Fault Feature Extraction and Enhancement of Rolling Element Bearings Based on Maximum Correlated Kurtosis Deconvolution and Improved Empirical Wavelet Transform
Abstract
:1. Introduction
2. The Basic Theory
2.1. MCKD Algorithm
- Step 1.
- Set the period of deconvolution T, the number of shift M, and the maximum iteration number n;
- Step 2.
- Compute , , and by using the acquired signal x;
- Step 3.
- Initiate the filter coefficients f with L samples; Generally, the initial filter coefficients equals to prevent the MCKD algorithm from converging to the local solution. The difference is in the center.
- Step 4.
- Calculate the output signal by Equation (1);
- Step 5.
- Compute and based on signal y;
- Step 6.
- Update the filter coefficient f by Equation (5);
- Step 7.
- Determine whether the difference between iterations () is less than the given threshold. If this criterion is satisfied or the iteration equals n, then terminate the iteration progress; otherwise, the process is reverted back to Step 4;
- Step 8.
- Once the optimal filter coefficients f is obtained, the final output signal can be calculated using Equation (1).
2.2. Empirical Wavelet Transform
3. Fault Feature Extraction and Enhancement of Rolling Element Bearing Based on MCKD-EWT
3.1. Improved Empirical Wavelet Transform
- Step 1.
- Conduct FFT and obtain the Fourier spectrum of the signal .
- Step 2.
- Calculate the following threshold according to the method of literature [35].
- Step 3.
- Calculate the envelope curve of the amplitude spectrum based on the local maxima and minima by linear interpolation method, then modify the envelope curve according to the threshold, and finally detect all the extras of the modified envelope curve. If the number of the local maxima points above the threshold is larger than the pre-defined number N of the components, then keep on calculating the envelope curve of the spectrum until the number of the local maxima points is equal or less than the pre-defined number N of the components. The number for the iteration process of the calculating envelope is always 5.
- Step 4.
- Obtain the segmentation boundaries. Detect all the local extrema points of the modified envelope curve, and then get the frequency bands of EWT modes by choosing the frequencies of local maxima points as the central frequencies of the modes. After that, locate the boundaries of each mode. Differing from the segmentation method, which uses the local minima points of the modified envelope curve as the boundaries of the EWT modes introduced in the literature, this method chooses the midpoints of the adjacent local minimum points as the segmentation boundaries. The detailed steps firstly calculate the midpoints of the adjacent local minimum points, and then sort them in ascending order, i.e., , which are used to segment the frequency spectrum. Finally, choose the between the adjacent local maximum points as the boundaries.
- Step 5.
- Segment the spectrum and choose the most meaningful component. Calculate the kurtosis value of each component and choose the component with the maximum kurtosis value to further detect the fault feature.
3.2. Program of the Proposed Method
- Step 1.
- De-noise the signal by MCKD. De-noise the single accidental impact and non-impact components by implementing the MCKD to x(t). The key issue of the MCKD for a given bearing signal with faults is choosing these four parameters: the period of deconvolution T, the length of filter L, the termination number of iterations, and the number of shift M. The parameter T can be easily calculated based on the theoretical fault characteristic frequency, whereas L and M should be subjectively set without any reference. Whether the extracted fault signal is desired or not is severely influenced by the selected parameters. Therefore, selecting the optimal parameters is extremely important in MCKD. This method often chooses the termination number of iterations as a range from 20 to 30, the length of filter as a range from 100 to 300, the period of deconvolution T depending on the actual signal, and the shift number as a range from 2 to 5. Additionally, a method for setting parameters is recommended in a previous study [44].
- Step 2.
- Spectrum segmentation. Calculate the envelope curve of the amplitude spectrum of the de-noising signal. Find the maxima and minima of the envelope curve and segment the spectrum on the modified envelope curve.
- Step 3.
- Signal decomposition. Design the wavelet filter banks based on the spectrum segmentation result and decompose the signal into several sub-signals.
- Step 4.
- IMF (Intrinsic Mode Function) selection. Calculate the kurtosis of each sub-signal and choose the best sub-signal with the maximum kurtosis.
- Step 5.
- Feature extraction. Calculate the squared envelope spectrum and teager energy operator spectrum of the chosen mode. Analyze the fault characteristic frequency of the spectrum to determine the fault type and location of the fault.
4. Simulation
4.1. Model of the Collected Vibration
4.2. Comparison of Improved EWT, EWT, and EMD
4.3. Outer Race Fault Simulation
4.4. Inner Race Fault Simulation
5. Experiment
5.1. Experiment Apparatus
5.2. Case 1: Outer Race Fault with Circumferential Width of 1 mm
5.3. Case 2: Outer Race Fault with Circumferential Width of 3 mm
5.4. Case 3: Inner Race Fault with Circumferential Width of 1 mm
5.5. Case 4: Inner Race Fault With Circumferential Width of 3 mm
6. Conclusions
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
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Li, Z.; Ming, A.; Zhang, W.; Liu, T.; Chu, F.; Li, Y. Fault Feature Extraction and Enhancement of Rolling Element Bearings Based on Maximum Correlated Kurtosis Deconvolution and Improved Empirical Wavelet Transform. Appl. Sci. 2019, 9, 1876. https://doi.org/10.3390/app9091876
Li Z, Ming A, Zhang W, Liu T, Chu F, Li Y. Fault Feature Extraction and Enhancement of Rolling Element Bearings Based on Maximum Correlated Kurtosis Deconvolution and Improved Empirical Wavelet Transform. Applied Sciences. 2019; 9(9):1876. https://doi.org/10.3390/app9091876
Chicago/Turabian StyleLi, Zheng, Anbo Ming, Wei Zhang, Tao Liu, Fulei Chu, and Yin Li. 2019. "Fault Feature Extraction and Enhancement of Rolling Element Bearings Based on Maximum Correlated Kurtosis Deconvolution and Improved Empirical Wavelet Transform" Applied Sciences 9, no. 9: 1876. https://doi.org/10.3390/app9091876
APA StyleLi, Z., Ming, A., Zhang, W., Liu, T., Chu, F., & Li, Y. (2019). Fault Feature Extraction and Enhancement of Rolling Element Bearings Based on Maximum Correlated Kurtosis Deconvolution and Improved Empirical Wavelet Transform. Applied Sciences, 9(9), 1876. https://doi.org/10.3390/app9091876