## 2 Rigid-Flexible Coupling Dynamic Model of Cylindrical Roller Bearings
### Rigid-Flexible Coupling Dynamics Theory
#### 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:  
\[q_{i}=\big{[}x,y,z,\psi,\theta,\varphi\big{]}_{i}^{T},\;q=\Big{[}q_{1}^{T},q_{ 2}^{T},\cdots,q_{n}^{T}\Big{]}^{T}.\]  
In this paper, the system motion equation is established with the Lagrange multiplier method. \[\frac{d}{d_{t}}\bigg{(}\frac{\partial T}{\partial\dot{q}}\bigg{)}^{T}-\bigg{(} \frac{\partial T}{\partial q}\bigg{)}^{T}+f_{q}^{T}\rho+g_{q}^{T}\mu-Q=0 \tag{17}\]  
Figure 5: Position of node P on flexible body. The constraint equation is as follows:  
\[g\big{(}q,\dot{q},t\big{)}=0 \tag{18}\]  
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. \[\frac{\partial f}{\partial q}|_{k}\Delta q_{k}=-f(q_{k},t_{n}) \tag{19}\]  
The instantaneous velocity and acceleration of the constraint equation at time \(t_{n}\) are as follows:  
\[\left\{ \begin{array}{l}\left(\frac{\partial f}{\partial q}\right)\dot{q}=- \frac{\partial f}{\partial t}\\ \left(\frac{\partial f}{\partial q}\right)\ddot{q}=-\left\{\left(\frac{ \partial^{2}f}{\partial t^{2}}\right)+\sum\limits_{j=1}^{n}\sum\limits_{t=1} ^{n}\frac{\partial^{2}f}{\partial q_{j}\partial q_{i}}\dot{q}_{i}\dot{q}_{j}+ \frac{\partial}{\partial t}\left(\frac{\partial f}{\partial q}\right)\dot{q}+ \frac{\partial}{\partial q}\left(\frac{\partial f}{\partial t}\right)\dot{q} \right\}\end{array} \right. \tag{20}\]  
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. Figure 6: Cage meshing. 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.