Study on the Time-Varying Stiffness Characteristics of Four-Point Contact Ball Bearings

Abstract

This paper takes a four-point contact ball bearing of a wind turbine as the research object, analyzes the force and deformation relationship under the combined action of axial load and radial load, obtains the load distribution of rolling elements, and establishes a time-varying stiffness model of four-point contact ball bearings without clearance. The stiffness variation law of the case bearing in one rolling period is analyzed, and the time-varying characteristics of stiffness are characterized by the average stiffness and stiffness amplitude variation rate. The influence laws of the number of rolling elements, initial contact angle, axial load, and radial load on the time-varying characteristics of bearing stiffness are analyzed. The results show that within one rolling period, the average value of axial stiffness is about 2.21 times that of radial stiffness, and the amplitude variation rates of radial stiffness and axial stiffness are 0.0047% and 0.002%, respectively. The time-varying characteristics of both are not obvious. The influence of the number of rolling elements on the two stiffnesses is almost linear, while the influence of axial load on stiffness is small; the initial contact angle is positively correlated with axial stiffness and negatively correlated with radial stiffness. With the increase in radial load, the two stiffnesses also increase. Finally, the stiffness test of four-point contact ball bearings was carried out, and the error between the test value and the theoretical value was less than 15%, which preliminarily verified the correctness of the stiffness model.

1. Introduction

Due to its unique geometric structure and force distribution characteristics, four-point contact ball bearings exhibit significant advantages in application scenarios with high precision and high load requirements [1,2]. Compared with traditional two-point or three-point contact ball bearings, four-point contact ball bearings can provide higher bearing capacity and stiffness in a more compact space, making them widely used in wind power pitch systems, aerospace equipment, high-speed precision machine tools, and high-performance automobiles [3,4]. The primary advantages of four-point contact ball bearings include compact structure design, high bearing capacity, excellent stiffness and durability, and the ability to withstand complex multi-axial loads. However, these bearings also have certain shortcomings, such as complex manufacturing processes, high costs, and the possibility of greater wear risks under high dynamic load conditions [5,6]. As a key performance parameter, stiffness directly affects its stability and response speed in actual operation. High stiffness not only reduces vibration and noise, enhancing system accuracy and reliability, but also effectively suppresses the transmission of external loads, improving the dynamic performance of the overall mechanical system [7]. Therefore, in-depth study into the stiffness characteristics of four-point contact ball bearings is of great significance for optimizing its design and improving its application performance.

