## 3 Comparison of Application Cases and Methods
### Comparison of Modal Decomposition Methods  
To assess the efficacy of the integrated approach combining VMD alignment entropy and the CSA-optimized LSTM algorithm, a comprehensive validation was conducted, particularly focusing on performance under variational mode decomposition techniques. To meticulously evaluate the effectiveness of the combined VMD and CSA-optimized LSTM algorithm, experimental data were derived from the vibration signals of gearboxes in three distinct states: normal gearing, worn gears, and broken gears. These signals were analyzed under the stringent condition of 100% rated lubricating oil pressure to ensure an exhaustive examination of the algorithm's performance across a spectrum of operational and wear conditions, thereby underscoring its versatility and robustness. The VMD algorithm initiates by pre-setting the number of decomposed modes, denoted as \(K\). Illustratively, taking the vibration signal of a gearbox in its normal state, a Fourier transform is applied. This process yields a detailed spectrogram, vividly illustrating the vibrational signal characteristics of the gearbox in its pristine operational state. Following this, VMD decomposition is executed, effectively distilling the intrinsic vibrational modes and their respective characteristics. The outcomes of this decomposition process are systematically illustrated in Figure 3, providing a visual and analytical insight into the vibrational dynamics of the gearbox. In this study, a random selection methodology was employed to curate the training set. Prior to initiating training, the original data from the gearbox underwent decomposition via VMD. Through iterative experimentation, it was observed that the VMD variational mode decomposition, when optimized using CSA, exhibited progressive convergence as the number of iterations increased. This methodological refinement ensures more accurate and consistent results, advancing our understanding of gearbox dynamics. IMF\({}_{1}\) is a trend component that reflects the overall trend of gear speed changes in the gearbox dataset. IMF\({}_{2}\)-IMF\({}_{n}\) are the remaining random components, resulting in very small prediction errors. When a transmission failure arises, alterations are observed not only in the fundamental frequency components but also in the octave components, each exhibiting varying degrees of suppression or enhancement. If the value of \(K\) is set at 3, the distinction between different models and the frequency components of the transmission vibration signal \"a\" becomes minimal. This makes the comprehensive extraction of fault-state information challenging. On the other hand, a \(K\) value exceeding 5 increases the complexity of thecalculation significantly, introducing the risk of over-decomposition. Such a scenario is not optimal for prompt fault diagnosis due to the increased computational demands and potential for muddled insights. To strike a balance, we opted for a \(K\) value of 5. This allowed us to efficiently capture the center frequency of different modes inherent in the normal-state vibration signal of the gearbox transmission without compromising computational efficiency or clarity of insights. The time-domain plot derived from the VMD decomposition is depicted in Figure 3a, and the corresponding frequency spectrum is presented in Figure 3b, offering a visual and analytical exploration of the transmission dynamics. As can be seen in Figure 3, the modal aliasing phenomenon of VMD is effectively solved, and the corresponding center frequency of each modal component is consistent with the overall characteristics of the frequency derived from the fast Fourier transform, which realistically restores the information contained in the original signal. Compared with the existing empirical mode decomposition (EMD), the time-domain diagram of the transmission system's normal-state vibration signal decomposed by EMD is shown in Figure 4.  
Figure 4: Original data and EMD decomposition results of each sequence. (**a**) Time-domain diagram of the EMD decomposition.