Skidding and spinning investigation for dry-lubricated angular contact ball bearing under combined loads

Sliding and spinning behaviors significantly affect the performance of rolling bearings, especially for dry-lubricated bearings, micro and macro sliding may lead to increased wear of the solid lubricating film. A unified rolling contact tribology analytical model is proposed for dry-lubricated angular contact ball bearings (ACBBs) considering the extreme conditions including high combined loads and rolling contact effects. A comprehensive solution framework is proposed to ensure the robustness of the model under different loading conditions. Equilibrium equations are solved to study the effects of friction coefficients, rotating speeds, and combined loads on the skidding and spinning characteristics of the ACBB. The results show that the rolling contact effects and combined loads significantly affect the skidding and spinning performance of the ACBB. Further analysis reveals that the skidding mechanism is related to the interaction between ball kinematical motion and traction forces. The developed analytical model is proved to more accurately predict the bearing kinematical and tribological behavior as it discards the raceway control hypothesis and considers the macro/micro-slipping, creepage, and self-spinning motions of the ball, which is validated using both the existing pure axial loading dry-lubricated ACBB model and the classical Jones—Harris model. The study would provide some guidance for the structure and lubrication design of dry-lubricated ACBBs.


Introduction
Dry-lubricated ball bearings are widely used in the aerospace field, especially benefit from the rapid development of coating technology [1]. For example, in a liquid rocket turbopump, angular contact ball bearings (ACBBs) are critical components for the stable operation of the rotor system, which works within cryogenic fluids and faces extreme loading conditions such as high speed, high loads as well as dry lubrication [2]. Under such extreme environments, micro and macro damages are readily induced due to the skidding and spinning movements of the ball [3]. Therefore, stable and reliable operation of the rolling bearing is particularly important for the key equipment. As its basic premise, it is very meaningful and necessary to understand the skidding and spinning characteristics of ACBBs under drylubricated, high rotating speed, and combined loading conditions.
In the last decades, with the continuous improvement of computational capacity and rolling bearing technology, great progress has been made in rolling bearing analysis theory [4,5]. Jones [6] pioneeringly developed the quasi-static analysis model for rolling bearing according to the raceway controlling hypothesis (RCH), which can consider the inertial effects such as centrifugal force and gyroscopic moment of the ball. Harris [7] further extended the quasi-static model by considering lubrication, clearance, heating,

