*Article* **Research on the Bearing Sliding Loss Based on Time-Varying Contact Angle between Ball and Raceway**

**Shuaijun Ma, Ke Yan \* , Mengnan Li, Yongsheng Zhu and Jun Hong**

Key Laboratory of Education Ministry for Modern Design and Rotor-Bearing System, Xi'an Jiaotong University, Xi'an 710049, China

**\*** Correspondence: yanke@mail.xjtu.edu.cn

**Abstract:** Based on the mechanical model, the friction loss between the ball and the raceway along the major axis of the contact ellipse is analyzed. The result shows that this part of the loss accounts for about 13.67% of the overall loss, which is mainly determined by the ball sliding length and cannot be ignored. The effects of the radial force, torque, rotational speed and groove curvature ratio on the sliding are all studied. Compared with other factors, radial force has the greatest influence on the sliding loss. As bearing speed gradually grows, the sliding on the inner raceway gradually increases while it gradually decreases on the outer raceway. Compared to the outer raceway curvature ratio, the sliding length is less sensitive to changes in the curvature ratio of the inner raceway. The paper provides theoretical guidance for the design and application of low-friction bearings.

**Keywords:** angular contact ball bearing; sliding length; contact ellipse major axis; friction loss; contact angle

#### **1. Introduction**

Owing to the characteristics of low friction and high precision, ball bearings are widely used in rotational systems, such as high-speed motors, precision machine tools and aero engines [1,2]. Regarding the complex internal structure of the bearing, relative motion and force conditions, etc., there are complex tribological behaviors between balls and inner/outer rings, which may cause bearing friction heat and power loss, and even accelerate bearing wear and reduce bearing operation accuracy, etc. [1]. Therefore, the bearing friction phenomenon hinders the further improvement of high-speed bearing performance [3]. Especially with the continuous improvement of energy consumption requirements of rotating equipment, the design of low-friction bearings has become a current research hot topic. It is preferred to analyze the various complex sources of bearing friction, such as the structural constraints, and to reduce their sizes in a targeted manner by improving the structural and external operating conditions. Accurate computation of bearing friction loss, operating in complex motion and force conditions, has become a prerequisite for this research.

Since the middle of the 20th century, the frictional properties of bearing are investigated by few scholars. Under different working conditions, the friction torque of different types of bearings was experimentally tested by Palmgren et al. [4,5]. The empirical formulas of bearing frictional torque were proposed based on the experimental data. However, the friction model by these empirical formulas is mainly suitable for light load and low speed conditions. Besides, the famous bearing company SKF proposed a set of models for calculating bearing frictional torque also based on experimental dates [6]. Compared to the former, the model accuracy has been improved, while it is mainly used for bearings in standard installation and standard load condition. With an experimental method, the influence of the raceway curvature radius on the frictional torque of angular contact ball bearings was experimentally analyzed by Todd et al. [7]. Similarly, Rodionov [8] studied

**Citation:** Ma, S.; Yan, K.; Li, M.; Zhu, Y.; Hong, J. Research on the Bearing Sliding Loss Based on Time-Varying Contact Angle between Ball and Raceway. *Lubricants* **2022**, *10*, 185. https://doi.org/10.3390/ lubricants10080185

Received: 30 June 2022 Accepted: 10 August 2022 Published: 15 August 2022

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**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

the effects of surface finish quality on bearing friction torque and it showed that, compared with the surface finish quality of rings, the rolling element significantly affected the friction torque. With the development of data processing methods, some new methods were applied to the study of bearing friction torque. Based on statistical correlation theory, the parametric-nonparametric fusion method was used to analyze the bearing friction torque by Xu [9]. By measuring the friction torque, Zhang et al. [10] modeled the data with gray theory to predict bearing friction torque. By summarizing the above work, it can be found that the present friction torque models were all based on experimental methods and dates, and then the overall friction performance of the bearing was investigated. However, the bearing friction torque is actually affected by many factors inside the bearing, such as bearing structural parameters, motion characteristics, load distribution etc. Especially for ball bearing, the ball's revolution, spin, gyro and other motion forms will affect its friction performance.

Snare, Li and Wang et al. [11–13] pointed out that the friction inside the bearing mainly comes from the elastic hysteresis, differential sliding, spinning, and friction agitation. Ye et al. [14] established a simple friction model for bearing in steady state. The differential sliding and spin friction were considered while the complex motion in the contact area was ignored. In order to explore the influence of the movement of the contact area on the friction performance of the bearing, by energy conservation law, the solution results of the mechanical model were used to calculate the friction torque by Deng [15]. The results show that the complex motion of the contact zone, such as balls sliding in the contact zone, has an important impact on the friction loss. On the basis of the above studies, Todd et al. [16] used the micro-slip theory to determine the pure rolling line in the contact zone. Then the frictional resistance was solved by integrating the complex motion of the contact zone. Unfortunately, the friction torque model and mechanical model are unidirectionally coupled, thus the effect of friction torque on bearing internal motion is missing. Houpert et al. [17] analyzed the motion of the contact zone and obtained the pure rolling line between ball and raceway. Then, the integral of the contact zone was calculated to obtain the contact friction. Finally, the bidirectionally coupled friction torque was achieved based on Cao's quasi-static model [18]. Because of the implementation of bidirectional coupling and the refinement of the contact zone, an obvious change of the friction results was observed. It indicates that the complex contact behavior between bearing components has a significant impact on its total friction loss. Therefore, it is necessary to study the internal friction characteristics of the bearing parts.

For bearing internal friction analysis, the current research mainly focuses on the steady-state friction. That is, the total friction is generated by the complex motion in the contact zone between the bearing parts, such as differential sliding and spin friction, etc. As mentioned earlier, due to the motion of the ball in the ball bearing being particularly special, the friction induced by the revolution is disregarded when the rolling elements move in the circumferential direction. In other words, the current friction calculation mainly considered the motions between the balls and the rings along the contact ellipse minor axis, while the sliding friction along the major axis is ignored [19]. In fact, in conditions of high speed and combined load, the internal forces of the bearing become very complicated. In addition to the friction caused by the gyroscopic torque, the friction generated by the contact angle cannot be neglected. Due to the inconsistent force of each rolling element, the contact angle between the balls and the ring is unequal, which is the main cause of the ball slides along the major axis of the contact ellipse during bearing operation. For instance, the technical report of SKF pointed out that a large variation in contact angle has an impact on the bearing friction [20]. As the ball slides along the major axis of the contact ellipse, the collision force between the cage pocket and the ball increased, which eventually causes the increases of bearing friction. Thus, it is of great significance to investigate the sliding of the rolling element of the bearing along the major axis of the contact zone. In this paper, taking the angular contact ball bearing as the research object, a quasi-static model is established. In condition of the combined load, it is found that the friction caused by

the sliding length of the bearing is discussed, such as operating parameters, structural parameters and contact angles. ‐

#### **2. Quasi-Static Model of Ball Bearing ‐**

#### *2.1. Establishment of Model*

‐

Due to the complex internal structure of angular contact ball bearings, the position relationship among the bearing parts is described. As shown in Figure 1, Figure 1b is the overall structure of an angular contact ball bearing, Figure 1a is a radial sectional view of the bearing, and Figure 1c is an axial sectional view. In Figure 1, the *x*-axis is defined as the bearing axial direction, and the radial plane is the *y*-*z* plane. As a whole, the bearing has five degrees of freedom (*δx*, *δy*, *δz*—movement along the *x*, *y*, and *z* axes, *θy*, *θz*—rotation around the *y*, *z* axes). In other words, under the action of load *F* (*Fx*, *Fy*, *Fz*, *My*, *Mz*), the bearing may produce a generalized displacement in five directions. When a bearing is used in engineering, the inner ring always rotates and the outer ring is fixed. Therefore, the external load and rotational speed are applied to the inner ring, while the outer ring is limited to six degrees of freedom. ‐ ‐ *δ δ δ θ θ* ‐

**Figure 1.** Internal structure of angular contact ball bearing. (**a**) Overall structure; (**b**) Radial sectional view; (**c**) Axial sectional view.

#### 2.1.1. Relative Position Analysis of Bearing Parts

−

*φ* ‐ After the ball at the azimuth position *ϕ<sup>j</sup>* is loaded, its position changes are revealed in Figure 2. Before being loaded, the three points of the curvature center of the inner raceway, the ball center and the curvature center of the outer raceway were collinear and there was no contact deformation. The distance between the curvature center, the inner raceway and the outer can be expressed as follows:

$$l = B \cdot D\_{w\_{\prime\prime}} \tag{1}$$

where *B = f<sup>i</sup> + f<sup>o</sup>* − 1, *f<sup>i</sup>* and *f<sup>o</sup>* are the curvature ratios of the inner raceway and the outer, respectively. *Dw* represents the ball diameter.

When the load *F* (*Fx*, *Fy*, *Fz*, *My*, *Mz*) and rotational speed *n<sup>i</sup>* are applied to the inner ring, the relative positions change among the curvature center of the inner raceway, the ball center and the curvature center of the outer raceway and the three are no longer collinear. As the rotational speed increases, the ball-inner raceway contact angle gradually enlarges, while the contact angle of the ball-outer raceway gradually decreases.

**Figure 2.** Position relationships among the ball center and the raceway curvature centers.

‐

At the azimuth position *ϕ<sup>j</sup>* , the distance between the curvature center of the inner raceway and the final position of the ball center is written as:

$$d\_{\vec{i}\vec{j}} = (f\_{\vec{i}} - 0.5) \cdot D\_w + \delta\_{\vec{i}\vec{j}}.\tag{2}$$

‐

*φ* Similarly, the distance of the outer ring is obtained by:

*δ*

$$d\_{o\circ} = (f\_o - 0.5) \cdot D\_w + \delta\_{o\circ} \tag{3}$$

where *δij* is the contact deformation between the ball and the inner raceway while the deformation of the outer is *δoj*.

 From Figure 2, the distances between the curvature center of the inner raceway and the outer in the horizontal and vertical directions are calculated as:

$$A\_{\mathbf{x}\mathbf{j}} = B \cdot D\_{\mathbf{w}} \cdot \sin \boldsymbol{\alpha}^{0} + B\_{\mathbf{x}\mathbf{j}} \tag{4}$$

$$A\_{yj} = B \cdot D\_w \cdot \cos \alpha^0 + B\_{yj\prime} \tag{5}$$

 where *α* <sup>0</sup> means the initial contact angle. *Bxj* and *Byj* are the variations of the curvature center of the inner raceway before and after loading on the *x*-axis and *y*-axis, respectively, which can be expressed as:

$$B\_{\rm xj} = \delta\_{\rm x} + R\_i \cdot \theta\_{\rm y} \cdot \sin \varphi\_{\rm j} + R\_i \cdot \theta\_{\rm z} \cdot \cos \varphi\_{\rm j} \tag{6}$$

$$B\_{\vec{y}\vec{j}} = \delta\_{\vec{y}} \cdot \sin \varphi\_{\vec{j}} + \delta\_{\vec{z}} \cdot \cos \varphi\_{\vec{j}\prime} \tag{7}$$

‐ ‐

 where *δx*, *δy*, *δz* are the displacements of the inner center relative to the outer center on the *x*, *y*, and *z* axes, respectively. *θy*, *θ<sup>z</sup>* denote the angular displacements around the *y* and *z* axes. *R<sup>i</sup>* represents the radius of the inner curvature center, which is determined by:

$$R\_i = d\_m/2 + (f\_i - 0.5) \cdot D\_w \cdot \cos \alpha^0 \,\tag{8}$$

*θ θ* where *dm* is the pitch diameter of the bearing.

