## 2 Rigid-Flexible Coupling Dynamic Model of Cylindrical Roller Bearings
### Rigid-Flexible Coupling Dynamics Theory
#### 2.3.3 Rigid-Flexible Coupling Motion Equations
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. 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.  
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. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Contact Stiffness N/mm** & **Stress Index** & **Damping N-sec/mm** & **Contact Depth/mm** & **Static Coefficient** & **Dynamic Coefficient** \\ \hline \(1.5\times 10^{5}\) & 10/9 & 750 & 0.1 & \(7.0\times 10^{-2}\) & \(6.0\times 10^{-2}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Contact collision parameters. Figure 7: Dynamic modeling process. 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  
Figure 8: Time domain curves of cage equivalent forces. Figure 9: Trajectory of cage motion of dimensional deviations in the length. 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. Figures 12 and 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.