# Dynamic Parameter Analysis of Spindle Bearing Using 3-Dimension Quasi-Dynamic Model

Authors: Jie Yu; Songsheng Li; Xiaoyang Chen; Lei He; Wei Yuan
Journal: Mathematical Problems in Engineering
 (2019)

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## Abstract

Spindle bearing with high rotational speed is the kernel of many rotating machines which plays massive important role in the performance of the whole machine. There are also some deficiencies of the present quasi-dynamic mathematical models of the rolling bearing in analyzing the bearing dynamic parameters. This paper develops the 3D quasi-dynamic model with ball’s 6-DOF, cage’s 6-DOF, and driving ring’s 5-DOF under steady working stage, based on the analysis of the relative position of each component to determine the normal forces between each component and the analysis of the relative velocity of the contacting point to determine the tangential force between each component. This model can analyze the dynamic parameters, such as the spin-to-roll ratio of inner and outer rings, the angular velocity of rolling elements iny-axis of the location coordinate, the normal contact force of cage pocket and ball, the cage angular velocity and ball revolution velocity, and the gyroscopic moment under different working condition. Compared with the numerical calculation results of ADORE program, the results of the proposed model are able to accurately describe the change laws of the spindle bearing parameters.


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## Body

## 1. Introduction

Development of a spindle bearing system can significantly improve the efficiency and accuracy of machine tools. The quasi-dynamic models of the spindle bearing are not only better capable of analyzing bearing dynamic parameters than quasi-static models and could also need less numerical calculation time than the dynamic model. Generally, the steady working duration of ball bearings would last 90 percent of the whole service life under normal condition. Therefore, establishment of perfect bearing quasi-dynamic model plays massive significant role on the analysis of the bearing operational performance [1–3], such as cage slipping, cage impacts, roller skidding, roller-flange contact, and roller skewing.Many contributions for spindle bearing system were conducted based on the bearing quasi-dynamic modeling, such as Harris [4] and Yuan [5], who established the equation groups about 6 degrees of rolling elements, 3 degrees of cages, and 5 degrees of rings by dropping the hypothesis of the ring control theory and considering the oil film drags of the contacting ellipse between rolling elements and raceways with oil film lubrication. Wang et al. [6] analyzed the boundary conditions of aeroengine bearing by the quasi-dynamic method and then discussed the relationship between status parameters associated with Failure modes (such as lubricating oil film thickness, etc.) and working parameters (such as rotation speed and external load). Ren et al. [7] established a new quasi-dynamic model of the counter-rotating cylindrical roller bearing based on elastohydrodynamic lubrication theory, hydrodynamic lubrication theory, and the quasi-dynamic analysis method to dynamic behaviors of counter-rotating cylindrical roller bearing. However, those researches have not considered the complete degree of freedom of the rolling bearing components which is very significant to determine their dynamic characteristics. Therefore, it is badly necessary to establish the bearing quasi-dynamic model for the analysis of the component complete degree of freedom.In order to reflect the real operation of high speed angular contact ball bearings, a quasi-dynamic calculation model for the high speed angular contact ball bearing was built by embedded Broyden-Fletcher-GoldFarb-Shanno mutative scale optimization in quasi-dynamic calculation program [8]. For the spindle bearing with a certain contact angle, gyroscopic moment will inevitably induct large sliding between the balls and raceways and increase the bearing friction moment, which are the main reasons for the bearing heat generation and wear [9–11]. Therefore, accurately analyzing the spin angular velocity between ball and raceway is benefit for the correct calculation of bearing power dissipation, assessment of the rate of heat generation, and appropriate decision of lubrication flow rate, etc. Meanwhile, the bearing cage is a critical component which is often the most Failure part for high speed bearings, typically. With the increase of bearing rotation speed, the analytical precision of the contact force between the cage and the balls or the guide rings is dramatically important for the force characteristics of bearing research [12–14].To analyze the operating characteristics of rolling bearings at steady working stage, the bearing quasi-dynamic model is developed in this paper. In the optimal model, (i) by the analysis of the relative positions of each bearing component, the normal forces between the bearing components are determined and (ii) the tangential forces between the bearing components are obtained according the study of the directions and magnitudes of the relative velocities of the contact points and then (iii) the force mathematical models of balls’ 6-DOF, cage’s 6-DOF, and the driving ring’s 5-DOF. On the base of those models, the gyroscopic moment model is established to analyze the pitching angles and yaw angles, and the cage forces and operating condition are studied by the contacting force models between the cage and the ball and guide ring. The results of the improved quasi-dynamic model are furthermore compared with the results of the typical rolling bearing force calculation example of Gupta [15]. Studies show that the model is able to accurately describe the dynamic variations of the parameter characteristics of spindle bearings.


## 2. Mechanics Modeling of Spindle Bearing

### 2.1. Techniques of Mechanics Modeling

Figure1 shows the interaction model of bearing components. The interaction forces between these parts are mainly the normal contact forces and the tangential forces. First, by means of determining the position vectors of part B and part A in space, judging whether part B in contact with part A and calculating the normal contact force, the normal force of part B relative to part A can be obtained. Second, according to the velocity vectors of part A and part B in space, the velocity direction of part B relative to part A at the contact point can be determined, and on the basic of Coulomb’s law, the tangential force of part B relative to part A is obtained. Finally, the influence of the lubricant on the force will be considered to solve the force of part B relative to part A. Equilibrium equations of part A and part B will be established and the numerical solution can be done by considering these three factors.Figure 1
Interaction model of each part in the bearing.


### 2.2. Interaction Model of Each Parts of Bearing

#### 2.2.1. Interaction of Ball and Ring

The vector position relationship of ball and ring in space is shown in Figure2. The points ofOi andOe are the centers of inner and outer raceway groove curvatures, respectively. The position vectors of  rboe  and  rboei  are from the center of the ball to the centers of groove curvature of the outer ring and the inner ring, respectively.Figure 2
Geometric relationship of balls and ring.In the vector triangle ofΔOOaOe (seeing Figure 2), the vector of  rboe  is the position vector from the ball center to the outer raceway groove curvature center and can be expressed by(1)rboe=rb-rOoeLikewise, the vector of  rboe  in the vector triangle ofΔOrOaOi is the position vector from the ball center to the inner raceway groove curvature center and can be described as(2)rboi=rb-rOoiThe elastic approaches of ball and ring raceways are expressed by(3)δk=rbok-fk-0.5dwwherek =i,e denote the inner and outer ring, respectively. Whenδk ≤ 0, the ball and raceway have no contact with each other and contact deformation will not occur, and the normal force is zero, whereas whenδk > 0, the contact deformation occurred, and the normal contact load can be obtained by (4) using Hertz theory of contact stress.(4)FkH=Kδkwherek is the stiffness coefficient of Hertz contact.As illustrated in Figure3, the local coordinate ofo_xy is established to analyze the contact ellipse between ball and inner raceway. Relative linear velocity can be described in the direction ofx-axis andy-axis as (5) and (6), respectively.(5)Vxi=-Ri2-xi2-Ri2-ai2+dw22-ai2ωiωbωicos⁡βsinβ′(6)Vyi=-ωi2Dm-Ri2-xi2-Ri2-ai2+dw22-ai2×ωbωicosβcosβ′cosαi+ωbωisinβsinαi-cos⁡αiωiFigure 3
Contact feature of ball and inner raceways.The angle between the direction of relative linear velocity and the direction of semimajor axis of the contact ellipse is(7)ϑbr=arctan⁡-VxiVyiAccording to Coulomb’s law, tangential friction with respect to semimajor axis and semiminor axis of the contact ellipse is stated as(8)Tix=-TisinϑbrTiy=TicosϑbrThe force of contact point between ball and inner raceway is then obtained as(9)Fi=Tix,Tiy,FiHAs illustrated by Figure4, the vectors from the ball center to the contact ellipse centers of inner and outer raceways are  robi  and  robe,  respectively. The friction torque between the ball and the inner raceway is as follows:(10)Mbri=robi×Tix,Tiy,0and the friction torque of the inner raceway is(11)Mrbi=robe+rori×Tix,Tiy,0where  rori  is the vector from the ball center to inner ring coordinate center. Similarly, the contact force between ball and outer raceway can be given as(12)Fe=Tex,Tey,FeHFigure 4
Relative position between the ball center and the inner and outer raceway.


