## 3 Numerical Analysis of Dynamic Contact Characteristics
### Variation of Dynamic Contact Stiffness
The reason for this is mainly that the screw and nut's normal contact stiffness decreases with the screw speed' increase. On the other hand, because the ball and raceway form a more close contact state which is caused by the changing of the contact angle, it leads to a stronger supporting ability along the lateral direction, and the axial support ability will decrease. Figure 20c exhibits the variation of the local oscillating stiffness with the screw speed and the position of the contact pair. It is different from lateral and axial stiffness that when the screw speed increases, the oscillating stiffness of the first three turns decreases, but the amplitude of the decrease gradually decreases with the phase fluctuation. The fourth half turn is a transition stage. From the fifth half turn, the contact pair oscillating stiffness increases gradually, and the amplitude of the increase gradually  
Figure 19: Variation of the local stiffness with the contact position and the axial force. (**a**) Local lateral stiffness of the x axis. (**b**) Local axial stiffness of the z axis. (**c**) Local oscillating stiffness around the x axis. increases with the phase fluctuation. Figure 21 shows the difference value between the local oscillating stiffness of each contact pair at screw speeds of 5000 and 500 rpm when the axial force is 1000 N.  
Substituting local stiffness into Equation (46), and the screw-nut pair global stiffness will be calculated. As shown in Figure 22, the global contact stiffness in all directions increased with the increase in the axial force. Compared with the lateral stiffness \(K_{x}\) and \(K_{y}\), the axial stiffness \(K_{z}\) and  
Figure 21: The difference value between the local oscillating stiffness of each contact pair at screw speeds of 5000 and 500 rpm. Figure 20: Variation of the local stiffness with the contact position and the screw rotational speed. (**a**) Local lateral stiffness of the x axis. (**b**) Local axial stiffness of the z axis. (**c**) Local oscillating stiffness around the x axis. the oscillating stiffness \(K_{\phi x}\), \(K_{\phi y}\) increase more significantly, which indicates that the effect of the axial force on the latter two is more significant. Figure 23 exhibits the global stiffness varying with screw speed when the axial force is 1000 N. It is clear that with the rotating speed of the screw increase, the lateral stiffness \(K_{x}\) and \(K_{y}\) increase monotonically, while the axial stiffness \(K_{z}\) decreases monotonically, which is in accordance with variation rule of the local contact stiffness. The oscillating stiffness \(K_{\phi x}\) and \(K_{\phi y}\) increase with screw speed increasing. Although the local oscillating stiffness of the first 1.5 turns decreases with screw speed increasing, it increases with the screw speed, increasing two cycles, and the increment is significant. So the total value of the oscillating stiffness increases monotonically, but it is clear that the amplitude of its increase is not very obvious. Sensorless Stiffness Estimating Method Based on Contact Stiffness Model and Experimental Verification