Effect of Rotational Speed Fluctuation Parameters on Dynamic Characteristics of Angular Contact Ball Bearings

Abstract

The fluctuation in the rotational speed of the inner ring can lead to significant instability in the motion of both the inner ring and the cage of rolling bearings. This instability seriously impacts the operational performance and service life of the bearings. In this paper, a nonlinear dynamic model of a fully flexible angular contact ball bearing was established by comprehensively considering various nonlinear factors, including elastic contact relationships, internal collisions, friction, and clearance. The dynamic characteristics of the inner ring and cage under sinusoidal rotational speed fluctuations were studied. The effects of amplitude and frequency of rotational speed fluctuation of the inner ring on the motion stability of the inner ring and cage were analyzed. The results show that a greater the fluctuation amplitude leads to a higher the fluctuation amplitude in the cage’s rotational speed curve, while a higher fluctuation frequency correlates with an increased frequency in the cage’s rotational speed curve. These results indicate that increases in both the amplitude and frequency of rotational speed fluctuations result in more pronounced oscillations of the inner ring. The validity of the model was confirmed by comparing the LS-DYNA results with the analytical results and experimental results. The research findings can provide a theoretical foundation for enhancing motion stability and optimizing design of the bearings.

1. Introduction

Angular contact ball bearings are crucial mechanical components supporting the lead screw in the feed system of CNC machine tools. The dynamic characteristics of internal components during the operation of the bearings have a significant impact on the accuracy and reliability of the entire machine. Especially under harsh working conditions, it is easy to cause faults such as cage breakage and bearing ring wear, thereby reducing the service life and operational performance of the bearings. Therefore, by analyzing the dynamic characteristics of bearing components to accurately understand the generation and variation mechanisms of unstable motions of the inner ring and cage of the bearing, it is of great engineering practical significance for improving the operational performance of the angular contact ball bearings.

Harris [1] proposed the calculation method of quasi-dynamics after quasi-statics and established a pseudo-dynamics model of cylindrical roller bearings based on the theory of elastohydrodynamic lubrication, predicting the slipping of the bearings under the action of radial loads. Polanski et al. [2] established a more comprehensive quasi-dynamic model of bearings considering the internal friction between rolling bearing elements. Kleckner et al. [3] developed a quasi-dynamic analysis software for cylindrical roller bearings that can be used to analyze the deformation of rings, load variations, and the inclination and skewing of rollers, etc. Creju et al. [4,5] established a quasi-dynamic model of tapered roller bearings that can be used to analyze load distribution, cage slipping, and the sliding speed between rollers and rings, etc. Nelias et al. [6] established a quasi-dynamic model of angular contact ball bearings considering the force between the cage and the rolling elements, analyzed its motion characteristics and operational performance under the combined radial and axial loads, and verified the correctness of the calculation results through experiments. Hagiu et al. [7] established a quasi-dynamic vibration model under lubrication conditions considering internal elastic deformation and oil film squeezing effects. However, the quasi-dynamic analysis is limited to the dynamic characteristics of bearings under absolute steady-state conditions, and cannot clearly reveal the time-varying dynamic characteristics of the internal components of the rolling bearings.

