## 3 Vibration Signal Preprocessing and Fault Feature Extraction
### Method Verification Experiments
From Figure 3, it can be seen that the white noise introduced many wide frequency band disturbances to \(s(t)\); and from Figure 4, the emulated fault feature frequency 20 Hz and its multi-fold frequencies (40 Hz, 60 Hz,... ) are the main components of envelope spectra. It is easy to find that the noise had obvious effects on the amplitude of these frequency components. For example, in the envelope spectrum of \(s(t)\), the emulated fault feature frequency component (20 Hz) has an amplitude of 0.2051, but after the white noise was imposed, the amplitude of the same feature frequency in \(x(t)\) became 0.1624; the amplitude decreased by 20.1%. Similar effects can also be found in other frequency components tagged in the figures. GWO was applied to search optimal values of \([K,\alpha]\) for VMD to minimize the objective function shown in Equation (6). The parameter configuration of GWO is shown in Table 3. The evolutionary process of objective function value, \(K\), and \(\alpha\) during the iterations are shown in Figure 5. After the iterations, the obtained optimization results were \(K\)=8 and \(\alpha\)=650. With these values, the decomposition results by VMD are shown in Figure 6.  
Figure 5: Evolutionary process of objective function values, \(K\), and \(\alpha\) during the GWO iteration. From Figure 6, it can be seen that the 8 IMFs had differentiable center frequencies, which were 140 Hz, 180 Hz, 220 Hz, 360 Hz, 480 Hz, 640 Hz, 771 Hz, and 938 Hz, respectively; the mode mixing problem is not obvious in the GWO-VMD-based signal decomposition. Correlation coefficients, \(CORR\), of the above 8 IMFs were calculated, and the results are shown in Table 4. IMFs with correlations higher than 0.25 were chosen; thus, IMF1-IMF4 were picked up for signal reconstruction with Equation (11). The reconstructed experimental signal and its spectrum are shown in Figure 7, and the envelop spectrum of the reconstructed signal is shown in Figure 8.  
Figure 6: IMFs and their spectra decomposed by GWO-VMD. From Table 5, it can be found that the Kurtosis of the emulated fault feature signal \(s(t)\) was 7.7708, but the imposed Gaussian white noise made the Kurtosis value of \(x(t)\) decrease to 5.3719, indicating the noise covered the fault and made its feature less sensible. After applying the GWO-VMD-based decomposition and the correlation coefficient-based reconstruction methods, part of the noise was removed and the Kurtosis value increased to 5.4677, higher than that of \(x(t)\). With the basic VMD having no parameter optimization, the Kurtosis value of the reconstructed \(x(t)\) was 4.7667, lower than 5.4677 and even lower than the value of \(x(t)\). Figure 3:0 Envelope spectrum of reconstructed experimental signals through VMD. Figure 3:1 Reconstructed experimental signal and its spectrum through VMD. For signal-noise ratio comparison, the SNR of \(x(t)\) was 5.0021; because the GWO-VMD-based method eliminated part of the noise from \(x(t)\), the SNR of the reconstructed signal increased to 8.7756, while the basic VMD-based method only increased the SNR value of the reconstructed signal to 8.1462, smaller than 8.7756. For RMSE comparison, smaller root mean squared error value indicates less diffused signals. The RMSE value of \(x(t)\) with noise was 0.1019; after noise elimination by GWO-VMD, the RMSE decreased to 0.0660; the VMD-based method only decreased the value to 0.0710, still higher than 0.0660. For Permutation Entropy (PE), a smaller value means more regular signals. The PE value of the emulated periodic fault impulsive signal \(s(t)\) was 0.4545, and due to the noise, the PE of \(x(t)\) increased to 0.9320. After the noise elimination by GWO-VMD and signal reconstruction, the PE was reduced to 0.6145, lower than that by VMD-based method. From the above comparison of the four signal characteristic parameters, it can be seen that GWO-VMD-based signal decomposition and correlation-based signal reconstruction can more effectively eliminate noise from the vibration signal than VMD does.