# An Improved Empirical Wavelet Transform and Its Applications in Rolling Bearing Fault Diagnosis
## 5 Applications
### Analysis of Bearing Inner Ring Fault Data  
In the fast kurtogram proposed by [PERSON] (1996), the spectrum kurtosis is calculated after decomposing the signal using the 1/3-binary tree filter bank. This section draws on the fast kurtogram to obtain a new kurtogram based on FEWT. Section 3.1 introduces the periodic impact when a rolling bearing fails and specifies the number of reconfiguration points as \(\mathrm{B}\in[10,60]\). It is necessary to study how to select the appropriate frequency component in the key function from this interval to reconstruct trend components and extract fault information. The fast kurtogram can accelerate the operation speed and show the position of periodic impulse signals in the frequency domain, and then get an estimation of center frequency and bandwidth. The kurtogram method based on FEWT inherits these advantages. The flowchart is shown in Figure 24.  
Figure 23: The EM8 and its envelope. Figure 22: The effect of spectrum segmentation based on the FEWT method (red lines). Step 1: Since the number of reconstructed points affects the shape of the trend components, the number of the first reconstructed point is set in this section, named ResCot, and the initial value is 5. Step 2: With ResCot as the boundary, the key function and trend component are obtained by FEWT. Finally, a series of signal components are obtained. Step 3: Determine whether the number of reconstructed points exceeds the scope. If ResCot \(\in[5,60]\), then set ResCot \(=\) ResCot \(+\) 5 and repeat step 2. If ResCot \(>\) 60, then stop cycling and calculate the kurtosis of all signal components and draw the kurtosis. Step 4: Extract the components with the maximum kurtosis in the kurtosis; calculate envelope spectrum and extract faults. In this experiment, the bearing fault test rig shown in Figure 25a is adopted to collect the vibration signal of the bearing inner ring fault data with type 6307. The motor speed is 1496 r/min, and the sampling frequency \(\mathrm{Fs}=15,360\) Hz. By calculation, the rotation frequency of the bearing inner ring is 24.9 Hz and the fault characteristic frequency is 122.5 Hz. Figure 25b shows the signal waveform and Figure 25c shows its spectrum. The experimental signal contains a large amount of noise. It is difficult to find the most suitable frequency band that may contain fault information from the spectrum. The signal processed by the FK  
Figure 24: A flowchart of the FEWT method. Figure 25: (**a**) The bearing fault test rig; (**b**) original signal; (**c**) spectrum of the signal. method can quickly find the frequency band with the maximum kurtosis and extract the information, as shown in Figure 26a. The eighth bands from the left side at level 4 with the maximum kurtosis are extracted. The noise is suppressed to a certain extent, and the periodicity of the impact is not obvious. Then the envelope is found, such as in Figure 26(a2). The characteristic frequency can be found from the envelope spectrum, but the high-frequency component cannot be found. The effect of extraction is not the most ideal. Signal processing is done by the method mentioned in this paper. The kurtosis is shown in Figure 26b. When the number of reconstruction points \(\text{ResPoint}\in[20;36]\), although the number of reconstructed components is different, the frequency band of the component with the largest kurtosis is basically fixed. When the number of reconstructed points is 24, the maximum component of kurtosis appears. Its kurtosis is \(\text{K}_{\text{max}}=10.5\); the central frequency is \(\text{f}_{\text{c}}=3628\) Hz; the boundaries are \([3101.125\) Hz; \(4155.125\) Hz); the bandwidth is \(\text{Bw}=1054\) Hz. After extracting this component, it can be found that the component has an obvious periodic shock. There are obvious characteristic frequencies and frequency doubling in the envelope. By comparing the analysis results of FEWT and FK, the results obtained by FEWT are more reasonable. The extracted components have less noise and a more obvious impact.