#### Abstract

Tapered roller bearings (TRBs) are applied extensively in the field of high-speed trains, machine tools, automobiles, etc. The motion prediction of main components of TRBs under grease lubrication will be beneficial to the design of bearings and the selection of lubricating grease. In this study, considering the dynamic contact relationship among the cage, rollers, and raceways, a multibody contact dynamic model of the TRB was established based on the geometric interaction models and grease lubrication theories. The impacts of load, grease rheological properties, and temperature on the roller tilt and skew and the bearing slip were simulated by using the fourth-order Runge–Kutta method. The results show that the roller tilt angle in the unloaded zone is obviously larger than that in the loaded zone, while the roller skew angle in the unloaded zone is smaller than that in the loaded zone. As the speed increases, the roller tilt and skew and the bearing slip become more serious. Bearing preload can effectively reduce the bearing slip but will make the roller tilt and skew angle increase. The roller skew angle and the bearing slip decrease with the increase of the grease plastic viscosity. The roller tilt angle increases with the increase of the plastic viscosity. The yield stress of the grease has little effect on motions of the roller and cage. The influence of temperature on the roller and cage motions varies with the type of grease used.

#### 1. Introduction

Tapered roller bearings, as the separable bearing, have the ability to withstand combined loads, large load-carrying capacity, well adjustability, and long service life. Nowadays, nearly 90% of rolling bearings are grease-lubricated [1]. Generally, compared to oil lubrication, lubricating grease in a rolling bearing has a wide operating temperature range and good extreme pressure (EP) property and adhesion property, and the construction of the lubricating device for the grease lubrication is sometimes relatively simple. However, due to the pressure difference inside the bearing contacts, the grease will flow to sides and next to the raceways over time. There may be very little reflow back into the raceways, and the bearing may suffer from starvation. Despite the abovementioned conditions, there also exists a film inside the bearing contacts at the beginning of bearing operation, formed by the combination of thickener and base oil [1, 2]. For the beginning of operation of the grease-lubricated TRB, the analysis of the TRB dynamic characteristics should be made to clarify the relation between the bearing dynamics and the lubricating grease. The bearing lubrication and dynamics not only affect each other but both have important impacts on the bearing failure, service life, and reliability. The analysis may have implications for the design of the bearing and the selection of the grease.

The dynamics of ball and cylindrical roller bearings have been extensively studied in the past few decades [3–5]. By considering the four degree-of-freedom balls and the six degree-of-freedom cage, Walters [3] firstly presented a comprehensive analysis for the balls and cage motions. After that, Gupta [4] and Meeks and Ng [5] carried out a series of research on dynamic problems of ball and cylindrical roller bearings. The movements of the rolling elements and cage were minutely described by classical differential equations of motion under specific operating conditions. For tapered roller bearings, compared with ball bearings (except angular contact ball bearings) and cylindrical roller bearings, the structure type and dynamics are more complicated. Gupta [6] studied the dynamic model and developed the dynamic analysis program ADORE for TRBs; the cage whirling and roller skew were analyzed under different cage clearances and traction-slip relations. However, the bearing slip and impacts of the lubricant on motions of bearing parts were not presented. Cretu et al. [7] proposed a quasidynamic model for the TRB, based on the Johnson–Tevaarwerk rheological model; the traction performance and other properties of the bearing were analyzed, but the translational motion of the cage was neglected. Deng [8] analyzed the dynamics of the TRB with oil lubrication by using the Adams-Bashforth–Moulton multistep method and studied the cage whirling and roller skew of the bearing. By considering six degrees of the cage motion, Sakaguchi and Harada [9, 10] simulated motions of the rigid or flexible cage in TRBs on the dynamic simulation software ADAMS. Bercea [11] proposed a comprehensive model to predict the roller skew motion in TRBs. He found that the roller skew is greatly influenced by traction at the flange/roller-end contact and by the roller-end geometry. A three-dimensional dynamic model of the double row TRB of a certain type of high-speed train was established by Gai and Zhang [12], and the roller contact stress and the cage stability were simulated. However, the lubrication state of axlebox bearings was not taken into account. Compared with other rolling bearings, the research on dynamics of grease-lubricated TRBs is relatively deficient.