OXYZ
Bearing inertial frame A s , A a Areas of sliding and adhesion ox b y b z b Ball-fixed frame P m,n Traction force of the point mass o i x i y i z i Inner contact frame P x , P y Traction force components Outer contact frame L, l Tangential deformation coefficients ω bx/by/bz Self-rotation components of ball u x , u y Tangential deformation components Translational displacements of ball s Contact particles relative sliding rate ω b Total self-rotational speed of ball and other factors to summarize the Jones-Harris (JH) model. The JH model not only is able to consider more factors but also has high computational efficiency, thus is widely used in industry and academia. However, the accuracy of the JH model is sometimes limited by the RCH. Gupta [8] broke the limitation of RCH and established a dynamic model that includes six degree-of-freedom (DOF) of all bearing components. Some scholars have also carried out researches by using the multi-body dynamics and finite element methods. Xu et al. [9] established the dynamic model of deep groove ball bearing based on 2D multi-body dynamics, and studied the vibrational response of bearing under waviness fault through this model. Hou et al. [10] studied the dynamic loading distribution of a roller bearing via a 3D finite element model, and designed an experimental system to verify the results. The advantages and disadvantages of different analysis methods are described in detail in Refs. [4,5]. For rolling bearings, rolling-sliding is defined when the rolling element not only rolls with respect to the raceway, but also slides in the contact area, which is also called skidding. Skidding is the main cause of wear and incipient failure of rolling bearings [11]. There is a large volume of published studies describing the skidding characteristics of rolling bearings. Xi et al. [12] developed a wear prediction model based on the rolling-sliding ratio through experimental data, and studied the influence of rolling-sliding ratio, load, and rotational speed on the wear of ball bearings. Wang et al. [13] proposed a dynamic model considering the combined radial and axial load and lubricant to understand the effect of high speed on skidding in ACBBs. Wang et al. [14] investigated the effect of skidding on surface roughness and sub-surface stress using a dynamic model, and later formulated a roughness optimization model for bearing rings. Tu et al. [15] studied the relationship between the friction force and vibrational response when the rolling element is sliding, and found that the friction force will increase vibration amplitudes of cylindrical roller bearings, which seriously affects the bearing service life. Xu et al. [16] presented a tribo-dynamic model to study the effect of skidding on vibrational www.Springer.com/journal/40544 | Friction response and surface wear of the bearing. Gao et al. [17][18][19] experimentally and theoretically investigated the skidding performance of the ball bearing in jet engine and developed a triboelectric based high-precision sensor to measure the skidding performance. Most of the existed studies are focused on oil-lubricated bearings and use hydrodynamic theory to model the traction force, however, the traction force of dry-lubricated rolling bearings must be modeled in a different way.
Different from the hydrodynamic lubrication mechanism of oil lubrication, rolling contact effect is the main factor of dry lubrication friction. Due to the local tangential deformation behavior of rolling contact surfaces, micro-slipping phenomenon, also called creepage, will appear. The micro-slipping affects the traction forces of the ball as well as the service life of the solid film [20]. The first scholar to study creepage in a rolling bearing is Hailing [21], who established a spinning-rolling model of ball in ACBBs by dividing the contact zone into two parts (slipping area and adhesion area), and then using the strip theory [22] to calculate the friction force. However, absence of the lateral creepage force causes the inadequacy of his model. Kalker [23,24] found that the ball spinning motion will result in a lateral traction force, and subsequently deduced the equilibrium equations of the ball, in which friction forces are calculated using the simplified algorithm of rolling contact. Later, Legrand et al. [25] employed the basic theories of rolling bearing [26,27] and the simplified rolling contact model [28] to develop a quasi-static model for investigating the friction torque of the bearing. Chen et al. [29,30] further considered the curvature of ball and raceway and built a differential slipping quasi-static model for the ACBB along with the consideration of the inertial effects of the ball. Using this model, the relationship between friction coefficient and contact angle of the ACBB under axial load is investigated. More recently, Zhang et al. [31] proposed a tribological model considering the roller skewing to predict the friction torque of dry-lubricated tapered roller bearing under axial force preload or axial displacement preload. Most of these studies are limited to analyze the bearing under pure axial loads (PALs). The main reason is that the PAL model assumes that the internal baring loading is central symmetry, thus only one of the rolling elements needs to be modeled, which greatly reduces the calculation efforts. However, the PAL model is not competent for the analysis of turbopump rolling bearing as it is subject to not only axial force, but also radial and moment loads at the same time.
As another important issue, many scholars have studied the influence of temperature rise on the skidding and dynamic characteristics of rolling bearings. Liu et al. [32] presented a dynamic model for ACBBs with considering the traction coefficient of the lubricating oil, and studied the effects of lubricating oil temperature and other parameters on the skidding behavior of the ball. Gao et al. [33] developed a comprehensive dynamic model to study the skidding and over-skidding mechanism of the rolling element and found that the oil temperature is the key influencing factor of over-skidding. The hydrodynamic theory based model is only applicable to fluid lubricated bearings. For turbopump cryogenic bearings, there are mainly two ways of heat generation: (a) friction between rolling elements, raceways and cages; (b) drag and churning losses of the cryogenic fluid [34]. Hiromitsu et al. [35] found that while the friction is the main source of bearing heat at low speeds, friction and fluid are equally important at high speed, in which the heat generated by the fluid churning accounts for more than 30% of the total heat. Although the local high temperature caused by ball skidding and spinning may produce dark oxidation areas on the surface of the rolling element, the bearing will not be damaged immediately under the cryogenic coolant (e.g., 20 K for the liquid hydrogen and 50 K for the liquid oxygen) [36]. According to the experimental results of NASA Bearing and Seal Material Tester (BSMT), when the LN2 flow rate is higher than 9.2 lbs/s, the bearing can still operate stably for more than 900 seconds even if temperature spikes and rolling element wear have occurred [37]. Therefore, unless severe wear or damage occurs, the temperature rise has limited effect on the steady-state skidding and spinning characteristics of the cryogenic bearing.
Based on the above analysis, in order to carry out the research on the turbopump ACBBs, this paper The rest of this paper is organized as follows: Section 2 is the derivation process of the theoretical model. Based on the established model and solving strategy, Section 3 validates the model and presents the analysis of the skidding and spinning characteristics of the ACBB. Finally, Section 4 summarizes the study.