*δ*

*α*

*δ δ δ*

 By observing Figure 2, it can be seen that these equilibrium equations, established in the horizontal and vertical directions, contain a larger number of trigonometric functions. Trigonometric functions are known to be unfavorable for numerical iterative solutions because they are periodic functions. However, refer to Jones's quasi-static model [21]; the new variables *X<sup>x</sup>* and *X<sup>y</sup>* are introduced to eliminate these functions to simplify the solution process. Among them, *X<sup>x</sup>* and *X<sup>y</sup>* denote the distance between the ball center and the inner curvature center in the horizontal and vertical directions, respectively. As

shown in Figure 2, The contact angles of the ball at the azimuth position *ϕ<sup>j</sup>* are written as Equations (9)–(12): 

*φ*

$$\sin \alpha\_{oj} = \frac{\ddot{X}\_{xj}}{(f\_o - 0.5) \cdot D\_w + \delta\_{oj}} \tag{9}$$

‐

$$\cos \mathfrak{a}\_{oj} = \frac{\mathfrak{A}\_{yj}}{(f\_o - 0.5) \cdot D\_w + \delta\_{oj}} \tag{10}$$

$$\sin \mathfrak{a}\_{\mathrm{ij}} = \frac{A\_{\mathrm{xj}} \frac{}{\delta} X\_{\mathrm{xj}}}{(f\_{\mathrm{i}} - 0.5) \cdot D\_{\mathrm{w}} + \delta\_{\mathrm{ij}}} \tag{11}$$

$$\cos \mathfrak{a}\_{ij} = \frac{A\_{yj} - X\_{yj}}{(f\_i - 0.5) \overleftarrow{\upbullet} \cdot D\_w + \delta\_{ij}}. \tag{12}$$

In Figure 2, the geometric compatibility equations of the ball at the *ϕ<sup>j</sup>* azimuth are given as Equations (13) and (14): *φ*

$$\left(\left(A\_{\rm xj} - X\_{\rm xj}\right)^2 + \left(A\_{\rm xj} - X\_{\rm xj}\right)^2 + \left(\left(f\_o - 0.5\right) \cdot D\_w + \delta\_{o\dot{j}}\right)^2 = 0\tag{13}$$

$$\left(X\_{x\circ}\,^2 + X\_{y\circ}\,^2 - \left(\left(f\_o - 0.5\right)\cdot \dot{\mathcal{D}}\_w + \delta\_{o\circ}\right)\right)^2 = 0.\tag{14}$$

#### 2.1.2. Interaction Force of Bearing Parts

Under high-speed operating conditions, the mechanical analysis of bearing components is complicated. The ball is applied to the centrifugal force, causing dissimilar ball-inner and ball-outer contact angles. The friction between the ball and the raceway is caused by the gyroscopic torque. The interaction of the bearing parts is displayed in Figure 3; Figure 3a is the force analysis of the entire bearing while Figure 3b is the *j*-th ball. When a load *F* (*Fx*, *Fy*, *Fz*, *My*, *Mz*) and rotational speed *n<sup>i</sup>* are applied to the inner ring, the corresponding displacement *δ* (*δx*, *δy*, *δz*, *θy*, *θz*) is generated. At this time, the inner ring is pressed against the ball, and a corresponding contact deformation occurs between them. The gyroscopic torque is generated because of the non-zero contact angle. ‐ ‐ ‐ ‐ ‐ ‐ *δ δ δ δ θ θ* ‐

**Figure 3.** Force analysis of bearing parts. (**a**) Entire bearing; ( ‐ **b**) *j*-th ball.

By observing Figures 2 and 3, the mechanical equilibrium equations of the ball at the azimuth position *ϕ<sup>j</sup>* are as follows:

$$\lambda\_o Q\_{oj} \cdot \cos a\_{oj} - \lambda\_o \frac{M\_{\text{gj}}}{D\_w} \cdot \sin a\_{oj} - Q\_{ij} \cdot \cos a\_{ij} + \lambda\_i \frac{M\_{\text{gj}}}{D\_w} \cdot \sin a\_{ij} - F\_{cj} = 0 \tag{15}$$

$$Q\_{oj} \cdot \sin \mathfrak{a}\_{oj} + \lambda\_o \frac{M\_{\mathfrak{g}j}}{D\_w} \cdot \cos \mathfrak{a}\_{oj} - Q\_{\bar{i}j} \cdot \sin \mathfrak{a}\_{i\bar{j}} - \lambda\_i \frac{M\_{\mathfrak{g}j}}{D\_w} \cdot \cos \mathfrak{a}\_{i\bar{j}} = 0,\tag{16}$$

where *Qi/oj* = *Ki/oj*·*δi/oj* 1.5 , *Ki/oj* is the contact deformation coefficient between the ball and the inner/outer ring, which can be calculated from [22]. *λi/o* represents the inner/outer raceway control coefficient. According to reference [23], if it is inner raceway control, *λ<sup>i</sup>* = 1, *λ<sup>o</sup>* = 1, otherwise it is outer raceway control, *λ<sup>i</sup>* = 0, *λ<sup>o</sup>* = 2.

Besides, the gyroscopic torque and centrifugal force of the ball are calculated with Equations (17) and (18):

$$M\_{\mathcal{g}\bar{j}} = f\_b \cdot \left(\frac{w\_R}{w\_i}\right)\_{\bar{j}} \cdot \left(\frac{w\_m}{w\_i}\right)\_{\bar{j}} \cdot w\_{\bar{i}}^2 \cdot \sin\beta\_{\bar{j}} \tag{17}$$

$$F\_{cj} = 0.5 \cdot m\_b \cdot d\_m^2 \cdot w\_i^2 \cdot \left(\frac{w\_m}{w\_i}\right)\_{j'}^2 \tag{18}$$

where *J<sup>b</sup>* is the ball moment of inertia, *w<sup>i</sup>* means the angular velocity of the inner ring, and *m<sup>b</sup>* denotes the ball mass.

According to Jones' model theory [24], *wm/w<sup>i</sup>* , *wR/w<sup>i</sup>* , and tan*β<sup>j</sup>* are related to the rotation speed and attitude of the ball, which can be expressed as the following formula:

$$\frac{w\_m}{w\_i} = \frac{1}{1 + \left(\frac{\cos \mathfrak{a}\_{oj} + \tan \mathfrak{f} \cdot \sin \mathfrak{a}\_{oj}}{\cos \mathfrak{a}\_{ij} + \tan \mathfrak{f} \cdot \sin \mathfrak{a}\_{ij}} \times \frac{1 + \gamma \cdot \cos \mathfrak{a}\_{oj}}{1 - \gamma \cdot \cos \mathfrak{a}\_{ij}}\right)}}\tag{19}$$

$$\frac{w\_R}{w\_i} = \frac{-1}{\left(\frac{\cos a\_{oj} + \tan \beta \cdot \sin a\_{oj}}{1 + \gamma \cdot \cos a\_{oj}} + \frac{\cos a\_{ij} + \tan \beta \cdot \sin a\_{ij}}{1 - \gamma \cdot \cos a\_{ij}}\right) \cdot \gamma \cdot \cos \beta} \tag{20}$$

$$\beta = \arctan\left(\frac{\sin \alpha\_{ij}}{\cos \alpha\_{ij} + \gamma}\right) \tag{21}$$

where *γ* represents the dimensionless constant, *γ* = *Dw*/*dm*.

Under high-speed operating conditions, the angular contact ball bearing can be regarded as a whole, and the load acting on the bearing should be balanced. In this paper, the inner ring is used as the carrier, and the load applied by the outside should be balanced with the load applied by the ball. The equilibrium equations of the inner can be established:

$$F\_x - \sum\_{j=1}^{Z} \left( Q\_{ij} \cdot \sin \mathfrak{a}\_{ij} + \lambda\_i \frac{M\_{\mathfrak{g}j}}{D\_w} \cdot \cos \mathfrak{a}\_{ij} \right) = 0 \tag{22}$$

$$F\_{\mathcal{Y}} - \sum\_{j=1}^{Z} \left( Q\_{ij} \cdot \cos \alpha\_{ij} - \lambda\_i \frac{M\_{\mathcal{G}j}}{D\_w} \cdot \sin \alpha\_{ij} \right) \cos \varphi\_{\mathcal{Y}} = 0 \tag{23}$$

$$F\_z - \sum\_{j=1}^{Z} \left( Q\_{ij} \cdot \cos a\_{ij} - \lambda\_i \frac{M\_{\mathcal{G}j}}{D\_w} \cdot \sin a\_{ij} \right) \sin \varphi\_j = 0 \tag{24}$$

$$M\_{\mathcal{Y}} - \sum\_{j=1}^{Z} \left( \left( Q\_{i\bar{j}} \cdot \sin \alpha\_{i\bar{j}} + \lambda\_i \frac{M\_{\mathcal{S}\bar{j}}}{D\_w} \cdot \cos \alpha\_{i\bar{j}} \right) \cdot \mathcal{R}\_i - \lambda\_i \cdot f\_i \cdot M\_{\mathcal{S}\bar{j}} \right) \sin \varrho\_{\bar{j}} = 0 \tag{25}$$

$$M\_z - \sum\_{j=1}^{Z} \left( \left( Q\_{\bar{i}j} \cdot \sin a\_{\bar{i}j} + \lambda\_i \frac{M\_{\bar{g}j}}{D\_w} \cdot \cos a\_{\bar{i}j} \right) \cdot R\_i - \lambda\_i \cdot f\_{\bar{i}} \cdot M\_{\bar{g}j} \right) \cos \varphi\_{\bar{j}} = 0,\tag{26}$$

where *Z* means the number of balls.

#### *2.2. Model Solution*

The quasi-static model of an angular contact ball bearing is composed of nonlinear equations, including the geometric compatibility Equations (13) and (14) of the ball, the force balance Equations (15) and (16), and the balance equation of the inner ring

Equations (22)–(26). The model has *4Z* + *5* nonlinear equations and *4Z* + 5 unknowns, such as *Xxj*, *Xyj*, *δij*, *δoj*, *δx*, *δy*, *δz*, *θy*, *θ<sup>z</sup>* (*j =* 1····*z*). Considering the solution accuracy and efficiency of non-linear equations, the Newton–Raphson iterative algorithm is adopted to solve. Because the initial value affects the accuracy of the iterative algorithm, and a large number of solution parameters is included in this model. The results of the statics model [21] are used as the initial value of the quasi-static and are transferred to the quasi-static model. The solution process of the above model is shown in Figure 4, and the iteration convergence accuracy (eps) sets to 10−<sup>5</sup> . *δ δ δ δ δ θ θ ∙∙∙∙* ‐ ‐ ‐ −

‐

 

 

 

 

 

   

 

‐

‐ ‐

‐ **Figure 4.** Solution process of the quasi-static model.

#### **3. Model Verification and Sliding Loss Analysis**

#### *3.1. Model Versatility Verification*

In this section, the mechanical model established above is verified by comparing the relationship between the force and displacement of the bearing and the contact characteristic. The displacement relation (stiffness) is verified by experiments, and the contact characteristic (contact angle) is compared with the published reference.

#### 3.1.1. The Relationship between Force and Displacement

 First, the bearing stiffness test system built by our team was used to verify the mechanical model above. As shown in Figure 5, the inner ring of bearing rotates with the precision mandrel 5 while the outer is fixed. The non-contact axial load is applied to the bearing through the air bearing plate 3, and the displacement sensor 6 and the force sensor 2 are used to test the relative displacement of the inner and outer rings and the axial force. The force and displacement signals are collected through the data acquisition system to measure the axial stiffness of the bearing.

> Based on the stiffness test system, the stiffness of a B7008C/P4 angular contact ball bearing is measured. The specific structural parameters of B7008C/P4 are shown in Table 1. We applied an axial force of 150 N and 300 N respectively, the rotational speed was gradually increased from 100 r/min to 1900 r/min. The change in axial stiffness of the bearing is shown in Figure 6.

