Stable distributions, generalised entropy, and fractal diagnostic models of mechanical vibration signals

Puchalski, A.; Komorska, I.

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Tytuł artykułu

Stable distributions, generalised entropy, and fractal diagnostic models of mechanical vibration signals

Autorzy

Puchalski A. , Komorska I.

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Warianty tytułu

Rozkłady stabilne, entropia uogólniona i fraktalne modele diagnostyczne sygnałów drgań mechanicznych

Języki publikacji

Abstrakty

Vibrodiagnostic analysis of wearing and/or defects of complex rotating systems confirms the presence of non-linear, nonstationary and multiscale properties as well as long-term correlations of real signals. The recorded time series of vibrations are often of an impulse character. Probability distributions are different than Gaussian distributions and exhibit heavy-tails. These are important sources of multifractal dynamics, requiring advanced, data-based modelling methods. The reliable numerical algorithms, used for calculations of functions of stable distributions and multifractal properties, were applied in the approach presented in the hereby paper. Relations between parameters of stable distributions and singularity spectra indicate the possibility of applying both methods for modelling mechanical vibrations signals in diagnostics of complex systems. The performed investigations confirmed the possibility of modelling and assessing the observed states of the powertrain of vehicles with SI engines, on the bases of parameters of alpha-stable distribution (ASD) and parameterised entropy of mechanical vibrations signals.

Wibrodiagnostyczna analiza zużycia i / lub wad złożonych układów wirujących potwierdza obecność nieliniowych, niestacjonarnych i wieloskalowych właściwości oraz długookresowe korelacje sygnalozywisych. Rejestrowane szeregi czasowe drgań mają często charakter impulsowy. Rozkłady prawdopodobieństwa odbiegają od rozkładów Gaussowskich i wykazują gruboogonowość. Są to ważne źródła dynamiki multifraktalnej, wymagające zaawansowanych metod modelowania bazującego na danych. W podejściu przedstawionym w pracy wykorzystano niezawodne algorytmy numeryczne służące do obliczania funkcji stabilnych dystrybucji i cech multifraktalnych. Relacje między parametrami stabilnych rozkładów i widmami osobliwości wskazują na możliwość zastosowania obu metod do modelowania sygnałów drgań mechanicznych w diagnostyce układów złożonych. Wykonane badania potwierdziły możliwość modelowania i oceny obserwowanych stanów układu napędowego pojazdu z silnikiem o zapłonie iskrowym, na podstawie parametrów rozkładów alfastabilnych (ASD) gęstości prawdopodobieństwa i entropii parametryzowanej sygnałów drgań mechanicznych.

Słowa kluczowe

vibrodiagnostics multifractality stable distributions parameterised entropy

wibrodiagnostyka multifraktalność rozkłady stabilne entropia parametryzowana

Wydawca

Polskie Towarzystwo Diagnostyki Technicznej

Czasopismo

Diagnostyka

Rocznik

2017

Tom

Vol. 18, No. 4

Strony

103--110

Opis fizyczny

Bibliogr. 40 poz., wykr., tab.

Twórcy

autor

Puchalski A.

andrzej.puchalski@uthrad.pl

University of Technology and Humanities in Radom, Radom, Poland

autor

Komorska I.

