# Abnormal Wear Mechanism and Improvement of High-Speed Cylindrical Roller Bearing

Authors: Wenhu Zhang; Sier Deng; Song Zhang; Jian Wang; Jintao Zheng
Journal: Shock and Vibration
 (2021)

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## Abstract

This paper presents the dynamic differential equations of cylindrical roller bearing, considering the dynamic unbalance mass of roller and the contact between the roller end face and the inner ring rib. The influence of bearing working condition and structural parameters on thePV value of roller end face and the slip speed of roller working surface were analysed. The theoretical analysis and experimental results both show that the large amount of roller’s dynamic unbalance mass due to the difference in chamfering size on both sides of the roller and the deviation of the bearing axial clearance are the main causes of abnormal wear on the roller working surface and end face.


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## Body

## 1. Introduction

The primary failure modes of high-speed cylindrical roller bearing under light load are the skidding of roller and the fracture of cage [1, 2]. There is a lack of research on bearing failures caused by the serious wear of roller working surface and end faces, named “abnormal wear of roller” in the paper, which cannot provide a theoretical support for bearing fault location and optimization design.Yang et al. [3] analysed the wear of roller end face in high-speed cylindrical roller bearing and obtained the interconnection between the influencing factors and solved the wear of roller end face. Based on discrete mathematics and computer technology, Shi et al. [4, 5] established a numerical wear simulation model of cylindrical roller bearing, considering the time-varying characteristics of contact pressure and spin speed, to predict its wear life and reliability. Yao et al. [6] analysed the bearing with different degrees of wear failure through the whole simulation model of bearing, revealed the failure process of bearing and the characteristic value of failure fault, and carried out the experiment verification. The results showed that the concentration of the base vibration signal spectrum can reflect the degree of bearing wear and tear. Yu et al. [7] analysed the wear phenomenon of the roller end face and inner ring rib during the operation of an aero-engine cylindrical roller bearing and deemed that the main reason is the insufficient precision control of roller and inner ring rib. Accordingly, improvement for reducing the tolerance range of roller chamfers and increasing the angle of inner ring rib was proposed. Based on the working conditions and structural characteristics of a certain type of aero-engine intermediary bearing, Wen [8] determined that the main reason for the wear of the roller end face is the unbalanced torque caused by the high-speed operation of the roller, and the frequent contact between the roller and the inner ring rib and the cage pocket leads to the torsion pendulum of the roller back and forth. The axial clearance and radial clearance range of bearing are improved reasonably based on the geometric relation of cylindrical roller bearing. Aditya et al. [9] developed an experimental test rig to investigate the wear in a grease-lubricated cylindrical roller bearing and studied the influence of bearing temperature, misalignment, and roughness on the lubricant film thickness. Han et al. [10] presented a nonlinear dynamic model for skidding behaviour of cylindrical roller bearing, considering the radial clearance, roller crown profile, and discontinuous contact between the roller and cage to study the roller’s skidding behaviour under the time-variable load condition. Zhang et al. [11] investigated the skidding damage dynamic behaviours of cylindrical roller bearing using a high-speed rolling bearing test rig and revealed the skidding damage mechanism and the failure characteristics. Li et al. [12] proposed a new damage-mechanics-based model for rolling contact fatigue analysis in terms of the amplitude of octahedral shear stress, considering the nonproportional variation of stress, and the material parameters and simulated the crack initiation, crack propagation, and spalling in the working process of cylindrical roller bearing. Tu et al. [13] proposed a dynamic model to investigate the vibration of cylindrical roller bearing considering skidding, of which the generating mechanism is explained, and it is found that the existence of skidding may lead to fluctuations and a significant increase of the friction force, which will increase the vibration level and introduce impact components into the vibration. In addition, according to the analysis of the dynamic characteristics of cylindrical roller bearing, many scholars have also discussed many aspects: The dynamic characteristics of load distribution, oil film thickness, roller and cage slip, roller skew, and axial movement were calculated, and the nonlinear dynamic model of cylindrical roller bearing was established by Leblanc et al. [14], considering the structural deformation of the ring and cage material, the force and torque produced by the roller raceway contact were analysed, and the nonlinear dynamic model of high-speed cylindrical roller bearing was established by Wang et al. [15]. Yongcun et al. [16, 17] established a dynamics model of cylindrical roller bearing considering roller’s dynamic unbalance mass and studied the effects of roller’s dynamic unbalance mass on cage nonlinear dynamic characteristics, especially on cage vibration. Sun et al. [18,19] established the nonlinear dynamic differential equations of cylindrical roller bearings with an elastic ring and studied the effect of the elastic ring on the slip ratio and stability of cage. Liu et al. [20] developed a coupled model of roller tilt and skew, considering thermal effects and non-Newtonian characteristic lubricants, to study the effect of roller tilt and skew on thermal elastohydrodynamic lubrication. Zhang et al. [21] proposed the dynamic differential equations of high-speed cylindrical roller bearing, considering the effect of roller convexity offset on roller mass centre and the effect of convexity offset on the tilt angle and skew angle of the roller under four different methods and different speeds and loads were studied. Patra et al. [22] used the Lagrange equation to establish a dynamic model of cylindrical roller bearing by using the improved Newmark-β method to solve the numerical problem; the dynamic characteristics of cylindrical roller bearing in an equilibrium and unbalanced state are studied. Aiming at the atypical failure problem of cylindrical roller bearing in an aero-engine, Zheng et al. [23] established the dynamic differential equations of high-speed cylindrical roller bearing by taking into account the contact behaviour between the roller with dynamic unbalanced mass and inner ring rib. The influence of the bearing parameters on the maximum skew angle of roller and the maximum collision force between the roller and rib were analysed. The above studies mainly focused on the failure analysis, improved design, and bearing dynamic performance of cylindrical roller bearing; however, the wear mechanisms of roller end faces and end surface have not been studied. The solution to the abnormal wear of the roller working surface and end face cannot be put forward from the root.In this paper, a dynamic simulation analysis model of cylindrical roller bearing is carried out by considering the dynamic unbalance mass of roller and contact between the roller end face and inner ring rib. The wear degree of roller end face and roller working surface are characterized by thePV value and roller’s slip speed, respectively. This study investigates the key factors affecting the wear of roller end face and working surface and reveals the formation mechanism of abnormal wear. It provides a theoretical support for the troubleshooting and structural optimization design of cylindrical roller bearing under high speed and light load.


