## 3 Case Study
### Implementation Details
Figure 9: The original signal, after undergoing VMD processing, yields various IMF signal plots, Figure (**a**) shows the time-domain waveforms of the original signal and each IMF component; Figure (**b**) shows All the IMF components; Figure (**c**) shows The Fast Fourier Transform (FFT) spectra of all IMF. It is worth mentioning that in subsequent experimental sections, where VMD decomposition is utilized as input for a neural network model, the final recognition performance further confirms the robustness of VMD's processing capabilities. Then, the data set was decomposed by variational mode to obtain the modal component IMF of the data, and the IMF data and noisy dataset were horizontally expanded to obtain the dataset to be trained with dimension 4096 \(\times\) 1232. To investigate the impact of acceleration response signal selection criteria on the effectiveness of structural damage identification in this experiment, three experimental schemes were adopted under impact excitation. These schemes involved the selection of three different operating conditions (i.e., extracting data from the dataset with matching labels of the corresponding operating condition number):  
1. Working conditions WC1, WC3, WC5, WC7, and WC9 are selected;
2. Working conditions WC2, 4, 6, 8, and 10 are selected;
3. All working conditions are selected. In different experimental schemes, the quantity of IMF components is determined by the decomposition scale parameter, \(k\), in VMD. Leveraging the inherent mode superposition property of VMD, the IMF components can be combined and averaged to accentuate the representation of crucial information, as demonstrated in Equation (13). \[X(t)=\sum_{k=1}^{K}C_{k}(t) \tag{13}\]  
In Equation (13), \(X(t)\) represents the original signal, and \(C_{k}(t)\) represents the \(k\) mode component. Through a trial-and-error approach, an evaluation is conducted on each segment obtained by directly concatenating IMF components with the original signal and on the new signal segment obtained by averaging several IMF components and concatenating it with the original signal. Evaluation criteria include assessing whether the IMF signals deviate significantly from the structural resonance characteristics, whether nonlinear breakpoints occur at the concatenation points of the new signal segments, and whether the overall smoothness of the new signal is reduced. After selecting effective components [44] suitable for horizontal concatenation with the original signal, the selected effective components and the new signal data obtained from the original dataset replace the original data. The final selection of effective IMF components related to structural characteristics and the total number of IMF components obtained after VMD decomposition for each of the three schemes are presented in Table 5. \begin{table}
\begin{tabular}{c c c c}
**Working Condition** & **Option** & **Option 1** & **Option 2** & **Option 3** \\ WC0 & \(\backslash\) & \(\backslash\) & 1/4 \\ WC1 & 2/4 & \(\backslash\) & 1/4 \\ WC2 & \(\backslash\) & 1/4 & 2/4 \\ WC3 & 2/4 & \(\backslash\) & 1/4 \\ WC4 & \(\backslash\) & 2/4 & 1/4 \\ WC5 & 2/4 & \(\backslash\) & 2/4 \\ WC6 & 1/4 & 2/4 & 1/4 \\ WC7 & 2/4 & \(\backslash\) & 2/4 \\ WC8 & \(\backslash\) & 1/4 & 1/4 \\ WC9 & 1/4 & \(\backslash\) & 1/4 \\ WC10 & \(\backslash\) & 2/4 & 2/4 \\ \end{tabular}
\end{table} 
Table 5: Screening of effective components and the total number of effective components after VMD decomposition of time series signals under the three schemes. The \(\backslash\) symbol indicates that the data has not been applied in this Option. The VMD-CSANN model in this paper consists of two sets of hyperparameters. The first set pertains to the parameters that need to be adjusted for VMD decomposition, as described in Table 4. The second set pertains to the hyperparameters that need to be adjusted for the CSANN model, such as the convolution layer size, dropout layer ratio, learning rate, etc.