#### 2.2.2. Interaction of Ball and Cage

As illustrated in Figure5, the vectors of rbp1p and rbp2p are the first and the second components of the vectors from the ball center to the cage pocket center in the pocket coordinatesop_xpypzp, respectively. The contact point between the ball and the pocket wall and the ball center and the pocket center is aligned in line. The minimum distance from pocket wall to the ball surFace can be expressed as(13)δbp=dp-dw2-rbp1p2+rbp2p2Whenδbp ≤ 0, the elastohydrodynamic lubrication state will occur between the ball and cage pocket, and the normal force component between the ball and cage in the contact coordinate system can be obtained based on Brewe formula by [16](14)Fbp=ULφ128αrRzhwhere L=0.131arctanαr/2+1.683, φ=1+2/3αr-1, and αr=Rx/RzU=ηou/E′Rz. And the corresponding tangential force component can be given using Coulomb Law as follows:(15)Tbc=μbpFbpwhereRx is the equivalent radius along semimajor axis of the contact area in the contact coordinate,Rz is the equivalent radius along semiminor axis of the contact area in the contact coordinate,η0 is the dynamic viscosity of lubricating oil,u is the relative sliding speed of the ball and the pocket wall, E′ is the equivalent elastic modulus of the ball and the cage, andμbp of 0.02 ~ 0.08 is the friction coefficient between the ball and the pocket wall.Figure 5
Geometric relationship of balls and cage.Whenδbp < 0, contact deformation will occur between the ball and the pocket wall, and the normal contact load can be obtained by using Hertz theory as(16)Fbp=Kcδbp3/2The relative line velocity at the contact point of the ball and the pocket wall is(17)vbp=vgb-vgpIn contact coordinate og_xgygzg, the angle betweenνbp with zg axis is(18)ϑbp=acrtan⁡vbp1vbp3The tangential friction forces along the directions of xg and zg are expressed as follows, respectively:(19)Tbcx=TbcsinϑbpTbcz=TbccosϑbpThe force and the moment subjected to the ball by cage pocket can be stated as follows, respectively:(20)Fbc=Tbcx,Fbpn,TbczT(21)Mbc=rgb×FbcAccording to Newton’s third law, the force and the moment subjected to the cage by the ball can be obtained as follows, respectively:(22)Fcb=-Fbc(23)Mcb=rgp+rpc×Fcbwhere rgp  is the position vector from the contact point of ball and pocket wall to the pocket center and rpc  is the position vector from the pocket center to the cage coordinate center.


#### 2.2.3. Interaction of Cage and Guide Ring

Because of the effect of inertial force during steadily working process, the cage and guide ring exist potential collision, which can be assessed by the minimum clearance between the cage and the guide ring. As shown in Figure6, the minimum clearance in the vector triangleΔOrOcg is given as(24)δrc=±Rr-rrg2r2+rrg3r2where the symbols “+” and “-” are for outer and inner guide rings, respectively. rrg2r and rrg3r are the second and third components of rrgr, respectively, which is the vector from the guide ring center to contact point of cage and ring in ring coordinate systemor_xryrzr.Figure 6
Geometric relationship of cage and guide ring.Whenδrc ≤ 0, hydrodynamic lubrication will occur between the cage and the guide ring, and the interaction condition can be regarded as short sliding bearing. In the contact coordinate system og_xgygzg (seeing in Figure 5), the interaction forces of each direction are calculated by the following formula [17, 18]:(25)Fcrxg=-2πηVcrxgRc2BcCgFcryg=±πηuBc3ε4Cg21-ε23/2Fcrzg=±ηuBc3ε2Cg21-ε22where Vcrxg is the relative speed of the cage with respect to the guide ring at the minimum clearance in the direction of xg axis,Bc is the width of the cage,u is drag speed of lubricating oil,u=Rc(ωr+ωc), Cg is the cage guide clearance, andε is relative offset of the cage center.(26)ε=eCge=xc2+yc2+zc2The friction moment of the moving cage caused by distribution pressure of fluid pressure oil film can be expressed as(27)Mcrxc=2πηVcrRgc2BcCg1-ε2whereVcr is the relative sliding speed of the cage rail with respect to the ring land and can be stated as follows:(28)Vcrg=Rcwr-wcWhenδrc > 0, contact deformation will occur between the cage and the guide ring, and the normal contact load can be obtained by using Hertz theory as (29)Fcrzg=Klδrc10/9whereKl is the Hertz contact stiffness coefficient with finite length.The relative sliding speed of the ring with respect to the cage at contact point is(30)vrc=vrg-vcgwhere vrg is the velocity at the contact point of the ring and vcg is the velocity at the contact point of the cage. vrc1 and vrc2 are the first and second components of the tangential relative velocity, respectively, and the angel along xg axis is(31)ϑcr=arctan⁡vrc2vrc1Applying Coulomb’s law, the tangential force at the contact point is(32)Fcr=μcrFcrzwhere ucr is the friction coefficient between the cage and the guide ring and is set 0.02 ~ 0.08.The tangential forces in the direction ofxg and yg axis of the contact coordinate og_xgygzg are as follows, respectively:(33)Tcrx=TcrcosϑcrTcry=TcrsinϑcrAccording to Newton’s third law, the force and the moment of the cage caused by the guide ring are(34)Fcr=Tcrx,Tcry,FcrzMcr=rgc×Fcrwhere  rgc  is the vector distance from the center of cage coordinate system to the contact point between the cage and the ring.


#### 2.2.4. Drag of Ball from Oil-Gas Mixture

The balls should overcome the viscous friction drag caused by the bearing internal lubrication during orbital motion. The viscous friction drag of a ball shown in [19] can be approximately expressed by(35)Fd=ρefπCDdw232gDmωm1.95whereρef is the lubricant density in the bearing free space divided by the volume of the free space and the drag coefficientsCD can be determined from [20].


### 2.3. The Establishment of Equilibrium Equations

The equilibrium equations of the ball are established in the azimuth coordinate system of the ball, in which there are six unknown parameters that need to be calculated, and they arexb,rb,ωm, ωbx,ωby, andωbz.The inertial forces on a ball are(36)Fx=0Fy=mbay=mbω˙mrbFy=mbaz=mbωmrb2and the inertial moments on a ball are(37)Mx=-Ibω˙bxMy=-Ibω˙by+ωmωbzMz=-Ibω˙bz-ωmωbywhereIb is the inertia moment and can be calculated by Ib=mbdw2/10.The equilibrium equations of ball’s force and moment are(38)Fbr1a+Fbc1a=0(39)Fbr2a+Fbc2a+Fd+Fy=0(40)Fbr3a+Fbc3a+Fz=0(41)Mbr1a+Mbc1a+Mx=0(42)Mbr2a+Mbc2a+My=0(43)Mbr2a+Mbc2a+My=0The force equilibrium equations of the cage are established in the inertial coordinate system of the bearing, and the moment equilibrium equation are established in the fixed coordinate system of the cage. There are the six parameters need to be determined asxc, yc, zc,ωcxθcy, andθcz.The force and moment equilibrium equations of the cage are(44)Fcr1i+Fcb1i=0(45)Fcr2i+Fcb2i=0(46)Fcr3i+Fcb3i=0(47)Mcr1c+Mcb1c=0(48)Mcr2c+Mcb2c=0(49)Mcr3c+Mcb3c=0The force equilibrium equations of the guide ring are established in the inertial coordinate system of the bearing, and the moment equilibrium equations are established in the ring fixed coordinate system. Similarly, there are also six parameters need to be solvedxr, yr, zr, θry, θrz. The force and moment equilibrium equations of the guide ring are(50)Fx-∑j=1zFrcj1i+Frbj1i=0Fy-∑j=1zFrcj2i+Frbj2i=0Fz-∑j=1zFrcj3i+Frbj3i=0My-∑j=1zMrc2r+Mrb2r=0Mz-∑j=1zMrc3r+Mrb3r=0


### 2.4. Numerical Solution

In this paper, six equations of the ball and cage and five equations of the guide ring were solved numerically by Matlab tools. The magnitude order of the displacement and angular velocity is significantly different; therefore, it is easy to get a nonconvergent solution by combining them to solve. In order to solve this problem, the method from first partial to whole is adopted in this paper. The displacement variables are determined first, and then the angular velocity variables are calculated. Thereafter, the initial value, the displacement, and angular velocity variables will be solved together. The solution process is shown in Figure7.Figure 7
Numerical calculation flow chart.


