Research ArticleAnalysis of Dynamic Characteristics of Grease-LubricatedTapered Roller Bearings

Zheng-Hai Wu 1 Ying-Qiang Xu 1 and Si-Er Deng2

1School of Mechatronical Engineering Northwestern Polytechnical University Xirsquoan 710072 China2School of Mechatronical Engineering Henan University of Science and Technology Luoyang 471023 China

Correspondence should be addressed to Ying-Qiang Xu xuyngqngnwpueducn

Received 7 June 2018 Accepted 4 October 2018 Published 13 November 2018

Academic Editor Miguel Neves

Copyright copy 2018 Zheng-HaiWu et al1is is an open access article distributed under the Creative CommonsAttribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Tapered roller bearings (TRBs) are applied extensively in the field of high-speed trains machine tools automobiles etc 1e motionprediction of main components of TRBs under grease lubrication will be beneficial to the design of bearings and the selection oflubricating grease In this study considering the dynamic contact relationship among the cage rollers and raceways a multibodycontact dynamic model of the TRB was established based on the geometric interaction models and grease lubrication theories 1eimpacts of load grease rheological properties and temperature on the roller tilt and skew and the bearing slip were simulated by usingthe fourth-order RungendashKutta method1e results show that the roller tilt angle in the unloaded zone is obviously larger than that inthe loaded zone while the roller skew angle in the unloaded zone is smaller than that in the loaded zone As the speed increases theroller tilt and skew and the bearing slip becomemore serious Bearing preload can effectively reduce the bearing slip but will make theroller tilt and skew angle increase 1e roller skew angle and the bearing slip decrease with the increase of the grease plastic viscosity1e roller tilt angle increases with the increase of the plastic viscosity 1e yield stress of the grease has little effect on motions of theroller and cage 1e influence of temperature on the roller and cage motions varies with the type of grease used

1 Introduction

Tapered roller bearings as the separable bearing have theability to withstand combined loads large load-carryingcapacity well adjustability and long service life Nowa-days nearly 90 of rolling bearings are grease-lubricated [1]Generally compared to oil lubrication lubricating grease ina rolling bearing has a wide operating temperature range andgood extreme pressure (EP) property and adhesion propertyand the construction of the lubricating device for the greaselubrication is sometimes relatively simple However due tothe pressure difference inside the bearing contacts the greasewill flow to sides and next to the raceways over time 1eremay be very little reflow back into the raceways and thebearing may suffer from starvation Despite the above-mentioned conditions there also exists a film inside thebearing contacts at the beginning of bearing operationformed by the combination of thickener and base oil [1 2]For the beginning of operation of the grease-lubricated TRBthe analysis of the TRB dynamic characteristics should be

made to clarify the relation between the bearing dynamicsand the lubricating grease 1e bearing lubrication anddynamics not only affect each other but both have importantimpacts on the bearing failure service life and reliability1e analysis may have implications for the design of thebearing and the selection of the grease

1e dynamics of ball and cylindrical roller bearings havebeen extensively studied in the past few decades [3ndash5] Byconsidering the four degree-of-freedom balls and the sixdegree-of-freedom cage Walters [3] firstly presenteda comprehensive analysis for the balls and cage motionsAfter that Gupta [4] and Meeks and Ng [5] carried outa series of research on dynamic problems of ball and cy-lindrical roller bearings 1e movements of the rolling el-ements and cage were minutely described by classicaldifferential equations of motion under specific operatingconditions For tapered roller bearings compared with ballbearings (except angular contact ball bearings) and cylin-drical roller bearings the structure type and dynamics aremore complicated Gupta [6] studied the dynamic model

HindawiShock and VibrationVolume 2018 Article ID 7183042 17 pageshttpsdoiorg10115520187183042

and developed the dynamic analysis program ADORE forTRBs the cage whirling and roller skew were analyzed underdierent cage clearances and traction-slip relations How-ever the bearing slip and impacts of the lubricant on mo-tions of bearing parts were not presented Cretu et al [7]proposed a quasidynamic model for the TRB based on theJohnsonndashTevaarwerk rheological model the traction per-formance and other properties of the bearing were analyzedbut the translational motion of the cage was neglected Deng[8] analyzed the dynamics of the TRB with oil lubrication byusing the Adams-BashforthndashMoulton multistep method andstudied the cage whirling and roller skew of the bearing Byconsidering six degrees of the cage motion Sakaguchi andHarada [9 10] simulated motions of the rigid or exible cagein TRBs on the dynamic simulation software ADAMSBercea [11] proposed a comprehensive model to predict theroller skew motion in TRBs He found that the roller skew isgreatly inuenced by traction at the angeroller-end con-tact and by the roller-end geometry A three-dimensionaldynamic model of the double row TRB of a certain type ofhigh-speed train was established by Gai and Zhang [12] andthe roller contact stress and the cage stability were simulatedHowever the lubrication state of axlebox bearings was nottaken into account Compared with other rolling bearingsthe research on dynamics of grease-lubricated TRBs isrelatively decient

e objective of this study was to present an accurateanalysis of dynamic response of the tapered roller bearing Forthe beginning operating stages of grease-lubricated taperedroller bearings considering dynamic interactions in thebearing contacts and grease lubrication theories a multibodycontact dynamic model of TRB under the grease-lubricatedcondition was established e bearing dynamics eg theroller tilt and skew and the bearing slip was analyzed eimpacts of speed preload temperature and grease rheo-logical properties on the bearing dynamics were studied

2 Dynamic Analysis Model

For the sake of accurately describing the relative position andmovement of each component of the TRB an inertial co-ordinate system o-xyz was established As shown in Figure 1the origin coincides with the mass center of the bearing andthe o-x axis coincides with the axis of the bearingsrsquo shaft Abody-xed coordinate system oc-xcyczc is dened for the cageand its origin is the mass center of the cage e roller body-xed coordinate system is or-xryrzr the origin is the rollermass center and the or-xr axis along the roller axis Becausethe roller is prone to bear unbalance moment causing theroller to rotate abnormally two harmful but inevitablemovements of rollers are emphasized tilt and skewe tilt isnormally referred to the roller rotation about or-yr axis andthe skew is the roller rotation about or-zr axis

21 RollerRaceway Interaction When the roller tilting orskewing the interaction between the roller and raceway willvary along the contact line As shown in Figure 2 the roller isdivided into several slices and each slice interaction with the

raceway is calculated independently en the total contactload between the roller and raceway can be obtained byintegrating these local interactions

In order to conrm the contact force at point P on slicesurface the gap or interference at point P should be de-termined rst In Figure 2 let rib and rir represent the masscenter positions of the race and roller in the inertial framerespectively And rrgm and rbgm are respectively the geometriccenter position vectors of the roller and race relative to theirmass centers en geometric center position rbbr of the kthslice relative to the race center can be expressed as

rbbr Tib rir minus rib + Tirprime r

rgm + Tirprime x

kr 0 0[ ]

T( )minus rbgm (1)

where xkr is the slice position at the roller axis Tib is thetransformation matrix (Eulerrsquos rotation matrix) between theinertial and race body-xed coordinate (ob-xbybzb) andTir isthe transformation matrix between the inertial and rollercoordinate system

e azimuth angle ψ of the slice relative to the race canbe dened by components rbbr2 and rbbr3 of rbbr us

ψ a tanminusrbbr2rbbr3

( ) (2)

en the transformationmatrixTba(ψ 0 0) between therace and slice azimuth coordinate system is known Assumec is the azimuth angle of the point P in the coordinate plane

x y

z xc

yc

zc

o oprime

orxr

zr

yr

Figure 1 Bearing coordinate system

oy

z

x

obyb

zb

xb

oryr

zrxr

xrk

αo

ααi

rgrm

rgbm

rri

rbi

ξ

zprime

yprimeP

Figure 2 Geometric interaction between the roller and raceway

2 Shock and Vibration

or-yrzr and ς is the roller radius at the axial position xkr In the

roller coordinate system the position of the point P relativeto the roller center is

rrpb xkr minusς sin c ς cos c1113960 1113961

T (3)

In the slice azimuth coordinate system the position ofthe point P relative to the race center is

rba Tba rbbr + TibTirprime Trpb1113960 1113961 (4)

1e c should satisfy the condition that the yba directioncomponent rba2 of rba is zero

rba2 rabr2 + T21xkr minusT22ς sin c + T23ς cos c 0 (5)

where rabr2 is the component of Tbarbbr in the yba directionand T21T22T23 are components of T TbaTibTirprime

If ϕ a tan(ςT23ςT22) is assumed the above formulacan be changed to

c ϕ + a sinrba2 + T21x

kr

ς T222 + T2

23( 111385712

or c π + ϕ + a sinrba2 + T21x

kr

ς T222 + T2

23( 111385712

(6)

For the roller is in contact with the inner raceway thevalue of c should make the value of rba3 smaller in the zbadirection if the roller contacts with the outer raceway the c

should make rba3 larger in the zba direction 1en the in-terference between the kth slice and the raceway can beconfirmed by subtracting the race radius from the aboveposition vector and the result may be transformed toa contact coordinate system to compute the interaction δnormal to the contact plane Symbolically

δ rba3 minus ξ rba1ψ( 1113857( 1113857cos αp (7)

where αp is the contact angle and ξ is the radius of the racewayat the point P If the value of δ is negative it indicates thatthere is no contact if not it means there is contact

1en the position of the point P can be rewritten as

rrpb xkr minus(ςminus 05δ)sin c (ςminus 05δ)cos c1113960 1113961

T (8)

Assume that vib[ _xb _yb _zb]T and ωbb[ωbxωbyωbz]

T arerespectively the velocity and the angular velocity of the raceand vir[ _x _r _θ]T and ωr

r[ωrxωryωrz]T are the velocity and

the angular velocity of the roller respectively In the contactcoordinate system the velocities at the point P on theraceway and roller are

vcb TacTia1113876vib + rir minus r

ib + Tirprime r

rpb + Tirprime r

rgm + Tibprime r

bgm1113872 1113873

middot Tibprime ωbb minus[ _θ 0 0]

T1113872 11138731113877

vcr Tac Tia Tirprime ωrr times Tirprime r

rpb + Tirprime r

rgm1113872 11138731113872 1113873 +[ _x 0 _r]

T1113960 1113961

(9)

where Tac(0 αp 0) is the transformation matrix between theazimuth and contact coordinate system

1en the velocity at the point P on the raceway relativeto the roller is

vcbr vcb minus vcr (10)

For the line contact between the kth slice and theraceway assuming that rollers behave as elastic half spacethe normal force q at the point P is

q kwδ109

+ cwvcbr3 (11)

where kw is the Hertzian contact stiffness kw

0356Eprimenminus1s l89e [13] Eprime is the equivalent elastic modulus ofthe roller and raceway ns is the total number of slices le isthe effective length of the roller cw is the viscous dampingcoefficient cw 15αekwδ109 [14 15] and αe is related to therestitution coefficient for steel bronze or ivory αe 008 d032 sm [14]

1e grease lubricated condition of the TRB is consideredhere In practice the grease properties change over time (byoverrolling oil bleeding starvation shearing etc) whichaffect the bearing performance In order to simplify thedynamic model it is assumed that an ideal elastohy-drodynamic lubrication state at the rollerraceway contactBased on the grease EHL theory the grease film tractionforces at the point P are [16]

fr πleεφ0Rprime

12

212nsh120

3vcb2 minus v

cr2( 1113857

fb πleεφ0Rprime

12

212nsh120

3vcr2 minus v

cb2( 1113857

(12)

where φ0 is the grease plastic viscosity at atmosphericpressure Rprime is the equivalent radius of the kth slice andthe raceway h0 is the minimum grease film thickness h0is calculated by using the model in [17] ε is related to theHertzian contact half width b film thickness h0 andradius Rprime

ε 2π

a tan212b

Rprime12

h120

⎛⎝ ⎞⎠minus212bRprime

12h120

Rprimeh0 + 2b2⎡⎢⎢⎣ ⎤⎥⎥⎦ (13)

In order to determine the total contact load between theroller and raceway the integration of local interactions isrequired along the contact line 1e total contact force andtorque acting on the jth roller are

Qaj 1113944

ns

k1TcaF

crjk

Nrj 1113944

ns

k1rrpb + rrgm1113872 1113873 times TcrF

crjk

(14)

1e contact loads acting on the race are

Qij 1113944

ns

k1Ticprime F

cbjk

Nbj 1113944

ns

k1rbpb + Tibr

irb1113872 1113873 times TibTicprime F

cbjk

(15)

Shock and Vibration 3

where

Fcrjkbjk frjkbjk sin αf frjkbjk cos αf plusmnqjk1113960 1113961T

αf a sin vcbr2v

cbr1( 1113857

(16)

22 FlangeRoller-End Interaction In race body-fixed co-ordinate system (ob-xbybzb) the roller-end curvature centerrelative to the race center is

rbbe Tib rir minus rib + Tirprime r

rgm + Tirprime x

re 0 01113858 1113859

T1113872 1113873minus rbgm (17)

where xre is the position of the roller-end curvature center at

the roller axisSince normal force will act on the flange surface and pass

through the roller-end curvature center the above centercan be transformed into the race azimuth coordinate systemby Tba(ψ 0 0) where ψ a tan(rbbe2r

bbe3)

rabe Tbarbbe (18)

As illustrated in Figure 2 the position of the roller-endcurvature center relative to the flange coordinate is

rfef Taf rabe minusTbarbf1113960 1113961 (19)

where Taf(0 β 0) is the transformation between the raceazimuth and flange coordinate β is the flange angle definedas a rotation about the o-y axis in accordance to the right-hand screw rule and rbf locates the flange origin in the racecoordinate

If the radius rs of the roller-end is known the geometricinteraction between the roller-end and flange is simply given by

δ rfef1 minus rs (20)

Definition re is the distance from the bearing apex tothe roller spherical end If re gt rs the flangeroller-endcontact is elliptical contact [11] 1e Hertzian contacttheory can be employed to determine the contact load qf atthe flangeroller-end contact

qf 3212δ32

3n32δ Eprimeρ12 (21)

where nδ is the contact deformation coefficient and 1113936 ρ is thesum of principal curvatures

1e friction at the flangeroller-end contact contains twoparts the friction induced bymicroasperities contact and thetraction force produced by the grease lubricant [18 19]

ff fasp + fgrease (22)

1e friction force induced by the asperities contact is

fasp μaqf expminusBλC

s0

p0qf expminusBλC

(23)

where B and C are related to morphology andmaterial properties of the contact surface s0 and p0 arerespectively the critical shear stress and the yield stressfor most metals s0p0 asymp 02 [18] λ is the ratio of the greasefilm thickness hc to surface roughness 1e film thicknesshc is approximated as [20]

hc 18035 times 10minus2Mminus05066Lminus11869

(24)

where

M qf

Eprimer2s

Eprimersφ0uf

1113888 1113889

34

L αEprimeφ0uf

Eprimers1113888 1113889

14

(25)

uf is the EHL entrainment speed uf (uflange + uend)2For the analysis of the grease traction force the

HerschelndashBulkley flow model is adopted

τ τy + φ|D|n (26)

where τ is the grease shear stress τy is the yield stress φ isthe plastic viscosity φ φ0eαp p is the contact pressure thedistribution of p can be approximated by the Hertziancontact pressure α is the viscosity-pressure coefficient nis the power law exponent and D is the shear rate D ≌(uflange minus uend)hc 1en the traction force due to the greaseshear stress is obtained by the integration

fgrease 1113946a

minusa1113946

b

minusbτ(x y)x dx dy (27)

1en the loads acting on the jth roller are

Qafj Tafprime F

fff Tafprime qfj ffj 01113960 1113961

T

Nrfj rrfr + rrgm1113872 1113873 times TfrF

ffj

(28)

1e loads acting on the flange are

Qifj minusTftF

ffj

Nbfj rbfb + rbgm1113872 1113873 times TibQ

ifj

(29)

where rrfr and rbfb are positions of the contact point relative to

the roller and inner-ring centers respectively

23 Forces Acting on the Cage During the cage and rollersrotation the master-slave relationship of rotating cage-rollerassembly will change over time due to different velocitiesand displacements As shown in Figures 2 and 3 the locationof the jth roller geometric center in the cage pocket co-ordinate system (op-xpypzp) is

rpcr Tcp Tic rir minus ric + Tirr

rgm1113872 1113873minus r01113872 1113873 (30)

where ric is the position of the cage mass center in theinertial coordinate system Tcp is the transformationmatrix from the cage-fixed coordinate system (oc-xcyczc) tothe pocket coordinate system r0 locates the pocket centerin the cage coordinate system and Tic is the trans-formation matrix between the inertial and cage coordinatesystem

Now similar to the rollerraceway interaction let rrglocates a point Pprime on the kth slice of the roller such that

rrpprime xkr minusς sin c ς cos c1113960 1113961

T (31)

4 Shock and Vibration

In the pocket coordinate system the position of the pointPprime is

rppprimec rpcr + TicTirprime rrpprime (32)

As illustrated in Figure 3 the clearance between thepoint P and the pocket crossbeam is

δ Δδ minus rppprimec2 (33)

where Δδ is the initial clearance between the roller andcrossbeam Δδ (lc minus dw)2 lc is the average width of thepocket and dw is the average diameter of the roller

For the modeling of the assembly interactions assumingthat there is an excess of the grease in the cage-roller as-sembly a critical value Δh0 of the grease lm thickness isassumed for the contact state transitionWhen δ ge Δh0 thereare only the hydrodynamic eect between the kth slice andpocket crossbeam and no Hertzian contact [9 21] eminimum grease lm thickness h0 δ e hydrodynamicpressure qc and the grease lm traction force fc between thekth slice and crossbeam are

qc qcehl keδ + ce _δ

fc fcehl 3πεφ0uleς12

212h120

(34)

where [17 22]

ke zf h0( )zh0

ce 6πεφ0leς32

212nsh320

f h0( ) 29παφnsξ1un

9EPrimeh6n+20 ςminus3le4 +

2n

( )n

+ξ2λnτyh

n0

ξ1φun[ ]

(35)

where u is the entrainment speed ξ1 ξ2 are the constantsassociated with the power law exponent n [17] and ce isgrease lm damping [22]

When δ lt Δh0 it means that the contact between the kthslice and crossbeam is in a boundary state So the Hertziancontact is included in the contact e minimum grease

lm thickness h0 Δh0 e Hertzian contact deformationis

δh Δh0 minus δ (36)

e contact force qc and the friction force fc between theroller and crossbeam are

qc khδh + ch _δh + qcehlfc μbdqc + fcehl

(37)

where kh is the linearized Hertzian contact stiness [23] ch isthe Herbert viscous damping coecurrencient [24] and μbd is thetraction coecurrencient under boundary lubrication [9]

en the load at the contact point Pprime on the kth slice inthe contact coordinate system is

Fccjk 0 qcjk fcjk[ ]T (38)

Using the processing method of the rollerracewayinteraction the load Fccjk can also be transferred tothe inertial coordinate and the rollercage body-xedcoordinate respectively

Owing to the smaller sliding speed between the roller-end and the side beam and the smaller curvature of rollerspherical end the EHL lubrication state cannot be eec-tively formed at this contact but the squeeze lm lubri-cation will be formed As shown in Figure 3 the Δδprime is theinitial pocket clearance and ra rpcr1 If dradt ge 0 thesmall end contacts with the side beam if dradt lt 0the large end contacts Neglecting the grease traction thesqueeze force qs is

qs ks Δδprime minus ra( ) (39)

where [25]

ks minusz

zha

dlsφ0k3s

h3a

dradt

( ) (40)

where ha is the distance from the side beam to the end dl andds are the diameters of the large and small end respectivelyand hs is the thickness of the side beam

en the loads acting on the jth roller are

αc

Δδprime

Δδ

δ

ha

ke

ce

ks

xr

yr

xp

ypop

or

Side beam

Cros

sbea

m

Figure 3 Interaction between the roller and cage

Shock and Vibration 5

Qasj qsj 0 0[ ]T

Nrsj rrsr + rrgm( ) times TarQ

asj

(41)

e loads acting on the cage areQi

sj minusTaiQasj

Ncsj r0 + Tcpprime r

pcr + Trc rrsr + rrgm( ) times TicQ

isj[ ]

(42)

where rrsr is the position of the contact point in the rollerbody-xed coordinate and Tar and Tai are respectivelythe transform from the azimuth to roller and inertialcoordinate

According to geometric characteristics of the cage andguide ring it can be considered as a nite journal bearinglubrication condition between the guide surface and cagecentering surface In Figure 4 the clearance δprime between thecage and guide ring is

δprime Δe minus e (43)

where Δe is the initial clearance Δe (ry minus rd)2 rd ryare respectively the radii of the centering surface andguide surface and e is cage displacement in the radial plane

Similarly assume that Δh0prime is the critical value of thegrease lm thickness When δprime ge Δh0prime the contact force qicand friction torque Nic on the cage are

qic qicehl keprimeδ + ceprime _δprime

Nic Nicehl πεφ0br

12ic rd

212h1203ωbxry minusωcxrd( )

(44)

where ric is the equivalent radius and ωcx ωbx arerespectively the angular velocities of the cage and guidering e calculation of the grease lm stiness keprime anddamping ceprime is the same as that at the rollercrossbeamcontact

When δprime lt Δh0prime the elastic deformation between the cageand guide ring is δhprime δprime minus h0 and the minimum grease lmthickness h0 Δh0prime e qic and Nic are

qic khprimeδhprime + chprimeδhprime + qicehlNic μbdqicrd +Nicehl

(45)

where khprime is the contact stiness and chprime is the viscousdamping coecurrencient e calculation of khprime and chprime is the sameas that in Equation (35) μbd is the traction coecurrencient underboundary lubrication

In order to simplify themodel the load of the cage on theguide ring is ignored in contrast to large external load on theguide ring en the loads acting on the cage are

Qiic 0 qic sinφic qic cosφic[ ]T

Ncic plusmnNic 0 0[ ]

(46)

If the TRB is lubricated by the oil the eect of oil-gasmixture on the cage and rollers may not be overlookedHowever if the bearing is lubricated by the grease it seemsimpossible to have the grease-gas mixture eect on the

dynamics of the cage and rollers So the eect of the grease-gas mixture is not considered in the dynamic model

24DynamicModel In real application also the surroundings(inertiadamping of shaft assembly and housing) are expectedto inuence the TRB dynamics ese eects are neglected inthe modeling Generally it is convenient to consider thetranslational moving of rollers in the cylindrical coordinatesystem while the Cartesian coordinate system is convenient forthe cage and racee translationalmoving of bearing parts canbe simply described by Newtonrsquos laws such that

Meuroδ Q (47)

e generalized force Q is obtained by superimposingthe forces from Section 2 e centrifugal force and thegyroscopic moment should be superimposed on the gen-eralized forceQ for the roller For the cage and race the massmatrices Mb and Mc are

Mbc diag mbc mbc mbc[ ] (48)

For the special case of conical rollers Mr is

Mr diag mr mrrrj mr[ ] (49)

Both rollers and the cage have six degrees-of-freedomUnlike the translational motion (in three directions) therotating of the bearing parts are described in their own body-xed coordinates For rollers and the cage using the Eulerdynamic equations such that

Ix _ωx minusωyωz Iy minus Iz( )

Iy _ωy minusωzωx Iz minus Ix( )

Iz _ωz minusωxωx Ix minus Iy( )

Nx

Ny

Nz

(50)

where Ix Iy and Iz are inertia principalmoments of the cage orrollers the total momentN is also obtained by superimposingthe forces from Section 2 ω[ωx ωy ωz] is the angular velocityin the inertial coordinate systeme relationship between theangular velocity ωxed in the body-xed coordinate systemand ω is as follows B is the Eulerrsquos rotation matrix

∆y

yb

zb

yc

ωcx

ωbx

φicNic

δprime

zc

oc

∆z eob

qic

Guidering

Cage

Roller

Figure 4 Interaction between the cage and guide ring (inner-ringguided)

6 Shock and Vibration

ωfixed Bω

_ωfixed dBdt

Bminus1 _ω

(51)

Admitting the outer ring as macroscopically stationaryand a rotating inner ring with no torque load it resultsfour degrees of freedom of inner ring moving in the x yand z directions and constant rotating about the shaftaxis (o-x)

3 Numerical Method

1e bearing geometry and material parameters and greasemain rheological parameters (30degC) are shown in Table 11e cage is an inner-ring guided type and its material ispolyamide 1e polyurea grease was adopted [26] 1efourth-order RungendashKutta method was used to dissolvetransient responses of the TRB on MATLAB To ensure theconvergence of simulation initial values of displacementvelocity and load are obtained by the quasidynamicmethod

4 Results and Discussion

For the complicated dynamic response of TRBs it is essentialto consider several typical and harmful movements of thebearing in the process of dynamic analysis such as the rollerskew and tilt bearing slip and so on 1e slip rate of theroller and cage is defined as

|theoretical speedminus actual speed|

theoretical speedtimes 100 (52)

For the roller the speed refers to the revolution speedfor the cage refers to the rotational speed

41 Mass Center Trajectory and Velocity 1e trajectoriesand velocities of mass centers of the main bearing parts areshown in Figure 5 1e axial preload is not applied to theinner ring but limited to the degree of freedom of the innerring in the axial direction 1e cage initial rotation speed isthe same as the roller initial revolution speed Due to thechanging position of rollers the distribution of rollers issymmetrical or asymmetric with respect to the o-z axis1erefore the inner ring vibrates slightly in the o-y di-rection 1e radial and axial displacement (relative to theinitial position) of the roller exhibit a periodic fluctuationfor the bearing is in a semicircle-loaded state under pureradial load conditions When the rollers alternately enter theloaded and unloaded zones the load on rollers will showperiodic changes resulting in a regular variation in dis-placement 1e cage whirling is approximately a circle Fora number of uncertain rollers acting on the cage the cagespeed does not show a periodic change

42 Rotational Speed Influence In this study the tilt andskew of conical rollers are regarded as the primary eval-uation variables for the dynamic performance of the TRB

1e roller skew is mainly generated from the tangentialfriction force on the flangeroller-end and rollerracewaycontacts while the tilt motion is mainly induced by thenormal contact force on those contacts According to thelubrication theory the bearing speed will be a key factoraffecting the load case in bearing contacts

1e influence of the bearing speed on the roller tilt andskew (relative to the initial position) is shown in Figure 61e degree of freedom of the inner ring in the axial directionis also limited At each speed level the roller tilt anglepresents a relatively regular periodic vibration 1e tilt anglein the unloaded zone is obviously larger than that in theloaded zone 1is is due to the larger gap between the rollerand raceways in the unloaded zone when only the radicalload is applied and rollers in unloaded zone seem to beldquorelaxedrdquo As the speed increases the maximum of the tiltangle almost does not change while the average tilt anglerises from minus014 times 10minus4 rad to minus040 times 10minus4 rad indicatingthat the bearing speed affected the roller tilt It can beconfirmed that the gyroscopic moment is one of the reasonsthat cause the rising of the roller tilt angle for the gyroscopicmotion is intensified with the bearing speed increasingAlthough the gyroscopic moment increases as the bearingspeed increases it is still relatively smaller than theforcetorque in the rollerraceway contact So for the radialload is constant the maximum tilt angle of the roller changeslittle with the speed

As it is shown in Figure 7 the skew angle both in theloaded zone and in the unloaded zone is relatively large at thespeed of 2 krpm In the entire speed range the roller skewangle in the loaded zone is greater than that in the unloadedzone For the rollers is ldquorelaxedrdquo in the unloaded zone the

Table 1 Geometry material and rheological parameters

Parameter ValueNumber of rollers 17Roller length lemm 4206Roller-end sphere radius rsmm 13620Diameter of roller large end dlmm 1785Diameter of roller small end dsmm 2476Outer contact angledeg 15Inner contact angledeg 5Flange contact angledeg 83Outer raceway radiusmm 9952Inner raceway radiusmm 7848Guide land clearancemm 05Guide land widthmm 493Cage outer diametermm 20576Cage inner diametermm 19610Cage thickness hsmm 5Pocket clearance Δδmm 027Elastic modulus of rollerraceGPa 205Poissonrsquos ratio of rollerrace 03Elastic modulus of cageGPa 83Poissonrsquos ratio of cage 028

Grease rheological parameters (30degC)Yield stress τyPa 3518Plastic viscosity φPamiddotsn 844Power law exponent n 07196Pressure-viscosity coefficient αm2middotNminus11 4662 times 10minus8

Shock and Vibration 7

ndash6 ndash4 ndash2 0 2 4 60

3

6

9

12

15

Trajectory of the inner ring

z-di

rect

ion

disp

lace

men

t (μm

)

y-direction displacement (μm)

(a)

Displacement of the roller

0 003 006 009 012 015 018ndash20

ndash10

0

10

Radi

al (μ

m)

t (s)

0 003 006 009 012 015 018ndash20

10

40

70

Axi

al (μ

m)

(b)

Trajectory of the cage

ndash06 ndash04 ndash02 00 02 04 06ndash06

ndash04

ndash02

00

02

04

06

z-di

rect

ion

disp

lace

men

t (m

m)

y-direction displacement (mm)

(c)

Cage rotation speed

0 003 006 009 012 015 018102104106108110

Cage

(krp

m)

t (s)

Roller revolution speed0 003 006 009 012 015 018

102104106108110

Rolle

r (kr

pm)

(d)

Figure 5 1e trajectory and velocity of mass center Fr 20 kN Fa 0 ωbx 24 krpm

000 003 006 009 012 015 018ndash9

ndash6

ndash3

0

3

6

9

101

Tilt angle

ωbx = 2krpm

t (s)

Tilt

angl

e (10

ndash4 ra

d)

(a)

Tilt angle

ωbx = 3krpm

000 003 006 009 012 015 018ndash9

ndash6

ndash3

0

3

6

9

101

Tilt

angl

e (10

ndash4 ra

d)

t (s)

(b)

Figure 6 Continued

8 Shock and Vibration

skewmoment of the raceway and flange acting on the roller issmaller than that in the loaded zone It shows obvious pe-riodicity when the speed reaches 3 krpm or more1e averageskew angle is increased from 306 times 10minus3 rad017 deg to 447 times

10minus3 rad025 deg 1e increase of bearing rotating speedenlarges the centrifugal force of rollers which makes contactforces at the rollerouter raceway and flangeroller-end

increase As a result of combined contribution of racewayfriction and the flange friction the roller skew angle is greatlyincreased Compared with the experimental result measuredby Falodi et al in which theymeasured the average skew anglebetween 015 and 045 deg [27] 1e average skew angle inFigure 7 is between 017 and 025 deg and at an order ofmagnitude compared to the experimental result 1e

Tilt angle

ωbx = 4krpm

Tilt

angl

e (10

ndash4 ra

d)

000 003 006 009 012 015 018ndash9

ndash6

ndash3

0

3

6

9

101

t (s)

(c)

Tilt angle

ωbx = 5krpm

Tilt

angl

e (10

ndash4 ra

d)

000 003 006 009 012 015 018ndash9

ndash6

ndash3

0

3

6

9

101

t (s)

(d)

Figure 6 1e tilt angle of the conical roller 1 loaded zone 0 unloaded zone Fr 20 kN Fa 0 ωbx 2sim5 krpm

000 003 006 009 012 015 018ndash8

ndash3

2

7

12

17

22

101

Skew

angl

e (10

ndash3 ra

d)

t (s)

ωbx = 2krpm

Skew angle

(a)

Skew

angl

e (10

ndash3 ra

d)

000 003 006 009 012 015 018ndash4

0

4

8

12

16

101

t (s)

ωbx = 3krpm

Skew angle

(b)

Skew

angl

e (10

ndash3 ra

d)

000 003 006 009 012 015 018ndash4

0

4

8

12

16

101

t (s)

ωbx = 4krpm

Skew angle

(c)

Skew

angl

e (10

ndash3 ra

d)

000 003 006 009 012 015 018ndash4

0

4

8

12

16

0 11

t (s)

ωbx = 5krpm

Skew angle

(d)

Figure 7 1e skew angle of the conical roller 1 loaded zone 0 unloaded zone Fr 20 kN Fa 0 ωbx 2sim5 krpm

Shock and Vibration 9

changing trend of the skew angle with speed is also consistentwith Yangrsquos results

1e bearing slip is a destructive motion that easily leadsto scuffing welding and overwear of bearing surfaces 1ebearing slip usually occurs at high-speed and light-loadconditions 1e influence of the bearing speed on the slipof the roller and cage is exhibited in Figures 8 and 9

1e diagrams depicted in Figure 8 indicate that theroller slip is becoming more and more serious as thebearing speed is raised 1is variation trend is similar tothat in [7] With the bearing speed increases the roller slipgradually presents a periodic change rule especially whenthe speed is up to 3 krpm or more 1e roller slip in loadedand unloaded zones is not the same In the unloaded zonethe roller slip is more serious than that in the loaded zoneAt the speed of 5 krpm the bearing is in a state of seriousslipping In Figure 9 the cage slip is aggravated with thebearing speed increase 1ere is no periodic change in thecage slip rate 1is is because when the cage is whirlinga number of uncertain rollers may act on the cage whichcan lead to an irregular motion of the cage At the samespeed level the maximum amplitude of the slip rate of thecage and roller is basically the same for the dependentrelationship between the cage and roller

43 Axial Preload Influence Axial preload is not only a keyfactor affecting the bearing dynamics but one of the mainmeans to ensure the normal service life of the bearing Inorder to familiarize the influence of the axial preload onthe roller tilt and skew different axial preloads wereapplied to the TRB under the condition that the radial loadis zero

As shown in Figures 10 and 11 after entering the smoothrunning phase as the axial preload varies from 10 μm to40 μm the average value of the roller tilt rises from minus027 times

10minus4 rad to minus096 times 10minus4 rad 1e greater the preload themore obvious the roller tends to tilt 1is is because as thepreload increases the negative moment of the flange actingon the roller-end becomes larger thus causing the roller tiltto assume a negative tendency

Except for the condition that the preload is 10 μm theroller skew has larger amplitude at the beginning and thenthe amplitude decreases gradually and tends to be stable Inthe smooth running phase as the preload increases theroller skew angle increases 1is trend is in line with theresults in [8 27] 1e average skew angle is increasedfrom 423 times 10minus3 rad024 deg to 550 times 10minus3 rad031 degBoth the roller tilt and skew present a more regular vibrationthroughout the bearing operation this is due to symmetry ofthe bearing on the radial plane and pure preloading con-dition the loaded and unloaded zones do not exist in thebearing and loads on rollers almost change a little in anyposition

1e changes of the slip rate of the roller and cage with thepreload are presented in Figures 12 and 13 As the bearingpreload increases the bearing slip decreases in the smoothrunning phase 1e average slip rate of the roller decreasesfrom 00189 to 00057 and that of the cage decreases from

00189 to 00055 indicating that the appropriate bearingpreload can effectively prevent bearing slipping failure 1eresult is similar with the conclusion in [7 28] 1e influenceof the preload on the bearing slip reduced with the increaseof the preload value 1e average slip rate of the roller andcage when the preload is 40 μm changes little compared with00073 and 00076 when the preload is 30 μm Obviouslyincreasing the amount of preload will help to decrease thebearing slip but excessive preload will inevitably reduce theservice life of bearings 1erefore to meet the bearing lifeand service requirements reasonable selection of preload issignificant

44Grease Physical Properties Influence In view the fact thatthe calculation of the traction force stiffness and damping atthe bearing contacts is mainly based on the plastic viscosity φand yield stress τy of the grease For a clearer understandingof the relationship between the bearing dynamics and thegrease used grease physical properties influence on bearingdynamics is analyzed 1e temperature pressure speed andother surroundings have significant effects on the greasephysical properties For the sake of simplification the im-pacts of these factors on the grease physical properties areignored In order to highlight the effect of a certain rheo-logical parameter on the bearing performance only oneparameter was given an appropriate change while otherrheological parameters remain unchanged 1e effects areshown in Figures 14 and 15 After the bearing enters thestable running state (after 01 s) the average value of theangle (tilt and skew) is used as an indicator of bearingoperating state

1e effect of the plastic viscosity on the roller and cagemotions are shown in Figure 14(a) As the plastic viscosityincreases the tilt angle and skew angle increases and de-creases respectively For the grease film thickness increaseswith the increase of the plastic viscosity the load ratio ofmicroasperities at the flangeroller-end contact is reduced1en it indirectly leads to the decrease of the skew mo-ment at the flangeroller-end contact and the roller skewangle Due to the smaller skew angle the roller misalignmentwill also be small 1erefore the normal force at therollerraceways and flangeroller-end contacts will havea greater contribution to the roller tilt 1erefore the resultshows that the roller tilt angle increases with the plasticviscosity increase In Figure 14(b) the bearing slip decreaseswith the grease plastic viscosity increase As the plasticviscosity increases the grease film thickness at therollerraceways and flangeroller-end contacts increase Sothe friction force at these contacts is reduced which maylead to the decrease of the bearing slip1e bearing slip is theresult of the combined effect of raceway friction and theflange friction

Compared with the impacts of the plastic viscosity of thegrease on the motions of the roller and cage the influence ofthe grease yield stress on the roller tilt roller skew andbearing slip is relatively smaller as shown in Figure 15 Forthe yield stress has a negligible effect on the film thickness inall practical cases [9]

10 Shock and Vibration

000 003 006 009 012 015 01800

05

10

15

20

25

30

Slip

rate

of t

he ro

ller (

)

t (s)

01000 01025 01050 01075 011ndash001

001

003

005

007ωbx = 2krpm

Slip rate of the roller

(a)

000 003 006 009 012 015 01800

02

04

06

08

Slip

rate

of t

he ro

ller (

)

t (s)

ωbx = 3krpm

Slip rate of the roller

(b)

000 003 006 009 012 015 01800

05

10

15

20

25

30

Slip

rate

of t

he ro

ller (

)

t (s)

ωbx = 4krpm

Slip rate of the roller

(c)

000 003 006 009 012 015 0180

2

4

6

8

10

Slip

rate

of t

he ro

ller (

)

t (s)

ωbx = 5krpm

Slip rate of the roller

(d)

Figure 8 1e slip rate of the conical roller Fr 20 kN Fa 0 ωbx 2sim5 krpm

000 003 006 009 012 015 01800

05

10

15

20

25

30

Slip

rate

of t

he ca

ge (

)

01000 01025 01050 01075 011ndash000

001

002

003

t (s)

Slip rate of the cage

ωbx = 2krpm

(a)

000 003 006 009 012 015 01800

05

10

15

20

25

30

Slip

rate

of t

he ca

ge (

)

t (s)

Slip rate of the cage

ωbx = 3krpm

(b)

Figure 9 Continued

Shock and Vibration 11

45 Temperature Influence Due to the improper in-stallation overwear or not timely cooling the operatingtemperature of TRBs will change more or less Sometimesit may lead to heat accumulation or thermal failure ofbearings 1e most direct effect of the temperature ischanges of the grease physical properties 1e rheologicalparameters of the greases at different temperatures in

Table 2 are quoted from the data in reference [26] 1eeffect of temperature on the dynamics of the bearing wasstudied 1e impact of temperature on the bearing struc-ture is not considered here

1ree typical greases were used such as the calciumgrease lithium grease and polyurea grease 1e largestdifference in the composition of three greases lies in the

000 003 006 009 012 015 01800

05

10

15

20

25

30

35

Slip

rate

of t

he ca

ge (

)

t (s)

Slip rate of the cage

ωbx = 4krpm

(c)

000 003 006 009 012 015 0180

2

4

6

8

10

Slip

rate

of t

he ca

ge (

)

t (s)

Slip rate of the cage

ωbx = 5krpm

(d)

Figure 9 1e slip rate of the cage Fr 20 kN Fa 0 ωbx 2sim5 krpm

000 003 006 009 012 015 018ndash2

0

2

4

6

8

10

12

0100 0101 0102ndash15ndash10ndash05

00051015

δa = 10μm

Tilt anglet (s)

Tilt

angl

e (10

ndash4 ra

d)

(a)

000 003 006 009 012 015 018ndash2

0

2

4

6

8

10

12

0100 0101 0102ndash20ndash15ndash10ndash05

000510

Tilt anglet (s)

Tilt

angl

e (10

ndash4 ra

d)δa = 20μm

(b)

000 003 006 009 012 015 018ndash3

0

3

6

9

12

15

0100 0101 0102ndash25ndash20ndash15ndash10ndash05

0005

Tilt angle

t (s)

Tilt

angl

e (10

ndash4 ra

d)

δa = 30μm

(c)

000 003 006 009 012 015 018ndash3

0

3

6

9

12

15

0100 0101 0102ndash30ndash23ndash16ndash09ndash02

05

Tilt angle

t (s)

Tilt

angl

e (10

ndash4 ra

d)

δa = 40μm

(d)

Figure 10 1e tilt angle of the conical roller Fr 0 δa 10sim40 μm ωbx 25 krpm

12 Shock and Vibration

000 003 006 009 012 015 01810

26

42

58

74

90

Skew angle

t (s)

Skew

angl

e (10

ndash3 ra

d)

δa = 10μm

(a)

000 003 006 009 012 015 01810

26

42

58

74

90

Skew angle

t (s)

Skew

angl

e (10

ndash3 ra

d)

δa = 40μm

(b)

000 003 006 009 012 015 01810

26

42

58

74

90

Skew angle

t (s)

Skew

angl

e (10

ndash3 ra

d)

δa = 30μm

(c)

000 003 006 009 012 015 01810

26

42

58

74

90

Skew angle

t (s)

Skew

angl

e (10

ndash3 ra

d)

δa = 40μm

(d)

Figure 11 1e skew angle of the conical roller Fr 0 δa 10sim40 μm ωbx 25 krpm

000 003 006 009 012 015 018000

002

004

006

008

Slip

rate

of t

he ro

ller (

)

Slip rate of the roller

t (s)

δa = 10μm

(a)

