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Hysteresis modelling and chaos prediction in one- and two-DOF hysteretic models

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Abstract

In the present work hysteresis is simulated by means of internal variables. The analytical models of different types of hysteresis loops allow the reproduction of major and minor loops and provide a high degree of correspondence with experimental data. In models of this type adding an external periodic excitation or increasing the number of dimensions can lead to the occurrence of chaotic behaviour. Using an effective algorithm based on numerical analysis of the wandering trajectories [1–7], the evolution of the chaotic behaviour regions of oscillators with hysteresis is presented in various parametric planes. The substantial influence of a hysteretic dissipation value on the form and location of these regions, as well as the restraining and generating effects of hysteretic dissipation on the occurrence of chaos, are ascertained. Conditions for pinched hysteresis are defined. Furthermore, autonomous coupled hysteretic oscillators under sliding friction are investigated. Conditions for the occurrence of chaotic behaviour in a two-degree-of-freedom (two-DOF) hysteretic system are found in the plane of maximal static friction forces of both oscillators versus belt velocity.

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References

  1. Awrejcewicz J. and Mosdorf R. (2003). Numerical Analysis of some Problems of Chaotic Dynamics (in Polish). WNT, Warsaw

    Google Scholar 

  2. Awrejcewicz J. and Dzyubak L. (2003). Stick-slip chaotic oscillations in a quasi-autonomous mechanical system. Int. J. Nonlin. Sci. Num. Simul. 4(2): 155–160

    Google Scholar 

  3. Awrejcewicz J., Dzyubak L. and Grebogi C. (2004). A direct numerical method for quantifying regular and chaotic orbits. Chaos Solit Fract. 19: 503–507

    Article  MATH  Google Scholar 

  4. Awrejcewicz J. and Dzyubak L. (2005). Quantifying smooth and non-smooth regular and chaotic dynamics. Int. J. Bifur. Chaos 15(6): 2041–2055

    Article  MATH  MathSciNet  Google Scholar 

  5. Awrejcewicz J. and Dzyubak L. (2005). Influence of hysteretic dissipation on chaotic responses. J. Sound Vib. 284: 513–519

    Article  MathSciNet  Google Scholar 

  6. Awrejcewicz J. and Dzyubak L. (2005). Evolution of chaotic regions in control parameter planes depending on hysteretic dissipation. Spec. Issue Nonlin. Anal. 63(5–7): 155–164

    Article  Google Scholar 

  7. Awrejcewicz J., Dzyubak L. and Grebogi C. (2005). Estimation of chaotic and regular (stick-slip and slip-slip) oscillations exhibited by coupled oscillators with dry friction. Nonlin. Dyn. 42(2): 383–394

    Article  MATH  MathSciNet  Google Scholar 

  8. Awrejcewicz J. and Lamarque C.-H. (2003). Bifurcations and Chaos in Nonsmooth Mechanical systems. World Scientific, New Jersey, London, Singapore, Hong Kong

    Google Scholar 

  9. Bastien J., Schatzman M. and Lamarque C.H. (2000). Study of some rheological models with a finite number of degrees of freedom. Eur. J. Mech. A Solids 19: 277–307

    Article  MATH  MathSciNet  Google Scholar 

  10. di Bernardo M., Kowalczyk P. and Nordmark A. (2003). Sliding bifurcation: a novel mechanism for the sudden onset of chaos in dry-friction oscillators. Int. J. Bifur. Chaos 13(10): 2935–2948

    Article  MATH  MathSciNet  Google Scholar 

  11. Brogliato B. (1999). Nonsmooth Mechanics. Springer, Berlin Heidelberg New York

    MATH  Google Scholar 

  12. Capecchi D. and Masiani R. (1996). Reduced phase space analysis for hysteretic oscillators of Masing type. Chaos, Solitons Fractals 7: 1583–1600