**Figure 5.** The Bearing stiffness test system. **1**—FESTO cylinder; **2**—The axial force sensor; **3**—Air bearing plate; **4**—Bearing gland; **5**—Precision mandrel; **6**—Capacitive displacement sensor; **7**—B&K.

‐

‐

‐

‐



‐ By observing Figure 6, it can be seen that the bearing stiffness decreases as the rotational speed increases. This is caused by the change in contact angle due to the variable rotational speed. The axial force has an obvious influence on the bearing stiffness, and the stiffness increases as the axial force grows at the same speed. By comparing the theoretical simulation with the experiment, the errors do not exceed 5%, which proves the model. The errors may be caused by the axial runout of the spindle.

‐

#### 3.1.2. The Contact Characteristic

Second, by comparing the deviation of the solution results of the model with that of Harris et al. [24], the model is proved. A 218 angular contact ball bearing was taken as the research object and the model was solved under given conditions. From reference [24], the initial contact angle of the 218 was 40◦ , and the rotational speed respectively was set to 6000, 10,000, 15,000 r/min, while the axial force was gradually increased from 17,500 N to 44,500 N since the focus of this study was the contact angle, which was the object of verification. Figure 7 is the result of the comparison.

‐

**Figure 7.** Comparison of the results [24] with the model. (**a**) Results of the model; (**b**) Results of the Ref. [24]; (**c**) Error between the reference and the model.

‐ ‐ From Figure 7, the contact angle is greatly affected by the axial force and the rotational speed. The ball-inner raceway contact angle gradually reduces as the axial force increases, while the contact angle of the outer becomes larger. With the increasing rotational speed, the ball-inner raceway contact angle continuously increases and the contact angle of the outer decreases. By comparison, the deviation between the results of this model and the results [24] is within 5%, which is considered to be caused by unit conversion, and the correct and reliable of the model was obtained. The error may be caused by unit conversion.

#### *3.2. Analysis of Sliding Loss Caused by Contact Angle Changes*

Based on the model above, the B7008C/P4 bearing is taken as the research object to investigate the sliding. Firstly, the contact characteristics among the bearing components under the combined load are analyzed.

The dynamic characteristics of the 7008 can be obtained by solving the quasi-static model. The constant combined load (*F<sup>x</sup>* = 500 N, *F<sup>z</sup>* = −300 N) and rotational speed (*n<sup>i</sup>* = 10,000 r/min) are considered. The force applied to each ball is not equal because of the radial load, which causes their contact angle values to be different. The variability of the contact angle will cause sliding between the ball and the raceway, as shown in Figure 8.

*φ*

*α φ α α*

**Figure 8.** Analysis of sliding under a combined load.

‐ Owing to the variable of the contact angle, the ball slides along the major axis of the contact ellipse relative to the raceway and the sliding length of the ball relative to the inner raceway at the azimuth position *ϕ<sup>j</sup>* is:

$$s\_{i\bar{j}} = r\_{\bar{i}} \cdot (\mathfrak{a}\_{i\bar{j}} - \mathfrak{a}\_{i\bar{m}\bar{n}}).\tag{27}$$

‐

‐

‐ ‐

−

 

 

*α*

The sliding length of the ball at this azimuth relative to the outer raceway is:

$$s\_{oj} = r\_o \cdot (\mathfrak{a}\_{oj} - \mathfrak{a}\_{o\text{min}})\_\prime \tag{28}$$

where *αij* is the contact angle of the ball at the azimuth *ϕ<sup>j</sup>* with the inner ring and the angle of the ball with the outer is *αij*. *αimin* expresses the minimum contact angle of the ball with the inner ring when it runs for a cycle while *αomin* means the minimum angle of the ball with the outer ring.

To clearly express the ball information of each azimuth angle, the ball is labeled. The ball mark in the opposite direction to the *z* axis is designated as 0, where the azimuth angle is 0◦ . The counterclockwise rotation around the *x*-axis is specified as the positive direction.

By observing Figures 8 and 9, when the ball azimuth is 0◦ , the contact angle between the ball and the inner ring or the outer is a minimum. At this location, the contact point between the ball and the inner ring is *a1*, and the contact point with the outer ring is *b1*. At the azimuth position 180◦ , the contact angle between the ball and the inner ring or the outer is a maximum. The contact point between the ball and the inner ring at the position is *a2*, and the contact point with the outer ring is *b2*. When the ball moves from 0◦ to 180◦ , the contact angle between the ball and the ring gradually becomes bigger and the contact point between the ball and the ring moves from 1 to 2 along the major axis of the contact ellipse. In contrast, the contact angle between the ball and the ring reduces, when the ball moves from 180◦ to 360◦ (0◦ ). The contact point between the ball and the ring moves from 2 to 1 along the major axis of the contact ellipse. In the dynamic operation process, when the ball runs for a week, the contact angle changes cyclically once (from small to large, and then from large to small). The contact point moves back and forth between 1 and 2 points. Figure 9 is the relationship between the change of the contact angle and the sliding length of the ball in a period. Under the working conditions, the maximum sliding length between the ball and the inner ring is 0.6987 mm for 7008, and the maximum sliding length of the outer ring is 0.5381 mm.

‐ ‐ ‐ **Figure 9.** Relationship between sliding length and contact angle. (**a**) Ball-Inner raceway contact angle; (**b**) Ball-Outer raceway contact angle. ‐ ‐ ‐

‐ ‐ During operation, the motion of the bearing parts is shown in Figure 10. Assuming no sliding occurs on the bearing, the driving rotational speed applied by the inner ring only rotates around its own *x*-axis. Thus, the velocity component of the ball is concentrated in the ellipse contact minor axis direction between the ball and the inner ring, while the component in the major axis direction is smaller under the circumstances. The motion in the minor axis direction is rolling, and the sliding occurs in the major axis direction. Since the rolling friction coefficient is very small, there is little heat generation. However, the sliding friction coefficient is large, and the friction heat is serious. The sliding in major axis direction is the focus of the investigation. ‐ ‐

**Figure 10.** The relative relationship between ball and raceway. (**a**) Inner raceway; (**b**) Outer raceway.

The bearing friction loss above occurs due to sliding caused by changes in the contact angle. The micro-element method is used to calculate the loss. Firstly, the ball is divided into finite parts for a week and the results are added up to obtain the average value, which is expressed as:

$$H = \frac{1}{n} \sum\_{j=1}^{n} Ff\_j \cdot \frac{\Delta s\_j}{\Delta t} \,\tag{29}$$

where *n* means that the operation week of the ball is divided into *n* stages. *Ff<sup>j</sup>* is the average friction force experienced by the ball in stage *j*. ∆*s<sup>j</sup>* represents the slip increment of the ball in stage *j*. ∆*t* denotes the time taken to pass stage *j* for the ball.

For the frictional force *Ff<sup>j</sup>* , assuming ignoring the slight rolling, the sliding is only considered in the major axis direction. Such force can be determined as follows:

$$Ff\_{\bar{j}} = \frac{\mu \left(Q\_{\bar{j}} + Q\_{\bar{j}+1}\right)}{2},\tag{30}$$

where *µ* is the sliding friction coefficient, which can be obtained in [25].

‐

Δ

*μ*

Based on the above theory, a combined-loaded bearing (*F<sup>x</sup>* = 500 N, *F<sup>z</sup>* = −300 N, *n<sup>i</sup>* = 10,000 r/min) is solved. Because of the ball sliding along the major axis, the heat generated on the inner raceway is 9.456 W, while the outer is 8.624 W. The integral heating of the bearing obtained by empirical equation [26] is 132.2792 W. The power loss by sliding approximately accounts for 13.67% of the total heat. Ignoring the sliding loss, the results of frictional heat will be seriously affected and the heat generated by the sliding loss is almost proportional to the sliding length of the ball along the major axis direction on the raceway. Thus, the sliding length is taken as the evaluation criterion to investigate the friction, and the effects of working conditions and curvature ratio are analyzed. ‐

‐ −

Δ

 

#### **4. Results and Discussion**

Based on the model in Section 2, a 7008 angular contact ball bearing is taken as the object to be solved. By the sliding length of the ball on the raceway, the friction loss of the bearing with different initial contact angles (15◦ , 25◦ ) is studied. The efforts of the working conditions and the structural parameters are discussed.

#### *4.1. Radial Loaded Bearing*

The influence of the radial force on the ball sliding is first studied, including the direction and magnitude. Without the torque (*M<sup>y</sup>* = *M<sup>z</sup>* = 0 N·mm), a constant axial force (*F<sup>x</sup>* = 500 N) and rotational speed (*n<sup>i</sup>* = 10,000 r/min) is considered. By changing the radial force, the trend of sliding with different contact angles is analyzed. ∙ ‐ ‐

‐

Firstly, the *y*-axis is applied to the radial force, while the force of the *z*-axis is 0. With the rising of radial force (*Fz*) on the *y*-axis, the sliding length of the ball with different contact angles is shown in Figure 11. ‐

**Figure 11.** *Fy*'s influence on the sliding of the inner ring and outer ring. (**a**) The sliding of inner raceway-15◦ ; (**b**) The sliding of outer raceway-15◦ ; (**c**) The sliding of inner raceway-25◦ ; (**d**) The sliding of outer raceway-25◦ .

The relationship between the sliding length and the radial force is shown in Figure 9. For bearings with the same initial contact angle, the effect of force is similar. The minimum

*α*

*α α*

*α*

*α*

*α α*

*α*

‐

*α*

*α α*

contact angle (*αmin*) appears when the ball is 90◦ , where the sliding length is 0. The maximum contact angle (*αmax*) is located at the ball azimuth of 270◦ , which is the largest sliding. As the radial force of *Fy* grows, the sliding length increases. The variation trend of the sliding is relatively slight when the contact angle of the ball near the *αmin* or *αmax*. The friction loss is quite small. On the contrary, the trend is relatively sharp in other positions, and the loss becomes large. As for bearings of different initial contact angles, the azimuth angles of *αmin* or *αmax* are the same. However, the sliding trend of the bearing with 25 ◦ is larger than that of the 15◦ under the same force, and the friction loss is also greater. When the ball azimuth closes 45◦ and 135◦ , the variation of sliding changes rapidly for the inner, while the larger change for outer appears near the azimuth of 225◦ and 315◦ . This means there is more friction. *α α* ‐ ‐

‐ ‐ ‐ ‐

*α* ‐

Secondly, setting *Fy* to 0, the *z*-axis is applied to the radial force. As shown in Figure 12, the relationship between the sliding with different contact angles and the radial force is analyzed. No matter whether the initial contact angle is 15◦ or 25◦ , the *αmin* appears at the position where the azimuth is 0◦ , and the azimuth of the *αmax* is 180◦ . The other effects of the *z*-axis are similar to the force applied in the *y*-axis. *α α* ‐ ‐

‐ ‐ ‐ ‐ ‐ **Figure 12.** *Fz*'s influence on the sliding of the inner ring and outer ring. (**a**) The sliding of inner raceway-15◦ ; (**b**) The sliding of outer raceway-15◦ ; (**c**) The sliding of inner raceway-25◦ ; (**d**) The sliding of outer raceway-25◦ .

Finally, the *y*-axis and the *z*-axis are applied to the radial force at the same time, which is gradually growing. Figure 13 shows the relationship between the radial force and the sliding with different initial contact angles and the radial force. It can be found that, regardless of the initial contact angle, the azimuths of *αmin* and *αmax* are constant and the *αmin* appears at the azimuth of 45◦ , while the 225◦ azimuth is the *αmax*. Other changes have similar effects to the radial force applied to the *y*-axis or the *z*-axis individually.

‐ ‐

‐ ‐ ‐ ‐ **Figure 13.** The combined effect of *Fy* and *Fz* on the sliding of the inner ring and outer ring. (**a**) The sliding of inner raceway-15◦ ; (**b**) The sliding of outer raceway-15◦ ; (**c**) The sliding of inner raceway-25◦ ; (**d**) The sliding of outer raceway-25◦ .

