## 4 Analysis and Discussion
### Frequency-Domain Analysis for Multi-Feature Extraction and Assessment
The peak amplitudes exhibited obvious change, and this was similar to that situation in the time domain. The two features were able to distinguish interlayer debonding, so the peak frequency in the frequency domain was used as a feature for feature fusion. It can be seen from Figure 13 and Table 4 that the vibration energy was concentrated in the frequency band of 0-30 Hz, and the peak frequencies were around 2.58 Hz and 3.68 Hz. The relationship between this phenomenon and the periodic excitation caused by the geometric characteristics of high-speed vehicles is explored in this part. The frequencies related to the geometric characteristics can be formulated as follows [63]:  
\[f=v/L_{m}\ (m=1,\ 2,\ 3,\ 4) \tag{13}\]  
where \(v\) is the running speed of a passing-by high-speed vehicle, and \(L_{m}\) is a geometric characteristic of the vehicle. As shown in Figure 14, the lengths \(L_{1}\)-\(L_{4}\) were the distance between the two wheelsets of one bogie, the central distance between two bogies in a carriage, the distance between the front and rear bogies of two adjacent carriages, and the carriage length of the selected train type. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Number** & **Feature** & **Segment 1** & **Segment 2** & **Difference** \\  & & **(_a_)** & **(_b_)** & **(\(1b-a1/a\))** \\ \hline \multirow{4}{*}{Sensor 1} & Peak frequency 1/Hz & 2.56 & 2.46 & 0.039 \\  & Peak frequency 2/Hz & 3.63 & 3.43 & 0.055 \\  & Peak amplitude 1/\(10^{-4}\) g & 4.32 & 15.8 & 2.66 \\  & Peak amplitude 2/\(10^{-4}\) g & 5.73 & 14.7 & 1.57 \\  & Peak frequency 1/Hz & 2.66 & 2.58 & 0.030 \\  & Peak frequency 2/Hz & 3.74 & 3.63 & 0.029 \\  & Peak amplitude 1/\(10^{-4}\) g & 3.01 & 23.2 & 6.71 \\  & Peak amplitude 2/\(10^{-4}\) g & 2.81 & 15.3 & 4.44 \\  & Peak frequency 1/Hz & 2.49 & 2.37 & 0.048 \\  & Peak frequency 2/Hz & 3.80 & 3.67 & 0.034 \\  & Peak amplitude 1/\(10^{-4}\) g & 4.61 & 10.9 & 1.36 \\  & Peak amplitude 2/\(10^{-4}\) g & 5.76 & 19.1 & 2.32 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Deviation of peak frequencies and amplitudes. Figure 14: Sketch of characteristic lengths of a high-speed train. When the running speed was 232 km/h (measured by tachometer), the frequencies associated with the geometric characteristics were calculated by applying Equation (13) and listed in Table 5. The peak frequencies (around 2.58 Hz and 3.68 Hz) in Table 4 are ascribed to the two dominant geometric characteristics \(L_{4}\) and \(L_{2}\) in Table 5. The geometric-characteristics-related frequencies, namely 2.58, 3.68, 8.59, and 25.78 Hz, contributed to the energy concentration in 0-30 Hz. The measured frequencies deviated from these characteristic frequencies, and this may result from the Doppler effect happening when a high-speed train is approaching or leaving the measurement positions. The vibration acceleration level [64] can reveal the vibration level of a structure. The formula of the vibration acceleration level VAL is shown in Equation (14):  
\[VAL=20\log\frac{a}{a_{0}} \tag{14}\]  
where VAL is vibration acceleration level (dB); \(a\) is the effective value of vibration acceleration (m/s\({}^{2}\)); \(a_{0}\) is the base acceleration, which is taken as \(10^{-6}\) m/s\({}^{2}\). As can be seen from Table 6, the vibration level of Segment 2 was larger than that of Segment 1, indicating that the vibration response of the track structure increased significantly when the track structure was in an interlayer debonding state. In addition, there was little difference in the vibration level between sensor No. 1 and sensor No. 3 regardless of the segment, owing to the fact that sensor No. 1 and sensor No. 3 were both located at the end of the viaduct. The vibration level of sensor No. 2 was greater than that of sensors No. 1 and 3 in that sensor No. 2, which was located in the middle of the viaduct, vibrated larger than the end of the viaduct when subjected to train loads. This also verifies that the monitoring system proposed in this manuscript is accurate.