# An Optimal Resonant Frequency Band Feature Extraction Method Based on Empirical Wavelet Transform
## 5 Application to Roller Bearings Testing
### Case 1: Analysis of CWRU Outer Ring Fault Vibration Signal  
As shown in Figure 9, the detailed bearing parameters are as follows: the bearing model is 6205-2 RSJEMSKF, and three acceleration sensors are installed at the motor drive end at the 3 o'clock, 6 o'clock and 12 o'clock positions of the bearing housing (in this paper, the vibration signal of the bearing outer ring at 6 o'clock is selected for the experimental analysis). The motor speed is 1797 rpm (rotational frequency is 1797/60 Hz = 29.95 Hz), the sampling frequency is 12 kHz, and the experimental data length is 2048.  
Figure 8: Illustration of the normal and degraded bearings.  
Figure 7: Results using spectral kurtosis: (**a**) kurtogram and (**b**) envelope spectrum.  
The outer ring vibration signal with a damage diameter of 0.007 inches (0.01778 cm) is used for experimental analysis. According to the calculation formula in [36], the characteristic frequency of bearing outer ring fault can be obtained:  
\[\text{BPFO}=\frac{1}{2}f_{s}(1-\frac{d_{r}}{D_{m}}\cos\theta)n_{r} \tag{18}\]  
Among them, the diameter of the roller element \(D_{m}=39\) mm, the pitch diameter \(d_{r}=7.938\) mm, the number of roller elements \(n_{r}=1\) and the contact angle \(\theta=0^{\circ}\). Therefore \(\text{BPFO}=107.3648\) Hz. The time domain waveform of the outer ring fault vibration original signal is shown in Figure 10. It can be seen that the noise interference in the signal is weak and the defective pulse characteristic is obvious. The proposed method is applied to the abnormal condition of the outer ring of rolling bearing in CWRU experimental platform to explore the effectiveness of the proposed method in a relatively pure environment.  
The Fourier spectrum of the outer ring fault signal is subjected to parameterless scale-space segmentation, and a total of nine modes are decomposed, as shown in Figure 11. The ASN of all modes, the correlation coefficient with the original signal, and the results of the AASN are given in Table 2. In these modes, the highest value of the AASN appears in mode 7. Among them, the highest value of ASN appears in mode 1, and the highest correlation coefficient appears in the model 7, so the decision result of the AASN is consistent with the decision result of a high correlation.  
Figure 10: Time-domain waveform of the original signal.  
Figure 9: Illustration of the bearing experiment platform.  
To verify the superiority of the proposed method, the same signal is also compared and analyzed through the EMD. The envelope spectrum in Figure 14 shows the most abundant mode of the fault characteristics in the decomposition results. It can be seen that both the kurtosis of the failure frequency and the number of harmonics are weakened due to modal mixing and other phenomena in the EMD. At the same time, in order to verify the rationality of the proposed method in reconstructing the resonant frequency band, the test signal is demodulation and analyzed by the spectral kurtosis method. After the three-layer kurtogram, the frequency band with the highest kurtosis is selected, with a center frequency of 3000 Hz, and a bandwidth of 2000 Hz. As shown in the dotted circle in Figure 15. By comparing this with Figure 10, it is determined that the center position and frequency bandwidth selected by SK is similar to the proposed method. This finding shows that both methods can locate the optimal filter band when the impulse interference in the system is less.  
Figure 14: The Hilbert envelope spectrum analysis results by the EMD method.  
Figure 13: The Hilbert envelope spectrum of the optimal merging mode.  
Figure 15: Results using spectral kurtosis.