# Study on Load Distribution and Fatigue Elastic Life of Ball Screw under Ultimate Conditions
## 3 Result and Discussion
### The Ball Screw's Fatigue Elastic Life under the Ultimate Condition  
The screw length and nut position influence the distribution of loads and, thus, the plastic deformation of its surface. Due to the uneven distribution of the ball load, it is necessary to calculate the stress of the maximum bearing ball, study the plastic deformation of the ball, and take this as the plastic deformation of the life study of the ball screw under the ultimate condition. The selected ball screw parameters and applied load are shown in Table 1. As Ref. [2] does not give the screw length, it is assumed here that the screw length is the nut length plus 30 mm.  
Figure 8 shows the relationship between the accumulative plastic deformation calculated in this paper and compares it with [PERSON]'s test results [2]. The theoretical and experimental values of the accumulative plastic deformation of the four ball screws are in good agreement. The theoretical fatigue elastic life calculated in this paper and [PERSON]'s experimental results and relative errors are shown in Table 2. The errors between theoretical values and experimental values of the four ball screws are less than 8%, and the relative errors calculated by [PERSON] are less than 15%. Thus, the model introduced in the present study is correct and more accurate than [PERSON]'s model.  
Figure 6: The maximum load of the ball at different screw lengths.  
Figure 7: The plastic deformation of the ball screw.  
Figure 9 shows the effect of different lengths and nut positions on the fatigue elastic life of ball screws. The maximum fatigue elastic life decreases with the increase of screw length. Moreover, the closer the nut is to two sections of the ball screw, the shorter the fatigue elastic life is. Figure 10 shows the ratio of nut position to ball screw length when the fatigue elastic life is the maximum under different lengths, loads, and ball screw models. When the fatigue elastic life of the screw is at its maximum, the ratio of the nut position to the screw length is around 0.7, and is not affected by the screw length, load, and model. The nut should therefore work in this position whenever possible.  
\begin{table}
\begin{tabular}{c c c c} \hline
**Type of Ball** & **Axial Force** & **Theoretical** & **Experimental** & **Errors (\%)** \\
**Screws** & **(KN)** & **Results** & **Results** & **Errors (\%)** \\ \hline GD4010-6 & 35 & 22.35 & 24 & 6.88 \\ GD4010-6 & 40 & 14.59 & 15 & 2.73 \\ GD4010-6 & 45 & 11.51 & 11 & 4.6 \\ GD4016-5 & 29.3 & 29.16 & 27 & 8 \\ \hline \end{tabular}
\end{table}
Table 2: Theoretical and experimental results of the fatigue elastic life.  
Figure 8: The fatigue elastic life of ball screw in this paper and in [PERSON]’s result.  
Figure 11 shows the maximum fatigue elastic life of ball screws under different contact angles, ball diameters, and conformity ratios when the lead screw length is 100 mm and 500 mm, respectively. The influence of ball screw parameters on the maximum fatigue elastic life under different lengths is the same. The fatigue elastic life increases approximately linearly with the increase of contact angle, but the effect is insignificant. With the increase of ball diameter, the fatigue elastic life increases, and the influence of ball diameter is larger. The fatigue elastic life decreases with the increase of the conformity ratio. Moreover, when the conformity ratio is small, its influence on the fatigue elastic life is very obvious. Therefore, selecting a larger ball diameter and a smaller conformity ratio can significantly improve the fatigue elastic life of the ball screw.  
Figure 10: The ratio of the nut position to the screw length under different parameters. (**a**) GD4010-6; (**b**) GD4016-5.  
Figure 9: The fatigue elastic life of ball screw under different nut positions and screw lengths.