# Analysis of Dynamic Characteristics of Grease-Lubricated Tapered Roller Bearings

Authors: Zheng-Hai Wu; Ying-Qiang Xu; Si-Er Deng
Journal: Shock and Vibration
 (2018)

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## Abstract

Tapered roller bearings (TRBs) are applied extensively in the field of high-speed trains, machine tools, automobiles, etc. The motion prediction of main components of TRBs under grease lubrication will be beneficial to the design of bearings and the selection of lubricating grease. In this study, considering the dynamic contact relationship among the cage, rollers, and raceways, a multibody contact dynamic model of the TRB was established based on the geometric interaction models and grease lubrication theories. The impacts of load, grease rheological properties, and temperature on the roller tilt and skew and the bearing slip were simulated by using the fourth-order Runge–Kutta method. The results show that the roller tilt angle in the unloaded zone is obviously larger than that in the loaded zone, while the roller skew angle in the unloaded zone is smaller than that in the loaded zone. As the speed increases, the roller tilt and skew and the bearing slip become more serious. Bearing preload can effectively reduce the bearing slip but will make the roller tilt and skew angle increase. The roller skew angle and the bearing slip decrease with the increase of the grease plastic viscosity. The roller tilt angle increases with the increase of the plastic viscosity. The yield stress of the grease has little effect on motions of the roller and cage. The influence of temperature on the roller and cage motions varies with the type of grease used.


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## Body

## 1. Introduction

Tapered roller bearings, as the separable bearing, have the ability to withstand combined loads, large load-carrying capacity, well adjustability, and long service life. Nowadays, nearly 90% of rolling bearings are grease-lubricated [1]. Generally, compared to oil lubrication, lubricating grease in a rolling bearing has a wide operating temperature range and good extreme pressure (EP) property and adhesion property, and the construction of the lubricating device for the grease lubrication is sometimes relatively simple. However, due to the pressure difference inside the bearing contacts, the grease will flow to sides and next to the raceways over time. There may be very little reflow back into the raceways, and the bearing may suffer from starvation. Despite the abovementioned conditions, there also exists a film inside the bearing 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 dynamics and the lubricating grease. The bearing lubrication and dynamics not only affect each other but both have important impacts on the bearing failure, service life, and reliability. The analysis may have implications for the design of the bearing and the selection of the grease.The dynamics of ball and cylindrical roller bearings have been extensively studied in the past few decades [3–5]. By considering the four degree-of-freedom balls and the six degree-of-freedom cage, Walters [3] firstly presented a comprehensive analysis for the balls and cage motions. After that, Gupta [4] and Meeks and Ng [5] carried out a series of research on dynamic problems of ball and cylindrical roller bearings. The movements of the rolling elements and cage were minutely described by classical differential equations of motion under specific operating conditions. For tapered roller bearings, compared with ball bearings (except angular contact ball bearings) and cylindrical roller bearings, the structure type and dynamics are more complicated. Gupta [6] studied the dynamic model and developed the dynamic analysis program ADORE for TRBs; the cage whirling and roller skew were analyzed under different cage clearances and traction-slip relations. However, the bearing slip and impacts of the lubricant on motions of bearing parts were not presented. Cretu et al. [7] proposed a quasidynamic model for the TRB, based on the Johnson–Tevaarwerk rheological model; the traction performance 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 by using the Adams-Bashforth–Moulton multistep method and studied the cage whirling and roller skew of the bearing. By considering six degrees of the cage motion, Sakaguchi and Harada [9, 10] simulated motions of the rigid or flexible cage in TRBs on the dynamic simulation software ADAMS. Bercea [11] proposed a comprehensive model to predict the roller skew motion in TRBs. He found that the roller skew is greatly influenced by traction at the flange/roller-end contact and by the roller-end geometry. A three-dimensional dynamic model of the double row TRB of a certain type of high-speed train was established by Gai and Zhang [12], and the roller contact stress and the cage stability were simulated. However, the lubrication state of axlebox bearings was not taken into account. Compared with other rolling bearings, the research on dynamics of grease-lubricated TRBs is relatively deficient.The objective of this study was to present an accurate analysis of dynamic response of the tapered roller bearing. For the beginning operating stages of grease-lubricated tapered roller bearings, considering dynamic interactions in the bearing contacts and grease lubrication theories, a multibody contact dynamic model of TRB under the grease-lubricated condition was established. The bearing dynamics, e.g., the roller tilt and skew and the bearing slip, was analyzed. The impacts of speed, preload, temperature, and grease rheological properties on the bearing dynamics were studied.


## 2. Dynamic Analysis Model

For the sake of accurately describing the relative position and movement of each component of the TRB, an inertial coordinate systemo-xyz was established. As shown in Figure 1, the origin coincides with the mass center of the bearing, and the o-x axis coincides with the axis of the bearings’ shaft. A body-fixed coordinate system oc-xcyczc is defined for the cage, and its origin is the mass center of the cage. The roller body-fixed coordinate system is or-xryrzr, the origin is the roller mass center, and the or-xr axis along the roller axis. Because the roller is prone to bear unbalance moment, causing the roller to rotate abnormally, two harmful but inevitable movements of rollers are emphasized: tilt and skew. The tilt is normally referred to the roller rotation about or-yr axis, and the skew is the roller rotation about or-zr axis.Figure 1
Bearing coordinate system.


### 2.1. Roller/Raceway Interaction

When the roller tilting or skewing, the interaction between the roller and raceway will vary along the contact line. As shown in Figure2, the roller is divided into several slices, and each slice interaction with the raceway is calculated independently. Then, the total contact load between the roller and raceway can be obtained by integrating these local interactions.Figure 2
Geometric interaction between the roller and raceway.In order to confirm the contact force at pointP on slice surface, the gap or interference at point P should be determined first. In Figure 2, let rbi and rri represent the mass center positions of the race and roller in the inertial frame, respectively. And rgmr and rgmb are, respectively, the geometric center position vectors of the roller and race relative to their mass centers. Then, geometric center position rbrb of the kth slice relative to the race center can be expressed as(1)rbrb=Tibrri−rbi+Tir′rgmr+Tir′xrk,0,0T−rgmb,where xrk is the slice position at the roller axis; Tib is the transformation matrix (Euler’s rotation matrix) between the inertial and race body-fixed coordinate (ob-xbybzb); and Tir is the transformation matrix between the inertial and roller coordinate system.The azimuth angleψ of the slice relative to the race can be defined by components rbr2b and rbr3b of rbrb. Thus,(2)ψ=atan−rbr2brbr3b.Then, the transformation matrixTbaψ,0,0 between the race and slice azimuth coordinate system is known. Assume γ is the azimuth angle of the point P in the coordinate plane or-yrzr and ς is the roller radius at the axial position xrk. In the roller coordinate system, the position of the point P relative to the roller center is(3)rpbr=xrk,−ςsinγ,ςcosγT.In the slice azimuth coordinate system, the position of the pointP relative to the race center is(4)rba=Tbarbrb+TibTir′Tpbr.Theγ should satisfy the condition that the yba direction component rba2 of rba is zero:(5)rba2=rbr2a+T21xrk−T22ςsinγ+T23ςcosγ=0,where rbr2a is the component of Tbarbrb in the yba direction and T21,T22,T23 are components of T=TbaTibTir′.Ifϕ = atanςT23/ςT22 is assumed, the above formula can be changed to(6)γ=ϕ+asinrba2+T21xrkςT222+T2321/2orγ=π+ϕ+asinrba2+T21xrkςT222+T2321/2.For the roller is in contact with the inner raceway, the value ofγ should make the value of rba3 smaller in the zba direction; if the roller contacts with the outer raceway, the γ should make rba3 larger in the zba direction. Then, the interference between the kth slice and the raceway can be confirmed by subtracting the race radius from the above position vector, and the result may be transformed to a contact coordinate system to compute the interaction δ normal to the contact plane. Symbolically,(7)δ=rba3−ξrba1,ψcosαp,where αp is the contact angle and ξ is the radius of the raceway at the point P. If the value of δ is negative, it indicates that there is no contact; if not, it means there is contact.Then, the position of the pointP can be rewritten as(8)rpbr=xrk,−ς−0.5δsinγ,ς−0.5δcosγT.Assume thatvbix˙b,y˙b,z˙bTandωbbωbx,ωby,ωbzT are, respectively, the velocity and the angular velocity of the race and vrix˙,r˙,θ˙Tandωrrωrx,ωry,ωrzT are the velocity and the angular velocity of the roller, respectively. In the contact coordinate system, the velocities at the point P on the raceway and roller are(9)vbc=TacTiavbi+rri−rbi+Tir′rpbr+Tir′rgmr+Tib′rgmbTib′ωbb−θ˙,0,0T,vrc=TacTiaTir′ωrr×Tir′rpbr+Tir′rgmr+x˙,0,r˙T,where Tac0,αp,0 is the transformation matrix between the azimuth and contact coordinate system.Then, the velocity at the pointP on the raceway relative to the roller is(10)vbrc=vbc−vrc.For the line contact between thekth slice and the raceway, assuming that rollers behave as elastic half space, the normal force q at the point P is(11)q=kwδ10/9+cwvbr3c,where kw is the Hertzian contact stiffness, kw = 0.356E′ns−1le8/9 [13]; E′ is the equivalent elastic modulus of the roller and raceway; ns is the total number of slices; le is the effective length of the roller; cw is the viscous damping coefficient, cw = 1.5αekwδ10/9 [14, 15] and αe is related to the restitution coefficient, for steel, bronze, or ivory, αe = 0.08 d 0.32 s/m [14].The grease lubricated condition of the TRB is considered here. In practice, the grease properties change over time (by overrolling, oil bleeding, starvation, shearing, etc.), which affect the bearing performance. In order to simplify the dynamic model, it is assumed that an ideal elastohydrodynamic lubrication state at the roller/raceway contact. Based on the grease EHL theory, the grease film traction forces at the pointP are [16](12)fr=πleεφ0R′1/221/2nsh01/23vb2c−vr2c,fb=πleεφ0R′1/221/2nsh01/23vr2c−vb2c,where φ0 is the grease plastic viscosity at atmospheric pressure; R′ is the equivalent radius of the kth slice and the raceway; h0 is the minimum grease film thickness, h0 is calculated by using the model in [17]; ε is related to the Hertzian contact half width b, film thickness h0, and radius R′.(13)ε=2πatan21/2bR′1/2h01/2−21/2bR′1/2h01/2R′h0+2b2.In order to determine the total contact load between the roller and raceway, the integration of local interactions is required along the contact line. The total contact force and torque acting on thejth roller are(14)Qja=∑k=1nsTcaFrjkc,Njr=∑k=1nsrpbr+rgmr×TcrFrjkc.The contact loads acting on the race are(15)Qji=∑k=1nsTic′Fbjkc,Njb=∑k=1nsrpbb+Tibrrbi×TibTic′Fbjkc,where(16)Frjk,bjkc=frjk,bjksinαf,frjk,bjkcosαf,±qjkT,αf=asinvbr2c/vbr1c.