Theoretical model
In general, the analytical accuracy of kinematical and tribological performance of the bearing is dependent on the calculation of contact forces and traction forces. In this section, the fundamental theory of rolling contact traction is briefly introduced firstly; then, the equilibrium equations of the ball and inner ring under axial and combined loads are derived. Finally, a comprehensive solution strategy is proposed to ensure the efficient convergence of the model.
For the convenience of modeling, ignoring some less important factors, the assumptions made in this paper are as follows: (1) Only the interaction between the inner/outer raceway and the rolling element of the ACBB is considered, that is, the influence of cage, seal ring, dust cover, and other parts on the kinematic and mechanical properties of the bearing is ignored.
(2) Ignoring the influence of the surface microstructure of the contact bodies as well as that of the cryogenic liquid medium on the friction characteristics of the contact surfaces. It is also assumed that the rolling element to inner/outer raceway contact is dry contact which obeys the Amontons-Coulomb friction law.
(3) The influence of friction thermal effect between contact surfaces is neglected, and the material hysteresis effect of the bearings is ignored.
(4) Using the rolling contact simplification theory, the two rolling contact bodies (ball and inner/outer raceway) are assumed to be quasi-identical [28].

Sliding and spinning ratios
To describe the orbital and self-revolution movements of the ball, two Cartesian reference systems are established, as shown in Fig. 1(a). The first reference system is fixed at the bearing center, with its z-axis coinciding with the bearing axis. The second reference system is a moving coordinate system (o-x b y b z b ) with its z-axis parallel to the bearing axis and the origin attached to the geometry center of the ball, based on which the ball has three translational movements x b , y b , z b and three angular velocities b ,   cos sin sin cos cos where  and  are the attitude angle and the yaw angle of the ball, which can be calculated by Using the rotational components, the sliding and spinning ratios of the ball in both inner and outer contacts are formulated as Eqs. (4)-(9):  According to Eqs. (4)-(9), the total sliding ratio can be calculated as

Traction force and torque
Traction forces of dry-lubricated ball bearings are calculated by utilizing the simplified rolling contact theory [28]. According to the superposition principle of contact mechanics, the elastic deformation of any point is obtained from the integration of deformation of the whole contact area [38]. For reducing calculation effort, it is assumed in the simplified rolling contact theory that the elastic deformation of a mass point is only related to its own traction force [39], so that each point in the contact is independent, which is known as the "bristle" model [40], as shown in Fig. 2(a). In this way, the traction force of each point can be obtained through its own deformation. Based on the above assume, the tangential traction force can be given as [28]: where L is the tangential flexibility of the contact surface, u is the tangential deformation, P is the tangential friction force. The elliptical contact area is divided into two parts, namely, the sliding area (A s ) and the adhesion area (A a ), as illustrated in Fig. 2 It is reasonably assumed that the sliding at the front edge of the elliptical contact spot is zero [41]. Based on the FASTSIM algorithm [42], the contact area is divided into a finite number of small blocks, and each block is replaced by a mass point at its center, as shown in Fig. 2(c). The traction force of each mass point can be iteratively obtained by where s is the relative sliding rate of contact particles, and c is the creep rate. When traction forces for all mass points are obtained, the total traction force can be integrated by In addition, the concentrated traction torque can be formulated by The rolling contact theory and the FASTSIM algorithm can be found in detail in Refs. [23,24,28,41,42].

Equilibrium equations of ball and inner ring
As shown in Fig. 3(b), the ball is balanced by contact forces, traction forces, centrifugal forces, traction torques, and gyroscopic torques. Equilibrium equations for the ball in three translational directions are given as Eq. (15): where x T and y T are the longitudinal and lateral traction force, Q is the normal contact force and c F is the centrifugal force, subscripts i and o denote inner and outer contact, respectively. The ball equilibrium equations in three rotational directions can be given as where , However, if the inner ring is subjected to combined loads, it will move in five directions along the coordinate axis (x, y, z, x  , y  ), as shown in Fig. 3(a).
The equilibrium equations of the inner ring with five DOFs can be expressed in the form of vectors: where bi k T is the matrix that converts the force from the moving reference system to the inertial reference system; b i k F is the force acting on the inner raceway by the ball in the moving coordinate system. F is the 5-DOF load acting on the inner ring. These vectors are expressed as Eq. (23): where k  is the orbital positon of the k-th ball, i R represents the pitch radius of the inner ring.

