*Article* **Study on Distribution of Lubricating Oil Film in Contact Micro-Zone of Full Ceramic Ball Bearings and the Influence Mechanism on Service Performance**

**Jinmei Yao <sup>1</sup> , Yuhou Wu 1 , Jiaxing Yang 1 , Jian Sun 1,2, \*, Zhongxian Xia 1 , Junxing Tian 1 , Zhigang Bao 1 and Longfei Gao 1**


**Abstract:** Compared with metal ball bearings, full ceramic ball bearings have more outstanding service performance under extreme working conditions. In order to reveal the lubrication mechanism and improve the operation performance and service life of full ceramic ball bearings, in this paper, the friction, vibration, and temperature rise characteristics of 6208 silicon nitride full ceramic deep groove ball bearing, under the condition of oil lubrication, are studied experimentally. Based on the test results, and through theoretical calculation and simulation analysis, the distribution of the lubricating oil film in bearing contact micro-zone under different working conditions was simulated. After that, the surface of contact micro-zone of full ceramic ball bearing was analyzed. It was found that there is an optimal oil supply for full ceramic ball bearing oil lubrication in service. Under the optimal oil supply lubrication, full film lubrication can be achieved, and the bearing exhibits the best characteristics of friction, vibration, and temperature rise. Compared with the load, the rotational speed of the bearing has a decisive influence on the optimal oil supply. When the rotational speed and load are constant, the minimum oil film thickness and oil film pressure in the contact area of the rolling body decrease with the increase of angle *ψ* from the minimum stress point of the rolling body. Under the action of high contact stress, thin oil film will be formed in the bearing outer ring raceway. In the field of full ceramic ball bearings, the research content of this paper is innovative. The research results of this paper have an important guiding significance for revealing the oil lubrication mechanism of full ceramic ball bearing and enriching its lubrication theory and methods.

**Keywords:** full ceramic ball bearing; lubricant oil film; service performance; simulation model

#### **1. Introduction**

Full ceramic ball bearings refer to high-tech bearing products whose rings and rolling bodies are made of ceramic materials. It has excellent performance at high speed, compression resistance, high/low temperature resistance, wear resistance, corrosion resistance, and electromagnetic insulation, as well as other aspects, and can be widely used in aerospace, navigation, metallurgy, chemical and national defense, and military fields [1–3]. A good lubrication state is the prerequisite for the normal operation of full ceramic ball bearings. It is also an important factor affecting the performance of the bearings, such as friction, vibration, and temperature rise under complex working conditions [4]. At present, based on the lubrication theory of metal ball bearing, experts and scholars at home and abroad have carried out relevant research on the operation performance of ceramic ball bearings under oil lubrication conditions.

**Citation:** Yao, J.; Wu, Y.; Yang, J.; Sun, J.; Xia, Z.; Tian, J.; Bao, Z.; Gao, L. Study on Distribution of Lubricating Oil Film in Contact Micro-Zone of Full Ceramic Ball Bearings and the Influence Mechanism on Service Performance. *Lubricants* **2022**, *10*, 174. https://doi.org/10.3390/ lubricants10080174

Received: 29 May 2022 Accepted: 26 July 2022 Published: 1 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/).

Meyer L. D [5] studied the analytical method for the continuous and periodic changes of the ball contact force when the bearing rotates, based on the Lagrange equation for solving time-varying displacement of the bearing. N. Lynagh [6] established a detailed bearing vibration model by considering the influence of rolling body surface corrugation and ball size on radial clearance and deduced the vibration force and torque generated by bearing movement by using the formula. It provides a great reference for the subsequent research. Mohammed Alfares [7] studied the influence of heat generation on the performance of ball bearings during service, as well as the influence of this heat effect on system dynamics, by using transient thermal model. A set of differential equations was obtained by using thermal equilibrium. Jalali-vahid D. [8] studied and obtained the multi-stage multi-grid solution for isothermal elastohydrodynamic circular contact problem. By deploying a large number of units, the accuracy of the numerical solution is improved. The conclusion presented in this paper is of great prospective significance and has guiding value for the analysis of isothermal elastic hydrodynamics. As a pioneer in this field, Dowson D. [9] conducted a numerical evaluation on the analysis of point contact of isothermal elastohydrodynamic lubrication. In the elastic analysis, the contact area is divided into equal rectangular areas, and it is assumed that a uniform pressure is applied to each element to analyze the oil film thickness. This paper also carries out a more detailed study on this basis. Ioannides, E. [10] put forward an idea a long time ago to study an analytical model for predicting the life of rolling bearings under the consideration of the fatigue life, fatigue criterion, and fatigue limit of bearings. The viewpoints proposed at that time were forward-looking and provided great significance for the research of bearing field. Zhu Weibing [11] established a calculation model of oil injection lubrication and under-race lubrication for angular contact ball bearings. The results showed that: when the speed and load were low, it was more appropriate to use oil injection lubrication. When the speed and load are high, it is more advantageous to use under-race lubrication. Nagare [12] experimentally analyzed and investigated the effects of performance parameters, such as speed, eccentricity, load, and friction torque, on the performance of overloaded low speed bearings. Liming Lu [13] designed a set of independent devices to explore the impact of impact load on the lubrication performance of roller sliding bearings. The results showed that: the greater the impact load frequency, the greater the minimum oil film thickness, greater the impact load amplitude, and thinner the roller oil film thickness. The sliding of the rolling element may lead to the failure of the rolling bearing. Kang Jianxiong [14] considered the interaction between the ball and raceway, interaction between the cage and raceway, and elastohydrodynamic lubrication and other factors, in order to effectively analyze the sliding characteristics of the rolling bearing using the dynamic response. Antonio [15] studied the influence of lubricant film thickness on bearing service performance under hydrodynamic lubrication conditions. Cho [16] improved the Elrod algorithm based on the mass conservation boundary conditions and obtained the governing equation and the lubrication equation of the complete oil film region that can automatically determine the dynamic boundary. Biswas [17] studied the lubrication performance of medium-low speed bearings under different load conditions, as well as the changes regarding oil film thickness, oil film pressure, and oil film velocity. Dmitrichenko [18] established the dynamic model of ball bearings and studied the influence of different distribution models of lubricants and fluid dynamic pressure on the dynamic characteristics of bearings. Brizmer [19] studied the mechanism of micro-pitting resistance of hybrid ceramic bearings under reduced lubrication conditions, thus providing a new idea for tribology and performance of silicon nitride ceramic rolling bearings. Kang Li [20] analyzed the tribological properties of GCr15-GCr15/Si3N4-GCr15 materials under the condition of oil lubrication by using pin-disc friction and wear testing machine. Xiang Guo [21] studied and investigated the dynamic mixed elastohydrodynamic lubrication behavior of water-lubricated bearings with unbalanced rotors in the starting process.

From the above, it can be seen that most studies were based on metal ball bearings, only part of the research is suitable for the hybrid ceramic ball bearing. Additionally, a

small number of reports on the lubrication mechanism and method of full ceramic ball bearings are only described for the objective phenomena in the lubrication process. The objective laws and scientific problems behind it have not been further revealed. Therefore, this paper takes silicon nitride full ceramic ball bearing as the research object and studies its service performance, regarding friction, vibration, and temperature rise under the condition of oil lubrication. The effects of lubrication, rotational speed, and load on the performance of full ceramic ball bearings were revealed by means of experiments, and the optimal oil supply was determined. Based on the optimal oil supply, the distribution characteristics of lubricating oil film and its influence on service performance of full ceramic ball bearings were revealed by means of calculation and simulation. Finally, the surface characteristics of contact microzone after service of full ceramic ball bearing under oil lubrication condition were analyzed. The research results have important guiding significance for forming a lubrication theory and method suitable for full ceramic ball bearing, as well as improving its service performance and life.

### **2. Experimental Study on Oil Lubrication Characteristics of Full Ceramic Ball Bearing**

#### *2.1. Test Bearings and Components*

The test took 6208 silicon nitride full ceramic deep groove ball bearing with P4 accuracy as the research object, carried out the service performance test under oil lubrication condition, and revealed the distribution law of the full ceramic ball bearing lubricating oil film, as well as its influence mechanism on friction performance. The structure of the test bearing is shown in Figure 1. The clearance of the test bearing was C<sup>N</sup> standard clearance, and the cage guide mode was an outer ring guide. The lubricating oil used in the test was principal axis oil, with a viscosity value of 30 mm2/s. The material used for the inner and outer rings and rolling bodies was silicon nitride ceramic, and the powder was produced by Ube Group in Japan and formed by hot isostatic pressing sintering process. The material performance test results are shown in Table 1. The structural parameters of the test bearing are shown in Table 2.