### 2.2. Flange/Roller-End Interaction

In race body-fixed coordinate system (ob-xbybzb), the roller-end curvature center relative to the race center is(17)rbeb=Tibrri−rbi+Tir′rgmr+Tir′xer,0,0T−rgmb,where xer is the position of the roller-end curvature center at the roller axis.Since normal force will act on the flange surface and pass through the roller-end curvature center, the above center can be transformed into the race azimuth coordinate system byTbaψ,0,0, where ψ=atanrbe2b/rbe3b.(18)rbea=Tbarbeb.As illustrated in Figure2, the position of the roller-end curvature center relative to the flange coordinate is(19)reff=Tafrbea−Tbarfb,where Taf0,β,0 is the transformation between the race azimuth and flange coordinate; β is the flange angle defined as a rotation about the o-y axis in accordance to the right-hand screw rule; and rfb locates the flange origin in the race coordinate.If the radiusrs of the roller-end is known, the geometric interaction between the roller-end and flange is simply given by(20)δ=ref1f−rs.Definitionre is the distance from the bearing apex to the roller spherical end. If re > rs, the flange/roller-end contact is elliptical contact [11]. The Hertzian contact theory can be employed to determine the contact load qf at the flange/roller-end contact.(21)qf=321/2δ3/23nδ3/2E′ρ1/2,where nδ is the contact deformation coefficient and ∑ρ is the sum of principal curvatures.The friction at the flange/roller-end contact contains two parts: the friction induced by microasperities contact and the traction force produced by the grease lubricant [18, 19].(22)ff=fasp+fgrease.The friction force induced by the asperities contact is(23)fasp=μaqfexp−BλC=s0p0qfexp−BλC,where B and C are related to morphology and material 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 grease film thickness hc to surface roughness. The film thickness hc is approximated as [20](24)hc=1.8035×10−2M−0.5066L−1.1869,where(25)M=qfE′rs2E′rsφ0uf3/4,L=αE′φ0ufE′rs1/4,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:(26)τ=τy+φDn,where τ is the grease shear stress; τy is the yield stress; φ is the plastic viscosity, φ = φ0eαp; p is the contact pressure, the distribution of p can be approximated by the Hertzian contact pressure; α is the viscosity-pressure coefficient; n is the power law exponent; and D is the shear rate, D ≌ (uflange − uend)/hc. Then, the traction force due to the grease shear stress is obtained by the integration:(27)fgrease=∫−aa∫−bbτx,yxdxdy.Then, the loads acting on thejth roller are(28)Qfja=Taf′Ffff=Taf′qfj,ffj,0T,Nfjr=rfrr+rgmr×TfrFfjf.The loads acting on the flange are(29)Qfji=−TftFfjf,Nfjb=rfbb+rgmb×TibQfji,where rfrr and rfbb 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 rollers rotation, the master-slave relationship of rotating cage-roller assembly will change over time due to different velocities and displacements. As shown in Figures2 and 3, the location of the jth roller geometric center in the cage pocket coordinate system (op-xpypzp) is(30)rcrp=TcpTicrri−rci+Tirrgmr−r0,where rci is the position of the cage mass center in the inertial coordinate system; Tcp is the transformation matrix from the cage-fixed coordinate system (oc-xcyczc) to the pocket coordinate system; r0 locates the pocket center in the cage coordinate system; and Tic is the transformation matrix between the inertial and cage coordinate system.Figure 3
Interaction between the roller and cage.Now, similar to the roller/raceway interaction, letrgr locates a point P′ on the kth slice of the roller such that(31)rp′r=xrk,−ςsinγ,ςcosγT.In the pocket coordinate system, the position of the pointP′ is(32)rp′cp=rcrp+TicTir′rp′r.As illustrated in Figure3, the clearance between the point P and the pocket crossbeam is(33)δ=Δδ−rp′c2p,where Δδ is the initial clearance between the roller and crossbeam, Δδ = (lc − dw)/2; lc is the average width of the pocket; and dw is the average diameter of the roller.For the modeling of the assembly interactions, assuming that there is an excess of the grease in the cage-roller assembly, a critical value Δh0 of the grease film thickness is assumed for the contact state transition. When δ ≥ Δh0, there are only the hydrodynamic effect between the kth slice and pocket crossbeam, and no Hertzian contact [9, 21]. The minimum grease film thickness h0 = δ. The hydrodynamic pressure qc and the grease film traction force fc between the kth slice and crossbeam are(34)qc=qcehl=keδ+ceδ˙,fc=fcehl=3πεφ0uleς1/221/2h01/2,where [17, 22](35)ke=∂fh0∂h0,ce=6πεφ0leς3/221/2nsh03/2,fh0=29παφnsξ1un9E″h06n+2ς−3le4+2nn+ξ2λnτyh0nξ1φun,where u is the entrainment speed; ξ1, ξ2 are the constants associated with the power law exponent n [17]; and ce is grease film damping [22].Whenδ < Δh0, it means that the contact between the kth slice and crossbeam is in a boundary state. So, the Hertzian contact is included in the contact. The minimum grease film thickness h0 = Δh0. The Hertzian contact deformation is(36)δh=Δh0−δ.The contact forceqc and the friction force fc between the roller and crossbeam are(37)qc=khδh+chδ˙h+qcehl,fc=μbdqc+fcehl,where kh is the linearized Hertzian contact stiffness [23]; ch is the Herbert viscous damping coefficient [24]; and μbd is the traction coefficient under boundary lubrication [9].Then, the load at the contact pointP′ on the kth slice in the contact coordinate system is(38)Fcjkc=0,qcjk,fcjkT.Using the processing method of the roller/raceway interaction, the loadFcjkc can also be transferred to the inertial coordinate and the roller/cage body-fixed coordinate, respectively.Owing to the smaller sliding speed between the roller-end and the side beam and the smaller curvature of roller spherical end, the EHL lubrication state cannot be effectively formed at this contact, but the squeeze film lubrication will be formed. As shown in Figure3, the Δδ′ is the initial pocket clearance and ra = rcr1p. If dra/dt ≥ 0, the small end contacts with the side beam; if dra/dt < 0, the large end contacts. Neglecting the grease traction, the squeeze force qs is(39)qs=ksΔδ′−ra,where [25](40)ks=−∂∂hadl,sφ0ks3ha3dradt,where ha is the distance from the side beam to the end; dl and ds are the diameters of the large and small end, respectively; and hs is the thickness of the side beam.Then, the loads acting on thejth roller are(41)Qsja=qsj,0,0T,Nsjr=rsrr+rgmr×TarQsja.The loads acting on the cage are(42)Qsji=−TaiQsja,Nsjc=r0+Tcp′rcrp+Trcrsrr+rgmr×TicQsji,where rsrr is the position of the contact point in the roller body-fixed coordinate and Tar and Tai are, respectively, the transform from the azimuth to roller and inertial coordinate.According to geometric characteristics of the cage and guide ring, it can be considered as a finite journal bearing lubrication condition between the guide surface and cage centering surface. In Figure4, the clearance δ′ between the cage and guide ring is(43)δ′=Δe−e,where Δe is the initial clearance, Δe = (ry − rd)/2; rd, ry are, respectively, the radii of the centering surface and guide surface and e is cage displacement in the radial plane.Figure 4
Interaction between the cage and guide ring (inner-ring guided).Similarly, assume thatΔh0′ is the critical value of the grease film thickness. When δ′ ≥ Δh0′, the contact force qic and friction torque Nic on the cage are(44)qic=qicehl=ke′δ+ce′δ˙′,Nic=Nicehl=πεφ0bric1/2rd21/2h01/23ωbxry−ωcxrd,where ric is the equivalent radius and ωcx, ωbx are, respectively, the angular velocities of the cage and guide ring. The calculation of the grease film stiffness ke′ and damping ce′ is the same as that at the roller/crossbeam contact.Whenδ′ < Δh0′, the elastic deformation between the cage and guide ring is δh′ = δ′ − h0, and the minimum grease film thickness h0 = Δh0′. The qic and Nic are(45)qic=kh′δh′+ch′δh′+qicehl,Nic=μbdqicrd+Nicehl,where kh′ is the contact stiffness and ch′ is the viscous damping coefficient. The calculation of kh′ and ch′ is the same as that in Equation (35). μbd is the traction coefficient under boundary lubrication.In order to simplify the model, the load of the cage on the guide ring is ignored in contrast to large external load on the guide ring. Then, the loads acting on the cage are(46)Qici=0,qicsinφic,qiccosφicT,Nicc=±Nic,0,0.If the TRB is lubricated by the oil, the effect of oil-gas mixture on the cage and rollers may not be overlooked. However, if the bearing is lubricated by the grease, it seems impossible to have the grease-gas mixture effect on the dynamics of the cage and rollers. So, the effect of the grease-gas mixture is not considered in the dynamic model.