The objective of this study was to present an accurate analysis of dynamic response of the tapered roller bearing. For the beginning operating stages of grease-lubricated tapered roller bearings, considering dynamic interactions in the bearing contacts and grease lubrication theories, a multibody contact dynamic model of TRB under the grease-lubricated condition was established. The bearing dynamics, e.g., the roller tilt and skew and the bearing slip, was analyzed. The impacts of speed, preload, temperature, and grease rheological properties on the bearing dynamics were studied.

#### 2. Dynamic Analysis Model

For the sake of accurately describing the relative position and movement of each component of the TRB, an inertial coordinate system *o*-*xyz* was established. As shown in Figure 1, the origin coincides with the mass center of the bearing, and the *o*-*x* axis coincides with the axis of the bearings’ shaft. A body-fixed coordinate system *o*_{c}-*x*_{c}*y*_{c}*z*_{c} is defined for the cage, and its origin is the mass center of the cage. The roller body-fixed coordinate system is *o*_{r}-*x*_{r}*y*_{r}*z*_{r}, the origin is the roller mass center, and the *o*_{r}-*x*_{r} axis along the roller axis. Because the roller is prone to bear unbalance moment, causing the roller to rotate abnormally, two harmful but inevitable movements of rollers are emphasized: tilt and skew. The tilt is normally referred to the roller rotation about *o*_{r}-*y*_{r} axis, and the skew is the roller rotation about *o*_{r}-*z*_{r} axis.

##### 2.1. Roller/Raceway Interaction

When the roller tilting or skewing, the interaction between the roller and raceway will vary along the contact line. As shown in Figure 2, the roller is divided into several slices, and each slice interaction with the raceway is calculated independently. Then, the total contact load between the roller and raceway can be obtained by integrating these local interactions.

In order to confirm the contact force at point *P* on slice surface, the gap or interference at point *P* should be determined first. In Figure 2, let and represent the mass center positions of the race and roller in the inertial frame, respectively. And and are, respectively, the geometric center position vectors of the roller and race relative to their mass centers. Then, geometric center position of the *k*th slice relative to the race center can be expressed aswhere is the slice position at the roller axis; is the transformation matrix (Euler’s rotation matrix) between the inertial and race body-fixed coordinate (*o*_{b}-*x*_{b}*y*_{b}*z*_{b}); and is the transformation matrix between the inertial and roller coordinate system.

The azimuth angle *ψ* of the slice relative to the race can be defined by components and of . Thus,

Then, the transformation matrix between the race and slice azimuth coordinate system is known. Assume *γ* is the azimuth angle of the point *P* in the coordinate plane *o*_{r}-*y*_{r}*z*_{r} and *ς* is the roller radius at the axial position . In the roller coordinate system, the position of the point *P* relative to the roller center is

In the slice azimuth coordinate system, the position of the point *P* relative to the race center is

The *γ* should satisfy the condition that the direction component of is zero:where is the component of in the direction and are components of .

If *ϕ* = is assumed, the above formula can be changed to

For the roller is in contact with the inner raceway, the value of *γ* should make the value of *r*_{ba3} smaller in the *z*_{ba} direction; if the roller contacts with the outer raceway, the *γ* should make *r*_{ba3} larger in the *z*_{ba} direction. Then, the interference between the *k*th slice and the raceway can be confirmed by subtracting the race radius from the above position vector, and the result may be transformed to a contact coordinate system to compute the interaction *δ* normal to the contact plane. Symbolically,where is the contact angle and *ξ* is the radius of the raceway at the point *P*. If the value of *δ* is negative, it indicates that there is no contact; if not, it means there is contact.

Then, the position of the point *P* can be rewritten as

Assume that are, respectively, the velocity and the angular velocity of the race and are the velocity and the angular velocity of the roller, respectively. In the contact coordinate system, the velocities at the point *P* on the raceway and roller arewhere is the transformation matrix between the azimuth and contact coordinate system.

