## 3 Case Study
### Implementation Details  
The platform configuration used in this experiment is as follows: platform system--Windows11; hardware configuration--CPU is 11 th Gen Intel(R) Core(TM) i7-11800H, memory is 16 G; the graphics card is an NVIDIA GeForce RTX3050 Ti 8G*2; experimental framework--Tensorflow2.3.0. \begin{table}
\begin{tabular}{c c c c c c} \hline \multirow{2}{*}{**Working Condition**} & \multicolumn{3}{c}{**Damage Unit**} & \multicolumn{3}{c}{**Cutting Length**} \\ \cline{2-6}  & **No. 7** & **No. 7 Straight Belly Rod** & **No. 10** & **No. 35** & **10 cm** & **20 cm** \\ \hline WC0 & + & & & & + \\ WC1 & + & & & & + \\ WC2 & & & + & & + \\ WC3 & & & + & & + \\ WC4 & & + & & & + \\ WC5 & + & & + & & + \\ WC6 & + & & + & & + \\ WC7 & + & & + & No. 10 & No. 10 \\ WC8 & + & + & + & No. 10 & No. 7 straight belly rod \\ WC9 & + & + & & & + \\ WC10 & + & + & + & No. 10 & No. 7, No. 35 \\ \hline \end{tabular}
\end{table} 
Table 2: Setting table of experimental conditions. The meaning of the + signal indicating the location where the damage occurs. Figure 7: Sensor layout diagram. # Indicates that a sensor is placed at this position. The intergrated dataset was processed using z-score standardization for the preliminary processing of one-dimensional sequence data of each measurement point:  
\[z=\frac{x-u}{\sigma} \tag{11}\]  
In Formula (11), \(x\) is the single-signal data in the measuring point, \(u\) is the mean value of all signal values at the measuring point, and \(\sigma\) is the variance of all signal values at the measuring point. Basic attributes such as dimension, quantity and length of data segment do not change after z-score processing, and standardized processing can avoid extreme outliers in the data. The obtained new time series data is defined as a window by the sliding average window, and the sliding window is updated by the fixed sliding window. The sequential average of the window is calculated successively, and the obtained sequential average is connected to generate a new equilong time interval to obtain the dimensionality reduction dataset. To verify whether the VMD preprocessing step provides positive value to the subsequent neural network, this study first applies noise addition to the aforementioned dimensionality reduction dataset. Gaussian noise with a noise rate of 0.10 \(f_{n}(t)\), resulting in the noisy dataset. The \(f_{n}(t)\) as described in Equation (12). \[f_{n}(t)=\cos(4\pi t)+\frac{1}{4}\cos(48\pi t)+\frac{1}{16}\cos(576\pi t)+n \tag{12}\]  
To simulate the interference effects of external environmental factors on the signal data in real-world scenarios, noise is added to the data, and then both EMD and VMD are applied for processing. Figure 8a displays the time-domain waveforms of the original signal and the IMF components obtained after EMD decomposition, while Figure 8b shows the FFT spectra of the IMF components. During the actual experiment, it was observed that the waveforms of the modal components after IMF1 gradually disperse, and the FFT spectra beyond IMF9 become too boundary-dominated, losing their reference value. Therefore, only the FFT spectra of IMF1 to IMF9 are presented. In the decomposition process of VMD, the Lagrangian multiplier and quadratic penalty play crucial roles as smoothing mechanisms. The Lagrangian multiplier strengthens the constraints, while the quadratic penalty enhances convergence. The Lagrangian multiplier method transforms the constraints into penalty terms in the objective function and adjusts the weight of the penalty term using Lagrangian multipliers. During the hyperparameter tuning process, the regularization strength or weight of the regularization term can be flexibly controlled by setting the penalty factor of VMD, allowing for better control over the smoothness or sparsity of the decomposed mode functions. Additionally, the algorithm's