Abstract
The entropy-based complexity measurement tools have been widely used in extracting fault characteristics of rolling bearings. However, the fault information generally is hidden in both time and frequency domains, and thus one-dimensional entropy is unable to fully extract the comprehensive fault information from the measured vibration signals of rolling bearings. Focus on this shortcoming, a novel entropy-based complexity evaluation method called three-dimensional Holo–Hilbert spectral entropy (HHSE3D) is developed to extract the fault feature of rolling bearings, where the Holo–Hilbert spectral analysis is used to expand the one-dimensional signal to the three-dimensional relationship among time domain information, amplitude-modulated and frequency-modulated features. Meanwhile, to obtain a comprehensively nonlinear dynamic feature description in different scales, the proposed HHSE3D method is extended into the multiscale framework through the coarse-graining process, and thus the multiscale HHSE3D (MHHSE3D) method can be achieved. The robustness and effectiveness of MHHSE3D is verified using both simulated signals and experimental bearing data. The analysis results demonstrate that the proposed method exhibits the best feature extraction ability with highest diagnostic accuracy compared with the other four traditional entropy based diagnosis methods.
Similar content being viewed by others
Availability of data and materials
Data will be made available on reasonable request.
References
Li, Y., Wang, S., Yang, Y., et al.: Multiscale symbolic fuzzy entropy: an entropy denoising method for weak feature extraction of rotating machinery. Mech. Syst. Signal Process. 162, 108052 (2022)
Wang, X., Si, S., Li, Y.: Hierarchical diversity entropy for the early fault diagnosis of rolling bearing. Nonlinear Dyn. 108(2), 1447–1462 (2022)
Ying, W., Zheng, J., Pan, H., et al.: Permutation entropy-based improved uniform phase empirical mode decomposition for mechanical fault diagnosis. Digit. Signal Process. 117, 103167 (2021)
Wang, X., Zheng, J., Ni, Q., et al.: Traversal index enhanced-gram (TIEgram): a novel optimal demodulation frequency band selection method for rolling bearing fault diagnosis under non-stationary operating conditions. Mech. Syst. Signal Process. 172, 109017 (2022)
Feng, K., Smith, W.A., Randall, R.B., et al.: Vibration-based monitoring and prediction of surface profile change and pitting density in a spur gear wear process. Mech. Syst. Signal Process. 165, 108319 (2022)
Feng, K., Ji, J.C., Ni, Q., et al.: A review of vibration-based gear wear monitoring and prediction techniques. Mech. Syst. Signal Process. 182, 109605 (2023)
Martin, H.R., Honarvar, F.: Application of statistical moments to bearing failure detection. Appl. Acoust. 44(1), 67–77 (1995)
Heng, R.B.W., Nor, M.J.M.: Statistical analysis of sound and vibration signals for monitoring rolling element bearing condition. Appl. Acoust. 53(1–3), 211–226 (1998)
Mechefske, C.K., Mathew, J.: Fault detection and diagnosis in low speed rolling element bearings. Part I: the use of parametric spectra. Mech. Syst. Signal Process. 6(4), 297–307 (1992)
Logen, D., Mathew, J.: Using correlation dimension for vibration fault diagnosis of rolling element bearing-I. Basic concept. Mech. Syst. Signal Process. 10(3), 241–250 (1996)
Kang, J., Feng, C., Hu, H., et al.: Research on chatter prediction and monitor based on DHMM pattern recognition theory. In: 2007 IEEE International Conference on Automation and Logistics, pp. 1368–1372. IEEE (2007)
Frank, P.M., Ding, X.: Frequency domain approach to optimally robust residual generation and evaluation for model-based fault diagnosis. Automatica 30(5), 789–804 (1994)
Moshrefzadeh, A., Fasana, A., Antoni, J.: The spectral amplitude modulation: A nonlinear filtering process for diagnosis of rolling element bearings[J]. Mech. Syst. Signal Process. 132, 253–276 (2019)
