## 2 Theoretical Aspects
### Improved Variational Mode Decomposition (IVMD)  
Variational mode decomposition (VMD) is a new method of multi-component adaptive signal decomposition [15]. Compared to traditional signal decomposition methods, it effectively avoids modal aliasing and over-decomposition defects and has a higher utilization value [16]. VMD comprises two processes comprising the establishment of variational constraints and iteration to find the optimal solution. The specific operation process is as follows: VMD decomposes a given signal \(f\) into \(K\) modal functions using variational constraints \(m_{k}(t)\). The bandwidth of each IMF is limited, and each IMF is distributed around the central pulsating frequency. The variational constraint model is as follows [17]:\[\left\{ \begin{array}{l}\min\limits_{m_{k},w_{k}}\left\{\sum\limits_{k}\left\| \partial_{t}\Big{[}\Big{(}\sigma(t)+\frac{j}{\pi t}\Big{)}m_{k}(t)\Big{]}e^{-i \
u_{k}t}\right\|_{2}^{2}\right\}\\ s.t.\sum\limits_{k}m_{k}=f\end{array} \right. \tag{7}\]  
where \(\{m_{k}\}\) represents the decomposed K IMF components, \(\{m_{k}\}=\{m_{1},m_{2},\cdots,m_{k}\}\); \(\sigma(t)\) is a pulse function; \(\{w_{k}\}\) is the central frequency of each IMF, \(\{w_{k}\}=\{w_{1},\ldots w_{k}\}\). To complete the adaptive decomposition of input signals \(f\) and to obtain the IMFs with the minimum sum of bandwidth, the following expanded Lagrange expression is introduced:  
\[L(m_{k},w_{k},\lambda)= \tag{8}\]  
where \(\alpha\) is the penalty factor to ensure the accuracy of signal reconstruction; \(\lambda(t)\) is a Lagrange multiplier used to strengthen the constraint; \(\langle\rangle\) represents the inner product operation. To solve the above variational constraint problem, the dual decomposition and alternate direction multiplication sub-algorithm are used [18]. Keep updating \(m_{k}\), \(w_{k}\) and \(\lambda(t)\) to find the saddle point of Equation (8), that is, the optimal solution of Equation (7). The modal component function \(m_{k}\) and the central frequency \(w_{k}\) are  
\[m_{k}^{n+1}(w)=\frac{f(w)-\sum\limits_{i\
eq k}m_{i}(w)+\frac{\lambda(\omega) }{2}}{1+2\alpha(w-w_{k})^{2}} \tag{9}\]  
\[w_{k}^{n+1}=\frac{\int_{0}^{\infty}w\left|m_{k}(w)\right|^{2}dw}{\int_{0}^{ \infty}\left|m_{k}(w)\right|^{2}dw} \tag{10}\]  
\[\lambda^{n+1}=\lambda^{n}+\tau\Bigg{(}f(w)-\sum\limits_{k}m_{k}^{n+1}(w) \Bigg{)} \tag{11}\]  
When VMD decomposes the vibration response sequence, determining the total modal number is a crucial step. The selection of modal parameters \(K\) greatly affects the accuracy of the results [19]. A parameter \(K\) is usually challenging to determine. If \(K\) is greater than the number of useful components obtained by signal decomposition, information superposition will occur; if \(K\) is smaller than it, a part of the limited bandwidth of the solid modulus cannot be decomposed. An IVMD method based on the mutual information (MI) method is proposed for \(K\) selection. MI reflects the correlation between two random variables and allows better identification of the degree of correlation [15]. MI is as follows:  
\[I(X,Y)=H(Y)-H(X|Y) \tag{12}\]  
where \(H(Y)\) is the entropy of \(Y\), and \(H(Y|X)\) is the conditional entropy of \(Y\) when \(X\) is known. When \(I(X,Y)=0\), \(X\) and \(Y\) are independent of each other. The mutual information \(I_{k}\) of the original signal and each IMF obtained by the IVMD decomposition is calculated and normalised by Equation (13). Then the correlation between each modal component and the original signal is judged, that is, whether the original signal is completely decomposed. \[\sigma_{1}=\frac{I_{i}}{\max(I_{i})} \tag{13}\]  
where \(\sigma_{1}\) is the normalized mutual information value of each IMF, \(\text{i}=1,2,\ldots k\). Refer to reference [20], when \(\sigma_{1}\) is less than 0.02, it is considered that the IMF does not contain valid feature information. The original signal has been decomposed completely.