**Figure 1.** Full ceramic ball bearing for test. (**a**) Structure diagram (**b**) Test bearing diagram.


**Table 1.** Physical properties of Si3N<sup>4</sup> for full ceramic ball bearings.


**Table 2.** Structure parameters of 6208 full ceramic ball bearings.

The test bearing cage material was carbon dairy produce, PTFE, graphite reinforced PVX-based composite material, which was produced by Ensinger Group in Berlin, Germany. The specific properties of the material are shown in Table 3. The experimental verification and application experience show that the material can be applied to the wide temperature range of −70–200 ◦C. In addition, the material has the appropriate strength and toughness to withstand a certain load and impact, small friction coefficient, and good wear resistance. It also has smaller specific gravity and similar expansion coefficient to the rolling body, which is suitable for the test project.

**Table 3.** PVX composite enhanced cage material properties.


#### *2.2. Introduction to the Test Equipment*

Figure 2 shows the JH-200E rolling bearing testing machine (Shenyang Jianzhu University, Shenyang, China) and its structure. The testing machine is a horizontal bearing performance life testing machine, which is mainly composed of bearing test chamber, axial loading system, radial loading system, test shafting, oil supply system, force sensor, temperature sensor, vibration sensor, and control system. The main test parameters were: the outer diameter of the test bearing was 30–200 mm, the inner diameter was 10–170 mm, and the width was 10–30 mm. Oil supply system flow range: 0.001–10 mL/min, and flow control accuracy was ±0.001 mL/min. The radial loading range of test bearings was 100–30,000 N, and the axial loading range was 50–10,000 N. The maximum spindle rotational speed was 30,000 r/min.

**Figure 2.** Lubrication test of full ceramic ball bearing oil based on JH-200E (Shenyang Jianzhu University, Shenyang, China).

#### *2.3. Test Scheme Design*

The oil lubrication test of the full ceramic ball bearing was carried out at room temperature. The target value is measured by changing the bearing oil supply, bearing radial load, bearing rotational speed, and other working conditions. The specific experimental process was as follows. (1) Before the bearing was started, oil was supplied to the test bearing according to the target set value. (2) Under no-load condition, the test bearing speed was increased from 0 r/min to the set value of the target speed within 3 min. (3) After the oil supply and speed of the test bearing were stable, the radial loading of the test bearing was carried out according to the set value of the target load. (4) After the test lasted for 30 min and the temperature of the test chamber and outer ring of the bearing, vibration value of the bearing, friction value, and other target measurement values were stable, the measurement and recording data were started. The test scheme is shown in Table 4.

**Table 4.** The 6208 full ceramic ball bearing oil lubrication test scheme.


#### **3. Test Results and Analysis**

*3.1. Variation Characteristics of Friction*

The JH-200E rolling bearing testing machine was used to carry out the full ceramic ball bearing oil lubrication test. Based on the friction sensor, the friction change of the full ceramic ball bearing under variable working conditions was measured, as shown in Figure 3.

**Figure 3.** Variation law of friction of full ceramic ball bearing under oil lubrication condition. (**a**) n = 5000 r/min, (**b**) n = 10,000 r/min.

It can be seen from Figure 3 that, when the rotational speed is constant with the increase of oil supply, the friction value of the full ceramic ball bearing decreases first and then increases. When the bearing speed was 5000 r/min, the friction force had a minimum inflection point value in the range of 0.2–2.0 mL/min oil supply, and the value was about 1.2 mL/min oil supply. This indicates that in this particular working condition, when the oil supply was 1.2 mL/min, the full ceramic ball bearing showed the best friction characteristics. When the oil supply was less than 1.2 mL/min, the full ceramic ball bearing was in the state of lack of oil lubrication, and the lubricating oil in the bearing contact micro-zone had not formed the state of full film lubrication, so the friction force was large. In this range, with the continuous increase of oil supply, the lubrication state improved, lubricating oil film gradually thickened, friction decreased, and change trend was very obvious. When the oil supply was greater than 1.2 mL/min, due to the large amount of lubricating oil, the excess lubricating oil produced viscous resistance to the operation of the bearing under the action of viscosity, thus leading to the gradual increase of the friction of the bearing. However, the influence of oil viscosity resistance on the change of friction was relatively small, so when the oil supply was greater than the optimal oil supply, and with the continuous increase of oil supply, the increase trend of friction was relatively moderate. By comparing Figure 3a,b, it can be seen that the oil supply at the inflection point of friction of the full ceramic ball bearings increased gradually with the increase of rotational speed. The change of load had no obvious effect on the oil supply at the inflection point of friction.

#### *3.2. Variation Characteristics of Vibration Acceleration*

The JH-200E rolling bearing testing machine was used to carry out the full ceramic ball bearing oil lubrication test, and the vibration variation of the full ceramic ball bearing outer ring under variable working conditions was measured based on the acceleration sensor, as shown in Figure 4.

**Figure 4.** Vibration law of outer ring of full ceramic ball bearing under oil lubrication condition. (**a**) *n* = 5000 r/min, (**b**) *n* = 10,000 r/min.

According to Figure 4, when the rotational speed was constant, the vibration acceleration value of the outer ring of the full ceramic ball bearing decreased from large to small and then increased with the increase of oil supply. When the rotational speed was 5000 r/min, the vibration acceleration of outer ring had a minimum inflection point value, and the corresponding oil supply was also about 1.2 mL/min. This indicates that, when the rotational speed was 5000 r/min, the full ceramic ball bearing exhibited the best friction and vibration characteristics under the condition of 1.2 mL/min oil supply, which can be determined as its optimal oil supply. When the oil supply was less than the optimal oil supply, due to the influence of lack of oil lubrication, there was oil–solid mixed lubrication in the bearing contact micro-zone; the sliding roll ratio changed more frequently, and the vibration acceleration value of the bearing outer ring increased accordingly [22–24]. With the continuous increase of oil supply, the lubrication state improved; the vibration value of the bearing's outer ring decreased, and the change trend was very obvious. When the oil supply was greater than the optimal oil supply, the amount of lubricating oil was large, and the excess lubricating oil produced viscous resistance to the operation of the bearing under the action of viscosity, thus leading to a gradual increase in the vibration value of the bearing's outer ring, this phenomenon could be seen in Figure 4. However, the influence of oil viscosity resistance on the vibration change of the bearing's outer ring was relatively small, so when the oil supply was greater than the optimal oil supply, and with the continuous increase of oil supply, the increase trend of vibration acceleration value was relatively moderate [25,26]. In addition, when the radial load of the bearing increased, the vibration acceleration value of the bearing increased accordingly.

#### *3.3. Variation Characteristics of Outer Ring Temperature Rise*

The JH-200E rolling bearing testing machine was used to carry out the oil lubrication test of the full ceramic ball bearing, and the temperature rise change of the outer ring of the full ceramic ball bearing is shown in Figure 5, based on the temperature sensor measured under variable working conditions.

**Figure 5.** Temperature law of outer ring of full ceramic ball bearing under oil lubrication condition. (**a**) *n* = 5000 r/min, (**b**) *n* = 10,000 r/min.

According to the test data in Figure 5, with the increase of oil supply, the temperature rise of the outer ring gradually decreased. When the oil supply was lower than the optimal range, the temperature rise decreased obviously with the increase of oil supply. This is because, in the state of lack of oil lubrication, the lubrication effect of bearings is not good. The running state was mixed friction, and dry friction in some areas led to a significant rise in temperature. As the oil supply continued to increase, the lubrication state improved, and the temperature rise of bearing outer ring decreased. When the optimum oil supply was in place, the full ceramic ball bearing was in a state of full-film lubrication, which had a great improvement effect on the temperature rise of the bearing. When the oil supply continued to increase and was greater than the optimal oil supply, a large amount of lubricating oil removed the temperature rise generated by the operation of the bearing and caused a cooling effect, so the temperature rise of the bearing outer ring continued to decrease. The temperature rise of outer ring increased with the increase of bearing speed; it also increased with the increase of radial load. This is because, with the increase of the radial load and speed, the contact stress and contact frequency in the contact micro-zone of the full ceramic ball bearing became larger, and the heat generated by friction increased. The heat gradually accumulated in the outer ring of the bearing, and the temperature of the outer ring rose correspondingly.

#### **4. Theoretical Calculation and Analysis**

#### *4.1. Oil Lubrication Dynamics Model of the Full Ceramic Ball Bearing*

4.1.1. Establishment of Coordinate System of the Full Ceramic Ball Bearing

In the operation process of full ceramic ball bearings, the ceramic ball has the most complex force and contact with the inner and outer rings and cage, thus resulting in friction and impact. In addition, under the action of oil–gas lubrication, the ceramic ball was also affected by hydraulic force, which caused the ceramic ball to have a more complicated motion state [27,28]. In order to accurately describe the motion characteristics and interaction forces of the internal parts of the full ceramic ball bearing, a coordinate system was established, as shown in Figure 6.

