## 5 Experiment
### Case 4: Inner Race Fault With Circumferential Width of 3 mm  
The parameters of MCKD are set as follows: the period of deconvolution \(T\) is equal to 131, the length of filter \(L\) is equal to 100, the order of iteration is equal to 30, and the number of shift \(M\) is equal to 3.  
Figure 36 illustrates the original waveforms, spectrum, waveforms obtained by the MCKD, and the spectrum segmentation of the EWT. From Figure 36a,c, it can be seen that noise seriously disturbs the inner race fault diagnosis, and only rotational frequency and its harmonics can be found in the spectrum. Figure 36b shows the signal preprocessed by MCKD. It can be seen that MCKD helps clarify the impact of the original signal. The Fourier spectrum of the preprocessed signal is segmented by EWT in Figure 36d and the result can be seen in Figure 37.  
Figure 36: The collected signal of the bearing with 3 mm inner race fault and the result of the collected signal processed by the MCKD-EWT method: (**a**) the original signal; (**b**) the de-noised signal processed by MCKD method; (**c**) the spectrum of the original signal; (**d**) the Fourier spectrum segmentation result of the de-noised signal obtained by Improved EWT method.  
Figure 37: The IMFs of the collected signal of bearing with 3 mm inner race fault processed by the Improved EWT method: (**a**) IMF1; (**b**) IMF2; (**c**) IMF3; (**d**) IMF4.  
The kurtosis values of these modes are, respectively, 3.398, 3.323, 4.172, 3.658, and 3.725. Figure 38 displays the squared envelope spectrum and teager energy operator spectrum of the selected signal mode 3. In the squared envelope analysis and teager energy operator spectrum, it can be seen that both methods successfully extract the inner race fault feature. The \(2f_{i}\) and \((2f_{i}\pm f_{r})\) are obvious.  
The FK and corresponding squared envelope spectrum are presented in Figure 39. The recommended center frequency and bandwidth of the optimal filter are 16,000 Hz and 2133 Hz, respectively. Namely, the recommended frequency band of FK is \(\left[\begin{array}{cc}14.93&17.06\\ \end{array}\right]\) kHz. Both \(f_{i}\) and its harmonics, \(2f_{i}\), can be found in the squared envelope spectrum.  
The robust performance of the OWPT model is shown in Figure 40a. It can be seen that the mean fitness of the GA training is 0.0226. The algorithm tends to be stable at the 3 rd epoch and the fitness value reaches 0.0109 at the 51 th epoch. Figure 30b is the squared envelope spectrum obtained by OWPT. We can find that several discrete spectral lines protrude at the frequencies of \(f_{i}\), \(f_{i}\pm f_{r}\), and \(2f_{i}\).  
Figure 40: The diagnosis result of the collected signal of the bearing with 3 mm inner race fault processed by Optimal Wavelet Packet Transform (OWPT): (**a**) the training performance of the OWPT; (**b**) the squared envelope spectrum result.  
Figure 38: The diagnosis result obtained by IMF3 of the collected signal of the bearing with 3 mm inner race fault: (**a**) the squared envelope spectrum of IMF3; (**b**) the teager energy spectrum of IMF3.  
Figure 39: The diagnosis result of the collected signal of the bearing with 3 mm inner race fault processed by the Fast Kurtogram (FK) method: (**a**) the kurtogram result; (**b**) the squared envelope spectrum result.  
Considering both outer and inner race fault cases, it is reasonable to conclude that the proposed method is more effective than the FK and OWPT methods in the weak fault feature extraction and enhancement of the rolling element bearing in strong noise conditions.