# Study on Load Distribution and Fatigue Elastic Life of Ball Screw under Ultimate Conditions
## 3 Result and Discussion  
### The Load Distribution and Plastic Deformation of the Ball Screw  
The distribution of load is hard to measure directly in experiments due to the spatial constraints. To validate the accuracy of the proposed model, the parameters provided by [PERSON] [20] are used, and the screw model is BSS3220-5E. The proposed model was compared with [PERSON]'s results. In [PERSON]'s model [20], the nut is at the screw's midpoint; the screw's left end is fixed, allowing only axial displacement, while the right end remains free. The load-bearing balls within the nut are assumed to be uniformly distributed along the helix. Under a 5000 N axial load applied to the screw's left end, the load distributions calculated by the two are shown in Figure 4. It can be seen that the force on the balls inside the nut is uneven and the results of the proposed model match well with [PERSON]'s results, with a maximum error of 8.86%. The reason for the certain error between the two results may be that [PERSON] considered the variations in the contact angle with the different load.  
Figure 4: Load distribution for the models in this paper and in [PERSON]’s result [20].  
To study the static load distribution of the ball screw, it is assumed that there is no pre-load. The distribution of loads under different lengths and loads is shown in Figure 5. With the change in the nut position on the screw, the distribution of loads also changes. The load distribution becomes uneven when the nut is positioned at the screw's ends, whereas it becomes more uniform when the nut is situated at its midpoint. This is because the middle part of the screw is more deformable than the ends, so the distribution of loads at the screw's middle position is more uniform than when the nut is located at the ends of the screw. The sequence number of the ball with the maximum load in the distribution of loads and the position of the nut at this time will not change with the change in the length of the screw. The shape of the distribution curve changes with the axial load. Additionally, with the difference in axial contact load, the size of the load distribution will also change; however, it does not have a significant impact on the shape of the distribution curve.  
Figure 6 shows the effect of the screw length on the static load. Assuming that the ball screw has no preload, the maximum load borne by the ball at different screw lengths is calculated. As the screw length increases, the maximum load borne by the ball also increases. This is because the longer the screw, the more easily it deforms under axial force, which results in an increase in the unevenness of the load borne by the ball.  
Figure 7 shows the plastic deformation of the position when each ball contacts the ball screw raceway under different nut positions. It can be seen that the raceway's plastic deformation changes with the nut position and the ball serial number due to the uneven stress of the ball.  
Figure 5: The distribution of loads under different lengths and loads: (**a**) \(L=300\) mm \(F_{a}=5000\) N (**b**) \(L=1000\) mm \(F_{a}=5000\) N (**c**) \(L=1000\) mm \(F_{a}=2000\) N.