000 003 006 009 012 015 018000

002

004

006

008

Slip

rate

of t

he ro

ller (

)

Slip rate of the roller

t (s)

δa = 20μm

(b)

Figure 12 Continued

Shock and Vibration 13

types of thickeners In Figure 16(a) under the lubricationof the calcium and polyurea greases the roller tilt angledecreases with the temperature increase Under the lithiumgrease lubrication the roller tilt angle increases with theincrease in temperature In the whole temperature rangethe tilt angle under the lithium grease is generally larger

than that under others With the increase in temperaturethe roller skew angle under the polyurea grease increaseswhile the roller skew angle under the other two types ofgreases decreases 1at the tilt and skew angle under thedifferent greases and temperature show different trends isa result of multiple factors One of the factors is that the

000 003 006 009 012 015 018000

002

004

006

008

Slip

rate

of t

he ro

ller (

)

Slip rate of the roller

t (s)

δa = 30μm

(c)

000 003 006 009 012 015 018000

002

004

006

008

Slip

rate

of t

he ro

ller (

)

Slip rate of the roller

t (s)

δa = 40μm

(d)

Figure 12 1e slip rate of the conical roller Fr 0 δa 10sim40 μm ωbx 25 krpm

000 003 006 009 012 015 01800

05

10

15

20

25

30

Slip rate of the cage

δa = 10μm

01000 01023 01046 010690010

0015

0020

0025

0030

Slip

rate

of t

he ca

ge (

)

t (s)

(a)

Slip rate of the cage

δa = 20μm

000 003 006 009 012 015 01800

05

10

15

20

25

30

0110 0111 0112 0113 0114 0115ndash001

000

001

002

003

004

Slip

rate

of t

he ca

ge (

)

t (s)

(b)

Slip rate of the cage

δa = 30μm

000 003 006 009 012 015 01800

05

10

15

20

25

30

0110 0111 0112 0113 0114 0115000000030006000900120015

Slip

rate

of t

he ca

ge (

)

t (s)

(c)

Slip rate of the cage

δa = 40μm

000 003 006 009 012 015 01800

05

10

15

20

25

30

0110 0111 0112 0113 0114 0115000000030006000900120015

Slip

rate

of t

he ca

ge (

)

t (s)

(d)

Figure 13 1e slip rate of the cage Fr 0 δa 10sim40 μm ωbx 25 krpm

14 Shock and Vibration

244 844 1444 2044 2644 3244 3844ndash52

ndash51

ndash50

ndash49

ndash48

ndash47

Plastic viscosity (Pamiddotsn)

00

16

32

48

64

80

Tilt angleSkew angle

Tilt

angl

e (10

ndash5 ra

d)

Skew

angl

e (10

ndash3 ra

d)

(a)

844 1444 2044 2644 3244 38440000

0004

0008

0012

0016

0020

Slip

rate

of t

he ro

ller (

)

Plastic viscosity (Pamiddotsn)

ndash0006

0000

0006

0012

0018

Slip

rate

of t

he ca

ge (

)

Slip rate of the rollerSlip rate of the cage

(b)

Figure 14 Plastic viscosity influence Fr 0 Fa 15 kN ωbx 28 krpm (a) 1e conical roller tilt and skew (b) Slip rate of the conicalroller and cage

2518 3018 3518 4018 4518 5018ndash550

ndash536

ndash522

ndash508

ndash494

ndash480

Yield stress (Pa)

350

370

390

410

430

450

Tilt angleSkew angle

Tilt

angl

e (10

ndash5 ra

d)

Skew

angl

e (10

ndash3 ra

d)

(a)

Slip rate of the rollerSlip rate of the cage

2518 3018 3518 4018 4518 50180009

0010

0011

0012

0013

0014

0015Sl

ip ra

te o

f the

rolle

r (

)

Yield stress (Pa)

0009

0010

0011

0012

0013

0014

0015

Slip

rate

of t

he ca

ge (

)

(b)

Figure 15 Yield stress influence Fr 0 Fa 15 kN ωbx 28 krpm (a)1e conical roller tilt and skew (b) Slip rate of the conical roller andcage

Table 2 Relationships between grease physical properties and temperature [26]

Grease TemperaturedegC Yield stress τyPa Plastic viscosity φPamiddotsn Power law exponent n

Polyurea

15 5017 1187 0773130 3518 844 0719650 3055 616 0662170 1885 610 07378

Lithium

15 9045 5435 0305330 8052 4623 0312550 7104 4113 0315170 3398 5296 03203

Calcium

15 9908 6482 0334930 6495 4923 0534450 3361 3059 0525670 2960 3588 04356

Shock and Vibration 15

relationships between the grease rheological parametersand temperature are not simple linear For example theplastic viscosity and the power law exponent do not simplyincrease or decrease with temperature

In Figure 16(b) firstly the roller slip rate shows dif-ferent values due to the different greases secondly theroller slip rate shows a variety of trends with the varioustemperature Under the lubrication of the polyurea greasethe roller slip rate increases with the increase in temper-ature while the roller slip with the calcium and lithiumgreases lubricated shows a decreasing trend In real ap-plication the operating temperature of bearings may not beconstant 1e influence of the temperature on dynamiccharacteristics of grease lubricated bearings needs to befurther explored

5 Conclusions

For the beginning operating stages of grease-lubricatedTRBs the multibody contact dynamic model of TRBs wasestablished 1e impacts of preload temperature and greaserheological properties on the bearing dynamics were ana-lyzed Based on numerical results several conclusions can besummarized

(1) 1e roller tilt angle in the unloaded zone is obviouslylarger than that in the loaded zone while the rollerskew angle in the unloaded zone is smaller than thatin the loaded zone 1e effect of bearing speed on theroller skew is greater than that on the roller tilt Asthe speed increases the roller tilt and skew and thebearing slip become more serious

(2) Bearing preload can effectively reduce the bearingslip but will make the roller tilt angle increase In thestable operation stage of the bearing as the preloadincreases the roller skew angle increases

(3) 1e roller skew angle and the bearing slip rate de-crease with the increase of the grease plastic viscosity1e roller tilt angle increases with the increase of the

plastic viscosity 1e yield stress of the grease haslittle effect on motions of the roller and cage

(4) 1e influence of temperature onmotions of the rollerand cage varies with the type of grease used Whenthe TRB is lubricated by the lithium grease with theincrease of the temperature the roller tilt angle in-creases the roller skew angle and the bearing slip ratedecrease When the polyurea grease is adopted thechange trend of the roller tilt and skew and thebearing slip is just opposite to that when the lithiumgrease is used Under the calcium grease lubricationthe roller tilt and skew angle and the bearing slip ratedecrease with the temperature increase

Data Availability

1e figure data used to support the findings of this studyhave been deposited in the FigShare repository at httpsfigsharecoms15c2556bee83d1adf01d

Conflicts of Interest

1e authors declare that they have no conflicts of interest

Acknowledgments

1e paper was supported by the National Natural ScienceFoundation of China (Grant No 51675427)

References

[1] P M Lugt ldquoA review on grease lubrication in rollingbearingsrdquo Tribology Transactions vol 52 no 4 pp 470ndash4802009

[2] T Cousseau Film thickness and friction in grease lubricatedcontacts Application to rolling bearing torque loss PhDDepartamento De Engenharia Mecanica E Gestao Industrial2013

[3] C T Walters ldquo1e dynamics of ball bearingsrdquo Journal ofTribology vol 93 no 1 pp 1ndash10 1971

10 23 36 49 62 75ndash58

ndash56

ndash54

ndash52

ndash50

ndash48

ndash46

ndash44

CalciumLithiumPolyurea tilt angle skew angle

tilt angle skew angletilt angle skew angle

Temperature (degC)

10

30

50

70

90

110

130Ti

lt an

gle (

10ndash5

rad)

Skew

angl

e (10

ndash3 ra

d)

(a)

10 23 36 49 62 750000

0005

0010

0015

0020

0025

0030

Poly

urea

gre

ase (

)

Temperature (degC)

0000

0001

0002

0003

0004

0005

0006

Calc

ium

lith

ium

gre

ase (

)

CalciumLithiumPolyurea

(b)

Figure 16 Temperature influence Fr 0 Fa 15 kN ωbx 28 krpm (a) 1e conical roller tilt and skew (b) Slip rate of the conical roller

16 Shock and Vibration

or-yrzr and ς is the roller radius at the axial position xkr In the

roller coordinate system the position of the point P relativeto the roller center is

rrpb xkr minusς sin c ς cos c1113960 1113961

T (3)

In the slice azimuth coordinate system the position ofthe point P relative to the race center is

rba Tba rbbr + TibTirprime Trpb1113960 1113961 (4)

1e c should satisfy the condition that the yba directioncomponent rba2 of rba is zero

rba2 rabr2 + T21xkr minusT22ς sin c + T23ς cos c 0 (5)

where rabr2 is the component of Tbarbbr in the yba directionand T21T22T23 are components of T TbaTibTirprime

If ϕ a tan(ςT23ςT22) is assumed the above formulacan be changed to

c ϕ + a sinrba2 + T21x

kr

ς T222 + T2

23( 111385712

or c π + ϕ + a sinrba2 + T21x

kr

ς T222 + T2

23( 111385712

(6)

For the roller is in contact with the inner raceway thevalue of c should make the value of rba3 smaller in the zbadirection if the roller contacts with the outer raceway the c

should make rba3 larger in the zba direction 1en the in-terference between the kth slice and the raceway can beconfirmed by subtracting the race radius from the aboveposition vector and the result may be transformed toa contact coordinate system to compute the interaction δnormal to the contact plane Symbolically

δ rba3 minus ξ rba1ψ( 1113857( 1113857cos αp (7)

where αp is the contact angle and ξ is the radius of the racewayat the point P If the value of δ is negative it indicates thatthere is no contact if not it means there is contact

1en the position of the point P can be rewritten as

rrpb xkr minus(ςminus 05δ)sin c (ςminus 05δ)cos c1113960 1113961

T (8)

Assume that vib[ _xb _yb _zb]T and ωbb[ωbxωbyωbz]

T arerespectively the velocity and the angular velocity of the raceand vir[ _x _r _θ]T and ωr

r[ωrxωryωrz]T are the velocity and

the angular velocity of the roller respectively In the contactcoordinate system the velocities at the point P on theraceway and roller are

vcb TacTia1113876vib + rir minus r

ib + Tirprime r

rpb + Tirprime r

rgm + Tibprime r

bgm1113872 1113873

middot Tibprime ωbb minus[ _θ 0 0]

T1113872 11138731113877

vcr Tac Tia Tirprime ωrr times Tirprime r

rpb + Tirprime r

rgm1113872 11138731113872 1113873 +[ _x 0 _r]

T1113960 1113961

(9)

where Tac(0 αp 0) is the transformation matrix between theazimuth and contact coordinate system

1en the velocity at the point P on the raceway relativeto the roller is

vcbr vcb minus vcr (10)

For the line contact between the kth slice and theraceway assuming that rollers behave as elastic half spacethe normal force q at the point P is

q kwδ109

+ cwvcbr3 (11)

where kw is the Hertzian contact stiffness kw

0356Eprimenminus1s l89e [13] Eprime is the equivalent elastic modulus ofthe roller and raceway ns is the total number of slices le isthe effective length of the roller cw is the viscous dampingcoefficient cw 15αekwδ109 [14 15] and αe is related to therestitution coefficient for steel bronze or ivory αe 008 d032 sm [14]

1e grease lubricated condition of the TRB is consideredhere In practice the grease properties change over time (byoverrolling oil bleeding starvation shearing etc) whichaffect the bearing performance In order to simplify thedynamic model it is assumed that an ideal elastohy-drodynamic lubrication state at the rollerraceway contactBased on the grease EHL theory the grease film tractionforces at the point P are [16]

fr πleεφ0Rprime

12

212nsh120

3vcb2 minus v

cr2( 1113857

fb πleεφ0Rprime

12

212nsh120

3vcr2 minus v

cb2( 1113857

(12)

where φ0 is the grease plastic viscosity at atmosphericpressure Rprime is the equivalent radius of the kth slice andthe raceway h0 is the minimum grease film thickness h0is calculated by using the model in [17] ε is related to theHertzian contact half width b film thickness h0 andradius Rprime

ε 2π

a tan212b

Rprime12

h120

⎛⎝ ⎞⎠minus212bRprime

12h120

Rprimeh0 + 2b2⎡⎢⎢⎣ ⎤⎥⎥⎦ (13)

In order to determine the total contact load between theroller and raceway the integration of local interactions isrequired along the contact line 1e total contact force andtorque acting on the jth roller are

Qaj 1113944

ns

k1TcaF

crjk

Nrj 1113944

ns

k1rrpb + rrgm1113872 1113873 times TcrF

crjk

(14)

1e contact loads acting on the race are

Qij 1113944

ns

k1Ticprime F

cbjk

Nbj 1113944

ns

k1rbpb + Tibr

irb1113872 1113873 times TibTicprime F

cbjk

(15)

Shock and Vibration 3

where

Fcrjkbjk frjkbjk sin αf frjkbjk cos αf plusmnqjk1113960 1113961T

αf a sin vcbr2v

cbr1( 1113857

(16)

22 FlangeRoller-End Interaction In race body-fixed co-ordinate system (ob-xbybzb) the roller-end curvature centerrelative to the race center is

rbbe Tib rir minus rib + Tirprime r

rgm + Tirprime x

re 0 01113858 1113859

T1113872 1113873minus rbgm (17)

where xre is the position of the roller-end curvature center at

the roller axisSince normal force will act on the flange surface and pass

through the roller-end curvature center the above centercan be transformed into the race azimuth coordinate systemby Tba(ψ 0 0) where ψ a tan(rbbe2r

bbe3)

rabe Tbarbbe (18)

As illustrated in Figure 2 the position of the roller-endcurvature center relative to the flange coordinate is

rfef Taf rabe minusTbarbf1113960 1113961 (19)

where Taf(0 β 0) is the transformation between the raceazimuth and flange coordinate β is the flange angle definedas a rotation about the o-y axis in accordance to the right-hand screw rule and rbf locates the flange origin in the racecoordinate

If the radius rs of the roller-end is known the geometricinteraction between the roller-end and flange is simply given by

δ rfef1 minus rs (20)

Definition re is the distance from the bearing apex tothe roller spherical end If re gt rs the flangeroller-endcontact is elliptical contact [11] 1e Hertzian contacttheory can be employed to determine the contact load qf atthe flangeroller-end contact

qf 3212δ32

3n32δ Eprimeρ12 (21)

where nδ is the contact deformation coefficient and 1113936 ρ is thesum of principal curvatures

1e friction at the flangeroller-end contact contains twoparts the friction induced bymicroasperities contact and thetraction force produced by the grease lubricant [18 19]

ff fasp + fgrease (22)

1e friction force induced by the asperities contact is

fasp μaqf expminusBλC

s0

p0qf expminusBλC

(23)

where B and C are related to morphology andmaterial properties of the contact surface s0 and p0 arerespectively the critical shear stress and the yield stressfor most metals s0p0 asymp 02 [18] λ is the ratio of the greasefilm thickness hc to surface roughness 1e film thicknesshc is approximated as [20]

hc 18035 times 10minus2Mminus05066Lminus11869

(24)

where

M qf

Eprimer2s

Eprimersφ0uf

1113888 1113889

34

L αEprimeφ0uf

Eprimers1113888 1113889

14

(25)

uf is the EHL entrainment speed uf (uflange + uend)2For the analysis of the grease traction force the

HerschelndashBulkley flow model is adopted

τ τy + φ|D|n (26)

where τ is the grease shear stress τy is the yield stress φ isthe plastic viscosity φ φ0eαp p is the contact pressure thedistribution of p can be approximated by the Hertziancontact pressure α is the viscosity-pressure coefficient nis the power law exponent and D is the shear rate D ≌(uflange minus uend)hc 1en the traction force due to the greaseshear stress is obtained by the integration

fgrease 1113946a

minusa1113946

b

minusbτ(x y)x dx dy (27)

1en the loads acting on the jth roller are

Qafj Tafprime F

fff Tafprime qfj ffj 01113960 1113961

T

Nrfj rrfr + rrgm1113872 1113873 times TfrF

ffj

(28)

1e loads acting on the flange are

Qifj minusTftF

ffj

Nbfj rbfb + rbgm1113872 1113873 times TibQ

ifj

(29)

where rrfr and rbfb are positions of the contact point relative to

the roller and inner-ring centers respectively

23 Forces Acting on the Cage During the cage and rollersrotation the master-slave relationship of rotating cage-rollerassembly will change over time due to different velocitiesand displacements As shown in Figures 2 and 3 the locationof the jth roller geometric center in the cage pocket co-ordinate system (op-xpypzp) is

rpcr Tcp Tic rir minus ric + Tirr

rgm1113872 1113873minus r01113872 1113873 (30)

where ric is the position of the cage mass center in theinertial coordinate system Tcp is the transformationmatrix from the cage-fixed coordinate system (oc-xcyczc) tothe pocket coordinate system r0 locates the pocket centerin the cage coordinate system and Tic is the trans-formation matrix between the inertial and cage coordinatesystem

Now similar to the rollerraceway interaction let rrglocates a point Pprime on the kth slice of the roller such that

rrpprime xkr minusς sin c ς cos c1113960 1113961

T (31)

4 Shock and Vibration

In the pocket coordinate system the position of the pointPprime is

rppprimec rpcr + TicTirprime rrpprime (32)

As illustrated in Figure 3 the clearance between thepoint P and the pocket crossbeam is

δ Δδ minus rppprimec2 (33)

where Δδ is the initial clearance between the roller andcrossbeam Δδ (lc minus dw)2 lc is the average width of thepocket and dw is the average diameter of the roller

For the modeling of the assembly interactions assumingthat there is an excess of the grease in the cage-roller as-sembly a critical value Δh0 of the grease lm thickness isassumed for the contact state transitionWhen δ ge Δh0 thereare only the hydrodynamic eect between the kth slice andpocket crossbeam and no Hertzian contact [9 21] eminimum grease lm thickness h0 δ e hydrodynamicpressure qc and the grease lm traction force fc between thekth slice and crossbeam are

qc qcehl keδ + ce _δ

fc fcehl 3πεφ0uleς12

212h120

(34)

where [17 22]

ke zf h0( )zh0

ce 6πεφ0leς32

212nsh320

f h0( ) 29παφnsξ1un

9EPrimeh6n+20 ςminus3le4 +

2n

( )n

+ξ2λnτyh

n0

ξ1φun[ ]

(35)

where u is the entrainment speed ξ1 ξ2 are the constantsassociated with the power law exponent n [17] and ce isgrease lm damping [22]

When δ lt Δh0 it means that the contact between the kthslice and crossbeam is in a boundary state So the Hertziancontact is included in the contact e minimum grease

lm thickness h0 Δh0 e Hertzian contact deformationis

δh Δh0 minus δ (36)

e contact force qc and the friction force fc between theroller and crossbeam are

qc khδh + ch _δh + qcehlfc μbdqc + fcehl

(37)

where kh is the linearized Hertzian contact stiness [23] ch isthe Herbert viscous damping coecurrencient [24] and μbd is thetraction coecurrencient under boundary lubrication [9]

en the load at the contact point Pprime on the kth slice inthe contact coordinate system is

Fccjk 0 qcjk fcjk[ ]T (38)

Using the processing method of the rollerracewayinteraction the load Fccjk can also be transferred tothe inertial coordinate and the rollercage body-xedcoordinate respectively

Owing to the smaller sliding speed between the roller-end and the side beam and the smaller curvature of rollerspherical end the EHL lubrication state cannot be eec-tively formed at this contact but the squeeze lm lubri-cation will be formed As shown in Figure 3 the Δδprime is theinitial pocket clearance and ra rpcr1 If dradt ge 0 thesmall end contacts with the side beam if dradt lt 0the large end contacts Neglecting the grease traction thesqueeze force qs is

qs ks Δδprime minus ra( ) (39)

where [25]

ks minusz

zha

dlsφ0k3s

h3a

dradt

( ) (40)

where ha is the distance from the side beam to the end dl andds are the diameters of the large and small end respectivelyand hs is the thickness of the side beam

en the loads acting on the jth roller are

αc

Δδprime

Δδ

δ

ha

ke

ce

ks

xr

yr

xp

ypop

or

Side beam

Cros

sbea

m

Figure 3 Interaction between the roller and cage

Shock and Vibration 5

Qasj qsj 0 0[ ]T

Nrsj rrsr + rrgm( ) times TarQ

asj

(41)

e loads acting on the cage areQi

sj minusTaiQasj

Ncsj r0 + Tcpprime r

pcr + Trc rrsr + rrgm( ) times TicQ

isj[ ]

(42)

where rrsr is the position of the contact point in the rollerbody-xed coordinate and Tar and Tai are respectivelythe transform from the azimuth to roller and inertialcoordinate

According to geometric characteristics of the cage andguide ring it can be considered as a nite journal bearinglubrication condition between the guide surface and cagecentering surface In Figure 4 the clearance δprime between thecage and guide ring is

δprime Δe minus e (43)

where Δe is the initial clearance Δe (ry minus rd)2 rd ryare respectively the radii of the centering surface andguide surface and e is cage displacement in the radial plane

Similarly assume that Δh0prime is the critical value of thegrease lm thickness When δprime ge Δh0prime the contact force qicand friction torque Nic on the cage are

qic qicehl keprimeδ + ceprime _δprime

Nic Nicehl πεφ0br

12ic rd

212h1203ωbxry minusωcxrd( )

(44)

where ric is the equivalent radius and ωcx ωbx arerespectively the angular velocities of the cage and guidering e calculation of the grease lm stiness keprime anddamping ceprime is the same as that at the rollercrossbeamcontact

When δprime lt Δh0prime the elastic deformation between the cageand guide ring is δhprime δprime minus h0 and the minimum grease lmthickness h0 Δh0prime e qic and Nic are

qic khprimeδhprime + chprimeδhprime + qicehlNic μbdqicrd +Nicehl

(45)

where khprime is the contact stiness and chprime is the viscousdamping coecurrencient e calculation of khprime and chprime is the sameas that in Equation (35) μbd is the traction coecurrencient underboundary lubrication

In order to simplify themodel the load of the cage on theguide ring is ignored in contrast to large external load on theguide ring en the loads acting on the cage are

Qiic 0 qic sinφic qic cosφic[ ]T

Ncic plusmnNic 0 0[ ]

(46)

If the TRB is lubricated by the oil the eect of oil-gasmixture on the cage and rollers may not be overlookedHowever if the bearing is lubricated by the grease it seemsimpossible to have the grease-gas mixture eect on the

dynamics of the cage and rollers So the eect of the grease-gas mixture is not considered in the dynamic model

24DynamicModel In real application also the surroundings(inertiadamping of shaft assembly and housing) are expectedto inuence the TRB dynamics ese eects are neglected inthe modeling Generally it is convenient to consider thetranslational moving of rollers in the cylindrical coordinatesystem while the Cartesian coordinate system is convenient forthe cage and racee translationalmoving of bearing parts canbe simply described by Newtonrsquos laws such that

Meuroδ Q (47)

e generalized force Q is obtained by superimposingthe forces from Section 2 e centrifugal force and thegyroscopic moment should be superimposed on the gen-eralized forceQ for the roller For the cage and race the massmatrices Mb and Mc are

Mbc diag mbc mbc mbc[ ] (48)

For the special case of conical rollers Mr is

Mr diag mr mrrrj mr[ ] (49)

Both rollers and the cage have six degrees-of-freedomUnlike the translational motion (in three directions) therotating of the bearing parts are described in their own body-xed coordinates For rollers and the cage using the Eulerdynamic equations such that

Ix _ωx minusωyωz Iy minus Iz( )

Iy _ωy minusωzωx Iz minus Ix( )

Iz _ωz minusωxωx Ix minus Iy( )

Nx

Ny

Nz

(50)

where Ix Iy and Iz are inertia principalmoments of the cage orrollers the total momentN is also obtained by superimposingthe forces from Section 2 ω[ωx ωy ωz] is the angular velocityin the inertial coordinate systeme relationship between theangular velocity ωxed in the body-xed coordinate systemand ω is as follows B is the Eulerrsquos rotation matrix

∆y

yb

zb

yc

ωcx

ωbx

φicNic

δprime

zc

oc

∆z eob

qic

Guidering

Cage

Roller

Figure 4 Interaction between the cage and guide ring (inner-ringguided)

6 Shock and Vibration

ωfixed Bω

_ωfixed dBdt

Bminus1 _ω

(51)

Admitting the outer ring as macroscopically stationaryand a rotating inner ring with no torque load it resultsfour degrees of freedom of inner ring moving in the x yand z directions and constant rotating about the shaftaxis (o-x)

3 Numerical Method

1e bearing geometry and material parameters and greasemain rheological parameters (30degC) are shown in Table 11e cage is an inner-ring guided type and its material ispolyamide 1e polyurea grease was adopted [26] 1efourth-order RungendashKutta method was used to dissolvetransient responses of the TRB on MATLAB To ensure theconvergence of simulation initial values of displacementvelocity and load are obtained by the quasidynamicmethod

4 Results and Discussion

For the complicated dynamic response of TRBs it is essentialto consider several typical and harmful movements of thebearing in the process of dynamic analysis such as the rollerskew and tilt bearing slip and so on 1e slip rate of theroller and cage is defined as

|theoretical speedminus actual speed|

theoretical speedtimes 100 (52)

For the roller the speed refers to the revolution speedfor the cage refers to the rotational speed

41 Mass Center Trajectory and Velocity 1e trajectoriesand velocities of mass centers of the main bearing parts areshown in Figure 5 1e axial preload is not applied to theinner ring but limited to the degree of freedom of the innerring in the axial direction 1e cage initial rotation speed isthe same as the roller initial revolution speed Due to thechanging position of rollers the distribution of rollers issymmetrical or asymmetric with respect to the o-z axis1erefore the inner ring vibrates slightly in the o-y di-rection 1e radial and axial displacement (relative to theinitial position) of the roller exhibit a periodic fluctuationfor the bearing is in a semicircle-loaded state under pureradial load conditions When the rollers alternately enter theloaded and unloaded zones the load on rollers will showperiodic changes resulting in a regular variation in dis-placement 1e cage whirling is approximately a circle Fora number of uncertain rollers acting on the cage the cagespeed does not show a periodic change

42 Rotational Speed Influence In this study the tilt andskew of conical rollers are regarded as the primary eval-uation variables for the dynamic performance of the TRB

1e roller skew is mainly generated from the tangentialfriction force on the flangeroller-end and rollerracewaycontacts while the tilt motion is mainly induced by thenormal contact force on those contacts According to thelubrication theory the bearing speed will be a key factoraffecting the load case in bearing contacts

1e influence of the bearing speed on the roller tilt andskew (relative to the initial position) is shown in Figure 61e degree of freedom of the inner ring in the axial directionis also limited At each speed level the roller tilt anglepresents a relatively regular periodic vibration 1e tilt anglein the unloaded zone is obviously larger than that in theloaded zone 1is is due to the larger gap between the rollerand raceways in the unloaded zone when only the radicalload is applied and rollers in unloaded zone seem to beldquorelaxedrdquo As the speed increases the maximum of the tiltangle almost does not change while the average tilt anglerises from minus014 times 10minus4 rad to minus040 times 10minus4 rad indicatingthat the bearing speed affected the roller tilt It can beconfirmed that the gyroscopic moment is one of the reasonsthat cause the rising of the roller tilt angle for the gyroscopicmotion is intensified with the bearing speed increasingAlthough the gyroscopic moment increases as the bearingspeed increases it is still relatively smaller than theforcetorque in the rollerraceway contact So for the radialload is constant the maximum tilt angle of the roller changeslittle with the speed

As it is shown in Figure 7 the skew angle both in theloaded zone and in the unloaded zone is relatively large at thespeed of 2 krpm In the entire speed range the roller skewangle in the loaded zone is greater than that in the unloadedzone For the rollers is ldquorelaxedrdquo in the unloaded zone the

Table 1 Geometry material and rheological parameters

Parameter ValueNumber of rollers 17Roller length lemm 4206Roller-end sphere radius rsmm 13620Diameter of roller large end dlmm 1785Diameter of roller small end dsmm 2476Outer contact angledeg 15Inner contact angledeg 5Flange contact angledeg 83Outer raceway radiusmm 9952Inner raceway radiusmm 7848Guide land clearancemm 05Guide land widthmm 493Cage outer diametermm 20576Cage inner diametermm 19610Cage thickness hsmm 5Pocket clearance Δδmm 027Elastic modulus of rollerraceGPa 205Poissonrsquos ratio of rollerrace 03Elastic modulus of cageGPa 83Poissonrsquos ratio of cage 028

Grease rheological parameters (30degC)Yield stress τyPa 3518Plastic viscosity φPamiddotsn 844Power law exponent n 07196Pressure-viscosity coefficient αm2middotNminus11 4662 times 10minus8

Shock and Vibration 7

ndash6 ndash4 ndash2 0 2 4 60

3

6

9

12

15

Trajectory of the inner ring

z-di

rect

ion

disp

lace

men

t (μm

)

y-direction displacement (μm)

(a)

Displacement of the roller

0 003 006 009 012 015 018ndash20

ndash10

0

10

Radi

al (μ

m)

t (s)

0 003 006 009 012 015 018ndash20

10

40

70

Axi

al (μ

m)

(b)

Trajectory of the cage

ndash06 ndash04 ndash02 00 02 04 06ndash06

ndash04

ndash02

00

02

04

06

z-di

rect

ion

disp

lace

men

t (m

m)

y-direction displacement (mm)

(c)

Cage rotation speed

0 003 006 009 012 015 018102104106108110

Cage

(krp

m)

t (s)

Roller revolution speed0 003 006 009 012 015 018

102104106108110

Rolle

r (kr

pm)

(d)

Figure 5 1e trajectory and velocity of mass center Fr 20 kN Fa 0 ωbx 24 krpm

000 003 006 009 012 015 018ndash9

ndash6

ndash3

0

3

6

9

101

Tilt angle

ωbx = 2krpm

t (s)

Tilt

angl

e (10

ndash4 ra

d)

(a)

Tilt angle

ωbx = 3krpm

000 003 006 009 012 015 018ndash9

ndash6

ndash3

0

3

6

9

101

Tilt

angl

e (10

ndash4 ra

d)

t (s)

(b)

Figure 6 Continued

8 Shock and Vibration

skewmoment of the raceway and flange acting on the roller issmaller than that in the loaded zone It shows obvious pe-riodicity when the speed reaches 3 krpm or more1e averageskew angle is increased from 306 times 10minus3 rad017 deg to 447 times

10minus3 rad025 deg 1e increase of bearing rotating speedenlarges the centrifugal force of rollers which makes contactforces at the rollerouter raceway and flangeroller-end

increase As a result of combined contribution of racewayfriction and the flange friction the roller skew angle is greatlyincreased Compared with the experimental result measuredby Falodi et al in which theymeasured the average skew anglebetween 015 and 045 deg [27] 1e average skew angle inFigure 7 is between 017 and 025 deg and at an order ofmagnitude compared to the experimental result 1e

Tilt angle

ωbx = 4krpm

Tilt

angl

e (10

ndash4 ra

d)

000 003 006 009 012 015 018ndash9

ndash6

ndash3

0

3

6

9

101

t (s)

(c)

Tilt angle

ωbx = 5krpm

Tilt

angl

e (10

ndash4 ra

d)

000 003 006 009 012 015 018ndash9

ndash6

ndash3

0

3

6

9

101

t (s)

(d)

Figure 6 1e tilt angle of the conical roller 1 loaded zone 0 unloaded zone Fr 20 kN Fa 0 ωbx 2sim5 krpm

000 003 006 009 012 015 018ndash8

ndash3

2

7

12

17

22

101

Skew

angl

e (10

ndash3 ra

d)

t (s)

ωbx = 2krpm

Skew angle

(a)

Skew

angl

e (10

ndash3 ra

d)

000 003 006 009 012 015 018ndash4

0

4

8

12

16

101

t (s)

ωbx = 3krpm

Skew angle

(b)

Skew

angl

e (10

ndash3 ra

d)

000 003 006 009 012 015 018ndash4

0

4

8

12

16

101

t (s)

ωbx = 4krpm

Skew angle

(c)

Skew

angl

e (10

ndash3 ra

d)

000 003 006 009 012 015 018ndash4

0

4

8

12

16

0 11

t (s)

ωbx = 5krpm

Skew angle

(d)

Figure 7 1e skew angle of the conical roller 1 loaded zone 0 unloaded zone Fr 20 kN Fa 0 ωbx 2sim5 krpm

Shock and Vibration 9

changing trend of the skew angle with speed is also consistentwith Yangrsquos results

1e bearing slip is a destructive motion that easily leadsto scuffing welding and overwear of bearing surfaces 1ebearing slip usually occurs at high-speed and light-loadconditions 1e influence of the bearing speed on the slipof the roller and cage is exhibited in Figures 8 and 9

1e diagrams depicted in Figure 8 indicate that theroller slip is becoming more and more serious as thebearing speed is raised 1is variation trend is similar tothat in [7] With the bearing speed increases the roller slipgradually presents a periodic change rule especially whenthe speed is up to 3 krpm or more 1e roller slip in loadedand unloaded zones is not the same In the unloaded zonethe roller slip is more serious than that in the loaded zoneAt the speed of 5 krpm the bearing is in a state of seriousslipping In Figure 9 the cage slip is aggravated with thebearing speed increase 1ere is no periodic change in thecage slip rate 1is is because when the cage is whirlinga number of uncertain rollers may act on the cage whichcan lead to an irregular motion of the cage At the samespeed level the maximum amplitude of the slip rate of thecage and roller is basically the same for the dependentrelationship between the cage and roller

43 Axial Preload Influence Axial preload is not only a keyfactor affecting the bearing dynamics but one of the mainmeans to ensure the normal service life of the bearing Inorder to familiarize the influence of the axial preload onthe roller tilt and skew different axial preloads wereapplied to the TRB under the condition that the radial loadis zero

As shown in Figures 10 and 11 after entering the smoothrunning phase as the axial preload varies from 10 μm to40 μm the average value of the roller tilt rises from minus027 times

10minus4 rad to minus096 times 10minus4 rad 1e greater the preload themore obvious the roller tends to tilt 1is is because as thepreload increases the negative moment of the flange actingon the roller-end becomes larger thus causing the roller tiltto assume a negative tendency

Except for the condition that the preload is 10 μm theroller skew has larger amplitude at the beginning and thenthe amplitude decreases gradually and tends to be stable Inthe smooth running phase as the preload increases theroller skew angle increases 1is trend is in line with theresults in [8 27] 1e average skew angle is increasedfrom 423 times 10minus3 rad024 deg to 550 times 10minus3 rad031 degBoth the roller tilt and skew present a more regular vibrationthroughout the bearing operation this is due to symmetry ofthe bearing on the radial plane and pure preloading con-dition the loaded and unloaded zones do not exist in thebearing and loads on rollers almost change a little in anyposition

1e changes of the slip rate of the roller and cage with thepreload are presented in Figures 12 and 13 As the bearingpreload increases the bearing slip decreases in the smoothrunning phase 1e average slip rate of the roller decreasesfrom 00189 to 00057 and that of the cage decreases from

00189 to 00055 indicating that the appropriate bearingpreload can effectively prevent bearing slipping failure 1eresult is similar with the conclusion in [7 28] 1e influenceof the preload on the bearing slip reduced with the increaseof the preload value 1e average slip rate of the roller andcage when the preload is 40 μm changes little compared with00073 and 00076 when the preload is 30 μm Obviouslyincreasing the amount of preload will help to decrease thebearing slip but excessive preload will inevitably reduce theservice life of bearings 1erefore to meet the bearing lifeand service requirements reasonable selection of preload issignificant

44Grease Physical Properties Influence In view the fact thatthe calculation of the traction force stiffness and damping atthe bearing contacts is mainly based on the plastic viscosity φand yield stress τy of the grease For a clearer understandingof the relationship between the bearing dynamics and thegrease used grease physical properties influence on bearingdynamics is analyzed 1e temperature pressure speed andother surroundings have significant effects on the greasephysical properties For the sake of simplification the im-pacts of these factors on the grease physical properties areignored In order to highlight the effect of a certain rheo-logical parameter on the bearing performance only oneparameter was given an appropriate change while otherrheological parameters remain unchanged 1e effects areshown in Figures 14 and 15 After the bearing enters thestable running state (after 01 s) the average value of theangle (tilt and skew) is used as an indicator of bearingoperating state

1e effect of the plastic viscosity on the roller and cagemotions are shown in Figure 14(a) As the plastic viscosityincreases the tilt angle and skew angle increases and de-creases respectively For the grease film thickness increaseswith the increase of the plastic viscosity the load ratio ofmicroasperities at the flangeroller-end contact is reduced1en it indirectly leads to the decrease of the skew mo-ment at the flangeroller-end contact and the roller skewangle Due to the smaller skew angle the roller misalignmentwill also be small 1erefore the normal force at therollerraceways and flangeroller-end contacts will havea greater contribution to the roller tilt 1erefore the resultshows that the roller tilt angle increases with the plasticviscosity increase In Figure 14(b) the bearing slip decreaseswith the grease plastic viscosity increase As the plasticviscosity increases the grease film thickness at therollerraceways and flangeroller-end contacts increase Sothe friction force at these contacts is reduced which maylead to the decrease of the bearing slip1e bearing slip is theresult of the combined effect of raceway friction and theflange friction

Compared with the impacts of the plastic viscosity of thegrease on the motions of the roller and cage the influence ofthe grease yield stress on the roller tilt roller skew andbearing slip is relatively smaller as shown in Figure 15 Forthe yield stress has a negligible effect on the film thickness inall practical cases [9]

10 Shock and Vibration

000 003 006 009 012 015 01800

05

10

15

20

25

30

Slip

rate

of t

he ro

ller (

)

t (s)

01000 01025 01050 01075 011ndash001

001

003

005

007ωbx = 2krpm

Slip rate of the roller

(a)

000 003 006 009 012 015 01800

02

04

06

08

Slip

rate

of t

he ro

ller (

)

t (s)

ωbx = 3krpm

Slip rate of the roller

(b)

000 003 006 009 012 015 01800

05

10

15

20

25

30

Slip

rate

of t

he ro

ller (

)

t (s)

ωbx = 4krpm

Slip rate of the roller

(c)

000 003 006 009 012 015 0180

2

4

6

8

10

Slip

rate

of t

he ro

ller (

)

t (s)

ωbx = 5krpm

Slip rate of the roller

(d)

Figure 8 1e slip rate of the conical roller Fr 20 kN Fa 0 ωbx 2sim5 krpm

000 003 006 009 012 015 01800

05

10

15

20

25

30

Slip

rate

of t

he ca

ge (

)

01000 01025 01050 01075 011ndash000

001

002

003

t (s)

Slip rate of the cage

ωbx = 2krpm

(a)

000 003 006 009 012 015 01800

05

10

15

20

25

30

Slip

rate

of t

he ca

ge (

)

t (s)

Slip rate of the cage

ωbx = 3krpm

(b)

Figure 9 Continued

Shock and Vibration 11

45 Temperature Influence Due to the improper in-stallation overwear or not timely cooling the operatingtemperature of TRBs will change more or less Sometimesit may lead to heat accumulation or thermal failure ofbearings 1e most direct effect of the temperature ischanges of the grease physical properties 1e rheologicalparameters of the greases at different temperatures in

Table 2 are quoted from the data in reference [26] 1eeffect of temperature on the dynamics of the bearing wasstudied 1e impact of temperature on the bearing struc-ture is not considered here

1ree typical greases were used such as the calciumgrease lithium grease and polyurea grease 1e largestdifference in the composition of three greases lies in the

000 003 006 009 012 015 01800

05

10

15

20

25

30

35

Slip

rate

of t

he ca

ge (

)

t (s)

Slip rate of the cage

ωbx = 4krpm

(c)

000 003 006 009 012 015 0180

2

4

6

8

10

Slip

rate

of t

he ca

ge (

)

t (s)

Slip rate of the cage

ωbx = 5krpm

(d)

Figure 9 1e slip rate of the cage Fr 20 kN Fa 0 ωbx 2sim5 krpm

000 003 006 009 012 015 018ndash2

0

2

4

6

8

10

12

0100 0101 0102ndash15ndash10ndash05

00051015

δa = 10μm

Tilt anglet (s)

Tilt

angl

e (10

ndash4 ra

d)

(a)

000 003 006 009 012 015 018ndash2

0

2

4

6

8

10

12

0100 0101 0102ndash20ndash15ndash10ndash05

000510

Tilt anglet (s)

Tilt

angl

e (10

ndash4 ra

d)δa = 20μm

(b)

000 003 006 009 012 015 018ndash3

0

3

6

9

12

15

0100 0101 0102ndash25ndash20ndash15ndash10ndash05

0005

Tilt angle

t (s)

Tilt

angl

e (10

ndash4 ra

d)

δa = 30μm

(c)

000 003 006 009 012 015 018ndash3

0

3

6

9

12

15

0100 0101 0102ndash30ndash23ndash16ndash09ndash02

05

Tilt angle

t (s)

Tilt

angl

e (10

ndash4 ra

d)

δa = 40μm

(d)

Figure 10 1e tilt angle of the conical roller Fr 0 δa 10sim40 μm ωbx 25 krpm

12 Shock and Vibration

000 003 006 009 012 015 01810

26

42

58

74

90

Skew angle

t (s)

Skew

angl

e (10

ndash3 ra

d)

δa = 10μm

(a)

000 003 006 009 012 015 01810

26

42

58

74

90

Skew angle

t (s)

Skew

angl

e (10

ndash3 ra

d)

δa = 40μm

(b)

000 003 006 009 012 015 01810

26

42

58

74

90

Skew angle

t (s)

Skew

angl

e (10

ndash3 ra

d)

