## 3 Numerical Analysis of Dynamic Contact Characteristics
### Variation of Dynamic Contact Angle  
Figure 12 illustrates the variation of the dynamic contact angle of the first contact pair with axial force and screw speed.  
It is clear that under the same axial force, the screw dynamic contact angle increases with the screw speed increase. However, the variation of dynamic contact angle of the nut is just the opposite. The contact angle of the nut increases with the axial force increase. When the screw speed is lower than 2000 rpm, the contact angle of the screw increases with the increase in axial force. But when the speed of the screw is higher than 2500 rpm, it decreases first and then increases with the increase in axial force. This is because the factor of screw speed has a coupling effect on the dynamic contact angle. At the same time, it can be seen that the higher the screw speed is, and the smaller the axial force is, the greater the difference in the contact angle between the screw and the nut. However, the lower the screw speed is, and the smaller the axial force is, the closer the two dynamic contact angles.  
Figure 13 shows the variation of dynamic contact angle with the contact position and the screw speed when the axial force is 1000 N. At the same speed of screw, the dynamic contact angles of screw and nut both decrease from the first contact pair to the last one. However, the characteristics of the two  
Figure 11: Variation of the dynamic contact force with the contact position and the axial force. (**a**) Dynamic contact normal force of screw. (**b**) Dynamic contact normal force of the nut.  
Figure 12: Variation of the dynamic contact angle of first ball contact pairs with axial force and screw speed.  
are not the same. The uneven distribution of the dynamic contact angle of the screw is not obvious with the increase in the screw speed, while that of the nut is opposite.  
The reason is that the ball's spinning angle velocity (\(\omega_{R}\)) and revolutional angle velocity (\(\omega_{m}\)) are the two important factors which affect the dynamic contact angle. The bigger the two velocities are, the greater the screw contact angle is and the smaller the nut contact angle is. Figure 14a illustrates the variations of \(\omega_{R}\) and \(\omega_{m}\) with screw speed under different axial force. It is obvious that both two angle velocities are almost linear with screw speed. The axial force has little effect on the two angular velocities. Figure 14b shows variations of \(\omega_{R}\) with the contact position. It is clear that when the screw speed is higher than 2000 rpm, \(\omega_{R}\) increases from the first contact pair to the last one. When the screw speed is higher than 4000 rpm, \(\omega_{m}\) increases from the first contact pair to the last one. Normal contact force decreases with variation of the contact position, which is also the positive factor that makes the contact angle change with the screw speed. All of these factors make the distribution of the dynamic contact angle of the screw less obvious but that of the nut more obvious.  
Figure 14: _Cont._  
Figure 13: Variation of dynamic contact angle with the contact position and the screw speed.  
Figure 15 demonstrates the dynamic contact angle variation of the screw and nut with the contact position and axial force when the screw speed is 2000 rpm, respectively.  
For the contact pairs in the same position, both the screw and nut dynamic contact angles increase with the axial force increase. For contact pairs in different position, dynamic contact angle decreases with the contact pair position changing, which presents the phenomenon of uneven distribution. Under the same axial force, the uneven distribution of the screw dynamic contact angle is more significant. However, as the axial force increases, the uneven distribution of the dynamic contact angles of the lead screw will be more and more significant, while that of the nut is contrary.