## 2.1. Techniques of Mechanics Modeling

Figure1 shows the interaction model of bearing components. The interaction forces between these parts are mainly the normal contact forces and the tangential forces. First, by means of determining the position vectors of part B and part A in space, judging whether part B in contact with part A and calculating the normal contact force, the normal force of part B relative to part A can be obtained. Second, according to the velocity vectors of part A and part B in space, the velocity direction of part B relative to part A at the contact point can be determined, and on the basic of Coulomb’s law, the tangential force of part B relative to part A is obtained. Finally, the influence of the lubricant on the force will be considered to solve the force of part B relative to part A. Equilibrium equations of part A and part B will be established and the numerical solution can be done by considering these three factors.Figure 1
Interaction model of each part in the bearing.


## 2.2. Interaction Model of Each Parts of Bearing

### 2.2.1. Interaction of Ball and Ring

The vector position relationship of ball and ring in space is shown in Figure2. The points ofOi andOe are the centers of inner and outer raceway groove curvatures, respectively. The position vectors of  rboe  and  rboei  are from the center of the ball to the centers of groove curvature of the outer ring and the inner ring, respectively.Figure 2
Geometric relationship of balls and ring.In the vector triangle ofΔOOaOe (seeing Figure 2), the vector of  rboe  is the position vector from the ball center to the outer raceway groove curvature center and can be expressed by(1)rboe=rb-rOoeLikewise, the vector of  rboe  in the vector triangle ofΔOrOaOi is the position vector from the ball center to the inner raceway groove curvature center and can be described as(2)rboi=rb-rOoiThe elastic approaches of ball and ring raceways are expressed by(3)δk=rbok-fk-0.5dwwherek =i,e denote the inner and outer ring, respectively. Whenδk ≤ 0, the ball and raceway have no contact with each other and contact deformation will not occur, and the normal force is zero, whereas whenδk > 0, the contact deformation occurred, and the normal contact load can be obtained by (4) using Hertz theory of contact stress.(4)FkH=Kδkwherek is the stiffness coefficient of Hertz contact.As illustrated in Figure3, the local coordinate ofo_xy is established to analyze the contact ellipse between ball and inner raceway. Relative linear velocity can be described in the direction ofx-axis andy-axis as (5) and (6), respectively.(5)Vxi=-Ri2-xi2-Ri2-ai2+dw22-ai2ωiωbωicos⁡βsinβ′(6)Vyi=-ωi2Dm-Ri2-xi2-Ri2-ai2+dw22-ai2×ωbωicosβcosβ′cosαi+ωbωisinβsinαi-cos⁡αiωiFigure 3
Contact feature of ball and inner raceways.The angle between the direction of relative linear velocity and the direction of semimajor axis of the contact ellipse is(7)ϑbr=arctan⁡-VxiVyiAccording to Coulomb’s law, tangential friction with respect to semimajor axis and semiminor axis of the contact ellipse is stated as(8)Tix=-TisinϑbrTiy=TicosϑbrThe force of contact point between ball and inner raceway is then obtained as(9)Fi=Tix,Tiy,FiHAs illustrated by Figure4, the vectors from the ball center to the contact ellipse centers of inner and outer raceways are  robi  and  robe,  respectively. The friction torque between the ball and the inner raceway is as follows:(10)Mbri=robi×Tix,Tiy,0and the friction torque of the inner raceway is(11)Mrbi=robe+rori×Tix,Tiy,0where  rori  is the vector from the ball center to inner ring coordinate center. Similarly, the contact force between ball and outer raceway can be given as(12)Fe=Tex,Tey,FeHFigure 4
Relative position between the ball center and the inner and outer raceway.


### 2.2.2. Interaction of Ball and Cage

As illustrated in Figure5, the vectors of rbp1p and rbp2p are the first and the second components of the vectors from the ball center to the cage pocket center in the pocket coordinatesop_xpypzp, respectively. The contact point between the ball and the pocket wall and the ball center and the pocket center is aligned in line. The minimum distance from pocket wall to the ball surFace can be expressed as(13)δbp=dp-dw2-rbp1p2+rbp2p2Whenδbp ≤ 0, the elastohydrodynamic lubrication state will occur between the ball and cage pocket, and the normal force component between the ball and cage in the contact coordinate system can be obtained based on Brewe formula by [16](14)Fbp=ULφ128αrRzhwhere L=0.131arctanαr/2+1.683, φ=1+2/3αr-1, and αr=Rx/RzU=ηou/E′Rz. And the corresponding tangential force component can be given using Coulomb Law as follows:(15)Tbc=μbpFbpwhereRx is the equivalent radius along semimajor axis of the contact area in the contact coordinate,Rz is the equivalent radius along semiminor axis of the contact area in the contact coordinate,η0 is the dynamic viscosity of lubricating oil,u is the relative sliding speed of the ball and the pocket wall, E′ is the equivalent elastic modulus of the ball and the cage, andμbp of 0.02 ~ 0.08 is the friction coefficient between the ball and the pocket wall.Figure 5
Geometric relationship of balls and cage.Whenδbp < 0, contact deformation will occur between the ball and the pocket wall, and the normal contact load can be obtained by using Hertz theory as(16)Fbp=Kcδbp3/2The relative line velocity at the contact point of the ball and the pocket wall is(17)vbp=vgb-vgpIn contact coordinate og_xgygzg, the angle betweenνbp with zg axis is(18)ϑbp=acrtan⁡vbp1vbp3The tangential friction forces along the directions of xg and zg are expressed as follows, respectively:(19)Tbcx=TbcsinϑbpTbcz=TbccosϑbpThe force and the moment subjected to the ball by cage pocket can be stated as follows, respectively:(20)Fbc=Tbcx,Fbpn,TbczT(21)Mbc=rgb×FbcAccording to Newton’s third law, the force and the moment subjected to the cage by the ball can be obtained as follows, respectively:(22)Fcb=-Fbc(23)Mcb=rgp+rpc×Fcbwhere rgp  is the position vector from the contact point of ball and pocket wall to the pocket center and rpc  is the position vector from the pocket center to the cage coordinate center.


### 2.2.3. Interaction of Cage and Guide Ring

Because of the effect of inertial force during steadily working process, the cage and guide ring exist potential collision, which can be assessed by the minimum clearance between the cage and the guide ring. As shown in Figure6, the minimum clearance in the vector triangleΔOrOcg is given as(24)δrc=±Rr-rrg2r2+rrg3r2where the symbols “+” and “-” are for outer and inner guide rings, respectively. rrg2r and rrg3r are the second and third components of rrgr, respectively, which is the vector from the guide ring center to contact point of cage and ring in ring coordinate systemor_xryrzr.Figure 6
Geometric relationship of cage and guide ring.Whenδrc ≤ 0, hydrodynamic lubrication will occur between the cage and the guide ring, and the interaction condition can be regarded as short sliding bearing. In the contact coordinate system og_xgygzg (seeing in Figure 5), the interaction forces of each direction are calculated by the following formula [17, 18]:(25)Fcrxg=-2πηVcrxgRc2BcCgFcryg=±πηuBc3ε4Cg21-ε23/2Fcrzg=±ηuBc3ε2Cg21-ε22where Vcrxg is the relative speed of the cage with respect to the guide ring at the minimum clearance in the direction of xg axis,Bc is the width of the cage,u is drag speed of lubricating oil,u=Rc(ωr+ωc), Cg is the cage guide clearance, andε is relative offset of the cage center.(26)ε=eCge=xc2+yc2+zc2The friction moment of the moving cage caused by distribution pressure of fluid pressure oil film can be expressed as(27)Mcrxc=2πηVcrRgc2BcCg1-ε2whereVcr is the relative sliding speed of the cage rail with respect to the ring land and can be stated as follows:(28)Vcrg=Rcwr-wcWhenδrc > 0, contact deformation will occur between the cage and the guide ring, and the normal contact load can be obtained by using Hertz theory as (29)Fcrzg=Klδrc10/9whereKl is the Hertz contact stiffness coefficient with finite length.The relative sliding speed of the ring with respect to the cage at contact point is(30)vrc=vrg-vcgwhere vrg is the velocity at the contact point of the ring and vcg is the velocity at the contact point of the cage. vrc1 and vrc2 are the first and second components of the tangential relative velocity, respectively, and the angel along xg axis is(31)ϑcr=arctan⁡vrc2vrc1Applying Coulomb’s law, the tangential force at the contact point is(32)Fcr=μcrFcrzwhere ucr is the friction coefficient between the cage and the guide ring and is set 0.02 ~ 0.08.The tangential forces in the direction ofxg and yg axis of the contact coordinate og_xgygzg are as follows, respectively:(33)Tcrx=TcrcosϑcrTcry=TcrsinϑcrAccording to Newton’s third law, the force and the moment of the cage caused by the guide ring are(34)Fcr=Tcrx,Tcry,FcrzMcr=rgc×Fcrwhere  rgc  is the vector distance from the center of cage coordinate system to the contact point between the cage and the ring.