Then, the velocity at the point *P* on the raceway relative to the roller is

For the line contact between the *k*th slice and the raceway, assuming that rollers behave as elastic half space, the normal force *q* at the point *P* iswhere is the Hertzian contact stiffness, = 0.356 [13]; is the equivalent elastic modulus of the roller and raceway; *n*_{s} is the total number of slices; *l*_{e} is the effective length of the roller; *c*_{w} is the viscous damping coefficient, *c*_{w} = 1.5*α*_{e}*k*_{w}*δ*^{10/9} [14, 15] and *α*_{e} is related to the restitution coefficient, for steel, bronze, or ivory, *α*_{e} = 0.08 d 0.32 s/m [14].

The grease lubricated condition of the TRB is considered here. In practice, the grease properties change over time (by overrolling, oil bleeding, starvation, shearing, etc.), which affect the bearing performance. In order to simplify the dynamic model, it is assumed that an ideal elastohydrodynamic lubrication state at the roller/raceway contact. Based on the grease EHL theory, the grease film traction forces at the point *P* are [16]where *φ*_{0} is the grease plastic viscosity at atmospheric pressure; is the equivalent radius of the *k*th slice and the raceway; *h*_{0} is the minimum grease film thickness, *h*_{0} is calculated by using the model in [17]; *ε* is related to the Hertzian contact half width *b*, film thickness *h*_{0}, and radius .

In order to determine the total contact load between the roller and raceway, the integration of local interactions is required along the contact line. The total contact force and torque acting on the *j*th roller are

The contact loads acting on the race arewhere

##### 2.2. Flange/Roller-End Interaction

In race body-fixed coordinate system (*o*_{b}-*x*_{b}*y*_{b}*z*_{b}), the roller-end curvature center relative to the race center iswhere is the position of the roller-end curvature center at the roller axis.

Since normal force will act on the flange surface and pass through the roller-end curvature center, the above center can be transformed into the race azimuth coordinate system by , where .

As illustrated in Figure 2, the position of the roller-end curvature center relative to the flange coordinate iswhere is the transformation between the race azimuth and flange coordinate; *β* is the flange angle defined as a rotation about the *o*-*y* axis in accordance to the right-hand screw rule; and locates the flange origin in the race coordinate.

If the radius *r*_{s} of the roller-end is known, the geometric interaction between the roller-end and flange is simply given by

Definition *r*_{e} is the distance from the bearing apex to the roller spherical end. If *r*_{e} > *r*_{s}, the flange/roller-end contact is elliptical contact [11]. The Hertzian contact theory can be employed to determine the contact load *q*_{f} at the flange/roller-end contact.where *n*_{δ} is the contact deformation coefficient and is the sum of principal curvatures.

The friction at the flange/roller-end contact contains two parts: the friction induced by microasperities contact and the traction force produced by the grease lubricant [18, 19].

The friction force induced by the asperities contact iswhere *B* and *C* are related to morphology and material properties of the contact surface; *s*_{0} and *p*_{0} are, respectively, the critical shear stress and the yield stress, for most metals, *s*_{0}/*p*_{0} *≈* 0.2 [18]; *λ* is the ratio of the grease film thickness *h*_{c} to surface roughness. The film thickness *h*_{c} is approximated as [20]where*u*_{f} is the EHL entrainment speed, *u*_{f} = (*u*_{flange} + *u*_{end})/2.

For the analysis of the grease traction force, the Herschel–Bulkley flow model is adopted:where *τ* is the grease shear stress; *τ*_{y} is the yield stress; *φ* is the plastic viscosity, *φ* = *φ*_{0}*e*^{αp}; *p* is the contact pressure, the distribution of *p* can be approximated by the Hertzian contact pressure; *α* is the viscosity-pressure coefficient; *n* is the power law exponent; and *D* is the shear rate, *D* ≌ (*u*_{flange} − *u*_{end})/*h*_{c}. Then, the traction force due to the grease shear stress is obtained by the integration:

Then, the loads acting on the *j*th roller are

The loads acting on the flange arewhere and are positions of the contact point relative to the roller and inner-ring centers, respectively.

