## 3 Numerical Analysis of Dynamic Contact Characteristics
### Variation of Dynamic Contact Stiffness  
Substituting \(Q_{ci}\) into Equation (41), normal stiffness \(k_{ci}\) will be calculated. Figure 16 reveals the variation of the normal stiffness of the first contact pair with the axial force and the screw speed,  
Figure 14: Variation of ball’s spinning angle velocity and revolutional angle velocity. (**a**) Variations of \(\omega_{R}\) and \(\omega_{m}\) with screw speed. (**b**) Variations of \(\omega_{R}\) with the contact position. Figure 15: Variation of dynamic contact angle with contact position and axial force. respectively. It can be seen from the figure that the general variation rules of the screw and nut's normal contact stiffness are consistent. The normal contact stiffness increases as the axial force increases. The normal contact stiffness of the screw is less than that of the nut. With the screw speed increase, the normal contact stiffness decreases gradually, but the influence of the screw speed on the contact stiffness is less than that of the axial force. Figure 17 exhibits variation of the normal contact stiffness with the screw speed and the contact position when the axial force is 1000 N. It is obvious that as the contact position is moving backward, the normal contact stiffness of both the screw and nut decreases gradually. In the same position, the normal contact stiffness is gradually reduced with the screw speed increase. The nut normal contact stiffness is always greater than screw. In the static state, the ball screw local contact stiffness in five directions can be calculated through Equations (42), (43) and (45). Figure 18 exhibits the variation of the local stiffness with contact position; the local lateral stiffness and oscillating stiffness fluctuate with the period of \(\pi\) and the phase difference of \(\pi\)/2. The amplitudes of lateral stiffness decrease slightly with the contact pair position far away from the SNBSP load side, but that of oscillating stiffness increases, as illustrated in Figure 18a,c. The load axial stiffness decreases with the contact pair position far away from the SNBSP load side, as shown in Figure 18b. Figure 16: Variation of the first contact pair normal contact stiffness with the axial force and the screw speed. Figure 17: Variation of the normal contact stiffness with the screw speed and the contact position. Figure 19 demonstrates the variation of ball screw dynamic local contact stiffness with contact position and axial force when the speed of the screw rotational is 2000 rpm. The variation of dynamic contact stiffness with the contact position in all directions are consistent with them in A static state. It can be noted that single ball local stiffness depends upon the normal force, hence all of the local stiffness increases with the increase in the axial force. Figure 18: Local stiffness of contact pairs in static state. (**a**) Local lateral stiffness of the x and y axis. (**b**) Local axial stiffness of the z axis. (**c**) Local oscillating stiffness around the x axis and y axis. Figure 20 exhibits the variation of the local contact stiffness with the contact position and the screw speed when the axial force is 1000 N. From Figure 20a, it can be seen that the lateral stiffness increases gradually with the increase in the screw speed. The main reason for this phenomenon is that with the increase in screw rotational speed, the spinning and revolution angular velocities of the ball increase, and the ball moves to the outside gradually, which means that the nut contact angle decreases and the screw contact angle increases slightly. A closer contact state between the ball and the screw and nut raceway is generated, which leads to stronger support ability in the lateral direction. Therefore, lateral stiffness becomes bigger. Figure 20b illustrates the variation of the local axial stiffness with the screw speed and the contact position. As the contact position moves backwards, the local axial stiffness decreases gradually. However, as the screw speed increases, the axial local stiffness gradually decreases.