## 4 Dynamic Simulation and Experimental Verification
### Effect of the Number of Spalling Faulty Tenth on Dynamic Characteristics of Gears  
In this paper, we assume that the rotational speed of pinion is \(n\) = 1500 r/min, the rotational frequency of pinion is \(f_{\mathrm{s}}\) = 25 Hz, the rotating period of pinion is \(t_{\mathrm{s}}\) = 0.04 s, the meshing frequency is \(f_{m}\) = 400 Hz, and the meshing period is \(t_{m}\) = 0.0025 s. Three types of the spalling faulty tooth number are considered, namely health pinion, pinion with single tooth spalling, and pinion with adjacent double teeth spalling. Figure 7a, Figure 8a, and Figure 9a present the failure states of tooth surface spalling, respectively. It can be clearly seen that the spalling is near the pitch line. This paper we set that \(r\) = 0.75 mm, \(h\) = 1 mm, and \(s\) = 3, where \(r\), \(h\), and \(s\) are the radius, depth, and numbers of spalling point, respectively. According to the method described in Section 2, the mesh stiffness is simulated. It can be observed from Figure 7b that the mesh stiffness curve is periodically changing during the meshing process. The mesh stiffness decreases when the spalling faulty tooth participates in an engagement. The frequency of mesh stiffness reduction is related to the number of teeth with spalling fault, as shown in Figures 8b and 9b. Figure 8: Status of tooth surface and mesh stiffness curves (single tooth with spalling fault): (**a**) status of tooth surface and (**b**) mesh stiffness curves. Figure 7: Status of tooth surface and mesh stiffness curves (healthy pinion): (**a**) status of tooth surface and (**b**) mesh stiffness curves. The simulated vibration signal and the experimental measurement signal under a different number of spalling faulty tooth are displayed in Figures 10-12. The simulated results are close to the experimental results when the pinion is in a healthy state, as shown in Figure 10. There are no clear impulse vibrations in the time domain. Only tooth meshing frequency (TMF) and its harmonics are included in the frequency domain. In the frequency domain, the TMF is equal to the meshing frequency of gears, and the remaining harmonics are multiples of the TMF, as marked in Figure 10b,d. Figure 10: The simulated and experimental signal in the meshing line direction (healthy pinion): (**a**) simulated signal, (**b**) simulated frequency signal, (**c**) experimental signal, and (**d**) experimental frequency signal. Figure 9: Status of tooth surface and mesh stiffness curves (adjacent teeth with spalling fault): (**a**) status of tooth surface and (**b**) mesh stiffness curves. Figure 11: The simulated and experimental signal in the meshing line direction (single tooth with spalling fault): (**a**) simulated signal, (**b**) simulated frequency signal, (**c**) experimental signal, and (**d**) experimental frequency signal. Figure 12: The simulated and experimental signal along the line of action (adjacent teeth spalling fault): (**a**) simulated signal, (**b**) simulated frequency signal, (**c**) experimental signal, and (**d**) experimental frequency signal. Compared with the healthy gear, the spalling fault can generate the periodic impulse vibration, and sidebands are spaced by the rotational frequency of the faulty gears in the frequency domain. These features are successfully captured by the simulated signal. Figure 11 shows the simulated and experimental signal results of the pinion with single tooth spalling fault. As shown in Figure 11, the periodic impulse vibration can be recognized in the time domain, and the period of the impulse vibration is 0.04 s, which is equal to the rotating period of the pinion. The sideband appears in the frequency domain, and the spacing of the sideband is equal to the rotating frequency of the pinion. Figure 12 shows the simulated and experimental signal results of the pinion with an adjacent double tooth spalling fault. Compared with the time domain in Figure 11a, Figure 12a shows that the impulse vibration was excited twice with an interval of 0.0025 s in one rotational period of the pinion.