Analysis of Dynamic Characteristics of Grease-Lubricated … · 2019. 7. 30. · Analysis of Dynamic Characteristics of Grease-Lubricated Tapered Roller Bearings Zheng-Hai Wu ,1 Ying-Qiang

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, Xi’an 710072, China2School of Mechatronical Engineering, Henan University of Science and Technology, Luoyang 471023, China

Correspondence should be addressed to Ying-Qiang Xu; [email protected]

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

Academic Editor: Miguel Neves

Copyright © 2018 Zheng-HaiWu et al.1is is an open access article distributed under the Creative CommonsAttribution License,which 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 Runge–Kutta method.1e 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 viscosity.1e 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 property,and 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 operation,formed by the combination of thickener and base oil [1, 2].For the beginning of operation of the grease-lubricated TRB,the 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 reliability.1e 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 [3–5]. Byconsidering the four degree-of-freedom balls and the sixdegree-of-freedom cage, Walters [3] firstly presenteda comprehensive analysis for the balls and cage motions.After 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 pageshttps://doi.org/10.1155/2018/7183042

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and developed the dynamic analysis program ADORE forTRBs; the cage whirling and roller skew were analyzed underdi�erent 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 theJohnson–Tevaarwerk rheological model; the traction per-formance and other properties of the bearing were analyzed,but the translational motion of the cage was neglected. Deng[8] analyzed the dynamics of the TRB with oil lubrication byusing the Adams-Bashforth–Moulton 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 ADAMS.Bercea [11] proposed a comprehensive model to predict theroller skew motion in TRBs. He found that the roller skew isgreatly inuenced by traction at the ange/roller-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 simulated.However, the lubrication state of axlebox bearings was nottaken into account. Compared with other rolling bearings,the 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, e.g., 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 1,the origin coincides with the mass center of the bearing, andthe o-x axis coincides with the axis of the bearings’ shaft. Abody-xed coordinate system oc-xcyczc is dened for the cage,and 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 skew.e tilt isnormally referred to the roller rotation about or-yr axis, andthe skew is the roller rotation about or-zr axis.

2.1. Roller/Raceway 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 r

ir represent the mass

center positions of the race and roller in the inertial frame,respectively. And rrgm and r

bgm are, respectively, the geometric

center 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 − r

ib + Tir′ r

rgm + Tir′ x

kr , 0, 0[ ]

T( )− rbgm, (1)

where xkr is the slice position at the roller axis; Tib is thetransformation matrix (Euler’s 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 r

bbr3 of r

bbr. us,

ψ � a tan−rbbr2rbbr3

( ). (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 o′

orxr

zr

yr

Figure 1: Bearing coordinate system.

oy

z

x

obyb

zb

xb

oryr

zrxr

xrk

αoααi

rgrm

rgbm

rri

rbi

ξ

z′

y′P

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

rrpb � xkr ,−ς sin c, ς cos c

T. (3)

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

rba � Tba rbbr + TibTir′ T

rpb . (4)

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

rba2 � rabr2 + T21x

kr −T22ς sin c + T23ς cos c � 0, (5)

where rabr2 is the component of Tbarbbr in the yba direction

and T21,T22,T23 are components of T � TbaTibTir′ .If ϕ � a tan(ςT23/ςT22) is assumed, the above formula

can be changed to

c � ϕ + a sinrba2 + T21xkr

ς T222 + T223(

1/2

or c � π + ϕ + a sinrba2 + T21xkr

ς T222 + T223(

1/2.

(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 cshould 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 − ξ rba1,ψ( ( cos 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 ,−(ς− 0.5δ)sin c, (ς− 0.5δ)cos c

T. (8)

Assume that vib[ _xb, _yb, _zb]T and ωbb[ωbx,ωby,ωbz]

T are,respectively, the velocity and the angular velocity of the raceand vir[ _x, _r, _θ]

T and ωrr[ωrx,ωry,ωrz]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 � TacTiavib + r

ir − r

ib + Tir′ r

rpb + Tir′ r

rgm + Tib′ r

bgm

· Tib′ ωbb −[ _θ, 0, 0]

T ,

vcr � Tac Tia Tir′ ωrr × Tir′ r

rpb + Tir′ r

rgm +[ _x, 0, _r]

T ,

(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 − v

cr . (10)