In order to obtain more accurate dynamic characteristics of bearings, Walers [8] first proposed the dynamic analysis method in early 1970 and established a dynamic model of rolling bearings based on the motion equations of the six-degree-of-freedom cage and the four-degree-of-freedom rolling elements. Kannel et al. [9] developed an angular contact ball bearing analysis model based on elastohydrodynamic lubrication theory and analyzed the smoothness of cage plane motion under different lubrication conditions. Boesiger et al. [10] simplified the degrees of freedom of bearing components and established a two-dimensional dynamic model for deep groove ball bearings, and studied the relationship between friction and the stability of the cage through simulation and experiments. Neglia et al. [11] investigated the relationship between inertia force, impact force and the trajectory of the cage centroid. However, the above research has not yet deeply analyzed the effects of geometric parameters and operating conditions. Additionally, the above-mentioned model simplifies the contact behavior among the various components. Meeks [12,13] established the dynamic friction model of the bearing considering the six degrees of freedom of the cage, and wrote the calculation programs for the ball rollers, cage, and inner and outer rings under different load conditions. Moreover, a nonlinear contact bearing dynamic model was established, and the relationships between the inner ring speed, cage structure parameters, lubrication parameters and cage stability were analyzed. Choe et al. [14,15] analyzed the influence of pocket clearance and guiding clearance on the motion smoothness of the cage through experiments. In addition, some scholars have conducted relevant research on other factors that affect the motion stability of the cage. Ghaisas et al. [16] studied the effects of cage asymmetry, raceway misalignment and roller size on the stability of the cage. Pederson et al. [17] established a dynamic model of deep groove ball bearings based on the ANSYS flexible cage and analyzed the influences of circumferential stiffness, torsional stiffness and cage clearance on the motion stability of the cage. Cheng et al. [18,19,20] proposed a candidate fault frequencies-based blind deconvolution, which can enhance the fault characteristics of rolling bearings. The motion stability of the inner ring and the cage has become the focus of the performance analysis of the rolling bearings.

With the development of electromechanical equipment such as CNC machine tools towards high speed and high precision, higher requirements have been put forward for the operational stability of angular contact ball bearings. The research on the dynamic characteristics of the inner ring and cage of angular contact ball bearings under typical complex variable working conditions such as variable speed and variable load has become a research hotspot. Lu [21] established a vibration model of roll-slip bearings under cyclic load impact and studied the relationship between its amplitude and the rotational position of the inner ring, the transverse stiffness of the rollers, and the number of rollers. Jia et al. [22] and Li et al. [23], respectively, established dynamic models for cylindrical roller bearings during start-up and stoppage stages, and analyzed the influence of working condition parameters and structural parameters on their slip characteristics, and conducted experimental verification. Zhang et al. [24] established a multi-rigid body dynamic model of the main shaft bearing of wind turbine units using ADAMS, and analyzed the internal contact forces under three working conditions: the start-up process, the sudden change in rotational speed, and the emergency stop process. Li et al. [25] established a dynamic model of cylindrical roller bearings during the emergency stop process and used the angular velocity and collision force between the rollers and cage as boundary conditions of their finite element model to study the effects of inner ring angular acceleration, radial force, oil temperature, and radial clearance on cage stress. Ye [26] established a dynamic model of high-speed rolling bearings in aircraft engines and investigated the stability variation laws of the cage under three working conditions: starting, acceleration and loading, but did not study its influencing factors. Yao et al. [27] established a dynamic model of angular contact ball bearings considering the three-dimensional dynamic contact relationship inside the bearing, and studied the influence of clearance, friction and lubrication resistance on the stability of the cage under variable-speed, variable-load and preload conditions. Lioulios et al. [28] considered the effect of radial clearance, contact force between the balls and rings, and flexible contact stiffness, and analyzed the dynamic characteristics of the periodic, unstable periodic, and chaotic response of rotor bearings using frequency spectrum method, phase space method, high-order Poincare diagram method, and Lyapunov exponent method.

Currently, researchers have studied the dynamic characteristics of rolling bearings from various perspectives, particularly focusing on the motion stability of the inner ring and cage under different operating conditions. However, much of the existing research primarily focuses on steady-state operating conditions, and most analytical models overlook the nonlinear interactions among the components of rolling bearings. In reality, the dynamic interactions between the moving parts significantly influence the movement and performance of the inner ring and cage, especially under non-stationary working conditions. While some scholars have made notable advancements regarding the motion characteristics of rolling bearings under variable operating conditions—such as acceleration, deceleration, and impact loads—there is a lack of research on the impact of periodic variable speed working parameters on the motion stability of the inner ring and cage. Therefore, in this paper, a nonlinear dynamic model for a fully flexible angular contact ball bearing was established, comprehensively taking into account nonlinear factors such as elastic contact relationships, collisions between rolling elements and the cage, friction, and clearance. Based on this model, the dynamic characteristics of the inner ring and cage under sinusoidal rotational speed fluctuations were investigated. A comparative analysis was conducted regarding the rotational speed, centroid trajectory of the inner ring and cage, and the contact forces between single rolling element and the inner ring and cage under varying amplitudes and frequencies of rotational speed fluctuations. The aim is to uncover the intrinsic relationship between the instability of the inner ring and cage and the working conditions associated with rotational speed variations, thus providing theoretical references for the design optimization and failure analysis of angular contact ball bearings.

