Starved lubrication analysis of angular contact ball bearing based on a multi-degree-of-freedom tribo-dynamic model

When the oil supply is not adequate to maintain the ideal lubrication, angular contact ball bearing will enter into the starved lubrication regime resulting in the potential performance degradation and consequently the severe failures. To study the effects of starved lubrication on the performance of angular contact ball bearing, this paper first proposes a multi-degree-of-freedom (DOF) tribo-dynamic model by introducing five-DOF inner ring, six-DOF balls, and six-DOF cage. The model considers the starved lubrication in the ball-raceway contact and the full multi-body interactions between the bearing components. With different ball-raceway starvation degrees being analyzed, the effects of starved lubrication on the bearing tribo-dynamic performance are first revealed. By comparison, it is found that the oil film thickness, the skidding performance, and the traction forces in the ball-raceway contact are significantly influenced by the starvation degrees. It is also found that the starvation-induced change of the ball-pocket contact force is dramatical under combined loads, and the maximum contact force under this load condition increases with the increasing starvation degrees.


Introduction
As the core supporting component of much rotating machinery, angular contact ball bearings are frequently required to operate under various arduous conditions with long service life and high reliability. In the analysis of these conditions, it is often assumed that the lubricant supply for the bearing is adequate and the interacting surfaces are under fully flooded lubrication [1][2][3]. However, in the actual applications of the bearing, it is extremely difficult to guarantee this ideal lubrication condition especially for the higher pair with large entrainment velocity. When the bearing is under starved lubrication, the inlet pressure buildup in the contacts will be delayed and the oil film thickness will decrease to some extent [4]. This will result in the asperity contact, and consequently the serious wear and the increase of friction heat within the bearing.
For this reason, the starvation phenomenon in contact elements or rolling bearings has long been the concern of researchers both in experimental investigation and in theoretical analysis [5][6][7]. The ball-and-disc and roller-and-disc configurations with optical interferometry were the preferences for observation in the early stage, which were later extended to study the lubricant replenishment [5,8], the friction coefficient [9,10], and the film thickness decay [11][12][13]. Wedeven et al. [4] and Chiu [14] experimentally observed the oil film in the ball-flat race contact and found that the interference fringes remove to the locations of greater thickness owing to the starved lubrication. It was also found that the degree of fluid film starvation increases with both rolling speed and viscosity. Based on the further experiments from Wedeven, a starved film was observed to result in a greater traction than a flooded film for the same slide roll ratio [15]. To provide the starved conditions efficiently, Svoboda et al. [16] improved the optical test rig, in which the roller-disc contact is applied to supply the defined oil layer to the ball-disc contact. However, the oil supply and the oil layer distribution in these configurations are much different from those in actual applications of the bearings. Not only the mechanical structures but also the oil supply system can dramatically affect the starvation characteristics within the bearing [17,18]. For improvement, Hargreaves and Higginson [19] conducted the first experimental work on the actual rolling bearing under starved lubrication. The experiment results indicated that the friction torque and operating temperatures of the cylindrical roller bearing are much reduced at low lubricant supply rates. Later on, the film thickness [20], the cage gross skidding [18], the fully flooded and starved transition [21], and the rotating speeds [22] caused by starved lubrication were further discussed based on the bearing experiments. In the field of theoretical analysis, most of the researches focus on the starved elastohydrodynamic lubrication (EHL) in line and point contacts. In the early numerical solution, the Reynolds equation and the Reynolds boundary conditions were applied with the fluid inlet level describing the starved regime [23]. Some researchers considered the thermal effects, the non-Newtonian behavior, and the asperity contact in the subsequent analysis [24][25][26]. To model the starved regime more accurately, Lubrecht et al. [27] and Damiens et al. [28] presented a new starvation model for the starved EHL, in which the ratio of the gap filled with oil is considered in the Reynolds equation. van Zoelen et al. [12,13] further proposed a thin layer flow model which can be applied to predict the decrease of total available lubricant on the tracks and the film thickness decay. Apart from the above aspects, great efforts have been devoted to the determination of film thickness formulas under starved lubrication [5]. Generally, there are two methods to consider the effects of lubricant starvation in the formulas, which are modeled with the meniscus distance and the mass flow rate respectively. In McCool's presentation, the Archard-Cowking film thickness formula was modified by adding the inlet heating factor and the starvation reduction factor [29], and the starvation reduction factor was exactly expressed with the meniscus distance. By contrast, Hamrock and Dowson [30] proposed a more concise correction coefficient for starved elastohydrodynamic point contacts based upon their work under fully flooded lubrication. Olaru and Gafitanu [31] developed a complex analytical model to reveal the starvation mechanism in ball-race contact. Based on their theoretical results, the reduction factor of the film thickness was obtained which considers the influence of starvation and thermal effects together. Nijenbanning et al. [32] also proposed the formula to predict the central film thickness in elliptic contact, which incorporates asymptotic behaviors and can be applied for all conditions. However, in real applications, the position of the inlet meniscus is hard to be measured [33], which makes it inconvenient to evaluate the starvation characteristics. To study the effects of inlet supply starvation on the film thickness in EHL point contacts, Chevalier et al. [33] presented an alternative approach in which the amount of oil available on the surfaces is applied to define the degree of starvation. Masjedi and Khonsari [34] further considered the effects of asperity contact and proposed the formulas for starvation in the mixed EHL regime.