##### 2.3. Forces Acting on the Cage

During the cage and rollers rotation, the master-slave relationship of rotating cage-roller assembly will change over time due to different velocities and displacements. As shown in Figures 2 and 3, the location of the *j*th roller geometric center in the cage pocket coordinate system (*o*_{p}-*x*_{p}*y*_{p}*z*_{p}) iswhere is the position of the cage mass center in the inertial coordinate system; is the transformation matrix from the cage-fixed coordinate system (*o*_{c}-*x*_{c}*y*_{c}*z*_{c}) to the pocket coordinate system; **r**_{0} locates the pocket center in the cage coordinate system; and **T**_{ic} is the transformation matrix between the inertial and cage coordinate system.

Now, similar to the roller/raceway interaction, let locates a point on the *k*th slice of the roller such that

In the pocket coordinate system, the position of the point is

As illustrated in Figure 3, the clearance between the point *P* and the pocket crossbeam iswhere Δ*δ* is the initial clearance between the roller and crossbeam, Δ*δ* = (*l*_{c} − *d*_{w})/2; *l*_{c} is the average width of the pocket; and *d*_{w} is the average diameter of the roller.

For the modeling of the assembly interactions, assuming that there is an excess of the grease in the cage-roller assembly, a critical value Δ*h*_{0} of the grease film thickness is assumed for the contact state transition. When *δ* ≥ Δ*h*_{0}, there are only the hydrodynamic effect between the *k*th slice and pocket crossbeam, and no Hertzian contact [9, 21]. The minimum grease film thickness *h*_{0} = *δ*. The hydrodynamic pressure *q*_{c} and the grease film traction force *f*_{c} between the *k*th slice and crossbeam arewhere [17, 22]where *u* is the entrainment speed; *ξ*_{1}, *ξ*_{2} are the constants associated with the power law exponent *n* [17]; and *c*_{e} is grease film damping [22].

When *δ* < Δ*h*_{0}, it means that the contact between the *k*th slice and crossbeam is in a boundary state. So, the Hertzian contact is included in the contact. The minimum grease film thickness *h*_{0} = Δ*h*_{0}. The Hertzian contact deformation is

The contact force *q*_{c} and the friction force *f*_{c} between the roller and crossbeam arewhere *k*_{h} is the linearized Hertzian contact stiffness [23]; *c*_{h} is the Herbert viscous damping coefficient [24]; and *μ*_{bd} is the traction coefficient under boundary lubrication [9].

Then, the load at the contact point on the *k*th slice in the contact coordinate system is

Using the processing method of the roller/raceway interaction, the load can also be transferred to the inertial coordinate and the roller/cage body-fixed coordinate, respectively.

Owing to the smaller sliding speed between the roller-end and the side beam and the smaller curvature of roller spherical end, the EHL lubrication state cannot be effectively formed at this contact, but the squeeze film lubrication will be formed. As shown in Figure 3, the Δ*δ′* is the initial pocket clearance and *r*_{a} = . If *dr*_{a}/*dt* ≥ 0, the small end contacts with the side beam; if *dr*_{a}/*dt* < 0, the large end contacts. Neglecting the grease traction, the squeeze force *q*_{s} iswhere [25]where *h*_{a} is the distance from the side beam to the end; *d*_{l} and *d*_{s} are the diameters of the large and small end, respectively; and *h*_{s} is the thickness of the side beam.

Then, the loads acting on the *j*th roller are

The loads acting on the cage arewhere is the position of the contact point in the roller body-fixed coordinate and **T**_{ar} and **T**_{ai} are, respectively, the transform from the azimuth to roller and inertial coordinate.