2. Full Flexible Body Dynamic Model of the Angular Contact Ball Bearing

2.1. Fluctuation Curve of Inner Ring Rotational Speed

Periodic rotational speed fluctuations are common in rotating machinery, and any form of such fluctuations can be represented as the superposition of multiple or potentially infinite harmonic components of different frequencies. Thus, assuming a sinusoidal pattern for rotational speed fluctuations, studying the influence of fluctuation parameters on the dynamic characteristics of the bearings is highly representative, which can be described as [28]:

$$\{\begin{array}{l}{n}_{\mathrm{i}}=n_0\left[1+A\sin \left(N{\omega }_{0}t\right)\right]\\ {\omega }_0={\displaystyle \frac{2\pi n_0}{60}}\end{array}$$

(1)

where $n_{\mathrm{i}}$ is the rotational speed of the inner ring. $n_0$ is the average rotational speed. A is the fluctuation amplitude, $A=\left(n_{\max }-n_{\min }\right)/\left(2n_0\right)$. N is the number of fluctuations in each cycle.

The rotational speed curve of the inner ring under sinusoidal fluctuation condition is depicted in Figure 1. For the study of the impact of fluctuation amplitude A and fluctuation frequency Nω₀ on the angular contact ball bearing’s dynamic characteristics, the following parameters were selected: average rotational speed n₀ = 2000 r/min. For N = 1, the inner ring fluctuation amplitudes A are set at 3%, 6% and 9%, respectively. When A = 6%, N is selected as 1, 2, and 3, corresponding to inner ring fluctuation frequencies of ω = 209.44 rad/s, 418.88 rad/s and 628.32 rad/s, respectively.

2.2. Geometric Parameters of the Angular Contact Ball Bearing

The GSCK200A CNC lathe’s feed system employs HRB angular contact ball bearing 7603025 to support the ball screw, and its geometric relationships and primary parameters are shown in Figure 2.

2.3. Element Type and Meshing

In this study, all bearing components were constructed using the SOLID164 solid element type. Due to the lack of rotational degrees of freedom in solid element, a SHELL163 thin shell element was established on the inner surface of the inner ring, sharing nodes with the hexahedral elements on the inner ring surface to facilitate load application and rotational speed. Meshing is critical for executing dynamic analysis of rolling bearings in LS-DYNA. To determine the type and size of grid elements suitable for the analysis model, multiple factors such as calculation accuracy and solution time were carefully considered. To improve grid quality, small chamfers at the edges of the bearing model were eliminated. Generally, 8-node hexahedral elements were used with a mapped meshing approach, and finer grid sizes were used in contact areas. The full flexible body dynamic model of the angular contact ball bearing 7603025 is illustrated in Figure 3, comprising a total of 132,660 solid elements and 413,935 nodes.

2.4. Material Model

Because the plastic deformation of the angular contact ball bearing supporting the lead screw in the CNC lathe is minimal, the inner ring, outer ring, rolling elements and cage all employ the Isotropic Elastic Model (MAT1 MAT ELASTIC) in the Linear Elastic Material. The shell elements on the inner surface of the inner ring adopt the Rigid Material Model (MAT20 MAT RIGID). In practice, the inner ring, outer ring, and rolling elements of the angular contact ball bearing 7603025 are fabricated from GCr15 steel, and the cage is made of Nylon 66. The specific material parameters are summarized in

Table 1. Material parameters of the bearing components [29,30].

Materials	Density ρ/(kg/m3)	Elastic Modulus E/(kg/m3)	Poisson’s Ratio ν
GCr15 steel	7830	207	0.3
Nylon 66	1140	2.83	0.4

.