## 2. Characterization of Wear Failure of the Bearing

The wear of roller working surface and end faces after the bearing test is shown in Figure1. These failure modes are not common in the same type of bearing test and host application. Based on our experience, it is suspected that the difference in chamfering size on both sides of the roller and the deviation of bearing axial clearance are the main causes; however, the abnormal wear mechanism of roller and the optimization design of bearing are not clear. Bearing faults are difficult to solve.Figure 1
Examples of high-speed cylindrical roller bearing failures.
(a)(b)


## 3. Dynamic Model of High-Speed Cylindrical Roller Bearing

### 3.1. Coordinate System of the Cylindrical Roller Bearing

To accurately describe the motion state of each component of cylindrical roller bearing and the interaction relationship between the components, the coordinate systems of cylindrical roller bearing are shown in Figure2. The coordinate systems include the following:(1)
Inertial Coordinate SystemO;X,Y,Z. The origin O is located in the geometric centre of outer ring, the X-axis coincides with outer ring axis, and the YOZ plane is parallel to the radial plane of outer ring. The outer ring is fixed during the operation of bearing; that is, the inertial coordinate system is fixed in space.(2)
Inner Ring Centroid Coordinate Systemoi,xi,yi,zi. The origin oi coincides with inner ring centroid, the xi axis is parallel to the inertial coordinate system X-axis, the yioizi plane is parallel to the radial plane of inner ring, and this coordinate system moves and rotates with inner ring.(3)
Cage Centroid Coordinate Systemoc;xc,yc,zc. The origin oc coincides with the cage centroid, the xc axis is parallel to the inertial coordinate system X-axis, the ycoczc plane is parallel to the radial plane of cage, and this coordinate system moves and rotates with the cage.(4)
Cage Pocket Coordinate Systemopj;xpj,ypj,zpj. The origin opj coincides with the centre of the j pocket of cage, the xpj axis is parallel to the inertial coordinate system X-axis, the ypjopjzpj plane is parallel to the cage radial plane, the ypj axis passes through the cage centre, and this coordinate system moves and rotates with the cage.(5)
Roller Centre Coordinate Systemorj;xrj,yrj,zrj. The origin orj coincides with the jth roller geometric centre, the xrj axis is along the axis direction of the jth roller, the yrjorjzrj plane is parallel to the radial plane of the jth roller, and this coordinate system moves and rotates with the roller.(6)
Roller Reference Coordinate Systemorrj;xrrj,yrrj,zrrj. The origin orrj coincides with the jth roller geometric centre, the xrrj axis is parallel to the X-axis of the inertial coordinate system, the yrrjorrjzrrj plane is parallel to the YOZ plane of the inertial coordinate system, and this coordinate system moves and rotates with the roller.Figure 2
Coordinate systems of the cylindrical roller bearing.


### 3.2. Analysis on the Force between Roller End Face and Inner Ring Rib

The outer ring of cylindrical roller bearing has no rib, and the inner ring has double ribs. When the skew angle of roller is large, the contact between the roller end face and inner ring rib occurs [24], and the contact and geometric relationships are shown in Figure 3.Figure 3
Contact and geometric relationship between the roller end face and inner ring rib.From the geometric relationship [25], the elastic deformation between the roller end face and the unilateral rib of inner ring is given in the following equation:(1)δLK=1−cosαr⋅Rp−Lp2−Ca2+Caf+XrK⋅cosθf,where the calculation corresponds to the left side when K is equal to L and to the right side when K is equal to R; αr is the skew angle of roller; Ca is bearing axial clearance; Rp is the distance between the contact point and the crowned centre of roller end face in the xrj direction.(2)Rp=Rs⋅cosθf.Equation (3) gives the length of the roller Lp in the plane, parallel to the xrjorjzrj surface where the contact point is located.(3)Lp=Lr−2Rs−Rp,where Rs is the crowned radius of roller end face, Lr is the length of roller, and Caf is the unilateral axial clearance caused by the angle of rib θf.(4)Caf=Hrollercm−Dw2⋅tanθf,where Hrollercm is the distance between the centre of roller and the raceway of inner ring and Dw is the diameter of roller.XrK is the gap increment caused by the axial displacement of roller.(5)XrL=−Xr,XrR=Xr,where Xr is the displacement of roller relative to the mass centre of inner ring in the X-axis direction of inertial coordinate system.According to the Hertz point contact theory, the normal contacts force between ribs and roller end faces is determined [26, 27].(6)QezK=πE′k32ReffΠδLKΓ31/2,where E′ is the comprehensive elastic modulus of materials for the roller and inner ring and k is the ratio of the semimajor axis to the semiminor axis of contact ellipse, given by the following equation:(7)k=1.0339RyRx0.636.Reff is the equivalent radius of curvature of the contact point.(8)Reff=11/Rx+1/Ry.Γ and Π are the first kind of complete elliptic integral and the second kind of complete elliptic integral, respectively.(9)Γ=1.0003+0.5968Ry/Rx,Π=1.5277+0.6023lnRy/Rx,where Rx and Ry are the equivalent radii of curvature in the two orthogonal planes where the contact point is located.(10)Rx=11/Rs+1/di/2+Hcontact/tanθf,Ry=Rs,RyRx≥1,where Hcontact is the height of the contact point between roller end face and rib.(11)Hcontact=Hrollercm−Rs⋅sinθf+0.5Lr⋅sinαe,where αe is the tilt angle of roller.The friction force between roller end face and one rib is given in the following equation:(12)QeθK=C⋅FG+μQezK,where μ is the friction coefficient between the roller end face and inner ring rib and C is the direction of μ, which is determined by the following equation:(13)C=+1,vi≥vr,C=−1,vi<vr,where vi and vr are the linear velocities of rib and roller at the contact point, respectively.FG is the friction force generated by the lubricant on the roller end face.(14)FG=ωi−ω0jυρe∫RCadA,where ωi is the angular velocity of inner ring, ω0j is the orbital angular velocity of the jth roller, υ is the kinematic viscosity of lubricant, and ρe is the oil/gas density.(15)∫RCadA=43R0CahfDw22−Dw22,where R0 is the radius of the contact point and hf is the height of rib. In summation, the overall forces and torques acting on the jth roller’s end faces by double ribs of inner ring are determined.(16)QezX=QezL−QezRcosθf,QezY=QezL+QezRsinθf,Qeθ=QeθL+QeθR,Meθx=QeθL+QeθR⋅Hrollercm−Hcontact−2MG,Meθ=QeθL−QeθR⋅Lp2+Ca2,Mer=QezL+QezRcosθf⋅Rp−Lp2sinαr,Mee=QezL+QezRsinθf⋅Lp2sinαe,where QezX and QezY are the components along the xrrj and yrrj directions in orrj;xrrj,yrrj,zrrj, respectively; Qeθ is the friction force acting on roller end face; Meθx andMeθ are the additional moments generated by Qeθ; Mer and Mee are the additional moments generated by QezX and QezY; MG is the moment generated by the lubricant acting on roller end face.(17)MG=FG⋅Dw2−hf2.