Yan, W.J., Ren, W.X.: Operational modal parameter identification from power spectrum density transmissibility. Comput.-Aided Civil Infrastruct. Eng. 27(3), 202–217 (2012)
Feldman, M.: Hilbert transform in vibration analysis. Mech. Syst. Signal Process. 25(3), 735–802 (2011)
Lei, Y., He, Z., Zi, Y.: Application of the EEMD method to rotor fault diagnosis of rotating machinery. Mech. Syst. Signal Process. 23(4), 1327–1338 (2009)
Wang, Y., He, Z., Zi, Y.: A comparative study on the local mean decomposition and empirical mode decomposition and their applications to rotating machinery health diagnosis. J. Vib. Acoust. 132(2), 10–21 (2010)
Ni, Q., Ji, J.C., Feng, K., et al.: A fault information-guided variational mode decomposition (FIVMD) method for rolling element bearings diagnosis. Mech. Syst. Signal Process. 164, 108216 (2022)
Wang, X., Shi, J., Zhang, J.: A power information guided-variational mode decomposition (PIVMD) and its application to fault diagnosis of rolling bearing. Digit. Signal Process. 132, 103814 (2022)
Yu, G., Yu, M., Xu, C.: Synchroextracting transform. IEEE Trans. Ind. Electron. 64(10), 8042–8054 (2017)
Shannon, C.E.: A mathematical theory of communication. ACM SIGMOBILE Mob. Comput. Commun. Rev. 5(1), 3–55 (2001)
Cui, H., Zhang, L., Kang, R., et al.: Research on fault diagnosis for reciprocating compressor valve using information entropy and SVM method. J. Loss Prev. Process. Ind. 22(6), 864–867 (2009)
Ji, Y., Wang, X., Liu, Z., et al.: EEMD-based online milling chatter detection by fractal dimension and power spectral entropy. Int. J. Adv. Manuf. Technol. 92(1), 1185–1200 (2017)
Zhang, A., Yang, B., Huang, L.: Feature extraction of EEG signals using power spectral entropy. In: 2008 International Conference on BioMedical Engineering and Informatics, vol. 2, pp. 435–439. IEEE (2008)
Dai, Y., Zhang, H., Mao, X., et al.: Complexity–entropy causality plane based on power spectral entropy for complex time series. Phys. A 509, 501–514 (2018)
Li, X., Li, D., Liang, Z., et al.: Analysis of depth of anesthesia with Hilbert–Huang spectral entropy. Clin. Neurophysiol. 119(11), 2465–2475 (2008)
Huang, N.E., Shen, Z., Long, S.R., et al.: The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time series analysis. Proc. R. Soc. Lond. Ser. A: Math., Phys. Eng. Sci. 1998(454), 903–995 (1971)
Humeau-Heurtier, A., Wu, C.W., Wu, S.D., et al.: Refined multiscale Hilbert–Huang spectral entropy and its application to central and peripheral cardiovascular data. IEEE Trans. Biomed. Eng. 63(11), 2405–2415 (2016)
Hoseinzadeh, M.S., Khadem, S.E., Sadooghi, M.S.: Modifying the Hilbert–Huang transform using the nonlinear entropy-based features for early fault detection of ball bearings. Appl. Acoust. 150, 313–324 (2019)
Rosso, O.A., Blanco, S., Yordanova, J., et al.: Wavelet entropy: a new tool for analysis of short duration brain electrical signals. J. Neurosci. Methods 105(1), 65–75 (2001)
Yu, Y., Junsheng, C.: A roller bearing fault diagnosis method based on EMD energy entropy and ANN. J. Sound Vib. 294(1–2), 269–277 (2006)
Rényi, A.: On measures of entropy and information. In: Proceedings of the 4th Berkeley Symposium on Mathematical Statistics and Probability, Volume 1: Contributions to the Theory of Statistics, vol. 4, pp. 547–562. University of California Press (1961)
Pincus, S.M.: Approximate entropy as a measure of system complexity. Proc. Natl. Acad. Sci. 88(6), 2297–2301 (1991)
Richman, J.S., Moorman, J.R.: Physiological time-series analysis using approximate entropy and sample entropy. Am. J. Physiol.-Heart Circ. Physiol. 278(6), 2039–2049 (2000)
Zheng, J., Cheng, J., Yang, Y., et al.: A rolling bearing fault diagnosis method based on multi-scale fuzzy entropy and variable predictive model-based class discrimination. Mech. Mach. Theory 78, 187–200 (2014)
Li, Y., Yang, Y., Li, G., et al.: A fault diagnosis scheme for planetary gearboxes using modified multi-scale symbolic dynamic entropy and mRMR feature selection. Mech. Syst. Signal Process. 91, 295–312 (2017)