The above formulas provide an efficient method to analyze the influence of starvation between rolling elements and raceways. However, for the angular contact ball bearing, the inner lubrication behaviors are closely associated with the multi-degree-of-freedom (DOF) movements and the bearing components will interact with each other during operation. To analyze the bearing performance accurately, the multi-DOF mechanical models revealing the multi-body interactions are requisite. So far, the quasi-static model [35], the quasi-dynamic model [36], and the dynamic model [37,38] have been applied in the bearing analysis.
The quasi-static model shows high efficiency in predicting the load distribution but is not accurate in obtaining the motions of the bearing components [2]. By contrast, the quasi-dynamic model considers the influence of cage and all the necessary inertia items [36]. But the model is based on the force equilibrium and shows limitations in considering the time-varying characteristics. To describe the time-varying motions of the bearing components, the dynamic model has been further established and improved since Gupta conducted the fundamental work [39]. With the aid of these models, the bearing performance under various operation conditions can be predicted and many influencing factors, such as angular misalignments [36] and geometrical imperfections [38], can be analyzed.
However, to the best of the authors' knowledge, the applications of the multi-DOF mechanical models with oil lubrication are all based on the assumptions of fully flooded lubrication. The starved lubrication analysis presented by Masjedi and Khonsari [34] is limited to the pure point and line contacts, which cannot be applied to reveal the bearing performance. In a word, the numerical analysis of the complete bearing under starved lubrication has not been found yet. Under the circumstances, in this paper, the impacts of lubricant starvation in angular contact ball bearings are investigated. Considering the mutual influence of the lubrication behaviors and the multi-DOF movements, this paper proposes a multi-DOF tribodynamic model through mechanical analysis. The model is established by introducing five-DOF inner ring, six-DOF balls, and six-DOF cage, which considers the complete multi-body interactions between the bearing components. The lubrication between the cage pocket and the ball is modeled with the formulas in Shaberth software, while that between the guiding surface and the cage land is described with the short bearing theory [36,37]. In practice, the starved lubrication within the bearing can appear in all the friction pairs owing to the limited oil supply. For simplification, this study focuses on the lubricant starvation in the ball-raceway contacts. To model the starved lubrication regime in the ball-raceway contact, the Masjedi formula which combines the mass flow rate and the mixed EHL theory is applied to determine the oil film thickness [34]. By solving the proposed multi-DOF dynamic model under different situations, the tribo-dynamic performance of the bearing is obtained. After the comprehensive analysis with different starvation degrees, the effects of lubricant starvation are finally revealed.

Theoretical model
To establish the multi-DOF dynamic model, eight essential coordinate systems are defined as shown in Fig. 1. These coordinate systems are applied to describe the full multi-body interactions in the bearing, the details of which are given as follows: (1) The inertial coordinate system (O ) xyz  is fixed at the outer ring center, the x axis of which is along the bearing axis. Based on the inertial coordinate system, the equilibrium equations of the inner ring and the cage can be determined.
