## 5 Bearing Fault Diagnosis
### Engineering Example  
The test bench is ABLT-1A bearing fatigue testing machine used by bearing enterprises to ensure product quality, as shown in Figure 10. ABLT-1B can test four bearings at a time, and all four test bearings are 6309 type. The rotating speed of the shaft is 3000 r/min. The acceleration sensor directly contacts the bearing outer race, and the sampling frequency is 48 KHz. The main parameters of bearing 6309 are a pitch diameter of 72.5 mm, a ball diameter of 17.462 mm and a ball number of eight. According to the structural parameters, the fault frequencies of the inner and outer races can be calculated as: fi = 248.171 HZ and fo = 151.829 HZ.  
We typically analyze early fault signals during fatigue testing. Due to the vibration superposition effect of multiple bearings and the strong interference generated by a testing machine, it is basically difficult to obtain effective fault information at the early stage of the fault. Then, the proposed method is employed to better the detection result. Figure 11 shows the decomposition results using the parameters with \(k\) = 6, \(\alpha\) = 200. The mode u3 calculated by Equation (18) is selected as the sensitive mode to extract fault features. The de-noising signal by NSD and local envelope spectrum is displayed in Figure 12. It can be seen that the de-noising signal is considerably sparse and has a periodic structure in which the fault cycle period of the inner race and modulation components induced by the  
Figure 10: ABLT-1A test rig.  
Figure 9: Processing results by FSC.  
rotation period is clearly visible on the waveform. It should be pointed out that since the bearing inner race rotates with the shaft, when the inner race fails, the fault frequency fi will be modulated by the rotation frequency fr, and obvious modulation will naturally occur in the envelope spectrum. As shown in Figure 12, this modulation phenomenon is well reflected in the time and frequency domains, thus verifying the reliability and accuracy of the diagnosis results.  
Figure 11: Test signal and decomposition results by FE-guided VMD.  
Figure 12: (**a**) De-noising signal by NSD. (**b**) Local envelope spectrum.  
Figure 13 shows the results of envelope spectrum analysis and MED processing. It is not difficult to see that the traditional envelope spectrum struggles to display the fault frequency doubling and modulation components. Similarly, MED is also difficult to enhance the fault modulation component, and the frequency amplitude at the frequency multiplication is relatively weak, which is not conducive to fault diagnosis compared with the method proposed in this paper.  
To further verify the engineering practicability, subsequently, the proposed method is applied to analyze another phase of the fatigue test. The optimal VMD decomposition results with \(k=5\), \(\alpha=300\) are displayed in Figure 14. The de-noising mode by the NSD method and its local envelope spectrum is shown in Figure 15. We found the clear periodic impulse signals in Figure 15a with regular intervals, which agree with the theoretical value of the outer race fault. As expected, the fault frequency fo and its harmonic components 2 fo,