‐

*α α α*

*α* ‐

‐ *α α* To summarize, the sliding length is only affected by the magnitude of the radial force, independent of its direction. With the increasing of the radial force, the variation trend of sliding is increases. However, the direction of the radial force has an influence on the azimuth angles of *αmin* and *αmax*.

#### *4.2. Torque Component*

‐ ‐

In this section, the relationship between the torque and the sliding is investigated. The axial force (*F<sup>x</sup>* = 500 N), radial force (*F<sup>y</sup>* = 200 N, *F<sup>z</sup>* = 0 N) and rotational speed (*n<sup>i</sup>* = 10,000 r/min) are constant. The sliding of different initial contact angles is discussed by changing the torque.

‐ ∙ ∙ ‐ Firstly, the *y*-axis is only applied to the torque, which increases from 0 N·mm to 250 N·mm. The torque of the *z*-axis is 0. This means that the radial force and the torque are exerted to the same axis. Figure 14 shows the relationship between the torque and the sliding of different initial contact angles.

‐ ∙

‐ ‐

‐

‐

*α α α*

‐ ‐

‐ ‐ ‐

‐

*α α*

∙ ‐

*α* ‐

‐ ‐ ‐ ‐ ‐ **Figure 14.** *My*'s influence on the sliding of the inner ring and outer ring. (**a**) The sliding of inner raceway-15◦ ; (**b**) The sliding of outer raceway-15◦ ; (**c**) The sliding of inner raceway-25◦ ; (**d**) The sliding of outer raceway-25◦ . ‐ ‐ ‐

‐ ‐ ‐ ‐

*α α*

‐

‐

*α α*

As shown in Figure 15, this type of torque has an influence on the sliding length of the bearing with the same initial contact angle. As the torque rises, the sliding length gradually decreases and the azimuth angles of *αmin* and *αmax* are constant. As for the variation rate of sliding, it also gradually reduces. With regard to bearings with different initial contact angles, the variation rates are distinct. The rate of bearings with an initial contact angle of 15◦ is more sensitive to torque than the rate of 25◦ . However, the total sliding length of the 25◦ is generally greater than that of the 15◦ .

In summary, when the torque and radial load are applied to the same axis, the effect of torque on sliding is very weak. On the contrary, applying the torque and radial load to the different axes, the torque has a corresponding effect on the sliding. With the torque increasing, the sliding length gradually lessons. From Figures 14 and 15, regardless of whether the torque and the radial force are on the same axis, the sliding lengths of different initial contact angles are unequal and the sliding with an initial contact angle of 25◦ is always greater than that of 15◦ .

#### *4.3. Rotational Speed Bearings*

To reveal the influence of rotational speed on sliding length, speed gradually increases. Regardless of the torque (*M<sup>y</sup>* = *M<sup>z</sup>* = 0 N·mm), the axial force (*F<sup>x</sup>* = 500 N) and the radial force (*F<sup>y</sup>* = 200 N, *F<sup>z</sup>* = 0 N) are considered. Figure 16 shows the relationship between the sliding length and the rotational speed.

**Figure 16.** The effect of rotational speed on the sliding of the inner ring and outer ring. (**a**) The sliding of inner raceway-15◦ ; (**b**) The sliding of outer raceway-15◦ ; (**c**) The sliding of inner raceway-25◦ ; (**d**) The sliding of outer raceway-25◦ .

Figure 16 shows the curves of the sliding length changing with the rotational speed for the bearing. It can be seen that, as the rotational speed increases, the azimuth angles of *αmin* and *αmax* remain unchanged. For the inner ring, the higher the speed, the greater the sliding length, and the faster the variation rate of sliding. On the contrary, with the rising rotational speed, the sliding length of the outer ring reduces and the change rate of outer sliding gradually grows. Compared with the outer ring, the inner ring is more sensitive to

*α*

the speed. As for the bearing with different initial contact angles, the sliding with an initial contact angle of 25◦ changes more than that of 15◦ . Besides, for the bearing with an initial contact angle of 25◦ , the non-sliding area appears near the *αmin*.

#### *4.4. Groove Curvature Ratio*

In addition to working conditions, the structural parameters may also affect the sliding. The curvature ratio of the raceway has an important influence on the friction loss. Ignoring the torque (*M<sup>y</sup>* = *M<sup>z</sup>* = 0 N·mm), a constant combined force (*F<sup>x</sup>* = 500 N, *F<sup>y</sup>* = 200 N, *F<sup>z</sup>* = 0 N) and constant speed (*n<sup>i</sup>* = 10,000 r/min) are applied to the bearings. The effects of the curvature ratio of the raceway on the sliding are investigated. ‐

Firstly, the influence of the curvature ratio of the inner raceway on the sliding is considered. The curvature ratio of the outer raceway is constant (*fo* = 0.535), while the ratio of the inner (*f<sup>i</sup>* ) gradually increases from 0.515 to 0.565. The relationship between the ratio of the inner and the sliding lengths is shown in Figure 17.

‐ ‐ ‐ ‐ **Figure 17.** The effect of curvature ratios on the sliding of the inner ring and outer ring. (**a**) The sliding of inner raceway-15◦ ; (**b**) The sliding of outer raceway-15◦ ; (**c**) The sliding of inner raceway-25◦ ; (**d**) The sliding of outer raceway-25◦ .

‐ *α α* By observing Figure 17, as the curvature ratio of the inner raceway increases, the azimuths of *αmin* and *αmax* are constant and the sliding length of rings gradually decreases. However, once the ratio reaches a certain value, the sliding changes will be not obvious. The initial contact angle of the bearing is different, and the influence of the ratio on the sliding is different. The sliding length of a bearing with an initial contact angle of 15◦ is smaller than that of 25◦ .

‐ ‐ ‐ Then the relationship between the curvature ratio of the outer raceway and the sliding is analyzed. Keeping the ratio of the inner constant (*f<sup>i</sup>* = 0.535), the outer ratio (*fo*) gradually increases. The influence of the outer ratio on the sliding is presented in Figure 18. The outer ratio has a weak effect on the sliding length, but the overall trend gradually increases. When the outer ratio exceeds a certain value and continues to increase, the sliding length is almost constant. For bearings with different initial contact angles, the effects of the outer ratio on the sliding are disparate. For the bearing with an initial contact angle of 15◦ , the effect of the outer ratio on sliding is not distinct. However, the outer ratio has a significant

impact on the sliding for the bearing with 25◦ . As the outer ratio increases, not only the sliding length but also the variation rate of the sliding gradually grows.

‐ ‐ ‐ ‐ **Figure 18.** The effect of curvature ratios on the sliding of the inner ring and outer ring. (**a**) The sliding of inner raceway-15◦ ; (**b**) The sliding of outer raceway-15◦ ; (**c**) The sliding of inner raceway-25◦ ; (**d**) The sliding of outer raceway-25◦ .

To summarize, compared with the curvature ratio of the outer raceway, the sliding is more sensitive to the inner ratio. In addition, the larger the inner ratio, the smaller the sliding length. The influence of the outer ratio on the sliding is just the opposite; with an increasing outer ratio, the sliding gradually decreases.

#### **5. Conclusions**

‐ Taking the sliding length as the evaluation object, the influences of working conditions and structural parameters on bearing friction are investigated, and the conclusions are as follows:


**Author Contributions:** Conceptualization, S.M., K.Y. and J.H.; data curation, S.M. and M.L.; formal analysis, S.M. and Y.Z.; funding acquisition, J.H.; investigation, S.M., M.L. and K.Y.; methodology, S.M., M.L. and Y.Z.; project administration, K.Y. and J.H.; resources, J.H.; software, S.M.; supervision, M.L. and K.Y.; validation, M.L. and Y.Z.; visualization, Y.Z., Y.Z. and J.H.; writing—original draft, S.M. and K.Y.; writing—review & editing, K.Y., Y.Z. and J.H. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received was founded by the National Outstanding Youth Science Fund Project of the National Science Foundation of China (52022077).

**Data Availability Statement:** Not applicable.

**Acknowledgments:** The authors are very grateful for the editors of *Lubricants* and the anonymous reviewers for their work in processing this article.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**


**Yanfei Zhang 1,2,\*, Yunhao Li 1 , Lingfei Kong 1 , Zhenchao Yang 1, \* and Yue Si 1**


**\*** Correspondence: yfzhang@xaut.edu.cn (Y.Z.); zcyang@xaut.edu.cn (Z.Y.)

**Abstract:** In this paper, a quasi-static angular contact ball bearing model, considering assembly accuracy is constructed, while a numerical solution method for bearing stiffness under bad assembly state is established. A 7014C angular contact ball bearing is used as the research object and five groups of different spacer inclinations are designed to imitate the installation error of the spindle bearing. The bearing stiffness performance was comparatively analyzed, according to the five spacers. The effect of preload and rotation speed on bearing stiffness are systematically investigated, considering different parallelism errors, as induced by the spacers. The influence mechanism of the badly assembled bearing on the respective stiffness anisotropy is studied based on the proposed model. The results show that the variations of the inclination between the inner and outer rings of the bearing exhibit a very weak effect on the axial stiffness, while the influence on the radial and angular stiffness is more significant.

**Keywords:** angular contact ball bearing; spacer inclination; bearing stiffness

#### **1. Introduction**

The factors that affect the performance of a machine tool, over the course of a spindle system's whole life cycle, include not only the quality of the spindle components, but also the design of the spindle system, the component assembly process and the assembly technique. Angular contact ball bearings have a significant impact on the performance of the spindle system, as the core rotating support element, while bearing stiffness performance indicators have a direct impact on spindle system vibration, noise, rotational precision, end jump and service life [1–5]. Moreover, the accuracy of the bearing stiffness values is crucial in building a global model of the spindle system. Indeed, the frequency response function and, in particular, the critical eigenfrequencies are directly linked to the bearings' stiffness. As a result, additional quantitative study is required, to investigate the relationship between bearing assembly quality and bearing stiffness in all directions, in order to determine how the bearing assembly quality affects the stiffness performance of the bearing and even the spindle system.

Ball bearings, as a kind of rolling bearing, are one of the most important parts of spindle-bearing system. Many researchers [6–8] established that spindle performance changes dynamically due to the nonlinear effect of bearings stiffness. A great number of scholars have conducted studies on bearing stiffness, including bearing stiffness analysis and calculation, as well as bearing stiffness change affect factor analysis [9]. Jones, for example, was the first to develop the rolling bearing contact angle analysis mechanics model, which was the basis for the numerical assessment of the bearing load and change law, while Harris later enhanced the model and it was widely accepted by other researchers as Jones and Harris model [10,11].

**Citation:** Zhang, Y.; Li, Y.; Kong, L.; Yang, Z.; Si, Y. Research on the Mechanism of the Stiffness Performance of Rolling Bearings under Wrong Assembly State. *Lubricants* **2022**, *10*, 116. https:// doi.org/10.3390/lubricants10060116

Received: 11 April 2022 Accepted: 31 May 2022 Published: 5 June 2022

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

J. Jedrzejewski [12] researched the relationship between the rotation speed and bearing stiffness, deriving the "bearing stiffness softening phenomenon", where the bearing stiffness continues to drop, as the bearing rotational speed rises. Matti Rantatalo [13] held the same view and the respective research also pointed that the radial stiffness would drop to 40%, when the rotational speed was up to 20,000 rpm, based on the presented calculations. However, both of them did not give a detailed calculation model to explain the speed varying stiffness, dependent on the analysis of bearing contact force. In the work of Sheng et al. [14]., the detail notion of rolling bearing speed-varying stiffness is introduced and explained, based on the relations of load-deflection, according to the bearing dynamic model, also based on Jones and Harris's model.

