThe effect of waviness on vibrations associated withball bearingsrdquo Journal of Tribology vol 121 no 4 pp 667ndash6771999

[6] S P Harsha K Sandeep and R Prakash ldquoNon-linear dynamicbehaviors of rolling element bearings due to surface wavinessrdquoJournal of Sound and Vibration vol 272 no 3ndash5 pp 557ndash5802004

[7] J Sopanen and A Mikkola ldquoDynamic model of a deep-grooveball bearing including localized and distributed defects Part

12 Shock and Vibration

2 Implementation and resultsrdquo Proceedings of the Institution ofMechanical Engineers Part K vol 217 no 3 pp 213ndash223 2003

[8] S H Upadhyay S C Jain and S P Harsha ldquoChaotic dynamicsof high speed rotating shaft supported by ball bearings due todistributed defectsrdquo International Journal of Engineering Scienceand Technology vol 2 no 10 pp 5746ndash5794 2010

[9] S P Harsha ldquoNonlinear dynamic analysis of a high-speed rotorsupported by rolling element bearingsrdquo Journal of Sound andVibration vol 290 no 1-2 pp 65ndash100 2006

[10] C Villa J-J Sinou and F Thouverez ldquoStability and vibrationanalysis of a complex flexible rotor bearing systemrdquo Communi-cations in Nonlinear Science and Numerical Simulation vol 13no 4 pp 804ndash821 2008

[11] L Houpert ldquoCAGEDYN a contribution to roller bearingdynamic calculations Part III experimental validationrdquo ASMETribology Transactions vol 53 no 6 pp 848ndash859 2010

[12] S P Harsha ldquoNonlinear dynamic analysis of rolling elementbearings due to cage run-out and number of ballsrdquo Journal ofSound and Vibration vol 289 no 1-2 pp 360ndash381 2006

[13] I Bercea S Cretu M Bercea and D Olaru ldquoSimulating roller-cage pocket friction in a tapered roller bearingrdquo EuropeanJournal of Mechanical and Environmental Engineering vol 43no 4 pp 189ndash194 1998

[14] T Sakaguchi andKHarada ldquoDynamic analysis of cage behaviorin a tapered roller bearingrdquo Journal of Tribology vol 128 no 3pp 604ndash611 2006

[15] T A Harris Rolling Bearing Analysis John Wiley amp Sons 4thedition 2001

[16] C Spiridon ldquoMachined brass and pressed steel bearing cages acomparative studyrdquo RKB Technical Review vol 7 pp 1ndash13 2011

[17] R Riccardo ldquoChaos theory and some practical applications intechnical analysisrdquo in Proceedings of the 11th Annual Conferenceof the International Federation of Technical Analysts 1998

[18] S Tomoya and U Kaoru ldquoDynamic analysis of cage behavior ina cylindrical roller bearingrdquo NTN Technical Review 71 2004

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International Journal of

2 Shock and Vibration

Outer ring

Inner ring

Cage Rollers

Δ2 Δ2

Ca2 Ca2

Gr

Cr

(a) Outer-ring-rib guidance

Δ2 Δ2

Ca2Ca2

Gr

Cr

(b) Inner-ring-rib guidance

Δ2 Δ2

Ca2 Ca2

Gr

Cr

(c) Roller guidance

Figure 1 Cage guidance in cylindrical roller bearings

and nonlinear dynamic behavior of flexible rotor bearingsObviously one of themost importantmechanical elements tobe taken into account is bearings due to their large influenceon the dynamic behavior of rotating machinery (Tiwari et al[1] Adam Jr [2])

Many researchers developed special technique ondynamic behavior considering details of geometricparameters and applicable conditions of rolling bearingsAkturk et al [3] and Upadhyay et al [4] presented theoreticalinvestigations of varying preload the influence of the numberof the balls and ball diameters on vibration characteristicsof a rotor bearing system Akturk [5] and Harsha et al [6]researched the effect of surface waviness on vibrations of ballbearings Sopanen and Mikkola [7] and Upadhyay et al [8]investigated dynamic behaviors of high-speed rotation withlocalized and distributed defects Harsha [9] and Villa et al[10] presented a nonlinear dynamic analysis of a flexibleunbalanced rotor supported by ball bearings

The importance of energy efficiency has been increasingand has become a quality criterion for bearing producers andusers in recent years Hencemore andmore researchers drewtheir attention on dynamic behaviors of the cage in rollingbearing Houpert [11] developed a simulation software tosimulate cage behavior and carried out relative experimentalvalidation Harsha [12] analyzed the nonlinear dynamics ofball bearings due to cage run-out and varying number ofballs He presented the results in the form of fast Fouriertransformations (FFT) and phase trajectories It is impliedfrom the obtained FFT that due to the nonuniform spacingthe ball passage frequency is modulated with the cagefrequency Bercea et al [13] and Sakaguchi and Harada [14]investigated cage behavior of tapered roller bearings andmade a comparison between numerical and experimentalresults

In this paper a theoretical investigation considering allpossible contacts of cage is conducted to observe its dynamicbehavior Owing to the investigation object applied in high-speed working condition hydrodynamics in the contacts istaken into consideration for a more realistic calculation ofacting forces by simulation model with the hypothesis of

enough lubrication to supply all the contacts Brass is chosenas the raw material of the cage in the paper consideringthe working speed of the bearing Brass cages are machinedby lathe and drilling in processing technique in effect Thiscauses uneven quality distribution and error of the cageroundnessThendifferent cage eccentricity and cage gap drawour attention and they are analyzed In addition lubricationdynamic viscosity exponent is another key factor in theapplication of high-speed bearings Influence of differentlubrication dynamic viscosity exponents is investigated aswell Chaos theory is applied for analyzing above-mentionedparameters Both Poincare map and bifurcation diagram areobtained and the key factor which influences the dynamicbehavior is found

2 Static Displacements of the Cage

The bearing under study has the outer ring fixed to a rigidsupport and the inner ring fixed rigidly to the shaft Aconstant vertical radial load acts on the bearing and it shiftsthe inner ring to the outer ring through rollers The staticradial displacement is influenced by many factors such asclearance load and number of the rollers

21 Displacements of Different Guidance Types As illustratedin the instructions different guidance types mean differentcage structure and different contact position Then differentdisplacement caused by different guidance type is describedfirst here

211 Outer-Ring-Rib-Guidance Bearing As can be seen inFigure 1(a) radial gap between cage outer diameter and innerdiameter of outer ring rib is the only moveable space forouter-ring-rib-guidance cylindrical roller bearing Then thedisplacement is confirmed by cage guidance gap (119862

119903) and it

can be given by

119884119888119900=119862119903

2 (1)

Shock and Vibration 3

F

Supporting zone

Outer ring

Inner ring

Cage Rollers

R-C contact region

XY

Z

120596o

120596w

120596c

120596i

OOOD

OC

120593

i c o

2120579

Gr

Cr

(a) Schematic diagram

TT

XY

Z

T

T

120596o 120596c120596i

(b) Pocket clearance

Figure 2 Schematic diagram and pocket clearance in a cylindrical roller bearing

212 Inner-Ring-Rib-Guidance Bearing Inner ring displace-ment of inner-ring-rib-guided bearing is caused by the clear-ance (119866

119903) cage guidance gap (119862

119903) and the elastic deformation

(120575119903) in the contact lines of the raceways and the rollers

The value of elastic deformation depends on the value ofthe load and the number of the rollers under load Thensupporting zone as can be seen in Figure 2(a) should beconfirmed firstlyThe angular extent of the supporting zone isdetermined by the radial clearance (119866

119903) of a cylindrical roller

bearing such as

2120579 = cosminus1119866119903

2120575119903

(2)

Elastic deformation between the raceway and roller showsa nonlinear relation which is obtained by using the Hertziantheory [15] The local Hertzian contact force and deflectionrelationship for a bearing may be written as

119876119894= 119870119894120575119894

109

119876119900= 119870119900120575119900

109

(3)

The local Hertzian contact forces in (3) are determinedby external load (119865) number of rollers in supporting zone(119873) and the radial clearance (119866

119903) The calculation method is

a very standardized procedure and will not be described hereThe total normal approach between two raceways under loadseparated by a roller is the sum of the approaches between theroller and each raceway Hence

120575119903= 120575119903119894+ 120575119903119900 (4)

Then the radial displacement of the cage can be obtainedfrom above equations as

119884119888119894= 120575119903+1

2119866119903+1

2119862119903 (5)

213 Roller Guidance The normal approach between tworaceways under load influences the cage displacementdirectly in this kind of guidance type The cage displacementin roller-guidance bearing can be given as

119884119888119908= 120575119903+1

2119866119903+ 119879 (6)

22 Cage Displacement Caused by Rotating Bearings areused to bear various kinds of loads while keeping a shaftrotating Rollers are subjected to dynamic loading due tospeed effects As a general case it will be initially assumedthat both inner and outer rings are rotating in a bearing asillustrated in Figure 2 Consequently

V119894=1

2120596119894119889119894

V119900=1

2120596119900119889119900

(7)

Then the cage speed or the speed of rotation of the set ofrollers around the origin is

120596119888=1

2[120596119894(1 minus

119889119908

119889119898

) + 120596119900(1 +

119889119908

119889119898

)] (8)

The centrifugal force caused by the 119895th roller is calculatedas

119865119895cen= 119898119895

1

2120596119888

2

119889119898 (9)

According to design guidelines of rolling bearings thevalue of 119862

119886is larger than that of Δ the value of 119879 is larger

than that of 119862119903 and the value of 119862

119903is larger than that of

119866119903generally Thus the rollersrsquo centrifugal forces are acting

4 Shock and Vibration

X

Y

p

e

120596o

120596c

OO

OD

OC

2120579

120572

120588

F(jminus2)cw

fy

fx

Rg

RcCr

Fjcen

Figure 3 Forces generated by motion on the cage

on outer raceway Subsequently the centrifugal forces of therollers cause larger contact deformation It can be added tothe original deformation on the outer raceway as

120575119903119900= (

119865cen + 119876119900119870119900

)

910

(10)

Substituting (10) into (4) equations (5) and (6) willchange along with (4) According to different expressions ofcage displacements both inner ring guidance and roller guid-ance bearings will generate complex dynamic displacementswhen they are operating at high speedsThenwe can drawourconclusion that both inner ring guidance and roller guidanceare not applicable under high-speed conditions Outer-ring-rib-guidance bearing is chosen as the research object in thefollowing analysis naturally

3 Physical Model

The cage has two kinds of contacts in a rotating outer-ring-guided cylindrical bearing One is the cage-rib contact andthe other is the roller-pocket contactThe forces generated bythe contacts are marked in Figure 3 Cage-rib contact statusis determined by the relative rotational speed (120596

119888119900) and the

cage guidance gap (119862119903) The cage has physical contact with

guiding ring in the starting phase owing to a low relativespeed However the cage and the rib are separated by oilfilm eventually with the increasing of relative speed Thencage-rib contact is considered as a fluid-structure interactionproblem Similarly roller-cage contact status is determined byrotational speed of the roller around its own axil (120596

119888119908) and the

pocket clearance (119879) The cage and the rollers are separatedby oil film with the increasing of speed too Note that thecage is driven by the rollers in supporting zone and the rollersare driven by cage in nonloaded zone The driving forces byrollers on the cage can be seen in Figure 3 as well Differentdrivingmode of the rollers in different force directions causesimpact loads on the cage The impact between the cage andthe roller in a very short time results in large impact forces

31 Instantaneous Motion Forces Different contacts causedifferent motion forces and there is no strong relationshipamong the motion forces then linear superposition methodcan be adopted for calculating motion forces on the cage

The relative rotational velocity of the cage to the rib isdefined as

120596119888119900= 120596119888minus 120596119900 (11)

And the cage guidance gap determined by cage diameterand rib can be expressed as

119862119903= 119889119892minus 119889119888 (12)

Brass cage is usually machined from centrifugally casttubing and then nonuniform distribution of the mass causesstatic imbalance (119890) of the cage Nondimensional eccentricityof the cage to cage guidance gap can be given as

120588 =119890

119862119903

(13)

Spiridon [16] established an elastic system with twodegrees of freedom (DOF) that considered both cage-roller stiffness (119870

119888119908) and cage structural stiffness (119870

119888) for

cageroller contact As described in Figure 4 the dynamicequilibrium condition written for each of the two massesprovides the following set of two differential equations

119870119888119908(120575119908minus 120575119888)109

+119872119908

1198892

120575119908

1198891199052= 0

minus119870119888119908(120575119908minus 120575119888)109

+ 119870119888120575119888+119872119888

1198892

120575119908

1198891199052= 0

(14)

119870119888119908

can be derived from the Hertzian contact theorywhereas119870

119888can be obtained analyticallyThe cage-roller stress

(119876119888119908) can be obtained from (3) and its variation trend for

the specified roller is shown in Figure 5 when the bearing isrotating

Impact force of cageroller contact can be given as

119891ℎ119909= 119876119888119908[sin120596

119888119900119905 + sin (120587 minus 120596

119888119900119905)] cos 120579

119891ℎ119910= 119876119888119908[sin120596

119888119900119905 minus sin (120587 minus 120596

119888119900119905)] sin 120579

(15)

As for the cage-rib dynamic model it can encompassradial-plane orbital cage motion with 2-DOF equationsas well According to Figure 4 motion equation can beexpressed as

119872119863119863= minus2119870

119904(119874119862minus 119874119874) minus 2119861

119904(119862minus 119874)

+ 2119872119863120588119892119890119910

minus 119865119904(119883119863 119863 119863 119884119863 119863 119863)

119872119862119862= minus119870119904(119874119862minus 119874119874) minus 119861119904(119862minus 119874) + 119891119890119909sdot 119890119909

+ 119891119890119910sdot 119890119910minus119872119874120588119892119890119910

(16)

Shock and Vibration 5

Roller

Cage120575c(t)

120575w(t)

(a) Cageroller contact

120575c(t) 120575w(t)

kckcw

Mc Mw

(b) Model of cageroller contact

Figure 4 A 2-DOF model of cageroller contact of the 119895th roller

Load

(N)

0 1 2 3 4 5 6minus40

minus20

0

20

40

60

80

100

120

t(s) times 10minus4

Figure 5 Demonstration on variation trend of cage-roller contactforce

The position vector velocity vector and accelerationvector of the unbalanced gravitational center of the cage canbe given as

119874119863= (119883119888+ 120588 cos120596119905) sdot 119890

119909+ (119884119888+ 120588 sin120596119905) sdot 119890

119910

119863= (119888+ 120588 cos120596119905) sdot 119890

119909+ (119888+ 120588 sin120596119905) sdot 119890

119910

119863= (119888+ 120588 cos120596119905) sdot 119890

119909+ (119888+ 120588 sin120596119905) sdot 119890

119910

(17)

In the same way the position velocity and accelerationvector of geometry center of the cage are

119874119888= 119883119888sdot 119890119909+ 119884119888sdot 119890119910

119888= 119888sdot 119890119909+ 119888sdot 119890119910

119888= 119888sdot 119890119909+ 119888sdot 119890119910

(18)

Similarly using above-mentioned exhibition the posi-tion velocity and acceleration vector of the center of the ribare

119874119900= 119883119900sdot 119890119909+ 119884119900sdot 119890119910

119900= 119883119900sdot 119890119909+ 119900sdot 119890119910

119900= 119883119900sdot 119890119909+ 119900sdot 119890119910

(19)

Substituting (15) (16) and (17) into (14) then the follow-ing two equations described in 119909-119910 plane can be obtained

119872119863119874

= minus2119870119904(119883119862minus 119883119874) minus 2119861

119904(119862minus 119874)

+ 2119872119863120588120596119888minus119900

2 cos120596119888minus119900119905 minus 119870119909119910119884119863minus 1198701015840

119909119910119884119863

3

119872119863119874

= minus2119870119904(119884119862minus 119884119874) minus 2119861

119904(119862minus 119874)

+ 2119872119863120588120596119888minus119900

2 sin120596119888minus119900119905 minus 119870119909119910119883119863minus 1198701015840

119909119910119883119863

3

minus 2119872119863119892

(20)

119872119862119862minus 119891119890119909minus 119870119904(119883119874minus 119883119862) minus 119861119904(119874minus 119862) = 0

119872119862119862minus 119891119890119910minus 119870119904(119884119874minus 119884119862) minus 119861119904(119862minus 119874)

= minus119872119862119892

(21)

Considering (20)-(21) the results can be obtained as

119891119890119909= 1205921

2

(119874minus 119862) minus 1205922(119874minus 119862) minus 1205923(119909119874minus 119909119862)

119891119890119910= 1205922

2

( 119910119874minus 119910119862) minus 1205922( 119910119874minus 119910119862) minus 1205923(119910119874minus 119910119862)

+ 119870119891119889119891119889

(22)

Here 119909119874 119910119874 119909119862 and 119910

119862are nondimensional form of

rib center (119883119874 119884119874) and cage center (119883

119862 119884119862) and they are

obtained by original value divided by 119862119903 The other factors in

(22) can be expressed as

1205921=2 (119872119863120596119888119900119861)2

119882

1205922= 2119870119904119861

1205923=2119870119904

32

119861

11987212

119863

120596119888119900

(23)

6 Shock and Vibration

When the linear superposition method is adopted tocontact forces of the cage in 119909-119910 plane the total force ondifferent directions can be given as

119891119909= 119891119890119909

119891119910= 119891119886119910+ 119891119890119910

(24)

32 Hydrodynamic Fluid-Film Force Cage-rib contact is afluid-structure interaction problem with bearings rotatingat a high speed The main supporting force generated bylubricant squeeze effects can be approximated to a short-width journal bearing theory as illustrated in Figure 3 Thendynamic-deviation interactive force between cage and rib canbe described as a continuous function of cage position andvelocity components Reynolds lubrication equation (RLE)provides the basis of lubrication theory [15] and the forcegiven on the cage by the film is expressed as

120597

120597120572[(1 + 120576 cos120572)3

120597119901

120597120572] + 1198772

119888

120597

120597119909[(1 + 120576 cos120572)2

120597119901

120597119910]

= minus6120583(119877119888

119862119903

)

2

[(120596119888119900minus 2) 120576 sin120572 minus 2120576 cos120572]

(25)

And

119901 (120572) = 1198750+120596119888119900

119862119903

3[1198972

119888

4minus 1199102

]120576 sin120572

1 + 120576cos3120572 (26)

Solutions to the RLE are a nonlinear function of displace-ment and angle at cage center Hydrodynamic fluid-film forcecan be written as follows

119891119890= 120583119877119888119897119888(119897119888

119862119903

)

2

[(120596119888119900minus 2)

1205762

(1 minus 1205762)2

+120587

2

(1 + 21205762

) 120576

(1 minus 1205762)52

]

119891120572= 120583119877119888119897119888(119877119888

119862119903

)

2

sdot [(120596119888119900minus 2120572)

120587120576

4 (1 minus 1205762)32

+2120576 120576

(1 minus 1205762)2]

+ 21198771198881198971198881199010

(27)

The fluid-film force can also be represented as two com-ponent forces in horizontal and vertical force equilibrium

119891119909= 119891119890sdot cos120572 + 119891

120572sdot sin120572

119891119910= 119891119890sdot sin120572 minus 119891

120572sdot cos120572

(28)

33 Equilibrium of the Equations In hydrodynamic fluid-film bearing the fluid supported pressure is generated nor-mally by motion of the cage and depends on the dynamic

viscosity exponent of the lubricating fluid For the cagestructure and coordination please refer to Figure 6 Thegenerated forces can be expressed as

119891119890sdot cos120572 + 119891

120572sdot sin120572 = 120592

1

2

119862minus 1205922(119874minus 119862)

minus 1205923(119909119874minus 119909119862)

119891119890sdot sin120572 minus 119891

120572sdot cos120572 = 120592

1

2

119910119862minus 1205922( 119910119874minus 119910119862)

minus 1205923(119910119874minus 119910119862) + 119861119891119889119891119889

+ 119891119894119910

(29)

Displacement relations between the centers of cage andrib are

119909119888= 119909119900+ 120576 cos120593

119910119888= 119910119900+ 120576 sin120593

(30)

Similarly for the velocity

119888= 119900+ 120576 cos120593 minus 120576 sin120593 sdot

119910119888= 119910119900+ 120576 sin120593 minus 120576 cos120593 sdot

(31)

Substituting (27) (30) and (31) into (29) integration ofequation can be obtained

120576 (1 minus 1205762

)3

+ 120576 [1205922(1 minus 120576

2

)3

minus 1205921015840

1

120587

2(1 + 2120576

2

) (1 minus 1205762

)12

] minus 1205922120576 sin 2120572 sdot (1

minus 1205762

)3

+ 1205923120576 (1 minus 120576

2

)3

minus 120583 cos 2120572 minus 12059210158401(120596119888119900minus 2)

sdot 1205762

(1 minus 1205762

) = 0

[120576 (1 minus 1205762

)2

+ 120576 (1 minus 1205762

)2

2120583 sin 2120572 sdot (1 minus 1205762)2

]

+ [2 120576 (1 minus 1205762

)2

+ 1205922120576 (1 minus 120576

2

)2

]

+ 1205921015840

1[(120596119888119900minus 2120572) 120587120576 (1 minus 120576

2

)12

+ 2120576 120576]

+ 21198771198881198971198881198750 = 0

(32)

Equation (32) can be solved by modified Newmark-120573method to obtain the displacement and velocity of the cage

4 Chaos Analyses

Chaos Theory is a new way to analyze complexity whichHenri Poincare studied as the possibility of forecastingA small variation in the initial conditions can generate adifferent set of resultssolutions [17]

A bearing typed as NU2310 is chosen to analyze chaoticresponse Tomoya and Kaoru [18] tested the dynamic cagedisplacements of this bearing and the maximum displace-ments can be used to verify the validity of theoretical analysis

Shock and Vibration 7

e

x

y

X

Y

OO

OD

OC

Rg

Rc

120576

(a) Cage-rib contact

Mc

MD

xc(t)Kx

Kxy

Ky K998400xyBy

Bx

yc(t)

(b) Cage-rib contact model

Figure 6 2-DOF model of cagerib contact

The structural specifications of the bearing are listed inTable 1

The cage of the bearing is made of machined brass DINCuZn40Pb2F43 (GB ZCuZn38Mn2Pb2) and the rings of thebearing are made of through-harden rolling bearing steelDIN 100Cr6 (SAE 52100GB GCr15) The cage is guided bythe outer-ring rib Both static load rating and dynamic loadrating are 186KN According to the recommendation of themanufacturer reference speed rating is 6700 rpm and limitedspeed rating is 12000 rpm Then the weight of the cage anddamping and stiffness parameters can be obtained accordingto the relative parameters in Table 1 The displacement in119909 direction is zero and the displacement in 119910 directionis determined by the cage guidance gap according to (1)Bifurcation diagram of 119883

119862versus spin speed increases with

different cage guide gaps cage eccentricities and dynamicviscosity exponent of lubrication oil The change curvesare described in Figure 7 respectively All three cases aresimulated and variation parameters of the cases are listed inTable 2 An interesting phenomenon that all the cases havetwo bifurcation zones and two steady zones is found in allcurves In relative low rotation speed transient responses canbe found The dynamic response of cage center goes intothe first ordered stage after the end of the resonance Thenthe first bifurcation response occurs The second orderedstatus arises after the first bifurcation ends and then thecurve goes into the second bifurcation Figure 7(a) comparesthe bifurcation diagrams of cage with different guidancegaps The first chaotic response zone of the bearing withsmaller guidance gap (case 1a) is from 120596

119888119900= 9100 rpm

to 120596119888119900= 12000 rpm and it is from 120596

119888119900= 7100 rpm to

120596119888119900= 22000 rpm for the bearing with larger guidance gap

(case 1b) The second chaotic response zone of the bearingwith smaller guidance gap is from 120596

119888119900= 25500 rpm and

it is from 120596119888119900

= 26000 rpm for the bearing with largerguidance gap Chaotic response zone is wider when the gap islarger Figure 7(b) shows bifurcation diagrams of cage with

different eccentricities The first chaotic zone of case 2a isfrom 120596

119888119900= 7600 rpm to 120596

119888119900= 14000 rpm and it is from

120596119888119900= 7800 rpm to 120596

119888119900= 13000 rpm of case 2b The second

chaotic zone of case 2a is from 120596119888119900= 22500 rpm and it is

from 120596119888119900= 19800 rpm of case 2b Chaotic response zone

increaseswith eccentricity In addition transient oscillation ismore drastic when comparing case 1a and case 2a Figure 7(c)depicts bifurcation diagrams of cage with different dynamicviscosity exponents The first chaotic zones of case 3a andcase 3b are from 120596

119888119900= 7650 rpm to 120596

119888119900= 13400 rpm The

second chaotic zones of case 3a and case 3b are from 120596119888119900=

21000 rpmThe change trend of case 3a curve is approachingto that of case 3b But the variation curve of cage with VG78is lower than that with VG56 due to higher dynamic viscosityexponent

The first bifurcation point 119884119862versus spin speed zooming

is shown in Figure 8 Fluctuation can be found even in steadyzone owing to the impact forces between roller and cagepockets In low-speed zone the wave frequency is low and itincreases with rotational speedThe difference of steady zonerange is shown more clearly in the enlarged view Bifurcationdiagrams of each case are different obviously and this denotesthat the position of119883

119862varies timely as well

Cage response behaviors of chaos which are subjected tothe cases in Table 2 at120596 = 8500RPM Poincare section pointsof 119884119862displacement in vertical direction and time history for

4th revolution to 4000th revolution are shown in Figure 9

5 Conclusion and Discussion

Rolling bearing life is typically calculated on the basis of itsload ratings relative to the applied loads and the requirementsregarding bearing life and reliability Both dynamic loadrating and rating life equations neglect cage design Actuallythe cage design has evolved to bear higher and higher load byincorporating more rollers and to lower costs by adopting

8 Shock and Vibration

2500 7500 12500 17500 22500 27500

Rotation speed (rpm)

Case 1b

Case 1a

03

02

01

0

minus01

minus02

minus03

Vert

ical

disp

lace

men

t of

cage

cent

er (m

m)

(a) Different cage guidance gaps

2500 7500 12500 17500 22500 27500

Rotation speed (rpm)

Case 2b

Case 2a

03

02

01

0

minus01

minus02

minus03

Vert

ical

disp

lace

men

t of

cage

cent

er (m

m)