2.5. Boundary Conditions

Considering the actual working conditions of angular contact ball bearings, an interference fit is applied between the outer ring of the bearing and the inner hole of the housing. Fixed constraints were enforced on the outer ring, while the inner ring and cage retain three translational degrees of freedom and one rotational degree of freedom around the bearing axis. The rotational speed curves corresponding to the fluctuation conditions were applied to the rigid SHELL163 shell elements at the same nodes shared with the inner ring’s inner surface. The working condition parameters are outlined in

Table 2. Working conditions parameters [29,31].

Axial Load Fa/(N)	Radial Load Fr/(N)	Average Angular Speed n0/(r/min)
3500 N	1200 N	2000

.

The Automatic Surface To Surface (ASTS) is used when defining the contact between the bearing components to establish the contact relationships between the rolling elements and the inner ring, outer ring and cage. The ASTS adopts the Symmetrical Penalty Function Method.

$$F_{\mathrm{p}}=k\cdot \delta$$

(2)

where F_p is the Penalty Force. k is the Penalty Stiffness. δ is the Penetration Distance.

The calculation of Penalty Stiffness k adopts a Variable Stiffness Algorithm.

$$k=SLSFAC\cdot SF\cdot \left(K{S}^2/V\right)$$

(3)

where SLSFAC is the global penalty function scaling factor. SF is the scale factor of the local master surface and slave surface penalty factors. K is the Bulk Modulus of the material, representing its stiffness. S is the area of the main contact section. V is the volume of the element to which the slave node belongs.

The frictional force between contact interfaces is mainly calculated using the Coulomb Friction Model, supplemented by the Equivalent Elasto-Plastic Spring Model for numerical calculation. When the tangential force is less than (F_s)_max, the spring is in the stage of elastic deformation, simulating microscopic displacement without macroscopic sliding. When the tangential force exceeds the maximum static friction force, the spring enters a plastic flow stage, simulating macroscopic sliding. At this point, the magnitude of the friction force is F_d. The Friction coefficients of the ASTS contact model are shown in

Table 3. Friction coefficient of the ASTS contact model [29,30].

Contact Parts	Target Parts	fs	fd
Rolling elements	Inner ring	0.1	0.002
Rolling elements	Outer ring	0.1	0.002
Rolling elements	Cage	0.15	0.02

.

$$\{\begin{array}{l}{\left(F_{\mathrm{s}}\right)}_{\max }=f_s\cdot F_{\mathrm{n}}\\ F_{\mathrm{d}}=f_{\mathrm{d}}\cdot F_{\mathrm{n}}\end{array}$$

(4)

where (F_s)_max is the maximum static friction force. f_s is the static friction coefficient. F_n is the normal force between the contact surfaces. F_d is the sliding static friction force. f_d is the dynamic friction coefficient.

LS-DYNA is solved using the central difference method, and the maximum value of Δt that satisfies the stability condition is

$$\Delta t\le \Delta t_{\mathrm{cr}}={\displaystyle \frac{T_{\mathrm{n}}}{\pi }}={\displaystyle \frac{l_{\min }}{\sqrt{E\rho /(1-{\nu }^2)}}}$$

(5)

where T_n is the minimum natural period of the system. l_min is the minimum unit length. ρ is the density of the material. ν is the Poisson’s ratio of the material. E is the elastic modulus of the material.

This article adjusts the minimum time step through quality scaling (*CONTROL_TIMESTEP), which can greatly save CPU time while ensuring accuracy. And control the hourglass deformation of the structure by increasing the volume viscosity of the structure (*CONTROL_BULK_VISCOSITY) and increasing the elastic stiffness of the model (*CONTROL_HOURGLASS). The Solving control parameters of the model are shown in

Table 4. Solving control parameters of the model.

Parameter Names	Parameter Values
*CONTROL_TERMINATION	ENDTIM = 0.45 s
*CONTROL_TIMESTEP	TSSFAC = 0.9; DT2MS = −6.5 × 10−8
*CONTROL_BULK_VISCOSITY	Q1 = 1.5; Q2 = 0.06; TYPE = 1
*CONTROL_HOURGLASS	IHQ = 1; QH = 0.1
*DATABASE_BINARY_OPTION	D3PLOT = 0.001; D3THDT = 0.001

.