*ρ*

*ξ*

*ω*

**Figure 6.** Coordinate system of deep groove ball bearing. *ψ*


*α η ρ ξ η β η* The force between the balls, rings, and cage is shown in Figure 7, in which the subscripts *I* and *e* represent the inner and outer rings, respectively, *j* represents the *Jth* ball, *Q* represents the normal contact force between the ball and the ring raceway, *T<sup>η</sup>* and *T<sup>ξ</sup>* are the drag forces on the contact surface between the ball and the ring raceway, *Qc* represents the force of the cage pocket on the ball, and the angle between the cage pocket hole and the three directions of *Opxpypz<sup>p</sup>* in the coordinate system, *βx*, *βy*, *βz*. *PRη*(*ξ*) , and *PSη*(*ξ*) , are the rolling friction force and sliding friction force of the fluid at the entrance of the contact surface between the ball and cage pocket, respectively. *η ξ βx βy βz Rη <sup>ξ</sup> Sη <sup>ξ</sup>*

**Figure 7.** Mechanical model of full ceramic deep groove ball bearing. (**a**) Ball-ring contact force model, (**b**) Ball-cage contact force model.

4.1.2. Elastohydrodynamic Model of Full Ceramic Ball Bearing under Oil Lubrication

Under the action of oil–gas lubrication, each rolling body had two velocity directions of contact ellipse, i.e., the long and short axes, and the density and viscosity of lubrication medium changed along the direction of lubrication film thickness. Its steady-state generalized Reynolds lubrication equation can be written as follows:

$$\begin{cases} \frac{\partial}{\partial \mathbf{x}\_{\varepsilon}} \left[ \left( \frac{\rho}{\eta} \right) h^{3} \frac{\partial p}{\partial \mathbf{x}\_{\varepsilon}} \right] + \frac{\partial}{\partial y\_{\varepsilon}} \left[ \left( \frac{\rho}{\eta} \right) h^{3} \frac{\partial p}{\partial y\_{\varepsilon}} \right] = 12U \frac{\partial (\rho\_{\varepsilon}^{\*} h)}{\partial \mathbf{x}\_{\varepsilon}} + 12V \frac{\partial (\rho\_{\eta}^{\*} h)}{\partial y\_{\varepsilon}}\\\ U = \frac{1}{2} (u\_{1} + u\_{2}) \\\ V = \frac{1}{2} (v\_{1} + v\_{2}) \end{cases} \tag{1}$$

where *h* is the oil film thickness, *p* is oil film pressure distribution, and *U* and *V* are the entrainment velocity in *xc* and *yc* directions, respectively (the coiling speed in *yc* direction is much higher than that in *xc* direction), which is related to the actual contact speed between the inner and outer ring and the rolling bodies. *ρ* and *η* are density and viscosity coefficients, respectively, and *ρx* \* and *ρ<sup>y</sup>* \* are the equivalent densities related to the equivalent viscosity of contact extrusion.

The lubrication film thickness *h* considering elastic deformation can be expressed as:

$$h = h\_0 + \frac{\mathbf{x}\_c^2}{2R\_X} + \frac{y\_c^2}{2R\_Y} + \delta(\mathbf{x}\_c, y\_c) \tag{2}$$

where *h<sup>0</sup>* is the central film thickness of the elastic contact area, and *R<sup>x</sup>* and *R<sup>y</sup>* are the equivalent radius of curvature in the *xc* and *yc* directions, respectively. *δ*(*xc*, *yc*) is the elastic deformation in the contact region, and its expression is:

$$\delta(\mathbf{x}\_{\varepsilon}, y\_{\varepsilon}) = \frac{2}{\pi E} \iint \frac{p(\mathbf{s}, t)}{\sqrt{\left(\mathbf{x}\_{\varepsilon} - \mathbf{s}\right)^{2} + \left(y\_{\varepsilon} - t\right)^{2}}} \text{dsdt} \tag{3}$$

where *E* is the equivalent elastic model of two contact objects *x*.

#### 4.1.3. Friction Torque

The rolling friction torque *M<sup>E</sup>* generated when the ball rolls on the raceway was:

$$M\_E = 0.25 D\_{pw} \left[ \left( 1 - \gamma\_i^2 \right) \sum\_{j=1}^{z} \Phi\_{ij} + \left( 1 - \gamma\_e^2 \right) \sum\_{j=1}^{z} \Phi\_{ej} \right] \beta\_a \tag{4}$$

$$\gamma\_{\rm i(\varepsilon)} = D\_{\rm w} \cos \alpha\_{\rm i(\varepsilon)} / D\_{\rm pw} \tag{5}$$

where *β<sup>a</sup>* is the elastic hysteresis coefficient. Φ can be calculated by referring to reference. The friction torque *M*<sup>D</sup> caused by differential sliding was:

$$M\_{\rm D} = \frac{D\_{\rm pw}}{2D\_{\rm w}} \left[ \left( 1 - \gamma\_{\rm i}^2 \right) \sum\_{j=1}^{Z} M\_{\rm Dij} + \left( 1 - \gamma\_{\rm e}^2 \right) \sum\_{j=1}^{Z} M\_{\rm Dej} \right] f\_{\rm s} \tag{6}$$

where *f* s is the sliding friction factor between the ball and the raceway. The friction moment *M*<sup>s</sup> caused by the spin sliding of the ball was:

$$M\_{\rm s} = \frac{3}{8} f\_{\rm s} \left[ \sum\_{j=1}^{Z} \left( E\_{\rm W} E\_{\rm i} a\_{\rm i} Q\_{\rm ij} \sin a\_{\rm ij} \right) + \sum\_{j=1}^{Z} \left( E\_{\rm W} E\_{\rm e} a\_{\rm e} Q\_{\rm ej} \sin a\_{\rm ej} \right) \right] \tag{7}$$

where *E*<sup>w</sup> is the elastic modulus of the ball material. *E*<sup>i</sup> and *E*<sup>e</sup> are the elastic moduli of inner and outer ring materials, respectively. *a*<sup>i</sup> and *a<sup>e</sup>* are the long half axes of the contact ellipse of the inner and outer rings and the balls, respectively.

The friction torque *M*<sup>c</sup> caused by the friction between the ball and the cage was:

$$M\_{\rm C} = 0.25 D\_{\rm pw} \left( 1 - \gamma^2 \right) \times \sin \left( \varkappa\_0 + \arctan \frac{D\_{\rm w} \sin \varkappa\_0}{2 \gamma\_1} \right) m\_d \mu\_c \tag{8}$$

where *M*<sup>c</sup> is the cage mass and *µ<sup>c</sup>* is the sliding friction coefficient between the ball and cage.

The friction torque *M<sup>l</sup>* caused by the viscous resistance of lubricating oil in the running process of the bearing was:

$$M\_{l} = 6.53 \text{a}^{-1} \text{S}\_{1} D\_{pw} \times \left\{ 2 \sum\_{j=1}^{Z} \left[ \frac{h\_{ij} + h\_{ej}}{2} (a\_{ij} + a\_{ej}) \right] \text{S}\_{2} \right\}^{-1} \tag{9}$$

where *S*<sup>1</sup> is the sufficient lubrication coefficient, and the lubrication coefficient of oil film was taken. *h* is the oil film thickness in the center of the bearing contact area. *S*<sup>2</sup> is the side leakage coefficient of lubrication, and the value was 1 in the calculation.

In the process of rotational service, the total friction torque *M* of the bearing was:

$$M = M\_E + M\_D + M\_s + M\_c + M\_l \tag{10}$$

4.1.4. Influence of Temperature Rise on Structural Parameters of the Full Ceramic Ball Bearing under Oil Lubrication Condition

Due to the inconsistent deformation of bearing inner and outer rings and ball under the condition of temperature rise change, the clearance of deep groove ball bearings will change, and the specific calculation formula is as follows:

$$
\Delta p = \Delta p\_i - \Delta p\_o + 2\Delta p\_r \tag{11}
$$

$$\mathbf{C}\_{r} = \mathbf{C}\_{0} + \Delta p \tag{12}$$

where ∆*p* is the change of bearing clearance. ∆*p<sup>o</sup>* is the deformation of outer ring. ∆*p<sup>i</sup>* is the deformation of inner ring. ∆*p<sup>r</sup>* is the deformation amount of ceramic sphere. *C*<sup>0</sup> is the initial clearance of bearing. *Cr* is bearing clearance.

