# Impulse Signal Detection for Bearing Fault Diagnosis via Residual-Variational Mode Decomposition
## 3 Simulation and Case Study
### Case I  
In this section, the proposed method is applied to analyze some signals from the bearing fault experiment. The signals used in this section are obtained from the Society for Machinery Failure Prevention Technology (MFPT) [33]. On the basis of the description in [33], the fault kind of the test bearing includes outer and inner race fault conditions. The ball pass frequency on the inner race (BPFI) is 118 Hz, and the ball pass frequency on the outer race (BPFO) is 81 Hz with a test speed of 25 Hz. The sampling frequency is 48,828 Hz, and the data length applied in this section is 1 s. The information on the raw signal is displayed in Figure 7.  
Then, the proposed method is applied to process this signal, and the results are shown in Figures 8 and 9. The second sub-mode of the results from the RVMD for the inner race fault signal has the larger NEK, as shown in Figure 8. Accordingly, it should be processed by envelope analysis technology. On the basis of the results from the outer race fault signal, the first sub-mode should be analyzed further, and their results are shown in Figure 9. The results shown in Figure 9 are extremely similar to the raw SES shown in Figure 7b. To describe the differences clearly, the ratio of the largest amplitudes between the fault feature frequency and interference frequency (IF) is applied in this study. A larger ratio means a better fault feature extraction result. Based on the ratios shown in the two figures, we can say the inner race fault feature is efficiently enhanced by our method. By comparing the results shown in Figures 7d and 9b, only the primary fault frequency and 2\(\times\) can be found in the raw SES, but the primary fault frequency, 2\(\times\), 3\(\times\), 4\(\times\), 5\(\times\), and 6\(\times\) can be found from the result by the proposed method. Consequently, RVMD can efficiently extract the bearing fault feature. Figure 6: SES of the result shown in Figure 5b. Figure 7: Signals from MFPT: (**a**,**b**) correspond to the TDW and SES of the inner race fault bearing; (**c**,**d**) correspond to the TDW and SES of the outer race fault bearing. These signals are also processed by some other methods, including fast kurtogram (FK), ICF-VMD, and RVMD with the traditional kurtosis, to show the superiority of the proposed method. Figure 10 displays the results by FK. From Figure 10a, the optimal demodulation frequency band (ODFB) for the inner fault race signal is in level 3.5 with a center frequency of 21,362 Hz. The corresponding SES based on it is shown in Figure 10b. From it, we can see the fault is weakened by FK, which is consistent with the results shown by the ratio in the upright. From Figure 10c, the ODFB for the outer race fault is located at level 4.5 with the center frequency of 10,681 Hz. The corresponding SES is shown in Figure 10d. By comparing it with the result shown in Figure 9b, we can see there are more interferences in the result by FK. These results indicate our method can extract more fault features than FK. Figure 11 shows the results of ICF-VMD. By comparing the SES shown in Figure 11b with the result shown in Figure 9a, it is not difficult to see that there are more interferences in Figure 11b. Also, the fault features in the SES for the outer race fault are much weaker than the result by our method because the high-order harmonics (\(3\times\), \(4\times\), \(5\times\) and \(6\times\)) are difficult to observe in Figure 11d. Thus, we can say the performance of ICF-VMD for extracting periodic impulses blew our method. Figure 12 shows the results by RVMD with EK which is calculated by the traditional kurtosis. The fault features shown in Figure 12a are polluted by many other interference components. Figure 12b displays the result of the outer race fault signal. By comparing it with the result shown in Figure 9, it is easy to understand that our method extracts more fault features because the high-order harmonics (\(4\times\) and \(5\times\)) are difficult to be observed in Figure 12b.