According to geometric characteristics of the cage and guide ring, it can be considered as a finite journal bearing lubrication condition between the guide surface and cage centering surface. In Figure 4, the clearance *δ′* between the cage and guide ring iswhere Δ*e* is the initial clearance, Δ*e* = (*r*_{y} − *r*_{d})/2; *r*_{d}, *r*_{y} are, respectively, the radii of the centering surface and guide surface and **e** is cage displacement in the radial plane.

Similarly, assume that is the critical value of the grease film thickness. When ≥ , the contact force *q*_{ic} and friction torque *N*_{ic} on the cage arewhere *r*_{ic} is the equivalent radius and *ω*_{cx}, *ω*_{bx} are, respectively, the angular velocities of the cage and guide ring. The calculation of the grease film stiffness and damping is the same as that at the roller/crossbeam contact.

When < , the elastic deformation between the cage and guide ring is = − *h*_{0}, and the minimum grease film thickness *h*_{0} = . The *q*_{ic} and *N*_{ic} arewhere is the contact stiffness and is the viscous damping coefficient. The calculation of and is the same as that in Equation (35). *μ*_{bd} is the traction coefficient under boundary lubrication.

In order to simplify the model, the load of the cage on the guide ring is ignored in contrast to large external load on the guide ring. Then, the loads acting on the cage are

If the TRB is lubricated by the oil, the effect of oil-gas mixture on the cage and rollers may not be overlooked. However, if the bearing is lubricated by the grease, it seems impossible to have the grease-gas mixture effect on the dynamics of the cage and rollers. So, the effect of the grease-gas mixture is not considered in the dynamic model.

##### 2.4. Dynamic Model

In real application also, the surroundings (inertia/damping of shaft assembly and housing) are expected to influence the TRB dynamics. These effects are neglected in the modeling. Generally, it is convenient to consider the translational moving of rollers in the cylindrical coordinate system, while the Cartesian coordinate system is convenient for the cage and race. The translational moving of bearing parts can be simply described by Newton’s laws such that

The generalized force **Q** is obtained by superimposing the forces from Section 2. The centrifugal force and the gyroscopic moment should be superimposed on the generalized force **Q** for the roller. For the cage and race, the mass matrices **M**_{b} and **M**_{c} are

For the special case of conical rollers, **M**_{r} is

Both rollers and the cage have six degrees-of-freedom. Unlike the translational motion (in three directions), the rotating of the bearing parts are described in their own body-fixed coordinates. For rollers and the cage, using the Euler dynamic equations such thatwhere *I*_{x}, *I*_{y}, and *I*_{z} are inertia principal moments of the cage or rollers; the total moment **N** is also obtained by superimposing the forces from Section 2. **ω**[*ω*_{x}, *ω*_{y}, *ω*_{z}] is the angular velocity in the inertial coordinate system. The relationship between the angular velocity **ω**_{fixed} in the body-fixed coordinate system and **ω** is as follows. **B** is the Euler’s rotation matrix.

Admitting the outer ring as macroscopically stationary and a rotating inner ring with no torque load, it results four degrees of freedom of inner ring: moving in the *x*, *y*, and *z* directions and constant rotating about the shaft axis (*o*-*x*).

#### 3. Numerical Method

The bearing geometry and material parameters and grease main rheological parameters (30°C) are shown in Table 1. The cage is an inner-ring guided type, and its material is polyamide. The polyurea grease was adopted [26]. The fourth-order Runge–Kutta method was used to dissolve transient responses of the TRB on MATLAB. To ensure the convergence of simulation, initial values of displacement, velocity, and load are obtained by the quasidynamic method.

#### 4. Results and Discussion

For the complicated dynamic response of TRBs, it is essential to consider several typical and harmful movements of the bearing in the process of dynamic analysis, such as the roller skew and tilt, bearing slip, and so on. The slip rate of the roller and cage is defined as

For the roller, the speed refers to the revolution speed; for the cage, refers to the rotational speed.

