Numerical Study on Raceway Wear of Angular Contact Ball Bearings Considering Curvature Radius Variation

Abstract

Based on outer raceway control theory and a five-degree-of-freedom quasi-static model of angular contact ball bearings, a raceway wear model considering curvature radius variation is proposed, which couples the quasi-static model with a modified Archard wear formulation and a dynamic curvature radius update mechanism. As wear accumulates, the worn curvature radii are fed back into the quasi-static model to recalculate the raceway contact dynamic parameters. Taking the SKF 7012ACE/HCP4A spindle bearing as an example, the wear depth evolution and the variations of contact ellipse area, contact stress, sliding velocity, and wear coefficient with wear time are investigated under combined loads. The results indicate that as wear progresses, the raceway curvature radii increase, leading to a decrease in contact ellipse area but an increase in contact stress and sliding velocity, which in turn accelerates the wear process. The findings demonstrate that the degradation of raceway curvature radius has a cumulative and non-negligible influence on wear evolution and should be incorporated into bearing wear calculations for more accurate life prediction.

1. Introduction

Angular contact ball bearings (ACBBs) are critical components in precision rotating machinery. Under combined loads, involving axial force, radial force, and overturning moment, the dynamic parameters of the raceway exhibit complex nonlinear behavior, which affects the wear evolution of the bearing under long-term operation. An accurate understanding of the wear process and its interaction with bearing kinematics is the foundation for achieving reliable life prediction and condition-based maintenance.

The mechanical modeling of ACBBs has undergone continuous refinement since the seminal work of Jones [1], who first formulated the raceway control theory and established the classical quasi-static model for low- and high-speed conditions. Harris [2] systematically extended this framework by incorporating centrifugal force and gyroscopic moment effects, establishing the standard methodology for rolling bearing analysis that remains widely adopted. Subsequent developments introduced five degrees of freedom (5-DOF) in bearing modeling. De Mul et al. [3] presented an equilibrium and load distribution model for ball bearings loaded in five degrees of freedom, laying the foundation for multi-DOF analysis. Gunduz and Singh [4] derived an analytical stiffness matrix formulation for double-row ACBBs, and Lin and Jiang [5] further developed a 5-DOF quasi-static model for preloaded duplex ACBBs with experimental validation. Jiang and Mao [6] investigated variable optimum preload for machine tool spindles, and Guo and Parker [7] proposed a stiffness matrix calculation method using a finite element/contact mechanics model. He et al. [8] studied the thermo-mechanical coupling characteristics of ACBBs with fixed-position preload, while Lim and Singh [9] analyzed vibration transmission through rolling element bearings. More recently, Tu et al. [10] established a quasi-dynamic model that comprehensively considers the inclination angle due to external lateral moment, viscous drag, inertial force, and the spin-sliding behavior of the ball in the contact ellipse, emphasizing the contact characteristics under combined radial and moment loads. Wang et al. [11] proposed a quasi-static model without the raceway control hypothesis to improve simulation accuracy under combined loading conditions. Guo et al. [12] conducted dynamics modeling and analysis of rolling bearings with local faults in a variable stiffness system, and Zhao et al. [13] performed parameter research on the time-varying stiffness of ball bearing systems without the race control hypothesis. These quasi-static and quasi-dynamic models have been demonstrated to be capable of providing reliable raceway dynamic parameters—including contact angles, contact loads, and sliding velocities—that serve as necessary inputs for subsequent wear analysis.

Wear modeling for rolling bearings has been extensively studied over the past decades. The classical Archard wear model [14] has been widely adopted as the theoretical foundation for quantifying material removal in sliding contacts. El-Thalji and Jantunen [15] developed a descriptive model of wear evolution in rolling bearings and correlated vibration characteristics with wear progression. Olofsson et al. [16] simulated mild wear in boundary-lubricated spherical roller thrust bearings using an Archard-based approach. Janakiraman et al. [17] investigated the impacts of contact parameters on the wear coefficient, providing essential experimental data for model calibration. Liu et al. [18] applied the Archard model to estimate high-speed wear lifetime of instrument ball bearings, and Yu et al. [19] established a wear prediction model for ACBBs under mixed lubrication conditions by coupling the oil film and surface roughness effects. Yang et al. [20] focused on the effect of torque on spindle bearing wear, and Shen et al. [21] conducted numerical simulation of sliding wear for self-lubricating spherical plain bearings. Alfares and Elsharkawy [22] examined the effects of axial preloading of ACBBs on grinding machine spindle system dynamics, and Lu et al. [23] presented a probabilistic approach to wear lifetime prediction for hinge configurations based on numerical simulation. In parallel, research on the coupling between wear and system dynamics has gained increasing attention. Li et al. [24] introduced wear depth into the rotor-bearing system dynamics model as bearing clearance and analyzed the dynamic evolution of the system with bearing wear. Xi et al. [25] presented a dynamic wear simulation model for paired ACBBs and investigated the coupling effects of wear and rolling contact on contact stiffness and nonlinear dynamic characteristics. Liu and Tu [26] provided a comparative study of bearing wear in spindle systems at different working conditions. Tu et al. [27] further conducted a wear analysis of ACBBs in multiple-bearing spindle systems subjected to uncertain initial angular misalignment. Xiong et al. [28] studied the wear of tapered roller bearings under different service conditions. In terms of theoretical foundations, Brewe and Hamrock [29] provided a simplified solution for elliptical contact deformation between two elastic solids, and Wan [30] compiled the standard analytical method for rolling bearings. Wang et al. [31] presented a unified rolling contact tribology analytical model for dry-lubricated ACBBs under extreme conditions, while Sun et al. [32] recently performed a time-variant reliability analysis considering the coupled effect of rolling contact fatigue damage and wear. Houpert [33] compared analytical, numerical and experimental results for ball and tapered roller bearing torque, and Cann and Lubrecht [34] as well as Lugt [35] contributed to the understanding of lubricant replenishment and grease lubrication in rolling bearings.

The aforementioned research has made a series of advances in the wear modeling of bearings. However, in the modeling of raceway wear, the initial raceway parameters are retained throughout the calculation—that is, the raceway geometric profile, especially the raceway curvature radius, is generally assumed to remain unchanged during the entire wear process. The wear depth is typically calculated based on the initial bearing geometry, and the resulting geometric degradation is not fed back into the contact mechanics model. In actual operation, however, the raceway curvature radius varies continuously with time as wear progresses. Changes in the raceway groove curvature radius often affect the internal load distribution, contact stress, and fatigue life of the bearing [36,37]; likewise, variations in the raceway curvature also influence the contact state and wear behavior [38]. Nevertheless, the coupling between wear-induced curvature radius degradation and the time-varying contact parameters—including the contact ellipse area, contact stress distribution, sliding velocity, and wear coefficient—has so far received relatively little research attention.

