Dynamic Characteristics Analysis of Cylindrical Roller Bearing with Dimensional Deviations in Cage Pocket

Abstract

Dimensional deviations in the cage pocket of a roller bearing can significantly affect the bearing’s dynamic performance, directly determining the positional stability of the roller. These deviations can result in roller misalignment, increasing friction and wear. Deviations arise from machining errors and deformation during motion, etc. A dynamic model of a cylindrical roller bearing that accounts for cage flexibility was developed to explore the impact of deviations. The flexible cage provides a more realistic representation compared to the rigid cage. The effects of deviations in the length and width of the cage pocket on the bearing’s dynamic behavior were analyzed, and the results show that deviations in cage pocket dimensions lead to notable changes in bearing dynamics. Specifically, when the length deviation is negative and increasing, the amplitude of cage motion decreases, while both transitional and rotational speeds rise. It also causes greater fluctuations in the rotational speeds of the inner ring and rollers. Conversely, the cage’s equivalent stress and the contact load decrease and the amplitude of cage motion increases with increases in width deviation.

1. Introduction

The rolling bearing is one of the critical components in rotating machinates, in which the cage acts as a core tool to space apart and guide rollers, and to keep rollers under both uniform and alternating load conditions. Dimensional deviations in the cage pockets may alter the amplitude of cage motion, affecting the stress and load distribution across various bearing components, leading to increased wear, noise generation, and even premature bearing failure. Traditional bearing cage models often idealize the pocket dimensions, failing to account for the impact of pocket dimensional deviations on the bearing’s dynamic characteristics.

In the research field of bearing dynamics, Gupta [1] established a dynamic model of full degrees of freedom in bearing parts and obtained the dynamic performance of bearings, changing over time, by integral solution of the dynamic equation. However, there were problems such as the strict selection of initial values and long solving time. Based on Hertzian contact theory, Gunduz [2] proposed a calculation formula and an experimental study of the stiffness damping matrix of ball bearings using a dynamic model. Park [3] established a four-degree-of-freedom dynamic equation of rolling bearings, considering the clearance, and conducted a dynamic analysis of bearing stiffness changes under different working conditions. Sakaguchi and Kaoru [4] employed the ADAMS software to construct a dynamic model of the cylindrical roller bearing cage and carried out dynamic simulations to analyze the motion of the cage’s center of mass and the stress conditions. Deng Sier et al. [5] utilized the ADAMS system to develop a rigid–flexible multi-body dynamic analysis program for angular contact ball bearings and analyzed the dynamic performance of the high-speed angular contact ball bearing cage. Additionally, they [6] established the differential equations of the dynamics of high-speed cylindrical roller bearings. The differential equations were solved by combining the fine integration method and the predictive correction Adams–Bashforth–Moulton multistep method. Theoretical analyses were conducted on the dynamic characteristics of the cage.

In terms of research on bearing cage pockets, Gao [7] proposed a comprehensive dynamic model for analyzing the stability, skidding degree, ball–cage collision, wear distribution, and wear rate of four types of cage pocket. Yuan [8] studied the cage’s motion with different pocket shapes, such as spherical, square, and cylindrical, with an angular contact ball bearing under different operating conditions. Chen [9] established the dynamic models of spherical and cylindrical pocket cages, and studied the effect of cage pocket shape on the dynamics of angular contact ball bearings with different pocket clearances. Wen [10] conducted experiments on cage motion and wear to investigate the influence of clearance on ball bearings. Yang [11] experimentally investigated the stability of six cages with different pocket shapes and obtained the influence patterns of pocket shapes on cage stability. Deng [12] analyzed the influencing factors for different amounts of friction consumption at various structural sizes of the cage, and revealed the effects of these structural sizes on friction consumption. Wang [13] simulated the vibration response of the bearing-rotor system.

In the research field of bearing cage motion, Wen [14] experimented on cage motion and wear to investigate the influence of clearance on ball bearings. Chen [15] et al. explored the dynamic characteristics of the cage during the acceleration and deceleration of angular contact ball bearings under different radial loads through experiments, found that acceleration and deceleration significantly affected the dynamic characteristics of the cage, and proposed a model to measure the relationships between the cage sliding ratio, acceleration, and radial load. Schwarz [16] investigated the characteristics of different cage movements using multibody simulations and experimental investigations. Li [17] established a dynamic model of ball bearing contact, and analyzed the dynamic performance of the cage by using the eddy radius ratio and the eddy velocity deviation ratio of the cage’s centroid trajectory. The previous research primarily focused on bearing dynamics, cage pocket shapes, and cage motion, without considering the impact of dimensional deviations in the cage pocket on the dynamic characteristics of bearings, resulting in their influence remaining unclear.