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

q � kwδ10/9

+ cwvcbr3, (11)

where kw is the Hertzian contact stiffness, kw �0.356E′n−1s l

8/9e [13]; E′ is the equivalent elastic modulus of

the roller and raceway; ns is the total number of slices; le isthe effective length of the roller; cw is the viscous dampingcoefficient, cw � 1.5αekwδ10/9 [14, 15] and αe is related to therestitution coefficient, for steel, bronze, or ivory, αe � 0.08 d0.32 s/m [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 roller/raceway contact.Based on the grease EHL theory, the grease film tractionforces at the point P are [16]

fr �πleεφ0R′

1/2

21/2nsh1/20

3vcb2 − vcr2( ,

fb �πleεφ0R′

1/2

21/2nsh1/20

3vcr2 − vcb2( ,

(12)

where φ0 is the grease plastic viscosity at atmosphericpressure; R′ 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 R′.

ε �2π

a tan21/2b

R′1/2

h1/20

⎛⎝ ⎞⎠−21/2bR′1/2h1/20

R′h0 + 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 � ns

k�1TcaF

crjk,

Nrj � ns

k�1rrpb + r

rgm × TcrF

crjk.

(14)

1e contact loads acting on the race are

Qij � ns

k�1Tic′ F

cbjk,

Nbj � ns

k�1rbpb + Tibr

irb × TibTic′ F

cbjk,

(15)

Shock and Vibration 3

where

Fcrjk,bjk � frjk,bjk sin αf , frjk,bjk cos αf , ±qjk T,

αf � a sin vcbr2/v

cbr1( .

(16)

2.2. Flange/Roller-End Interaction. In race body-fixed co-ordinate system (ob-xbybzb), the roller-end curvature centerrelative to the race center is

rbbe � Tib rir − r

ib + Tir′ r

rgm + Tir′ x

re, 0, 0

T − rbgm, (17)

where xre is the position of the roller-end curvature center atthe roller axis.

Since normal force will act on the flange surface and passthrough the roller-end curvature center, the above centercan be transformed into the race azimuth coordinate systemby Tba(ψ, 0, 0), where ψ � a tan(rbbe2/r

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 −Tbar

bf , (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 − rs. (20)

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

qf �321/2δ3/2

3n3/2δ E′ρ1/2, (21)

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

1e friction at the flange/roller-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 exp−BλC

�s0

p0qf exp−BλC

, (23)

where B and C are related to morphology andmaterial properties of the contact surface; s0 and p0 are,respectively, the critical shear stress and the yield stress,for most metals, s0/p0 ≈ 0.2 [18]; λ is the ratio of the greasefilm thickness hc to surface roughness. 1e film thicknesshc is approximated as [20]

hc � 1.8035 × 10−2

M−0.5066

L−1.1869

, (24)

where

M �qf

E′r2s

E′rsφ0uf

3/4

,

L � αE′φ0ufE′rs

1/4

,

(25)

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

Herschel–Bulkley 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 − uend)/hc. 1en, the traction force due to the greaseshear stress is obtained by the integration:

fgrease � a

−a

b

−bτ(x, y)x dx dy. (27)

1en, the loads acting on the jth roller are

Qafj � Taf′ Ffff � Taf′ qfj, ffj, 0

T,

Nrfj � rrfr + r

rgm × TfrF

ffj.

(28)

1e loads acting on the flange are

Qifj � −TftFffj,

Nbfj � rbfb + r

bgm × TibQ

ifj,

(29)

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

the roller and inner-ring centers, respectively.

2.3. 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 − r

ic + Tirr

rgm − r0 , (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 roller/raceway interaction, let rrglocates a point P′ on the kth slice of the roller such that

rrp′ � xkr ,−ς sin c, ς cos c

T. (31)


In the pocket coordinate system, the position of the pointP′ is

rpp′c � rpcr + TicTir′ r

rp′ . (32)

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

δ � Δδ − rpp′c2, (33)

where Δδ is the initial clearance between the roller andcrossbeam, Δδ � (lc − 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 transition.When δ ≥ Δh0, thereare only the hydrodynamic e�ect 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 �3πεφ0uleς1/2

21/2h1/20,

(34)

where [17, 22]

ke �zf h0( )zh0

,

ce �6πεφ0leς3/2

21/2nsh3/20,

f h0( ) �29παφnsξ1un

9E″h6n+20 ς−3le4 +

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 < 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 − δ. (36)

