# Feature Extraction of Bearing Weak Fault Based on Sparse Coding Theory and Adaptive EWT
## 5 Simulation and Experimental Verification
### Experimental Verification  
The experimental data are from the NASA diagnostic database and the rolling bearing life test data provided by the IMS center of the University of Cincinnati [25]. The test bench device is shown in Figure 10, and related parameters are shown in Table 1.  
The sensor used in the test is a PCB353b3 piezoelectric acceleration sensor, and there are 20,480 data points in each run. Three groups of tests were carried out, and the whole life \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Number of Rollers** & **Diameter of Roller** & **Pitch Diameter** & **Contact Angle** & **Sampling Frequency** \\ \hline
16 & 8.407 mm & 71.501 mm & 15.17\({}^{\circ}\) & 20 KZ \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Relevant parameters of the test bearing. Figure 10: Illustration of the test device: (**a**) Testbed of rolling element bearings; and (**b**) Schematic diagram of the bearing test bench. Figure 9: The result of SNR = \(-\)17 dB simulation signal processing based on the LWFSS-AEWT algorithm: (**a**) Time domain waveform of reconstructed signal; and (**b**) Envelope spectrum of the reconstructed signal. data of bearing three in data set 1 was selected as the experimental data. In the experiment, an inner ring-pitting fault occurred in the bearing. Similarly, the theoretical characteristic fault frequency \(f_{i}=296.63\) Hz is calculated according to relevant parameters. The data collected in the horizontal direction of bearing 3, that is, the data collected in channel 5, were selected for analysis. There are 2156 groups of data, and the trend chart of root mean square (RMS) value in bearing 3 life cycle is drawn, as shown in Figure 10(a). RMS is a common monitoring index in engineering and represents the energy of the vibration signal. By observing the change trend of the RMS value, it can be seen that the life cycle of the bearing inner ring consisted of four stages (trouble-free stage, initial failure stage, complete failure stage and catastrophic damage). When the bearing is in the initial fault stage, its amplitude does not change much; but over time, the vibration signal power spectrum will reflect a large increase in the amplitude of some frequency bands, which can be considered as the frequency band of the resonance frequency excited by the impact of the pitting fault. When the bearing is in the stage of complete failure, the impact may excite more high-frequency resonances unrelated to the impact until catastrophic bearing damage occurs. Therefore, for the whole life of bearing fault signals, we study the signal within the scope of the initial failure stage through KSES and CN to select the components, including periodic impact, then determine the value of the parameter \(f\) of the Laplace dictionary related to the resonance frequency generated by the shock. We then select the 2056 th group of data belonging to the initial failure stage to verify the algorithm, as shown in Figure 10(b). The waveform of the 2056 th set of data is shown in Figure 12, and the diagram of the corresponding envelope spectrum is shown in Figure 13. FFT is performed to obtain the spectrum, and then EWT is performed based on the scale space method to obtain the spectrum segmentation diagram, as shown in Figure 14. Eighteen components are obtained by EWT decomposition, whose KSES value and CN value are then calculated, as shown in Figure 15.  
From the KSES and CN values, we can conclude that the 10 th component is the one we need. Looking at the spectrum division (Figure 14), we can see that the component lies in the frequency band of about 5100 Hz. In addition, the resonant frequency band is not divided, so it is not necessary to combine the adjacent components. As can be seen from Figure 16, the kurtosis of the whole frequency band is the highest, and the best frequency band cannot be selected.