##### 4.1. Mass Center Trajectory and Velocity

The trajectories and velocities of mass centers of the main bearing parts are shown in Figure 5. The axial preload is not applied to the inner ring but limited to the degree of freedom of the inner ring in the axial direction. The cage initial rotation speed is the same as the roller initial revolution speed. Due to the changing position of rollers, the distribution of rollers is symmetrical or asymmetric with respect to the *o*-*z* axis. Therefore, the inner ring vibrates slightly in the *o*-*y* direction. The radial and axial displacement (relative to the initial position) of the roller exhibit a periodic fluctuation, for the bearing is in a semicircle-loaded state under pure radial load conditions. When the rollers alternately enter the loaded and unloaded zones, the load on rollers will show periodic changes, resulting in a regular variation in displacement. The cage whirling is approximately a circle. For a number of uncertain rollers acting on the cage, the cage speed does not show a periodic change.

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##### 4.2. Rotational Speed Influence

In this study, the tilt and skew of conical rollers are regarded as the primary evaluation variables for the dynamic performance of the TRB. The roller skew is mainly generated from the tangential friction force on the flange/roller-end and roller/raceway contacts, while the tilt motion is mainly induced by the normal contact force on those contacts. According to the lubrication theory, the bearing speed will be a key factor affecting the load case in bearing contacts.

The influence of the bearing speed on the roller tilt and skew (relative to the initial position) is shown in Figure 6. The degree of freedom of the inner ring in the axial direction is also limited. At each speed level, the roller tilt angle presents a relatively regular periodic vibration. The tilt angle in the unloaded zone is obviously larger than that in the loaded zone. This is due to the larger gap between the roller and raceways in the unloaded zone when only the radical load is applied, and rollers in unloaded zone seem to be “relaxed.” As the speed increases, the maximum of the tilt angle almost does not change, while the average tilt angle rises from −0.14 × 10^{−4} rad to −0.40 × 10^{−4} rad, indicating that the bearing speed affected the roller tilt. It can be confirmed that the gyroscopic moment is one of the reasons that cause the rising of the roller tilt angle, for the gyroscopic motion is intensified with the bearing speed increasing. Although the gyroscopic moment increases as the bearing speed increases, it is still relatively smaller than the force/torque in the roller/raceway contact. So, for the radial load is constant, the maximum tilt angle of the roller changes little with the speed.

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As it is shown in Figure 7, the skew angle, both in the loaded zone and in the unloaded zone, is relatively large at the speed of 2 krpm. In the entire speed range, the roller skew angle in the loaded zone is greater than that in the unloaded zone. For the rollers is “relaxed” in the unloaded zone, the skew moment of the raceway and flange acting on the roller is smaller than that in the loaded zone. It shows obvious periodicity when the speed reaches 3 krpm or more. The average skew angle is increased from 3.06 × 10^{−3} rad/0.17 deg to 4.47 × 10^{−3} rad/0.25 deg. The increase of bearing rotating speed enlarges the centrifugal force of rollers, which makes contact forces at the roller/outer raceway and flange/roller-end increase. As a result of combined contribution of raceway friction and the flange friction, the roller skew angle is greatly increased. Compared with the experimental result measured by Falodi et al. in which they measured the average skew angle between 0.15 and 0.45 deg [27]. The average skew angle in Figure 7 is between 0.17 and 0.25 deg and at an order of magnitude compared to the experimental result. The changing trend of the skew angle with speed is also consistent with Yang’s results.

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The bearing slip is a destructive motion that easily leads to scuffing, welding, and overwear of bearing surfaces. The bearing slip usually occurs at high-speed and light-load conditions. The influence of the bearing speed on the slip of the roller and cage is exhibited in Figures 8 and 9.

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The diagrams depicted in Figure 8 indicate that the roller slip is becoming more and more serious as the bearing speed is raised. This variation trend is similar to that in [7]. With the bearing speed increases, the roller slip gradually presents a periodic change rule, especially when the speed is up to 3 krpm or more. The roller slip in loaded and unloaded zones is not the same. In the unloaded zone, the roller slip is more serious than that in the loaded zone. At the speed of 5 krpm, the bearing is in a state of serious slipping. In Figure 9, the cage slip is aggravated with the bearing speed increase. There is no periodic change in the cage slip rate. This is because when the cage is whirling, a number of uncertain rollers may act on the cage, which can lead to an irregular motion of the cage. At the same speed level, the maximum amplitude of the slip rate of the cage and roller is basically the same, for the dependent relationship between the cage and roller.