δa = 30μm

(c)

000 003 006 009 012 015 01810

26

42

58

74

90

Skew angle

t (s)

Skew

angl

e (10

ndash3 ra

d)

δa = 40μm

(d)

Figure 11 1e skew angle of the conical roller Fr 0 δa 10sim40 μm ωbx 25 krpm

000 003 006 009 012 015 018000

002

004

006

008

Slip

rate

of t

he ro

ller (

)

Slip rate of the roller

t (s)

δa = 10μm

(a)

000 003 006 009 012 015 018000

002

004

006

008

Slip

rate

of t

he ro

ller (

)

Slip rate of the roller

t (s)

δa = 20μm

(b)

Figure 12 Continued

Shock and Vibration 13

types of thickeners In Figure 16(a) under the lubricationof the calcium and polyurea greases the roller tilt angledecreases with the temperature increase Under the lithiumgrease lubrication the roller tilt angle increases with theincrease in temperature In the whole temperature rangethe tilt angle under the lithium grease is generally larger

than that under others With the increase in temperaturethe roller skew angle under the polyurea grease increaseswhile the roller skew angle under the other two types ofgreases decreases 1at the tilt and skew angle under thedifferent greases and temperature show different trends isa result of multiple factors One of the factors is that the

000 003 006 009 012 015 018000

002

004

006

008

Slip

rate

of t

he ro

ller (

)

Slip rate of the roller

t (s)

δa = 30μm

(c)

000 003 006 009 012 015 018000

002

004

006

008

Slip

rate

of t

he ro

ller (

)

Slip rate of the roller

t (s)

δa = 40μm

(d)

Figure 12 1e slip rate of the conical roller Fr 0 δa 10sim40 μm ωbx 25 krpm

000 003 006 009 012 015 01800

05

10

15

20

25

30

Slip rate of the cage

δa = 10μm

01000 01023 01046 010690010

0015

0020

0025

0030

Slip

rate

of t

he ca

ge (

)

t (s)

(a)

Slip rate of the cage

δa = 20μm

000 003 006 009 012 015 01800

05

10

15

20

25

30

0110 0111 0112 0113 0114 0115ndash001

000

001

002

003

004

Slip

rate

of t

he ca

ge (

)

t (s)

(b)

Slip rate of the cage

δa = 30μm

000 003 006 009 012 015 01800

05

10

15

20

25

30

0110 0111 0112 0113 0114 0115000000030006000900120015

Slip

rate

of t

he ca

ge (

)

t (s)

(c)

Slip rate of the cage

δa = 40μm

000 003 006 009 012 015 01800

05

10

15

20

25

30

0110 0111 0112 0113 0114 0115000000030006000900120015

Slip

rate

of t

he ca

ge (

)

t (s)

(d)

Figure 13 1e slip rate of the cage Fr 0 δa 10sim40 μm ωbx 25 krpm

14 Shock and Vibration

244 844 1444 2044 2644 3244 3844ndash52

ndash51

ndash50

ndash49

ndash48

ndash47

Plastic viscosity (Pamiddotsn)

00

16

32

48

64

80

Tilt angleSkew angle

Tilt

angl

e (10

ndash5 ra

d)

Skew

angl

e (10

ndash3 ra

d)

(a)

844 1444 2044 2644 3244 38440000

0004

0008

0012

0016

0020

Slip

rate

of t

he ro

ller (

)

Plastic viscosity (Pamiddotsn)

ndash0006

0000

0006

0012

0018

Slip

rate

of t

he ca

ge (

)

Slip rate of the rollerSlip rate of the cage

(b)

Figure 14 Plastic viscosity influence Fr 0 Fa 15 kN ωbx 28 krpm (a) 1e conical roller tilt and skew (b) Slip rate of the conicalroller and cage

2518 3018 3518 4018 4518 5018ndash550

ndash536

ndash522

ndash508

ndash494

ndash480

Yield stress (Pa)

350

370

390

410

430

450

Tilt angleSkew angle

Tilt

angl

e (10

ndash5 ra

d)

Skew

angl

e (10

ndash3 ra

d)

(a)

Slip rate of the rollerSlip rate of the cage

2518 3018 3518 4018 4518 50180009

0010

0011

0012

0013

0014

0015Sl

ip ra

te o

f the

rolle

r (

)

Yield stress (Pa)

0009

0010

0011

0012

0013

0014

0015

Slip

rate

of t

he ca

ge (

)

(b)

Figure 15 Yield stress influence Fr 0 Fa 15 kN ωbx 28 krpm (a)1e conical roller tilt and skew (b) Slip rate of the conical roller andcage

Table 2 Relationships between grease physical properties and temperature [26]

Grease TemperaturedegC Yield stress τyPa Plastic viscosity φPamiddotsn Power law exponent n

Polyurea

15 5017 1187 0773130 3518 844 0719650 3055 616 0662170 1885 610 07378

Lithium

15 9045 5435 0305330 8052 4623 0312550 7104 4113 0315170 3398 5296 03203

Calcium

15 9908 6482 0334930 6495 4923 0534450 3361 3059 0525670 2960 3588 04356

Shock and Vibration 15

relationships between the grease rheological parametersand temperature are not simple linear For example theplastic viscosity and the power law exponent do not simplyincrease or decrease with temperature

In Figure 16(b) firstly the roller slip rate shows dif-ferent values due to the different greases secondly theroller slip rate shows a variety of trends with the varioustemperature Under the lubrication of the polyurea greasethe roller slip rate increases with the increase in temper-ature while the roller slip with the calcium and lithiumgreases lubricated shows a decreasing trend In real ap-plication the operating temperature of bearings may not beconstant 1e influence of the temperature on dynamiccharacteristics of grease lubricated bearings needs to befurther explored

5 Conclusions

For the beginning operating stages of grease-lubricatedTRBs the multibody contact dynamic model of TRBs wasestablished 1e impacts of preload temperature and greaserheological properties on the bearing dynamics were ana-lyzed Based on numerical results several conclusions can besummarized

(1) 1e roller tilt angle in the unloaded zone is obviouslylarger than that in the loaded zone while the rollerskew angle in the unloaded zone is smaller than thatin the loaded zone 1e effect of bearing speed on theroller skew is greater than that on the roller tilt Asthe speed increases the roller tilt and skew and thebearing slip become more serious

(2) Bearing preload can effectively reduce the bearingslip but will make the roller tilt angle increase In thestable operation stage of the bearing as the preloadincreases the roller skew angle increases

(3) 1e roller skew angle and the bearing slip rate de-crease with the increase of the grease plastic viscosity1e roller tilt angle increases with the increase of the

plastic viscosity 1e yield stress of the grease haslittle effect on motions of the roller and cage

(4) 1e influence of temperature onmotions of the rollerand cage varies with the type of grease used Whenthe TRB is lubricated by the lithium grease with theincrease of the temperature the roller tilt angle in-creases the roller skew angle and the bearing slip ratedecrease When the polyurea grease is adopted thechange trend of the roller tilt and skew and thebearing slip is just opposite to that when the lithiumgrease is used Under the calcium grease lubricationthe roller tilt and skew angle and the bearing slip ratedecrease with the temperature increase

Data Availability

1e figure data used to support the findings of this studyhave been deposited in the FigShare repository at httpsfigsharecoms15c2556bee83d1adf01d

Conflicts of Interest

1e authors declare that they have no conflicts of interest

Acknowledgments

1e paper was supported by the National Natural ScienceFoundation of China (Grant No 51675427)

References

[1] P M Lugt ldquoA review on grease lubrication in rollingbearingsrdquo Tribology Transactions vol 52 no 4 pp 470ndash4802009

[2] T Cousseau Film thickness and friction in grease lubricatedcontacts Application to rolling bearing torque loss PhDDepartamento De Engenharia Mecanica E Gestao Industrial2013

[3] C T Walters ldquo1e dynamics of ball bearingsrdquo Journal ofTribology vol 93 no 1 pp 1ndash10 1971

10 23 36 49 62 75ndash58

ndash56

ndash54

ndash52

ndash50

ndash48

ndash46

ndash44

CalciumLithiumPolyurea tilt angle skew angle

tilt angle skew angletilt angle skew angle

Temperature (degC)

10

30

50

70

90

110

130Ti

lt an

gle (

10ndash5

rad)

Skew

angl

e (10

ndash3 ra

d)

(a)

10 23 36 49 62 750000

0005

0010

0015

0020

0025

0030

Poly

urea

gre

ase (

)

Temperature (degC)

0000

0001

0002

0003

0004

0005

0006

Calc

ium

lith

ium

gre

ase (

)

CalciumLithiumPolyurea

(b)

Figure 16 Temperature influence Fr 0 Fa 15 kN ωbx 28 krpm (a) 1e conical roller tilt and skew (b) Slip rate of the conical roller

16 Shock and Vibration

where

Fcrjkbjk frjkbjk sin αf frjkbjk cos αf plusmnqjk1113960 1113961T

αf a sin vcbr2v

cbr1( 1113857

(16)

22 FlangeRoller-End Interaction In race body-fixed co-ordinate system (ob-xbybzb) the roller-end curvature centerrelative to the race center is

rbbe Tib rir minus rib + Tirprime r

rgm + Tirprime x

re 0 01113858 1113859

T1113872 1113873minus rbgm (17)

where xre is the position of the roller-end curvature center at

the roller axisSince normal force will act on the flange surface and pass

through the roller-end curvature center the above centercan be transformed into the race azimuth coordinate systemby Tba(ψ 0 0) where ψ a tan(rbbe2r

bbe3)

rabe Tbarbbe (18)

As illustrated in Figure 2 the position of the roller-endcurvature center relative to the flange coordinate is

rfef Taf rabe minusTbarbf1113960 1113961 (19)

where Taf(0 β 0) is the transformation between the raceazimuth and flange coordinate β is the flange angle definedas a rotation about the o-y axis in accordance to the right-hand screw rule and rbf locates the flange origin in the racecoordinate

If the radius rs of the roller-end is known the geometricinteraction between the roller-end and flange is simply given by

δ rfef1 minus rs (20)

Definition re is the distance from the bearing apex tothe roller spherical end If re gt rs the flangeroller-endcontact is elliptical contact [11] 1e Hertzian contacttheory can be employed to determine the contact load qf atthe flangeroller-end contact

qf 3212δ32

3n32δ Eprimeρ12 (21)

where nδ is the contact deformation coefficient and 1113936 ρ is thesum of principal curvatures

1e friction at the flangeroller-end contact contains twoparts the friction induced bymicroasperities contact and thetraction force produced by the grease lubricant [18 19]

ff fasp + fgrease (22)

1e friction force induced by the asperities contact is

fasp μaqf expminusBλC

s0

p0qf expminusBλC

(23)

where B and C are related to morphology andmaterial properties of the contact surface s0 and p0 arerespectively the critical shear stress and the yield stressfor most metals s0p0 asymp 02 [18] λ is the ratio of the greasefilm thickness hc to surface roughness 1e film thicknesshc is approximated as [20]

hc 18035 times 10minus2Mminus05066Lminus11869

(24)

where

M qf

Eprimer2s

Eprimersφ0uf

1113888 1113889

34

L αEprimeφ0uf

Eprimers1113888 1113889

14

(25)

uf is the EHL entrainment speed uf (uflange + uend)2For the analysis of the grease traction force the

HerschelndashBulkley flow model is adopted

τ τy + φ|D|n (26)

where τ is the grease shear stress τy is the yield stress φ isthe plastic viscosity φ φ0eαp p is the contact pressure thedistribution of p can be approximated by the Hertziancontact pressure α is the viscosity-pressure coefficient nis the power law exponent and D is the shear rate D ≌(uflange minus uend)hc 1en the traction force due to the greaseshear stress is obtained by the integration

fgrease 1113946a

minusa1113946

b

minusbτ(x y)x dx dy (27)

1en the loads acting on the jth roller are

Qafj Tafprime F

fff Tafprime qfj ffj 01113960 1113961

T

Nrfj rrfr + rrgm1113872 1113873 times TfrF

ffj

(28)

1e loads acting on the flange are

Qifj minusTftF

ffj

Nbfj rbfb + rbgm1113872 1113873 times TibQ

ifj

(29)

where rrfr and rbfb are positions of the contact point relative to

the roller and inner-ring centers respectively

23 Forces Acting on the Cage During the cage and rollersrotation the master-slave relationship of rotating cage-rollerassembly will change over time due to different velocitiesand displacements As shown in Figures 2 and 3 the locationof the jth roller geometric center in the cage pocket co-ordinate system (op-xpypzp) is

rpcr Tcp Tic rir minus ric + Tirr

rgm1113872 1113873minus r01113872 1113873 (30)

where ric is the position of the cage mass center in theinertial coordinate system Tcp is the transformationmatrix from the cage-fixed coordinate system (oc-xcyczc) tothe pocket coordinate system r0 locates the pocket centerin the cage coordinate system and Tic is the trans-formation matrix between the inertial and cage coordinatesystem

Now similar to the rollerraceway interaction let rrglocates a point Pprime on the kth slice of the roller such that

rrpprime xkr minusς sin c ς cos c1113960 1113961

T (31)

4 Shock and Vibration

In the pocket coordinate system the position of the pointPprime is

rppprimec rpcr + TicTirprime rrpprime (32)

As illustrated in Figure 3 the clearance between thepoint P and the pocket crossbeam is

δ Δδ minus rppprimec2 (33)

where Δδ is the initial clearance between the roller andcrossbeam Δδ (lc minus dw)2 lc is the average width of thepocket and dw is the average diameter of the roller

For the modeling of the assembly interactions assumingthat there is an excess of the grease in the cage-roller as-sembly a critical value Δh0 of the grease lm thickness isassumed for the contact state transitionWhen δ ge Δh0 thereare only the hydrodynamic eect between the kth slice andpocket crossbeam and no Hertzian contact [9 21] eminimum grease lm thickness h0 δ e hydrodynamicpressure qc and the grease lm traction force fc between thekth slice and crossbeam are

qc qcehl keδ + ce _δ

fc fcehl 3πεφ0uleς12

212h120

(34)

where [17 22]

ke zf h0( )zh0

ce 6πεφ0leς32

212nsh320

f h0( ) 29παφnsξ1un

9EPrimeh6n+20 ςminus3le4 +

2n

( )n

+ξ2λnτyh

n0

ξ1φun[ ]

(35)

where u is the entrainment speed ξ1 ξ2 are the constantsassociated with the power law exponent n [17] and ce isgrease lm damping [22]

When δ lt Δh0 it means that the contact between the kthslice and crossbeam is in a boundary state So the Hertziancontact is included in the contact e minimum grease

lm thickness h0 Δh0 e Hertzian contact deformationis

δh Δh0 minus δ (36)

e contact force qc and the friction force fc between theroller and crossbeam are

qc khδh + ch _δh + qcehlfc μbdqc + fcehl

(37)

where kh is the linearized Hertzian contact stiness [23] ch isthe Herbert viscous damping coecurrencient [24] and μbd is thetraction coecurrencient under boundary lubrication [9]

en the load at the contact point Pprime on the kth slice inthe contact coordinate system is

Fccjk 0 qcjk fcjk[ ]T (38)

Using the processing method of the rollerracewayinteraction the load Fccjk can also be transferred tothe inertial coordinate and the rollercage body-xedcoordinate respectively

Owing to the smaller sliding speed between the roller-end and the side beam and the smaller curvature of rollerspherical end the EHL lubrication state cannot be eec-tively formed at this contact but the squeeze lm lubri-cation will be formed As shown in Figure 3 the Δδprime is theinitial pocket clearance and ra rpcr1 If dradt ge 0 thesmall end contacts with the side beam if dradt lt 0the large end contacts Neglecting the grease traction thesqueeze force qs is

qs ks Δδprime minus ra( ) (39)

where [25]

ks minusz

zha

dlsφ0k3s

h3a

dradt

( ) (40)

where ha is the distance from the side beam to the end dl andds are the diameters of the large and small end respectivelyand hs is the thickness of the side beam

en the loads acting on the jth roller are

αc

Δδprime

Δδ

δ

ha

ke

ce

ks

xr

yr

xp

ypop

or

Side beam

Cros

sbea

m

Figure 3 Interaction between the roller and cage

Shock and Vibration 5

Qasj qsj 0 0[ ]T

Nrsj rrsr + rrgm( ) times TarQ

asj

(41)

e loads acting on the cage areQi

sj minusTaiQasj

Ncsj r0 + Tcpprime r

pcr + Trc rrsr + rrgm( ) times TicQ

isj[ ]

(42)

where rrsr is the position of the contact point in the rollerbody-xed coordinate and Tar and Tai are respectivelythe transform from the azimuth to roller and inertialcoordinate

According to geometric characteristics of the cage andguide ring it can be considered as a nite journal bearinglubrication condition between the guide surface and cagecentering surface In Figure 4 the clearance δprime between thecage and guide ring is

δprime Δe minus e (43)

where Δe is the initial clearance Δe (ry minus rd)2 rd ryare respectively the radii of the centering surface andguide surface and e is cage displacement in the radial plane

Similarly assume that Δh0prime is the critical value of thegrease lm thickness When δprime ge Δh0prime the contact force qicand friction torque Nic on the cage are

qic qicehl keprimeδ + ceprime _δprime

Nic Nicehl πεφ0br

12ic rd

212h1203ωbxry minusωcxrd( )

(44)

where ric is the equivalent radius and ωcx ωbx arerespectively the angular velocities of the cage and guidering e calculation of the grease lm stiness keprime anddamping ceprime is the same as that at the rollercrossbeamcontact

When δprime lt Δh0prime the elastic deformation between the cageand guide ring is δhprime δprime minus h0 and the minimum grease lmthickness h0 Δh0prime e qic and Nic are

qic khprimeδhprime + chprimeδhprime + qicehlNic μbdqicrd +Nicehl

(45)

where khprime is the contact stiness and chprime is the viscousdamping coecurrencient e calculation of khprime and chprime is the sameas that in Equation (35) μbd is the traction coecurrencient underboundary lubrication

In order to simplify themodel the load of the cage on theguide ring is ignored in contrast to large external load on theguide ring en the loads acting on the cage are

Qiic 0 qic sinφic qic cosφic[ ]T

Ncic plusmnNic 0 0[ ]

(46)

If the TRB is lubricated by the oil the eect of oil-gasmixture on the cage and rollers may not be overlookedHowever if the bearing is lubricated by the grease it seemsimpossible to have the grease-gas mixture eect on the

dynamics of the cage and rollers So the eect of the grease-gas mixture is not considered in the dynamic model

24DynamicModel In real application also the surroundings(inertiadamping of shaft assembly and housing) are expectedto inuence the TRB dynamics ese eects are neglected inthe modeling Generally it is convenient to consider thetranslational moving of rollers in the cylindrical coordinatesystem while the Cartesian coordinate system is convenient forthe cage and racee translationalmoving of bearing parts canbe simply described by Newtonrsquos laws such that

Meuroδ Q (47)

e generalized force Q is obtained by superimposingthe forces from Section 2 e centrifugal force and thegyroscopic moment should be superimposed on the gen-eralized forceQ for the roller For the cage and race the massmatrices Mb and Mc are

Mbc diag mbc mbc mbc[ ] (48)

For the special case of conical rollers Mr is

Mr diag mr mrrrj mr[ ] (49)

Both rollers and the cage have six degrees-of-freedomUnlike the translational motion (in three directions) therotating of the bearing parts are described in their own body-xed coordinates For rollers and the cage using the Eulerdynamic equations such that

Ix _ωx minusωyωz Iy minus Iz( )

Iy _ωy minusωzωx Iz minus Ix( )

Iz _ωz minusωxωx Ix minus Iy( )

Nx

Ny

Nz

(50)

where Ix Iy and Iz are inertia principalmoments of the cage orrollers the total momentN is also obtained by superimposingthe forces from Section 2 ω[ωx ωy ωz] is the angular velocityin the inertial coordinate systeme relationship between theangular velocity ωxed in the body-xed coordinate systemand ω is as follows B is the Eulerrsquos rotation matrix

∆y

yb

zb

yc

ωcx

ωbx

φicNic

δprime

zc

oc

∆z eob

qic

Guidering

Cage

Roller

Figure 4 Interaction between the cage and guide ring (inner-ringguided)

6 Shock and Vibration

ωfixed Bω

_ωfixed dBdt

Bminus1 _ω

(51)

Admitting the outer ring as macroscopically stationaryand a rotating inner ring with no torque load it resultsfour degrees of freedom of inner ring moving in the x yand z directions and constant rotating about the shaftaxis (o-x)

3 Numerical Method

1e bearing geometry and material parameters and greasemain rheological parameters (30degC) are shown in Table 11e cage is an inner-ring guided type and its material ispolyamide 1e polyurea grease was adopted [26] 1efourth-order RungendashKutta method was used to dissolvetransient responses of the TRB on MATLAB To ensure theconvergence of simulation initial values of displacementvelocity and load are obtained by the quasidynamicmethod

4 Results and Discussion

For the complicated dynamic response of TRBs it is essentialto consider several typical and harmful movements of thebearing in the process of dynamic analysis such as the rollerskew and tilt bearing slip and so on 1e slip rate of theroller and cage is defined as

|theoretical speedminus actual speed|

theoretical speedtimes 100 (52)

For the roller the speed refers to the revolution speedfor the cage refers to the rotational speed

41 Mass Center Trajectory and Velocity 1e trajectoriesand velocities of mass centers of the main bearing parts areshown in Figure 5 1e axial preload is not applied to theinner ring but limited to the degree of freedom of the innerring in the axial direction 1e cage initial rotation speed isthe same as the roller initial revolution speed Due to thechanging position of rollers the distribution of rollers issymmetrical or asymmetric with respect to the o-z axis1erefore the inner ring vibrates slightly in the o-y di-rection 1e radial and axial displacement (relative to theinitial position) of the roller exhibit a periodic fluctuationfor the bearing is in a semicircle-loaded state under pureradial load conditions When the rollers alternately enter theloaded and unloaded zones the load on rollers will showperiodic changes resulting in a regular variation in dis-placement 1e cage whirling is approximately a circle Fora number of uncertain rollers acting on the cage the cagespeed does not show a periodic change

42 Rotational Speed Influence In this study the tilt andskew of conical rollers are regarded as the primary eval-uation variables for the dynamic performance of the TRB

1e roller skew is mainly generated from the tangentialfriction force on the flangeroller-end and rollerracewaycontacts while the tilt motion is mainly induced by thenormal contact force on those contacts According to thelubrication theory the bearing speed will be a key factoraffecting the load case in bearing contacts

1e influence of the bearing speed on the roller tilt andskew (relative to the initial position) is shown in Figure 61e degree of freedom of the inner ring in the axial directionis also limited At each speed level the roller tilt anglepresents a relatively regular periodic vibration 1e tilt anglein the unloaded zone is obviously larger than that in theloaded zone 1is is due to the larger gap between the rollerand raceways in the unloaded zone when only the radicalload is applied and rollers in unloaded zone seem to beldquorelaxedrdquo As the speed increases the maximum of the tiltangle almost does not change while the average tilt anglerises from minus014 times 10minus4 rad to minus040 times 10minus4 rad indicatingthat the bearing speed affected the roller tilt It can beconfirmed that the gyroscopic moment is one of the reasonsthat cause the rising of the roller tilt angle for the gyroscopicmotion is intensified with the bearing speed increasingAlthough the gyroscopic moment increases as the bearingspeed increases it is still relatively smaller than theforcetorque in the rollerraceway contact So for the radialload is constant the maximum tilt angle of the roller changeslittle with the speed

As it is shown in Figure 7 the skew angle both in theloaded zone and in the unloaded zone is relatively large at thespeed of 2 krpm In the entire speed range the roller skewangle in the loaded zone is greater than that in the unloadedzone For the rollers is ldquorelaxedrdquo in the unloaded zone the

Table 1 Geometry material and rheological parameters

Parameter ValueNumber of rollers 17Roller length lemm 4206Roller-end sphere radius rsmm 13620Diameter of roller large end dlmm 1785Diameter of roller small end dsmm 2476Outer contact angledeg 15Inner contact angledeg 5Flange contact angledeg 83Outer raceway radiusmm 9952Inner raceway radiusmm 7848Guide land clearancemm 05Guide land widthmm 493Cage outer diametermm 20576Cage inner diametermm 19610Cage thickness hsmm 5Pocket clearance Δδmm 027Elastic modulus of rollerraceGPa 205Poissonrsquos ratio of rollerrace 03Elastic modulus of cageGPa 83Poissonrsquos ratio of cage 028

Grease rheological parameters (30degC)Yield stress τyPa 3518Plastic viscosity φPamiddotsn 844Power law exponent n 07196Pressure-viscosity coefficient αm2middotNminus11 4662 times 10minus8

Shock and Vibration 7

ndash6 ndash4 ndash2 0 2 4 60

3

6

9

12

15

Trajectory of the inner ring

z-di

rect

ion

disp

lace

men

t (μm

)

y-direction displacement (μm)

(a)

Displacement of the roller

0 003 006 009 012 015 018ndash20

ndash10

0

10

Radi

al (μ

m)

t (s)

0 003 006 009 012 015 018ndash20

10

40

70

Axi

al (μ

m)

(b)

Trajectory of the cage

ndash06 ndash04 ndash02 00 02 04 06ndash06

ndash04

ndash02

00

02

04

06

z-di

rect

ion

disp

lace

men

t (m

m)

y-direction displacement (mm)

(c)

Cage rotation speed

0 003 006 009 012 015 018102104106108110

Cage

(krp

m)

t (s)

Roller revolution speed0 003 006 009 012 015 018

102104106108110

Rolle

r (kr

pm)

(d)

Figure 5 1e trajectory and velocity of mass center Fr 20 kN Fa 0 ωbx 24 krpm

000 003 006 009 012 015 018ndash9

ndash6

ndash3

0

3

6

9

101

Tilt angle

ωbx = 2krpm

t (s)

Tilt

angl

e (10

ndash4 ra

d)

(a)

Tilt angle

ωbx = 3krpm

000 003 006 009 012 015 018ndash9

ndash6

ndash3

0

3

6

9

101

Tilt

angl

e (10

ndash4 ra

d)

t (s)

(b)

Figure 6 Continued

8 Shock and Vibration

skewmoment of the raceway and flange acting on the roller issmaller than that in the loaded zone It shows obvious pe-riodicity when the speed reaches 3 krpm or more1e averageskew angle is increased from 306 times 10minus3 rad017 deg to 447 times

10minus3 rad025 deg 1e increase of bearing rotating speedenlarges the centrifugal force of rollers which makes contactforces at the rollerouter raceway and flangeroller-end

increase As a result of combined contribution of racewayfriction and the flange friction the roller skew angle is greatlyincreased Compared with the experimental result measuredby Falodi et al in which theymeasured the average skew anglebetween 015 and 045 deg [27] 1e average skew angle inFigure 7 is between 017 and 025 deg and at an order ofmagnitude compared to the experimental result 1e

Tilt angle

ωbx = 4krpm

Tilt

angl

e (10

ndash4 ra

d)

000 003 006 009 012 015 018ndash9

ndash6

ndash3

0

3

6

9

101

t (s)

(c)

Tilt angle

ωbx = 5krpm

Tilt

angl

e (10

ndash4 ra

d)

000 003 006 009 012 015 018ndash9

ndash6

ndash3

0

3

6

9

101

t (s)

(d)

Figure 6 1e tilt angle of the conical roller 1 loaded zone 0 unloaded zone Fr 20 kN Fa 0 ωbx 2sim5 krpm

000 003 006 009 012 015 018ndash8

ndash3

2

7

12

17

22

101

Skew

angl

e (10

ndash3 ra

d)

t (s)

ωbx = 2krpm

Skew angle

(a)

Skew

angl

e (10

ndash3 ra

d)

000 003 006 009 012 015 018ndash4

0

4

8

12

16

101

t (s)

ωbx = 3krpm

Skew angle

(b)

Skew

angl

e (10

ndash3 ra

d)

000 003 006 009 012 015 018ndash4

0

4

8

12

16

101

t (s)

ωbx = 4krpm

Skew angle

(c)

Skew

angl

e (10

ndash3 ra

d)

000 003 006 009 012 015 018ndash4

0

4

8

12

16

0 11

t (s)

ωbx = 5krpm

Skew angle

(d)

Figure 7 1e skew angle of the conical roller 1 loaded zone 0 unloaded zone Fr 20 kN Fa 0 ωbx 2sim5 krpm

Shock and Vibration 9

changing trend of the skew angle with speed is also consistentwith Yangrsquos results

1e bearing slip is a destructive motion that easily leadsto scuffing welding and overwear of bearing surfaces 1ebearing slip usually occurs at high-speed and light-loadconditions 1e influence of the bearing speed on the slipof the roller and cage is exhibited in Figures 8 and 9

1e diagrams depicted in Figure 8 indicate that theroller slip is becoming more and more serious as thebearing speed is raised 1is variation trend is similar tothat in [7] With the bearing speed increases the roller slipgradually presents a periodic change rule especially whenthe speed is up to 3 krpm or more 1e roller slip in loadedand unloaded zones is not the same In the unloaded zonethe roller slip is more serious than that in the loaded zoneAt the speed of 5 krpm the bearing is in a state of seriousslipping In Figure 9 the cage slip is aggravated with thebearing speed increase 1ere is no periodic change in thecage slip rate 1is is because when the cage is whirlinga number of uncertain rollers may act on the cage whichcan lead to an irregular motion of the cage At the samespeed level the maximum amplitude of the slip rate of thecage and roller is basically the same for the dependentrelationship between the cage and roller

43 Axial Preload Influence Axial preload is not only a keyfactor affecting the bearing dynamics but one of the mainmeans to ensure the normal service life of the bearing Inorder to familiarize the influence of the axial preload onthe roller tilt and skew different axial preloads wereapplied to the TRB under the condition that the radial loadis zero

As shown in Figures 10 and 11 after entering the smoothrunning phase as the axial preload varies from 10 μm to40 μm the average value of the roller tilt rises from minus027 times

10minus4 rad to minus096 times 10minus4 rad 1e greater the preload themore obvious the roller tends to tilt 1is is because as thepreload increases the negative moment of the flange actingon the roller-end becomes larger thus causing the roller tiltto assume a negative tendency

Except for the condition that the preload is 10 μm theroller skew has larger amplitude at the beginning and thenthe amplitude decreases gradually and tends to be stable Inthe smooth running phase as the preload increases theroller skew angle increases 1is trend is in line with theresults in [8 27] 1e average skew angle is increasedfrom 423 times 10minus3 rad024 deg to 550 times 10minus3 rad031 degBoth the roller tilt and skew present a more regular vibrationthroughout the bearing operation this is due to symmetry ofthe bearing on the radial plane and pure preloading con-dition the loaded and unloaded zones do not exist in thebearing and loads on rollers almost change a little in anyposition

1e changes of the slip rate of the roller and cage with thepreload are presented in Figures 12 and 13 As the bearingpreload increases the bearing slip decreases in the smoothrunning phase 1e average slip rate of the roller decreasesfrom 00189 to 00057 and that of the cage decreases from

00189 to 00055 indicating that the appropriate bearingpreload can effectively prevent bearing slipping failure 1eresult is similar with the conclusion in [7 28] 1e influenceof the preload on the bearing slip reduced with the increaseof the preload value 1e average slip rate of the roller andcage when the preload is 40 μm changes little compared with00073 and 00076 when the preload is 30 μm Obviouslyincreasing the amount of preload will help to decrease thebearing slip but excessive preload will inevitably reduce theservice life of bearings 1erefore to meet the bearing lifeand service requirements reasonable selection of preload issignificant

44Grease Physical Properties Influence In view the fact thatthe calculation of the traction force stiffness and damping atthe bearing contacts is mainly based on the plastic viscosity φand yield stress τy of the grease For a clearer understandingof the relationship between the bearing dynamics and thegrease used grease physical properties influence on bearingdynamics is analyzed 1e temperature pressure speed andother surroundings have significant effects on the greasephysical properties For the sake of simplification the im-pacts of these factors on the grease physical properties areignored In order to highlight the effect of a certain rheo-logical parameter on the bearing performance only oneparameter was given an appropriate change while otherrheological parameters remain unchanged 1e effects areshown in Figures 14 and 15 After the bearing enters thestable running state (after 01 s) the average value of theangle (tilt and skew) is used as an indicator of bearingoperating state

1e effect of the plastic viscosity on the roller and cagemotions are shown in Figure 14(a) As the plastic viscosityincreases the tilt angle and skew angle increases and de-creases respectively For the grease film thickness increaseswith the increase of the plastic viscosity the load ratio ofmicroasperities at the flangeroller-end contact is reduced1en it indirectly leads to the decrease of the skew mo-ment at the flangeroller-end contact and the roller skewangle Due to the smaller skew angle the roller misalignmentwill also be small 1erefore the normal force at therollerraceways and flangeroller-end contacts will havea greater contribution to the roller tilt 1erefore the resultshows that the roller tilt angle increases with the plasticviscosity increase In Figure 14(b) the bearing slip decreaseswith the grease plastic viscosity increase As the plasticviscosity increases the grease film thickness at therollerraceways and flangeroller-end contacts increase Sothe friction force at these contacts is reduced which maylead to the decrease of the bearing slip1e bearing slip is theresult of the combined effect of raceway friction and theflange friction

Compared with the impacts of the plastic viscosity of thegrease on the motions of the roller and cage the influence ofthe grease yield stress on the roller tilt roller skew andbearing slip is relatively smaller as shown in Figure 15 Forthe yield stress has a negligible effect on the film thickness inall practical cases [9]

10 Shock and Vibration

000 003 006 009 012 015 01800

05

10

15

20

25

30

Slip

rate

of t

he ro

ller (

)

t (s)

01000 01025 01050 01075 011ndash001

001

003

005

007ωbx = 2krpm

Slip rate of the roller

(a)

000 003 006 009 012 015 01800

02

04

06

08

Slip

rate

of t

he ro

ller (

)

t (s)

ωbx = 3krpm

Slip rate of the roller

(b)

000 003 006 009 012 015 01800

05

10

15

20

25

30

Slip

rate

of t

he ro

ller (

)

t (s)

ωbx = 4krpm

Slip rate of the roller

(c)

000 003 006 009 012 015 0180

2

4

6

8

10

Slip

rate

of t

he ro

ller (

)

t (s)

ωbx = 5krpm

Slip rate of the roller

(d)

Figure 8 1e slip rate of the conical roller Fr 20 kN Fa 0 ωbx 2sim5 krpm

000 003 006 009 012 015 01800

05

10

15

20

25

30

Slip

rate

of t

he ca

ge (

)

01000 01025 01050 01075 011ndash000

001

002

003

t (s)

Slip rate of the cage

ωbx = 2krpm

(a)

000 003 006 009 012 015 01800

05

10

15

20

25

30

Slip

rate

of t

he ca

ge (

)

t (s)

Slip rate of the cage

ωbx = 3krpm

(b)

Figure 9 Continued

Shock and Vibration 11

45 Temperature Influence Due to the improper in-stallation overwear or not timely cooling the operatingtemperature of TRBs will change more or less Sometimesit may lead to heat accumulation or thermal failure ofbearings 1e most direct effect of the temperature ischanges of the grease physical properties 1e rheologicalparameters of the greases at different temperatures in

Table 2 are quoted from the data in reference [26] 1eeffect of temperature on the dynamics of the bearing wasstudied 1e impact of temperature on the bearing struc-ture is not considered here

1ree typical greases were used such as the calciumgrease lithium grease and polyurea grease 1e largestdifference in the composition of three greases lies in the

000 003 006 009 012 015 01800

05

10

15

20

25

30

35

Slip

rate

of t

he ca

ge (

)

t (s)

Slip rate of the cage

ωbx = 4krpm

(c)

000 003 006 009 012 015 0180

2

4

6

8

10

Slip

rate

of t

he ca

ge (

)

t (s)

Slip rate of the cage

ωbx = 5krpm

(d)

Figure 9 1e slip rate of the cage Fr 20 kN Fa 0 ωbx 2sim5 krpm

000 003 006 009 012 015 018ndash2

0

2

4

6

8

10

12

0100 0101 0102ndash15ndash10ndash05

00051015

δa = 10μm

Tilt anglet (s)

Tilt

angl

e (10

ndash4 ra

d)

(a)

000 003 006 009 012 015 018ndash2

0

2

4

6

8

10

12

0100 0101 0102ndash20ndash15ndash10ndash05

000510

Tilt anglet (s)

Tilt

angl

e (10

ndash4 ra

d)δa = 20μm

(b)

000 003 006 009 012 015 018ndash3

0

3

6

9

12

15

0100 0101 0102ndash25ndash20ndash15ndash10ndash05

0005

Tilt angle

t (s)

Tilt

angl

e (10

ndash4 ra

d)

δa = 30μm

(c)

000 003 006 009 012 015 018ndash3

0

3

6

9

12

15

0100 0101 0102ndash30ndash23ndash16ndash09ndash02

05

Tilt angle

t (s)

Tilt

angl

e (10

ndash4 ra

d)

δa = 40μm

(d)

Figure 10 1e tilt angle of the conical roller Fr 0 δa 10sim40 μm ωbx 25 krpm

12 Shock and Vibration

000 003 006 009 012 015 01810

26

42

58

74

90

Skew angle

t (s)

Skew

angl

e (10

ndash3 ra

d)

δa = 10μm

(a)

000 003 006 009 012 015 01810

26

42

58

74

90

Skew angle

t (s)

Skew

angl

e (10

ndash3 ra

d)

δa = 40μm

(b)

000 003 006 009 012 015 01810

26

42

58

74

90

Skew angle

t (s)

Skew

angl

e (10

ndash3 ra

d)

δa = 30μm

(c)

000 003 006 009 012 015 01810

26

42

58

74

90

Skew angle

t (s)

Skew

angl

e (10

ndash3 ra

d)

δa = 40μm

(d)

Figure 11 1e skew angle of the conical roller Fr 0 δa 10sim40 μm ωbx 25 krpm

000 003 006 009 012 015 018000

002

004

006

008

Slip

rate

of t

he ro

ller (

)

Slip rate of the roller

t (s)

δa = 10μm

(a)

000 003 006 009 012 015 018000

002

004

006

008

Slip

rate

of t

he ro

ller (

)

Slip rate of the roller

t (s)

δa = 20μm

(b)

Figure 12 Continued

Shock and Vibration 13

types of thickeners In Figure 16(a) under the lubricationof the calcium and polyurea greases the roller tilt angledecreases with the temperature increase Under the lithiumgrease lubrication the roller tilt angle increases with theincrease in temperature In the whole temperature rangethe tilt angle under the lithium grease is generally larger

than that under others With the increase in temperaturethe roller skew angle under the polyurea grease increaseswhile the roller skew angle under the other two types ofgreases decreases 1at the tilt and skew angle under thedifferent greases and temperature show different trends isa result of multiple factors One of the factors is that the

000 003 006 009 012 015 018000

002

004

006

008

Slip

rate

of t

he ro

ller (

)

Slip rate of the roller

t (s)

δa = 30μm

(c)

000 003 006 009 012 015 018000

002

004

006

008

Slip

rate

of t

he ro

ller (

)

Slip rate of the roller

t (s)

δa = 40μm

(d)

Figure 12 1e slip rate of the conical roller Fr 0 δa 10sim40 μm ωbx 25 krpm

000 003 006 009 012 015 01800

05

10

15

20

25

30

Slip rate of the cage

δa = 10μm

01000 01023 01046 010690010

0015

0020

0025

0030

Slip

rate

of t

he ca

ge (

)

t (s)

(a)

Slip rate of the cage

δa = 20μm

000 003 006 009 012 015 01800

05

10

15

20

25

30

0110 0111 0112 0113 0114 0115ndash001

000

001

002

003

004

Slip

rate

of t

he ca

ge (

)

t (s)

(b)

Slip rate of the cage

δa = 30μm

000 003 006 009 012 015 01800

05

10

15

20

25

30

0110 0111 0112 0113 0114 0115000000030006000900120015

Slip

rate

of t

he ca

ge (

)

t (s)

(c)

Slip rate of the cage

δa = 40μm

000 003 006 009 012 015 01800

05

10

15

20

25

30

0110 0111 0112 0113 0114 0115000000030006000900120015

Slip

rate

of t

he ca

ge (

)

t (s)

(d)

Figure 13 1e slip rate of the cage Fr 0 δa 10sim40 μm ωbx 25 krpm

14 Shock and Vibration

244 844 1444 2044 2644 3244 3844ndash52

ndash51

ndash50

ndash49

ndash48

ndash47

Plastic viscosity (Pamiddotsn)

00

16

32

48

64

80

Tilt angleSkew angle

Tilt

angl

e (10

ndash5 ra

d)

Skew

angl

e (10

ndash3 ra

d)

(a)

844 1444 2044 2644 3244 38440000

0004

0008

0012

0016

0020

Slip

rate

of t

he ro

ller (

)

Plastic viscosity (Pamiddotsn)

ndash0006

0000

0006

0012

0018

Slip

rate

of t

he ca

ge (

)

Slip rate of the rollerSlip rate of the cage

(b)

Figure 14 Plastic viscosity influence Fr 0 Fa 15 kN ωbx 28 krpm (a) 1e conical roller tilt and skew (b) Slip rate of the conicalroller and cage

2518 3018 3518 4018 4518 5018ndash550

ndash536

ndash522

ndash508

ndash494

ndash480

Yield stress (Pa)

350

370

390

410

430

450

Tilt angleSkew angle

Tilt

angl

e (10

ndash5 ra

d)

Skew

angl

e (10

ndash3 ra

d)

(a)

Slip rate of the rollerSlip rate of the cage

2518 3018 3518 4018 4518 50180009

0010

0011

0012

0013

0014

0015Sl

ip ra

te o

f the

rolle

r (

)

Yield stress (Pa)

0009

0010

0011

0012

0013

0014

0015

Slip

rate

of t

he ca

ge (

)