The deformation of bearing ring and cage affected by temperature change can be expressed as:

$$
\Delta \mu = \Gamma\_{\text{S}} D\_{\text{C}} \Delta T \tag{13}
$$

where ∆*u* refers to the inner and outer diameters of bearing ring and cage. Γ is the expansion coefficient of the corresponding material. *D<sup>c</sup>* represents the bearing inner and outer rings, as well as the cage's inner and outer diameter sizes. ∆*T* is the temperature difference.

#### 4.1.5. Boundary Conditions

In order to ensure the convergence and accuracy of the contact elastohydrodynamic model of the full ceramic deep groove ball bearing, the value range of boundary coordinates was *x*in = 2*a*, *x*out = 2*a*, *y*in = 3.5*b*, *y*out = 1.5*b*. Thus, the *x<sup>c</sup>* and *y<sup>c</sup>* directions to solve the area was defined as: {(*xc*,*yc*)|−2*a* ≤ *x<sup>c</sup>* ≤ 2*a*, −3.5*b* ≤ *y<sup>c</sup>* ≤ 1.5*b*} or less or less. In this paper, the region is divided into 50 × 50 grids in two directions.

Boundary conditions of Reynolds lubrication equation: boundary pressure was 0. The pressure of the whole bearing area was greater than or equal to 0. Considering the type of lubricating oil was isothermal solution, Thermo elastohydrodynamic lubrication(TEHL) method should be used to solve the problem. The pressure gradient in the oil film rupture zone was 0, as follows:

$$\begin{cases} p(\mathbf{x}\_{\text{in}}, y\_c) = p(\mathbf{x}\_{\text{out}}, y\_c) = 0 \\ p(\mathbf{x}\_c, y\_{\text{in}}) = p(\mathbf{x}\_c, y\_{\text{out}}) = 0 \\ p(\mathbf{x}\_c, y\_c) \ge 0 (\mathbf{x}\_{\text{in}} < \mathbf{x}\_c < \mathbf{x}\_{\text{out}}, y\_{\text{in}} < y\_c < y\_{\text{out}}) \\ \frac{\partial p(\mathbf{x}\_{\text{out}}, y\_c)}{\partial \mathbf{x}\_c} = \frac{\partial p(\mathbf{x}\_c, y\_{\text{out}})}{\partial y\_c} = 0 \end{cases} \tag{14}$$

#### *4.2. Numerical Solution Process*

Figure 8 shows the data coupling and solving process of the above equations.

( ) ( )

≥ << <<

( ) ( ) ( ) ( )

= =

= =

**Figure 8.** Flow chart of elastohydrodynamic coupling calculation of full ceramic ball bearings.

#### *4.3. Calculation Results and Analysis*

In 6208CE silicon nitride ceramic ball bearings, and combining the elastohydrodynamic lubrication equation shown in the Figure 2 flow calculation program, the test process to get the best oil for lubrication conditions, performing lubrication, and contact characteristics analysis were concluded, and the ceramic ball bearing lubrication under the condition of different speed and load of oil film thickness distribution of the simulation results are shown in Figure 9.

By comparing the simulation results in Figure 9a–f, it can be seen that the minimum liquid film thickness in the contact area of the rolling body decreased gradually with the increase of the angle *ψ* from the distance to the minimum stress point of the rolling body. This is because the squeezing effect of liquid film decreases with the increase of *ψ*. The minimum liquid film thickness of the rolling body at the same position increased with the increase of rotational speed and load. This was caused by the enhancement of dynamic pressure effect of oil film with the increase of rotational speed. Furthermore, the influence of rotational speed on the minimum liquid film thickness at different locations was similar, and the influence of rotational speed on the liquid film at different locations had no obvious change.

Under the condition of optimal oil supply lubrication for the full ceramic ball bearings, the pressure distribution of lubricating oil film of different rolling bodies under different rotational speeds and loads is shown in Figure 10.

*ψ ψ ψ ψ ψ ψ*  **Figure 9.** Oil film thickness distribution of ceramic ball bearings under different working conditions. (**a**) *ψ* = 0◦ , *F* = 3000 N, *n* = 10,000 r/min, (**b**) *ψ* = 0◦ , *F* = 900 N, *n* = 5000 r/min, (**c**) *ψ* = 80◦ , *F* = 3000 N, *n* = 10,000 r/min, (**d**) *ψ* = 80◦ , *F* = 900 N, *n* = 5000 r/min, (**e**) *ψ* = 160◦ , *F* = 3000 N, *n* = 10,000 r/min, (**f**) *ψ* = 160◦ , *F* = 900 N, *n* = 5000 r/min.

*ψ* By comparing the simulation results of Figure 10a–f, it can be seen that the maximum pressure of the rolling body increased with the increment of *ψ*. This is because the maximum Hertz contact pressure of the rolling body increased with the increase of *ψ*. It can be seen from the figure that the maximum pressure of lubrication film decreased with the decrease of load contact pressure for different rolling bodies. The maximum oil film pressure corresponding to the rolling body at the same position increased with the increase of rotational speed and load. This was due to the enhanced dynamic pressure effect of oil film, caused by the increase of rotational speed. At the same speed, the angle between the rolling body and maximum stress position of the rolling body at different positions of the same bearing were larger, and the corresponding load was smaller, so the maximum oil film pressure caused by it also decreases correspondingly.

*ψ*

) ψ = 0°, F = 3000N, n = 10,000 r/min, ( *ψ ψ ψ ψ ψ*  **Figure 10.** Oil film pressure distribution of ceramic ball bearings under different working conditions. (**a**) ψ = 0◦ , F = 3000N, n = 10,000 r/min, (**b**) *ψ* = 0◦ , *F* = 900 N, *n* = 5000 r/min, (**c**) *ψ* = 80◦ , *F* = 3000 N, *n* = 10,000 r/min, (**d**) *ψ* = 80◦ , *F* = 900 N, *n* = 5000 r/min, (**e**) *ψ* = 160◦ , *F* = 3000 N, *n* = 10,000 r/min, (**f**) *ψ* = 160◦ , *F* = 900 N, *n* = 5000 r/min.

*ψ* By comparing the simulation and experimental results, it can be found that the thickness and pressure of the bearing oil film in the simulation model changed with the change of working conditions, and the change trend had a good consistency in the experimental results. In the case of large lubricating oil film thickness, the bearings showed excellent characteristics in the experiment. The reliability and correctness of the experimental data were verified.

*ψ*

#### **5. Morphological Characteristics and Microstructure Properties of Contact Micro-Zone of Full Ceramic Ball Bearings under Oil Lubrication**

*5.1. Test Analysis of Full Ceramic Ball Bearings*

The full ceramic ball bearings, tested under different lubrication conditions and working conditions, were tested and analyzed. Among them, the bearing in the figure below was tested under the condition of 900 N load, 5000 rpm, and 0.2 mL/min oil supply. After disassembling the tested bearing assembly, no obvious failure was found in the observation, as shown in Figure 11.

**Figure 11.** Removed silicon nitride full ceramic ball bearing and its components.

However, in Figure 11, the surface of the PVX cage side beam (the outer circular surface of the cage) showed obvious scratching and rubbing areas. According to the scratches, it can be judged that the surface of the outer ring and inner circle contact and collide when the bearing rotates. The reason for this phenomenon is that the cage was made of composite materials. According to the material properties in Tables 1 and 3, the thermal expansion coefficient of the cage was greater than that of silicon nitride. In the service process, due to the influence of temperature rise, the cage deformation was relatively large, and the cage was guided by the outer diameter. In this guidance mode, the coupling effects of temperature rise, load, and impact made the cage produce elliptic deformation, which then caused friction with the outer ring and inner circle, thus resulting in the cage's rub. In addition, PVX abrasive chips have certain lubricity, which helps to lubricate bearings in service conditions, to some extent.

#### *5.2. Full Ceramic Ball Bearing Contact Area Surface and Surface Quality Testing*

The contact area between the rolling body and the raceway was observed, as shown in Figure 12.

**Figure 12.** Outer ring raceway and rolling bodies of full ceramic ball bearing after the test. (**a**) Bearing outer ring raceway, (**b**) Rolling bodies.

It can be seen from Figure 12 that there was no damage phenomenon on the surface of the contact area of the full ceramic ball bearing after the test. However, black film appeared in some areas of outer ring raceway and ball bearing surfaces. In order to further reveal the chemical composition of the film, the mechanism of the film generation and its influence on the surface friction and wear quality of the bearing contact micro-zone were analyzed from the microscopic point of view. We performed non-destructive cutting of test bearing ring and cage. They were tested together with ceramic balls by SEM, XRD, and other instruments.

