Feature Enhancement Method of Rolling Bearing Based on K-Adaptive VMD and RBF-Fuzzy Entropy
Abstract
:1. Introduction
2. Basic Theory
2.1. VMD
2.2. Fuzzy Entropy
- Step 1. Get the coarse-grained data. Work out from the original signal sequence x based on Equation (7).
- Step 2. Calculate the maximum absolute distance between X(i) and X(j).
- Step 3. Work out similarity between X(i) and X(j) through fuzzy function based on maximum absolute distance .
- Step 4. Take m = m + 1, and repeat steps 2 and 3.
- Step 5. Calculate fuzzy entropy based on Equations (10) and (11).
3. K-Adaptive VMD and RBF-FuzzyEn
3.1. K-Adaptive VMD
3.1.1. Determination of K
- Step 1. Initialization parameters: Ktemp = 2, α = 2000, τ = 0, ε = 10−7, num = 0.
- Step 2. Perform VMD on raw signal and get IMFs.
- Step 3. If ADCF occurs, go to step 4, otherwise num = 0 and go to step 5.
- Step 4. num = num + 1. If num is 2, that means ADCF occurs twice continuously, go to step 6, otherwise go to step 5.
- Step 5. Ktemp = Ktemp + 1, go to step 2.
- Step 6. Number of optimized modes K = Ktemp − 2.
3.1.2. Selection of IMFs
- Step 1. Calculate autocorrelation operation of original signal by Equation (12) and perform fast Fourier transform to work out the autocorrelation operation frequency spectrum, i = 1.
- Step 2. Compute CFBi of IMFi. CFBi is centered on the center frequency ωi of IMFi, and the energy of IMFi frequency spectrum in CFBi is b times of the total energy of this component.
- Step 3. Calculate the energy En(i) of the ampcorr for raw signal in CFBi.
- Step 4. Repeat steps 2 and 3 to get CFBi (i = 1, …, K) and En = [En(1), En(2), …,En(K)] of all IMFs.
- Step 5. Determine whether prior knowledge is needed to assist in selecting IMF. If so, update En according to the knowledge and then go to step 6, otherwise go to step 6.
- Step 6. Select all components corresponding to En index greater than 0.03 × ∑En, and reconstruct the signal.
- Prior knowledge one: delete the low-frequency components. Bearing vibration signals frequency bands can be divided into low-frequency, mid-frequency, and high-frequency in frequency domain. Low-frequency signals are usually components related to rotating speed [26]. These low-frequency components will affect the noise reduction effect of K-adaptive VMD. Therefore, the low-frequency components should not be contained in when reconstructing the signal by IMFs, so we can ignore these IMFs by setting En value of the components in low frequency to 0. We set En value of the IMF1 to IMF[K/3] components to 0 (IMF is sorted according to its center frequency from low to high, and [•] means rounded to 0).
- Prior knowledge two: delete the components containing interference frequencies. In the actual engineering environment, bearing vibration signal is often interfered by natural frequency and rotation frequency of the equipment. In order to remove the influence of these interference frequencies, it is necessary to set En corresponding to the CFBs with these interference frequencies to 0.
3.2. RBF-FuzzyEn
- Step 1. Coarse the original data.
- Step 2. Calculate the maximum absolute distance and between X(i) and X(j) respectively.
- Step 3. Compute r according to Equation (16).
- Step 4. Determine the similarity and based on Equation (15).
- Step 5. Calculate fuzzy entropy according to Equations (10) and (11).
