## 2 Rigid-Flexible Coupling Dynamic Model of Cylindrical Roller Bearings
### Pocket Dimensional Deviation Characterization
#### 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:  
Figure 4: Interaction forces in bearings. Figure 3: Rigid–flexible coupling dynamic model of cylindrical roller bearing. \[\left\{ \begin{array}{l}m_{c}\ddot{y}_{c}=\sum\limits_{j=1}^{Z}\left(Q_{Bj}\sin \theta_{j}-Q_{Fj}\sin\theta_{j}-f_{TBFj}\cos\theta_{j}+f_{TFRj}\cos\theta_{j} \right)-Q_{c}\cos\theta_{j}+f_{c}\sin\theta_{j}\\ m_{c}\ddot{z}_{c}=\sum\limits_{j=1}^{Z}\left(-Q_{Bj}\cos\theta_{j}+Q_{Fj}\cos \theta_{j}-f_{TBFj}\sin\theta_{j}+f_{TFRj}\sin\theta_{j}\right)-Q_{c}\sin\theta _{j}+f_{c}\cos\theta_{j}\\ l_{cx}\dot{cu}_{cx}=\frac{1}{2}d_{m}\sum\limits_{j=1}^{Z}\left(-Q_{Bj}+Q_{Fj}- f_{c}\right)\\ l_{cy}\dot{cu}_{cy}=\frac{1}{2}L_{r}\sum\limits_{j=1}^{Z}\left(Q_{Bj}-Q_{Fj} \right)\cos\theta_{j}\\ l_{cz}\dot{cu}_{cz}=\frac{1}{2}L_{r}\sum\limits_{j=1}^{Z}\left(Q_{Fj}-Q_{Bj} \right)\sin\theta_{j}\\ \end{array} \right. \tag{3}\]  
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, \(f_{efj}\) 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, \(l_{cx}\), \(l_{cy}\), \(l_{cz}\) 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:  
\[Q_{n}=\left\{ \begin{array}{cc}K_{n}(\delta_{0}-\delta)^{e}-C_{max}(\frac{d \delta}{dt})\times step(\delta,\delta_{0}-d,1,\delta_{0},0),&\delta\leq\delta _{0}\\ 0,&\delta>\delta_{0}\end{array} \right. \tag{4}\]  
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:  
\[f_{n}=\mu Q_{n} \tag{5}\]  
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\{