Solution strategy of the CMDL model
The above mentioned equilibrium equations define the complete motion of an ACBB. Equations (15)- (21) forms the PAL model, which includes only seven equations and can be iteratively solved by calling the FSOLVE function in MATLAB ® . However, when it comes to a combined load, each ball will have its own six equilibrium equations. So the CMDL model will totally contain (6N b +5) equations and variables. For instance, bearing B7218 has 16 balls, so there will be in total 101 equations to be solved, which is difficult to handle in a traditional way. Therefore, a comprehensive solution strategy is proposed here to ensure the efficient convergence, as shown in Fig. 4. The solving process mainly consists of three parts: (1) At the beginning, judge whether the load is  (2) For a combined load, a progressive strategy is adopted. Because the combined load has a great influence on the displacement of the inner ring, the convergence is very sensitive to the initial value, thus the radial and moment loads must be gradually increased from zero, with the initial value coming from the PAL model. In the iterative increase of loads, the solution obtained under the previous load F should be used as the initial value of the increased load F' iteratively, until the increased load is equal to the given load F 0 .
(3) As for the CMDL equilibrium equations, the solving process is divided into two stages: the ball-balancing step and the race-balancing stage. In the first stage, the balls are solved one by one, so that only six equations need to be handled at one step. In the second stage, equilibrium equations of the inner ring are solved after all balls are balanced, so that only five equations are solved at one step. In doing so, the influence induced by the ball number can be completely eliminated.

Compare with the PAL model
To verify the proposed model, the 7218B angular contact ball bearing is analyzed. At first, calculations are performed under an axial load of F a = 5,000 N and a rotational speed of 2,000 rpm. The bearing parameters are listed in Table 1 and the results are listed in Table 2. It can be observed that the rotational speeds of the ball and the cage, as well as the inner/outer contact angles are in good agreement with the reference. The error of spin-to-roll (STR) ratios are below 14% and the yaw angle obtained by the proposed model approaches zero, while that of the literature is 0.0076°.
As shown in Eq. (3), the yaw angle  of the ball is directly related to the spinning component velocity b , x  which indicates the pivot movement of the ball.
Under stable working conditions,  is very small as bx  generally approaches zero, so the RCH theory, which assumes  to be zero, is reasonable most of the time. Note that under PAL conditions, as the results of Ref. [30] are calculated without consideration of the gyroscopic torque and the centrifugal force, the ball will not have any axial movement, thus  should be zero in theory. As a result, by comparison of the yaw angle, it is demonstrated that the proposed model has a higher accuracy in kinematic analysis of the ball. Generally, under the effect of radial or moment loads, the yaw angle of the ball is not equal to zero. At the occurrence of insufficient preload or serious skidding, the yaw angle of the ball may experience a sharp increase, resulting in instability of the bearing. This phenomenon is usually caused by the non-uniform internal loading distribution which will be discussed in subsequent sections.  The above comparison has proved the effectiveness and reliability of the proposed model under axial loading conditions. In Section 3.2, calculations are performed under a combined load by using both the proposed model and the classical JH model, based on both the inner raceway controlling (ICTL) method and the outer raceway controlling (OCTL) method, to demonstrate the superiority of the proposed model under combined loads. obtained from the JH model is slightly smaller than the proposed model, and the outer contact angle obtained from the JH model is slightly larger than the proposed model, which means that the JH model may underestimate the inner contact angle and overestimate the outer contact angle. The attitude angle is in good agreement when μ=0.1-0.5, as shown in Fig. 6(a). It can be found in Fig. 6(c) that the yaw angle obtained from the JH model is always equal to zero, this is because the raceway controlling hypothesis assumes that the ball has no pivot movement. Note that the normal contact force results are almost the same at any friction coefficient or raceway controlling methods, which demonstrates that while the JH model is accurate enough to analyze the kinematical performance and contact forces of the ACBB, it is insufficient to study the spinning and pivot movement (°) (°) for the ball of the ACBBs, which is the limitation of RCH.

Kinematic analysis of ball
The forces acting on the ball of an ACBB are dependent on its kinematic motion, which can be characterized generally by the contact angle, attitude angle, and yaw angle. It can be found in Fig. 5 that the inner/outer contact angle is larger at 90°-270° region. In the presence of radial and moment loads, the ball is subjected to a non-uniform internal loading distribution in a complete rotation, which significantly influences the kinematical performance of the ball. Moreover, it can be seen from Fig. 5(b) that the outer contact angle is less affected by the non-uniform internal loading distribution.
As shown in Fig. 6, it can be found that the friction coefficient has a great influence on attitude and yaw angles. The attitude angle approaches zero at low friction coefficients, which means that the contact force is not large enough to provide traction force that can offset the drag force of the ball, thus skidding occurs. It should be noted that the yaw angle increases incredibly and experiences a great variation only at μ=0.01 as shown in Fig. 6(b). The yaw angle will increase the axial rotation component of the ball, leading to the pivot movement of the ball and results in unstable performance of the bearing, so we must take measures to avoid this situation.
In Fig. 7, the normal contact force circles are biased to the right side, which means that the -90°(270°)-90° region is the heavy loading zone, and the 90°-270° region is the slight loading zone. Moreover, due to the centrifugal effect, the outer contact force is larger than the inner contact force, as shown in Fig. 7(b). The friction coefficient hardly affects the contact force, but leads to a bigger inner contact angle and a smaller outer contact angle, which highlights the effect of the traction force. Similar phenomenon of the internal load distribution is also discussed in Ref. [13] through an oil-lubricated ball bearing model.

