## 3 Numerical Analysis of Dynamic Contact Characteristics
### Experiment
Figure 27 shows the configuration of the z-axis feed system and the dynamic model. The mounting method of ball screw is fixed-fixed, as shown in Figure 27a. The stiffness values of the frame and worktable of the experimental machine tool are all large. Compared with the worktable mass, the mass of the screw can be neglected. Especially, low screw lead reduces the influence of the torsional stiffness  
Figure 26: Acceleration signal and torque current of servo motor (TCSM) signal. (**a**) Acceleration signal. (**b**) TCSM signal. and moments of inertia of the screw and other rotating parts. Therefore, the dynamic model of the worktable is simplified as in Figure 27b. The stiffness of the system is expressed as follows:  
\[\frac{1}{K_{f}}=\frac{1}{\frac{1}{\frac{1}{K_{b}}+\frac{1}{K_{d0}}+\frac{1}{K_{b }}}}+\frac{1}{\frac{1}{K_{b}}+\frac{1}{K_{d1}}}}+\frac{1}{K_{p}} \tag{49}\]  
where \(K_{b}\) denotes the axial stiffness of fixed end bearing. \(K_{p}\) denotes the axial stiffness of the screw-nut pair, which means that \(K_{p}=K_{z}\). \(K_{s0}\) and \(K_{s1}\) denote the axial stiffness of the screw shaft on both sides of the nut, with  
\[K_{s\chi}\doteq\frac{\pi(d_{m}-d_{B})^{2}E_{s}}{L_{\chi}} \tag{50}\]  
where \(L_{\chi}\) denotes the distance between the nut and the two fixed ends, \(\chi=\)0 or 1. Then the feed system natural frequency is expressed as [24]:  
\[f_{\rm n}=\frac{1}{2\pi}\sqrt{\frac{K_{f}}{m_{t}}} \tag{51}\]  
The axial stiffness of the feed system can be determined by numeric iteration of Equations (49) and (50). It is clear from Equation (50) that the screw shaft axial stiffness varies with the position of the worktable, so the axial stiffness and natural frequency of the feed system are also time-varying. The theoretical value of the axial stiffness and natural frequency of the feed system variation with the position of the worktable under the experimental conditions are shown in Figure 28. It can be seen that although the axial stiffness and natural frequency vary with the position of the worktable, the variation is not obvious. For example, when the screw speed is 1000 rpm, the axial stiffness variation ratio is 0.087%, while the natural frequency variation ratio is only 0.046%. Therefore, the axial stiffness and natural frequency of the Z axis of the feed system can be considered to be approximately the same in different positions. The feed system natural frequency will be calculated by Equation (51). Herein, the contact stiffness and the natural frequency determined above are considered to be theoretical values of them, respectively. Figure 27: Configuration of the \(z\)-axis feed system and the dynamic model. (**a**) Configuration of the \(z\)-axis feed system. (**b**) The dynamic model. On the other hand, the FFT algorithm is carried out to investigate the spectrum characteristics of the worktable acceleration. In Figure 29, the FFT results of the worktable acceleration are represented in the stable running state in Figure 26a. It is obvious that the 16.66 Hz and its two and three times harmonic frequency components are the frequency components corresponding to the screw speed of 1000 rpm, and there are several frequency components at the low frequency section. In fact, the feed system has shapes of many modes [10], and so the FFT results of its vibration contain multiple frequencies. However, the first mode of the feed system is generally axial [25]. Herein, the frequency in Figure 29, 108.6 Hz, should be the natural frequency, whose corresponding mode should be the axial vibration of the worktable. Therefore, 108.6 Hz corresponds to the natural frequency \(f_{\rm n}\) in Equation (51). The theoretical and experiment values of the feed system dynamic parameters under the two screw speeds in Section 4.1 are exhibited in Table 3. The results show that the theoretical values of natural frequency are consistent with the measured values. This fact verifies the effectiveness of the whole rolling elements dynamic contact model and sensorless stiffness estimating method proposed in this paper.