(b) Different eccentricities

2500 7500 12500 17500 22500 27500

Rotation speed (rpm)

Case 3b

Case 3a

03

02

01

0

minus01

minus02

minus03

Vert

ical

disp

lace

men

t of

cage

cent

er (m

m)

(c) Different dynamic viscosity exponents

Figure 7 Bifurcation diagram points119883119862versus spin speed

Table 1 Specifications of the bearing structure and size

Bearing typeBoundary size Roller size Rib diameter Clearance

Boremm

Outsidemm

Widthmm Number Diameter

mmLengthmm mm Radial

120583mPocketmm

NU2310 Φ50 Φ110 40 13 Φ16 27 Φ921 40 03

Table 2 Case study

CaseVariation parameters

Guidance gap Eccentricity Dynamic viscosity exponent(mm) (mm) (Pasdots)

1a 045 23 VG561b 060 23 VG562a 045 35 VG562b 045 57 VG563a 045 42 VG783b 045 42 VG32

different material and improve high-speed performance byusing optimized geometry structure This paper pays closeattention to influence from cage on speed characteristics ofcylindrical roller bearings The most suitable cage guidancetype is confirmed by mathematical derivation firstly Twomajor factors originating from manufacturing process andone involved in applying process are considered to establishdynamical mathematical model of cage with appropriate cageguidance type Analysis of real-time dynamic behavior of

a cage on radial plane was carried out using chaos theorybased on the theoretical mathematical model The chaoticanalysis is limited by the initial values of the system Hencethe initial values are determined firstly in analytical processPreliminary conclusions are listed as follows

(1) There are two bifurcation zones of cage dynamiccurves and the bearing should be away from these twozones in practiceThe range of the 1st bifurcation zoneismainly influenced by the value of cage guidance gapand eccentricity especially the cage guidance gapThelubrication dynamic viscosity exponent has relativelylittle influence on cage dynamic

(2) Numerical simulations show that the roller tends toacceleratedecelerate in switching process of support-ing zoneloaded zone which leads to single or mul-tiple roller-cage bridge impacts Impact force causesperiodic small oscillation on the curves Oscillationamplitude is determined by the pocket gap

(3) Eccentricity of the cage causes dramatic transientresponses in low rotational speed zone The transientresponses are easily tending to cause knocking and

Shock and Vibration 9

7000 10000 13000 16000 19000 22000

Rotation speed (rpm)

Case 1b

Case 1a

002

001

0

minus001

minus002

minus003

minus004

minus005

minus006

minus007

minus008

Vert

ical

disp

lace

men

t of

cage

cent

er (m

m)

(a) Different cage guidance gaps

Rotation speed (rpm)

Case 2b

Case 2a

0

minus001

minus002

minus003

006

005

004

003

002

001

7500 8800 10100 11400 12700 14000

Vert

ical

disp

lace

men

t of

cage

cent

er (m

m)

(b) Different eccentricities

Case 3b

Case 3a

Rotation speed (rpm)7500 8800 10100 11400 12700 14000

0

minus001

minus002

minus003

minus004

005

004

003

002

001

Vert

ical

disp

lace

men

t of

cage

cent

er (m

m)

(c) Different dynamic viscosity exponents

Figure 8 The first bifurcation zone from Figure 7 in enlarged scale

faster wear of cage pocket in terms of worse lubrica-tion film forming The steady zone is narrower wheneccentricity increases In addition divergent trendcan be found in the second bifurcation zonewith lagereccentricity

(4) According to the zooming of the first bifurcationzone variations of any parameters cause totally dif-ferent bifurcations We can draw our conclusion thatmovement of cage is a chaotic motion

(5) The Poincare maps show that position and rotationalspeed of the occurrence bifurcation zones are verydifferent with varied parameters

(6) The change of fluid dynamic viscosity exponent willnot affect the region of bifurcation due to the weaknonlinear of fluid film force

Considering cage dynamics of high-speed cylindricalroller bearing the values of cage guidance gap and eccen-tricity are crucial factors From the point of view of themanufacturers the very small cage guidance gap is difficult toobtain owing to the material (brass) of the cage used in high-speed bearingThe authors strongly suggest that the designershould minimize the gap within the confine of processingcapacity As for the cage eccentricity dynamic balance ofcage is widely used in ultra-high-speed bearings Analysesfrom the paper show that cage eccentricity brings violent

oscillation in low-speed zone and oscillation amplitude inchaotic zone is higher with the increase of cage eccentricityHence dynamic balance is another important factor inmanufacturing process With respect to lubrication dynamicviscosity exponent this study indicates that influence fromdynamic viscosity exponent on cage dynamics is not signifi-cant as the prediction

As a matter of fact researches in the paper are merelyrestricted to cage dynamic response What is to be pointedout is that dynamic response of a cylindrical roller bearingis a complex and interrelated problem among componentsHence there are many imperfect points in the paper whichshould be noted here Preliminary discussions are listed asfollows

(1) The displacements of the cage discussed in the paperare only in 119909-119910 plane During the rotation of thebearing rollers will inevitably have a certain slip-page towards the rings due to insufficient tractionfriction outside the load-zone churning momentfrom lubricant the friction loss between roller-pocketcontact and roller-rib contacts in the actual runningprocess Rollerraceways contact has not been takeninto account in the paper for the sake of simplifyingthe calculation

(2) Damping coefficient and contact stiffness involvedin the calculation are obtained according to material

10 Shock and Vibration

minus03 minus02 minus01 0 01 02minus400

minus200

0

200

400

Vertical displacement of cage center (mm)

Vert

ical

velo

city

(mm

s)

(a) Smaller cage guidance

minus03 minus02 minus01 0 01 02Vertical displacement of cage center (mm)

minus400

minus200

0

200

400

Vert

ical

velo

city

(mm

s)

(b) Bigger cage guidance

minus02 minus01 0 01Vertical displacement of cage center (mm)

minus300

minus200

0

200

300

Vert

ical

velo

city

(mm

s)

(c) Smaller eccentricity

minus03 minus02 minus01 0 01 02Vertical displacement of cage center (mm)

minus400

minus200

0

200

400

Vert

ical

velo

city

(mm

s)

(d) Bigger eccentricity

minus03 minus02 minus01 0 01 02Vertical displacement of cage center (mm)

minus400

minus200

0

200

400

Vert

ical

velo

city

(mm

s)

(e) Higher dynamic viscosity exponent

minus03 minus02 minus01 0 01 02Vertical displacement of cage center (mm)

minus400

minus200

0

200

400

Vert

ical

velo

city

(mm

s)

(f) Lower dynamic viscosity exponent

Figure 9 Poincare section points of 119884119862displacement in vertical direction

property and geometry characteristic parametersActually the damping coefficient and contact stiff-ness can be described more accurately by immediatemovement and velocity Namely contact stiffness isthe first-order derivative of immediate movementand damping coefficient is the first-order derivative ofimmediate velocity For the same sake of simplifyingcalculation the accurate expressions are not substi-tuted into the equations

(3) The bearing under study has the outer ring fixed to arigid support and the inner ring fixed rigidly to theshaft Bearing house and the shaft are truly nonlinearDOF systems The influence of bearing house andshaft on cage dynamic is relatively small so it can besimplified in the calculation

(4) As illustrated in Section 3 cage is driven by the rollersin supporting zone and the rollers are driven by cage

Shock and Vibration 11

in nonloaded zone Driving forces on the total cageare balanced and they are ignored except the impactforces in switching zones of driven modes

Symbols

120572 Angle at cage center in rad Angular velocity in tangential direction in rads Angular acceleration in tangential direction in

rads2119861 Damping coefficient119862 Cage gap in mm119889 Diameter in mmsΔ Roller gap in mm120575 Deformation in mm119890 Cage eccentricity in mm120576 Displacement in normal direction in mm120576 Velocity in normal direction in mms120576 Acceleration in normal direction in mms2119865 Load in N119891 Load caused by motion in N119866 Clearance in mm120579 Angle in ∘119870 Hertzian stiffness119897 Length in mm119872 Mass in kg120583 Fluid dynamic viscosity in Pasdots119873 Number of the rollers120592 Dimensionless parameter of impact factor119874 Position of the center in mm Velocity vector of the center in mms Acceleration vector of the center in mms2119901 Hydrodynamic pressure in Pa1199010 Initial hydrodynamic pressure in Pa

119876 Hertz stress in Nmm2119877 Radius in mm120588 Dimensionless parameter of cage eccentricity to cage

gap119905 Time in sec119879 Pocket clearance in mmV Surface velocity in mms119883 Coordinate119883 direction distance in mm119909 Position in coordinate direction in mm Velocity in 119909 direction in mms Acceleration in 119909 direction in mms2119884 Coordinate 119884 direction distance in mm119910 Dimensionless parameter in coordinate direction119910 Velocity in 119910 direction in mms119910 Acceleration in 119910 direction in mms2120596 Rotation speed in rads

Subscripts

119886 Axial direction120572 Tangential direction119862 Cage center119888 Cagecen Centrifugal

119888119894 Relative relation between cage and innerring

119888119900 Relative relation between cage and outerring

119888119908 Relative relation between cage and roller119863 Unbalanced gravitational center119890 Normal direction119891119889 Added fluid

119892 Ribℎ Impact119894 Inner raceway119895 The 119895th roller119895cen Centrifugal force of the 119895th roller119898 Mean value119874 Rib center119900 Outer raceway119903 Radial direction119903119894 Inner raceway on radial direction119903119900 Outer raceway on radial direction119904 Polar coordinate119908 Roller119909 119909 direction119910 119910 direction

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The project is supported by National Natural Science Foun-dation of China (Grant no 51475144) and the Foundation ofInnovation and Research Team of Science and Technology inUniversities in Henan Province (Grant no 13IRTSTHN025)

References

[1] M Tiwari K Gupta and O Prakash ldquoDynamic response of anunbalanced rotor supported on ball bearingsrdquo Journal of Soundand Vibration vol 238 no 5 pp 757ndash779 2000

[2] M L Adam Jr Rotating Machinery VibrationmdashFrom Analysisto Trouble Shooting Marcel Dekker New York NY USA 2001

[3] N Akturk M Uneeb and R Gohar ldquoThe effects of number ofballs and preload on vibrations associated with ball bearingsrdquoJournal of Tribology vol 119 no 4 pp 747ndash753 1997

[4] S H Upadhyay S C Jain and S P Harsha ldquoNon-linearvibration signature analysis of a high-speed rotating shaft due toball size variations and varying number of ballsrdquo Proceedings ofthe Institution of Mechanical Engineers Part K Journal of Multi-body Dynamics vol 223 no 2 pp 83ndash105 2009

[5] N Akturk ldquoThe effect of waviness on vibrations associated withball bearingsrdquo Journal of Tribology vol 121 no 4 pp 667ndash6771999

[6] S P Harsha K Sandeep and R Prakash ldquoNon-linear dynamicbehaviors of rolling element bearings due to surface wavinessrdquoJournal of Sound and Vibration vol 272 no 3ndash5 pp 557ndash5802004

[7] J Sopanen and A Mikkola ldquoDynamic model of a deep-grooveball bearing including localized and distributed defects Part

12 Shock and Vibration

2 Implementation and resultsrdquo Proceedings of the Institution ofMechanical Engineers Part K vol 217 no 3 pp 213ndash223 2003

[8] S H Upadhyay S C Jain and S P Harsha ldquoChaotic dynamicsof high speed rotating shaft supported by ball bearings due todistributed defectsrdquo International Journal of Engineering Scienceand Technology vol 2 no 10 pp 5746ndash5794 2010

[9] S P Harsha ldquoNonlinear dynamic analysis of a high-speed rotorsupported by rolling element bearingsrdquo Journal of Sound andVibration vol 290 no 1-2 pp 65ndash100 2006

[10] C Villa J-J Sinou and F Thouverez ldquoStability and vibrationanalysis of a complex flexible rotor bearing systemrdquo Communi-cations in Nonlinear Science and Numerical Simulation vol 13no 4 pp 804ndash821 2008

[11] L Houpert ldquoCAGEDYN a contribution to roller bearingdynamic calculations Part III experimental validationrdquo ASMETribology Transactions vol 53 no 6 pp 848ndash859 2010

[12] S P Harsha ldquoNonlinear dynamic analysis of rolling elementbearings due to cage run-out and number of ballsrdquo Journal ofSound and Vibration vol 289 no 1-2 pp 360ndash381 2006

[13] I Bercea S Cretu M Bercea and D Olaru ldquoSimulating roller-cage pocket friction in a tapered roller bearingrdquo EuropeanJournal of Mechanical and Environmental Engineering vol 43no 4 pp 189ndash194 1998

[14] T Sakaguchi andKHarada ldquoDynamic analysis of cage behaviorin a tapered roller bearingrdquo Journal of Tribology vol 128 no 3pp 604ndash611 2006

[15] T A Harris Rolling Bearing Analysis John Wiley amp Sons 4thedition 2001

[16] C Spiridon ldquoMachined brass and pressed steel bearing cages acomparative studyrdquo RKB Technical Review vol 7 pp 1ndash13 2011

[17] R Riccardo ldquoChaos theory and some practical applications intechnical analysisrdquo in Proceedings of the 11th Annual Conferenceof the International Federation of Technical Analysts 1998

[18] S Tomoya and U Kaoru ldquoDynamic analysis of cage behavior ina cylindrical roller bearingrdquo NTN Technical Review 71 2004

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Shock and Vibration 3

F

Supporting zone

Outer ring

Inner ring

Cage Rollers

R-C contact region

XY

Z

120596o

120596w

120596c

120596i

OOOD

OC

120593

i c o

2120579

Gr

Cr

(a) Schematic diagram

TT

XY

Z

T

T

120596o 120596c120596i

(b) Pocket clearance

Figure 2 Schematic diagram and pocket clearance in a cylindrical roller bearing

212 Inner-Ring-Rib-Guidance Bearing Inner ring displace-ment of inner-ring-rib-guided bearing is caused by the clear-ance (119866

119903) cage guidance gap (119862

119903) and the elastic deformation

(120575119903) in the contact lines of the raceways and the rollers

The value of elastic deformation depends on the value ofthe load and the number of the rollers under load Thensupporting zone as can be seen in Figure 2(a) should beconfirmed firstlyThe angular extent of the supporting zone isdetermined by the radial clearance (119866

119903) of a cylindrical roller

bearing such as

2120579 = cosminus1119866119903

2120575119903

(2)

Elastic deformation between the raceway and roller showsa nonlinear relation which is obtained by using the Hertziantheory [15] The local Hertzian contact force and deflectionrelationship for a bearing may be written as

119876119894= 119870119894120575119894

109

119876119900= 119870119900120575119900

109

(3)

The local Hertzian contact forces in (3) are determinedby external load (119865) number of rollers in supporting zone(119873) and the radial clearance (119866

119903) The calculation method is

a very standardized procedure and will not be described hereThe total normal approach between two raceways under loadseparated by a roller is the sum of the approaches between theroller and each raceway Hence

120575119903= 120575119903119894+ 120575119903119900 (4)

Then the radial displacement of the cage can be obtainedfrom above equations as

119884119888119894= 120575119903+1

2119866119903+1

2119862119903 (5)

213 Roller Guidance The normal approach between tworaceways under load influences the cage displacementdirectly in this kind of guidance type The cage displacementin roller-guidance bearing can be given as

119884119888119908= 120575119903+1

2119866119903+ 119879 (6)

22 Cage Displacement Caused by Rotating Bearings areused to bear various kinds of loads while keeping a shaftrotating Rollers are subjected to dynamic loading due tospeed effects As a general case it will be initially assumedthat both inner and outer rings are rotating in a bearing asillustrated in Figure 2 Consequently

V119894=1

2120596119894119889119894

V119900=1

2120596119900119889119900

(7)

Then the cage speed or the speed of rotation of the set ofrollers around the origin is

120596119888=1

2[120596119894(1 minus

119889119908

119889119898

) + 120596119900(1 +

119889119908

119889119898

)] (8)

The centrifugal force caused by the 119895th roller is calculatedas

119865119895cen= 119898119895

1

2120596119888

2

119889119898 (9)

According to design guidelines of rolling bearings thevalue of 119862

119886is larger than that of Δ the value of 119879 is larger

than that of 119862119903 and the value of 119862

119903is larger than that of

119866119903generally Thus the rollersrsquo centrifugal forces are acting

4 Shock and Vibration

X

Y

p

e

120596o

120596c

OO

OD

OC

2120579

120572

120588

F(jminus2)cw

fy

fx

Rg

RcCr

Fjcen

Figure 3 Forces generated by motion on the cage

on outer raceway Subsequently the centrifugal forces of therollers cause larger contact deformation It can be added tothe original deformation on the outer raceway as

120575119903119900= (

119865cen + 119876119900119870119900

)

910

(10)

Substituting (10) into (4) equations (5) and (6) willchange along with (4) According to different expressions ofcage displacements both inner ring guidance and roller guid-ance bearings will generate complex dynamic displacementswhen they are operating at high speedsThenwe can drawourconclusion that both inner ring guidance and roller guidanceare not applicable under high-speed conditions Outer-ring-rib-guidance bearing is chosen as the research object in thefollowing analysis naturally

3 Physical Model

The cage has two kinds of contacts in a rotating outer-ring-guided cylindrical bearing One is the cage-rib contact andthe other is the roller-pocket contactThe forces generated bythe contacts are marked in Figure 3 Cage-rib contact statusis determined by the relative rotational speed (120596

119888119900) and the

cage guidance gap (119862119903) The cage has physical contact with

guiding ring in the starting phase owing to a low relativespeed However the cage and the rib are separated by oilfilm eventually with the increasing of relative speed Thencage-rib contact is considered as a fluid-structure interactionproblem Similarly roller-cage contact status is determined byrotational speed of the roller around its own axil (120596

119888119908) and the

pocket clearance (119879) The cage and the rollers are separatedby oil film with the increasing of speed too Note that thecage is driven by the rollers in supporting zone and the rollersare driven by cage in nonloaded zone The driving forces byrollers on the cage can be seen in Figure 3 as well Differentdrivingmode of the rollers in different force directions causesimpact loads on the cage The impact between the cage andthe roller in a very short time results in large impact forces

31 Instantaneous Motion Forces Different contacts causedifferent motion forces and there is no strong relationshipamong the motion forces then linear superposition methodcan be adopted for calculating motion forces on the cage

The relative rotational velocity of the cage to the rib isdefined as

120596119888119900= 120596119888minus 120596119900 (11)

And the cage guidance gap determined by cage diameterand rib can be expressed as

119862119903= 119889119892minus 119889119888 (12)

Brass cage is usually machined from centrifugally casttubing and then nonuniform distribution of the mass causesstatic imbalance (119890) of the cage Nondimensional eccentricityof the cage to cage guidance gap can be given as

120588 =119890

119862119903

(13)

Spiridon [16] established an elastic system with twodegrees of freedom (DOF) that considered both cage-roller stiffness (119870

119888119908) and cage structural stiffness (119870

119888) for

cageroller contact As described in Figure 4 the dynamicequilibrium condition written for each of the two massesprovides the following set of two differential equations

119870119888119908(120575119908minus 120575119888)109

+119872119908

1198892

120575119908

1198891199052= 0

minus119870119888119908(120575119908minus 120575119888)109

+ 119870119888120575119888+119872119888

1198892

120575119908

1198891199052= 0

(14)

119870119888119908

can be derived from the Hertzian contact theorywhereas119870

119888can be obtained analyticallyThe cage-roller stress

(119876119888119908) can be obtained from (3) and its variation trend for

the specified roller is shown in Figure 5 when the bearing isrotating

Impact force of cageroller contact can be given as

119891ℎ119909= 119876119888119908[sin120596

119888119900119905 + sin (120587 minus 120596

119888119900119905)] cos 120579

119891ℎ119910= 119876119888119908[sin120596

119888119900119905 minus sin (120587 minus 120596

119888119900119905)] sin 120579

(15)

As for the cage-rib dynamic model it can encompassradial-plane orbital cage motion with 2-DOF equationsas well According to Figure 4 motion equation can beexpressed as

119872119863119863= minus2119870

119904(119874119862minus 119874119874) minus 2119861

119904(119862minus 119874)

+ 2119872119863120588119892119890119910

minus 119865119904(119883119863 119863 119863 119884119863 119863 119863)

119872119862119862= minus119870119904(119874119862minus 119874119874) minus 119861119904(119862minus 119874) + 119891119890119909sdot 119890119909

+ 119891119890119910sdot 119890119910minus119872119874120588119892119890119910

(16)

Shock and Vibration 5

Roller

Cage120575c(t)

120575w(t)

(a) Cageroller contact

120575c(t) 120575w(t)

kckcw

Mc Mw

(b) Model of cageroller contact

Figure 4 A 2-DOF model of cageroller contact of the 119895th roller

Load

(N)

0 1 2 3 4 5 6minus40

minus20

0

20

40

60

80

100

120

t(s) times 10minus4

Figure 5 Demonstration on variation trend of cage-roller contactforce

The position vector velocity vector and accelerationvector of the unbalanced gravitational center of the cage canbe given as

119874119863= (119883119888+ 120588 cos120596119905) sdot 119890

119909+ (119884119888+ 120588 sin120596119905) sdot 119890

119910

119863= (119888+ 120588 cos120596119905) sdot 119890

119909+ (119888+ 120588 sin120596119905) sdot 119890

119910

119863= (119888+ 120588 cos120596119905) sdot 119890

119909+ (119888+ 120588 sin120596119905) sdot 119890

119910

(17)

In the same way the position velocity and accelerationvector of geometry center of the cage are

119874119888= 119883119888sdot 119890119909+ 119884119888sdot 119890119910

119888= 119888sdot 119890119909+ 119888sdot 119890119910

119888= 119888sdot 119890119909+ 119888sdot 119890119910

(18)

Similarly using above-mentioned exhibition the posi-tion velocity and acceleration vector of the center of the ribare

119874119900= 119883119900sdot 119890119909+ 119884119900sdot 119890119910

119900= 119883119900sdot 119890119909+ 119900sdot 119890119910

119900= 119883119900sdot 119890119909+ 119900sdot 119890119910

(19)

Substituting (15) (16) and (17) into (14) then the follow-ing two equations described in 119909-119910 plane can be obtained

119872119863119874

= minus2119870119904(119883119862minus 119883119874) minus 2119861

119904(119862minus 119874)

+ 2119872119863120588120596119888minus119900

2 cos120596119888minus119900119905 minus 119870119909119910119884119863minus 1198701015840

119909119910119884119863

3

119872119863119874

= minus2119870119904(119884119862minus 119884119874) minus 2119861

119904(119862minus 119874)

+ 2119872119863120588120596119888minus119900

2 sin120596119888minus119900119905 minus 119870119909119910119883119863minus 1198701015840

119909119910119883119863

3

minus 2119872119863119892

(20)

119872119862119862minus 119891119890119909minus 119870119904(119883119874minus 119883119862) minus 119861119904(119874minus 119862) = 0

119872119862119862minus 119891119890119910minus 119870119904(119884119874minus 119884119862) minus 119861119904(119862minus 119874)

= minus119872119862119892

(21)

Considering (20)-(21) the results can be obtained as

119891119890119909= 1205921

2

(119874minus 119862) minus 1205922(119874minus 119862) minus 1205923(119909119874minus 119909119862)

119891119890119910= 1205922

2

( 119910119874minus 119910119862) minus 1205922( 119910119874minus 119910119862) minus 1205923(119910119874minus 119910119862)

+ 119870119891119889119891119889

(22)

Here 119909119874 119910119874 119909119862 and 119910

119862are nondimensional form of

rib center (119883119874 119884119874) and cage center (119883

119862 119884119862) and they are

obtained by original value divided by 119862119903 The other factors in

(22) can be expressed as

1205921=2 (119872119863120596119888119900119861)2

119882

1205922= 2119870119904119861

1205923=2119870119904

32

119861

11987212

119863

120596119888119900

(23)

6 Shock and Vibration

When the linear superposition method is adopted tocontact forces of the cage in 119909-119910 plane the total force ondifferent directions can be given as

119891119909= 119891119890119909

119891119910= 119891119886119910+ 119891119890119910

(24)

32 Hydrodynamic Fluid-Film Force Cage-rib contact is afluid-structure interaction problem with bearings rotatingat a high speed The main supporting force generated bylubricant squeeze effects can be approximated to a short-width journal bearing theory as illustrated in Figure 3 Thendynamic-deviation interactive force between cage and rib canbe described as a continuous function of cage position andvelocity components Reynolds lubrication equation (RLE)provides the basis of lubrication theory [15] and the forcegiven on the cage by the film is expressed as

120597

120597120572[(1 + 120576 cos120572)3

120597119901

120597120572] + 1198772

119888

120597

120597119909[(1 + 120576 cos120572)2

120597119901

120597119910]

= minus6120583(119877119888

119862119903

)

2

[(120596119888119900minus 2) 120576 sin120572 minus 2120576 cos120572]

(25)

And

119901 (120572) = 1198750+120596119888119900

119862119903

3[1198972

119888

4minus 1199102

]120576 sin120572

1 + 120576cos3120572 (26)

Solutions to the RLE are a nonlinear function of displace-ment and angle at cage center Hydrodynamic fluid-film forcecan be written as follows

119891119890= 120583119877119888119897119888(119897119888

119862119903

)

2

[(120596119888119900minus 2)

1205762

(1 minus 1205762)2

+120587

2

(1 + 21205762

) 120576

(1 minus 1205762)52

]

119891120572= 120583119877119888119897119888(119877119888

119862119903

)

2

sdot [(120596119888119900minus 2120572)

120587120576

4 (1 minus 1205762)32

+2120576 120576

(1 minus 1205762)2]

+ 21198771198881198971198881199010

(27)

The fluid-film force can also be represented as two com-ponent forces in horizontal and vertical force equilibrium

119891119909= 119891119890sdot cos120572 + 119891

120572sdot sin120572

119891119910= 119891119890sdot sin120572 minus 119891

120572sdot cos120572

(28)

33 Equilibrium of the Equations In hydrodynamic fluid-film bearing the fluid supported pressure is generated nor-mally by motion of the cage and depends on the dynamic

viscosity exponent of the lubricating fluid For the cagestructure and coordination please refer to Figure 6 Thegenerated forces can be expressed as

