Motion stability analysis of cage of rolling bearing under the variable-speed condition

The fluctuation of the cage rotating speed will lead to the strong instability of the cage movement of the rolling bearing, which will seriously affect the operation performance and service life of the bearing. In this paper, a nonlinear dynamic model of a cylindrical roller bearing is established considering the dynamic contact relationship between the rollers, the raceways, and the cage. The variation of cage motion characteristics and its internal mechanism under the uniform, acceleration, deceleration, and harmonic speed fluctuation are discussed, respectively. The effects of angular acceleration, angular deceleration, amplitude, and period of fluctuation on cage stability are analyzed. The results show the unstable motion of the cage mainly happens at the initial stage of the acceleration process and the later stage of the deceleration process. Appropriate increases in angular acceleration and deceleration can help to improve the stability of the cage movement, but excessive speed fluctuations should be avoided to adversely affect the stability of the cage. The guiding force between the cage and the outer raceway plays an important role in the stability of the cage under variable-speed conditions. The results of this paper can provide a theoretical basis for the design and failure analysis of the cage.


Introduction
Cage is one of the indispensable components of rolling bearing, which plays an important role in uniformly isolating the rollers and improving the dynamic characteristics of the bearing. Due to the particularity of its structure, there will be frequent collisions among the cage, the rollers, and the raceways, respectively, during the operation of the bearing, which will cause the cage vibration. The above phenomenon is called ''cage instability'' [1]. The instability of cage was first proposed by Kingsbury [2]. Stevens [3] divided the instability of cage into three categories: radial instability, axial instability, and instability caused by the deviation of cage operation position. This paper mainly studies cage instability in the radial motion plane. In addition, poor working conditions will cause cage fracture, bearing wear, or other faults, thus reducing the service life and operation performance of bearings. Therefore, it is of great engineering significance to control the instability of the cage, reduce cage failure, and improve bearing performance by analyzing the motion characteristics and stability of the cage under variable-speed conditions.
In the 1960s, Kingsbury [2] first investigated the relationship between the movement of ball bearing cage and the moment fluctuation through experimental studies. He defined four modes of cage movement: (1) whirl or howl model (the whirl radius and whirl velocity of cage are constant or change irregularly with time); (2) synchronous vortex (cage vortex speed is equal to the ring speed, only when the bearing preload is low and the ring speed is high); (3) stability model (the vortex angular velocity of cage is equal to the rotation angular velocity); (4) ball jump model (bearing inner and outer rings are relatively inclined). Since then, cage stability has received more and more attention in the literature.
Based on the assumption of static equilibrium and sufficient lubrication, Walters [4] innovatively proposed the cage dynamic model under the steady motion of rolling bearings and studied the transient motion characteristics of the cage, which lays a good foundation for the further development of cage dynamics research. Kannel et al. [5] developed an analytical model for the planar motion of the cage of angular contact ball bearings by referring to the elastohydrodynamic lubrication theory. The changes of cage motion characteristics in the zone of elastohydrodynamic lubrication and the influence of different lubrication degrees on the cage motion stability were intensively studied. They concluded that the reduction of the friction coefficient between the ball and cage or lubricant viscosity is conducive to cage stability. The lubrication of the ball and raceways has significant effects on cage stability. Gupta [6] studied the mechanism of cage movement variation and concluded that the coupling effect of friction, geometry, and working conditions is the main reason affecting cage stability. However, due to the consideration of many factors, the solving process of these models is complex and the convergence rate is low. Therefore, Boesiger et al. [7] proposed a two-dimensional dynamic model for the cage of deep groove ball bearing based on Gupta's work by considering only the planar motion of bearing components and carried out simulation analysis and experimental studies on the relationship between the friction effect in the radial plane and the unstable motion of cage. The results show that the unstable motion of the cage is highly sensitive to the friction between the ball and the cage, and presents a constant characteristic frequency. The characteristic frequency of instability is independent of the bearing speed and load. Nogi et al. [8] proposed a simplified quasi-dynamic model for the cage of ball bearings under an axial load and pointed out that the high-frequency eddy caused by the friction between the ball and the cage is the main reason for the unstable motion of cages. Niu et al. [9] presented a model for stable cage vorticity by considering the influence of centrifugal force and ball spacing. The analysis shows that the vortex radius of the cage is maintained by the normal force and friction force of the ball and cage pocket, the normal force of the cage and the rim of the raceway, and the centrifugal force of the cage. Neglia et al. [10] analyzed the influence of the collision force and the inertia force on the trajectory of the cage's centroid, and found that the collision contact between the cage and the roller has an obvious influence on the stable motion of the cage. However, the influences of geometric parameters and working conditions have not