# Rotor Fault Diagnosis Method Based on VMD Symmetrical Polar Image and Fuzzy Neural Network
## 4 Experimental Analysis
### Feature Extraction of the Snowflake Image Based on VMD Reconstruction Signal  
Figures 6 and 7 show the time-domain waveform and spectrum of a group of rotor signals randomly selected under different states. From Figures 6 and 7, noise-disturbing components are clearly shown in the signal and the time-domain waveform of the rotor signal in some states have similar patterns, such as BBLF and MF. Therefore, the rotor fault state could not be effectually distinguished by the time-frequency waveform or signal spectrum. For removing noise and extracting the characteristics of the signal accurately, it is very necessary to preprocess the signal. The center frequency of each IMF component of an MF signal obtained by VMD under different \(K\) values is shown in Table 5.  
From Table 5, when \(K=4\), the IMF component center frequency obtains the maximum for the first time. When \(K=5\), the center frequency 686 Hz and 688 Hz appear, indicating that over-decomposition occurs during the VMD decomposition process, and the maximum growth rate \(\Delta\) is defined here to evaluate the fluctuation of the center frequency:  
\[\Delta=\frac{\omega_{k}-\omega_{k-1}}{\omega_{k-1}}\times 100\% \tag{23}\]  
where, when \(K=k\), \(\omega_{k}\) and \(\omega_{k}\)-1 are the \(k\)-th and \((k-1)\)th IMF components of the original rotor vibration signal obtained by VMD. Figure 8 shows the change of maximum growth rate \(\Delta\) under different values of \(K\). From Figure 8 we can know that when \(K=4\), \(\Delta=52.33\%\). With the increase of \(K\), the value of \(\Delta\) does not increase significantly. For example, when \(K=5\), \(\Delta\) is only 0.29%, indicating that the maximum center frequency does not increase obviously. Therefore, it can be considered that the optimal value of \(K\) is 4. For each scale factor \(\tau\), the fuzzy entropy of the signal \(y_{i}(\tau)\) at each scale is calculated first, and then the mean value of the fuzzy entropy is obtained. When \(K=4\) and with different values of \(\alpha\), the mean value of MFE of the reconstructed MF signal is shown in Figure 9, where, \(m=2\), \(r=0.15\)-SD (standard deviation of the signal to be decomposed), scale factor \(s=10\). Figure 8: The change of maximum growth rate \(\Delta\) under different values of \(K\). Figure 7: The spectrum of a group of rotor signals: (**a**) Nor, (**b**) BBLF, (**c**) MF, (**d**) IF, (**e**) LIRIF. From Figure 9, when \(\alpha=2500\), the MFE mean value of the reconstructed signal after the VMD decomposition is the smallest, which indicates that the impact component related to the fault feature in the reconstructed signal contains the most, and also shows strong regularity and self-similarity. So the penalty factor \(\alpha=2500\) is taken to decompose the MF signal. When \(K=4\) and \(\alpha=2500\), the VMD decomposition results of the MF signal are shown in Figure 10.  
It could be seen that the VMD method overcomes the modal mixing problem from Figure 10. The frequency of each IMF component, with the pattern of concentrating near its center frequency, shows that this method has effectively reduced information leakage between the modal components, which proved its effectiveness. Calculate \(P_{k}\) for the determination of the false IMF components. The calculation results are displayed in Figure 11.  
As shown in Figure 11, the sensitive components obtained from the discrimination results of comprehensive evaluation factor Pk of the IMF components are the IMF1 and IMF2. Following the previous analysis, IMF3 and IMF4 are the noise disturbing component. Therefore, IMF1 and IMF2 are selected as sensitive components to reconstruct the MF signal. According to the above methods, the rotor vibration signals in other states in Figure 6 can be reconstructed. Figures 12 and 13 show the time-domain waveform and spectrum of the rotor reconstructed signal.