## 3.1. Coordinate System of the Cylindrical Roller Bearing

To accurately describe the motion state of each component of cylindrical roller bearing and the interaction relationship between the components, the coordinate systems of cylindrical roller bearing are shown in Figure2. The coordinate systems include the following:(1)
Inertial Coordinate SystemO;X,Y,Z. The origin O is located in the geometric centre of outer ring, the X-axis coincides with outer ring axis, and the YOZ plane is parallel to the radial plane of outer ring. The outer ring is fixed during the operation of bearing; that is, the inertial coordinate system is fixed in space.(2)
Inner Ring Centroid Coordinate Systemoi,xi,yi,zi. The origin oi coincides with inner ring centroid, the xi axis is parallel to the inertial coordinate system X-axis, the yioizi plane is parallel to the radial plane of inner ring, and this coordinate system moves and rotates with inner ring.(3)
Cage Centroid Coordinate Systemoc;xc,yc,zc. The origin oc coincides with the cage centroid, the xc axis is parallel to the inertial coordinate system X-axis, the ycoczc plane is parallel to the radial plane of cage, and this coordinate system moves and rotates with the cage.(4)
Cage Pocket Coordinate Systemopj;xpj,ypj,zpj. The origin opj coincides with the centre of the j pocket of cage, the xpj axis is parallel to the inertial coordinate system X-axis, the ypjopjzpj plane is parallel to the cage radial plane, the ypj axis passes through the cage centre, and this coordinate system moves and rotates with the cage.(5)
Roller Centre Coordinate Systemorj;xrj,yrj,zrj. The origin orj coincides with the jth roller geometric centre, the xrj axis is along the axis direction of the jth roller, the yrjorjzrj plane is parallel to the radial plane of the jth roller, and this coordinate system moves and rotates with the roller.(6)
Roller Reference Coordinate Systemorrj;xrrj,yrrj,zrrj. The origin orrj coincides with the jth roller geometric centre, the xrrj axis is parallel to the X-axis of the inertial coordinate system, the yrrjorrjzrrj plane is parallel to the YOZ plane of the inertial coordinate system, and this coordinate system moves and rotates with the roller.Figure 2
Coordinate systems of the cylindrical roller bearing.


## 3.2. Analysis on the Force between Roller End Face and Inner Ring Rib

The outer ring of cylindrical roller bearing has no rib, and the inner ring has double ribs. When the skew angle of roller is large, the contact between the roller end face and inner ring rib occurs [24], and the contact and geometric relationships are shown in Figure 3.Figure 3
Contact and geometric relationship between the roller end face and inner ring rib.From the geometric relationship [25], the elastic deformation between the roller end face and the unilateral rib of inner ring is given in the following equation:(1)δLK=1−cosαr⋅Rp−Lp2−Ca2+Caf+XrK⋅cosθf,where the calculation corresponds to the left side when K is equal to L and to the right side when K is equal to R; αr is the skew angle of roller; Ca is bearing axial clearance; Rp is the distance between the contact point and the crowned centre of roller end face in the xrj direction.(2)Rp=Rs⋅cosθf.Equation (3) gives the length of the roller Lp in the plane, parallel to the xrjorjzrj surface where the contact point is located.(3)Lp=Lr−2Rs−Rp,where Rs is the crowned radius of roller end face, Lr is the length of roller, and Caf is the unilateral axial clearance caused by the angle of rib θf.(4)Caf=Hrollercm−Dw2⋅tanθf,where Hrollercm is the distance between the centre of roller and the raceway of inner ring and Dw is the diameter of roller.XrK is the gap increment caused by the axial displacement of roller.(5)XrL=−Xr,XrR=Xr,where Xr is the displacement of roller relative to the mass centre of inner ring in the X-axis direction of inertial coordinate system.According to the Hertz point contact theory, the normal contacts force between ribs and roller end faces is determined [26, 27].(6)QezK=πE′k32ReffΠδLKΓ31/2,where E′ is the comprehensive elastic modulus of materials for the roller and inner ring and k is the ratio of the semimajor axis to the semiminor axis of contact ellipse, given by the following equation:(7)k=1.0339RyRx0.636.Reff is the equivalent radius of curvature of the contact point.(8)Reff=11/Rx+1/Ry.Γ and Π are the first kind of complete elliptic integral and the second kind of complete elliptic integral, respectively.(9)Γ=1.0003+0.5968Ry/Rx,Π=1.5277+0.6023lnRy/Rx,where Rx and Ry are the equivalent radii of curvature in the two orthogonal planes where the contact point is located.(10)Rx=11/Rs+1/di/2+Hcontact/tanθf,Ry=Rs,RyRx≥1,where Hcontact is the height of the contact point between roller end face and rib.(11)Hcontact=Hrollercm−Rs⋅sinθf+0.5Lr⋅sinαe,where αe is the tilt angle of roller.The friction force between roller end face and one rib is given in the following equation:(12)QeθK=C⋅FG+μQezK,where μ is the friction coefficient between the roller end face and inner ring rib and C is the direction of μ, which is determined by the following equation:(13)C=+1,vi≥vr,C=−1,vi<vr,where vi and vr are the linear velocities of rib and roller at the contact point, respectively.FG is the friction force generated by the lubricant on the roller end face.(14)FG=ωi−ω0jυρe∫RCadA,where ωi is the angular velocity of inner ring, ω0j is the orbital angular velocity of the jth roller, υ is the kinematic viscosity of lubricant, and ρe is the oil/gas density.(15)∫RCadA=43R0CahfDw22−Dw22,where R0 is the radius of the contact point and hf is the height of rib. In summation, the overall forces and torques acting on the jth roller’s end faces by double ribs of inner ring are determined.(16)QezX=QezL−QezRcosθf,QezY=QezL+QezRsinθf,Qeθ=QeθL+QeθR,Meθx=QeθL+QeθR⋅Hrollercm−Hcontact−2MG,Meθ=QeθL−QeθR⋅Lp2+Ca2,Mer=QezL+QezRcosθf⋅Rp−Lp2sinαr,Mee=QezL+QezRsinθf⋅Lp2sinαe,where QezX and QezY are the components along the xrrj and yrrj directions in orrj;xrrj,yrrj,zrrj, respectively; Qeθ is the friction force acting on roller end face; Meθx andMeθ are the additional moments generated by Qeθ; Mer and Mee are the additional moments generated by QezX and QezY; MG is the moment generated by the lubricant acting on roller end face.(17)MG=FG⋅Dw2−hf2.


