## 4 Dynamic Characteristics and Fault Mechanism Analysis
### Dynamic Characteristics Analysis
Moreover, the CF amplitude is still the most sensitive but it decreases nearly 3% more than the vertical amplitude at the maximum spalling length of 20 mm. Similarly, we select the most sensitive indicator CF to evaluate the feature evolution trends at different speed stages. Figure 11 shows the variation of CF at varying times under traction conditions, where these data points are calculated every 2 s. Figure 11a describes the longitudinal indicator changes without track irregularities, which show the ideal denoising results. All curves can orderly represent the indicator rise characteristics due to the spalling propagation. Unlike the case of vertical vibration, these curves exhibit more significant oscillations as the vehicle speed changes. This indicates that longitudinal vibration makes it is more challenging to measure the disturbance of the indicator by velocity changes than vertical vibration and determines the fault threshold with difficulty. Figure 11b describes the longitudinal indicator changes in railway vehicles with track irregularities. In contrast, the changes in these indicators exhibit a strong irregularity. It proves that even longitudinal vibration acceleration can significantly disturb the indicator due to the impact of track irregularity caused by the vibration coupling effect. In addition, we use a time-frequency analysis tool to analyze the characteristic frequencies caused by gear tooth spalling. Figure 12 shows the time-frequency spectrum of  
Figure 11: Feature evolution of longitudinal vibration of the first gearbox housing under traction condition. Figure 10: The variations of the RMS, kurtosis, CF, and VAR of the first gearbox housing longitudinal vibration at different spalling lengths. the vertical vibration of the first gearbox housing with health conditions and a spalling length of 20 mm. The natural frequency, meshing, and harmonic components are evident in both the time-frequency spectra. Theoretically, local faults such as cracks and spalling can lead to significant sidebands in the frequency spectrum around the meshing frequency, which is the central frequency. However, there is no significant difference between the two time-frequency spectra. The reason is that the excitation generated by gear tooth spalling is weak for railway vehicles and insufficient to excite significant characteristic frequencies. In contrast, it excites a more substantial natural frequency of the system. The research result indicates that it is difficult to apply a time-frequency analysis tool for fault diagnosis of spalling gear of railway vehicles under strong noise and non-stationary environments. Theoretically, as a random excitation with a wavelength of more than 1 m, the track spectrum has a low-frequency effect on the vehicle vibration compared to gear engagement. Therefore, we designed a simple high-pass filter to suppress the impact of the track spectrum, thereby using indicators of interest for the health assessment. To evaluate the effect of the passband frequency selection, we show the comparison of time histories. Figure 13 shows the vertical acceleration comparison of the raw signal without a track spectrum, the raw signal with a track spectrum, and the filtered signal with a track spectrum at a passing frequency of 350 Hz. The filtered signal is consistent with the raw signal without a track spectrum in the later stage of the traction condition. However, the low-frequency characteristics caused by resonance in the early stage are ignored. Figure 12: Comparison of time–frequency spectrum of healthy and maximum spalling length gears without track irregularities. Figure 6:0 The comparison of the time histories of the first gearbox housing under traction condition. Finally, we show the variation in the four time-domain indicators of the filtered signal, as shown in Figure 6:1 Even when applying the ideal filtering techniques, RMS and VAR present clear classifications for different severity levels. However, the indicators under different severities show little difference.