### 2.2.4. Drag of Ball from Oil-Gas Mixture

The balls should overcome the viscous friction drag caused by the bearing internal lubrication during orbital motion. The viscous friction drag of a ball shown in [19] can be approximately expressed by(35)Fd=ρefπCDdw232gDmωm1.95whereρef is the lubricant density in the bearing free space divided by the volume of the free space and the drag coefficientsCD can be determined from [20].


## 2.2.1. Interaction of Ball and Ring

The vector position relationship of ball and ring in space is shown in Figure2. The points ofOi andOe are the centers of inner and outer raceway groove curvatures, respectively. The position vectors of  rboe  and  rboei  are from the center of the ball to the centers of groove curvature of the outer ring and the inner ring, respectively.Figure 2
Geometric relationship of balls and ring.In the vector triangle ofΔOOaOe (seeing Figure 2), the vector of  rboe  is the position vector from the ball center to the outer raceway groove curvature center and can be expressed by(1)rboe=rb-rOoeLikewise, the vector of  rboe  in the vector triangle ofΔOrOaOi is the position vector from the ball center to the inner raceway groove curvature center and can be described as(2)rboi=rb-rOoiThe elastic approaches of ball and ring raceways are expressed by(3)δk=rbok-fk-0.5dwwherek =i,e denote the inner and outer ring, respectively. Whenδk ≤ 0, the ball and raceway have no contact with each other and contact deformation will not occur, and the normal force is zero, whereas whenδk > 0, the contact deformation occurred, and the normal contact load can be obtained by (4) using Hertz theory of contact stress.(4)FkH=Kδkwherek is the stiffness coefficient of Hertz contact.As illustrated in Figure3, the local coordinate ofo_xy is established to analyze the contact ellipse between ball and inner raceway. Relative linear velocity can be described in the direction ofx-axis andy-axis as (5) and (6), respectively.(5)Vxi=-Ri2-xi2-Ri2-ai2+dw22-ai2ωiωbωicos⁡βsinβ′(6)Vyi=-ωi2Dm-Ri2-xi2-Ri2-ai2+dw22-ai2×ωbωicosβcosβ′cosαi+ωbωisinβsinαi-cos⁡αiωiFigure 3
Contact feature of ball and inner raceways.The angle between the direction of relative linear velocity and the direction of semimajor axis of the contact ellipse is(7)ϑbr=arctan⁡-VxiVyiAccording to Coulomb’s law, tangential friction with respect to semimajor axis and semiminor axis of the contact ellipse is stated as(8)Tix=-TisinϑbrTiy=TicosϑbrThe force of contact point between ball and inner raceway is then obtained as(9)Fi=Tix,Tiy,FiHAs illustrated by Figure4, the vectors from the ball center to the contact ellipse centers of inner and outer raceways are  robi  and  robe,  respectively. The friction torque between the ball and the inner raceway is as follows:(10)Mbri=robi×Tix,Tiy,0and the friction torque of the inner raceway is(11)Mrbi=robe+rori×Tix,Tiy,0where  rori  is the vector from the ball center to inner ring coordinate center. Similarly, the contact force between ball and outer raceway can be given as(12)Fe=Tex,Tey,FeHFigure 4
Relative position between the ball center and the inner and outer raceway.


## 2.2.2. Interaction of Ball and Cage

As illustrated in Figure5, the vectors of rbp1p and rbp2p are the first and the second components of the vectors from the ball center to the cage pocket center in the pocket coordinatesop_xpypzp, respectively. The contact point between the ball and the pocket wall and the ball center and the pocket center is aligned in line. The minimum distance from pocket wall to the ball surFace can be expressed as(13)δbp=dp-dw2-rbp1p2+rbp2p2Whenδbp ≤ 0, the elastohydrodynamic lubrication state will occur between the ball and cage pocket, and the normal force component between the ball and cage in the contact coordinate system can be obtained based on Brewe formula by [16](14)Fbp=ULφ128αrRzhwhere L=0.131arctanαr/2+1.683, φ=1+2/3αr-1, and αr=Rx/RzU=ηou/E′Rz. And the corresponding tangential force component can be given using Coulomb Law as follows:(15)Tbc=μbpFbpwhereRx is the equivalent radius along semimajor axis of the contact area in the contact coordinate,Rz is the equivalent radius along semiminor axis of the contact area in the contact coordinate,η0 is the dynamic viscosity of lubricating oil,u is the relative sliding speed of the ball and the pocket wall, E′ is the equivalent elastic modulus of the ball and the cage, andμbp of 0.02 ~ 0.08 is the friction coefficient between the ball and the pocket wall.Figure 5
Geometric relationship of balls and cage.Whenδbp < 0, contact deformation will occur between the ball and the pocket wall, and the normal contact load can be obtained by using Hertz theory as(16)Fbp=Kcδbp3/2The relative line velocity at the contact point of the ball and the pocket wall is(17)vbp=vgb-vgpIn contact coordinate og_xgygzg, the angle betweenνbp with zg axis is(18)ϑbp=acrtan⁡vbp1vbp3The tangential friction forces along the directions of xg and zg are expressed as follows, respectively:(19)Tbcx=TbcsinϑbpTbcz=TbccosϑbpThe force and the moment subjected to the ball by cage pocket can be stated as follows, respectively:(20)Fbc=Tbcx,Fbpn,TbczT(21)Mbc=rgb×FbcAccording to Newton’s third law, the force and the moment subjected to the cage by the ball can be obtained as follows, respectively:(22)Fcb=-Fbc(23)Mcb=rgp+rpc×Fcbwhere rgp  is the position vector from the contact point of ball and pocket wall to the pocket center and rpc  is the position vector from the pocket center to the cage coordinate center.


## 2.2.3. Interaction of Cage and Guide Ring

Because of the effect of inertial force during steadily working process, the cage and guide ring exist potential collision, which can be assessed by the minimum clearance between the cage and the guide ring. As shown in Figure6, the minimum clearance in the vector triangleΔOrOcg is given as(24)δrc=±Rr-rrg2r2+rrg3r2where the symbols “+” and “-” are for outer and inner guide rings, respectively. rrg2r and rrg3r are the second and third components of rrgr, respectively, which is the vector from the guide ring center to contact point of cage and ring in ring coordinate systemor_xryrzr.Figure 6
Geometric relationship of cage and guide ring.Whenδrc ≤ 0, hydrodynamic lubrication will occur between the cage and the guide ring, and the interaction condition can be regarded as short sliding bearing. In the contact coordinate system og_xgygzg (seeing in Figure 5), the interaction forces of each direction are calculated by the following formula [17, 18]:(25)Fcrxg=-2πηVcrxgRc2BcCgFcryg=±πηuBc3ε4Cg21-ε23/2Fcrzg=±ηuBc3ε2Cg21-ε22where Vcrxg is the relative speed of the cage with respect to the guide ring at the minimum clearance in the direction of xg axis,Bc is the width of the cage,u is drag speed of lubricating oil,u=Rc(ωr+ωc), Cg is the cage guide clearance, andε is relative offset of the cage center.(26)ε=eCge=xc2+yc2+zc2The friction moment of the moving cage caused by distribution pressure of fluid pressure oil film can be expressed as(27)Mcrxc=2πηVcrRgc2BcCg1-ε2whereVcr is the relative sliding speed of the cage rail with respect to the ring land and can be stated as follows:(28)Vcrg=Rcwr-wcWhenδrc > 0, contact deformation will occur between the cage and the guide ring, and the normal contact load can be obtained by using Hertz theory as (29)Fcrzg=Klδrc10/9whereKl is the Hertz contact stiffness coefficient with finite length.The relative sliding speed of the ring with respect to the cage at contact point is(30)vrc=vrg-vcgwhere vrg is the velocity at the contact point of the ring and vcg is the velocity at the contact point of the cage. vrc1 and vrc2 are the first and second components of the tangential relative velocity, respectively, and the angel along xg axis is(31)ϑcr=arctan⁡vrc2vrc1Applying Coulomb’s law, the tangential force at the contact point is(32)Fcr=μcrFcrzwhere ucr is the friction coefficient between the cage and the guide ring and is set 0.02 ~ 0.08.The tangential forces in the direction ofxg and yg axis of the contact coordinate og_xgygzg are as follows, respectively:(33)Tcrx=TcrcosϑcrTcry=TcrsinϑcrAccording to Newton’s third law, the force and the moment of the cage caused by the guide ring are(34)Fcr=Tcrx,Tcry,FcrzMcr=rgc×Fcrwhere  rgc  is the vector distance from the center of cage coordinate system to the contact point between the cage and the ring.