(b)

Figure 15 Yield stress influence Fr 0 Fa 15 kN ωbx 28 krpm (a)1e conical roller tilt and skew (b) Slip rate of the conical roller andcage

Table 2 Relationships between grease physical properties and temperature [26]

Grease TemperaturedegC Yield stress τyPa Plastic viscosity φPamiddotsn Power law exponent n

Polyurea

15 5017 1187 0773130 3518 844 0719650 3055 616 0662170 1885 610 07378

Lithium

15 9045 5435 0305330 8052 4623 0312550 7104 4113 0315170 3398 5296 03203

Calcium

15 9908 6482 0334930 6495 4923 0534450 3361 3059 0525670 2960 3588 04356

Shock and Vibration 15

relationships between the grease rheological parametersand temperature are not simple linear For example theplastic viscosity and the power law exponent do not simplyincrease or decrease with temperature

In Figure 16(b) firstly the roller slip rate shows dif-ferent values due to the different greases secondly theroller slip rate shows a variety of trends with the varioustemperature Under the lubrication of the polyurea greasethe roller slip rate increases with the increase in temper-ature while the roller slip with the calcium and lithiumgreases lubricated shows a decreasing trend In real ap-plication the operating temperature of bearings may not beconstant 1e influence of the temperature on dynamiccharacteristics of grease lubricated bearings needs to befurther explored

5 Conclusions

For the beginning operating stages of grease-lubricatedTRBs the multibody contact dynamic model of TRBs wasestablished 1e impacts of preload temperature and greaserheological properties on the bearing dynamics were ana-lyzed Based on numerical results several conclusions can besummarized

(1) 1e roller tilt angle in the unloaded zone is obviouslylarger than that in the loaded zone while the rollerskew angle in the unloaded zone is smaller than thatin the loaded zone 1e effect of bearing speed on theroller skew is greater than that on the roller tilt Asthe speed increases the roller tilt and skew and thebearing slip become more serious

(2) Bearing preload can effectively reduce the bearingslip but will make the roller tilt angle increase In thestable operation stage of the bearing as the preloadincreases the roller skew angle increases

(3) 1e roller skew angle and the bearing slip rate de-crease with the increase of the grease plastic viscosity1e roller tilt angle increases with the increase of the

plastic viscosity 1e yield stress of the grease haslittle effect on motions of the roller and cage

(4) 1e influence of temperature onmotions of the rollerand cage varies with the type of grease used Whenthe TRB is lubricated by the lithium grease with theincrease of the temperature the roller tilt angle in-creases the roller skew angle and the bearing slip ratedecrease When the polyurea grease is adopted thechange trend of the roller tilt and skew and thebearing slip is just opposite to that when the lithiumgrease is used Under the calcium grease lubricationthe roller tilt and skew angle and the bearing slip ratedecrease with the temperature increase

Data Availability

1e figure data used to support the findings of this studyhave been deposited in the FigShare repository at httpsfigsharecoms15c2556bee83d1adf01d

Conflicts of Interest

1e authors declare that they have no conflicts of interest

Acknowledgments

1e paper was supported by the National Natural ScienceFoundation of China (Grant No 51675427)

References

[1] P M Lugt ldquoA review on grease lubrication in rollingbearingsrdquo Tribology Transactions vol 52 no 4 pp 470ndash4802009

[2] T Cousseau Film thickness and friction in grease lubricatedcontacts Application to rolling bearing torque loss PhDDepartamento De Engenharia Mecanica E Gestao Industrial2013

[3] C T Walters ldquo1e dynamics of ball bearingsrdquo Journal ofTribology vol 93 no 1 pp 1ndash10 1971

10 23 36 49 62 75ndash58

ndash56

ndash54

ndash52

ndash50

ndash48

ndash46

ndash44

CalciumLithiumPolyurea tilt angle skew angle

tilt angle skew angletilt angle skew angle

Temperature (degC)

10

30

50

70

90

110

130Ti

lt an

gle (

10ndash5

rad)

Skew

angl

e (10

ndash3 ra

d)

(a)

10 23 36 49 62 750000

0005

0010

0015

0020

0025

0030

Poly

urea

gre

ase (

)

Temperature (degC)

0000

0001

0002

0003

0004

0005

0006

Calc

ium

lith

ium

gre

ase (

)

CalciumLithiumPolyurea

(b)

Figure 16 Temperature influence Fr 0 Fa 15 kN ωbx 28 krpm (a) 1e conical roller tilt and skew (b) Slip rate of the conical roller

16 Shock and Vibration

In the pocket coordinate system the position of the pointPprime is

rppprimec rpcr + TicTirprime rrpprime (32)

As illustrated in Figure 3 the clearance between thepoint P and the pocket crossbeam is

δ Δδ minus rppprimec2 (33)

where Δδ is the initial clearance between the roller andcrossbeam Δδ (lc minus dw)2 lc is the average width of thepocket and dw is the average diameter of the roller

For the modeling of the assembly interactions assumingthat there is an excess of the grease in the cage-roller as-sembly a critical value Δh0 of the grease lm thickness isassumed for the contact state transitionWhen δ ge Δh0 thereare only the hydrodynamic eect between the kth slice andpocket crossbeam and no Hertzian contact [9 21] eminimum grease lm thickness h0 δ e hydrodynamicpressure qc and the grease lm traction force fc between thekth slice and crossbeam are

qc qcehl keδ + ce _δ

fc fcehl 3πεφ0uleς12

212h120

(34)

where [17 22]

ke zf h0( )zh0

ce 6πεφ0leς32

212nsh320

f h0( ) 29παφnsξ1un

9EPrimeh6n+20 ςminus3le4 +

2n

( )n

+ξ2λnτyh

n0

ξ1φun[ ]

(35)

where u is the entrainment speed ξ1 ξ2 are the constantsassociated with the power law exponent n [17] and ce isgrease lm damping [22]

When δ lt Δh0 it means that the contact between the kthslice and crossbeam is in a boundary state So the Hertziancontact is included in the contact e minimum grease

lm thickness h0 Δh0 e Hertzian contact deformationis

δh Δh0 minus δ (36)

e contact force qc and the friction force fc between theroller and crossbeam are

qc khδh + ch _δh + qcehlfc μbdqc + fcehl

(37)

where kh is the linearized Hertzian contact stiness [23] ch isthe Herbert viscous damping coecurrencient [24] and μbd is thetraction coecurrencient under boundary lubrication [9]

en the load at the contact point Pprime on the kth slice inthe contact coordinate system is

Fccjk 0 qcjk fcjk[ ]T (38)

Using the processing method of the rollerracewayinteraction the load Fccjk can also be transferred tothe inertial coordinate and the rollercage body-xedcoordinate respectively

Owing to the smaller sliding speed between the roller-end and the side beam and the smaller curvature of rollerspherical end the EHL lubrication state cannot be eec-tively formed at this contact but the squeeze lm lubri-cation will be formed As shown in Figure 3 the Δδprime is theinitial pocket clearance and ra rpcr1 If dradt ge 0 thesmall end contacts with the side beam if dradt lt 0the large end contacts Neglecting the grease traction thesqueeze force qs is

qs ks Δδprime minus ra( ) (39)

where [25]

ks minusz

zha

dlsφ0k3s

h3a

dradt

( ) (40)

where ha is the distance from the side beam to the end dl andds are the diameters of the large and small end respectivelyand hs is the thickness of the side beam

en the loads acting on the jth roller are

αc

Δδprime

Δδ

δ

ha

ke

ce

ks

xr

yr

xp

ypop

or

Side beam

Cros

sbea

m

Figure 3 Interaction between the roller and cage

Shock and Vibration 5

Qasj qsj 0 0[ ]T

Nrsj rrsr + rrgm( ) times TarQ

asj

(41)

e loads acting on the cage areQi

sj minusTaiQasj

Ncsj r0 + Tcpprime r

pcr + Trc rrsr + rrgm( ) times TicQ

isj[ ]

(42)

where rrsr is the position of the contact point in the rollerbody-xed coordinate and Tar and Tai are respectivelythe transform from the azimuth to roller and inertialcoordinate

According to geometric characteristics of the cage andguide ring it can be considered as a nite journal bearinglubrication condition between the guide surface and cagecentering surface In Figure 4 the clearance δprime between thecage and guide ring is

δprime Δe minus e (43)

where Δe is the initial clearance Δe (ry minus rd)2 rd ryare respectively the radii of the centering surface andguide surface and e is cage displacement in the radial plane

Similarly assume that Δh0prime is the critical value of thegrease lm thickness When δprime ge Δh0prime the contact force qicand friction torque Nic on the cage are

qic qicehl keprimeδ + ceprime _δprime

Nic Nicehl πεφ0br

12ic rd

212h1203ωbxry minusωcxrd( )

(44)

where ric is the equivalent radius and ωcx ωbx arerespectively the angular velocities of the cage and guidering e calculation of the grease lm stiness keprime anddamping ceprime is the same as that at the rollercrossbeamcontact

When δprime lt Δh0prime the elastic deformation between the cageand guide ring is δhprime δprime minus h0 and the minimum grease lmthickness h0 Δh0prime e qic and Nic are

qic khprimeδhprime + chprimeδhprime + qicehlNic μbdqicrd +Nicehl

(45)

where khprime is the contact stiness and chprime is the viscousdamping coecurrencient e calculation of khprime and chprime is the sameas that in Equation (35) μbd is the traction coecurrencient underboundary lubrication

In order to simplify themodel the load of the cage on theguide ring is ignored in contrast to large external load on theguide ring en the loads acting on the cage are

Qiic 0 qic sinφic qic cosφic[ ]T

Ncic plusmnNic 0 0[ ]

(46)

If the TRB is lubricated by the oil the eect of oil-gasmixture on the cage and rollers may not be overlookedHowever if the bearing is lubricated by the grease it seemsimpossible to have the grease-gas mixture eect on the

dynamics of the cage and rollers So the eect of the grease-gas mixture is not considered in the dynamic model

24DynamicModel In real application also the surroundings(inertiadamping of shaft assembly and housing) are expectedto inuence the TRB dynamics ese eects are neglected inthe modeling Generally it is convenient to consider thetranslational moving of rollers in the cylindrical coordinatesystem while the Cartesian coordinate system is convenient forthe cage and racee translationalmoving of bearing parts canbe simply described by Newtonrsquos laws such that

Meuroδ Q (47)

e generalized force Q is obtained by superimposingthe forces from Section 2 e centrifugal force and thegyroscopic moment should be superimposed on the gen-eralized forceQ for the roller For the cage and race the massmatrices Mb and Mc are

Mbc diag mbc mbc mbc[ ] (48)

For the special case of conical rollers Mr is

Mr diag mr mrrrj mr[ ] (49)

Both rollers and the cage have six degrees-of-freedomUnlike the translational motion (in three directions) therotating of the bearing parts are described in their own body-xed coordinates For rollers and the cage using the Eulerdynamic equations such that

Ix _ωx minusωyωz Iy minus Iz( )

Iy _ωy minusωzωx Iz minus Ix( )

Iz _ωz minusωxωx Ix minus Iy( )

Nx

Ny

Nz

(50)

where Ix Iy and Iz are inertia principalmoments of the cage orrollers the total momentN is also obtained by superimposingthe forces from Section 2 ω[ωx ωy ωz] is the angular velocityin the inertial coordinate systeme relationship between theangular velocity ωxed in the body-xed coordinate systemand ω is as follows B is the Eulerrsquos rotation matrix

∆y

yb

zb

yc

ωcx

ωbx

φicNic

δprime

zc

oc

∆z eob

qic

Guidering

Cage

Roller

Figure 4 Interaction between the cage and guide ring (inner-ringguided)

6 Shock and Vibration

ωfixed Bω

_ωfixed dBdt

Bminus1 _ω

(51)

Admitting the outer ring as macroscopically stationaryand a rotating inner ring with no torque load it resultsfour degrees of freedom of inner ring moving in the x yand z directions and constant rotating about the shaftaxis (o-x)

3 Numerical Method

1e bearing geometry and material parameters and greasemain rheological parameters (30degC) are shown in Table 11e cage is an inner-ring guided type and its material ispolyamide 1e polyurea grease was adopted [26] 1efourth-order RungendashKutta method was used to dissolvetransient responses of the TRB on MATLAB To ensure theconvergence of simulation initial values of displacementvelocity and load are obtained by the quasidynamicmethod

4 Results and Discussion

For the complicated dynamic response of TRBs it is essentialto consider several typical and harmful movements of thebearing in the process of dynamic analysis such as the rollerskew and tilt bearing slip and so on 1e slip rate of theroller and cage is defined as

|theoretical speedminus actual speed|

theoretical speedtimes 100 (52)

For the roller the speed refers to the revolution speedfor the cage refers to the rotational speed

41 Mass Center Trajectory and Velocity 1e trajectoriesand velocities of mass centers of the main bearing parts areshown in Figure 5 1e axial preload is not applied to theinner ring but limited to the degree of freedom of the innerring in the axial direction 1e cage initial rotation speed isthe same as the roller initial revolution speed Due to thechanging position of rollers the distribution of rollers issymmetrical or asymmetric with respect to the o-z axis1erefore the inner ring vibrates slightly in the o-y di-rection 1e radial and axial displacement (relative to theinitial position) of the roller exhibit a periodic fluctuationfor the bearing is in a semicircle-loaded state under pureradial load conditions When the rollers alternately enter theloaded and unloaded zones the load on rollers will showperiodic changes resulting in a regular variation in dis-placement 1e cage whirling is approximately a circle Fora number of uncertain rollers acting on the cage the cagespeed does not show a periodic change

42 Rotational Speed Influence In this study the tilt andskew of conical rollers are regarded as the primary eval-uation variables for the dynamic performance of the TRB

1e roller skew is mainly generated from the tangentialfriction force on the flangeroller-end and rollerracewaycontacts while the tilt motion is mainly induced by thenormal contact force on those contacts According to thelubrication theory the bearing speed will be a key factoraffecting the load case in bearing contacts

1e influence of the bearing speed on the roller tilt andskew (relative to the initial position) is shown in Figure 61e degree of freedom of the inner ring in the axial directionis also limited At each speed level the roller tilt anglepresents a relatively regular periodic vibration 1e tilt anglein the unloaded zone is obviously larger than that in theloaded zone 1is is due to the larger gap between the rollerand raceways in the unloaded zone when only the radicalload is applied and rollers in unloaded zone seem to beldquorelaxedrdquo As the speed increases the maximum of the tiltangle almost does not change while the average tilt anglerises from minus014 times 10minus4 rad to minus040 times 10minus4 rad indicatingthat the bearing speed affected the roller tilt It can beconfirmed that the gyroscopic moment is one of the reasonsthat cause the rising of the roller tilt angle for the gyroscopicmotion is intensified with the bearing speed increasingAlthough the gyroscopic moment increases as the bearingspeed increases it is still relatively smaller than theforcetorque in the rollerraceway contact So for the radialload is constant the maximum tilt angle of the roller changeslittle with the speed

As it is shown in Figure 7 the skew angle both in theloaded zone and in the unloaded zone is relatively large at thespeed of 2 krpm In the entire speed range the roller skewangle in the loaded zone is greater than that in the unloadedzone For the rollers is ldquorelaxedrdquo in the unloaded zone the

Table 1 Geometry material and rheological parameters

Parameter ValueNumber of rollers 17Roller length lemm 4206Roller-end sphere radius rsmm 13620Diameter of roller large end dlmm 1785Diameter of roller small end dsmm 2476Outer contact angledeg 15Inner contact angledeg 5Flange contact angledeg 83Outer raceway radiusmm 9952Inner raceway radiusmm 7848Guide land clearancemm 05Guide land widthmm 493Cage outer diametermm 20576Cage inner diametermm 19610Cage thickness hsmm 5Pocket clearance Δδmm 027Elastic modulus of rollerraceGPa 205Poissonrsquos ratio of rollerrace 03Elastic modulus of cageGPa 83Poissonrsquos ratio of cage 028

Grease rheological parameters (30degC)Yield stress τyPa 3518Plastic viscosity φPamiddotsn 844Power law exponent n 07196Pressure-viscosity coefficient αm2middotNminus11 4662 times 10minus8

Shock and Vibration 7

ndash6 ndash4 ndash2 0 2 4 60

3

6

9

12

15

Trajectory of the inner ring

z-di

rect

ion

disp

lace

men

t (μm

)

y-direction displacement (μm)

(a)

Displacement of the roller

0 003 006 009 012 015 018ndash20

ndash10

0

10

Radi

al (μ

m)

t (s)

0 003 006 009 012 015 018ndash20

10

40

70

Axi

al (μ

m)

(b)

Trajectory of the cage

ndash06 ndash04 ndash02 00 02 04 06ndash06

ndash04

ndash02

00

02

04

06

z-di

rect

ion

disp

lace

men

t (m

m)

y-direction displacement (mm)

(c)

Cage rotation speed

0 003 006 009 012 015 018102104106108110

Cage

(krp

m)

t (s)

Roller revolution speed0 003 006 009 012 015 018

102104106108110

Rolle

r (kr

pm)

(d)

Figure 5 1e trajectory and velocity of mass center Fr 20 kN Fa 0 ωbx 24 krpm

000 003 006 009 012 015 018ndash9

ndash6

ndash3

0

3

6

9

101

Tilt angle

ωbx = 2krpm

t (s)

Tilt

angl

e (10

ndash4 ra

d)

(a)

Tilt angle

ωbx = 3krpm

000 003 006 009 012 015 018ndash9

ndash6

ndash3

0

3

6

9

101

Tilt

angl

e (10

ndash4 ra

d)

t (s)

(b)

Figure 6 Continued

8 Shock and Vibration

skewmoment of the raceway and flange acting on the roller issmaller than that in the loaded zone It shows obvious pe-riodicity when the speed reaches 3 krpm or more1e averageskew angle is increased from 306 times 10minus3 rad017 deg to 447 times

10minus3 rad025 deg 1e increase of bearing rotating speedenlarges the centrifugal force of rollers which makes contactforces at the rollerouter raceway and flangeroller-end

increase As a result of combined contribution of racewayfriction and the flange friction the roller skew angle is greatlyincreased Compared with the experimental result measuredby Falodi et al in which theymeasured the average skew anglebetween 015 and 045 deg [27] 1e average skew angle inFigure 7 is between 017 and 025 deg and at an order ofmagnitude compared to the experimental result 1e

Tilt angle

ωbx = 4krpm

Tilt

angl

e (10

ndash4 ra

d)

000 003 006 009 012 015 018ndash9

ndash6

ndash3

0

3

6

9

101

t (s)

(c)

Tilt angle

ωbx = 5krpm

Tilt

angl

e (10

ndash4 ra

d)

000 003 006 009 012 015 018ndash9

ndash6

ndash3

0

3

6

9

101

t (s)

(d)

Figure 6 1e tilt angle of the conical roller 1 loaded zone 0 unloaded zone Fr 20 kN Fa 0 ωbx 2sim5 krpm

000 003 006 009 012 015 018ndash8

ndash3

2

7

12

17

22

101

Skew

angl

e (10

ndash3 ra

d)

t (s)

ωbx = 2krpm

Skew angle

(a)

Skew

angl

e (10

ndash3 ra

d)

000 003 006 009 012 015 018ndash4

0

4

8

12

16

101

t (s)

ωbx = 3krpm

Skew angle

(b)

Skew

angl

e (10

ndash3 ra

d)

000 003 006 009 012 015 018ndash4

0

4

8

12

16

101

t (s)

ωbx = 4krpm

Skew angle

(c)

Skew

angl

e (10

ndash3 ra

d)

000 003 006 009 012 015 018ndash4

0

4

8

12

16

0 11

t (s)

ωbx = 5krpm

Skew angle

(d)

Figure 7 1e skew angle of the conical roller 1 loaded zone 0 unloaded zone Fr 20 kN Fa 0 ωbx 2sim5 krpm

Shock and Vibration 9

changing trend of the skew angle with speed is also consistentwith Yangrsquos results

1e bearing slip is a destructive motion that easily leadsto scuffing welding and overwear of bearing surfaces 1ebearing slip usually occurs at high-speed and light-loadconditions 1e influence of the bearing speed on the slipof the roller and cage is exhibited in Figures 8 and 9

1e diagrams depicted in Figure 8 indicate that theroller slip is becoming more and more serious as thebearing speed is raised 1is variation trend is similar tothat in [7] With the bearing speed increases the roller slipgradually presents a periodic change rule especially whenthe speed is up to 3 krpm or more 1e roller slip in loadedand unloaded zones is not the same In the unloaded zonethe roller slip is more serious than that in the loaded zoneAt the speed of 5 krpm the bearing is in a state of seriousslipping In Figure 9 the cage slip is aggravated with thebearing speed increase 1ere is no periodic change in thecage slip rate 1is is because when the cage is whirlinga number of uncertain rollers may act on the cage whichcan lead to an irregular motion of the cage At the samespeed level the maximum amplitude of the slip rate of thecage and roller is basically the same for the dependentrelationship between the cage and roller

43 Axial Preload Influence Axial preload is not only a keyfactor affecting the bearing dynamics but one of the mainmeans to ensure the normal service life of the bearing Inorder to familiarize the influence of the axial preload onthe roller tilt and skew different axial preloads wereapplied to the TRB under the condition that the radial loadis zero

As shown in Figures 10 and 11 after entering the smoothrunning phase as the axial preload varies from 10 μm to40 μm the average value of the roller tilt rises from minus027 times

10minus4 rad to minus096 times 10minus4 rad 1e greater the preload themore obvious the roller tends to tilt 1is is because as thepreload increases the negative moment of the flange actingon the roller-end becomes larger thus causing the roller tiltto assume a negative tendency

Except for the condition that the preload is 10 μm theroller skew has larger amplitude at the beginning and thenthe amplitude decreases gradually and tends to be stable Inthe smooth running phase as the preload increases theroller skew angle increases 1is trend is in line with theresults in [8 27] 1e average skew angle is increasedfrom 423 times 10minus3 rad024 deg to 550 times 10minus3 rad031 degBoth the roller tilt and skew present a more regular vibrationthroughout the bearing operation this is due to symmetry ofthe bearing on the radial plane and pure preloading con-dition the loaded and unloaded zones do not exist in thebearing and loads on rollers almost change a little in anyposition

1e changes of the slip rate of the roller and cage with thepreload are presented in Figures 12 and 13 As the bearingpreload increases the bearing slip decreases in the smoothrunning phase 1e average slip rate of the roller decreasesfrom 00189 to 00057 and that of the cage decreases from

00189 to 00055 indicating that the appropriate bearingpreload can effectively prevent bearing slipping failure 1eresult is similar with the conclusion in [7 28] 1e influenceof the preload on the bearing slip reduced with the increaseof the preload value 1e average slip rate of the roller andcage when the preload is 40 μm changes little compared with00073 and 00076 when the preload is 30 μm Obviouslyincreasing the amount of preload will help to decrease thebearing slip but excessive preload will inevitably reduce theservice life of bearings 1erefore to meet the bearing lifeand service requirements reasonable selection of preload issignificant

44Grease Physical Properties Influence In view the fact thatthe calculation of the traction force stiffness and damping atthe bearing contacts is mainly based on the plastic viscosity φand yield stress τy of the grease For a clearer understandingof the relationship between the bearing dynamics and thegrease used grease physical properties influence on bearingdynamics is analyzed 1e temperature pressure speed andother surroundings have significant effects on the greasephysical properties For the sake of simplification the im-pacts of these factors on the grease physical properties areignored In order to highlight the effect of a certain rheo-logical parameter on the bearing performance only oneparameter was given an appropriate change while otherrheological parameters remain unchanged 1e effects areshown in Figures 14 and 15 After the bearing enters thestable running state (after 01 s) the average value of theangle (tilt and skew) is used as an indicator of bearingoperating state

1e effect of the plastic viscosity on the roller and cagemotions are shown in Figure 14(a) As the plastic viscosityincreases the tilt angle and skew angle increases and de-creases respectively For the grease film thickness increaseswith the increase of the plastic viscosity the load ratio ofmicroasperities at the flangeroller-end contact is reduced1en it indirectly leads to the decrease of the skew mo-ment at the flangeroller-end contact and the roller skewangle Due to the smaller skew angle the roller misalignmentwill also be small 1erefore the normal force at therollerraceways and flangeroller-end contacts will havea greater contribution to the roller tilt 1erefore the resultshows that the roller tilt angle increases with the plasticviscosity increase In Figure 14(b) the bearing slip decreaseswith the grease plastic viscosity increase As the plasticviscosity increases the grease film thickness at therollerraceways and flangeroller-end contacts increase Sothe friction force at these contacts is reduced which maylead to the decrease of the bearing slip1e bearing slip is theresult of the combined effect of raceway friction and theflange friction

Compared with the impacts of the plastic viscosity of thegrease on the motions of the roller and cage the influence ofthe grease yield stress on the roller tilt roller skew andbearing slip is relatively smaller as shown in Figure 15 Forthe yield stress has a negligible effect on the film thickness inall practical cases [9]

10 Shock and Vibration

000 003 006 009 012 015 01800

05

10

15

20

25

30

Slip

rate

of t

he ro

ller (

)

t (s)

01000 01025 01050 01075 011ndash001

001

003

005

007ωbx = 2krpm

Slip rate of the roller

(a)

000 003 006 009 012 015 01800

02

04

06

08

Slip

rate

of t

he ro

ller (

)

t (s)

ωbx = 3krpm

Slip rate of the roller

(b)

000 003 006 009 012 015 01800

05

10

15

20

25

30

Slip

rate

of t

he ro

ller (

)

t (s)

ωbx = 4krpm

Slip rate of the roller

(c)

000 003 006 009 012 015 0180

2

4

6

8

10

Slip

rate

of t

he ro

ller (

)

t (s)

ωbx = 5krpm

Slip rate of the roller

(d)

Figure 8 1e slip rate of the conical roller Fr 20 kN Fa 0 ωbx 2sim5 krpm

000 003 006 009 012 015 01800

05

10

15

20

25

30

Slip

rate

of t

he ca

ge (

)

01000 01025 01050 01075 011ndash000

001

002

003

t (s)

Slip rate of the cage

ωbx = 2krpm

(a)

000 003 006 009 012 015 01800

05

10

15

20

25

30

Slip

rate

of t

he ca

ge (

)

t (s)

Slip rate of the cage

ωbx = 3krpm

(b)

Figure 9 Continued

Shock and Vibration 11

45 Temperature Influence Due to the improper in-stallation overwear or not timely cooling the operatingtemperature of TRBs will change more or less Sometimesit may lead to heat accumulation or thermal failure ofbearings 1e most direct effect of the temperature ischanges of the grease physical properties 1e rheologicalparameters of the greases at different temperatures in

Table 2 are quoted from the data in reference [26] 1eeffect of temperature on the dynamics of the bearing wasstudied 1e impact of temperature on the bearing struc-ture is not considered here

1ree typical greases were used such as the calciumgrease lithium grease and polyurea grease 1e largestdifference in the composition of three greases lies in the

000 003 006 009 012 015 01800

05

10

15

20

25

30

35

Slip

rate

of t

he ca

ge (

)

t (s)

Slip rate of the cage

ωbx = 4krpm

(c)

000 003 006 009 012 015 0180

2

4

6

8

10

Slip

rate

of t

he ca

ge (

)

t (s)

Slip rate of the cage

ωbx = 5krpm

(d)

Figure 9 1e slip rate of the cage Fr 20 kN Fa 0 ωbx 2sim5 krpm

000 003 006 009 012 015 018ndash2

0

2

4

6

8

10

12

0100 0101 0102ndash15ndash10ndash05

00051015

δa = 10μm

Tilt anglet (s)

Tilt

angl

e (10

ndash4 ra

d)

(a)

000 003 006 009 012 015 018ndash2

0

2

4

6

8

10

12

0100 0101 0102ndash20ndash15ndash10ndash05

000510

Tilt anglet (s)

Tilt

angl

e (10

ndash4 ra

d)δa = 20μm

(b)

000 003 006 009 012 015 018ndash3

0

3

6

9

12

15

0100 0101 0102ndash25ndash20ndash15ndash10ndash05

0005

Tilt angle

t (s)

Tilt

angl

e (10

ndash4 ra

d)

δa = 30μm

(c)

000 003 006 009 012 015 018ndash3

0

3

6

9

12

15

0100 0101 0102ndash30ndash23ndash16ndash09ndash02

05

Tilt angle

t (s)

Tilt

angl

e (10

ndash4 ra

d)

δa = 40μm

(d)

Figure 10 1e tilt angle of the conical roller Fr 0 δa 10sim40 μm ωbx 25 krpm

12 Shock and Vibration

000 003 006 009 012 015 01810

26

42

58

74

90

Skew angle

t (s)

Skew

angl

e (10

ndash3 ra

d)

δa = 10μm

(a)

000 003 006 009 012 015 01810

26

42

58

74

90

Skew angle

t (s)

Skew

angl

e (10

ndash3 ra

d)

δa = 40μm

(b)

000 003 006 009 012 015 01810

26

42

58

74

90

Skew angle

t (s)

Skew

angl

e (10

ndash3 ra

d)

δa = 30μm

(c)

000 003 006 009 012 015 01810

26

42

58

74

90

Skew angle

t (s)

Skew

angl

e (10

ndash3 ra

d)

δa = 40μm

(d)

Figure 11 1e skew angle of the conical roller Fr 0 δa 10sim40 μm ωbx 25 krpm

000 003 006 009 012 015 018000

002

004

006

008

Slip

rate

of t

he ro

ller (

)

Slip rate of the roller

t (s)

δa = 10μm

(a)

000 003 006 009 012 015 018000

002

004

006

008

Slip

rate

of t

he ro

ller (

)

Slip rate of the roller

t (s)

δa = 20μm

(b)

Figure 12 Continued

Shock and Vibration 13

types of thickeners In Figure 16(a) under the lubricationof the calcium and polyurea greases the roller tilt angledecreases with the temperature increase Under the lithiumgrease lubrication the roller tilt angle increases with theincrease in temperature In the whole temperature rangethe tilt angle under the lithium grease is generally larger

than that under others With the increase in temperaturethe roller skew angle under the polyurea grease increaseswhile the roller skew angle under the other two types ofgreases decreases 1at the tilt and skew angle under thedifferent greases and temperature show different trends isa result of multiple factors One of the factors is that the

000 003 006 009 012 015 018000

002

004

006

008

Slip

rate

of t

he ro

ller (

)

Slip rate of the roller

t (s)

δa = 30μm

(c)

000 003 006 009 012 015 018000

002

004

006

008

Slip

rate

of t

he ro

ller (

)

Slip rate of the roller

t (s)

δa = 40μm

(d)

Figure 12 1e slip rate of the conical roller Fr 0 δa 10sim40 μm ωbx 25 krpm

000 003 006 009 012 015 01800

05

10

15

20

25

30

Slip rate of the cage

δa = 10μm

01000 01023 01046 010690010

0015

0020

0025

0030

Slip

rate

of t

he ca

ge (

)

t (s)

(a)

Slip rate of the cage

δa = 20μm

000 003 006 009 012 015 01800

05

10

15

20

25

30

0110 0111 0112 0113 0114 0115ndash001

000

001

002

003

004

Slip

rate

of t

he ca

ge (

)

t (s)

(b)

Slip rate of the cage

δa = 30μm

000 003 006 009 012 015 01800

05

10

15

20

25

30

0110 0111 0112 0113 0114 0115000000030006000900120015

Slip

rate

of t

he ca

ge (

)

t (s)

(c)

Slip rate of the cage

δa = 40μm

000 003 006 009 012 015 01800

05

10

15

20

25

30

0110 0111 0112 0113 0114 0115000000030006000900120015

Slip

rate

of t

he ca

ge (

)

t (s)

(d)

Figure 13 1e slip rate of the cage Fr 0 δa 10sim40 μm ωbx 25 krpm

14 Shock and Vibration

244 844 1444 2044 2644 3244 3844ndash52

ndash51

ndash50

ndash49

ndash48

ndash47

Plastic viscosity (Pamiddotsn)

00

16

32

48

64

80

Tilt angleSkew angle

Tilt

angl

e (10

ndash5 ra

d)

Skew

angl

e (10

ndash3 ra

d)

(a)

844 1444 2044 2644 3244 38440000

0004

0008

0012

0016

0020

Slip

rate

of t

he ro

ller (

)

Plastic viscosity (Pamiddotsn)

ndash0006

0000

0006

0012

0018

Slip

rate

of t

he ca

ge (

)

Slip rate of the rollerSlip rate of the cage

(b)

Figure 14 Plastic viscosity influence Fr 0 Fa 15 kN ωbx 28 krpm (a) 1e conical roller tilt and skew (b) Slip rate of the conicalroller and cage

2518 3018 3518 4018 4518 5018ndash550

ndash536

ndash522

ndash508

ndash494

ndash480

Yield stress (Pa)

350

370

390

410

430

450

Tilt angleSkew angle

Tilt

angl

e (10

ndash5 ra

d)

Skew

angl

e (10

ndash3 ra

d)

(a)

Slip rate of the rollerSlip rate of the cage

2518 3018 3518 4018 4518 50180009

0010

0011

0012

0013

0014

0015Sl

ip ra

te o

f the

rolle

r (

)

Yield stress (Pa)

0009

0010

0011

0012

0013

0014

0015

Slip

rate

of t

he ca

ge (

)

(b)

Figure 15 Yield stress influence Fr 0 Fa 15 kN ωbx 28 krpm (a)1e conical roller tilt and skew (b) Slip rate of the conical roller andcage

Table 2 Relationships between grease physical properties and temperature [26]

Grease TemperaturedegC Yield stress τyPa Plastic viscosity φPamiddotsn Power law exponent n

Polyurea

15 5017 1187 0773130 3518 844 0719650 3055 616 0662170 1885 610 07378

Lithium

15 9045 5435 0305330 8052 4623 0312550 7104 4113 0315170 3398 5296 03203

Calcium

15 9908 6482 0334930 6495 4923 0534450 3361 3059 0525670 2960 3588 04356

Shock and Vibration 15

relationships between the grease rheological parametersand temperature are not simple linear For example theplastic viscosity and the power law exponent do not simplyincrease or decrease with temperature

In Figure 16(b) firstly the roller slip rate shows dif-ferent values due to the different greases secondly theroller slip rate shows a variety of trends with the varioustemperature Under the lubrication of the polyurea greasethe roller slip rate increases with the increase in temper-ature while the roller slip with the calcium and lithiumgreases lubricated shows a decreasing trend In real ap-plication the operating temperature of bearings may not beconstant 1e influence of the temperature on dynamiccharacteristics of grease lubricated bearings needs to befurther explored

5 Conclusions

For the beginning operating stages of grease-lubricatedTRBs the multibody contact dynamic model of TRBs wasestablished 1e impacts of preload temperature and greaserheological properties on the bearing dynamics were ana-lyzed Based on numerical results several conclusions can besummarized

(1) 1e roller tilt angle in the unloaded zone is obviouslylarger than that in the loaded zone while the rollerskew angle in the unloaded zone is smaller than thatin the loaded zone 1e effect of bearing speed on theroller skew is greater than that on the roller tilt Asthe speed increases the roller tilt and skew and thebearing slip become more serious

(2) Bearing preload can effectively reduce the bearingslip but will make the roller tilt angle increase In thestable operation stage of the bearing as the preloadincreases the roller skew angle increases

(3) 1e roller skew angle and the bearing slip rate de-crease with the increase of the grease plastic viscosity1e roller tilt angle increases with the increase of the

plastic viscosity 1e yield stress of the grease haslittle effect on motions of the roller and cage

(4) 1e influence of temperature onmotions of the rollerand cage varies with the type of grease used Whenthe TRB is lubricated by the lithium grease with theincrease of the temperature the roller tilt angle in-creases the roller skew angle and the bearing slip ratedecrease When the polyurea grease is adopted thechange trend of the roller tilt and skew and thebearing slip is just opposite to that when the lithiumgrease is used Under the calcium grease lubricationthe roller tilt and skew angle and the bearing slip ratedecrease with the temperature increase

Data Availability

1e figure data used to support the findings of this studyhave been deposited in the FigShare repository at httpsfigsharecoms15c2556bee83d1adf01d

Conflicts of Interest

1e authors declare that they have no conflicts of interest

Acknowledgments

1e paper was supported by the National Natural ScienceFoundation of China (Grant No 51675427)

References

[1] P M Lugt ldquoA review on grease lubrication in rollingbearingsrdquo Tribology Transactions vol 52 no 4 pp 470ndash4802009

[2] T Cousseau Film thickness and friction in grease lubricatedcontacts Application to rolling bearing torque loss PhDDepartamento De Engenharia Mecanica E Gestao Industrial2013

[3] C T Walters ldquo1e dynamics of ball bearingsrdquo Journal ofTribology vol 93 no 1 pp 1ndash10 1971

10 23 36 49 62 75ndash58

ndash56

ndash54

ndash52

ndash50

ndash48

ndash46

ndash44

CalciumLithiumPolyurea tilt angle skew angle

tilt angle skew angletilt angle skew angle

Temperature (degC)

10

30

50

70

90

110

130Ti

lt an

gle (

10ndash5

rad)

Skew

angl

e (10

ndash3 ra

d)

(a)

10 23 36 49 62 750000

0005

0010

0015

0020

0025

0030

Poly

urea

gre

ase (

)

Temperature (degC)

0000

0001

0002

0003

0004

0005

0006

Calc

ium

lith

ium

gre

ase (

)

CalciumLithiumPolyurea

(b)

Figure 16 Temperature influence Fr 0 Fa 15 kN ωbx 28 krpm (a) 1e conical roller tilt and skew (b) Slip rate of the conical roller

16 Shock and Vibration

Qasj qsj 0 0[ ]T

Nrsj rrsr + rrgm( ) times TarQ

asj

(41)

e loads acting on the cage areQi

sj minusTaiQasj

Ncsj r0 + Tcpprime r

pcr + Trc rrsr + rrgm( ) times TicQ

isj[ ]

(42)

where rrsr is the position of the contact point in the rollerbody-xed coordinate and Tar and Tai are respectivelythe transform from the azimuth to roller and inertialcoordinate

According to geometric characteristics of the cage andguide ring it can be considered as a nite journal bearinglubrication condition between the guide surface and cagecentering surface In Figure 4 the clearance δprime between thecage and guide ring is

δprime Δe minus e (43)

where Δe is the initial clearance Δe (ry minus rd)2 rd ryare respectively the radii of the centering surface andguide surface and e is cage displacement in the radial plane

Similarly assume that Δh0prime is the critical value of thegrease lm thickness When δprime ge Δh0prime the contact force qicand friction torque Nic on the cage are

qic qicehl keprimeδ + ceprime _δprime

Nic Nicehl πεφ0br

12ic rd

212h1203ωbxry minusωcxrd( )

(44)

where ric is the equivalent radius and ωcx ωbx arerespectively the angular velocities of the cage and guidering e calculation of the grease lm stiness keprime anddamping ceprime is the same as that at the rollercrossbeamcontact

When δprime lt Δh0prime the elastic deformation between the cageand guide ring is δhprime δprime minus h0 and the minimum grease lmthickness h0 Δh0prime e qic and Nic are

qic khprimeδhprime + chprimeδhprime + qicehlNic μbdqicrd +Nicehl

(45)

where khprime is the contact stiness and chprime is the viscousdamping coecurrencient e calculation of khprime and chprime is the sameas that in Equation (35) μbd is the traction coecurrencient underboundary lubrication

In order to simplify themodel the load of the cage on theguide ring is ignored in contrast to large external load on theguide ring en the loads acting on the cage are

Qiic 0 qic sinφic qic cosφic[ ]T

Ncic plusmnNic 0 0[ ]

(46)

If the TRB is lubricated by the oil the eect of oil-gasmixture on the cage and rollers may not be overlookedHowever if the bearing is lubricated by the grease it seemsimpossible to have the grease-gas mixture eect on the

dynamics of the cage and rollers So the eect of the grease-gas mixture is not considered in the dynamic model

24DynamicModel In real application also the surroundings(inertiadamping of shaft assembly and housing) are expectedto inuence the TRB dynamics ese eects are neglected inthe modeling Generally it is convenient to consider thetranslational moving of rollers in the cylindrical coordinatesystem while the Cartesian coordinate system is convenient forthe cage and racee translationalmoving of bearing parts canbe simply described by Newtonrsquos laws such that

Meuroδ Q (47)

e generalized force Q is obtained by superimposingthe forces from Section 2 e centrifugal force and thegyroscopic moment should be superimposed on the gen-eralized forceQ for the roller For the cage and race the massmatrices Mb and Mc are

Mbc diag mbc mbc mbc[ ] (48)

For the special case of conical rollers Mr is

Mr diag mr mrrrj mr[ ] (49)

Both rollers and the cage have six degrees-of-freedomUnlike the translational motion (in three directions) therotating of the bearing parts are described in their own body-xed coordinates For rollers and the cage using the Eulerdynamic equations such that

Ix _ωx minusωyωz Iy minus Iz( )

Iy _ωy minusωzωx Iz minus Ix( )

Iz _ωz minusωxωx Ix minus Iy( )

Nx

Ny

Nz

(50)

where Ix Iy and Iz are inertia principalmoments of the cage orrollers the total momentN is also obtained by superimposingthe forces from Section 2 ω[ωx ωy ωz] is the angular velocityin the inertial coordinate systeme relationship between theangular velocity ωxed in the body-xed coordinate systemand ω is as follows B is the Eulerrsquos rotation matrix

∆y

yb

zb

yc

ωcx

ωbx

φicNic

δprime

zc

oc

∆z eob

qic

Guidering

Cage

Roller

Figure 4 Interaction between the cage and guide ring (inner-ringguided)

6 Shock and Vibration

ωfixed Bω

_ωfixed dBdt

Bminus1 _ω

(51)

