# An Improved Empirical Wavelet Transform and Its Applications in Rolling Bearing Fault Diagnosis
## 5 Applications
### Analysis of Bearing Outer Ring Fault Data  
This experiment uses the fault bearing test data collected by the laboratory at Xi'an Jiaotong University. The Spectra Quest, Inc test bench is shown in Figure 18. The actual motor frequency is 33.6 Hz, the motor speed is 2000.9 r/min, and the sampling frequency is \(\rm{Fs}=15,000\) Hz. The sensor is connected to the outer ring of the motor end bearing. After calculation, the fault characteristic frequency of the inner ring of the bearing can be obtained as \(\rm{f_{q}}=102.4\) Hz.  
Figure 19a shows the original signal and its spectrum, and Figure 19b is the envelope. There is no obvious periodic shock component in the original signal. In the envelope, there is neither a distinct peak nor a frequency multiplier with the same interval. It is difficult to find useful information to diagnose a fault.  
Figure 19: (**a**) The original signal and its spectrum; (**b**) the envelope of the signal.  
Figure 18: The Spectra Quest, Inc. test bench.  
EWT is used to process signals, and the segmentation results are shown in Figure 20. The blue dotted line is the effect of spectrum segmentation based on the EWT method. As can be seen from the graph, this method divides many boundaries and gets a lot of empirical modes (EMs), which brings a lot of work to the fault analysis. The kurtosis of each EM is extracted and a broken line graph is drawn.  
The four largest components of kurtosis, A, B, C, and D, are extracted and their waveforms and envelopes are shown in Figure 21. In the envelope of A, B, and C with large kurtosis, the characteristic frequency with a prominent peak value and its frequency doubling cannot be found. Only component D has a characteristic frequency but no frequency doubling information.  
FEWT is used to process data, with 45 points of the key function selected to reconstruct the trend component. The threshold \(\lambda=0.0251\). The segmentation results are shown in Figure 22. The red dotted line is the result of spectrum segmentation based on the FEWT method. The number of boundaries is much less than with the EWT method; the calculation efficiency is greatly improved. The blue line is the kurtosis of each EM component. The spectrum of the maximum kurtosis is located at \([3869\) Hz, \(5355\) Hz\(]\).  
Figure 21: The EM7, EM11, EM12, and EM23 and their envelopes.  
Figure 20: The effect of spectrum segmentation based on the EWT method (blue lines); kurtosis of each EM (red line).  
The maximum EM component of kurtosis is taken out, as shown in Figure 23. Large noise can be seen in the waveform of EM8, but the periodic impact is not obvious. The envelope has a characteristic frequency of 102 Hz and 2-7 frequency doubling, and the fault characteristics are obvious. In summary, the bearing outer ring has a fault and the FEWT algorithm can classify fewer EM components than EWT when dealing with the fault of the outer ring of the rolling bearing, thus reducing the diagnosis time. Meanwhile, the maximum kurtosis component of EMs obtained by FEWT contains more characteristic frequencies and their frequency doubling. Therefore, the FEWT method can diagnose outer ring faults of rolling bearings and has obvious advantages over the EWT method.