## 4 Time-Domain and Time-Frequency-Domain Feature Extraction Based on the LMD Method
### Time-Frequency-Domain Feature Extraction  
When damage occurs to the track structure, the vibration energy distribution of the structure will change, and, therefore, the energy of the characteristic PFs of the track structure will change simultaneously. A certain location of the track structure at which damage occurs can be considered the location with reduced frequency softening to absorb more vibration energy, which leads to an increase in the energy entropy of the vibration response. Therefore, the energy entropy of characteristic PFs can be selected as one of the damage identification features [59]. The energy entropy of characteristic PFs is calculated by the following steps:  
1. Calculate the energy of the \(q\)th characteristic PF: \[E_{q}=\sum\limits_{j=1}^{l}\abs{PF_{qj}}^{2}\] (13) where \(l\) is the length of a characteristic PF.
2. Calculate the total energy of these \(r\) efficient characteristic PFs: \[E=\sum\limits_{q=1}^{r}E_{q}\] (14)
3. Calculate the energy entropy of characteristic PFs: \[H{=}-\sum\limits_{q=1}^{r}p_{q}\log(p_{q})\] (15) where \(p_{q}=\frac{E_{q}}{L}\) is the percentage of the energy of the \(q\)th characteristic PF relative to the total energy entropy, and \(\sum\limits_{q=1}^{r}p_{q}=1\).  
To better understand the model formulation, the variables and parameters that appear in this section are classified and named. The results are shown in Table 2.  
Figure 6 is a scatter plot of the energy entropy of the vibration response around track slab deformation. The black squares and red dots in the figure represent the energy entropy of the characteristic PFs in the normal and deformed state, respectively. Figure 6 indicates that the energy entropy of the characteristic PFs is larger in the deformed state, which is due to the damage of the track slab. When the track slab is in the deformation state, the  
Figure 5: Scatter plot of the skewness of the vibration response.  
structural stiffness decreases and the position needs to absorb more energy to achieve the equilibrium state; thus, the energy entropy increases. This is consistent with the existing literature results [62].  
In this section, LMD is performed on the vibration data into several PFs, and the characteristic PFs are then selected with Pearson correlation coefficients. Six time-domain statistics and one time-frequency-domain feature are extracted from the characteristic PFs, which form a characteristic parameter vector matrix.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Category** & **Symbol** & **Name** \\ \hline \multirow{8}{*}{variables} & \(x\) & Vibration signal \\  & \(t\) & Time \\  & \(N\) & Local extremum point \\  & \(i\) & Local extremum point variable \\  & \(m\) & Average of local extremum point \\  & \(a\) & Local amplitude \\  & \(h\) & Intermediate process variable \\  & \(s\) & Pure frequency modulation signal \\  & \(PF\) & Product function \\  & \(m\) & Pure frequency modulation signal iteration \\  & \(m\) & count variable \\  & \(kk\) & Monotonic signal iteration count variable \\  & \(u\) & Residual signal \\  & \(q\) & Number of characteristic product function \\  & \(j\) & Length of characteristic product function \\  & \(E\) & Total energy \\  & \(p\) & Normalized energy \\  & \(H\) & Energy entropy \\ \hline \multirow{4}{*}{parameters} & \(n\) & Pure frequency modulation signal iteration count \\  & \(k\) & Monotonic signal iteration count \\ \cline{1-1}  & \(r\) & Number of characteristic product function \\ \cline{1-1}  & \(l\) & Length of characteristic product function \\ \hline \hline \end{tabular}
\end{table}
Table 2: Nomenclature table of the model variables and parameters.  
Figure 6: Scatter plot of the energy entropy of the vibration response.