(2) The azimuth coordinate system is defined to establish the equilibrium equations of the balls. In this coordinate system, the b x axis is parallel to the x axis and the b y axis points radially outward.  x axis is along the semimajor axis of the contact ellipse, the  y axis is along the semi-minor axis of the contact ellipse, and the  z axis is along the normal direction.
(4) The cage coordinate system are defined to describe the angular displacements of the shaft and the cage respectively.

Starvation in ball-raceway contact
To describe the starvation in the ball-raceway contact (see Fig. 2), the oil film thickness and the traction in the starved regime need to be calculated. In the lubrication analysis, the oil film thickness and the traction forces in the starved cases can be obtained by solving the Reynolds equation, the deformation equation, the asperity contact equation, and the oil supply equation [34,40]. However, this method is too time-consuming to be coupled with the dynamic model directly. For this reason, the oil film thickness in the starved point contact is evaluated with the approach proposed by Masjedi and Khonsari [34,41].
In their approach, the Zhao-Maietta-Chang (ZMC) model is applied to consider the influence of asperity contact and the starvation degree is employed to where  represents the dimensionless surface roughness and V denotes the dimensionless hardness number. When the lubricant starvation occurs in the ball-raceway contact, the oil film thicknesses will decrease to some extent. According to Masjedi and Khonsari's numerical analysis, the oil film thicknesses and the asperity load ratio in the starved cases can be modified from those under fully flooded lubrication condition, where  c h represent the ratio of starved to fully-flooded film thicknesses.  denotes the starvation degree and can be described with the oil flow rates: where S m  represents the mass flow rate in the starved condition and F m  represents the mass flow rate in the fully flooded condition.
When the oil film thickness under starved lubrication is calculated with the above analysis, the ball-raceway interaction can be further determined. As shown in Fig. 2, the contact force exists in the normal direction, while the traction appears in the tangent plane. The contact force and the pressure distribution can be evaluated with the Hertz contact theory [36,42], is the maximum contact pressure, a and b represent the semimajor and semi-minor axes,   ( , ) x y is the point position in the contact ellipse.
Under starved lubrication, the oil film thickness in the ball-raceway contact will decrease to some extent, which will result in the mixed lubrication [34]. In this case, the traction forces and the traction moments need to be modified from the authors's previous work [36,37]. In this study, the oil film thickness throughout the contact area is assumed to be constant [42]. The traction forces and the traction moments can be evaluated as [37,43]: where  represents the oil viscosity,  / , p and a p denote the hydrodynamic pressure and the asperity contact pressure respectively, which can be calculated as [43,44]:

Interaction between cage and other components
The cage in the bearing not only interacts with each ball through the cage pocket, but also interacts with the guiding surface (see Figs. 3 and 4). In the authors' previous work, only the translational displacements of the cage in the three directions are considered in the dynamic model. To improve the model accuracy, six DOFs of the cage are considered in this study and the cage displacement vector is described as Based on the analysis of multi-body movements, the displacement vector of the pocket center relative to the ball center in the pocket coordinate system where pj T represents the transform matrix from the coordinate system  cos cos sin sin sin cos sin cos cos cos sin cos cos sin sin cos sin  (19) where pj  represents the transformation angle from the coordinate system i  to the coordinate system  is the ball position angle, ora R is the radius of locus of outer raceway groove curvature centers, aj X and rj X are the axial and radial distances between the ball center and the outer raceway groove center.
