# Study on Ring Deformation and Contact Characteristics of Thin-Walled Bearing for RV Reducer
## 2 Theoretical Analysis  
### Mathematical Model  
In this study, the bearing outer ring is in a fixed state, while the inner ring is rotating. Figure 1 illustrates the relative displacement relationship of the curvature centers of the inner and outer raceways before and after the bearing is loaded. Before the bearing is loaded, there is no contact deformation between the ball and the raceway, and the distance \(A\) between the curvature centers of the inner and outer raceways can be expressed as  
\[A=(f_{i}+f_{o}-1)D_{w} \tag{1}\]  
where \(f_{i}\) and \(f_{o}\) are the curvature coefficients of the inner and outer raceways, respectively, and \(D_{w}\) is the diameter of the ball. After the bearing is loaded, the curvature center of the inner raceway changes relative to the curvature center of the outer raceway, and the curvature center of the inner raceway moves from \(O_{i}\) to \(O_{i}^{\prime}\). According to the geometric relationship in Figure 1, the distance \(S\) between the curvature centers of the inner and outer raceways at any ball position \(\varphi\) can be expressed as  
\[S=\sqrt{\left(A\sin\alpha^{o}+\delta_{a}+\Re_{i}\theta\cos\varphi\right)^{2}+ \left(A\cos\alpha^{o}+\delta_{r}\cos\varphi\right)^{2}} \tag{2}\]  
where \(\alpha^{o}\) is the initial contact angle; \(\delta_{a}\), \(\delta_{r}\), \(\theta\) are the axial, radial, and angular relative displacements of the inner and outer ring; \(\Re=d_{m}/2+(f_{i}-0.5)D_{w}{\rm cos}\alpha^{o}\); \(d_{m}\) is the pitch diameter of the bearing. According to Equations (1) and (2), the ball-raceway total contact deformation \(\delta_{n}\) can be expressed as  
\[\delta_{n}=\delta_{i}+\delta_{o}=S-A=\sqrt{\left(A\sin\alpha^{o}+\delta_{a}+ \Re_{i}\theta\cos\varphi\right)^{2}+\left(A\cos\alpha^{o}+\delta_{r}\cos \varphi\right)^{2}-A} \tag{3}\]  
where \(\delta_{i}\), \(\delta_{o}\) are the contact deformations of the ball-inner raceway and the ball-outer raceway. According to the load-displacement relationship of the ball-raceway contact, the expression of the ball-raceway contact load \(Q\) is as follows:  
\[\left\{ \begin{array}{l}Q=0,\ \delta^{n}\leq 0\\ Q=K_{n}\delta^{n},\delta^{n}>0\end{array} \right. \tag{4}\]  
where \(K_{n}\) is the load-displacement coefficient, and \(n=3/2\) for ball bearings. When \(\delta^{n}\leq 0\), the ball separates from the raceway, while when \(\delta^{n}>0\), the ball makes contact with the  
Figure 1: The position relationships between the ball center and the raceway curvature centers. raceway. The specific expression for the ball-raceway contact load at any azimuth angle \(\varphi\) can be obtained by substituting Equation (3) into Equation (4):  
\[Q=K_{n}\left(\sqrt{\left(A\sin\alpha^{o}+\delta_{a}+\Re_{i}\theta\cos\varphi \right)^{2}+\left(A\cos\alpha^{o}+\delta_{r}\cos\varphi\right)^{2}}-A\right)^{1.5} \tag{5}\]  
According to the geometric relationship in Figure 1, the expression of the ball-raceway contact angle at any azimuth angle \(\varphi\) is as follows:  
\[\left\{ \begin{array}{l}\sin\alpha=\frac{A\sin\alpha^{o}+\delta_{a}+\Re_{i} \theta\cos\varphi}{\sqrt{\left(A\sin\alpha^{o}+\delta_{a}+\Re_{i}\theta\cos \varphi\right)^{2}+\left(A\cos\alpha^{o}+\delta_{r}\cos\varphi\right)^{2}}}\\ \cos\alpha=\frac{A\cos\alpha^{o}+\delta_{r}\cos\varphi}{\sqrt{\left(A\sin\alpha ^{o}+\delta_{r}+\Re_{i}\theta\cos\varphi\right)^{2}+\left(A\cos\alpha^{o}+ \delta_{r}\cos\varphi\right)^{2}}}\\ \tan\alpha=\frac{A\sin\alpha^{o}+\delta_{a}+\Re_{i}\theta\cos\varphi}{A\cos \alpha^{o}+\delta_{r}\cos\varphi}\end{array} \right. \tag{6}\]  
Assuming that the axial force, radial force, and moment applied to the bearing are \(F_{a}\), \(F_{r}\), and \(M\), respectively, and that the external load applied to the inner ring should be balanced with the load applied to the inner ring by the balls, then, the balance equation of the inner ring can be written as  
\[\left\{ \begin{array}{l}F_{a}=\sum Q_{k}\sin\alpha_{k}\\ F_{r}=\sum Q_{k}\cos\varphi_{k}\cos\alpha_{k}\\ M=\frac{d_{m}}{2}\sum Q_{k}\cos\varphi_{k}\sin\alpha_{k}\end{array} \right.