## 2 Rigid-Flexible Coupling Dynamic Model of Cylindrical Roller Bearings
### Rigid-Flexible Coupling Dynamics Theory
#### 2.3.3 Rigid-Flexible Coupling Motion Equations
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. Figures 19 and 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 7:0 Time domain curves of cage equivalent forces for dimensional deviation. Figure 7:1 Trajectory of cage centroid of dimensional deviations in the width. 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. Figure 7:2 Inner ring displacement of dimensional deviations in the width. Figure 7:3 Cage speed of dimensional deviations in the width. 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  
Figure 7:4 Acceleration of inner ring of dimensional deviations in the width. Figure 7:5 Inner circle centroid trajectory of dimensional deviations in the width. tolerances in the manufacturing process to ensure that deviations are minimized and controlled, particularly in high-performance applications.