David Noel [15] proposed a new method for the computation of the stiffness matrix: a complete analytical expression is presented, including dynamic effects, in order to ensure accuracy at high shaft speed. This new method is particularly relevant in the case of lightly loaded bearings in critical applications (both high values of shaft speed and ball orbital diameter). Yi Guo [16] developed a finite element/contact mechanics model for rolling element bearings, with the focus on obtaining accurate bearing stiffness for a wide range of bearing types and parameters. The presented fully-populated stiffness matrix demonstrates the coupling between bearing radial, axial and tilting bearing deflections. However, this method did not consider the rotational speed effect on the bearing stiffness.

Cao [17] et al. tested the influence of bearing positioning preload and constant pressure preload on spindle-bearing dynamic stiffness, concluding that positioning preload is more effective than constant pressure preload in preserving spindle dynamic stiffness, under cutting loads at high speeds. Aramaki [18] studied bearing stiffness under the influence of preload force and confirmed the mapping link between preload force and bearing stiffness features. Yang [19] used the proposed static model of rolling bearing, to calculate the dynamic stiffness of the bearing and determined its operational stiffness, considering the influence of rotational speed, initial preload force, thermal preload force and oil film thickness on bearing stiffness. Liu [20] suggested a methodology, based on finite element analysis, in order to study the influence of preload on bearing stiffness and discover the bearing's equilibrium state while under load.

The experiment about bearing stiffness has also been conducted by many researchers. Walford et al. [21] developed a spindle-bearing test platform, to estimate the bearing's radial stiffness and damping, by measuring the response of the spindle, as well as to determine the effect of temperature on the stiffness of the bearings. Similarly, Kraus et al. [22] pointed out that, static bearing stiffness is very close to the stiffness measured while the bearing is running, based on stiffness and damping characteristics of a radial ball bearing, as derived by experimental modal tests. Marsh E R et al. [23] carried out an experimental measurement of precision bearing dynamic stiffness, where it was pointed out that an analysis of the method is sensitive to errors in sensor location, while several practical advantages of the proposed approach, over traditional static testing, were demonstrated. However, the majority of the above studies are based on a mechanical model of the bearing, under ideal operating conditions. Nonetheless, in engineering applications, various errors and even installation eccentricity in the bearing-spindle coupling system are inherent, while the bearing operation is affected by the additional moment.

Recently, ring misalignment occurrence has attracted much attention from the academia. Many researchers are devoted to studying the effect of ring misalignment error on the operating characteristics of rolling bearings. Zhang [24] studied the effect of ring misalignment on the service characteristics of ball bearing and rotor system; however, the analysis of bearing stiffness is not the key research content of this study. Xu [25] researched the effect of angular misalignment of inner ring on the surface contact characteristics and stiffness coefficients of duplex angular contact ball bearing. It was pointed out that, the ring misalignment considerably changes the surface contact characteristics of ball-raceway and causes uneven load distribution. As a result, Zhang [26] built a model that included the

bearing's deflection angle and assessed the effect of axial and radial loads on the contact angle, but the change law of bearing stiffness performance was not investigated. change is studied, in the context of poor assembly. Furthermore, the influence of rotational speed, preload and assembly quality on bearing stiffness, in all directions, is systemati-

loads on the contact angle, but the change law of bearing stiffness performance was not

This issue is addressed in this paper, where the mechanism of bearing stiffness

This issue is addressed in this paper, where the mechanism of bearing stiffness change is studied, in the context of poor assembly. Furthermore, the influence of rotational speed, preload and assembly quality on bearing stiffness, in all directions, is systematically analyzed (the relative tilt amount of the inner and outer ring of the bearing), while the quantitative description of the bearing, from the assembly error expression to bearing stiffness performance change, is realized. cally analyzed (the relative tilt amount of the inner and outer ring of the bearing), while the quantitative description of the bearing, from the assembly error expression to bearing stiffness performance change, is realized. **2. Bearing Stiffness Solution Model Construction** 

#### **2. Bearing Stiffness Solution Model Construction** *2.1. Equivalent Transformation of Bearing Spacer Non-Parallel*

#### *2.1. Equivalent Transformation of Bearing Spacer Non-Parallel* A common way to adjust the initial assembly preload of spindle bearings is the use

*Lubricants* **2022**, *10*, 116 3 of 16

A common way to adjust the initial assembly preload of spindle bearings is the use of spacers. In the case where the spacer end faces are not parallel, there is a greater impact on the bearing's initial assembly state, causing the bearing to operate in a relatively tilted attitude, between the inner and outer rings, which affects the bearing's mechanical and stiffness characteristics. This approach allows for the simulation of various bearing spacer tilting circumstances. Figure 1 shows the tilting of the spacer and the subsequent changes in the forces on the bearing's inner and outer rings. of spacers. In the case where the spacer end faces are not parallel, there is a greater impact on the bearing's initial assembly state, causing the bearing to operate in a relatively tilted attitude, between the inner and outer rings, which affects the bearing's mechanical and stiffness characteristics. This approach allows for the simulation of various bearing spacer tilting circumstances. Figure 1 shows the tilting of the spacer and the subsequent changes in the forces on the bearing's inner and outer rings.

investigated.

**Figure 1.** Diagram of the forces on the bearing assembly: (**a**) Bearing spacer titling diagram, (**b**) Schematic diagram of the Load on the inner/outer rings of the bearing. **Figure 1.** Diagram of the forces on the bearing assembly: (**a**) Bearing spacer titling diagram, (**b**) Schematic diagram of the Load on the inner/outer rings of the bearing.

The bearing outer ring force equilibrium equation is determined using the force analysis diagram in Figure 1: The bearing outer ring force equilibrium equation is determined using the force analysis diagram in Figure 1:

$$
\begin{bmatrix} F\_a \\ F\_r \end{bmatrix} = \begin{bmatrix} \begin{bmatrix} \sum\_j \left( Q\_{oj} \sin \mathfrak{a}\_{oj} - \lambda\_{oj} \frac{M\_{\text{cj}}}{D\_b} \cos \mathfrak{a}\_{oj} \right) \\\ \sum\_j \left( Q\_{oj} \sin \mathfrak{a}\_{oj} - \lambda\_{oj} \frac{M\_{\text{cj}}}{D\_b} \cos \mathfrak{a}\_{oj} \right) \\\ \vdots \\\ \sum\_j \end{bmatrix} \end{bmatrix} \begin{bmatrix} \\\\ \\\\ \\\\ \end{bmatrix} \tag{1}
$$
 
$$\text{the axial load applied on the horizon (D): } \mathbb{E} \text{ is the radial load amplitude}$$

sin cos *oj oj oj oj j b Q D* where, *F<sup>a</sup>* represents the axial load applied on the bearing (N); *Fr* is the radial load applied on bearing (N); *Z* is the number of the rolling balls; *Qoj* is the contact load between *j*th ball where, *Fa* represents the axial load applied on the bearing (N); *Fr* is the radial load applied on bearing (N); *Z* is the number of the rolling balls; *Qoj* is the contact load between *j*th ball and outer ring (); *αoj* is the contact angle between the *j*th ball and the outer ring (rad); *Mgj* is the gyroscopic moment of the *j*th rolling element (N·mm); *D<sup>b</sup>* is the diameter of the rolling ball; *λoj* is the load distribution coefficient of outer raceway.

and outer ring (); *oj* is the contact angle between the *j*th ball and the outer ring (rad); *Mgj* is the gyroscopic moment of the *j*th rolling element (N·mm); *Db* is the diameter of the Furthermore, in the case of bearing spacer non-parallelism, caused by the relative tilt of the bearing's inner and outer rings, the contact force created on the rolling body and

Furthermore, in the case of bearing spacer non-parallelism, caused by the relative tilt of the bearing's inner and outer rings, the contact force created on the rolling body and raceway contact region produces a moment *M*, relative to the inner and outer ring of the

$$\mathcal{N}\_{\psi}$$

rolling ball;

raceway contact region produces a moment *M*, relative to the inner and outer ring of the bearing tilt, whose value is calculated as follows: sin cos cos *oj gj oj oj oj o Z b j M Q R D M*  (2)

*Lubricants* **2022**, *10*, 116 4 of 16

$$M = \sum\_{j}^{Z} \left[ \begin{pmatrix} Q\_{oj} \sin \mathfrak{a}\_{oj} - \frac{\lambda\_{oj} M\_{\text{gj}}}{D\_b} \cos \mathfrak{a}\_{oj} \end{pmatrix} \mathbf{R}\_o \right] \begin{pmatrix} \mathbf{r}\_o \\\\ \cos \mathfrak{p}\_j \\\\ \end{pmatrix} \tag{2}$$

where, *R<sup>o</sup>* is the radius of the outer raceway curvature center (mm); *r* is the distance from the point of load force application to the axis (mm); *ϕ<sup>j</sup>* is the orientation of the *j*th rolling element (rad). the point of load force application to the axis (mm); *j* is the orientation of the *j*th rolling element (rad). It is evident that, due to the non-parallelism of the spindle spacer and the resulting

It is evident that, due to the non-parallelism of the spindle spacer and the resulting bearing inner ring, the outer ring experiences a relative tilt, causing non-uniformity in the rolling body and raceway contact area, while under load, which triggers the additional bending moment and has an impact on the bearing stiffness performance. bearing inner ring, the outer ring experiences a relative tilt, causing non-uniformity in the rolling body and raceway contact area, while under load, which triggers the additional bending moment and has an impact on the bearing stiffness performance.

#### *2.2. Bearing Stiffness Solution 2.2. Bearing Stiffness Solution*

The 5th order square matrix, created by taking the partial derivative of the external load f applied to the bearing, against the relative displacement d of the inner and outer rings of the bearing, is the analytical way of solving the bearing stiffness matrix. The Jones model is used to calculate the three-degree-of-freedom hydrostatic parameters, used as the initial value of the iterative algorithm to obtain the proposed three-degree-of-freedom hydrostatic solution parameters. Next, the derived three-degree-of-freedom hydrostatic parameters are used as the initial value of the Newton-Raphson iterative solution method, used to obtain the proposed five-degree-of-freedom hydrostatic parameters of the bearing and solve the 4Z + 5 equations. In the solution process, each rolling element equation is a local equation, whereas the overall bearing force equation is a global equation. The solution variables for each rolling element are: **X** = {*X*1, *X*2, *δ<sup>i</sup>* , *δo*}*<sup>j</sup>* , *j* = 1, 2 . . . *Z* where *Z* is the number of scrolling bodies and is called the local solution variable. **f** = {*Fx, Fy, Fz, My, Mz*} is called a global load variable, d = {*δx*, *δy*, *δz*, *θy*, *θz*} is called a universal displacement variable. According to the global equations, the overall bearing force and each rolling body force are related, so when solving the overall bearing force equation, each rolling body equilibrium equation must also be solved, joining each rolling body solution formula and force equilibrium equation. The specific solution flow is shown in Figure 2. The 5th order square matrix, created by taking the partial derivative of the external load f applied to the bearing, against the relative displacement d of the inner and outer rings of the bearing, is the analytical way of solving the bearing stiffness matrix. The Jones model is used to calculate the three-degree-of-freedom hydrostatic parameters, used as the initial value of the iterative algorithm to obtain the proposed three-degree-of-freedom hydrostatic solution parameters. Next, the derived three-degree-of-freedom hydrostatic parameters are used as the initial value of the Newton-Raphson iterative solution method, used to obtain the proposed five-degree-of-freedom hydrostatic parameters of the bearing and solve the 4Z + 5 equations. In the solution process, each rolling element equation is a local equation, whereas the overall bearing force equation is a global equation. The solution variables for each rolling element are: **X** = {*X*1, *X*2, *δi*, *δo*}*j*, *j* = 1, 2…*Z* where *Z* is the number of scrolling bodies and is called the local solution variable. **f** = {*Fx, Fy, Fz, My, Mz*} is called a global load variable, d = {*δx*, *δy*, *δz*, *θy*, *θz*} is called a universal displacement variable. According to the global equations, the overall bearing force and each rolling body force are related, so when solving the overall bearing force equation, each rolling body equilibrium equation must also be solved, joining each rolling body solution formula and force equilibrium equation. The specific solution flow is shown in Figure 2.

