# Weak Multiple Fault Detection Based on Weighted
## 5 Experimental Verification
### Application to a Synthesized Signal Mixture  
Rolling bearing data were taken from the Case West Reserve University (CWRU) [37], wherein inner and outer races each had a single point fault generated by electro-discharge machining, and the experimental platform is shown in Figure 12. The fault sizes of the inner race and outer race were 0.007 inch (width) \(\times\) 0.011 inch (depth). The motor speed was 1797 rpm (29.95 Hz), the sampling frequency was 12,000 Hz, and the number of sampling points was 12,000. The type of rolling bearing was SKF62055, and its structural parameters are shown in Table 2. According to the structural parameters of the rolling bearing, the characteristic frequencies of rolling bearing can be calculated; characteristic frequencies of the outer and inner races are about 162.2 Hz and 107.4 Hz respectively. The mixed signal was synthesized by the superposition matrix for the inner and outer race fault data of Case West Reserve University. In order to further weaken the fault characteristics, the inner and outer faults were superimposed with \(-\)5 dB noise. Figure 13 shows the time-domain waveform and frequency spectra of the inner race fault, outer race fault, and mixed signal. In the envelope spectrum of the mixed signal Figure 13f, the characteristic frequency of inner race 107.4 Hz can be extracted. However, the characteristic frequency of outer race and its harmonic components cannot be observed.  
Figure 14 shows the denoised results based on the proposed WOGS algorithm with a group size \(6\times 3\). In the envelope spectrum, Figure 14b, the characteristic frequencies of inner and outer race faults are clearly extracted. Thus, the fault features of the two defects are clearly revealed by the proposed algorithm. We adopt TQWT, OGS, and the soft threshold algorithm for the same synthesized signal for improved verification and comparison. The denoised results are shown in Figures 15-17, where the characteristic frequencies and its harmonics are not obvious. Similarly, \(R_{f}\) and \(T_{f}\) are used for comparative analysis of four algorithms, and the results are shown in Table 3. It can be seen from the table that the energy ratio of WOGS algorithm is the largest, and the CPU running time of soft threshold noise reduction is the shortest.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Type** & **Diameter of Balls** & **Pitch Diameter** & **Pitch Diameter** & **Contact Angle** \\ \hline SKF62055 & 7.94 (mm) & 39.04 (mm) & 9 (mm) & 0\({}^{\circ}\) (\({}^{\circ}\)) \\ \hline \hline \end{tabular}
\end{table}
Table 2: Structural parameters of rolling bearing.  
Figure 12: Experimental setup.  
Figure 13: _Cont._  
Figure 14: Results of the proposed weighted WOGS algorithm: (**a**) Time-domain waveform. (**b**) Envelope spectrum.  
Figure 13: Time-domain waveform and frequency spectrum of the single fault signals and the mixed signal: (**a**) Time-domain waveform of inner race fault signal. (**b**) Envelope spectrum of inner race fault signal. (**c**) Time-domain waveform of outer race fault signal. (**d**) Envelope spectrum of outer race fault signal. (**e**) Time-domain waveform of the mixed signal. (**f**) Envelope spectrum of the mixed signal.  
Figure 16: Results of soft threshold: (**a**) Time-domain waveform. (**b**) Envelope spectrum.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Algorithms** & \(R_{f}\) & \(T_{f}\) \\ \hline TQWT & 0.0143 & 2.25 (s) \\ OGS & 0.0194 & 1.43 (s) \\ Soft threshold & 0.0143 & 0.51 (s) \\ WOGS & 0.0553 & 1.48 (s) \\ \hline \hline \end{tabular}
\end{table}
Table 3: Comparison of \(R_{f}\) and \(T_{f}\) of four algorithms.  
Figure 15: Results of TQWT: (**a**) High oscillatory component. (**b**) Low oscillatory component.