## 4 Dynamic Characteristics and Fault Mechanism Analysis
### TVMS Analysis  
This subsection discusses the effects of TVMS on spalling length, width, and height, respectively. When studying the effects of a particular geometric dimension, the other two variables are fixed. Therefore, we will analyze the impact of the spalling defects' length, width, and depth sequentially. Figure 4 compares meshing stiffness under different spalling lengths when the width and depth are set at 4 mm. As can be seen from the figure, the spalling length causes a decrease in the TVMS of the gear fault location, and the influence of the contact position is obvious, indicating that the decline in the Hertzian stiffness is relatively significant. As the fault severity increases, the loss range of TVMS does not expand, only reducing the stiffness. In addition, with the meshing progress, the loss of the TVMS near the pitch circle presents a maximum value. Figure 5 shows a comparison of TVMS for different spalling widths with a spalling length of 10 mm and a spalling depth of 4 mm. Unlike the effect of the spalling length, the increase in the spalling width gradually expands the range of a loss of TVMS centered on the pitch circle, but does not lead to any further losses of the stiffness amplitude. Therefore, the impact of the spalling width on the amplitude loss of TVMS on localized defects is limited. It has a lesser effect on the dynamic response than in the case of a large reduction in the localized defect. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Symbol** & **Value** & **Symbol** & **Value** \\ \hline \(M_{c}\) & 35,400 kg & \(l_{c}\) & 1,970,300 kg\(\cdot\)m\({}^{2}\) \\ \(M_{b}\) & 3188 kg & \(l_{b}\) & 2710 kg\(\cdot\)m\({}^{2}\) \\ \(M_{a}\) & 85 kg & \(l_{a}\) & 2.45 kg\(\cdot\)m\({}^{2}\) \\ \(M_{w}\) & 1024 kg & \(l_{w}\) & 78 kg\(\cdot\)m\({}^{2}\) \\ \(M_{p}\) & 5.15 kg & \(l_{p}\) & 0.007 kg\(\cdot\)m\({}^{2}\) \\ \(M_{glt}\) & 149.75 kg & \(l_{gh}\) & 10.45 kg\(\cdot\)m\({}^{2}\) \\ \(M_{mr}\) & 178.36 kg & \(l_{mr}\) & 1.9 kg\(\cdot\)m\({}^{2}\) \\ \(M_{mlt}\) & 422.82 kg & \(l_{mlt}\) & 24.7 kg\(\cdot\)m\({}^{2}\) \\ \(K_{ss}\) & 4.41 \(\times\) 10\({}^{5}\) N/m & \(K_{ss}\) & 2.06 \(\times\)10\({}^{5}\) N/m \\ \(C_{\text{rp}}\) & 1.25 \(\times\) 10\({}^{3}\) N\(\cdot\)s/m & \(C_{\text{zp}}\) & 2 \(\times\) 10\({}^{3}\) N\(\cdot\)s/m \\ \(C_{\text{rs}}\) & 2.5 \(\times\) 10\({}^{3}\) N\(\cdot\)s/m & \(C_{\text{zs}}\) & 6 \(\times\) 10\({}^{4}\) N\(\cdot\)s/m \\ \(K_{b}\) & 1 \(\times\) 10\({}^{8}\) N/m & \(C_{b}\) & 1 \(\times\) 10\({}^{3}\) N\(\cdot\)s/m \\ \(K_{zgh}\) & 5 \(\times\) 10\({}^{6}\) N/m & \(C_{\text{zm}h}\) & 1 \(\times\) 10\({}^{3}\) N\(\cdot\)s/m \\ \(K_{xmlh}\) & 1 \(\times\) 10\({}^{9}\) N/m & \(C_{\text{zm}h}\) & 1 \(\times\) 10\({}^{3}\) N\(\cdot\)s/m \\ \(l_{\text{zc}}\) & 6.3 m & \(l_{oplb}\) & 1.375 m \\ \(l_{ipb}\) & 0.825 m & \(l_{opla}\) & 0.275 m \\ \(l_{ipka}\) & 0.275 m & \(l_{pbgh}\) & 0.178 m \\ \(l_{gbgh}\) & 0.183 m & \(l_{mlmh}\) & 0.329 m \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Railway vehicle parameters. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameters** & **Pinion** & **Gear** \\ \hline Tooth number & 22 & 133 \\ Module (mm) & 5 & 5 \\ Pressure angle (\({}^{\circ}\)) & 20 & 20 \\ Young’s modulus (Gpa) & 206 & 206 \\ Poisson’s ratio & 0.3 & 0.3 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Gear system parameters. Figure 6 shows the comparison of TVMS at different spalling depths with a spalling length of 10 mm and a spalling width of 4 mm. Similar to the effect of spalling length, the spalling depth decreases the overall meshing stiffness at the gear fault location. Compared to the spalling length, the spalling depth does not affect the change in the Hertzian stiffness, so its impact effect is smaller. Thus, it can be seen that the most significant impact on the peak value of the loss of TVMS is the spalling length, while the largest impact on the loss range of the TVMS is the spalling width. Figure 4: Comparison of TVMS of gears with different spalling lengths under fixed width and depths.