## 4. Dynamic Differential Equations of the Roller

The outer ring is fixed, while the inner ring rotates at a constant speed. The cage is guided by the inner ring, and the inner ring only bears a pure radial load during the operation. When bearing runs at a high speed, the dynamic unbalance mass of roller causes it to bear the additional dynamic load, which causes roller skew and tilt. As shown in Figure4, suppose that the dynamic unbalance mass of roller Omj is located at an angle θmj in the axial direction xrj and an angle φmj in the radial direction yrj, the distance between Omj and orj is emj, Fcvj and Fcwj are the centrifugal forces generated by the jth roller’s dynamic unbalance mass, and the calculated expressions can be found in the literature [16, 17].Figure 4
Schematic diagram of thejth roller’s dynamic unbalance mass.
(a)(b)During the operation of bearing, the roller is acted by the ring, cage, and lubricant. As shown in Figure5, the forces and torques between roller end face and inner ring rib are obtained by equation (17). Nrji and Nrjo are the normal contact forces between the inner and outer raceways and the jth roller; Trji and Trjo are the drag forces between the inner and outer raceways and the jth roller. MNrji and MNrjo are the additional moments between the inner and outer raceways and the jth roller generated by Nrji and Nrjo; MTrji and MTrjo are the additional moments between the inner and outer raceways and the jth roller generated by Trji and Trjo; Nrjc and Frjc are the normal contact force and tangential friction force between the cage beam and the jth roller; MNrjc and MFrjc are the additional moments between the jth roller and the cage beam generated by Nrjc andFrjc; Frjm is the jth roller centrifugal force; Nrjmi and Nrjmo are the contact forces between the inner and outer raceways and the mth slice; Trjmi and Trjmo are the drag forces between the inner and outer raceways and the mth slice; Nrjmc and Frjmc are the normal contact force and tangential friction force between the cage cross beam and the mth slice; and αrj and αej are the skew angle and tilt angle of the jth roller. The expressions of the other symbols are in the literature [23].Figure 5
Schematic diagram of forces acting on thejth roller.
(a)(b)The nonlinear dynamic differential equations of thejth roller are(18)mrjx¨rj=QezjX−Fcwxj+μsjiNrji+μsjoNrjo,mrjy¨rj=−Nrjicosφj+Nrjocosφj−Nrjcsinφj−Trjisinφj+Trjosinφj−Frjccosφj−Frjmcosφj−QezjYcosφj−Qeθjsinφj+Fcwyjcosφj+Fcwzjsinφj−Fcvjcosφj,mrjz¨rj=Nrjisinφj−Nrjosinφj−Nrjccosφj−Trjicosφj+Trjocosφj+Frjcsinφj+Frjmsinφj+QezjYsinφj−Qeθjcosφj−Fcwyjsinφj+Fcwzjcosφj+Fcvjsinφj,Jrxjω˙rxj=−TrjiDw2−TrjoDw2+FrjcDw2−Meθxj+ΔMxmj,Jryjω˙ryj=MNrjisinφj+MNrjosinφj−MTrjisinφj−MTrjosinφj+MFrjccosφj+Merjcosφj+Meejsinφj−Meθjcosφj−ΔMymjcosφj+ΔMzmjsinφj,rzjω˙rzj=MNrjicosφj+MNrjocosφj−MTrjicosφj−MTrjocosφj−MFrjcsinφj−Merjsinφj+Meejcosφj+Meθjsinφj+ΔMymjsinφj+ΔMzmjcosφj,where mrj is the jth roller mass; φj is the azimuth angle of the jth roller; y¨rj and z¨rj are the jth roller’s mass centres in the inertial coordinate system of displacement acceleration; Jrxj, Jryj, and Jrzj are the rotational inertias of the jth roller in the inertial coordinate system; ω˙rxj, μsji, and μsjo are the angular accelerations of the jth roller in the inertial coordinate system; μsji and μsjo are the transverse friction coefficients between the jth roller and the raceway; Fcwxj, Fcwyj, and Fcwzj are the components of xrj along xrj, yrj, and zrj directions in the roller reference coordinate system; and ΔMmxj, ΔMmyj, and ΔMmzj are the components of additional moments in the inertial coordinate system due to Fcvj and Fcwj.


## 5. Dynamic Differential Equations of the Cage

The cage is acted by roller’s collision and friction, gravity, guiding surface’s collision and friction, and the dragging of lubricant acting on the nonguiding surface and end faces, as shown in Figure6.Figure 6
Schematic diagram of cage forces.In Figure6, O;Y,Z is the inertial coordinate system; oc;yc,zc is the cage centre-of-mass coordinate system; e is the offset of the cage centre of mass relative to the centre of mass of the inner ring, where e=Δcy2+Δcz2; Ψc′ is the eccentricity of the force between the guiding surface and the overall coordinate system of the cage; when the inner ring guides the cage, the eccentricity is Ψc, where Ψc=Ψc′+π; Fcy and Fcz′ are the two orthogonal components of the force between the guiding surface of the inner ring and the cage cantering surface because of the fluid dynamic pressure effect, respectively; Mcx′ is the frictional moment generated on the cage surface by the distributed pressure of the lubricant fluid dynamic pressure film; MCDO and MCDS are the resistances of the outer surface and end face of the cage caused by the shear of the surrounding air-oil mist mixture; their expressions for the forces are given in the literature [26]. The nonlinear dynamic differential equations of the cage are(19)mcy¨c=∑jRNNrjcsinφj+Frjccosφj+Fcy′cosψc+Fcz′sinψc−Gc,mcz¨c=∑jRNNrjccosφj−Frjcsinφj+Fcy′sinψc−Fcz′cosψc,Jcxω˙cx=∑jRNFrjcDw2−Mcx′−MCDO−MCDS,Jcyω˙cy=∑jRN−Mrjccosφj,Jczω˙cz=∑jRNMrjcsinφj,where mc is the mass of cage; y¨c and z¨c are the displacement accelerations of the cage’s mass centre in the inertial coordinate system; Jcx, Jcy, and Jcz are the rotational inertias of cage in the inertial coordinate system; ω˙cx, ω˙cy, and ω˙cz are the angular accelerations of the cage in the inertial coordinate system; Gc is the gravity of cage; and RNis the number of rollers.


## 6. Dynamic Differential Equations of the Inner Ring

In addition to the normal force, friction force, and torque of the internal elements, the inner ring is subjected to an external radial load. The nonlinear dynamic differential equations of the inner ring are(20)mix¨i=∑jRNQezjX−Fcwxj+μsjiNrji,miy¨i=∑jRNNrjicosφj+Trjisinφj+QezjYcosφj+Qeθjsinφj−Fr−Gi,miz¨i=∑jRN−Nrjisinφj+Trjicosφj−QezjYsinφj+Qeθjcosφj,Jixω˙ix=∑jRNTrjiDw2+Mezj,Jiyω˙iy=∑jRNMTrjisinφj−Merjcosφj−Meejsinφj+Meθjcosφj,Jizω˙iz=∑jRNMTrjicosφj+Merjsinφj−Meejcosφj−Meθjsinφj,where mi is the mass of the inner ring; y¨i and z¨i are the displacement accelerations of the inner ring mass centre in the inertial coordinate system; Jix,Jiy, and Jiz are the rotational inertias of the inner ring in the inertial coordinate system; ω˙ix, ω˙iy, and ω˙iz are the angular accelerations of the inner ring in the inertial coordinate system; Fr is the radial load; and Gi is the gravity of inner ring.


## 7. Calculation of thePV Value and Slip Speed of the Roller

### 7.1. Calculation of thePV Value of the Roller End Face

If the roller does not exhibit the dynamic unbalance mass, its skew angle in the operation process is very small, and the friction between roller end face and inner ring rib is not considered to occur. However, when the roller exhibits a certain amount of dynamic unbalance mass, its oscillation in the raceway causes a significant contact between the roller end face and inner ring rib. According to the Archard wear model [28], the wear rate of the material is proportional to the contact pressure p and sliding velocity V. Therefore, the PV value between the roller end face and inner ring rib is used to evaluate the wear degree of roller end face. According to the contact geometry and relative motion relationship between roller end face and inner ring rib, the PV value of the jth roller end face can be obtained as follows:(21)PVerj=PrjVerj′,where Prj is the contact stress for the jth roller end face and inner ring rib, and it can be determined using the following equation:(22)Prj=3Qez2πab,where a and b are the long semiaxis and the short semiaxis of contact ellipse, respectively:(23)a=6k2ΓReffQezπE′1/3,b=6ΓReffQezπkE′1/3.Verj′ is the relative speed of contact point between the jth roller end face and inner ring rib:(24)Vrj′=vi−vr.


