# Study on Ring Deformation and Contact Characteristics of Thin-Walled Bearing for RV Reducer
## 2 Theoretical Analysis
### Mathematical Model
\tag{7}\]  
where \(Q_{k}\) and \(\alpha_{k}\) represent the ball-raceway contact load and contact angle at the angular position \(\varphi_{k}\), respectively, and subscript \(k\) represents the \(k\)th ball. However, due to the influence of substantial external forces, the bending deformation of thin-walled bearing rings is particularly significant, approaching or even exceeding the Hertzian contact deformation. In this case, the influence of radial deflection at arbitrary ring positions on the contact deformation must be considered. Figure 2 shows the radial deflections \(w_{\varphi o}\) and \(w_{qi}\) caused by the ring deformation at the arbitrary azimuth \(\varphi\). Then, the ball-raceway radial deflection is  
\[w_{\varphi}=w_{\varphi o}-w_{qi} \tag{8}\]  
Figure 2: Effect of ring elastic deformation on ball–raceway contact deformation. According to Equations (2) and (8), the ball-raceway total radial deformation \(\delta_{qr}\) at any position \(\varphi\) of the ring under the flexible ring model is  
\[\delta_{qr}=\overline{\delta_{qr}}-w_{\varphi}=A\cos\alpha^{o}+\delta_{r}\cos \varphi-w_{\varphi} \tag{9}\]  
where \(\overline{\delta_{qr}}\) is the radial displacement of the curvature center of the inner ring relative to the curvature center of the outer ring after the bearing is loaded under the rigid ring condition. According to Equation (9), the key to obtaining the bearing contact characteristics under the flexible ring condition is to solve the ball-raceway radial deflection \(w_{\varphi}\) at any position. Bearing ring bending deformation can be considered a planar problem. According to the thin-walled ring plane bending theory [25], the differential equation for the radial deflection deformation of the ring at the arbitrary azimuth \(\varphi\) is given by  
\[\frac{\mathrm{d}^{2}w}{\mathrm{d}\varphi^{2}}+w=-\frac{M}{EI}R^{2} \tag{10}\]  
where \(w\) is the bending radial displacement of the thin-walled ring relative to the center of the circle; \(M\) is the bending moment that causes the curvature of the circle to change (\(M=\frac{Q}{2}\left[\frac{\cos\varphi}{\sin(\Delta\varphi/2)}-\frac{2}{\Delta \varphi}\right]R\); \(Q\) is the ball-raceway contact load; \(\Delta\varphi\) is the ball equidistant angle; \(E\) is the elastic modulus of the material; \(I\) is the section moment of inertia in bending (\(I=\pi/64\left(D^{4}-D^{4}\right)\); \(D\) is the outer diameter of the outer ring; \(D^{\prime}\) is the inner diameter of the outer ring), and \(R\) is the pitch radius of the ring. The thin-walled ring produces small deflection deformation under external force, and the displacement at any point on the center line of the ring can be decomposed into a radial displacement \(w\) and a tangential displacement \(v\), as illustrated in Figure 3. \(P_{r}\) is the radial force; \(P_{t}\) is the tangential force; \(\phi_{k}^{Q}\) is the azimuthal angle of the load, and the subscript \(k\) represents different load positions in Figure 3. The expression for the trigonometric series of radial displacement and tangential displacement is given by  
\[ \begin{array}{l}w(\varphi)=\sum\limits_{n=2}^{n=\infty}(a_{n}\cos(n\varphi) +b_{n}\sin(n\varphi))\\ v(\varphi)=\sum\limits_{n=2}^{n=\infty}(a_{n}\cos(n\varphi)-b_{n}\sin(n\varphi) )\end{array}  \tag{11}\]  
where \(a\) and \(b\) are the major and minor axes of contact deformation caused by deflection deformation, respectively, and \(n\) is a constant. Thin-walled ring  
Figure 3: The force and displacement of the ring. The ring is initially in equilibrium, and with the effect of the radial force \(P_{r}\) and the tangential force \(P_{t}\), the ring deviates from the equilibrium position and produces an imaginary displacement.