4. Simulation Analysis and Verification
4.1. Analysis and Verification of K-adaptive VMD
4.1.1. Analysis Based on Nonlinear AM-FM Simulation Signal
4.1.2. Analysis Based on Nonlinear AM-FM Simulation Signal with Close Frequencies
4.1.3. Analysis Based on Rolling Bearing Fault Simulation Signal
4.1.4. Verification of Noise Reduction Effect
4.2. Analysis and Verification of RBF-FuzzyEn
4.2.1. Analysis of the Distribution for Maximum Absolute Distance
4.2.2. Analysis and Verification under Different Noise Level
4.2.3. Analysis and Verification with Various Data Length
5. Application in Bearing Feature Enhancing
5.1. Case 1: Bearing Data of CWRU
5.1.1. Noise Reduction
5.1.2. Feature Extraction
5.2. Case 2: Bearing Data of IMS
6. Conclusions
- An algorithm of K-adaptive VMD was developed, which can obtain K adaptively based on ADCF, and select the optimal IMFs by a coefficient En. The analysis of reconstructed signal of five types of signals (nonlinear AM-FM simulation signal, nonlinear AM-FM simulation signal with close frequencies, inner race fault simulation signal, CWRU rolling bearing inner and outer race fault signal and IMS bearing outer race fault signal) demonstrated that compared with FBE-VMD and TVMD, the K-adaptive VMD has good noise reduction ability. The envelope spectrum of simulation and experimental bearing fault signals proved that K-adaptive VMD can realize the effective extraction of fault impulses.
- RBF-FuzzyEn was proposed, which introduced an innovative fuzzy function and carried out a specific way for determination of parameter r in fuzzy function. To verify the entropy feature extraction method proposed in this paper, RBF-FuzzyEn, FuzzyEn1, FuzzyEn2, ApEn, SampEn and MFE are used to extract the entropy of bearing fault simulation signals under different SNR and data length, as well as bearing fault experiment signals. The conclusion can be obtained from the results that RBF-FuzzyEn outperformed FuzzyEn1, FuzzyEn2, ApEn, SampEn and MFE in distinction between different sates of bearing. Meanwhile, the proposed RBF-FuzzyEn shows a marked noise robustness than other entropy methods and it is independent on data length N.
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
References
- Cerrada, M.; Sánchez, R.; Li, C.; Pacheco, F.; Cabrera, D.; Valente De Oliveira, J.; Vásquez, R.E. A review on data-driven fault severity assessment in rolling bearings. Mech. Syst. Signal Pr. 2018, 99, 169–196. [Google Scholar] [CrossRef]
- Adamczak, S.; Stępień, K.; Wrzochal, M. Comparative Study of Measurement Systems Used to Evaluate Vibrations of Rolling Bearings. Procedia Eng. 2017, 192, 971–975. [Google Scholar] [CrossRef]
- Song, L.; Wang, H.; Chen, P. Vibration-Based Intelligent Fault Diagnosis for Roller Bearings in Low-Speed Rotating Machinery. IEEE Trans. Instrum. Meas. 2018, 67, 1887–1899. [Google Scholar] [CrossRef]
- Ben Ali, J.; Fnaiech, N.; Saidi, L.; Chebel-Morello, B.; Fnaiech, F. Application of empirical mode decomposition and artificial neural network for automatic bearing fault diagnosis based on vibration signals. Appl. Acoust. 2015, 89, 16–27. [Google Scholar] [CrossRef]
- Zhao, H.; Sun, M.; Deng, W.; Yang, X. A New Feature Extraction Method Based on EEMD and Multi-Scale Fuzzy Entropy for Motor Bearing. Entropy 2017, 19, 14. [Google Scholar] [CrossRef] [Green Version]
- Wang, L.; Liu, Z.; Miao, Q.; Zhang, X. Time–frequency analysis based on ensemble local mean decomposition and fast kurtogram for rotating machinery fault diagnosis. Mech. Syst. Signal Pr. 2018, 103, 60–75. [Google Scholar] [CrossRef]
- Zhao, M.; Jia, X. A novel strategy for signal denoising using reweighted SVD and its applications to weak fault feature enhancement of rotating machinery. Mech. Syst. Signal Pr. 2017, 94, 129–147. [Google Scholar] [CrossRef]
- Qiao, Z.; Lei, Y.; Li, N. Applications of stochastic resonance to machinery fault detection: A review and tutorial. Mech. Syst. Signal Pr. 2019, 122, 502–536. [Google Scholar] [CrossRef]