### 2.4. Dynamic Model

In real application also, the surroundings (inertia/damping of shaft assembly and housing) are expected to influence the TRB dynamics. These effects are neglected in the modeling. Generally, it is convenient to consider the translational moving of rollers in the cylindrical coordinate system, while the Cartesian coordinate system is convenient for the cage and race. The translational moving of bearing parts can be simply described by Newton’s laws such that(47)Mδ¨=Q.The generalized forceQ is obtained by superimposing the forces from Section 2. The centrifugal force and the gyroscopic moment should be superimposed on the generalized force Q for the roller. For the cage and race, the mass matrices Mb and Mc are(48)Mb,c=diagmb,c,mb,c,mb,c.For the special case of conical rollers,Mr is(49)Mr=diagmr,mrrrj,mr.Both rollers and the cage have six degrees-of-freedom. Unlike the translational motion (in three directions), the rotating of the bearing parts are described in their own body-fixed coordinates. For rollers and the cage, using the Euler dynamic equations such that(50)Ixω˙x−ωyωzIy−IzIyω˙y−ωzωxIz−IxIzω˙z−ωxωxIx−Iy=NxNyNz,where Ix, Iy, and Iz are inertia principal moments of the cage or rollers; the total moment N is also obtained by superimposing the forces from Section 2. ω[ωx, ωy, ωz] is the angular velocity in the inertial coordinate system. The relationship between the angular velocity ωfixed in the body-fixed coordinate system and ω is as follows. B is the Euler’s rotation matrix.(51)ωfixed=Bω,ω˙fixed=dBdtB−1ω˙.Admitting the outer ring as macroscopically stationary and a rotating inner ring with no torque load, it results four degrees of freedom of inner ring: moving in thex, y, and z directions and constant rotating about the shaft axis (o-x).


## 2.1. Roller/Raceway Interaction

When the roller tilting or skewing, the interaction between the roller and raceway will vary along the contact line. As shown in Figure2, the roller is divided into several slices, and each slice interaction with the raceway is calculated independently. Then, the total contact load between the roller and raceway can be obtained by integrating these local interactions.Figure 2
Geometric interaction between the roller and raceway.In order to confirm the contact force at pointP on slice surface, the gap or interference at point P should be determined first. In Figure 2, let rbi and rri represent the mass center positions of the race and roller in the inertial frame, respectively. And rgmr and rgmb are, respectively, the geometric center position vectors of the roller and race relative to their mass centers. Then, geometric center position rbrb of the kth slice relative to the race center can be expressed as(1)rbrb=Tibrri−rbi+Tir′rgmr+Tir′xrk,0,0T−rgmb,where xrk is the slice position at the roller axis; Tib is the transformation matrix (Euler’s rotation matrix) between the inertial and race body-fixed coordinate (ob-xbybzb); and Tir is the transformation matrix between the inertial and roller coordinate system.The azimuth angleψ of the slice relative to the race can be defined by components rbr2b and rbr3b of rbrb. Thus,(2)ψ=atan−rbr2brbr3b.Then, the transformation matrixTbaψ,0,0 between the race and slice azimuth coordinate system is known. Assume γ is the azimuth angle of the point P in the coordinate plane or-yrzr and ς is the roller radius at the axial position xrk. In the roller coordinate system, the position of the point P relative to the roller center is(3)rpbr=xrk,−ςsinγ,ςcosγT.In the slice azimuth coordinate system, the position of the pointP relative to the race center is(4)rba=Tbarbrb+TibTir′Tpbr.Theγ should satisfy the condition that the yba direction component rba2 of rba is zero:(5)rba2=rbr2a+T21xrk−T22ςsinγ+T23ςcosγ=0,where rbr2a is the component of Tbarbrb in the yba direction and T21,T22,T23 are components of T=TbaTibTir′.Ifϕ = atanςT23/ςT22 is assumed, the above formula can be changed to(6)γ=ϕ+asinrba2+T21xrkςT222+T2321/2orγ=π+ϕ+asinrba2+T21xrkςT222+T2321/2.For the roller is in contact with the inner raceway, the value ofγ should make the value of rba3 smaller in the zba direction; if the roller contacts with the outer raceway, the γ should make rba3 larger in the zba direction. Then, the interference between the kth slice and the raceway can be confirmed by subtracting the race radius from the above position vector, and the result may be transformed to a contact coordinate system to compute the interaction δ normal to the contact plane. Symbolically,(7)δ=rba3−ξrba1,ψcosαp,where αp is the contact angle and ξ is the radius of the raceway at the point P. If the value of δ is negative, it indicates that there is no contact; if not, it means there is contact.Then, the position of the pointP can be rewritten as(8)rpbr=xrk,−ς−0.5δsinγ,ς−0.5δcosγT.Assume thatvbix˙b,y˙b,z˙bTandωbbωbx,ωby,ωbzT are, respectively, the velocity and the angular velocity of the race and vrix˙,r˙,θ˙Tandωrrωrx,ωry,ωrzT are the velocity and the angular velocity of the roller, respectively. In the contact coordinate system, the velocities at the point P on the raceway and roller are(9)vbc=TacTiavbi+rri−rbi+Tir′rpbr+Tir′rgmr+Tib′rgmbTib′ωbb−θ˙,0,0T,vrc=TacTiaTir′ωrr×Tir′rpbr+Tir′rgmr+x˙,0,r˙T,where Tac0,αp,0 is the transformation matrix between the azimuth and contact coordinate system.Then, the velocity at the pointP on the raceway relative to the roller is(10)vbrc=vbc−vrc.For the line contact between thekth slice and the raceway, assuming that rollers behave as elastic half space, the normal force q at the point P is(11)q=kwδ10/9+cwvbr3c,where kw is the Hertzian contact stiffness, kw = 0.356E′ns−1le8/9 [13]; E′ is the equivalent elastic modulus of the roller and raceway; ns is the total number of slices; le is the effective length of the roller; cw is the viscous damping coefficient, cw = 1.5αekwδ10/9 [14, 15] and αe is related to the restitution coefficient, for steel, bronze, or ivory, αe = 0.08 d 0.32 s/m [14].The grease lubricated condition of the TRB is considered here. In practice, the grease properties change over time (by overrolling, oil bleeding, starvation, shearing, etc.), which affect the bearing performance. In order to simplify the dynamic model, it is assumed that an ideal elastohydrodynamic lubrication state at the roller/raceway contact. Based on the grease EHL theory, the grease film traction forces at the pointP are [16](12)fr=πleεφ0R′1/221/2nsh01/23vb2c−vr2c,fb=πleεφ0R′1/221/2nsh01/23vr2c−vb2c,where φ0 is the grease plastic viscosity at atmospheric pressure; R′ is the equivalent radius of the kth slice and the raceway; h0 is the minimum grease film thickness, h0 is calculated by using the model in [17]; ε is related to the Hertzian contact half width b, film thickness h0, and radius R′.(13)ε=2πatan21/2bR′1/2h01/2−21/2bR′1/2h01/2R′h0+2b2.In order to determine the total contact load between the roller and raceway, the integration of local interactions is required along the contact line. The total contact force and torque acting on thejth roller are(14)Qja=∑k=1nsTcaFrjkc,Njr=∑k=1nsrpbr+rgmr×TcrFrjkc.The contact loads acting on the race are(15)Qji=∑k=1nsTic′Fbjkc,Njb=∑k=1nsrpbb+Tibrrbi×TibTic′Fbjkc,where(16)Frjk,bjkc=frjk,bjksinαf,frjk,bjkcosαf,±qjkT,αf=asinvbr2c/vbr1c.


