# Fault-Diagnosis Method for Rotating Machinery Based on SVMD Entropy and Machine Learning
## 2 Signal-Decomposition Methods
### Simulated Signal Analysis  
Create the simulation signal \(x(t)\) as shown in Equations (4) and (5):  
\[\left\{\begin{array}{c}x1(t)=3*\sin(2\pi 35t)\\ x2(t)=2*\sin(2\pi 65t)\\ x3(t)=1/2*\cos(2\pi 105t)\end{array}\right\} \tag{4}\]  
\[x(t)=x1(t)+x2(t)+x3(t) \tag{5}\]The time-domain and frequency-domain plots of the simulated signal are shown in Figure 1.  
The decomposition of the simulated signal by VMD and SVMD is shown in Figures 2 and 3, respectively. The information on the decomposition of the simulated signal by other signal-decomposition methods is shown in Table 1. It can be seen from the figures and tables that VMD had better performance for the simulated signal, SVMD was the second best, and the other methods all showed mode mixing. However, when the value of \(K\) was set to 4, mode aliasing occurred.  
Figure 1: Time-domain–frequency-domain diagram of the simulated signal: (**a**) signal time-domain waveform; (**b**) signal frequency-domain diagram.  
Figure 2: VMD signal-decomposition-component time-domain–frequency-domain diagram: (**a**) time-domain diagram of each component; (**b**) frequency-domain diagram of each component.  
In the simulated signal \(x(t)\), based on the random function, randn(\(\cdot\)) is superimposed on a random signal composition \(X(t)\). The random signal and \(X(t)\) are shown in Figure 4.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Signal-Decomposition Method** & **Running Time (s)** & **Decomposition Fraction (pcs)** & **Problem** \\ \hline EMD & 1.270 & 7 & Mode mixing \\ LMD & 1.857 & 4 & Mode mixing \\ ITD & 1.666 & 5 & Mode mixing \\ VMD & 1.100 & 3 & None \\ SVMD & 1.277 & 4 & One more invalid component \\ \hline \hline \end{tabular}
\end{table}
Table 1: Decomposition of the simulated signal by several signal-decomposition algorithms.  
Figure 4: Time-domain–frequency-domain diagram of the simulated signal with superimposed random signal: (**a**) time-domain diagram of the random signal; (**b**) \(X(t)\) signal time-domain frequency-domain diagram.  
Figure 3: SVMD signal-decomposition-component time-domain–frequency-domain diagram: (**a**) time-domain diagram of each component; (**b**) frequency-domain diagram of each component.  
Figures 5 and 6 show the \(X(t)\) signal-decomposition-component of the time-domain-frequency-domain plots for VMD and SVMD, respectively. The statistical results are shown in Table 2. From the experimental results, it can be seen that VMD was the most efficient way to achieve results, but one of the characteristic frequencies could not be separated effectively. The third characteristic frequencies could be separated when the \(k\) value was set to 4. SVMD took slightly less time and could separate each feature frequency at one time, which had better robustness. The other methods took more time, had serious mode overlap, and could not separate the feature frequencies effectively.  
Figure 5: VMD signal-decomposition-component time-domain–frequency-domain diagram: (**a**) time-domain diagram of each component; (**b**) frequency-domain diagram of each component.  
Figure 6: SVMD signal-decomposition-component time-domain–frequency-domain plots: (**a**) time-domain plots of each component; (**b**) frequency-domain plots of each component.