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

qc � khδh + ch _δh + qcehl,fc � μbdqc + fcehl,

(37)

where kh is the linearized Hertzian contact sti�ness [23]; ch isthe Herbert viscous damping coe¤cient [24]; and μbd is thetraction coe¤cient under boundary lubrication [9].

en, the load at the contact point P′ on the kth slice inthe contact coordinate system is

Fccjk � 0, qcjk, fcjk[ ]T. (38)

Using the processing method of the roller/racewayinteraction, the load Fccjk can also be transferred tothe inertial coordinate and the roller/cage 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 e�ec-tively formed at this contact, but the squeeze lm lubri-cation will be formed. As shown in Figure 3, the Δδ′ is theinitial pocket clearance and ra � r

pcr1. If dra/dt ≥ 0, the

small end contacts with the side beam; if dra/dt < 0,the large end contacts. Neglecting the grease traction, thesqueeze force qs is

qs � ks Δδ′ − ra( ), (39)

where [25]

ks � −z

zha

dl,sφ0k3s

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, respectively;and hs is the thickness of the side beam.

en, the loads acting on the jth roller are

αc

Δδ′

Δδ

δ

ha

ke

ce

ks

xryr

xp

ypop

or

Side beam

Cros

sbea

m

Figure 3: Interaction between the roller and cage.


Qasj � qsj, 0, 0[ ]T,

Nrsj � rrsr + r

rgm( ) × TarQ

asj.

(41)

e loads acting on the cage areQisj � −TaiQ

asj,

Ncsj � r0 + Tcp′ rpcr + Trc r

rsr + r

rgm( ) × TicQ

isj[ ],

(42)

where rrsr is the position of the contact point in the rollerbody-xed coordinate and Tar and Tai are, respectively,the 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 δ′ between thecage and guide ring is

δ′ � Δe − ‖e‖, (43)

where Δe is the initial clearance, Δe � (ry − 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 Δh0′ is the critical value of thegrease lm thickness. When δ′ ≥ Δh0′, the contact force qicand friction torque Nic on the cage are

qic � qicehl � ke′δ + ce′ _δ′,

Nic � Nicehl �πεφ0br1/2ic rd21/2h1/20

3ωbxry −ωcxrd( ),(44)

where ric is the equivalent radius and ωcx, ωbx are,respectively, the angular velocities of the cage and guidering. e calculation of the grease lm sti�ness ke′ anddamping ce′ is the same as that at the roller/crossbeamcontact.

When δ′ < Δh0′, the elastic deformation between the cageand guide ring is δh′ � δ′ − h0, and the minimum grease lmthickness h0 � Δh0′. e qic and Nic are

qic � kh′δh′ + ch′δh′ + qicehl,Nic � μbdqicrd +Nicehl,

(45)

where kh′ is the contact sti�ness and ch′ is the viscousdamping coe¤cient. e calculation of kh′ and ch′ is the sameas that in Equation (35). μbd is the traction coe¤cient 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 � ±Nic, 0, 0[ ].(46)

If the TRB is lubricated by the oil, the e�ect of oil-gasmixture on the cage and rollers may not be overlooked.However, if the bearing is lubricated by the grease, it seemsimpossible to have the grease-gas mixture e�ect on the

dynamics of the cage and rollers. So, the e�ect of the grease-gas mixture is not considered in the dynamic model.

2.4.DynamicModel. In real application also, the surroundings(inertia/damping of shaft assembly and housing) are expectedto inuence the TRB dynamics. ese e�ects 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 race.e translationalmoving of bearing parts canbe simply described by Newton’s laws such that

M€δ � 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

Mb,c � diag mb,c, mb,c, mb,c[ ]. (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-freedom.Unlike 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 −ωyωz Iy − Iz( )

Iy _ωy −ωzωx Iz − Ix( )

Iz _ωz −ωxωx Ix − 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 system.e relationship between theangular velocity ωxed in the body-xed coordinate systemand ω is as follows. B is the Euler’s rotation matrix.

∆y

yb

zb

yc

ωcx

ωbx

φicNic

δ′

zc

oc∆z e

obqic

Guidering

Cage

Roller

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


ωfixed � Bω,

_ωfixed �dBdt

B−1 _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, y,and z directions and constant rotating about the shaftaxis (o-x).