Skidding and spinning mechanism
To study the rolling contact skidding and spinning mechanism, the slipping distributions within the elliptical contact spots are drawn. As it can be seen in Figs. 5 and 6 that the ball experiences symmetrical loads in an entire rotation, for the sake of convenience, the slipping distributions are only given at angular positions of 0°, 90°, and 180°. Red arrows in the figures represent sliding vectors of the mass points within the contact spot.
In Fig. 8, under μ=0.001-0.01, all points at the contact areas are sliding and point of the upper and lower areas slides in opposite directions, which indicates the spinning movement of the ball. Moreover, it can be observed that the ball spins counterclockwise relative to the inner raceway and clockwise relative to the outer raceway. Interestingly, with the change of friction coefficient μ from 0.001 to 0.01, the outer relative spin direction begins to change in some of the angular positions ( Fig. 8(b)), indicating the occurrence of macro-slipping. As μ continues to increase, adhesion begins to appear (Fig. 8(c)), which implies the appearance of micro-slipping (creepage). When μ=0.1, the outer relative spinning direction at all angular positions turns to be counterclockwise, and the adhesion area gets larger as μ continues to increase (Fig. 8(d)). With the increase of friction coefficient, the inner contact keeps almost always spinning, in the same direction, while the outer contact changes from spinning to skidding and finally to creepage.
The main feature of creepage phenomenon is the adhesive area in the contact spot between the rolling element and the raceway, which indicates the attenuation of the micro-slipping of the ball relative to the raceway, that is, the enhancement of the control ability of the raceway to the ball. This situation usually occurs under the condition of large axial force, high friction coefficient, or high speed. Compared with the hydrodynamic traction model, the area of adhesive area can well indicate the control degree of the raceway to the ball, instead of only applying the sliding and spinning ratios to predict the ball slipping state, e.g., Refs. [13] and [33]. It can be concluded that skidding can be eliminated by increasing the friction coefficient. However, under a medium friction coefficient (e.g. 0.01), the ball spins in an unstable way as its outer relative spin axis switches rapidly when it runs from 0° to 90°, which is unfavorable for the stable operation of the ACBB, as shown in Fig. 8(b). This phenomenon is related to the abnormal increase of the yaw angle in Fig. 6(b).
Based on the above analysis, here we focus on the skidding and spinning mechanism for μ=0.01. Note that although the relative spin axis changes in different loading zones, adhesion area is not observed in the contact spots, which implies that the unstable state cannot be easily explained as the transition from skidding to creepage. The interaction mechanism of traction force and kinematical movement is described in Fig. 9. Due to the non-axial loads, it can be seen from Fig. 7 that the ball bears the maximum internal load at 0° and as it gradually enters to the 180° position, the internal load decreases to minimum. The friction coefficient and the contact force are the major factors that affect the ball motion. When the friction coefficient is relatively low, the traction force provided by the contact area is too small to prevent the ball from skidding. Note that the change of outer contact angle in a whole rotation is not obvious, as shown in Fig. 5(b), as the ball is at the critical situation of skidding, according to Eqs. (1) and (9), a slight change of attitude angle and yaw angle at this condition may lead to the change of direction of the ball-outer raceway relative spin. And the change of spin direction will significantly affect the direction of the traction force of the ball, and greatly change the equilibrium state of the ball. Finally, the change of the equilibrium state results in a large variation in the yaw angle. It should be noted that the spin axis reversal phenomenon will lead to a rapid change of spinning friction direction of the ball relative to  | https://mc03.manuscriptcentral.com/friction the outer raceway, which is unconducive to the performance of the ball and will destroy the solid lubricating film.