- Dragomiretskiy, K.; Zosso, D. Variational Mode Decomposition. IEEE Trans. Signal Proces. 2014, 62, 531–544. [Google Scholar] [CrossRef]
- Li, H.; Liu, T.; Wu, X.; Chen, Q. An optimized VMD method and its applications in bearing fault diagnosis. Measurement 2020, 166, 108185. [Google Scholar] [CrossRef]
- Dibaj, A.; Hassannejad, R.; Ettefagh, M.M.; Ehghaghi, M.B. Incipient fault diagnosis of bearings based on parameter-optimized VMD and envelope spectrum weighted kurtosis index with a new sensitivity assessment threshold. ISA Trans. 2020, 114, 413–433. [Google Scholar] [CrossRef] [PubMed]
- Gong, T.; Yuan, X.; Yuan, Y.; Lei, X.; Wang, X. Application of tentative variational mode decomposition in fault feature detection of rolling element bearing. Measurement 2019, 135, 481–492. [Google Scholar] [CrossRef]
- Li, J.; Yao, X.; Wang, H.; Zhang, J. Periodic impulses extraction based on improved adaptive VMD and sparse code shrinkage denoising and its application in rotating machinery fault diagnosis. Mech. Syst. Signal Pr. 2019, 126, 568–589. [Google Scholar] [CrossRef]
- Chen, S.; Yang, Y.; Dong, X.; Xing, G.; Peng, Z.; Zhang, W. Warped Variational Mode Decomposition with Application to Vibration Signals of Varying-Speed Rotating Machineries. IEEE Trans. Instrum. Meas. 2019, 68, 2755–2767. [Google Scholar] [CrossRef]
- Wu, S.; Feng, F.; Wu, C.; Li, B. Research on fault diagnosis method of tank planetary gearbox based on VMD-DE. J. Vib. Shock 2020, 39, 170–179. [Google Scholar] [CrossRef]
- Qi, Y.; Bai, Y.; Gao, S.; Li, Y. Fault diagnosis of wind turbine bearing based on AVMD and spectral correlation analysis. ACTA Energ. Sol. Sin. 2019, 48, 2053–2063. [Google Scholar]
- Jiang, X.; Shen, C.; Shi, J.; Zhu, Z. Initial center frequency-guided VMD for fault diagnosis of rotating machines. J. Sound Vib. 2018, 435, 36–55. [Google Scholar] [CrossRef]
- Li, X.; Ma, Z.; Kang, D.; Li, X. Fault diagnosis for rolling bearing based on VMD-FRFT. Measurement 2020, 155, 107554. [Google Scholar] [CrossRef]
- Zhang, C.; Wang, Y.; Deng, W. Fault diagnosis for rolling bearings using optimized variational mode decomposition and resonance demodulation. Entropy 2020, 22, 739. [Google Scholar] [CrossRef]
- Liu, L.; Zhi, Z.; Zhang, H.; Guo, Q.; Peng, Y.; Liu, D. Related entropy theories application in condition monitoring of rotating machineries. Entropy 2019, 21, 1061. [Google Scholar] [CrossRef] [Green Version]
- Pincus, S.M. Approximate entropy as a measure of system complexity. Proc. Natl. Acad. Sci. USA 1991, 88, 2297–2301. [Google Scholar] [CrossRef] [PubMed] [Green Version]
- Richman, J.S.; Moorman, J.R. Physiological time-series analysis using approximate entropy and sample entropy. Am. J. Physiol. Heart Circ. Physiol. 2000, 278, H2039–H2049. [Google Scholar] [CrossRef] [PubMed] [Green Version]
- Chen, W.; Wang, Z.; Xie, H.; Yu, W. Characterization of surface EMG signal based on fuzzy entropy. IEEE Trans. Neural Syst. Rehabil. Eng. 2007, 15, 266–272. [Google Scholar] [CrossRef]
- Chen, W.; Zhuang, J.; Yu, W.; Wang, Z. Measuring complexity using FuzzyEn, ApEn, and SampEn. Med. Eng. Phys. 2009, 31, 61–68. [Google Scholar] [CrossRef] [PubMed]
- Yang, H.; Liu, S.; Zhang, H. Adaptive estimation of VMD modes number based on cross correlation coefficient. J. Vibroeng. 2017, 19, 1185–1196. [Google Scholar] [CrossRef]
- Maliuk, A.S.; Prosvirin, A.E.; Ahmad, Z.; Kim, C.H.; Kim, J. Novel bearing fault diagnosis using Gaussian mixture Model-based fault band selection. Sensors 2021, 21, 6579. [Google Scholar] [CrossRef]
- Zheng, J.; Cheng, J.; Yang, Y.; Luo, S. A rolling bearing fault diagnosis method based on multi-scale fuzzy entropy and variable predictive model-based class discrimination. Mech. Mach. Theory 2014, 78, 187–200. [Google Scholar] [CrossRef]
- Case Western Reserve University Bearing Data Center Website. Available online: https://engineering.case.edu/bearingdatacenter (accessed on 25 December 2019).