In summary, based on previous research, this paper proposes a bearing raceway wear model that couples a five-degree-of-freedom quasi-static bearing model, the Archard wear model, and a dynamic curvature radius update mechanism. The raceway curvature radius variation induced by wear is incorporated into the wear calculation loop: as wear time accumulates, the updated curvature radii are fed back into the quasi-static model to recalculate the contact dynamic parameters, which are then used to determine the wear rate for the next time step. Using this model and taking the SKF 7012ACE/HCP4A spindle bearing as the research object, this paper investigates the effects of operating conditions on the wear coefficient and sliding velocity distribution on the raceway elliptical contact surface; the variation of raceway curvature radii under prolonged wear and its influence on the contact ellipse area, contact stress, sliding velocity, and wear coefficient; and the evolution of the wear depth of the inner and outer raceways with and without considering curvature radius degradation.

2. Theoretical Model of Angular Contact Ball Bearing

2.1. Geometric Analysis

Figure 1 shows the static-state geometric structure of the bearing. The center of the ball $O_b$, the inner curvature centers $O_i$, and the outer curvature center $O_o$ are collinear in the static state. The inner and outer contact angles, ${\alpha }_i$, ${\alpha }_o$, are the same with the initial contact angles ${\alpha }_0$. The distance between $O_i$ and $O_o$ is $BD$ [1], and the component of the distance between $O_i$ and the bearing center in the radial plane is $R_i$ [1].

$$BD=(f_i+f_o-1)D_b$$

(1)

$$R_i=0.5d_m+(f_i-0.5)D_b\cos {\alpha }_0$$

(2)

The inner curvature radius coefficient is $f_i=r_i/D_b$, and the inner curvature radius is $r_i$; the outer curvature radius coefficient is $f_o=r_o/D_b$, and the outer curvature radius is $r_o$; and the ball’s diameter is $D_b$, and the pitch circle diameter is $d_m$, $\gamma =D_b/d_m$.

Under combined load and speed, the displacement vector between the inner and outer rings is ${\delta }_b=[{\delta }_x{\delta }_y{\delta }_z{\theta }_x{\theta }_y]$; ${\delta }_x$, ${\delta }_y$ are radial displacement values perpendicular to each other; ${\delta }_z$ is the axial displacement; and ${\theta }_x$ and ${\theta }_y$ are inclination angles generated by the $X$ and $Y$ axes. In Figure 2, ${\phi }_j$ is the $j$-th ball’s position angle along the raceway, and the ball’s number is $Z$; then, ${\phi }_j=2\pi (j-1)/Z$.

High speed, and affected by centrifugal and gyroscopic effects, causes the ball at each position angle to deviate from the geometric position in the static state. $O_i$, $O_b$ and $O_o$ change from three points and one line in the static state to the broken-line state as shown in Figure 3. All displacements are defined relative to the stationary outer ring, independent of the operating conditions.

In Figure 3, $f_d$ is the radial component of the motion from $O_i$ to $O_i^{\prime }$, and the axial component is $f_z$. Under the influence of elastic deformation, The distance between the ball’s center to the inner raceway’s center and outer raceway’s center becomes $\overline{O_b^{\prime }{O}_i^{\prime }}$, $\overline{O_b^{\prime }{O}_o}$. The radial and axial distances from $O_i$ to $O_o$ become $A_{dj}$, $A_{zj}$; The inner and outer contact angles become ${\alpha }_{ij}$ and ${\alpha }_{oj}$. The radial and axial distances from $O_b$ to $O_i$, $O_o$ become $X_{dj}$, $X_{zj}$.

$$f_d={\delta }_x\sin {\phi }_j+{\delta }_y\cos {\phi }_j,f_z={\delta }_z+R_i({\theta }_x\cos {\phi }_j-{\theta }_y\sin {\phi }_j)$$

(3)

$$\overline{O_b^{\prime }{O}_i^{\prime }}=(f_i-0.5)\cdot D_b+{\delta }_{ij},\overline{O_b^{\prime }{O}_o}=(f_o-0.5)\cdot D_b+{\delta }_{oj}$$

(4)

$$A_{dj}=BD\cos {\alpha }_0+f_d,A_{dj}=\overline{O_b^{\prime }{O}_o}\cos {\alpha }_{oj}+\overline{O_b^{\prime }{O}_i^{\prime }}\cos {\alpha }_{ij}$$

(5)

$$A_{zj}=BD\sin {\alpha }_0+{\delta }_0+f_z,A_{zj}=\overline{O_b^{\prime }{O}_o}\sin {\alpha }_{oj}+\overline{O_b^{\prime }{O}_i^{\prime }}\sin {\alpha }_{ij}$$

(6)

$$\cos {\alpha }_{ij}={\displaystyle \frac{A_{dj}-X_{dj}}{\overline{O_b^{\prime }{O}_i^{\prime }}}},\sin {\alpha }_{ij}={\displaystyle \frac{A_{zj}-X_{zj}}{\overline{O_b^{\prime }{O}_i^{\prime }}}},\cos {\alpha }_{oj}={\displaystyle \frac{X_{dj}}{\overline{O_b^{\prime }{O}_i^{\prime }}}},\sin {\alpha }_{oj}={\displaystyle \frac{X_{zj}}{\overline{O_b^{\prime }{O}_i^{\prime }}}}$$

(7)

It is assumed in this paper that the bearing operates under axial preload and the initial radial clearance has been eliminated. In Equation (6), ${\delta }_0$ is the axial displacement of the inner ring under the preload $F_0$, which can be calculated by the following formula:

$${\delta }_0=BD{\displaystyle \frac{\sin ({\alpha }_f-{\alpha }_0)}{\cos {\alpha }_f}}$$

(8)

In the expression, ${\alpha }_f$ is the contact angle after the load is applied, which can be calculated by the following formula:

$${\displaystyle \frac{F_0}{Z_b{K}_{m}B{D}_m^{1.5}}}=\sin {\alpha }_f{\left({\displaystyle \frac{\cos {\alpha }_0}{\cos {\alpha }_f}}-1\right)}^{1.5}$$

(9)

where $K_m$ is a function of the changed contact angle. Equation (9) can be solved using the Newton–Raphson method [39] to obtain the changed contact angle ${\alpha }_f$ of the bearing under the initial preload $F_0$.