Effects of friction coefficient
The total inner/outer contact sliding ratios under different friction coefficients are calculated by Eq. (10), as shown in Fig. 10. One can easily find in Fig. 10(a) that the sliding ratio of μ=0.01 and μ=0.1 are much larger than those of others, implying the occurrence of skidding. Moreover, the variation trends of them are opposite to those of others, which means that the skidding behavior is more obvious in the heavy loading zone. Similar phenomena can also be found in the yaw angle performance in Fig. 6(b). Note that the ball has a larger axial displacement in the heavy loading zone, so this will result in a bigger value of the yaw angle, then change the ball rotational components and finally influence the ball sliding behaviors. Figure 11 illustrates the effect of friction coefficient on spinning ratio of the ball in a complete rotation. when μ=0.1, with the increase of friction coefficient, the magnitude of the inner spinning ratio reduces to minimum first and then increases with the further increase of μ, as shown in Fig. 11(a). The opposite phenomenon can be observed in the outer spin velocity in Fig. 11(b) for μ=0.1. It is interesting that the outer spinning ratio is negative when μ=0.1 to μ=0.5, while it is positive when μ=0.01 and μ=0.001. As explained in Fig. 9, this is because skidding happens www.Springer.com/journal/40544 | Friction as low friction coefficients greatly change the attitude and yaw angles of the ball. From the above results, it can be concluded that the friction coefficient has a significant influence on skidding and spinning characteristics of the ball. Moreover, ball skidding can be suppressed by increasing the friction coefficient; however, the spinning ratio is not always decreased with the increase of friction coefficient. In the case of this study, the optimum friction coefficient to get the minimum spinning ratio is μ=0.1. Figure 12 shows the variation rule of the sliding ratio of the bearing with the speed, from which it can be seen that the sliding ratio under low friction coefficient is larger at all speeds, indicating that the friction coefficient has a greater impact than the speed. With the increase of rotating speed, the sliding ratio of inner and outer contacts will first increase to a peak and then drop. The difference is that the sliding ratio of the inner contact gradually increases after the peak value, while that of the outer contact approaches zero. The curve of the inner contact sliding ratio can be divided into three stages: AB, BC, and CD, as shown in Fig. 12(a). In order to further analyze the influence mechanism of rotating speed on skidding, the slipping distributions at four rotating speeds, namely A, B, C, and D, are drawn, as shown in Fig. 13. According to Fig. 13(a), when the friction coefficient is equal to 0.001, skidding exists during the whole speed range. By observing the distribution law of internal contact sliding speed when μ=0.001, it can be found that when the rotating speed is 1,000 rpm, the sliding speed of the upper and lower parts of each mass point in the contact spot is inconsistent. With the increase of rotating speed, the sliding speed of the mass points in the upper half increases gradually, forming an obvious spinning performance. This proves that in Fig. 12(a), starting from point B, spin gradually becomes the main cause of inner contact slipping. The main reason for the spinning is the gyroscopic motion of the rolling element, so the above analysis shows that the gyroscopic motion is the main reason for the inner contact skidding at high speeds. Furthermore, the sliding ratio curves for skidding (µ is equal to 0.001 and 0.01) and creepage (µ is equal to 0.1, 0.3, and 0.5) eventually converge to curves with the same slope as the speed increases. Thus, a predicting line can be drawn to help predict the slipping behavior under high speed (larger than 6,000 rpm), as shown in Fig. 12(a), i.e., when the sliding rate is above the prediction line, it indicates skidding otherwise it indicates creepage.