3. Model Validation

Rolling bearings typically operate with the outer ring stationary and the inner ring rotating. Consequently, an axial force of 3500 N, a radial force of 1200 N, and a constant rotational speed of 2000 r/min are applied to the inner ring. The analytical solutions for the kinematic parameters of the angular contact ball bearing 7603025 were derived from Equations (6)–(10). The full flexible body dynamic model analyzed the same operational conditions, extracting corresponding parameters from the calculation results for comparison with the analytical solutions.

The rotational speed of the cage is given by:

$$n_{\mathrm{c}}={\displaystyle \frac{1}{2}}{n}_{\mathrm{i}}\left(1-{\displaystyle \frac{D_{\mathrm{w}}\cos \alpha }{d_{\mathrm{m}}}}\right)$$

(6)

where $n_{\mathrm{i}}$ is the rotational speed of the inner ring.

The linear velocity at the contact point between the inner ring and the rolling elements is calculated as:

$$v_{\mathrm{i}}={\displaystyle \frac{\pi n_{\mathrm{i}}}{60}}\left(d_{\mathrm{m}}-D_{\mathrm{w}}\cos \alpha \right)$$

(7)

The linear velocity at the center of the rolling element is expressed as:

$$v_{\mathrm{r}}={\displaystyle \frac{2\pi n_{\mathrm{c}}{d}_{\mathrm{m}}}{120}}$$

(8)

During one rotation of the inner ring, the number of rolling elements passing a specific point on the inner and outer rings can be calculated using:

$$N_{\mathrm{ir}}={\displaystyle \frac{1}{2}}Z\left(1+{\displaystyle \frac{D_{\mathrm{w}}\cos \alpha }{d_{\mathrm{m}}}}\right)$$

(9)

$$N_{\mathrm{or}}={\displaystyle \frac{1}{2}}Z\left(1-{\displaystyle \frac{D_{\mathrm{w}}\cos \alpha }{d_{\mathrm{m}}}}\right)$$

(10)

The ADAMS simulation results and experimental results for cage rotational speed are illustrated in Figure 4. Figure 5 presents the LS-DYNA results compared to experimental results for the cage rotational speed, and

Table 5. Comparison of Kinematic Parameters of the Bearings.

Kinematic Parameters	Analytical Solutions	LS-DYNA Solutions	Relative Errors (%)
Rotational Speed of the Cage (r/min)	935	922	1.39
Linear Velocity of the Contact Point on the Inner Ring (mm/s)	4150	4135	0.36
Linear Velocity at the Center of the Rolling Element (mm/s)	2075	2044	1.49
The Number of Contacts between the Rolling Elements and the Inner Ring Per Unit Time (min−1)	18,112	18,545	2.39
The Number of Contacts between the Rolling Elements and the Outer Ring Per Unit Time (min−1)	15,888	15,692	1.23

provides a comparison of the LS-DYNA solutions with the analytical solutions.

As can be seen from Figure 4 and Figure 5, and Table 5, relative errors between LS-DYNA solutions and analytical solutions of the kinematic parameters remain within 3%. The LS-DYNA results for the cage rotational speed are consistent with both the ADAMS result (928 r/min) and the experimental finding (927 r/min) [29,30]. This strong agreement confirms the accuracy of the full flexible body dynamic model of the angular contact ball bearing developed in this paper.

4. Calculation Results and Analysis

4.1. Influence of Fluctuation Amplitude on Dynamic Characteristics

The relationship curves between cage rotational speed and time under various fluctuation amplitudes are depicted in Figure 6, with corresponding parameter values summarized in

Table 6. Rotational speed parameter values of the cage under different fluctuation amplitude conditions.

Fluctuation Amplitude of the Inner Ring Speed ${\mathit{A}}_{\mathbf{i}}$ (%)	Average Value of the Cage ${\overline{\mathit{n}}}_{\mathbf{c}}$ (r/min)	Fluctuation Amplitude of the Cage Speed ${\mathit{A}}_{\mathbf{c}}$ (%)
0	922.42	2.89
3	918.60	3.95
6	918.66	6.48
9	919.16	9.96

.