According to the geometric relation of the ball’s center $O_b^{\prime }$ in Figure 3, the displacement coordination equation of the $j$-th ball can be obtained as

$${(A_{dj}-X_{dj})}^2+{(A_{zj}-X_{zj})}^2=\overline{O_b^{\prime }{O_i^{\prime }}^2},{X_{dj}}^2+{X_{zj}}^2=\overline{O_b^{\prime }{O_o^{\prime }}^2}$$

(10)

2.2. Force Analysis of Angular Contact Ball Bearing

Figure 4 shows the ball’s state under stress, where $F_{cj}$ [2] is the ball’s centrifugal force and $M_{gj}$ [2] is the gyro moment generated by the constant change in the rolling body’s rotation axis. $Q_{ij}$, $Q_{oj}$ is the normal force acting on the ball. $F_{ij}$ and $F_{oj}$ are, respectively, the ball’s surface friction caused by the inner and outer raceways, which is used to balance the gyro-torque generated by the ball.

$$F_{cj}={\displaystyle \frac{1}{2}}m{d}_m{\omega }^2{\left({\displaystyle \frac{{\omega }_m}{\omega }}\right)}_j^2$$

(11)

$$M_{gj}=J{\omega }^2\sin \beta {\left({\displaystyle \frac{{\omega }_b}{\omega }}\right)}_j{\left({\displaystyle \frac{{\omega }_m}{\omega }}\right)}_j$$

(12)

$$Q_{ij}=K_{ij}{\delta }_{ij}^{3/2},Q_{oj}=K_{oj}{\delta }_{oj}^{3/2}$$

(13)

$$F_{ij}=2(1-{\lambda }_j)M_{gj}/D_b,F_{oj}=2{\lambda }_j{M}_{gj}/D_b$$

(14)

In Formula (11), the ball’s mass is $m$, and ${\omega }_m/\omega$ is the ratio of the ball’s revolution velocity to the inner ring’s velocity, which can be calculated according to Formula (15). In Formula (12), $J=\rho \pi D_b^5/60$ is the ball’s rotational inertia. The ball’s density is $\rho$, and $\beta$ is the ball’s rotation axis angle. This paper uses D’Alembert’s principle from References [31,40] to calculate $\beta$. ${\omega }_b/\omega$ is the ratio of the ball’s rotation velocity to the inner ring’s velocity, which can be calculated according to Formula (17). In Formula (13), $K_{ij}$ and $K_{oj}$ are the contact stiffness values between ball and the inner and outer rings; the calculation method in reference [41] can be referred to. The value of ${\lambda }_j$ in Equation (14) is determined by the raceway integrated control method found in literature reference [42].

$${\left({\displaystyle \frac{{\omega }_m}{\omega }}\right)}_j={\displaystyle \frac{1-\gamma \cos {\alpha }_{ij}}{1+\cos ({\alpha }_{ij}-{\alpha }_{oj})}}$$

(15)

$$\tan \beta ={\displaystyle \frac{\frac{Q_{ij}{a}_{ij}{L}_{ij}}{Q_{oj}{a}_{oj}{L}_{oj}}\left(\frac{1+\gamma \cos {\alpha }_{oj}}{1-\cos {\alpha }_{ij}}+1\right)\sin {\alpha }_{ij}+2\sin {\alpha }_{oj}}{\frac{Q_{ij}{a}_{ij}{L}_{ij}}{Q_{oj}{a}_{oj}{L}_{oj}}\left(\frac{1+\gamma \cos {\alpha }_{oj}}{1-\cos {\alpha }_{ij}}+1\right)\cos {\alpha }_{ij}+2(\cos {\alpha }_{oj}+\gamma )+\gamma \frac{Q_{ij}{a}_{ij}{L}_{ij}}{Q_{oj}{a}_{oj}{L}_{oj}}\left[\cos ({\alpha }_{ij}-{\alpha }_{oj})-\frac{1+\gamma \cos {\alpha }_{oj}}{1-\cos {\alpha }_{ij}}\right]}}$$

(16)

$${\left({\displaystyle \frac{{\omega }_b}{\omega }}\right)}_j={\displaystyle \frac{-1}{(C_{oj}+C_{ij})\gamma \cos {\beta }_j}}$$

(17)

$$C_{oj}={\displaystyle \frac{\cos {\alpha }_{oj}+\tan {\beta }_j\sin {\alpha }_{oj}}{1+\gamma \cos {\alpha }_{oj}}},C_{ij}={\displaystyle \frac{\cos {\alpha }_{ij}+\tan {\beta }_j\sin {\alpha }_{ij}}{1-\gamma \cos {\alpha }_{ij}}}$$

(18)

Reference [40] unified the calculation method of β under both low-speed and high-speed conditions. By establishing a friction force equilibrium equation for the ball and then incorporating it into the overall bearing equilibrium through Equations (9)–(12), this approach avoids the limitation that, under the outer raceway control method, all spin occurs at the inner raceway contact. Using the bearing design parameters defined in Reference [40], the variation of β was calculated under the same operating conditions at both low and high speeds, and the results are shown in Figure 5. The calculated results in Figure 5 exhibit good agreement with those in Reference [40] in terms of numerical trends and accuracy.

Based on the contact loads $Q_{ij}$ and $Q_{oj}$, gyro moment $M_{gj}$, and friction forces $F_{ij}$ and $F_{oj}$ acting on the rolling element at the inner and outer raceways shown in Figure 4, and also considering the centrifugal force $F_{cj}$ acting on the rolling element, the force equilibrium equations for the rolling element in the horizontal and vertical directions are established as follows:

$$\left\{\begin{array}{l}{Q}_{ij}\sin {\alpha }_{ij}-F_{oj}\cos {\alpha }_{ij}/2=Q_{oj}\sin {\alpha }_{oj}-F_{oj}\cos {\alpha }_{oj}/2\hfill \\ Q_{ij}\cos {\alpha }_{ij}+F_{oj}\cos {\alpha }_{ij}/2+F_{cj}=Q_{oj}\cos {\alpha }_{oj}+F_{oj}\sin {\alpha }_{oj}/2\hfill \end{array}\right.$$

(19)