been further analyzed in the above research. In addition, the models used above are relatively simple to deal with the contact behavior of each component. Meeks et al. [11] simulated the contact among cage, rolling body, and ring in an incomplete linear contact mode and studied the variation rule of cage stability under different rotational speeds, cage structural parameters, and lubricating oil properties. The results show that when the cage clearance ratio (the ratio of hole clearance to guide clearance) is 1, howling occurs; and when the cage clearance ratio is greater than 1, the cage centroid motion is irregular. Yang et al. [12] and Liu et al. [13] defined the occurrence conditions of collision contact between the rolling elements and the cage by using the fuzzy collision theory. They found that cage guide clearance and pocket clearance play a decisive role in the motion stability of cage. Liu [14] developed a cage dynamic model of high-speed rolling bearings considering viscous damping and oil film damping and studied the relationship between structural parameters, working conditions, and the stability of cage motion. Zhang et al. [15] developed a numerical model to simulate the dynamic wear of high-speed angular contact ball bearings and analyzed the cage centroid movement under different load conditions and cage clearance ratios. Yao et al. [16] proposed a multi-body dynamic analysis model of cylindrical roller bearing based on the ANSYS/LS-DYNA software and explored the influence of working conditions, geometry, and structural elastic deformation on the trajectory of the cage's centroid. Choe et al. [17,18] experimentally studied the variation of cage stability with the change of guide clearance, pocket clearance, and uneven distribution of cage mass under a lowtemperature environment. Zeng et al. [19] presented an explicit dynamic model of angular contact ball bearings with the help of the ABAQUS software and investigated the changes of cage stability under different guide clearance and pocket clearance. Tan et al. [20] used the ADMAS software to analyze the various characteristics of the stability of the cage's centroid motion under different parts' surface corrugations. Moreover, some scholars have also carried out relevant studies on other possible factors affecting the kinematic stability of the cage. For example, Ghaisas et al. [21] analyzed the variation rule of cage movement stability under the condition of raceway misalignment, asymmetric cage, and different roller sizes. Pederson et al. [22] coupled the ANSYS flexible cage finite element model with the dynamics model of the deep groove ball bearing. The kinematic stability of the cage with different torsional stiffness, circumferential stiffness, and cage clearance was investigated. Deng et al. [23] and Chen et al. [24] studied the transient nonlinear dynamic behavior of cage of highspeed cylindrical roller bearings under different cage guidance modes to improve cage stability. Sharma et al. [25][26][27][28] built a rotor-bearing dynamic model considering the effect of rotor unbalance, varying number of rollers, and rolling element defects on the bearing dynamics. They found the presence of instability and chaos in the dynamic response as the speed increases.
With the development of rolling bearings toward high speed, heavy load, and high-precision transmission, the phenomenon of cage failure is becoming more and more obvious. The research on the motion characteristics of the cage under typical complex variable working conditions such as variable speed, variable acceleration, and variable load has become a hot issue yet to be investigated. Ye [29] presented the dynamic model of aero-engine high-speed rolling bearings based on the ring balance control theory and studied the cage stability under three transition states including startup, acceleration, and loading. However, the influence of the change in working conditions on cage stability was not studied. Yao et al. [30] developed a dynamic model of angular contact ball bearing considering the three-dimensional dynamic contact relationship between components in the bearing. The effects of clearance friction and lubrication drag on the stability of the cage under variable speed, incremental loading, and preload conditions are analyzed, but the comparative study between different variable conditions was not involved. Qu et al. [31] comparatively analyzed the variation characteristics of centroid motion of angular contact ball bearing cage under uniform and variable deceleration operation conditions. However, the research is not comprehensive due to the lack of quantitative analysis of cage stability. Zhang et al. [32] presented a multi-body dynamics model of ball bearing with the bearing clearance and lubrication ignored. The influence of bearing starting acceleration, axial load, and the presence of gravity field on cage stability is studied. Tu et al. [33] and Luo et al. [34], respectively, studied the effects of different external loads and angular accelerations on the dynamic parameters such as cage force and slip of diagonal contact ball bearing under acceleration and deceleration conditions through simulation analysis and experimental verification.
At present, researchers have analyzed the dynamic characteristics of the cage during the operation of the rolling bearing from different perspectives and explored the variation principles of the cage's movement stability under the action of different factors. However, current work mainly focuses on stationary condition, and most of the analysis models are established based on the static equilibrium equation. The dynamic effects of the rollers, raceways on cage movement are rarely considered comprehensively. In addition, although some scholars have made some achievements in the research on the motion characteristics of the cage under variable working conditions, they mainly use dynamic software (e.g., ANSYS and ADMAS) for simulation, and the detailed comparative study of influence degree under different variable working conditions is not involved.
