## 3 Comparison of Application Cases and Methods
### Comparison of Modal Decomposition Methods
(**b**) Spectrogram of the EMD decomposition. Figure 3: Original data and VMD decomposition results of each sequence. (**a**) Time-domain diagram of the VMD decomposition. (**b**) Spectrogram of the VMD decomposition. Figure 4 illustrates that as the quantity of decomposed modes escalates, overlapping of modes commences from mode 6 onward. This overlap gives rise to invalid components, which are ineffectual in representing information pertinent to the transmission's wear state. To mitigate the influence of these non-representative components, we introduced correlation number analysis. This technique calculates the correlation between each modal component and the chosen transmission vibration signal, ensuring that only valid and informative components are retained for subsequent analysis, enhancing the precision and reliability of the diagnostic insights. The correlation between each modal component and the selected transmission vibration signal is calculated to distinguish valid from invalid components based on their ability to reflect the state of the transmission. Modal components with correlation coefficients exceeding 0.1 are deemed valid, as they effectively encapsulate the transmission-state information and are, therefore, selected for further analysis. Conversely, components with correlation coefficients below 0.1 are classified as invalid, lacking the capacity to serve as feature quantities indicative of the transmission's state. These low-correlation components are unable to accurately portray the variations in vibration signals distinguishing normal and fault conditions of the transmission and are consequently excluded from further consideration and analysis. This ensures that the subsequent evaluation and diagnostics are rooted in the most informative and representative data, enhancing the accuracy and reliability of the findings. VMD-PE and EMD-PE are used to extract features from the vibration signals. m and \(\tau\) are required to set the embedding dimension and delay time in advance for PE calculation, and \(m=6\) and \(\tau=1\) are generally chosen to better reflect the dynamics of the time sequence, so \(m=6\) and \(\tau=1\) are chosen. PE is calculated as the feature quantity of the modal components of the three states of transmission, with a total of 750 sets of data with 250 sets for each state of the transmission. We randomly select 200 sets of normal data, 200 sets of incomplete loosening data, and 200 sets of complete loosening data as the training set and the rest as the test set and carry out the training of the model and the classification of the results through the CSA-optimized LSTM algorithm. The diagnostic results are shown in Tables 1 and 2. Tables 1 and 2 clearly delineate the superior efficacy of the proposed method out- lined in this paper. It boasts a fault identification rate of 100% for diagnosing transmission wear associated with varying degrees of loosening. This impressive accuracy eclipses the 96.67% fault identification rate achieved by the EMD-PE method. This stark contrast in diagnostic precision attests to the robust capabilities of the VMD method in accurately discerning \begin{table}
\begin{tabular}{c c c c c} \hline
**Typology** & **Training Sample** & **Test Sample** & **Correct SAMPLE** & **Test Set Correctness/\%** \\ \hline Healthy & 200 & 50 & 50 & 100 \\ \hline Rear wear & 200 & 50 & 50 & 100 \\ \hline Broken gear teeth & 200 & 50 & 50 & 100 \\ \hline Aggregate & 600 & 150 & 150 & 100 \\ \hline \end{tabular}
\end{table} 
Table 1: VMD-PE fault diagnosis results. \begin{table}
\begin{tabular}{c c c c c} \hline
**Typology** & **Training Sample** & **Test Sample** & **Correct Sample** & **Test Set Correctness/\%** \\ \hline Healthy & 200 & 50 & 50 & 100 \\ \hline Rear wear & 200 & 50 & 50 & 100 \\ \hline Repeat & 200 & 50 & 47 & 94 \\ \hline Aggregate & 600 & 150 & 145 & 96.67 \\ \hline \end{tabular}
\end{table} 
Table 2: EMD-PE fault diagnosis results. and isolating fault information related to gearbox transmission.