3. Numerical Method

1e bearing geometry and material parameters and greasemain rheological parameters (30°C) are shown in Table 1.1e cage is an inner-ring guided type, and its material ispolyamide. 1e polyurea grease was adopted [26]. 1efourth-order Runge–Kutta method was used to dissolvetransient responses of the TRB on MATLAB. To ensure theconvergence of simulation, initial values of displacement,velocity, 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 speed− actual speed|theoretical speed

× 100%. (52)

For the roller, the speed refers to the revolution speed;for the cage, refers to the rotational speed.

4.1. 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 axis.1erefore, 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 fluctuation,for 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.

4.2. 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 flange/roller-end and roller/racewaycontacts, 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 6.1e 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 be“relaxed.” As the speed increases, the maximum of the tiltangle almost does not change, while the average tilt anglerises from −0.14 × 10−4 rad to −0.40 × 10−4 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 increasing.Although the gyroscopic moment increases as the bearingspeed increases, it is still relatively smaller than theforce/torque in the roller/raceway 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 “relaxed” in the unloaded zone, the

Table 1: Geometry, material and rheological parameters.

Parameter ValueNumber of rollers 17Roller length le/mm 42.06Roller-end sphere radius rs/mm 136.20Diameter of roller large end dl/mm 17.85Diameter of roller small end ds/mm 24.76Outer contact angle/deg 15Inner contact angle/deg 5Flange contact angle/deg 83Outer raceway radius/mm 99.52Inner raceway radius/mm 78.48Guide land clearance/mm 0.5Guide land width/mm 4.93Cage outer diameter/mm 205.76Cage inner diameter/mm 196.10Cage thickness hs/mm 5Pocket clearance Δδ/mm 0.27Elastic modulus of roller/race/GPa 205Poisson’s ratio of roller/race 0.3Elastic modulus of cage/GPa 8.3Poisson’s ratio of cage 0.28

Grease rheological parameters (30°C)Yield stress τy/Pa 351.8Plastic viscosity φ/Pa·sn 8.44Power law exponent n 0.7196Pressure-viscosity coefficient α/m2·N−11 4.662 × 10−8


–6 –4 –2 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 0.03 0.06 0.09 0.12 0.15 0.18–20

–10

0

10

Radi

al (μ

m)

t (s)

0 0.03 0.06 0.09 0.12 0.15 0.18–20

10

40

70

Axi

al (μ

m)

(b)

Trajectory of the cage

–0.6 –0.4 –0.2 0.0 0.2 0.4 0.6–0.6

–0.4

–0.2

0.0

0.2

0.4

0.6

z-di

rect

ion

disp

lace

men

t (m

m)

y-direction displacement (mm)

(c)

Cage rotation speed

0 0.03 0.06 0.09 0.12 0.15 0.181.021.041.061.081.10

Cage

(krp

m)

t (s)

Roller revolution speed0 0.03 0.06 0.09 0.12 0.15 0.18

1.021.041.061.081.10

Rolle

r (kr

pm)

(d)

Figure 5: 1e trajectory and velocity of mass center. Fr � 20 kN; Fa � 0; ωbx � 2.4 krpm.

0.00 0.03 0.06 0.09 0.12 0.15 0.18–9

–6

–3

0

3

6

9

101

Tilt angle

ωbx = 2krpm

t (s)

Tilt

angl

e (10

–4 ra

d)

(a)

Tilt angle

ωbx = 3krpm

0.00 0.03 0.06 0.09 0.12 0.15 0.18–9

–6

–3

0

3

6

9

101

Tilt

angl

e (10

–4 ra

d)

t (s)

(b)

Figure 6: Continued.


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 more.1e averageskew angle is increased from 3.06 × 10−3 rad/0.17 deg to 4.47 ×10−3 rad/0.25 deg. 1e increase of bearing rotating speedenlarges the centrifugal force of rollers, which makes contactforces at the roller/outer raceway and flange/roller-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 0.15 and 0.45 deg [27]. 1e average skew angle inFigure 7 is between 0.17 and 0.25 deg and at an order ofmagnitude compared to the experimental result. 1e

Tilt angle

ωbx = 4krpm

Tilt

angl

e (10

–4 ra

d)

0.00 0.03 0.06 0.09 0.12 0.15 0.18–9

–6

–3

0

3

6

9

101

t (s)

(c)

Tilt angle

ωbx = 5krpm

Tilt

angl

e (10

–4 ra

d)

0.00 0.03 0.06 0.09 0.12 0.15 0.18–9

–6

–3

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 � 2∼5 krpm.