Dimensional deviations in both the length and width of the cage pocket were considered in our dynamic model of bearings, and the influence of these deviations on the dynamic behavior of cylindrical roller bearings was obtained through simulation analysis. This will provide valuable insights for the dimensional regulation of cage pockets and for bearing design.

2. Rigid–Flexible Coupling Dynamic Model of Cylindrical Roller Bearings

2.1. Pocket Dimensional Deviation Characterization

The dimensional deviation of pockets can induce alterations in the clearance between the cage and the rollers, consequently impacting the interaction between the cage and the rolling elements. These alterations may result in changes in cage motion and stress. Dimensional deviations can be present in both the length and width dimensions of pockets, as shown in Figure 1. $L_1$ and $L_2$ are the actual and ideal dimensions in length.

The dimensional deviation in length, $L$ can be obtained with the following:

$$L=L_1-L_2.$$

(1)

As shown in Figure 2, $S_1$ and $S_2$ represent the actual and ideal dimensions in width.

The dimensional deviation in width, $S$ can be obtained with the following:

$$S=S_1-S_2.$$

(2)

2.2. Rigid–Flexible Coupling Dynamic Model of Cylindrical Roller Bearings, Considering Pocket Dimensional Deviation

Based on rigid multi-body dynamics and the finite element method, a rigid–flexible coupling dynamic model of cylindrical roller bearings was established, in which dimensional deviation of the cage pocket was present.

Building upon the rigid body dynamics model of the bearing, the dynamic model incorporates a finite element representation of the flexible cage with pocket size deviation, thus establishing a coupled rigid–flexible model of cylindrical roller bearings that integrates pocket size deviation. The inner ring, outer ring, and rollers are depicted as rigid components, while the cage is modeled as a flexible component, as shown in Figure 3. $Fr$ is the radial force borne by the bearing’s inner ring, ${\omega }_i$ is the bearing’s inner ring speed. The inner ring of the bearing rotates around the x-axis. According to the characteristics of the bearing’s outer ring, to fix the contact surface between the bearing’s outer ring and the bearing’s pedestal and impose the radial force in the bearing’s inner ring, setting the contact pair between the body parts—such as the inner ring and rolling body, the outer ring and rolling body, or the rolling body and the cage—is necessary.

2.2.1. Model Assumptions

The following assumptions are used to establish the rigid–flexible coupling dynamic model of cylindrical roller bearings:

(1) The center of mass of each part in the bearing coincides with the geometric center (except for the dimensional deviation of the cage).

(2) The bearing cage is considered as a flexible body, and the rest of the parts are considered as rigid bodies.

(3) In the model, the inner ring and roller have six degrees of freedom, the cage has five degrees of freedom, and the outer ring is fixed.

(4) The influence of internal temperature changes on the bearing is not considered.

2.2.2. Interaction Forces in Bearing

The interaction forces in the bearing components during operation are shown in Figure 4. Contact emerges between the steel balls and both the inner and outer rings, along with the balls and the cage. The cage assumes the role of guiding the outer ring of the bearing, with contact established between the cage and the guiding surface of the outer ring during bearing rotation.

In summary, the differential equation of the cage dynamics is established as follows:

$$\left\{\begin{array}{l}{m}_c{\ddot{y}}_c={\displaystyle \sum _{j=1}^Z\left(Q_{Bj}\sin {\theta }_j-Q_{Fj}\sin {\theta }_j-f_{TBFj}\cos {\theta }_j+f_{TFRj}\cos {\theta }_j\right)}-Q_c\cos {\theta }_j+f_c\sin {\theta }_j\\ m_c{\ddot{z}}_c={\displaystyle \sum _{j=1}^Z\left(-Q_{Bj}\cos {\theta }_j+Q_{Fj}\cos {\theta }_j-f_{TBFj}\sin {\theta }_j+f_{TFRj}\sin {\theta }_j\right)}-Q_c\sin {\theta }_j+f_c\cos {\theta }_j\\ I_{cx}{\dot{\omega }}_{cx}={\displaystyle \frac{1}{2}}{d}_m{\displaystyle \sum _{j=1}^Z\left(-Q_{Bj}+Q_{Fj}-f_c\right)}\\ I_{cy}{\dot{\omega }}_{cy}={\displaystyle \frac{1}{2}}{L}_r{\displaystyle \sum _{j=1}^Z\left(Q_{Bj}-Q_{Fj}\right)\cos {\theta }_j}\\ I_{cz}{\dot{\omega }}_{cz}={\displaystyle \frac{1}{2}}{L}_r{\displaystyle \sum _{j=1}^Z\left(Q_{Fj}-Q_{Bj}\right)\sin {\theta }_j}\end{array}\right.$$

(3)

where $Q_{ij}$ is the normal contact force between the roller and the inner raceways, $Q_{ej}$ is the normal contact force between the roller and the outer raceways, and $Q_{Bj}$ and $Q_{Fj}$ are the normal force between the cage pocket and the roller. $Q_c$ is the force between the cage and the outer ring guide surface. $f_{ij}$ is the drag force between the roller and the inner raceway, $f_{ej}$ is the drag force between the roller and the outer raceway, $f_{TBRj}$ and $f_{TFRj}$ are the tangential friction force between the cage pocket and the roller, and $f_c$ is the tangential friction force between the cage and the outer ring guide surface. $m_c$ is the cage mass,${\ddot{y}}_c$ and ${\ddot{z}}_c$ are the acceleration of the cage in inertial coordinate system, ${\theta }_j$ is the azimuth of the j-th roller, $Z$ is the number of rollers, $I_{cx},I_{cy},I_{cz}$ is the moment of inertia of the cage in the inertial coordinate system, ${\omega }_{cx},{\omega }_{cy},{\omega }_{cz}$ is the angular acceleration of the cage in the inertial coordinate system, $d_m$ is the pitch circle diameter, and $L_r$ is the length of the roller.

The contact force, $Q_n$, is calculated by the impact function provided by the function library. The contact force of the bearing is composed of the contact normal force, generated by the collision deformation between the parts, and the tangential friction force, generated by the relative speed between the parts. The function expression is as follows:

$$Q_n=\left\{\begin{array}{cc}\hfill K_n{\left({\delta }_0-\delta \right)}^e-C_{max}({\displaystyle \frac{d\delta }{dt}})\times step(\delta ,{\delta }_0-d,1,{\delta }_0,0),& \hfill \delta \le {\delta }_0\\ \hfill 0,& \hfill \delta >{\delta }_0\end{array}\right.$$

(4)

where $v_r^i$ is the stiffness coefficient, $v_r^i$ is the initial distance before the collision of two objects, $v_r^i$ is the distance after the collision of two objects, $v_r^i$ is the collision coefficient of the object, $v_r^i$ is the damping coefficient, $v_r^i$ is the relative motion speed between the two collision objects, and $v_r^i$ is the penetration depth of the object.

The tangential friction, $f_n$, is calculated by the Coulomb formula, and the mathematical expression is as follows:

$$f_n=\mu Q_n$$

(5)

where $\mu$ is the friction coefficient. The friction coefficient is determined by the relative sliding speed, $v$, between the two contact parts. The calculation formula is as follows:

$$\mu =\left\{\begin{array}{cc}-sign\left(v\right)\cdot d,& \left|v\right|>v_d\\ -step\left(\left|v\right|,v_s,{\mu }_s,v_d,{\mu }_d\right)\cdot sign\left(v\right),& v_s\le \left|v\right|\le v_d\\ step\left(v,-v_s,{\mu }_s,v_s,-{\mu }_s\right),& -v_s<v<v_s\end{array}\right.$$

(6)

where $v_s$ is the static friction transition speed, $v_d$ is the dynamic friction transition speed, ${\mu }_s$ is the maximum static friction coefficient, and ${\mu }_d$ is the dynamic friction coefficient.

2.2.3. Contact Stiffness between the Rollers and the Ring Raceway

During operation, an oil film will be formed between the roller and the raceway. In this paper, the stiffness between the roller and the ring includes the contact stiffness between the roller and the raceway and the oil film stiffness between the roller and the raceway [18].