Admitting the outer ring as macroscopically stationaryand a rotating inner ring with no torque load it resultsfour degrees of freedom of inner ring moving in the x yand z directions and constant rotating about the shaftaxis (o-x)

3 Numerical Method

1e bearing geometry and material parameters and greasemain rheological parameters (30degC) are shown in Table 11e cage is an inner-ring guided type and its material ispolyamide 1e polyurea grease was adopted [26] 1efourth-order RungendashKutta method was used to dissolvetransient responses of the TRB on MATLAB To ensure theconvergence of simulation initial values of displacementvelocity and load are obtained by the quasidynamicmethod

4 Results and Discussion

For the complicated dynamic response of TRBs it is essentialto consider several typical and harmful movements of thebearing in the process of dynamic analysis such as the rollerskew and tilt bearing slip and so on 1e slip rate of theroller and cage is defined as

|theoretical speedminus actual speed|

theoretical speedtimes 100 (52)

For the roller the speed refers to the revolution speedfor the cage refers to the rotational speed

41 Mass Center Trajectory and Velocity 1e trajectoriesand velocities of mass centers of the main bearing parts areshown in Figure 5 1e axial preload is not applied to theinner ring but limited to the degree of freedom of the innerring in the axial direction 1e cage initial rotation speed isthe same as the roller initial revolution speed Due to thechanging position of rollers the distribution of rollers issymmetrical or asymmetric with respect to the o-z axis1erefore the inner ring vibrates slightly in the o-y di-rection 1e radial and axial displacement (relative to theinitial position) of the roller exhibit a periodic fluctuationfor the bearing is in a semicircle-loaded state under pureradial load conditions When the rollers alternately enter theloaded and unloaded zones the load on rollers will showperiodic changes resulting in a regular variation in dis-placement 1e cage whirling is approximately a circle Fora number of uncertain rollers acting on the cage the cagespeed does not show a periodic change

42 Rotational Speed Influence In this study the tilt andskew of conical rollers are regarded as the primary eval-uation variables for the dynamic performance of the TRB

1e roller skew is mainly generated from the tangentialfriction force on the flangeroller-end and rollerracewaycontacts while the tilt motion is mainly induced by thenormal contact force on those contacts According to thelubrication theory the bearing speed will be a key factoraffecting the load case in bearing contacts

1e influence of the bearing speed on the roller tilt andskew (relative to the initial position) is shown in Figure 61e degree of freedom of the inner ring in the axial directionis also limited At each speed level the roller tilt anglepresents a relatively regular periodic vibration 1e tilt anglein the unloaded zone is obviously larger than that in theloaded zone 1is is due to the larger gap between the rollerand raceways in the unloaded zone when only the radicalload is applied and rollers in unloaded zone seem to beldquorelaxedrdquo As the speed increases the maximum of the tiltangle almost does not change while the average tilt anglerises from minus014 times 10minus4 rad to minus040 times 10minus4 rad indicatingthat the bearing speed affected the roller tilt It can beconfirmed that the gyroscopic moment is one of the reasonsthat cause the rising of the roller tilt angle for the gyroscopicmotion is intensified with the bearing speed increasingAlthough the gyroscopic moment increases as the bearingspeed increases it is still relatively smaller than theforcetorque in the rollerraceway contact So for the radialload is constant the maximum tilt angle of the roller changeslittle with the speed

As it is shown in Figure 7 the skew angle both in theloaded zone and in the unloaded zone is relatively large at thespeed of 2 krpm In the entire speed range the roller skewangle in the loaded zone is greater than that in the unloadedzone For the rollers is ldquorelaxedrdquo in the unloaded zone the

Table 1 Geometry material and rheological parameters

Parameter ValueNumber of rollers 17Roller length lemm 4206Roller-end sphere radius rsmm 13620Diameter of roller large end dlmm 1785Diameter of roller small end dsmm 2476Outer contact angledeg 15Inner contact angledeg 5Flange contact angledeg 83Outer raceway radiusmm 9952Inner raceway radiusmm 7848Guide land clearancemm 05Guide land widthmm 493Cage outer diametermm 20576Cage inner diametermm 19610Cage thickness hsmm 5Pocket clearance Δδmm 027Elastic modulus of rollerraceGPa 205Poissonrsquos ratio of rollerrace 03Elastic modulus of cageGPa 83Poissonrsquos ratio of cage 028

Grease rheological parameters (30degC)Yield stress τyPa 3518Plastic viscosity φPamiddotsn 844Power law exponent n 07196Pressure-viscosity coefficient αm2middotNminus11 4662 times 10minus8

Shock and Vibration 7

ndash6 ndash4 ndash2 0 2 4 60

3

6

9

12

15

Trajectory of the inner ring

z-di

rect

ion

disp

lace

men

t (μm

)

y-direction displacement (μm)

(a)

Displacement of the roller

0 003 006 009 012 015 018ndash20

ndash10

0

10

Radi

al (μ

m)

t (s)

0 003 006 009 012 015 018ndash20

10

40

70

Axi

al (μ

m)

(b)

Trajectory of the cage

ndash06 ndash04 ndash02 00 02 04 06ndash06

ndash04

ndash02

00

02

04

06

z-di

rect

ion

disp

lace

men

t (m

m)

y-direction displacement (mm)

(c)

Cage rotation speed

0 003 006 009 012 015 018102104106108110

Cage

(krp

m)

t (s)

Roller revolution speed0 003 006 009 012 015 018

102104106108110

Rolle

r (kr

pm)

(d)

Figure 5 1e trajectory and velocity of mass center Fr 20 kN Fa 0 ωbx 24 krpm

000 003 006 009 012 015 018ndash9

ndash6

ndash3

0

3

6

9

101

Tilt angle

ωbx = 2krpm

t (s)

Tilt

angl

e (10

ndash4 ra

d)

(a)

Tilt angle

ωbx = 3krpm

000 003 006 009 012 015 018ndash9

ndash6

ndash3

0

3

6

9

101

Tilt

angl

e (10

ndash4 ra

d)

t (s)

(b)

Figure 6 Continued

8 Shock and Vibration

skewmoment of the raceway and flange acting on the roller issmaller than that in the loaded zone It shows obvious pe-riodicity when the speed reaches 3 krpm or more1e averageskew angle is increased from 306 times 10minus3 rad017 deg to 447 times

10minus3 rad025 deg 1e increase of bearing rotating speedenlarges the centrifugal force of rollers which makes contactforces at the rollerouter raceway and flangeroller-end

increase As a result of combined contribution of racewayfriction and the flange friction the roller skew angle is greatlyincreased Compared with the experimental result measuredby Falodi et al in which theymeasured the average skew anglebetween 015 and 045 deg [27] 1e average skew angle inFigure 7 is between 017 and 025 deg and at an order ofmagnitude compared to the experimental result 1e

Tilt angle

ωbx = 4krpm

Tilt

angl

e (10

ndash4 ra

d)

000 003 006 009 012 015 018ndash9

ndash6

ndash3

0

3

6

9

101

t (s)

(c)

Tilt angle

ωbx = 5krpm

Tilt

angl

e (10

ndash4 ra

d)

000 003 006 009 012 015 018ndash9

ndash6

ndash3

0

3

6

9

101

t (s)

(d)

Figure 6 1e tilt angle of the conical roller 1 loaded zone 0 unloaded zone Fr 20 kN Fa 0 ωbx 2sim5 krpm

000 003 006 009 012 015 018ndash8

ndash3

2

7

12

17

22

101

Skew

angl

e (10

ndash3 ra

d)

t (s)

ωbx = 2krpm

Skew angle

(a)

Skew

angl

e (10

ndash3 ra

d)

000 003 006 009 012 015 018ndash4

0

4

8

12

16

101

t (s)

ωbx = 3krpm

Skew angle

(b)

Skew

angl

e (10

ndash3 ra

d)

000 003 006 009 012 015 018ndash4

0

4

8

12

16

101

t (s)

ωbx = 4krpm

Skew angle

(c)

Skew

angl

e (10

ndash3 ra

d)

000 003 006 009 012 015 018ndash4

0

4

8

12

16

0 11

t (s)

ωbx = 5krpm

Skew angle

(d)

Figure 7 1e skew angle of the conical roller 1 loaded zone 0 unloaded zone Fr 20 kN Fa 0 ωbx 2sim5 krpm

Shock and Vibration 9

changing trend of the skew angle with speed is also consistentwith Yangrsquos results

1e bearing slip is a destructive motion that easily leadsto scuffing welding and overwear of bearing surfaces 1ebearing slip usually occurs at high-speed and light-loadconditions 1e influence of the bearing speed on the slipof the roller and cage is exhibited in Figures 8 and 9

1e diagrams depicted in Figure 8 indicate that theroller slip is becoming more and more serious as thebearing speed is raised 1is variation trend is similar tothat in [7] With the bearing speed increases the roller slipgradually presents a periodic change rule especially whenthe speed is up to 3 krpm or more 1e roller slip in loadedand unloaded zones is not the same In the unloaded zonethe roller slip is more serious than that in the loaded zoneAt the speed of 5 krpm the bearing is in a state of seriousslipping In Figure 9 the cage slip is aggravated with thebearing speed increase 1ere is no periodic change in thecage slip rate 1is is because when the cage is whirlinga number of uncertain rollers may act on the cage whichcan lead to an irregular motion of the cage At the samespeed level the maximum amplitude of the slip rate of thecage and roller is basically the same for the dependentrelationship between the cage and roller

43 Axial Preload Influence Axial preload is not only a keyfactor affecting the bearing dynamics but one of the mainmeans to ensure the normal service life of the bearing Inorder to familiarize the influence of the axial preload onthe roller tilt and skew different axial preloads wereapplied to the TRB under the condition that the radial loadis zero

As shown in Figures 10 and 11 after entering the smoothrunning phase as the axial preload varies from 10 μm to40 μm the average value of the roller tilt rises from minus027 times

10minus4 rad to minus096 times 10minus4 rad 1e greater the preload themore obvious the roller tends to tilt 1is is because as thepreload increases the negative moment of the flange actingon the roller-end becomes larger thus causing the roller tiltto assume a negative tendency

Except for the condition that the preload is 10 μm theroller skew has larger amplitude at the beginning and thenthe amplitude decreases gradually and tends to be stable Inthe smooth running phase as the preload increases theroller skew angle increases 1is trend is in line with theresults in [8 27] 1e average skew angle is increasedfrom 423 times 10minus3 rad024 deg to 550 times 10minus3 rad031 degBoth the roller tilt and skew present a more regular vibrationthroughout the bearing operation this is due to symmetry ofthe bearing on the radial plane and pure preloading con-dition the loaded and unloaded zones do not exist in thebearing and loads on rollers almost change a little in anyposition

1e changes of the slip rate of the roller and cage with thepreload are presented in Figures 12 and 13 As the bearingpreload increases the bearing slip decreases in the smoothrunning phase 1e average slip rate of the roller decreasesfrom 00189 to 00057 and that of the cage decreases from

00189 to 00055 indicating that the appropriate bearingpreload can effectively prevent bearing slipping failure 1eresult is similar with the conclusion in [7 28] 1e influenceof the preload on the bearing slip reduced with the increaseof the preload value 1e average slip rate of the roller andcage when the preload is 40 μm changes little compared with00073 and 00076 when the preload is 30 μm Obviouslyincreasing the amount of preload will help to decrease thebearing slip but excessive preload will inevitably reduce theservice life of bearings 1erefore to meet the bearing lifeand service requirements reasonable selection of preload issignificant

44Grease Physical Properties Influence In view the fact thatthe calculation of the traction force stiffness and damping atthe bearing contacts is mainly based on the plastic viscosity φand yield stress τy of the grease For a clearer understandingof the relationship between the bearing dynamics and thegrease used grease physical properties influence on bearingdynamics is analyzed 1e temperature pressure speed andother surroundings have significant effects on the greasephysical properties For the sake of simplification the im-pacts of these factors on the grease physical properties areignored In order to highlight the effect of a certain rheo-logical parameter on the bearing performance only oneparameter was given an appropriate change while otherrheological parameters remain unchanged 1e effects areshown in Figures 14 and 15 After the bearing enters thestable running state (after 01 s) the average value of theangle (tilt and skew) is used as an indicator of bearingoperating state

1e effect of the plastic viscosity on the roller and cagemotions are shown in Figure 14(a) As the plastic viscosityincreases the tilt angle and skew angle increases and de-creases respectively For the grease film thickness increaseswith the increase of the plastic viscosity the load ratio ofmicroasperities at the flangeroller-end contact is reduced1en it indirectly leads to the decrease of the skew mo-ment at the flangeroller-end contact and the roller skewangle Due to the smaller skew angle the roller misalignmentwill also be small 1erefore the normal force at therollerraceways and flangeroller-end contacts will havea greater contribution to the roller tilt 1erefore the resultshows that the roller tilt angle increases with the plasticviscosity increase In Figure 14(b) the bearing slip decreaseswith the grease plastic viscosity increase As the plasticviscosity increases the grease film thickness at therollerraceways and flangeroller-end contacts increase Sothe friction force at these contacts is reduced which maylead to the decrease of the bearing slip1e bearing slip is theresult of the combined effect of raceway friction and theflange friction

Compared with the impacts of the plastic viscosity of thegrease on the motions of the roller and cage the influence ofthe grease yield stress on the roller tilt roller skew andbearing slip is relatively smaller as shown in Figure 15 Forthe yield stress has a negligible effect on the film thickness inall practical cases [9]

10 Shock and Vibration

000 003 006 009 012 015 01800

05

10

15

20

25

30

Slip

rate

of t

he ro

ller (

)

t (s)

01000 01025 01050 01075 011ndash001

001

003

005

007ωbx = 2krpm

Slip rate of the roller

(a)

000 003 006 009 012 015 01800

02

04

06

08

Slip

rate

of t

he ro

ller (

)

t (s)

ωbx = 3krpm

Slip rate of the roller

(b)

000 003 006 009 012 015 01800

05

10

15

20

25

30

Slip

rate

of t

he ro

ller (

)

t (s)

ωbx = 4krpm

Slip rate of the roller

(c)

000 003 006 009 012 015 0180

2

4

6

8

10

Slip

rate

of t

he ro

ller (

)

t (s)

ωbx = 5krpm

Slip rate of the roller

(d)

Figure 8 1e slip rate of the conical roller Fr 20 kN Fa 0 ωbx 2sim5 krpm

000 003 006 009 012 015 01800

05

10

15

20

25

30

Slip

rate

of t

he ca

ge (

)

01000 01025 01050 01075 011ndash000

001

002

003

t (s)

Slip rate of the cage

ωbx = 2krpm

(a)

000 003 006 009 012 015 01800

05

10

15

20

25

30

Slip

rate

of t

he ca

ge (

)

t (s)

Slip rate of the cage

ωbx = 3krpm

(b)

Figure 9 Continued

Shock and Vibration 11

45 Temperature Influence Due to the improper in-stallation overwear or not timely cooling the operatingtemperature of TRBs will change more or less Sometimesit may lead to heat accumulation or thermal failure ofbearings 1e most direct effect of the temperature ischanges of the grease physical properties 1e rheologicalparameters of the greases at different temperatures in

Table 2 are quoted from the data in reference [26] 1eeffect of temperature on the dynamics of the bearing wasstudied 1e impact of temperature on the bearing struc-ture is not considered here

1ree typical greases were used such as the calciumgrease lithium grease and polyurea grease 1e largestdifference in the composition of three greases lies in the

000 003 006 009 012 015 01800

05

10

15

20

25

30

35

Slip

rate

of t

he ca

ge (

)

t (s)

Slip rate of the cage

ωbx = 4krpm

(c)

000 003 006 009 012 015 0180

2

4

6

8

10

Slip

rate

of t

he ca

ge (

)

t (s)

Slip rate of the cage

ωbx = 5krpm

(d)

Figure 9 1e slip rate of the cage Fr 20 kN Fa 0 ωbx 2sim5 krpm

000 003 006 009 012 015 018ndash2

0

2

4

6

8

10

12

0100 0101 0102ndash15ndash10ndash05

00051015

δa = 10μm

Tilt anglet (s)

Tilt

angl

e (10

ndash4 ra

d)

(a)

000 003 006 009 012 015 018ndash2

0

2

4

6

8

10

12

0100 0101 0102ndash20ndash15ndash10ndash05

000510

Tilt anglet (s)

Tilt

angl

e (10

ndash4 ra

d)δa = 20μm

(b)

000 003 006 009 012 015 018ndash3

0

3

6

9

12

15

0100 0101 0102ndash25ndash20ndash15ndash10ndash05

0005

Tilt angle

t (s)

Tilt

angl

e (10

ndash4 ra

d)

δa = 30μm

(c)

000 003 006 009 012 015 018ndash3

0

3

6

9

12

15

0100 0101 0102ndash30ndash23ndash16ndash09ndash02

05

Tilt angle

t (s)

Tilt

angl

e (10

ndash4 ra

d)

δa = 40μm

(d)

Figure 10 1e tilt angle of the conical roller Fr 0 δa 10sim40 μm ωbx 25 krpm

12 Shock and Vibration

000 003 006 009 012 015 01810

26

42

58

74

90

Skew angle

t (s)

Skew

angl

e (10

ndash3 ra

d)

δa = 10μm

(a)

000 003 006 009 012 015 01810

26

42

58

74

90

Skew angle

t (s)

Skew

angl

e (10

ndash3 ra

d)

δa = 40μm

(b)

000 003 006 009 012 015 01810

26

42

58

74

90

Skew angle

t (s)

Skew

angl

e (10

ndash3 ra

d)

δa = 30μm

(c)

000 003 006 009 012 015 01810

26

42

58

74

90

Skew angle

t (s)

Skew

angl

e (10

ndash3 ra

d)

δa = 40μm

(d)

Figure 11 1e skew angle of the conical roller Fr 0 δa 10sim40 μm ωbx 25 krpm

000 003 006 009 012 015 018000

002

004

006

008

Slip

rate

of t

he ro

ller (

)

Slip rate of the roller

t (s)

δa = 10μm

(a)

000 003 006 009 012 015 018000

002

004

006

008

Slip

rate

of t

he ro

ller (

)

Slip rate of the roller

t (s)

δa = 20μm

(b)

Figure 12 Continued

Shock and Vibration 13

types of thickeners In Figure 16(a) under the lubricationof the calcium and polyurea greases the roller tilt angledecreases with the temperature increase Under the lithiumgrease lubrication the roller tilt angle increases with theincrease in temperature In the whole temperature rangethe tilt angle under the lithium grease is generally larger

than that under others With the increase in temperaturethe roller skew angle under the polyurea grease increaseswhile the roller skew angle under the other two types ofgreases decreases 1at the tilt and skew angle under thedifferent greases and temperature show different trends isa result of multiple factors One of the factors is that the

000 003 006 009 012 015 018000

002

004

006

008

Slip

rate

of t

he ro

ller (

)

Slip rate of the roller

t (s)

δa = 30μm

(c)

000 003 006 009 012 015 018000

002

004

006

008

Slip

rate

of t

he ro

ller (

)

Slip rate of the roller

t (s)

δa = 40μm

(d)

Figure 12 1e slip rate of the conical roller Fr 0 δa 10sim40 μm ωbx 25 krpm

000 003 006 009 012 015 01800

05

10

15

20

25

30

Slip rate of the cage

δa = 10μm

01000 01023 01046 010690010

0015

0020

0025

0030

Slip

rate

of t

he ca

ge (

)

t (s)

(a)

Slip rate of the cage

δa = 20μm

000 003 006 009 012 015 01800

05

10

15

20

25

30

0110 0111 0112 0113 0114 0115ndash001

000

001

002

003

004

Slip

rate

of t

he ca

ge (

)

t (s)

(b)

Slip rate of the cage

δa = 30μm

000 003 006 009 012 015 01800

05

10

15

20

25

30

0110 0111 0112 0113 0114 0115000000030006000900120015

Slip

rate

of t

he ca

ge (

)

t (s)

(c)

Slip rate of the cage

δa = 40μm

000 003 006 009 012 015 01800

05

10

15

20

25

30

0110 0111 0112 0113 0114 0115000000030006000900120015

Slip

rate

of t

he ca

ge (

)

t (s)

(d)

Figure 13 1e slip rate of the cage Fr 0 δa 10sim40 μm ωbx 25 krpm

14 Shock and Vibration

244 844 1444 2044 2644 3244 3844ndash52

ndash51

ndash50

ndash49

ndash48

ndash47

Plastic viscosity (Pamiddotsn)

00

16

32

48

64

80

Tilt angleSkew angle

Tilt

angl

e (10

ndash5 ra

d)

Skew

angl

e (10

ndash3 ra

d)

(a)

844 1444 2044 2644 3244 38440000

0004

0008

0012

0016

0020

Slip

rate

of t

he ro

ller (

)

Plastic viscosity (Pamiddotsn)

ndash0006

0000

0006

0012

0018

Slip

rate

of t

he ca

ge (

)

Slip rate of the rollerSlip rate of the cage

(b)

Figure 14 Plastic viscosity influence Fr 0 Fa 15 kN ωbx 28 krpm (a) 1e conical roller tilt and skew (b) Slip rate of the conicalroller and cage

2518 3018 3518 4018 4518 5018ndash550

ndash536

ndash522

ndash508

ndash494

ndash480

Yield stress (Pa)

350

370

390

410

430

450

Tilt angleSkew angle

Tilt

angl

e (10

ndash5 ra

d)

Skew

angl

e (10

ndash3 ra

d)

(a)

Slip rate of the rollerSlip rate of the cage

2518 3018 3518 4018 4518 50180009

0010

0011

0012

0013

0014

0015Sl

ip ra

te o

f the

rolle

r (

)

Yield stress (Pa)

0009

0010

0011

0012

0013

0014

0015

Slip

rate

of t

he ca

ge (

)

(b)

Figure 15 Yield stress influence Fr 0 Fa 15 kN ωbx 28 krpm (a)1e conical roller tilt and skew (b) Slip rate of the conical roller andcage

Table 2 Relationships between grease physical properties and temperature [26]

Grease TemperaturedegC Yield stress τyPa Plastic viscosity φPamiddotsn Power law exponent n

Polyurea

15 5017 1187 0773130 3518 844 0719650 3055 616 0662170 1885 610 07378

Lithium

15 9045 5435 0305330 8052 4623 0312550 7104 4113 0315170 3398 5296 03203

Calcium

15 9908 6482 0334930 6495 4923 0534450 3361 3059 0525670 2960 3588 04356

Shock and Vibration 15

relationships between the grease rheological parametersand temperature are not simple linear For example theplastic viscosity and the power law exponent do not simplyincrease or decrease with temperature

In Figure 16(b) firstly the roller slip rate shows dif-ferent values due to the different greases secondly theroller slip rate shows a variety of trends with the varioustemperature Under the lubrication of the polyurea greasethe roller slip rate increases with the increase in temper-ature while the roller slip with the calcium and lithiumgreases lubricated shows a decreasing trend In real ap-plication the operating temperature of bearings may not beconstant 1e influence of the temperature on dynamiccharacteristics of grease lubricated bearings needs to befurther explored

5 Conclusions

For the beginning operating stages of grease-lubricatedTRBs the multibody contact dynamic model of TRBs wasestablished 1e impacts of preload temperature and greaserheological properties on the bearing dynamics were ana-lyzed Based on numerical results several conclusions can besummarized

(1) 1e roller tilt angle in the unloaded zone is obviouslylarger than that in the loaded zone while the rollerskew angle in the unloaded zone is smaller than thatin the loaded zone 1e effect of bearing speed on theroller skew is greater than that on the roller tilt Asthe speed increases the roller tilt and skew and thebearing slip become more serious

(2) Bearing preload can effectively reduce the bearingslip but will make the roller tilt angle increase In thestable operation stage of the bearing as the preloadincreases the roller skew angle increases

(3) 1e roller skew angle and the bearing slip rate de-crease with the increase of the grease plastic viscosity1e roller tilt angle increases with the increase of the

plastic viscosity 1e yield stress of the grease haslittle effect on motions of the roller and cage

(4) 1e influence of temperature onmotions of the rollerand cage varies with the type of grease used Whenthe TRB is lubricated by the lithium grease with theincrease of the temperature the roller tilt angle in-creases the roller skew angle and the bearing slip ratedecrease When the polyurea grease is adopted thechange trend of the roller tilt and skew and thebearing slip is just opposite to that when the lithiumgrease is used Under the calcium grease lubricationthe roller tilt and skew angle and the bearing slip ratedecrease with the temperature increase

Data Availability

1e figure data used to support the findings of this studyhave been deposited in the FigShare repository at httpsfigsharecoms15c2556bee83d1adf01d

Conflicts of Interest

1e authors declare that they have no conflicts of interest

Acknowledgments

1e paper was supported by the National Natural ScienceFoundation of China (Grant No 51675427)

References

[1] P M Lugt ldquoA review on grease lubrication in rollingbearingsrdquo Tribology Transactions vol 52 no 4 pp 470ndash4802009

[2] T Cousseau Film thickness and friction in grease lubricatedcontacts Application to rolling bearing torque loss PhDDepartamento De Engenharia Mecanica E Gestao Industrial2013

[3] C T Walters ldquo1e dynamics of ball bearingsrdquo Journal ofTribology vol 93 no 1 pp 1ndash10 1971

10 23 36 49 62 75ndash58

ndash56

ndash54

ndash52

ndash50

ndash48

ndash46

ndash44

CalciumLithiumPolyurea tilt angle skew angle

tilt angle skew angletilt angle skew angle

Temperature (degC)

10

30

50

70

90

110

130Ti

lt an

gle (

10ndash5

rad)

Skew

angl

e (10

ndash3 ra

d)

(a)

10 23 36 49 62 750000

0005

0010

0015

0020

0025

0030

Poly

urea

gre

ase (

)

Temperature (degC)

0000

0001

0002

0003

0004

0005

0006

Calc

ium

lith

ium

gre

ase (

)

CalciumLithiumPolyurea

(b)

Figure 16 Temperature influence Fr 0 Fa 15 kN ωbx 28 krpm (a) 1e conical roller tilt and skew (b) Slip rate of the conical roller

16 Shock and Vibration

ωfixed Bω

_ωfixed dBdt

Bminus1 _ω

(51)

Admitting the outer ring as macroscopically stationaryand a rotating inner ring with no torque load it resultsfour degrees of freedom of inner ring moving in the x yand z directions and constant rotating about the shaftaxis (o-x)

3 Numerical Method

1e bearing geometry and material parameters and greasemain rheological parameters (30degC) are shown in Table 11e cage is an inner-ring guided type and its material ispolyamide 1e polyurea grease was adopted [26] 1efourth-order RungendashKutta method was used to dissolvetransient responses of the TRB on MATLAB To ensure theconvergence of simulation initial values of displacementvelocity and load are obtained by the quasidynamicmethod

4 Results and Discussion

For the complicated dynamic response of TRBs it is essentialto consider several typical and harmful movements of thebearing in the process of dynamic analysis such as the rollerskew and tilt bearing slip and so on 1e slip rate of theroller and cage is defined as

|theoretical speedminus actual speed|

theoretical speedtimes 100 (52)

For the roller the speed refers to the revolution speedfor the cage refers to the rotational speed

41 Mass Center Trajectory and Velocity 1e trajectoriesand velocities of mass centers of the main bearing parts areshown in Figure 5 1e axial preload is not applied to theinner ring but limited to the degree of freedom of the innerring in the axial direction 1e cage initial rotation speed isthe same as the roller initial revolution speed Due to thechanging position of rollers the distribution of rollers issymmetrical or asymmetric with respect to the o-z axis1erefore the inner ring vibrates slightly in the o-y di-rection 1e radial and axial displacement (relative to theinitial position) of the roller exhibit a periodic fluctuationfor the bearing is in a semicircle-loaded state under pureradial load conditions When the rollers alternately enter theloaded and unloaded zones the load on rollers will showperiodic changes resulting in a regular variation in dis-placement 1e cage whirling is approximately a circle Fora number of uncertain rollers acting on the cage the cagespeed does not show a periodic change

42 Rotational Speed Influence In this study the tilt andskew of conical rollers are regarded as the primary eval-uation variables for the dynamic performance of the TRB

1e roller skew is mainly generated from the tangentialfriction force on the flangeroller-end and rollerracewaycontacts while the tilt motion is mainly induced by thenormal contact force on those contacts According to thelubrication theory the bearing speed will be a key factoraffecting the load case in bearing contacts

1e influence of the bearing speed on the roller tilt andskew (relative to the initial position) is shown in Figure 61e degree of freedom of the inner ring in the axial directionis also limited At each speed level the roller tilt anglepresents a relatively regular periodic vibration 1e tilt anglein the unloaded zone is obviously larger than that in theloaded zone 1is is due to the larger gap between the rollerand raceways in the unloaded zone when only the radicalload is applied and rollers in unloaded zone seem to beldquorelaxedrdquo As the speed increases the maximum of the tiltangle almost does not change while the average tilt anglerises from minus014 times 10minus4 rad to minus040 times 10minus4 rad indicatingthat the bearing speed affected the roller tilt It can beconfirmed that the gyroscopic moment is one of the reasonsthat cause the rising of the roller tilt angle for the gyroscopicmotion is intensified with the bearing speed increasingAlthough the gyroscopic moment increases as the bearingspeed increases it is still relatively smaller than theforcetorque in the rollerraceway contact So for the radialload is constant the maximum tilt angle of the roller changeslittle with the speed

As it is shown in Figure 7 the skew angle both in theloaded zone and in the unloaded zone is relatively large at thespeed of 2 krpm In the entire speed range the roller skewangle in the loaded zone is greater than that in the unloadedzone For the rollers is ldquorelaxedrdquo in the unloaded zone the

Table 1 Geometry material and rheological parameters

Parameter ValueNumber of rollers 17Roller length lemm 4206Roller-end sphere radius rsmm 13620Diameter of roller large end dlmm 1785Diameter of roller small end dsmm 2476Outer contact angledeg 15Inner contact angledeg 5Flange contact angledeg 83Outer raceway radiusmm 9952Inner raceway radiusmm 7848Guide land clearancemm 05Guide land widthmm 493Cage outer diametermm 20576Cage inner diametermm 19610Cage thickness hsmm 5Pocket clearance Δδmm 027Elastic modulus of rollerraceGPa 205Poissonrsquos ratio of rollerrace 03Elastic modulus of cageGPa 83Poissonrsquos ratio of cage 028

Grease rheological parameters (30degC)Yield stress τyPa 3518Plastic viscosity φPamiddotsn 844Power law exponent n 07196Pressure-viscosity coefficient αm2middotNminus11 4662 times 10minus8

Shock and Vibration 7

ndash6 ndash4 ndash2 0 2 4 60

3

6

9

12

15

Trajectory of the inner ring

z-di

rect

ion

disp

lace

men

t (μm

)

y-direction displacement (μm)

(a)

Displacement of the roller

0 003 006 009 012 015 018ndash20

ndash10

0

10

Radi

al (μ

m)

t (s)

0 003 006 009 012 015 018ndash20

10

40

70

Axi

al (μ

m)

(b)

Trajectory of the cage

ndash06 ndash04 ndash02 00 02 04 06ndash06

ndash04

ndash02

00

02

04

06

z-di

rect

ion

disp

lace

men

t (m

m)

y-direction displacement (mm)

(c)

Cage rotation speed

0 003 006 009 012 015 018102104106108110

Cage

(krp

m)

t (s)

Roller revolution speed0 003 006 009 012 015 018

102104106108110

Rolle

r (kr

pm)

(d)

Figure 5 1e trajectory and velocity of mass center Fr 20 kN Fa 0 ωbx 24 krpm

000 003 006 009 012 015 018ndash9

ndash6

ndash3

0

3

6

9

101

Tilt angle

ωbx = 2krpm

t (s)

Tilt

angl

e (10

ndash4 ra

d)

(a)

Tilt angle

ωbx = 3krpm

000 003 006 009 012 015 018ndash9

ndash6

ndash3

0

3

6

9

101

Tilt

angl

e (10

ndash4 ra

d)

t (s)

(b)

Figure 6 Continued

8 Shock and Vibration

skewmoment of the raceway and flange acting on the roller issmaller than that in the loaded zone It shows obvious pe-riodicity when the speed reaches 3 krpm or more1e averageskew angle is increased from 306 times 10minus3 rad017 deg to 447 times

10minus3 rad025 deg 1e increase of bearing rotating speedenlarges the centrifugal force of rollers which makes contactforces at the rollerouter raceway and flangeroller-end

increase As a result of combined contribution of racewayfriction and the flange friction the roller skew angle is greatlyincreased Compared with the experimental result measuredby Falodi et al in which theymeasured the average skew anglebetween 015 and 045 deg [27] 1e average skew angle inFigure 7 is between 017 and 025 deg and at an order ofmagnitude compared to the experimental result 1e

Tilt angle

ωbx = 4krpm

Tilt

angl

e (10

ndash4 ra

d)

000 003 006 009 012 015 018ndash9

ndash6

ndash3

0

3

6

9

101

t (s)

(c)

Tilt angle

ωbx = 5krpm

Tilt

angl

e (10

ndash4 ra

d)

000 003 006 009 012 015 018ndash9

ndash6

ndash3

0

3

6

9

101

t (s)

(d)

Figure 6 1e tilt angle of the conical roller 1 loaded zone 0 unloaded zone Fr 20 kN Fa 0 ωbx 2sim5 krpm

000 003 006 009 012 015 018ndash8

ndash3

2

7

12

17

22

101

Skew

angl

e (10

ndash3 ra

d)

t (s)

ωbx = 2krpm

Skew angle

(a)

Skew

angl

e (10

ndash3 ra

d)

000 003 006 009 012 015 018ndash4

0

4

8

12

16

101

t (s)

ωbx = 3krpm

Skew angle

(b)

Skew

angl

e (10

ndash3 ra

d)

000 003 006 009 012 015 018ndash4

0

4

8

12

16

101

t (s)

ωbx = 4krpm

Skew angle

(c)

Skew

angl

e (10

ndash3 ra

d)

000 003 006 009 012 015 018ndash4

0

4

8

12

16

0 11

t (s)

ωbx = 5krpm

Skew angle

(d)

Figure 7 1e skew angle of the conical roller 1 loaded zone 0 unloaded zone Fr 20 kN Fa 0 ωbx 2sim5 krpm

Shock and Vibration 9

changing trend of the skew angle with speed is also consistentwith Yangrsquos results

1e bearing slip is a destructive motion that easily leadsto scuffing welding and overwear of bearing surfaces 1ebearing slip usually occurs at high-speed and light-loadconditions 1e influence of the bearing speed on the slipof the roller and cage is exhibited in Figures 8 and 9

1e diagrams depicted in Figure 8 indicate that theroller slip is becoming more and more serious as thebearing speed is raised 1is variation trend is similar tothat in [7] With the bearing speed increases the roller slipgradually presents a periodic change rule especially whenthe speed is up to 3 krpm or more 1e roller slip in loadedand unloaded zones is not the same In the unloaded zonethe roller slip is more serious than that in the loaded zoneAt the speed of 5 krpm the bearing is in a state of seriousslipping In Figure 9 the cage slip is aggravated with thebearing speed increase 1ere is no periodic change in thecage slip rate 1is is because when the cage is whirlinga number of uncertain rollers may act on the cage whichcan lead to an irregular motion of the cage At the samespeed level the maximum amplitude of the slip rate of thecage and roller is basically the same for the dependentrelationship between the cage and roller

43 Axial Preload Influence Axial preload is not only a keyfactor affecting the bearing dynamics but one of the mainmeans to ensure the normal service life of the bearing Inorder to familiarize the influence of the axial preload onthe roller tilt and skew different axial preloads wereapplied to the TRB under the condition that the radial loadis zero

As shown in Figures 10 and 11 after entering the smoothrunning phase as the axial preload varies from 10 μm to40 μm the average value of the roller tilt rises from minus027 times

10minus4 rad to minus096 times 10minus4 rad 1e greater the preload themore obvious the roller tends to tilt 1is is because as thepreload increases the negative moment of the flange actingon the roller-end becomes larger thus causing the roller tiltto assume a negative tendency

Except for the condition that the preload is 10 μm theroller skew has larger amplitude at the beginning and thenthe amplitude decreases gradually and tends to be stable Inthe smooth running phase as the preload increases theroller skew angle increases 1is trend is in line with theresults in [8 27] 1e average skew angle is increasedfrom 423 times 10minus3 rad024 deg to 550 times 10minus3 rad031 degBoth the roller tilt and skew present a more regular vibrationthroughout the bearing operation this is due to symmetry ofthe bearing on the radial plane and pure preloading con-dition the loaded and unloaded zones do not exist in thebearing and loads on rollers almost change a little in anyposition

1e changes of the slip rate of the roller and cage with thepreload are presented in Figures 12 and 13 As the bearingpreload increases the bearing slip decreases in the smoothrunning phase 1e average slip rate of the roller decreasesfrom 00189 to 00057 and that of the cage decreases from

00189 to 00055 indicating that the appropriate bearingpreload can effectively prevent bearing slipping failure 1eresult is similar with the conclusion in [7 28] 1e influenceof the preload on the bearing slip reduced with the increaseof the preload value 1e average slip rate of the roller andcage when the preload is 40 μm changes little compared with00073 and 00076 when the preload is 30 μm Obviouslyincreasing the amount of preload will help to decrease thebearing slip but excessive preload will inevitably reduce theservice life of bearings 1erefore to meet the bearing lifeand service requirements reasonable selection of preload issignificant

44Grease Physical Properties Influence In view the fact thatthe calculation of the traction force stiffness and damping atthe bearing contacts is mainly based on the plastic viscosity φand yield stress τy of the grease For a clearer understandingof the relationship between the bearing dynamics and thegrease used grease physical properties influence on bearingdynamics is analyzed 1e temperature pressure speed andother surroundings have significant effects on the greasephysical properties For the sake of simplification the im-pacts of these factors on the grease physical properties areignored In order to highlight the effect of a certain rheo-logical parameter on the bearing performance only oneparameter was given an appropriate change while otherrheological parameters remain unchanged 1e effects areshown in Figures 14 and 15 After the bearing enters thestable running state (after 01 s) the average value of theangle (tilt and skew) is used as an indicator of bearingoperating state

1e effect of the plastic viscosity on the roller and cagemotions are shown in Figure 14(a) As the plastic viscosityincreases the tilt angle and skew angle increases and de-creases respectively For the grease film thickness increaseswith the increase of the plastic viscosity the load ratio ofmicroasperities at the flangeroller-end contact is reduced1en it indirectly leads to the decrease of the skew mo-ment at the flangeroller-end contact and the roller skewangle Due to the smaller skew angle the roller misalignmentwill also be small 1erefore the normal force at therollerraceways and flangeroller-end contacts will havea greater contribution to the roller tilt 1erefore the resultshows that the roller tilt angle increases with the plasticviscosity increase In Figure 14(b) the bearing slip decreaseswith the grease plastic viscosity increase As the plasticviscosity increases the grease film thickness at therollerraceways and flangeroller-end contacts increase Sothe friction force at these contacts is reduced which maylead to the decrease of the bearing slip1e bearing slip is theresult of the combined effect of raceway friction and theflange friction

Compared with the impacts of the plastic viscosity of thegrease on the motions of the roller and cage the influence ofthe grease yield stress on the roller tilt roller skew andbearing slip is relatively smaller as shown in Figure 15 Forthe yield stress has a negligible effect on the film thickness inall practical cases [9]

10 Shock and Vibration

000 003 006 009 012 015 01800

05

10

15

20

25

30

Slip

rate

of t

he ro

ller (

)

t (s)

01000 01025 01050 01075 011ndash001

001

003

005

007ωbx = 2krpm

Slip rate of the roller

(a)

000 003 006 009 012 015 01800

02

04

06

08

Slip

rate

of t

he ro

ller (

)

t (s)

ωbx = 3krpm

Slip rate of the roller

(b)

000 003 006 009 012 015 01800

05

10

15

20

25

30

Slip

rate

of t

he ro

ller (

)

t (s)

ωbx = 4krpm

Slip rate of the roller

(c)

000 003 006 009 012 015 0180

2

4

6

8

10

Slip

rate

of t

he ro

ller (

)

t (s)

ωbx = 5krpm

Slip rate of the roller

(d)

Figure 8 1e slip rate of the conical roller Fr 20 kN Fa 0 ωbx 2sim5 krpm

000 003 006 009 012 015 01800

05

10

15

20

25

30

Slip

rate

of t

he ca

ge (

)

01000 01025 01050 01075 011ndash000

001

002

003

t (s)

Slip rate of the cage

ωbx = 2krpm

(a)

000 003 006 009 012 015 01800

05

10

15

20

25

30

Slip

rate

of t

he ca

ge (

)

t (s)

Slip rate of the cage

ωbx = 3krpm

(b)

Figure 9 Continued

Shock and Vibration 11

45 Temperature Influence Due to the improper in-stallation overwear or not timely cooling the operatingtemperature of TRBs will change more or less Sometimesit may lead to heat accumulation or thermal failure ofbearings 1e most direct effect of the temperature ischanges of the grease physical properties 1e rheologicalparameters of the greases at different temperatures in

Table 2 are quoted from the data in reference [26] 1eeffect of temperature on the dynamics of the bearing wasstudied 1e impact of temperature on the bearing struc-ture is not considered here

1ree typical greases were used such as the calciumgrease lithium grease and polyurea grease 1e largestdifference in the composition of three greases lies in the

000 003 006 009 012 015 01800

05

10

15

20

25

30

35

Slip

rate

of t

he ca

ge (

)

t (s)

Slip rate of the cage

ωbx = 4krpm

(c)

000 003 006 009 012 015 0180

2

4

6

8

10

Slip

rate

of t

he ca

ge (

)

t (s)

Slip rate of the cage

ωbx = 5krpm

(d)

