## 5 Variational Modal Decomposition
### Variational Modal Decomposition Process  
VMD (variational mode decomposition) is a signal feature extraction algorithm with excellent performance. It can decompose the signal into several components with limited bandwidth. Compared with EMD (empirical mode decomposition), this method has the advantage of effectively reducing frequency aliasing [27]. For weak fault signals of early bearings, if frequency aliasing occurs, the energy of the periodic signal in the test signal will be dispersed into the adjacent two IMFs, making it more difficult for the double-coupled Duffing system to detect periodic signals. However, VMD also has certain defects. The adaptability of its algorithm is weaker than other algorithms. The number of mode components (IMF) generated by its decomposition is usually determined by manual setting. If the number of IMFs is too small, frequency aliasing will occur. If the number of IMFs is too much, More useless components will be generated, so the number of IMFs is particularly critical. The eigen modal components obtained by VMD algorithm decomposition can be expressed by Equation (10) [28]:  
\[u_{k}(t)=A_{k}(t)\text{cos}\varphi_{k}(t) \tag{10}\]  
where \(u_{k}(t)\) is the kth eigenmode function obtained by decomposition, \(u_{k}(t)\) satisfys \(\sum_{k=1}^{K}u_{k}=f(t)\), \(f(t)\) is the signal before decomposition, \(K\) is the total number of eigenmode functions to be decomposed, \(A_{k}(t)\) is the instantaneous amplitude, and \(\varphi_{k}(t)\) is the instantaneous phase. The instantaneous frequency is the derivative of the instantaneous phase, and the values of the instantaneous amplitude and the instantaneous frequency are positive real numbers. The VMD algorithm has completed the solution of a variation problem: \(u_{k}(t)\) is transformed by Hilbert transform to obtain its unilateral spectrum, and each IMF is modulated to the response fundamental frequency band via exponential mixed modulation. Finally, Gaussian smoothing is used to demodulate, and the bandwidth of \(u_{k}(t)\) is estimated. The variation model to be solved is as follows:  
\[\min_{\{u_{k}\},\{\omega_{k}\}}\left\{\sum_{k=1}^{K}\left\|\partial_{t}\left[ \left(\delta(t)+\frac{j}{\pi t}\right)*u_{k}(t)\right]e^{-j\omega_{k}t}\right\| _{2}^{2}\right\} \tag{11}\]  
where \(\omega_{k}\) is the center frequency corresponding to each \(u_{k}\), and \(\delta(t)\) is the unit pulse function. The penalty factor \(\alpha\) and the Lagrange multiplier \(\lambda\) are introduced into the model. By introducing the penalty factor, the fidelity of the reconstructed signal can be guaranteed. The larger the value is, the smaller the bandwidth of the decomposed frequency band will be. If the bandwidth is too small, the signal will be lost. By introducing Lagrangemultipliers, the strictness of the constraint conditions can be guaranteed. After modification, the constrained variational problem is changed into an unconstrained variational problem:  
\[L(\{u_{k}\},\{\omega_{k}\},\lambda(t))\] \[= \alpha\sum\limits_{k=1}^{K}\left\|\partial_{t}\Big{[}(\delta(t)+ \tfrac{j}{n!})*u_{k}(t)\Big{]}e^{-i\omega_{k}t}\right\|_{2}^{2}\] \[+ \left\|f(t)-\sum\limits_{k=1}^{K}u_{k}(t)\right\|_{2}^{2}+\langle \lambda(t),f(t)-\sum\limits_{k=1}^{K}u_{k}(t)\rangle\]  
where \(\alpha\) is the bandwidth coefficient, and \(\lambda(t)\) is the Lagrange multiplier. The ADMM method is used to update \(\hat{u}_{k}^{n+1},\hat{\mathcal{O}}_{k}^{n+1},\hat{\mathcal{I}}_{k}^{n+1}\) iteratively. After meeting the iteration termination conditions, K IMF components obtained by decomposition can be output. The process is as follows:  
1. Initialize \(\{\hat{a}_{k}^{1}\},\{\hat{\mathcal{O}}_{k}^{1}\},\hat{\lambda}^{1}\) and set \(n\) to 0;
2.