# Impulse Signal Detection for Bearing Fault Diagnosis via Residual-Variational Mode Decomposition
## 3 Simulation and Case Study
### Case II  
In the last section, two signals are applied to verify the effectiveness of the proposed method. However, the two signals cannot highlight the whole superiority of RVMD because interference impulses are not included in these two signals. Thus, another signal that includes some large interference impulses is applied to show the superiority of our method.  
The signal comes from the Curtin University [34]. The type of test bearing is MB ER-16K, and a local defect exists in its outer race. This signal is marked as CU-O in this study for convenience. The shaft speed is 1740 rpm, and the BPFO is 103.6 Hz. The sampling frequency is 51.2 kHz, and the signal length applied in this study is 1 s. The TDW and SES of the raw signal are shown in Figure 13. Some large interference impulses (marked by red points) are included in the measured signal, as shown in Figure 13a. The fault feature frequency and its harmonics are difficult to observe in the SES shown in Figure 13b. Then, the proposed method is applied to analyze this signal, and the results are shown in Figure 14. From it, the second sub-mode should be analyzed by envelope analysis technology to show the fault feature. The responding result is shown in Figure 15. The fundamental fault feature frequency and its harmonics are clearly shown in it, indicating that the proposed method has succeeded in extracting the bearing fault feature from the signal polluted by some large impulses.  
This signal is also processed by FK, ICF-VMD, and RVMD with the traditional EK. The corresponding results are shown in Figures 16-18, respectively. From Figure 16a, the ODFB  
Figure 14: Results by RVMD for the signal from the Curtin University: (**a**) the first and (**b**) second sub-modes.  
Figure 13: The signal from Curtin University: (**a**) TDW and (**b**) SES.  
Figure 15: SES of the results by RVMD for the signal from the Curtin University.  
for this signal is in level 3 with the center frequency of 8000 Hz. The corresponding SES is shown in Figure 16b, where only the fundamental fault feature frequency can be observed clearly. Figure 17a shows the TDW of the result by ICF-VMD. From it, there are still some large interference impulses in the filtered signal, which means ICF-VMD fails to suppress the interference impulses. Meanwhile, the SES shown in Figure 17b indicates ICF-VMD fails to extract the periodic impulses. Figure 18 displays the result by RVMD with the traditional definition of kurtosis, where the fault features are seriously interfered with due to the existence of noise. By comparing its two sub-modes, shown in Figure 19, we find the reason is that EK with traditional kurtosis fails to identify the sub-mode which includes the periodic impulses. This means the redefined kurtosis in this study can suppress the interferences from large aperiodic impulses effectively. Consequently, we can say our method can accurately extract bearing fault features and has a strong resistance against interference from aperiodic impulses.  
Figure 16: Results by FK for the signal from the Curtin University: (**a**) Kurtogram and (**b**) SES.  
Figure 17: Results by the ICF-VMD for the signal from the Curtin University: (**a**) TDW and (**b**) SES.  
Figure 18: Result by RVMD with the traditional definition of kurtosis for the signal from the Curtin University.