Figure 9 1e slip rate of the cage Fr 20 kN Fa 0 ωbx 2sim5 krpm

000 003 006 009 012 015 018ndash2

0

2

4

6

8

10

12

0100 0101 0102ndash15ndash10ndash05

00051015

δa = 10μm

Tilt anglet (s)

Tilt

angl

e (10

ndash4 ra

d)

(a)

000 003 006 009 012 015 018ndash2

0

2

4

6

8

10

12

0100 0101 0102ndash20ndash15ndash10ndash05

000510

Tilt anglet (s)

Tilt

angl

e (10

ndash4 ra

d)δa = 20μm

(b)

000 003 006 009 012 015 018ndash3

0

3

6

9

12

15

0100 0101 0102ndash25ndash20ndash15ndash10ndash05

0005

Tilt angle

t (s)

Tilt

angl

e (10

ndash4 ra

d)

δa = 30μm

(c)

000 003 006 009 012 015 018ndash3

0

3

6

9

12

15

0100 0101 0102ndash30ndash23ndash16ndash09ndash02

05

Tilt angle

t (s)

Tilt

angl

e (10

ndash4 ra

d)

δa = 40μm

(d)

Figure 10 1e tilt angle of the conical roller Fr 0 δa 10sim40 μm ωbx 25 krpm

12 Shock and Vibration

000 003 006 009 012 015 01810

26

42

58

74

90

Skew angle

t (s)

Skew

angl

e (10

ndash3 ra

d)

δa = 10μm

(a)

000 003 006 009 012 015 01810

26

42

58

74

90

Skew angle

t (s)

Skew

angl

e (10

ndash3 ra

d)

δa = 40μm

(b)

000 003 006 009 012 015 01810

26

42

58

74

90

Skew angle

t (s)

Skew

angl

e (10

ndash3 ra

d)

δa = 30μm

(c)

000 003 006 009 012 015 01810

26

42

58

74

90

Skew angle

t (s)

Skew

angl

e (10

ndash3 ra

d)

δa = 40μm

(d)

Figure 11 1e skew angle of the conical roller Fr 0 δa 10sim40 μm ωbx 25 krpm

000 003 006 009 012 015 018000

002

004

006

008

Slip

rate

of t

he ro

ller (

)

Slip rate of the roller

t (s)

δa = 10μm

(a)

000 003 006 009 012 015 018000

002

004

006

008

Slip

rate

of t

he ro

ller (

)

Slip rate of the roller

t (s)

δa = 20μm

(b)

Figure 12 Continued

Shock and Vibration 13

types of thickeners In Figure 16(a) under the lubricationof the calcium and polyurea greases the roller tilt angledecreases with the temperature increase Under the lithiumgrease lubrication the roller tilt angle increases with theincrease in temperature In the whole temperature rangethe tilt angle under the lithium grease is generally larger

than that under others With the increase in temperaturethe roller skew angle under the polyurea grease increaseswhile the roller skew angle under the other two types ofgreases decreases 1at the tilt and skew angle under thedifferent greases and temperature show different trends isa result of multiple factors One of the factors is that the

000 003 006 009 012 015 018000

002

004

006

008

Slip

rate

of t

he ro

ller (

)

Slip rate of the roller

t (s)

δa = 30μm

(c)

000 003 006 009 012 015 018000

002

004

006

008

Slip

rate

of t

he ro

ller (

)

Slip rate of the roller

t (s)

δa = 40μm

(d)

Figure 12 1e slip rate of the conical roller Fr 0 δa 10sim40 μm ωbx 25 krpm

000 003 006 009 012 015 01800

05

10

15

20

25

30

Slip rate of the cage

δa = 10μm

01000 01023 01046 010690010

0015

0020

0025

0030

Slip

rate

of t

he ca

ge (

)

t (s)

(a)

Slip rate of the cage

δa = 20μm

000 003 006 009 012 015 01800

05

10

15

20

25

30

0110 0111 0112 0113 0114 0115ndash001

000

001

002

003

004

Slip

rate

of t

he ca

ge (

)

t (s)

(b)

Slip rate of the cage

δa = 30μm

000 003 006 009 012 015 01800

05

10

15

20

25

30

0110 0111 0112 0113 0114 0115000000030006000900120015

Slip

rate

of t

he ca

ge (

)

t (s)

(c)

Slip rate of the cage

δa = 40μm

000 003 006 009 012 015 01800

05

10

15

20

25

30

0110 0111 0112 0113 0114 0115000000030006000900120015

Slip

rate

of t

he ca

ge (

)

t (s)

(d)

Figure 13 1e slip rate of the cage Fr 0 δa 10sim40 μm ωbx 25 krpm

14 Shock and Vibration

244 844 1444 2044 2644 3244 3844ndash52

ndash51

ndash50

ndash49

ndash48

ndash47

Plastic viscosity (Pamiddotsn)

00

16

32

48

64

80

Tilt angleSkew angle

Tilt

angl

e (10

ndash5 ra

d)

Skew

angl

e (10

ndash3 ra

d)

(a)

844 1444 2044 2644 3244 38440000

0004

0008

0012

0016

0020

Slip

rate

of t

he ro

ller (

)

Plastic viscosity (Pamiddotsn)

ndash0006

0000

0006

0012

0018

Slip

rate

of t

he ca

ge (

)

Slip rate of the rollerSlip rate of the cage

(b)

Figure 14 Plastic viscosity influence Fr 0 Fa 15 kN ωbx 28 krpm (a) 1e conical roller tilt and skew (b) Slip rate of the conicalroller and cage

2518 3018 3518 4018 4518 5018ndash550

ndash536

ndash522

ndash508

ndash494

ndash480

Yield stress (Pa)

350

370

390

410

430

450

Tilt angleSkew angle

Tilt

angl

e (10

ndash5 ra

d)

Skew

angl

e (10

ndash3 ra

d)

(a)

Slip rate of the rollerSlip rate of the cage

2518 3018 3518 4018 4518 50180009

0010

0011

0012

0013

0014

0015Sl

ip ra

te o

f the

rolle

r (

)

Yield stress (Pa)

0009

0010

0011

0012

0013

0014

0015

Slip

rate

of t

he ca

ge (

)

(b)

Figure 15 Yield stress influence Fr 0 Fa 15 kN ωbx 28 krpm (a)1e conical roller tilt and skew (b) Slip rate of the conical roller andcage

Table 2 Relationships between grease physical properties and temperature [26]

Grease TemperaturedegC Yield stress τyPa Plastic viscosity φPamiddotsn Power law exponent n

Polyurea

15 5017 1187 0773130 3518 844 0719650 3055 616 0662170 1885 610 07378

Lithium

15 9045 5435 0305330 8052 4623 0312550 7104 4113 0315170 3398 5296 03203

Calcium

15 9908 6482 0334930 6495 4923 0534450 3361 3059 0525670 2960 3588 04356

Shock and Vibration 15

relationships between the grease rheological parametersand temperature are not simple linear For example theplastic viscosity and the power law exponent do not simplyincrease or decrease with temperature

In Figure 16(b) firstly the roller slip rate shows dif-ferent values due to the different greases secondly theroller slip rate shows a variety of trends with the varioustemperature Under the lubrication of the polyurea greasethe roller slip rate increases with the increase in temper-ature while the roller slip with the calcium and lithiumgreases lubricated shows a decreasing trend In real ap-plication the operating temperature of bearings may not beconstant 1e influence of the temperature on dynamiccharacteristics of grease lubricated bearings needs to befurther explored

5 Conclusions

For the beginning operating stages of grease-lubricatedTRBs the multibody contact dynamic model of TRBs wasestablished 1e impacts of preload temperature and greaserheological properties on the bearing dynamics were ana-lyzed Based on numerical results several conclusions can besummarized

(1) 1e roller tilt angle in the unloaded zone is obviouslylarger than that in the loaded zone while the rollerskew angle in the unloaded zone is smaller than thatin the loaded zone 1e effect of bearing speed on theroller skew is greater than that on the roller tilt Asthe speed increases the roller tilt and skew and thebearing slip become more serious

(2) Bearing preload can effectively reduce the bearingslip but will make the roller tilt angle increase In thestable operation stage of the bearing as the preloadincreases the roller skew angle increases

(3) 1e roller skew angle and the bearing slip rate de-crease with the increase of the grease plastic viscosity1e roller tilt angle increases with the increase of the

plastic viscosity 1e yield stress of the grease haslittle effect on motions of the roller and cage

(4) 1e influence of temperature onmotions of the rollerand cage varies with the type of grease used Whenthe TRB is lubricated by the lithium grease with theincrease of the temperature the roller tilt angle in-creases the roller skew angle and the bearing slip ratedecrease When the polyurea grease is adopted thechange trend of the roller tilt and skew and thebearing slip is just opposite to that when the lithiumgrease is used Under the calcium grease lubricationthe roller tilt and skew angle and the bearing slip ratedecrease with the temperature increase

Data Availability

1e figure data used to support the findings of this studyhave been deposited in the FigShare repository at httpsfigsharecoms15c2556bee83d1adf01d

Conflicts of Interest

1e authors declare that they have no conflicts of interest

Acknowledgments

1e paper was supported by the National Natural ScienceFoundation of China (Grant No 51675427)

References

[1] P M Lugt ldquoA review on grease lubrication in rollingbearingsrdquo Tribology Transactions vol 52 no 4 pp 470ndash4802009

[2] T Cousseau Film thickness and friction in grease lubricatedcontacts Application to rolling bearing torque loss PhDDepartamento De Engenharia Mecanica E Gestao Industrial2013

[3] C T Walters ldquo1e dynamics of ball bearingsrdquo Journal ofTribology vol 93 no 1 pp 1ndash10 1971

10 23 36 49 62 75ndash58

ndash56

ndash54

ndash52

ndash50

ndash48

ndash46

ndash44

CalciumLithiumPolyurea tilt angle skew angle

tilt angle skew angletilt angle skew angle

Temperature (degC)

10

30

50

70

90

110

130Ti

lt an

gle (

10ndash5

rad)

Skew

angl

e (10

ndash3 ra

d)

(a)

10 23 36 49 62 750000

0005

0010

0015

0020

0025

0030

Poly

urea

gre

ase (

)

Temperature (degC)

0000

0001

0002

0003

0004

0005

0006

Calc

ium

lith

ium

gre

ase (

)

CalciumLithiumPolyurea

(b)

Figure 16 Temperature influence Fr 0 Fa 15 kN ωbx 28 krpm (a) 1e conical roller tilt and skew (b) Slip rate of the conical roller

16 Shock and Vibration

ndash6 ndash4 ndash2 0 2 4 60

3

6

9

12

15

Trajectory of the inner ring

z-di

rect

ion

disp

lace

men

t (μm

)

y-direction displacement (μm)

(a)

Displacement of the roller

0 003 006 009 012 015 018ndash20

ndash10

0

10

Radi

al (μ

m)

t (s)

0 003 006 009 012 015 018ndash20

10

40

70

Axi

al (μ

m)

(b)

Trajectory of the cage

ndash06 ndash04 ndash02 00 02 04 06ndash06

ndash04

ndash02

00

02

04

06

z-di

rect

ion

disp

lace

men

t (m

m)

y-direction displacement (mm)

(c)

Cage rotation speed

0 003 006 009 012 015 018102104106108110

Cage

(krp

m)

t (s)

Roller revolution speed0 003 006 009 012 015 018

102104106108110

Rolle

r (kr

pm)

(d)

Figure 5 1e trajectory and velocity of mass center Fr 20 kN Fa 0 ωbx 24 krpm

000 003 006 009 012 015 018ndash9

ndash6

ndash3

0

3

6

9

101

Tilt angle

ωbx = 2krpm

t (s)

Tilt

angl

e (10

ndash4 ra

d)

(a)

Tilt angle

ωbx = 3krpm

000 003 006 009 012 015 018ndash9

ndash6

ndash3

0

3

6

9

101

Tilt

angl

e (10

ndash4 ra

d)

t (s)

(b)

Figure 6 Continued

8 Shock and Vibration

skewmoment of the raceway and flange acting on the roller issmaller than that in the loaded zone It shows obvious pe-riodicity when the speed reaches 3 krpm or more1e averageskew angle is increased from 306 times 10minus3 rad017 deg to 447 times

10minus3 rad025 deg 1e increase of bearing rotating speedenlarges the centrifugal force of rollers which makes contactforces at the rollerouter raceway and flangeroller-end

increase As a result of combined contribution of racewayfriction and the flange friction the roller skew angle is greatlyincreased Compared with the experimental result measuredby Falodi et al in which theymeasured the average skew anglebetween 015 and 045 deg [27] 1e average skew angle inFigure 7 is between 017 and 025 deg and at an order ofmagnitude compared to the experimental result 1e

Tilt angle

ωbx = 4krpm

Tilt

angl

e (10

ndash4 ra

d)

000 003 006 009 012 015 018ndash9

ndash6

ndash3

0

3

6

9

101

t (s)

(c)

Tilt angle

ωbx = 5krpm

Tilt

angl

e (10

ndash4 ra

d)

000 003 006 009 012 015 018ndash9

ndash6

ndash3

0

3

6

9

101

t (s)

(d)

Figure 6 1e tilt angle of the conical roller 1 loaded zone 0 unloaded zone Fr 20 kN Fa 0 ωbx 2sim5 krpm

000 003 006 009 012 015 018ndash8

ndash3

2

7

12

17

22

101

Skew

angl

e (10

ndash3 ra

d)

t (s)

ωbx = 2krpm

Skew angle

(a)

Skew

angl

e (10

ndash3 ra

d)

000 003 006 009 012 015 018ndash4

0

4

8

12

16

101

t (s)

ωbx = 3krpm

Skew angle

(b)

Skew

angl

e (10

ndash3 ra

d)

000 003 006 009 012 015 018ndash4

0

4

8

12

16

101

t (s)

ωbx = 4krpm

Skew angle

(c)

Skew

angl

e (10

ndash3 ra

d)

000 003 006 009 012 015 018ndash4

0

4

8

12

16

0 11

t (s)

ωbx = 5krpm

Skew angle

(d)

Figure 7 1e skew angle of the conical roller 1 loaded zone 0 unloaded zone Fr 20 kN Fa 0 ωbx 2sim5 krpm

Shock and Vibration 9

changing trend of the skew angle with speed is also consistentwith Yangrsquos results

1e bearing slip is a destructive motion that easily leadsto scuffing welding and overwear of bearing surfaces 1ebearing slip usually occurs at high-speed and light-loadconditions 1e influence of the bearing speed on the slipof the roller and cage is exhibited in Figures 8 and 9

1e diagrams depicted in Figure 8 indicate that theroller slip is becoming more and more serious as thebearing speed is raised 1is variation trend is similar tothat in [7] With the bearing speed increases the roller slipgradually presents a periodic change rule especially whenthe speed is up to 3 krpm or more 1e roller slip in loadedand unloaded zones is not the same In the unloaded zonethe roller slip is more serious than that in the loaded zoneAt the speed of 5 krpm the bearing is in a state of seriousslipping In Figure 9 the cage slip is aggravated with thebearing speed increase 1ere is no periodic change in thecage slip rate 1is is because when the cage is whirlinga number of uncertain rollers may act on the cage whichcan lead to an irregular motion of the cage At the samespeed level the maximum amplitude of the slip rate of thecage and roller is basically the same for the dependentrelationship between the cage and roller

43 Axial Preload Influence Axial preload is not only a keyfactor affecting the bearing dynamics but one of the mainmeans to ensure the normal service life of the bearing Inorder to familiarize the influence of the axial preload onthe roller tilt and skew different axial preloads wereapplied to the TRB under the condition that the radial loadis zero

As shown in Figures 10 and 11 after entering the smoothrunning phase as the axial preload varies from 10 μm to40 μm the average value of the roller tilt rises from minus027 times

10minus4 rad to minus096 times 10minus4 rad 1e greater the preload themore obvious the roller tends to tilt 1is is because as thepreload increases the negative moment of the flange actingon the roller-end becomes larger thus causing the roller tiltto assume a negative tendency

Except for the condition that the preload is 10 μm theroller skew has larger amplitude at the beginning and thenthe amplitude decreases gradually and tends to be stable Inthe smooth running phase as the preload increases theroller skew angle increases 1is trend is in line with theresults in [8 27] 1e average skew angle is increasedfrom 423 times 10minus3 rad024 deg to 550 times 10minus3 rad031 degBoth the roller tilt and skew present a more regular vibrationthroughout the bearing operation this is due to symmetry ofthe bearing on the radial plane and pure preloading con-dition the loaded and unloaded zones do not exist in thebearing and loads on rollers almost change a little in anyposition

1e changes of the slip rate of the roller and cage with thepreload are presented in Figures 12 and 13 As the bearingpreload increases the bearing slip decreases in the smoothrunning phase 1e average slip rate of the roller decreasesfrom 00189 to 00057 and that of the cage decreases from

00189 to 00055 indicating that the appropriate bearingpreload can effectively prevent bearing slipping failure 1eresult is similar with the conclusion in [7 28] 1e influenceof the preload on the bearing slip reduced with the increaseof the preload value 1e average slip rate of the roller andcage when the preload is 40 μm changes little compared with00073 and 00076 when the preload is 30 μm Obviouslyincreasing the amount of preload will help to decrease thebearing slip but excessive preload will inevitably reduce theservice life of bearings 1erefore to meet the bearing lifeand service requirements reasonable selection of preload issignificant

44Grease Physical Properties Influence In view the fact thatthe calculation of the traction force stiffness and damping atthe bearing contacts is mainly based on the plastic viscosity φand yield stress τy of the grease For a clearer understandingof the relationship between the bearing dynamics and thegrease used grease physical properties influence on bearingdynamics is analyzed 1e temperature pressure speed andother surroundings have significant effects on the greasephysical properties For the sake of simplification the im-pacts of these factors on the grease physical properties areignored In order to highlight the effect of a certain rheo-logical parameter on the bearing performance only oneparameter was given an appropriate change while otherrheological parameters remain unchanged 1e effects areshown in Figures 14 and 15 After the bearing enters thestable running state (after 01 s) the average value of theangle (tilt and skew) is used as an indicator of bearingoperating state

1e effect of the plastic viscosity on the roller and cagemotions are shown in Figure 14(a) As the plastic viscosityincreases the tilt angle and skew angle increases and de-creases respectively For the grease film thickness increaseswith the increase of the plastic viscosity the load ratio ofmicroasperities at the flangeroller-end contact is reduced1en it indirectly leads to the decrease of the skew mo-ment at the flangeroller-end contact and the roller skewangle Due to the smaller skew angle the roller misalignmentwill also be small 1erefore the normal force at therollerraceways and flangeroller-end contacts will havea greater contribution to the roller tilt 1erefore the resultshows that the roller tilt angle increases with the plasticviscosity increase In Figure 14(b) the bearing slip decreaseswith the grease plastic viscosity increase As the plasticviscosity increases the grease film thickness at therollerraceways and flangeroller-end contacts increase Sothe friction force at these contacts is reduced which maylead to the decrease of the bearing slip1e bearing slip is theresult of the combined effect of raceway friction and theflange friction

Compared with the impacts of the plastic viscosity of thegrease on the motions of the roller and cage the influence ofthe grease yield stress on the roller tilt roller skew andbearing slip is relatively smaller as shown in Figure 15 Forthe yield stress has a negligible effect on the film thickness inall practical cases [9]

10 Shock and Vibration

000 003 006 009 012 015 01800

05

10

15

20

25

30

Slip

rate

of t

he ro

ller (

)

t (s)

01000 01025 01050 01075 011ndash001

001

003

005

007ωbx = 2krpm

Slip rate of the roller

(a)

000 003 006 009 012 015 01800

02

04

06

08

Slip

rate

of t

he ro

ller (

)

t (s)

ωbx = 3krpm

Slip rate of the roller

(b)

000 003 006 009 012 015 01800

05

10

15

20

25

30

Slip

rate

of t

he ro

ller (

)

t (s)

ωbx = 4krpm

Slip rate of the roller

(c)

000 003 006 009 012 015 0180

2

4

6

8

10

Slip

rate

of t

he ro

ller (

)

t (s)

ωbx = 5krpm

Slip rate of the roller

(d)

Figure 8 1e slip rate of the conical roller Fr 20 kN Fa 0 ωbx 2sim5 krpm

000 003 006 009 012 015 01800

05

10

15

20

25

30

Slip

rate

of t

he ca

ge (

)

01000 01025 01050 01075 011ndash000

001

002

003

t (s)

Slip rate of the cage

ωbx = 2krpm

(a)

000 003 006 009 012 015 01800

05

10

15

20

25

30

Slip

rate

of t

he ca

ge (

)

t (s)

Slip rate of the cage

ωbx = 3krpm

(b)

Figure 9 Continued

Shock and Vibration 11

45 Temperature Influence Due to the improper in-stallation overwear or not timely cooling the operatingtemperature of TRBs will change more or less Sometimesit may lead to heat accumulation or thermal failure ofbearings 1e most direct effect of the temperature ischanges of the grease physical properties 1e rheologicalparameters of the greases at different temperatures in

Table 2 are quoted from the data in reference [26] 1eeffect of temperature on the dynamics of the bearing wasstudied 1e impact of temperature on the bearing struc-ture is not considered here

1ree typical greases were used such as the calciumgrease lithium grease and polyurea grease 1e largestdifference in the composition of three greases lies in the

000 003 006 009 012 015 01800

05

10

15

20

25

30

35

Slip

rate

of t

he ca

ge (

)

t (s)

Slip rate of the cage

ωbx = 4krpm

(c)

000 003 006 009 012 015 0180

2

4

6

8

10

Slip

rate

of t

he ca

ge (

)

t (s)

Slip rate of the cage

ωbx = 5krpm

(d)

Figure 9 1e slip rate of the cage Fr 20 kN Fa 0 ωbx 2sim5 krpm

000 003 006 009 012 015 018ndash2

0

2

4

6

8

10

12

0100 0101 0102ndash15ndash10ndash05

00051015

δa = 10μm

Tilt anglet (s)

Tilt

angl

e (10

ndash4 ra

d)

(a)

000 003 006 009 012 015 018ndash2

0

2

4

6

8

10

12

0100 0101 0102ndash20ndash15ndash10ndash05

000510

Tilt anglet (s)

Tilt

angl

e (10

ndash4 ra

d)δa = 20μm

(b)

000 003 006 009 012 015 018ndash3

0

3

6

9

12

15

0100 0101 0102ndash25ndash20ndash15ndash10ndash05

0005

Tilt angle

t (s)

Tilt

angl

e (10

ndash4 ra

d)

δa = 30μm

(c)

000 003 006 009 012 015 018ndash3

0

3

6

9

12

15

0100 0101 0102ndash30ndash23ndash16ndash09ndash02

05

Tilt angle

t (s)

Tilt

angl

e (10

ndash4 ra

d)

δa = 40μm

(d)

Figure 10 1e tilt angle of the conical roller Fr 0 δa 10sim40 μm ωbx 25 krpm

12 Shock and Vibration

000 003 006 009 012 015 01810

26

42

58

74

90

Skew angle

t (s)

Skew

angl

e (10

ndash3 ra

d)

δa = 10μm

(a)

000 003 006 009 012 015 01810

26

42

58

74

90

Skew angle

t (s)

Skew

angl

e (10

ndash3 ra

d)

δa = 40μm

(b)

000 003 006 009 012 015 01810

26

42

58

74

90

Skew angle

t (s)

Skew

angl

e (10

ndash3 ra

d)

δa = 30μm

(c)

000 003 006 009 012 015 01810

26

42

58

74

90

Skew angle

t (s)

Skew

angl

e (10

ndash3 ra

d)

δa = 40μm

(d)

Figure 11 1e skew angle of the conical roller Fr 0 δa 10sim40 μm ωbx 25 krpm

000 003 006 009 012 015 018000

002

004

006

008

Slip

rate

of t

he ro

ller (

)

Slip rate of the roller

t (s)

δa = 10μm

(a)

000 003 006 009 012 015 018000

002

004

006

008

Slip

rate

of t

he ro

ller (

)

Slip rate of the roller

t (s)

δa = 20μm

(b)

Figure 12 Continued

Shock and Vibration 13

types of thickeners In Figure 16(a) under the lubricationof the calcium and polyurea greases the roller tilt angledecreases with the temperature increase Under the lithiumgrease lubrication the roller tilt angle increases with theincrease in temperature In the whole temperature rangethe tilt angle under the lithium grease is generally larger

than that under others With the increase in temperaturethe roller skew angle under the polyurea grease increaseswhile the roller skew angle under the other two types ofgreases decreases 1at the tilt and skew angle under thedifferent greases and temperature show different trends isa result of multiple factors One of the factors is that the

000 003 006 009 012 015 018000

002

004

006

008

Slip

rate

of t

he ro

ller (

)

Slip rate of the roller

t (s)

δa = 30μm

(c)

000 003 006 009 012 015 018000

002

004

006

008

Slip

rate

of t

he ro

ller (

)

Slip rate of the roller

t (s)

δa = 40μm

(d)

Figure 12 1e slip rate of the conical roller Fr 0 δa 10sim40 μm ωbx 25 krpm

000 003 006 009 012 015 01800

05

10

15

20

25

30

Slip rate of the cage

δa = 10μm

01000 01023 01046 010690010

0015

0020

0025

0030

Slip

rate

of t

he ca

ge (

)

t (s)

(a)

Slip rate of the cage

δa = 20μm

000 003 006 009 012 015 01800

05

10

15

20

25

30

0110 0111 0112 0113 0114 0115ndash001

000

001

002

003

004

Slip

rate

of t

he ca

ge (

)

t (s)

(b)

Slip rate of the cage

δa = 30μm

000 003 006 009 012 015 01800

05

10

15

20

25

30

0110 0111 0112 0113 0114 0115000000030006000900120015

Slip

rate

of t

he ca

ge (

)

t (s)

(c)

Slip rate of the cage

δa = 40μm

000 003 006 009 012 015 01800

05

10

15

20

25

30

0110 0111 0112 0113 0114 0115000000030006000900120015

Slip

rate

of t

he ca

ge (

)

t (s)

(d)

Figure 13 1e slip rate of the cage Fr 0 δa 10sim40 μm ωbx 25 krpm

14 Shock and Vibration

244 844 1444 2044 2644 3244 3844ndash52

ndash51

ndash50

ndash49

ndash48

ndash47

Plastic viscosity (Pamiddotsn)

00

16

32

48

64

80

Tilt angleSkew angle

Tilt

angl

e (10

ndash5 ra

d)

Skew

angl

e (10

ndash3 ra

d)

(a)

844 1444 2044 2644 3244 38440000

0004

0008

0012

0016

0020

Slip

rate

of t

he ro

ller (

)

Plastic viscosity (Pamiddotsn)

ndash0006

0000

0006

0012

0018

Slip

rate

of t

he ca

ge (

)

Slip rate of the rollerSlip rate of the cage

(b)

Figure 14 Plastic viscosity influence Fr 0 Fa 15 kN ωbx 28 krpm (a) 1e conical roller tilt and skew (b) Slip rate of the conicalroller and cage

2518 3018 3518 4018 4518 5018ndash550

ndash536

ndash522

ndash508

ndash494

ndash480

Yield stress (Pa)

350

370

390

410

430

450

Tilt angleSkew angle

Tilt

angl

e (10

ndash5 ra

d)

Skew

angl

e (10

ndash3 ra

d)

(a)

Slip rate of the rollerSlip rate of the cage

2518 3018 3518 4018 4518 50180009

0010

0011

0012

0013

0014

0015Sl

ip ra

te o

f the

rolle

r (

)

Yield stress (Pa)

0009

0010

0011

0012

0013

0014

0015

Slip

rate

of t

he ca

ge (

)

(b)

Figure 15 Yield stress influence Fr 0 Fa 15 kN ωbx 28 krpm (a)1e conical roller tilt and skew (b) Slip rate of the conical roller andcage

Table 2 Relationships between grease physical properties and temperature [26]

Grease TemperaturedegC Yield stress τyPa Plastic viscosity φPamiddotsn Power law exponent n

Polyurea

15 5017 1187 0773130 3518 844 0719650 3055 616 0662170 1885 610 07378

Lithium

15 9045 5435 0305330 8052 4623 0312550 7104 4113 0315170 3398 5296 03203

Calcium

15 9908 6482 0334930 6495 4923 0534450 3361 3059 0525670 2960 3588 04356

Shock and Vibration 15

relationships between the grease rheological parametersand temperature are not simple linear For example theplastic viscosity and the power law exponent do not simplyincrease or decrease with temperature

In Figure 16(b) firstly the roller slip rate shows dif-ferent values due to the different greases secondly theroller slip rate shows a variety of trends with the varioustemperature Under the lubrication of the polyurea greasethe roller slip rate increases with the increase in temper-ature while the roller slip with the calcium and lithiumgreases lubricated shows a decreasing trend In real ap-plication the operating temperature of bearings may not beconstant 1e influence of the temperature on dynamiccharacteristics of grease lubricated bearings needs to befurther explored

5 Conclusions

For the beginning operating stages of grease-lubricatedTRBs the multibody contact dynamic model of TRBs wasestablished 1e impacts of preload temperature and greaserheological properties on the bearing dynamics were ana-lyzed Based on numerical results several conclusions can besummarized

(1) 1e roller tilt angle in the unloaded zone is obviouslylarger than that in the loaded zone while the rollerskew angle in the unloaded zone is smaller than thatin the loaded zone 1e effect of bearing speed on theroller skew is greater than that on the roller tilt Asthe speed increases the roller tilt and skew and thebearing slip become more serious

(2) Bearing preload can effectively reduce the bearingslip but will make the roller tilt angle increase In thestable operation stage of the bearing as the preloadincreases the roller skew angle increases

(3) 1e roller skew angle and the bearing slip rate de-crease with the increase of the grease plastic viscosity1e roller tilt angle increases with the increase of the

plastic viscosity 1e yield stress of the grease haslittle effect on motions of the roller and cage

(4) 1e influence of temperature onmotions of the rollerand cage varies with the type of grease used Whenthe TRB is lubricated by the lithium grease with theincrease of the temperature the roller tilt angle in-creases the roller skew angle and the bearing slip ratedecrease When the polyurea grease is adopted thechange trend of the roller tilt and skew and thebearing slip is just opposite to that when the lithiumgrease is used Under the calcium grease lubricationthe roller tilt and skew angle and the bearing slip ratedecrease with the temperature increase

Data Availability

1e figure data used to support the findings of this studyhave been deposited in the FigShare repository at httpsfigsharecoms15c2556bee83d1adf01d

Conflicts of Interest

1e authors declare that they have no conflicts of interest

Acknowledgments

1e paper was supported by the National Natural ScienceFoundation of China (Grant No 51675427)

References

[1] P M Lugt ldquoA review on grease lubrication in rollingbearingsrdquo Tribology Transactions vol 52 no 4 pp 470ndash4802009

[2] T Cousseau Film thickness and friction in grease lubricatedcontacts Application to rolling bearing torque loss PhDDepartamento De Engenharia Mecanica E Gestao Industrial2013

[3] C T Walters ldquo1e dynamics of ball bearingsrdquo Journal ofTribology vol 93 no 1 pp 1ndash10 1971

10 23 36 49 62 75ndash58

ndash56

ndash54

ndash52

ndash50

ndash48

ndash46

ndash44

CalciumLithiumPolyurea tilt angle skew angle

tilt angle skew angletilt angle skew angle

Temperature (degC)

10

30

50

70

90

110

130Ti

lt an

gle (

10ndash5

rad)

Skew

angl

e (10

ndash3 ra

d)

(a)

10 23 36 49 62 750000

0005

0010

0015

0020

0025

0030

Poly

urea

gre

ase (

)

Temperature (degC)

0000

0001

0002

0003

0004

0005

0006

Calc

ium

lith

ium

gre

ase (

)

CalciumLithiumPolyurea

(b)

Figure 16 Temperature influence Fr 0 Fa 15 kN ωbx 28 krpm (a) 1e conical roller tilt and skew (b) Slip rate of the conical roller

16 Shock and Vibration

skewmoment of the raceway and flange acting on the roller issmaller than that in the loaded zone It shows obvious pe-riodicity when the speed reaches 3 krpm or more1e averageskew angle is increased from 306 times 10minus3 rad017 deg to 447 times

10minus3 rad025 deg 1e increase of bearing rotating speedenlarges the centrifugal force of rollers which makes contactforces at the rollerouter raceway and flangeroller-end

increase As a result of combined contribution of racewayfriction and the flange friction the roller skew angle is greatlyincreased Compared with the experimental result measuredby Falodi et al in which theymeasured the average skew anglebetween 015 and 045 deg [27] 1e average skew angle inFigure 7 is between 017 and 025 deg and at an order ofmagnitude compared to the experimental result 1e

Tilt angle

ωbx = 4krpm

Tilt

angl

e (10

ndash4 ra

d)

000 003 006 009 012 015 018ndash9

ndash6

ndash3

0

3

6

9

101

t (s)

(c)

Tilt angle

ωbx = 5krpm

Tilt

angl

e (10

ndash4 ra

d)

000 003 006 009 012 015 018ndash9

ndash6

ndash3

0

3

6

9

101

t (s)

(d)

Figure 6 1e tilt angle of the conical roller 1 loaded zone 0 unloaded zone Fr 20 kN Fa 0 ωbx 2sim5 krpm

000 003 006 009 012 015 018ndash8

ndash3

2

7

12

17

22

101

Skew

angl

e (10

ndash3 ra

d)

t (s)

ωbx = 2krpm

Skew angle

(a)

Skew

angl

e (10

ndash3 ra

d)

000 003 006 009 012 015 018ndash4

0

4

8

12

16

101

t (s)

ωbx = 3krpm

Skew angle

(b)

Skew

angl

e (10

ndash3 ra

d)

000 003 006 009 012 015 018ndash4

0

4

8

12

16

101

t (s)

ωbx = 4krpm

Skew angle

(c)

Skew

angl

e (10

ndash3 ra

d)

000 003 006 009 012 015 018ndash4

0

4

8

12

16

0 11

t (s)

ωbx = 5krpm

Skew angle

(d)

Figure 7 1e skew angle of the conical roller 1 loaded zone 0 unloaded zone Fr 20 kN Fa 0 ωbx 2sim5 krpm

Shock and Vibration 9

changing trend of the skew angle with speed is also consistentwith Yangrsquos results

1e bearing slip is a destructive motion that easily leadsto scuffing welding and overwear of bearing surfaces 1ebearing slip usually occurs at high-speed and light-loadconditions 1e influence of the bearing speed on the slipof the roller and cage is exhibited in Figures 8 and 9

1e diagrams depicted in Figure 8 indicate that theroller slip is becoming more and more serious as thebearing speed is raised 1is variation trend is similar tothat in [7] With the bearing speed increases the roller slipgradually presents a periodic change rule especially whenthe speed is up to 3 krpm or more 1e roller slip in loadedand unloaded zones is not the same In the unloaded zonethe roller slip is more serious than that in the loaded zoneAt the speed of 5 krpm the bearing is in a state of seriousslipping In Figure 9 the cage slip is aggravated with thebearing speed increase 1ere is no periodic change in thecage slip rate 1is is because when the cage is whirlinga number of uncertain rollers may act on the cage whichcan lead to an irregular motion of the cage At the samespeed level the maximum amplitude of the slip rate of thecage and roller is basically the same for the dependentrelationship between the cage and roller

43 Axial Preload Influence Axial preload is not only a keyfactor affecting the bearing dynamics but one of the mainmeans to ensure the normal service life of the bearing Inorder to familiarize the influence of the axial preload onthe roller tilt and skew different axial preloads wereapplied to the TRB under the condition that the radial loadis zero

As shown in Figures 10 and 11 after entering the smoothrunning phase as the axial preload varies from 10 μm to40 μm the average value of the roller tilt rises from minus027 times

10minus4 rad to minus096 times 10minus4 rad 1e greater the preload themore obvious the roller tends to tilt 1is is because as thepreload increases the negative moment of the flange actingon the roller-end becomes larger thus causing the roller tiltto assume a negative tendency

Except for the condition that the preload is 10 μm theroller skew has larger amplitude at the beginning and thenthe amplitude decreases gradually and tends to be stable Inthe smooth running phase as the preload increases theroller skew angle increases 1is trend is in line with theresults in [8 27] 1e average skew angle is increasedfrom 423 times 10minus3 rad024 deg to 550 times 10minus3 rad031 degBoth the roller tilt and skew present a more regular vibrationthroughout the bearing operation this is due to symmetry ofthe bearing on the radial plane and pure preloading con-dition the loaded and unloaded zones do not exist in thebearing and loads on rollers almost change a little in anyposition

1e changes of the slip rate of the roller and cage with thepreload are presented in Figures 12 and 13 As the bearingpreload increases the bearing slip decreases in the smoothrunning phase 1e average slip rate of the roller decreasesfrom 00189 to 00057 and that of the cage decreases from

00189 to 00055 indicating that the appropriate bearingpreload can effectively prevent bearing slipping failure 1eresult is similar with the conclusion in [7 28] 1e influenceof the preload on the bearing slip reduced with the increaseof the preload value 1e average slip rate of the roller andcage when the preload is 40 μm changes little compared with00073 and 00076 when the preload is 30 μm Obviouslyincreasing the amount of preload will help to decrease thebearing slip but excessive preload will inevitably reduce theservice life of bearings 1erefore to meet the bearing lifeand service requirements reasonable selection of preload issignificant

44Grease Physical Properties Influence In view the fact thatthe calculation of the traction force stiffness and damping atthe bearing contacts is mainly based on the plastic viscosity φand yield stress τy of the grease For a clearer understandingof the relationship between the bearing dynamics and thegrease used grease physical properties influence on bearingdynamics is analyzed 1e temperature pressure speed andother surroundings have significant effects on the greasephysical properties For the sake of simplification the im-pacts of these factors on the grease physical properties areignored In order to highlight the effect of a certain rheo-logical parameter on the bearing performance only oneparameter was given an appropriate change while otherrheological parameters remain unchanged 1e effects areshown in Figures 14 and 15 After the bearing enters thestable running state (after 01 s) the average value of theangle (tilt and skew) is used as an indicator of bearingoperating state

1e effect of the plastic viscosity on the roller and cagemotions are shown in Figure 14(a) As the plastic viscosityincreases the tilt angle and skew angle increases and de-creases respectively For the grease film thickness increaseswith the increase of the plastic viscosity the load ratio ofmicroasperities at the flangeroller-end contact is reduced1en it indirectly leads to the decrease of the skew mo-ment at the flangeroller-end contact and the roller skewangle Due to the smaller skew angle the roller misalignmentwill also be small 1erefore the normal force at therollerraceways and flangeroller-end contacts will havea greater contribution to the roller tilt 1erefore the resultshows that the roller tilt angle increases with the plasticviscosity increase In Figure 14(b) the bearing slip decreaseswith the grease plastic viscosity increase As the plasticviscosity increases the grease film thickness at therollerraceways and flangeroller-end contacts increase Sothe friction force at these contacts is reduced which maylead to the decrease of the bearing slip1e bearing slip is theresult of the combined effect of raceway friction and theflange friction

Compared with the impacts of the plastic viscosity of thegrease on the motions of the roller and cage the influence ofthe grease yield stress on the roller tilt roller skew andbearing slip is relatively smaller as shown in Figure 15 Forthe yield stress has a negligible effect on the film thickness inall practical cases [9]

10 Shock and Vibration

000 003 006 009 012 015 01800

05

10

15

20

25

30

Slip

rate

of t

he ro

ller (

)

t (s)

01000 01025 01050 01075 011ndash001

001

003

005

007ωbx = 2krpm

Slip rate of the roller

(a)

000 003 006 009 012 015 01800

02

04

06

08

Slip

rate

of t

he ro

ller (

)

t (s)

ωbx = 3krpm

Slip rate of the roller

(b)

000 003 006 009 012 015 01800

05

10

15

20

25

30

Slip

rate

of t

he ro

ller (

)

t (s)

ωbx = 4krpm

Slip rate of the roller

(c)

000 003 006 009 012 015 0180

2

4

6

8

10

Slip

rate

of t

he ro

ller (

)

t (s)

ωbx = 5krpm

Slip rate of the roller

(d)

Figure 8 1e slip rate of the conical roller Fr 20 kN Fa 0 ωbx 2sim5 krpm

000 003 006 009 012 015 01800

05

10

15

20

25

30

Slip

rate

of t

he ca

ge (

)

01000 01025 01050 01075 011ndash000

001

002

003

t (s)

Slip rate of the cage

ωbx = 2krpm

(a)

000 003 006 009 012 015 01800

05

10

15

20

25

30

Slip

rate

of t

he ca

ge (

)

t (s)

Slip rate of the cage

ωbx = 3krpm

(b)

Figure 9 Continued

Shock and Vibration 11

45 Temperature Influence Due to the improper in-stallation overwear or not timely cooling the operatingtemperature of TRBs will change more or less Sometimesit may lead to heat accumulation or thermal failure ofbearings 1e most direct effect of the temperature ischanges of the grease physical properties 1e rheologicalparameters of the greases at different temperatures in

Table 2 are quoted from the data in reference [26] 1eeffect of temperature on the dynamics of the bearing wasstudied 1e impact of temperature on the bearing struc-ture is not considered here

1ree typical greases were used such as the calciumgrease lithium grease and polyurea grease 1e largestdifference in the composition of three greases lies in the

000 003 006 009 012 015 01800

05

10

15

20

25

30

35

Slip

rate

of t

he ca

ge (

)

t (s)

Slip rate of the cage

ωbx = 4krpm

(c)

000 003 006 009 012 015 0180

2

4

6

8

10

Slip

rate

of t

he ca

ge (

)

t (s)

Slip rate of the cage

ωbx = 5krpm

(d)