119891119890sdot cos120572 + 119891

120572sdot sin120572 = 120592

1

2

119862minus 1205922(119874minus 119862)

minus 1205923(119909119874minus 119909119862)

119891119890sdot sin120572 minus 119891

120572sdot cos120572 = 120592

1

2

119910119862minus 1205922( 119910119874minus 119910119862)

minus 1205923(119910119874minus 119910119862) + 119861119891119889119891119889

+ 119891119894119910

(29)

Displacement relations between the centers of cage andrib are

119909119888= 119909119900+ 120576 cos120593

119910119888= 119910119900+ 120576 sin120593

(30)

Similarly for the velocity

119888= 119900+ 120576 cos120593 minus 120576 sin120593 sdot

119910119888= 119910119900+ 120576 sin120593 minus 120576 cos120593 sdot

(31)

Substituting (27) (30) and (31) into (29) integration ofequation can be obtained

120576 (1 minus 1205762

)3

+ 120576 [1205922(1 minus 120576

2

)3

minus 1205921015840

1

120587

2(1 + 2120576

2

) (1 minus 1205762

)12

] minus 1205922120576 sin 2120572 sdot (1

minus 1205762

)3

+ 1205923120576 (1 minus 120576

2

)3

minus 120583 cos 2120572 minus 12059210158401(120596119888119900minus 2)

sdot 1205762

(1 minus 1205762

) = 0

[120576 (1 minus 1205762

)2

+ 120576 (1 minus 1205762

)2

2120583 sin 2120572 sdot (1 minus 1205762)2

]

+ [2 120576 (1 minus 1205762

)2

+ 1205922120576 (1 minus 120576

2

)2

]

+ 1205921015840

1[(120596119888119900minus 2120572) 120587120576 (1 minus 120576

2

)12

+ 2120576 120576]

+ 21198771198881198971198881198750 = 0

(32)

Equation (32) can be solved by modified Newmark-120573method to obtain the displacement and velocity of the cage

4 Chaos Analyses

Chaos Theory is a new way to analyze complexity whichHenri Poincare studied as the possibility of forecastingA small variation in the initial conditions can generate adifferent set of resultssolutions [17]

A bearing typed as NU2310 is chosen to analyze chaoticresponse Tomoya and Kaoru [18] tested the dynamic cagedisplacements of this bearing and the maximum displace-ments can be used to verify the validity of theoretical analysis

Shock and Vibration 7

e

x

y

X

Y

OO

OD

OC

Rg

Rc

120576

(a) Cage-rib contact

Mc

MD

xc(t)Kx

Kxy

Ky K998400xyBy

Bx

yc(t)

(b) Cage-rib contact model

Figure 6 2-DOF model of cagerib contact

The structural specifications of the bearing are listed inTable 1

The cage of the bearing is made of machined brass DINCuZn40Pb2F43 (GB ZCuZn38Mn2Pb2) and the rings of thebearing are made of through-harden rolling bearing steelDIN 100Cr6 (SAE 52100GB GCr15) The cage is guided bythe outer-ring rib Both static load rating and dynamic loadrating are 186KN According to the recommendation of themanufacturer reference speed rating is 6700 rpm and limitedspeed rating is 12000 rpm Then the weight of the cage anddamping and stiffness parameters can be obtained accordingto the relative parameters in Table 1 The displacement in119909 direction is zero and the displacement in 119910 directionis determined by the cage guidance gap according to (1)Bifurcation diagram of 119883

119862versus spin speed increases with

different cage guide gaps cage eccentricities and dynamicviscosity exponent of lubrication oil The change curvesare described in Figure 7 respectively All three cases aresimulated and variation parameters of the cases are listed inTable 2 An interesting phenomenon that all the cases havetwo bifurcation zones and two steady zones is found in allcurves In relative low rotation speed transient responses canbe found The dynamic response of cage center goes intothe first ordered stage after the end of the resonance Thenthe first bifurcation response occurs The second orderedstatus arises after the first bifurcation ends and then thecurve goes into the second bifurcation Figure 7(a) comparesthe bifurcation diagrams of cage with different guidancegaps The first chaotic response zone of the bearing withsmaller guidance gap (case 1a) is from 120596

119888119900= 9100 rpm

to 120596119888119900= 12000 rpm and it is from 120596

119888119900= 7100 rpm to

120596119888119900= 22000 rpm for the bearing with larger guidance gap

(case 1b) The second chaotic response zone of the bearingwith smaller guidance gap is from 120596

119888119900= 25500 rpm and

it is from 120596119888119900

= 26000 rpm for the bearing with largerguidance gap Chaotic response zone is wider when the gap islarger Figure 7(b) shows bifurcation diagrams of cage with

different eccentricities The first chaotic zone of case 2a isfrom 120596

119888119900= 7600 rpm to 120596

119888119900= 14000 rpm and it is from

120596119888119900= 7800 rpm to 120596

119888119900= 13000 rpm of case 2b The second

chaotic zone of case 2a is from 120596119888119900= 22500 rpm and it is

from 120596119888119900= 19800 rpm of case 2b Chaotic response zone

increaseswith eccentricity In addition transient oscillation ismore drastic when comparing case 1a and case 2a Figure 7(c)depicts bifurcation diagrams of cage with different dynamicviscosity exponents The first chaotic zones of case 3a andcase 3b are from 120596

119888119900= 7650 rpm to 120596

119888119900= 13400 rpm The

second chaotic zones of case 3a and case 3b are from 120596119888119900=

21000 rpmThe change trend of case 3a curve is approachingto that of case 3b But the variation curve of cage with VG78is lower than that with VG56 due to higher dynamic viscosityexponent

The first bifurcation point 119884119862versus spin speed zooming

is shown in Figure 8 Fluctuation can be found even in steadyzone owing to the impact forces between roller and cagepockets In low-speed zone the wave frequency is low and itincreases with rotational speedThe difference of steady zonerange is shown more clearly in the enlarged view Bifurcationdiagrams of each case are different obviously and this denotesthat the position of119883

119862varies timely as well

Cage response behaviors of chaos which are subjected tothe cases in Table 2 at120596 = 8500RPM Poincare section pointsof 119884119862displacement in vertical direction and time history for

4th revolution to 4000th revolution are shown in Figure 9

5 Conclusion and Discussion

Rolling bearing life is typically calculated on the basis of itsload ratings relative to the applied loads and the requirementsregarding bearing life and reliability Both dynamic loadrating and rating life equations neglect cage design Actuallythe cage design has evolved to bear higher and higher load byincorporating more rollers and to lower costs by adopting

8 Shock and Vibration

2500 7500 12500 17500 22500 27500

Rotation speed (rpm)

Case 1b

Case 1a

03

02

01

0

minus01

minus02

minus03

Vert

ical

disp

lace

men

t of

cage

cent

er (m

m)

(a) Different cage guidance gaps

2500 7500 12500 17500 22500 27500

Rotation speed (rpm)

Case 2b

Case 2a

03

02

01

0

minus01

minus02

minus03

Vert

ical

disp

lace

men

t of

cage

cent

er (m

m)

(b) Different eccentricities

2500 7500 12500 17500 22500 27500

Rotation speed (rpm)

Case 3b

Case 3a

03

02

01

0

minus01

minus02

minus03

Vert

ical

disp

lace

men

t of

cage

cent

er (m

m)

(c) Different dynamic viscosity exponents

Figure 7 Bifurcation diagram points119883119862versus spin speed

Table 1 Specifications of the bearing structure and size

Bearing typeBoundary size Roller size Rib diameter Clearance

Boremm

Outsidemm

Widthmm Number Diameter

mmLengthmm mm Radial

120583mPocketmm

NU2310 Φ50 Φ110 40 13 Φ16 27 Φ921 40 03

Table 2 Case study

CaseVariation parameters

Guidance gap Eccentricity Dynamic viscosity exponent(mm) (mm) (Pasdots)

1a 045 23 VG561b 060 23 VG562a 045 35 VG562b 045 57 VG563a 045 42 VG783b 045 42 VG32

different material and improve high-speed performance byusing optimized geometry structure This paper pays closeattention to influence from cage on speed characteristics ofcylindrical roller bearings The most suitable cage guidancetype is confirmed by mathematical derivation firstly Twomajor factors originating from manufacturing process andone involved in applying process are considered to establishdynamical mathematical model of cage with appropriate cageguidance type Analysis of real-time dynamic behavior of

a cage on radial plane was carried out using chaos theorybased on the theoretical mathematical model The chaoticanalysis is limited by the initial values of the system Hencethe initial values are determined firstly in analytical processPreliminary conclusions are listed as follows

(1) There are two bifurcation zones of cage dynamiccurves and the bearing should be away from these twozones in practiceThe range of the 1st bifurcation zoneismainly influenced by the value of cage guidance gapand eccentricity especially the cage guidance gapThelubrication dynamic viscosity exponent has relativelylittle influence on cage dynamic

(2) Numerical simulations show that the roller tends toacceleratedecelerate in switching process of support-ing zoneloaded zone which leads to single or mul-tiple roller-cage bridge impacts Impact force causesperiodic small oscillation on the curves Oscillationamplitude is determined by the pocket gap

(3) Eccentricity of the cage causes dramatic transientresponses in low rotational speed zone The transientresponses are easily tending to cause knocking and

Shock and Vibration 9

7000 10000 13000 16000 19000 22000

Rotation speed (rpm)

Case 1b

Case 1a

002

001

0

minus001

minus002

minus003

minus004

minus005

minus006

minus007

minus008

Vert

ical

disp

lace

men

t of

cage

cent

er (m

m)

(a) Different cage guidance gaps

Rotation speed (rpm)

Case 2b

Case 2a

0

minus001

minus002

minus003

006

005

004

003

002

001

7500 8800 10100 11400 12700 14000

Vert

ical

disp

lace

men

t of

cage

cent

er (m

m)

(b) Different eccentricities

Case 3b

Case 3a

Rotation speed (rpm)7500 8800 10100 11400 12700 14000

0

minus001

minus002

minus003

minus004

005

004

003

002

001

Vert

ical

disp

lace

men

t of

cage

cent

er (m

m)

(c) Different dynamic viscosity exponents

Figure 8 The first bifurcation zone from Figure 7 in enlarged scale

faster wear of cage pocket in terms of worse lubrica-tion film forming The steady zone is narrower wheneccentricity increases In addition divergent trendcan be found in the second bifurcation zonewith lagereccentricity

(4) According to the zooming of the first bifurcationzone variations of any parameters cause totally dif-ferent bifurcations We can draw our conclusion thatmovement of cage is a chaotic motion

(5) The Poincare maps show that position and rotationalspeed of the occurrence bifurcation zones are verydifferent with varied parameters

(6) The change of fluid dynamic viscosity exponent willnot affect the region of bifurcation due to the weaknonlinear of fluid film force

Considering cage dynamics of high-speed cylindricalroller bearing the values of cage guidance gap and eccen-tricity are crucial factors From the point of view of themanufacturers the very small cage guidance gap is difficult toobtain owing to the material (brass) of the cage used in high-speed bearingThe authors strongly suggest that the designershould minimize the gap within the confine of processingcapacity As for the cage eccentricity dynamic balance ofcage is widely used in ultra-high-speed bearings Analysesfrom the paper show that cage eccentricity brings violent

oscillation in low-speed zone and oscillation amplitude inchaotic zone is higher with the increase of cage eccentricityHence dynamic balance is another important factor inmanufacturing process With respect to lubrication dynamicviscosity exponent this study indicates that influence fromdynamic viscosity exponent on cage dynamics is not signifi-cant as the prediction

As a matter of fact researches in the paper are merelyrestricted to cage dynamic response What is to be pointedout is that dynamic response of a cylindrical roller bearingis a complex and interrelated problem among componentsHence there are many imperfect points in the paper whichshould be noted here Preliminary discussions are listed asfollows

(1) The displacements of the cage discussed in the paperare only in 119909-119910 plane During the rotation of thebearing rollers will inevitably have a certain slip-page towards the rings due to insufficient tractionfriction outside the load-zone churning momentfrom lubricant the friction loss between roller-pocketcontact and roller-rib contacts in the actual runningprocess Rollerraceways contact has not been takeninto account in the paper for the sake of simplifyingthe calculation

(2) Damping coefficient and contact stiffness involvedin the calculation are obtained according to material

10 Shock and Vibration

minus03 minus02 minus01 0 01 02minus400

minus200

0

200

400

Vertical displacement of cage center (mm)

Vert

ical

velo

city

(mm

s)

(a) Smaller cage guidance

minus03 minus02 minus01 0 01 02Vertical displacement of cage center (mm)

minus400

minus200

0

200

400

Vert

ical

velo

city

(mm

s)

(b) Bigger cage guidance

minus02 minus01 0 01Vertical displacement of cage center (mm)

minus300

minus200

0

200

300

Vert

ical

velo

city

(mm

s)

(c) Smaller eccentricity

minus03 minus02 minus01 0 01 02Vertical displacement of cage center (mm)

minus400

minus200

0

200

400

Vert

ical

velo

city

(mm

s)

(d) Bigger eccentricity

minus03 minus02 minus01 0 01 02Vertical displacement of cage center (mm)

minus400

minus200

0

200

400

Vert

ical

velo

city

(mm

s)

(e) Higher dynamic viscosity exponent

minus03 minus02 minus01 0 01 02Vertical displacement of cage center (mm)

minus400

minus200

0

200

400

Vert

ical

velo

city

(mm

s)

(f) Lower dynamic viscosity exponent

Figure 9 Poincare section points of 119884119862displacement in vertical direction

property and geometry characteristic parametersActually the damping coefficient and contact stiff-ness can be described more accurately by immediatemovement and velocity Namely contact stiffness isthe first-order derivative of immediate movementand damping coefficient is the first-order derivative ofimmediate velocity For the same sake of simplifyingcalculation the accurate expressions are not substi-tuted into the equations

(3) The bearing under study has the outer ring fixed to arigid support and the inner ring fixed rigidly to theshaft Bearing house and the shaft are truly nonlinearDOF systems The influence of bearing house andshaft on cage dynamic is relatively small so it can besimplified in the calculation

(4) As illustrated in Section 3 cage is driven by the rollersin supporting zone and the rollers are driven by cage

Shock and Vibration 11

in nonloaded zone Driving forces on the total cageare balanced and they are ignored except the impactforces in switching zones of driven modes

Symbols

120572 Angle at cage center in rad Angular velocity in tangential direction in rads Angular acceleration in tangential direction in

rads2119861 Damping coefficient119862 Cage gap in mm119889 Diameter in mmsΔ Roller gap in mm120575 Deformation in mm119890 Cage eccentricity in mm120576 Displacement in normal direction in mm120576 Velocity in normal direction in mms120576 Acceleration in normal direction in mms2119865 Load in N119891 Load caused by motion in N119866 Clearance in mm120579 Angle in ∘119870 Hertzian stiffness119897 Length in mm119872 Mass in kg120583 Fluid dynamic viscosity in Pasdots119873 Number of the rollers120592 Dimensionless parameter of impact factor119874 Position of the center in mm Velocity vector of the center in mms Acceleration vector of the center in mms2119901 Hydrodynamic pressure in Pa1199010 Initial hydrodynamic pressure in Pa

119876 Hertz stress in Nmm2119877 Radius in mm120588 Dimensionless parameter of cage eccentricity to cage

gap119905 Time in sec119879 Pocket clearance in mmV Surface velocity in mms119883 Coordinate119883 direction distance in mm119909 Position in coordinate direction in mm Velocity in 119909 direction in mms Acceleration in 119909 direction in mms2119884 Coordinate 119884 direction distance in mm119910 Dimensionless parameter in coordinate direction119910 Velocity in 119910 direction in mms119910 Acceleration in 119910 direction in mms2120596 Rotation speed in rads

Subscripts

119886 Axial direction120572 Tangential direction119862 Cage center119888 Cagecen Centrifugal

119888119894 Relative relation between cage and innerring

119888119900 Relative relation between cage and outerring

119888119908 Relative relation between cage and roller119863 Unbalanced gravitational center119890 Normal direction119891119889 Added fluid

119892 Ribℎ Impact119894 Inner raceway119895 The 119895th roller119895cen Centrifugal force of the 119895th roller119898 Mean value119874 Rib center119900 Outer raceway119903 Radial direction119903119894 Inner raceway on radial direction119903119900 Outer raceway on radial direction119904 Polar coordinate119908 Roller119909 119909 direction119910 119910 direction

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The project is supported by National Natural Science Foun-dation of China (Grant no 51475144) and the Foundation ofInnovation and Research Team of Science and Technology inUniversities in Henan Province (Grant no 13IRTSTHN025)

References

[1] M Tiwari K Gupta and O Prakash ldquoDynamic response of anunbalanced rotor supported on ball bearingsrdquo Journal of Soundand Vibration vol 238 no 5 pp 757ndash779 2000

[2] M L Adam Jr Rotating Machinery VibrationmdashFrom Analysisto Trouble Shooting Marcel Dekker New York NY USA 2001

[3] N Akturk M Uneeb and R Gohar ldquoThe effects of number ofballs and preload on vibrations associated with ball bearingsrdquoJournal of Tribology vol 119 no 4 pp 747ndash753 1997

[4] S H Upadhyay S C Jain and S P Harsha ldquoNon-linearvibration signature analysis of a high-speed rotating shaft due toball size variations and varying number of ballsrdquo Proceedings ofthe Institution of Mechanical Engineers Part K Journal of Multi-body Dynamics vol 223 no 2 pp 83ndash105 2009

[5] N Akturk ldquoThe effect of waviness on vibrations associated withball bearingsrdquo Journal of Tribology vol 121 no 4 pp 667ndash6771999

[6] S P Harsha K Sandeep and R Prakash ldquoNon-linear dynamicbehaviors of rolling element bearings due to surface wavinessrdquoJournal of Sound and Vibration vol 272 no 3ndash5 pp 557ndash5802004

[7] J Sopanen and A Mikkola ldquoDynamic model of a deep-grooveball bearing including localized and distributed defects Part

12 Shock and Vibration

2 Implementation and resultsrdquo Proceedings of the Institution ofMechanical Engineers Part K vol 217 no 3 pp 213ndash223 2003

[8] S H Upadhyay S C Jain and S P Harsha ldquoChaotic dynamicsof high speed rotating shaft supported by ball bearings due todistributed defectsrdquo International Journal of Engineering Scienceand Technology vol 2 no 10 pp 5746ndash5794 2010

[9] S P Harsha ldquoNonlinear dynamic analysis of a high-speed rotorsupported by rolling element bearingsrdquo Journal of Sound andVibration vol 290 no 1-2 pp 65ndash100 2006

[10] C Villa J-J Sinou and F Thouverez ldquoStability and vibrationanalysis of a complex flexible rotor bearing systemrdquo Communi-cations in Nonlinear Science and Numerical Simulation vol 13no 4 pp 804ndash821 2008

[11] L Houpert ldquoCAGEDYN a contribution to roller bearingdynamic calculations Part III experimental validationrdquo ASMETribology Transactions vol 53 no 6 pp 848ndash859 2010

[12] S P Harsha ldquoNonlinear dynamic analysis of rolling elementbearings due to cage run-out and number of ballsrdquo Journal ofSound and Vibration vol 289 no 1-2 pp 360ndash381 2006

[13] I Bercea S Cretu M Bercea and D Olaru ldquoSimulating roller-cage pocket friction in a tapered roller bearingrdquo EuropeanJournal of Mechanical and Environmental Engineering vol 43no 4 pp 189ndash194 1998

[14] T Sakaguchi andKHarada ldquoDynamic analysis of cage behaviorin a tapered roller bearingrdquo Journal of Tribology vol 128 no 3pp 604ndash611 2006

[15] T A Harris Rolling Bearing Analysis John Wiley amp Sons 4thedition 2001

[16] C Spiridon ldquoMachined brass and pressed steel bearing cages acomparative studyrdquo RKB Technical Review vol 7 pp 1ndash13 2011

[17] R Riccardo ldquoChaos theory and some practical applications intechnical analysisrdquo in Proceedings of the 11th Annual Conferenceof the International Federation of Technical Analysts 1998

[18] S Tomoya and U Kaoru ldquoDynamic analysis of cage behavior ina cylindrical roller bearingrdquo NTN Technical Review 71 2004

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International Journal of

4 Shock and Vibration

X

Y

p

e

120596o

120596c

OO

OD

OC

2120579

120572

120588

F(jminus2)cw

fy

fx

Rg

RcCr

Fjcen

Figure 3 Forces generated by motion on the cage

on outer raceway Subsequently the centrifugal forces of therollers cause larger contact deformation It can be added tothe original deformation on the outer raceway as

120575119903119900= (

119865cen + 119876119900119870119900

)

910

(10)

Substituting (10) into (4) equations (5) and (6) willchange along with (4) According to different expressions ofcage displacements both inner ring guidance and roller guid-ance bearings will generate complex dynamic displacementswhen they are operating at high speedsThenwe can drawourconclusion that both inner ring guidance and roller guidanceare not applicable under high-speed conditions Outer-ring-rib-guidance bearing is chosen as the research object in thefollowing analysis naturally

3 Physical Model

The cage has two kinds of contacts in a rotating outer-ring-guided cylindrical bearing One is the cage-rib contact andthe other is the roller-pocket contactThe forces generated bythe contacts are marked in Figure 3 Cage-rib contact statusis determined by the relative rotational speed (120596

119888119900) and the

cage guidance gap (119862119903) The cage has physical contact with

guiding ring in the starting phase owing to a low relativespeed However the cage and the rib are separated by oilfilm eventually with the increasing of relative speed Thencage-rib contact is considered as a fluid-structure interactionproblem Similarly roller-cage contact status is determined byrotational speed of the roller around its own axil (120596

119888119908) and the

pocket clearance (119879) The cage and the rollers are separatedby oil film with the increasing of speed too Note that thecage is driven by the rollers in supporting zone and the rollersare driven by cage in nonloaded zone The driving forces byrollers on the cage can be seen in Figure 3 as well Differentdrivingmode of the rollers in different force directions causesimpact loads on the cage The impact between the cage andthe roller in a very short time results in large impact forces

31 Instantaneous Motion Forces Different contacts causedifferent motion forces and there is no strong relationshipamong the motion forces then linear superposition methodcan be adopted for calculating motion forces on the cage

The relative rotational velocity of the cage to the rib isdefined as

120596119888119900= 120596119888minus 120596119900 (11)

And the cage guidance gap determined by cage diameterand rib can be expressed as

119862119903= 119889119892minus 119889119888 (12)

Brass cage is usually machined from centrifugally casttubing and then nonuniform distribution of the mass causesstatic imbalance (119890) of the cage Nondimensional eccentricityof the cage to cage guidance gap can be given as

120588 =119890

119862119903

(13)

Spiridon [16] established an elastic system with twodegrees of freedom (DOF) that considered both cage-roller stiffness (119870

119888119908) and cage structural stiffness (119870

119888) for

cageroller contact As described in Figure 4 the dynamicequilibrium condition written for each of the two massesprovides the following set of two differential equations

119870119888119908(120575119908minus 120575119888)109

+119872119908

1198892

120575119908

1198891199052= 0

minus119870119888119908(120575119908minus 120575119888)109

+ 119870119888120575119888+119872119888

1198892

120575119908

1198891199052= 0

(14)

119870119888119908

can be derived from the Hertzian contact theorywhereas119870

119888can be obtained analyticallyThe cage-roller stress

(119876119888119908) can be obtained from (3) and its variation trend for

the specified roller is shown in Figure 5 when the bearing isrotating

Impact force of cageroller contact can be given as

119891ℎ119909= 119876119888119908[sin120596

119888119900119905 + sin (120587 minus 120596

119888119900119905)] cos 120579

119891ℎ119910= 119876119888119908[sin120596

119888119900119905 minus sin (120587 minus 120596

119888119900119905)] sin 120579

(15)

As for the cage-rib dynamic model it can encompassradial-plane orbital cage motion with 2-DOF equationsas well According to Figure 4 motion equation can beexpressed as

119872119863119863= minus2119870

119904(119874119862minus 119874119874) minus 2119861

119904(119862minus 119874)

+ 2119872119863120588119892119890119910

minus 119865119904(119883119863 119863 119863 119884119863 119863 119863)

119872119862119862= minus119870119904(119874119862minus 119874119874) minus 119861119904(119862minus 119874) + 119891119890119909sdot 119890119909

+ 119891119890119910sdot 119890119910minus119872119874120588119892119890119910

(16)

Shock and Vibration 5

Roller

Cage120575c(t)

120575w(t)

(a) Cageroller contact

120575c(t) 120575w(t)

kckcw

Mc Mw

(b) Model of cageroller contact

Figure 4 A 2-DOF model of cageroller contact of the 119895th roller

Load

(N)

0 1 2 3 4 5 6minus40

minus20

0

20

40

60

80

100

120

t(s) times 10minus4

Figure 5 Demonstration on variation trend of cage-roller contactforce

The position vector velocity vector and accelerationvector of the unbalanced gravitational center of the cage canbe given as

119874119863= (119883119888+ 120588 cos120596119905) sdot 119890

119909+ (119884119888+ 120588 sin120596119905) sdot 119890

119910

119863= (119888+ 120588 cos120596119905) sdot 119890

119909+ (119888+ 120588 sin120596119905) sdot 119890

119910

119863= (119888+ 120588 cos120596119905) sdot 119890

119909+ (119888+ 120588 sin120596119905) sdot 119890

119910

(17)

In the same way the position velocity and accelerationvector of geometry center of the cage are

119874119888= 119883119888sdot 119890119909+ 119884119888sdot 119890119910

119888= 119888sdot 119890119909+ 119888sdot 119890119910

119888= 119888sdot 119890119909+ 119888sdot 119890119910

(18)

Similarly using above-mentioned exhibition the posi-tion velocity and acceleration vector of the center of the ribare

119874119900= 119883119900sdot 119890119909+ 119884119900sdot 119890119910

119900= 119883119900sdot 119890119909+ 119900sdot 119890119910

119900= 119883119900sdot 119890119909+ 119900sdot 119890119910

(19)

Substituting (15) (16) and (17) into (14) then the follow-ing two equations described in 119909-119910 plane can be obtained

119872119863119874

= minus2119870119904(119883119862minus 119883119874) minus 2119861

119904(119862minus 119874)

+ 2119872119863120588120596119888minus119900

2 cos120596119888minus119900119905 minus 119870119909119910119884119863minus 1198701015840