Based on Eq. (14), the contact force between the ball and the cage pocket can be calculated as where c K and n K are the lubricant stiffness and the contact force-deformation coefficient respectively, p C denotes the pocket clearance. cj Z represents the distance normal to the contact plane and can be obtained as Furtherly, as shown in Fig. 3, the vector form of the contact force between the ball and the cage pocket in different coordinate systems can be expressed as where a i T represents the transform matrix from the to the inertial coordinate system and can be calculated as  (26) Apart from the normal force, the traction forces and the traction moments in the tangent plane also need to be determined. For convenience, a new coordinate system marked in dark blue is temporally established for the ball-pocket contact as shown in Fig. 3 and the transformation angle between it and the coordinate system The rotation speed in this coordinate system is described as where b b ω j represents the rotation speed in the azimuth and the model used in the authors' previous work, the traction forces can be obtained [36,37]. Then, the traction forces and the traction moments acting on the ball can be expressed as (32) where R j F  and R j F  represent the rolling components, S j F  and S j F  represent the sliding components, N j F  and N j F  denote the normal forces, d is the ball diameter. The traction forces and the traction moments acting on the cage can be expressed as During operation, the cage with the rotation speed c  is supported by the guiding surface, which will greatly affect the dynamic performance of the cage (see Fig. 4). However, the solution of the lubrication area between the cage land and the guiding surface is time-consuming. To improve the calculation efficiency of the dynamic model, the short bearing theory is adopted. According to the authors' previous work, the oil film forces  cy F ,  cz F , and the friction moment  cx M in the cage coordinate system can be calculated [37]. To establish the dynamic differential equations in the inertial coordinate system (O ) xyz  and the body-fixed coordinate systems , the forces and the moment need to be transformed [36,37],

Differential equations of motion
In the current dynamic model, six DOFs of the ball, five DOFs of the inner ring, and six DOFs of the cage are considered. The ball not only interacts with the raceways and the cage pocket, but also interacts with the oil-air mixture, resulting in the drag force in the cavity of the bearing. Owing to the orbit speed of the ball, the centrifugal force will appear and influence the ball-raceway contact. In addition, the non-parallel relationship between the rotation speed and orbit speed of the ball will further result in the gyroscopic moments. In terms of the inner ring, the angular misalignments (angular displacements) will occur owing to the external loads which can dramatically change the load characteristics. The angular displacements of the cage will also change the interaction between the cage pocket and the ball.
Considering the above influencing factors, the dynamic differential equations of the j th ball can be expressed as [37]: where m represents the mass of the ball, c represents the damping coefficient, J is the rotational inertia, ij  and oj  denote the contact angles between balls and raceways, cj F is the centrifugal force, dj F is the drag force, , gy M and , gz M represent the gyroscopic moments.
The dynamic differential equations of the inner ring are given by [37]:  The dynamic differential equations of the outer ring riding cage are given by [37]: where c m is the mass of the cage, cage G is the gravity of the cage, c c c ( , , ) x y z I I I are the principal moments of inertia,

Solution procedure
Based on above analysis, the multi-DOF dynamic model is established which considers 6N+11 DOFs of the bearing. The fourth-order Runge-Kutta method is employed to solve the model and a fixed time step is adopted for convenience. After the sensitive test, the time step t  is set as 5 × 10 -7 s. The C# code is developed for the simulations and the solution will be finished when the time t reaches to the designated value. As explained in the authors' previous work, the solution of the dynamic model is extremely sensitive to the initial values and only the appropriate initial values can guarantee the success of the calculation [37]. For this reason, the static model, the quasi-static model, and the quasi-dynamic model are solved to provide the initial conditions before the dynamic model. In the simulations, the angular contact ball bearing with an outer ring riding cage in the highspeed motorized spindle and the PAO base oil are employed. The lubricant density is regarded as a constant and the lubricant viscosity is evaluated with the Roelands equation [43]. Some important parameters of the bearing and the lubricant oil are presented in Table 1, and the flowchart of the solution is depicted in Fig. 5. The complete presentation about the bearing is made in the authors' previous work [37]. In Table 1, the guiding clearance C 1 represents the difference between the guiding surface radius and the cage land radius.

Results and discussion
In this section, the effects of the lubricant starvation in the ball-raceway contact on the tribo-dynamic performance of the bearing are presented. In the www.Springer.com/journal/40544 | Friction authors' previous work, the numerical models are verified with the experimental data [36,37]. Compared with the dynamic model in Ref. [37], the current model considers more DOFs of the bearing and the lubricant starvation in the ball-raceway contacts. For this reason, the verification of the model with the same experimental data under fully flooded lubrication is no longer presented. As the bearing temperature is essential in the simulations, the temperature measurement of the outer ring with proper preload is conducted through the thermal resistances under the ambient temperature of 23 °C. According to the experimental results, the temperature of the outer ring is generally between 20 and 40 °C even at the speed of 6 × 10 4 rpm. Considering the change of the ambient temperature, this paper mainly focuses on the cases of T = 20 °C. When the influence of temperature needs to be discussed, the oil temperature is set as 20, 40, and 60 °C, respectively. In addition, as the occurrences of the starvation on different raceways will result in different consequences, the starvation degree i  in the ball-inner raceway contact and the starvation degree o  in the ball-outer raceway contact are discussed separately. In Masjedi's theory about the starvation in the mixed EHL, the appropriate range of the starvation degree is between 0 and 0.3 for the point contact [34]. Therefore, this paper considers the influence of starvation degrees i  and o  in the same scope. Based on above considerations, various simulations are conducted under different rotation speeds, oil temperatures, and external loads. Finally, the impacts of the ball-raceway starvation on the oil film thickness, the skidding behavior, the ballraceway traction, and the ball-pocket contact force are revealed.