The forces acting on the inner ring of the bearing are balanced in the five degrees of freedom. The five-degree-of-freedom force equilibrium equations of the bearing are given in Equations (20) and (21). By taking the partial derivatives of the bearing load vector formed in Equations (20) and (21) with respect to the displacements, the 5 × 5 stiffness matrix [3] of the bearing can be obtained.

$$\left\{\begin{array}{l}{F}_x={\displaystyle \sum _{j=1}^z(Q_{ij}\cos {\alpha }_{ij}+F_{oj}\sin {\alpha }_{ij}/2)\sin {\phi }_j}\hfill \\ F_y={\displaystyle \sum _{j=1}^z(Q_{ij}\cos {\alpha }_{ij}+F_{oj}\sin {\alpha }_{ij}/2)\cos {\phi }_j}\hfill \\ F_z={\displaystyle \sum _{j=1}^z(Q_{ij}\sin {\alpha }_{ij}-F_{oj}\cos {\alpha }_{ij}/2)}\hfill \end{array}\right.$$

(20)

$$\left\{\begin{array}{l}{M}_x={\displaystyle \sum _{j=1}^z\left((Q_{ij}\right.\sin {\alpha }_{ij}-F_{oj}\cos {\alpha }_{ij}/2)R_i\cos {\phi }_j\left.+F_{oj}{r}_i\cos {\phi }_j\right)}\hfill \\ M_y={\displaystyle \sum _{j=1}^z-\left((Q_{ij}\right.\sin {\alpha }_{ij}-F_{oj}\cos {\alpha }_{ij}/2)R_i\sin {\phi }_j\left.+F_{oj}{r}_i\sin {\phi }_j\right)}\hfill \end{array}\right.$$

(21)

2.3. Theoretical Model Solving and Verification

The spindle bearing 7012ACE/HCP4A is selected as the research object of this study, and its design parameters are listed in

Table 1. Design parameters of the SKF 7012ACE/HCP4A bearing.

Bearing Design Parameters	Value
Inner raceway curvature radius ri/o /(mm)	4.08807
Outer raceway curvature radius ri/o /(mm)	4.16745
Initial contact angle α0/(°)	25
Number of rolling elements z	25
Rolling element diameter D/(mm)	7.938
Pitch circle diameter Dm/(mm)	77.591
Density (kg/m3)	7800
Elastic modulus (Mpa)	2.07 × 105
Poisson’s ratio	0.28

, including basic design parameters and material parameters.

To verify the validity of the quasi-static calculation, the obtained results are compared with the numerical results from Reference [43]. The comparison focuses on the variations of the raceway contact angles and contact loads with the radial load. The bearing used in Reference [43] is a 7011C angular contact ball bearing, and its design parameters are listed in

Table 2. Design parameters of the 7011C bearing [43].

Bearing Design Parameters	Value
Inner raceway curvature radius ri/o /(mm)	5.72
outer raceway curvature radius ri/o /(mm)	5.72
Initial contact angle α0/(°)	15
Number of rolling elements z	17
Rolling element diameter D/(mm)	11
Pitch circle diameter Dm/(mm)	72.5
Density (kg/m3)	7850
Elastic modulus (Mpa)	2 × 105
Poisson’s ratio	0.3

. The calculation results for the 7011C bearing obtained in this study are shown in Figure 6.

As shown in Figure 6, the contact angle and contact load solved in this paper are generally consistent with those in Reference [43] in terms of the curve variation trends. The “W-shaped” phenomenon presented in the calculation is consistent with that in Reference [43]. The “W-shaped” phenomenon typically appears in the direction opposite to the radial load and represents the variation pattern of the rolling elements driven by centrifugal force in the unloaded zone. Here, the radial load acts at an azimuthal angle of 90°; therefore, the center of the unloaded zone is located at a position angle of 270°.

To further verify the validity of the model solution, the calculated stiffness values are compared with the experimentally measured stiffness data from Reference [44], including the axial stiffness $k_z$, radial stiffness $k_x$ and $k_y$, and angular stiffness $k_{\theta x}$ and $k_{\theta y}$. The bearing model used in the experiment is RPF7039, whose design parameters are listed in

Table 3. Design parameters of the RPF7039 bearing [44].

Bearing Design Parameters	Value
Inner/outer raceway curvature radius ri/o /(mm)	9.16/9.38
Initial contact angle α0/(°)	40
Number of rolling elements z	12
Rolling element diameter D/(mm)	17.7
Pitch circle diameter Dm/(mm)	72.5

. The experimental operating conditions were $F_z=2000 \mathrm{N}$, $F_x=F_y=500 \mathrm{N}$, $M_x=M_y=1 \mathrm{N}\cdot \mathrm{m}$, $\omega =0 \mathrm{r}/\min$. A comparison between the stiffness values obtained in this study and the experimental values is presented in

Table 4. Verification of the bearing stiffness solution.

Stiffness/(N∙mm−1)	Experimental Result [44]	Present Result	Error/(%)
Axial stiffness $k_z$	226,011.1	230,175.1	1.8
Radial stiffness $k_x$	172,131.6	166,519.9	−3.3
Radial stiffness $k_y$	155,520.7	166,504.6	7.1
Angular stiffness $k_{\theta x}$	139,304,365.4	143,212,104.9	2.8
Angular stiffness $k_{\theta y}$	157,160,958.5	153,228,230.2	−2.5

.

According to the comparison results in Table 4, the errors between the stiffness values calculated by the present model and the experimentally measured values reported in the literature are within 8%, which demonstrates the validity of the proposed model.

3. Sliding Wear Model of Angular Contact Ball Bearing Raceway

3.1. Raceway Wear Volume Model

We consider the non-uniform distribution of sliding speed and stress in the inner and outer raceways’ contact ellipse. Based on the Archard original expression $V=KFL$ [14], the tiny contact area within the contact ellipse $dA$ is defined as the wear surface, while $dF$ is the contact pressure acting on $dA$. The micro-sliding distance on $dA$ is $dL$; then, the small wear volume $dV$ on $dA$ is

$$dV=kdFdL$$

(22)

where $k$ stands for wear coefficient; $dF=pdA$, where $p$ stands for contact stress; and $dL=vdt$, where $v$ stands for sliding speed and $dt$ is the sliding time. The calculation model of raceway wear volume of the bearing is shown in Equation (21) [15].

$$V={\displaystyle \int dV={\displaystyle \iint kpv}}dAdt$$

(23)