Chen, W., Zhuang, J., Yu, W., et al.: Measuring complexity using fuzzyen, apen, and sampen. Med. Eng. Phys. 31(1), 61–68 (2009)
Bandt, C., Pompe, B.: Permutation entropy: a natural complexity measure for time series. Phys. Rev. Lett. 88(17), 174102 (2002)
Wang, H., Chen, J., Zhou, Y., et al.: Early fault diagnosis of rolling bearing based on noise-assisted signal feature enhancement and stochastic resonance for intelligent manufacturing. Int. J. Adv. Manuf. Technol. 107(3), 1017–1023 (2020)
Rostaghi, M., Azami, H.: Dispersion entropy: a measure for time-series analysis. IEEE Signal Process. Lett. 23(5), 610–614 (2016)
Huang, N.E., Hu, K., Yang, A.C.C., et al.: On Holo-Hilbert spectral analysis: a full informational spectral representation for nonlinear and non-stationary data. Philos. Trans. R. Soc. A: Math., Phys. Eng. Sci. 374(2065), 20150206 (2016)
Liang, W.K., Tseng, P., Yeh, J.R., et al.: Frontoparietal beta amplitude modulation and its interareal cross-frequency coupling in visual working memory. Neuroscience 460, 69–87 (2021)
Wu, S.D., Wu, P.H., Wu, C.W., et al.: Bearing fault diagnosis based on multiscale permutation entropy and support vector machine. Entropy 14(8), 1343–1356 (2012)
Azami, H., Kinney-Lang, E., Ebied, A., et al.: Multiscale dispersion entropy for the regional analysis of resting-state magnetoencephalogram complexity in Alzheimer's disease. In: 2017 39th Annual International Conference of the IEEE Engineering in Medicine and Biology Society (EMBC), pp. 3182–3185. IEEE (2017)
Ying, W., Tong, J., Dong, Z., et al.: Composite multivariate multi-scale permutation entropy and Laplacian score based fault diagnosis of rolling bearing. Entropy 24(2), 160 (2022)
Huang, N.E., Wu, Z., Long, S.R., et al.: On instantaneous frequency. Adv. Adapt. Data Anal. 1(02), 177–229 (2009)
Ying, W., Zheng, J., Pan, H., et al.: Use of Holo-Hilbert spectral analysis to reveal the amplitude modulation features of faulty rolling bearing signals. J. Vib. Control 2022, 10775463221130820 (2022)
Costa, M., Goldberger, A.L., Peng, C.K.: Multiscale entropy analysis of biological signals. Phys. Rev. E 71(2), 021906 (2005)
Yang, X.S., He, X.: Bat algorithm: literature review and applications. Int. J. Bio-inspired Comput. 5(3), 141–149 (2013)
Daga, A.P., Fasana, A., Marchesiello, S., et al.: The Politecnico di Torino rolling bearing test rig: description and analysis of open access data. Mech. Syst. Signal Process. 120, 252–273 (2019)
Moshrefzadeh, A.: Condition monitoring and intelligent diagnosis of rolling element bearings under constant/variable load and speed conditions. Mech. Syst. Signal Process. 149, 107153 (2021)
Wang, X., Wang, T., Ming, A., et al.: Semi-supervised hierarchical attribute representation learning via multi-layer matrix factorization for machinery fault diagnosis. Mech. Mach. Theory 167, 104445 (2022)
Acknowledgements
This work is supported by the National Natural Science Foundation of China (No. 51975004), the Natural Science Foundation of Anhui Provence of China (No. 2008085QE215) and the State Key Laboratory of Mechanical Transmissions (SKLMT-MSKFKT-202107).
Funding
The funding was provided by the National Natural Science Foundation of China (No. 51975004), the Natural Science Foundation of Anhui Provence of China (No. 2008085QE215) and the State Key Laboratory of Mechanical Transmissions (SKLMT-MSKFKT-202107).
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Ethical approval
The authors declare that they have adhered to the ethical standards of research execution.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Zheng, J., Ying, W., Tong, J. et al. Multiscale three-dimensional Holo–Hilbert spectral entropy: a novel complexity-based early fault feature representation method for rotating machinery. Nonlinear Dyn 111, 10309–10330 (2023). https://doi.org/10.1007/s11071-023-08392-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11071-023-08392-z