Oil film thickness between balls and raceways
According to the EHL theory and Eqs. (3) and (4), both the external loads and the starvation degrees can influence the oil film thickness in the ball-raceway contact. In addition, the lubricant starvation can change the orbit and rotation speeds of the balls. As a consequence, the lubricant starvation on one raceway (for example, the inner raceway) can also affect the oil film thickness in the opposite ball-raceway contact (for example, the outer raceway).
The change of the oil film thickness in the ball-raceway contact under pure axial load is depicted in Fig. 6, in which the bearing speed and the oil temperature are set as 15,000 rpm and 20 °C, respectively. As illustrated in Figs. 6(a) and 6(b), the oil film thicknesses between the balls and the raceways decrease with the increase of the axial load from 25 to 400 N. However, the variation range caused by the load is not very large and the change rate is no longer obvious at relatively heavy load. By contrast, the impacts of the ball-raceway starvation are much greater, which will dramatically change the variation range of the oil film thickness (see Fig. 6(a)). As can be predicted, the lubricant starvation in the ball-inner raceway contact even has some influence on the film thickness in the ball-outer raceway contact at light load (see Fig. 6 decreases with the increase of  o , which is different from that in Fig. 6(b). In addition, compared to the influence of o  , the effects of the axial load on the oil film thickness co, j h also seem to be very limited.
Under combined axial and radial loads, the oil film thickness in the ball-raceway contact will also be influenced by the radial load. According to the results presented in the authors' previous work [37], when the radial load reaches to 200 N, the ball-raceway contact force will change appreciably with the ball position angle. combining with the results under pure axial load, it can be obtained that the starvation degree has a greater influence on the oil film thickness in the ball-raceway contact compared to the two types of loads. In Fig. 7(b), there is nearly no difference between the results when the starvation degree i  is set as different values.
It indicates that, the impacts of the ball-raceway starvation on the oil film thickness between the ball and the opposite raceway are negligible under this load condition.
Based on the mixed EHL theory, the change of the oil film thickness will result in the risk of asperity contact and consequently the varying asperity load ratio. To reveal the characteristics of the asperity load ratio a starved L  , Fig. 8 gives the results under the axial load of 400 N. It can be obviously observed that, the a starved L  in the ball-inner raceway contact increases greatly with the increasing starvation degree  i . Besides, the a starved L  is much larger at higher lubricant temperature and lower bearing speed. In terms of the  Hence, in view of the measured outer ring temperature, the hydrodynamic lubrication dominates in operation for the bearing studied even under starved regime.

Skidding characteristics
The skidding velocity i/o, j U between balls and raceways results from the translational sliding velocity si / o, j U and the spinning velocity si / o , j  [36,42], which can also be influenced by the lubricant starvation. Actually, the lubricant viscosity, the oil film thickness, and the skidding speed are the three basic influencing factors to produce the ball-raceway traction. As the effects of lubricant starvation on the ball-raceway load distribution are not significant, the skidding performance and the oil film thickness are closely associated to provide sufficient traction.
