## 3 Proposed Method  
### EMD Construction  
The EMD, due to its adaptive nature, is able to effectively decompose non-stationary and nonlinear signals. Unlike methods that rely on pre-defined basis functions, EMD performs stepwise decomposition based on the extreme points of the local signal, allowing it to better capture the intrinsic modes within the signal. Compared to wavelet transform and VMD, EMD offers greater flexibility and adaptability, making it particularly suitable for complex dynamic vibration signals. The basis of EMD lies in adaptively decomposing nonlinear and non-stationary signals in vibration data into several intrinsic mode functions (IMFs) [22; 23]. Each IMF reflects the dynamic characteristics of the signal at different time scales. For the selection of IMFs, a criterion based on the energy contribution of the decomposed modes was employed in this study. Specifically, modes with significant energy contributions were retained, while those with minimal energy contributions were discarded. Additionally, when processing the residual components after decomposition, optimization was performed based on the frequency characteristics of the residuals. Typically, the low-frequency component is considered noise, and it was filtered out to improve the signal quality and enhance the stability of the model.  
This method emphasizes time-frequency analysis by extracting local features through an iterative process, effectively capturing instantaneous frequency and amplitude variations within the vibration signals. EMD is employed as a data processing method for vibration signals due to its ability to decompose based on the inherent characteristics of the signal, in contrast to other decomposition methods. It effectively extracts dynamic features across various time scales and possesses time-frequency analysis capabilities, allowing for a more comprehensive characterization of the signal. Specifically, the EMD method can decompose a non-stationary signal into a trend component and several meaningful IMFs with instantaneous frequencies [24; 25].  
The processing of vibration signal data is conducted based on three assumptions: (1) The signal must have at least two extreme points, including one maximum and one minimum. (2) The time scale of signal is defined by the time between the two extreme points. (3) If the signal lacks extreme points but contains inflection points, extreme points may be obtained by differentiating the data once or multiple times, followed by integration to achieve the decomposition result. The EMD algorithm, as illustrated in Figure 1, separates different frequency components from the original signal, thereby enhancing feature extraction. The main steps are detailed as follows:  
_Step 1_: For the input signal (x(\(t\))), the maximum and minimum points of the original signal are fitted using spline interpolation to construct the upper and lower envelope curves. The mean envelope (x\({}_{0}(t)\)) is then calculated, and the original signal is subtracted from this mean to obtain a new signal (h\({}_{1}(t)\)). This process can be expressed as follows:  
\[{\rm h}_{1}(t)={\rm x}(t)-{\rm x}_{0}(t) \tag{7}\]  
_Step 2_: The new signal (h\({}_{1}(t)\)) is evaluated to determine whether it satisfies the conditions of an IMF. If the conditions are satisfied, h\({}_{1}(t)\) is designated as the first IMF component. If h\({}_{1}(t)\) does not meet the conditions, the process continues in a loop using _Step 1_ until h\({}_{1}(t)\) fulfills the feature conditions of an IMF.  
_Step 3_: The first IMF (imf\({}_{1}\)) is separated from the original vibration signal to obtain (r\({}_{1}(t)\)). The new signal can be expressed as r\({}_{1}(t)={\rm x}(t)-{\rm imf}_{1}\).  
_Step 4_: The process is repeated until the \(K\)-th IMF is obtained. The final result of the EMD can be expressed as follows:  
\[{\rm x}(t)=\sum_{j=1}^{n}{\rm imf}_{j}+r_{n}(t) \tag{8}\]  
where r\({}_{n}(t)\) denotes the residue, which represents the central tendency of the vibration signal.