##### 4.3. Axial Preload Influence

Axial preload is not only a key factor affecting the bearing dynamics, but one of the main means to ensure the normal service life of the bearing. In order to familiarize the influence of the axial preload on the roller tilt and skew, different axial preloads were applied to the TRB under the condition that the radial load is zero.

As shown in Figures 10 and 11, after entering the smooth running phase, as the axial preload varies from 10 *μ*m to 40 *μ*m, the average value of the roller tilt rises from −0.27 × 10^{−4} rad to −0.96 × 10^{−4} rad. The greater the preload, the more obvious the roller tends to tilt. This is because as the preload increases, the negative moment of the flange acting on the roller-end becomes larger, thus causing the roller tilt to assume a negative tendency.

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Except for the condition that the preload is 10 *μ*m, the roller skew has larger amplitude at the beginning, and then the amplitude decreases gradually and tends to be stable. In the smooth running phase, as the preload increases, the roller skew angle increases. This trend is in line with the results in [8, 27]. The average skew angle is increased from 4.23 × 10^{−3} rad/0.24 deg to 5.50 × 10^{−3} rad/0.31 deg. Both the roller tilt and skew present a more regular vibration throughout the bearing operation, this is due to symmetry of the bearing on the radial plane and pure preloading condition; the loaded and unloaded zones do not exist in the bearing, and loads on rollers almost change a little in any position.

The changes of the slip rate of the roller and cage with the preload are presented in Figures 12 and 13. As the bearing preload increases, the bearing slip decreases in the smooth running phase. The average slip rate of the roller decreases from 0.0189% to 0.0057% and that of the cage decreases from 0.0189% to 0.0055%, indicating that the appropriate bearing preload can effectively prevent bearing slipping failure. The result is similar with the conclusion in [7, 28]. The influence of the preload on the bearing slip reduced with the increase of the preload value. The average slip rate of the roller and cage when the preload is 40 *μ*m changes little compared with 0.0073% and 0.0076% when the preload is 30 *μ*m. Obviously, increasing the amount of preload will help to decrease the bearing slip, but excessive preload will inevitably reduce the service life of bearings. Therefore, to meet the bearing life and service requirements, reasonable selection of preload is significant.

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##### 4.4. Grease Physical Properties Influence

In view the fact that the calculation of the traction force, stiffness and damping at the bearing contacts is mainly based on the plastic viscosity *φ* and yield stress *τ*_{y} of the grease. For a clearer understanding of the relationship between the bearing dynamics and the grease used, grease physical properties influence on bearing dynamics is analyzed. The temperature, pressure, speed, and other surroundings have significant effects on the grease physical properties. For the sake of simplification, the impacts of these factors on the grease physical properties are ignored. In order to highlight the effect of a certain rheological parameter on the bearing performance, only one parameter was given an appropriate change while other rheological parameters remain unchanged. The effects are shown in Figures 14 and 15. After the bearing enters the stable running state (after 0.1 s), the average value of the angle (tilt and skew) is used as an indicator of bearing operating state.

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The effect of the plastic viscosity on the roller and cage motions are shown in Figure 14(a). As the plastic viscosity increases, the tilt angle and skew angle increases and decreases, respectively. For the grease film thickness increases with the increase of the plastic viscosity, the load ratio of microasperities at the flange/roller-end contact is reduced. Then, it indirectly leads to the decrease of the skew moment at the flange/roller-end contact and the roller skew angle. Due to the smaller skew angle, the roller misalignment will also be small. Therefore, the normal force at the roller/raceways and flange/roller-end contacts will have a greater contribution to the roller tilt. Therefore, the result shows that the roller tilt angle increases with the plastic viscosity increase. In Figure 14(b), the bearing slip decreases with the grease plastic viscosity increase. As the plastic viscosity increases, the grease film thickness at the roller/raceways and flange/roller-end contacts increase. So, the friction force at these contacts is reduced, which may lead to the decrease of the bearing slip. The bearing slip is the result of the combined effect of raceway friction and the flange friction.