Considering the impacts of lubricant starvation, the sliding and spinning characteristics of the ball-raceway contact under pure axial load are presented in Fig. 9. As illustrated in Fig. 9(a), the starvation degree i  has significant influence on the sliding velocity si , j U in the center of the contact ellipse. Under light axial load, the change of si , j U caused by starvation is remarkable and the more serious starvation results in the smaller sliding speed. With the increase of the axial load, the sliding velocity si , j U gradually decreases to zero and the influence of starvation is no longer obvious. Figure 9(b) gives the impacts of the starvation degree i  on the sliding velocity so , j U in the ball-outer raceway contact. It can be observed that the sliding velocity so , j U is nearly not affected by the ball-inner raceway starvation. The same relationship can also be found between the sliding velocity si , j U and the starvation degree o  . By contrast, the occurrence of lubricant starvation has a completely different influence on the spinning velocity and the change trends of the spinning velocity resulting from starvation can be inconsistent. As shown in Fig. 9 To reveal the influence of lubricant starvation on the spinning velocities in detail, the relative changes of the absolute values under pure axial load are discussed here. Figure 10 gives   The changes of the maximum and the minimum skidding speeds in the ball-raceway contacts under pure axial load are given in Fig. 12. As shown in the figure, the maximum and the minimum skidding speeds decrease consistently with the external load. As illustrated in Figs. 12(a) and 12(b), the starvation degree i  dramatically influences the maximum and minimum of , i j U at light axial load and they decrease greatly with the increase of  i . Under heavy axial load, the sliding in the ball-raceway contact is prevented to some extent. Though the spinning velocity is still influenced by i  , the effect of the lubricant starvation on the maximum and minimum skidding speeds is no longer obvious even when i  reaches to 0.3. To reveal the mutual influence of the ball-inner raceway and ball-outer raceway contacts, the impacts of i  on the maximum and minimum of , o j U are presented in Figs. 12(c) and 12(d). It can be observed that, whether the axial load is light or heavy, the impacts of i  on the maximum and the minimum of , o j U are not dramatical. As a matter of fact, the similar trends can also be found for the cases considering the impacts of the starvation degree o  .
To detect the skidding characteristics more comprehensively, the maximum of the skidding speed , i j U under combined loads is presented in Fig. 13. Unlike the radial load-induced characteristics of the oil film thickness, the influence of the radial load on the skidding is very obvious. As shown in the figures, the maximum of , i j U changes greatly in one cycle and the variation range increases with the increasing radial load and bearing speed. By comparison, it can also be found that the changes of the maximum of

Ball-raceway traction
The ball-raceway traction is related to the lubricant viscosity, the skidding velocity, and the oil film thickness. In view of the above analysis, it can be inferred that the lubricant starvation between balls and raceways may have significant influence on the ball-raceway traction. Based on Eqs. in the transverse and rolling directions can be obtained, and then the total traction force / i o F can be further determined. As presented in the authors' previous work [37], the ball-raceway traction forces under pure axial load remain constant in the whole cycle, while it can change with the position angle under combined loads.
The characteristics of the ball-raceway traction forces under pure axial load are depicted in Fig. 14. It can be observed that the traction forces in the ball-inner raceway and ball-outer raceway contacts show some different variation trends with the axial load. As shown in Fig. 14(a), with the increase of the axial load, the positive traction force ,, ix j F increases dramatically at first and then decreases gradually under heavy load. However, the traction force ,, ox j F remains negative with the increase of the axial load, the absolute value of which increases at light load, decreases at medium load, and begins to increase at relatively heavy load (see Fig. 14(b)). In terms of the traction forces in the rolling direction, both ,,  The results of the traction forces under combined loads are given in Fig. 15 and the force components varying with the starvation degrees are presented in Fig. 16. Under this load condition, the variation ranges of the traction forces in the rolling direction are much larger than those in the transverse direction (see Fig. 15(a)). As shown in Fig. 15(b), when the lubricant starvation occurs, the total traction force i F changes obviously, especially for the maximum values. To reveal the influence of the lubricant starvation on the traction forces in detail, it is necessary to study the force components under starved lubrication. As described in Fig. 16, the traction forces in the transverse and rolling directions are all influenced by the lubricant starvation. However, considering the relative variation range of the traction force in one cycle, the impacts of

Ball-pocket contact force