3.1.1. Contact Stress Calculation on the Raceway Contact Ellipse

Under the action of contact load, the contact area between the rolling element and the raceway is an elliptical surface, and the stress distribution on this elliptical surface can be expressed as:

$$p(x,y)=p_0\sqrt{1-{({\displaystyle \frac{x}{a}})}^2-{({\displaystyle \frac{y}{b}})}^2}$$

(24)

where $a$ is the semi-major axis of the contact ellipse, $x$ is the coordinate along the major axis direction, $b$ is the semi-minor axis of the contact ellipse, $y$ is the coordinate along the minor axis direction, $p_0=3Q/(2\pi ab)$ is the contact stress at the center of the contact ellipse, and $p(x,y)$ is the contact stress distribution on the elliptical surface. Under the operating conditions of an axial load of 300 N, a radial load of 300 N, and a rotational speed of 10,000 r/min, the rolling element located in the radial load direction is selected to calculate the contact stress distribution on the elliptical surfaces between the rolling element and the inner and outer raceways. The calculation results are shown in Figure 7. The contact angle and contact load data of the selected rolling element are listed in

Table 5. Numerical calculation results of contact angle and contact load.

Calculation Parameter	Inner Raceway	Outer Raceway
Contact angle	13.344°	10.696°
Contact load	99.812 N	117.194 N

.

As shown in Figure 7, the contact stress distributions on the elliptical surfaces between the rolling element and the inner and outer raceways both exhibit a semi-ellipsoidal shape. The stress value increases gradually from the boundary of the ellipse toward the center and reaches its maximum at the center. Under the current operating conditions, the maximum contact stresses on the inner and outer raceway contact ellipses are 1016 MPa and 1126 MPa, respectively, with a difference of 110 MPa between them. This phenomenon is caused by the centrifugal fling-out effect, which results in a higher contact stress between the rolling element and the outer raceway than that on the inner raceway at the same azimuthal angle.

3.1.2. Sliding Velocity Calculation on the Raceway Contact Ellipse

As shown in Figure 8, the differential sliding velocity $v_{yi/o}$ at minor axis b and spin sliding velocity ${\omega }_{si/o}$ of the ball form the sliding velocity on the contact ellipse. The expressions for $v_{yi/o}$, ${\omega }_{si/o}$ and total sliding velocity $v{(x,y)}_{i/o}$ are shown in Equations (25)–(27).

$$\left\{\begin{array}{l}{v}_{yi}=-\omega d_m/2-\left\{{(r_i^2-x_i^2)}^{1/2}-{(r_i^2-a_i^2)}^{1/2}+\right.[{(D_b/2)}^2\left.-a_i^2{]}^{1/2}\right\}\times ({\omega }_b\sin \beta \sin {\alpha }_i-\omega \cos {\alpha }_i)\hfill \\ v_{yo}=\left\{{(r_o^2-x_o^2)}^{1/2}-{(r_o^2-a_o^2)}^{1/2}+\right.[{(D_b/2)}^2\left.-a_o^2{]}^{1/2}\right\}\times ({\omega }_b\cos (\beta -{\alpha }_o))\hfill \end{array}\right.$$

(25)

$$\left\{\begin{array}{l}{\omega }_{si}={\omega }_b\sin (\beta -{\alpha }_i)+\omega \sin {\alpha }_i\hfill \\ {\omega }_{so}={\omega }_b\sin ({\alpha }_o-\beta )\hfill \end{array}\right.$$

(26)

$$v{(x,y)}_{i/o}={[{(v_{yi/o}-{\omega }_{si/o}y)}^2+{({\omega }_{si/o}x)}^2]}^{1/2}$$

(27)

Under the operating conditions of an axial load of 300 N, a radial load of 300 N, and a rotational speed of 10,000 r/min, The sliding velocity distributions on the contact areas of the inner and outer raceways were first calculated without considering spin, and the results are presented in Figure 9. Subsequently, the sliding velocity distributions for gradually increasing spin speed were calculated, and the corresponding results are shown in Figure 10, Figure 11 and Figure 12. The detailed numerical results are presented in

Table 6. Numerical calculation results of the sliding velocity on the contact ellipse.

Calculation Parameter	Inner Contact Ellipse	Outer Contact Ellipse
Semi-major axis	0.6907 mm	0.5789 mm
Semi-minor axis	0.0659 mm	0.0858 mm
Max sliding (ωsi = 0 rad/s)	26,295 mm/s	12,894 mm/s
Min sliding (ωsi = 0 rad/s)	−13,148 mm/s	−25,789 mm/s
Max sliding (ωsi = 2000 rad/s)	26,300 mm/s	14,610 mm/s
Min sliding (ωsi = 2000 rad/s)	−14,466 mm/s	−25,808 mm/s
Max sliding (ωsi = 5000 rad/s)	26,360 mm/s	17,184 mm/s
Min sliding (ωsi = 5000 rad/s)	−16,443 mm/s	−25,908 mm/s
Max sliding (ωsi = 10,000 rad/s)	26,566 mm/s	21,474 mm/s
Min sliding (ωsi = 10,000 rad/s)	−19,738 mm/s	−26,265 mm/s

. The contact angle and load data of the selected rolling element are listed in Table 5.

As shown in Figure 9, Figure 10, Figure 11 and Figure 12 and Table 6, when spin is not considered, two pure rolling lines appear in the contact zones of the inner and outer raceways. As the spin speed gradually increases, these pure rolling lines shift, and the higher the spin speed, the larger the shift. Meanwhile, the maximum sliding velocity within the contact zone also increases progressively.

3.1.3. Wear Coefficient Calculation for the Raceway Contact Ellipse

Considering the material hardness, the expression for $k$ is

$$k=({\alpha }^{\ast }\times {\Lambda }^{{\beta }^{\ast }})/H$$

(28)

where ${\alpha }^{\ast }$ and ${\beta }^{\ast }$ are the characteristic parameters of the contact auxiliary material and lubricant; $H$ is Brinell hardness of bearing material; $\mathsf{\Lambda }=h_{\min }/\sqrt{{\sigma }_r^2+{\sigma }_g^2}$ is the oil film parameter, $h_{\min }$ stands for minimum film thickness, and ${\sigma }_r$, ${\sigma }_b$ stand for the surface roughness parameters; $h_{\min }$ [45] can be expressed as

$$h_{\min }=3.63U^{0.68}{G}^{0.49}{W}^{-0.073}(1-e^{-0.68\kappa })R_x$$

(29)