To sum up, based on work in Ref. [35], an analytical model of rolling bearing was established to study the cage stability under different variable-speed condition. The innovation of the proposed model lies in the further consideration of the masses of the roller, the cage, and the ring as well as the slipping effect between the roller and the cage, and the introduction of the collision contact and hydrodynamic lubrication among the cage the roller and the guide ring. The motion stability of the cage under the conditions of uniform, acceleration, deceleration, and simple harmonic speed fluctuation is studied. The stability changes of the cage under different angular acceleration, deceleration, fluctuation amplitude, and fluctuation period are compared and analyzed. The purpose of this paper is to reveal the internal relationship between the instability of the cage and the characteristics of working conditions, to provide theoretical reference for improving the motion stability and optimal design of the cage.

Dynamic model of rolling bearing
Due to the limitation of structural size and working conditions, the contact action among the cage, the raceways, and the rolling elements changes frequently in the actual working process, and the motion state of the cage is complex. Therefore, to focus on analyzing the cage stability under variable-speed working conditions, this paper introduces the elastic support function of shaft and housing, and makes the following reasonable assumptions for the bearing system: (1) Each element of rolling bearing only moves in the radial direction; (2) the inner raceway is fixed with a rotating shaft, driven by input load and speed. The outer raceway is rigidly fixed with housing. The outer raceway's speed is zero, and the cage is guided by the outer raceway; (3) each element of rolling bearing is a rigid body. Thus, the structural deformation is ignored, and only local deformation occurs when each component contacts with the cage and raceways; (4) the lubrication among the rollers, the raceways, and the cage is simplified as equivalent lubrication drag coefficient and drag force.
In this paper, the elastic support action of the shaft and housing is described by a spring-damper model. The contact action between the roller and raceway is simulated by the Hertzian elastic contact and the traction-lubrication model [36]. The collision contact between the cage pocket and the roller is considered as the equivalent action of elastic spring stiffness and damping, as shown in Fig. 1. In the figure, the stiffness and damping coefficients of shaft and housing are represented by K sh , c s , and c h , respectively; m in , m r , m out , and m c represent the masses of the inner raceway (including shaft), rollers, outer raceway (including housing), and cage, respectively; x i , x aj , x bj , and x c , respectively, represent the inner ring angular velocity, the jth roller's revolution angular velocity, the rotation angular velocity, and the cage's angular velocity. In the following, the subscripts i, o, j, and c, respectively, represent the inner raceway, outer raceway, jth rolling element, and the cage. The subscripts x and y represent the components of each element on the X-axis and Y-axis.

The interactions between the rollers and the raceways
The contact deformation and contact force between the jth roller and the inner and outer raceways are mainly related to their relative position. The positional relationship between the jth roller and the raceways at any angular position W j under the action of a radial force W is shown in Fig. 2. The dotted lines in Fig. 2 indicate the initial position of the raceways and the jth roller before loading. At this time, the inner raceway and the outer raceway are concentric (O), and the jth roller is located in the middle of the inner ring raceway and the outer ring raceway. The solid lines are the positions of the jth roller and raceways after loading. Due to the radial load and the influence of the dynamic effect, the centers of the inner raceway and outer raceway move from O to O i and O o , respectively, and their displacements in the vertical and horizontal directions are x i , x o , and y i , y o , respectively. The center of the jth roller moves from O r1 to O r2 , and its displacement in the vertical and the horizontal directions is x j and y j , respectively. Angle position of the jth roller W j can be expressed as Eq. (1): where N b is the number of the rollers; h j (t) is the rotation angle of the jth roller.
Considering the radial clearance of the bearing and oil film lubrication, the contact deformation between the jth rolling element and the inner ring can be expressed as [35]: where C r represents the radial clearance of the rollering bearing. ''[•] ? '' indicates that when the value inside the brackets is less than zero, then the contact deformation d ij is 0 as the contact deformation cannot be a negative value. h i is the oil film thickness, and relevant calculation methods can be referred to Ref. [37]. The contact deformation between the jth roller and the outer raceway depends on the radial clearance and the relative displacement between the jth roller and the outer raceway, which can be expressed as: According to the Hertz contact theory, the contact force between the jth roller and the inner and outer raceways can be expressed as [38]: where K i and K o , respectively, refer to the contact stiffness coefficients between the roller and the inner and outer raceways under the action of oil film lubrication, and the detailed calculation process is shown in Ref. [35]. n is the contact coefficient between the rollers and the inner and outer raceways. For the cylindrical roller bearings, n is 10/9.
The contact damping force between the jth roller and the inner and outer raceways can be expressed as: where c i and c o are the viscous damping coefficients between the rollers and the inner and outer rings, respectively; k j is the coefficient to judge whether the contact damping exists. When the contact deformation exists, it is taken as 1; otherwise, it is taken as 0.
According to Coulomb's friction law, the friction force is the product of the friction coefficient and the normal force between the contact surfaces. Thus, the friction between the jth roller and the inner and outer race raceways can be expressed as: where l i and l o are the traction lubrication friction coefficients between the rollers and inner and outer raceways, which change with the relative sliding velocity DV. Relevant calculation methods can be referred to Ref. [23].