## 2.2.4. Drag of Ball from Oil-Gas Mixture

The balls should overcome the viscous friction drag caused by the bearing internal lubrication during orbital motion. The viscous friction drag of a ball shown in [19] can be approximately expressed by(35)Fd=ρefπCDdw232gDmωm1.95whereρef is the lubricant density in the bearing free space divided by the volume of the free space and the drag coefficientsCD can be determined from [20].


## 2.3. The Establishment of Equilibrium Equations

The equilibrium equations of the ball are established in the azimuth coordinate system of the ball, in which there are six unknown parameters that need to be calculated, and they arexb,rb,ωm, ωbx,ωby, andωbz.The inertial forces on a ball are(36)Fx=0Fy=mbay=mbω˙mrbFy=mbaz=mbωmrb2and the inertial moments on a ball are(37)Mx=-Ibω˙bxMy=-Ibω˙by+ωmωbzMz=-Ibω˙bz-ωmωbywhereIb is the inertia moment and can be calculated by Ib=mbdw2/10.The equilibrium equations of ball’s force and moment are(38)Fbr1a+Fbc1a=0(39)Fbr2a+Fbc2a+Fd+Fy=0(40)Fbr3a+Fbc3a+Fz=0(41)Mbr1a+Mbc1a+Mx=0(42)Mbr2a+Mbc2a+My=0(43)Mbr2a+Mbc2a+My=0The force equilibrium equations of the cage are established in the inertial coordinate system of the bearing, and the moment equilibrium equation are established in the fixed coordinate system of the cage. There are the six parameters need to be determined asxc, yc, zc,ωcxθcy, andθcz.The force and moment equilibrium equations of the cage are(44)Fcr1i+Fcb1i=0(45)Fcr2i+Fcb2i=0(46)Fcr3i+Fcb3i=0(47)Mcr1c+Mcb1c=0(48)Mcr2c+Mcb2c=0(49)Mcr3c+Mcb3c=0The force equilibrium equations of the guide ring are established in the inertial coordinate system of the bearing, and the moment equilibrium equations are established in the ring fixed coordinate system. Similarly, there are also six parameters need to be solvedxr, yr, zr, θry, θrz. The force and moment equilibrium equations of the guide ring are(50)Fx-∑j=1zFrcj1i+Frbj1i=0Fy-∑j=1zFrcj2i+Frbj2i=0Fz-∑j=1zFrcj3i+Frbj3i=0My-∑j=1zMrc2r+Mrb2r=0Mz-∑j=1zMrc3r+Mrb3r=0


## 2.4. Numerical Solution

In this paper, six equations of the ball and cage and five equations of the guide ring were solved numerically by Matlab tools. The magnitude order of the displacement and angular velocity is significantly different; therefore, it is easy to get a nonconvergent solution by combining them to solve. In order to solve this problem, the method from first partial to whole is adopted in this paper. The displacement variables are determined first, and then the angular velocity variables are calculated. Thereafter, the initial value, the displacement, and angular velocity variables will be solved together. The solution process is shown in Figure7.Figure 7
Numerical calculation flow chart.


## 3. Validation

In order to verify the accuracy of the updated mathematical model of the angular contact ball bearing, the results of a typical example is illustrated to compare the results with ADORE calculation. The values of bearing parameters are listed in Table1.Table 1
Geometry parameters of ball bearing.
Bearing parameters value Pitch diameter(mm) 31 Ball diameter(mm) 8 The number of balls 6 Initial contact angle(°) 24 Inner rolling groove curvature 0.56 outer rolling groove curvature 0.52 Bearing bore diameter(mm) 20 Bearing outside diameter (mm) 42 Cage outside diameter(mm) 35 Cage core diameter (mm) 30 Cage width(mm) 14 Elasticity modulus (MPa) 2e5Bearings working condition is set asFa of 2000 N, Fr of 400 N,ni of 120000 r/min with the fixed outer ring. Comparing the results of this paper’s program with Gupta’s ADORE dynamics program at 12.87 ms, the results are compared in Table 2.Table 2
Comparison of the computation results at 12.87ms.
Calculating parameter The result of Gupta The result of this paper Contact angle of ball and inner ring(°) 28.29 33.20 Contact angle of ball and outer ring (°) 13.87 14.14 Maximum contact stress of ball and inner ring (Gpa) 2.799 2.883 Maximum contact stress of ball and outer ring (Gpa) 2.253 2.308 Semi-major axis of the contact area of ball and inner ring (mm) 0.7971 0.8018 Semi-major axis of the contact area of ball and outer ring (mm) 1.478 1.512 Spin-to-roll ratio of ball and inner ring 0.3482 0.4072 Revolution velocity of ball(r/min) 46660 48961 Rotation velocity of ball(r/min) 228100 240172 Revolution velocity of cage(r/min) 47660 47950As listed in Table2, the results of this paper and Gupta’s show that the maximum error is about 10%. Such phenomenon is due to the fact that (i) the interference fit generated by bearing installed on the spindle was not considered, which would lead the initial contact angle to decrease, (ii) Gupta’s ADORE dynamics program considered the acceleration influence, but the proposed mechanical model of ball bearing in this paper did not considered such factor.


## 4. Dynamic Analysis of Spindle Bearing

### 4.1. Influence of Working Condition Parameters

The changes ofFa,Fr, andni have significant influence on the dynamic characteristics of bearing. Based on the variation of these three working conditions (Table 3), the variation rule of bearing dynamic parameters is analyzed.Table 3
Three different conditions.
operating condition constant operating condition constant operating condition F a=1000:500:3000 N F r=400N ni=120000r/min F r=1000:100:1500N F a=1000N ni=100000r/min ni=100000:10000:140000r/min F a=2500N F r=400N