As can be seen from Figure 6 and Table 6, when the fluctuation amplitude of the inner ring’s rotational speed is small, the periodic fluctuation pattern of the cage’s speed is not pronounced. Conversely, larger fluctuation amplitudes result in a more clearly defined periodic fluctuation pattern for the cage’s rotational speed, and the fluctuation frequency remains relatively consistent. It indicates that the fluctuation amplitude of the inner ring speed does not affect the fluctuation frequency of the cage speed curve. Furthermore, the fluctuation amplitude of the cage is close to that of the corresponding inner ring rotational speed. As the fluctuation amplitude of the inner ring’s rotational speed increases, the maximum speed of the cage rises and its minimum speed declines, while the average speed remains largely unchanged, consequently broadening the fluctuation range of the cage rotational speed curve.

The centroid trajectory curves of the cage under various fluctuation amplitudes are shown in Figure 7. The variation curves of the contact force between the single rolling element and the cage over time under various fluctuation amplitudes are shown in Figure 8. The curves of maximum centroid distance and maximum contact force of the cage varying with fluctuation amplitude are shown in Figure 9.

From Figure 7, Figure 8 and Figure 9, it is apparent that at small fluctuation amplitudes, the centroid trajectory of the cage is approximately elliptical. However, it becomes irregular at larger amplitudes. As the fluctuation amplitude of the inner ring’s rotational speed increases, the motion range of the centroid trajectory of the cage slightly expands, and the trajectory becomes increasingly disordered. Concurrently, the contact force between the single rolling element and the cage rises. This indicates that an increase in the fluctuation amplitude of the inner ring’s rotational speed intensifies collisions between the rolling elements and the cage to a certain extent. Consequently, this reduces the motion stability of the cage and accelerates wear and deformation failure of the cage.

The centroid trajectory curves of the inner ring under various fluctuation amplitudes are shown in Figure 10. The curve of maximum distance of the inner ring centroid varying with fluctuation amplitude is shown in Figure 11. The variation curves of the contact force between the single rolling element and the inner ring over time under different fluctuation amplitude conditions are shown in Figure 12.

Figure 10, Figure 11 and Figure 12 illustrate that the contact force between the rolling element and the inner ring increases and decreases periodically over time. This periodic behavior occurs because the rolling elements experience uneven loading due to the combined effects of radial and axial loads. A greater fluctuation amplitude in the inner ring’s rotational speed leads to an enlarged motion range of the inner ring’s centroid trajectory, and the appearance of sharp angles away from the center of the trajectory becomes more likely. Consequently, the fluctuations in the contact force curve between the rolling element and the inner ring intensify. This is attributed to the increased fluctuation amplitude of the inner ring’s rotational speed, which intensifies the sliding between the rolling element and the inner ring, resulting in a deterioration of their contact state and increased wear and deformation.

4.2. Influence of Fluctuation Frequency on Dynamic Characteristics

The relationship curves between cage rotational speed and time under various fluctuation frequencies are depicted in Figure 13, with corresponding parameter values summarized in

Table 7. Rotational speed parameter values of the cage under different fluctuation frequency conditions.

Fluctuation Frequency of the Inner Ring Speed N (Times)	Average Value of the Cage ${\overline{\mathit{n}}}_{\mathbf{c}}$ (r/min)	Fluctuation Amplitude of the Cage Speed ${\mathit{A}}_{\mathbf{c}}$ (%)
0	922.42	2.89
1	921.13	6.40
2	918.66	6.48
3	918.55	6.10

.

As shown in Figure 13 and Table 7, even when the inner ring’s rotational speed remains stable, there is still some degree of fluctuation in the cage’s rotational speed. When substantial fluctuations occur in the rotational speed of the inner ring, the fluctuation period of the cage rotational speed curve closely aligns with that of the inner ring rotational speed curve. In other words, a greater fluctuation frequency of the inner ring’s rotational speed corresponds to a larger fluctuation frequency in the cage rotational speed curve. Under varying fluctuation frequency conditions, the maximum, minimum, and average values of the cage rotational speed curves exhibit minimal differences. Therefore, the fluctuation frequency of the inner ring’s rotational speed has little effect on the fluctuation amplitude of the cage rotational speed curve.