When the bearing is loaded, the normal stiffness of the roller unit and the inner and outer ring raceway units is the series of normal contact stiffness and oil film stiffness, which is expressed as follows:

$$K_i={\displaystyle \frac{K_{ri}{K}_{oi}}{K_{ri}+K_{oi}}}$$

(7)

$$K_e={\displaystyle \frac{K_{re}{K}_{oe}}{K_{re}+K_{oe}}}$$

(8)

where $K_{ri}$ and $K_{re}$ are the normal contact stiffness of the roller and the inner and outer ring raceways, respectively. $K_{oi}$ and $K_{oe}$ are expressed as the normal oil film stiffness of the roller and the inner and outer ring raceways, respectively.

(1) Contact stiffness between roller and raceway.

The cylindrical roller bearing only bears the radial load. The contact stiffness between the roller and the raceway is the normal contact stiffness caused by the radial load. The normal contact stiffness of the cylindrical roller bearing is expressed as follows:

$$K_{ri}={\displaystyle \frac{d{Q}_i}{d{\delta }_{ri}}}$$

(9)

$$K_{re}={\displaystyle \frac{d{Q}_e}{d{\delta }_{re}}}$$

(10)

where $Q_i$ and $Q_e$ are, respectively, expressed as the normal contact load between the roller and the inner and outer ring raceways; ${\delta }_{ri}$ and ${\delta }_{re}$ are expressed as the normal elastic deformation of the contact between the roller and the inner and outer ring raceways, respectively.

(2) Oil film stiffness of roller and raceway.

The partial derivative of the contact load between the roller and the raceway and the minimum oil film thickness is the oil film stiffness. According to the theory of elastic fluid lubrication, the oil film stiffness of the cylindrical roller bearing is as follows:

$$K_{oi}={\displaystyle \frac{d{Q}_i}{d{h}_{\min }}}=-13860.986\times {\eta }_0^{5.3846}{u}^{5.3846}{\alpha }^{4.1538}{E}^{-0.2307}{R}^{3.3076}{h}_{\min }^{-8.6923}$$

(11)

$$K_{oe}={\displaystyle \frac{d{Q}_e}{d{h}_{\min }}}=-13860.986\times {\eta }_0^{5.3846}{u}^{5.3846}{\alpha }^{4.1538}{E}^{-0.2307}{R}^{3.3076}{h}_{\min }^{-8.6923}$$

(12)

where ${\eta }_0$ represents the dynamic viscosity at an atmospheric pressure of 20 °C; $u$ is the fluid velocity of the roller and the raceway inlet; $\alpha$ represents the coefficient in the Barau viscosity–pressure formula of base oil contained in lubricating oil or grease; $E$ is the equivalent elastic modulus of contact between the roller and the inner and outer ring raceways; $R$ is the equivalent radius of curvature perpendicular to the moving direction of the rolling element; and $h_{\min }$ is the minimum oil film thickness between the roller and the inner and outer ring raceways [19].

2.3. Rigid–Flexible Coupling Dynamics Theory

2.3.1. Position of Arbitrary Points on Flexible Body

Figure 5 shows the position coordinate system of the node $P$ in the flexible body, where e^r is the fixed inertial coordinate system and e^b is the dynamic coordinate system that changes with the movement of the flexible body. The dynamic coordinate system changes with the displacement and rotation of the flexible body, and the position and direction of the dynamic coordinate system relative to the inertial coordinate system are defined as the reference coordinates.

2.3.2. Velocity and Acceleration of Flexible Body at Arbitrary Points

A flexible body moves from the P1 position to the P2 position, including rigid translation to rigid rotation and deformation motion. The position vector of the node P on the flexible body in space can be expressed as follows:

$$r=r_0+A\left(s_p+u_p\right)$$

(13)

where $r$ is the position vector of point P in the bearing’s inertial coordinate system. $r_0$ is the position vector of the cage’s floating coordinate origin in the bearing’s inertial coordinate system. $A$ is the direction cosine matrix. $s_p$ is the position vector in the floating coordinate system of point P when the flexible cage is not deformed. $u_p$ is the relative deformation vector, and its expression is as follows:

$$u_p={\mathsf{\Phi }}_p{q}_f$$

(14)

where ${\mathsf{\Phi }}_p$ is the assumed deformation mode matrix of a point conforming to the Ritz basis vector condition. $q_f$ is the generalized coordinate of deformation.

