# Study on Ring Deformation and Contact Characteristics of Thin-Walled Bearing for RV Reducer
## 2 Theoretical Analysis
### Mathematical Model
Thus, the imaginary displacement along the radial direction at the point of force action can be described as  
\[\Delta w=\sum\limits_{n=2}^{n=\infty}\left(\Delta a_{n}\cos\left(n\phi_{k}^{Q} \right)+\Delta b_{n}\sin\left(n\phi_{k}^{Q}\right)\right) \tag{12}\]  
\[\Delta v=\frac{1}{n}\sum\limits_{n=2}^{n=\infty}\left(\Delta a_{n}\cos\left(n \phi_{k}^{Q}\right)-\Delta b_{n}\sin\left(n\phi_{k}^{Q}\right)\right) \tag{13}\]  
Suppose the ring receives \(l\) radial force and \(j\) tangential force; then the work executed by all external forces is as follows:  
\[\Delta W= \begin{array}{c}\sum\limits_{k=1}^{k=l}P_{rk}\left(\Delta a_{n} \cos\left(n\phi_{k}^{Q}\right)+\Delta b_{n}\sin\left(n\phi_{k}^{Q}\right) \right)+\\ \frac{1}{n}\sum\limits_{k=1}^{k=l}P_{lk}\left(\Delta a_{n}\cos\left(n\phi_{k} ^{Q}\right)-\Delta b_{n}\sin\left(n\phi_{k}^{Q}\right)\right)\end{array}  \tag{14}\]  
The strain energy of the ring due to deflecting deformation can be written as  
\[U=\int_{0}^{2\pi}\frac{M^{2}R}{2 EI}\;\mathrm{d}\varphi \tag{15}\]  
By substituting Equations (10) and (12) into Equation (15), the strain energy can be represented by the following equation:  
\[U=\frac{\pi EI}{2R^{3}}\sum\limits_{n=2}^{n=\infty}\left(n^{2}-1\right)\left( a_{n}^{2}+b_{n}^{2}\right) \tag{16}\]  
According to the principle of imaginary displacement, the change in strain energy of the ring is equal to the work carried out by the external force; then,  
\[\frac{\partial U}{\partial a_{n}}\Delta a_{n}+\frac{\partial U}{\partial b_{n} }\Delta b_{n}=\Delta W \tag{17}\]  
By incorporating Equations (14) and (16) into the equation provided above, the expression of \(a_{n}\) and \(b_{n}\) can be derived as follows:  
\[ \begin{array}{c}a_{n}=\frac{R^{3}}{\pi EI(n^{2}-1)^{2}}\left(\sum\limits_{k =1}^{l}P_{rk}\cos\left(n\phi_{k}^{Q}\right)+\frac{1}{n}\sum\limits_{k=1}^{l}P _{tk}\sin\left(n\phi_{k}^{Q}\right)\right)\\ b_{n}=\frac{R^{3}}{\pi EI(n^{2}-1)^{2}}\left(\sum\limits_{k=1}^{l}P_{rk}\sin \left(n\phi_{k}^{Q}\right)-\frac{1}{n}\sum\limits_{k=1}^{l}P_{tk}\cos\left(n \phi_{k}^{Q}\right)\right)\end{array}  \tag{18}\]  
The radial displacement \(w_{p}\) caused by the radial force and the radial displacement \(w_{t}\) caused by the tangential force are, respectively,  
\[w_{p}(\varphi)=\frac{R^{3}}{\pi EI}\sum\limits_{k=1}^{l}\sum\limits_{n=2}^{ \infty}\frac{P_{rk}\cos\left(n\phi_{k}^{Q}-n\varphi\right)}{\left(n^{2}-1 \right)^{2}} \tag{19}\]  
\[w_{t}(\varphi)=\frac{R^{3}}{\pi EI}\sum\limits_{k=1}^{l}\sum\limits_{n=2}^{ \infty}\frac{P_{tk}\sin\left(n\phi_{k}^{Q}-n\varphi\right)}{n\left(n^{2}-1 \right)^{2}} \tag{20}\]  
From the above analysis, the radial deformation \(w_{po}\) and \(w_{to}\) of the ball-outer ring and the radial deformation \(w_{pi}\) and \(w_{ti}\) of the ball-inner ring, which are caused by the ring deformation, can be obtained. Therefore, the total contact deformation from the ring deformation at arbitrary azimuth angle \(\varphi\) is  
\[ \begin{array}{rcl}w_{\varphi}&=&w_{\varphi o}-w_{qi}=w_{po}+w_{to}-w_{pi}-w_{ti }\\ &=&\frac{R^{3}}{\pi EL_{o}}\left[\sum\limits_{k=1}^{l}\sum\limits_{n=2}^{\infty} \frac{P_{nk}^{o}\cos\left(n\varphi_{k}^{\beta^{\beta^{\beta}}}-n\varphi\right) }{\left(n^{2}-1\right)^{2}}+\sum\limits_{k=1}^{j}\sum\limits_{n=2}^{\infty} \frac{P_{nk}^{o}\sin\left(n\varphi_{k}^{\beta^{\beta^{\beta}}}-n\varphi\right) }{n\left(n^{2}-1\right)^{2}}\right]-\\ &&\frac{R^{3}}{\pi EI_{i}}\left[\sum\limits_{k=1}^{l}\sum\limits_{n=2}^{\infty} \frac{P_{nk}^{i}\cos\left(n\varphi_{k}^{\beta^{i}}-n\varphi\right)}{\left(n^{ 2}-1\right)^{2}}+\sum\limits_{k=1}^{j}\sum\limits_{n=2}^{\infty}\frac{P_{nk}^ {i}\sin\left(n\varphi_{k}^{\beta^{i}}-n\varphi\right)}{n\left(n^{2}-1\right)^ {2}}\right]\end{array}  \tag{21}\]  
where \(I_{o}\) and \(I_{i}\) are the bending moments of inertia of the outer and inner rings, respectively; \(P_{rk}^{o}\), \(P_{lk}^{o}\), \(P_{rk}^{i}\), and \(P_{lk}^{i}\) represent the radial and tangential forces on the outer and inner rings.