## 4 Analysis and Discussion
### Time-Frequency-Domain Analysis for Multi-Feature Extraction and Assessment  
Time-frequency analysis of current data was conducted by using the HHT method described in Section 3.3. The IMFs of a typical response of Segment 1 and those of Segment 2 are shown in Figures 15 and 16. The correlation coefficient is a quantity that examines the degree of linear correlation between variables. The formula for the correlation coefficient between one IMF and the original signal is:  
\[r(\mathbf{x}_{i},\mathbf{x})=\frac{Cov(\mathbf{x}_{i},\mathbf{x})}{\sqrt{Var( \mathbf{x}_{i})}\sqrt{Var(\mathbf{x})}} \tag{15}\]  
where \(\mathbf{x}_{i}\) denotes the \(i\)th IMF vector, \(\mathbf{x}\) denotes the vector of the original signal, \(Cov(\mathbf{x}_{i}\mathbf{x})\) is the covariance of \(\mathbf{x}_{i}\) and \(\mathbf{x}\), \(Var(\mathbf{x}_{i})\) is the variance of \(\mathbf{x}_{i}\), and \(Var(\mathbf{x})\) is the variance of \(\mathbf{x}\). \begin{table}
\begin{tabular}{c c c} \hline \hline
**Category** & **Segment 1/dB** & **Segment 2/dB** \\ \hline Sensor No 1 & 283.63 & 357.44 \\ Sensor No 2 & 298.42 & 375.63 \\ Sensor No 3 & 284.51 & 357.55 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Values of vibration level of the two segments in the current stage of monitoring. \begin{table}
\begin{tabular}{c c c c c} \hline \hline Characteristic & \(\mathbf{L_{1}}\) & \(\mathbf{L_{2}}\) & \(\mathbf{L_{3}}\) & \(\mathbf{L_{4}}\) \\ \cline{2-5} length/m & 2.5 & 17.5 & 7.5 & 25.0 \\ \hline Characteristic & 25.78 & 3.68 & 8.59 & 2.58 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Geometric characteristics and the associated frequencies when the running speed is 232 km/h. In Figures 15 and 16, IMF1-IMF8 represents the eight IMFs obtained by using EMD of the original signal, and the residual are the remaining components. The correlation coefficients between IMF1-IMF8 and the original signal were \(-\)0.0639, 0.1293, 0.5204, 0.3890, 0.6243, 0.0028, \(-\)1.1562 \(\times\) 10\({}^{-4}\), and \(-\)1.9159 \(\times\) 10\({}^{-4}\). Since the correlation coefficients of the first six orders of IMF were larger, the Hilbert transform was performed by using IMF1-IMF6. In Figure 16, the correlation coefficients of eight IMFs were 0.0586, 0.3286, 0.6378, 0.4109, 0.0200, \(-\)7.541 \(\times\) 10\({}^{-4}\), \(-\)3.8681 \(\times\) 10\({}^{-4}\), and 1.9866 \(\times\) 10\({}^{-5}\), respectively, so the time-frequency response was obtained by the Hilbert transform with the use of IMF1-IMF5. The results of HHT are depicted in Figures 17-19. The changes in local peaks and energy in different frequency bands shown in these figures are summarized in Table 6. Energy is represented by the colored area above the coordinate axis. The local peak is the maximal value of amplitude in a frequency band. It can be seen that the energy distribution of Segment 2 was significantly different from that of Segment 1 at all measurement positions. Energy-concentration frequency bands changed, and local peaks and energy were higher due to the debonding. Specifically, the main energy-concentration area of sensor No. 1 was shifted from 0-10 Hz to 0-35 Hz, and the local peak and energy of this sensory signal had a drastic rise with the occurrence of the debonding. Similar changes happened in the response of sensor No. 3. The main energy-concentration area of sensor No. 2 was shifted from 0-80 Hz to 36-80 Hz, and the local peak and energy of this sensor also witnessed significant increases, accompanied by the occurrence of debonding. In the main energy-concentration area, the local peak and energy of Segment 2 (in the debonding state) were 10 and 14 times higher than those of Segment 1 (in normal state), respectively. The features in Table 7 include the maximum local peak and the maximum local energy from one of the three sensors, and their relative differences is listed in the last column in this table. According to Table 7, it was found that the local peak, local energy, maximum local  
Figure 16: IMFs of a typical response of Segment 2.