119909119910119884119863

3

119872119863119874

= minus2119870119904(119884119862minus 119884119874) minus 2119861

119904(119862minus 119874)

+ 2119872119863120588120596119888minus119900

2 sin120596119888minus119900119905 minus 119870119909119910119883119863minus 1198701015840

119909119910119883119863

3

minus 2119872119863119892

(20)

119872119862119862minus 119891119890119909minus 119870119904(119883119874minus 119883119862) minus 119861119904(119874minus 119862) = 0

119872119862119862minus 119891119890119910minus 119870119904(119884119874minus 119884119862) minus 119861119904(119862minus 119874)

= minus119872119862119892

(21)

Considering (20)-(21) the results can be obtained as

119891119890119909= 1205921

2

(119874minus 119862) minus 1205922(119874minus 119862) minus 1205923(119909119874minus 119909119862)

119891119890119910= 1205922

2

( 119910119874minus 119910119862) minus 1205922( 119910119874minus 119910119862) minus 1205923(119910119874minus 119910119862)

+ 119870119891119889119891119889

(22)

Here 119909119874 119910119874 119909119862 and 119910

119862are nondimensional form of

rib center (119883119874 119884119874) and cage center (119883

119862 119884119862) and they are

obtained by original value divided by 119862119903 The other factors in

(22) can be expressed as

1205921=2 (119872119863120596119888119900119861)2

119882

1205922= 2119870119904119861

1205923=2119870119904

32

119861

11987212

119863

120596119888119900

(23)

6 Shock and Vibration

When the linear superposition method is adopted tocontact forces of the cage in 119909-119910 plane the total force ondifferent directions can be given as

119891119909= 119891119890119909

119891119910= 119891119886119910+ 119891119890119910

(24)

32 Hydrodynamic Fluid-Film Force Cage-rib contact is afluid-structure interaction problem with bearings rotatingat a high speed The main supporting force generated bylubricant squeeze effects can be approximated to a short-width journal bearing theory as illustrated in Figure 3 Thendynamic-deviation interactive force between cage and rib canbe described as a continuous function of cage position andvelocity components Reynolds lubrication equation (RLE)provides the basis of lubrication theory [15] and the forcegiven on the cage by the film is expressed as

120597

120597120572[(1 + 120576 cos120572)3

120597119901

120597120572] + 1198772

119888

120597

120597119909[(1 + 120576 cos120572)2

120597119901

120597119910]

= minus6120583(119877119888

119862119903

)

2

[(120596119888119900minus 2) 120576 sin120572 minus 2120576 cos120572]

(25)

And

119901 (120572) = 1198750+120596119888119900

119862119903

3[1198972

119888

4minus 1199102

]120576 sin120572

1 + 120576cos3120572 (26)

Solutions to the RLE are a nonlinear function of displace-ment and angle at cage center Hydrodynamic fluid-film forcecan be written as follows

119891119890= 120583119877119888119897119888(119897119888

119862119903

)

2

[(120596119888119900minus 2)

1205762

(1 minus 1205762)2

+120587

2

(1 + 21205762

) 120576

(1 minus 1205762)52

]

119891120572= 120583119877119888119897119888(119877119888

119862119903

)

2

sdot [(120596119888119900minus 2120572)

120587120576

4 (1 minus 1205762)32

+2120576 120576

(1 minus 1205762)2]

+ 21198771198881198971198881199010

(27)

The fluid-film force can also be represented as two com-ponent forces in horizontal and vertical force equilibrium

119891119909= 119891119890sdot cos120572 + 119891

120572sdot sin120572

119891119910= 119891119890sdot sin120572 minus 119891

120572sdot cos120572

(28)

33 Equilibrium of the Equations In hydrodynamic fluid-film bearing the fluid supported pressure is generated nor-mally by motion of the cage and depends on the dynamic

viscosity exponent of the lubricating fluid For the cagestructure and coordination please refer to Figure 6 Thegenerated forces can be expressed as

119891119890sdot cos120572 + 119891

120572sdot sin120572 = 120592

1

2

119862minus 1205922(119874minus 119862)

minus 1205923(119909119874minus 119909119862)

119891119890sdot sin120572 minus 119891

120572sdot cos120572 = 120592

1

2

119910119862minus 1205922( 119910119874minus 119910119862)

minus 1205923(119910119874minus 119910119862) + 119861119891119889119891119889

+ 119891119894119910

(29)

Displacement relations between the centers of cage andrib are

119909119888= 119909119900+ 120576 cos120593

119910119888= 119910119900+ 120576 sin120593

(30)

Similarly for the velocity

119888= 119900+ 120576 cos120593 minus 120576 sin120593 sdot

119910119888= 119910119900+ 120576 sin120593 minus 120576 cos120593 sdot

(31)

Substituting (27) (30) and (31) into (29) integration ofequation can be obtained

120576 (1 minus 1205762

)3

+ 120576 [1205922(1 minus 120576

2

)3

minus 1205921015840

1

120587

2(1 + 2120576

2

) (1 minus 1205762

)12

] minus 1205922120576 sin 2120572 sdot (1

minus 1205762

)3

+ 1205923120576 (1 minus 120576

2

)3

minus 120583 cos 2120572 minus 12059210158401(120596119888119900minus 2)

sdot 1205762

(1 minus 1205762

) = 0

[120576 (1 minus 1205762

)2

+ 120576 (1 minus 1205762

)2

2120583 sin 2120572 sdot (1 minus 1205762)2

]

+ [2 120576 (1 minus 1205762

)2

+ 1205922120576 (1 minus 120576

2

)2

]

+ 1205921015840

1[(120596119888119900minus 2120572) 120587120576 (1 minus 120576

2

)12

+ 2120576 120576]

+ 21198771198881198971198881198750 = 0

(32)

Equation (32) can be solved by modified Newmark-120573method to obtain the displacement and velocity of the cage

4 Chaos Analyses

Chaos Theory is a new way to analyze complexity whichHenri Poincare studied as the possibility of forecastingA small variation in the initial conditions can generate adifferent set of resultssolutions [17]

A bearing typed as NU2310 is chosen to analyze chaoticresponse Tomoya and Kaoru [18] tested the dynamic cagedisplacements of this bearing and the maximum displace-ments can be used to verify the validity of theoretical analysis

Shock and Vibration 7

e

x

y

X

Y

OO

OD

OC

Rg

Rc

120576

(a) Cage-rib contact

Mc

MD

xc(t)Kx

Kxy

Ky K998400xyBy

Bx

yc(t)

(b) Cage-rib contact model

Figure 6 2-DOF model of cagerib contact

The structural specifications of the bearing are listed inTable 1

The cage of the bearing is made of machined brass DINCuZn40Pb2F43 (GB ZCuZn38Mn2Pb2) and the rings of thebearing are made of through-harden rolling bearing steelDIN 100Cr6 (SAE 52100GB GCr15) The cage is guided bythe outer-ring rib Both static load rating and dynamic loadrating are 186KN According to the recommendation of themanufacturer reference speed rating is 6700 rpm and limitedspeed rating is 12000 rpm Then the weight of the cage anddamping and stiffness parameters can be obtained accordingto the relative parameters in Table 1 The displacement in119909 direction is zero and the displacement in 119910 directionis determined by the cage guidance gap according to (1)Bifurcation diagram of 119883

119862versus spin speed increases with

different cage guide gaps cage eccentricities and dynamicviscosity exponent of lubrication oil The change curvesare described in Figure 7 respectively All three cases aresimulated and variation parameters of the cases are listed inTable 2 An interesting phenomenon that all the cases havetwo bifurcation zones and two steady zones is found in allcurves In relative low rotation speed transient responses canbe found The dynamic response of cage center goes intothe first ordered stage after the end of the resonance Thenthe first bifurcation response occurs The second orderedstatus arises after the first bifurcation ends and then thecurve goes into the second bifurcation Figure 7(a) comparesthe bifurcation diagrams of cage with different guidancegaps The first chaotic response zone of the bearing withsmaller guidance gap (case 1a) is from 120596

119888119900= 9100 rpm

to 120596119888119900= 12000 rpm and it is from 120596

119888119900= 7100 rpm to

120596119888119900= 22000 rpm for the bearing with larger guidance gap

(case 1b) The second chaotic response zone of the bearingwith smaller guidance gap is from 120596

119888119900= 25500 rpm and

it is from 120596119888119900

= 26000 rpm for the bearing with largerguidance gap Chaotic response zone is wider when the gap islarger Figure 7(b) shows bifurcation diagrams of cage with

different eccentricities The first chaotic zone of case 2a isfrom 120596

119888119900= 7600 rpm to 120596

119888119900= 14000 rpm and it is from

120596119888119900= 7800 rpm to 120596

119888119900= 13000 rpm of case 2b The second

chaotic zone of case 2a is from 120596119888119900= 22500 rpm and it is

from 120596119888119900= 19800 rpm of case 2b Chaotic response zone

increaseswith eccentricity In addition transient oscillation ismore drastic when comparing case 1a and case 2a Figure 7(c)depicts bifurcation diagrams of cage with different dynamicviscosity exponents The first chaotic zones of case 3a andcase 3b are from 120596

119888119900= 7650 rpm to 120596

119888119900= 13400 rpm The

second chaotic zones of case 3a and case 3b are from 120596119888119900=

21000 rpmThe change trend of case 3a curve is approachingto that of case 3b But the variation curve of cage with VG78is lower than that with VG56 due to higher dynamic viscosityexponent

The first bifurcation point 119884119862versus spin speed zooming

is shown in Figure 8 Fluctuation can be found even in steadyzone owing to the impact forces between roller and cagepockets In low-speed zone the wave frequency is low and itincreases with rotational speedThe difference of steady zonerange is shown more clearly in the enlarged view Bifurcationdiagrams of each case are different obviously and this denotesthat the position of119883

119862varies timely as well

Cage response behaviors of chaos which are subjected tothe cases in Table 2 at120596 = 8500RPM Poincare section pointsof 119884119862displacement in vertical direction and time history for

4th revolution to 4000th revolution are shown in Figure 9

5 Conclusion and Discussion

Rolling bearing life is typically calculated on the basis of itsload ratings relative to the applied loads and the requirementsregarding bearing life and reliability Both dynamic loadrating and rating life equations neglect cage design Actuallythe cage design has evolved to bear higher and higher load byincorporating more rollers and to lower costs by adopting

8 Shock and Vibration

2500 7500 12500 17500 22500 27500

Rotation speed (rpm)

Case 1b

Case 1a

03

02

01

0

minus01

minus02

minus03

Vert

ical

disp

lace

men

t of

cage

cent

er (m

m)

(a) Different cage guidance gaps

2500 7500 12500 17500 22500 27500

Rotation speed (rpm)

Case 2b

Case 2a

03

02

01

0

minus01

minus02

minus03

Vert

ical

disp

lace

men

t of

cage

cent

er (m

m)

(b) Different eccentricities

2500 7500 12500 17500 22500 27500

Rotation speed (rpm)

Case 3b

Case 3a

03

02

01

0

minus01

minus02

minus03

Vert

ical

disp

lace

men

t of

cage

cent

er (m

m)

(c) Different dynamic viscosity exponents

Figure 7 Bifurcation diagram points119883119862versus spin speed

Table 1 Specifications of the bearing structure and size

Bearing typeBoundary size Roller size Rib diameter Clearance

Boremm

Outsidemm

Widthmm Number Diameter

mmLengthmm mm Radial

120583mPocketmm

NU2310 Φ50 Φ110 40 13 Φ16 27 Φ921 40 03

Table 2 Case study

CaseVariation parameters

Guidance gap Eccentricity Dynamic viscosity exponent(mm) (mm) (Pasdots)

1a 045 23 VG561b 060 23 VG562a 045 35 VG562b 045 57 VG563a 045 42 VG783b 045 42 VG32

different material and improve high-speed performance byusing optimized geometry structure This paper pays closeattention to influence from cage on speed characteristics ofcylindrical roller bearings The most suitable cage guidancetype is confirmed by mathematical derivation firstly Twomajor factors originating from manufacturing process andone involved in applying process are considered to establishdynamical mathematical model of cage with appropriate cageguidance type Analysis of real-time dynamic behavior of

a cage on radial plane was carried out using chaos theorybased on the theoretical mathematical model The chaoticanalysis is limited by the initial values of the system Hencethe initial values are determined firstly in analytical processPreliminary conclusions are listed as follows

(1) There are two bifurcation zones of cage dynamiccurves and the bearing should be away from these twozones in practiceThe range of the 1st bifurcation zoneismainly influenced by the value of cage guidance gapand eccentricity especially the cage guidance gapThelubrication dynamic viscosity exponent has relativelylittle influence on cage dynamic

(2) Numerical simulations show that the roller tends toacceleratedecelerate in switching process of support-ing zoneloaded zone which leads to single or mul-tiple roller-cage bridge impacts Impact force causesperiodic small oscillation on the curves Oscillationamplitude is determined by the pocket gap

(3) Eccentricity of the cage causes dramatic transientresponses in low rotational speed zone The transientresponses are easily tending to cause knocking and

Shock and Vibration 9

7000 10000 13000 16000 19000 22000

Rotation speed (rpm)

Case 1b

Case 1a

002

001

0

minus001

minus002

minus003

minus004

minus005

minus006

minus007

minus008

Vert

ical

disp

lace

men

t of

cage

cent

er (m

m)

(a) Different cage guidance gaps

Rotation speed (rpm)

Case 2b

Case 2a

0

minus001

minus002

minus003

006

005

004

003

002

001

7500 8800 10100 11400 12700 14000

Vert

ical

disp

lace

men

t of

cage

cent

er (m

m)

(b) Different eccentricities

Case 3b

Case 3a

Rotation speed (rpm)7500 8800 10100 11400 12700 14000

0

minus001

minus002

minus003

minus004

005

004

003

002

001

Vert

ical

disp

lace

men

t of

cage

cent

er (m

m)

(c) Different dynamic viscosity exponents

Figure 8 The first bifurcation zone from Figure 7 in enlarged scale

faster wear of cage pocket in terms of worse lubrica-tion film forming The steady zone is narrower wheneccentricity increases In addition divergent trendcan be found in the second bifurcation zonewith lagereccentricity

(4) According to the zooming of the first bifurcationzone variations of any parameters cause totally dif-ferent bifurcations We can draw our conclusion thatmovement of cage is a chaotic motion

(5) The Poincare maps show that position and rotationalspeed of the occurrence bifurcation zones are verydifferent with varied parameters

(6) The change of fluid dynamic viscosity exponent willnot affect the region of bifurcation due to the weaknonlinear of fluid film force

Considering cage dynamics of high-speed cylindricalroller bearing the values of cage guidance gap and eccen-tricity are crucial factors From the point of view of themanufacturers the very small cage guidance gap is difficult toobtain owing to the material (brass) of the cage used in high-speed bearingThe authors strongly suggest that the designershould minimize the gap within the confine of processingcapacity As for the cage eccentricity dynamic balance ofcage is widely used in ultra-high-speed bearings Analysesfrom the paper show that cage eccentricity brings violent

oscillation in low-speed zone and oscillation amplitude inchaotic zone is higher with the increase of cage eccentricityHence dynamic balance is another important factor inmanufacturing process With respect to lubrication dynamicviscosity exponent this study indicates that influence fromdynamic viscosity exponent on cage dynamics is not signifi-cant as the prediction

As a matter of fact researches in the paper are merelyrestricted to cage dynamic response What is to be pointedout is that dynamic response of a cylindrical roller bearingis a complex and interrelated problem among componentsHence there are many imperfect points in the paper whichshould be noted here Preliminary discussions are listed asfollows

(1) The displacements of the cage discussed in the paperare only in 119909-119910 plane During the rotation of thebearing rollers will inevitably have a certain slip-page towards the rings due to insufficient tractionfriction outside the load-zone churning momentfrom lubricant the friction loss between roller-pocketcontact and roller-rib contacts in the actual runningprocess Rollerraceways contact has not been takeninto account in the paper for the sake of simplifyingthe calculation

(2) Damping coefficient and contact stiffness involvedin the calculation are obtained according to material

10 Shock and Vibration

minus03 minus02 minus01 0 01 02minus400

minus200

0

200

400

Vertical displacement of cage center (mm)

Vert

ical

velo

city

(mm

s)

(a) Smaller cage guidance

minus03 minus02 minus01 0 01 02Vertical displacement of cage center (mm)

minus400

minus200

0

200

400

Vert

ical

velo

city

(mm

s)

(b) Bigger cage guidance

minus02 minus01 0 01Vertical displacement of cage center (mm)

minus300

minus200

0

200

300

Vert

ical

velo

city

(mm

s)

(c) Smaller eccentricity

minus03 minus02 minus01 0 01 02Vertical displacement of cage center (mm)

minus400

minus200

0

200

400

Vert

ical

velo

city

(mm

s)

(d) Bigger eccentricity

minus03 minus02 minus01 0 01 02Vertical displacement of cage center (mm)

minus400

minus200

0

200

400

Vert

ical

velo

city

(mm

s)

(e) Higher dynamic viscosity exponent

minus03 minus02 minus01 0 01 02Vertical displacement of cage center (mm)

minus400

minus200

0

200

400

Vert

ical

velo

city

(mm

s)

(f) Lower dynamic viscosity exponent

Figure 9 Poincare section points of 119884119862displacement in vertical direction

property and geometry characteristic parametersActually the damping coefficient and contact stiff-ness can be described more accurately by immediatemovement and velocity Namely contact stiffness isthe first-order derivative of immediate movementand damping coefficient is the first-order derivative ofimmediate velocity For the same sake of simplifyingcalculation the accurate expressions are not substi-tuted into the equations

(3) The bearing under study has the outer ring fixed to arigid support and the inner ring fixed rigidly to theshaft Bearing house and the shaft are truly nonlinearDOF systems The influence of bearing house andshaft on cage dynamic is relatively small so it can besimplified in the calculation

(4) As illustrated in Section 3 cage is driven by the rollersin supporting zone and the rollers are driven by cage

Shock and Vibration 11

in nonloaded zone Driving forces on the total cageare balanced and they are ignored except the impactforces in switching zones of driven modes

Symbols

120572 Angle at cage center in rad Angular velocity in tangential direction in rads Angular acceleration in tangential direction in

rads2119861 Damping coefficient119862 Cage gap in mm119889 Diameter in mmsΔ Roller gap in mm120575 Deformation in mm119890 Cage eccentricity in mm120576 Displacement in normal direction in mm120576 Velocity in normal direction in mms120576 Acceleration in normal direction in mms2119865 Load in N119891 Load caused by motion in N119866 Clearance in mm120579 Angle in ∘119870 Hertzian stiffness119897 Length in mm119872 Mass in kg120583 Fluid dynamic viscosity in Pasdots119873 Number of the rollers120592 Dimensionless parameter of impact factor119874 Position of the center in mm Velocity vector of the center in mms Acceleration vector of the center in mms2119901 Hydrodynamic pressure in Pa1199010 Initial hydrodynamic pressure in Pa

119876 Hertz stress in Nmm2119877 Radius in mm120588 Dimensionless parameter of cage eccentricity to cage

gap119905 Time in sec119879 Pocket clearance in mmV Surface velocity in mms119883 Coordinate119883 direction distance in mm119909 Position in coordinate direction in mm Velocity in 119909 direction in mms Acceleration in 119909 direction in mms2119884 Coordinate 119884 direction distance in mm119910 Dimensionless parameter in coordinate direction119910 Velocity in 119910 direction in mms119910 Acceleration in 119910 direction in mms2120596 Rotation speed in rads

Subscripts

119886 Axial direction120572 Tangential direction119862 Cage center119888 Cagecen Centrifugal

119888119894 Relative relation between cage and innerring

119888119900 Relative relation between cage and outerring

119888119908 Relative relation between cage and roller119863 Unbalanced gravitational center119890 Normal direction119891119889 Added fluid

119892 Ribℎ Impact119894 Inner raceway119895 The 119895th roller119895cen Centrifugal force of the 119895th roller119898 Mean value119874 Rib center119900 Outer raceway119903 Radial direction119903119894 Inner raceway on radial direction119903119900 Outer raceway on radial direction119904 Polar coordinate119908 Roller119909 119909 direction119910 119910 direction

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The project is supported by National Natural Science Foun-dation of China (Grant no 51475144) and the Foundation ofInnovation and Research Team of Science and Technology inUniversities in Henan Province (Grant no 13IRTSTHN025)

References

[1] M Tiwari K Gupta and O Prakash ldquoDynamic response of anunbalanced rotor supported on ball bearingsrdquo Journal of Soundand Vibration vol 238 no 5 pp 757ndash779 2000

[2] M L Adam Jr Rotating Machinery VibrationmdashFrom Analysisto Trouble Shooting Marcel Dekker New York NY USA 2001

[3] N Akturk M Uneeb and R Gohar ldquoThe effects of number ofballs and preload on vibrations associated with ball bearingsrdquoJournal of Tribology vol 119 no 4 pp 747ndash753 1997

[4] S H Upadhyay S C Jain and S P Harsha ldquoNon-linearvibration signature analysis of a high-speed rotating shaft due toball size variations and varying number of ballsrdquo Proceedings ofthe Institution of Mechanical Engineers Part K Journal of Multi-body Dynamics vol 223 no 2 pp 83ndash105 2009

[5] N Akturk ldquoThe effect of waviness on vibrations associated withball bearingsrdquo Journal of Tribology vol 121 no 4 pp 667ndash6771999

[6] S P Harsha K Sandeep and R Prakash ldquoNon-linear dynamicbehaviors of rolling element bearings due to surface wavinessrdquoJournal of Sound and Vibration vol 272 no 3ndash5 pp 557ndash5802004

[7] J Sopanen and A Mikkola ldquoDynamic model of a deep-grooveball bearing including localized and distributed defects Part

12 Shock and Vibration

2 Implementation and resultsrdquo Proceedings of the Institution ofMechanical Engineers Part K vol 217 no 3 pp 213ndash223 2003

[8] S H Upadhyay S C Jain and S P Harsha ldquoChaotic dynamicsof high speed rotating shaft supported by ball bearings due todistributed defectsrdquo International Journal of Engineering Scienceand Technology vol 2 no 10 pp 5746ndash5794 2010

[9] S P Harsha ldquoNonlinear dynamic analysis of a high-speed rotorsupported by rolling element bearingsrdquo Journal of Sound andVibration vol 290 no 1-2 pp 65ndash100 2006

[10] C Villa J-J Sinou and F Thouverez ldquoStability and vibrationanalysis of a complex flexible rotor bearing systemrdquo Communi-cations in Nonlinear Science and Numerical Simulation vol 13no 4 pp 804ndash821 2008

[11] L Houpert ldquoCAGEDYN a contribution to roller bearingdynamic calculations Part III experimental validationrdquo ASMETribology Transactions vol 53 no 6 pp 848ndash859 2010

[12] S P Harsha ldquoNonlinear dynamic analysis of rolling elementbearings due to cage run-out and number of ballsrdquo Journal ofSound and Vibration vol 289 no 1-2 pp 360ndash381 2006

[13] I Bercea S Cretu M Bercea and D Olaru ldquoSimulating roller-cage pocket friction in a tapered roller bearingrdquo EuropeanJournal of Mechanical and Environmental Engineering vol 43no 4 pp 189ndash194 1998

[14] T Sakaguchi andKHarada ldquoDynamic analysis of cage behaviorin a tapered roller bearingrdquo Journal of Tribology vol 128 no 3pp 604ndash611 2006

[15] T A Harris Rolling Bearing Analysis John Wiley amp Sons 4thedition 2001

[16] C Spiridon ldquoMachined brass and pressed steel bearing cages acomparative studyrdquo RKB Technical Review vol 7 pp 1ndash13 2011

[17] R Riccardo ldquoChaos theory and some practical applications intechnical analysisrdquo in Proceedings of the 11th Annual Conferenceof the International Federation of Technical Analysts 1998

[18] S Tomoya and U Kaoru ldquoDynamic analysis of cage behavior ina cylindrical roller bearingrdquo NTN Technical Review 71 2004

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Shock and Vibration

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International Journal of

Shock and Vibration 5

Roller

Cage120575c(t)

120575w(t)

(a) Cageroller contact

120575c(t) 120575w(t)

kckcw

Mc Mw

(b) Model of cageroller contact

Figure 4 A 2-DOF model of cageroller contact of the 119895th roller

Load

(N)

0 1 2 3 4 5 6minus40

minus20

0

20

40

60

80

100

120

t(s) times 10minus4

Figure 5 Demonstration on variation trend of cage-roller contactforce

The position vector velocity vector and accelerationvector of the unbalanced gravitational center of the cage canbe given as

119874119863= (119883119888+ 120588 cos120596119905) sdot 119890

119909+ (119884119888+ 120588 sin120596119905) sdot 119890

119910

119863= (119888+ 120588 cos120596119905) sdot 119890

119909+ (119888+ 120588 sin120596119905) sdot 119890

119910

119863= (119888+ 120588 cos120596119905) sdot 119890

119909+ (119888+ 120588 sin120596119905) sdot 119890

119910

(17)

In the same way the position velocity and accelerationvector of geometry center of the cage are

119874119888= 119883119888sdot 119890119909+ 119884119888sdot 119890119910

119888= 119888sdot 119890119909+ 119888sdot 119890119910

119888= 119888sdot 119890119909+ 119888sdot 119890119910

(18)

Similarly using above-mentioned exhibition the posi-tion velocity and acceleration vector of the center of the ribare

119874119900= 119883119900sdot 119890119909+ 119884119900sdot 119890119910

119900= 119883119900sdot 119890119909+ 119900sdot 119890119910

119900= 119883119900sdot 119890119909+ 119900sdot 119890119910

(19)

Substituting (15) (16) and (17) into (14) then the follow-ing two equations described in 119909-119910 plane can be obtained

119872119863119874

= minus2119870119904(119883119862minus 119883119874) minus 2119861

119904(119862minus 119874)

+ 2119872119863120588120596119888minus119900

2 cos120596119888minus119900119905 minus 119870119909119910119884119863minus 1198701015840

119909119910119884119863

3

119872119863119874

= minus2119870119904(119884119862minus 119884119874) minus 2119861

119904(119862minus 119874)

+ 2119872119863120588120596119888minus119900

2 sin120596119888minus119900119905 minus 119870119909119910119883119863minus 1198701015840

119909119910119883119863

3

minus 2119872119863119892

(20)