Jones [8] studied the displacement of the inner and outer rings under load by an iterative method and successfully solved the load distribution of the rolling element. Building on Jones’s work, Harris [9] extended the model on the basis of Jones to further explore the characteristics of the load distribution inside the bearing. He ZS [10] proposed an integrated model to analyze the effects of the number of rolling elements, contact angle, and the number of bolts on the comprehensive performance of the bearing, including stiffness variations. Chen et al. [11] developed the time-varying stiffness model of angular contact ball bearings under zero-clearance conditions. The time-varying characteristics were quantified by the average stiffness and amplitude change rate, and the variation law of stiffness with rolling period was revealed. Su et al. [12] proposed a dynamic analysis method of a flexible cage considering multi-point contact behavior, analyzing the impact of operational conditions and manufacturing errors on the rotational stability and wear degree of the cage, and tested its influence on bearing stiffness. In addition, Escanciano et al. [13] proposed a new method to evaluate the friction torque of spherical slewing bearings. This method takes into account the distribution of the preload, and by simulating the dynamic change of the contact point of the sphere, the results are correlated with the experimental data, thus revealing the variation law of bearing stiffness. Liu et al. [14] established a finite element model of a single-row four-point contact ball bearing using ABAQUS software and designed a static loading experiment to discuss discrepancies between numerical and experimental stiffness results. Yao et al. [15] established a parametric three-dimensional assembly model of a thin-walled four-point contact ball bearing using the ADAMS macro program. The penalty formula and unilateral nonlinear spring-damper model were applied to the bearing, and their influence on the stiffness characteristics was analyzed. He et al. [16] optimized the structural parameters of four-point contact ball bearings using numerical methods and analyzed the stiffness performance under different parameters. Han et al. [17] used the finite element method to calculate the contact stress between the rolling element and the raceway and verified the effectiveness of the bearing stress analysis method based on the spring element in the stiffness calculation. Li et al. [18] performed the mechanical model of the internal load of the four-point contact ball bearing through the secondary development of ADAMS software and further analyzed the influence of load change on stiffness. Halpin et al. [19] constructed an analysis model for four-point contact ball bearings. The model is solved analytically by the standard Newton–Raphson method, and the performance parameters, including stiffness, are calculated. The accuracy of the results is verified by comparison with the existing analysis programs. Mireia et al. [20] considered the load distribution algorithm of four-point contact ball bearings considering structural elasticity, iteratively obtained a highly nonlinear elastic model, and obtained variations in stiffness under different load conditions. According to the performance characteristics of double row four-point contact ball bearings under negative clearance condition, the Newton–Raphson method was used to calculate the load distribution and stiffness variations of double row four-point contact ball bearings under combined load [21]. Based on the potential energy formula and minimization principle of four-point contact ball bearings, Iker et al. [22] proposed a method for calculating the load distribution and stiffness by considering the preload, error, and ring flexibility. Li et al. [23] investigated the geometric compatibility equation of a four-point contact ball bearing and measured the axial load–displacement curve and inclined load–displacement curve of the sample bearing under five different design parameters to evaluate its stiffness characteristics. Based on the rolling element and four-way traction-contact equation, Ma et al. [24] studied the change in contact characteristics of four-point contact ball bearings under complex working conditions and discussed its influence on bearing stiffness. Gilbert et al. [25] proposed a quasi-static model under five-degree-of-freedom load and displacement and evaluated the stiffness performance under different working conditions, aiming to address issues in the application of four-point contact ball bearings. Aiming at the multi-point contact problem of a QJ214 four-point contact ball bearing, Wang et al. [26] established a steel ball-channel contact model and analyzed the influence of its causes on stiffness.

Despite the significant progress made in stiffness calculation and testing, most studies are still based on Hertz contact theory [27], assuming that the internal stiffness of the bearing is constant, and the dynamic change in stiffness in actual operation is not fully considered. In addition, existing experimental methods are mostly carried out under static or quasi-static conditions [25], and it is difficult to fully reflect the stiffness characteristics of the bearing under dynamic load, which limits the accuracy and applicability of the model to a certain extent. To address these challenges, this study systematically investigates four-point contact ball bearings with the following objectives:

(1) Based on the combined effects of axial and radial loads, the force and deformation relationships of rolling elements are analyzed, and a time-varying stiffness model for four-point contact ball bearings is established.

(2) The stiffness variation over a complete rolling period is examined, along with a systematic analysis of the influence of parameters, such as the number of rolling elements, initial contact angle, axial load, and radial load, on the time-varying stiffness characteristics of the bearing. The time-varying stiffness characteristics are quantified using average stiffness and stiffness amplitude variation rates.

(3) A stiffness test of a four-point contact ball bearing is designed and conducted. The experimental results align well with the theoretical model predictions, which verifies the accuracy and applicability of the established time-varying stiffness model.

2. Time-Varying Stiffness Model of Four-Point Contact Ball Bearing

2.1. Load Distribution of Rolling Element Under Combined Load

The internal contact of the bearing is elastic deformation, which conforms to the Hertz contact theory [28]. At low speeds, the influence of inertia force on bearing stiffness is negligible [29]. Under the conditions of low speed and heavy load, the influence of friction on bearing stiffness is negligible. Based on this simplified assumption, a mechanical model is constructed, as shown in Figure 1 and Figure 2.