## 2.2. Flange/Roller-End Interaction

In race body-fixed coordinate system (ob-xbybzb), the roller-end curvature center relative to the race center is(17)rbeb=Tibrri−rbi+Tir′rgmr+Tir′xer,0,0T−rgmb,where xer is the position of the roller-end curvature center at the roller axis.Since normal force will act on the flange surface and pass through the roller-end curvature center, the above center can be transformed into the race azimuth coordinate system byTbaψ,0,0, where ψ=atanrbe2b/rbe3b.(18)rbea=Tbarbeb.As illustrated in Figure2, the position of the roller-end curvature center relative to the flange coordinate is(19)reff=Tafrbea−Tbarfb,where Taf0,β,0 is the transformation between the race azimuth and flange coordinate; β is the flange angle defined as a rotation about the o-y axis in accordance to the right-hand screw rule; and rfb locates the flange origin in the race coordinate.If the radiusrs of the roller-end is known, the geometric interaction between the roller-end and flange is simply given by(20)δ=ref1f−rs.Definitionre is the distance from the bearing apex to the roller spherical end. If re > rs, the flange/roller-end contact is elliptical contact [11]. The Hertzian contact theory can be employed to determine the contact load qf at the flange/roller-end contact.(21)qf=321/2δ3/23nδ3/2E′ρ1/2,where nδ is the contact deformation coefficient and ∑ρ is the sum of principal curvatures.The friction at the flange/roller-end contact contains two parts: the friction induced by microasperities contact and the traction force produced by the grease lubricant [18, 19].(22)ff=fasp+fgrease.The friction force induced by the asperities contact is(23)fasp=μaqfexp−BλC=s0p0qfexp−BλC,where B and C are related to morphology and material 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 grease film thickness hc to surface roughness. The film thickness hc is approximated as [20](24)hc=1.8035×10−2M−0.5066L−1.1869,where(25)M=qfE′rs2E′rsφ0uf3/4,L=αE′φ0ufE′rs1/4,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:(26)τ=τy+φDn,where τ is the grease shear stress; τy is the yield stress; φ is the plastic viscosity, φ = φ0eαp; p is the contact pressure, the distribution of p can be approximated by the Hertzian contact pressure; α is the viscosity-pressure coefficient; n is the power law exponent; and D is the shear rate, D ≌ (uflange − uend)/hc. Then, the traction force due to the grease shear stress is obtained by the integration:(27)fgrease=∫−aa∫−bbτx,yxdxdy.Then, the loads acting on thejth roller are(28)Qfja=Taf′Ffff=Taf′qfj,ffj,0T,Nfjr=rfrr+rgmr×TfrFfjf.The loads acting on the flange are(29)Qfji=−TftFfjf,Nfjb=rfbb+rgmb×TibQfji,where rfrr and rfbb 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 rollers rotation, the master-slave relationship of rotating cage-roller assembly will change over time due to different velocities and displacements. As shown in Figures2 and 3, the location of the jth roller geometric center in the cage pocket coordinate system (op-xpypzp) is(30)rcrp=TcpTicrri−rci+Tirrgmr−r0,where rci is the position of the cage mass center in the inertial coordinate system; Tcp is the transformation matrix from the cage-fixed coordinate system (oc-xcyczc) to the pocket coordinate system; r0 locates the pocket center in the cage coordinate system; and Tic is the transformation matrix between the inertial and cage coordinate system.Figure 3
Interaction between the roller and cage.Now, similar to the roller/raceway interaction, letrgr locates a point P′ on the kth slice of the roller such that(31)rp′r=xrk,−ςsinγ,ςcosγT.In the pocket coordinate system, the position of the pointP′ is(32)rp′cp=rcrp+TicTir′rp′r.As illustrated in Figure3, the clearance between the point P and the pocket crossbeam is(33)δ=Δδ−rp′c2p,where Δδ is the initial clearance between the roller and crossbeam, Δδ = (lc − dw)/2; lc is the average width of the pocket; and dw is the average diameter of the roller.For the modeling of the assembly interactions, assuming that there is an excess of the grease in the cage-roller assembly, a critical value Δh0 of the grease film thickness is assumed for the contact state transition. When δ ≥ Δh0, there are only the hydrodynamic effect between the kth slice and pocket crossbeam, and no Hertzian contact [9, 21]. The minimum grease film thickness h0 = δ. The hydrodynamic pressure qc and the grease film traction force fc between the kth slice and crossbeam are(34)qc=qcehl=keδ+ceδ˙,fc=fcehl=3πεφ0uleς1/221/2h01/2,where [17, 22](35)ke=∂fh0∂h0,ce=6πεφ0leς3/221/2nsh03/2,fh0=29παφnsξ1un9E″h06n+2ς−3le4+2nn+ξ2λnτyh0nξ1φun,where u is the entrainment speed; ξ1, ξ2 are the constants associated with the power law exponent n [17]; and ce is grease film damping [22].Whenδ < Δh0, it means that the contact between the kth slice and crossbeam is in a boundary state. So, the Hertzian contact is included in the contact. The minimum grease film thickness h0 = Δh0. The Hertzian contact deformation is(36)δh=Δh0−δ.The contact forceqc and the friction force fc between the roller and crossbeam are(37)qc=khδh+chδ˙h+qcehl,fc=μbdqc+fcehl,where kh is the linearized Hertzian contact stiffness [23]; ch is the Herbert viscous damping coefficient [24]; and μbd is the traction coefficient under boundary lubrication [9].Then, the load at the contact pointP′ on the kth slice in the contact coordinate system is(38)Fcjkc=0,qcjk,fcjkT.Using the processing method of the roller/raceway interaction, the loadFcjkc can also be transferred to the inertial coordinate and the roller/cage body-fixed coordinate, respectively.Owing to the smaller sliding speed between the roller-end and the side beam and the smaller curvature of roller spherical end, the EHL lubrication state cannot be effectively formed at this contact, but the squeeze film lubrication will be formed. As shown in Figure3, the Δδ′ is the initial pocket clearance and ra = rcr1p. If dra/dt ≥ 0, the small end contacts with the side beam; if dra/dt < 0, the large end contacts. Neglecting the grease traction, the squeeze force qs is(39)qs=ksΔδ′−ra,where [25](40)ks=−∂∂hadl,sφ0ks3ha3dradt,where ha is the distance from the side beam to the end; dl and ds are the diameters of the large and small end, respectively; and hs is the thickness of the side beam.Then, the loads acting on thejth roller are(41)Qsja=qsj,0,0T,Nsjr=rsrr+rgmr×TarQsja.The loads acting on the cage are(42)Qsji=−TaiQsja,Nsjc=r0+Tcp′rcrp+Trcrsrr+rgmr×TicQsji,where rsrr is the position of the contact point in the roller body-fixed coordinate and Tar and Tai are, respectively, the transform from the azimuth to roller and inertial coordinate.According to geometric characteristics of the cage and guide ring, it can be considered as a finite journal bearing lubrication condition between the guide surface and cage centering surface. In Figure4, the clearance δ′ between the cage and guide ring is(43)δ′=Δe−e,where Δe is the initial clearance, Δe = (ry − rd)/2; rd, ry are, respectively, the radii of the centering surface and guide surface and e is cage displacement in the radial plane.Figure 4
Interaction between the cage and guide ring (inner-ring guided).Similarly, assume thatΔh0′ is the critical value of the grease film thickness. When δ′ ≥ Δh0′, the contact force qic and friction torque Nic on the cage are(44)qic=qicehl=ke′δ+ce′δ˙′,Nic=Nicehl=πεφ0bric1/2rd21/2h01/23ωbxry−ωcxrd,where ric is the equivalent radius and ωcx, ωbx are, respectively, the angular velocities of the cage and guide ring. The calculation of the grease film stiffness ke′ and damping ce′ is the same as that at the roller/crossbeam contact.Whenδ′ < Δh0′, the elastic deformation between the cage and guide ring is δh′ = δ′ − h0, and the minimum grease film thickness h0 = Δh0′. The qic and Nic are(45)qic=kh′δh′+ch′δh′+qicehl,Nic=μbdqicrd+Nicehl,where kh′ is the contact stiffness and ch′ is the viscous damping coefficient. The calculation of kh′ and ch′ is the same as that in Equation (35). μbd is the traction coefficient under boundary lubrication.In order to simplify the model, the load of the cage on the guide ring is ignored in contrast to large external load on the guide ring. Then, the loads acting on the cage are(46)Qici=0,qicsinφic,qiccosφicT,Nicc=±Nic,0,0.If the TRB is lubricated by the oil, the effect of oil-gas mixture on the cage and rollers may not be overlooked. However, if the bearing is lubricated by the grease, it seems impossible to have the grease-gas mixture effect on the dynamics of the cage and rollers. So, the effect of the grease-gas mixture is not considered in the dynamic model.


