## 2 Materials and Methods' Theoretical Basis and Diagnostic Model  
### RSSD Optimized Based on ASO  
In mechanical systems, noise and harmonic components that generate a large number of oscillations possess a high-quality factor, and a high-quality factor exhibits strong frequency clustering. Conversely, transient shock signals with fewer vibrations exhibit a lower-quality factor, and a low-quality factor demonstrates greater time clustering [17]. Essentially, resonance sparsity involves the use of two wavelet basis functions with distinct quality factors to sparsely represent a complex signal, thus overcoming the limitations inherent in traditional filters that rely on frequency band partitioning. The specific steps of the algorithm are as follows. The Tunable Q-factor Wavelet Transformation (TQWT) decomposes and reconstructs the original signal through a dual-channel filter bank. The appropriate dual quality factor Q1 and Q2 is selected based on the original vibration signal. TQWT generates wavelet basis function libraries \(S_{1}\) and \(S_{2}\). with corresponding different resonance properties. The objective function is established as shown through Morphological Component Analysis (MCA) in Equation (1) to determine the appropriate input \(\lambda_{1}\) and \(\lambda_{2}\)[18]. The Split Augmented Lagrangian Shrinkage Algorithm (SALSA) is used to iteratively calculate Equation (1). When the objective function is minimized, the optimal matching between the wavelet basis function library and the dissipation function can be achieved, and the obtained optimal coefficient matrices are \(W_{1}\) and \(W_{2}\). \[J(W_{1},W_{2})=\left\|x-S_{1}W_{1}-S_{2}W_{2}\right\|_{2}^{2}+\sum_{j=1}^{I_{ 1}+1}\lambda_{1,j}\left\|W_{1,j}\right\|_{1}+\sum_{j=1}^{I_{1}+1}\lambda_{2,j }\left\|W_{2,j}\right\|_{1} \tag{1}\]  
The original signal is reconstructed using \(W_{1}^{*}\) and \(W_{2}^{*}\) to achieve effective separation of high- and low-resonance components, whose approximate values are represented as \(\hat{x_{1}}=S_{1}W_{1}^{*}\) and \(\hat{x_{2}}=S_{2}W_{2}^{*}\). In resonance-based sparse signal decomposition, it is crucial that the selection of quality factors Q1 and Q2 plays a decisive role in the final decomposition result. In the traditional resonant sparse decomposition method, the selection of quality factor parameters often requires the support of prior knowledge, significantly escalating operational costs. To address the aforementioned issues, a quality factor optimization scheme is presented leveraging the ASO. The kurtosis of the low-resonance component, decomposed by RSSD, is utilized as the objective function. Kurtosis can be effectively applied as an important indicator to measure the pulse component in signals [19]. ASO is a molecular dynamic-based meta-inspired optimization algorithm that simulates the movement of atoms in natural molecular systems under the combined action of interaction forces and system constraints, in order to find the optimal position. Compared with traditional intelligent algorithms for optimizing biological populations, it has the advantages of strong convergence and fewer parameters [20]. In the d-dimensional space, s atoms are distributed, and the position of the \(i\)th atom at the \(t\)th iteration is represented as  
\[X_{i}^{d}(t)=(X_{i,1},X_{i,2},\cdots X_{i,d})\left(i\in[1,s],t\in[1,t_{max}]\right. \tag{2}\]  
where \(t_{max}\) is the maximum number of iterations. The laws of atomic motion in space follow [PERSON]'s second law. \[F_{i}^{d}(t)+G_{i}^{d}(t)=m_{i}^{d}(t)a_{i}^{d}(t) \tag{3}\]  
where \(F_{i}^{d}(t)\) is the resultant force exerted on the atom by other atoms, \(G_{i}^{d}(t)\) is the geometric constraint force exerted by the globally optimal atom, \(m_{i}^{d}(t)\) is the mass of the atom, and \(a_{i}^{d}(t)\) is the acceleration of the atom's motion. The speed and displacement of atomic motion change under acceleration, thereby updating the atomic position and ultimately outputting the optimal atomic position \(X_{best}^{d}(t)\).