Effects of high rotational speed
According to Fig. 12(b), it can be inferred that under different friction coefficients, the sliding ratio of the outer contact will reach maximum at a certain speed, then gradually approaches a small value at high speeds, results in a threshold peak peed that indicates the skidding behavior. In this process, the greater the friction coefficient, the greater the threshold speed is. For relatively high friction coefficients, the  | https://mc03.manuscriptcentral.com/friction slope of outer contact sliding ratio curves are close to each other, mainly influenced by the size of adhesive area, as shown in Fig. 8(c). The above phenomenon shows that spinning in the outer contact is suppressed at high speed. According to Fig. 13, under low friction coefficients and at high speeds, the sliding directions of the mass points in the upper and lower parts of the outer contact spot become inconsistent again, confirming the above inference. In particular, when the creepage phenomenon occurs, the sliding velocity directions of the upper and lower mass points of the outer contact are almost the same, indicating the disappearance of the spinning motion.
Under low friction coefficients, the initial increase of rotating speed leads to a significant increase of spinning ratio, as shown in Fig. 14. With the increase of rotating speed, the amplitude of inner spinning ratio continues to increase, while the outer spinning ratio gradually approaches zero, which agrees with the hypothesis of OCTL. This is the reason why Harris [27] suggests that the OCTL is more appropriate at high rotating speed. The main reason for this phenomenon is that the outer contact force is greatly enhanced by the centrifugal effect at high speed. However, the controlling ability of the raceway is greatly affected by the friction coefficient as well as the loads. When the friction coefficient is not bigger than 0.01, the rotating speed must be as high as 40,000 rpm to get a small outer spinning ratio, as shown in Fig. 14(b).
Previous studies have shown that the rotating speed of ball bearing cage may be higher than the theoretical value obtained under pure rolling assume, which is often called "negative-skidding" or "over-skidding" phenomenon. The over-skidding phenomenon can be roughly detected by using the cage to rotor speed ratio (CSR) [33]. With the proposed model, the cage speed is represented by the average orbital speed of all balls, and the CSR results with increasing inner ring speed under different friction coefficients are given in Fig. 15(a). As shown in Fig. 15(a), when the inner ring speed is lower than 5,000 rpm, the CSR curves are very close to each other, approximating the pure rolling value. Nevertheless, all curves show an increasing trend with the further increase of inner ring speed, indicating the appearance of over-skidding phenomenon. The established equilibrium equations mainly describe the steady-state characteristics of the bearing, and the drag effect of cage is not considered, thus the cage can finally reach the theoretical speed even if skidding may have happened. In Ref. [33], the author used the dynamic model called KH-THD to analyze the time history response of the cage speed under different axial forces at a constant working speed, as shown in Fig. 15(b). Under an axial load that is larger than 50% of the maximum axial load (MAL), the rotating speed of bearing cage will gradually converge from the value lower than the theoretical one to a higher value than the theoretical one. It is illustrated that although there will be skidding phenomenon during operation, the CSR will eventually convergence to the over-skidding state, which proves the results of Fig. 15(a).
The rotational speed of the cage is mainly affected   by the orbital motion of the rolling element, which is related to the two augular speed components of the rolling element, i.e., ω by and ω bz . In order to further study the over-skidding mechanism for dry-lubricated ACBBs under different friction coefficients and speeds, the calculation results of the two augular speed components of the rolling element are given in Fig. 16. With the increase of inner ring speed, ω by firstly increases gradually, as shown in Fig. 16(a). At low friction coefficient, it decreases sharply after reaching a peak, just as the peak phenomenon observed in Fig. 12(a). According to Eq. (1), ω by is positively correlated with the attitude angle β, and the sharp decrease implies the ball skidding behavior.
According to the conservation of angular momentum, the reduction of ω by is bound to bring the increase of ω bz (ω bx is latetively small and can be ignored), so ω bz obtained under low friction coefficients will be significantly larger than the result of high friction coefficients, as shown in Fig. 16(b). As it can be seen from Fig. 16(a) that ω by of high friction coefficients (creepage) also decreases slowly after reaching a certain peak, resulting in a slight increase of the slope of ω bz curve, as shown in Fig. 16(b). Because the external contact angle and the outer contact slipping are very small at high speed, the increase of ω bz eventually leads to a higher rolling speed on the outer raceway, and finally results in the increase of cage speed. The peak effect at high friction coefficients indicates the transfer of creepage phenomenon of ball, that is, from internal contact creepage to external contact creepage, as shown in Fig. 13. The transfer of creepage can better explain how the rolling element is controlled by the raceway under high friction coefficients, which is not available in the oil-lubricated model. In summary, the phenomena of skidding and over-skidding of ACBBs are mainly attributed to the gyroscopic effect and the conservation of angular momentum of the ball. To be specific, under low friction coefficients, the traction force cannot offset the gyroscopic torque of the ball, resulting in macro slipping of the ball and a sharp decrease of the attitude angle, which consequently leads to over-skidding. Under larger friction coefficients, though traction force is able to offset the gyroscopic torque and no macro-slipping occur, micro-slipping of the ball due to creepage and spinning also leads to a slight reduction of the attitude angle, and resulting in over-skidding.

Effects of combined radial and moment loads
In order to investigate the comprehensive effect of radial and moment loads on skidding and spinning performance, the analysis results of sliding and spinning ratios under different radial and moment loads are calculated, and the curves and contour plots are drawn. According to the previous analysis, since the skidding characteristics of inner and outer contacts are almost the same, in order to save space, only the www.Springer.com/journal/40544 | Friction outer contact results are given here. Except for radial and moment loads, other working condition parameters are the same as that in Section 3.2.

