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Oscillations Under Hysteretic Conditions: From Simple Oscillator to Discrete Sine-Gordon Model

  • Mikhail E. SemenovEmail author
  • Olga O. Reshetova
  • Akim V. Tolkachev
  • Andrey M. Solovyov
  • Peter A. Meleshenko
Conference paper
Part of the Springer Proceedings in Physics book series (SPPHY, volume 228)

Abstract

In this paper we study the resonance properties of oscillating system in the case when the energy pumping is made by external source of hysteretic nature. We investigate the unbounded solutions of autonomous oscillating system with hysteretic block with a negative spin. The influence of a hysteretic block on an oscillator in the presence of Coulomb and viscous friction is also investigated. Namely, we establish the appearance of self-oscillating regimes for both kinds of friction. A separate part of this work is devoted to synchronization of periodic self-oscillations by a harmonic external force. Using the small parameter approach it is shown that the width of “trapping” band depends on the intensity (amplitude) of the external impact. Also in this work we introduce the novel class of hysteretic operators with random parameters. We consider the definition of these operators in terms of the “input-output” relations, namely: for all permissible continuous inputs corresponds the output in the form of stochastic Markovian process. The properties of such operators are also considered and discussed on the example of a non-ideal relay with random parameters. Application of hysteretic operator with stochastic parameters is demonstrated on the example of simple oscillating system and the results of numerical simulations are presented. We consider also a nonlinear dynamical system which is a set of nonlinear oscillators coupled by springs with hysteretic blocks (modified sine-Gordon system or hysteretic sine-Gordon model where the hysteretic nonlinearity is simulated by the Bouc-Wen model). We investigate the wave processes (namely, the solitonic solutions) in such a system taking into account the hysteretic nonlinearity in the coupling.

Keywords

Hysteresis Oscillator Non-ideal relay Random parameters Sine-gordon model Solitonic solutions Bouc-wen model 

Notes

Acknowledgements

The works of authors (Introduction, Oscillator under hysteretic force and Oscillator under force with random parameters (Sects. 12.112.3)) was supported by the RFBR (Grants 17-01-00251-a, 18-08-00053-a, and 19-08-00158-a). The work of M.E. Semenov and P.A. Meleshenko (Hysteresis in discrete sine-Gordon model (Sect. 12.4)) was supported by the RSF grant No. 19-11-0197.