0.00 0.03 0.06 0.09 0.12 0.15 0.18–8

–3

2

7

12

17

22

101

Skew

angl

e (10

–3 ra

d)

t (s)

ωbx = 2krpm

Skew angle

(a)

Skew

angl

e (10

–3 ra

d)

0.00 0.03 0.06 0.09 0.12 0.15 0.18–4

0

4

8

12

16

101

t (s)

ωbx = 3krpm

Skew angle

(b)

Skew

angl

e (10

–3 ra

d)

0.00 0.03 0.06 0.09 0.12 0.15 0.18–4

0

4

8

12

16

101

t (s)

ωbx = 4krpm

Skew angle

(c)

Skew

angl

e (10

–3 ra

d)

0.00 0.03 0.06 0.09 0.12 0.15 0.18–4

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 � 2∼5 krpm.


changing trend of the skew angle with speed is also consistentwith Yang’s 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 zone,the roller slip is more serious than that in the loaded zone.At 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 whirling,a 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.

4.3. 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 −0.27 ×10−4 rad to −0.96 × 10−4 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 4.23 × 10−3 rad/0.24 deg to 5.50 × 10−3 rad/0.31 deg.Both 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 0.0189% to 0.0057% and that of the cage decreases from

0.0189% to 0.0055%, 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 with0.0073% and 0.0076% when the preload is 30 μm. Obviously,increasing 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.

4.4.Grease 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 0.1 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 flange/roller-end contact is reduced.1en, it indirectly leads to the decrease of the skew mo-ment at the flange/roller-end contact and the roller skewangle. Due to the smaller skew angle, the roller misalignmentwill also be small. 1erefore, the normal force at theroller/raceways and flange/roller-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 theroller/raceways and flange/roller-end contacts increase. So,the friction force at these contacts is reduced, which maylead to the decrease of the bearing slip.1e 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].


0.00 0.03 0.06 0.09 0.12 0.15 0.180.0

0.5

1.0

1.5

2.0

2.5

3.0

Slip

rate

of t

he ro

ller (

%)

t (s)

0.1000 0.1025 0.1050 0.1075 0.11–0.01

0.01

0.03

0.05

0.07ωbx = 2krpm

Slip rate of the roller

(a)

0.00 0.03 0.06 0.09 0.12 0.15 0.180.0

0.2

0.4

0.6

0.8

Slip

rate

of t

he ro

ller (

%)

t (s)

ωbx = 3krpm


(b)

0.00 0.03 0.06 0.09 0.12 0.15 0.180.0

0.5

1.0

1.5

2.0

2.5

3.0

Slip

rate

of t

he ro

ller (

%)

t (s)

ωbx = 4krpm


(c)

0.00 0.03 0.06 0.09 0.12 0.15 0.180

2

4

6

8

10

Slip

rate

of t

he ro

ller (

%)

t (s)

ωbx = 5krpm


(d)

Figure 8: 1e slip rate of the conical roller. Fr � 20 kN; Fa � 0; ωbx � 2∼5 krpm.

0.00 0.03 0.06 0.09 0.12 0.15 0.180.0

0.5

1.0

1.5

2.0

2.5

3.0

Slip

rate

of t

he ca

ge (%

)

0.1000 0.1025 0.1050 0.1075 0.11–0.00

0.01

0.02

0.03

t (s)

Slip rate of the cage

ωbx = 2krpm

(a)

0.00 0.03 0.06 0.09 0.12 0.15 0.180.0

0.5

1.0

1.5

2.0

2.5

3.0

Slip

rate

of t

he ca

ge (%

)

t (s)Slip rate of the cage

ωbx = 3krpm

(b)



4.5. Temperature Influence. Due to the improper in-stallation, overwear or not timely cooling, the operatingtemperature of TRBs will change more or less. Sometimes,it 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

0.00 0.03 0.06 0.09 0.12 0.15 0.180.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

Slip

rate

of t

he ca

ge (%

)

t (s)


ωbx = 4krpm

(c)

0.00 0.03 0.06 0.09 0.12 0.15 0.180

2

4

6

8

10

Slip

rate

of t

he ca

ge (%

)

t (s)


ωbx = 5krpm

(d)

Figure 9: 1e slip rate of the cage. Fr � 20 kN; Fa � 0; ωbx � 2∼5 krpm.