Effects of radial and moment loads on skidding and spinning performance
Since the non-axial loads have a significant impact on the internal load distribution of ACBBs, the analysis results of the ball in both heavy loading zone (0°) and slight loading zone (180°) are given in this section. The non-uniform internal load distribution increases the load in the heavily loaded area and meantime reduces the load in the lightly loaded area, thus the curves present opposite trends. In order to prolong the service life of the bearing, effort should be made to avoid large changes in the sliding and spinning performance between heavy and slight loading zones. Therefore, a trade-off between the performance of heavy loading zone and slight loading zone becomes crucial for the design of loads. The best radial load can be found as the intersections of the curves. Corresponding to μ=0.001 and μ=0.01, the radial loads to get an equal sliding ratio are 16,000 N and 19,500 N respectively, as shown in Fig. 17(a). In addition, the radial loads for equal spinning ratios are 16,000 N and 19,500 N respectively, as shown in Fig. 17(b). It can be seen from Fig. 18 that the difference between the curves of heavy loading zone and slight loading zone is more obvious, indicating that the effect of non-uniform internal loading distribution caused by the moment load is more significant. According to the intersection of the curves, the moment loads corresponding to the most stable sliding ratios are 350 N·m and 35 N·m, respectively, and that of the spinning ratios are 480 N·m and 700 N·m, respectively.   Although the above analysis is already able to find the best load to avoid unstable skidding or spinning, different analysis results may be contradictory. Therefore, it is necessary to study the interaction mechanism between radial and moment loads.

The effect of combined radial and moment loads
To study the combined effect of the radial and moment loads, different combinations of radial and moment loads normalized to the maximum radial load (40 kN) and the maximum moment load (1 kN·m) are applied, and contour diagrams of ball sliding and spinning ratios are drawn in Fig. 19. Figure 19 shows the contour maps of the sliding ratio and spinning ratio of the outer contact with μ=0.001, in which subplots (a) and (b) are the results at 0° position corresponding with Fig. 7 (representing the heavy loading zone) and subplots (c) and (d) are that at 180° position corresponding with Fig. 7 (representing the slight loading zone). It can be found that the sliding ratio in the heavy loading zone increases first and then decreases with the radial force, and the peak position is affected by the moment load to a certain extent, as shown in Fig. 19(a). In addition, the spinning ratio of the heavy loading region decreases linearly with the increase of both radial and moment loads, as shown in Fig. 19(b). It can be found in Figs. 19(c) and 19(d) that the increase of radial and moment loads can both increase the sliding and spinning ratios to a certain degree, but the position of the maximum value is different. In terms of design, the combined load should avoid the points with large sliding and spinning ratio, as well as large variation between heavy and slight loading zones. According to the red colored regions in Figs. 19(a)-19(d), a relatively stable sliding/spinning range, which can be recognized as the optimum loading range, is obtained through the composition of all diagrams, as shown in Fig. 19(d).

Conclusions
In the present study, an analytical model of dry-lubricated ACBB under combined load is established, in which the bearing traction forces are calculated utilizing the simplified rolling contact theories. Using the proposed model, effects of friction coefficient, rotational speed, and the combined loads Fig. 19 Effects of combined radial and moment loads: (a) sliding ratio in heavy loading zone, (b) spinning ratio in heavy loading zone, (c) sliding ratio in slight loading zone, (d) spinning ratio in slight loading zone.
www.Springer.com/journal/40544 | Friction on the skidding and spinning characteristics of the ACBB are investigated. Some meaningful conclusions can be summarized as follows: (1) Compared with the existing pure axial force model and the JH model, the proposed model is proved to be able to accurately calculate the kinematical performance of the ball and effectively analyze the macro/micro sliding and spinning characteristics of the bearing, which is helpful for the design and research of the dry-lubricated angular contact ball bearing.
(2) The friction coefficient significantly affects the skidding and spinning characteristics of the ball. The attitude angle approaches zero at low friction coefficients, implying the occurrence of skidding, which can be eliminated by increasing the friction coefficient. However, under a specific friction coefficient, the non-uniform internal load distribution could result in a reverse of the outer spinning direction, which is unfavorable to the stable operation of the bearing.
(3) The rotational speed has a great influence on the sliding ratio and spinning ratio. With the increase of rotating speed, the sliding ratio will experience a nonlinear increasing trend, which is mainly attributed to the insufficient traction force. Spinning ratio gets larger in the inner contact and approaches zero in the outer contact due to the gyroscopic effect and centrifugal effect at high speeds, which coincides with the outer raceway controlling hypothesis.
(4) The sliding ratio changes nonlinearly with the increase of radial loads. With the increased radial or moment load, the sliding ratio and spinning ratio generally decrease in the heavy loading zone, and increase in the slight loading zone. Through the study of skidding and spinning ratios under combined loads, the most favorable combined loading condition for the bearing can be found, which can provide some guidance for the tribological design of dry-lubricated ACBBs.