## 2 Dynamic Model Analysis
### Comparative Analysis of Simulation Results and Experimental Results  
Let nominal frequency \(\omega_{n}=\sqrt{k_{m}/m_{e}}(k_{m}\) is the average meshing stiffness), dimensionless displacement \(q=x/l\) (\(l\) is the nominal dimension), dimensionless time \(\tau=\omega_{n}t\), dimensionless frequency \(\Omega_{h}=\omega/\omega_{n}(\omega\) is meshing frequency), damping ratio \(\zeta=c_{m}(\tau)/2m_{e}\omega_{n}\). The Equation (3) can be nondimensionalized as follows:  
\[\tilde{q}(\tau)+2\zeta\tilde{q}(\tau)+\frac{\overline{W_{m}}(\tau)}{k_{m}}= \frac{F_{m}}{k_{m}l} \tag{7}\]  
where \(\overline{W_{m}}(\tau)\) is dimensionless comprehensive internal incentive. For the convenience of analysis, it is assumed that the meshing damping of each pair of teeth is the same and constant. Hence, Equation (7) can be written as:  
\[\tilde{q}(\tau)+2\zeta\overline{\rho}(\tau)\tilde{q}(\tau)+\frac{\overline{W_ {m}}(\tau)}{k_{m}}=\frac{F_{m}}{k_{m}l} \tag{8}\]  
where  
\[\overline{\rho}(\tau)= \begin{cases}2&w_{n}nt_{z}\leq\tau<w_{n}[nt_{z}+(t_{h}-t _{z})]\\ 1&w_{n}[nt_{z}+(t_{h}-t_{z})]\leq\tau<w_{n}(nt_{z}+t_{z})\\ 2&w_{n}(nt_{z}+t_{z})\leq\tau<w_{n}(nt_{z}+t_{h})\end{cases} n=0,1,2,\ldots \tag{9}\]  
To verify the correctness of the above dynamic model, the simulation results are analyzed based on the experimental data in Reference [14]. In this literature, [PERSON] et al., carried out a series of dynamic experiments based on a pair of gears with basic parameters as shown in Table 1 and obtained the equivalent root-mean-square amplitude \(A_{rms}\) of dynamic transmission error varying with frequency \(\Omega_{h}\) under a certain torque. According to the References [14; 15], the expression of \(A_{rms}\) can be written as follows:  
\[A_{rms}=\sqrt{\sum_{r=1}^{3}A_{r}^{2}} \tag{10}\]  
where \(A_{r}\) is \(r\)th mesh harmonic amplitude of dynamic transmission error, which can be determined from the resulting Fourier spectra according to  
\[A_{r}=\sqrt{\sum_{s_{i}=N_{i}-BW/2}^{N_{i}+BW/2}W(rs_{i})} \tag{11}\]Here, \(W\) is the one-sided discrete autopower spectra of dynamic transmission error, \(s_{i}\) is a shaft order index, \(N_{i}\) is the number of teeth on gear \(i\) and \(BW\) is the analysis bandwidth in shaft orders. Based on the Reference [14], let \(\overline{\zeta}=0.02,T=340N\cdot m,\overline{e}(\tau)=0\), the equivalent root-mean-square amplitude \(A_{rms}\) with frequency \(\Omega_{h}\) can be obtained. The comparison between the simulation results and the experimental results is shown in Figure 3. When \(\Omega_{h}\) changes from 0.18 to 0.23, the simulation result is compatible with the experimental result. When \(\Omega_{h}>0.25\), the dynamic model can effectively reflect the change trend of the equivalent root-mean-square amplitude \(A_{rms}\) with frequency \(\Omega_{h}\). With \(\Omega_{h}\) varying from 0.25 to 0.32, 0.3 to 0.43, 0.37 to 0.83, \(A_{rms}\) decreases sharply and then increases slowly. In the whole region, the value of \(A_{rms}\) obtained by the simulation is very close to that of the experiment, especially in the decline stage. The frequency of each inflection point (near \(\Omega_{h}=0.28,0.38\) and 0.6, respectively) reflected by the simulation results and the experimental results is almost consistent. Additionally, the simulation results can accurately predict the transition frequencies where the jump of \(A_{rms}\) occurs. It can be seen that though there are some differences between the simulation results and the experimental results due to the neglect of the influence of some factors such as elastic deformation of shaft and bearing, lubrication, friction, etc., we can obtain a great similarity between them. Therefore, the dynamic model established in this paper is reasonable.