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Bouc Hysteresis Under White Noise Excitation Using Nonlinear Mapping and the Beta Distribution

Christian H. Kasess

http://orcid.org/0000-0002-5644-7229

Acoustics Research Institute, Austrian Academy of Sciences, Vienna 1040, Austria

E-mail Address: christian.kasess@oeaw.ac

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and

Holger Waubke

Acoustics Research Institute, Austrian Academy of Sciences, Vienna 1040, Austria

E-mail Address: holger.waubke@oeaw.ac.at

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https://doi.org/10.1142/S2591728518500421Cited by:2 (Source: Crossref)

This article is part of the issue:

Special Issue on Advances in Finite and Boundary Element Methods; Edited by: M. Kaltenbacher, S. Marburg, M. Ochmann and R. Piscoya

Abstract

The Bouc model describes the behavior of hysteretic materials using three-coupled differential equations. Under white noise excitation, the probability density of the state variables can be approximated using closure methods. Here, a new closure approach is presented that utilizes the beta distribution for hysteretic softening together with a nonlinear mapping using the inverse hyperbolic tangent. The beta distribution properly models the distribution of the force-proportional displacement, the required nonlinear relation between displacement, velocity, and force-proportional displacement is, however, not entirely captured for large excitation levels leading to deviations from the Monte-Carlo simulations.

Keywords:

References

1. R. Bouc, Forced vibration of mechanical systems with hysteresis, in Proc. Fourth Conf. Non-linear Oscillation, Prague (1967). Google Scholar
2. Y. K. Wen, Comparision of Gaussian closure technique and equivalent linearization method, J. Appl. Mech. ASME 47 (1980) 150–154. Crossref, Google Scholar
3. R. Bouc and D. Boussaa, Drifting response of hysteretic oscillators to stochastic excitation, Int. J. Non-Linear Mech. 37 (2002) 1397–1406. Crossref, Google Scholar
4. H. Grundmann, C. Hartmann and H. Waubke, Buildings subjected to stochastic loadings. Preliminary assessment by statistical linearization combined with an evolutionary algorithm, Comput. Struct. 67 (1998) 53–64. Crossref, Google Scholar
5. A. Giaralis and P. D. Spanos, Derivation of equivalent linear properties of Bouc–Wen hysteretic systems for seismic response spectrum analysis via statistical linearization, 10th HSTAM Int. Congr. Mechanics, Chania, Crete, Greece (2013). Google Scholar
6. J. B. Roberts and P. D. Spanos, Random Vibration and Statistical Linearization (Dover Publications Minela, New York, 2003). Google Scholar
7. A. Kolmogorov, Ueber die analytischen Methoden in der Wahrscheinlichkeitsrechnung, Math. Ann. 104 (1931) 415–458. Crossref, Google Scholar
8. L. A. Bergman Jr. and B. F. Spencer, Robust numerical solution of the transient Fokker–Planck equation for nonlinear dynamical systems, in Nonlinear Stochastic Mechanics, IUTAM Symposium (Springer, Berlin, 1992), pp. 49–56. Google Scholar
9. L. A. Bergman Jr. and B. F. Spencer, Numerical solution of the Fokker–Planck equation: The movie, in Proc. Sixth Speciality Conf. (ASCE, New York, 1992), pp. 519–522. Google Scholar
10. L. A. Bergman Jr. and B. F. Spencer, On the numerical solution of the Fokker–Planck equations for nonlinear stochastic systems, Nonlinear Dyn. 4 (1993) 357–372. Crossref, Google Scholar
11. L. A. Bergman Jr., B. F. Spencer, S. F. Wojtkiewicz and E. A. Johnson, Robust numerical solution of the Fokker–Planck equation for second-order dynamical systems under parametric and external white noise excitations, in Nonlinear Dynamics in Stochastic Mechanics, Fields Institute Communications, Vol. 109 (American Mathematical Society, Providence, 1996), pp. 23–37. Google Scholar
12. T. K. Caughey and J. K. Dienes, Analysis of a nonlinear first-order system with a white noise input, J. Appl. Phys. 33(11) (1961) 2476–2479. Crossref, Google Scholar
13. L. Isserlis, On certain probable errors and correlation coefficients of multiple frequency distributions with skew regression, Biometrika 11 (1916) 185–190. Crossref, Google Scholar
14. L. Isserlis, On a formula for the product-moment coefficient of any order of a normal frequency distribution in any number of variables, Biometrika 12 (1918) 134–139. Crossref, Google Scholar
15. H. Waubke, Moment closure technique for materials with Bouc hysteresis and prestressing, 11th Int. Congr. Sound and Vibration (2004). Google Scholar
16. H. Waubke and C. H. Kasess, Gaussian closure technique applied to the hysteretic Bouc model with non-zero mean white noise excitation, J. Sound Vib. 382 (2016) 258–273. Crossref, Google Scholar
17. M. Abramowitz and I. Stegun, Handbook of Mathematical Functions with Formulas Graphs and Mathematical Tables (Dover Publications, New York, 1970). Google Scholar
18. Maxima, A Computer Algebra System. Version 5.40.0 (2014), http://maxima.sourceforge.net. Google Scholar
19. R. J. Boik and J. F. Robinson-Cox, Derivatives of the incomplete beta function, J. Stat. Softw. 3 (1998) 1–19. Crossref, Google Scholar
20. R Core Team, R: A Language and Environment for Statistical Computing, R Foundation for Statistical Computing, Vienna (2014). Google Scholar
21. A. Kronrod, Nodes and Weights of Quadrature Formulas: Sixteen-Place Tables (Consultants Bureau, New York, 1965), Authorized translation from the Russian. Google Scholar
22. H. W. Borchers, Pracma: Practical numerical math functions, R package version 1.8.3 (2015). Google Scholar