iwona.komorska@uthrad.pl

University of Technology and Humanities in Radom, Radom, Poland

Bibliografia

1. Ahmad N, Kamal A, Khan M, Hushnud M, Tufail A. A Study of Multifractal Spectra and Renyi Dimensions in 14.5A GeV/c 28Si-Nucleus Collisions. Journal of Modern Physics 2014; 5: 1288-1293. Published Online August 2014 in SciRes. http://www.scirp.org/journal/jmp; http://dx.doi.org/10.4236/jmp.2014.514129.
2. Bashana A, Bartsch R, Kantelhardt JW, Havlin S. Comparison of detrending methods for fluctuation analysis. Physica A 2008; 387:5080-5090.
3. Batko W, Dąbrowski Z, Engel Z, Kiciński J, Weyna S. (2005) Modern methods of investigations of vibroacoustis processes. Ed.by ITE - PIB Radom 2005. Polish.
4. Batko W, Dąbrowski Z, Kiciński J. Nonlinear Effects In Technical Diagnostics. Ed. by ITE - PIB; Radom; 2008.
5. Borak Sz, Härdle W, Weron R. Stable Distributions. Humboldt-Universität zu Berlin; 2005; http://prac.im.pwr.edu.pl/~hugo/publ/SFB2005- 008_Borak_Haerdle_Weron.pdf.
6. Bromiley PA, Thacker NA, Bouhova-Thacker E. Shannon Entropy, Renyi Entropy, and Information. http://www.tina-vision.net/docs/memos/2004-004.pdf.
7. Bruninx K, Delarue E, D’haeseleer W. Statistical description of the error on wind power forecasts via a L´evy -stable distribution. Energy and Environment 2013. http://www.mech.kuleuven.be/tme/research.
8. Butar FB, Kale M. (2011) Fractal analysis of time series and distribution properties of Hurst exponent. Journal of Mathematical Science and Mathematical Education 2011; 6(1):8-19.
9. Hurst HE. Long term storage capacity of reservoirs. Transactions of American Society of Civil Engineering 1951; 116: 770-799.
10. Jianbo G, Feiyan L, Jianfang Z, Jing H, Yinhe C. Information Entropy As a Basic Building Block of Complexity Theory. Entropy 2013; 15:3396-3418; DOI:10.3390/e15093396.
11. Jizbaa P, Korbel J. Multifractal Diffusion Entropy Analysis: Optimal Bin Width of Probability Histograms. Physica A 00 2014: p.1-18.
12. Kantelhardt IW. Fractal and Multifractal Time Series. Mathematics of Complexity and Dynamical Systems 2011: 463-487.
13. Kantelhardt JW , Zschiegner SA, Koscielny-Bunde E, Havlin S, Bunde A, Stanley HE. Multifractal detrended fluctuation analysis of nonstationary time series. Physica A 2002; 316:87-114.
14. Kantz H, Schreiber T. Nonlinear time series analysis. University Press; Cambridge; 2004.
15. Komorska I, Puchalski A. On-board diagnostics of mechanical defects of the vehicle drive system based on the vibration signal reference model. Journal of Vibroegineering 2013; 15(1):450-458.
16. Komorska I, Puchalski A. On-line diagnosis of mechanical defects of the combustion engine with principal components analysis. Journal of Vibroegineering 2015; 17(8):4279 – 4288.
17. Korbicz J, Kościelny JM, Kowalczuk Z, Cholewa W (Eds.). Fault diagnosis Models, Artificial Intelligence, Applications. Springer-Verlag; Berlin; 2004.
18. Li J, Du Q, Sun C. An improved box-counting method for image fractal dimension estimation. Pattern Recognition 2009; 42:2460 - 2469.
19. Lin J, Chen Q. Fault diagnosis of rolling bearings based on multifractal detrended fluctuation analysis and Mahalanobis distance criterion. Mechanical Systems and Signal Processing 2013; 38:515-533.
20. Lin J, Chen Q. A novel method for feature extraction using crossover characteristics of nonlinear data and its application to fault diagnosis of rotary machinery. Mechanical Systems and Signal Processing 2014; 48:174-187.
21. Liu H, Wang X, Lu C. Rolling bearing fault diagnosis based on LCD–TEO and multifractal detrended fluctuation analysis. Mechanical Systems and Signal Processing 2015; 60-61:273-288.
22. Maszczyk T, Duch W. Comparison of Shannon, Renyi and Tsallis Entropy used in Decision Trees. Lecture Notes in Computer Science 2008; 5097:643- 651.