The first derivative and the second derivative of time are obtained by the Formula (13), and the velocity vector and acceleration vector of any point on the flexible body can be obtained as shown below.

$${\dot{r}}^p={\dot{r}}_0+\dot{A}\left(s_p+u_p\right)+A{\mathsf{\Phi }}_P{\dot{q}}_f$$

(15)

$${\ddot{r}}^p={\ddot{r}}_0+\ddot{A}\left(s_p+u_p\right)+2\dot{A}{\mathsf{\Phi }}_P{\dot{q}}_f+A{\mathsf{\Phi }}_P{\ddot{q}}_f$$

(16)

2.3.3. Rigid–Flexible Coupling Motion Equations

In this paper, the rigid cage is set up as a flexible cage, and the model is set to the corresponding boundary conditions. When solving the rigid body system in rigid–flexible coupling, it is necessary to establish the corresponding generalized coordinate system. In Adams2018, the Cartesian coordinate system of the center of mass of the rigid body and the generalized Euler angle reflecting the position of the rigid body are used as the generalized coordinates, which can be expressed as follows:

$$q_i={\left[x,y,z,\psi ,\theta ,\phi \right]}_i^T, q={\left[q_1^T,q_2^T,\cdot \cdot \cdot ,q_n^T\right]}^T.$$

In this paper, the system motion equation is established with the Lagrange multiplier method.

$${\displaystyle \frac{d}{d_t}}{\left({\displaystyle \frac{\mathsf{\partial }T}{\mathsf{\partial }\dot{q}}}\right)}^T-{\left({\displaystyle \frac{\mathsf{\partial }T}{\mathsf{\partial }q}}\right)}^T+f_q^T\rho +g_q^T\mu -Q=0$$

(17)

The constraint equation is as follows:

$$g\left(q,\dot{q},t\right)=0$$

(18)

where $T$ is the kinetic energy of the system, $Q$ is the generalized force of the system, $\dot{q}$ is the generalized velocity array, and $\rho$ and $\mu$ are the complete constraint and the incomplete constraint matrix, respectively.

In the simulation of the bearing’s motion process, the Newton–Raphson algorithm is used to iteratively calculate the position of $t_n$ at a certain time under the algebraic equation of the complete constraint.

$${\displaystyle \frac{\mathsf{\partial }f}{\mathsf{\partial }q}}{|}_k\Delta q_k=-f\left(q_k,t_n\right)$$

(19)

The instantaneous velocity and acceleration of the constraint equation at time $t_n$ are as follows:

$$\left\{\begin{array}{l}\left({\displaystyle \frac{\mathsf{\partial }f}{\mathsf{\partial }q}}\right)\dot{q}=-{\displaystyle \frac{\mathsf{\partial }f}{\mathsf{\partial }t}}\\ \left({\displaystyle \frac{\mathsf{\partial }f}{\mathsf{\partial }q}}\right)\ddot{q}=-\left\{\left({\displaystyle \frac{{\mathsf{\partial }}^{2}f}{\mathsf{\partial }{t}^2}}\right)+{\displaystyle \sum _{j=1}^n{\displaystyle \sum _{t=1}^n\frac{{\mathsf{\partial }}^{2}f}{\mathsf{\partial }{q}_j\mathsf{\partial }{q}_i}{\dot{q}}_i{\dot{q}}_j+\frac{\mathsf{\partial }}{\mathsf{\partial }t}\left(\frac{\mathsf{\partial }f}{\mathsf{\partial }q}\right)\dot{q}+\frac{\mathsf{\partial }}{\mathsf{\partial }q}\left(\frac{\mathsf{\partial }f}{\mathsf{\partial }t}\right)\dot{q}}}\right\}\end{array}\right.$$

(20)

The flexibility of the cage was modeled using ANSYS18.0 analysis software. The model utilized SOLID185 elements, with the mesh divided using standard tetrahedral elements. In the flexible model of the normal cage, a total of 30,608 mesh elements were generated, with 126,742 nodes. In the models with length and width deviations, the number of mesh elements was 28,887 and 27,928, with 118,170 and 114,026 nodes, respectively. The finite element mesh division results are shown in Figure 6.