In Figure 1, the number of left and right load rollers is represented by m and n, respectively, while the position angle of the roller φ_j varies in the range of 0° to ψ. Fa and Fr represent the axial and radial external loads, respectively, and δ_a and δ_r represent the corresponding displacements. For the contact pairs AB and CD, the normal loads acting on them under combined loads are expressed as Q_lm and Q_rn, and Q’_lm and Q’_rn, respectively. The sum of the horizontal components generated by these contact pairs under normal load should be equal to F_a, and the sum of the vertical components should be equal to F_r. The contact angle of the AB contact pair is α_u, while the contact angle of the CD contact pair is α’_u. The whole force balance equation is established as follows:

$$\left\{\begin{array}{l}{F}_r={\displaystyle \sum _{i=1}^m\left\{Q_{li}\cos {\alpha }_u\cos \left[\psi (i-1)+{\phi }_j\right]+{Q_{li}}^{\prime }\cos {{\alpha }^{\prime }}_u\cos \left[\psi (i-1)+{\phi }_j\right]\right\}}\\ +{\displaystyle \sum _{j=1}^n\left\{Q_{rj}\cos {\alpha }_u\cos (j\psi -{\phi }_j)+Q_{rj}\cos {{\alpha }^{\prime }}_u\cos (j\psi -{\phi }_j)\right\}}\\ F_a={\displaystyle \sum _{i=1}^m\left(Q_{li}\sin {\alpha }_u+{Q_{li}}^{\prime }\sin {{\alpha }^{\prime }}_u\right)}+{\displaystyle \sum _{j=1}^n\left(Q_{rj}\sin {\alpha }_u+{Q_{rj}}^{\prime }\sin {{\alpha }^{\prime }}_u\right)}\end{array}\right.$$

(1)

$$\left\{\begin{array}{l}{\alpha }_u=\arctan \left({\displaystyle \frac{s\sin {\alpha }_0+{\delta }_a}{s\sin {\alpha }_0+{\delta }_r}}\right)\\ {{\alpha }_u}^{\prime }=\arctan \left({\displaystyle \frac{s^{\prime }\sin {\alpha }_0-{\delta }_a}{s^{\prime }\sin {\alpha }_0+{\delta }_r}}\right)\end{array}\right.$$

(2)

In the formula, s is the curvature center distance of the AB contact to the inner and outer ring channel, unit mm; s’ is the curvature center distance of the CD contact to the inner and outer ring channels, unit mm.

Here, m and n represent, respectively, the number of load rolling elements on the left and right sides of the bearing:

$$\left\{\begin{array}{l}\psi (m-1)+{\phi }_j\le \phi /2\\ n\psi -{\phi }_j\le \phi /2\end{array}\right.$$

(3)

In the formula, j is the bearing angle of the combined load, which can be calculated according to the following formula:

$$\phi =2\arccos \left(-{\displaystyle \frac{{\delta }_a\tan {\alpha }_u}{{\delta }_r}}\right)$$

(4)

If ${\delta }_a\tan {\alpha }_u<1$ in Equation (4), then φ is less than 2π, reflecting the bearing part of the rolling element, and when ${\delta }_a\tan {\alpha }_u>1$, φ = 2π, indicating that all rolling elements are subjected to load.

According to the relationship between load and displacement, the normal load of the AB contact pair on each rolling element on the left and right sides of a single- row four-point contact ball bearing is derived [8]:

$$\left\{\begin{array}{l}{Q}_{lm}=K_n{{\delta }_{lm}}^{1.5}\\ Q_{rn}=K_n{{\delta }_{rn}}^{1.5}\end{array}\right.$$

(5)

The normal load of the CD contact pair of each rolling element is [8]

$$\left\{\begin{array}{l}{Q_{lm}}^{\prime }=K_n{{{\delta }^{\prime }}_{lm}}^{1.5}\\ {Q_{rn}}^{\prime }=K_n{{{\delta }^{\prime }}_{rn}}^{1.5}\end{array}\right.$$

(6)