#### 4.1.1. Influence of Working Conditions on the Spin-To-Roll Ratio

As an important parameter of the angular contact bearing at high speed, spin-to-roll ratio represents the state of the ball rolling in the raceway. Friction and heating are produced during the spinning of the raceway relative to the ball, and the rotation sliding is more serious with large spin-to-roll ratio, which will lead to the severe heat and wear. Therefore, the spin-to-roll ratio should be reduced as much as possible during operation. The change rule of spin-to-roll ratio is analyzed under three working conditions as tabulated in Table3.Figures8 and 9 show that the variations of the spin-to-roll ratio of ball- inner and ball-outer raceway with the change ofFa. It can be illustrated that (i) the spin-to-roll ratio of the ball-inner raceway is much larger than that of the ball-outer raceway under the constantFr andni, (ii) the ball-raceway spin-to-roll ratio decreases with the increase ofFa, and such phenomena are due to the fact thatFa can reduce axial clearance and the spin angular velocity of the ball-inner and ball-outer raceways and increase the spindle bearing axial preload, which will result in spin-to-roll ratio lower, (iii) the ball spin-to-roll ratio is different at any ball azimuth location with the influence ofFr, (iv) there exists a certain spin-to-roll ratio between the ball and the outer raceway, which indicates that the high speed spindle bearing of oil lubrication does not conform to the hypothesis of outer raceway control, and (v) whenFa=1000 N andFa=1500 N, the ball at the location angle of 120° has the maximum spin-to-roll ratio; with the increase of the load, the spin-to-roll ratio of the ball in this position gradually becomes the minimum. The reason for this phenomenon is that when bearing axial load is relatively small, there is axial clearance between ball and outer ring, which cannot make the ball press outer ring tightly that results in high spin angular velocity.Figure 8
Spin-to-roll ratio of ball and inner raceway under differentFa.Figure 9
Spin-to-roll ratio of ball and outer raceway under differentFa.Figures10 and 11 show the variation curves of the spin-to-roll ratio of ball-inner and ball-outer raceways with respect to the radial load. Results show that, with constantFa andni, the spin-to-roll ratio of the ball-inner raceway decreases in the loading area and increases in the nonloading area, because the radial load can decrease the radial clearance in the loading area and then the ball and the raceway are tightly pressed, which resulted in the spin-to-roll ratio reducing. On the contrary, the spin-to-roll ratio will increase in the nonloading area. It also can be found that the radial load has less effect on the spin-to-roll ratio of the ball-outer raceway than of the ball-inner raceway.Figure 10
Spin-to-roll ratio of ball and inner raceway under differentFr.Figure 11
Spin-to-roll ratio of ball and outer raceway under differentFr.As shown in Figures12 and 13, under the constant loadsFa andFr, the spin-to-roll ratio of the ball-inner raceway increases obviously withni increase; comparatively, the spin-to-roll ratio of the ball-outer raceway increases gradually. Such phenomena are due to the fact that the centrifugal force increased withni increasing, which leads to the decrease of the contact load of the ball-inner raceway and the increase of the sliding of the ball-inner raceway and the spin-to-roll ratio obviously. The increase of the centrifugal force can also increase the contact load of the ball and the raceway, restrain the sliding between the ball and the raceway, and slow down the spin-to-roll ratio increase. The ball at location angle of 120° has the maximum spin-to roll ratio with the inner ring and the minimum spin-to-roll ratio with the outer ring; the reason for this phenomenon is that the ball at location angle of 120° is affected by the maximum centrifugal force.Figure 12
Spin-to-roll ratio of ball and inner raceway with differentni.Figure 13
Spin-to-roll ratio of ball and outer raceway with differentni.


#### 4.1.2. Influence of Working Conditions onωby

Figure14 shows the ωby curves in the direction ofy-axis in the azimuth coordinate system with the change ofFa. Under the condition loadFr and the constant rotational speedni, ωby gradually increases withFa increasing. WhenFa=3000 N, ωby is basically stable at any ball azimuth location. Such phenomena suggest thatFa can reduce the axial clearance and decrease the influence ofFr on the ball, and the load condition of the ball tends to be steady.Figure 14
ω b y curves under differentFa.The changing curves ofωby alongy-axis in the azimuth coordinate system with the change ofFr are shown in Figure 15. It can be illustrated that whenFa andni are constant, ωby gradually decreases with the increase ofFr, which is larger in the loading area. Such phenomena are due to the fact thatFr can reduce the radial clearance and inhibits the gyroscopic motion between the ball and the raceway and typically lead to ωby decreasing significantly in loading area.Figure 15
ω b y curves under differentFr.As shown in the Figure16 of ωby changing curves with the change ofni, under the constant loadFa andFr, the value of ωby gradually increases withni increasing, and the ωby of the ball varies obviously with different ball azimuth location. ωby increases slowly with the increase ofni in the loading area, whereas the increase is obvious in nonloading areas. Such phenomena are due to the fact that the radial load holds the ball and the raceway tight in the loading area and inhibits the changes of the ball in the space position, which causes ωby increasing slowly. Whenni is greater than 110000r/min, at location angle of 120°, the ωby value of the ball reaches the minimum. The reason for this phenomenon is that the ball at this position bear the maximum load because of radial load Fr, and the increased centrifugal force compacts the ball and the raceway, resulting in the minimum ωby here.Figure 16
ω b y curves with differentni.


### 4.2. Influence of Working Conditions on Cage

#### 4.2.1. Cage Forces

There exits three types of interaction conditions between the ball and the pocket of cage, such as (i) the pocket colliding the balls, (ii) the balls colliding the pocket, and (iii) no contact between the ball and the pocket. In order to study the relative motions between the ball and the pocket, the three working condition as tabulated in Table2 were analyzed.Figures17–19 illustrate the Fbc curves between the ball and the pocket of cage with differentFa,Fr, andni. In these Figures, the negative value represents the pocket colliding the ball and conversely the positive is the ball colliding the pocket. (i) It can be illustrated that the normal load between the ball and the pocket of cage is very small during the steady-state working period; (ii) the ball and the pocket collide with each other. It will last until the dynamic equilibrium state is achieved during the bearing steady working state. (iii) Figure 19 shows that asni increases, the normal load between the cage and the ball gradually decreases in the loading area. The reason for this phenomenon is that the centrifugal force increases withni increasing in the loading area; it leads to the fact that the ball and the raceway are compressed; and the result is that the ball rotation angular velocity decreases, making the contact force between the ball and the cage decrease. In no-load areas, the opposite is true.Figure 17
F b c curves under differentFa.Figure 18
F b c curves under differentFr.Figure 19
F b c curves under differentni.


#### 4.2.2. Cage Rotation Speed

In this paper, the relationship between the cage rotational angular speed and the ball revolution angular speed is studied from the three working conditions listed in Table2.ω c and ωm curves of with differentFa, Fr, andni are shown in Figures 20–22, which suggest that ωc and ωm decrease with the increase ofFa, and increase with the increase ofni, while the change ofFr has little effect on ωc and ωm. The value of ωc lies in the middle position of the values of ωm. Therefore, it is reasonable to take the average speed value of the balls as the cage rotational angular speed.Figure 20
ω c and ωm curves with differentFa.Figure 21
ω c and ωm curves with differentFr.Figure 22
ω c and ωm curves with differentni.


### 4.3. Influence of Working Conditions on Gyroscopic Moment

The gyroscopic moment is a dynamic inertial force produced during the process of bearing operation, which is harmful for the bearing operation condition. Especially, under the condition of high speed and light load, it causes the sliding, which will lead to serious friction, makes the bearing internal fever, heats up, and accelerates the bearing degeneration.Figures23 and 24 show the curves ofMy andMz with the change ofFa. Under the constant values ofFr andni, (i)My andMz increase obviously with the increase ofFa, (ii) whenFa=3000 N,Mz tends to be stable at each ball azimuth location. Such phenomena are due to the fact thatFa reduces the axial clearance and the impact ofFr on balls and results in the same load condition for balls.Figure 23
M y curves with differentFa.Figure 24
M z curves with differentFa.Figures25 and 26 show the curves ofMy andMz with the change ofFr. Under the constant values ofFa andni,My andMz gradually decrease with the increase ofFr. It is because that the increase ofFr will decrease the radial clearance of the bearing; therefore, the rolling body and the ring are pressed tightly, and the gyroscopic moment is suppressed and reduced.Figure 25
M y curves with differentFr.Figure 26
M z curves with differentFr.Figures27 and 28 show the curves ofMy andMz with the change ofni in the direction ofy andz of the azimuth coordinate system of the ball. It is illustrated thatMy andMz increase significantly with the increase ofni under the constant load ofFa andFr.Figure 27
M y curves with differentni.Figure 28
M z curves with differentni.


