## 5 Bearing Fault Diagnosis  
### Case Study  
The test signal used in this case comes from the rolling bearing fault test bench (Figure 5) in the intelligent maintenance system (IMS) center [33]. The test bearings were Rexnord za-2115 double-row bearings, and the shaft speed was fixed at 2000 rpm. The radial load applied to the bearing was 26.7 kN. The sampling frequency was 20 KHZ. At the end of the experiment, local defects were found on the outer race of bearing 1, and the characteristic fault frequency of the outer race is fo = 236 Hz.  
According to the research results on health monitoring and degradation assessment for the IMS bearing dataset in the literature [34; 35], we give the measure values under four different fault conditions in Table 1. These four fault conditions occur at the time points around 5310 min, 6000 min, 6500 min and 7000 min, respectively, representing four increasing fault degrees. It is not difficult to see that FE increases in turn with the increase of fault degree, showing good monotonicity, while PE has poor monotonicity. This clearly proves that FE is very effective in measuring fault information.  
For early fault indication, Figure 6 provides the decomposition result of the original signal using the VMD method for optimal parameters with \(k\) = 3, \(\alpha\) = 800. As can be seen, the proposed VMD method successfully extracts the mode components, and the high-frequency mode component (u3) reveals a certain periodic impulse. Applying the NSD-based filtering, the result shown in Figure 7 clearly lets us notice the presence of the fault period (T = 1/236 s) and the significant noise reduction. In contrast, for this kind of sparse smooth signal, the Donoho threshold is very unsatisfactory, as shown in Figure 7b. This is because the Donoho threshold [36; 37; 38] is often too large in engineering applications and lacks the morphology and statistical rules of bearing fault signals.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Condition** & **1** & **2** & **3** & **4** \\ \hline FE & 1.9049 & 1.8867 & 1.7810 & 1.6414 \\ PE & 6.2004 & 6.1112 & 6.1134 & 5.8567 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Measures under different fault conditions.  
Figure 5: Rolling bearing experimental setup at IMS.  
Figure 6: The case signal and decomposition results by FE guided-VMD.  
Figure 7: (**a**) De-noising signal by NSD (**b**) Envelope spectrum of the de-noising signal (**c**) De-noising signal by Donoho threshold.  
In addition, we make a comparative study to specially evaluate the advantages of the optimal VMD method. In the conventional VMD method, the determination of mode number and balance parameter is not considered to maximize the fault information. In this case, the mode number is specified as \(k\) = 5, and the balance parameter adopts the arbitrary value \(\alpha\) = 800. Figure 8 shows the corresponding envelope spectrum of traditional VMD decomposition. The first-order fault frequency can be identified through the spectrum, but the amplitude of the fault frequency is smaller than that in Figure 7, and the higher harmonic frequency cannot be clearly observed. Figure 9 shows the analysis results using the fast spectral correlation (FSC) [39]. Although the first-order fault frequency can be observed through the envelope spectrum of the filtered signal, a large number of interference components around the cyclic frequency band make the peak value of the frequency spectrum and the higher harmonics frequency very fuzzy. Therefore, the proposed method exhibit promising superiority for fault information extraction and feature enhancement.  
Figure 8: Envelope spectrum of arbitrary parameters guided-VMD.