0.00 0.03 0.06 0.09 0.12 0.15 0.18–2

0

2

4

6

8

10

12

0.100 0.101 0.102–1.5–1.0–0.5

0.00.51.01.5

δa = 10μm

Tilt anglet (s)

Tilt

angl

e (10

–4 ra

d)

(a)

0.00 0.03 0.06 0.09 0.12 0.15 0.18–2

0

2

4

6

8

10

12

0.100 0.101 0.102–2.0–1.5–1.0–0.5

0.00.51.0

Tilt anglet (s)

Tilt

angl

e (10

–4 ra

d)δa = 20μm

(b)

0.00 0.03 0.06 0.09 0.12 0.15 0.18–3

0

3

6

9

12

15

0.100 0.101 0.102–2.5–2.0–1.5–1.0–0.5

0.00.5

Tilt angle

t (s)

Tilt

angl

e (10

–4 ra

d)

δa = 30μm

(c)

0.00 0.03 0.06 0.09 0.12 0.15 0.18–3

0

3

6

9

12

15

0.100 0.101 0.102–3.0–2.3–1.6–0.9–0.2

0.5

Tilt angle

t (s)

Tilt

angl

e (10

–4 ra

d)

δa = 40μm

(d)

Figure 10: 1e tilt angle of the conical roller. Fr � 0; δa � 10∼40 μm; ωbx � 2.5 krpm.


0.00 0.03 0.06 0.09 0.12 0.15 0.181.0

2.6

4.2

5.8

7.4

9.0

Skew angle

t (s)

Skew

angl

e (10

–3 ra

d)

δa = 10μm

(a)

0.00 0.03 0.06 0.09 0.12 0.15 0.181.0

2.6

4.2

5.8

7.4

9.0

Skew angle

t (s)

Skew

angl

e (10

–3 ra

d)

δa = 40μm

(b)

0.00 0.03 0.06 0.09 0.12 0.15 0.181.0

2.6

4.2

5.8

7.4

9.0

Skew angle

t (s)

Skew

angl

e (10

–3 ra

d)

δa = 30μm

(c)

0.00 0.03 0.06 0.09 0.12 0.15 0.181.0

2.6

4.2

5.8

7.4

9.0

Skew angle

t (s)

Skew

angl

e (10

–3 ra

d)

δa = 40μm

(d)

Figure 11: 1e skew angle of the conical roller. Fr � 0; δa � 10∼40 μm; ωbx � 2.5 krpm.

0.00 0.03 0.06 0.09 0.12 0.15 0.180.00

0.02

0.04

0.06

0.08

Slip

rate

of t

he ro

ller (

%)


t (s)

δa = 10μm

(a)

0.00 0.03 0.06 0.09 0.12 0.15 0.180.00

0.02

0.04

0.06

0.08

Slip

rate

of t

he ro

ller (

%)


t (s)

δa = 20μm

(b)



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 range,the tilt angle under the lithium grease is generally larger

than that under others. With the increase in temperature,the roller skew angle under the polyurea grease increases,while 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

0.00 0.03 0.06 0.09 0.12 0.15 0.180.00

0.02

0.04

0.06

0.08

Slip

rate

of t

he ro

ller (

%)


t (s)

δa = 30μm

(c)

0.00 0.03 0.06 0.09 0.12 0.15 0.180.00

0.02

0.04

0.06

0.08

Slip

rate

of t

he ro

ller (

%)


t (s)

δa = 40μm

(d)

Figure 12: 1e slip rate of the conical roller. Fr � 0; δa � 10∼40 μm; ωbx � 2.5 krpm.

0.00 0.03 0.06 0.09 0.12 0.15 0.180.0

0.5

1.0

1.5

2.0

2.5

3.0


δa = 10μm

0.1000 0.1023 0.1046 0.10690.010

0.015

0.020

0.025

0.030

Slip

rate

of t

he ca

ge (%

)

t (s)

(a)


δa = 20μm

0.00 0.03 0.06 0.09 0.12 0.15 0.180.0

0.5

1.0

1.5

2.0

2.5

3.0

0.110 0.111 0.112 0.113 0.114 0.115–0.01

0.00

0.01

0.02

0.03

0.04

Slip

rate

of t

he ca

ge (%

)

t (s)

(b)