## 4.1. Influence of Working Condition Parameters

The changes ofFa,Fr, andni have significant influence on the dynamic characteristics of bearing. Based on the variation of these three working conditions (Table 3), the variation rule of bearing dynamic parameters is analyzed.Table 3
Three different conditions.
operating condition constant operating condition constant operating condition F a=1000:500:3000 N F r=400N ni=120000r/min F r=1000:100:1500N F a=1000N ni=100000r/min ni=100000:10000:140000r/min F a=2500N F r=400N


### 4.1.1. Influence of Working Conditions on the Spin-To-Roll Ratio

As an important parameter of the angular contact bearing at high speed, spin-to-roll ratio represents the state of the ball rolling in the raceway. Friction and heating are produced during the spinning of the raceway relative to the ball, and the rotation sliding is more serious with large spin-to-roll ratio, which will lead to the severe heat and wear. Therefore, the spin-to-roll ratio should be reduced as much as possible during operation. The change rule of spin-to-roll ratio is analyzed under three working conditions as tabulated in Table3.Figures8 and 9 show that the variations of the spin-to-roll ratio of ball- inner and ball-outer raceway with the change ofFa. It can be illustrated that (i) the spin-to-roll ratio of the ball-inner raceway is much larger than that of the ball-outer raceway under the constantFr andni, (ii) the ball-raceway spin-to-roll ratio decreases with the increase ofFa, and such phenomena are due to the fact thatFa can reduce axial clearance and the spin angular velocity of the ball-inner and ball-outer raceways and increase the spindle bearing axial preload, which will result in spin-to-roll ratio lower, (iii) the ball spin-to-roll ratio is different at any ball azimuth location with the influence ofFr, (iv) there exists a certain spin-to-roll ratio between the ball and the outer raceway, which indicates that the high speed spindle bearing of oil lubrication does not conform to the hypothesis of outer raceway control, and (v) whenFa=1000 N andFa=1500 N, the ball at the location angle of 120° has the maximum spin-to-roll ratio; with the increase of the load, the spin-to-roll ratio of the ball in this position gradually becomes the minimum. The reason for this phenomenon is that when bearing axial load is relatively small, there is axial clearance between ball and outer ring, which cannot make the ball press outer ring tightly that results in high spin angular velocity.Figure 8
Spin-to-roll ratio of ball and inner raceway under differentFa.Figure 9
Spin-to-roll ratio of ball and outer raceway under differentFa.Figures10 and 11 show the variation curves of the spin-to-roll ratio of ball-inner and ball-outer raceways with respect to the radial load. Results show that, with constantFa andni, the spin-to-roll ratio of the ball-inner raceway decreases in the loading area and increases in the nonloading area, because the radial load can decrease the radial clearance in the loading area and then the ball and the raceway are tightly pressed, which resulted in the spin-to-roll ratio reducing. On the contrary, the spin-to-roll ratio will increase in the nonloading area. It also can be found that the radial load has less effect on the spin-to-roll ratio of the ball-outer raceway than of the ball-inner raceway.Figure 10
Spin-to-roll ratio of ball and inner raceway under differentFr.Figure 11
Spin-to-roll ratio of ball and outer raceway under differentFr.As shown in Figures12 and 13, under the constant loadsFa andFr, the spin-to-roll ratio of the ball-inner raceway increases obviously withni increase; comparatively, the spin-to-roll ratio of the ball-outer raceway increases gradually. Such phenomena are due to the fact that the centrifugal force increased withni increasing, which leads to the decrease of the contact load of the ball-inner raceway and the increase of the sliding of the ball-inner raceway and the spin-to-roll ratio obviously. The increase of the centrifugal force can also increase the contact load of the ball and the raceway, restrain the sliding between the ball and the raceway, and slow down the spin-to-roll ratio increase. The ball at location angle of 120° has the maximum spin-to roll ratio with the inner ring and the minimum spin-to-roll ratio with the outer ring; the reason for this phenomenon is that the ball at location angle of 120° is affected by the maximum centrifugal force.Figure 12
Spin-to-roll ratio of ball and inner raceway with differentni.Figure 13
Spin-to-roll ratio of ball and outer raceway with differentni.


### 4.1.2. Influence of Working Conditions onωby

Figure14 shows the ωby curves in the direction ofy-axis in the azimuth coordinate system with the change ofFa. Under the condition loadFr and the constant rotational speedni, ωby gradually increases withFa increasing. WhenFa=3000 N, ωby is basically stable at any ball azimuth location. Such phenomena suggest thatFa can reduce the axial clearance and decrease the influence ofFr on the ball, and the load condition of the ball tends to be steady.Figure 14
ω b y curves under differentFa.The changing curves ofωby alongy-axis in the azimuth coordinate system with the change ofFr are shown in Figure 15. It can be illustrated that whenFa andni are constant, ωby gradually decreases with the increase ofFr, which is larger in the loading area. Such phenomena are due to the fact thatFr can reduce the radial clearance and inhibits the gyroscopic motion between the ball and the raceway and typically lead to ωby decreasing significantly in loading area.Figure 15
ω b y curves under differentFr.As shown in the Figure16 of ωby changing curves with the change ofni, under the constant loadFa andFr, the value of ωby gradually increases withni increasing, and the ωby of the ball varies obviously with different ball azimuth location. ωby increases slowly with the increase ofni in the loading area, whereas the increase is obvious in nonloading areas. Such phenomena are due to the fact that the radial load holds the ball and the raceway tight in the loading area and inhibits the changes of the ball in the space position, which causes ωby increasing slowly. Whenni is greater than 110000r/min, at location angle of 120°, the ωby value of the ball reaches the minimum. The reason for this phenomenon is that the ball at this position bear the maximum load because of radial load Fr, and the increased centrifugal force compacts the ball and the raceway, resulting in the minimum ωby here.Figure 16
ω b y curves with differentni.


## 4.1.1. Influence of Working Conditions on the Spin-To-Roll Ratio

As an important parameter of the angular contact bearing at high speed, spin-to-roll ratio represents the state of the ball rolling in the raceway. Friction and heating are produced during the spinning of the raceway relative to the ball, and the rotation sliding is more serious with large spin-to-roll ratio, which will lead to the severe heat and wear. Therefore, the spin-to-roll ratio should be reduced as much as possible during operation. The change rule of spin-to-roll ratio is analyzed under three working conditions as tabulated in Table3.Figures8 and 9 show that the variations of the spin-to-roll ratio of ball- inner and ball-outer raceway with the change ofFa. It can be illustrated that (i) the spin-to-roll ratio of the ball-inner raceway is much larger than that of the ball-outer raceway under the constantFr andni, (ii) the ball-raceway spin-to-roll ratio decreases with the increase ofFa, and such phenomena are due to the fact thatFa can reduce axial clearance and the spin angular velocity of the ball-inner and ball-outer raceways and increase the spindle bearing axial preload, which will result in spin-to-roll ratio lower, (iii) the ball spin-to-roll ratio is different at any ball azimuth location with the influence ofFr, (iv) there exists a certain spin-to-roll ratio between the ball and the outer raceway, which indicates that the high speed spindle bearing of oil lubrication does not conform to the hypothesis of outer raceway control, and (v) whenFa=1000 N andFa=1500 N, the ball at the location angle of 120° has the maximum spin-to-roll ratio; with the increase of the load, the spin-to-roll ratio of the ball in this position gradually becomes the minimum. The reason for this phenomenon is that when bearing axial load is relatively small, there is axial clearance between ball and outer ring, which cannot make the ball press outer ring tightly that results in high spin angular velocity.Figure 8
Spin-to-roll ratio of ball and inner raceway under differentFa.Figure 9
Spin-to-roll ratio of ball and outer raceway under differentFa.Figures10 and 11 show the variation curves of the spin-to-roll ratio of ball-inner and ball-outer raceways with respect to the radial load. Results show that, with constantFa andni, the spin-to-roll ratio of the ball-inner raceway decreases in the loading area and increases in the nonloading area, because the radial load can decrease the radial clearance in the loading area and then the ball and the raceway are tightly pressed, which resulted in the spin-to-roll ratio reducing. On the contrary, the spin-to-roll ratio will increase in the nonloading area. It also can be found that the radial load has less effect on the spin-to-roll ratio of the ball-outer raceway than of the ball-inner raceway.Figure 10
Spin-to-roll ratio of ball and inner raceway under differentFr.Figure 11
Spin-to-roll ratio of ball and outer raceway under differentFr.As shown in Figures12 and 13, under the constant loadsFa andFr, the spin-to-roll ratio of the ball-inner raceway increases obviously withni increase; comparatively, the spin-to-roll ratio of the ball-outer raceway increases gradually. Such phenomena are due to the fact that the centrifugal force increased withni increasing, which leads to the decrease of the contact load of the ball-inner raceway and the increase of the sliding of the ball-inner raceway and the spin-to-roll ratio obviously. The increase of the centrifugal force can also increase the contact load of the ball and the raceway, restrain the sliding between the ball and the raceway, and slow down the spin-to-roll ratio increase. The ball at location angle of 120° has the maximum spin-to roll ratio with the inner ring and the minimum spin-to-roll ratio with the outer ring; the reason for this phenomenon is that the ball at location angle of 120° is affected by the maximum centrifugal force.Figure 12
Spin-to-roll ratio of ball and inner raceway with differentni.Figure 13
Spin-to-roll ratio of ball and outer raceway with differentni.