Compared with the impacts of the plastic viscosity of the grease on the motions of the roller and cage, the influence of the grease yield stress on the roller tilt, roller skew, and bearing slip is relatively smaller, as shown in Figure 15. For the yield stress has a negligible effect on the film thickness in all practical cases [9].

##### 4.5. Temperature Influence

Due to the improper installation, overwear or not timely cooling, the operating temperature of TRBs will change more or less. Sometimes, it may lead to heat accumulation or thermal failure of bearings. The most direct effect of the temperature is changes of the grease physical properties. The rheological parameters of the greases at different temperatures in Table 2 are quoted from the data in reference. [26] The effect of temperature on the dynamics of the bearing was studied. The impact of temperature on the bearing structure is not considered here.

Three typical greases were used, such as the calcium grease, lithium grease, and polyurea grease. The largest difference in the composition of three greases lies in the types of thickeners. In Figure 16(a), under the lubrication of the calcium and polyurea greases, the roller tilt angle decreases with the temperature increase. Under the lithium grease lubrication, the roller tilt angle increases with the increase in temperature. In the whole temperature range, the tilt angle under the lithium grease is generally larger than that under others. With the increase in temperature, the roller skew angle under the polyurea grease increases, while the roller skew angle under the other two types of greases decreases. That the tilt and skew angle under the different greases and temperature show different trends is a result of multiple factors. One of the factors is that the relationships between the grease rheological parameters and temperature are not simple linear. For example, the plastic viscosity and the power law exponent do not simply increase or decrease with temperature.

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In Figure 16(b), firstly, the roller slip rate shows different values due to the different greases; secondly, the roller slip rate shows a variety of trends with the various temperature . Under the lubrication of the polyurea grease, the roller slip rate increases with the increase in temperature, while the roller slip with the calcium and lithium greases lubricated shows a decreasing trend. In real application, the operating temperature of bearings may not be constant. The influence of the temperature on dynamic characteristics of grease lubricated bearings needs to be further explored.

#### 5. Conclusions

For the beginning operating stages of grease-lubricated TRBs, the multibody contact dynamic model of TRBs was established. The impacts of preload, temperature, and grease rheological properties on the bearing dynamics were analyzed. Based on numerical results, several conclusions can be summarized:(1)The roller tilt angle in the unloaded zone is obviously larger than that in the loaded zone, while the roller skew angle in the unloaded zone is smaller than that in the loaded zone. The effect of bearing speed on the roller skew is greater than that on the roller tilt. As the speed increases, the roller tilt and skew and the bearing slip become more serious.(2)Bearing preload can effectively reduce the bearing slip but will make the roller tilt angle increase. In the stable operation stage of the bearing, as the preload increases, the roller skew angle increases.(3)The roller skew angle and the bearing slip rate decrease with the increase of the grease plastic viscosity. The roller tilt angle increases with the increase of the plastic viscosity. The yield stress of the grease has little effect on motions of the roller and cage.(4)The influence of temperature on motions of the roller and cage varies with the type of grease used. When the TRB is lubricated by the lithium grease, with the increase of the temperature, the roller tilt angle increases, the roller skew angle and the bearing slip rate decrease. When the polyurea grease is adopted, the change trend of the roller tilt and skew and the bearing slip is just opposite to that when the lithium grease is used. Under the calcium grease lubrication, the roller tilt and skew angle and the bearing slip rate decrease with the temperature increase.

#### Data Availability

The figure data used to support the findings of this study have been deposited in the FigShare repository at https://figshare.com/s/15c2556bee83d1adf01d.

#### Conflicts of Interest

The authors declare that they have no conflicts of interest.

#### Acknowledgments

The paper was supported by the National Natural Science Foundation of China (Grant No. 51675427).