$G=\xi E$ stands for viscosity parameter, $\xi$ stands for oil film pressure parameters, and $E$ stands for effective elastic modulus. $W=Q_{i/j}/{(R_x)}^{2}E$ is the dimensionless load parameter, while $R_x$ and $R_y$ are the contact ellipse equivalent radii. $e=\sqrt{1-{(b/a)}^2}$ is the eccentricity of the ellipse, and $a$, $b$ are the contact ellipse’s major and minor axes. $\kappa =1.0339{(R_y/R_x)}^{0.636}$ is the ellipticity parameter, $U={\eta }_{0}u/E{R}_x$ is the dimensionless speed parameter, ${\eta }_0$ is the lubricating oil viscosity parameter, and $u$ stands for rolling speed in the raceway contact ellipse. The expressions of the inner and outer raceway parameters $R_x$, $R_y$ and $u$ are

$$\left\{\begin{array}{l}{R}_{xij}=(D_b/2)(1-D_b\cos {\alpha }_{ij}/d_m)\hfill \\ R_{yi}=D_b{f}_i/(2f_i-1)\hfill \\ u_{ij}=(\omega -{\omega }_{mj})(d_m-D_b\cos {\alpha }_{ij})/2\hfill \end{array}\right.$$

(30)

$$\left\{\begin{array}{l}{R}_{xoj}=(D_b/2)(1+D_b\cos {\alpha }_{oj}/d_m)\hfill \\ R_{yo}=D_b{f}_o/(2f_o-1)\hfill \\ u_{oj}=-{\omega }_{mj}(d_m+D_b\cos {\alpha }_{oj})/2\hfill \end{array}\right.$$

(31)

It should be noted that the wear coefficient calculation model adopted in this paper is based on classical elastohydrodynamic lubrication (EHL) theory and assumes a fully flooded condition. The lubricant is treated as a Newtonian fluid, and thermal effects as well as lubricant starvation are neglected. The SKF 7012ACE/HCP4A bearing used here has a grease-lubricated attainable speed of 22,000 r/min, and the maximum rotational speed in the present calculations (10,000 r/min) is approximately 45% of this value, well below the regime where significant starvation is typically expected [46]. Therefore, the present model is applicable to moderate-speed and moderate-load conditions where the spindle speed does not exceed approximately 10,000 r/min and the load remains below 30% of the basic dynamic load rating. Under high-speed or extreme heavy-load conditions where starvation or thermal effects become non-negligible, the model may introduce certain calculation errors.

The values of the raceway lubrication parameters, raceway surface topography parameters, and other parameters used in the calculation of the wear coefficient are shown in

Table 7. Parameter values used for the calculation of the wear coefficient [47].

Calculation Parameters for the Wear Coefficient	Literature Result
Contact pair material parameters α *	3.433 × 10−8
Lubricant characteristic parameters β *	−1.032
Brinell hardness of bearing material H/(Mpa)	8470
RMS roughness of rolling element surface σr /(μm)	0.18
RMS roughness of raceway surface σr/(μm)	0.18
Pressure–viscosity coefficient of lubricant ξ/(m2/N)	2.2 × 10−8
Absolute viscosity of lubricant η0/(Pa·s)	0.04
Effective elastic modulus E/(Mpa)	1.12 × 105

* indicates the lubricant property parameters.

.

During the wear process between the rolling element and the raceway, the wear coefficient may vary with the rotational speed and load. The variation of the wear coefficient with rotational speed is calculated under the conditions of an axial load $F_z=300 \mathrm{N}$ and a radial load $F_x=300 \mathrm{N}$; the variation with the radial load $F_x$ is calculated under the conditions of an axial load $F_z=300 \mathrm{N}$ and a rotational speed $n=8000 \mathrm{r}/\min$. The calculation results and the variations of the contact angle and contact load are shown in Figure 13, Figure 14, Figure 15 and Figure 16, respectively.

The specific values of the peak wear coefficients of the inner and outer raceways are presented in

Table 8. Variation of the peak wear coefficient of the inner raceway with rotational speed.

Rotational Speed	${\mathit{F}}_{\mathit{x}}$ Direction Wear Coefficients	$-{\mathit{F}}_{\mathit{x}}$ Direction Wear Coefficients
8000 r/min	3.4482 × 10−13	3.0193 × 10−13
9000 r/min	3.1717 × 10−13	2.7879 × 10−13
10,000 r/min	2.9441 × 10−13	2.6184 × 10−13

and

Table 9. Variation of the peak wear coefficient of the outer raceway with rotational speed.

Rotational Speed	${\mathit{F}}_{\mathit{x}}$ Direction Wear Coefficients	$-{\mathit{F}}_{\mathit{x}}$ Direction Wear Coefficients
8000 r/min	2.1701 × 10−13	1.9053 × 10−13
9000 r/min	2.0001 × 10−13	1.7205 × 10−13
10,000 r/min	1.8597 × 10−13	1.5697 × 10−13

. It can be observed that the peak wear coefficient of the inner raceway is larger than that of the outer raceway, indicating that the higher sliding velocity on the inner raceway leads to a larger wear coefficient. Meanwhile, as can be seen from Figure 14, when the rotational speed increases, the wear coefficients of both the inner and outer raceways gradually decrease over the entire orbital angular range of the rolling elements. This is because the oil film thickness gradually increases with the rotational speed, thereby causing the wear coefficient to decrease gradually. It should be noted that lubricant starvation effects [47] are not considered in the calculation. The rotational speed used in this study is controlled below 50% of the bearing’s limiting speed. If a higher rotational speed were adopted, lubricant starvation might occur, in which case the oil film thickness would gradually decrease with increasing speed, leading to a progressive increase in the wear coefficient.

The variations of the peak wear coefficients of the inner and outer raceways with radial load are presented in

Table 10. Variation of the peak wear coefficient of the inner raceway with radial load.

Rotational Speed	${\mathit{F}}_{\mathit{x}}$ Direction Wear Coefficients	$-{\mathit{F}}_{\mathit{x}}$ Direction Wear Coefficients
100 N	3.1881 × 10−13	3.0052 × 10−13
200 N	3.3459 × 10−13	3.0125 × 10−13
300 N	3.4482 × 10−13	3.0193 × 10−13

and

Table 11. Variation of the peak wear coefficient of the outer raceway with radial load.

