## 2 FE Guided-VMD  
The VMD algorithm first assumes that each mode component \(u_{k}\) has a different center frequency and limited bandwidth and then uses the alternating direction multiplier method to continuously update each mode and its center frequency to minimize the sum of the estimated bandwidth of each component. Finally, each mode is demodulated to the corresponding fundamental frequency band step by step so that the signal can be effectively separated from low frequency to high frequency. The construction process of the corresponding variational problem is as follows [10]:  
\[\min\Biggl{\{}\sum_{k}\biggl{\|}\partial_{t}\biggl{[}\biggl{(}\delta(t)+\frac{ j}{\pi t}\biggr{)}*u_{k}(t)\biggr{]}e^{-j\omega_{k}t}\biggr{\|}_{2}^{2} \Biggr{\}}\quad s.t.\sum_{k}u_{k}=f \tag{1}\]  
Reconstruction constraints can be resolved in different ways. It is suggested to introduce quadratic penalty function term and Lagrange operator for constraint. The augmented Lagrange function is described as follows:  
\[L(u_{k},\omega_{k},\lambda)=\alpha\frac{\sum_{k}\biggl{\|}\partial_{t}\biggl{[} \biggl{(}\delta(t)+\frac{j}{\pi t}\biggr{)}*u_{k}(t)\biggr{]}e^{-j\omega_{k}t} \biggr{\|}_{2}^{2}+\biggl{\|}f-\sum_{k}u_{k}\biggr{\|}_{2}^{2}+\biggl{\langle} \lambda,f-\sum_{k}u_{k}\biggr{\rangle} \tag{2}\]  
The center frequency \(\omega_{k}\) does not appear in the reconstruction term but only in the previous bandwidth. The relevant problem thus writes in the Fourier domain:  
\[\omega_{k}^{n+1}=\underset{\omega_{k}}{\operatorname{argmin}}\biggl{\{}\int_ {0}^{\infty}\left(\omega-\omega_{k}\right)^{2}\lvert\hat{u}_{k}(\omega)\rvert ^{2}d\omega\biggr{\}} \tag{3}\]  
Performing numerical calculations, the decomposition mode number \(k\), balance parameter \(\alpha\) and Lagrange multiplier \(\hat{\lambda}\) are specified in advance, and other original conditions are initialized to zero. Subsequently, the mode \(\hat{u}_{k}^{n+1}\) and center frequency \(\hat{\omega}_{k}^{n+1}\) are updated iteratively by Equations (4) and (5), respectively.  
\[\hat{u}_{k}^{n+1}\leftarrow\frac{\hat{f}-\sum_{i<k}\hat{u}_{i}^{n+1}-\sum_{i> k}\hat{u}_{i}^{n}+\frac{\hat{\lambda}^{n}}{2}}{1+2\alpha\left(\omega-\omega_{k}^{n} \right)^{2}} \tag{4}\]  
\[\omega_{k}^{n+1}\leftarrow\frac{\int_{0}^{\infty}\omega\Bigl{\lvert}\hat{u}_{ k}^{n+1}(\omega)\Bigr{\rvert}^{2}d\omega}{\int_{0}^{\infty}\Bigl{\lvert}\hat{u}_{ k}^{n+1}(\omega)\Bigr{\rvert}^{2}d\omega} \tag{5}\]  
Dual ascent for all \(\omega\geq 0\):  
\[\hat{\lambda}^{n+1}\leftarrow\hat{\lambda}^{n}+\tau\left(\hat{f}-\hat{u}_{k}^{ n+1}\right) \tag{6}\]  
Until convergence:  
\[\sum_{k}\Bigl{\lVert}\hat{u}_{k}^{n+1}-\hat{u}_{k}^{n}\Bigr{\rVert}_{2}^{2}/ \lVert\hat{u}_{k}^{n}\rVert_{2}^{2}<\varepsilon \tag{7}\]  
It can be seen from the above description that VMD needs to specify two important parameters in advance: mode number \(k\) and balance parameter \(\alpha\). In fact, these two parameters have a great influence on vibration signal decomposition. For rolling bearings, this requires that the parameters can well match the transient and fluctuating attributes of the impact signal synchronously so as to obtain the mode component containing the most fault information.