## 2.4. Dynamic Model

In real application also, the surroundings (inertia/damping of shaft assembly and housing) are expected to influence the TRB dynamics. These effects are neglected in the modeling. Generally, it is convenient to consider the translational moving of rollers in the cylindrical coordinate system, while the Cartesian coordinate system is convenient for the cage and race. The translational moving of bearing parts can be simply described by Newton’s laws such that(47)Mδ¨=Q.The generalized forceQ is obtained by superimposing the forces from Section 2. The centrifugal force and the gyroscopic moment should be superimposed on the generalized force Q for the roller. For the cage and race, the mass matrices Mb and Mc are(48)Mb,c=diagmb,c,mb,c,mb,c.For the special case of conical rollers,Mr is(49)Mr=diagmr,mrrrj,mr.Both rollers and the cage have six degrees-of-freedom. Unlike the translational motion (in three directions), the rotating of the bearing parts are described in their own body-fixed coordinates. For rollers and the cage, using the Euler dynamic equations such that(50)Ixω˙x−ωyωzIy−IzIyω˙y−ωzωxIz−IxIzω˙z−ωxωxIx−Iy=NxNyNz,where Ix, Iy, and Iz are inertia principal moments of the cage or rollers; the total moment N is also obtained by superimposing the forces from Section 2. ω[ωx, ωy, ωz] is the angular velocity in the inertial coordinate system. The relationship between the angular velocity ωfixed in the body-fixed coordinate system and ω is as follows. B is the Euler’s rotation matrix.(51)ωfixed=Bω,ω˙fixed=dBdtB−1ω˙.Admitting the outer ring as macroscopically stationary and a rotating inner ring with no torque load, it results four degrees of freedom of inner ring: moving in thex, y, and z directions and constant rotating about the shaft axis (o-x).


## 3. Numerical Method

The bearing geometry and material parameters and grease main rheological parameters (30°C) are shown in Table1. The cage is an inner-ring guided type, and its material is polyamide. The polyurea grease was adopted [26]. The fourth-order Runge–Kutta method was used to dissolve transient responses of the TRB on MATLAB. To ensure the convergence of simulation, initial values of displacement, velocity, and load are obtained by the quasidynamic method.Table 1
Geometry, material and rheological parameters.
Parameter
Value
Number of rollers
17
Roller lengthle/mm
42.06
Roller-end sphere radiusrs/mm
136.20
Diameter of roller large enddl/mm
17.85
Diameter of roller small endds/mm
24.76
Outer contact angle/deg
15
Inner contact angle/deg
5
Flange contact angle/deg
83
Outer raceway radius/mm
99.52
Inner raceway radius/mm
78.48
Guide land clearance/mm
0.5
Guide land width/mm
4.93
Cage outer diameter/mm
205.76
Cage inner diameter/mm
196.10
Cage thicknesshs/mm
5
Pocket clearance Δδ/mm
0.27
Elastic modulus of roller/race/GPa
205
Poisson’s ratio of roller/race
0.3
Elastic modulus of cage/GPa
8.3
Poisson’s ratio of cage
0.28
Grease rheological parameters (30°C)
Yield stressτy/Pa
351.8
Plastic viscosityφ/Pa·sn
8.44
Power law exponentn
0.7196
Pressure-viscosity coefficientα/m2·N−11
4.662 × 10−8


## 4. Results and Discussion

For the complicated dynamic response of TRBs, it is essential to consider several typical and harmful movements of the bearing in the process of dynamic analysis, such as the roller skew and tilt, bearing slip, and so on. The slip rate of the roller and cage is defined as(52)theoreticalspeed−actualspeedtheoreticalspeed×100%.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

The trajectories and velocities of mass centers of the main bearing parts are shown in Figure5. The axial preload is not applied to the inner ring but limited to the degree of freedom of the inner ring in the axial direction. The cage initial rotation speed is the same as the roller initial revolution speed. Due to the changing position of rollers, the distribution of rollers is symmetrical or asymmetric with respect to the o-z axis. Therefore, the inner ring vibrates slightly in the o-y direction. The radial and axial displacement (relative to the initial position) of the roller exhibit a periodic fluctuation, for the bearing is in a semicircle-loaded state under pure radial load conditions. When the rollers alternately enter the loaded and unloaded zones, the load on rollers will show periodic changes, resulting in a regular variation in displacement. The cage whirling is approximately a circle. For a number of uncertain rollers acting on the cage, the cage speed does not show a periodic change.Figure 5
The trajectory and velocity of mass center.Fr = 20 kN; Fa = 0; ωbx = 2.4 krpm.
(a)
(b)
(c)
(d)