The normal displacement of the AB contact pair of each rolling element of the bearing is

$$\left\{\begin{array}{l}{\delta }_{lm}={\delta }_r\cos \left[\psi \left(i-1\right)+\theta \right]\cos {\alpha }_u+{\delta }_a\sin {\alpha }_u\\ {\delta }_{rn}={\delta }_r\cos \left(j\psi -\theta \right)\cos {\alpha }_u+{\delta }_a\sin {\alpha }_u\end{array}\right.$$

(7)

The normal displacement of the CD contact pair of each rolling element is

$$\left\{\begin{array}{l}{\delta }_{lm}^{\prime }={\delta }_r\cos \left[\psi \left(i-1\right)+\theta \right]\cos {\alpha }_u^{\prime }+{\delta }_a\sin {\alpha }_u^{\prime }\\ {\delta }_{rn}^{\prime }={\delta }_r\cos \left(j\psi -\theta \right)\cos {\alpha }_u^{\prime }+{\delta }_a\sin {\alpha }_u^{\prime }\end{array}\right.$$

(8)

Equations (7) and (8) are the normal displacements of AB contact pairs obtained from the geometric relationship between the axial displacement, radial displacement, and contact angle of the bearing. Figure 2 clearly expresses the relationship between contact and deformation.

2.2. Stiffness Model and Verification

2.2.1. Time-Varying Stiffness Model Solution

The flow chart for solving the stiffness model in this paper is shown in Figure 3.

Axial and radial stiffness are used to describe the resistance of the bearing to the load in the axial and radial directions, respectively. The calculation formula is

$$\left\{\begin{array}{l}{K}_r={\displaystyle \sum _{i=1}^m}\left\{{\displaystyle \frac{Q_{li}}{{\delta }_{li}}}\cos {\alpha }_u\cos \left[\psi (i-1)+{\phi }_j\right]+{\displaystyle \frac{Q_{li}^{\prime }}{{\delta }_{li}^{\prime }}}\cos {\alpha }_u^{\prime }\cos \left[\psi (i-1)+{\phi }_j\right]\right\}\\ +{\displaystyle \sum _{j=1}^n}\left\{{\displaystyle \frac{Q_{rj}}{{\delta }_{rj}}}\cos {\alpha }_u\cos (j\psi -{\phi }_j)+{\displaystyle \frac{Q_{rj}^{\prime }}{{\delta }_{rj}^{\prime }}}\cos {\alpha }_u^{\prime }\cos (j\psi -{\phi }_j)\right\}\\ K_a={\displaystyle \sum _{i=1}^m}\left({\displaystyle \frac{Q_{li}}{{\delta }_{li}}}\sin {\alpha }_u+{\displaystyle \frac{Q_{li}^{\prime }}{{\delta }_{li}^{\prime }}}\sin {\alpha }_u^{\prime }\right)+{\displaystyle \sum _{j=1}^n}\left({\displaystyle \frac{Q_{rj}}{{\delta }_{rj}}}\sin {\alpha }_u+{\displaystyle \frac{Q_{rj}^{\prime }}{{\delta }_{rj}^{\prime }}}\sin {\alpha }_u^{\prime }\right)\end{array}\right.$$

(9)

2.2.2. Time-Varying Stiffness Model Verification

To study the time-varying stiffness characteristics of single-row four-point contact ball bearings, the corresponding calculation model is established by the MATLAB(V2022b) program. Under the condition of low-speed operation, the influence of rotational speed on bearing stiffness is small, so it is not considered in this study.

Table 1. Parameters of the bearing used for model verification.

Bearing Parameter	Numerical Value
Ball diameter, Dw/mm	50.8
Bearing pitch circle diameter, dm/mm	1300
Number of rolling elements, Z	70
Inner ring curvature coefficient, fi	0.525
Curvature coefficient of outer ring, fo	0.525
Initial contact angle, α/(°)	45

shows the parameters of the bearings used for model verification.

Table 2. The comparison results of axial stiffness between the validated bearing model and reference [30].

