# Sparse Envelope Spectra for Feature Extraction of Bearing Faults Based on NMF
## 3 Results
### Experimental Verification
#### 3.2.2 Case 2  
Another experiment is carried out on the test bench of mechanical equipment failure simulation, and its structure is shown in Figure 17. The rotating shaft is driven by an alternating current (AC) motor, and fault simulation can be conducted by replacing the fault motors. The motor bearing is 6203. The inner ring peeling fault is simulated on the inner ring of the motor drive end bearing. When the rotation frequency \(f_{r}\) is 20 Hz, the fault characteristic frequency \(f_{i}\) is 99.5 Hz. The accelerometer is installed at the vertical direction of the motor end bearing. The vibration signal is acquired with the sampling frequency of 20,480 Hz and the sample size of 4096. Figure 18 shows the waveform and frequency spectrum of the acquired signal. In Figure 18a, weak impulse features are buried in interference and noise, while it is shown that from the spectrum shown in Figure 18b, there are multiple resonance frequency bands, and the rotating characteristic frequencies are submerged.  
frequency band from 1400-3800 Hz. In addition to the frequency basis \(\mathbf{w_{3}}\), the other filtering bands cannot extract the corresponding fault feature frequency.  
Furthermore, the band-pass filtering for the five bands and the fast spectral kurtosis method are also analyzed. The results are shown in Figures 21 and 22, respectively. In the ranges of the five decomposed frequency bands, the fault component is not displayed in \(\mathbf{w_{1}}\), \(\mathbf{w_{3}}\), \(\mathbf{w_{4}}\), and \(\mathbf{w_{5}}\) due to the noise component, and the feature frequency in \(\mathbf{w_{2}}\) is also inconspicuous. From the result extracted by the fast spectral kurtosis method, it is difficult to detect the existence of fault characteristic frequencies because of its large bandwidth and interference components. Compared with Figure 20d, the advantage of the proposed sparse envelope spectrum is shown again. Besides the characteristic frequency, the spectrum of band-pass filtering also contains a large number of other components, which are easy to interfere with the judgment of fault. Additionally, the inaccurate frequency band extracts the small amplitude of the characteristic frequency. Obviously, the sparse envelope spectrum has an excellent visualization effect, so the product of time weight and filtering envelopes also can reduce the requirement of parameter selection for band-pass filtering.  
Figure 19: Spectral basis \(\mathbf{W}\) decomposed by NMF.  
Figure 18: Outer ring fault bearing: (**a**) Waveform; (**b**) Frequency spectrum of the fault signal.  
Figure 20: Obtained sparse spectrum with decomposed sub-matrix: (**a**) Basis \(\mathbf{w}_{1}\) and weight vector \(\mathbf{h}_{1}\); (**b**) Basis \(\mathbf{w}_{2}\) and weight vector \(\mathbf{h}_{2}\); (**c**) Basis \(\mathbf{w}_{3}\) and weight vector \(\mathbf{h}_{3}\); (**d**) Basis \(\mathbf{w}_{4}\) and weight vector \(\mathbf{h}_{4}\); (**e**) Basis \(\mathbf{w}_{5}\) and weight vector \(\mathbf{h}_{5}\).  
Figure 21: Envelope spectrum with band-pass filtering: (**a**) Filtered with \(\mathbf{w}_{1}\); (**b**) Filtered with \(\mathbf{w}_{2}\); (**c**) Filter with \(\mathbf{w}_{3}\); (**d**) Filtered with \(\mathbf{w}_{4}\); (**e**) Filtered with \(\mathbf{w}_{5}\).