## 4 Dynamic Characteristics and Fault Mechanism Analysis
### Dynamic Characteristics Analysis  
This section mainly studies the law of the most significant impact of spalling length on vehicle dynamic characteristics. Due to the most significant impact of spalling length on TVMS, we fixed the spalling width and depth to 4 mm and set TVMS with spalling lengths of 4 mm, 8 mm, 12 mm, 16 mm, and 20 mm, respectively, as a research group. Firstly, we chose the non-stationary process of the traction condition as the simulation environment, where the strong traction force better excites the characteristic frequency of gear meshing. We set the railway vehicle to an initial speed of 10 km/h and accelerated for 20 s after a stable dynamic period of 5 s. In this simulation, the Shanghai track spectrum is used as an input to the vehicle system because of its fidelity to urban rail excitation [27]. Specifically, the driving torque can be represented by the mechanical characteristics of the traction motor in Figure 7.  
Then, we chose the root mean square (RMS), kurtosis, crest factor (CF), and variance (VAR) as candidate indicators to measure the health of the gear system. The mathematical expressions for these indicators can be obtained from Ref. [45]. To compare the sensitivity of these three indicators, we selected the numerical data from the last 2 s of the traction process to analyze the fault severity. Additionally, the vertical vibration on the first gearbox housing is carried out as research data of interest. To describe the differences more accurately in  
Figure 6: Comparison of TVMS of gears with different spalling depths under fixed length and width. Figure 7: Mechanical characteristics of traction motors. these indicators, we chose the ratio of the differences with the health gear, which can be written as follows:  
\[P_{Y_{i}}=\frac{Y_{i}-Y_{1}}{Y_{1}}\times 100\%, \tag{30}\]  
where \(Y\) denotes the relative indicator, the subscript is fault severity, and one represents the health gear. Figure 8 shows the comparison of the four time-domain statistical indicators for the last 2 s of vertical vibration without track irregularity. All indicators show excellent monotonicity so they can describe the severity of the fault. The CF amplitude at 20 mm spalling is approximately 8% higher than the health gear, while the RMS, kurtosis, and VAR variations are still less than 1% at the maximum spalling length of 20 mm. The CF curve exhibits a strong linear relationship with the spalling length after the fault occurs. Due to its sensitivity and linearity, CF is considered the most suitable indicator for analyzing spalling evolution. Further, we select the most sensitive indicator CF to evaluate the feature evolution trends at different speed stages. Figure 9 shows the variation of CF at varying times under traction conditions, where these data points are calculated every 2 s. Figure 9 describes the indicator changes in the railway vehicles without track irregularities, which shows the ideal denoising results. All curves can orderly represent the indicator rise characteristics due to the spalling propagation. However, regardless of the fault severity, the indicators at low speeds are lower than the amplitude at high speeds. It demonstrates that the impact of vehicle speed on the indicator is more significant than the fault severity. Figure 9 describes the indicator changes in railway vehicles with track irregularities. In contrast, the changes in these indicators exhibit a strong irregularity. This indicates that the strong noise caused by track irregularities significantly impacts the indicator. Figure 8: The variations of the RMS, kurtosis, CF, and VAR of the first gearbox housing vertical vibration at different spalling lengths. Figure 9: Feature evolution of vertical vibration of the first gearbox housing under traction condition. At the same time, the four time-domain indicators of longitudinal acceleration for the last 2 s are also used for comparative research, as shown in Figure 10. Similarly to the vertical indicators results, the four indicators exhibit excellent monotonicity.