Hitachi S-4800 scanning electron microscope (SEM), which was made by Hitachi in Japan, was used to detect the raceway and black film covered area on ball surface after the test in Figure 12, as shown in Figures 13 and 14. As can be seen from scanning electron microscopy, the black film area was the covered area in Figures 13 and 14. This indicates that a thin film was formed on the surface of the contact micro-zone between the raceway and ball bearings under the coupling effects of temperature rise, heavy load, sliding effect, and cage collision. By comparing the pictures in Figures 12–14, it can be seen that both the macro- and micro-observation results show that the film formed on the surface of outer ring raceway had a better and more significant effect. Meanwhile, observe the removed bearing inner ring in Figure 11—the black film was not visible on its raceway surface.

**Figure 13.** SEM imaging of full ceramic ball bearing after the test. (**a**) Measuring point 1, (**b**) Measuring point 2.

**Figure 14.** SEM imaging of outer raceway of full ceramic ball bearing after the test. (**a**) Measuring point 1, (**b**) Measuring point 2.

The preliminary analysis of the reasons for the above phenomenon was that, when the full ceramic ball bearing ran in oil lubrication conditions, the debris on the outer ring of the cage and surface of the pockets were involved in the bearing contact micro area under the action of bearing rotation.

#### *5.3. Chemical Composition and Qualitative Analysis of Surface Layer of Bearing Contact Area*

The black film and cage on the outer racing surface of ceramic ball bearing were analyzed by Raman spectroscopy. In order to ensure the accuracy of the test, the components of the cage and raceway surface film were measured and analyzed twice, and the test results are shown in Figure 15.

**Figure 15.** Raman spectra peaks of black film between cage and raceway after the test. 1–3. correspond to 3 characteristic peaks, respectively.

By comparing the four groups of measured curves in the figure, it can be seen that the spectral distribution trend of the film on the surface of the bearing cage and raceway were different after the test, that is, the 1/2 distribution trend of the curve is different from the 3/4 distribution trend of the curve. However, by comparing the three characteristic peaks of the four curves, it was found that the Raman wavelengths of the characteristic scattering peaks were almost the same. This indicates that the crystal interface between the cage and raceway surface film material was basically the same; we can preliminarily judge that the chemical compositions of the two groups of substances were similar.

XRD analysis was carried out on the powder of the outer ring raceway of the bearing and film layer on the surface of the ball and cage powder. The test results were shown in Figure 16.

**Figure 16.** XRD spectrogram of cage and raceway black film. 1–6. correspond to 6 characteristic peaks, respectively.

The XRD results show that the phase diffraction peaks of the bearing cage and raceway film were basically the same before and after the test. Combined with the above analysis, it was preliminarily shown that PVX did not generate new substances by chemical reaction with the medium in the environment under the service conditions of extreme low variable temperature conditions. In Figure 15, the reason the distribution trend of curve 1/2 was different from that of curve 3/4 was that the film components formed by the retener powder under rolling action contain a very small amount of worn silicon nitride and sintering agent materials.

By comparing the above analysis results, it can be inferred that the powder formed by the cage under the action of physical conditions became the lubricating medium of the full ceramic ball bearing oil, which played a promoting role in the service of full ceramic ball bearings.

#### **6. Conclusions**


**Author Contributions:** Data curation, Z.B.; resources, Y.W.; software, Z.X., J.T. and L.G.; writing original draft, J.Y. (Jinmei Yao); writing—review and editing, J.Y. (Jiaxing Yang) and J.S. All authors have read and agreed to the published version of the manuscript.

**Funding:** The authors acknowledge the collective support granted by the National Natural Science Foundation of China (grant No. 52105196), Department of Science and Technology of Liaoning Province (grant No. 2020-BS-159), and Young and Middle-aged Innovation Team of Shenyang (grant No. RC210343).

**Data Availability Statement:** Not applicable.

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

#### **References**


**Yeteng Li <sup>1</sup> , Wenchao Li 1,2 , Yongsheng Zhu 1, \* , Gaobo He 1 , Shuaijun Ma <sup>1</sup> and Jun Hong 1**


<sup>2</sup> Luoyang Bearing Research Institute Co., Ltd., Luoyang 471039, China **\*** Correspondence: yszhu@mail.xjtu.edu.cn; Tel.: +86-13991149360

**Abstract:** Due to the special structure of double-half inner rings, four-point contact ball bearings are prone to uneven forces in the inner raceway during movement, which affects the dynamic performance of the rolling element and cage, and even leads to cage sliding. Dynamic performance of the cage is an important factor affecting the working stability of bearings. In this paper, in order to grasp the operation law of the cage so as to guide the application of four-point contact ball bearings, the dynamic model of four-point contact ball bearings is established by the secondary development of Automatic Dynamic Analysis of Mechanical Systems (ADAMS). The dynamic performance of the cage is analyzed and evaluated with the indexes of vortex radius ratio and vortex velocity deviation ratio of the cage centroid trajectory. The results show the following: the cage stability increases and then decreases to a certain degree with rotating speed-rise; it increases and then decreases with the increase in the pure axial load; under a combination of axial and radial load, the cage moves more smoothly with smaller radial force. Rotating speed has little effect on cage stability, while radial force has a great influence on cage stability, followed by axial load. In order to verify the simulation results, a test bench for rolling bearing cages is developed, and the accuracy of the simulation results is verified by the test results.

**Keywords:** load distribution; four-point contact ball bearing; bearing dynamics; cage motions experiment

#### **1. Introduction**

As an important part of the high-speed railways, the bogie plays a role in connecting the body and the track, thus its motion stability affects the safety and comfort of the train [1,2]. Figure 1 shows a bogie. The four-point contact ball bearing (FPCBB) is one of the important components of the bogie, which could support axial loads in both directions due to its double-half structure. It is usually used in conjunction with cylindrical roller bearings for locating and supporting. Compared to a traditional paired configuration angular contact ball bearing, FCPBB requires less coaxiality and space between bore and shaft. However, in the motion process, the force between the rolling elements and the double-half raceway will change, inevitably making a four-point contact, three-point contact or two-point contact inside the bearing, which may lead to cage instability. In case of serious instability, it will even cause abnormal heat generation and the early failure of the bearing. Therefore, investigation of its dynamic characteristics is desired.

Research on FPCBB focuses more on wind turbines, aero-engines and robots joints, and most of them are static analysis. For example, Zhang [3] has analyzed the influence of radial load and overturning moment on the load distribution of FPCBB; Li [4] has analyzed the influence of positive and negative clearance on the load distribution of FPCBB; Li and Tang [5] have studied the influence of different parameters on the load–displacement relationship theoretically by establishing the geometric coordination equation of FPCBB.

**Citation:** Li, Y.; Li, W.; Zhu, Y.; He, G.; Ma, S.; Hong, J. Dynamic Performance Analysis of Cage in Four-Point Contact Ball Bearing. *Lubricants* **2022**, *10*, 149. https:// doi.org/10.3390/lubricants10070149

Received: 14 April 2022 Accepted: 29 June 2022 Published: 11 July 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/).

In the research on the dynamic characteristics of FPCBB, researchers mostly pay more attention to the wind turbine bearings that mainly bear radial load and overturning torque, as well as the thin-walled FPCBB of industrial robots that prefer structural optimization and lightweight design. Wu [6] has analyzed the influence of different speed and preload on the dynamic characteristics of FPCBB of wind turbine generator; Shi [7] has studied the influence of different structural parameters on the dynamic characteristics of four point contact ball bearing under the combined action of axial load and overturning moment; Yang [8] has studied the precise elliptical contact area shape and contact state of thinwalled FPCBB based on the finite element method; Liu [9] has explored the optimization of structural parameters of thin-walled FPCBB from the aspects of contact force between ball and cage and its influence on axial vibration intensity of cage centroid by multi-body dynamics and Hertz contact theory. These studies provide great help for improving the service performance and the efficiency of bearing design.

**Figure 1.** The bogie of high-speed railway train.

In terms of dynamic analysis of bearings, the establishment of motion differential equations was started by Walters [10]. His model with a 4 degrees of freedom (4-DOF) ball and a 6-DOF cage laid the foundation for dynamic analysis of ball bearing. Based on the study of Walters, Gupta [11–13] studied the complex motion and contact state between the rollers and raceway, and comprehensively analyzed the steady and transient dynamic characteristics of the bearing. Hagiu [14] programmed the calculation of the interaction between bearing components, but the model did not take into account the lubrication. Wijnant [15] constructed the ball bearing dynamic model of considering lubrication film, but the calculation accuracy is dependent on the step size and is difficult to guarantee. Weinzapel [16] established a flexible cage and obtained the rigid–flexible coupling model in order to be closer to the actual working conditions, which makes the calculation more complex. In addition, some scholars [17–20] applied the finite element method to analyze contact characteristics of FPCBB, but due to the complex meshing, these studies are mostly limited to the analysis of large bearings under steady low speed conditions.

