## 2 Rigid-Flexible Coupling Dynamic Model of Cylindrical Roller Bearings
### Rigid-Flexible Coupling Dynamics Theory
#### 2.3.3 Rigid-Flexible Coupling Motion Equations
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. Figure 11: Cage speed of dimensional deviations in the length. Figure 10: Transnational velocity of cage of dimensional deviations in the length. 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. Figure 12: Inner circle centroid trajectory of dimensional deviations in the length. Figure 13: Inner ring speed of dimensional deviations in the length. 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 14: Roller speed of dimensional deviations in the length. Figure 15: Roller slip rate of dimensional deviations in the length. 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.