119872119862119862minus 119891119890119909minus 119870119904(119883119874minus 119883119862) minus 119861119904(119874minus 119862) = 0

119872119862119862minus 119891119890119910minus 119870119904(119884119874minus 119884119862) minus 119861119904(119862minus 119874)

= minus119872119862119892

(21)

Considering (20)-(21) the results can be obtained as

119891119890119909= 1205921

2

(119874minus 119862) minus 1205922(119874minus 119862) minus 1205923(119909119874minus 119909119862)

119891119890119910= 1205922

2

( 119910119874minus 119910119862) minus 1205922( 119910119874minus 119910119862) minus 1205923(119910119874minus 119910119862)

+ 119870119891119889119891119889

(22)

Here 119909119874 119910119874 119909119862 and 119910

119862are nondimensional form of

rib center (119883119874 119884119874) and cage center (119883

119862 119884119862) and they are

obtained by original value divided by 119862119903 The other factors in

(22) can be expressed as

1205921=2 (119872119863120596119888119900119861)2

119882

1205922= 2119870119904119861

1205923=2119870119904

32

119861

11987212

119863

120596119888119900

(23)

6 Shock and Vibration

When the linear superposition method is adopted tocontact forces of the cage in 119909-119910 plane the total force ondifferent directions can be given as

119891119909= 119891119890119909

119891119910= 119891119886119910+ 119891119890119910

(24)

32 Hydrodynamic Fluid-Film Force Cage-rib contact is afluid-structure interaction problem with bearings rotatingat a high speed The main supporting force generated bylubricant squeeze effects can be approximated to a short-width journal bearing theory as illustrated in Figure 3 Thendynamic-deviation interactive force between cage and rib canbe described as a continuous function of cage position andvelocity components Reynolds lubrication equation (RLE)provides the basis of lubrication theory [15] and the forcegiven on the cage by the film is expressed as

120597

120597120572[(1 + 120576 cos120572)3

120597119901

120597120572] + 1198772

119888

120597

120597119909[(1 + 120576 cos120572)2

120597119901

120597119910]

= minus6120583(119877119888

119862119903

)

2

[(120596119888119900minus 2) 120576 sin120572 minus 2120576 cos120572]

(25)

And

119901 (120572) = 1198750+120596119888119900

119862119903

3[1198972

119888

4minus 1199102

]120576 sin120572

1 + 120576cos3120572 (26)

Solutions to the RLE are a nonlinear function of displace-ment and angle at cage center Hydrodynamic fluid-film forcecan be written as follows

119891119890= 120583119877119888119897119888(119897119888

119862119903

)

2

[(120596119888119900minus 2)

1205762

(1 minus 1205762)2

+120587

2

(1 + 21205762

) 120576

(1 minus 1205762)52

]

119891120572= 120583119877119888119897119888(119877119888

119862119903

)

2

sdot [(120596119888119900minus 2120572)

120587120576

4 (1 minus 1205762)32

+2120576 120576

(1 minus 1205762)2]

+ 21198771198881198971198881199010

(27)

The fluid-film force can also be represented as two com-ponent forces in horizontal and vertical force equilibrium

119891119909= 119891119890sdot cos120572 + 119891

120572sdot sin120572

119891119910= 119891119890sdot sin120572 minus 119891

120572sdot cos120572

(28)

33 Equilibrium of the Equations In hydrodynamic fluid-film bearing the fluid supported pressure is generated nor-mally by motion of the cage and depends on the dynamic

viscosity exponent of the lubricating fluid For the cagestructure and coordination please refer to Figure 6 Thegenerated forces can be expressed as

119891119890sdot cos120572 + 119891

120572sdot sin120572 = 120592

1

2

119862minus 1205922(119874minus 119862)

minus 1205923(119909119874minus 119909119862)

119891119890sdot sin120572 minus 119891

120572sdot cos120572 = 120592

1

2

119910119862minus 1205922( 119910119874minus 119910119862)

minus 1205923(119910119874minus 119910119862) + 119861119891119889119891119889

+ 119891119894119910

(29)

Displacement relations between the centers of cage andrib are

119909119888= 119909119900+ 120576 cos120593

119910119888= 119910119900+ 120576 sin120593

(30)

Similarly for the velocity

119888= 119900+ 120576 cos120593 minus 120576 sin120593 sdot

119910119888= 119910119900+ 120576 sin120593 minus 120576 cos120593 sdot

(31)

Substituting (27) (30) and (31) into (29) integration ofequation can be obtained

120576 (1 minus 1205762

)3

+ 120576 [1205922(1 minus 120576

2

)3

minus 1205921015840

1

120587

2(1 + 2120576

2

) (1 minus 1205762

)12

] minus 1205922120576 sin 2120572 sdot (1

minus 1205762

)3

+ 1205923120576 (1 minus 120576

2

)3

minus 120583 cos 2120572 minus 12059210158401(120596119888119900minus 2)

sdot 1205762

(1 minus 1205762

) = 0

[120576 (1 minus 1205762

)2

+ 120576 (1 minus 1205762

)2

2120583 sin 2120572 sdot (1 minus 1205762)2

]

+ [2 120576 (1 minus 1205762

)2

+ 1205922120576 (1 minus 120576

2

)2

]

+ 1205921015840

1[(120596119888119900minus 2120572) 120587120576 (1 minus 120576

2

)12

+ 2120576 120576]

+ 21198771198881198971198881198750 = 0

(32)

Equation (32) can be solved by modified Newmark-120573method to obtain the displacement and velocity of the cage

4 Chaos Analyses

Chaos Theory is a new way to analyze complexity whichHenri Poincare studied as the possibility of forecastingA small variation in the initial conditions can generate adifferent set of resultssolutions [17]

A bearing typed as NU2310 is chosen to analyze chaoticresponse Tomoya and Kaoru [18] tested the dynamic cagedisplacements of this bearing and the maximum displace-ments can be used to verify the validity of theoretical analysis

Shock and Vibration 7

e

x

y

X

Y

OO

OD

OC

Rg

Rc

120576

(a) Cage-rib contact

Mc

MD

xc(t)Kx

Kxy

Ky K998400xyBy

Bx

yc(t)

(b) Cage-rib contact model

Figure 6 2-DOF model of cagerib contact

The structural specifications of the bearing are listed inTable 1

The cage of the bearing is made of machined brass DINCuZn40Pb2F43 (GB ZCuZn38Mn2Pb2) and the rings of thebearing are made of through-harden rolling bearing steelDIN 100Cr6 (SAE 52100GB GCr15) The cage is guided bythe outer-ring rib Both static load rating and dynamic loadrating are 186KN According to the recommendation of themanufacturer reference speed rating is 6700 rpm and limitedspeed rating is 12000 rpm Then the weight of the cage anddamping and stiffness parameters can be obtained accordingto the relative parameters in Table 1 The displacement in119909 direction is zero and the displacement in 119910 directionis determined by the cage guidance gap according to (1)Bifurcation diagram of 119883

119862versus spin speed increases with

different cage guide gaps cage eccentricities and dynamicviscosity exponent of lubrication oil The change curvesare described in Figure 7 respectively All three cases aresimulated and variation parameters of the cases are listed inTable 2 An interesting phenomenon that all the cases havetwo bifurcation zones and two steady zones is found in allcurves In relative low rotation speed transient responses canbe found The dynamic response of cage center goes intothe first ordered stage after the end of the resonance Thenthe first bifurcation response occurs The second orderedstatus arises after the first bifurcation ends and then thecurve goes into the second bifurcation Figure 7(a) comparesthe bifurcation diagrams of cage with different guidancegaps The first chaotic response zone of the bearing withsmaller guidance gap (case 1a) is from 120596

119888119900= 9100 rpm

to 120596119888119900= 12000 rpm and it is from 120596

119888119900= 7100 rpm to

120596119888119900= 22000 rpm for the bearing with larger guidance gap

(case 1b) The second chaotic response zone of the bearingwith smaller guidance gap is from 120596

119888119900= 25500 rpm and

it is from 120596119888119900

= 26000 rpm for the bearing with largerguidance gap Chaotic response zone is wider when the gap islarger Figure 7(b) shows bifurcation diagrams of cage with

different eccentricities The first chaotic zone of case 2a isfrom 120596

119888119900= 7600 rpm to 120596

119888119900= 14000 rpm and it is from

120596119888119900= 7800 rpm to 120596

119888119900= 13000 rpm of case 2b The second

chaotic zone of case 2a is from 120596119888119900= 22500 rpm and it is

from 120596119888119900= 19800 rpm of case 2b Chaotic response zone

increaseswith eccentricity In addition transient oscillation ismore drastic when comparing case 1a and case 2a Figure 7(c)depicts bifurcation diagrams of cage with different dynamicviscosity exponents The first chaotic zones of case 3a andcase 3b are from 120596

119888119900= 7650 rpm to 120596

119888119900= 13400 rpm The

second chaotic zones of case 3a and case 3b are from 120596119888119900=

21000 rpmThe change trend of case 3a curve is approachingto that of case 3b But the variation curve of cage with VG78is lower than that with VG56 due to higher dynamic viscosityexponent

The first bifurcation point 119884119862versus spin speed zooming

is shown in Figure 8 Fluctuation can be found even in steadyzone owing to the impact forces between roller and cagepockets In low-speed zone the wave frequency is low and itincreases with rotational speedThe difference of steady zonerange is shown more clearly in the enlarged view Bifurcationdiagrams of each case are different obviously and this denotesthat the position of119883

119862varies timely as well

Cage response behaviors of chaos which are subjected tothe cases in Table 2 at120596 = 8500RPM Poincare section pointsof 119884119862displacement in vertical direction and time history for

4th revolution to 4000th revolution are shown in Figure 9

5 Conclusion and Discussion

Rolling bearing life is typically calculated on the basis of itsload ratings relative to the applied loads and the requirementsregarding bearing life and reliability Both dynamic loadrating and rating life equations neglect cage design Actuallythe cage design has evolved to bear higher and higher load byincorporating more rollers and to lower costs by adopting

8 Shock and Vibration

2500 7500 12500 17500 22500 27500

Rotation speed (rpm)

Case 1b

Case 1a

03

02

01

0

minus01

minus02

minus03

Vert

ical

disp

lace

men

t of

cage

cent

er (m

m)

(a) Different cage guidance gaps

2500 7500 12500 17500 22500 27500

Rotation speed (rpm)

Case 2b

Case 2a

03

02

01

0

minus01

minus02

minus03

Vert

ical

disp

lace

men

t of

cage

cent

er (m

m)

(b) Different eccentricities

2500 7500 12500 17500 22500 27500

Rotation speed (rpm)

Case 3b

Case 3a

03

02

01

0

minus01

minus02

minus03

Vert

ical

disp

lace

men

t of

cage

cent

er (m

m)

(c) Different dynamic viscosity exponents

Figure 7 Bifurcation diagram points119883119862versus spin speed

Table 1 Specifications of the bearing structure and size

Bearing typeBoundary size Roller size Rib diameter Clearance

Boremm

Outsidemm

Widthmm Number Diameter

mmLengthmm mm Radial

120583mPocketmm

NU2310 Φ50 Φ110 40 13 Φ16 27 Φ921 40 03

Table 2 Case study

CaseVariation parameters

Guidance gap Eccentricity Dynamic viscosity exponent(mm) (mm) (Pasdots)

1a 045 23 VG561b 060 23 VG562a 045 35 VG562b 045 57 VG563a 045 42 VG783b 045 42 VG32

different material and improve high-speed performance byusing optimized geometry structure This paper pays closeattention to influence from cage on speed characteristics ofcylindrical roller bearings The most suitable cage guidancetype is confirmed by mathematical derivation firstly Twomajor factors originating from manufacturing process andone involved in applying process are considered to establishdynamical mathematical model of cage with appropriate cageguidance type Analysis of real-time dynamic behavior of

a cage on radial plane was carried out using chaos theorybased on the theoretical mathematical model The chaoticanalysis is limited by the initial values of the system Hencethe initial values are determined firstly in analytical processPreliminary conclusions are listed as follows

(1) There are two bifurcation zones of cage dynamiccurves and the bearing should be away from these twozones in practiceThe range of the 1st bifurcation zoneismainly influenced by the value of cage guidance gapand eccentricity especially the cage guidance gapThelubrication dynamic viscosity exponent has relativelylittle influence on cage dynamic

(2) Numerical simulations show that the roller tends toacceleratedecelerate in switching process of support-ing zoneloaded zone which leads to single or mul-tiple roller-cage bridge impacts Impact force causesperiodic small oscillation on the curves Oscillationamplitude is determined by the pocket gap

(3) Eccentricity of the cage causes dramatic transientresponses in low rotational speed zone The transientresponses are easily tending to cause knocking and

Shock and Vibration 9

7000 10000 13000 16000 19000 22000

Rotation speed (rpm)

Case 1b

Case 1a

002

001

0

minus001

minus002

minus003

minus004

minus005

minus006

minus007

minus008

Vert

ical

disp

lace

men

t of

cage

cent

er (m

m)

(a) Different cage guidance gaps

Rotation speed (rpm)

Case 2b

Case 2a

0

minus001

minus002

minus003

006

005

004

003

002

001

7500 8800 10100 11400 12700 14000

Vert

ical

disp

lace

men

t of

cage

cent

er (m

m)

(b) Different eccentricities

Case 3b

Case 3a

Rotation speed (rpm)7500 8800 10100 11400 12700 14000

0

minus001

minus002

minus003

minus004

005

004

003

002

001

Vert

ical

disp

lace

men

t of

cage

cent

er (m

m)

(c) Different dynamic viscosity exponents

Figure 8 The first bifurcation zone from Figure 7 in enlarged scale

faster wear of cage pocket in terms of worse lubrica-tion film forming The steady zone is narrower wheneccentricity increases In addition divergent trendcan be found in the second bifurcation zonewith lagereccentricity

(4) According to the zooming of the first bifurcationzone variations of any parameters cause totally dif-ferent bifurcations We can draw our conclusion thatmovement of cage is a chaotic motion

(5) The Poincare maps show that position and rotationalspeed of the occurrence bifurcation zones are verydifferent with varied parameters

(6) The change of fluid dynamic viscosity exponent willnot affect the region of bifurcation due to the weaknonlinear of fluid film force

Considering cage dynamics of high-speed cylindricalroller bearing the values of cage guidance gap and eccen-tricity are crucial factors From the point of view of themanufacturers the very small cage guidance gap is difficult toobtain owing to the material (brass) of the cage used in high-speed bearingThe authors strongly suggest that the designershould minimize the gap within the confine of processingcapacity As for the cage eccentricity dynamic balance ofcage is widely used in ultra-high-speed bearings Analysesfrom the paper show that cage eccentricity brings violent

oscillation in low-speed zone and oscillation amplitude inchaotic zone is higher with the increase of cage eccentricityHence dynamic balance is another important factor inmanufacturing process With respect to lubrication dynamicviscosity exponent this study indicates that influence fromdynamic viscosity exponent on cage dynamics is not signifi-cant as the prediction

As a matter of fact researches in the paper are merelyrestricted to cage dynamic response What is to be pointedout is that dynamic response of a cylindrical roller bearingis a complex and interrelated problem among componentsHence there are many imperfect points in the paper whichshould be noted here Preliminary discussions are listed asfollows

(1) The displacements of the cage discussed in the paperare only in 119909-119910 plane During the rotation of thebearing rollers will inevitably have a certain slip-page towards the rings due to insufficient tractionfriction outside the load-zone churning momentfrom lubricant the friction loss between roller-pocketcontact and roller-rib contacts in the actual runningprocess Rollerraceways contact has not been takeninto account in the paper for the sake of simplifyingthe calculation

(2) Damping coefficient and contact stiffness involvedin the calculation are obtained according to material

10 Shock and Vibration

minus03 minus02 minus01 0 01 02minus400

minus200

0

200

400

Vertical displacement of cage center (mm)

Vert

ical

velo

city

(mm

s)

(a) Smaller cage guidance

minus03 minus02 minus01 0 01 02Vertical displacement of cage center (mm)

minus400

minus200

0

200

400

Vert

ical

velo

city

(mm

s)

(b) Bigger cage guidance

minus02 minus01 0 01Vertical displacement of cage center (mm)

minus300

minus200

0

200

300

Vert

ical

velo

city

(mm

s)

(c) Smaller eccentricity

minus03 minus02 minus01 0 01 02Vertical displacement of cage center (mm)

minus400

minus200

0

200

400

Vert

ical

velo

city

(mm

s)

(d) Bigger eccentricity

minus03 minus02 minus01 0 01 02Vertical displacement of cage center (mm)

minus400

minus200

0

200

400

Vert

ical

velo

city

(mm

s)

(e) Higher dynamic viscosity exponent

minus03 minus02 minus01 0 01 02Vertical displacement of cage center (mm)

minus400

minus200

0

200

400

Vert

ical

velo

city

(mm

s)

(f) Lower dynamic viscosity exponent

Figure 9 Poincare section points of 119884119862displacement in vertical direction

property and geometry characteristic parametersActually the damping coefficient and contact stiff-ness can be described more accurately by immediatemovement and velocity Namely contact stiffness isthe first-order derivative of immediate movementand damping coefficient is the first-order derivative ofimmediate velocity For the same sake of simplifyingcalculation the accurate expressions are not substi-tuted into the equations

(3) The bearing under study has the outer ring fixed to arigid support and the inner ring fixed rigidly to theshaft Bearing house and the shaft are truly nonlinearDOF systems The influence of bearing house andshaft on cage dynamic is relatively small so it can besimplified in the calculation

(4) As illustrated in Section 3 cage is driven by the rollersin supporting zone and the rollers are driven by cage

Shock and Vibration 11

in nonloaded zone Driving forces on the total cageare balanced and they are ignored except the impactforces in switching zones of driven modes

Symbols

120572 Angle at cage center in rad Angular velocity in tangential direction in rads Angular acceleration in tangential direction in

rads2119861 Damping coefficient119862 Cage gap in mm119889 Diameter in mmsΔ Roller gap in mm120575 Deformation in mm119890 Cage eccentricity in mm120576 Displacement in normal direction in mm120576 Velocity in normal direction in mms120576 Acceleration in normal direction in mms2119865 Load in N119891 Load caused by motion in N119866 Clearance in mm120579 Angle in ∘119870 Hertzian stiffness119897 Length in mm119872 Mass in kg120583 Fluid dynamic viscosity in Pasdots119873 Number of the rollers120592 Dimensionless parameter of impact factor119874 Position of the center in mm Velocity vector of the center in mms Acceleration vector of the center in mms2119901 Hydrodynamic pressure in Pa1199010 Initial hydrodynamic pressure in Pa

119876 Hertz stress in Nmm2119877 Radius in mm120588 Dimensionless parameter of cage eccentricity to cage

gap119905 Time in sec119879 Pocket clearance in mmV Surface velocity in mms119883 Coordinate119883 direction distance in mm119909 Position in coordinate direction in mm Velocity in 119909 direction in mms Acceleration in 119909 direction in mms2119884 Coordinate 119884 direction distance in mm119910 Dimensionless parameter in coordinate direction119910 Velocity in 119910 direction in mms119910 Acceleration in 119910 direction in mms2120596 Rotation speed in rads

Subscripts

119886 Axial direction120572 Tangential direction119862 Cage center119888 Cagecen Centrifugal

119888119894 Relative relation between cage and innerring

119888119900 Relative relation between cage and outerring

119888119908 Relative relation between cage and roller119863 Unbalanced gravitational center119890 Normal direction119891119889 Added fluid

119892 Ribℎ Impact119894 Inner raceway119895 The 119895th roller119895cen Centrifugal force of the 119895th roller119898 Mean value119874 Rib center119900 Outer raceway119903 Radial direction119903119894 Inner raceway on radial direction119903119900 Outer raceway on radial direction119904 Polar coordinate119908 Roller119909 119909 direction119910 119910 direction

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The project is supported by National Natural Science Foun-dation of China (Grant no 51475144) and the Foundation ofInnovation and Research Team of Science and Technology inUniversities in Henan Province (Grant no 13IRTSTHN025)

References

[1] M Tiwari K Gupta and O Prakash ldquoDynamic response of anunbalanced rotor supported on ball bearingsrdquo Journal of Soundand Vibration vol 238 no 5 pp 757ndash779 2000

[2] M L Adam Jr Rotating Machinery VibrationmdashFrom Analysisto Trouble Shooting Marcel Dekker New York NY USA 2001

[3] N Akturk M Uneeb and R Gohar ldquoThe effects of number ofballs and preload on vibrations associated with ball bearingsrdquoJournal of Tribology vol 119 no 4 pp 747ndash753 1997

[4] S H Upadhyay S C Jain and S P Harsha ldquoNon-linearvibration signature analysis of a high-speed rotating shaft due toball size variations and varying number of ballsrdquo Proceedings ofthe Institution of Mechanical Engineers Part K Journal of Multi-body Dynamics vol 223 no 2 pp 83ndash105 2009

[5] N Akturk ldquoThe effect of waviness on vibrations associated withball bearingsrdquo Journal of Tribology vol 121 no 4 pp 667ndash6771999

[6] S P Harsha K Sandeep and R Prakash ldquoNon-linear dynamicbehaviors of rolling element bearings due to surface wavinessrdquoJournal of Sound and Vibration vol 272 no 3ndash5 pp 557ndash5802004

[7] J Sopanen and A Mikkola ldquoDynamic model of a deep-grooveball bearing including localized and distributed defects Part

12 Shock and Vibration

2 Implementation and resultsrdquo Proceedings of the Institution ofMechanical Engineers Part K vol 217 no 3 pp 213ndash223 2003

[8] S H Upadhyay S C Jain and S P Harsha ldquoChaotic dynamicsof high speed rotating shaft supported by ball bearings due todistributed defectsrdquo International Journal of Engineering Scienceand Technology vol 2 no 10 pp 5746ndash5794 2010

[9] S P Harsha ldquoNonlinear dynamic analysis of a high-speed rotorsupported by rolling element bearingsrdquo Journal of Sound andVibration vol 290 no 1-2 pp 65ndash100 2006

[10] C Villa J-J Sinou and F Thouverez ldquoStability and vibrationanalysis of a complex flexible rotor bearing systemrdquo Communi-cations in Nonlinear Science and Numerical Simulation vol 13no 4 pp 804ndash821 2008

[11] L Houpert ldquoCAGEDYN a contribution to roller bearingdynamic calculations Part III experimental validationrdquo ASMETribology Transactions vol 53 no 6 pp 848ndash859 2010

[12] S P Harsha ldquoNonlinear dynamic analysis of rolling elementbearings due to cage run-out and number of ballsrdquo Journal ofSound and Vibration vol 289 no 1-2 pp 360ndash381 2006

[13] I Bercea S Cretu M Bercea and D Olaru ldquoSimulating roller-cage pocket friction in a tapered roller bearingrdquo EuropeanJournal of Mechanical and Environmental Engineering vol 43no 4 pp 189ndash194 1998

[14] T Sakaguchi andKHarada ldquoDynamic analysis of cage behaviorin a tapered roller bearingrdquo Journal of Tribology vol 128 no 3pp 604ndash611 2006

[15] T A Harris Rolling Bearing Analysis John Wiley amp Sons 4thedition 2001

[16] C Spiridon ldquoMachined brass and pressed steel bearing cages acomparative studyrdquo RKB Technical Review vol 7 pp 1ndash13 2011

[17] R Riccardo ldquoChaos theory and some practical applications intechnical analysisrdquo in Proceedings of the 11th Annual Conferenceof the International Federation of Technical Analysts 1998

[18] S Tomoya and U Kaoru ldquoDynamic analysis of cage behavior ina cylindrical roller bearingrdquo NTN Technical Review 71 2004

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International Journal of

6 Shock and Vibration

When the linear superposition method is adopted tocontact forces of the cage in 119909-119910 plane the total force ondifferent directions can be given as

119891119909= 119891119890119909

119891119910= 119891119886119910+ 119891119890119910

(24)

32 Hydrodynamic Fluid-Film Force Cage-rib contact is afluid-structure interaction problem with bearings rotatingat a high speed The main supporting force generated bylubricant squeeze effects can be approximated to a short-width journal bearing theory as illustrated in Figure 3 Thendynamic-deviation interactive force between cage and rib canbe described as a continuous function of cage position andvelocity components Reynolds lubrication equation (RLE)provides the basis of lubrication theory [15] and the forcegiven on the cage by the film is expressed as

120597

120597120572[(1 + 120576 cos120572)3

120597119901

120597120572] + 1198772

119888

120597

120597119909[(1 + 120576 cos120572)2

120597119901

120597119910]

= minus6120583(119877119888

119862119903

)

2

[(120596119888119900minus 2) 120576 sin120572 minus 2120576 cos120572]

(25)

And

119901 (120572) = 1198750+120596119888119900

119862119903

3[1198972

119888

4minus 1199102

]120576 sin120572

1 + 120576cos3120572 (26)

Solutions to the RLE are a nonlinear function of displace-ment and angle at cage center Hydrodynamic fluid-film forcecan be written as follows

119891119890= 120583119877119888119897119888(119897119888

119862119903

)

2

[(120596119888119900minus 2)

1205762

(1 minus 1205762)2

+120587

2

(1 + 21205762

) 120576

(1 minus 1205762)52

]

119891120572= 120583119877119888119897119888(119877119888

119862119903

)

2

sdot [(120596119888119900minus 2120572)

120587120576

4 (1 minus 1205762)32

+2120576 120576

(1 minus 1205762)2]

+ 21198771198881198971198881199010

(27)

The fluid-film force can also be represented as two com-ponent forces in horizontal and vertical force equilibrium

119891119909= 119891119890sdot cos120572 + 119891

120572sdot sin120572

119891119910= 119891119890sdot sin120572 minus 119891

120572sdot cos120572

(28)

33 Equilibrium of the Equations In hydrodynamic fluid-film bearing the fluid supported pressure is generated nor-mally by motion of the cage and depends on the dynamic

viscosity exponent of the lubricating fluid For the cagestructure and coordination please refer to Figure 6 Thegenerated forces can be expressed as

119891119890sdot cos120572 + 119891

120572sdot sin120572 = 120592

1

2

119862minus 1205922(119874minus 119862)

minus 1205923(119909119874minus 119909119862)

119891119890sdot sin120572 minus 119891

120572sdot cos120572 = 120592

1

2

119910119862minus 1205922( 119910119874minus 119910119862)

minus 1205923(119910119874minus 119910119862) + 119861119891119889119891119889

+ 119891119894119910

(29)