### 7.2. Calculation of Slip Speed of the Roller Working Surface

When the roller has a dynamic unbalance mass, the roller oscillates significantly, resulting in a large relative sliding between its working surface and the raceway, intensifying the frictional wear of the roller. However, the additional contact stress between the roller and raceway is relatively small due to the roller’s dynamic unbalance mass; therefore, the slip speed is used to evaluate the wear degree of roller’s working surface, and it can be expressed as(25)Vrj=absωix×di2−ωrxj×Rro−ωrsxj×Dw2,where Vrj is the jth roller’s slip speed; ωix and ωrxj are the angular velocities of the inner ring and the jth roller in the inertial coordinate system, respectively; ωrsxj is the spin angular velocity of the jth roller; Rro is the distance between the jth roller’s centre and the inner ring’s centre along the radial direction in the inertial coordinate system; and di is the inner ring raceway diameter.


## 7.1. Calculation of thePV Value of the Roller End Face

If the roller does not exhibit the dynamic unbalance mass, its skew angle in the operation process is very small, and the friction between roller end face and inner ring rib is not considered to occur. However, when the roller exhibits a certain amount of dynamic unbalance mass, its oscillation in the raceway causes a significant contact between the roller end face and inner ring rib. According to the Archard wear model [28], the wear rate of the material is proportional to the contact pressure p and sliding velocity V. Therefore, the PV value between the roller end face and inner ring rib is used to evaluate the wear degree of roller end face. According to the contact geometry and relative motion relationship between roller end face and inner ring rib, the PV value of the jth roller end face can be obtained as follows:(21)PVerj=PrjVerj′,where Prj is the contact stress for the jth roller end face and inner ring rib, and it can be determined using the following equation:(22)Prj=3Qez2πab,where a and b are the long semiaxis and the short semiaxis of contact ellipse, respectively:(23)a=6k2ΓReffQezπE′1/3,b=6ΓReffQezπkE′1/3.Verj′ is the relative speed of contact point between the jth roller end face and inner ring rib:(24)Vrj′=vi−vr.


## 7.2. Calculation of Slip Speed of the Roller Working Surface

When the roller has a dynamic unbalance mass, the roller oscillates significantly, resulting in a large relative sliding between its working surface and the raceway, intensifying the frictional wear of the roller. However, the additional contact stress between the roller and raceway is relatively small due to the roller’s dynamic unbalance mass; therefore, the slip speed is used to evaluate the wear degree of roller’s working surface, and it can be expressed as(25)Vrj=absωix×di2−ωrxj×Rro−ωrsxj×Dw2,where Vrj is the jth roller’s slip speed; ωix and ωrxj are the angular velocities of the inner ring and the jth roller in the inertial coordinate system, respectively; ωrsxj is the spin angular velocity of the jth roller; Rro is the distance between the jth roller’s centre and the inner ring’s centre along the radial direction in the inertial coordinate system; and di is the inner ring raceway diameter.


## 8. Interpretation of Results

The dynamic differential equations (19)–(21) of cylindrical roller bearing are solved by GSTIFF (Gear Stiff) integer algorithm with variable step [29]. The PV value and the slip speed were obtained. The main parameters of cylindrical roller bearing are listed in Table 1.Table 1
Main parameters of bearing.
ParameterParameter valuesBearing inner diameter (mm)50Bearing outer diameter (mm)75Bearing width (mm)16Roller diameter (mm)6Cage guide methodInner ring guideDue to the influence of machining factors, the size of roller’s chamfer is the most likely reason to cause the dynamic unbalance mass. Therefore, the dynamic unbalance mass of the roller was placed in this position of the roller’s chamfer, and the dynamic unbalance mass was analysed in the range of 0.06–0.11 g. It was assumed that the dynamic unbalance mass is located at the right chamfer position of the roller.


## 9. Effect of Bearing Condition on thePV Value and Slip Speed

Assuming that the axial clearance of bearing was 0.03 mm, the circumferential clearance of cage pocket was 0.2 mm, the angle of the inner ring rib was 0.3°, and the crowned radius of roller end face was 300 mm. The following bearing operating parameters were varied in order to study their effect on thePV value and its slip speed under the different dynamic unbalance mass.


### 9.1. Influence of Radial Load on thePV Value and Slip Speed

The effect of radial load on thePV value and slip speed was analysed. For the inner ring speed of 30000 r/min, the PV value and its slip speed varying with the radial load are shown in Figure 7, where L and R represent the PV values of the left end face and the right end face of the roller, respectively.Figure 7
Variation ofPV values and its slip speed with the radial load. (a) Change of PV value. (b) Change of roller slip speed.
(a)(b)Figure7 indicates that the PV value and slip speed increase with the increase in roller’s dynamic unbalance mass, and the PV value of the right end face is larger. This is because of the increase in centrifugal force, the increase in roller skew angle, the increase in roller swing speed, and aggravation of roller slip. In addition, the increase in the skew angle of the roller increases the force between roller end face and the inner ring’s rib increase. This caused the PV value to increase with an increase in the roller’s dynamic unbalance mass. Because the roller is subjected to the unilateral centrifugal force caused by the dynamic unbalanced mass, the PV value of the roller end face where the dynamic unbalanced mass is located is larger.When the dynamic unbalance mass of the roller is eliminated, thePV value of the roller end face does not change noticeably with the increase in radial load, and the slip speed of the roller decreases with the increase in radial load and tends to be moderate when the load exceeds 5000 N. This is because the bearing with a pure radial load has both a bearing area and a nonbearing area during operation. In the nonbearing zone, the correlation between the running state of the roller itself and the radial load is small, and the limit of the roller swing is weak; that is, the change in the PV value is not clear. In the bearing area, the larger the radial load is, the greater the force the ring is subjected to and the stronger the drag force of the inner ring on the roller is. This leads to a higher cage speed, that is, a higher roller rotation speed and a relative decrease in the roller’s slip speed. After the radial load exceeded 5000 N, the actual speed of the roller was close to the theoretical value, and the roller’s slip speed tended to be stable.


### 9.2. Effect of the Inner Ring Speed on Roller EndPV and Slip Speed

For the radial load of 3000 N, thePV value and its slip speed varying with the bearing speed are shown in Figure 8.Figure 8
Variation ofPV value and its slip speed with the inner ring speed. (a) Change of the PV value. (b) Change of the roller’s slip speed.
(a)(b)From Figure8, when the roller’s dynamic unbalance mass is certain, the PV value and the slip speed increase with a rise in the inner ring speed. With an increase in the inner ring speed, the rotational speed of roller increases, the centrifugal force produced by the dynamic unbalance mass rises, the degree of roller skew increases, the force between the roller end face and the inner ring rib will increase, and the roller slip speed will be faster. Therefore, the PV value increases with the inner ring speed and the dynamic unbalance mass, and the impact is more significant. In contrast, with an increase in the inner ring speed and centrifugal force, the drag ability of the inner ring to the roller is weakened, the roller swing ability is enhanced, the contact force and friction force between the roller end face and inner ring rib increases, and the slip speed of the roller increases.


