## 2 Theory and Model
### The Separate Model of Multiple Low-Rank Constraints  
The vibration signal obtained by the test generally contains multiple vibration source signals. The separation model of Equation (2) can only be used to separate one vibration source signal. Therefore, the separation requirement cannot be satisfied. In order to separate multiple vibration source signals at the same time, a multi-low-rank constrained vibration source signal separation model is proposed, as follows:  
\[ \begin{split}&\underset{L,A}{argmin}\quad\sum\limits_{i=1}^{n} \lambda_{i}\|L_{i}\|_{*}+\beta\times kur(N)\\ & s.t.\quad x=\sum\limits_{i=1}^{n}l_{i}+N\end{split}  \tag{3}\]  
where \(n\) is the number of vibration source signals, which need to be separated, \(\lambda_{i}\) is the weight of the \(i\)-th low-rank matrix, and \(N\) is the noise term. If the length of a one-dimensional vibration signal \(x(t)\) is \(m\), the low-rank matrix \(L_{i}\) can be obtained by \(L_{i}=w(x,o_{i},p_{i})\):R\({}^{m}\rightarrow\) R\({}^{o_{i}\times p_{i}}\). \(w\) is a block operator, and the calculation method is shown in Figure 1. The block operator \(w\) can realize the transformation from one-dimensional vector \(x(t)\) to two-dimensional matrix \(L_{i}\). The vibration source signal is divided into blocks according to the characteristic frequency of the characteristic waveform, where \(o\) is the length of a low-rank matrix column, \(p\) is the number of low-rank matrix rows, \(o\) and \(p\) can be calculated by Equations (4) and (5), and \(l_{i}\) is an inverse operation of \(L_{i}\) by block operator. \[o_{i}=\frac{f_{s}}{f_{source}i} \tag{4}\]  
\[p_{i}=\frac{m}{o_{i}} \tag{5}\]  
where \(f_{s}\) is the sampling frequency, and \(f_{source}i\) is the characteristic frequency of the vibration source signal. The model is a can convex optimization problem, and it can be solved by the augmented Lagrange multiplier method. First, we constructed the augmented Lagrange function \(\mathrm{La}\):  
\[ \begin{split}\mathrm{La}(l_{i},N,Y,\mu_{k})=\sum\limits_{i=1}^{n} \lambda_{i}\|L_{i}\|_{*}&+\beta\times kur(S)+\left\langle Y,x- \sum\limits_{i=1}^{k}l_{i}-N\right\rangle\\ &+\frac{\mu_{k}}{2}\left\|x-\sum\limits_{i=1}^{k}l_{i}-N\right\| _{F}^{2}\end{split}  \tag{6}\]  
In Equation (6), \(\mathrm{La}\) is the augmented Lagrange function, \(Y\) is the Lagrange multiplier, \(\mu\) is the iteration step size, and \(\left\langle\right\rangle\) is the point product computation. The constrained objective optimization problem is transformed into the minimum value problem of the augmented Lagrange function \(\mathrm{La}\), and the expression of the optimal solution is as follows:  
\[(l_{i}^{*},N^{*})=\underset{\tilde{l}_{i},N}{argmin}\left(\mathrm{La}(l_{i},N, Y,\mu_{k})\right)\,i=1,2,\cdots,n \tag{7}\]  
where \(\left(l_{i}^{*},N^{*}\right)\) is the optimal solution of \(\left(l_{i},N\right)\). The optimization problem is solved by the alternating direction multiplier method. Each step of the iterative process can be decomposed into the optimization problem for  
Figure 1: The calculation method of the block operator \(w\). finding the minimum nuclear norm of each low-rank matrix \(L_{i}\) and the minimum kurtosis of the noise term. The iteration process is as follows:  
\[l_{i;k+1}^{s}=\underset{l_{i}}{arg}\left\{\frac{\lambda_{i}}{\mu_{k}}\|L_{i}\|_{ s}+\frac{1}{2}\right\| \begin{array}{l}x-\sum\limits_{j=1}^{i-1}l_{j,k}-l_{i}- \\ \sum\limits_{j=i+1}^{n}l_{j,k}-N_{k}+\frac{1}{\mu_{k}}Y_{k}\end{array}