## 4.1.2. Influence of Working Conditions onωby

Figure14 shows the ωby curves in the direction ofy-axis in the azimuth coordinate system with the change ofFa. Under the condition loadFr and the constant rotational speedni, ωby gradually increases withFa increasing. WhenFa=3000 N, ωby is basically stable at any ball azimuth location. Such phenomena suggest thatFa can reduce the axial clearance and decrease the influence ofFr on the ball, and the load condition of the ball tends to be steady.Figure 14
ω b y curves under differentFa.The changing curves ofωby alongy-axis in the azimuth coordinate system with the change ofFr are shown in Figure 15. It can be illustrated that whenFa andni are constant, ωby gradually decreases with the increase ofFr, which is larger in the loading area. Such phenomena are due to the fact thatFr can reduce the radial clearance and inhibits the gyroscopic motion between the ball and the raceway and typically lead to ωby decreasing significantly in loading area.Figure 15
ω b y curves under differentFr.As shown in the Figure16 of ωby changing curves with the change ofni, under the constant loadFa andFr, the value of ωby gradually increases withni increasing, and the ωby of the ball varies obviously with different ball azimuth location. ωby increases slowly with the increase ofni in the loading area, whereas the increase is obvious in nonloading areas. Such phenomena are due to the fact that the radial load holds the ball and the raceway tight in the loading area and inhibits the changes of the ball in the space position, which causes ωby increasing slowly. Whenni is greater than 110000r/min, at location angle of 120°, the ωby value of the ball reaches the minimum. The reason for this phenomenon is that the ball at this position bear the maximum load because of radial load Fr, and the increased centrifugal force compacts the ball and the raceway, resulting in the minimum ωby here.Figure 16
ω b y curves with differentni.


## 4.2. Influence of Working Conditions on Cage

### 4.2.1. Cage Forces

There exits three types of interaction conditions between the ball and the pocket of cage, such as (i) the pocket colliding the balls, (ii) the balls colliding the pocket, and (iii) no contact between the ball and the pocket. In order to study the relative motions between the ball and the pocket, the three working condition as tabulated in Table2 were analyzed.Figures17–19 illustrate the Fbc curves between the ball and the pocket of cage with differentFa,Fr, andni. In these Figures, the negative value represents the pocket colliding the ball and conversely the positive is the ball colliding the pocket. (i) It can be illustrated that the normal load between the ball and the pocket of cage is very small during the steady-state working period; (ii) the ball and the pocket collide with each other. It will last until the dynamic equilibrium state is achieved during the bearing steady working state. (iii) Figure 19 shows that asni increases, the normal load between the cage and the ball gradually decreases in the loading area. The reason for this phenomenon is that the centrifugal force increases withni increasing in the loading area; it leads to the fact that the ball and the raceway are compressed; and the result is that the ball rotation angular velocity decreases, making the contact force between the ball and the cage decrease. In no-load areas, the opposite is true.Figure 17
F b c curves under differentFa.Figure 18
F b c curves under differentFr.Figure 19
F b c curves under differentni.


### 4.2.2. Cage Rotation Speed

In this paper, the relationship between the cage rotational angular speed and the ball revolution angular speed is studied from the three working conditions listed in Table2.ω c and ωm curves of with differentFa, Fr, andni are shown in Figures 20–22, which suggest that ωc and ωm decrease with the increase ofFa, and increase with the increase ofni, while the change ofFr has little effect on ωc and ωm. The value of ωc lies in the middle position of the values of ωm. Therefore, it is reasonable to take the average speed value of the balls as the cage rotational angular speed.Figure 20
ω c and ωm curves with differentFa.Figure 21
ω c and ωm curves with differentFr.Figure 22
ω c and ωm curves with differentni.


## 4.2.1. Cage Forces

There exits three types of interaction conditions between the ball and the pocket of cage, such as (i) the pocket colliding the balls, (ii) the balls colliding the pocket, and (iii) no contact between the ball and the pocket. In order to study the relative motions between the ball and the pocket, the three working condition as tabulated in Table2 were analyzed.Figures17–19 illustrate the Fbc curves between the ball and the pocket of cage with differentFa,Fr, andni. In these Figures, the negative value represents the pocket colliding the ball and conversely the positive is the ball colliding the pocket. (i) It can be illustrated that the normal load between the ball and the pocket of cage is very small during the steady-state working period; (ii) the ball and the pocket collide with each other. It will last until the dynamic equilibrium state is achieved during the bearing steady working state. (iii) Figure 19 shows that asni increases, the normal load between the cage and the ball gradually decreases in the loading area. The reason for this phenomenon is that the centrifugal force increases withni increasing in the loading area; it leads to the fact that the ball and the raceway are compressed; and the result is that the ball rotation angular velocity decreases, making the contact force between the ball and the cage decrease. In no-load areas, the opposite is true.Figure 17
F b c curves under differentFa.Figure 18
F b c curves under differentFr.Figure 19
F b c curves under differentni.


## 4.2.2. Cage Rotation Speed

In this paper, the relationship between the cage rotational angular speed and the ball revolution angular speed is studied from the three working conditions listed in Table2.ω c and ωm curves of with differentFa, Fr, andni are shown in Figures 20–22, which suggest that ωc and ωm decrease with the increase ofFa, and increase with the increase ofni, while the change ofFr has little effect on ωc and ωm. The value of ωc lies in the middle position of the values of ωm. Therefore, it is reasonable to take the average speed value of the balls as the cage rotational angular speed.Figure 20
ω c and ωm curves with differentFa.Figure 21
ω c and ωm curves with differentFr.Figure 22
ω c and ωm curves with differentni.


## 4.3. Influence of Working Conditions on Gyroscopic Moment

The gyroscopic moment is a dynamic inertial force produced during the process of bearing operation, which is harmful for the bearing operation condition. Especially, under the condition of high speed and light load, it causes the sliding, which will lead to serious friction, makes the bearing internal fever, heats up, and accelerates the bearing degeneration.Figures23 and 24 show the curves ofMy andMz with the change ofFa. Under the constant values ofFr andni, (i)My andMz increase obviously with the increase ofFa, (ii) whenFa=3000 N,Mz tends to be stable at each ball azimuth location. Such phenomena are due to the fact thatFa reduces the axial clearance and the impact ofFr on balls and results in the same load condition for balls.Figure 23
M y curves with differentFa.Figure 24
M z curves with differentFa.Figures25 and 26 show the curves ofMy andMz with the change ofFr. Under the constant values ofFa andni,My andMz gradually decrease with the increase ofFr. It is because that the increase ofFr will decrease the radial clearance of the bearing; therefore, the rolling body and the ring are pressed tightly, and the gyroscopic moment is suppressed and reduced.Figure 25
M y curves with differentFr.Figure 26
M z curves with differentFr.Figures27 and 28 show the curves ofMy andMz with the change ofni in the direction ofy andz of the azimuth coordinate system of the ball. It is illustrated thatMy andMz increase significantly with the increase ofni under the constant load ofFa andFr.Figure 27
M y curves with differentni.Figure 28
M z curves with differentni.


## 5. Conclusions

(1)
The change ofFa,Fr, andni can affect the spin-to-roll ratio of ball-inner raceway and ball-outer raceway. In different working conditions, the spin-to-roll ratio of the ball-inner raceway is greater than that of the ball-outer raceway. In high speed, the outer raceway control hypothesis is only a special case. In general, there is a spin angular velocity between the ball and the outer raceway. (2)
The increase ofFa andni of spindle bearing can cause ωby increase; however, the increase ofFr will cause ωby decrease. A specificFa can reduce the axial clearance of the bearing, which will make the azimuth of the ball tend to be stable in the space. (3)
The interaction between the ball and the pocket of cage is that not only does the ball collide with the pocket but also the pocket collides with the ball, which can achieve the dynamic equilibrium state of the bearing steady-state work. (4)
It is reasonable to take the average speed value of the balls as the cage rotational angular speed. (5)
Gyroscope moment increases significantly with the increase ofFa andni. The increase ofFr can suppress the gyroscopic movement.



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*Source: 7050151-2019-01-17.xml*