## 9.1. Influence of Radial Load on thePV Value and Slip Speed

The effect of radial load on thePV value and slip speed was analysed. For the inner ring speed of 30000 r/min, the PV value and its slip speed varying with the radial load are shown in Figure 7, where L and R represent the PV values of the left end face and the right end face of the roller, respectively.Figure 7
Variation ofPV values and its slip speed with the radial load. (a) Change of PV value. (b) Change of roller slip speed.
(a)(b)Figure7 indicates that the PV value and slip speed increase with the increase in roller’s dynamic unbalance mass, and the PV value of the right end face is larger. This is because of the increase in centrifugal force, the increase in roller skew angle, the increase in roller swing speed, and aggravation of roller slip. In addition, the increase in the skew angle of the roller increases the force between roller end face and the inner ring’s rib increase. This caused the PV value to increase with an increase in the roller’s dynamic unbalance mass. Because the roller is subjected to the unilateral centrifugal force caused by the dynamic unbalanced mass, the PV value of the roller end face where the dynamic unbalanced mass is located is larger.When the dynamic unbalance mass of the roller is eliminated, thePV value of the roller end face does not change noticeably with the increase in radial load, and the slip speed of the roller decreases with the increase in radial load and tends to be moderate when the load exceeds 5000 N. This is because the bearing with a pure radial load has both a bearing area and a nonbearing area during operation. In the nonbearing zone, the correlation between the running state of the roller itself and the radial load is small, and the limit of the roller swing is weak; that is, the change in the PV value is not clear. In the bearing area, the larger the radial load is, the greater the force the ring is subjected to and the stronger the drag force of the inner ring on the roller is. This leads to a higher cage speed, that is, a higher roller rotation speed and a relative decrease in the roller’s slip speed. After the radial load exceeded 5000 N, the actual speed of the roller was close to the theoretical value, and the roller’s slip speed tended to be stable.


## 9.2. Effect of the Inner Ring Speed on Roller EndPV and Slip Speed

For the radial load of 3000 N, thePV value and its slip speed varying with the bearing speed are shown in Figure 8.Figure 8
Variation ofPV value and its slip speed with the inner ring speed. (a) Change of the PV value. (b) Change of the roller’s slip speed.
(a)(b)From Figure8, when the roller’s dynamic unbalance mass is certain, the PV value and the slip speed increase with a rise in the inner ring speed. With an increase in the inner ring speed, the rotational speed of roller increases, the centrifugal force produced by the dynamic unbalance mass rises, the degree of roller skew increases, the force between the roller end face and the inner ring rib will increase, and the roller slip speed will be faster. Therefore, the PV value increases with the inner ring speed and the dynamic unbalance mass, and the impact is more significant. In contrast, with an increase in the inner ring speed and centrifugal force, the drag ability of the inner ring to the roller is weakened, the roller swing ability is enhanced, the contact force and friction force between the roller end face and inner ring rib increases, and the slip speed of the roller increases.


## 10. Effect of Bearing Structure Parameters on thePV Value and Slip Speed

When the inner ring speed was 30000 r/min, the radial load was 3000 N, and the influence of the bearing structure parameters on thePV value and slip speed under the different roller’s dynamic unbalance mass was analysed.


### 10.1. Effect of Axial Clearance of Bearing on thePV Value and Slip Speed

For the cage pocket circumferential clearance of 0.2 mm, the angle of inner ring rib of 0.3°, and the crowned radius of roller end face of 300 mm, thePV value and its slip speed changing with bearing axial clearance are shown in Figure 9.Figure 9
Variation ofPV value and its slip speed with the axial clearance. (a) Change of the PV value. (b) Change of the roller slip speed.
(a)(b)Figure9 shows that when the roller’s dynamic unbalance mass is certain, the PV value increases with the increase of the axial clearance, and the roller’s slip speed decreases with the increase of the axial clearance, both of which tend to flatten after the axial clearance exceeds 0.05 mm. This was because when the axial clearance is small, the roller contact with the inner ring rib greatly inhibits the skew of the roller and reduces the speed of the roller swing. As a result, the PV value was reduced, but this inhibition will weaken the roller's normal rolling ability and increase the roller’s slip speed. When the clearance exceeds 0.05 mm, the inhibition of the roller skew by the inner ring rib is weakened, and the inhibition ability of the cage to the roller skew was relatively improved; that is, the force between the roller and the cage was increased (which may cause the violent friction between the cage and the roller) [22]. The roller swing will not continue to enhance the PV value and slip speed to maintain a steady change. Considering the dynamic unbalance mass of roller, there was a reasonable axial clearance range of bearing (0.03–0.05 mm) which makes the roller slip speed smaller when the PV value is not too large.


### 10.2. Effect of Circumferential Clearance of Cage Pocket on thePV Value and Slip Speed

For the bearing axial clearance of 0.03 mm, the angle of inner ring rib of 0.3°, and the crowned radius of roller end face of 300 mm, thePV value and its slip speed with the cage pocket circumferential clearance change as shown in Figure 10.Figure 10
Variation ofPV value and its slip speed with the cage pocket circumferential clearance. (a) Change of the PV value of the roller end face. (b) Change of the roller slip speed.
(a)(b)Figure10 shows that when the roller’s dynamic unbalance mass was certain, the PV value and the slip speed increased with the circumferential clearance of cage pocket and tended to flatten out after the circumferential clearance exceeded approximately 0.35 mm. When the circumferential clearance was small, the swing of the roller in the pocket was limited by the front and rear beam of the cage, its inhibition on the swing ability of the roller was greater, and the contact deformation between the roller and the inner ring rib was smaller. At this stage, with the increase of circumferential clearance, the ability of the roller swing was significantly enhanced, and the force between roller end face and inner ring rib increased gradually, making the PV value and its slip speed increase significantly. When the circumferential clearance of the pocket exceeded 0.35 mm, the swing of the roller was limited by the unilateral beam, the inhibition effect of the cage on the skew of the roller was weakened, and the swing ability of the roller was restored. The change of the PV value and slip speed tended to stabilize. Therefore, considering the roller’s dynamic unbalance mass, to ensure that the roller and the cage beam collision force would not be too large, smaller cage pocket circumferential clearance should be chosen.