The process of dynamic modeling for cylindrical roller bearings is illustrated in Figure 7. First, a geometric model of the bearing is created and imported into the Adams2018 dynamic simulation software. Then, the cage is made flexible using the ANSYS18.0 finite element software, and the flexible cage with hole size deviation is subsequently imported back into the Adams2018 dynamic simulation software. In Adams2018, material properties are assigned to all bearing components, and the necessary constraints are defined, such as applying a drive and radial force to the inner ring and fixing the outer ring to the bearing housing. Additionally, the contact interactions between the bearing components are considered, and contact definitions are added. Finally, the system is iteratively solved using the Adams/Solver. The results of the simulation can be viewed in the post-processing module.

3. The Influence of Pocket Size Deviation on the Dynamic Characteristics of Cylindrical Roller Bearings

Pocket dimensional deviations induce alterations in the clearance between the cage and the rollers, subsequently affecting the interactions between the cage and the rolling elements, and further impacting dynamic characteristics such as cage motion and stress. Based on this, the coupled rigid–flexible dynamic model of cylindrical roller bearings was proposed, considering pocket dimensional deviations, and the impacts of dimensional deviations in cage pocket length and width on the dynamic characteristics of bearings were investigated.

3.1. The Influence of Dimensional Deviations in the Length of the Cage Pocket on the Dynamic Characteristics of the Bearing

To examine the impact of dimensional deviations in pocket length on the dynamic characteristics of cylindrical roller bearings, dynamic analyses were conducted under the conditions of dimensional deviations in length of −0.25 mm, 0 mm, and 0.25 mm, respectively. The bearing’s inner ring, outer ring, and roller material were set to GCr15, and the cage material was set to Q235. For the inner ring, the speed of the bearing was set to 6000 r/min, and it was subjected to a radial load of 1000 N. The corresponding contact collision parameters are shown in

Table 1. Contact collision parameters.

Contact Stiffness N/mm	Stress Index	Damping N-sec/mm	Contact Depth/mm	Static Coefficient	Dynamic Coefficient
1.5 × 105	10/9	750	0.1	7.0 × 10−2	6.0 × 10−2

.

Figure 8 illustrates the time domain curves of equivalent stress on the cage for three cases: negative (−0.25 mm), neutral (0 mm), and positive (0.25 mm) dimensional deviations in pocket length. The results indicate that negative deviations lead to significantly lower stress on the cage compared to neutral and positive deviations. This occurs due to reduced clearance between the cage and rollers, resulting in tighter contact and limited freedom of motion for the cage, ultimately reducing stress. However, as the deviation becomes positive, the increased clearance reduces the contact forces and lowers the stress on the cage.

Figure 9 presents the motion trajectory of the cage for the three different deviations. As dimensional deviations increase, the range of cage motion becomes more constrained. Under negative deviations, cage motion exhibits a complex pattern with a reduced range. This observation is significant, as it indicates that negative deviations impose stricter constraints on the cage, hindering the smooth operation of the bearing. In contrast, positive deviations provide a wider range of motion, indicating greater freedom and reduced friction between components.

Figure 10 illustrates the translational velocity of the cage for the same three cases. When the pocket length deviation is negative, the velocity fluctuation range is minimized. This is attributed to the tight constraint caused by reduced clearance, which limits the axial motion of the cage. As the deviation becomes positive, the increased clearance reduces friction and allows the cage to move more freely, resulting in larger velocity fluctuations.

Figure 11 presents the speed of the cage in the y and z directions. Negative deviations result in smaller speed fluctuations compared to neutral and positive ones. This is consistent with the reduced clearance hypothesis, where tight constraints limit the cage’s freedom to rotate. As the deviation increases, reduced interaction between the cage and the rollers allows for a more significant speed increase, making the cage’s movement less stable.

Figure 12 and Figure 13 illustrate the trajectory of the center of mass and the speed of the inner ring, respectively. The results indicate that the inner ring’s motion remains relatively stable, even with deviations in the cage pocket length. However, speed fluctuations in the y and z directions increase slightly with greater positive deviations. This suggests that while the inner ring is less affected by pocket size deviations, extreme negative deviations still contribute to greater instability.