Axial Load/(N)	Reference [30] Results/(N/μm)	Calculation Results of This Paper/(N/μm)	Error (%)
20,000	2000.00	2028.486	1.424
40,000	2500.00	2560.449	2.418
60,000	3000.00	2935.465	2.151
80,000	3200.00	3235.258	1.101

shows that the theoretical calculation results are compared with the calculation results of reference [30] when the bearing is subjected to a pure axial load of 20,000~80,000 N.

According to the comparison results, the error range between the calculation results of the time-varying stiffness model established in this chapter and the results of reference [30] is 1.101~2.418%, not more than 5%.

Table 3. The comparison results of radial stiffness between the validated bearing model and reference [30].

Axial Load/(N)	Reference [30] Results/(N/μm)	Calculation Results of This Paper/(N/μm)	Error (%)
20,000	1176.47	1208.815	2.749
40,000	1538.46	1528.777	0.629
60,000	1714.28	1755.507	2.405
80,000	1904.76	1937.528	1.720

shows the comparison between the theoretical calculation results and the calculation results of reference [30] when the bearing is subjected to a pure radial load of 20,000~80,000 N.

Based on the comparison results, the time-varying stiffness model for single-row four-point contact ball bearings developed in this chapter demonstrates a calculation error range of 0.629% to 2.749% under different load conditions compared with the results reported in the reference, with all errors remaining below 5%. In summary, the axial and radial stiffness calculation results of the four-point contact ball bearing time-varying stiffness model constructed in this chapter align closely with the data from the reference, with small error values consistently controlled within 5%. This result verifies the accuracy and reliability of the developed model.

2.3. Characterization Parameters of Time-Varying Stiffness

In order to accurately describe the dynamic characteristics of time-varying stiffness in a complete rolling period, two key parameters, average stiffness K_m and stiffness amplitude change rate x, are introduced [31]:

$$\left\{\begin{array}{l}{K}_m=\left({\displaystyle \sum _{i=1}^p{K}_i}\right)/p\\ \xi ={\displaystyle \frac{K_{\max }-K_{\min }}{K_m}}\times 100\%\end{array}\right.$$

(10)

In the given formula, the variables are defined as follows: K_min represents the minimum stiffness value in the same revolution angle range, with units of N/μm; K_max is the maximum stiffness value in a certain range of the revolution angle, with units of N/μm; p represents the total number of data points corresponding to the revolution angle; K_i represents the stiffness value of the bearing under different revolution angles, with units of N/μm.

3. Analysis of Time-Varying Stiffness Characteristics of Four-Point Contact Ball Bearings

3.1. The Structural Parameters and Working Conditions of the Case Bearing

The case bearing is the pitch bearing of a wind turbine.

Table 4. Case analysis of bearing structure and working condition parameters.

Structural Parameter of Bearing	Numerical Value
Ball diameter/mm	53.7
Bearing outside diameter/mm	2376
Bearing width/mm	235
Bearing pitch circle diameter/mm	2174
Number of rolling elements	68
Inner ring curvature coefficient	0.525
initial contact angle/°	45
axial load/N	20,000
radial load/N	20,000

shows its structure and working condition parameters.

3.2. The Stiffness Variations of the Bearing in a Complete Period

The periodic change of the position caused by the revolution of the rolling element around the spindle is the main factor causing time-varying stiffness. In this paper, when rolling element 1 is at the bottom of the initial position, the axial load and the radial load are 20,000 N and there are 68 rolling elements. The rotation angle θ varies periodically (360°/Z). The load borne by each rolling element in the vertical direction is shown in Figure 4.

As shown in Figure 4, in a complete rolling period, the axial stiffness and radial stiffness show a sine-like curve, and the phase difference between the two is about 180°. The axial average stiffness is 2882.92 N/μm, and the radial average stiffness is 1305.08 N/μm. The axial stiffness is about 2.2 times the radial stiffness. This is because the axial load is borne by all the rolling elements, while the radial load is only borne by the rolling elements at the bearing angle. The change rate of axial stiffness amplitude is 0.0047%, and the change rate of radial stiffness amplitude is 0.002%. The time-varying characteristics are not obvious. This is because the number of rollers in the case bearing is large, and the load distribution of the rollers changes little in a rolling period [32].