### 4.2. Rotational Speed Influence

In this study, the tilt and skew of conical rollers are regarded as the primary evaluation variables for the dynamic performance of the TRB. The roller skew is mainly generated from the tangential friction force on the flange/roller-end and roller/raceway contacts, while the tilt motion is mainly induced by the normal contact force on those contacts. According to the lubrication theory, the bearing speed will be a key factor affecting the load case in bearing contacts.The influence of the bearing speed on the roller tilt and skew (relative to the initial position) is shown in Figure6. The degree of freedom of the inner ring in the axial direction is also limited. At each speed level, the roller tilt angle presents a relatively regular periodic vibration. The tilt angle in the unloaded zone is obviously larger than that in the loaded zone. This is due to the larger gap between the roller and raceways in the unloaded zone when only the radical load is applied, and rollers in unloaded zone seem to be “relaxed.” As the speed increases, the maximum of the tilt angle almost does not change, while the average tilt angle rises from −0.14 × 10−4 rad to −0.40 × 10−4 rad, indicating that the bearing speed affected the roller tilt. It can be confirmed that the gyroscopic moment is one of the reasons that cause the rising of the roller tilt angle, for the gyroscopic motion is intensified with the bearing speed increasing. Although the gyroscopic moment increases as the bearing speed increases, it is still relatively smaller than the force/torque in the roller/raceway contact. So, for the radial load is constant, the maximum tilt angle of the roller changes little with the speed.Figure 6
The tilt angle of the conical roller. 1: loaded zone; 0: unloaded zone.Fr = 20 kN; Fa = 0; ωbx = 2∼5 krpm.
(a)
(b)
(c)
(d)As it is shown in Figure7, the skew angle, both in the loaded zone and in the unloaded zone, is relatively large at the speed of 2 krpm. In the entire speed range, the roller skew angle in the loaded zone is greater than that in the unloaded zone. For the rollers is “relaxed” in the unloaded zone, the skew moment of the raceway and flange acting on the roller is smaller than that in the loaded zone. It shows obvious periodicity when the speed reaches 3 krpm or more. The average skew angle is increased from 3.06 × 10−3 rad/0.17 deg to 4.47 × 10−3 rad/0.25 deg. The increase of bearing rotating speed enlarges the centrifugal force of rollers, which makes contact forces at the roller/outer raceway and flange/roller-end increase. As a result of combined contribution of raceway friction and the flange friction, the roller skew angle is greatly increased. Compared with the experimental result measured by Falodi et al. in which they measured the average skew angle between 0.15 and 0.45 deg [27]. The average skew angle in Figure 7 is between 0.17 and 0.25 deg and at an order of magnitude compared to the experimental result. The changing trend of the skew angle with speed is also consistent with Yang’s results.Figure 7
The skew angle of the conical roller. 1: loaded zone; 0: unloaded zone.Fr = 20 kN; Fa = 0; ωbx = 2∼5 krpm.
(a)
(b)
(c)
(d)The bearing slip is a destructive motion that easily leads to scuffing, welding, and overwear of bearing surfaces. The bearing slip usually occurs at high-speed and light-load conditions. The influence of the bearing speed on the slip of the roller and cage is exhibited in Figures8 and 9.Figure 8
The slip rate of the conical roller.Fr = 20 kN; Fa = 0; ωbx = 2∼5 krpm.
(a)
(b)
(c)
(d)Figure 9
The slip rate of the cage.Fr = 20 kN; Fa = 0; ωbx = 2∼5 krpm.
(a)
(b)
(c)
(d)The diagrams depicted in Figure8 indicate that the roller slip is becoming more and more serious as the bearing speed is raised. This variation trend is similar to that in [7]. With the bearing speed increases, the roller slip gradually presents a periodic change rule, especially when the speed is up to 3 krpm or more. The roller slip in loaded and 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 serious slipping. In Figure 9, the cage slip is aggravated with the bearing speed increase. There is no periodic change in the cage slip rate. This is because when the cage is whirling, a number of uncertain rollers may act on the cage, which can lead to an irregular motion of the cage. At the same speed level, the maximum amplitude of the slip rate of the cage and roller is basically the same, for the dependent relationship between the cage and roller.


### 4.3. Axial Preload Influence

Axial preload is not only a key factor affecting the bearing dynamics, but one of the main means to ensure the normal service life of the bearing. In order to familiarize the influence of the axial preload on the roller tilt and skew, different axial preloads were applied to the TRB under the condition that the radial load is zero.As shown in Figures10 and 11, after entering the smooth running phase, as the axial preload varies from 10 μm to 40 μm, the average value of the roller tilt rises from −0.27 × 10−4 rad to −0.96 × 10−4 rad. The greater the preload, the more obvious the roller tends to tilt. This is because as the preload increases, the negative moment of the flange acting on the roller-end becomes larger, thus causing the roller tilt to assume a negative tendency.Figure 10
The tilt angle of the conical roller.Fr = 0; δa = 10∼40 μm; ωbx = 2.5 krpm.
(a)
(b)
(c)
(d)Figure 11
The skew angle of the conical roller.Fr = 0; δa = 10∼40 μm; ωbx = 2.5 krpm.
(a)
(b)
(c)
(d)Except for the condition that the preload is 10μm, the roller skew has larger amplitude at the beginning, and then the amplitude decreases gradually and tends to be stable. In the smooth running phase, as the preload increases, the roller skew angle increases. This trend is in line with the results in [8, 27]. The average skew angle is increased from 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 vibration throughout the bearing operation, this is due to symmetry of the bearing on the radial plane and pure preloading condition; the loaded and unloaded zones do not exist in the bearing, and loads on rollers almost change a little in any position.The changes of the slip rate of the roller and cage with the preload are presented in Figures12 and 13. As the bearing preload increases, the bearing slip decreases in the smooth running phase. The average slip rate of the roller decreases from 0.0189% to 0.0057% and that of the cage decreases from 0.0189% to 0.0055%, indicating that the appropriate bearing preload can effectively prevent bearing slipping failure. The result is similar with the conclusion in [7, 28]. The influence of the preload on the bearing slip reduced with the increase of the preload value. The average slip rate of the roller and cage when the preload is 40 μm changes little compared with 0.0073% and 0.0076% when the preload is 30 μm. Obviously, increasing the amount of preload will help to decrease the bearing slip, but excessive preload will inevitably reduce the service life of bearings. Therefore, to meet the bearing life and service requirements, reasonable selection of preload is significant.Figure 12
The slip rate of the conical roller.Fr = 0; δa = 10∼40 μm; ωbx = 2.5 krpm.
(a)
(b)
(c)
(d)Figure 13
The slip rate of the cage.Fr = 0; δa = 10∼40 μm; ωbx = 2.5 krpm.
(a)
(b)
(c)
(d)


### 4.4. Grease Physical Properties Influence

In view the fact that the calculation of the traction force, stiffness and damping at the bearing contacts is mainly based on the plastic viscosityφ and yield stress τy of the grease. For a clearer understanding of the relationship between the bearing dynamics and the grease used, grease physical properties influence on bearing dynamics is analyzed. The temperature, pressure, speed, and other surroundings have significant effects on the grease physical properties. For the sake of simplification, the impacts of these factors on the grease physical properties are ignored. In order to highlight the effect of a certain rheological parameter on the bearing performance, only one parameter was given an appropriate change while other rheological parameters remain unchanged. The effects are shown in Figures 14 and 15. After the bearing enters the stable running state (after 0.1 s), the average value of the angle (tilt and skew) is used as an indicator of bearing operating state.Figure 14
Plastic viscosity influence.Fr = 0; Fa = 15 kN; ωbx = 2.8 krpm. (a) The conical roller tilt and skew. (b) Slip rate of the conical roller and cage.
(a)
(b)Figure 15
Yield stress influence.Fr = 0; Fa = 15 kN; ωbx = 2.8 krpm. (a) The conical roller tilt and skew. (b) Slip rate of the conical roller and cage.
(a)
(b)The effect of the plastic viscosity on the roller and cage motions are shown in Figure14(a). As the plastic viscosity increases, the tilt angle and skew angle increases and decreases, respectively. For the grease film thickness increases with the increase of the plastic viscosity, the load ratio of microasperities at the flange/roller-end contact is reduced. Then, it indirectly leads to the decrease of the skew moment at the flange/roller-end contact and the roller skew angle. Due to the smaller skew angle, the roller misalignment will also be small. Therefore, the normal force at the roller/raceways and flange/roller-end contacts will have a greater contribution to the roller tilt. Therefore, the result shows that the roller tilt angle increases with the plastic viscosity increase. In Figure 14(b), the bearing slip decreases with the grease plastic viscosity increase. As the plastic viscosity increases, the grease film thickness at the roller/raceways and flange/roller-end contacts increase. So, the friction force at these contacts is reduced, which may lead to the decrease of the bearing slip. The bearing slip is the result of the combined effect of raceway friction and the flange friction.Compared with the impacts of the plastic viscosity of the grease on the motions of the roller and cage, the influence of the grease yield stress on the roller tilt, roller skew, and bearing slip is relatively smaller, as shown in Figure15. For the yield stress has a negligible effect on the film thickness in all practical cases [9].


