# Study on Ring Deformation and Contact Characteristics of Thin-Walled Bearing for RV Reducer
## 3 Results and Discussion
### The Effect of Axial Force  
Figure 12 illustrates the effect of axial force on the outer ring deformation under constant radial force. Noticeable differences in ring deformation and variation tendency are observed between the vertical and horizontal directions. When under pure radial force, the ball-raceway contact loads concentrate in the 90\({}^{\circ}\) azimuth area, causing an expansion in the vertical diameter and a reduction in the horizontal diameter of the ring. As the axial force increases, the vertical diameter of the ring increases while the horizontal diameter decreases further. This is because the increase in the axial force causes the balls at the 270\({}^{\circ}\) azimuth angle area to be loaded, which in turn causes the vertical diameter of the ring to increase, while the unloaded zone in the horizontal area leads to a further decrease in the ring diameter. Figure 13 depicts the variation trend in the loaded zones under the action of axial force. When all the balls are loaded, the decrease in ring diameter tends to the maximum. However, with further increases in axial force, the horizontal diameter of the ring begins to increase. As seen in Figure 14, there is a critical axial force on the deformation curve of the horizontal diameter of the ring. The trend of ring deformation on both sides of the critical axial force is different, and the critical axial force values are variable. When the radial force is 3.1 kN, the critical axial force is 3 kN. Meanwhile, when the radial force is 3.5 kN and 3.9 kN, the critical axial force values change to 3.4 kN and 3.6 kN, respectively. As the radial force value increases, the critical axial force value increases. The primary reason is that a larger radial force makes a larger unloaded zone and a larger reduction in ring diameter in the horizontal direction at the same axial force. Therefore, a larger axial force is required to load the balls in the horizontal area and then change the tendency of ring deformation in that area. Figure 11: Effect of ring deformation on contact characteristics: (**a**) ball–raceway contact load of the left bearing; (**b**) ball–raceway contact load of the right bearing; (**c**) contact angles of the left bearing; (**d**) contact angles of the right bearing. of the maximum load is shown in Figure 15b. The relative ratio of the maximum load is defined as the maximum ball-raceway contact load in a flexible ring divided by that in a rigid ring. A smaller relative ratio corresponds to a smaller maximum contact load in the flexible ring, indicating a more pronounced optimization effect in load distribution due to the ring deformation. However, as the axial force increases, the relative ratio also rises, suggesting that the load optimization effect becomes less evident under higher axial force. This can be attributed to the increase in axial force reducing the difference between the ball-raceway contact loads, leading to the ring deformation tending to a standard circle, thus reducing the degree of load optimization. As axial force increases, the maximum contact stress initially decreases and then increases. This occurs because the increased number of loaded balls initially reduces contact stress. However, once all balls are loaded, the contact stress increases as the axial force increases. Figure 15c illustrates how the axial force affects the radial displacement of the inner ring. The displacement of the inner ring under flexible conditions is greater than that under rigid conditions. The reason is that under rigid conditions, the shaft and inner ring can only deflect at an angle to balance the external load. However, the bearing rings and supporting structures can also produce elastic deformation to balance the external load under flexible conditions [27]. Figure 15d shows the influence of axial force on ball-raceway contact angles. The contact angles near 90\({}^{\circ}\) azimuth increase with axial force, while the contact angles near 270\({}^{\circ}\) azimuth change conversely.