Figure 14 presents the roller speed. Similar to that of the inner ring, the roller speed exhibits minimal fluctuation under negative deviations in pocket length. However, as the deviation increases, the fluctuation range expands, reflecting reduced constraints on the rollers due to increased clearance. This suggests that extreme negative deviations in pocket length can disrupt the roller’s motion, compromising the bearing’s stability.

Figure 15 presents the roller slip rate. The bar chart illustrates the slip ratios, $S_r$, of rollers under different length deviations, $L$, of the cage pocket, specifically at three levels: −0.25 mm, 0 mm, and 0.25 mm. It is observed that the slip ratio decreases as the pocket length deviation increases. The highest slip ratio occurs at −0.25 mm, while the lowest is recorded at 0.25 mm.

This trend can be attributed to the fact that a negative deviation (−0.25 mm) in pocket length reduces the clearance between the cage and the rollers, resulting in more constrained motion and increased friction. Consequently, the rollers face greater resistance, leading to a higher slip ratio. In contrast, a positive deviation (0.25 mm) increases the clearance, diminishing the contact forces and friction, which allows the rollers to move more freely, thereby resulting in a lower slip ratio.

The neutral case (0 mm) presents a moderate slip ratio, suggesting that the nominal pocket length provides a balanced roller motion, with neither excessive constraint nor excessive clearance. This finding underscores the importance of controlling the dimensional deviation of pocket length to minimize roller slip and ensure smooth bearing operation.

3.2. The Influence of Dimensional Deviations in the Width of the Cage Pocket on the Dynamic Characteristics of Cylindrical Roller Bearings

In order to study the influence of dimensional deviations in the width of the cage pocket on the dynamic characteristics of the cylindrical roller bearing, a dynamic analysis was conducted under the conditions of dimensional deviations in width of −0.2 mm, 0 mm, and 0.2 mm, respectively. The inner ring speed of the bearing was set to 6000 r/min, and it was subjected to a radial load of 1000 N.

Figure 16 illustrates the equivalent stress for different dimensional deviations in pocket width. Negative deviations lead to significantly higher stress levels compared to neutral and positive deviations. As the width deviation becomes positive, stress decreases significantly. This is because a reduced pocket width (negative deviation) increases contact pressure and friction between the cage and the rollers, thereby increasing stress. Positive deviations, with larger clearances, reduce interaction forces and result in lower stress levels.

Figure 17 presents the cage’s motion trajectory under different width deviations. Under negative deviations, the cage’s motion range increases as the deviation grows. This occurs because a smaller pocket width more tightly constrains the rollers, leading to increased instability in the cage’s motion. As the deviation becomes positive, the cage’s motion becomes less constrained, allowing freer movement.

Figure 18 illustrates the cage speed for different width deviations. Negative deviations result in reduced cage speed, as tight pocket clearance restricts motion. As the deviation increases, the speed grows, reflecting reduced constraints and smoother motion. Beyond a certain point of positive deviation, further increases in clearance do not significantly affect the cage’s speed, indicating that the system has reached a stable dynamic state.

Figure 19 and Figure 20 illustrate the radial displacement and trajectory of the inner ring’s center of mass under different pocket width deviations. Negative deviations impose tighter constraints on the inner ring, resulting in less fluctuation in radial displacement. As the deviation becomes positive, the inner ring gains more freedom in motion, resulting in smoother displacement and a more stable trajectory.

Figure 21 presents the acceleration of the inner ring for different width deviations. Acceleration decreases as the deviation becomes more negative. As the deviation increases, the interaction forces between the cage and the rollers decrease, reducing the overall system’s acceleration. Positive deviations result in stable acceleration patterns, indicating smoother motion.

Figure 22 illustrates the roller speed for different width deviations. Negative deviations result in significant speed fluctuations, as the rollers experience increased interaction forces. As the width deviation increases, the roller speed stabilizes, with positive deviations facilitating smoother motion due to reduced interaction with the cage.

Figure 23 presents the roller slip rate. The bar chart illustrates the slip ratio, $S_r$, of the rollers under different width deviations, $S$, of the cage pocket, specifically at values of −0.2 mm, 0 mm, and 0.2 mm. It is evident that the slip ratio is highest for the negative deviation of −0.2 mm, while both the neutral (0 mm) and positive (0.2 mm) deviations exhibit significantly lower slip ratios, nearly negligible in comparison.