- Qiu, H.; Lee, J.; Lin, J.; Yu, G. Wavelet filter-based weak signature detection method and its application on rolling element bearing prognostics. J. Sound Vib. 2006, 289, 1066–1090. [Google Scholar] [CrossRef]
Number of Samples | K | Selected IMFs | Center Frequency of the Selected IMFs (Hz) | SNR of Reconstructed Signal (dB) |
---|---|---|---|---|
500 | 31 | 1, 2, 6 | 7.04, 29.19, 101.64 | 5.00 |
800 | 34 | 1, 2, 5, 6 | 7.38, 30.19, 98.92, 107.53 | 4.09 |
1000 | 35 | 1, 2, 6 | 8.23, 29.42, 100.71 | 5.69 |
1200 | 32 | 1, 2, 5 | 7.19, 29.88, 99.01 | 4.02 |
1500 | 32 | 1, 2, 5 | 7.10, 29.92, 99.35 | 3.95 |
2000 | 32 | 1, 2, 5 | 7.68, 29.53, 99.40 | 3.77 |
Signal | Method | SNR of Reconstructed Signal (dB) | K | Selected IMFs | CF of the Selected IMFs (Hz) | Running Time(s) | Complexity |
---|---|---|---|---|---|---|---|
fisg1 (6 dB) | FBE-VMD | −0.29 | 12 | 7, 8, 9, 10, 12 | 244.05, 293.80, 338.51, 377.83, 481.32 | —— | high |
TVMD | −0.07 | 3 | 3 | 361.3 | 2.45 | low | |
K-adaptive VMD | 13.44 | 17 | 1, 2, 4 | 7.29, 30.37, 100.13 | 35.44 | middle | |
fsig1 (−6 dB) | FBE-VMD | −1.21 | 15 | 11, 13 | 344.63, 406.17 | —— | high |
TVMD | −0.76 | 3 | 2 | 165.57 | 0.62 | low | |
K-adaptive VMD | 5.69 | 35 | 1, 2, 6 | 8.23, 29.42, 100.71 | 162.45 | middle |
SNR of Original Signal (dB) | Method | K | Selected IMFs | SNR of Reconstructed Signal (dB) |
---|---|---|---|---|
−6 | FBE-VMD | 9 | 8 | −0.82 |
TVMD | 3 | 3 | 4.01 | |
K-adaptive VMD | 37 | 14, 15, 16, 17 | 3.38 | |
−13 | FBE-VMD | 14 | 6, 8, 9 | −5.25 |
TVMD | 5 | 3 4 | −4.51 | |
K-adaptive VMD | 35 | 15, 16, 26 | −1.44 |
Data Set | State | Fault Diameter | Motor Load | Motor Speed (rpm) | Motor Ratational Frequency fr (Hz) | Fault Frequencies (Hz) |
---|---|---|---|---|---|---|
169 | Inner race fault | 0.014″ | 0 | 1796 | 29.93 | 162.09 |
158 | Outer race fault | 0.007″ | 1 | 1773 | 29.55 | 105.94 |
No | CFB (Hz) | No | CFB (Hz) | No | CFB (Hz) | No | CFB (Hz) |
---|---|---|---|---|---|---|---|
IMF1 | [22, 162] | IMF5 | [1920, 2004] | IMF9 | [4138, 4146] | IMF13 | [5314, 5350] |
IMF2 | [444, 528] | IMF6 | [2252, 2304] | IMF10 | [4260, 4264] | ||
IMF3 | [868, 924] | IMF7 | [2692, 2736] | IMF11 | [4384, 4384] | ||
IMF4 | [1380, 1436] | IMF8 | [3102, 3194] | IMF12 | [4786, 4862] |
No | CFB (Hz) | No | CFB (Hz) | No | CFB (Hz) | No | CFB (Hz) |
---|---|---|---|---|---|---|---|
IMF1 | [162, 166] | IMF5 | [1438, 1442] | IMF9 | [3214, 3258] | IMF13 | [4254, 4258] |
