## 3 Case Study
### Implementation Details
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Operating Condition Number** & **Working Condition** & **Damage Condition** \\ \hline
0 & WC0 & No. 7 10 cm \\
1 & WC1 & No. 7 20 cm \\
2 & WC2 & No. 10 10 cm \\
3 & WC3 & No. 10 20 cm \\
4 & WC4 & No. 7 straight belty rod 20 cm \\
5 & WC5 & No. 7 10 cm + No. 10 10 cm \\
6 & WC6 & No. 7 20 cm + No. 10 20 cm \\
7 & WC7 & No. 7 20 cm + No. 10 10 cm \\
8 & WC8 & No. 7 20 cm + No. 7 straight \\  & & belly rod 20 cm + No. 10 10 cm \\
9 & WC9 & No. 7 20 cm + No. 7 straight \\  & & belly rod 20 cm \\
10 & WC10 & No. 7 20 cm + No. 10 \\  & & 10 cm + No. 35 20 cm \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Damage conditions and data labels. noise tolerance is set to 0. The mode functions are uniformly initialized, with their initial values set as random numbers from a uniform distribution. The convergence of the algorithm is controlled by setting the tolerance parameter. The decomposition scale is shown in Table 4, with the center frequencies of f_1 to f_4 set as 2, 24, 128, and 288, respectively. The core hyperparameters of VMD are listed in Table 4. Figure 9a shows the time-domain waveforms of the original signal and the IMF components obtained after VMD decomposition. Figure 9b displays all the IMF components, Figure 9c shows the FFT spectra of each IMF component. Figure 8: The original signal, after undergoing EMD processing, yields various IMF signal plots, Figure (**a**) shows the time domain waveforms of the original signal and each IMF component; Figure (**b**) shows The fast Fourier transform (FFT) spectra of IMF1 to IMF9. \begin{table}
\begin{tabular}{c c} \hline \hline
**VMD** & \\ \hline Penalty Factor (alpha) & 200 \\ Noise Tolerance (tau) & 0 \\ Decomposition Scale (K) & 4 \\ Direct Current Part (DC) & 0 (None) \\ Uniform Initialization & 1 \\ Tolerance & \(1\times 10^{-7}\) \\ Noise Rate & 0.10 \\ Time Domain (T) & 4096 \\ f\_1 Center Frequency & 2 \\ f\_2 Central Frequency & 24 \\ f\_3 Central Frequency & 128 \\ f\_4 Central Frequency & 288 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: The important hyperparameters related to VMD decomposition. Figure 9: _Cont_. By comparing the signal representations of intrinsic mode functions (IMFs) obtained through empirical mode decomposition (EMD) and variational mode decomposition (VMD), it is evident that EMD, due to its inability to set the number of decomposed IMFs as hyperparameters, produces multiple signal components that deviate significantly from the original data. In contrast, VMD, with controllable parameters and decomposition guided by the central frequency \(\omega_{t}\), yields IMFs that are relatively uniformly dispersed around the original signal. Additionally, analyzing the FFT spectra of the IMF components obtained from EMD and VMD reveals distinct differences. The FFT spectrum of EMD exhibits gradual marginalization from IMF1 onwards, with IMF10 and IMF11 being severely marginalized, resulting in poor visualization. Therefore, these components are not included in the analysis. In contrast, the FFT spectra of the VMD-decomposed IMFs demonstrate distinct frequency distributions for each IMF component, indicating their potential value for data analysis and experimentation. Furthermore, in accordance with the theoretical framework outlined in reference [10], this study incorporates Gaussian noise with a 10% noise level into the data, under the prior setting of Lagrangian multipliers and quadratic penalty factors. The purpose is to simulate VMD's ability to withstand external disturbances and achieve noise robustness under generalized conditions, through its inherent central constraint and non-smooth band decomposition. The resulting decomposition plots clearly indicate that the IMF components generated by VMD do not exhibit sparsity.