**Figure 2.** Flow diagram for calculation of bearing stiffness characteristics in all directions. **Figure 2.** Flow diagram for calculation of bearing stiffness characteristics in all directions.

#### **3. Bearing Stiffness Solution Condition Settings** *fi* (mm) 0.52

In this paper, the well-known commercially available precision angular contact ball bearing 7014C is employed, with parameters as listed in Table 1. *fo* (mm) 0.52 Z 20

In this paper, the well-known commercially available precision angular contact ball

**Parameter Value**  *Db* (mm) 11.9

*Lubricants* **2022**, *10*, 116 5 of 16

bearing 7014C is employed, with parameters as listed in Table 1.

**3. Bearing Stiffness Solution Condition Settings** 


**Table 1.** Basic bearing parameters.


The end face of the spacer is always machined with one end face as the reference surface and the other end face is ground. Due to manufacturing errors, the machined surface is not perfectly flat. In other words, there is a parallelism error between the machined end face of the spacer and the reference surface, which is depicted in Figure 2. To facilitate analysis and calculation, this paper uses the inclination angle *θ* of the spacer end face to characterize the non-parallelism of the spacer end face, expressed as: face is not perfectly flat. In other words, there is a parallelism error between the machined end face of the spacer and the reference surface, which is depicted in Figure 2. To facilitate analysis and calculation, this paper uses the inclination angle *θ* of the spacer end face to characterize the non-parallelism of the spacer end face, expressed as: arctan (3)

$$\theta = \arctan \frac{\Delta}{L} \tag{3}$$

where *L* is the outer diameter dimension of the outer spacer (mm), in relation to the mating bearing. This paper is based on the analysis of the 7210C bearing, so the value of *L* is 90 mm; ∆: The height difference between the highest point and the lowest point of the spacer end face (µm) is shown in Figure 3. bearing. This paper is based on the analysis of the 7210C bearing, so the value of *L* is 90 mm; : The height difference between the highest point and the lowest point of the spacer end face (μm) is shown in Figure 3.

**Figure 3.** Graphical depiction of spacer end face non-parallelism. **Figure 3.** Graphical depiction of spacer end face non-parallelism.

For theoretical analysis purposes, several groups of spacers with parallelism errors are set up in this study, which are denoted as: = 10 μm, 20 μm, 30 μm, 40 μm, while the corresponding tilt angle values of these four groups of outer spacers with parallelism errors are: <sup>1</sup> 0.0052 , <sup>2</sup> 0.0104 , <sup>3</sup> 0.0156 , <sup>4</sup> 0.0208 as listed in Table 2. According to the NSK bearing manual, the spacer face runout within 3 μm is within the design error tolerance, so the spacer mark within the design error range is: = 0μm, which indicates a well parallel spacer. The angle of tilt is noted as: <sup>0</sup> 0 For theoretical analysis purposes, several groups of spacers with parallelism errors are set up in this study, which are denoted as: ∆ = 10 µm, 20 µm, 30 µm, 40 µm, while the corresponding tilt angle values of these four groups of outer spacers with parallelism errors are: *θ*<sup>1</sup> = 0.0052◦ , *θ*<sup>2</sup> = 0.0104◦ , *θ*<sup>3</sup> = 0.0156◦ , *θ*<sup>4</sup> = 0.0208◦as listed in Table 2. According to the NSK bearing manual, the spacer face runout within 3 µm is within the design error tolerance, so the spacer mark within the design error range is: ∆ = 0 µm, which indicates a well parallel spacer. The angle of tilt is noted as: *θ*<sup>0</sup> = 0 ◦ . In contrast with the tilting of a bearing, caused by an individually applied bending moment, when the bearing is tilted due to a dimensional error in the outer spacer, the bearing is also tilted to a certain extent by the geometric constraints of the spacer end face.

. In contrast


with the tilting of a bearing, caused by an individually applied bending moment, when the bearing is tilted due to a dimensional error in the outer spacer, the bearing is also tilted

**Table 2.** Description of bearing stiffness analysis conditions. **Table 2.** Description of bearing stiffness analysis conditions.

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to a certain extent by the geometric constraints of the spacer end face.

The analysis shows that, in the assembly of the spindle, there will be a tilting gap between the outer spacer and the bearing outer ring, due to the existence of the tilting angle of the outer spacer end face. As shown in Figure 4, the bearing and housing is usually a clearance fit, resulting in clearance inside the bearing. There is a possibility of the bearing outer ring being tilted, during the transfer of force, under the action of the axial assembly force and the contact between the tilted end face of the outer spacer and the bearing outer ring. The analysis shows that, in the assembly of the spindle, there will be a tilting gap between the outer spacer and the bearing outer ring, due to the existence of the tilting angle of the outer spacer end face. As shown in Figure 4, the bearing and housing is usually a clearance fit, resulting in clearance inside the bearing. There is a possibility of the bearing outer ring being tilted, during the transfer of force, under the action of the axial assembly force and the contact between the tilted end face of the outer spacer and the bearing outer ring.

**Figure 4.** Force analysis diagram of bearing outer ring under inclined spacer ring: (**a**) contact between bearing and spacer, (**b**) bearing force, (**c**) ball orientation diagram. **Figure 4.** Force analysis diagram of bearing outer ring under inclined spacer ring: (**a**) contact between bearing and spacer, (**b**) bearing force, (**c**) ball orientation diagram.

The notion of bearing mounting quality, as opposed to the ideal load bearing stiffness change law, can more accurately reflect the actual engineering application of bearing stiffness performance, bearing mounting quality analysis, bearing joint parts quality on the bearing and even spindle stiffness service performance of the mechanism. In this view, it is the bearing mounting quality and recommended assembly parameters in real-world engineering applications that are considered in this paper; as well as a number of working scenarios for the investigation of bearing stiffness performance, with exact factors (Table 2). Different speed values (at 1000 rpm increments), preload conditions and bearing deflection circumstances are all used to compute stiffness variations. The notion of bearing mounting quality, as opposed to the ideal load bearing stiffness change law, can more accurately reflect the actual engineering application of bearing stiffness performance, bearing mounting quality analysis, bearing joint parts quality on the bearing and even spindle stiffness service performance of the mechanism. In this view, it is the bearing mounting quality and recommended assembly parameters in real-world engineering applications that are considered in this paper; as well as a number of working scenarios for the investigation of bearing stiffness performance, with exact factors (Table 2). Different speed values (at 1000 rpm increments), preload conditions and bearing deflection circumstances are all used to compute stiffness variations.

#### **4. Analysis of Factors Influencing Bearing Stiffness**

#### **4. Analysis of Factors Influencing Bearing Stiffness**  *4.1. Bearing Stiffness Variation Law under Different Working Conditions*

*4.1. Bearing Stiffness Variation Law under Different Working Conditions*  The variation of axial, radial and angular stiffness values on a macro scale reflects the bearing stiffness characteristics. The diagram in Figure 5 demonstrates that there are five surfaces (five surfaces indicate five different tilt conditions: 0 μm −10 μm −20 μm −30 μm −40 μm). It is harder to determine that this is a superposition of five surfaces, based on the three-dimensional surface diagram. More specifically, the effect of tilting the inner and The variation of axial, radial and angular stiffness values on a macro scale reflects the bearing stiffness characteristics. The diagram in Figure 5 demonstrates that there are five surfaces (five surfaces indicate five different tilt conditions: 0 µm −10 µm −20 µm −30 µm −40 µm). It is harder to determine that this is a superposition of five surfaces, based on the three-dimensional surface diagram. More specifically, the effect of tilting the inner and outer rings of the bearing on its overall axial stiffness *Kxx*, radial stiffness *Kyy*, radial stiffness *Kzz*, angular stiffness *Kθ<sup>y</sup>* and angular stiffness *Kθ<sup>z</sup>* does not cause significant value fluctuation. The reason for this could be that some of the local differences in surface changes are difficult to discern in 3D views, while the changes in bearing stiffness are so miniature that they are difficult to reflect on the overall stiffness change relationship graph.

graph.

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outer rings of the bearing on its overall axial stiffness *Kxx*, radial stiffness *Kyy*, radial stiffness *Kzz*, angular stiffness *Kθy* and angular stiffness *Kθz* does not cause significant value fluctuation. The reason for this could be that some of the local differences in surface changes are difficult to discern in 3D views, while the changes in bearing stiffness are so miniature that they are difficult to reflect on the overall stiffness change relationship

**Figure 5.** Variation of spindle bearing stiffness in all directions in relation to tilt angle. **Figure 5.** Variation of spindle bearing stiffness in all directions in relation to tilt angle.

As a result, a view is transformed to analyze the stiffness variation rule (Figure 6). It is observed that the bearing stiffness varies significantly under different preloads. Specifically, raising the bearing preload can effectively improve bearing stiffness performance, whereas as preload and speed increase, bearing stiffness reduces significantly (bearing heavy preload versus light preload conditions). In addition, the axial stiffness is one order of magnitude lower, compared to the radial stiffness, while no significant variation is found between the horizontal radial stiffness and the vertical radial stiffness, as well as between the two angular stiffness components *Kθy*. The bearing stiffness variations in all directions, in relation to its inner and outer ring tilt angle fluctuations, are not obvious but exhibit a certain regularity. That is, stiffness increases and decreases in relation to spindle As a result, a view is transformed to analyze the stiffness variation rule (Figure 6). It is observed that the bearing stiffness varies significantly under different preloads. Specifically, raising the bearing preload can effectively improve bearing stiffness performance, whereas as preload and speed increase, bearing stiffness reduces significantly (bearing heavy preload versus light preload conditions). In addition, the axial stiffness is one order of magnitude lower, compared to the radial stiffness, while no significant variation is found between the horizontal radial stiffness and the vertical radial stiffness, as well as between the two angular stiffness components *Kθ<sup>y</sup>* . The bearing stiffness variations in all directions, in relation to its inner and outer ring tilt angle fluctuations, are not obvious but exhibit a certain regularity. That is, stiffness increases and decreases in relation to spindle speed and preload force, while it is not a linear relationship.

#### *4.2. Analysis of the Variation of Bearing Stiffness in All Directions in Relation to Speed*

The stiffness—speed—preload relationship graph, based on variations in the bearing's inner and outer ring tilt, does not adequately reflect the influence of imperfect mounting on bearing stiffness, as shown in Section 4.1. In engineering, the tilting of inner and outer rings inevitably leads to a change in the internal contact parameters of the bearing. If this tilting angle is known, the change in contact parameters will lead to a change in contact stiffness, until it is reflected in the isotropic stiffness of the bearing. As a result, the effect of the additional bending moment, induced by the relative tilting of the inner and outer rings, on bearing stiffness, will be investigated, in order to further confirm it, based on a two-dimensional plane perspective.

speed and preload force, while it is not a linear relationship.

**Figure 6.** Variation of spindle bearing stiffness in all directions under preload conditions. **Figure 6.** Variation of spindle bearing stiffness in all directions under preload conditions.