3.3. The Influence of the Number of Rolling Elements on the Time-Varying Characteristics of Stiffness

Under a constant axial load of 20,000 N and a constant radial load of 20,000 N, the stiffness characteristics of the bearing change with the number of rolling elements, as shown in Figure 5.

It can be seen from Figure 5 that with the increase in the number of rolling elements, the axial and radial average stiffness of the bearing increases approximately linearly. This is because as the number of rolling elements increases, the number of bearing rolling elements increases, which increases the bearing stiffness. When the number of rolling elements is small (between 50 and 60), the influence on the change rate of stiffness amplitude is slightly larger. When the number of rolling elements continues to increase, the change rate of stiffness amplitude continues to decrease, but the change is very small. This is because with the increase in the number of rolling elements, the corresponding angle of a rolling bearing decreases, and the variation in the load distribution of the rolling element also decreases, so the change rate of the stiffness amplitude also decreases.

3.4. The Influence of Initial Contact Angle on the Time-Varying Characteristics of Stiffness

Under simultaneous axial and radial loads of 20,000 N, with the number of rolling elements fixed at 68, the time-varying stiffness parameters of the bearing exhibit specific trends with changes in the initial contact angle, as shown in Figure 5.

It can be seen from Figure 6 that an increase in the initial contact angle makes the axial average stiffness increase almost linearly, while the radial average stiffness decreases monotonously. This is because the increase in the contact angle can improve the axial bearing capacity of the bearing, while weakening the radial bearing capacity of the bearing, resulting in a decrease in axial deformation and an increase in radial deformation. When the contact angle changes from 30° to 36°, the axial and radial stiffness amplitude change rates are significantly reduced. When the initial contact angle continues to increase to 50°, the amplitude change rates of the two stiffnesses are almost unchanged. This is because a larger contact angle makes more contact pairs of rolling elements bear load, and the load distribution is more uniform. The load of each rolling element is almost unchanged during the rolling bearing period.

3.5. The Influence of Axial Load on the Time-Varying Characteristics of Stiffness

When the number of rolling elements is 68 and the radial load is 60,000 N, the variation in the average stiffness and the change rate of the stiffness amplitude of the bearing with the axial load are shown in Figure 7.

It can be seen from Figure 7 that under the condition of constant radial load, with an increase in axial load, the axial average stiffness of the bearing gradually decreases, while the radial average stiffness gradually increases. Because the increase in axial load increases the contact angle and the number of bearing rolling elements increases, the influence of the two on the bearing stiffness is the opposite, so the change in the two is very small.

3.6. The Influence of Radial Load on the Time-Varying Characteristics of Stiffness

When the number of rolling elements is fixed at 68 and subjected to a constant axial load of 20,000 N, the time-varying characteristic parameters of the bearing are studied in depth as the radial load changes, as shown in Figure 7.

It can be seen from Figure 8 that when the axial load is constant, the axial average stiffness and radial average increase with an increase in radial load. This is because as the radial load increases, the bearing angle of the bearing increases, and the number of rolling elements participating in the load increases, which also leads to a decrease in the amplitude change rate of the axial and radial stiffness of the bearing.

4. Four-Point Contact Ball Bearing Stiffness Test

4.1. Test Bench and Test Scheme

The bearing stiffness test was carried out using the four-point contact ball bearing stiffness test bench shown in Figure 9. The motor was securely fixed to the base using bolts, ensuring stability throughout the experiment, and its shaft end was connected with the spindle flange by nylon rope. The test bench applied an external radial load in the middle of the rotor. To balance the axial derived force caused by the load, identical test bearings were mounted face-to-face at both ends of the main shaft. A central loading bearing was placed at the midpoint of the shaft to enable precise load distribution. Radial loads were applied to the central loading bearing using bolts, enabling controlled loading of the overall structure. In addition, although the theoretical model does not consider the influence of lubrication on bearing stiffness, in order to ensure the smooth operation of the bearing in the test, grease is used for lubrication, and the lubrication state of the bearing has little effect on its stiffness and can be ignored [33,34].

