## 3 Results and Discussion
### Effects of Design Parameters on Axial Vibration of the Feed System  
The initial contact angle, length of screw shaft and number of loaded balls are the main parameters in the design of the ball screw. According to Equations (17), (30) and (55), the restoring force of the ball screws is a function of these parameters. Obviously, the vibration characteristics of the feed system are influenced by these parameters.  
When the axial excitation amplitude is fixed at 1000 N, the amplitude-frequency curves of the worktable vibration with the initial contact angle variation (\(\alpha_{0}=30^{\circ}\), \(45^{\circ}\), \(60^{\circ}\)) are shown are Figure 14. It can be seen that these curves almost coincide in the low and high frequency regions, and the vibration amplitude of the worktable decreases slightly and the main resonance frequency moves to the high frequency region with the initial contact angle increasing. This may be that the initial contact angle increasing makes the increase of axial internal force at the screw-nut interface, which leads to the increase of natural frequency for the feed system. To analyze the influence of the initial contact angle on the nonlinear vibration at the resonance point, the time histories, spectrum, phase diagrams and Poincare sections are investigated, as shown in Figure 15. With the initial contact angle variation, the Phase diagrams and Poincare sections are a circle and a single point, respectively, and this shows that the vibration responses with the initial contact angle variation are periodic-1 motions.  
With the length of screw shaft variation, the amplitude-frequency curves of the worktable vibration for \(F_{a}=1000\) N are shown in Figure 16. The variation tendency of vibration amplitude in the resonance region is opposite to that with the variation of initial contact angle, and the stiffness of the screw shaft at the middle position decreases with the increase of the length of screw shaft, leading to the main resonance frequency of vibration response  
Figure 14: Amplitude-frequency curves of the feed system for different initial contact angles.  
Figure 15: Vibration responses in the resonance region: (**a**) Time history; (**b**) Spectrum; (**c**) Phase diagram and Poincaré section.  
moving to the low frequency region, and the axial displacement of the worktable increases. The time history, spectrum, phase diagram and Poincare section are investigated with the length of screw shaft variation, as shown in Figure 17, and the excitation frequency at resonance points increases equivalently with the increase of the length of screw shaft. The difference of the adjacent frequencies is about 20 rad/s; the Phase diagrams and Poincare sections are a circle and a single point, respectively, and this shows that the vibration responses with the length of screw shaft variation are periodic-1 motions.  
With the increase of the number of loaded balls, the amplitude-frequency curves of the worktable vibration for \(F_{a}=1000\) N are shown in Figure 18. Similar to the amplitude-frequency curves with the initial contact angle variation, the vibration amplitude of the worktable decreases and the resonance frequency moves to the high frequency region with the increase of the number of loaded balls. As shown in the Figure 19, the frequency components\(f\) and 3\(f\) appear in the spectrum, and the phase diagrams and Poincare sections are a circle and a single point, respectively; this illustrates that the vibration responses with the number of loaded balls variation are periodic-1 motions. From the analysis above, the initial contact angle and the number of loaded balls have the same effects on the vibration behaviors of the feed system, but opposite to that of the length of the screw shaft.  
Figure 16: Amplitude-frequency curves of the feed system for different length of screw shafts.  
Figure 17: Vibration responses in the resonance region: (**a**) Time history; (**b**) Spectrum; (**c**) Phase diagram and Poincaré section.