Figure 9 1e slip rate of the cage Fr 20 kN Fa 0 ωbx 2sim5 krpm

000 003 006 009 012 015 018ndash2

0

2

4

6

8

10

12

0100 0101 0102ndash15ndash10ndash05

00051015

δa = 10μm

Tilt anglet (s)

Tilt

angl

e (10

ndash4 ra

d)

(a)

000 003 006 009 012 015 018ndash2

0

2

4

6

8

10

12

0100 0101 0102ndash20ndash15ndash10ndash05

000510

Tilt anglet (s)

Tilt

angl

e (10

ndash4 ra

d)δa = 20μm

(b)

000 003 006 009 012 015 018ndash3

0

3

6

9

12

15

0100 0101 0102ndash25ndash20ndash15ndash10ndash05

0005

Tilt angle

t (s)

Tilt

angl

e (10

ndash4 ra

d)

δa = 30μm

(c)

000 003 006 009 012 015 018ndash3

0

3

6

9

12

15

0100 0101 0102ndash30ndash23ndash16ndash09ndash02

05

Tilt angle

t (s)

Tilt

angl

e (10

ndash4 ra

d)

δa = 40μm

(d)

Figure 10 1e tilt angle of the conical roller Fr 0 δa 10sim40 μm ωbx 25 krpm

12 Shock and Vibration

000 003 006 009 012 015 01810

26

42

58

74

90

Skew angle

t (s)

Skew

angl

e (10

ndash3 ra

d)

δa = 10μm

(a)

000 003 006 009 012 015 01810

26

42

58

74

90

Skew angle

t (s)

Skew

angl

e (10

ndash3 ra

d)

δa = 40μm

(b)

000 003 006 009 012 015 01810

26

42

58

74

90

Skew angle

t (s)

Skew

angl

e (10

ndash3 ra

d)

δa = 30μm

(c)

000 003 006 009 012 015 01810

26

42

58

74

90

Skew angle

t (s)

Skew

angl

e (10

ndash3 ra

d)

δa = 40μm

(d)

Figure 11 1e skew angle of the conical roller Fr 0 δa 10sim40 μm ωbx 25 krpm

000 003 006 009 012 015 018000

002

004

006

008

Slip

rate

of t

he ro

ller (

)

Slip rate of the roller

t (s)

δa = 10μm

(a)

000 003 006 009 012 015 018000

002

004

006

008

Slip

rate

of t

he ro

ller (

)

Slip rate of the roller

t (s)

δa = 20μm

(b)

Figure 12 Continued

Shock and Vibration 13

types of thickeners In Figure 16(a) under the lubricationof the calcium and polyurea greases the roller tilt angledecreases with the temperature increase Under the lithiumgrease lubrication the roller tilt angle increases with theincrease in temperature In the whole temperature rangethe tilt angle under the lithium grease is generally larger

than that under others With the increase in temperaturethe roller skew angle under the polyurea grease increaseswhile the roller skew angle under the other two types ofgreases decreases 1at the tilt and skew angle under thedifferent greases and temperature show different trends isa result of multiple factors One of the factors is that the

000 003 006 009 012 015 018000

002

004

006

008

Slip

rate

of t

he ro

ller (

)

Slip rate of the roller

t (s)

δa = 30μm

(c)

000 003 006 009 012 015 018000

002

004

006

008

Slip

rate

of t

he ro

ller (

)

Slip rate of the roller

t (s)

δa = 40μm

(d)

Figure 12 1e slip rate of the conical roller Fr 0 δa 10sim40 μm ωbx 25 krpm

000 003 006 009 012 015 01800

05

10

15

20

25

30

Slip rate of the cage

δa = 10μm

01000 01023 01046 010690010

0015

0020

0025

0030

Slip

rate

of t

he ca

ge (

)

t (s)

(a)

Slip rate of the cage

δa = 20μm

000 003 006 009 012 015 01800

05

10

15

20

25

30

0110 0111 0112 0113 0114 0115ndash001

000

001

002

003

004

Slip

rate

of t

he ca

ge (

)

t (s)

(b)

Slip rate of the cage

δa = 30μm

000 003 006 009 012 015 01800

05

10

15

20

25

30

0110 0111 0112 0113 0114 0115000000030006000900120015

Slip

rate

of t

he ca

ge (

)

t (s)

(c)

Slip rate of the cage

δa = 40μm

000 003 006 009 012 015 01800

05

10

15

20

25

30

0110 0111 0112 0113 0114 0115000000030006000900120015

Slip

rate

of t

he ca

ge (

)

t (s)

(d)

Figure 13 1e slip rate of the cage Fr 0 δa 10sim40 μm ωbx 25 krpm

14 Shock and Vibration

244 844 1444 2044 2644 3244 3844ndash52

ndash51

ndash50

ndash49

ndash48

ndash47

Plastic viscosity (Pamiddotsn)

00

16

32

48

64

80

Tilt angleSkew angle

Tilt

angl

e (10

ndash5 ra

d)

Skew

angl

e (10

ndash3 ra

d)

(a)

844 1444 2044 2644 3244 38440000

0004

0008

0012

0016

0020

Slip

rate

of t

he ro

ller (

)

Plastic viscosity (Pamiddotsn)

ndash0006

0000

0006

0012

0018

Slip

rate

of t

he ca

ge (

)

Slip rate of the rollerSlip rate of the cage

(b)

Figure 14 Plastic viscosity influence Fr 0 Fa 15 kN ωbx 28 krpm (a) 1e conical roller tilt and skew (b) Slip rate of the conicalroller and cage

2518 3018 3518 4018 4518 5018ndash550

ndash536

ndash522

ndash508

ndash494

ndash480

Yield stress (Pa)

350

370

390

410

430

450

Tilt angleSkew angle

Tilt

angl

e (10

ndash5 ra

d)

Skew

angl

e (10

ndash3 ra

d)

(a)

Slip rate of the rollerSlip rate of the cage

2518 3018 3518 4018 4518 50180009

0010

0011

0012

0013

0014

0015Sl

ip ra

te o

f the

rolle

r (

)

Yield stress (Pa)

0009

0010

0011

0012

0013

0014

0015

Slip

rate

of t

he ca

ge (

)

(b)

Figure 15 Yield stress influence Fr 0 Fa 15 kN ωbx 28 krpm (a)1e conical roller tilt and skew (b) Slip rate of the conical roller andcage

Table 2 Relationships between grease physical properties and temperature [26]

Grease TemperaturedegC Yield stress τyPa Plastic viscosity φPamiddotsn Power law exponent n

Polyurea

15 5017 1187 0773130 3518 844 0719650 3055 616 0662170 1885 610 07378

Lithium

15 9045 5435 0305330 8052 4623 0312550 7104 4113 0315170 3398 5296 03203

Calcium

15 9908 6482 0334930 6495 4923 0534450 3361 3059 0525670 2960 3588 04356

Shock and Vibration 15

relationships between the grease rheological parametersand temperature are not simple linear For example theplastic viscosity and the power law exponent do not simplyincrease or decrease with temperature

In Figure 16(b) firstly the roller slip rate shows dif-ferent values due to the different greases secondly theroller slip rate shows a variety of trends with the varioustemperature Under the lubrication of the polyurea greasethe roller slip rate increases with the increase in temper-ature while the roller slip with the calcium and lithiumgreases lubricated shows a decreasing trend In real ap-plication the operating temperature of bearings may not beconstant 1e influence of the temperature on dynamiccharacteristics of grease lubricated bearings needs to befurther explored

5 Conclusions

For the beginning operating stages of grease-lubricatedTRBs the multibody contact dynamic model of TRBs wasestablished 1e impacts of preload temperature and greaserheological properties on the bearing dynamics were ana-lyzed Based on numerical results several conclusions can besummarized

(1) 1e roller tilt angle in the unloaded zone is obviouslylarger than that in the loaded zone while the rollerskew angle in the unloaded zone is smaller than thatin the loaded zone 1e effect of bearing speed on theroller skew is greater than that on the roller tilt Asthe speed increases the roller tilt and skew and thebearing slip become more serious

(2) Bearing preload can effectively reduce the bearingslip but will make the roller tilt angle increase In thestable operation stage of the bearing as the preloadincreases the roller skew angle increases

(3) 1e roller skew angle and the bearing slip rate de-crease with the increase of the grease plastic viscosity1e roller tilt angle increases with the increase of the

plastic viscosity 1e yield stress of the grease haslittle effect on motions of the roller and cage

(4) 1e influence of temperature onmotions of the rollerand cage varies with the type of grease used Whenthe TRB is lubricated by the lithium grease with theincrease of the temperature the roller tilt angle in-creases the roller skew angle and the bearing slip ratedecrease When the polyurea grease is adopted thechange trend of the roller tilt and skew and thebearing slip is just opposite to that when the lithiumgrease is used Under the calcium grease lubricationthe roller tilt and skew angle and the bearing slip ratedecrease with the temperature increase

Data Availability

1e figure data used to support the findings of this studyhave been deposited in the FigShare repository at httpsfigsharecoms15c2556bee83d1adf01d

Conflicts of Interest

1e authors declare that they have no conflicts of interest

Acknowledgments

1e paper was supported by the National Natural ScienceFoundation of China (Grant No 51675427)

References

[1] P M Lugt ldquoA review on grease lubrication in rollingbearingsrdquo Tribology Transactions vol 52 no 4 pp 470ndash4802009

[2] T Cousseau Film thickness and friction in grease lubricatedcontacts Application to rolling bearing torque loss PhDDepartamento De Engenharia Mecanica E Gestao Industrial2013

[3] C T Walters ldquo1e dynamics of ball bearingsrdquo Journal ofTribology vol 93 no 1 pp 1ndash10 1971

10 23 36 49 62 75ndash58

ndash56

ndash54

ndash52

ndash50

ndash48

ndash46

ndash44

CalciumLithiumPolyurea tilt angle skew angle

tilt angle skew angletilt angle skew angle

Temperature (degC)

10

30

50

70

90

110

130Ti

lt an

gle (

10ndash5

rad)

Skew

angl

e (10

ndash3 ra

d)

(a)

10 23 36 49 62 750000

0005

0010

0015

0020

0025

0030

Poly

urea

gre

ase (

)

Temperature (degC)

0000

0001

0002

0003

0004

0005

0006

Calc

ium

lith

ium

gre

ase (

)

CalciumLithiumPolyurea

(b)

Figure 16 Temperature influence Fr 0 Fa 15 kN ωbx 28 krpm (a) 1e conical roller tilt and skew (b) Slip rate of the conical roller

16 Shock and Vibration

changing trend of the skew angle with speed is also consistentwith Yangrsquos results

1e bearing slip is a destructive motion that easily leadsto scuffing welding and overwear of bearing surfaces 1ebearing slip usually occurs at high-speed and light-loadconditions 1e influence of the bearing speed on the slipof the roller and cage is exhibited in Figures 8 and 9

1e diagrams depicted in Figure 8 indicate that theroller slip is becoming more and more serious as thebearing speed is raised 1is variation trend is similar tothat in [7] With the bearing speed increases the roller slipgradually presents a periodic change rule especially whenthe speed is up to 3 krpm or more 1e roller slip in loadedand unloaded zones is not the same In the unloaded zonethe roller slip is more serious than that in the loaded zoneAt the speed of 5 krpm the bearing is in a state of seriousslipping In Figure 9 the cage slip is aggravated with thebearing speed increase 1ere is no periodic change in thecage slip rate 1is is because when the cage is whirlinga number of uncertain rollers may act on the cage whichcan lead to an irregular motion of the cage At the samespeed level the maximum amplitude of the slip rate of thecage and roller is basically the same for the dependentrelationship between the cage and roller

43 Axial Preload Influence Axial preload is not only a keyfactor affecting the bearing dynamics but one of the mainmeans to ensure the normal service life of the bearing Inorder to familiarize the influence of the axial preload onthe roller tilt and skew different axial preloads wereapplied to the TRB under the condition that the radial loadis zero

As shown in Figures 10 and 11 after entering the smoothrunning phase as the axial preload varies from 10 μm to40 μm the average value of the roller tilt rises from minus027 times

10minus4 rad to minus096 times 10minus4 rad 1e greater the preload themore obvious the roller tends to tilt 1is is because as thepreload increases the negative moment of the flange actingon the roller-end becomes larger thus causing the roller tiltto assume a negative tendency

Except for the condition that the preload is 10 μm theroller skew has larger amplitude at the beginning and thenthe amplitude decreases gradually and tends to be stable Inthe smooth running phase as the preload increases theroller skew angle increases 1is trend is in line with theresults in [8 27] 1e average skew angle is increasedfrom 423 times 10minus3 rad024 deg to 550 times 10minus3 rad031 degBoth the roller tilt and skew present a more regular vibrationthroughout the bearing operation this is due to symmetry ofthe bearing on the radial plane and pure preloading con-dition the loaded and unloaded zones do not exist in thebearing and loads on rollers almost change a little in anyposition

1e changes of the slip rate of the roller and cage with thepreload are presented in Figures 12 and 13 As the bearingpreload increases the bearing slip decreases in the smoothrunning phase 1e average slip rate of the roller decreasesfrom 00189 to 00057 and that of the cage decreases from

00189 to 00055 indicating that the appropriate bearingpreload can effectively prevent bearing slipping failure 1eresult is similar with the conclusion in [7 28] 1e influenceof the preload on the bearing slip reduced with the increaseof the preload value 1e average slip rate of the roller andcage when the preload is 40 μm changes little compared with00073 and 00076 when the preload is 30 μm Obviouslyincreasing the amount of preload will help to decrease thebearing slip but excessive preload will inevitably reduce theservice life of bearings 1erefore to meet the bearing lifeand service requirements reasonable selection of preload issignificant

44Grease Physical Properties Influence In view the fact thatthe calculation of the traction force stiffness and damping atthe bearing contacts is mainly based on the plastic viscosity φand yield stress τy of the grease For a clearer understandingof the relationship between the bearing dynamics and thegrease used grease physical properties influence on bearingdynamics is analyzed 1e temperature pressure speed andother surroundings have significant effects on the greasephysical properties For the sake of simplification the im-pacts of these factors on the grease physical properties areignored In order to highlight the effect of a certain rheo-logical parameter on the bearing performance only oneparameter was given an appropriate change while otherrheological parameters remain unchanged 1e effects areshown in Figures 14 and 15 After the bearing enters thestable running state (after 01 s) the average value of theangle (tilt and skew) is used as an indicator of bearingoperating state

1e effect of the plastic viscosity on the roller and cagemotions are shown in Figure 14(a) As the plastic viscosityincreases the tilt angle and skew angle increases and de-creases respectively For the grease film thickness increaseswith the increase of the plastic viscosity the load ratio ofmicroasperities at the flangeroller-end contact is reduced1en it indirectly leads to the decrease of the skew mo-ment at the flangeroller-end contact and the roller skewangle Due to the smaller skew angle the roller misalignmentwill also be small 1erefore the normal force at therollerraceways and flangeroller-end contacts will havea greater contribution to the roller tilt 1erefore the resultshows that the roller tilt angle increases with the plasticviscosity increase In Figure 14(b) the bearing slip decreaseswith the grease plastic viscosity increase As the plasticviscosity increases the grease film thickness at therollerraceways and flangeroller-end contacts increase Sothe friction force at these contacts is reduced which maylead to the decrease of the bearing slip1e bearing slip is theresult of the combined effect of raceway friction and theflange friction

Compared with the impacts of the plastic viscosity of thegrease on the motions of the roller and cage the influence ofthe grease yield stress on the roller tilt roller skew andbearing slip is relatively smaller as shown in Figure 15 Forthe yield stress has a negligible effect on the film thickness inall practical cases [9]

10 Shock and Vibration

000 003 006 009 012 015 01800

05

10

15

20

25

30

Slip

rate

of t

he ro

ller (

)

t (s)

01000 01025 01050 01075 011ndash001

001

003

005

007ωbx = 2krpm

Slip rate of the roller

(a)

000 003 006 009 012 015 01800

02

04

06

08

Slip

rate

of t

he ro

ller (

)

t (s)

ωbx = 3krpm

Slip rate of the roller

(b)

000 003 006 009 012 015 01800

05

10

15

20

25

30

Slip

rate

of t

he ro

ller (

)

t (s)

ωbx = 4krpm

Slip rate of the roller

(c)

000 003 006 009 012 015 0180

2

4

6

8

10

Slip

rate

of t

he ro

ller (

)

t (s)

ωbx = 5krpm

Slip rate of the roller

(d)

Figure 8 1e slip rate of the conical roller Fr 20 kN Fa 0 ωbx 2sim5 krpm

000 003 006 009 012 015 01800

05

10

15

20

25

30

Slip

rate

of t

he ca

ge (

)

01000 01025 01050 01075 011ndash000

001

002

003

t (s)

Slip rate of the cage

ωbx = 2krpm

(a)

000 003 006 009 012 015 01800

05

10

15

20

25

30

Slip

rate

of t

he ca

ge (

)

t (s)

Slip rate of the cage

ωbx = 3krpm

(b)

Figure 9 Continued

Shock and Vibration 11

45 Temperature Influence Due to the improper in-stallation overwear or not timely cooling the operatingtemperature of TRBs will change more or less Sometimesit may lead to heat accumulation or thermal failure ofbearings 1e most direct effect of the temperature ischanges of the grease physical properties 1e rheologicalparameters of the greases at different temperatures in

Table 2 are quoted from the data in reference [26] 1eeffect of temperature on the dynamics of the bearing wasstudied 1e impact of temperature on the bearing struc-ture is not considered here

1ree typical greases were used such as the calciumgrease lithium grease and polyurea grease 1e largestdifference in the composition of three greases lies in the

000 003 006 009 012 015 01800

05

10

15

20

25

30

35

Slip

rate

of t

he ca

ge (

)

t (s)

Slip rate of the cage

ωbx = 4krpm

(c)

000 003 006 009 012 015 0180

2

4

6

8

10

Slip

rate

of t

he ca

ge (

)

t (s)

Slip rate of the cage

ωbx = 5krpm

(d)

Figure 9 1e slip rate of the cage Fr 20 kN Fa 0 ωbx 2sim5 krpm

000 003 006 009 012 015 018ndash2

0

2

4

6

8

10

12

0100 0101 0102ndash15ndash10ndash05

00051015

δa = 10μm

Tilt anglet (s)

Tilt

angl

e (10

ndash4 ra

d)

(a)

000 003 006 009 012 015 018ndash2

0

2

4

6

8

10

12

0100 0101 0102ndash20ndash15ndash10ndash05

000510

Tilt anglet (s)

Tilt

angl

e (10

ndash4 ra

d)δa = 20μm

(b)

000 003 006 009 012 015 018ndash3

0

3

6

9

12

15

0100 0101 0102ndash25ndash20ndash15ndash10ndash05

0005

Tilt angle

t (s)

Tilt

angl

e (10

ndash4 ra

d)

δa = 30μm

(c)

000 003 006 009 012 015 018ndash3

0

3

6

9

12

15

0100 0101 0102ndash30ndash23ndash16ndash09ndash02

05

Tilt angle

t (s)

Tilt

angl

e (10

ndash4 ra

d)

δa = 40μm

(d)

Figure 10 1e tilt angle of the conical roller Fr 0 δa 10sim40 μm ωbx 25 krpm

12 Shock and Vibration

000 003 006 009 012 015 01810

26

42

58

74

90

Skew angle

t (s)

Skew

angl

e (10

ndash3 ra

d)

δa = 10μm

(a)

000 003 006 009 012 015 01810

26

42

58

74

90

Skew angle

t (s)

Skew

angl

e (10

ndash3 ra

d)

δa = 40μm

(b)

000 003 006 009 012 015 01810

26

42

58

74

90

Skew angle

t (s)

Skew

angl

e (10

ndash3 ra

d)

δa = 30μm

(c)

000 003 006 009 012 015 01810

26

42

58

74

90

Skew angle

t (s)

Skew

angl

e (10

ndash3 ra

d)

δa = 40μm

(d)

Figure 11 1e skew angle of the conical roller Fr 0 δa 10sim40 μm ωbx 25 krpm

000 003 006 009 012 015 018000

002

004

006

008

Slip

rate

of t

he ro

ller (

)

Slip rate of the roller

t (s)

δa = 10μm

(a)

000 003 006 009 012 015 018000

002

004

006

008

Slip

rate

of t

he ro

ller (

)

Slip rate of the roller

t (s)

δa = 20μm

(b)

Figure 12 Continued

Shock and Vibration 13

types of thickeners In Figure 16(a) under the lubricationof the calcium and polyurea greases the roller tilt angledecreases with the temperature increase Under the lithiumgrease lubrication the roller tilt angle increases with theincrease in temperature In the whole temperature rangethe tilt angle under the lithium grease is generally larger

than that under others With the increase in temperaturethe roller skew angle under the polyurea grease increaseswhile the roller skew angle under the other two types ofgreases decreases 1at the tilt and skew angle under thedifferent greases and temperature show different trends isa result of multiple factors One of the factors is that the

000 003 006 009 012 015 018000

002

004

006

008

Slip

rate

of t

he ro

ller (

)

Slip rate of the roller

t (s)

δa = 30μm

(c)

000 003 006 009 012 015 018000

002

004

006

008

Slip

rate

of t

he ro

ller (

)

Slip rate of the roller

t (s)

δa = 40μm

(d)

Figure 12 1e slip rate of the conical roller Fr 0 δa 10sim40 μm ωbx 25 krpm

000 003 006 009 012 015 01800

05

10

15

20

25

30

Slip rate of the cage

δa = 10μm

01000 01023 01046 010690010

0015

0020

0025

0030

Slip

rate

of t

he ca

ge (

)

t (s)

(a)

Slip rate of the cage

δa = 20μm

000 003 006 009 012 015 01800

05

10

15

20

25

30

0110 0111 0112 0113 0114 0115ndash001

000

001

002

003

004

Slip

rate

of t

he ca

ge (

)

t (s)

(b)

Slip rate of the cage

δa = 30μm

000 003 006 009 012 015 01800

05

10

15

20

25

30

0110 0111 0112 0113 0114 0115000000030006000900120015

Slip

rate

of t

he ca

ge (

)

t (s)

(c)

Slip rate of the cage

δa = 40μm

000 003 006 009 012 015 01800

05

10

15

20

25

30

0110 0111 0112 0113 0114 0115000000030006000900120015

Slip

rate

of t

he ca

ge (

)

t (s)

(d)

Figure 13 1e slip rate of the cage Fr 0 δa 10sim40 μm ωbx 25 krpm

14 Shock and Vibration

244 844 1444 2044 2644 3244 3844ndash52

ndash51

ndash50

ndash49

ndash48

ndash47

Plastic viscosity (Pamiddotsn)

00

16

32

48

64

80

Tilt angleSkew angle

Tilt

angl

e (10

ndash5 ra

d)

Skew

angl

e (10

ndash3 ra

d)

(a)

844 1444 2044 2644 3244 38440000

0004

0008

0012

0016

0020

Slip

rate

of t

he ro

ller (

)

Plastic viscosity (Pamiddotsn)

ndash0006

0000

0006

0012

0018

Slip

rate

of t

he ca

ge (

)

Slip rate of the rollerSlip rate of the cage

(b)

Figure 14 Plastic viscosity influence Fr 0 Fa 15 kN ωbx 28 krpm (a) 1e conical roller tilt and skew (b) Slip rate of the conicalroller and cage

2518 3018 3518 4018 4518 5018ndash550

ndash536

ndash522

ndash508

ndash494

ndash480

Yield stress (Pa)

350

370

390

410

430

450

Tilt angleSkew angle

Tilt

angl

e (10

ndash5 ra

d)

Skew

angl

e (10

ndash3 ra

d)

(a)

Slip rate of the rollerSlip rate of the cage

2518 3018 3518 4018 4518 50180009

0010

0011

0012

0013

0014

0015Sl

ip ra

te o

f the

rolle

r (

)

Yield stress (Pa)

0009

0010

0011

0012

0013

0014

0015

Slip

rate

of t

he ca

ge (

)

(b)

Figure 15 Yield stress influence Fr 0 Fa 15 kN ωbx 28 krpm (a)1e conical roller tilt and skew (b) Slip rate of the conical roller andcage

Table 2 Relationships between grease physical properties and temperature [26]

Grease TemperaturedegC Yield stress τyPa Plastic viscosity φPamiddotsn Power law exponent n

Polyurea

15 5017 1187 0773130 3518 844 0719650 3055 616 0662170 1885 610 07378

Lithium

15 9045 5435 0305330 8052 4623 0312550 7104 4113 0315170 3398 5296 03203

Calcium

15 9908 6482 0334930 6495 4923 0534450 3361 3059 0525670 2960 3588 04356

Shock and Vibration 15

relationships between the grease rheological parametersand temperature are not simple linear For example theplastic viscosity and the power law exponent do not simplyincrease or decrease with temperature

In Figure 16(b) firstly the roller slip rate shows dif-ferent values due to the different greases secondly theroller slip rate shows a variety of trends with the varioustemperature Under the lubrication of the polyurea greasethe roller slip rate increases with the increase in temper-ature while the roller slip with the calcium and lithiumgreases lubricated shows a decreasing trend In real ap-plication the operating temperature of bearings may not beconstant 1e influence of the temperature on dynamiccharacteristics of grease lubricated bearings needs to befurther explored

5 Conclusions

For the beginning operating stages of grease-lubricatedTRBs the multibody contact dynamic model of TRBs wasestablished 1e impacts of preload temperature and greaserheological properties on the bearing dynamics were ana-lyzed Based on numerical results several conclusions can besummarized

(1) 1e roller tilt angle in the unloaded zone is obviouslylarger than that in the loaded zone while the rollerskew angle in the unloaded zone is smaller than thatin the loaded zone 1e effect of bearing speed on theroller skew is greater than that on the roller tilt Asthe speed increases the roller tilt and skew and thebearing slip become more serious

(2) Bearing preload can effectively reduce the bearingslip but will make the roller tilt angle increase In thestable operation stage of the bearing as the preloadincreases the roller skew angle increases

(3) 1e roller skew angle and the bearing slip rate de-crease with the increase of the grease plastic viscosity1e roller tilt angle increases with the increase of the

plastic viscosity 1e yield stress of the grease haslittle effect on motions of the roller and cage

(4) 1e influence of temperature onmotions of the rollerand cage varies with the type of grease used Whenthe TRB is lubricated by the lithium grease with theincrease of the temperature the roller tilt angle in-creases the roller skew angle and the bearing slip ratedecrease When the polyurea grease is adopted thechange trend of the roller tilt and skew and thebearing slip is just opposite to that when the lithiumgrease is used Under the calcium grease lubricationthe roller tilt and skew angle and the bearing slip ratedecrease with the temperature increase

Data Availability

1e figure data used to support the findings of this studyhave been deposited in the FigShare repository at httpsfigsharecoms15c2556bee83d1adf01d

Conflicts of Interest

1e authors declare that they have no conflicts of interest

Acknowledgments

1e paper was supported by the National Natural ScienceFoundation of China (Grant No 51675427)

References

[1] P M Lugt ldquoA review on grease lubrication in rollingbearingsrdquo Tribology Transactions vol 52 no 4 pp 470ndash4802009

[2] T Cousseau Film thickness and friction in grease lubricatedcontacts Application to rolling bearing torque loss PhDDepartamento De Engenharia Mecanica E Gestao Industrial2013

[3] C T Walters ldquo1e dynamics of ball bearingsrdquo Journal ofTribology vol 93 no 1 pp 1ndash10 1971

10 23 36 49 62 75ndash58

ndash56

ndash54

ndash52

ndash50

ndash48

ndash46

ndash44

CalciumLithiumPolyurea tilt angle skew angle

tilt angle skew angletilt angle skew angle

Temperature (degC)

10

30

50

70

90

110

130Ti

lt an

gle (

10ndash5

rad)

Skew

angl

e (10

ndash3 ra

d)

(a)

10 23 36 49 62 750000

0005

0010

0015

0020

0025

0030

Poly

urea

gre

ase (

)

Temperature (degC)

0000

0001

0002

0003

0004

0005

0006

Calc

ium

lith

ium

gre

ase (

)

CalciumLithiumPolyurea

(b)

Figure 16 Temperature influence Fr 0 Fa 15 kN ωbx 28 krpm (a) 1e conical roller tilt and skew (b) Slip rate of the conical roller

16 Shock and Vibration

000 003 006 009 012 015 01800

05

10

15

20

25

30

Slip

rate

of t

he ro

ller (

)

t (s)

01000 01025 01050 01075 011ndash001

001

003

005

007ωbx = 2krpm

Slip rate of the roller

(a)

000 003 006 009 012 015 01800

02

04

06

08

Slip

rate

of t

he ro

ller (

)

t (s)

ωbx = 3krpm

Slip rate of the roller

(b)

000 003 006 009 012 015 01800

05

10

15

20

25

30

Slip

rate

of t

he ro

ller (

)

t (s)

ωbx = 4krpm

Slip rate of the roller

(c)

000 003 006 009 012 015 0180

2

4

6

8

10

Slip

rate

of t

he ro

ller (

)

t (s)

ωbx = 5krpm

Slip rate of the roller

(d)

Figure 8 1e slip rate of the conical roller Fr 20 kN Fa 0 ωbx 2sim5 krpm

000 003 006 009 012 015 01800

05

10

15

20

25

30

Slip

rate

of t

he ca

ge (

)

01000 01025 01050 01075 011ndash000

001

002

003

t (s)

Slip rate of the cage

ωbx = 2krpm

(a)

000 003 006 009 012 015 01800

05

10

15

20

25

30

Slip

rate

of t

he ca

ge (

)

t (s)

Slip rate of the cage

ωbx = 3krpm

(b)

Figure 9 Continued

Shock and Vibration 11

45 Temperature Influence Due to the improper in-stallation overwear or not timely cooling the operatingtemperature of TRBs will change more or less Sometimesit may lead to heat accumulation or thermal failure ofbearings 1e most direct effect of the temperature ischanges of the grease physical properties 1e rheologicalparameters of the greases at different temperatures in

Table 2 are quoted from the data in reference [26] 1eeffect of temperature on the dynamics of the bearing wasstudied 1e impact of temperature on the bearing struc-ture is not considered here

1ree typical greases were used such as the calciumgrease lithium grease and polyurea grease 1e largestdifference in the composition of three greases lies in the

000 003 006 009 012 015 01800

05

10

15

20

25

30

35

Slip

rate

of t

he ca

ge (

)

t (s)

Slip rate of the cage

ωbx = 4krpm

(c)

000 003 006 009 012 015 0180

2

4

6

8

10

Slip

rate

of t

he ca

ge (

)

t (s)

Slip rate of the cage

ωbx = 5krpm

(d)

Figure 9 1e slip rate of the cage Fr 20 kN Fa 0 ωbx 2sim5 krpm

000 003 006 009 012 015 018ndash2

0

2

4

6

8

10

12

0100 0101 0102ndash15ndash10ndash05

00051015

δa = 10μm

Tilt anglet (s)

Tilt

angl

e (10

ndash4 ra

d)

(a)

000 003 006 009 012 015 018ndash2

0

2

4

6

8

10

12

0100 0101 0102ndash20ndash15ndash10ndash05

000510

Tilt anglet (s)

Tilt

angl

e (10

ndash4 ra

d)δa = 20μm

(b)

000 003 006 009 012 015 018ndash3

0

3

6

9

12

15

0100 0101 0102ndash25ndash20ndash15ndash10ndash05

0005

Tilt angle

t (s)

Tilt

angl

e (10

ndash4 ra

d)

δa = 30μm

(c)

000 003 006 009 012 015 018ndash3

0

3

6

9

12

15

0100 0101 0102ndash30ndash23ndash16ndash09ndash02

05

Tilt angle

t (s)

Tilt

angl

e (10

ndash4 ra

d)

δa = 40μm

(d)

Figure 10 1e tilt angle of the conical roller Fr 0 δa 10sim40 μm ωbx 25 krpm

12 Shock and Vibration

000 003 006 009 012 015 01810

26

42

58

74

90

Skew angle

t (s)

Skew

angl

e (10

ndash3 ra

d)

δa = 10μm

(a)

000 003 006 009 012 015 01810

26

42

58

74

90

Skew angle

t (s)

Skew

angl

e (10

ndash3 ra

d)

δa = 40μm

(b)

000 003 006 009 012 015 01810

26

42

58

74

90

Skew angle

t (s)

Skew

angl

e (10

ndash3 ra

d)

δa = 30μm

(c)

000 003 006 009 012 015 01810

26

42

58

74

90

Skew angle

t (s)

Skew

angl

e (10

ndash3 ra

d)

δa = 40μm

(d)

Figure 11 1e skew angle of the conical roller Fr 0 δa 10sim40 μm ωbx 25 krpm

000 003 006 009 012 015 018000

002

004

006

008

Slip

rate

of t

he ro

ller (

)

Slip rate of the roller

t (s)

δa = 10μm

(a)

000 003 006 009 012 015 018000

002

004

006

008

Slip

rate

of t

he ro

ller (

)

Slip rate of the roller

t (s)

δa = 20μm

(b)

Figure 12 Continued

Shock and Vibration 13

types of thickeners In Figure 16(a) under the lubricationof the calcium and polyurea greases the roller tilt angledecreases with the temperature increase Under the lithiumgrease lubrication the roller tilt angle increases with theincrease in temperature In the whole temperature rangethe tilt angle under the lithium grease is generally larger

than that under others With the increase in temperaturethe roller skew angle under the polyurea grease increaseswhile the roller skew angle under the other two types ofgreases decreases 1at the tilt and skew angle under thedifferent greases and temperature show different trends isa result of multiple factors One of the factors is that the

000 003 006 009 012 015 018000

002

004

006

008

Slip

rate

of t

he ro

ller (

)

Slip rate of the roller

t (s)

δa = 30μm

(c)

000 003 006 009 012 015 018000

002

004

006

008

Slip

rate

of t

he ro

ller (

)

Slip rate of the roller

t (s)

δa = 40μm

(d)

Figure 12 1e slip rate of the conical roller Fr 0 δa 10sim40 μm ωbx 25 krpm

000 003 006 009 012 015 01800

05

10

15

20

25

30

Slip rate of the cage

δa = 10μm

01000 01023 01046 010690010

0015

0020

0025

0030

Slip

rate

of t

he ca

ge (

)

t (s)

(a)

Slip rate of the cage

δa = 20μm

000 003 006 009 012 015 01800

05

10

15

20

25

30

0110 0111 0112 0113 0114 0115ndash001

000

001

002

003

004

Slip

rate

of t

he ca

ge (

)

t (s)

(b)

Slip rate of the cage

δa = 30μm

000 003 006 009 012 015 01800

05

10

15

20

25

30

0110 0111 0112 0113 0114 0115000000030006000900120015

Slip

rate

of t

he ca

ge (

)

t (s)

(c)

Slip rate of the cage

δa = 40μm

000 003 006 009 012 015 01800

05

10

15

20

25

30

0110 0111 0112 0113 0114 0115000000030006000900120015

Slip

rate

of t

he ca

ge (

)

t (s)

(d)

Figure 13 1e slip rate of the cage Fr 0 δa 10sim40 μm ωbx 25 krpm

14 Shock and Vibration

244 844 1444 2044 2644 3244 3844ndash52

ndash51

ndash50

ndash49

ndash48

ndash47

Plastic viscosity (Pamiddotsn)

00

16

32

48

64

80

Tilt angleSkew angle

Tilt

angl

e (10

ndash5 ra

d)

Skew

angl

e (10

ndash3 ra

d)

(a)

844 1444 2044 2644 3244 38440000

0004

0008

0012

0016

0020

Slip

rate

of t

he ro

ller (

)

Plastic viscosity (Pamiddotsn)

ndash0006

0000

0006

0012

0018

Slip

rate

of t

he ca

ge (

)

Slip rate of the rollerSlip rate of the cage

(b)

Figure 14 Plastic viscosity influence Fr 0 Fa 15 kN ωbx 28 krpm (a) 1e conical roller tilt and skew (b) Slip rate of the conicalroller and cage

2518 3018 3518 4018 4518 5018ndash550

ndash536

ndash522

ndash508

ndash494

ndash480

Yield stress (Pa)

350

370

390

410

430

450

Tilt angleSkew angle

Tilt

angl

e (10

ndash5 ra

d)

Skew

angl

e (10

ndash3 ra

d)

(a)

Slip rate of the rollerSlip rate of the cage

2518 3018 3518 4018 4518 50180009

0010

0011

0012

0013

0014

0015Sl

ip ra

te o

f the

rolle

r (

)

Yield stress (Pa)

0009

0010

0011

0012

0013

0014

0015

Slip

rate

of t

he ca

ge (

)

(b)

Figure 15 Yield stress influence Fr 0 Fa 15 kN ωbx 28 krpm (a)1e conical roller tilt and skew (b) Slip rate of the conical roller andcage

Table 2 Relationships between grease physical properties and temperature [26]

Grease TemperaturedegC Yield stress τyPa Plastic viscosity φPamiddotsn Power law exponent n

Polyurea

15 5017 1187 0773130 3518 844 0719650 3055 616 0662170 1885 610 07378

Lithium

15 9045 5435 0305330 8052 4623 0312550 7104 4113 0315170 3398 5296 03203

Calcium

15 9908 6482 0334930 6495 4923 0534450 3361 3059 0525670 2960 3588 04356

Shock and Vibration 15

relationships between the grease rheological parametersand temperature are not simple linear For example theplastic viscosity and the power law exponent do not simplyincrease or decrease with temperature

In Figure 16(b) firstly the roller slip rate shows dif-ferent values due to the different greases secondly theroller slip rate shows a variety of trends with the varioustemperature Under the lubrication of the polyurea greasethe roller slip rate increases with the increase in temper-ature while the roller slip with the calcium and lithiumgreases lubricated shows a decreasing trend In real ap-plication the operating temperature of bearings may not beconstant 1e influence of the temperature on dynamiccharacteristics of grease lubricated bearings needs to befurther explored

5 Conclusions

For the beginning operating stages of grease-lubricatedTRBs the multibody contact dynamic model of TRBs wasestablished 1e impacts of preload temperature and greaserheological properties on the bearing dynamics were ana-lyzed Based on numerical results several conclusions can besummarized

(1) 1e roller tilt angle in the unloaded zone is obviouslylarger than that in the loaded zone while the rollerskew angle in the unloaded zone is smaller than thatin the loaded zone 1e effect of bearing speed on theroller skew is greater than that on the roller tilt Asthe speed increases the roller tilt and skew and thebearing slip become more serious

(2) Bearing preload can effectively reduce the bearingslip but will make the roller tilt angle increase In thestable operation stage of the bearing as the preloadincreases the roller skew angle increases

(3) 1e roller skew angle and the bearing slip rate de-crease with the increase of the grease plastic viscosity1e roller tilt angle increases with the increase of the

plastic viscosity 1e yield stress of the grease haslittle effect on motions of the roller and cage

(4) 1e influence of temperature onmotions of the rollerand cage varies with the type of grease used Whenthe TRB is lubricated by the lithium grease with theincrease of the temperature the roller tilt angle in-creases the roller skew angle and the bearing slip ratedecrease When the polyurea grease is adopted thechange trend of the roller tilt and skew and thebearing slip is just opposite to that when the lithiumgrease is used Under the calcium grease lubricationthe roller tilt and skew angle and the bearing slip ratedecrease with the temperature increase

Data Availability

1e figure data used to support the findings of this studyhave been deposited in the FigShare repository at httpsfigsharecoms15c2556bee83d1adf01d

Conflicts of Interest

1e authors declare that they have no conflicts of interest

Acknowledgments

1e paper was supported by the National Natural ScienceFoundation of China (Grant No 51675427)

References

[1] P M Lugt ldquoA review on grease lubrication in rollingbearingsrdquo Tribology Transactions vol 52 no 4 pp 470ndash4802009

[2] T Cousseau Film thickness and friction in grease lubricatedcontacts Application to rolling bearing torque loss PhDDepartamento De Engenharia Mecanica E Gestao Industrial2013

[3] C T Walters ldquo1e dynamics of ball bearingsrdquo Journal ofTribology vol 93 no 1 pp 1ndash10 1971

10 23 36 49 62 75ndash58

ndash56

ndash54

ndash52

ndash50

ndash48

ndash46

ndash44

CalciumLithiumPolyurea tilt angle skew angle

tilt angle skew angletilt angle skew angle

Temperature (degC)

10

30

50

70

90

110

130Ti

lt an

gle (

10ndash5

rad)

Skew

angl

e (10

ndash3 ra

d)

(a)

10 23 36 49 62 750000

0005

0010

0015

0020

0025

0030

Poly

urea

gre

ase (

)

Temperature (degC)

0000

0001

0002

0003

0004

0005

0006

Calc

ium

lith

ium

gre

ase (

)

CalciumLithiumPolyurea

(b)

Figure 16 Temperature influence Fr 0 Fa 15 kN ωbx 28 krpm (a) 1e conical roller tilt and skew (b) Slip rate of the conical roller

16 Shock and Vibration

45 Temperature Influence Due to the improper in-stallation overwear or not timely cooling the operatingtemperature of TRBs will change more or less Sometimesit may lead to heat accumulation or thermal failure ofbearings 1e most direct effect of the temperature ischanges of the grease physical properties 1e rheologicalparameters of the greases at different temperatures in

Table 2 are quoted from the data in reference [26] 1eeffect of temperature on the dynamics of the bearing wasstudied 1e impact of temperature on the bearing struc-ture is not considered here

1ree typical greases were used such as the calciumgrease lithium grease and polyurea grease 1e largestdifference in the composition of three greases lies in the

000 003 006 009 012 015 01800

05

10

15

20

25

30

35

Slip

rate

of t

he ca

ge (

)

t (s)

Slip rate of the cage

ωbx = 4krpm

(c)

000 003 006 009 012 015 0180

2

4

6

8

10

Slip

rate

of t

he ca

ge (

)

t (s)

Slip rate of the cage

ωbx = 5krpm

(d)