Displacement relations between the centers of cage andrib are

119909119888= 119909119900+ 120576 cos120593

119910119888= 119910119900+ 120576 sin120593

(30)

Similarly for the velocity

119888= 119900+ 120576 cos120593 minus 120576 sin120593 sdot

119910119888= 119910119900+ 120576 sin120593 minus 120576 cos120593 sdot

(31)

Substituting (27) (30) and (31) into (29) integration ofequation can be obtained

120576 (1 minus 1205762

)3

+ 120576 [1205922(1 minus 120576

2

)3

minus 1205921015840

1

120587

2(1 + 2120576

2

) (1 minus 1205762

)12

] minus 1205922120576 sin 2120572 sdot (1

minus 1205762

)3

+ 1205923120576 (1 minus 120576

2

)3

minus 120583 cos 2120572 minus 12059210158401(120596119888119900minus 2)

sdot 1205762

(1 minus 1205762

) = 0

[120576 (1 minus 1205762

)2

+ 120576 (1 minus 1205762

)2

2120583 sin 2120572 sdot (1 minus 1205762)2

]

+ [2 120576 (1 minus 1205762

)2

+ 1205922120576 (1 minus 120576

2

)2

]

+ 1205921015840

1[(120596119888119900minus 2120572) 120587120576 (1 minus 120576

2

)12

+ 2120576 120576]

+ 21198771198881198971198881198750 = 0

(32)

Equation (32) can be solved by modified Newmark-120573method to obtain the displacement and velocity of the cage

4 Chaos Analyses

Chaos Theory is a new way to analyze complexity whichHenri Poincare studied as the possibility of forecastingA small variation in the initial conditions can generate adifferent set of resultssolutions [17]

A bearing typed as NU2310 is chosen to analyze chaoticresponse Tomoya and Kaoru [18] tested the dynamic cagedisplacements of this bearing and the maximum displace-ments can be used to verify the validity of theoretical analysis

Shock and Vibration 7

e

x

y

X

Y

OO

OD

OC

Rg

Rc

120576

(a) Cage-rib contact

Mc

MD

xc(t)Kx

Kxy

Ky K998400xyBy

Bx

yc(t)

(b) Cage-rib contact model

Figure 6 2-DOF model of cagerib contact

The structural specifications of the bearing are listed inTable 1

The cage of the bearing is made of machined brass DINCuZn40Pb2F43 (GB ZCuZn38Mn2Pb2) and the rings of thebearing are made of through-harden rolling bearing steelDIN 100Cr6 (SAE 52100GB GCr15) The cage is guided bythe outer-ring rib Both static load rating and dynamic loadrating are 186KN According to the recommendation of themanufacturer reference speed rating is 6700 rpm and limitedspeed rating is 12000 rpm Then the weight of the cage anddamping and stiffness parameters can be obtained accordingto the relative parameters in Table 1 The displacement in119909 direction is zero and the displacement in 119910 directionis determined by the cage guidance gap according to (1)Bifurcation diagram of 119883

119862versus spin speed increases with

different cage guide gaps cage eccentricities and dynamicviscosity exponent of lubrication oil The change curvesare described in Figure 7 respectively All three cases aresimulated and variation parameters of the cases are listed inTable 2 An interesting phenomenon that all the cases havetwo bifurcation zones and two steady zones is found in allcurves In relative low rotation speed transient responses canbe found The dynamic response of cage center goes intothe first ordered stage after the end of the resonance Thenthe first bifurcation response occurs The second orderedstatus arises after the first bifurcation ends and then thecurve goes into the second bifurcation Figure 7(a) comparesthe bifurcation diagrams of cage with different guidancegaps The first chaotic response zone of the bearing withsmaller guidance gap (case 1a) is from 120596

119888119900= 9100 rpm

to 120596119888119900= 12000 rpm and it is from 120596

119888119900= 7100 rpm to

120596119888119900= 22000 rpm for the bearing with larger guidance gap

(case 1b) The second chaotic response zone of the bearingwith smaller guidance gap is from 120596

119888119900= 25500 rpm and

it is from 120596119888119900

= 26000 rpm for the bearing with largerguidance gap Chaotic response zone is wider when the gap islarger Figure 7(b) shows bifurcation diagrams of cage with

different eccentricities The first chaotic zone of case 2a isfrom 120596

119888119900= 7600 rpm to 120596

119888119900= 14000 rpm and it is from

120596119888119900= 7800 rpm to 120596

119888119900= 13000 rpm of case 2b The second

chaotic zone of case 2a is from 120596119888119900= 22500 rpm and it is

from 120596119888119900= 19800 rpm of case 2b Chaotic response zone

increaseswith eccentricity In addition transient oscillation ismore drastic when comparing case 1a and case 2a Figure 7(c)depicts bifurcation diagrams of cage with different dynamicviscosity exponents The first chaotic zones of case 3a andcase 3b are from 120596

119888119900= 7650 rpm to 120596

119888119900= 13400 rpm The

second chaotic zones of case 3a and case 3b are from 120596119888119900=

21000 rpmThe change trend of case 3a curve is approachingto that of case 3b But the variation curve of cage with VG78is lower than that with VG56 due to higher dynamic viscosityexponent

The first bifurcation point 119884119862versus spin speed zooming

is shown in Figure 8 Fluctuation can be found even in steadyzone owing to the impact forces between roller and cagepockets In low-speed zone the wave frequency is low and itincreases with rotational speedThe difference of steady zonerange is shown more clearly in the enlarged view Bifurcationdiagrams of each case are different obviously and this denotesthat the position of119883

119862varies timely as well

Cage response behaviors of chaos which are subjected tothe cases in Table 2 at120596 = 8500RPM Poincare section pointsof 119884119862displacement in vertical direction and time history for

4th revolution to 4000th revolution are shown in Figure 9

5 Conclusion and Discussion

Rolling bearing life is typically calculated on the basis of itsload ratings relative to the applied loads and the requirementsregarding bearing life and reliability Both dynamic loadrating and rating life equations neglect cage design Actuallythe cage design has evolved to bear higher and higher load byincorporating more rollers and to lower costs by adopting

8 Shock and Vibration

2500 7500 12500 17500 22500 27500

Rotation speed (rpm)

Case 1b

Case 1a

03

02

01

0

minus01

minus02

minus03

Vert

ical

disp

lace

men

t of

cage

cent

er (m

m)

(a) Different cage guidance gaps

2500 7500 12500 17500 22500 27500

Rotation speed (rpm)

Case 2b

Case 2a

03

02

01

0

minus01

minus02

minus03

Vert

ical

disp

lace

men

t of

cage

cent

er (m

m)

(b) Different eccentricities

2500 7500 12500 17500 22500 27500

Rotation speed (rpm)

Case 3b

Case 3a

03

02

01

0

minus01

minus02

minus03

Vert

ical

disp

lace

men

t of

cage

cent

er (m

m)

(c) Different dynamic viscosity exponents

Figure 7 Bifurcation diagram points119883119862versus spin speed

Table 1 Specifications of the bearing structure and size

Bearing typeBoundary size Roller size Rib diameter Clearance

Boremm

Outsidemm

Widthmm Number Diameter

mmLengthmm mm Radial

120583mPocketmm

NU2310 Φ50 Φ110 40 13 Φ16 27 Φ921 40 03

Table 2 Case study

CaseVariation parameters

Guidance gap Eccentricity Dynamic viscosity exponent(mm) (mm) (Pasdots)

1a 045 23 VG561b 060 23 VG562a 045 35 VG562b 045 57 VG563a 045 42 VG783b 045 42 VG32

different material and improve high-speed performance byusing optimized geometry structure This paper pays closeattention to influence from cage on speed characteristics ofcylindrical roller bearings The most suitable cage guidancetype is confirmed by mathematical derivation firstly Twomajor factors originating from manufacturing process andone involved in applying process are considered to establishdynamical mathematical model of cage with appropriate cageguidance type Analysis of real-time dynamic behavior of

a cage on radial plane was carried out using chaos theorybased on the theoretical mathematical model The chaoticanalysis is limited by the initial values of the system Hencethe initial values are determined firstly in analytical processPreliminary conclusions are listed as follows

(1) There are two bifurcation zones of cage dynamiccurves and the bearing should be away from these twozones in practiceThe range of the 1st bifurcation zoneismainly influenced by the value of cage guidance gapand eccentricity especially the cage guidance gapThelubrication dynamic viscosity exponent has relativelylittle influence on cage dynamic

(2) Numerical simulations show that the roller tends toacceleratedecelerate in switching process of support-ing zoneloaded zone which leads to single or mul-tiple roller-cage bridge impacts Impact force causesperiodic small oscillation on the curves Oscillationamplitude is determined by the pocket gap

(3) Eccentricity of the cage causes dramatic transientresponses in low rotational speed zone The transientresponses are easily tending to cause knocking and

Shock and Vibration 9

7000 10000 13000 16000 19000 22000

Rotation speed (rpm)

Case 1b

Case 1a

002

001

0

minus001

minus002

minus003

minus004

minus005

minus006

minus007

minus008

Vert

ical

disp

lace

men

t of

cage

cent

er (m

m)

(a) Different cage guidance gaps

Rotation speed (rpm)

Case 2b

Case 2a

0

minus001

minus002

minus003

006

005

004

003

002

001

7500 8800 10100 11400 12700 14000

Vert

ical

disp

lace

men

t of

cage

cent

er (m

m)

(b) Different eccentricities

Case 3b

Case 3a

Rotation speed (rpm)7500 8800 10100 11400 12700 14000

0

minus001

minus002

minus003

minus004

005

004

003

002

001

Vert

ical

disp

lace

men

t of

cage

cent

er (m

m)

(c) Different dynamic viscosity exponents

Figure 8 The first bifurcation zone from Figure 7 in enlarged scale

faster wear of cage pocket in terms of worse lubrica-tion film forming The steady zone is narrower wheneccentricity increases In addition divergent trendcan be found in the second bifurcation zonewith lagereccentricity

(4) According to the zooming of the first bifurcationzone variations of any parameters cause totally dif-ferent bifurcations We can draw our conclusion thatmovement of cage is a chaotic motion

(5) The Poincare maps show that position and rotationalspeed of the occurrence bifurcation zones are verydifferent with varied parameters

(6) The change of fluid dynamic viscosity exponent willnot affect the region of bifurcation due to the weaknonlinear of fluid film force

Considering cage dynamics of high-speed cylindricalroller bearing the values of cage guidance gap and eccen-tricity are crucial factors From the point of view of themanufacturers the very small cage guidance gap is difficult toobtain owing to the material (brass) of the cage used in high-speed bearingThe authors strongly suggest that the designershould minimize the gap within the confine of processingcapacity As for the cage eccentricity dynamic balance ofcage is widely used in ultra-high-speed bearings Analysesfrom the paper show that cage eccentricity brings violent

oscillation in low-speed zone and oscillation amplitude inchaotic zone is higher with the increase of cage eccentricityHence dynamic balance is another important factor inmanufacturing process With respect to lubrication dynamicviscosity exponent this study indicates that influence fromdynamic viscosity exponent on cage dynamics is not signifi-cant as the prediction

As a matter of fact researches in the paper are merelyrestricted to cage dynamic response What is to be pointedout is that dynamic response of a cylindrical roller bearingis a complex and interrelated problem among componentsHence there are many imperfect points in the paper whichshould be noted here Preliminary discussions are listed asfollows

(1) The displacements of the cage discussed in the paperare only in 119909-119910 plane During the rotation of thebearing rollers will inevitably have a certain slip-page towards the rings due to insufficient tractionfriction outside the load-zone churning momentfrom lubricant the friction loss between roller-pocketcontact and roller-rib contacts in the actual runningprocess Rollerraceways contact has not been takeninto account in the paper for the sake of simplifyingthe calculation

(2) Damping coefficient and contact stiffness involvedin the calculation are obtained according to material

10 Shock and Vibration

minus03 minus02 minus01 0 01 02minus400

minus200

0

200

400

Vertical displacement of cage center (mm)

Vert

ical

velo

city

(mm

s)

(a) Smaller cage guidance

minus03 minus02 minus01 0 01 02Vertical displacement of cage center (mm)

minus400

minus200

0

200

400

Vert

ical

velo

city

(mm

s)

(b) Bigger cage guidance

minus02 minus01 0 01Vertical displacement of cage center (mm)

minus300

minus200

0

200

300

Vert

ical

velo

city

(mm

s)

(c) Smaller eccentricity

minus03 minus02 minus01 0 01 02Vertical displacement of cage center (mm)

minus400

minus200

0

200

400

Vert

ical

velo

city

(mm

s)

(d) Bigger eccentricity

minus03 minus02 minus01 0 01 02Vertical displacement of cage center (mm)

minus400

minus200

0

200

400

Vert

ical

velo

city

(mm

s)

(e) Higher dynamic viscosity exponent

minus03 minus02 minus01 0 01 02Vertical displacement of cage center (mm)

minus400

minus200

0

200

400

Vert

ical

velo

city

(mm

s)

(f) Lower dynamic viscosity exponent

Figure 9 Poincare section points of 119884119862displacement in vertical direction

property and geometry characteristic parametersActually the damping coefficient and contact stiff-ness can be described more accurately by immediatemovement and velocity Namely contact stiffness isthe first-order derivative of immediate movementand damping coefficient is the first-order derivative ofimmediate velocity For the same sake of simplifyingcalculation the accurate expressions are not substi-tuted into the equations

(3) The bearing under study has the outer ring fixed to arigid support and the inner ring fixed rigidly to theshaft Bearing house and the shaft are truly nonlinearDOF systems The influence of bearing house andshaft on cage dynamic is relatively small so it can besimplified in the calculation

(4) As illustrated in Section 3 cage is driven by the rollersin supporting zone and the rollers are driven by cage

Shock and Vibration 11

in nonloaded zone Driving forces on the total cageare balanced and they are ignored except the impactforces in switching zones of driven modes

Symbols

120572 Angle at cage center in rad Angular velocity in tangential direction in rads Angular acceleration in tangential direction in

rads2119861 Damping coefficient119862 Cage gap in mm119889 Diameter in mmsΔ Roller gap in mm120575 Deformation in mm119890 Cage eccentricity in mm120576 Displacement in normal direction in mm120576 Velocity in normal direction in mms120576 Acceleration in normal direction in mms2119865 Load in N119891 Load caused by motion in N119866 Clearance in mm120579 Angle in ∘119870 Hertzian stiffness119897 Length in mm119872 Mass in kg120583 Fluid dynamic viscosity in Pasdots119873 Number of the rollers120592 Dimensionless parameter of impact factor119874 Position of the center in mm Velocity vector of the center in mms Acceleration vector of the center in mms2119901 Hydrodynamic pressure in Pa1199010 Initial hydrodynamic pressure in Pa

119876 Hertz stress in Nmm2119877 Radius in mm120588 Dimensionless parameter of cage eccentricity to cage

gap119905 Time in sec119879 Pocket clearance in mmV Surface velocity in mms119883 Coordinate119883 direction distance in mm119909 Position in coordinate direction in mm Velocity in 119909 direction in mms Acceleration in 119909 direction in mms2119884 Coordinate 119884 direction distance in mm119910 Dimensionless parameter in coordinate direction119910 Velocity in 119910 direction in mms119910 Acceleration in 119910 direction in mms2120596 Rotation speed in rads

Subscripts

119886 Axial direction120572 Tangential direction119862 Cage center119888 Cagecen Centrifugal

119888119894 Relative relation between cage and innerring

119888119900 Relative relation between cage and outerring

119888119908 Relative relation between cage and roller119863 Unbalanced gravitational center119890 Normal direction119891119889 Added fluid

119892 Ribℎ Impact119894 Inner raceway119895 The 119895th roller119895cen Centrifugal force of the 119895th roller119898 Mean value119874 Rib center119900 Outer raceway119903 Radial direction119903119894 Inner raceway on radial direction119903119900 Outer raceway on radial direction119904 Polar coordinate119908 Roller119909 119909 direction119910 119910 direction

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The project is supported by National Natural Science Foun-dation of China (Grant no 51475144) and the Foundation ofInnovation and Research Team of Science and Technology inUniversities in Henan Province (Grant no 13IRTSTHN025)

References

[1] M Tiwari K Gupta and O Prakash ldquoDynamic response of anunbalanced rotor supported on ball bearingsrdquo Journal of Soundand Vibration vol 238 no 5 pp 757ndash779 2000

[2] M L Adam Jr Rotating Machinery VibrationmdashFrom Analysisto Trouble Shooting Marcel Dekker New York NY USA 2001

[3] N Akturk M Uneeb and R Gohar ldquoThe effects of number ofballs and preload on vibrations associated with ball bearingsrdquoJournal of Tribology vol 119 no 4 pp 747ndash753 1997

[4] S H Upadhyay S C Jain and S P Harsha ldquoNon-linearvibration signature analysis of a high-speed rotating shaft due toball size variations and varying number of ballsrdquo Proceedings ofthe Institution of Mechanical Engineers Part K Journal of Multi-body Dynamics vol 223 no 2 pp 83ndash105 2009

[5] N Akturk ldquoThe effect of waviness on vibrations associated withball bearingsrdquo Journal of Tribology vol 121 no 4 pp 667ndash6771999

[6] S P Harsha K Sandeep and R Prakash ldquoNon-linear dynamicbehaviors of rolling element bearings due to surface wavinessrdquoJournal of Sound and Vibration vol 272 no 3ndash5 pp 557ndash5802004

[7] J Sopanen and A Mikkola ldquoDynamic model of a deep-grooveball bearing including localized and distributed defects Part

12 Shock and Vibration

2 Implementation and resultsrdquo Proceedings of the Institution ofMechanical Engineers Part K vol 217 no 3 pp 213ndash223 2003

[8] S H Upadhyay S C Jain and S P Harsha ldquoChaotic dynamicsof high speed rotating shaft supported by ball bearings due todistributed defectsrdquo International Journal of Engineering Scienceand Technology vol 2 no 10 pp 5746ndash5794 2010

[9] S P Harsha ldquoNonlinear dynamic analysis of a high-speed rotorsupported by rolling element bearingsrdquo Journal of Sound andVibration vol 290 no 1-2 pp 65ndash100 2006

[10] C Villa J-J Sinou and F Thouverez ldquoStability and vibrationanalysis of a complex flexible rotor bearing systemrdquo Communi-cations in Nonlinear Science and Numerical Simulation vol 13no 4 pp 804ndash821 2008

[11] L Houpert ldquoCAGEDYN a contribution to roller bearingdynamic calculations Part III experimental validationrdquo ASMETribology Transactions vol 53 no 6 pp 848ndash859 2010

[12] S P Harsha ldquoNonlinear dynamic analysis of rolling elementbearings due to cage run-out and number of ballsrdquo Journal ofSound and Vibration vol 289 no 1-2 pp 360ndash381 2006

[13] I Bercea S Cretu M Bercea and D Olaru ldquoSimulating roller-cage pocket friction in a tapered roller bearingrdquo EuropeanJournal of Mechanical and Environmental Engineering vol 43no 4 pp 189ndash194 1998

[14] T Sakaguchi andKHarada ldquoDynamic analysis of cage behaviorin a tapered roller bearingrdquo Journal of Tribology vol 128 no 3pp 604ndash611 2006

[15] T A Harris Rolling Bearing Analysis John Wiley amp Sons 4thedition 2001

[16] C Spiridon ldquoMachined brass and pressed steel bearing cages acomparative studyrdquo RKB Technical Review vol 7 pp 1ndash13 2011

[17] R Riccardo ldquoChaos theory and some practical applications intechnical analysisrdquo in Proceedings of the 11th Annual Conferenceof the International Federation of Technical Analysts 1998

[18] S Tomoya and U Kaoru ldquoDynamic analysis of cage behavior ina cylindrical roller bearingrdquo NTN Technical Review 71 2004

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Shock and Vibration 7

e

x

y

X

Y

OO

OD

OC

Rg

Rc

120576

(a) Cage-rib contact

Mc

MD

xc(t)Kx

Kxy

Ky K998400xyBy

Bx

yc(t)

(b) Cage-rib contact model

Figure 6 2-DOF model of cagerib contact

The structural specifications of the bearing are listed inTable 1

The cage of the bearing is made of machined brass DINCuZn40Pb2F43 (GB ZCuZn38Mn2Pb2) and the rings of thebearing are made of through-harden rolling bearing steelDIN 100Cr6 (SAE 52100GB GCr15) The cage is guided bythe outer-ring rib Both static load rating and dynamic loadrating are 186KN According to the recommendation of themanufacturer reference speed rating is 6700 rpm and limitedspeed rating is 12000 rpm Then the weight of the cage anddamping and stiffness parameters can be obtained accordingto the relative parameters in Table 1 The displacement in119909 direction is zero and the displacement in 119910 directionis determined by the cage guidance gap according to (1)Bifurcation diagram of 119883

119862versus spin speed increases with

different cage guide gaps cage eccentricities and dynamicviscosity exponent of lubrication oil The change curvesare described in Figure 7 respectively All three cases aresimulated and variation parameters of the cases are listed inTable 2 An interesting phenomenon that all the cases havetwo bifurcation zones and two steady zones is found in allcurves In relative low rotation speed transient responses canbe found The dynamic response of cage center goes intothe first ordered stage after the end of the resonance Thenthe first bifurcation response occurs The second orderedstatus arises after the first bifurcation ends and then thecurve goes into the second bifurcation Figure 7(a) comparesthe bifurcation diagrams of cage with different guidancegaps The first chaotic response zone of the bearing withsmaller guidance gap (case 1a) is from 120596

119888119900= 9100 rpm

to 120596119888119900= 12000 rpm and it is from 120596

119888119900= 7100 rpm to

120596119888119900= 22000 rpm for the bearing with larger guidance gap

(case 1b) The second chaotic response zone of the bearingwith smaller guidance gap is from 120596

119888119900= 25500 rpm and

it is from 120596119888119900

= 26000 rpm for the bearing with largerguidance gap Chaotic response zone is wider when the gap islarger Figure 7(b) shows bifurcation diagrams of cage with

different eccentricities The first chaotic zone of case 2a isfrom 120596

119888119900= 7600 rpm to 120596

119888119900= 14000 rpm and it is from

120596119888119900= 7800 rpm to 120596

119888119900= 13000 rpm of case 2b The second

chaotic zone of case 2a is from 120596119888119900= 22500 rpm and it is

from 120596119888119900= 19800 rpm of case 2b Chaotic response zone

increaseswith eccentricity In addition transient oscillation ismore drastic when comparing case 1a and case 2a Figure 7(c)depicts bifurcation diagrams of cage with different dynamicviscosity exponents The first chaotic zones of case 3a andcase 3b are from 120596

119888119900= 7650 rpm to 120596

119888119900= 13400 rpm The

second chaotic zones of case 3a and case 3b are from 120596119888119900=

21000 rpmThe change trend of case 3a curve is approachingto that of case 3b But the variation curve of cage with VG78is lower than that with VG56 due to higher dynamic viscosityexponent

The first bifurcation point 119884119862versus spin speed zooming

is shown in Figure 8 Fluctuation can be found even in steadyzone owing to the impact forces between roller and cagepockets In low-speed zone the wave frequency is low and itincreases with rotational speedThe difference of steady zonerange is shown more clearly in the enlarged view Bifurcationdiagrams of each case are different obviously and this denotesthat the position of119883

119862varies timely as well

Cage response behaviors of chaos which are subjected tothe cases in Table 2 at120596 = 8500RPM Poincare section pointsof 119884119862displacement in vertical direction and time history for

4th revolution to 4000th revolution are shown in Figure 9

5 Conclusion and Discussion

Rolling bearing life is typically calculated on the basis of itsload ratings relative to the applied loads and the requirementsregarding bearing life and reliability Both dynamic loadrating and rating life equations neglect cage design Actuallythe cage design has evolved to bear higher and higher load byincorporating more rollers and to lower costs by adopting

8 Shock and Vibration

2500 7500 12500 17500 22500 27500

Rotation speed (rpm)

Case 1b

Case 1a

03

02

01

0

minus01

minus02

minus03

Vert

ical

disp

lace

men

t of

cage

cent

er (m

m)

(a) Different cage guidance gaps

2500 7500 12500 17500 22500 27500

Rotation speed (rpm)

Case 2b

Case 2a

03

02

01

0

minus01

minus02

minus03

Vert

ical

disp

lace

men

t of

cage

cent

er (m

m)

(b) Different eccentricities

2500 7500 12500 17500 22500 27500

Rotation speed (rpm)

Case 3b

Case 3a

03

02

01

0

minus01

minus02

minus03

Vert

ical

disp

lace

men

t of

cage

cent

er (m

m)

(c) Different dynamic viscosity exponents

Figure 7 Bifurcation diagram points119883119862versus spin speed

Table 1 Specifications of the bearing structure and size

Bearing typeBoundary size Roller size Rib diameter Clearance

Boremm

Outsidemm

Widthmm Number Diameter

mmLengthmm mm Radial

120583mPocketmm

NU2310 Φ50 Φ110 40 13 Φ16 27 Φ921 40 03

Table 2 Case study

CaseVariation parameters

Guidance gap Eccentricity Dynamic viscosity exponent(mm) (mm) (Pasdots)

1a 045 23 VG561b 060 23 VG562a 045 35 VG562b 045 57 VG563a 045 42 VG783b 045 42 VG32

different material and improve high-speed performance byusing optimized geometry structure This paper pays closeattention to influence from cage on speed characteristics ofcylindrical roller bearings The most suitable cage guidancetype is confirmed by mathematical derivation firstly Twomajor factors originating from manufacturing process andone involved in applying process are considered to establishdynamical mathematical model of cage with appropriate cageguidance type Analysis of real-time dynamic behavior of

a cage on radial plane was carried out using chaos theorybased on the theoretical mathematical model The chaoticanalysis is limited by the initial values of the system Hencethe initial values are determined firstly in analytical processPreliminary conclusions are listed as follows

(1) There are two bifurcation zones of cage dynamiccurves and the bearing should be away from these twozones in practiceThe range of the 1st bifurcation zoneismainly influenced by the value of cage guidance gapand eccentricity especially the cage guidance gapThelubrication dynamic viscosity exponent has relativelylittle influence on cage dynamic