*4.2. Analysis of the Variation of Bearing Stiffness in All Directions in Relation to Speed*  The stiffness—speed—preload relationship graph, based on variations in the bearing's inner and outer ring tilt, does not adequately reflect the influence of imperfect mounting on bearing stiffness, as shown in Section 4.1. In engineering, the tilting of inner and outer rings inevitably leads to a change in the internal contact parameters of the bearing. If this tilting angle is known, the change in contact parameters will lead to a change in contact stiffness, until it is reflected in the isotropic stiffness of the bearing. As a result, the effect of the additional bending moment, induced by the relative tilting of the inner and outer rings, on bearing stiffness, will be investigated, in order to further confirm it, based on a two-dimensional plane perspective. Figure 7 shows the change in the axial stiffness of the bearing, as a function of speed, under various preload forces. It also illustrates a comparison and analysis of the change in axial stiffness of the bearing, as a function of the inner and outer rings inclination value. Based on Figure 5, from a macroscopic point of view, bearing inner and outer ring relative tilt angle variations did not produce too high influence on the axial stiffness. In the operating conditions of Extremely Light preload (EL), Light preload (L), Medium preload (M), bearing stiffness change curve shows obvious consistency, as the five curves almost overlap. In the Heavy preload (H) operating conditions, the axial stiffness difference in the local magnification chart is slightly obvious. However, compared to the bearing axial stiffness value of 1–2 magnitude orders lower, this difference can be ignored. At the same Figure 7 shows the change in the axial stiffness of the bearing, as a function of speed, under various preload forces. It also illustrates a comparison and analysis of the change in axial stiffness of the bearing, as a function of the inner and outer rings inclination value. Based on Figure 5, from a macroscopic point of view, bearing inner and outer ring relative tilt angle variations did not produce too high influence on the axial stiffness. In the operating conditions of Extremely Light preload (EL), Light preload (L), Medium preload (M), bearing stiffness change curve shows obvious consistency, as the five curves almost overlap. In the Heavy preload (H) operating conditions, the axial stiffness difference in the local magnification chart is slightly obvious. However, compared to the bearing axial stiffness value of 1–2 magnitude orders lower, this difference can be ignored. At the same time, another occurrence was specified, namely the considerable change in axial stiffness of the bearing, caused by the preload force. In the mild preload situation, when the speed approaches 1800 rpm, the axial rigidity gradually drops, until stabilizing at 5000 rpm. The slope of the drop is less than that, in the slight preload conditions. Similarly, in the bearing under preload conditions, the stiffness decreases more obviously after the speed reaches about 3000 rpm, while even more obviously in heavy preload conditions, when the axial stiffness only shows a significant decreasing trend, after the speed reaches about 4000 rpm. The investigation of the axial stiffness of bearings, at various degrees of inner and outer ring tilting, reveals that bearing mounting quality has only a minor impact on axial stiffness, as well as confirms the efficacy and significance of bearing preload in enhancing bearing stiffness.

time, another occurrence was specified, namely the considerable change in axial stiffness of the bearing, caused by the preload force. In the mild preload situation, when the speed approaches 1800 rpm, the axial rigidity gradually drops, until stabilizing at 5000 rpm. The slope of the drop is less than that, in the slight preload conditions. Similarly, in the bearing

under preload conditions, the stiffness decreases more obviously after the speed reaches about 3000 rpm, while even more obviously in heavy preload conditions, when the axial stiffness only shows a significant decreasing trend, after the speed reaches about 4000 rpm. The investigation of the axial stiffness of bearings, at various degrees of inner and outer ring tilting, reveals that bearing mounting quality has only a minor impact on axial stiffness, as well as confirms the efficacy and significance of bearing preload in enhancing

**Figure 7.** Variation of axial bearing stiffness *Kxx* in relation to speed. **Figure 7.** Variation of axial bearing stiffness *Kxx* in relation to speed.

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bearing stiffness.

Figure 8 shows how the bearing's horizontal radial stiffness varies, with relation to preload force, at various tilting angles, while systematically represented in four operating conditions: extremely light preload, light preload, medium preload and heavy preload. The radial stiffness of the bearing exhibits the same regularity as the axial stiffness, while the influence of the preload on the radial stiffness is also significant. At a higher preload, the bearing stiffness becomes more resistant to the "softening phenomenon", as the speed increases, whereas the radial stiffness is an order of magnitude higher than the axial stiffness. It should be noted that changing the bearing's inner and outer ring tilt has a significantly greater impact on radial stiffness, i.e., under different preload circumstances, the bearing's radial stiffness increases with a tilt angle of roughly 2.5% to 3%. This probably occurs due to the bearing inner and outer ring tilt orientation (see Figure 1 for a schematic depiction). As a result, a thorough understanding of the tilt angle's influence on the radial stiffness, as well as the general and specific laws of bearing stiffness, is critical, not only for accurate bearing stiffness exploration, but also for providing a solid theoretical foun-Figure 8 shows how the bearing's horizontal radial stiffness varies, with relation to preload force, at various tilting angles, while systematically represented in four operating conditions: extremely light preload, light preload, medium preload and heavy preload. The radial stiffness of the bearing exhibits the same regularity as the axial stiffness, while the influence of the preload on the radial stiffness is also significant. At a higher preload, the bearing stiffness becomes more resistant to the "softening phenomenon", as the speed increases, whereas the radial stiffness is an order of magnitude higher than the axial stiffness. It should be noted that changing the bearing's inner and outer ring tilt has a significantly greater impact on radial stiffness, i.e., under different preload circumstances, the bearing's radial stiffness increases with a tilt angle of roughly 2.5% to 3%. This probably occurs due to the bearing inner and outer ring tilt orientation (see Figure 1 for a schematic depiction). As a result, a thorough understanding of the tilt angle's influence on the radial stiffness, as well as the general and specific laws of bearing stiffness, is critical, not only for accurate bearing stiffness exploration, but also for providing a solid theoretical foundation for subsequent bearing service performance accuracy.

dation for subsequent bearing service performance accuracy. Similar to Figures 8 and 9 shows how the bearing's vertical radial stiffness varies with relation to preload force, at different tilting angles, represented consistently under four operating conditions: EL, L, M and H. It is proven that the vertical radial stiffness of the bearing shows almost the same regularity as the horizontal radial stiffness, while the difference is reflected in a closer local view, where the horizontal radial stiffness of the bearing appears slightly increased, in an analogy to the increase of the inner and outer ring tilt. In regards to the vertical radial stiffness, the exact opposite relationship holds; that is, the vertical radial stiffness of the bearing decreases with the increase of the tilt angle, at about 2~2.5%. The bearing spacer tilt orientation, generated by this common difference, Similar to Figures 8 and 9 shows how the bearing's vertical radial stiffness varies with relation to preload force, at different tilting angles, represented consistently under four operating conditions: EL, L, M and H. It is proven that the vertical radial stiffness of the bearing shows almost the same regularity as the horizontal radial stiffness, while the difference is reflected in a closer local view, where the horizontal radial stiffness of the bearing appears slightly increased, in an analogy to the increase of the inner and outer ring tilt. In regards to the vertical radial stiffness, the exact opposite relationship holds; that is, the vertical radial stiffness of the bearing decreases with the increase of the tilt angle, at about 2~2.5%. The bearing spacer tilt orientation, generated by this common difference, tilts the bearing about the horizontal axis, according to the analysis. In order to accurately describe the law of tilt angle influence on bearing radial stiffness, besides the bearing tilt angle changes, the bearing tilt process of relative azimuth changes should be considered in a comprehensive analysis, which will provide results that are more in line with the actual engineering reality, while also more accurate.

tilts the bearing about the horizontal axis, according to the analysis. In order to accurately describe the law of tilt angle influence on bearing radial stiffness, besides the bearing tilt angle changes, the bearing tilt process of relative azimuth changes should be considered in a comprehensive analysis, which will provide results that are more in line with the

tilts the bearing about the horizontal axis, according to the analysis. In order to accurately describe the law of tilt angle influence on bearing radial stiffness, besides the bearing tilt angle changes, the bearing tilt process of relative azimuth changes should be considered in a comprehensive analysis, which will provide results that are more in line with the

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actual engineering reality, while also more accurate.

actual engineering reality, while also more accurate.

**Figure 8.** Variation of horizontal radial bearing stiffness *K*yy with relation to speed. **Figure 8.** Variation of horizontal radial bearing stiffness *K*yy with relation to speed. **Figure 8.** Variation of horizontal radial bearing stiffness *K*yy with relation to speed.

**Figure 9.** Bearing vertical radial stiffness *K*zz variation with relation to speed. **Figure 9.** Bearing vertical radial stiffness *K*zz variation with relation to speed. **Figure 9.** Bearing vertical radial stiffness *K*zz variation with relation to speed.

Figures 7–9 illustrate the three directions of the bearing plain stiffness change law. Since angular contact ball bearings also have an angular stiffness, an in-depth analysis Figures 7–9 illustrate the three directions of the bearing plain stiffness change law. Since angular contact ball bearings also have an angular stiffness, an in-depth analysis Figures 7–9 illustrate the three directions of the bearing plain stiffness change law. Since angular contact ball bearings also have an angular stiffness, an in-depth analysis should explore whether the bearing inner and outer ring relative tilt will have what effect on the angular stiffness and whether it is still similar to the plain stiffness change law, so as to accurately grasp the overall stiffness change mechanism.

Figure 10 shows that, the angular stiffness of the bearing changes in a similar way to the flatness of the bearing. From a macroscopic perspective, bearing speed and initial preload are the most important factors in determining bearing stiffness. Furthermore,

angular stiffness and plain stiffness decrease as frequency increases, and they rise as preload decreases. The fact that the tilt angle of the inner and outer rings vary around the *z* axis, makes the angular stiffness of the bearing changes, with relation to the tilt angle of the inner and outer rings, a little more visible (Figure 1), so the angular stiffness *K*θ<sup>y</sup> decreases as the tilt angle increases. stiffness and plain stiffness decrease as frequency increases, and they rise as preload decreases. The fact that the tilt angle of the inner and outer rings vary around the *z* axis, makes the angular stiffness of the bearing changes, with relation to the tilt angle of the inner and outer rings, a little more visible (Figure 1), so the angular stiffness *K*θy decreases as the tilt angle increases.

should explore whether the bearing inner and outer ring relative tilt will have what effect on the angular stiffness and whether it is still similar to the plain stiffness change law, so

Figure 10 shows that, the angular stiffness of the bearing changes in a similar way to the flatness of the bearing. From a macroscopic perspective, bearing speed and initial preload are the most important factors in determining bearing stiffness. Furthermore, angular

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as to accurately grasp the overall stiffness change mechanism.

**Figure 10.** Variation of angular bearing stiffness *Kθy* with relation to speed. **Figure 10.** Variation of angular bearing stiffness *Kθ<sup>y</sup>* with relation to speed.