Furthermore, some researchers used the commercial software of multi-body dynamics to numerically analyze dynamic characteristics of bearings. For example, Ji [21] used ADAMS to analyze the dynamic characteristics of the instantaneous response of the cage, and studied cage centroid trajectory characteristics under different rotating speeds, loads, radial clearances and the number of steel balls. Hong [22] and Chan [23] also used ADAMS to establish the bearing dynamic model, so as to carry out corresponding researches. Hou [24] built a rigid–flexible hybrid dynamic model of the bearing based on ANSYS and ADAMS, and obtained the motion law of rigid components and the influence of vibration. However, there are relatively few analyses on the dynamic performance for bogie fourpoint contact ball bearings, a load condition in which axial load is dominant and radial load

is supplementary. Wu [6] analyzed the bearing dynamic feature under main radial load. However, it did not put forward the corresponding evaluation index, and only analyzed the motion characteristics of the bearings under different working conditions. Zhang [25] analyzed the influence of cage structure shape on bearing dynamic characteristics under combined loads by ADAMS. Based on Hertz contact theory, Zhao [26] used ADAMS to establish a bearing dynamic model considering gear meshing, axial load, speed and overturning torque on bearing vibration and contact force.

As mentioned above, no clear evaluation indicators were given for the motion stability analysis of four-point contact ball bearings. The common method for bearing dynamic analysis is based on the multi-body dynamics commercial software, which is greatly affected by the limitation of the software itself and can only consider limited factors. In order to discuss the operation stability of bearings, a general methodology for dynamic performance analysis of cage will be proposed in this work. A dynamic model of FPCBB will be established based on ADAMS secondary development, and the cage centroid trajectory will be discussed under the conditions of radial load and axial load in the following sections. The variation of cage motion stability is described by vortex velocity deviation ratio and vortex radius ratio, the latter referring to the ratio of the difference between the maximum and minimum vortex radius to the average vortex radius. Finally, a test bench will be developed to collect the real speed and cage motion trajectory at different rotating speeds and load conditions to validate the simulation results.

#### **2. Bearing Dynamic Model**

#### *2.1. Solid Model Creation*

The bearing model is QJ215, which is generally used in the high-speed railway bogie. The structure is shown in Figure 2. The bearing inner rings, outer ring and rolling elements are made of bearing steel, and the cage is made of brass. Specific size and material parameters are shown in Table 1. In this paper, the parametric design method is used to model the FPCBB in ADAMS/View by macro commands and CMD program [27,28]. The 3D model, as shown in Figure 3, can be obtained by inputting corresponding structural parameters.

**Figure 2.** The structure of QJ215. (**a**) A physical picture of QJ215; (**b**) A dimension drawing of QJ215.


**Table 1.** Dimensional parameters and material parameters of the four-point contact ball bearing.

**Figure 3.** 3D model of QJ215.

#### *2.2. Bearing Dynamics Model Construction Based on GFOSUB*

The contact force between bearing components is greatly affected by the recognition accuracy of ADAMS solver on the solid model, and the results are prone to problems such as large values and burrs [29]. For more realistic results, the bearing contact force can be solved directly by GFOSUB subroutine, which can bypass the contact stiffness and ADAMS solver, reducing the negative effects of contact stiffness and other parameters. The process of building a bearing dynamics model based on Gforce Subroutine (GFOSUB) is shown in Figure 4.

**Figure 4.** The process of building the bearing dynamics model based on GFOSUB.

Firstly, the geometric model of FPCBB is established by structural parameters, and boundary conditions such as materials, constraints and loads are set to obtain the state information of bearing components, including displacement, speed and acceleration, etc. Meanwhile, the GFOSUB bypasses the Adams contact solver to complete the calculation of force, finally the output results are solved by the differential equation solver. The command box of GFOSUB is shown in Figure 5. The parameters in the red box are ball diameter, groove curvature radius of outer raceway, outer raceway groove bottom diameter, inner raceway groove bottom diameter, dynamic friction coefficient, static friction coefficient, outer ring outer diameter, mass marker point of the inner ring center and mass marker point of the outer ring center. The yellow box is the name of the .dll file where GFOSUB is located. GFOSUB can be called by entering the corresponding parameters.


**Figure 5.** The command box of GFOSUB.

#### *2.3. Boundary Condition Settings*

Boundary condition settings mainly include adding freedom constraint, load, driving speed, contact force, etc.

1. Addition of freedom constraints;

In the unconstrained state, the bearing has six degrees of freedom for translation and rotation in X, Y and Z directions. In the actual working process, the outer ring is in a fixed state with 0 degrees of freedom, so it is necessary to add the corresponding restraint sub to restrain its motion. The inner ring has 4 degrees of freedom, i.e., translational movement in X, Y and Z directions and axial rotation: *δx*, *δy*, *δ<sup>z</sup>* and *θy*; the rolling elements and the cage retain 6 degrees of freedom: *δx*, *δy*, *δz*, *θx*, *θy* and *θz*. The above freedom relationship is shown in Figure 6.

2. Addition of load and driving speed;

For the FPCBB model, it is only subjected to axial and radial loads in the working process, and the load acts on the center of the inner ring. In Figure 6, *F*a is the axial load, *F*r is the radial load. In addition, the driving speed is set in the direction of the axial rotation of the inner ring, and according to the design requirements, the rated speed of the bearing is 5988 r/min.

#### 3. Addition of contact force.

The interaction between the components of the bearing is mainly transmitted according to the contact force. FPCBB has a wide range of contact elements; according to the contact elements, they can be divided into rolling elements and inner rings, rolling elements and outer rings, rolling elements and cages, cages and guide rings, etc. When using ADAMS for dynamic simulation, the contact parameters between the two elements mainly include stiffness, damping and related friction coefficient.

For the FPCBB, the contact model between the rolling elements and the inner and outer rings is point contact, and the bearing contact stiffness *K*<sup>j</sup> (j = i,o) can be calculated by empirical formula [30], such as Equation (1):

$$K\_{\circ} = 2.15 \times 10^{5} \left(\sum \rho\right)^{-1/2} \left(n\_{\delta}\right)^{-3/2} \tag{1}$$

where ∑ *ρ* is the curvature of the contact point, and *n<sup>δ</sup>* is the contact deformation coefficient of elastomer.

The calculation of ∑ *ρ*<sup>i</sup> and ∑ *ρ*<sup>o</sup> is as follows:

$$
\sum \rho\_{\mathbf{i}} = \frac{4}{D\_w} + \frac{2}{d\_m - D\_w} - \frac{1}{r\_{\mathbf{i}}} \tag{2}
$$

$$
\sum \rho\_{\text{o}} = \frac{4}{D\_{\text{w}}} - \frac{2}{d\_{\text{m}} - D\_{\text{w}}} - \frac{1}{r\_{\text{o}}} \tag{3}
$$

where *dm* is the equivalent diameter of bearing, which can be calculated by Equation (4):

$$d\_m = (d+D)/2\tag{4}$$

Replace the bearing parameters to obtain the value of *K*<sup>j</sup> : *<sup>K</sup>*<sup>i</sup> = 1.25 <sup>×</sup> <sup>10</sup><sup>6</sup> , *<sup>K</sup>*<sup>o</sup> = 1.1 <sup>×</sup> <sup>10</sup><sup>6</sup> . In ADAMS, the bearing contact damping is taken as 0.1~0.01% times the contact stiffness and the friction coefficient is about 0.1 [31].

At this point, the dynamics model of FPCBB is completed in ADAMS.

#### **3. Verification of FPCBB Model**

For the created bearing model, it is necessary to verify its accuracy to improve the reliability of the simulation results. In this paper, the correctness of the dynamics model is judged by the theoretical calculation results as well as the bearing load distribution curve under the static simulation and dynamics simulation.

Based on the above-mentioned parametric model, an axial load of 2000 N, a radial load of 500 N and a rotating angular speed of 1000 rpm are applied to the center of inner rings. The dynamic simulation of the bearing is implemented in ADAMS software, the velocities and dynamic contact forces of the bearing are shown in Figure 7.

**Figure 7.** Velocities and dynamic contact forces of QJ215. (**a**) Rotating speed of the cage; (**b**) Rotating speed of the ball; (**c**) Contact force between ball and inner ring; (**d**) Contact force between ball and outer ring.

According to empirical formula, the bearing cage speed *n*<sup>c</sup> is

$$n\_{\mathbb{C}} = \frac{1}{2} [n\_i(1 - \gamma) + n\_o(1 + \gamma)] \tag{5}$$

where *n<sup>i</sup>* is the rotating speed of inner rings and *no* is outer rings speed.

$$\gamma = \frac{D\_w \cos \alpha}{d\_m} \tag{6}$$

where *α* is ball connect angle.