The interactions between the cage and rollers
The contact deformation between the cage and the jth roller is mainly caused by the difference between the rotational speed of the cage and the revolution speed of the jth roller. The position relationship between the jth roller and the cage at any angular position W j is shown in Fig. 3; O bj and O pj are centers of the jth roller and the jth pocket, respectively. At the same time, when the difference between the circumferential displacement of the jth roller and the cage is greater than the pocket clearance of the cage, the roller contacts the front end of the pocket of the cage, and the normal contact, and tangential friction forces are expressed by N c1j and F c1j , respectively. When the difference between the circumferential displacements of the jth roller and the cage is greater than the pocket clearance of the cage, the roller contacts the rear end of the pocket of the cage, the normal contact and tangential friction forces are expressed by N c2j and F c2j , respectively.
The contact deformation between the jth roller and the cage pocket can be expressed as [39] dcj ¼ zcj where z cj is the difference between the circumferential displacements of the jth roller center and the cage pocket center; C p is the pocket clearance of the cage. Considering the viscous damping effect at the contact, the normal contact force and tangential friction force between the jth rolling element and the cage pocket can be expressed as [40]: where K c is the equivalent contact stiffness coefficient between the roller and the cage, and the detailed calculation process is shown in Ref. [38]; a e is the relevant coefficient of elastic recovery; t p cjN is the normal relative velocity of the contact point; l c is the friction coefficient. Because the surface of the cage is rough and the relative sliding speed between the rolling element and the cage is large, l c is taken as a constant value of 0.002. When the rollering bearing is running under the action of lubricating oil, the hydrodynamic force will be generated at the minimum clearance between the guide surface of the outer raceway and the centering surface of the cage, which will affect the motion state of the cage and the outer raceway. The schematic diagram of the positional relationship between the cage and the guide raceway(outer) is shown in Fig. 4. The local coordinate system S c = {O c , x c , y c } in the figure is the cage coordinate system. Axis x c is in the direction of the line connecting the cage center and the point where the minimum oil film thickness h is located. W c is the included angle between the axis x c and the global axis X.
The minimum oil film clearance between the centering surface of the cage and the guide surface of the outer raceway is given by [14] h¼C g À e ð9Þ where C g is the guide clearance between the cage and the guide raceway(outer). e is the eccentricity of the cage center relative to the guide raceway(outer) , and Dx co and Dy co are the displacements of the cage center relative to the guide raceway(outer) in the X and Y directions, respectively.
After the contact conversion threshold Dh is set, when h [ Dh, the state between the cage and the guide surface is a hydrodynamic lubrication state. Otherwise, it is a Hertz contact state. In this paper, it is assumed that there is sufficient lubrication between the cage and the guide surface, and the influence of roughness is ignored. Thus, the contact conversion threshold Dh can be set as 0 to simplify the calculation.
When h [ Dh, the equivalent short sliding bearing model is used to describe the interaction between the guide raceway(outer) and the cage, as shown in Fig. 4a. In the local coordinate system S c , the resultant force F co acting on the cage generated by the distributed pressure of the hydrodynamic oil film can be described by two orthogonal components F cox and F coy , whose values are, respectively, given as [35] where g 0 is the dynamic viscosity of lubricating oil under atmospheric pressure. u 1 is the lubricating oil drag speed, and . e is the relative eccentricity of the cage center, and e = e/C g . L 1 is the width of the centering surface of the cage. C 1 is the cage guide clearance. The distributed pressure generated by the hydrodynamic oil film also produces the following friction torque M co on the moving cage surface: where V 1 is the relative sliding speed between the guide surface and centering surface, and The above forces and torques are projected into the bearing inertial coordinate system {O, X, Y}, where W c = arctan(Dy co /Dx c ).
When h \ Dh, the influence of the damping moment is ignored. The collision contact between the centering surface and the guide surface is considered as the equivalent action of normal rubbing force and tangential friction force, and such local contact friction is assumed to follow Coulomb's law, as shown in Fig. 4b. The contact force is calculated as follows [14]: where E' is the equivalent elastic modulus of the Hertz contact between the cage and the guide raceway(outer). L is the contact length. d co is the deformation at the contact point, and d co = e -C g . f is the boundary friction coefficient between the cage and the guide raceway(outer) without considering the effect of speed. The radial rubbing force P cox and tangential rubbing force P coy are decomposed into the bearing inertial coordinate system {O, X, Y},

Nonlinear differential equations of motion
According to Newton's law, the differential equation of the cage motion is described as [41] where I c is the rotational inertia of the cage around its own central axis. h c is the rotational angular displacement of the cage. F w is the eccentric force of the cage.