The centroid trajectory curves of the cage under various fluctuation frequencies are shown in Figure 14. The variation curves of the contact force between the single rolling element and the cage over time under various fluctuation frequencies are shown in Figure 15. The curves of maximum centroid distance and maximum contact force of the cage varying with fluctuation frequency are shown in Figure 16.

Figure 14, Figure 15 and Figure 16 illustrate that as the fluctuation frequency of the inner ring’s rotational speed increases, the motion range of the cage centroid trajectory enlarges, and the trajectory becomes relatively more chaotic, while the contact force between the single rolling element and the cage also increases. This suggests that an increase in the fluctuation frequency of the inner ring’s rotational speed leads to decreased stability in the cage’s motion. The heightened fluctuation frequency results in more frequent collisions between the rolling elements and the cage, which in turn causes increased vibration, wear, and deformation of the cage.

The centroid trajectory curves of the inner ring under various fluctuation frequencies are shown in Figure 17. The curve of maximum distance of inner ring centroid varying with fluctuation frequency is shown in Figure 18. The variation curves of the contact force between the single rolling element and the inner ring over time under various fluctuation frequencies are shown in Figure 19.

Figure 17, Figure 18 and Figure 19 indicate that higher fluctuation frequencies in the inner ring’s rotational speed lead to larger amplitudes and more frequent fluctuations in the contact force curve between the single rolling element and the inner ring. Additionally, the motion range of the centroid trajectory of the inner ring expands, and sharper angles away from the center of the trajectory area become apparent. This suggests that increasing the fluctuation frequency of the inner ring’s rotational speed exacerbates the vibration and wear of the bearing’s inner ring.

5. Conclusions

Rotational speed fluctuations are common in practice, which imposes stringent requirements on the dynamic characteristics of angular contact ball bearings under non-stationary conditions. This paper focused on the nonlinear dynamics of the inner ring and cage of angular contact ball bearings subjected to rotational speed fluctuations, investigating the mechanisms behind the generation and variation of unstable motion in these components under different fluctuation conditions. The objective of this research is to mitigate the impact of variable rotational speed conditions on the dynamic performance of the inner ring and cage, thereby enhancing the motion stability and service reliability of the bearings.

Taking the angular contact ball bearing 7603025 from a machine tool feed system as the research object, we have comprehensively considered the nonlinear factors inherent to the bearing. The contact relationships among the internal components were simulated using the Linear Elastic Material Model and the Symmetrical Penalty Function Method, leading to the establishment of a full flexible body dynamic model of the angular contact ball bearing. Based on this model, we investigated the effects of two parameters—the fluctuation amplitude and frequency of rotational speed—on the rotational speed, centroid trajectories of the inner ring and cage, as well as the contact force between the rolling element and the inner ring and cage under sinusoidal rotational speed fluctuation conditions. This study reveals the variation laws of the stability of the inner ring and cage under different rotational speed fluctuation parameters, providing valuable insights for enhancing the operational rotation and reliability of rolling bearings. The main conclusions of this research work are as follows:

• The relative errors between the LS-DYNA solutions and analytical solutions for the kinematic parameters are all within 3%, confirming the accuracy of the established dynamic model for the angular contact ball bearing.
• When the frequency of the rotational speed fluctuations is constant, an increase in the fluctuation amplitude of the inner ring’s rotational speed results in a corresponding increase in the fluctuation amplitude of the cage’s rotational speed curve, while the fluctuation frequency remains essentially unchanged.
• When the fluctuation amplitude of the rotational speed is constant, a higher fluctuation frequency of the inner ring’s rotational speed leads to a greater fluctuation frequency in the cage’s rotational speed curve, although the fluctuation amplitude remains largely unchanged.
• Increased fluctuation amplitude and frequency of the inner ring’s rotational speed lead to intensified vibrations of the inner ring, increased contact forces between the rolling elements and the cage, a more disordered centroid trajectory of the cage, and an expanded motion range of the cage, consequently reducing the motion stability of both the inner ring and the cage.