23. Moura EP et al. Classification of imbalance levels in a scaled wind turbine through detrended fluctuation analysis of vibration signals. Renewable Energy 2016; 96:993-1002.
24. Moura EP, Souto CR, Silva AA, Irmao MAS. Evaluation of principal component analysis and neural network performance for bearing fault diagnosis from vibration signal processed by RS and DF analyses. Mechanical Systems and Signal Processing 2011; 25:1765-1772.
25. Nolan JP. Stable Distributions Models for Heavy Tailed Data. Math/Stat Dep American University; 2015; http://fs2.american.edu/jpnolan/www/stable/chap1.pdf.
26. Puchalski A. A technique for the vibration signal analysis in vehicle diagnostics. Mechanical Systems and Signal Processing 2015; 56-57:173-180.
27. Puchalski A. Multiscale analysis of vibration signals in engine valve system. Journal of Vibroegineering 2015; 17(7):3586-3593.
28. Puchalski A, Komorska I. Stable Distributions and Fractal Diagnostic Models of Vibration Signals of Rotating Systems, Advances in Condition Monitoring of Machinery in Non-Stationary Operations, Applied Condition Monitoring; A. Timofiejczuk et al. (eds.); Springer International Publishing AG; (9)2018 (in press).
29. Rényi A. On Measures of Entropy and Information. Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability. University of California Press; 1961; 1:547-561.
30. Shannon CE. A Mathematical Theory of Communication. Bell System Technical Journal 1948; 27:379-423.
31. Sundaram S, McDonald K. Stable Distributions for Heavy-Tailed Data and Their Application in Asset Health Monitoring. Proc. of VII International Conference on Condition Monitoring and Machinery Failure Prevention Technologies; 2010; http://www.robots.ox.ac.uk/~davidc/pubs/stable_dists .pdf.
32. Thurner S, Hanel R. Is There a World Behind Shannon? Entropies for Complex Systems. A. Sanayei et al. (eds.); ISCS 2013: Interdisciplinary Symposium on Complex Systems; 9 Emergence, Complexity and Computation 8; Springer-Verlag; Berlin- Heidelberg; 2014. DOI: 10.1007/978-3-642- 45438-7_2.
33. Xiong Q, Zhang W, Lu T, Mei G, Liang S. A Fault Diagnosis Method for Rolling Bearings Based on Feature Fusion of Multifractal Detrended Fluctuation Analysis and Alpha Stable Distribution. Hindawi Publishing Corporation Shock and Vibration; 2016. DOI:10.1155/2016/1232893.
34. Yu G, Li Ch, Zhang J. A new statistical modeling and detection method for rolling element bearing faults based on alpha–stable distribution. Mechanical Systems and Signal Processing 2013; 41:155-175.
35. Żak G, Obuchowski J, Wyłomańska A, Zimroz R Application of ARMA modelling and alpha-stable distribution for local damage detection in bearings, Diagnostyka 2014; 15(3): 3-11.
36. Żak G, Wyłomańska A, Zimroz R Application of alpha-stable distribution approach for local damage detection in rotating machines, Journal of Vibroengineering 2015; 17(6): 2987-3002.
37. Żak G, Wyłomańska A, Zimroz R. Data-driven vibration signal filtering procedure based on the α- stable distribution. Journal of Vibroengineering 2016; 18(2): 826-837.
38. Żak G, Wyłomańska A, Zimroz R Periodically impulsive behaviour detection in noisy observation based on generalised fractional order dependency map, Applied Acoustics 2017; https://doi.org/10.1016/j.apacoust.2017.05.003.
39. Zhang S, He Y, Zhang J, Zhao Y. Multi-fractal Based Fault Diagnosis Method of Rotating Machinery. Trans Tech Publications; Switzerland; 2012.
40. Zmeskal O, Dzik P, Vesely M. Entropy of fractal systems. Computers and Mathematics with Applications 2013; 66:135-146.

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Bibliografia

Identyfikator YADDA

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