The differential equation of the inner raceway (including shaft) motion can be expressed as [41] where Q cix and Q ciy are the components of the resistance forces of the rollers acting on the inner raceway in the X and Y directions, respectively. N ix and N iy are the components of the total contact forces of the rollers acting on the inner raceway in the X and Y directions, respectively. The differential equation of outer raceway motion can be written as [41] where Q cox and Q coy are the components of the resistance forces of the rollers acting on the outer raceway in the X and Y directions, respectively. N ox and N oy are the components of the total contact forces of the rollers acting on the outer raceway in the X and Y directions, respectively. For the jth roller, the differential equation of motion is given as [41] m r € x j þN ojx þQ cojx ÀN ijx ÀQ cijx ÀF aj ÂcosW j Àm r g¼0 m r € y j þN ojy þQ cojy ÀN ijy ÀQ cijy ÀF aj ÂsinW j ¼0 where R m and r b are the pitch radius of a roller and the radius of a roller, respectively. I a and I b are the rotational inertia of a roller around the bearing axis and its own axis, respectively. U j is the angular displacement of the jth roller. F aj is the centrifugal force of the jth roller. In this paper, the fourth-order fixed step Runge-Kutta method is used to solve the above differential equations of motion simultaneously. Considering the calculation efficiency, the initial integration step Dt is taken as 1 9 10 -6 s. The calculation process flow is shown in Fig. 5.

Verification of the dynamic model
Taking a cylindrical roller bearing as the object, the dynamic simulation results, experimental results, and analytical solutions (under the pure rolling conditions) of the cage speed under the same working conditions are compared and analyzed. The geometric parameters of the bearing and the experimental set-up to measure the cage speed are shown in Table1 and Fig. 6. The cage of the bearing under test is uniformly pasted with a reflective plate in the circumference. In the experiment, photoelectric signal fed back by the movement of the reflective plate is collected and converted by the photoelectric speed sensor (Monar-Ros-W) installed in the axial direction of the bearing and the acquisition card, and finally input into the computer for processing and calculation to obtain the cage speed curve with time. The signals are acquired with a sampling frequency of 16 kHz. The main performance parameters of the sensor are shown in Table 2. In addition, when the radial load is large and the inner raceway speed is low, the cage rotates according to the theoretical speed. Therefore, in this paper, the radial load W is set as 3000N. Considering the limitation of the speed range in the experimental conditions, the inner raceway speed is selected as 500, 1000, and 2000 rpm, respectively.
When the outer raceway is fixed and the inner raceway rotates, the theoretical speed of the cage x cm is described as [38] xcm Figure 7 shows the comparisons between the simulation results and experimental results of cage speed under different inner raceway (including shaft) speeds. Table 3 shows the comparison between the mean value of simulation results, the mean value of experimental results, and the theoretical speed (i.e.,  Roller radius (R r ) 5.5 9 10 -3 m Equation (19)) of cage speed. It can be seen that the simulation results and experimental results of the cage's speed generally fluctuate around the theoretical speed value. Since the dynamic model established in this paper is mainly based on the guiding effect of the ring on the cage, the guiding effect increases with the increase of rotating speed. Therefore, when the radial load is constant, the relative error between the simulation value and the analytical solution of the cage average speed decreases with the increase of the inner ring speed. However, under experimental conditions, the increase of speed makes the state of internal components complex and changeable. Affected by a variety of nonstationary factors, the relative error between the experimental value of the average cage speed and the analytical solution becomes larger. Based on the dynamic model established in this paper, the average cage speed calculated by the simulation is generally in good agreement with the analytical solution and experimental results, and the relative errors between the average cage speeds of the simulation value, the experimental value and the analytical solution are less than 1%, which verifies the dynamic model built in this paper. It is worth noting that at certain points for 2000 rpm, there is a sudden drop in experimental results. This is because it is difficult to ensure strictly uniform distribution of the reflective plate in the circumference direction when attaching them to the bearing cage. Besides, when the bearing speed is high, the occurrence possibility of cage skidding and instability increases, and their influences on the instantaneous cage speed are significant.
In addition, the movement of the cage center forms special trajectories under different rotational speeds. Especially under high-speed conditions, the centroid trajectory of the cage is close to a circle. Therefore, this paper also refers to the results of the cage centroid trajectory from Gupta's classic calculation examples and experiments [41] (W = 1000N, x i = 12000 rpm) to verify the correctness of the cage centroid trajectory yielded by the proposed model, and the results are shown in Fig. 8. It can be seen that the simulation trajectory of the cage centroid obtained by the proposed model is consistent with that from Gupta's simulation and experimental results, which further verifies that the proposed model is reasonable and effective to study cage motion.

Dynamic response analysis
In this paper, a cylindrical roller bearing is taken as the simulation object. The main structural parameters of the simulated bearing are shown in Table 4. The dynamic viscosity of lubricating oil g 0 at atmospheric pressure was 0.033 PaÁs, and the viscosity pressure index a was 1.28 9 10 -8 Pa -1 .