### 10.3. Effect of Angle of Inner Ring Rib on thePV Value and Slip Speed

For the bearing axial clearance of 0.03 mm, the frame pocket hole circumferential clearance of 0.2 mm, and the crowned radius of roller end face of 300 mm, thePV value and its slip speed with the angle of inner ring rib are shown in Figure 11.Figure 11
Variation of thePV value and its slip speed with the angle of inner ring rib. (a) Change of the PV value. (b) Change of the roller slip speed.
(a)(b)Figure11 shows that when the roller’s dynamic unbalance mass is certain, the PV value decreases first and then increases with the increase of the angle of inner ring rib, and the minimum value appears when the angle is 0.55°. The slip speed of the roller increases when the angle of inner ring rib increases, and the slip speed increases sharply when the angle is 0.4°–0.7°. This was because when the angle of the inner ring rib is small, the more the angle increases, the farther the contact point between the roller end face and the rib is from the roller axis; that is, the smaller the distance between the contact point and the crowned centre of the roller end face in the direction of the roller axis, the larger the skewing angle of the roller; at this time, the axial clearance and the circumferential clearance of the cage pocket are unchanged. Under the limitation of the cage beam, the contact force between the roller end face and the inner ring rib decreases, and the PV value also decreases. When the angle of inner ring rib was more than 0.55°, the force acted on the roller end face was weaker, the roller swing speed increases, and the PV value is not only affected by the contact stress but also related to the relative velocity of the contact point. The PV value also begins to increase. In contrast, in the process of increasing the angle, the roller’s slip speed increased continuously because the roller swing ability was always improving. When the angle was in the range of 0.4°–0.7°, the force of the inner ring acting on the end face of the roller was weak, the limitation ability of the cage beam to the skew of the roller was moderate, and the slip speed of the roller increased sharply. When the angle was large, the limitation ability of cage beam to roller’s skew was stronger, and the increase of roller’s slip speed was slowed down. Therefore, considering the dynamic unbalance mass of roller, when the height of the inner ring rib was certain, there was a reasonable angle of the inner ring rib. The bearing studied in this paper is 0.4°–0.45°, which makes the PV value and the slip speed smaller.


### 10.4. Effect of the Crowned Radius of Roller End Face on thePV Value and Slip Speed

For the axial clearance of the bearing of 0.03 mm, the circumferential clearance of the cage pocket was 0.2 mm and the angle of inner ring rib was 0.3°. ThePV value and its slip speed varying with the crowned radius of roller end face are shown in Figure 12.Figure 12
Variation ofPV value and its slip speed with the crowned radius of the roller end face. (a) Change of the PV value. (b) Change of the roller slip speed.
(a)(b)Figure12 shows that when the dynamic unbalance mass of roller was fixed, the PV value decreased first and then increased with the increase of the crowned radius of roller end face, and the minimum value appeared when the crowned radius was 550 mm. The roller’s slip speed increases with the increase of the crowned radius, and the roller’s slip speed increases sharply when the crowned radius is 500–600 mm. As the crowned radius increases, the greater the distance between the contact point and the crowned centre of the roller in the direction of the roller axis, the smaller the theoretical skewing angle of the roller, and the increase of the contact load between the roller end face and the inner ring rib will decrease the contact stress, so the PV value will decrease with the increase of the crowned radius. In contrast, the effect of inner ring rib on roller’s skew was increased with the increase of crowned radius, and the drag force of inner ring rib on roller end face will inhibit roller rotation. Therefore, the roller slip speed always increases in the process of increasing the crowned radius. When the crowned radius was in the range of 500–600 mm, the limiting ability of the cage beam to the roller skew was moderate, and the slip speed of the roller increased sharply. When the crowned radius was large, the contact load between roller end face and inner ring rib was larger. Due to the contact angle being small, the collision between the cage beam and the roller will play the role of correcting roller attitude. The increase of roller slip speed is slowed down. Therefore, considering the dynamic unbalance mass of the roller, synthesising the geometric relationship between the angle of inner ring rib and the crowned radius, the crowned radius of roller end face in the range of 450–500 mm is selected.


## 10.1. Effect of Axial Clearance of Bearing on thePV Value and Slip Speed

For the cage pocket circumferential clearance of 0.2 mm, the angle of inner ring rib of 0.3°, and the crowned radius of roller end face of 300 mm, thePV value and its slip speed changing with bearing axial clearance are shown in Figure 9.Figure 9
Variation ofPV value and its slip speed with the axial clearance. (a) Change of the PV value. (b) Change of the roller slip speed.
(a)(b)Figure9 shows that when the roller’s dynamic unbalance mass is certain, the PV value increases with the increase of the axial clearance, and the roller’s slip speed decreases with the increase of the axial clearance, both of which tend to flatten after the axial clearance exceeds 0.05 mm. This was because when the axial clearance is small, the roller contact with the inner ring rib greatly inhibits the skew of the roller and reduces the speed of the roller swing. As a result, the PV value was reduced, but this inhibition will weaken the roller's normal rolling ability and increase the roller’s slip speed. When the clearance exceeds 0.05 mm, the inhibition of the roller skew by the inner ring rib is weakened, and the inhibition ability of the cage to the roller skew was relatively improved; that is, the force between the roller and the cage was increased (which may cause the violent friction between the cage and the roller) [22]. The roller swing will not continue to enhance the PV value and slip speed to maintain a steady change. Considering the dynamic unbalance mass of roller, there was a reasonable axial clearance range of bearing (0.03–0.05 mm) which makes the roller slip speed smaller when the PV value is not too large.


## 10.2. Effect of Circumferential Clearance of Cage Pocket on thePV Value and Slip Speed

For the bearing axial clearance of 0.03 mm, the angle of inner ring rib of 0.3°, and the crowned radius of roller end face of 300 mm, thePV value and its slip speed with the cage pocket circumferential clearance change as shown in Figure 10.Figure 10
Variation ofPV value and its slip speed with the cage pocket circumferential clearance. (a) Change of the PV value of the roller end face. (b) Change of the roller slip speed.
(a)(b)Figure10 shows that when the roller’s dynamic unbalance mass was certain, the PV value and the slip speed increased with the circumferential clearance of cage pocket and tended to flatten out after the circumferential clearance exceeded approximately 0.35 mm. When the circumferential clearance was small, the swing of the roller in the pocket was limited by the front and rear beam of the cage, its inhibition on the swing ability of the roller was greater, and the contact deformation between the roller and the inner ring rib was smaller. At this stage, with the increase of circumferential clearance, the ability of the roller swing was significantly enhanced, and the force between roller end face and inner ring rib increased gradually, making the PV value and its slip speed increase significantly. When the circumferential clearance of the pocket exceeded 0.35 mm, the swing of the roller was limited by the unilateral beam, the inhibition effect of the cage on the skew of the roller was weakened, and the swing ability of the roller was restored. The change of the PV value and slip speed tended to stabilize. Therefore, considering the roller’s dynamic unbalance mass, to ensure that the roller and the cage beam collision force would not be too large, smaller cage pocket circumferential clearance should be chosen.