Figure 9 1e slip rate of the cage Fr 20 kN Fa 0 ωbx 2sim5 krpm

000 003 006 009 012 015 018ndash2

0

2

4

6

8

10

12

0100 0101 0102ndash15ndash10ndash05

00051015

δa = 10μm

Tilt anglet (s)

Tilt

angl

e (10

ndash4 ra

d)

(a)

000 003 006 009 012 015 018ndash2

0

2

4

6

8

10

12

0100 0101 0102ndash20ndash15ndash10ndash05

000510

Tilt anglet (s)

Tilt

angl

e (10

ndash4 ra

d)δa = 20μm

(b)

000 003 006 009 012 015 018ndash3

0

3

6

9

12

15

0100 0101 0102ndash25ndash20ndash15ndash10ndash05

0005

Tilt angle

t (s)

Tilt

angl

e (10

ndash4 ra

d)

δa = 30μm

(c)

000 003 006 009 012 015 018ndash3

0

3

6

9

12

15

0100 0101 0102ndash30ndash23ndash16ndash09ndash02

05

Tilt angle

t (s)

Tilt

angl

e (10

ndash4 ra

d)

δa = 40μm

(d)

Figure 10 1e tilt angle of the conical roller Fr 0 δa 10sim40 μm ωbx 25 krpm

12 Shock and Vibration

000 003 006 009 012 015 01810

26

42

58

74

90

Skew angle

t (s)

Skew

angl

e (10

ndash3 ra

d)

δa = 10μm

(a)

000 003 006 009 012 015 01810

26

42

58

74

90

Skew angle

t (s)

Skew

angl

e (10

ndash3 ra

d)

δa = 40μm

(b)

000 003 006 009 012 015 01810

26

42

58

74

90

Skew angle

t (s)

Skew

angl

e (10

ndash3 ra

d)

δa = 30μm

(c)

000 003 006 009 012 015 01810

26

42

58

74

90

Skew angle

t (s)

Skew

angl

e (10

ndash3 ra

d)

δa = 40μm

(d)

Figure 11 1e skew angle of the conical roller Fr 0 δa 10sim40 μm ωbx 25 krpm

000 003 006 009 012 015 018000

002

004

006

008

Slip

rate

of t

he ro

ller (

)

Slip rate of the roller

t (s)

δa = 10μm

(a)

000 003 006 009 012 015 018000

002

004

006

008

Slip

rate

of t

he ro

ller (

)

Slip rate of the roller

t (s)

δa = 20μm

(b)

Figure 12 Continued

Shock and Vibration 13

types of thickeners In Figure 16(a) under the lubricationof the calcium and polyurea greases the roller tilt angledecreases with the temperature increase Under the lithiumgrease lubrication the roller tilt angle increases with theincrease in temperature In the whole temperature rangethe tilt angle under the lithium grease is generally larger

than that under others With the increase in temperaturethe roller skew angle under the polyurea grease increaseswhile the roller skew angle under the other two types ofgreases decreases 1at the tilt and skew angle under thedifferent greases and temperature show different trends isa result of multiple factors One of the factors is that the

000 003 006 009 012 015 018000

002

004

006

008

Slip

rate

of t

he ro

ller (

)

Slip rate of the roller

t (s)

δa = 30μm

(c)

000 003 006 009 012 015 018000

002

004

006

008

Slip

rate

of t

he ro

ller (

)

Slip rate of the roller

t (s)

δa = 40μm

(d)

Figure 12 1e slip rate of the conical roller Fr 0 δa 10sim40 μm ωbx 25 krpm

000 003 006 009 012 015 01800

05

10

15

20

25

30

Slip rate of the cage

δa = 10μm

01000 01023 01046 010690010

0015

0020

0025

0030

Slip

rate

of t

he ca

ge (

)

t (s)

(a)

Slip rate of the cage

δa = 20μm

000 003 006 009 012 015 01800

05

10

15

20

25

30

0110 0111 0112 0113 0114 0115ndash001

000

001

002

003

004

Slip

rate

of t

he ca

ge (

)

t (s)

(b)

Slip rate of the cage

δa = 30μm

000 003 006 009 012 015 01800

05

10

15

20

25

30

0110 0111 0112 0113 0114 0115000000030006000900120015

Slip

rate

of t

he ca

ge (

)

t (s)

(c)

Slip rate of the cage

δa = 40μm

000 003 006 009 012 015 01800

05

10

15

20

25

30

0110 0111 0112 0113 0114 0115000000030006000900120015

Slip

rate

of t

he ca

ge (

)

t (s)

(d)

Figure 13 1e slip rate of the cage Fr 0 δa 10sim40 μm ωbx 25 krpm

14 Shock and Vibration

244 844 1444 2044 2644 3244 3844ndash52

ndash51

ndash50

ndash49

ndash48

ndash47

Plastic viscosity (Pamiddotsn)

00

16

32

48

64

80

Tilt angleSkew angle

Tilt

angl

e (10

ndash5 ra

d)

Skew

angl

e (10

ndash3 ra

d)

(a)

844 1444 2044 2644 3244 38440000

0004

0008

0012

0016

0020

Slip

rate

of t

he ro

ller (

)

Plastic viscosity (Pamiddotsn)

ndash0006

0000

0006

0012

0018

Slip

rate

of t

he ca

ge (

)

Slip rate of the rollerSlip rate of the cage

(b)

Figure 14 Plastic viscosity influence Fr 0 Fa 15 kN ωbx 28 krpm (a) 1e conical roller tilt and skew (b) Slip rate of the conicalroller and cage

2518 3018 3518 4018 4518 5018ndash550

ndash536

ndash522

ndash508

ndash494

ndash480

Yield stress (Pa)

350

370

390

410

430

450

Tilt angleSkew angle

Tilt

angl

e (10

ndash5 ra

d)

Skew

angl

e (10

ndash3 ra

d)

(a)

Slip rate of the rollerSlip rate of the cage

2518 3018 3518 4018 4518 50180009

0010

0011

0012

0013

0014

0015Sl

ip ra

te o

f the

rolle

r (

)

Yield stress (Pa)

0009

0010

0011

0012

0013

0014

0015

Slip

rate

of t

he ca

ge (

)

(b)

Figure 15 Yield stress influence Fr 0 Fa 15 kN ωbx 28 krpm (a)1e conical roller tilt and skew (b) Slip rate of the conical roller andcage

Table 2 Relationships between grease physical properties and temperature [26]

Grease TemperaturedegC Yield stress τyPa Plastic viscosity φPamiddotsn Power law exponent n

Polyurea

15 5017 1187 0773130 3518 844 0719650 3055 616 0662170 1885 610 07378

Lithium

15 9045 5435 0305330 8052 4623 0312550 7104 4113 0315170 3398 5296 03203

Calcium

15 9908 6482 0334930 6495 4923 0534450 3361 3059 0525670 2960 3588 04356

Shock and Vibration 15

relationships between the grease rheological parametersand temperature are not simple linear For example theplastic viscosity and the power law exponent do not simplyincrease or decrease with temperature

In Figure 16(b) firstly the roller slip rate shows dif-ferent values due to the different greases secondly theroller slip rate shows a variety of trends with the varioustemperature Under the lubrication of the polyurea greasethe roller slip rate increases with the increase in temper-ature while the roller slip with the calcium and lithiumgreases lubricated shows a decreasing trend In real ap-plication the operating temperature of bearings may not beconstant 1e influence of the temperature on dynamiccharacteristics of grease lubricated bearings needs to befurther explored

5 Conclusions

For the beginning operating stages of grease-lubricatedTRBs the multibody contact dynamic model of TRBs wasestablished 1e impacts of preload temperature and greaserheological properties on the bearing dynamics were ana-lyzed Based on numerical results several conclusions can besummarized

(1) 1e roller tilt angle in the unloaded zone is obviouslylarger than that in the loaded zone while the rollerskew angle in the unloaded zone is smaller than thatin the loaded zone 1e effect of bearing speed on theroller skew is greater than that on the roller tilt Asthe speed increases the roller tilt and skew and thebearing slip become more serious

(2) Bearing preload can effectively reduce the bearingslip but will make the roller tilt angle increase In thestable operation stage of the bearing as the preloadincreases the roller skew angle increases

(3) 1e roller skew angle and the bearing slip rate de-crease with the increase of the grease plastic viscosity1e roller tilt angle increases with the increase of the

plastic viscosity 1e yield stress of the grease haslittle effect on motions of the roller and cage

(4) 1e influence of temperature onmotions of the rollerand cage varies with the type of grease used Whenthe TRB is lubricated by the lithium grease with theincrease of the temperature the roller tilt angle in-creases the roller skew angle and the bearing slip ratedecrease When the polyurea grease is adopted thechange trend of the roller tilt and skew and thebearing slip is just opposite to that when the lithiumgrease is used Under the calcium grease lubricationthe roller tilt and skew angle and the bearing slip ratedecrease with the temperature increase

Data Availability

1e figure data used to support the findings of this studyhave been deposited in the FigShare repository at httpsfigsharecoms15c2556bee83d1adf01d

Conflicts of Interest

1e authors declare that they have no conflicts of interest

Acknowledgments

1e paper was supported by the National Natural ScienceFoundation of China (Grant No 51675427)

References

[1] P M Lugt ldquoA review on grease lubrication in rollingbearingsrdquo Tribology Transactions vol 52 no 4 pp 470ndash4802009

[2] T Cousseau Film thickness and friction in grease lubricatedcontacts Application to rolling bearing torque loss PhDDepartamento De Engenharia Mecanica E Gestao Industrial2013

[3] C T Walters ldquo1e dynamics of ball bearingsrdquo Journal ofTribology vol 93 no 1 pp 1ndash10 1971

10 23 36 49 62 75ndash58

ndash56

ndash54

ndash52

ndash50

ndash48

ndash46

ndash44

CalciumLithiumPolyurea tilt angle skew angle

tilt angle skew angletilt angle skew angle

Temperature (degC)

10

30

50

70

90

110

130Ti

lt an

gle (

10ndash5

rad)

Skew

angl

e (10

ndash3 ra

d)

(a)

10 23 36 49 62 750000

0005

0010

0015

0020

0025

0030

Poly

urea

gre

ase (

)

Temperature (degC)

0000

0001

0002

0003

0004

0005

0006

Calc

ium

lith

ium

gre

ase (

)

CalciumLithiumPolyurea

(b)

Figure 16 Temperature influence Fr 0 Fa 15 kN ωbx 28 krpm (a) 1e conical roller tilt and skew (b) Slip rate of the conical roller

16 Shock and Vibration

000 003 006 009 012 015 01810

26

42

58

74

90

Skew angle

t (s)

Skew

angl

e (10

ndash3 ra

d)

δa = 10μm

(a)

000 003 006 009 012 015 01810

26

42

58

74

90

Skew angle

t (s)

Skew

angl

e (10

ndash3 ra

d)

δa = 40μm

(b)

000 003 006 009 012 015 01810

26

42

58

74

90

Skew angle

t (s)

Skew

angl

e (10

ndash3 ra

d)

δa = 30μm

(c)

000 003 006 009 012 015 01810

26

42

58

74

90

Skew angle

t (s)

Skew

angl

e (10

ndash3 ra

d)

δa = 40μm

(d)

Figure 11 1e skew angle of the conical roller Fr 0 δa 10sim40 μm ωbx 25 krpm

000 003 006 009 012 015 018000

002

004

006

008

Slip

rate

of t

he ro

ller (

)

Slip rate of the roller

t (s)

δa = 10μm

(a)

000 003 006 009 012 015 018000

002

004

006

008

Slip

rate

of t

he ro

ller (

)

Slip rate of the roller

t (s)

δa = 20μm

(b)

Figure 12 Continued

Shock and Vibration 13

types of thickeners In Figure 16(a) under the lubricationof the calcium and polyurea greases the roller tilt angledecreases with the temperature increase Under the lithiumgrease lubrication the roller tilt angle increases with theincrease in temperature In the whole temperature rangethe tilt angle under the lithium grease is generally larger

than that under others With the increase in temperaturethe roller skew angle under the polyurea grease increaseswhile the roller skew angle under the other two types ofgreases decreases 1at the tilt and skew angle under thedifferent greases and temperature show different trends isa result of multiple factors One of the factors is that the

000 003 006 009 012 015 018000

002

004

006

008

Slip

rate

of t

he ro

ller (

)

Slip rate of the roller

t (s)

δa = 30μm

(c)

000 003 006 009 012 015 018000

002

004

006

008

Slip

rate

of t

he ro

ller (

)

Slip rate of the roller

t (s)

δa = 40μm

(d)

Figure 12 1e slip rate of the conical roller Fr 0 δa 10sim40 μm ωbx 25 krpm

000 003 006 009 012 015 01800

05

10

15

20

25

30

Slip rate of the cage

δa = 10μm

01000 01023 01046 010690010

0015

0020

0025

0030

Slip

rate

of t

he ca

ge (

)

t (s)

(a)

Slip rate of the cage

δa = 20μm

000 003 006 009 012 015 01800

05

10

15

20

25

30

0110 0111 0112 0113 0114 0115ndash001

000

001

002

003

004

Slip

rate

of t

he ca

ge (

)

t (s)

(b)

Slip rate of the cage

δa = 30μm

000 003 006 009 012 015 01800

05

10

15

20

25

30

0110 0111 0112 0113 0114 0115000000030006000900120015

Slip

rate

of t

he ca

ge (

)

t (s)

(c)

Slip rate of the cage

δa = 40μm

000 003 006 009 012 015 01800

05

10

15

20

25

30

0110 0111 0112 0113 0114 0115000000030006000900120015

Slip

rate

of t

he ca

ge (

)

t (s)

(d)

Figure 13 1e slip rate of the cage Fr 0 δa 10sim40 μm ωbx 25 krpm

14 Shock and Vibration

244 844 1444 2044 2644 3244 3844ndash52

ndash51

ndash50

ndash49

ndash48

ndash47

Plastic viscosity (Pamiddotsn)

00

16

32

48

64

80

Tilt angleSkew angle

Tilt

angl

e (10

ndash5 ra

d)

Skew

angl

e (10

ndash3 ra

d)

(a)

844 1444 2044 2644 3244 38440000

0004

0008

0012

0016

0020

Slip

rate

of t

he ro

ller (

)

Plastic viscosity (Pamiddotsn)

ndash0006

0000

0006

0012

0018

Slip

rate

of t

he ca

ge (

)

Slip rate of the rollerSlip rate of the cage

(b)

Figure 14 Plastic viscosity influence Fr 0 Fa 15 kN ωbx 28 krpm (a) 1e conical roller tilt and skew (b) Slip rate of the conicalroller and cage

2518 3018 3518 4018 4518 5018ndash550

ndash536

ndash522

ndash508

ndash494

ndash480

Yield stress (Pa)

350

370

390

410

430

450

Tilt angleSkew angle

Tilt

angl

e (10

ndash5 ra

d)

Skew

angl

e (10

ndash3 ra

d)

(a)

Slip rate of the rollerSlip rate of the cage

2518 3018 3518 4018 4518 50180009

0010

0011

0012

0013

0014

0015Sl

ip ra

te o

f the

rolle

r (

)

Yield stress (Pa)

0009

0010

0011

0012

0013

0014

0015

Slip

rate

of t

he ca

ge (

)

(b)

Figure 15 Yield stress influence Fr 0 Fa 15 kN ωbx 28 krpm (a)1e conical roller tilt and skew (b) Slip rate of the conical roller andcage

Table 2 Relationships between grease physical properties and temperature [26]

Grease TemperaturedegC Yield stress τyPa Plastic viscosity φPamiddotsn Power law exponent n

Polyurea

15 5017 1187 0773130 3518 844 0719650 3055 616 0662170 1885 610 07378

Lithium

15 9045 5435 0305330 8052 4623 0312550 7104 4113 0315170 3398 5296 03203

Calcium

15 9908 6482 0334930 6495 4923 0534450 3361 3059 0525670 2960 3588 04356

Shock and Vibration 15

relationships between the grease rheological parametersand temperature are not simple linear For example theplastic viscosity and the power law exponent do not simplyincrease or decrease with temperature

In Figure 16(b) firstly the roller slip rate shows dif-ferent values due to the different greases secondly theroller slip rate shows a variety of trends with the varioustemperature Under the lubrication of the polyurea greasethe roller slip rate increases with the increase in temper-ature while the roller slip with the calcium and lithiumgreases lubricated shows a decreasing trend In real ap-plication the operating temperature of bearings may not beconstant 1e influence of the temperature on dynamiccharacteristics of grease lubricated bearings needs to befurther explored

5 Conclusions

For the beginning operating stages of grease-lubricatedTRBs the multibody contact dynamic model of TRBs wasestablished 1e impacts of preload temperature and greaserheological properties on the bearing dynamics were ana-lyzed Based on numerical results several conclusions can besummarized

(1) 1e roller tilt angle in the unloaded zone is obviouslylarger than that in the loaded zone while the rollerskew angle in the unloaded zone is smaller than thatin the loaded zone 1e effect of bearing speed on theroller skew is greater than that on the roller tilt Asthe speed increases the roller tilt and skew and thebearing slip become more serious

(2) Bearing preload can effectively reduce the bearingslip but will make the roller tilt angle increase In thestable operation stage of the bearing as the preloadincreases the roller skew angle increases

(3) 1e roller skew angle and the bearing slip rate de-crease with the increase of the grease plastic viscosity1e roller tilt angle increases with the increase of the

plastic viscosity 1e yield stress of the grease haslittle effect on motions of the roller and cage

(4) 1e influence of temperature onmotions of the rollerand cage varies with the type of grease used Whenthe TRB is lubricated by the lithium grease with theincrease of the temperature the roller tilt angle in-creases the roller skew angle and the bearing slip ratedecrease When the polyurea grease is adopted thechange trend of the roller tilt and skew and thebearing slip is just opposite to that when the lithiumgrease is used Under the calcium grease lubricationthe roller tilt and skew angle and the bearing slip ratedecrease with the temperature increase

Data Availability

1e figure data used to support the findings of this studyhave been deposited in the FigShare repository at httpsfigsharecoms15c2556bee83d1adf01d

Conflicts of Interest

1e authors declare that they have no conflicts of interest

Acknowledgments

1e paper was supported by the National Natural ScienceFoundation of China (Grant No 51675427)

References

[1] P M Lugt ldquoA review on grease lubrication in rollingbearingsrdquo Tribology Transactions vol 52 no 4 pp 470ndash4802009

[2] T Cousseau Film thickness and friction in grease lubricatedcontacts Application to rolling bearing torque loss PhDDepartamento De Engenharia Mecanica E Gestao Industrial2013

[3] C T Walters ldquo1e dynamics of ball bearingsrdquo Journal ofTribology vol 93 no 1 pp 1ndash10 1971

10 23 36 49 62 75ndash58

ndash56

ndash54

ndash52

ndash50

ndash48

ndash46

ndash44

CalciumLithiumPolyurea tilt angle skew angle

tilt angle skew angletilt angle skew angle

Temperature (degC)

10

30

50

70

90

110

130Ti

lt an

gle (

10ndash5

rad)

Skew

angl

e (10

ndash3 ra

d)

(a)

10 23 36 49 62 750000

0005

0010

0015

0020

0025

0030

Poly

urea

gre

ase (

)

Temperature (degC)

0000

0001

0002

0003

0004

0005

0006

Calc

ium

lith

ium

gre

ase (

)

CalciumLithiumPolyurea

(b)

Figure 16 Temperature influence Fr 0 Fa 15 kN ωbx 28 krpm (a) 1e conical roller tilt and skew (b) Slip rate of the conical roller

16 Shock and Vibration

types of thickeners In Figure 16(a) under the lubricationof the calcium and polyurea greases the roller tilt angledecreases with the temperature increase Under the lithiumgrease lubrication the roller tilt angle increases with theincrease in temperature In the whole temperature rangethe tilt angle under the lithium grease is generally larger

than that under others With the increase in temperaturethe roller skew angle under the polyurea grease increaseswhile the roller skew angle under the other two types ofgreases decreases 1at the tilt and skew angle under thedifferent greases and temperature show different trends isa result of multiple factors One of the factors is that the

000 003 006 009 012 015 018000

002

004

006

008

Slip

rate

of t

he ro

ller (

)

Slip rate of the roller

t (s)

δa = 30μm

(c)

000 003 006 009 012 015 018000

002

004

006

008

Slip

rate

of t

he ro

ller (

)

Slip rate of the roller

t (s)

δa = 40μm

(d)

Figure 12 1e slip rate of the conical roller Fr 0 δa 10sim40 μm ωbx 25 krpm

000 003 006 009 012 015 01800

05

10

15

20

25

30

Slip rate of the cage

δa = 10μm

01000 01023 01046 010690010

0015

0020

0025

0030

Slip

rate

of t

he ca

ge (

)

t (s)

(a)

Slip rate of the cage

δa = 20μm

000 003 006 009 012 015 01800

05

10

15

20

25

30

0110 0111 0112 0113 0114 0115ndash001

000

001

002

003

004

Slip

rate

of t

he ca

ge (

)

t (s)

(b)

Slip rate of the cage

δa = 30μm

000 003 006 009 012 015 01800

05

10

15

20

25

30

0110 0111 0112 0113 0114 0115000000030006000900120015

Slip

rate

of t

he ca

ge (

)

t (s)

(c)

Slip rate of the cage

δa = 40μm

000 003 006 009 012 015 01800

05

10

15

20

25

30

0110 0111 0112 0113 0114 0115000000030006000900120015

Slip

rate

of t

he ca

ge (

)

t (s)

(d)

Figure 13 1e slip rate of the cage Fr 0 δa 10sim40 μm ωbx 25 krpm

14 Shock and Vibration

244 844 1444 2044 2644 3244 3844ndash52

ndash51

ndash50

ndash49

ndash48

ndash47

Plastic viscosity (Pamiddotsn)

00

16

32

48

64

80

Tilt angleSkew angle

Tilt

angl

e (10

ndash5 ra

d)

Skew

angl

e (10

ndash3 ra

d)

(a)

844 1444 2044 2644 3244 38440000

0004

0008

0012

0016

0020

Slip

rate

of t

he ro

ller (

)

Plastic viscosity (Pamiddotsn)

ndash0006

0000

0006

0012

0018

Slip

rate

of t

he ca

ge (

)

Slip rate of the rollerSlip rate of the cage

(b)

Figure 14 Plastic viscosity influence Fr 0 Fa 15 kN ωbx 28 krpm (a) 1e conical roller tilt and skew (b) Slip rate of the conicalroller and cage

2518 3018 3518 4018 4518 5018ndash550

ndash536

ndash522

ndash508

ndash494

ndash480

Yield stress (Pa)

350

370

390

410

430

450

Tilt angleSkew angle

Tilt

angl

e (10

ndash5 ra

d)

Skew

angl

e (10

ndash3 ra

d)

(a)

Slip rate of the rollerSlip rate of the cage

2518 3018 3518 4018 4518 50180009

0010

0011

0012

0013

0014

0015Sl

ip ra

te o

f the

rolle

r (

)

Yield stress (Pa)

0009

0010

0011

0012

0013

0014

0015

Slip

rate

of t

he ca

ge (

)

(b)

Figure 15 Yield stress influence Fr 0 Fa 15 kN ωbx 28 krpm (a)1e conical roller tilt and skew (b) Slip rate of the conical roller andcage

Table 2 Relationships between grease physical properties and temperature [26]

Grease TemperaturedegC Yield stress τyPa Plastic viscosity φPamiddotsn Power law exponent n

Polyurea

15 5017 1187 0773130 3518 844 0719650 3055 616 0662170 1885 610 07378

Lithium

15 9045 5435 0305330 8052 4623 0312550 7104 4113 0315170 3398 5296 03203

Calcium

15 9908 6482 0334930 6495 4923 0534450 3361 3059 0525670 2960 3588 04356

Shock and Vibration 15

relationships between the grease rheological parametersand temperature are not simple linear For example theplastic viscosity and the power law exponent do not simplyincrease or decrease with temperature

In Figure 16(b) firstly the roller slip rate shows dif-ferent values due to the different greases secondly theroller slip rate shows a variety of trends with the varioustemperature Under the lubrication of the polyurea greasethe roller slip rate increases with the increase in temper-ature while the roller slip with the calcium and lithiumgreases lubricated shows a decreasing trend In real ap-plication the operating temperature of bearings may not beconstant 1e influence of the temperature on dynamiccharacteristics of grease lubricated bearings needs to befurther explored

5 Conclusions

For the beginning operating stages of grease-lubricatedTRBs the multibody contact dynamic model of TRBs wasestablished 1e impacts of preload temperature and greaserheological properties on the bearing dynamics were ana-lyzed Based on numerical results several conclusions can besummarized

(1) 1e roller tilt angle in the unloaded zone is obviouslylarger than that in the loaded zone while the rollerskew angle in the unloaded zone is smaller than thatin the loaded zone 1e effect of bearing speed on theroller skew is greater than that on the roller tilt Asthe speed increases the roller tilt and skew and thebearing slip become more serious

(2) Bearing preload can effectively reduce the bearingslip but will make the roller tilt angle increase In thestable operation stage of the bearing as the preloadincreases the roller skew angle increases

(3) 1e roller skew angle and the bearing slip rate de-crease with the increase of the grease plastic viscosity1e roller tilt angle increases with the increase of the

plastic viscosity 1e yield stress of the grease haslittle effect on motions of the roller and cage

(4) 1e influence of temperature onmotions of the rollerand cage varies with the type of grease used Whenthe TRB is lubricated by the lithium grease with theincrease of the temperature the roller tilt angle in-creases the roller skew angle and the bearing slip ratedecrease When the polyurea grease is adopted thechange trend of the roller tilt and skew and thebearing slip is just opposite to that when the lithiumgrease is used Under the calcium grease lubricationthe roller tilt and skew angle and the bearing slip ratedecrease with the temperature increase

Data Availability

1e figure data used to support the findings of this studyhave been deposited in the FigShare repository at httpsfigsharecoms15c2556bee83d1adf01d

Conflicts of Interest

1e authors declare that they have no conflicts of interest

Acknowledgments

1e paper was supported by the National Natural ScienceFoundation of China (Grant No 51675427)

References

[1] P M Lugt ldquoA review on grease lubrication in rollingbearingsrdquo Tribology Transactions vol 52 no 4 pp 470ndash4802009

[2] T Cousseau Film thickness and friction in grease lubricatedcontacts Application to rolling bearing torque loss PhDDepartamento De Engenharia Mecanica E Gestao Industrial2013

[3] C T Walters ldquo1e dynamics of ball bearingsrdquo Journal ofTribology vol 93 no 1 pp 1ndash10 1971

10 23 36 49 62 75ndash58

ndash56

ndash54

ndash52

ndash50

ndash48

ndash46

ndash44

CalciumLithiumPolyurea tilt angle skew angle

tilt angle skew angletilt angle skew angle

Temperature (degC)

10

30

50

70

90

110

130Ti

lt an

gle (

10ndash5

rad)

Skew

angl

e (10

ndash3 ra

d)

(a)

10 23 36 49 62 750000

0005

0010

0015

0020

0025

0030

Poly

urea

gre

ase (

)

Temperature (degC)

0000

0001

0002

0003

0004

0005

0006

Calc

ium

lith

ium

gre

ase (

)

CalciumLithiumPolyurea

(b)

Figure 16 Temperature influence Fr 0 Fa 15 kN ωbx 28 krpm (a) 1e conical roller tilt and skew (b) Slip rate of the conical roller

16 Shock and Vibration

244 844 1444 2044 2644 3244 3844ndash52

ndash51

ndash50

ndash49

ndash48

ndash47

Plastic viscosity (Pamiddotsn)

00

16

32

48

64

80

Tilt angleSkew angle

Tilt

angl

e (10

ndash5 ra

d)

Skew

angl

e (10

ndash3 ra

d)

(a)

844 1444 2044 2644 3244 38440000

0004

0008

0012

0016

0020

Slip

rate

of t

he ro

ller (

)

Plastic viscosity (Pamiddotsn)

ndash0006

0000

0006

0012

0018

Slip

rate

of t

he ca

ge (

)

Slip rate of the rollerSlip rate of the cage

(b)

Figure 14 Plastic viscosity influence Fr 0 Fa 15 kN ωbx 28 krpm (a) 1e conical roller tilt and skew (b) Slip rate of the conicalroller and cage

2518 3018 3518 4018 4518 5018ndash550

ndash536

ndash522

ndash508

ndash494

ndash480

Yield stress (Pa)

350

370

390

410

430

450

Tilt angleSkew angle

Tilt

angl

e (10

ndash5 ra

d)

Skew

angl

e (10

ndash3 ra

d)

(a)

Slip rate of the rollerSlip rate of the cage

2518 3018 3518 4018 4518 50180009

0010

0011

0012

0013

0014

0015Sl

ip ra

te o

f the

rolle

r (

)

Yield stress (Pa)

0009

0010

0011

0012

0013

0014

0015

Slip

rate

of t

he ca

ge (

)

(b)

Figure 15 Yield stress influence Fr 0 Fa 15 kN ωbx 28 krpm (a)1e conical roller tilt and skew (b) Slip rate of the conical roller andcage

Table 2 Relationships between grease physical properties and temperature [26]

Grease TemperaturedegC Yield stress τyPa Plastic viscosity φPamiddotsn Power law exponent n

Polyurea

15 5017 1187 0773130 3518 844 0719650 3055 616 0662170 1885 610 07378

Lithium

15 9045 5435 0305330 8052 4623 0312550 7104 4113 0315170 3398 5296 03203

Calcium

15 9908 6482 0334930 6495 4923 0534450 3361 3059 0525670 2960 3588 04356

Shock and Vibration 15

relationships between the grease rheological parametersand temperature are not simple linear For example theplastic viscosity and the power law exponent do not simplyincrease or decrease with temperature

In Figure 16(b) firstly the roller slip rate shows dif-ferent values due to the different greases secondly theroller slip rate shows a variety of trends with the varioustemperature Under the lubrication of the polyurea greasethe roller slip rate increases with the increase in temper-ature while the roller slip with the calcium and lithiumgreases lubricated shows a decreasing trend In real ap-plication the operating temperature of bearings may not beconstant 1e influence of the temperature on dynamiccharacteristics of grease lubricated bearings needs to befurther explored

5 Conclusions

For the beginning operating stages of grease-lubricatedTRBs the multibody contact dynamic model of TRBs wasestablished 1e impacts of preload temperature and greaserheological properties on the bearing dynamics were ana-lyzed Based on numerical results several conclusions can besummarized

(1) 1e roller tilt angle in the unloaded zone is obviouslylarger than that in the loaded zone while the rollerskew angle in the unloaded zone is smaller than thatin the loaded zone 1e effect of bearing speed on theroller skew is greater than that on the roller tilt Asthe speed increases the roller tilt and skew and thebearing slip become more serious

(2) Bearing preload can effectively reduce the bearingslip but will make the roller tilt angle increase In thestable operation stage of the bearing as the preloadincreases the roller skew angle increases

(3) 1e roller skew angle and the bearing slip rate de-crease with the increase of the grease plastic viscosity1e roller tilt angle increases with the increase of the

plastic viscosity 1e yield stress of the grease haslittle effect on motions of the roller and cage

(4) 1e influence of temperature onmotions of the rollerand cage varies with the type of grease used Whenthe TRB is lubricated by the lithium grease with theincrease of the temperature the roller tilt angle in-creases the roller skew angle and the bearing slip ratedecrease When the polyurea grease is adopted thechange trend of the roller tilt and skew and thebearing slip is just opposite to that when the lithiumgrease is used Under the calcium grease lubricationthe roller tilt and skew angle and the bearing slip ratedecrease with the temperature increase

Data Availability

1e figure data used to support the findings of this studyhave been deposited in the FigShare repository at httpsfigsharecoms15c2556bee83d1adf01d

Conflicts of Interest

1e authors declare that they have no conflicts of interest

Acknowledgments

1e paper was supported by the National Natural ScienceFoundation of China (Grant No 51675427)

References

[1] P M Lugt ldquoA review on grease lubrication in rollingbearingsrdquo Tribology Transactions vol 52 no 4 pp 470ndash4802009

[2] T Cousseau Film thickness and friction in grease lubricatedcontacts Application to rolling bearing torque loss PhDDepartamento De Engenharia Mecanica E Gestao Industrial2013

[3] C T Walters ldquo1e dynamics of ball bearingsrdquo Journal ofTribology vol 93 no 1 pp 1ndash10 1971

10 23 36 49 62 75ndash58

ndash56

ndash54

ndash52

ndash50

ndash48

ndash46

ndash44

CalciumLithiumPolyurea tilt angle skew angle

tilt angle skew angletilt angle skew angle

Temperature (degC)

10

30

50

70

90

110

130Ti

lt an

gle (

10ndash5

rad)

Skew

angl

e (10

ndash3 ra

d)

(a)

10 23 36 49 62 750000

0005

0010

0015

0020

0025

0030

Poly

urea

gre

ase (

)

Temperature (degC)

0000

0001

0002

0003

0004

0005

0006

Calc

ium

lith

ium

gre

ase (

)

CalciumLithiumPolyurea

(b)

Figure 16 Temperature influence Fr 0 Fa 15 kN ωbx 28 krpm (a) 1e conical roller tilt and skew (b) Slip rate of the conical roller

16 Shock and Vibration

relationships between the grease rheological parametersand temperature are not simple linear For example theplastic viscosity and the power law exponent do not simplyincrease or decrease with temperature

In Figure 16(b) firstly the roller slip rate shows dif-ferent values due to the different greases secondly theroller slip rate shows a variety of trends with the varioustemperature Under the lubrication of the polyurea greasethe roller slip rate increases with the increase in temper-ature while the roller slip with the calcium and lithiumgreases lubricated shows a decreasing trend In real ap-plication the operating temperature of bearings may not beconstant 1e influence of the temperature on dynamiccharacteristics of grease lubricated bearings needs to befurther explored

5 Conclusions

For the beginning operating stages of grease-lubricatedTRBs the multibody contact dynamic model of TRBs wasestablished 1e impacts of preload temperature and greaserheological properties on the bearing dynamics were ana-lyzed Based on numerical results several conclusions can besummarized

(1) 1e roller tilt angle in the unloaded zone is obviouslylarger than that in the loaded zone while the rollerskew angle in the unloaded zone is smaller than thatin the loaded zone 1e effect of bearing speed on theroller skew is greater than that on the roller tilt Asthe speed increases the roller tilt and skew and thebearing slip become more serious

(2) Bearing preload can effectively reduce the bearingslip but will make the roller tilt angle increase In thestable operation stage of the bearing as the preloadincreases the roller skew angle increases

(3) 1e roller skew angle and the bearing slip rate de-crease with the increase of the grease plastic viscosity1e roller tilt angle increases with the increase of the

plastic viscosity 1e yield stress of the grease haslittle effect on motions of the roller and cage

(4) 1e influence of temperature onmotions of the rollerand cage varies with the type of grease used Whenthe TRB is lubricated by the lithium grease with theincrease of the temperature the roller tilt angle in-creases the roller skew angle and the bearing slip ratedecrease When the polyurea grease is adopted thechange trend of the roller tilt and skew and thebearing slip is just opposite to that when the lithiumgrease is used Under the calcium grease lubricationthe roller tilt and skew angle and the bearing slip ratedecrease with the temperature increase

Data Availability

1e figure data used to support the findings of this studyhave been deposited in the FigShare repository at httpsfigsharecoms15c2556bee83d1adf01d

Conflicts of Interest

1e authors declare that they have no conflicts of interest

Acknowledgments

1e paper was supported by the National Natural ScienceFoundation of China (Grant No 51675427)

References

[1] P M Lugt ldquoA review on grease lubrication in rollingbearingsrdquo Tribology Transactions vol 52 no 4 pp 470ndash4802009

[2] T Cousseau Film thickness and friction in grease lubricatedcontacts Application to rolling bearing torque loss PhDDepartamento De Engenharia Mecanica E Gestao Industrial2013

[3] C T Walters ldquo1e dynamics of ball bearingsrdquo Journal ofTribology vol 93 no 1 pp 1ndash10 1971

10 23 36 49 62 75ndash58

ndash56

ndash54

ndash52

ndash50

ndash48

ndash46

ndash44

CalciumLithiumPolyurea tilt angle skew angle

tilt angle skew angletilt angle skew angle

Temperature (degC)

10

30

50

70

90

110

130Ti

lt an

gle (

10ndash5

rad)

Skew

angl

e (10

ndash3 ra

d)

(a)

10 23 36 49 62 750000

0005

0010

0015

0020

0025

0030

Poly

urea

gre

ase (

)

Temperature (degC)

0000

0001

0002

0003

0004

0005

0006

Calc

ium

lith

ium

gre

ase (

)

CalciumLithiumPolyurea

(b)

Figure 16 Temperature influence Fr 0 Fa 15 kN ωbx 28 krpm (a) 1e conical roller tilt and skew (b) Slip rate of the conical roller

16 Shock and Vibration

[4] P K Gupta ldquoDynamic of rolling element bearings Part IndashIVrdquo Journal of Tribology vol 10 no 3 pp 293ndash326 1979

[5] C R Meeks and K O Ng ldquo1e dynamics of ball separators inball bearings Part 1-2 analysis and results of optimizationstudyrdquo Tribology Transaction vol 28 no 3 pp 277ndash295 1985

[6] P K Gupta ldquoOn the dynamics of a tapered roller bearingrdquoJournal of Tribology vol 111 no 2 pp 278ndash287 1989

[7] S Cretu I Bercea and N Mitu ldquoA dynamic analysis oftapered roller bearing under fully flooded conditions Part 1-2rdquo Wear vol 188 no 1-2 pp 1ndash18 1995

[8] S Deng J Gu Y Cui et al ldquoDynamic analysis of a taperedroller bearingrdquo Industrial Lubrication and Tribology vol 70no 1 pp 191ndash200 2017

[9] T Sakaguchi ldquoDynamic analysis of a high-load capacity ta-pered roller bearingrdquo Special Issue Special Supplement toIndustrial Machines 2005

[10] T Sakaguchi and K Harada ldquoDynamic analysis of cage stressin tapered roller bearings using component-mode-synthesismethodrdquo Journal of Tribology vol 131 no 1 pp 1ndash9 2009

[11] I Bercea ldquoPrediction of Roller Skewing in tapered roller bear-ingsrdquo Tribology Transactions vol 51 no 2 pp 128ndash139 2008

[12] L S Gai and W H Zhang ldquoDynamic analysis of axle boxbearing of high-speed trainrdquo Journal of Harbin Bearingvol 37 no 1 pp 3ndash7 2016 in Chinese

[13] A Palmgren Ball and Roller Bearing EngineeringSKF In-dustries Inc 3rd edition Philadelphia PA USA 1959

[14] K H Hunt and F R E Crossley ldquoCoefficient of restitutioninterpreted as damping in vibroimpactrdquo Journal of AppliedMechanics vol 42 no 2 pp 440ndash445 1975

[15] D P Jin andH Y Hu Impact Oscillation and Control SciencePress Beijing China 2005 in Chinese

[16] T Sasaki H Mori and N Okino ldquoFluid lubrication theory ofroller bearing Part I-IIrdquo Journal of Fluids Engineering vol 84no 1 pp 166ndash180 1962

[17] J Kauzlarich and J A Greenwood ldquoElastohydrodynamiclubrication with HerschelndashBulkley model greasesrdquo ASLETransactions vol 15 no 4 pp 269ndash277 2012

[18] R S Zhou ldquoTorque of tapered roller bearingsrdquo ASME Journalof Tribology vol 113 no 3 pp 590ndash597 1991

[19] T Kiekbusch and B Sauer ldquoDynamic simulation of the in-teraction between raceway and rib contact of cylindrical rollerbearingsrdquo in Proceedings of STLE Meeting Atlanta GA USA2017

[20] V Chacko B K Karthikeyan and I Publication ldquoAnalysis ofgrease lubricated isoviscous-elastic point contactsrdquo In-ternational Journal of Advanced Research in Engineering andTechnology vol 976ndash6499 no 4 pp 207ndash217 2013

[21] T Q Yao Y L Chi L H Wang et al ldquoContact vibration andflexible-multibody dynamic analysis of a ball bearingrdquo Journalof Vibration and Shock vol 28 no 10 pp 158ndash162 2009 inChinese

[22] T L H Walford and B J Stone ldquo1e sources of damping inrolling element bearings under oscillating conditionsrdquo Pro-ceedings of the Institution of Mechanical Engineers Part CJournal of Mechanical Engineering Science vol 197 no 4pp 225ndash232 1983

[23] J Alves N Peixinho M T D Silva et al ldquoA comparativestudy of the viscoelastic constitutive models for frictionlesscontact interfaces in solidsrdquoMechanism and Machine Oeoryvol 85 pp 172ndash188 2015

[24] R G Herbert and D C Mcwhannell ldquoShape and frequencycomposition of pulses from an impact pairrdquo Journal ofEngineering for Industry vol 99 no 3 pp 513ndash518 1977

[25] S Z Wen and P Huang Principles of Tribology TsinghuaUniversity Press Beijing China 2012 in Chinese

[26] X L Wang G L Gui and T B Zhu ldquoStudy on rheologicalparameters of domestic lubricating greaserdquo Tribology vol 17no 3 pp 232ndash235 1997 in Chinese

[27] A Falodi Y K Chen M Caspall B Earthrowl and D DellldquoOn the measurement of roller skew of tapered roller bear-ingsrdquo Applied Mechanics and Materials vol 217ndash219pp 2328ndash2331 2012

[28] A Selvaraj and R Marappan ldquoExperimental analysis offactors influencing the cage slip in cylindrical roller bearingrdquoInternational Journal of Advanced Manufacturing Technologyvol 53 no 5ndash8 pp 635ndash644 2011

Shock and Vibration 17

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Submit your manuscripts atwwwhindawicom

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

VLSI Design

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

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