References

  1. 1.
    R. Bouc, Modèle mathématique d’hystérésis: application aux systèmes à un degré de liberté. Acustica 24, 16–25 (1971)zbMATHGoogle Scholar
  2. 2.
    A.E. Charalampakis, The response and dissipated energy of Bouc-Wen hysteretic model revisited. Arch. Appl. Mech. 85(9), 1209–1223 (2015)ADSCrossRefGoogle Scholar
  3. 3.
    H. Haken, Quantum Field Theory of Solids: An Introduction (North-Holland, 1976)Google Scholar
  4. 4.
    V. Hassani, T. Tjahjowidodo, T. Nho Do, A survey on hysteresis modeling, identification and control. Mech. Syst. Signal Process. 49(1–2), 209–233 (2014).  https://doi.org/10.1016/j.ymssp.2014.04.012ADSCrossRefGoogle Scholar
  5. 5.
    F. Ikhouane, J.E. Hurtado, J. Rodellar, Variation of the hysteresis loop with the Bouc-Wen model parameters. Nonlinear Dyn. 48(4), 361–380 (2007)MathSciNetCrossRefGoogle Scholar
  6. 6.
    F. Ikhouane, V. Mañosa, J. Rodellar, Dynamic properties of the hysteretic Bouc-Wen model. Syst. Control. Lett. 56(3), 197–205 (2007)MathSciNetCrossRefGoogle Scholar
  7. 7.
    F. Ikhouane, J. Rodellar, On the hysteretic Bouc-Wen model. Nonlinear Dyn. 42(1), 63–78 (2005)CrossRefGoogle Scholar
  8. 8.
    A. Krasnosel’skii, A. Pokrovskii, Dissipativity of a nonresonant pendulum with ferromagnetic friction. Autom. Remote. Control. 67, 221–232 (2006)MathSciNetCrossRefGoogle Scholar
  9. 9.
    M.A. Krasnosel’skii, A.V. Pokrovskii, Systems with Hysteresis (Springer, Berlin, 1989)CrossRefGoogle Scholar
  10. 10.
    L.D. Landau, E.M. Lifshitz, Course of Theoretical Physics, vol. 1, Mechanics (Pergamon Press, 1960)Google Scholar
  11. 11.
    A.J. Lichtenberg, R. Livi, M. Pettini, S. Ruffo, Dynamics of Oscillator Chains (Springer, Berlin, 2007), pp. 21–121zbMATHGoogle Scholar
  12. 12.
    P.A. Meleshenko, A.V. Tolkachev, M.E. Semenov, A.V. Perova, A.I. Barsukov, A.F. Klinskikh, Discrete hysteretic sine-Gordon model: soliton versus hysteresis, in MATEC Web of Conferences, vol. 241 (2018), p. 01027.  https://doi.org/10.1051/matecconf/201824101027CrossRefGoogle Scholar
  13. 13.
    B. Øksendal, Stochastic Differential Equations. An Introduction with Applications (Springer, Berlin, 2003)zbMATHGoogle Scholar
  14. 14.
    Y. Rochdi, F. Giri, F. Ikhouane, F.Z. Chaoui, J. Rodellar, Parametric identification of nonlinear hysteretic systems. Nonlinear Dyn. 58(1), 393–404 (2009).  https://doi.org/10.1007/s11071-009-9487-yMathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    A. Scott, A nonlinear Klein-Gordon equation. Am. J. Phys. 37(1), 52–61 (1969)ADSCrossRefGoogle Scholar
  16. 16.
    A.C. Scott, Active and Nonlinear Wave Propagation in Electronics (Wiley-Interscience, New-York, 1970)Google Scholar
  17. 17.
    A.C. Scott, Nonlinear Science. Emergence and Dynamics of Coherent Structures (Oxford University Press, 1999)Google Scholar
  18. 18.
    M.E. Semenov, P.A. Meleshenko, A.M. Solovyov, A.M. Semenov, Hysteretic Nonlinearity in Inverted Pendulum Problem (Springer International Publishing, 2015), pp. 463–506Google Scholar
  19. 19.
    M.E. Semenov, A.M. Solovyov, P.A. Meleshenko, Elastic inverted pendulum with backlash in suspension: stabilization problem. Nonlinear Dyn. 82(1), 677–688 (2015).  https://doi.org/10.1007/s11071-015-2186-yMathSciNetCrossRefGoogle Scholar
  20. 20.
    M.E. Semenov, A.M. Solovyov, M.A. Popov, P.A. Meleshenko, Coupled inverted pendulums: stabilization problem. Arch. Appl. Mech. 88(4), 517–524 (2018)ADSCrossRefGoogle Scholar
  21. 21.
    M.E. Semenov, A.M. Solovyov, A.G. Rukavitsyn, V.A. Gorlov, P.A. Meleshenko, Hysteretic damper based on the Ishlinsky–Prandtl model, in MATEC Web of Conferences, vol. 83 (2016), p. 01008CrossRefGoogle Scholar
  22. 22.
    J. Sieber, T. Kalmár-Nagy, Stability of a chain of phase oscillators. Phys. Rev. E 84, 016227 (2011)ADSCrossRefGoogle Scholar
  23. 23.
    A. Solovyov, M. Semenov, P. Meleshenko, A. Barsukov, Bouc-Wen model of hysteretic damping. Procedia Eng. 201, 549–555 (2017)CrossRefGoogle Scholar
  24. 24.
    A.M. Solovyov, M.E. Semenov, P.A. Meleshenko, O.O. Reshetova, M.A. Popov, E.G. Kabulova, Hysteretic nonlinearity and unbounded solutions in oscillating systems. Procedia Eng. 201, 578–583 (2017)CrossRefGoogle Scholar
  25. 25.
    A. Tolkachev, M. Semenov, P. Meleshenko, O. Reshetova, A. Klinskikh, E. Karpov, Sine-Gordon system with hysteretic links. J. Phys. Conf. Ser. 1096, 012072 (2018).  https://doi.org/10.1088/1742-6596/1096/1/012072CrossRefGoogle Scholar
  26. 26.
    C.G. Torre, Foundations of Wave Phenomena (Utah State University, 2015)Google Scholar
  27. 27.
    D.I. Trubetskov, A.G. Rozhnev, Lineynyye kolebaniya i volny (Fizmatlit, Moscow, 2001). (in Russian)Google Scholar
  28. 28.
    J.Y. Tu, P.Y. Lin, T.Y. Cheng, Continuous hysteresis model using Duffing-like equation. Nonlinear Dyn. 80(1), 1039–1049 (2015).  https://doi.org/10.1007/s11071-015-1926-3MathSciNetCrossRefGoogle Scholar
  29. 29.
    Y.K. Wen, Method for random vibration of hysteretic systems. J. Eng. Mech. 102(2), 249–263 (1976)Google Scholar
  30. 30.
    V.G. Zadorozhniy, Linear chaotic resonance in vortex motion. Comput. Math. Math. Phys. 53(4), 486–502 (2013)MathSciNetCrossRefGoogle Scholar
  31. 31.
    V.G. Zadorozhniy, S.S. Khrebtova, First moment functions of the solution to the heat equation with random coefficients. Comput. Math. Math. Phys. 49(11), 1853 (2009)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Singapore Pte Ltd. 2019

Authors and Affiliations

  • Mikhail E. Semenov
    • 1
    • 2
    • 3
    • 4
    Email author
  • Olga O. Reshetova
    • 5
  • Akim V. Tolkachev
    • 6
  • Andrey M. Solovyov
    • 5
  • Peter A. Meleshenko
    • 7
  1. 1.Geophysical Survey of Russia Academy of SciencesObninskRussia
  2. 2.Meteorology DepartmentZhukovsky–Gagarin Air Force AcademyVoronezhRussia
  3. 3.Digital Technologies DepartmentVoronezh State UniversityVoronezhRussia
  4. 4.Mathematics DepartmentVoronezh State University of Architecture and Civil EngineeringVoronezhRussia
  5. 5.Digital Technologies DepartmentVoronezh State UniversityVoronezhRussia
  6. 6.Zhukovsky–Gagarin Air Force AcademyVoronezhRussia
  7. 7.Communication DepartmentZhukovsky–Gagarin Air Force AcademyVoronezhRussia

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