Rotational Speed	${\mathit{F}}_{\mathit{x}}$ Direction Wear Coefficients	$-{\mathit{F}}_{\mathit{x}}$ Direction Wear Coefficients
100 N	2.0296 × 10−13	1.8983 × 10−13
200 N	2.1144 × 10−13	1.9001 × 10−13
300 N	2.1701 × 10−13	1.9053 × 10−13

, and the overall variation trends are shown in Figure 16. It can be observed that in the $F_x$ direction, as the load increases, the oil film is squeezed, resulting in a decrease in film thickness; consequently, the wear coefficient gradually increases with $F_x$. In the $-F_x$ direction, the contact between the rolling element and the raceway is primarily influenced by centrifugal force; therefore, when the radial load alone is increased while the rotational speed remains constant, the wear coefficient exhibits only a very small change.

3.2. Raceway Wear Depth Model

The wear degradation process of the raceway involves an increase in wear depth and a change in wear area. When the influence of the wear process on the raceway is not considered—for an operating condition with a stationary outer ring and a rotating inner ring—under bearing loads, due to the rotation of the inner ring, each azimuthal position of the inner raceway experiences an identical and complete load history, which manifests as uniform wear in the numerical calculation. In contrast, because the outer raceway remains stationary, its azimuthal positions experience different load histories and exhibit distinct nonlinear variations according to the direction of the applied load. The average wear area and wear depth of the inner raceway can be expressed as:

$${\overline{S}}_i={\displaystyle \frac{1}{Z}}{\displaystyle \sum _{j=0}^{Z}2\pi a_{ij}(d_m-D\cos {\alpha }_{ij})}$$

(32)

$$h_{ti}={\displaystyle \frac{1}{{\overline{S}}_i}}{\displaystyle \sum _{j=0}^Z{\displaystyle \iint t{k}_{ij}{p}_{ij}(x,y)v_{ij}(x,y)dxdy}}$$

(33)

The wear area and wear depth at each azimuthal angle $j$ of the outer raceway can be expressed as:

$$S_{oj}=2\pi a_{oj}(d_m+D\cos {\alpha }_{oj})$$

(34)

$$h_{to}={\displaystyle \frac{Z}{S_{oj}}}{\displaystyle \iint t{k}_{oj}{p}_{oj}(x,y)v_{oj}(x,y)dxdy}$$

(35)

Based on a total bearing operating time of one year, the influence of radial load on the inner raceway wear depth is investigated under an axial load of $F_z=300 \mathrm{N}$ and a rotational speed of $n=\mathrm{10,000} \mathrm{r}/\min$, while the influence of rotational speed on the inner raceway wear depth is investigated under an axial load of $F_z=300 \mathrm{N}$ and a radial load of $F_x=500 \mathrm{N}$. The calculation results are presented in Figure 11. Under an axial load of $F_z=300 \mathrm{N}$ and a rotational speed of $n=\mathrm{10,000} \mathrm{r}/\min$, the influence of the radial load on the outer raceway wear depth is investigated; under a radial load of $F_x=300 \mathrm{N}$ and a rotational speed of $n=\mathrm{10,000} \mathrm{r}/\min$, the influence of the axial load on the outer raceway wear depth is investigated. The calculation results and the variations of the contact angle and contact load are shown in Figure 17, Figure 18, Figure 19, Figure 20 and Figure 21, respectively.

As shown in Figure 22, uniform wear is observed at all orbital angles of the inner raceway, and the wear depth of the inner raceway gradually increases with increasing radial load and rotational speed. As shown in Figure 22a, in the $F_x$ direction, the wear depth of the outer raceway gradually increases with increasing radial load, whereas in the $-F_x$ direction, the wear depth of the outer raceway is only slightly affected by the radial load. As shown in Figure 22b, increasing the axial load causes the wear depth to increase at all orbital angles; however, no sharp increase is observed in the $F_x$ direction. The calculation results indicate that compared with the inner raceway, the wear of the outer raceway is more sensitive to the load direction and is more prone to concentrated wear.

4. Study on Raceway Wear Considering Curvature Radius Degradation

When the influence of wear degradation on the raceway is considered, the curvature radii of the inner and outer raceways change as the wear amount increases, leading to variations in the wear condition of the bearing under the same operating conditions. These variations manifest as changes in the elliptical contact surface, as well as variations in the contact stress, sliding velocity, and wear coefficient. Based on the wear volumes of the inner and outer raceways $V_i$ and $V_o$ after the bearing has operated for a time $t$, the worn curvature radii of the inner and outer raceways ${r^{\prime }}_i$ and ${r^{\prime }}_o$ can be expressed as:

$${r^{\prime }}_i=r_i+V_i/[2\pi a_i(d_m-D_b\cos {\alpha }_i)]$$

(36)

$${r^{\prime }}_o=r_o+V_o/[2\pi a_o(d_m+D_b\cos {\alpha }_o)]$$

(37)

In the numerical solution of the wear model considering raceway curvature radius variation, the wear depth model is first solved. Then, by combining Equations (36) and (37), the worn curvature radii of the raceways are obtained. The updated curvature radii are subsequently fed back into the wear depth model for further solving. The convergence tolerance is set to 10⁻⁶ and the maximum number of iterations to 1000. When convergence is reached, the updated raceway dynamic parameters are output and used in the next iteration.

Assuming a total bearing operating time of 3000 h, the variation of the raceway curvature radii is calculated at 1000 h intervals. The operating parameters are as follows: an axial load of $F_z=300 \mathrm{N}$, a radial load of $F_x=300 \mathrm{N}$, and a rotational speed increasing from the rated speed of $n=5000 \mathrm{r}/\min$ to the limiting speed of $n=\mathrm{10,000} \mathrm{r}/\min$. The calculation results are shown in Figure 23. The variations of the contact angle and contact load of the rolling element are shown in Figure 24.

Meanwhile, under constant operating conditions of an axial load of $F_z=300 \mathrm{N}$, a radial load of $F_x=300 \mathrm{N}$, and a rotational speed of $n=\mathrm{10,000} \mathrm{r}/\min$, variations in the raceway contact ellipse area, the contact stress at the center of the contact ellipse, the central sliding velocity, and the wear coefficient with wear time are investigated. The calculation results are shown in Figure 25, Figure 26, Figure 27 and Figure 28, respectively, and the peak value statistics of each parameter are presented in

Table 12. Variation of the peak contact ellipse area with time.

Operating Time	${\mathit{F}}_{\mathit{x}}$ Direction (Inner Raceway/Outer Raceway)	$-{\mathit{F}}_{\mathit{x}}$ Direction (Inner Raceway/Outer Raceway)
1000 h	0.1441 mm2/0.1571 mm2	0.0500 mm2/0.0741 mm2
2000 h	0.1435 mm2/0.1561 mm2	0.0496 mm2/0.0732 mm2
3000 h	0.1430 mm2/0.1553 mm2	0.0492 mm2/0.0725 mm2

,

Table 13. Variation of the peak contact stress at the center of the contact ellipse with time.