The motion process of rotating machinery can be roughly summarized into three stages: starting, stable operation, and stopping. In the process of starting and stopping, the rolling bearing generally has a large acceleration, and the sharp change of motion state will make the cage motion unstable. In addition, in practice, the running speed of bearings presents  Considering the ideal condition that the variation of the inner raceway speed will approximate the linear change when the lubricating oil density and viscosity are relatively small under the accelerating and decelerating conditions, without loss of generality, the variation of the inner raceway (including shaft) speed under the acceleration and deceleration conditions is simplified as a linear line in the following analysis.

Analysis of cage stability under the linear acceleration condition
It is assumed that the applied radial load W is 1000 N, and the inner raceway (including shaft) angular velocity x i accelerates linearly from 0 to x 1 with  [41]; c Gupta test [41] angular acceleration a and then maintains the speed. The inner raceway (including shaft) speed x i in the whole process can be expressed as Taking t 1 = 3.6 s, x 1 = 1200 rad/s, and the angular acceleration a = 400, 800, and 1200 rad/s 2 , respectively, the speed change curve of the inner raceway (including shaft) is shown in Fig. 9.
The change of centroid motion of cage during linear acceleration is first studied, and the influence of angular acceleration on centroid motion of cage is explored. The simulation results of the displacement of cage in the X and Y directions at any time under the three acceleration conditions are shown in Fig. 10. It can be seen from the figure that the centroid of the cage is difficult to form a completely stable vortex motion trajectory under the acceleration condition because the motion state changes at any time. The trajectory of the cage center of mass first deviates radially under the action of gravity and centrifugal force during linear acceleration. As the speed increases, the centrifugal force of the cage increases, while the trajectory oscillates and tends to be a vortex. These simulation results are consistent with the findings in Refs. [32,42,43], which says ''the cage center of mass has an evolutionary process of oscillating before entering a circular rotational orbit during acceleration''.
To more intuitively and quantitatively analyze the changing characteristics of cage stability in the process of changing speed, Ghaisas's method [17] is adopted in this paper to determine cage stability. The velocity deviation ratio r v (the ratio of the standard deviation of the moving speed of the cage centroid to the average speed) is adopted as the determination of the cage instability. The smaller the ratio, the higher the cage stability.
Considering that the velocity deviation ratio r v is a statistical parameter, it can be seen from the above analysis that the cage may show different operating stability at different times under the condition of variable speed. Therefore, the time interval t 0 is used in this paper to divide the whole speed change process and r v /t 0 is defined as the ratio of centroid velocity deviation of the cage within a single time interval t 0 . In the process of speed change, the variation principle   with time and the influence of the change of working conditions on r v /t 0 are explored. Based on the expression of velocity deviation ratio r v in Ref. [15], the expression of r v /t 0 is shown below, where v i /t 0 andv t0 are, respectively, the instantaneous velocity of cage centroid and the average velocity of cage centroid at each time point in t 0. n t0 is the number of original sample points sampled in t 0 , and n t0 = t 0 /Dt. The unit time interval variable t 0 was set as 0.1 s, and the variation curves of the velocity deviation ratio r v /t 0 of the cage centroid with time under different angular accelerations are shown in Fig. 11.
As can be seen from Fig. 11, compared to the uniform process, the velocity deviation ratio r v /t 0 of the cage centroid in the linear acceleration process is much larger than that in general and changes with time. The velocity deviation ratio r v /t 0 shows an obvious jitter rising trend in the initial acceleration. After reaching a maximum value, the velocity deviation ratio r v /t 0 gradually decreases with the acceleration process. This indicates that the cage stability first weakens and then strengthens in the process of linear acceleration. This phenomenon may be caused by the nonlinear change of the force between the cage and the guide raceway (outer). Since the changes of the stress and motion state of the bearing internal components tend to be stable, the velocity deviation ratio r v /t 0 drops sharply in the process of changing from acceleration to uniform speed, and the instability degree of the cage motion is weakened. This phenomenon is similar to the conclusion obtained in Ref [28] that the cage stability is worse during acceleration, but the increase in rotational speed will increase the cage stability. In addition, different from the acceleration process, the velocity deviation ratio r v /t 0 in the uniform speed process has no obvious trend with time. This may be due to the existence of strong instability factors in the internal moving components of the bearing, which leads to the strong randomness of the cage stability. With the increase of angular acceleration, the maximum value of the velocity deviation ratio r v /t 0 decreases in the linear acceleration process, and the fluctuation degree of centroid velocity deviation ratio over time weakens, which indicates that the stability of cage becomes gentle. This may be because the increase of angular acceleration makes the moving element inside the bearing affected by sufficient inertial force. When the angular acceleration is low, the applied force will easily destroy the relative stable state of each component once it changes, thus weakening the stability of the cage and producing obvious fluctuations.