### 4.5. Temperature Influence

Due to the improper installation, overwear or not timely cooling, the operating temperature of TRBs will change more or less. Sometimes, it may lead to heat accumulation or thermal failure of bearings. The most direct effect of the temperature is changes of the grease physical properties. The rheological parameters of the greases at different temperatures in Table2 are quoted from the data in reference. [26] The effect of temperature on the dynamics of the bearing was studied. The impact of temperature on the bearing structure is not considered here.Table 2
Relationships between grease physical properties and temperature [26].
Grease
Temperature/°C
Yield stressτy/Pa
Plastic viscosityφ/Pa·sn
Power law exponentn
Polyurea
15
501.7
11.87
0.7731
30
351.8
8.44
0.7196
50
305.5
6.16
0.6621
70
188.5
6.10
0.7378
Lithium
15
904.5
54.35
0.3053
30
805.2
46.23
0.3125
50
710.4
41.13
0.3151
70
339.8
52.96
0.3203
Calcium
15
990.8
64.82
0.3349
30
649.5
49.23
0.5344
50
336.1
30.59
0.5256
70
296.0
35.88
0.4356Three typical greases were used, such as the calcium grease, lithium grease, and polyurea grease. The largest difference in the composition of three greases lies in the types of thickeners. In Figure16(a), under the lubrication of the calcium and polyurea greases, the roller tilt angle decreases with the temperature increase. Under the lithium grease lubrication, the roller tilt angle increases with the increase 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 of greases decreases. That the tilt and skew angle under the different greases and temperature show different trends is a result of multiple factors. One of the factors is that the relationships between the grease rheological parameters and temperature are not simple linear. For example, the plastic viscosity and the power law exponent do not simply increase or decrease with temperature.Figure 16
Temperature influence.Fr = 0; Fa = 15 kN; ωbx = 2.8 krpm. (a) The conical roller tilt and skew. (b) Slip rate of the conical roller.
(a)
(b)In Figure16(b), firstly, the roller slip rate shows different values due to the different greases; secondly, the roller slip rate shows a variety of trends with the various temperature . Under the lubrication of the polyurea grease, the roller slip rate increases with the increase in temperature, while the roller slip with the calcium and lithium greases lubricated shows a decreasing trend. In real application, the operating temperature of bearings may not be constant. The influence of the temperature on dynamic characteristics of grease lubricated bearings needs to be further explored.


## 4.1. Mass Center Trajectory and Velocity

The trajectories and velocities of mass centers of the main bearing parts are shown in Figure5. The axial preload is not applied to the inner ring but limited to the degree of freedom of the inner ring in the axial direction. The cage initial rotation speed is the same as the roller initial revolution speed. Due to the changing position of rollers, the distribution of rollers is symmetrical or asymmetric with respect to the o-z axis. Therefore, the inner ring vibrates slightly in the o-y direction. The radial and axial displacement (relative to the initial position) of the roller exhibit a periodic fluctuation, for the bearing is in a semicircle-loaded state under pure radial load conditions. When the rollers alternately enter the loaded and unloaded zones, the load on rollers will show periodic changes, resulting in a regular variation in displacement. The cage whirling is approximately a circle. For a number of uncertain rollers acting on the cage, the cage speed does not show a periodic change.Figure 5
The trajectory and velocity of mass center.Fr = 20 kN; Fa = 0; ωbx = 2.4 krpm.
(a)
(b)
(c)
(d)


## 4.2. Rotational Speed Influence

In this study, the tilt and skew of conical rollers are regarded as the primary evaluation variables for the dynamic performance of the TRB. The roller skew is mainly generated from the tangential friction force on the flange/roller-end and roller/raceway contacts, while the tilt motion is mainly induced by the normal contact force on those contacts. According to the lubrication theory, the bearing speed will be a key factor affecting the load case in bearing contacts.The influence of the bearing speed on the roller tilt and skew (relative to the initial position) is shown in Figure6. The degree of freedom of the inner ring in the axial direction is also limited. At each speed level, the roller tilt angle presents a relatively regular periodic vibration. The tilt angle in the unloaded zone is obviously larger than that in the loaded zone. This is due to the larger gap between the roller and raceways in the unloaded zone when only the radical load is applied, and rollers in unloaded zone seem to be “relaxed.” As the speed increases, the maximum of the tilt angle almost does not change, while the average tilt angle rises from −0.14 × 10−4 rad to −0.40 × 10−4 rad, indicating that the bearing speed affected the roller tilt. It can be confirmed that the gyroscopic moment is one of the reasons that cause the rising of the roller tilt angle, for the gyroscopic motion is intensified with the bearing speed increasing. Although the gyroscopic moment increases as the bearing speed increases, it is still relatively smaller than the force/torque in the roller/raceway contact. So, for the radial load is constant, the maximum tilt angle of the roller changes little with the speed.Figure 6
The tilt angle of the conical roller. 1: loaded zone; 0: unloaded zone.Fr = 20 kN; Fa = 0; ωbx = 2∼5 krpm.
(a)
(b)
(c)
(d)As it is shown in Figure7, the skew angle, both in the loaded zone and in the unloaded zone, is relatively large at the speed of 2 krpm. In the entire speed range, the roller skew angle in the loaded zone is greater than that in the unloaded zone. For the rollers is “relaxed” in the unloaded zone, the skew moment of the raceway and flange acting on the roller is smaller than that in the loaded zone. It shows obvious periodicity when the speed reaches 3 krpm or more. The average skew angle is increased from 3.06 × 10−3 rad/0.17 deg to 4.47 × 10−3 rad/0.25 deg. The increase of bearing rotating speed enlarges the centrifugal force of rollers, which makes contact forces at the roller/outer raceway and flange/roller-end increase. As a result of combined contribution of raceway friction and the flange friction, the roller skew angle is greatly increased. Compared with the experimental result measured by Falodi et al. in which they measured the average skew angle between 0.15 and 0.45 deg [27]. The average skew angle in Figure 7 is between 0.17 and 0.25 deg and at an order of magnitude compared to the experimental result. The changing trend of the skew angle with speed is also consistent with Yang’s results.Figure 7
The skew angle of the conical roller. 1: loaded zone; 0: unloaded zone.Fr = 20 kN; Fa = 0; ωbx = 2∼5 krpm.
(a)
(b)
(c)
(d)The bearing slip is a destructive motion that easily leads to scuffing, welding, and overwear of bearing surfaces. The bearing slip usually occurs at high-speed and light-load conditions. The influence of the bearing speed on the slip of the roller and cage is exhibited in Figures8 and 9.Figure 8
The slip rate of the conical roller.Fr = 20 kN; Fa = 0; ωbx = 2∼5 krpm.
(a)
(b)
(c)
(d)Figure 9
The slip rate of the cage.Fr = 20 kN; Fa = 0; ωbx = 2∼5 krpm.
(a)
(b)
(c)
(d)The diagrams depicted in Figure8 indicate that the roller slip is becoming more and more serious as the bearing speed is raised. This variation trend is similar to that in [7]. With the bearing speed increases, the roller slip gradually presents a periodic change rule, especially when the speed is up to 3 krpm or more. The roller slip in loaded and 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 serious slipping. In Figure 9, the cage slip is aggravated with the bearing speed increase. There is no periodic change in the cage slip rate. This is because when the cage is whirling, a number of uncertain rollers may act on the cage, which can lead to an irregular motion of the cage. At the same speed level, the maximum amplitude of the slip rate of the cage and roller is basically the same, for the dependent relationship between the cage and roller.