It is obtained that the cage speed is 430.21 r/min while the simulation value is 432.92 r/min, and the difference is 0.63%, so the simulation results can be considered reasonable.

For ball bearings subjected to combined axial and radial loads, due to the influence of centrifugal force and gyro torque, the indirect contact angle between the rolling element and raceway at different phase angles is different, and the contact load between the rolling element and the inner and outer rings is also different. If the applied radial load is recorded as *Fr*, axial load is *Fa*, and the included angle between adjacent rolling elements is *φ*, the rolling element at *φ* = 0 will bear the maximum load *Q*max [32].

The maximum contact load of ball bearings under combined load is:

$$Q\_{\text{max}} = \frac{Fr}{Zl\_{\text{f}}(\varepsilon)\cos\mathfrak{a}}\tag{7}$$

where *Jr*(*ε*) can be obtained by referring to Table 7.4 of the reference [33].

The load distribution function at different phase angles is:

$$Q\_{\phi i} = Q\_{\text{max}} \left[ 1 - \frac{1}{2\varepsilon} (1 - \cos \phi) \right]^{1.5} \tag{8}$$

Applying radial and axial forces of 500 N to bearing, the result calculated by empirical formulas is shown in Figure 2.

The bearing static load distribution curve reflects the bearing load distribution in the stationary or ultra-low speed conditions of the bearings. For bearings running at medium and high speeds, collisions are generated between the components during the motion, resulting in instantaneous increase in load values between the rolling element and raceway or cage. Considering that the load curve in dynamic analysis is more confusing, the obtained dynamic load curve needs to be filtered to eliminate the burr of the signal. Static simulation and dynamic simulation were carried out by applying the same load conditions to the bearing model in ADAMS, and the load magnitudes of the rolling element under different phase angles were obtained. Accordingly, the static load distribution curve and dynamic load distribution curve are plotted and the results are shown in Figure 8.

**Figure 8.** Comparison of dynamic simulated, static simulated and theoretical values of combined load.

According to Figure 8, the dynamic load curve is roughly similar to the static simulation results and the load curve profile obtained from theoretical calculation, which proves that the bearing dynamic model in Section 2 is correct. In the real service process, the dynamic load profile is obviously not as stable as the other two static load profiles, due to the random collision between the rolling element and inner and outer rings, rolling element and cage, cage and inner and outer rings during the bearing motion. Additionally, due to the existence of inertia moment and centrifugal force, the non-load-bearing area where the contact load between rolling body and raceway is 0 under theoretical or static simulation conditions, and the same contact load exists under dynamic simulation conditions. Therefore, it is reasonable that the dynamic load curves are partially different from the static simulation curves in the validation analysis of the dynamics model.

#### **4. Analysis of Dynamic Characteristics of the Cage**

The dynamic characteristics of the cage have different performances with the change in working conditions. In previous studies [30,33,34], it is usually judged according to the motion state of the centroid: when the trajectory of the cage centroid is one point, it can be considered as a completely stable state; when the cage centroid trajectory is not a point and a vortex occurs, the motion stability of the cage is judged according to the centroid vortex trajectory and the vortex velocity deviation ratio. When the cage vortex trajectory is almost circular, it indicates that the cage centroid is in a stable vortex state. When the value of the cage vortex velocity deviation ratio is small, it indicates that its motion is relatively stable.

The deviation ratio of cage vortex velocity is expressed as *σ*v, and the calculation formula is as follows:

$$\sigma\_{\rm V} = \frac{\sqrt{\sum\_{i=1}^{n} \left(v\_{i} - \overline{v}\_{\rm m}\right)^{2} / (n-1)}}{\overline{v}\_{\rm m}} \tag{9}$$

where *v<sup>i</sup>* is the speed of the cage centroid at any moment, *v*<sup>m</sup> is the average speed of the cage centroid, and *n* is the number of sampling points of the cage speed time domain curve.

#### *4.1. Stability Variation of Cage with Rotating Speed*

Apply 1000 N axial load to the bearing, observe the simulation results at driving speeds of 1000, 2000, 3000, 4000, 5000, 6000, 7000 and 8000 r/min, and analyze the influence of speed on the stability of cage.

The trajectory of the cage centroid after stabilization obtained by simulation analysis at the above 8 different rotating speeds is shown in Figure 9.

From Figure 9, it can be seen that the vortex radius is close to the guiding gap after stabilization. When the rotating speed is less than 3000 r/min, the cage centroid trajectory tends to change steadily and regularly with the increase in rotating speed; after exceeding 3000 r/min, the centroid trajectory tends to become cluttered with the increase in rotating speed; after exceeding 6000 r/min, it changes relatively invisibly.

Figure 10 shows the vortex radius ratio and vortex velocity deviation ratio of the cage centroid trajectory. The vortex radius ratio is the ratio of the difference between the maximum and minimum vortex radius to the average vortex radius, and it reflects the divergence degree of the cage centroid vortex trajectory. It can be seen from the figure that the ratio value decreases gradually with the increase in speed when the speed is less than 3000 r/min, when the speed exceeds 3000 r/min, the deviation ratio of centroid vortex velocity increases gradually with the increase in speed, and finally tends to be level off. It is noteworthy that both ratios almost have the same trend, but the vortex radius ratio has a greater magnitude of change. Even so, its range of variation with rotating speed is only 0.15, so the effect of speed on cage stability can be considered small.

**Figure 9.** Cage centroid trajectory at different speeds. (**a**) 1000 r/min; (**b**) 2000 r/min; (**c**) 3000 r/min; (**d**) 4000 r/min; (**e**) 5000 r/min; (**f**) 6000 r/min; (**g**) 7000 r/min; (**h**) 8000 r/min.

**Figure 10.** Effect of rotating speed on the velocity deviation ratio and radius ratio of cage centroid vortex velocity.

As can be seen from Figures 9 and 10, the motion stability of the cage gradually stabilizes with an increase in speed and then gradually deteriorates to a certain level. This is due to the fact that with the increase in rotating speed, the unbalanced force of the cage itself and the interaction with the steel ball gradually increase, but the increasing amplitude of the cage unbalanced force is large and plays a major role, prompting the cage to push

towards the guide surface of the guiding ring and produce a stable vortex. When the rotating speed exceeds 3000 r/min, the interaction between the cage pocket and the steel ball increases, gradually changes the motion state of the cage, presents a trend of gradual confusion, and the motion stability becomes worse. Compared with the simulation results of Wen [34], the stability trend of the bearing cage in this paper is in good agreement with the simulation results of the angular contact ball bearing 7103AC, which verifies the correctness of the simulation results.

#### *4.2. Stability Variation of Cage with Pure Axial Load*

Apply the driving speed of 1000 r/min to the bearing and select the simulation analysis results of five different working conditions of 1000, 2000, 3000, 4000 and 5000 N to analyze the influence of axial load on the stability of cage. The trajectory of the cage centroid obtained under the action of the above five different axial forces is shown in Figure 11.

**Figure 11.** Cage centroid trajectory at different axial load. (**a**) Cage centroid trajectory at 1000 N; (**b**) Cage centroid trajectory at 2000 N; (**c**) Cage centroid trajectory at 3000 N; (**d**) Cage centroid trajectory at 4000 N; (**e**) Cage centroid trajectory at 5000 N.

It can be seen from Figure 11 that when the axial force is less than 3000 N, the trajectory of the cage centroid tends to be stable and regular with the increase in the axial force; after more than 3000 N, the trajectory of the centroid tends to be more complex with the increase in axial load.

The vortex radius ratio and vortex velocity deviation ratio are shown in Figure 12. It can be seen that when the axial force is less than 3000 N, the ratios gradually decrease with the increase in axial force, and when the axial force exceeds 3000 N, they enhance with the increase in axial force. The vortex radius ratio has a variation of 0.45, so the effect of axial force on cage stability is significant.

**Figure 12.** Effect of pure axial load on the velocity deviation ratio and radius ratio of the cage centroid trajectory.

It can be seen from Figures 11 and 12 that within a certain axial force range, the motion stability of the cage tends to be stable with the increase in the axial force. After exceeding the critical value, the stability of the cage gradually deteriorates. This is because within a certain range of axial force, with the increase in axial load, the load borne by each steel ball is gradually uniform. Therefore, the difference in the guiding ring drag force leads to the reduction in the fluctuation of rotating speed when the steel ball rotates to different azimuth angles, which reduces the collision between the steel ball and the cage pocket. The unbalanced force of the cage is dominant, which makes the cage show a stable vortex, which is conducive to the stability of the movement of the bearing cage. When the axial force is too large, the collision force fluctuation between the steel ball and the cage pocket hole increases, making the stability of the cage worse.