The guiding effect of the outer raceway plays an important role in the stability of the cage under variable-speed conditions. In order to further explore the internal causes of the above phenomena, the radial forces between the cage and the guide raceway(outer) are calculated based on Eqs. (9) * (14). The curves of the radial force between the cage and the guide raceway(outer) with time under different angular accelerations in the above inner raceway speed variation process are shown in Fig. 12.  As can be seen from the figure, during the initial period of acceleration, the interaction force between cage and guide raceway(outer) is very small and tends to be zero. According to Eq. (10), this is caused by the low speed and the small drag speed u 1 of the lubricating oil at the initial acceleration stage. At the same time, under the eccentric force, the eccentric distance e of the cage's center relative to the center of the guide ring is also less than the guide clearance C g (see Fig. 10). In combination with Eq. (15), it can be seen that the cage movement is mainly affected by the rollers and its inertia. Due to the increase in rotational speed, the unstable movements of the rollers increase, so the stability of the cage gradually weakens. As the acceleration process continues, the lubricating drag speed u 1 increases, the distance between the cage centering surface and the ring guide becomes closer and closer (see Fig. 10), the relative eccentricity e of the cage center increases, the guiding effect of the raceway guide (outer) surface is enhanced, and the interaction force fluctuates and increases with time. At this time, the unstable movement of the cage is inhibited and the stability of the cage is gradually enhanced.

Analysis of cage stability under the linear deceleration condition
It is assumed that the applied radial load W is 1000 N. The inner raceway (including shaft) angular velocity is x 1 during stable operation, and after t 2 , the inner raceway (including shaft) angular velocity x i is linearly decelerated to 0. The whole process can be expressed as Suppose t 2 = 2 s, x 1 = 1200 rad/s, the angular deceleration b is 400 rad/s 2 , 800 rad/s 2 , and 1200 rad/s 2 , respectively. The speed change curves of the inner raceway (including shaft) are shown in Fig. 13. Figure 14 shows the change curves of the centroid position of the cage with time when the above inner raceway speed changes. It can be seen from the figure that the cage still maintains the whirling motion in a relatively stable state for a long time after the linear deceleration process begins, and the motion trajectory curve has no obvious fluctuation. With the progress of deceleration, the reduction of rotational speed weakens the restraint effects of rollers and  raceways on the unstable motion of cage to some extent, and the motion trajectory of the cage centroid fluctuates slightly. At the late stage of deceleration, the relatively stable state between the moving elements is destroyed. There is an obvious oblique displacement at the mass center of the cage, and the fluctuation degree of the motion trajectory increases significantly. These results are consistent with the findings in Ref. [44]. In addition, the variations of the cage centroid motion trajectory are consistent with the experimental measurement results of the cage centroid in the deceleration process in Ref. [42], which says that the centroid of the cage first stays in a stable state, and then the rotating radius decreases continuously until the cage swings.
The unit time interval variable t 0 was set as 0.1 s. The variation curves of the velocity deviation ratio r v / t 0 of the cage with time under different angular deceleration conditions are shown in Fig. 15. As can be seen from the figure, the velocity deviation ratio r v / t 0 in the linear deceleration process first increases to a maximum value and then decreases, indicating that the cage stability in this process is first weakened and then enhanced. However, different from the linear acceleration process, the velocity deviation ratio r v /t 0 fluctuates less with time in the linear deceleration process. This phenomenon may be caused by the relatively flat change of dynamic effects of bearing internal components in the deceleration process. With the completion of deceleration process, the interaction between bearing components is gradually weakened as the speed decreases, and the cage stability is enhanced as the guiding effect of the ring still exists (see Fig. 16). Therefore, compared to the linear acceleration process, the maximum velocity deviation ratio r v /t 0 occurs in the later stage of deceleration.
The radial force between cage and guide raceway(outer) varies with time under different angular velocity reduction during the above inner raceway(including shaft) change process, as shown in Fig. 16. Combined with Eq/ (10), it can be seen that in the whole linear deceleration process, since the centering surface of the cage and the guide distance of the raceway(outer) are always in close contact (see Fig. 14), the relative eccentricity e of the cage center is larger. Thus, the interaction force between the cage and the raceway(outer) is more obvious, and the fluctuation degree of the force curve is smaller compared to that of the linear acceleration process. As the deceleration process progresses, the cage speed decreases gradually, the lubricating oil drag speed u 1 decreases, and the force between the cage and the guide raceway (outer) fluctuates and decreases on the whole. When the deceleration entered the later stage, the relatively stable motion state of the cage was destroyed. The collision contact between the cage and the guide raceway (outer), and the fluctuation degree of the force curve increase significantly.

