## 4 CRH5 Gearbox Degradation Feature Extraction and Analysis
### Hybrid Feature Domain Sensitivity Analysis  
Taking a 1500 rpm working condition as an example, there are 3000 sampling points of vibration acceleration in the normal state and degradation process of the key components of the gearbox, and every 100 sampling points of each state type are intercepted as a sample, for a total of 30 samples for each state type. The six characteristic indexes in time and frequency domains are calculated, respectively, and, after normalizing the calculation results, the comparison diagrams of normal state and degradation positions are shown in Figure 11. Among them, the RMS, kurtosis index, and RMS frequency have better sensitivity to component degradation, but they cannot be accurately differentiated for fault location.  
The wavelet packet decomposition feature indexes of 30 samples for each state are calculated, and the feature indexes are compared in Figure 12. Among them, W(3,3), W(3,4), and W(3,7) are more sensitive to degradation and can be effectively separated from different locations by combining the RMS, kurtosis index, and RMS frequency. Therefore, these six feature indicators are selected to construct a six-dimensional hybrid feature domain. The remaining feature indicators with low sensitivity are eliminated.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Designation** & **Expression (Math.)** & **Designation** & **Expression (Math.)** \\ \hline RMS & \(X_{rms}=\sqrt{\frac{\sum\limits_{i=1}^{N}x^{2}(n)}{N}}\) & Pulse index & \(I=\frac{\max\left|x(n)\right|}{\overline{x}}\) \\ \hline Margin index & \(L=\frac{\max\left|x(n)\right|}{X_{rms}}\) & Gravity frequency & \(F_{c}=\frac{\sum\limits_{i=1}^{N}f_{i}s(n)}{\sum\limits_{i=1}^{N}s(n)}\) \\ \hline Kurtosis index & \(K=\frac{\sum\limits_{i=1}^{N}\left[x(n)-\overline{x}\right]^{4}}{(N-1)\sigma_{ s}^{4}}\) & RMS frequency & \(F_{r}=\sqrt{\frac{\sum\limits_{i=1}^{N}\widehat{\sigma_{s}}^{2}s(n)}{\sum \limits_{i=1}^{N}s(n)}}\) \\ \hline \hline \end{tabular} where \(x(n)\) is the time-domain sequence of the vibration signal, \(\overline{x}\) is the mean value of \(x(n)\), \(s(n)\) is the spectrum of the signal \(x(n)\), and \(f_{n}\) is the frequency value of the nth spectral line.  
\end{table}
Table 1: Time-domain and frequency-domain vibration characterization metrics.  
The full lifecycle degraded hybrid feature domain of CRH5 gearbox key components is set in the preprocessing stage of the GRU-HMM fault prediction model. Before fault prediction, the operation monitoring data are noise reduced and the six-dimensional hybrid feature domain is calculated. The RMS, kurtosis index, and RMS frequency during operation are compared with the full lifecycle degradation hybrid feature domain to determine the initial probability of failure \(\pi\). Then, the eigenvalues of W(3,3), W(3,4), and W(3,7) are used to determine the location of the hidden failure, and the initial probability parameter is updated by the GRU to perform the failure prediction.  
Figure 11: Comparison of time and frequency domain feature metrics. (**a**) RMS; (**b**) margin index; (**c**) kurtosis index; (**d**) pulse index; (**e**) gravity frequency; (**f**) RMS frequency.  
Figure 12: Comparison of wavelet packet decomposition feature metrics. (**a**) W(3,0); (**b**) W(3,1); (**c**) W(3,2); (**d**) W(3,3); (**e**) W(3,4); (**f**) W(3,5); (**g**) W(3,6); (**h**) W(3,7).