## 10.3. Effect of Angle of Inner Ring Rib on thePV Value and Slip Speed

For the bearing axial clearance of 0.03 mm, the frame pocket hole circumferential clearance of 0.2 mm, and the crowned radius of roller end face of 300 mm, thePV value and its slip speed with the angle of inner ring rib are shown in Figure 11.Figure 11
Variation of thePV value and its slip speed with the angle of inner ring rib. (a) Change of the PV value. (b) Change of the roller slip speed.
(a)(b)Figure11 shows that when the roller’s dynamic unbalance mass is certain, the PV value decreases first and then increases with the increase of the angle of inner ring rib, and the minimum value appears when the angle is 0.55°. The slip speed of the roller increases when the angle of inner ring rib increases, and the slip speed increases sharply when the angle is 0.4°–0.7°. This was because when the angle of the inner ring rib is small, the more the angle increases, the farther the contact point between the roller end face and the rib is from the roller axis; that is, the smaller the distance between the contact point and the crowned centre of the roller end face in the direction of the roller axis, the larger the skewing angle of the roller; at this time, the axial clearance and the circumferential clearance of the cage pocket are unchanged. Under the limitation of the cage beam, the contact force between the roller end face and the inner ring rib decreases, and the PV value also decreases. When the angle of inner ring rib was more than 0.55°, the force acted on the roller end face was weaker, the roller swing speed increases, and the PV value is not only affected by the contact stress but also related to the relative velocity of the contact point. The PV value also begins to increase. In contrast, in the process of increasing the angle, the roller’s slip speed increased continuously because the roller swing ability was always improving. When the angle was in the range of 0.4°–0.7°, the force of the inner ring acting on the end face of the roller was weak, the limitation ability of the cage beam to the skew of the roller was moderate, and the slip speed of the roller increased sharply. When the angle was large, the limitation ability of cage beam to roller’s skew was stronger, and the increase of roller’s slip speed was slowed down. Therefore, considering the dynamic unbalance mass of roller, when the height of the inner ring rib was certain, there was a reasonable angle of the inner ring rib. The bearing studied in this paper is 0.4°–0.45°, which makes the PV value and the slip speed smaller.


## 10.4. Effect of the Crowned Radius of Roller End Face on thePV Value and Slip Speed

For the axial clearance of the bearing of 0.03 mm, the circumferential clearance of the cage pocket was 0.2 mm and the angle of inner ring rib was 0.3°. ThePV value and its slip speed varying with the crowned radius of roller end face are shown in Figure 12.Figure 12
Variation ofPV value and its slip speed with the crowned radius of the roller end face. (a) Change of the PV value. (b) Change of the roller slip speed.
(a)(b)Figure12 shows that when the dynamic unbalance mass of roller was fixed, the PV value decreased first and then increased with the increase of the crowned radius of roller end face, and the minimum value appeared when the crowned radius was 550 mm. The roller’s slip speed increases with the increase of the crowned radius, and the roller’s slip speed increases sharply when the crowned radius is 500–600 mm. As the crowned radius increases, the greater the distance between the contact point and the crowned centre of the roller in the direction of the roller axis, the smaller the theoretical skewing angle of the roller, and the increase of the contact load between the roller end face and the inner ring rib will decrease the contact stress, so the PV value will decrease with the increase of the crowned radius. In contrast, the effect of inner ring rib on roller’s skew was increased with the increase of crowned radius, and the drag force of inner ring rib on roller end face will inhibit roller rotation. Therefore, the roller slip speed always increases in the process of increasing the crowned radius. When the crowned radius was in the range of 500–600 mm, the limiting ability of the cage beam to the roller skew was moderate, and the slip speed of the roller increased sharply. When the crowned radius was large, the contact load between roller end face and inner ring rib was larger. Due to the contact angle being small, the collision between the cage beam and the roller will play the role of correcting roller attitude. The increase of roller slip speed is slowed down. Therefore, considering the dynamic unbalance mass of the roller, synthesising the geometric relationship between the angle of inner ring rib and the crowned radius, the crowned radius of roller end face in the range of 450–500 mm is selected.


## 11. Test Verification

Based on the aforedescribed theoretical analysis, it can be concluded that the roller dynamic unbalance mass caused by the inconsistent chamfering size at both ends of the roller and the excessive axial clearance of the bearing and are the main causes of abnormal wear on the working surface and end face of the roller; therefore, the axial clearance of the bearing is controlled within 0.03–0.05 mm, the angle of the inner ring rib is controlled within 0.4°–0.45°, the chamfering size of the roller is controlled at 0.5 ± 0.05 (0.5 ± 0.1 mm is the origin value), and the roller end face is still flat.Performance test of four sets of improved bearings using a ZYS-103 high-speed and high-temperature bearing performance testing machine is in Figure13. The test load, velocity, and time of bearing are 3000 N, 30000 r/min, and 500 hours, respectively. The appearance of the improved bearing after the test is shown in Figure 14 and it can be compared with that of the unimproved bearing shown in Figure 1. Some accuracy indices are listed in Table 2, it can be seen from Table 2 that the main precision index of the bearing after the test is still within the range of the design standard value, and some indexes are slightly improved. The improved bearing passed the performance test.Figure 13
ZYS-103 high-speed and high-temperature bearing performance testing machine.Figure 14
Appearance of the tested bearing.Table 2
Accuracy item of the tested bearing.
Serial numberRadial clearance (μm)Axial clearance (μm)Outer raceway (μm)Roller (μm)RoundnessRoughnessRoundnessRoughnessDiameter deviationStandard value60–8030–50≤20.0630.40.125−10–+516340–500.70.040.060.074−1-−226440–500.650.0610.080.058−1-–−236240–500.450.0450.120.053−2-−346340–500.610.0530.200.046−2-−3


## 12. Conclusion

(1)
The large amount of roller’s dynamic unbalance mass and the deviation of the bearing axial clearance are the main causes of abnormal wear on the roller working surface and end faces of cylindrical roller bearing. The larger roller’s dynamic unbalance mass leads to the largerPV value and slip speed; therefore the roller’s dynamic unbalance mass must be confined in a low level.(2)
ThePV value and its slip speed increased with the increase in roller dynamic unbalance mass, and the PV value of the roller end face where the dynamic unbalance mass is located in was larger. With an increase in the radial load, the PV value did not change significantly, the slip speed of roller decreased, and it tended to be gentle after the load exceeded 5000 N. With an increase in the inner ring speed, the PV value and its slip speed increased.(3)
A smaller axial clearance was found to effectively suppress thePV value, but it did cause an increase in the roller slip speed. A reasonable axial clearance range was found to exist. The axial clearance of bearing studied in this paper is 0.03–0.05 mm, which makes the roller slip speed smaller when the PV value is not too large.(4)
Both thePV value and the roller slip speed increased with the increase of the circumferential clearance of the cage pocket. When the circumferential clearance of the pocket exceeds 0.35 mm, the PV value and slip speed change more smoothly with the clearance. Considering the roller dynamic unbalance mass, a smaller circumferential clearance of cage pocket should be chosen.(5)
ThePV value decreased and then increased with an increase in the angle of the inner ring rib and the crowned radius of the roller end face, reaching a minimum at 0.55° and 550 mm, respectively. The roller slip speed increased with the increase in the angle of the inner ring rib and the crowned radius of the roller end face, increasing sharply at 0.4°–0.7° and 500–600 mm, respectively. A reasonable range of the angle of the inner ring rib and the crowned radius of the roller end face was found to exist. The bearings studied in this paper were 0.4°–0.45° and 450–500 mm, respectively, which reduced the PV value and the roller slip speed.



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*Source: 2522386-2021-07-26.xml*