    Article  MATH  MathSciNet  Google Scholar 

  13. Coddington E.A. and Levinson N. (1955). Theory of Ordinary Differential Equations. McGraw-Hill, New York

    MATH  Google Scholar 

  14. Lacarbonara W. and Vestroni F. (2003). Nonclassical responses of oscillators with hysteresis. Nonlin. Dyn. 32(3): 235–238

    Article  MATH  Google Scholar 

  15. Li H.G., Zhang J.W. and Wen B.C. (2002). Chaotic behaviors of a bilinear hysteretic oscillator. Mech. Res. Commun. 29: 283–289

    Article  MATH  Google Scholar 

  16. Leine, R.I., van de Vrande, B.L., van Campen, D.H.: Bifurcations in Nonlinear Discontinuous Systems. Report WFW 99.010, Vakgroep Fundamentele Werktuigkunde, Eindhoven (1999)

  17. Levinson N (1949). A second order differential equation with singular solutions. Ann. Math. 50: 127–153

    Article  MathSciNet  Google Scholar 

  18. Lyapunov, A.M.: Obshaya zadacha ob ustoichivosti dvizhenya, ONTI, Leningrad-Moskva, in Russian. (Liapunov, A.M.~(1966) Stability of Motion, Academic, New York) (1935)

  19. Masiani R., Capecchi D. and Vestroni F. (2002). Resonant and coupled response of hysteretic two-degree-of-freedom systems using harmonic balance method. Int. J. Nonlin. Mech. 37: 1421–1434

    Article  MATH  Google Scholar 

  20. Movchan A.A. (1960). Stability of processes with respect to two metrics (Ustoichivost’ protsessov po dvum metrikam) (in Russian). Prikl. Mat. Mekh. 24(6): 988–1001

    Google Scholar 

  21. Movchan A.A. (1965). Theoretical foundation of some criteria of equilibrium stability of plates (Obosnovanie nekotoryh kriteriev ustoichivosti ravnovesiia plastin). Inzhenernyi Z. 5(4): 773–777

    Google Scholar 

  22. Niemytzki, V.: Über vollstāunstabile dynamische Systeme. Annalidi Mat., Ser. IV, t.14 (1935–36)

  23. Ortin J. and Delaey L. (2002). Hysteresis in shape-memory alloys. Int. J. Nonlin. Mech. 37: 1275–1281

    Article  MATH  Google Scholar 

  24. Sapinski B. (2003). Dynamic characteristics of an experimental MR fluid. Eng. Trans. 51(4): 399–418

    Google Scholar 

  25. Sapinski B. and Filus J. (2003). Analysis of parametric models of MR linear damper. J. Theor. Appl. Mech. 41(2): 215–240

    Google Scholar 

  26. Schiehlen, W.: Nonlinear oscillations in multibody systems – modeling and stability assessment. In: Proceedings of 1st European Nonlinear Oscillations Conference, Hamburg (Mathematical Research, vol.72). Akademie Verlag, Berlin pp. 85–106 1993

  27. Wolf A., Jack B., Swinney H.L. and Vastano J.A. (1985). Determining Lyapunov exponents from a time series. Physica 16D: 285–317

    Google Scholar 

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Authors and Affiliations

  1. Department of Automatics and Biomechanics, Technical University of Łódź, Stefanowskiego 1/15, 90-924, Łódź, Poland

    Jan Awrejcewicz & Larisa Dzyubak

  2. Department of Applied Mathematics, National Technical University “Kharkov Polytechnic Institute”, 21 Frunze St., 61002, Kharkov, Ukraine

    Larisa Dzyubak

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  1. Jan Awrejcewicz
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  2. Larisa Dzyubak
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Correspondence to Jan Awrejcewicz.

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Awrejcewicz, J., Dzyubak, L. Hysteresis modelling and chaos prediction in one- and two-DOF hysteretic models. Arch Appl Mech 77, 261–279 (2007). https://doi.org/10.1007/s00419-006-0101-1

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  • DOI: https://doi.org/10.1007/s00419-006-0101-1

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