IMF2 | [644, 688] | IMF6 | [1970, 1998] | IMF10 | [3996, 4020] | IMF14 | [4380, 4400] |
IMF3 | [868, 936] | IMF7 | [2262, 2290] | IMF11 | [4136, 4140] | IMF15 | [4830, 4882] |
IMF4 | [1084, 1108] | IMF8 | [2762, 2806] | IMF12 | [4246, 4258] | IMF16 | [5318, 5338] |
State of Bearing | N | IF | OF | |||
---|---|---|---|---|---|---|
r | Entropy | r | Entropy | r | Entropy | |
RBF-FuzzyEn | 0.0131 | 1.0523 | 0.0131 | 0.5963 | 0.0131 | 0.4784 |
FuzzyEn1 | 0.0155 | 0.1336 | 0.0038 | 0.1325 | 0.0025 | 0.1055 |
FuzzyEn2 | 0.0155 | 1.1785 | 0.0038 | 1.8773 | 0.0025 | 2.1545 |
ApEn | 0.0155 | 1.1797 | 0.0038 | 1.4369 | 0.0025 | 1.6343 |
SampEn | 0.0155 | 1.1520 | 0.0038 | 1.3676 | 0.0025 | 1.5938 |
MFE | 0.0116 | 1.7887 | 0.0029 | 0.2171 | 0.0155 | 0.2502 |
No | CFB (Hz) | No | CFB (Hz) | No | CFB (Hz) | No | CFB (Hz) |
---|---|---|---|---|---|---|---|
IMF1 | [50, 126] | IMF8 | [3208, 3248] | IMF15 | [5076, 5124] | IMF22 | [7574, 7630] |
IMF2 | [464, 528] | IMF9 | [3488, 3532] | IMF16 | [5368, 5432] | IMF23 | [7902, 7966] |
IMF3 | [982, 986] | IMF10 | [3758, 3810] | IMF17 | [5690, 5738] | IMF24 | [8266, 8338] |
IMF4 | [1448, 1532] | IMF11 | [4096, 4140] | IMF18 | [5970, 6022] | IMF25 | [8662, 8746] |
IMF5 | [1870, 1918] | IMF12 | [4336, 4376] | IMF19 | [6296, 6352] | IMF26 | [9226, 9286] |
IMF6 | [2356, 2444] | IMF13 | [4582, 4602] | IMF20 | [6554, 6718] | IMF27 | [9528, 9580] |
IMF7 | [2866, 2942] | IMF14 | [4830, 4854] | IMF21 | [7152, 7240] | IMF28 | [9832, 9896] |
Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |
© 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).
Share and Cite
Jiao, J.; Yue, J.; Pei, D. Feature Enhancement Method of Rolling Bearing Based on K-Adaptive VMD and RBF-Fuzzy Entropy. Entropy 2022, 24, 197. https://doi.org/10.3390/e24020197
Jiao J, Yue J, Pei D. Feature Enhancement Method of Rolling Bearing Based on K-Adaptive VMD and RBF-Fuzzy Entropy. Entropy. 2022; 24(2):197. https://doi.org/10.3390/e24020197
Chicago/Turabian StyleJiao, Jing, Jianhai Yue, and Di Pei. 2022. "Feature Enhancement Method of Rolling Bearing Based on K-Adaptive VMD and RBF-Fuzzy Entropy" Entropy 24, no. 2: 197. https://doi.org/10.3390/e24020197
APA StyleJiao, J., Yue, J., & Pei, D. (2022). Feature Enhancement Method of Rolling Bearing Based on K-Adaptive VMD and RBF-Fuzzy Entropy. Entropy, 24(2), 197. https://doi.org/10.3390/e24020197