## 5 Experiment Validation
### Case 2: Bearing Rolling Element Fault  
In order to further research the advantage and effectiveness of the proposed method in bearing fault diagnosis, another artificial fault test is operated. As seen in Figure 11a, a groove shape is made on the rolling element of bearing by electrical discharge machining. The defect is in the form of 0.2 mm width and 0.1 mm depth. When the input shaft is rotating at the speed of 900 rpm, the defective roller passing frequency over the bearing race is calculated to be 62.7 Hz based on Equation (28). The waveform of the raw roller fault signal in time domain and its envelope spectrum are separately plotted in Figure 19. There is no visible impact in the waveform and the fault signature is submerged in background noise. It can merely figure out the mesh frequency 255 Hz from the envelope spectrum. In the following, the proposed method is also employed to the roller defective vibration signal. Then the entropy value sequences of the _MEVM_ in different scales are illustrated in Figure 20 and the waveforms present visibly similar impulses. Figure 21 is the comparison result of the fault bearing signal by different entropy methods. As seen in Figure 21a, the feature information of the traditional vector entropy sequence without sparse reconstruction is buried by heavy noise. The waveform of the proposed MEVS of 1-scale is displayed in Figure 21b, where some clear periodic impulses can be observed. Besides, the _MFEVS_ in Figure 21c shows a better ability to extract impulse features than Figure 21a,b through visual inspection. Then the optimal short sliding time series with the minimum MDErms is plotted in Figure 22a. According to the envelope spectrum in Figure 22b, the corresponding resonance frequency can be found (3400 Hz) and employed to the designing of FIR bandpass filter. As plotted in Figure 23a, the defective impulses of the filtered signal by the proposed method appear obvious periodically. The fault characteristic frequency and its several harmonics can be observed clearly from the envelope spectrum in Figure 23b, which indicates that the bearing rolling element is broken. Moreover, the corresponding envelope waveform of the filtered signal is displayed in Figure 24, where \(T_{bp}\) is the interval period of the roller fault. The waveforms of the MFEVS in Figure 21c and the filtered signal in Figure 24 show visibly similar periodicity, indicating that the proposed MFEVS has good abilities in fault identification and feature extraction. To further extract and analyze the fault information, three different methods are applied to the roller fault signal respectively and the comparison result is shown in Figure 23. As shown in Figure 23c-h, the other three methods all appear to have poor performance and have their own limitations. The waveform of the EEMD method in Figure 23c is full of noise. And several harmonics of the fault characteristic frequency can be roughly identified from the corresponding envelope spectrum. As seen in Figure 23e,f, the WP method just removes some noise segment, and the fault feature is still hard to recognize. The fast kurtogram of roller fault signal in Figure 25 shows that the selected center frequency and bandwidth are 2400 Hz and 64 Hz. Then the signal filtered by the kurtogram and its envelope spectrum are plotted in Figure 23g,h, respectively. It can be concluded that the fast kurtogram method shows a normal performance in eliminating background noise and fault identification. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & **Raw Signal** & **Proposed Method** & **EEMD** & **WP** & **Fast Kurtogram** \\ \hline
**SNR (\(f_{op}\))/dB** & \(-24.1\) & \(-15.9\) & \(-21.3\) & \(-22.5\) & \(-17.4\) \\
**Kurtosis** & 3.54 & 14.1 & 4.7 & 4.1 & 4.8 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: The comparison results of the outer-race fault signal by different methods. Figure 19: The bearing signal with roller defect: (**a**) the waveform in the time domain; (**b**) the envelope spectrum.