Operating Time	${\mathit{F}}_{\mathit{x}}$ Direction (Inner Raceway/Outer Raceway)	$-{\mathit{F}}_{\mathit{x}}$ Direction (Inner Raceway/Outer Raceway)
1000 h	1050.00 Mpa/1129.16 Mpa	601.86 Mpa/776.92 Mpa
2000 h	1062.68 Mpa/1144.05 Mpa	607.49 Mpa/785.33 Mpa
3000 h	1074.59 Mpa/1157.87 Mpa	612.86 Mpa/793.12 Mpa

,

Table 14. Variation of the peak sliding velocity at the center of the contact ellipse with time.

Operating Time	${\mathit{F}}_{\mathit{x}}$ Direction (Inner Raceway/Outer Raceway)	$-{\mathit{F}}_{\mathit{x}}$ Direction (Inner Raceway/Outer Raceway)
1000 h	12,953 mm/s/4099 mm/s	12,356 mm/s/3579 mm/s
2000 h	12,957 mm/s/4102 mm/s	12,346 mm/s/3572 mm/s
3000 h	12,960 mm/s/4103 mm/s	12,337 mm/s/3566 mm/s

and

Table 15. Variation of the peak wear coefficient with time.

Operating Time	${\mathit{F}}_{\mathit{x}}$ Direction (Inner Raceway/Outer Raceway)	$-{\mathit{F}}_{\mathit{x}}$ Direction (Inner Raceway/Outer Raceway)
1000 h	3.0771 × 10−13/1.9437 × 10−13	2.7367 × 10−13/1.6406 × 10−13
2000 h	3.0841 × 10−13/1.9487 × 10−13	2.7430 × 10−13/1.6449 × 10−13
3000 h	3.0909 × 10−13/1.9538 × 10−13	2.7493 × 10−13/1.6492 × 10−13

, respectively.

As shown in Figure 25, Figure 26, Figure 27 and Figure 28 and Table 12, Table 13, Table 14 and Table 15, with increasing operating time, the curvature radii of the inner and outer raceways increase, while the contact ellipse areas of the inner and outer raceways decrease. Meanwhile, the maximum contact stress at the center of the contact ellipse and the sliding velocity both increase, and the wear coefficient gradually increases. The changes in these parameters reduce the elliptical contact area between the rolling elements and the raceways while increasing the contact stress, sliding velocity, and wear coefficient over the elliptical contact region, making the raceways more susceptible to wear failure.

To verify the validity of the raceway wear calculation presented in this paper, the results are compared with the bearing wear life calculations in Reference [48]. In Reference [48], the wear life at each orbital angle position of the raceway is determined by specifying an allowable wear depth threshold. Using the same bearing design parameters and operating conditions as those in Reference [48], the calculated results are shown in Figure 29. Because Reference [48] employs a mixed lubrication model to calculate the raceway oil film thickness, the present results are slightly lower than those in Reference [48]; nevertheless, they are consistent in both order of magnitude and variation trend.

The raceway wear depths of bearings that have already undergone 1000, 2000, and 3000 h of wear, respectively, are calculated after one further year of operation under the same operating conditions. The operating parameters are an axial load of $F_z=300 \mathrm{N}$, a radial load of $F_x=100 \mathrm{N}$, a moment load of $M_y=1 \mathrm{N}\cdot \mathrm{m}$, and a rotational speed of $n=\mathrm{10,000} \mathrm{r}/\min$. The variations of the contact angle and contact load of the rolling element are shown in Figure 30. The calculation results are shown in Figure 31.

As shown in Figure 31 and

Table 16. Average wear depth of the inner raceway and peak wear depth of the outer raceway.

Operating Time	Inner Raceway Average Wear Depth	Outer Raceway Peak Wear Depth
1000 h	1.39 × 10−4 mm	6.22 × 10−4 mm
2000 h	1.47 × 10−4mm	7.83 × 10−4 mm
3000 h	1.57 × 10−4mm	9.83 × 10−4mm

, for bearings that have undergone different wear durations, when they are subsequently subjected to the same operating conditions for the same period of time, the wear depth increases progressively with the accumulated prior wear time. This indicates that the degradation of the raceway curvature radius has a cumulative and non-negligible influence on the wear evolution. Consequently, the effect of raceway curvature radius degradation should be taken into account in bearing wear calculations.

5. Conclusions

In this paper, a five-degree-of-freedom quasi-static model of angular contact ball bearings is established and validated against both numerical and experimental results from the literature. A modified Archard wear model incorporating a dynamic curvature radius update mechanism is developed to investigate raceway wear evolution under combined loads. The main conclusions are as follows:

• The wear coefficient is negatively correlated with rotational speed. For the inner raceway, the peak wear coefficient decreases from 3.4482 × 10⁻¹³ at 8000 r/min to 2.9441 × 10⁻¹³ at 10,000 r/min under the given loading conditions, owing to the increase in oil film thickness at higher speeds.
• Radial load causes concentrated wear on the outer raceway, whereas axial load promotes a more circumferentially uniform wear distribution. For example, under a radial load of 300 N, the outer raceway peak wear depth reaches 6.22 × 10⁻⁴ mm after 1000 h of operation, while the inner raceway exhibits uniform wear with an average depth of 1.39 × 10⁻⁴ mm.
• Raceway curvature radii increase with wear time. Over 3000 h of operation, the contact ellipse area in the loaded direction decreases (e.g., from 0.1441 mm² to 0.1430 mm² for the inner raceway), while the peak contact stress increases (from 1050.00 MPa to 1074.59 MPa) and the sliding velocity also rises. These coupled changes lead to a progressively increasing wear coefficient, further accelerating material removal.
• The cumulative effect of raceway curvature radius degradation is significant. A bearing with 3000 h of prior wear exhibits a subsequent annual wear depth approximately 1.6 times greater than that of a bearing with only 1000 h of prior wear under identical operating conditions. This demonstrates that neglecting curvature radius variation leads to an underestimation of wear depth, and the coupling between geometry degradation and contact mechanics must be considered for reliable bearing life prediction.

These quantitative findings provide a theoretical basis for the condition monitoring and maintenance scheduling of spindle bearings subjected to combined loads.