## 4.3. Axial Preload Influence

Axial preload is not only a key factor affecting the bearing dynamics, but one of the main means to ensure the normal service life of the bearing. In order to familiarize the influence of the axial preload on the roller tilt and skew, different axial preloads were applied to the TRB under the condition that the radial load is zero.As shown in Figures10 and 11, after entering the smooth running phase, as the axial preload varies from 10 μm to 40 μm, the average value of the roller tilt rises from −0.27 × 10−4 rad to −0.96 × 10−4 rad. The greater the preload, the more obvious the roller tends to tilt. This is because as the preload increases, the negative moment of the flange acting on the roller-end becomes larger, thus causing the roller tilt to assume a negative tendency.Figure 10
The tilt angle of the conical roller.Fr = 0; δa = 10∼40 μm; ωbx = 2.5 krpm.
(a)
(b)
(c)
(d)Figure 11
The skew angle of the conical roller.Fr = 0; δa = 10∼40 μm; ωbx = 2.5 krpm.
(a)
(b)
(c)
(d)Except for the condition that the preload is 10μm, the roller skew has larger amplitude at the beginning, and then the amplitude decreases gradually and tends to be stable. In the smooth running phase, as the preload increases, the roller skew angle increases. This trend is in line with the results in [8, 27]. The average skew angle is increased from 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 vibration throughout the bearing operation, this is due to symmetry of the bearing on the radial plane and pure preloading condition; the loaded and unloaded zones do not exist in the bearing, and loads on rollers almost change a little in any position.The changes of the slip rate of the roller and cage with the preload are presented in Figures12 and 13. As the bearing preload increases, the bearing slip decreases in the smooth running phase. The average slip rate of the roller decreases from 0.0189% to 0.0057% and that of the cage decreases from 0.0189% to 0.0055%, indicating that the appropriate bearing preload can effectively prevent bearing slipping failure. The result is similar with the conclusion in [7, 28]. The influence of the preload on the bearing slip reduced with the increase of the preload value. The average slip rate of the roller and cage when the preload is 40 μm changes little compared with 0.0073% and 0.0076% when the preload is 30 μm. Obviously, increasing the amount of preload will help to decrease the bearing slip, but excessive preload will inevitably reduce the service life of bearings. Therefore, to meet the bearing life and service requirements, reasonable selection of preload is significant.Figure 12
The slip rate of the conical roller.Fr = 0; δa = 10∼40 μm; ωbx = 2.5 krpm.
(a)
(b)
(c)
(d)Figure 13
The slip rate of the cage.Fr = 0; δa = 10∼40 μm; ωbx = 2.5 krpm.
(a)
(b)
(c)
(d)


## 4.4. Grease Physical Properties Influence

In view the fact that the calculation of the traction force, stiffness and damping at the bearing contacts is mainly based on the plastic viscosityφ and yield stress τy of the grease. For a clearer understanding of the relationship between the bearing dynamics and the grease used, grease physical properties influence on bearing dynamics is analyzed. The temperature, pressure, speed, and other surroundings have significant effects on the grease physical properties. For the sake of simplification, the impacts of these factors on the grease physical properties are ignored. In order to highlight the effect of a certain rheological parameter on the bearing performance, only one parameter was given an appropriate change while other rheological parameters remain unchanged. The effects are shown in Figures 14 and 15. After the bearing enters the stable running state (after 0.1 s), the average value of the angle (tilt and skew) is used as an indicator of bearing operating state.Figure 14
Plastic viscosity influence.Fr = 0; Fa = 15 kN; ωbx = 2.8 krpm. (a) The conical roller tilt and skew. (b) Slip rate of the conical roller and cage.
(a)
(b)Figure 15
Yield stress influence.Fr = 0; Fa = 15 kN; ωbx = 2.8 krpm. (a) The conical roller tilt and skew. (b) Slip rate of the conical roller and cage.
(a)
(b)The effect of the plastic viscosity on the roller and cage motions are shown in Figure14(a). As the plastic viscosity increases, the tilt angle and skew angle increases and decreases, respectively. For the grease film thickness increases with the increase of the plastic viscosity, the load ratio of microasperities at the flange/roller-end contact is reduced. Then, it indirectly leads to the decrease of the skew moment at the flange/roller-end contact and the roller skew angle. Due to the smaller skew angle, the roller misalignment will also be small. Therefore, the normal force at the roller/raceways and flange/roller-end contacts will have a greater contribution to the roller tilt. Therefore, the result shows that the roller tilt angle increases with the plastic viscosity increase. In Figure 14(b), the bearing slip decreases with the grease plastic viscosity increase. As the plastic viscosity increases, the grease film thickness at the roller/raceways and flange/roller-end contacts increase. So, the friction force at these contacts is reduced, which may lead to the decrease of the bearing slip. The bearing slip is the result of the combined effect of raceway friction and the flange friction.Compared with the impacts of the plastic viscosity of the grease on the motions of the roller and cage, the influence of the grease yield stress on the roller tilt, roller skew, and bearing slip is relatively smaller, as shown in Figure15. For the yield stress has a negligible effect on the film thickness in all practical cases [9].


## 4.5. Temperature Influence

Due to the improper installation, overwear or not timely cooling, the operating temperature of TRBs will change more or less. Sometimes, it may lead to heat accumulation or thermal failure of bearings. The most direct effect of the temperature is changes of the grease physical properties. The rheological parameters of the greases at different temperatures in Table2 are quoted from the data in reference. [26] The effect of temperature on the dynamics of the bearing was studied. The impact of temperature on the bearing structure is not considered here.Table 2
Relationships between grease physical properties and temperature [26].
Grease
Temperature/°C
Yield stressτy/Pa
Plastic viscosityφ/Pa·sn
Power law exponentn
Polyurea
15
501.7
11.87
0.7731
30
351.8
8.44
0.7196
50
305.5
6.16
0.6621
70
188.5
6.10
0.7378
Lithium
15
904.5
54.35
0.3053
30
805.2
46.23
0.3125
50
710.4
41.13
0.3151
70
339.8
52.96
0.3203
Calcium
15
990.8
64.82
0.3349
30
649.5
49.23
0.5344
50
336.1
30.59
0.5256
70
296.0
35.88
0.4356Three typical greases were used, such as the calcium grease, lithium grease, and polyurea grease. The largest difference in the composition of three greases lies in the types of thickeners. In Figure16(a), under the lubrication of the calcium and polyurea greases, the roller tilt angle decreases with the temperature increase. Under the lithium grease lubrication, the roller tilt angle increases with the increase 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 of greases decreases. That the tilt and skew angle under the different greases and temperature show different trends is a result of multiple factors. One of the factors is that the relationships between the grease rheological parameters and temperature are not simple linear. For example, the plastic viscosity and the power law exponent do not simply increase or decrease with temperature.Figure 16
Temperature influence.Fr = 0; Fa = 15 kN; ωbx = 2.8 krpm. (a) The conical roller tilt and skew. (b) Slip rate of the conical roller.
(a)
(b)In Figure16(b), firstly, the roller slip rate shows different values due to the different greases; secondly, the roller slip rate shows a variety of trends with the various temperature . Under the lubrication of the polyurea grease, the roller slip rate increases with the increase in temperature, while the roller slip with the calcium and lithium greases lubricated shows a decreasing trend. In real application, the operating temperature of bearings may not be constant. The influence of the temperature on dynamic characteristics of grease lubricated bearings needs to be further explored.


## 5. Conclusions

For the beginning operating stages of grease-lubricated TRBs, the multibody contact dynamic model of TRBs was established. The impacts of preload, temperature, and grease rheological properties on the bearing dynamics were analyzed. Based on numerical results, several conclusions can be summarized:(1)
The roller tilt angle in the unloaded zone is obviously larger than that in the loaded zone, while the roller skew angle in the unloaded zone is smaller than that in the loaded zone. The effect of bearing speed on the roller skew is greater than that on the roller tilt. As the speed increases, the roller tilt and skew and the bearing slip become more serious.(2)
Bearing preload can effectively reduce the bearing slip but will make the roller tilt angle increase. In the stable operation stage of the bearing, as the preload increases, the roller skew angle increases.(3)
The roller skew angle and the bearing slip rate decrease with the increase of the grease plastic viscosity. The roller tilt angle increases with the increase of the plastic viscosity. The yield stress of the grease has little effect on motions of the roller and cage.(4)
The influence of temperature on motions of the roller and cage varies with the type of grease used. When the TRB is lubricated by the lithium grease, with the increase of the temperature, the roller tilt angle increases, the roller skew angle and the bearing slip rate decrease. When the polyurea grease is adopted, the change trend of the roller tilt and skew and the bearing slip is just opposite to that when the lithium grease is used. Under the calcium grease lubrication, the roller tilt and skew angle and the bearing slip rate decrease with the temperature increase.



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*Source: 7183042-2018-11-13.xml*