Based on Figure 11, the angular stiffness *Kθz* variation characteristics with relation to speed are almost identical to the respective angular stiffness *Kθy* variation characteristics, whereas minor differences are only reflected in the local view enlargements; however, the effect of the bearing tilt angle is evident in this difference in variation. A comparison of the local enlargements in Figure 8 shows that the angular stiffness *Kθy* tends to decrease as the angle of inclination increases, while the angular stiffness *Kθz* tends to increase as the angle of inclination increases, while the greater the preload the more significant the dif-Based on Figure 11, the angular stiffness *Kθ<sup>z</sup>* variation characteristics with relation to speed are almost identical to the respective angular stiffness *Kθ<sup>y</sup>* variation characteristics, whereas minor differences are only reflected in the local view enlargements; however, the effect of the bearing tilt angle is evident in this difference in variation. A comparison of the local enlargements in Figure 8 shows that the angular stiffness *Kθ<sup>y</sup>* tends to decrease as the angle of inclination increases, while the angular stiffness *Kθ<sup>z</sup>* tends to increase as the angle of inclination increases, while the greater the preload the more significant the difference is.

ference is. Throughout the bearing stiffness curve with relation to speed and the preload force change law, it was discovered that the bearing's initial preload force, working speed and stiffness performance play dominant roles. However, the bearing inner and outer ring tilt angle, as well as the bearing stiffness in all directions show a significant correlation and a significant regularity difference. Although this variability may have a minor effect on the performance of rough machining spindles, it cannot be overlooked in precision machining spindles. Consequently, while mounting spindle bearings, special attention should be Throughout the bearing stiffness curve with relation to speed and the preload force change law, it was discovered that the bearing's initial preload force, working speed and stiffness performance play dominant roles. However, the bearing inner and outer ring tilt angle, as well as the bearing stiffness in all directions show a significant correlation and a significant regularity difference. Although this variability may have a minor effect on the performance of rough machining spindles, it cannot be overlooked in precision machining spindles. Consequently, while mounting spindle bearings, special attention should be paid to the quality of the installation, ensuring that their stiffness characteristics and even the service performance of the spindle system are carefully controlled.

paid to the quality of the installation, ensuring that their stiffness characteristics and even the service performance of the spindle system are carefully controlled. Due to the numerous elements affecting bearing stiffness fluctuation, a study focused just on the total stiffness under these five tilting scenarios would still yield traditional Due to the numerous elements affecting bearing stiffness fluctuation, a study focused just on the total stiffness under these five tilting scenarios would still yield traditional regularity results. Thus, the bearing stiffness influence must be decomposed into the local area of each rolling element and raceway contact, i.e., the overall rigidity. The bearing's performance is determined by the specific rolling element, while the stiffness of the inner and outer ring raceways follow a certain law combination, which requires a more complete examination of the local contact unit area.

regularity results. Thus, the bearing stiffness influence must be decomposed into the local area of each rolling element and raceway contact, i.e., the overall rigidity. The bearing's performance is determined by the specific rolling element, while the stiffness of the inner and outer ring raceways follow a certain law combination, which requires a more com-

**Figure 11.** Variation of angular bearing stiffness *Kθz* with relation to speed. **Figure 11.** Variation of angular bearing stiffness *Kθ<sup>z</sup>* with relation to speed.

#### *4.3. Analysis of Bearing Stiffness Variability in All Directions 4.3. Analysis of Bearing Stiffness Variability in All Directions*

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plete examination of the local contact unit area.

The bearing stiffness in all directions was studied, as well as bearing speed, preload, inner and outer ring tilt working circumstances, which derived that the bearing stiffness is predominantly represented in the radial stiffness. This paper further analyses the inner contact angle *αii*, outer contact angle *αoo*, inner ring contact force *Qi* and outer ring contact force *Qo* of the rolling element, in addition to the raceway in the local area of the bearing, carrying out numerical research on the internal contact mechanical properties of the bearing. Since the general law of the bearing is the same for different preload values, only the medium preload condition, which is typically employed in engineering, is selected as a representative case for the specific analysis, whereas the variation of the bearing's internal The bearing stiffness in all directions was studied, as well as bearing speed, preload, inner and outer ring tilt working circumstances, which derived that the bearing stiffness is predominantly represented in the radial stiffness. This paper further analyses the inner contact angle *αi<sup>i</sup>* , outer contact angle *αoo*, inner ring contact force *Q<sup>i</sup>* and outer ring contact force *Qo* of the rolling element, in addition to the raceway in the local area of the bearing, carrying out numerical research on the internal contact mechanical properties of the bearing. Since the general law of the bearing is the same for different preload values, only the medium preload condition, which is typically employed in engineering, is selected as a representative case for the specific analysis, whereas the variation of the bearing's internal contact characteristics is also investigated.

contact characteristics is also investigated. The characteristics of the 7014C bearing, under the influence of the inner and outer ring tilt rolling inner contact angle change, are shown in Figure 12. The inner contact angle gradually increases, as the speed rises, which is mostly due to the bearing's centrifugal force gradually increasing along the speed rise. At a constant speed, the analysis of the curve can provide certain results. Specifically, when the bearing inner and outer ring tilt is 0, each ball contact angle is a constant value (the impact of the bearing rolling body own weight is not considered). When the inner and outer rings show a certain tilt, the contact angle is then changed and some rolling body at the contact angle increased and some become smaller, while with the bearing inner and outer ring tilt degree of intensification, the difference of inner contact angle on individual balls is also increased. As the contact angle variations will affect the bearing performance, these will inevitably have a certain The characteristics of the 7014C bearing, under the influence of the inner and outer ring tilt rolling inner contact angle change, are shown in Figure 12. The inner contact angle gradually increases, as the speed rises, which is mostly due to the bearing's centrifugal force gradually increasing along the speed rise. At a constant speed, the analysis of the curve can provide certain results. Specifically, when the bearing inner and outer ring tilt is 0, each ball contact angle is a constant value (the impact of the bearing rolling body own weight is not considered). When the inner and outer rings show a certain tilt, the contact angle is then changed and some rolling body at the contact angle increased and some become smaller, while with the bearing inner and outer ring tilt degree of intensification, the difference of inner contact angle on individual balls is also increased. As the contact angle variations will affect the bearing performance, these will inevitably have a certain impact on the overall bearing rigidity.

impact on the overall bearing rigidity. Figure 13 shows the characteristics of the 7014C shaft at four different speed values, demonstrating the effect of the inner and outer ring tilt on the rolling element's outer contact angle. It becomes evident that the outer contact angle of the bearing decreases as the speed increases; a phenomenon due to the centrifugal effect, caused by the speed change. Moreover, the difference between the outer contact angles of each rolling element shows an obvious tendency.

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**Figure 12.** 7014C bearing rolling body and raceway internal contact angle *αii.* **Figure 12.** 7014C bearing rolling body and raceway internal contact angle *αii*. an obvious tendency.

**Figure 13.** 7014C bearing rolling body and the inner contact angle of the raceway *αoo*. **Figure 13.** 7014C bearing rolling body and the inner contact angle of the raceway *αoo*.

**Figure 13.** 7014C bearing rolling body and the inner contact angle of the raceway *αoo*. In Figure 14, the variation of the local contact force, between the rolling element and raceway of the inner ring of the bearing, is depicted. It can be seen that when the bearing speed increases, the inner ring local maximum contact force changes with the inner ring contact angle, while with the increase of speed the inner ring maximum local contact force significantly decreases. The main reason is that the contact angle increases along with the In Figure 14, the variation of the local contact force, between the rolling element and raceway of the inner ring of the bearing, is depicted. It can be seen that when the bearing speed increases, the inner ring local maximum contact force changes with the inner ring contact angle, while with the increase of speed the inner ring maximum local contact force significantly decreases. The main reason is that the contact angle increases along with the speed, which reduces the maximum local contact force. Under the same speed, the maximum local contact force changes more significantly with the increase of the inner and outer ring tilt angle. This indicates that when the tilt amount increases, the impact on the internal contact parameters of the bearing is more significant, intensifying the uneven load

In Figure 14, the variation of the local contact force, between the rolling element and raceway of the inner ring of the bearing, is depicted. It can be seen that when the bearing

contact angle, while with the increase of speed the inner ring maximum local contact force significantly decreases. The main reason is that the contact angle increases along with the

on the contact area of each rolling element of the bearing, whereas the unevenly distributed contact force on the circumferential radial is one of the reasons for the non-uniformity of the radial stiffness of the bearing, which may have a great impact on the accuracy and even the service life of the bearing. on the contact area of each rolling element of the bearing, whereas the unevenly distributed contact force on the circumferential radial is one of the reasons for the non-uniformity of the radial stiffness of the bearing, which may have a great impact on the accuracy and even the service life of the bearing.

speed, which reduces the maximum local contact force. Under the same speed, the maximum local contact force changes more significantly with the increase of the inner and outer ring tilt angle. This indicates that when the tilt amount increases, the impact on the internal contact parameters of the bearing is more significant, intensifying the uneven load

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**Figure 14.** 7014C bearing rolling body and raceway internal contact force *Qi*. **Figure 14.** 7014C bearing rolling body and raceway internal contact force *Q<sup>i</sup>* .

Figure 15 illustrates the variation pattern of the bearing outer ring contact force. It can be seen that, the bearing outer ring contact force exhibits a similar pattern to the bearing inner ring contact force. The difference lies in the speed rise causing the contact force on the outer ring of the bearing to increase. The change is still related to the increasing speed, caused by the effect of centrifugal force. At a certain speed, the outer ring local contact force changes. This means that the contact force on the outer ring of the bearing is non-uniform, along the circumferential direction, while this non-uniformity increases with the amount of skew. Figure 15 illustrates the variation pattern of the bearing outer ring contact force. It can be seen that, the bearing outer ring contact force exhibits a similar pattern to the bearing inner ring contact force. The difference lies in the speed rise causing the contact force on the outer ring of the bearing to increase. The change is still related to the increasing speed, caused by the effect of centrifugal force. At a certain speed, the outer ring local contact force changes. This means that the contact force on the outer ring of the bearing is non-uniform, along the circumferential direction, while this non-uniformity increases with the amount of skew. *Lubricants* **2022**, *10*, 116 15 of 16

**Figure 15.** 7014C bearing rolling body and raceway external contact force *Q*o. **Figure 15.** 7014C bearing rolling body and raceway external contact force *Q*o.

In view of the common problems of bearing mounting in actual working conditions,

the stiffness, along all directions, is examined the bearing stiffness (radial, axial and angular) is comparatively discussed under different assembly conditions, as well as the bearing

Bearing inner and outer ring tilt angle has a greater impact on bearing radial stiffness

 The inner and outer rings of the bearing changes lead to variations of the inner ring contact load, revealing the reasons for the bearing's anisotropic stiffness. Among the three influence factors of bearing preload, rotational speed and spacer inclination, although the inclination has the smallest influence on the macroscopic stiffness of the bearing, it will cause uneven radial stiffness of the bearing, which is

 Moreover, the anisotropy of radial stiffness, caused by the inclination of the bearing spacer, will be verified through experiments in the follow-up research. In the future, the study on the precision spindle system stiffness, induced by bearing anisotropy of

**Author Contributions:** Conceptualization, Y.Z. and Y.L.; methodology, Y.L.; validation, Y.Z., Y.L. and L.K.; formal analysis, Z.Y. and Y.S.; investigation, L.K. and Y.Z.; data curation, Y.L.; writing original draft preparation, Y.S.; writing—review and editing, Y.L.; visualization, Y.S.; supervision, Y.S.; project administration, Z.Y.; and funding acquisition, Y.S. All authors have read and agreed to

**Funding:** This research was funded by National Natural Science Foundation of China, grant number 52005405, General project of Shaanxi Natural Science Foundation, grant number 2022JM-244 and

Natural Science Basic Research Program of Shaanxi Province, grant number 2020JQ-639.

**Data Availability Statement:** Detailed data are contained within the article.

preload and rotation speed. The main conclusions of this paper are as follows:

**5. Conclusions** 

than on axial stiffness.

very important for a precision spindle.

radial stiffness, will be exhaustive.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

the published version of the manuscript.

#### **5. Conclusions**

In view of the common problems of bearing mounting in actual working conditions, the bad assembly state is simplified and represented by the relative tilting of the bearing spacer. Then, the influence of bearing preload, rotation speed and tilting of the spacer on the stiffness, along all directions, is examined the bearing stiffness (radial, axial and angular) is comparatively discussed under different assembly conditions, as well as the bearing preload and rotation speed. The main conclusions of this paper are as follows:


**Author Contributions:** Conceptualization, Y.Z. and Y.L.; methodology, Y.L.; validation, Y.Z., Y.L. and L.K.; formal analysis, Z.Y. and Y.S.; investigation, L.K. and Y.Z.; data curation, Y.L.; writing—original draft preparation, Y.S.; writing—review and editing, Y.L.; visualization, Y.S.; supervision, Y.S.; project administration, Z.Y.; and funding acquisition, Y.S. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by National Natural Science Foundation of China, grant number 52005405, General project of Shaanxi Natural Science Foundation, grant number 2022JM-244 and Natural Science Basic Research Program of Shaanxi Province, grant number 2020JQ-639.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Detailed data are contained within the article.

**Conflicts of Interest:** The authors declared no potential conflict of interest with respect to the research, authorship, and/or publication of this article.

#### **References**