δa = 30μm

0.00 0.03 0.06 0.09 0.12 0.15 0.180.0

0.5

1.0

1.5

2.0

2.5

3.0

0.110 0.111 0.112 0.113 0.114 0.1150.0000.0030.0060.0090.0120.015

Slip

rate

of t

he ca

ge (%

)

t (s)

(c)


δa = 40μm

0.00 0.03 0.06 0.09 0.12 0.15 0.180.0

0.5

1.0

1.5

2.0

2.5

3.0

0.110 0.111 0.112 0.113 0.114 0.1150.0000.0030.0060.0090.0120.015

Slip

rate

of t

he ca

ge (%

)

t (s)

(d)

Figure 13: 1e slip rate of the cage. Fr � 0; δa � 10∼40 μm; ωbx � 2.5 krpm.


2.44 8.44 14.44 20.44 26.44 32.44 38.44–5.2

–5.1

–5.0

–4.9

–4.8

–4.7

Plastic viscosity (Pa·sn)

0.0

1.6

3.2

4.8

6.4

8.0

Tilt angleSkew angle

Tilt

angl

e (10

–5 ra

d)

Skew

angl

e (10

–3 ra

d)

(a)

8.44 14.44 20.44 26.44 32.44 38.440.000

0.004

0.008

0.012

0.016

0.020

Slip

rate

of t

he ro

ller (

%)

Plastic viscosity (Pa·sn)

–0.006

0.000

0.006

0.012

0.018

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 � 2.8 krpm. (a) 1e conical roller tilt and skew. (b) Slip rate of the conicalroller and cage.

251.8 301.8 351.8 401.8 451.8 501.8–5.50

–5.36

–5.22

–5.08

–4.94

–4.80

Yield stress (Pa)

3.50

3.70

3.90

4.10

4.30

4.50

Tilt angleSkew angle

Tilt

angl

e (10

–5 ra

d)

Skew

angl

e (10

–3 ra

d)

(a)

Slip rate of the rollerSlip rate of the cage

251.8 301.8 351.8 401.8 451.8 501.80.009

0.010

0.011

0.012

0.013

0.014

0.015Sl

ip ra

te o

f the

rolle

r (%

)

Yield stress (Pa)

0.009

0.010

0.011

0.012

0.013

0.014

0.015

Slip

rate

of t

he ca

ge (%

)

(b)

Figure 15: Yield stress influence. Fr � 0; Fa � 15 kN; ωbx � 2.8 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 Temperature/°C Yield stress τy/Pa Plastic viscosity φ/Pa·sn Power law exponent n

Polyurea

15 501.7 11.87 0.773130 351.8 8.44 0.719650 305.5 6.16 0.662170 188.5 6.10 0.7378

Lithium

15 904.5 54.35 0.305330 805.2 46.23 0.312550 710.4 41.13 0.315170 339.8 52.96 0.3203

Calcium

15 990.8 64.82 0.334930 649.5 49.23 0.534450 336.1 30.59 0.525670 296.0 35.88 0.4356


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 grease,the 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 viscosity.1e 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 lubrication,the 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 https://figshare.com/s/15c2556bee83d1adf01d.

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).

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10 23 36 49 62 75–5.8

–5.6

–5.4

–5.2

–5.0

–4.8

–4.6

–4.4

Calcium:Lithium:Polyurea: tilt angle skew angle

tilt angle skew angletilt angle skew angle

Temperature (°C)

1.0

3.0

5.0

7.0

9.0

11.0

13.0Ti

lt an

gle (

10–5

rad)

Skew

angl

e (10

–3 ra

d)

(a)

10 23 36 49 62 750.000

0.005

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Poly

urea

gre

ase (

%)

Temperature (°C)

0.000

0.001

0.002

0.003

0.004

0.005

0.006

Calc

ium

/lith

ium

gre

ase (

%)

CalciumLithiumPolyurea

(b)

Figure 16: Temperature influence. Fr � 0; Fa � 15 kN; ωbx � 2.8 krpm. (a) 1e conical roller tilt and skew. (b) Slip rate of the conical roller.


https://figshare.com/s/15c2556bee83d1adf01dhttps://figshare.com/s/15c2556bee83d1adf01d

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Analysis of Dynamic Characteristics of Grease-Lubricated … · 2019. 7. 30. · Analysis of Dynamic Characteristics of Grease-Lubricated Tapered Roller Bearings Zheng-Hai Wu ,1 Ying-Qiang