# Study on Ring Deformation and Contact Characteristics of Thin-Walled Bearing for RV Reducer
## 2 Theoretical Analysis
### Mathematical Model
To sum up, considering the ring deformation, the ball-raceway total radial contact deformation is  
\[ \begin{array}{rcl}\delta_{\varphi r}=&\overline{\delta_{\varphi r}}-w_{ \varphi}=A\cos\alpha^{o}+\delta_{r}\cos\varphi-\\ &&\frac{R^{3}}{\pi EI_{o}}\left[\sum\limits_{k=1}^{l}\sum\limits_{n=2}^{\infty} \frac{P_{nk}^{o}\cos\left(n\varphi_{k}^{\beta^{\beta}}-n\varphi\right)}{\left( n^{2}-1\right)^{2}}+\sum\limits_{k=1}^{j}\sum\limits_{n=2}^{\infty}\frac{P_{nk}^{ o}\sin\left(n\varphi_{k}^{\beta^{\beta}}-n\varphi\right)}{n\left(n^{2}-1\right)^{2}} \right]+\\ &&\frac{R^{3}}{\pi EI_{i}}\left[\sum\limits_{k=1}^{l}\sum\limits_{n=2}^{\infty} \frac{P_{nk}^{i}\cos\left(n\varphi_{k}^{\beta^{i}}-n\varphi\right)}{\left(n^{2 }-1\right)^{2}}+\sum\limits_{k=1}^{j}\sum\limits_{n=2}^{\infty}\frac{P_{nk}^ {i}\sin\left(n\varphi_{k}^{\beta^{i}}-n\varphi\right)}{n\left(n^{2}-1\right)^ {2}}\right]\end{array}  \tag{22}\]  
The ring deformation leads to a change in the ball-raceway total contact deformation. Based on Equations (21) and (22), the geometric equation in the bearing rigid ring model is updated, where the expression of the distance between the curvature centers of the inner and outer raceway can be updated as follows:  
\[S^{\prime}=\sqrt{\left(A\sin\alpha^{o}+\delta_{a}+\Re_{i}\theta\cos\varphi \right)^{2}+\left(A\cos\alpha^{o}+\delta_{r}\cos\varphi-w_{\varphi}\right)^{2}} \tag{23}\]  
The expression of the ball-raceway total contact deformation can be updated as follows:  
\[\delta^{\prime}{}_{n}=\delta^{\prime}{}_{i}+\delta^{\prime}{}_{o}=\sqrt{\left( A\sin\alpha^{o}+\delta_{a}+\Re_{i}\theta\cos\varphi\right)^{2}+\left(A\cos \alpha^{o}+\delta_{r}\cos\varphi-w_{\varphi}\right)^{2}}-A \tag{24}\]  
The expression of the ball-raceway contact angle can be updated as follows:  
\[\left\{\begin{array}{rcl}\sin\alpha^{\prime}=\frac{A\sin\alpha^{o}+\delta_{a }+\Re_{i}\theta\cos\varphi}{\left(A\sin\alpha^{o}+\delta_{a}+\Re_{i}\theta \cos\varphi\right)^{2}+\left(A\cos\alpha^{o}+\delta_{r}\cos\varphi-w_{\varphi} \right)^{2}}\\ \cos\alpha^{\prime}=\frac{A\cos\alpha^{o}+\delta_{r}\cos\varphi-w_{\varphi 2}}{ \left(A\sin\alpha^{o}+\delta_{r}\Re_{i}\theta\cos\varphi\right)^{2}+\left(A\cos \alpha^{o}+\delta_{r}\cos\varphi-w_{\varphi}\right)^{2}}\\ \tan\alpha^{\prime}=\frac{A\sin\Model Solution  
The rigid ring model is composed of geometric Equations (1)-(3), Hertz contact Equation (5), and inner ring equilibrium Equation (7). This model is a set of simultaneous nonlinear equations with unknown variables (axial displacement \(\delta_{a}\), radial displacement \(\delta_{r}\), and angular displacement \(\theta\)). The flexible ring model should also consider the ring bending deformation \(w_{\varphi}\) of the ring at each rolling element position and its influence on the bearing contact characteristics. This model is solved by the Newton-Raphson iterative algorithm, and the detailed iterative process is shown in Figure 4. Firstly, the geometric parameters, material parameters, and working condition parameters of the bearing are input, and then the global displacement iteration variables are initialized. The ball-raceway contact deformation is calculated according to the geometric relationship, and the ball-raceway contact state is judged. Then, the ball-raceway contact load is calculated according to the Hertz contact theory. The above results are substituted into the inner ring equilibrium equations, and the Newton-Raphson iterative algorithm is used to solve the inner ring equilibrium equations. Finally, the bearing contact characteristics under the rigid ring conditions are obtained. Based on the rigidity results, the ring bending deformation is analyzed. Then, the contact deformation, contact load, contact angle, and inner ring equilibrium equations are updated. Finally, the above iterative steps are repeated to obtain the ring deformation and bearing contact characteristics under the flexible ring condition.