# Study on Ring Deformation and Contact Characteristics of Thin-Walled Bearing for RV Reducer
## 3 Results and Discussion
### Comparison of Rigid and Flexible Ring Model Results  
The ring deformation and contact characteristics of the two main bearings are comparative analyses under the normal operating conditions of the RV reducer. As seen in Figure 9, to facilitate the characterization of the ring deformation, the \(Y\)-axis denotes the vertical direction, while the \(X\)-axis represents the horizontal direction. The ring is deformed irregularly under the action of the ball-raceway contact loads, and the deformation level is in the same micrometer scale as the ball-raceway contact deformation, which further proves the necessity and correctness of the flexible ring model, as shown in Figure 10.  
\begin{table}
\begin{tabular}{c c c c} \hline
**Component** & **Modulus of Elasticity \(E\) (GPa)** & **Poisson’s Ratio \(\
u\)** & **Density \(\rho\) (kg/m\({}^{3}\))** \\ \hline Planetary frame & 210 & 0.29 & 7850 \\ \hline Inner and outer ring & 205 & 0.3 & 7806 \\ \hline Ball & 203 & 0.29 & 7810 \\ \hline Pin gear housing & 169 & 0.286 & 7120 \\ \hline \end{tabular}
\end{table}
Table 2: Material parameters of the thin-walled angular contact ball bearing.  
Figure 8: Comparison of results between the flexible and rigid ring hypothesis models: (**a**) ring deformation; (**b**) ball–raceway contact loads.  
The forces in the \(y\) and \(z\) directions in Figure 10 are external load conditions for these specific results. Compared with the standard circle of the rigid ring model, the outer ring deformation of the left bearing tends to be an elliptical shape, and the horizontal and vertical diameters become larger, with the maximum increment of the horizontal diameter being 2.4 \(\upmu\)m and the maximum increment of the vertical diameter being 4.08 \(\upmu\)m. The outer ring deformation of the right bearing appears to be a cam shape, with a decrease in the horizontal diameter and an increase in the vertical diameter. The maximum reduction in the horizontal diameter measures 2.46 \(\upmu\)m, while the maximum increase in the vertical diameter is 3.14 \(\upmu\)m. The two trends are closely linked to the distribution of the ball-raceway contact load. All balls of the left bearing are loaded, causing the ring diameter to increase, while the contact loads of the right bearing are concentrated in the 90\({}^{\circ}\) azimuth area, causing the ring diameter to increase in that direction and decrease in the horizontal direction.  
The ring deformation inevitably induces alterations in the bearing contact characteristics. In Figure 11a,b, compared with the rigid ring hypothesis, the ring deformation reduces the maximum contact load of the left bearing from 1294.7 N to 1250 N, representing a decrease of 3.4%, and the maximum contact load of the right bearing from 1382 N to 1296 N, representing a decrease of 6.2%. In Figure 11c,d, the ring deformation leads to a decrease in the contact angles near the 90\({}^{\circ}\) azimuth of the left bearing and an increase in the contact angles near the 270\({}^{\circ}\) azimuth, resulting in a decreased contact angles within the loaded zone of the right bearing. According to the above content and the analysis of reference [5], compared with the left bearing, the right bearing has more complex ring deformation and worse load distribution due to the different constraint positions and load conditions of the two bearings. Therefore, the subsequent analysis mainly focuses on the right bearing.  
Figure 10: Ring deformation: (**a**) left bearing; (**b**) right bearing.  
Figure 9: Horizontal and vertical areas of the bearing.