## 5 Application of the Proposed Method
### Experimental Verification  
Figures 11 and 12 shows the test rig to collect the vibration data of the bearing. The test rig consisted of a motor driving the shaft through a flexible coupling, HG3528A data collection instrument and laptop. The vibration acceleration of the damaged bearing in the vertical planes were measured using an accelerometer, the bearing model was SKF6307. The output from the accelerometer was sampled at 15360 Hz. The motor was run at 1496 rpm, and the collected data was uploaded to the laptop for data processing and analysis. Figure 11: Schematic diagram of the experimental bench system: 1 Three-phase asynchronous motor; 2 flexible shaft coupling; 3 the bearing with compound fault; 4 bearing rotor; 5 the normal bearing. Figure 12: The experimental bench system. Figure 10: The analyzed results of case 1 with the proposed method: (**a**) T\({}_{o}\) = 248, low resonance component; (**b**) T\({}_{o}\) = 248, Hilbert envelope spectrum; (**c**) T\({}_{i}\) = 51, low resonance component; (**d**) T\({}_{i}\) = 51, Hilbert envelope spectrum. The bearing with compound fault (Figure 13) was machined as a pitting defect on the outer race and inner race of the bearing with the size of 1 mm in diameter and 0.2 mm in depth. The fault characteristic frequencies of the outer race and inner race were \(f_{o}\) = 76.7 Hz and \(f_{i}\) = 122.7 Hz respectively. The experimental data of the compound fault was selected for analysis. The time-domain signal and the frequency spectrum are shown in Figure 14. The impact components in the graph were not obvious, so the fault features needed to be further extracted. The sparse decomposition of the parallel dual-Q-factor bases was applied to extract and sparsely represent the compound fault components. The decomposition parameters are shown in Table 5, and the analysis results are shown in Figure 15. Figure 15a,b are the low resonance components which were obtained by using the sparse decomposition method and the Hilbert envelope spectra of decomposed components respectively. It can be seen from Figure 15b that peaks were obvious at \(f_{o}\) (76.88 Hz), \(f_{i}\) (121.9 Hz) and their harmonics. Hence, the results indicate that the proposed method can identify and extract features of the compound impact components of the inner race and outer race simultaneously. However, the two fault impact signals are overlapped with each other, and the two fault characteristic frequencies and their harmonics are interlaced with each other. Therefore, it is difficult to accurately extract the two-fault characteristic information from the envelope spectrum. Figure 14: Compound fault vibration signal and its frequency spectrum. Figure 13: The bearing with compound fault. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Attribute** & \(\mathbf{Q_{1}}\), \(\mathbf{Q_{2}}\) & **Redundancy (\(\mathbf{r_{1}}\), \(\mathbf{r_{2}}\))** & **Number of Layers (\(\mathbf{J_{1}}\), \(\mathbf{J_{2}}\))** \\ \hline Low resonance component & 1 & 4 & 25 \\ High resonance component & 6 & 6 & 11 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: The sparse decomposition parameters of case 2. To further separate the inner and outer race fault features from the composite fault signals, the improved deconvolution techniques were applied to extract the periodic features from the low resonance component. The deconvolution period \(T_{*}\) (* is \(i\), \(o\)) of the signal is shown in the Table 6.  
Then, we selected the optimal parameter (\(M\), \(L\)) as mentioned in Section 3.2.2  
(1) When T\({}_{o}\)=200, Figure 16a,b show the relationship between \(M\) and MTEK, and the relationship between the filter length \(L\) and the TEK by iterative calculation respectively. It was found that when \(M=1\), the MTEK value was the largest from Figure 16a, and when \(L=260\), the TEK value was the largest from Figure 16b, so the optimal parameters \(T_{o}\), \(M_{o}\), and \(L_{o}\) were determined [200, 1, 260].