## 4 Simulation
### Outer Race Fault Simulation  
In the simulation, the rotational frequency of the bearing system is 60 Hz, and the ratio of the ball pass frequency on the outer race (BPFO) to the rotational frequency is 3.57. The force caused by the fault is \((1+\sin(2\pi f_{\mathrm{t}}\mathrm{t}))\)\(N\). The resonant frequency of the system is 10 kHz and the relative damping ratio \(\zeta\) is 0.1. Sampled at a frequency of 51.2 kHz and pre-amplified at 80 dB, the vibration signal with the length of 512,000 samplings is collected. Moreover, 0.5% degree of random fluctuation is introduced and \(-\)10 dB noise is added to make the simulation closer to the physical condition. The parameters of MCKD are set as follows: the period of deconvolution \(T\) is equal to 239, the length of filter \(L\) is equal to 90, the order of iteration is equal to 30, and the number of shift \(M\) is equal to 3, according to the method mentioned in Section 3.  
The waveform and spectrum are shown in Figure 8. It can be seen that the impact responses are covered by noises and only rotational frequency can be found in the spectrum.  
Figure 9a shows the signal obtained by MCKD. It can be seen that MCKD helps clarify the impacts of the signal. In Figure 9b, the Fourier spectrum of the pre-processed signal is segmented by EWT and the result can be seen in Figure 10.  
The kurtosis values of these modes are, respectively, 3.0495, 3.216, 3.004, and 3.493. Figure 11 displays the squared envelope spectrum and teager energy operator spectrum of the selected signal mode 4. More than five dominant frequency spectrum lines located at the harmonics of the characteristic frequency (i.e., \(f_{o}\), \(2f_{o}\), \(3f_{o}\), \(4f_{o}\), and \(5f_{o}\)) can be recognized in Figure 11.  
To validate the efficacy of the proposed method, both fast Kurtogram and OWPT are conducted on the simulated signal. What's more, both the squared envelope spectrum and teager energy spectrum are used to show the detailed fault characteristic frequency. Figure 12 illustrated the fast Kurtogram (FK) and the squared envelope spectrum of the filtered signal. The FK result implies that the center frequency and the bandwidth of the optimal filter equal 19,200 Hz and 12,800 Hz, respectively. Compared to the original signal, the filtered signal cannot yet clearly characterize the fault feature of the simulated signal and no fault features can be noticed in Figure 12b.  
Figure 8: The simulation signal of bearing with outer race fault: (**a**) waveform; (**b**) spectrum.  
Figure 9: The result of the simulation signal of the bearing with outer race fault processed by the MCKD-EWT method: (**a**) waveform of signal processed by MCKD; (**b**) Fourier spectrum segmentation results of Improved EWT method.  
Figure 11: The diagnosis result obtained by IMF4 of the simulation signal of the bearings with the outer race fault: **(a)** the squared envelope spectrum of IMF4; **(b)** the teager energy spectrum of IMF4.  
Figure 12: The diagnosis result of the simulation signal of the bearing with the outer race fault processed by the Fast Kurtogram (FK) method: **(a)** the kurtogram result; **(b)** the squared envelope spectrum result.  
Figure 10: The IMFs of the simulation signal of bearing with outer race fault processed by the Improved EWT method: **(a)** IMF1; **(b)** IMF2; **(c)** IMF3; **(d)** IMF4.  
The robust performance of the OWPT model is shown in Figure 13a. It can be seen that the mean fitness of the Genetic Algorithm (GA) training is 0.2835. The algorithm tends to be stable at 175 th epoch and the fitness value reaches 0.178 at the 200 th epoch. Figure 13b is the squared envelope spectrum obtained by OWPT. It implies that the architecture and parameters selected in the OWPT failed in the outer race fault feature extraction of the simulated signal.