To accurately obtain the force and displacement data required for stiffness calculation, a comprehensive test scheme was designed, as shown in Figure 10.

4.2. Experiment Results and Analyses

The four-point contact ball bearing shown in Figure 11 is used for constant speed and variable load test. The test bearing model is QJ206, and the structural parameters are shown in

Table 5. Test bearing parameter.

Structural Parameter	Numerical Value
Bearing outside diameter/mm	62
Bearing bore diameter/mm	30
Bearing pitch circle diameter/mm	46
Ball diameter/mm	9.525
Number of rolling elements	9
Initial contact angle/°	45

.

In the test, the rotation rate of the motor was constant at 1500 revolutions per minute, and the bearing is fixed by face-to-face installation. When the load gradually changed in the range of 100 N to 2000 N, five sets of tests were performed to obtain data under different load conditions. The test displacement data are shown in Figure 12.

In the test, it was found that there was a significant deviation between the results of the third experiment and the theoretical calculation. In order to further explore this phenomenon, the actual measurement data of bearing 1 and bearing 2 in the third test are compared with the theoretical results in detail. The results are shown in

Table 6. Comparison between the test value and the theoretical value of the test bearing stiffness.

	Stiffness	Theoretical Stiffness	Test Bearing 1 Test Stiffness	Test Bearing 2 Test Stiffness	Test Bearing 1 Error	Test Bearing 2 Error
N/μm Load/N
1000		66.478	60.351	58.017	10.15%	14.58%
1200		70.904	65.479	63.537	8.29%	11.59%
1400		74.902	68.742	66.712	8.96%	12.28%
1600		78.568	71.663	71.999	9.64%	9.12%
1800		81.967	73.993	74.651	10.78%	9.80%
2000		85.146	76.560	75.125	11.21%	13.34%

.

From the comparison of the data in Table 6, it is evident that the stiffness value obtained by theoretical calculation is generally higher than that of the test bearing, and the stiffness error of test bearing 2 is slightly larger than that of test bearing 1. The main reason may be due to the different forces at both ends of the spindle and the installation error of the bearing. The error range between the test results and the theoretical results of bearing 1 is 8.29–11.21%, and the error range between the test results and the theoretical results of bearing 2 is 9.12–14.58%. The test values are in good agreement with the theoretical values, which verifies the correctness of the stiffness model.

5. Conclusions

In this paper, the four-point contact ball bearing is taken as the research object, its time-varying stiffness characteristics are theoretically modeled and analyzed, and the stiffness test is carried out. The conclusions are as follows:

(1) A four-point contact ball bearing stiffness model considering time-varying characteristics is established. The variation in axial stiffness and radial stiffness in the rolling period is analyzed. It is found that the axial stiffness is significantly greater than the radial stiffness, and the change in the two is sinusoidal, but the overall time-varying characteristics are not obvious.

(2) The influence of bearing structure parameters and external load on the time-varying characteristics of stiffness is analyzed. With an increase in the number of rolling elements, the average value of axial and radial stiffness increases. When the contact angle increases, the axial stiffness increases and the radial stiffness decreases. The influence of load change on stiffness is also different. When the radial load increases, the change in stiffness amplitude decreases first and then tends to be stable.

(3) The stiffness of a four-point contact ball bearing is tested. The error between the experimental data and the theoretical calculation is less than 15%, which verifies the correctness of the stiffness model.

In this study, we simplified the model and ignored the influence of the deformation of the inner and outer rings of the bearing and lubrication state on bearing stiffness. In the future, we intend to establish a bearing stiffness model considering the deformation of the inner and outer rings of the bearing and incorporate the bearing lubrication state into the model to analyze the influence of lubrication state on bearing stiffness.