(2) Numerical simulations show that the roller tends toacceleratedecelerate in switching process of support-ing zoneloaded zone which leads to single or mul-tiple roller-cage bridge impacts Impact force causesperiodic small oscillation on the curves Oscillationamplitude is determined by the pocket gap

(3) Eccentricity of the cage causes dramatic transientresponses in low rotational speed zone The transientresponses are easily tending to cause knocking and

Shock and Vibration 9

7000 10000 13000 16000 19000 22000

Rotation speed (rpm)

Case 1b

Case 1a

002

001

0

minus001

minus002

minus003

minus004

minus005

minus006

minus007

minus008

Vert

ical

disp

lace

men

t of

cage

cent

er (m

m)

(a) Different cage guidance gaps

Rotation speed (rpm)

Case 2b

Case 2a

0

minus001

minus002

minus003

006

005

004

003

002

001

7500 8800 10100 11400 12700 14000

Vert

ical

disp

lace

men

t of

cage

cent

er (m

m)

(b) Different eccentricities

Case 3b

Case 3a

Rotation speed (rpm)7500 8800 10100 11400 12700 14000

0

minus001

minus002

minus003

minus004

005

004

003

002

001

Vert

ical

disp

lace

men

t of

cage

cent

er (m

m)

(c) Different dynamic viscosity exponents

Figure 8 The first bifurcation zone from Figure 7 in enlarged scale

faster wear of cage pocket in terms of worse lubrica-tion film forming The steady zone is narrower wheneccentricity increases In addition divergent trendcan be found in the second bifurcation zonewith lagereccentricity

(4) According to the zooming of the first bifurcationzone variations of any parameters cause totally dif-ferent bifurcations We can draw our conclusion thatmovement of cage is a chaotic motion

(5) The Poincare maps show that position and rotationalspeed of the occurrence bifurcation zones are verydifferent with varied parameters

(6) The change of fluid dynamic viscosity exponent willnot affect the region of bifurcation due to the weaknonlinear of fluid film force

Considering cage dynamics of high-speed cylindricalroller bearing the values of cage guidance gap and eccen-tricity are crucial factors From the point of view of themanufacturers the very small cage guidance gap is difficult toobtain owing to the material (brass) of the cage used in high-speed bearingThe authors strongly suggest that the designershould minimize the gap within the confine of processingcapacity As for the cage eccentricity dynamic balance ofcage is widely used in ultra-high-speed bearings Analysesfrom the paper show that cage eccentricity brings violent

oscillation in low-speed zone and oscillation amplitude inchaotic zone is higher with the increase of cage eccentricityHence dynamic balance is another important factor inmanufacturing process With respect to lubrication dynamicviscosity exponent this study indicates that influence fromdynamic viscosity exponent on cage dynamics is not signifi-cant as the prediction

As a matter of fact researches in the paper are merelyrestricted to cage dynamic response What is to be pointedout is that dynamic response of a cylindrical roller bearingis a complex and interrelated problem among componentsHence there are many imperfect points in the paper whichshould be noted here Preliminary discussions are listed asfollows

(1) The displacements of the cage discussed in the paperare only in 119909-119910 plane During the rotation of thebearing rollers will inevitably have a certain slip-page towards the rings due to insufficient tractionfriction outside the load-zone churning momentfrom lubricant the friction loss between roller-pocketcontact and roller-rib contacts in the actual runningprocess Rollerraceways contact has not been takeninto account in the paper for the sake of simplifyingthe calculation

(2) Damping coefficient and contact stiffness involvedin the calculation are obtained according to material

10 Shock and Vibration

minus03 minus02 minus01 0 01 02minus400

minus200

0

200

400

Vertical displacement of cage center (mm)

Vert

ical

velo

city

(mm

s)

(a) Smaller cage guidance

minus03 minus02 minus01 0 01 02Vertical displacement of cage center (mm)

minus400

minus200

0

200

400

Vert

ical

velo

city

(mm

s)

(b) Bigger cage guidance

minus02 minus01 0 01Vertical displacement of cage center (mm)

minus300

minus200

0

200

300

Vert

ical

velo

city

(mm

s)

(c) Smaller eccentricity

minus03 minus02 minus01 0 01 02Vertical displacement of cage center (mm)

minus400

minus200

0

200

400

Vert

ical

velo

city

(mm

s)

(d) Bigger eccentricity

minus03 minus02 minus01 0 01 02Vertical displacement of cage center (mm)

minus400

minus200

0

200

400

Vert

ical

velo

city

(mm

s)

(e) Higher dynamic viscosity exponent

minus03 minus02 minus01 0 01 02Vertical displacement of cage center (mm)

minus400

minus200

0

200

400

Vert

ical

velo

city

(mm

s)

(f) Lower dynamic viscosity exponent

Figure 9 Poincare section points of 119884119862displacement in vertical direction

property and geometry characteristic parametersActually the damping coefficient and contact stiff-ness can be described more accurately by immediatemovement and velocity Namely contact stiffness isthe first-order derivative of immediate movementand damping coefficient is the first-order derivative ofimmediate velocity For the same sake of simplifyingcalculation the accurate expressions are not substi-tuted into the equations

(3) The bearing under study has the outer ring fixed to arigid support and the inner ring fixed rigidly to theshaft Bearing house and the shaft are truly nonlinearDOF systems The influence of bearing house andshaft on cage dynamic is relatively small so it can besimplified in the calculation

(4) As illustrated in Section 3 cage is driven by the rollersin supporting zone and the rollers are driven by cage

Shock and Vibration 11

in nonloaded zone Driving forces on the total cageare balanced and they are ignored except the impactforces in switching zones of driven modes

Symbols

120572 Angle at cage center in rad Angular velocity in tangential direction in rads Angular acceleration in tangential direction in

rads2119861 Damping coefficient119862 Cage gap in mm119889 Diameter in mmsΔ Roller gap in mm120575 Deformation in mm119890 Cage eccentricity in mm120576 Displacement in normal direction in mm120576 Velocity in normal direction in mms120576 Acceleration in normal direction in mms2119865 Load in N119891 Load caused by motion in N119866 Clearance in mm120579 Angle in ∘119870 Hertzian stiffness119897 Length in mm119872 Mass in kg120583 Fluid dynamic viscosity in Pasdots119873 Number of the rollers120592 Dimensionless parameter of impact factor119874 Position of the center in mm Velocity vector of the center in mms Acceleration vector of the center in mms2119901 Hydrodynamic pressure in Pa1199010 Initial hydrodynamic pressure in Pa

119876 Hertz stress in Nmm2119877 Radius in mm120588 Dimensionless parameter of cage eccentricity to cage

gap119905 Time in sec119879 Pocket clearance in mmV Surface velocity in mms119883 Coordinate119883 direction distance in mm119909 Position in coordinate direction in mm Velocity in 119909 direction in mms Acceleration in 119909 direction in mms2119884 Coordinate 119884 direction distance in mm119910 Dimensionless parameter in coordinate direction119910 Velocity in 119910 direction in mms119910 Acceleration in 119910 direction in mms2120596 Rotation speed in rads

Subscripts

119886 Axial direction120572 Tangential direction119862 Cage center119888 Cagecen Centrifugal

119888119894 Relative relation between cage and innerring

119888119900 Relative relation between cage and outerring

119888119908 Relative relation between cage and roller119863 Unbalanced gravitational center119890 Normal direction119891119889 Added fluid

119892 Ribℎ Impact119894 Inner raceway119895 The 119895th roller119895cen Centrifugal force of the 119895th roller119898 Mean value119874 Rib center119900 Outer raceway119903 Radial direction119903119894 Inner raceway on radial direction119903119900 Outer raceway on radial direction119904 Polar coordinate119908 Roller119909 119909 direction119910 119910 direction

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The project is supported by National Natural Science Foun-dation of China (Grant no 51475144) and the Foundation ofInnovation and Research Team of Science and Technology inUniversities in Henan Province (Grant no 13IRTSTHN025)

References

[1] M Tiwari K Gupta and O Prakash ldquoDynamic response of anunbalanced rotor supported on ball bearingsrdquo Journal of Soundand Vibration vol 238 no 5 pp 757ndash779 2000

[2] M L Adam Jr Rotating Machinery VibrationmdashFrom Analysisto Trouble Shooting Marcel Dekker New York NY USA 2001

[3] N Akturk M Uneeb and R Gohar ldquoThe effects of number ofballs and preload on vibrations associated with ball bearingsrdquoJournal of Tribology vol 119 no 4 pp 747ndash753 1997

[4] S H Upadhyay S C Jain and S P Harsha ldquoNon-linearvibration signature analysis of a high-speed rotating shaft due toball size variations and varying number of ballsrdquo Proceedings ofthe Institution of Mechanical Engineers Part K Journal of Multi-body Dynamics vol 223 no 2 pp 83ndash105 2009

[5] N Akturk ldquoThe effect of waviness on vibrations associated withball bearingsrdquo Journal of Tribology vol 121 no 4 pp 667ndash6771999

[6] S P Harsha K Sandeep and R Prakash ldquoNon-linear dynamicbehaviors of rolling element bearings due to surface wavinessrdquoJournal of Sound and Vibration vol 272 no 3ndash5 pp 557ndash5802004

[7] J Sopanen and A Mikkola ldquoDynamic model of a deep-grooveball bearing including localized and distributed defects Part

12 Shock and Vibration

2 Implementation and resultsrdquo Proceedings of the Institution ofMechanical Engineers Part K vol 217 no 3 pp 213ndash223 2003

[8] S H Upadhyay S C Jain and S P Harsha ldquoChaotic dynamicsof high speed rotating shaft supported by ball bearings due todistributed defectsrdquo International Journal of Engineering Scienceand Technology vol 2 no 10 pp 5746ndash5794 2010

[9] S P Harsha ldquoNonlinear dynamic analysis of a high-speed rotorsupported by rolling element bearingsrdquo Journal of Sound andVibration vol 290 no 1-2 pp 65ndash100 2006

[10] C Villa J-J Sinou and F Thouverez ldquoStability and vibrationanalysis of a complex flexible rotor bearing systemrdquo Communi-cations in Nonlinear Science and Numerical Simulation vol 13no 4 pp 804ndash821 2008

[11] L Houpert ldquoCAGEDYN a contribution to roller bearingdynamic calculations Part III experimental validationrdquo ASMETribology Transactions vol 53 no 6 pp 848ndash859 2010

[12] S P Harsha ldquoNonlinear dynamic analysis of rolling elementbearings due to cage run-out and number of ballsrdquo Journal ofSound and Vibration vol 289 no 1-2 pp 360ndash381 2006

[13] I Bercea S Cretu M Bercea and D Olaru ldquoSimulating roller-cage pocket friction in a tapered roller bearingrdquo EuropeanJournal of Mechanical and Environmental Engineering vol 43no 4 pp 189ndash194 1998

[14] T Sakaguchi andKHarada ldquoDynamic analysis of cage behaviorin a tapered roller bearingrdquo Journal of Tribology vol 128 no 3pp 604ndash611 2006

[15] T A Harris Rolling Bearing Analysis John Wiley amp Sons 4thedition 2001

[16] C Spiridon ldquoMachined brass and pressed steel bearing cages acomparative studyrdquo RKB Technical Review vol 7 pp 1ndash13 2011

[17] R Riccardo ldquoChaos theory and some practical applications intechnical analysisrdquo in Proceedings of the 11th Annual Conferenceof the International Federation of Technical Analysts 1998

[18] S Tomoya and U Kaoru ldquoDynamic analysis of cage behavior ina cylindrical roller bearingrdquo NTN Technical Review 71 2004

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

8 Shock and Vibration

2500 7500 12500 17500 22500 27500

Rotation speed (rpm)

Case 1b

Case 1a

03

02

01

0

minus01

minus02

minus03

Vert

ical

disp

lace

men

t of

cage

cent

er (m

m)

(a) Different cage guidance gaps

2500 7500 12500 17500 22500 27500

Rotation speed (rpm)

Case 2b

Case 2a

03

02

01

0

minus01

minus02

minus03

Vert

ical

disp

lace

men

t of

cage

cent

er (m

m)

(b) Different eccentricities

2500 7500 12500 17500 22500 27500

Rotation speed (rpm)

Case 3b

Case 3a

03

02

01

0

minus01

minus02

minus03

Vert

ical

disp

lace

men

t of

cage

cent

er (m

m)

(c) Different dynamic viscosity exponents

Figure 7 Bifurcation diagram points119883119862versus spin speed

Table 1 Specifications of the bearing structure and size

Bearing typeBoundary size Roller size Rib diameter Clearance

Boremm

Outsidemm

Widthmm Number Diameter

mmLengthmm mm Radial

120583mPocketmm

NU2310 Φ50 Φ110 40 13 Φ16 27 Φ921 40 03

Table 2 Case study

CaseVariation parameters

Guidance gap Eccentricity Dynamic viscosity exponent(mm) (mm) (Pasdots)

1a 045 23 VG561b 060 23 VG562a 045 35 VG562b 045 57 VG563a 045 42 VG783b 045 42 VG32

different material and improve high-speed performance byusing optimized geometry structure This paper pays closeattention to influence from cage on speed characteristics ofcylindrical roller bearings The most suitable cage guidancetype is confirmed by mathematical derivation firstly Twomajor factors originating from manufacturing process andone involved in applying process are considered to establishdynamical mathematical model of cage with appropriate cageguidance type Analysis of real-time dynamic behavior of

a cage on radial plane was carried out using chaos theorybased on the theoretical mathematical model The chaoticanalysis is limited by the initial values of the system Hencethe initial values are determined firstly in analytical processPreliminary conclusions are listed as follows

(1) There are two bifurcation zones of cage dynamiccurves and the bearing should be away from these twozones in practiceThe range of the 1st bifurcation zoneismainly influenced by the value of cage guidance gapand eccentricity especially the cage guidance gapThelubrication dynamic viscosity exponent has relativelylittle influence on cage dynamic

(2) Numerical simulations show that the roller tends toacceleratedecelerate in switching process of support-ing zoneloaded zone which leads to single or mul-tiple roller-cage bridge impacts Impact force causesperiodic small oscillation on the curves Oscillationamplitude is determined by the pocket gap

(3) Eccentricity of the cage causes dramatic transientresponses in low rotational speed zone The transientresponses are easily tending to cause knocking and

Shock and Vibration 9

7000 10000 13000 16000 19000 22000

Rotation speed (rpm)

Case 1b

Case 1a

002

001

0

minus001

minus002

minus003

minus004

minus005

minus006

minus007

minus008

Vert

ical

disp

lace

men

t of

cage

cent

er (m

m)

(a) Different cage guidance gaps

Rotation speed (rpm)

Case 2b

Case 2a

0

minus001

minus002

minus003

006

005

004

003

002

001

7500 8800 10100 11400 12700 14000

Vert

ical

disp

lace

men

t of

cage

cent

er (m

m)

(b) Different eccentricities

Case 3b

Case 3a

Rotation speed (rpm)7500 8800 10100 11400 12700 14000

0

minus001

minus002

minus003

minus004

005

004

003

002

001

Vert

ical

disp

lace

men

t of

cage

cent

er (m

m)

(c) Different dynamic viscosity exponents

Figure 8 The first bifurcation zone from Figure 7 in enlarged scale

faster wear of cage pocket in terms of worse lubrica-tion film forming The steady zone is narrower wheneccentricity increases In addition divergent trendcan be found in the second bifurcation zonewith lagereccentricity

(4) According to the zooming of the first bifurcationzone variations of any parameters cause totally dif-ferent bifurcations We can draw our conclusion thatmovement of cage is a chaotic motion

(5) The Poincare maps show that position and rotationalspeed of the occurrence bifurcation zones are verydifferent with varied parameters

(6) The change of fluid dynamic viscosity exponent willnot affect the region of bifurcation due to the weaknonlinear of fluid film force

Considering cage dynamics of high-speed cylindricalroller bearing the values of cage guidance gap and eccen-tricity are crucial factors From the point of view of themanufacturers the very small cage guidance gap is difficult toobtain owing to the material (brass) of the cage used in high-speed bearingThe authors strongly suggest that the designershould minimize the gap within the confine of processingcapacity As for the cage eccentricity dynamic balance ofcage is widely used in ultra-high-speed bearings Analysesfrom the paper show that cage eccentricity brings violent

oscillation in low-speed zone and oscillation amplitude inchaotic zone is higher with the increase of cage eccentricityHence dynamic balance is another important factor inmanufacturing process With respect to lubrication dynamicviscosity exponent this study indicates that influence fromdynamic viscosity exponent on cage dynamics is not signifi-cant as the prediction

As a matter of fact researches in the paper are merelyrestricted to cage dynamic response What is to be pointedout is that dynamic response of a cylindrical roller bearingis a complex and interrelated problem among componentsHence there are many imperfect points in the paper whichshould be noted here Preliminary discussions are listed asfollows

(1) The displacements of the cage discussed in the paperare only in 119909-119910 plane During the rotation of thebearing rollers will inevitably have a certain slip-page towards the rings due to insufficient tractionfriction outside the load-zone churning momentfrom lubricant the friction loss between roller-pocketcontact and roller-rib contacts in the actual runningprocess Rollerraceways contact has not been takeninto account in the paper for the sake of simplifyingthe calculation

(2) Damping coefficient and contact stiffness involvedin the calculation are obtained according to material

10 Shock and Vibration

minus03 minus02 minus01 0 01 02minus400

minus200

0

200

400

Vertical displacement of cage center (mm)

Vert

ical

velo

city

(mm

s)

(a) Smaller cage guidance

minus03 minus02 minus01 0 01 02Vertical displacement of cage center (mm)

minus400

minus200

0

200

400

Vert

ical

velo

city

(mm

s)

(b) Bigger cage guidance

minus02 minus01 0 01Vertical displacement of cage center (mm)

minus300

minus200

0

200

300

Vert

ical

velo

city

(mm

s)

(c) Smaller eccentricity

minus03 minus02 minus01 0 01 02Vertical displacement of cage center (mm)

minus400

minus200

0

200

400

Vert

ical

velo

city

(mm

s)

(d) Bigger eccentricity

minus03 minus02 minus01 0 01 02Vertical displacement of cage center (mm)

minus400

minus200

0

200

400

Vert

ical

velo

city

(mm

s)

(e) Higher dynamic viscosity exponent

minus03 minus02 minus01 0 01 02Vertical displacement of cage center (mm)

minus400

minus200

0

200

400

Vert

ical

velo

city

(mm

s)

(f) Lower dynamic viscosity exponent

Figure 9 Poincare section points of 119884119862displacement in vertical direction

property and geometry characteristic parametersActually the damping coefficient and contact stiff-ness can be described more accurately by immediatemovement and velocity Namely contact stiffness isthe first-order derivative of immediate movementand damping coefficient is the first-order derivative ofimmediate velocity For the same sake of simplifyingcalculation the accurate expressions are not substi-tuted into the equations

(3) The bearing under study has the outer ring fixed to arigid support and the inner ring fixed rigidly to theshaft Bearing house and the shaft are truly nonlinearDOF systems The influence of bearing house andshaft on cage dynamic is relatively small so it can besimplified in the calculation

(4) As illustrated in Section 3 cage is driven by the rollersin supporting zone and the rollers are driven by cage

Shock and Vibration 11

in nonloaded zone Driving forces on the total cageare balanced and they are ignored except the impactforces in switching zones of driven modes

Symbols

120572 Angle at cage center in rad Angular velocity in tangential direction in rads Angular acceleration in tangential direction in

rads2119861 Damping coefficient119862 Cage gap in mm119889 Diameter in mmsΔ Roller gap in mm120575 Deformation in mm119890 Cage eccentricity in mm120576 Displacement in normal direction in mm120576 Velocity in normal direction in mms120576 Acceleration in normal direction in mms2119865 Load in N119891 Load caused by motion in N119866 Clearance in mm120579 Angle in ∘119870 Hertzian stiffness119897 Length in mm119872 Mass in kg120583 Fluid dynamic viscosity in Pasdots119873 Number of the rollers120592 Dimensionless parameter of impact factor119874 Position of the center in mm Velocity vector of the center in mms Acceleration vector of the center in mms2119901 Hydrodynamic pressure in Pa1199010 Initial hydrodynamic pressure in Pa

119876 Hertz stress in Nmm2119877 Radius in mm120588 Dimensionless parameter of cage eccentricity to cage

gap119905 Time in sec119879 Pocket clearance in mmV Surface velocity in mms119883 Coordinate119883 direction distance in mm119909 Position in coordinate direction in mm Velocity in 119909 direction in mms Acceleration in 119909 direction in mms2119884 Coordinate 119884 direction distance in mm119910 Dimensionless parameter in coordinate direction119910 Velocity in 119910 direction in mms119910 Acceleration in 119910 direction in mms2120596 Rotation speed in rads

Subscripts

119886 Axial direction120572 Tangential direction119862 Cage center119888 Cagecen Centrifugal

119888119894 Relative relation between cage and innerring

119888119900 Relative relation between cage and outerring

119888119908 Relative relation between cage and roller119863 Unbalanced gravitational center119890 Normal direction119891119889 Added fluid

119892 Ribℎ Impact119894 Inner raceway119895 The 119895th roller119895cen Centrifugal force of the 119895th roller119898 Mean value119874 Rib center119900 Outer raceway119903 Radial direction119903119894 Inner raceway on radial direction119903119900 Outer raceway on radial direction119904 Polar coordinate119908 Roller119909 119909 direction119910 119910 direction

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The project is supported by National Natural Science Foun-dation of China (Grant no 51475144) and the Foundation ofInnovation and Research Team of Science and Technology inUniversities in Henan Province (Grant no 13IRTSTHN025)

References

[1] M Tiwari K Gupta and O Prakash ldquoDynamic response of anunbalanced rotor supported on ball bearingsrdquo Journal of Soundand Vibration vol 238 no 5 pp 757ndash779 2000

[2] M L Adam Jr Rotating Machinery VibrationmdashFrom Analysisto Trouble Shooting Marcel Dekker New York NY USA 2001

[3] N Akturk M Uneeb and R Gohar ldquoThe effects of number ofballs and preload on vibrations associated with ball bearingsrdquoJournal of Tribology vol 119 no 4 pp 747ndash753 1997

[4] S H Upadhyay S C Jain and S P Harsha ldquoNon-linearvibration signature analysis of a high-speed rotating shaft due toball size variations and varying number of ballsrdquo Proceedings ofthe Institution of Mechanical Engineers Part K Journal of Multi-body Dynamics vol 223 no 2 pp 83ndash105 2009

[5] N Akturk ldquoThe effect of waviness on vibrations associated withball bearingsrdquo Journal of Tribology vol 121 no 4 pp 667ndash6771999

[6] S P Harsha K Sandeep and R Prakash ldquoNon-linear dynamicbehaviors of rolling element bearings due to surface wavinessrdquoJournal of Sound and Vibration vol 272 no 3ndash5 pp 557ndash5802004

[7] J Sopanen and A Mikkola ldquoDynamic model of a deep-grooveball bearing including localized and distributed defects Part

12 Shock and Vibration

2 Implementation and resultsrdquo Proceedings of the Institution ofMechanical Engineers Part K vol 217 no 3 pp 213ndash223 2003

[8] S H Upadhyay S C Jain and S P Harsha ldquoChaotic dynamicsof high speed rotating shaft supported by ball bearings due todistributed defectsrdquo International Journal of Engineering Scienceand Technology vol 2 no 10 pp 5746ndash5794 2010

[9] S P Harsha ldquoNonlinear dynamic analysis of a high-speed rotorsupported by rolling element bearingsrdquo Journal of Sound andVibration vol 290 no 1-2 pp 65ndash100 2006

[10] C Villa J-J Sinou and F Thouverez ldquoStability and vibrationanalysis of a complex flexible rotor bearing systemrdquo Communi-cations in Nonlinear Science and Numerical Simulation vol 13no 4 pp 804ndash821 2008

[11] L Houpert ldquoCAGEDYN a contribution to roller bearingdynamic calculations Part III experimental validationrdquo ASMETribology Transactions vol 53 no 6 pp 848ndash859 2010

[12] S P Harsha ldquoNonlinear dynamic analysis of rolling elementbearings due to cage run-out and number of ballsrdquo Journal ofSound and Vibration vol 289 no 1-2 pp 360ndash381 2006

[13] I Bercea S Cretu M Bercea and D Olaru ldquoSimulating roller-cage pocket friction in a tapered roller bearingrdquo EuropeanJournal of Mechanical and Environmental Engineering vol 43no 4 pp 189ndash194 1998

[14] T Sakaguchi andKHarada ldquoDynamic analysis of cage behaviorin a tapered roller bearingrdquo Journal of Tribology vol 128 no 3pp 604ndash611 2006

[15] T A Harris Rolling Bearing Analysis John Wiley amp Sons 4thedition 2001

[16] C Spiridon ldquoMachined brass and pressed steel bearing cages acomparative studyrdquo RKB Technical Review vol 7 pp 1ndash13 2011

[17] R Riccardo ldquoChaos theory and some practical applications intechnical analysisrdquo in Proceedings of the 11th Annual Conferenceof the International Federation of Technical Analysts 1998

[18] S Tomoya and U Kaoru ldquoDynamic analysis of cage behavior ina cylindrical roller bearingrdquo NTN Technical Review 71 2004

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Shock and Vibration 9

7000 10000 13000 16000 19000 22000

Rotation speed (rpm)

Case 1b

Case 1a

002

001

0

minus001

minus002

minus003

minus004

minus005

minus006

minus007

minus008

Vert

ical

disp

lace

men

t of

cage

cent

er (m

m)

(a) Different cage guidance gaps

Rotation speed (rpm)

Case 2b

Case 2a

0

minus001

minus002

minus003

006

005

004

003

002

001

7500 8800 10100 11400 12700 14000

Vert

ical

disp

lace

men

t of

cage

cent

er (m

m)

(b) Different eccentricities

Case 3b

Case 3a

Rotation speed (rpm)7500 8800 10100 11400 12700 14000

0

minus001

minus002

minus003

minus004

005

004

003

002

001

Vert

ical

disp

lace

men

t of

cage

cent

er (m

m)

(c) Different dynamic viscosity exponents

Figure 8 The first bifurcation zone from Figure 7 in enlarged scale

faster wear of cage pocket in terms of worse lubrica-tion film forming The steady zone is narrower wheneccentricity increases In addition divergent trendcan be found in the second bifurcation zonewith lagereccentricity