#### *4.3. Stability Variation of Cage with Radial Load*

FPCBB cannot bear pure radial force, the size of radial force it can bear should be less than 0.7 times of axial force. The simulation analysis results of 6 different radial force working conditions of 100, 200, 300, 400, 500 and 600 N are selected to analyze the influence of radial force on the stability of the cage under the combined load condition. The trajectory of the cage centroid obtained under the above six different working conditions is shown in Figure 13, from which it can be seen that the cage centroid trajectory is gradually complicated with the increase in radial force.

The vortex radius ratio and vortex velocity deviation ratio under the above six different load conditions are shown in Figure 14. It can be seen that the ratios have an upward trend and the trend tend to expand. The amplitude of the radius ratio exceeds 1.0 when the radial force is 600 N. It can be considered that the radial force has the greatest influence on the stability of the cage, and the influence enhances with the increase in the radial force.

**Figure 13.** Cage centroid trajectory at different radial load under combined load. (**a**) 100 N; (**b**) 200 N; (**c**) 300 N; (**d**) 400 N; (**e**) 500 N; (**f**) 600 N.

**Figure 14.** Effect of radial load on the speed deviation ratio and radius ratio of the cage centroid trajectory under combined load.

From Figures 13 and 14, it can be seen that the increase in radial force will prompt the cage centroid motion trajectory tends to be chaotic, and the stability becomes worse. This is because with the increase in radial force, the load of each steel ball is more and more uneven, which increases the frequency and amplitude of the collision between the steel ball and the cage pocket hole, thus changing the cage trajectory, making the trajectory irregular and reducing the stability of the cage.

#### *4.4. Experimental Verification*

In order to verify the accuracy of the analysis results of the bearing dynamic model established according to the parametric method in this paper, a rolling bearing cage performance analysis test bed is built. The test bed is composed of motorized spindle, coupling, support shaft, test shaft, loading device and test system. The main structure is shown in Figure 15. The test bearing is QJ215, its structure is shown in Figure 2. The structural parameters are consistent with the bearing used for simulation.

**Figure 15.** Test bed of rolling bearing cage performance analysis.

The test system includes a rotating speed measurement system and a motion displacement measurement system, as shown in Figure 16. The rotating speed measurement system includes laser speed sensor, reflector, signal acquisition system and computer, etc., and the motion displacement measurement system includes 2D laser profilers and processor.

**Figure 16.** Cage speed and displacement measuring device. (**a**) Cage speed measuring device; (**b**) Cage displacement measuring device.

The principle of cage speed measurement is to put a reflector on the end surface of the cage, and align the laser to the reflector. When the reflector returns to the laser with one revolution of the cage, the laser sensor returns a pulse signal, which is converted into a rotational speed signal by the acquisition system, and finally, the curve of rotating speed of the cage is obtained. The laser sensor is Laser Tacho Probe-MM0360 from Denmark, and the signal acquisition device is PAK Mobile MK II data acquisition system from Millerbem, Germany.

The principle of cage trajectory measurement is shown in Figure 17. The vibration displacement information of the cage is monitored in real time by 2D laser profiler, and the positions of the cage profile edge points at different moments are subtracted to obtain the vibration displacement signal of the cage. Two sensors are used in the test: sensor 1 is placed in the direction parallel to the test bench, for measuring the displacement in the *X*-axis direction; sensor 2 is placed in the direction perpendicular to the test bench, to measure the displacement in the *Z*-axis direction. The two sensors are synchronous and the signal is transmitted to the monitor through a switch. The 2D laser profiler is the LJ-X8020 model from Keyence.

**Figure 17.** Measurement principle of cage trajectory.

The rotating speed of the cage was measured using the rolling bearing cage motion test bench. The test rotating speed of the electric spindle changes from 600 r/min to 2000 r/min under constant load. The radial force loading device kept 0 N, and the output values of the axial force loading device were from 1000 N to 2000 N. The comparison results between simulation and experiment are shown in Figure 18.

**Figure 18.** Simulation and experimental error curve.

The experimental speed is higher than the simulated speed, and the maximum error between them is 10.4%. It shows that the rotating speed has little influence on the error, and the error decreases with the increase in the axial load. The main reason for the error is that the test bearing adopts SKF's LGMT 3/1 grease, whose base oil viscosity is 125 mm2/s. Its typical application is bearings with > 100mm inner diameter. It is larger than the viscosity recommended in the manual of QJ215. Thus, the drag force between the rolling body and the inner and outer rings is larger, which causes the cage speed to increase. Therefore, the simulation results can be considered to be consistent with the experimental results, and the accuracy of the simulation is verified.

The radial vibration signal of the cage is collected at the driving speeds of 1000, 1500 and 1800 r/min, as well as 1000 N pure axial load condition. The collected vibration displacement signals are filtered to eliminate the rotating frequency and external interference,

and the centroid trajectory is obtained by synthesizing the two vibration displacement signals. The synthetic centroid trajectory of the cage is shown in Figure 19.

**Figure 19.** Cage centroid trajectory at different driving speeds. (**a**) 1000 r/min; (**b**) 1500 r/min; (**c**) 1800 r/min.

From Figure 19, it can be found that the centroid trajectory is elliptical, rather than the circle shown in the simulation result of Figure 9, this is caused by the difference in the stiffness of the bearing in all directions due to the actual bearing installation error or the deflection of the loading force. As shown in the figure, in the speed range of 1000–1800 r/min, the centroid trajectory of the cage shows vortex motion and tends to change steadily and regularly with increasing speed. It is consistent with the trend of simulation results of Figure 9.

Similarly, with regard to a change in the operating conditions, the centroid trajectory was analyzed at the speed of 1000 r/min and pure axial loads of 1000, 1500 and 1800 N, respectively, and the results are shown in Figure 15.

It can be seen from the figure that the centroid trajectory with axial force of 1500N and 1800N is better concentrated than that at 1000N, and the cage vortex situation is more stable, which is consistent with the trend of simulation results of Figure 11. From Figures 19 and 20, it can be seen that the change rules of the cage centroid trajectory measured by the experiment is consistent with the change in the simulation results, indicating that the change rules of the centroid trajectory summarized before may be correct, which also verifies the correctness of the simulation. There are still many shortcomings of the experiment. For example, only pure axial force loading is realized. Due to the limitation of conditions, there is no way to stably load radial force and axial force at the same time. High speed makes the test bench vibrate seriously, so for safety and accuracy, only the operation at low speed is simulated.

**Figure 20.** Cage centroid trajectory at different axial forces. (**a**) 1000 N; (**b**) 1500 N; (**c**) 1800 N.

#### **5. Conclusions**

In this paper, the dynamic model of FPCBB is established according to the multi-body dynamics simulation software ADAMS and its subroutine GFOSUB, which makes full use of the advantages of ADAMS in solving dynamic equations with high stability. The correctness of the dynamic model is verified by comparing the theoretical calculation results with the static simulation results of the cage speed and bearing load distribution. The effects of rotating speed and load on the stability of the cage are evaluated with the indexes of vortex radius ratio and vortex speed deviation ratio of the cage centroid trajectory. A test bench is established, and measurement methods of the cage speed and centroid vortex trajectory are proposed. The motion stability of cage reflects the stability of bearings; therefore, it is of practical significance to master the operation law of cage and guide the application of four contact point ball bearing. Some conclusions have been given as follows:

The dynamic bearing load curve is similar to the static load curve in overall trend, but the dynamic load curve is less stable than the static load curve, due to the existence of collision force between components.

The cage stability is affected by rotating speed and load conditions. The vortex radius ratio and the vortex velocity decrease and then increase as the inner ring getting faster. When the axial load is small, these two indicators are relatively stable, but after the axial force exceeds 3000 N, the cage becomes more and more unstable. Under both axial load and radial load, the increased radial force reduces cage stability.

The experimental results verify the rationality of the simulation results to a certain extent. It also verifies the feasibility of the speed and centroid vortex trajectory measurement method proposed in this paper. For different four-point contact ball bearings, further simulation and experiments are needed to verify the conclusions of this paper.

**Author Contributions:** Conceptualization, Y.Z. and J.H.; methodology, Y.L., S.M. and Y.Z.; software, S.M. and G.H.; validation, W.L.; formal analysis, G.H.; funding acquisition, J.H.; data curation, W.L., Y.L. and G.H.; writing—original draft preparation, Y.L. and Y.Z.; writing—review and editing, Y.L. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received was funded by the National Science Foundation of China (52175250).

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

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Not applicable.

**Acknowledgments:** The authors would like to thank the National Science Foundation of China (52175250) for the financial support.

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

#### **References**