For the angular contact ball bearings, the wear and plastic deformation of the cage pocket frequently occur under high speeds. Besides, the collisions between the balls and the cage pocket can greatly influence the bearing dynamic performance. Therefore, the impacts of the lubricant starvation on the ball-pocket contact force are worthy of being studied. The results of the ball-pocket contact force under pure axial load are presented in Fig. 18. In this load condition, the ball center is in the front of the pocket center and the ball will interact with the front surface of the cage pocket. As shown in the figures, the ball-pocket contact force cj Q slowly increases at first  and then becomes stable with the increasing axial load. By comparing the results under fully flooded and starved regimes, it can be detected that the impacts of the starvation degrees on cj Q are negligible. By contrast, the effects of the lubricant temperature and the rotation speed are more obvious (see Fig. 18(b)), and the higher oil temperature and the lower bearing speed result in the smaller contact force cj Q . The results of the ball-pocket contact force under combined loads are given in Fig. 19, in which both the starvation degrees and the oil temperature are taken into account. Unlike the cases under pure axial load, the contact force cj Q is dramatically influenced by the starvation degrees at this load condition, which is also affected by the radial load and the bearing speed (see Figs. 19(a)-19(c)). Both i  and o  can increase the contact force cj Q at the large position scales, especially for the maximum values. As can also be clearly observed, the starvation degree o  has greater effects on the contact force cj Q than the starvation degree  .   In this paper, the impacts of the lubricant starvation in ball-raceway contacts on the tribo-dynamic performance of angular contact ball bearings are explored. Considering the mutual influence of the lubrication behaviors and the multi-degree-of-freedom (DOF) movements, a multi-DOF tribo-dynamic model is first established in this paper by introducing five-DOF inner ring, six-DOF balls, and six-DOF cage. The model not only considers the starved lubrication in the ball-raceway contacts, but also considers the full multi-body interactions between the bearing components. By applying Masjedi's theory in the mixed elastohydrodynamic lubrication (EHL), the starvation degrees describing the changes of the mass flow rates are adopted in the model to evaluate the severity of starvation. Based on the fourth-order Runge-Kutta method, the dynamic simulations considering different starvation degrees are conducted under two types of loads. Finally, the influence of the ball-raceway starvation on the tribo-dynamic performance of the bearing is revealed. On the one hand, the tribo-dynamic model considering full multi-body interactions and starvation degrees can provide a simulation tool for the bearing analysis under starved lubrication. On the other hand, the discussions and conclusions presented in this paper can help scholars and engineers have a better understanding of the impacts of starved lubrication in ball-raceway contacts. The specific conclusions are as follows: 1) The ball-raceway lubricant starvation can dramatically change the oil film thickness associated with the same raceway. In addition, the influence of the lubricant starvation on the oil film thickness is more obvious than the two types of loads and the more serious starvation results in the smaller oil film thickness. However, the lubricant starvation can only affect the oil film thickness associated with the opposite raceway at light pure axial load.
2) The impacts of the starved lubrication on the skidding behaviors show typical characteristics under the two types of loads. Under light pure axial load, the ball-raceway starvation can only have great influence on the maximum and the minimum skidding velocities of the same raceway. Under heavy pure axial load, the skidding performance is prevented to some extent and the effects of the lubricant starvation are no longer obvious. In addition, the sliding velocity decreases greatly with the increasing starvation degree only under light pure axial load, while the influence of the lubricant starvation on the spinning velocity continues  to be significant with the increasing axial load. Under combined loads, the ball-inner and ball-outer raceways have obviously mutual influence in terms of the skidding performance, and the lubricant starvation on the two raceways can seriously change the maximum skidding speed on one raceway.
3) The occurrence of the ball-raceway lubricant starvation has significant influence on the ball-raceway traction forces. Under pure axial load, the starved regimes on the two raceways result in the opposite changes of the total traction forces, and the lubricant starvation on one raceway also has the opposite influence on the total traction forces for different raceways. Under combined loads, the relative changes of the traction forces in the transverse direction caused by the lubricant starvation are more obvious than those in the rolling direction. As a consequence, the fractions of the gyroscopic moment vary obviously with the starvation degrees, which are much different from those in the race control hypothesis.
4) The effects of the starvation degrees on the ball-pocket contact force are negligible under pure axial load, but are extremely obvious under combined loads. Under combined loads, the occurrence of the ball-outer raceway starvation results in the greater change of the ball-pocket contact force than that of the ball-inner raceway starvation. Besides, the maximum of the ball-pocket contact force increases with the increasing ball-inner raceway and ball-outer raceway starvation degrees.