# Fault-Diagnosis Method for Rotating Machinery Based on SVMD Entropy and Machine Learning
## 3 Fault Feature-Extraction Method  
### Energy Entropy  
The operation of rolling bearings with different faults produces different resonant-frequency components in the vibration signal. The energy of the fault signal changes in different frequency bands [30]. The IMF components are obtained by decomposing the signal through the adaptive-decomposition method. The energy distribution of each IMF component is different. Therefore, the fault characteristics of the equipment can be characterized by finding the energy entropy of each IMF component.  
For the \(n\) IMF components decomposed from the vibration signal, \(c_{1}(t),c_{2}(t),\cdots,c_{n}(t)\), and the energy of each component is calculated using Equation (6).  
\[E_{i}=\int_{-\infty}^{+\infty}|c_{i}(t)|^{2}\mathrm{d}t \tag{6}\]  
Define the energy entropy expressed as:  
\[H_{EN}=-\sum_{i=1}^{n}p_{i}\mathrm{log}_{2}\,p_{i} \tag{7}\]  
where \(p_{i}=E_{i}/E\), \(E=\sum_{i=1}^{n}E_{i}\), \(x(t)\) is the simulated signal of Equation (5), and \(X(t)\) is a superimposed random signal by the function randn(\(\cdot\)).  
Decomposition of VMD and SVMD is performed to obtain the component-energy entropy, as shown in Figure 7. Figure 7a illustrates that the component-energy entropy of VMD decomposition is basically consistent with that of SVMD decomposition. In addition, it can be seen that the correlations of the IMF1, IMF2, and IMF3 components of VMD decomposition and the correlations of the IMF2, IMF3, and IMF4 components of SVMD decomposition also remained the same. Similarly, the energy entropies of the first two components of \(X(t)\) decomposed by VMD and SVMD from Figure 7b were basically the same, but the IMF3 component of the VMD decomposition was more complex and the eigenfrequencies were masked, at which point its energy-entropy value was half that of the IMF3 component of the SVMD decomposition.  
Figures 8 and 9 show the correlation analysis of the simulation signal and components. From Figure 8, it can be seen that the correlation between the components of the VMD decomposition and the simulated signal \(x(t)\) was the same size as the correlation between the components of the SVMD decomposition (except IMF1) and the simulated signal \(x(t)\), which also indirectly indicates that there was not much difference between the two methods. Similarly, it can be seen from Figure 9 that the correlation between the VMD decomposition components and the simulation signal \(X(t)\) remained basically the same as the correlation between the SVMD decomposition components (the first three components), but the IMF3 characteristic frequency of VMD was hidden by other information and it is not easy to determine the effective components if only correlation analysis is performed.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Signal-Decomposition Method** & **Running Time (s)** & **Decomposition Fraction (pcs)** & **Problem** \\ \hline EMD & 1.692 & 9 & Mode mixing \\ LMD & 2.077 & 5 & All three eigenfrequencies are in the same component \\ ITD & 1.940 & 7 & Mode mixing \\ VMD & 1.558 & 3 & The third characteristic frequency is not effectively separated \\ SVMD & 1.674 & 5 & Invalid fraction exists \\ \hline \hline \end{tabular}
\end{table}
Table 2: Decomposition of simulated signals with superimposed random signals by several signal-decomposition algorithms.  
Figure 8: Correlation-confusion matrix of \(x(t)\): (**a**) VMD decomposition correlation of \(x(t)\); (**b**) SVMD decomposition correlation of \(x(t)\).  
Figure 7: Simulated signal-component energy entropy: (**a**) component energy entropy of \(x(t)\); (**b**) component energy entropy of \(X(t)\).  
Figure 9: Correlation-confusion matrix of \(X(t)\): (**a**) VMD decomposition correlation of \(X(t)\); (**b**) SVMD decomposition correlation of \(X(t)\).