# Study on Ring Deformation and Contact Characteristics of Thin-Walled Bearing for RV Reducer
## 3 Results and Discussion
### The Effect of the Outer Ring Thickness  
The influence of the outer ring thickness on the ring deformation is shown in Figure 19. It is evident that with the increase in ring thickness, both the horizontal and vertical diameters of the ring decrease. The ring thickness only affects the ring deformation and does not change the deformation trend of the ring. However, there is a significant distance between the fixed constraint surface of the reducer and the outer ring surface, and this distance leads to significant ring deformation, making it difficult for the ring to approximate a standard circle, even with larger ring thicknesses. Figure 18: Effect of radial force on contact characteristics: (**a**) number of loaded balls; (**b**) the relative ratio of the maximum load and ball–raceway contact stress; (**c**) radial displacements of the inner ring; (**d**) ball–raceway contact angles. Figure 20a shows the number of loaded balls with different ring thicknesses, which increases from 29 under the rigid ring model to 37 under the flexible ring model. It can also be concluded that the thinner the ring thickness, the more the contact load is reduced, and the load optimization effect is more pronounced. However, the thinner ring thickness increases the failure risk of plastic deformation. The bearing fatigue life based on [28] was calculated, as shown in Figure 20b. The calculation equation of the bearing fatigue life is as follows:  
\[L_{n}=a_{1}a_{2}a_{3}L_{10} \tag{30}\]  
where \(L_{n}\) is the bearing fatigue life modified by [28]; \(a_{1}\) is the reliability coefficient; \(a_{2}\) is the material coefficient, and \(a_{3}\) is the service condition coefficient. The fatigue life \(L_{10}\) is determined by the service life of the ball-faceway contact, which is calculated as follows [29]:  
\[L_{10}=\left(L_{i}^{-10/9}+L_{o}^{-10/9}\right)^{-0.9} \tag{31}\]  
where \(L_{i}\) and \(L_{o}\) are the contact life of the ball-inner raceway and the ball-outer raceway, respectively, and the calculation method is as follows:  
\[L_{i,\rho}=\left(\frac{C_{i,\rho}}{Q_{i,\rho}}\right)^{3} \tag{32}\]  
where \(C_{i,\rho}\) is the basic dynamic load rating of the inner and outer raceway; \(Q_{i,\rho}\) is the dynamic equivalent rolling element load of the inner and outer raceway, and the calculation method of \(C_{i,\rho}\) is as follows:  
\[\left\{ \begin{array}{l}C_{i}=\frac{C_{r}}{0.407Z(\cos{a})^{0.7}}\left\{1+ \left[1.04\left(\frac{1-\gamma}{1+\gamma}\right)^{1.72}\left(\frac{2f_{i}f_{0} -f_{i}}{2f_{i}/b-f_{o}}\right)^{0.41}\right]^{10/3}\right\}^{0.3}\\ C_{o}=\frac{C_{r}}{0.389Z(\cos{a})^{0.7}}\left\{1+\left[1.04\left(\frac{1- \gamma}{1+\gamma}\right)^{1.72}\left(\frac{2f_{i}f_{0}-f_{i}}{2f_{i}/b-f_{o}} \right)^{0.41}\right]^{-10/3}\right\}^{0.3}\end{array} \right. \tag{33}\]  
where \(C_{r}\) is the basic dynamic load rating of the bearing, which is calculated as follows:  
\[C_{r}=b_{m}f_{c}(i\cos{\alpha})^{0.7}Z^{2/3}D_{w}^{1.8} \tag{34}\]  
where \(b_{m}\) is the dynamic load coefficient; \(i\) is the number of the bearing rows; \(\alpha\) is the bearing contact angle; \(Z\) is the number of balls; \(D_{w}\) is the diameter of the ball, and the expression of \(f_{c}\) is  
Figure 19: Effect of the ring thickness on ring deformation: (**a**) ring deformation; (**b**) diameter augment. \[f_{\rm c}=39.9\Bigg{\{}1+\bigg{[}1.04\bigg{(}\frac{1-\gamma}{1+\gamma}\bigg{)}^{1.72 }\bigg{(}\frac{2f_{\rm f}f_{\rm o}-f_{\rm i}}{2f_{\rm f}f_{\rm o}-f_{\rm o}} \bigg{)}^{0.41}\bigg{]}^{10/3}\Bigg{\}}^{-0.3}\times\frac{\gamma^{0.3}(1- \gamma)^{1.39}}{(1+\gamma)^{1/3}}\bigg{(}\frac{2f_{\rm i}}{2f_{\rm i}-1}\bigg{)} ^{0.41} \tag{35}\]  
where \(\gamma\) is a dimensionless parameter with the expression  
\[\gamma=\frac{D_{w}\cos\alpha}{d_{m}} \tag{36}\]  
In this study, the outer ring is fixed while the inner ring rotates. Therefore, the calculation methods for \(Q_{i\rho}\) are as follows:  
\[\left\{