## 4 Simulation
### Inner Race Fault Simulation  
In this section, cyclic impulse responses produced by an inner race fault are simulated based on the SDOF (Single Degree of Freedom) system, whose resonant frequency is 10 kHz, the relative damping ratio is 0.1, and the ball pass frequency on the inner race (BPFI) is 5.57 multiples of the rotational frequency. In this case, the force caused by the inner race fault is \((1+\sin(2\pi f_{\text{r}}\text{t}))\) N, where \(f_{\text{r}}=60\) Hz denotes the rotational frequency. Sampled at the frequency of 51.2 kHz, the signal is pre-amplified at 80 dB and lasts 10 s. 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 153, the length of filter \(L\) is equal to 100, the order of iteration is equal to 20, and the number of shift \(M\) is equal to 3, according to the method mentioned in Section 3.  
The waveforms and spectrum of the simulated signal are illustrated in Figure 14. It can be seen that the impact responses are covered by noises and only rotational frequency can be found in the spectrum.  
Figure 15a shows the signal obtained by MCKD. It can be seen that MCKD helps clarify the impact of the signal. Figure 15b shows the spectrum segmentation by improved EWT and the decomposed modes are shown in Figure 16.  
Figure 14: The simulation signal of the bearing with the inner race fault: (**a**) waveform; (**b**) spectrum.  
Figure 13: The diagnosis result of the simulation signal of the bearing with the outer race fault processed by the Optimal Wavelet Packet Transform (OWPT): (**a**) the training performance of the OWPT; (**b**) the squared envelope spectrum result.  
The kurtosis values of these modes are, respectively, 2.943, 3.029, 3.512, and 2.925. Figure 17 displays the squared envelope spectrum and teager energy operator spectrum of mode 3. It shows that the frequency 334.2 Hz and its harmonics (i.e., \(f_{i}\), \(2f_{i}\), \(3f_{i}\), \(4f_{i}\), and \(5f_{i}\)) can be recognized.  
Figure 18 illustrates the FK and the squared envelope spectrum of the filtered inner race fault simulation signal. The Kurtogram implies that the center frequency and the bandwidth of the optimal  
Figure 16: The IMFs of the simulation signal of the bearing with the inner race fault processed by the Improved EWT method: (**a**) IMF1; (**b**) IMF2; (**c**) IMF3; (**d**) IMF4.  
Figure 17: The diagnosis result obtained by IMF3 of the simulation signal of the bearing with the inner race fault: (**a**) the squared envelope spectrum of IMF3; (**b**) the teager energy spectrum of IMF3.  
Figure 15: The result of simulation signal of bearing with inner race fault processed by the MCKD-EWT method: (**a**) waveform of signal processed by MCKD; (**b**) Fourier spectrum segmentation results of Improved EWT method.  
filter equal 6400 Hz and 12,800 Hz, respectively. However, the squared envelope spectrum of the filtered signal is still dominated by the rotational frequency and noises and no fault features can be noticed.  
The robust performance of the OWPT model is shown in Figure 19a. It can be seen that the mean fitness of the GA training is 0.2645. The algorithm tends to be stable at the 110 th epoch and the fitness value reaches 0.148 at the 167 th epoch. Figure 19b is the squared envelope spectrum obtained by OWPT. Obviously, it is similar to the outer race fault simulation case and no fault features can be involved in the squared envelope spectrum. It implies that the architecture and parameters selected in the OWPT are not effective for the inner race fault simulation signal with heavy noise.