Analysis of cage stability under the condition of harmonic speed fluctuation
It is assumed that the applied radial load W is 1000N, and the inner ring speed changes according to the simple harmonic function. The inner ring speed x i in the whole process can be expressed as  where x 0 is the average speed of the inner raceway(including shaft), which is 1200 rad/s. A is the fluctuation range, and A = (x max -x min )/(2x 0 ), which is set as 5, 10, 20, and 40%, respectively. T is the speed fluctuation period. Since the dynamic effect of the moving parts of the rolling bearing changes obviously when the period is large, it is easy to observe and analyze the motion stability of the cage. Thus, it is taken as 0.2, 0.4, and 0.5 s, respectively. The speed change curve of the inner raceway (including shaft) is shown in Fig. 17. Figure 18 shows the change curve of cage centroid position with time under different fluctuation amplitudes and fluctuation periods. It can be seen from the figure that under the influence of speed change, the change curve of the cage centroid position with time under the harmonic speed fluctuation condition has obvious jitter compared with the constant speed condition, and the cage has a greater degree of shaking and poor stability. The change of fluctuation amplitude has a great influence on the motion of the center of mass of the cage. When the fluctuation amplitude is small, the motion of the center of mass of the cage is generally in a dynamic equilibrium state, and the motion of the center of mass follows the circular stable vortex trajectory. Under the large fluctuation amplitude, the stability of the cage is weakened, and a rhomboidal center trajectory appears, which may be caused by the sudden change of centroid position after the cage is subjected to a large impact force. With the decrease of the fluctuation period, the impact times between the cage and roller and guide raceway(outer) shorten. The motion instability increases, and the jitter times and amplitude of centroid motion trajectory increase.
The unit time interval variable t 0 was set as 0.1 s, and the variation curve of the velocity deviation ratio r v /t 0 of the cage centroid with time under different fluctuation amplitudes is shown in Fig. 19. With the increase of fluctuation amplitude, the dynamic effect of various components in the bearing is enhanced. In addition, the degree of unstable movement of the cage increases, the velocity deviation ratio r v /t 0 increases, and the stability of the cage is weakened.
Considering that the speed change time of each stage is different under different fluctuation periods, in order to avoid the superposition of data in different change stages which affects the calculation accuracy of the evaluation index of cage motion stability, the unit time interval variables t 0 in fluctuation periods of 0.2, 0.4, and 0.5 s are selected as 0.05, 0.1, and 0.125 s, respectively. The variation curve of the velocity deviation ratio r v /t 0 of cage centroid with time under different fluctuation periods is shown in Fig. 20.
It can be seen from the figure that when the fluctuation period is short, the velocity deviation ratio c is relatively large, and there is no obvious periodic rule in the curve change with time. In this case, the external force on the cage changes frequently and its stability is poor. With the increase of the fluctuation period, the fluctuation process of inner raceway(including shaft) speed gradually slows down, the stability of the cage increases, the velocity deviation ratio r v /t 0 decreases, and the periodic variation rule of the curve with time becomes more and more obvious.
The variation curves of the radial force between the cage and the guide raceway(outer) with time under different fluctuation amplitudes and fluctuation periods are shown in Fig. 21. As can be seen from Fig. 21a, when the fluctuation amplitude is small, the contact force between the cage and the guide raceway(outer) is relatively small, and the fluctuation degree of the force curve with time is small. When the fluctuation amplitude increases, the harmonic rule of the contact force curve changes obviously, the contact force increases, and there is a collision impact force with more frequency and larger amplitude.
In addition, under the condition of harmonic speed fluctuation, the force between the cage and the guide raceway(outer) presents a harmonic change which is consistent with the speed change of the inner ring. The decrease of the fluctuation period will accelerate the change rate of the motion state of the cage and aggravate the unstable motion of the cage. This significantly increases the change range of the indirect contact force between the cage and the guide raceway(outer). The contact frequency between them will be reduced due to the enhancement of the contact effect, as shown in Fig. 21b.

Conclusions
In this paper, we focused on the issue of the nonlinear dynamic modeling of the cage motion under the variable-speed condition. The generation and variation mechanism of the unstable motion of the cage under different variable-speed conditions were explored. A comprehensive dynamic model of the rolling bearing was established to study the cage stability with the experimental validation. The stable motion characteristics of the cage are studied from the aspects of the motion track of the cage centroid and the velocity deviation ratio of the cage centroid. The variation principles of the cage stability under different working conditions are analyzed, which provides strong support for the analysis and control of the unstable motion of the cage. The main conclusions of this work can be drawn as follows: (1) The unstable motion of the cage mainly happens at the initial stage of the acceleration process and the later stage of the deceleration process. The fluctuation of rotational speed results in jitters of the cage centroid, which has a great effect on cage stability. (2) The increase of angular acceleration (deceleration) under the linear acceleration (deceleration) condition can restrain the cage instability to a certain extent. Thus, appropriate increases in angular acceleration and deceleration can help to improve the stability of the cage movement, but excessive speed fluctuations should be It needs to be noted that although some progresses have been made in the study of cage stability under variable-speed conditions, the model established in this paper is a plane model, which cannot reflect the instability of cage axial motion. In future studies, we will further improve the model.