The pronounced increase in slip ratio for the negative deviation can be attributed to the reduction in clearance between the cage and the rollers, which results in excessive contact forces and friction. This tight contact hinders the smooth rolling of the rollers, leading to a significant increase in slippage. In contrast, the neutral and positive deviations offer adequate clearance, reducing friction and allowing the rollers to rotate more freely, thereby resulting in minimal slippage.

These findings suggest that negative deviations in pocket width exert a disproportionately large impact on roller performance, highlighting the importance of precise manufacturing to avoid negative dimensional deviations that can severely compromise bearing efficiency.

The results indicate that dimensional deviations in both the length and width of the cage pocket significantly affect the dynamic characteristics of cylindrical roller bearings. Negative deviations in both dimensions result in increased stress, greater speed fluctuations, and tighter constraints on cage motion, the inner ring, and the rollers. In contrast, positive deviations smooth the system’s dynamics by reducing contact forces and allowing freer motion.

These results indicate that precision in the dimensional control of the cage pocket is essential for bearing design. Designers should aim to minimize negative deviations, particularly in pocket length, as such deviations lead to higher stress and greater system instability. Positive deviations, within acceptable limits, may provide a more stable dynamic response by reducing interaction forces. These findings offer clear guidelines for setting tolerances in the manufacturing process to ensure that deviations are minimized and controlled, particularly in high-performance applications.

In manufacturing, strict quality control measures must be implemented to ensure that dimensional deviations in the cage pocket remain within optimal ranges. This may involve improved machining accuracy and stricter inspections throughout the production process. Given the pronounced effects of negative deviations, manufacturing processes should prioritize their avoidance, to maintain optimal bearing performance and prolong service life.

4. Conclusions

The influence of pocket size deviation on the dynamic characteristics of cylindrical roller bearings is studied in this paper, and the influence of dimensional deviations in cage pocket length and width on bearing motion is obtained. The conclusions are as follows.

(1) Dimensional deviations in the cage pocket length significantly affect the dynamic performance of the bearing. Negative length deviations result in reduced amplitudes of equivalent stress on the cage, indicating more constrained motion. As the deviation increases, the range of cage motion decreases, while fluctuations in the speed of both the inner ring and the rollers increase. The bearing’s overall dynamic response becomes more sensitive under negative deviations, primarily due to reduced clearance between the cage and the rollers, which amplifies friction and contact forces. This constraint on cage motion leads to imbalances in rotational speeds, contributing to increased fluctuations.

(2) Dimensional deviations in the cage pocket width exhibit a different influence pattern. Negative width deviations significantly increase the equivalent stress. As the width deviation becomes positive, the stress gradually decreases and stabilizes, reflecting the reduced interaction between the cage and the rollers. A wider pocket allows for greater freedom in cage movement, resulting in increased displacement and smoother dynamic behavior of the inner ring and the rollers. However, under negative width deviations, increased contact stress can lead to potential jamming and higher interaction forces, disrupting smooth operation.

(3) Negative deviations more significantly affect the dynamics of cylindrical roller bearings by reducing the clearances between the rolling elements and the cage, resulting in tighter contact. This intensifies the friction and collision forces between components, particularly in the case of length deviations. Increased stress, constrained movement, and altered interaction dynamics under negative deviations significantly disturb bearing balance, causing greater irregularities in speed and motion. In contrast, positive deviations increase clearance, thereby facilitating smoother interactions, reducing stress, and stabilizing the overall system.

(4) In practical bearing design, minimizing negative deviations is crucial for avoiding excessive friction, stress, and speed fluctuations, which can compromise the bearing’s performance and longevity. From a manufacturing perspective, this research highlights the necessity of precision in machining and assembling cage pockets to maintain optimal clearances. Specifically, stricter tolerances should be enforced for the length dimension to prevent significant performance degradation due to negative deviations. Insights from this study offer valuable guidelines for improving design and manufacturing processes, ensuring more reliable and durable cylindrical roller bearings.

(5) The influence of pocket size deviations on the dynamic characteristics of cylindrical roller bearings is analyzed in this paper. Future research could explore the impact of pocket position deviations on bearing friction and lubrication performance. Furthermore, the dynamic behavior of bearings under pocket position deviations could also be analyzed.