(4) According to the zooming of the first bifurcationzone variations of any parameters cause totally dif-ferent bifurcations We can draw our conclusion thatmovement of cage is a chaotic motion

(5) The Poincare maps show that position and rotationalspeed of the occurrence bifurcation zones are verydifferent with varied parameters

(6) The change of fluid dynamic viscosity exponent willnot affect the region of bifurcation due to the weaknonlinear of fluid film force

Considering cage dynamics of high-speed cylindricalroller bearing the values of cage guidance gap and eccen-tricity are crucial factors From the point of view of themanufacturers the very small cage guidance gap is difficult toobtain owing to the material (brass) of the cage used in high-speed bearingThe authors strongly suggest that the designershould minimize the gap within the confine of processingcapacity As for the cage eccentricity dynamic balance ofcage is widely used in ultra-high-speed bearings Analysesfrom the paper show that cage eccentricity brings violent

oscillation in low-speed zone and oscillation amplitude inchaotic zone is higher with the increase of cage eccentricityHence dynamic balance is another important factor inmanufacturing process With respect to lubrication dynamicviscosity exponent this study indicates that influence fromdynamic viscosity exponent on cage dynamics is not signifi-cant as the prediction

As a matter of fact researches in the paper are merelyrestricted to cage dynamic response What is to be pointedout is that dynamic response of a cylindrical roller bearingis a complex and interrelated problem among componentsHence there are many imperfect points in the paper whichshould be noted here Preliminary discussions are listed asfollows

(1) The displacements of the cage discussed in the paperare only in 119909-119910 plane During the rotation of thebearing rollers will inevitably have a certain slip-page towards the rings due to insufficient tractionfriction outside the load-zone churning momentfrom lubricant the friction loss between roller-pocketcontact and roller-rib contacts in the actual runningprocess Rollerraceways contact has not been takeninto account in the paper for the sake of simplifyingthe calculation

(2) Damping coefficient and contact stiffness involvedin the calculation are obtained according to material

10 Shock and Vibration

minus03 minus02 minus01 0 01 02minus400

minus200

0

200

400

Vertical displacement of cage center (mm)

Vert

ical

velo

city

(mm

s)

(a) Smaller cage guidance

minus03 minus02 minus01 0 01 02Vertical displacement of cage center (mm)

minus400

minus200

0

200

400

Vert

ical

velo

city

(mm

s)

(b) Bigger cage guidance

minus02 minus01 0 01Vertical displacement of cage center (mm)

minus300

minus200

0

200

300

Vert

ical

velo

city

(mm

s)

(c) Smaller eccentricity

minus03 minus02 minus01 0 01 02Vertical displacement of cage center (mm)

minus400

minus200

0

200

400

Vert

ical

velo

city

(mm

s)

(d) Bigger eccentricity

minus03 minus02 minus01 0 01 02Vertical displacement of cage center (mm)

minus400

minus200

0

200

400

Vert

ical

velo

city

(mm

s)

(e) Higher dynamic viscosity exponent

minus03 minus02 minus01 0 01 02Vertical displacement of cage center (mm)

minus400

minus200

0

200

400

Vert

ical

velo

city

(mm

s)

(f) Lower dynamic viscosity exponent

Figure 9 Poincare section points of 119884119862displacement in vertical direction

property and geometry characteristic parametersActually the damping coefficient and contact stiff-ness can be described more accurately by immediatemovement and velocity Namely contact stiffness isthe first-order derivative of immediate movementand damping coefficient is the first-order derivative ofimmediate velocity For the same sake of simplifyingcalculation the accurate expressions are not substi-tuted into the equations

(3) The bearing under study has the outer ring fixed to arigid support and the inner ring fixed rigidly to theshaft Bearing house and the shaft are truly nonlinearDOF systems The influence of bearing house andshaft on cage dynamic is relatively small so it can besimplified in the calculation

(4) As illustrated in Section 3 cage is driven by the rollersin supporting zone and the rollers are driven by cage

Shock and Vibration 11

in nonloaded zone Driving forces on the total cageare balanced and they are ignored except the impactforces in switching zones of driven modes

Symbols

120572 Angle at cage center in rad Angular velocity in tangential direction in rads Angular acceleration in tangential direction in

rads2119861 Damping coefficient119862 Cage gap in mm119889 Diameter in mmsΔ Roller gap in mm120575 Deformation in mm119890 Cage eccentricity in mm120576 Displacement in normal direction in mm120576 Velocity in normal direction in mms120576 Acceleration in normal direction in mms2119865 Load in N119891 Load caused by motion in N119866 Clearance in mm120579 Angle in ∘119870 Hertzian stiffness119897 Length in mm119872 Mass in kg120583 Fluid dynamic viscosity in Pasdots119873 Number of the rollers120592 Dimensionless parameter of impact factor119874 Position of the center in mm Velocity vector of the center in mms Acceleration vector of the center in mms2119901 Hydrodynamic pressure in Pa1199010 Initial hydrodynamic pressure in Pa

119876 Hertz stress in Nmm2119877 Radius in mm120588 Dimensionless parameter of cage eccentricity to cage

gap119905 Time in sec119879 Pocket clearance in mmV Surface velocity in mms119883 Coordinate119883 direction distance in mm119909 Position in coordinate direction in mm Velocity in 119909 direction in mms Acceleration in 119909 direction in mms2119884 Coordinate 119884 direction distance in mm119910 Dimensionless parameter in coordinate direction119910 Velocity in 119910 direction in mms119910 Acceleration in 119910 direction in mms2120596 Rotation speed in rads

Subscripts

119886 Axial direction120572 Tangential direction119862 Cage center119888 Cagecen Centrifugal

119888119894 Relative relation between cage and innerring

119888119900 Relative relation between cage and outerring

119888119908 Relative relation between cage and roller119863 Unbalanced gravitational center119890 Normal direction119891119889 Added fluid

119892 Ribℎ Impact119894 Inner raceway119895 The 119895th roller119895cen Centrifugal force of the 119895th roller119898 Mean value119874 Rib center119900 Outer raceway119903 Radial direction119903119894 Inner raceway on radial direction119903119900 Outer raceway on radial direction119904 Polar coordinate119908 Roller119909 119909 direction119910 119910 direction

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The project is supported by National Natural Science Foun-dation of China (Grant no 51475144) and the Foundation ofInnovation and Research Team of Science and Technology inUniversities in Henan Province (Grant no 13IRTSTHN025)

References

[1] M Tiwari K Gupta and O Prakash ldquoDynamic response of anunbalanced rotor supported on ball bearingsrdquo Journal of Soundand Vibration vol 238 no 5 pp 757ndash779 2000

[2] M L Adam Jr Rotating Machinery VibrationmdashFrom Analysisto Trouble Shooting Marcel Dekker New York NY USA 2001

[3] N Akturk M Uneeb and R Gohar ldquoThe effects of number ofballs and preload on vibrations associated with ball bearingsrdquoJournal of Tribology vol 119 no 4 pp 747ndash753 1997

[4] S H Upadhyay S C Jain and S P Harsha ldquoNon-linearvibration signature analysis of a high-speed rotating shaft due toball size variations and varying number of ballsrdquo Proceedings ofthe Institution of Mechanical Engineers Part K Journal of Multi-body Dynamics vol 223 no 2 pp 83ndash105 2009

[5] N Akturk ldquoThe effect of waviness on vibrations associated withball bearingsrdquo Journal of Tribology vol 121 no 4 pp 667ndash6771999

[6] S P Harsha K Sandeep and R Prakash ldquoNon-linear dynamicbehaviors of rolling element bearings due to surface wavinessrdquoJournal of Sound and Vibration vol 272 no 3ndash5 pp 557ndash5802004

[7] J Sopanen and A Mikkola ldquoDynamic model of a deep-grooveball bearing including localized and distributed defects Part

12 Shock and Vibration

2 Implementation and resultsrdquo Proceedings of the Institution ofMechanical Engineers Part K vol 217 no 3 pp 213ndash223 2003

[8] S H Upadhyay S C Jain and S P Harsha ldquoChaotic dynamicsof high speed rotating shaft supported by ball bearings due todistributed defectsrdquo International Journal of Engineering Scienceand Technology vol 2 no 10 pp 5746ndash5794 2010

[9] S P Harsha ldquoNonlinear dynamic analysis of a high-speed rotorsupported by rolling element bearingsrdquo Journal of Sound andVibration vol 290 no 1-2 pp 65ndash100 2006

[10] C Villa J-J Sinou and F Thouverez ldquoStability and vibrationanalysis of a complex flexible rotor bearing systemrdquo Communi-cations in Nonlinear Science and Numerical Simulation vol 13no 4 pp 804ndash821 2008

[11] L Houpert ldquoCAGEDYN a contribution to roller bearingdynamic calculations Part III experimental validationrdquo ASMETribology Transactions vol 53 no 6 pp 848ndash859 2010

[12] S P Harsha ldquoNonlinear dynamic analysis of rolling elementbearings due to cage run-out and number of ballsrdquo Journal ofSound and Vibration vol 289 no 1-2 pp 360ndash381 2006

[13] I Bercea S Cretu M Bercea and D Olaru ldquoSimulating roller-cage pocket friction in a tapered roller bearingrdquo EuropeanJournal of Mechanical and Environmental Engineering vol 43no 4 pp 189ndash194 1998

[14] T Sakaguchi andKHarada ldquoDynamic analysis of cage behaviorin a tapered roller bearingrdquo Journal of Tribology vol 128 no 3pp 604ndash611 2006

[15] T A Harris Rolling Bearing Analysis John Wiley amp Sons 4thedition 2001

[16] C Spiridon ldquoMachined brass and pressed steel bearing cages acomparative studyrdquo RKB Technical Review vol 7 pp 1ndash13 2011

[17] R Riccardo ldquoChaos theory and some practical applications intechnical analysisrdquo in Proceedings of the 11th Annual Conferenceof the International Federation of Technical Analysts 1998

[18] S Tomoya and U Kaoru ldquoDynamic analysis of cage behavior ina cylindrical roller bearingrdquo NTN Technical Review 71 2004

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

10 Shock and Vibration

minus03 minus02 minus01 0 01 02minus400

minus200

0

200

400

Vertical displacement of cage center (mm)

Vert

ical

velo

city

(mm

s)

(a) Smaller cage guidance

minus03 minus02 minus01 0 01 02Vertical displacement of cage center (mm)

minus400

minus200

0

200

400

Vert

ical

velo

city

(mm

s)

(b) Bigger cage guidance

minus02 minus01 0 01Vertical displacement of cage center (mm)

minus300

minus200

0

200

300

Vert

ical

velo

city

(mm

s)

(c) Smaller eccentricity

minus03 minus02 minus01 0 01 02Vertical displacement of cage center (mm)

minus400

minus200

0

200

400

Vert

ical

velo

city

(mm

s)

(d) Bigger eccentricity

minus03 minus02 minus01 0 01 02Vertical displacement of cage center (mm)

minus400

minus200

0

200

400

Vert

ical

velo

city

(mm

s)

(e) Higher dynamic viscosity exponent

minus03 minus02 minus01 0 01 02Vertical displacement of cage center (mm)

minus400

minus200

0

200

400

Vert

ical

velo

city

(mm

s)

(f) Lower dynamic viscosity exponent

Figure 9 Poincare section points of 119884119862displacement in vertical direction

property and geometry characteristic parametersActually the damping coefficient and contact stiff-ness can be described more accurately by immediatemovement and velocity Namely contact stiffness isthe first-order derivative of immediate movementand damping coefficient is the first-order derivative ofimmediate velocity For the same sake of simplifyingcalculation the accurate expressions are not substi-tuted into the equations

(3) The bearing under study has the outer ring fixed to arigid support and the inner ring fixed rigidly to theshaft Bearing house and the shaft are truly nonlinearDOF systems The influence of bearing house andshaft on cage dynamic is relatively small so it can besimplified in the calculation

(4) As illustrated in Section 3 cage is driven by the rollersin supporting zone and the rollers are driven by cage

Shock and Vibration 11

in nonloaded zone Driving forces on the total cageare balanced and they are ignored except the impactforces in switching zones of driven modes

Symbols

120572 Angle at cage center in rad Angular velocity in tangential direction in rads Angular acceleration in tangential direction in

rads2119861 Damping coefficient119862 Cage gap in mm119889 Diameter in mmsΔ Roller gap in mm120575 Deformation in mm119890 Cage eccentricity in mm120576 Displacement in normal direction in mm120576 Velocity in normal direction in mms120576 Acceleration in normal direction in mms2119865 Load in N119891 Load caused by motion in N119866 Clearance in mm120579 Angle in ∘119870 Hertzian stiffness119897 Length in mm119872 Mass in kg120583 Fluid dynamic viscosity in Pasdots119873 Number of the rollers120592 Dimensionless parameter of impact factor119874 Position of the center in mm Velocity vector of the center in mms Acceleration vector of the center in mms2119901 Hydrodynamic pressure in Pa1199010 Initial hydrodynamic pressure in Pa

119876 Hertz stress in Nmm2119877 Radius in mm120588 Dimensionless parameter of cage eccentricity to cage

gap119905 Time in sec119879 Pocket clearance in mmV Surface velocity in mms119883 Coordinate119883 direction distance in mm119909 Position in coordinate direction in mm Velocity in 119909 direction in mms Acceleration in 119909 direction in mms2119884 Coordinate 119884 direction distance in mm119910 Dimensionless parameter in coordinate direction119910 Velocity in 119910 direction in mms119910 Acceleration in 119910 direction in mms2120596 Rotation speed in rads

Subscripts

119886 Axial direction120572 Tangential direction119862 Cage center119888 Cagecen Centrifugal

119888119894 Relative relation between cage and innerring

119888119900 Relative relation between cage and outerring

119888119908 Relative relation between cage and roller119863 Unbalanced gravitational center119890 Normal direction119891119889 Added fluid

119892 Ribℎ Impact119894 Inner raceway119895 The 119895th roller119895cen Centrifugal force of the 119895th roller119898 Mean value119874 Rib center119900 Outer raceway119903 Radial direction119903119894 Inner raceway on radial direction119903119900 Outer raceway on radial direction119904 Polar coordinate119908 Roller119909 119909 direction119910 119910 direction

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The project is supported by National Natural Science Foun-dation of China (Grant no 51475144) and the Foundation ofInnovation and Research Team of Science and Technology inUniversities in Henan Province (Grant no 13IRTSTHN025)

References

[1] M Tiwari K Gupta and O Prakash ldquoDynamic response of anunbalanced rotor supported on ball bearingsrdquo Journal of Soundand Vibration vol 238 no 5 pp 757ndash779 2000

[2] M L Adam Jr Rotating Machinery VibrationmdashFrom Analysisto Trouble Shooting Marcel Dekker New York NY USA 2001

[3] N Akturk M Uneeb and R Gohar ldquoThe effects of number ofballs and preload on vibrations associated with ball bearingsrdquoJournal of Tribology vol 119 no 4 pp 747ndash753 1997

[4] S H Upadhyay S C Jain and S P Harsha ldquoNon-linearvibration signature analysis of a high-speed rotating shaft due toball size variations and varying number of ballsrdquo Proceedings ofthe Institution of Mechanical Engineers Part K Journal of Multi-body Dynamics vol 223 no 2 pp 83ndash105 2009

[5] N Akturk ldquoThe effect of waviness on vibrations associated withball bearingsrdquo Journal of Tribology vol 121 no 4 pp 667ndash6771999

[6] S P Harsha K Sandeep and R Prakash ldquoNon-linear dynamicbehaviors of rolling element bearings due to surface wavinessrdquoJournal of Sound and Vibration vol 272 no 3ndash5 pp 557ndash5802004

[7] J Sopanen and A Mikkola ldquoDynamic model of a deep-grooveball bearing including localized and distributed defects Part

12 Shock and Vibration

2 Implementation and resultsrdquo Proceedings of the Institution ofMechanical Engineers Part K vol 217 no 3 pp 213ndash223 2003

[8] S H Upadhyay S C Jain and S P Harsha ldquoChaotic dynamicsof high speed rotating shaft supported by ball bearings due todistributed defectsrdquo International Journal of Engineering Scienceand Technology vol 2 no 10 pp 5746ndash5794 2010

[9] S P Harsha ldquoNonlinear dynamic analysis of a high-speed rotorsupported by rolling element bearingsrdquo Journal of Sound andVibration vol 290 no 1-2 pp 65ndash100 2006

[10] C Villa J-J Sinou and F Thouverez ldquoStability and vibrationanalysis of a complex flexible rotor bearing systemrdquo Communi-cations in Nonlinear Science and Numerical Simulation vol 13no 4 pp 804ndash821 2008

[11] L Houpert ldquoCAGEDYN a contribution to roller bearingdynamic calculations Part III experimental validationrdquo ASMETribology Transactions vol 53 no 6 pp 848ndash859 2010

[12] S P Harsha ldquoNonlinear dynamic analysis of rolling elementbearings due to cage run-out and number of ballsrdquo Journal ofSound and Vibration vol 289 no 1-2 pp 360ndash381 2006

[13] I Bercea S Cretu M Bercea and D Olaru ldquoSimulating roller-cage pocket friction in a tapered roller bearingrdquo EuropeanJournal of Mechanical and Environmental Engineering vol 43no 4 pp 189ndash194 1998

[14] T Sakaguchi andKHarada ldquoDynamic analysis of cage behaviorin a tapered roller bearingrdquo Journal of Tribology vol 128 no 3pp 604ndash611 2006

[15] T A Harris Rolling Bearing Analysis John Wiley amp Sons 4thedition 2001

[16] C Spiridon ldquoMachined brass and pressed steel bearing cages acomparative studyrdquo RKB Technical Review vol 7 pp 1ndash13 2011

[17] R Riccardo ldquoChaos theory and some practical applications intechnical analysisrdquo in Proceedings of the 11th Annual Conferenceof the International Federation of Technical Analysts 1998

[18] S Tomoya and U Kaoru ldquoDynamic analysis of cage behavior ina cylindrical roller bearingrdquo NTN Technical Review 71 2004

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Shock and Vibration 11

in nonloaded zone Driving forces on the total cageare balanced and they are ignored except the impactforces in switching zones of driven modes

Symbols

120572 Angle at cage center in rad Angular velocity in tangential direction in rads Angular acceleration in tangential direction in

rads2119861 Damping coefficient119862 Cage gap in mm119889 Diameter in mmsΔ Roller gap in mm120575 Deformation in mm119890 Cage eccentricity in mm120576 Displacement in normal direction in mm120576 Velocity in normal direction in mms120576 Acceleration in normal direction in mms2119865 Load in N119891 Load caused by motion in N119866 Clearance in mm120579 Angle in ∘119870 Hertzian stiffness119897 Length in mm119872 Mass in kg120583 Fluid dynamic viscosity in Pasdots119873 Number of the rollers120592 Dimensionless parameter of impact factor119874 Position of the center in mm Velocity vector of the center in mms Acceleration vector of the center in mms2119901 Hydrodynamic pressure in Pa1199010 Initial hydrodynamic pressure in Pa

119876 Hertz stress in Nmm2119877 Radius in mm120588 Dimensionless parameter of cage eccentricity to cage

gap119905 Time in sec119879 Pocket clearance in mmV Surface velocity in mms119883 Coordinate119883 direction distance in mm119909 Position in coordinate direction in mm Velocity in 119909 direction in mms Acceleration in 119909 direction in mms2119884 Coordinate 119884 direction distance in mm119910 Dimensionless parameter in coordinate direction119910 Velocity in 119910 direction in mms119910 Acceleration in 119910 direction in mms2120596 Rotation speed in rads

Subscripts

119886 Axial direction120572 Tangential direction119862 Cage center119888 Cagecen Centrifugal

119888119894 Relative relation between cage and innerring

119888119900 Relative relation between cage and outerring

119888119908 Relative relation between cage and roller119863 Unbalanced gravitational center119890 Normal direction119891119889 Added fluid

119892 Ribℎ Impact119894 Inner raceway119895 The 119895th roller119895cen Centrifugal force of the 119895th roller119898 Mean value119874 Rib center119900 Outer raceway119903 Radial direction119903119894 Inner raceway on radial direction119903119900 Outer raceway on radial direction119904 Polar coordinate119908 Roller119909 119909 direction119910 119910 direction

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The project is supported by National Natural Science Foun-dation of China (Grant no 51475144) and the Foundation ofInnovation and Research Team of Science and Technology inUniversities in Henan Province (Grant no 13IRTSTHN025)

References

[1] M Tiwari K Gupta and O Prakash ldquoDynamic response of anunbalanced rotor supported on ball bearingsrdquo Journal of Soundand Vibration vol 238 no 5 pp 757ndash779 2000

[2] M L Adam Jr Rotating Machinery VibrationmdashFrom Analysisto Trouble Shooting Marcel Dekker New York NY USA 2001

[3] N Akturk M Uneeb and R Gohar ldquoThe effects of number ofballs and preload on vibrations associated with ball bearingsrdquoJournal of Tribology vol 119 no 4 pp 747ndash753 1997

[4] S H Upadhyay S C Jain and S P Harsha ldquoNon-linearvibration signature analysis of a high-speed rotating shaft due toball size variations and varying number of ballsrdquo Proceedings ofthe Institution of Mechanical Engineers Part K Journal of Multi-body Dynamics vol 223 no 2 pp 83ndash105 2009

[5] N Akturk ldquoThe effect of waviness on vibrations associated withball bearingsrdquo Journal of Tribology vol 121 no 4 pp 667ndash6771999

[6] S P Harsha K Sandeep and R Prakash ldquoNon-linear dynamicbehaviors of rolling element bearings due to surface wavinessrdquoJournal of Sound and Vibration vol 272 no 3ndash5 pp 557ndash5802004

[7] J Sopanen and A Mikkola ldquoDynamic model of a deep-grooveball bearing including localized and distributed defects Part

12 Shock and Vibration

2 Implementation and resultsrdquo Proceedings of the Institution ofMechanical Engineers Part K vol 217 no 3 pp 213ndash223 2003

[8] S H Upadhyay S C Jain and S P Harsha ldquoChaotic dynamicsof high speed rotating shaft supported by ball bearings due todistributed defectsrdquo International Journal of Engineering Scienceand Technology vol 2 no 10 pp 5746ndash5794 2010

[9] S P Harsha ldquoNonlinear dynamic analysis of a high-speed rotorsupported by rolling element bearingsrdquo Journal of Sound andVibration vol 290 no 1-2 pp 65ndash100 2006

[10] C Villa J-J Sinou and F Thouverez ldquoStability and vibrationanalysis of a complex flexible rotor bearing systemrdquo Communi-cations in Nonlinear Science and Numerical Simulation vol 13no 4 pp 804ndash821 2008

[11] L Houpert ldquoCAGEDYN a contribution to roller bearingdynamic calculations Part III experimental validationrdquo ASMETribology Transactions vol 53 no 6 pp 848ndash859 2010

[12] S P Harsha ldquoNonlinear dynamic analysis of rolling elementbearings due to cage run-out and number of ballsrdquo Journal ofSound and Vibration vol 289 no 1-2 pp 360ndash381 2006

[13] I Bercea S Cretu M Bercea and D Olaru ldquoSimulating roller-cage pocket friction in a tapered roller bearingrdquo EuropeanJournal of Mechanical and Environmental Engineering vol 43no 4 pp 189ndash194 1998

[14] T Sakaguchi andKHarada ldquoDynamic analysis of cage behaviorin a tapered roller bearingrdquo Journal of Tribology vol 128 no 3pp 604ndash611 2006

[15] T A Harris Rolling Bearing Analysis John Wiley amp Sons 4thedition 2001

[16] C Spiridon ldquoMachined brass and pressed steel bearing cages acomparative studyrdquo RKB Technical Review vol 7 pp 1ndash13 2011

[17] R Riccardo ldquoChaos theory and some practical applications intechnical analysisrdquo in Proceedings of the 11th Annual Conferenceof the International Federation of Technical Analysts 1998

[18] S Tomoya and U Kaoru ldquoDynamic analysis of cage behavior ina cylindrical roller bearingrdquo NTN Technical Review 71 2004

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

12 Shock and Vibration

2 Implementation and resultsrdquo Proceedings of the Institution ofMechanical Engineers Part K vol 217 no 3 pp 213ndash223 2003

[8] S H Upadhyay S C Jain and S P Harsha ldquoChaotic dynamicsof high speed rotating shaft supported by ball bearings due todistributed defectsrdquo International Journal of Engineering Scienceand Technology vol 2 no 10 pp 5746ndash5794 2010

[9] S P Harsha ldquoNonlinear dynamic analysis of a high-speed rotorsupported by rolling element bearingsrdquo Journal of Sound andVibration vol 290 no 1-2 pp 65ndash100 2006

[10] C Villa J-J Sinou and F Thouverez ldquoStability and vibrationanalysis of a complex flexible rotor bearing systemrdquo Communi-cations in Nonlinear Science and Numerical Simulation vol 13no 4 pp 804ndash821 2008

[11] L Houpert ldquoCAGEDYN a contribution to roller bearingdynamic calculations Part III experimental validationrdquo ASMETribology Transactions vol 53 no 6 pp 848ndash859 2010

[12] S P Harsha ldquoNonlinear dynamic analysis of rolling elementbearings due to cage run-out and number of ballsrdquo Journal ofSound and Vibration vol 289 no 1-2 pp 360ndash381 2006

[13] I Bercea S Cretu M Bercea and D Olaru ldquoSimulating roller-cage pocket friction in a tapered roller bearingrdquo EuropeanJournal of Mechanical and Environmental Engineering vol 43no 4 pp 189ndash194 1998

[14] T Sakaguchi andKHarada ldquoDynamic analysis of cage behaviorin a tapered roller bearingrdquo Journal of Tribology vol 128 no 3pp 604ndash611 2006

[15] T A Harris Rolling Bearing Analysis John Wiley amp Sons 4thedition 2001

[16] C Spiridon ldquoMachined brass and pressed steel bearing cages acomparative studyrdquo RKB Technical Review vol 7 pp 1ndash13 2011

[17] R Riccardo ldquoChaos theory and some practical applications intechnical analysisrdquo in Proceedings of the 11th Annual Conferenceof the International Federation of Technical Analysts 1998

[18] S Tomoya and U Kaoru ldquoDynamic analysis of cage behavior ina cylindrical roller bearingrdquo NTN Technical Review 71 2004

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of