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Generalized Play-Operator Under Stochastic Perturbations: An Analytic Approach

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Abstract

A new class of models of hysteretic converters which generalizes the classical play-operator with stochastic properties (defining curves of this operator are treated as non-deterministic and have a random distribution) is proposed. In this case an output of stochastic converter is defined as a random process. The correctness of the definition of the corresponding converter in terms of a special limit construction is proved. Within this definition an output of the corresponding converter is determined at arbitrary continuous inputs. Properties of introduced converters are investigated and illustrative examples are presented.

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Notes

  1. Here we follow the terminology of the classical book [19].

  2. Here we recall the global Lipschitz condition for the function f(z): a real-valued function \(f:{\mathbb {R}}\rightarrow {\mathbb {R}}\) is called Lipschitz continuous if there exists a positive real constant K such that, for all real \(x_1\) and \(x_2\):

    $$\begin{aligned} \left| f(x_{1})-f(x_{2})\right| \leqslant K\left| x_{1}-x_{2}\right| . \end{aligned}$$
    (1)

  3. The notation \(\xi _n{\mathop {\rightarrow }\limits ^{d}}\xi\) can be used in this case. Recall also the definition of the convergence in distribution: the sequence \(\xi _{1},\xi _{2},\ldots\) of random variables converges in distribution to the random variable \(\xi\) if

    $$\begin{aligned} \langle f(\xi _{n})\rangle \rightarrow \langle f(\xi )\rangle \,\,\hbox {when}\,\,n\rightarrow \infty . \end{aligned}$$

  4. Here we use the standard definitions for the moment functions:

    $$\begin{aligned} \langle u\rangle = \int \limits _{-\infty }^{\infty } u \psi (u)\mathrm {d}u,\quad \langle u^{2}\rangle =\int \limits _{-\infty }^{\infty } u^2 \psi (u)\mathrm {d}u, \end{aligned}$$

    where \(\psi (u)=\frac{\mathrm {d}P\{u(t) < u\}}{\mathrm {d}u}\) is the probability density for the random process determined in Theorem 1.

References

  1. Belbas SA (2005) New hysteresis operators with applications to counterterrorism. Appl Math Comput 170:425–439

    MathSciNet  MATH  Google Scholar 

  2. Belhaq M, Bichri A, Der Hogapian J, Mahfoud J (2011) Effect of electromagnetic actuations on the dynamics of a harmonically excited cantilever beam. Int J Non-Linear Mech 46:828–833

    Article  Google Scholar 

  3. Borzunov SV, Semenov ME, Sel’vesyuk NI, Meleshenko PA (2020) Hysteretic converters with stochastic parameters. Math Models Comput Simul 12:164–175

    Article  MathSciNet  Google Scholar 

  4. Bouc R (1967) Forced vibration of mechanical systems with hysteresis. In: Proceedings of the fourth conference on nonlinear oscillation. Prague, Czechoslovakia, p 315

  5. Bouc R (1971) Modèle mathématique d’hystérésis: application aux systèmes à un degrè de liberté. Acustica 24:16–25 (in French)

    MATH  Google Scholar 

  6. Brokate M, Sprekels J (1996) Hysteresis and phase transitions. Springer, New York

    Book  Google Scholar 

  7. Carboni B, Lacarbonara W (2016) Nonlinear dynamic characterization of a new hysteretic device: experiments and computations. Nonlinear Dyn 83:23–39

    Article  Google Scholar 

  8. Charalampakis AE, Koumousis VK (2008) Identification of Bouc-Wen hysteretic systems by a hybrid evolutionary algorithm. J Sound Vib 314:571–585

    Article  Google Scholar 

  9. Charalampakis AE, Koumousis VK (2009) A Bouc-Wen model compatible with plasticity postulates. J Sound Vib 322:954–968

    Article  Google Scholar 

  10. Cross R, McNamara H, Pokrovskii A, Rachinskii D (2008) A new paradigm for modelling hysteresis in macroeconomic flows. Phys B 403:231–236

    Article  Google Scholar 

  11. Fahsi A, Belhaq M, Lakrad F (2009) Suppression of hysteresis in a forced van der Pol-Duffing oscillator. Commun Nonlinear Sci Numer Simul 14:1609–1616

    Article  MathSciNet  Google Scholar 

  12. Ikhouane F, Rodellar J (2005) On the hysteretic Bouc-Wen model. Nonlinear Dyn 42:63–78

    Article  Google Scholar 

  13. Ikhouane F, Rodellar J (2007) Systems with hysteresis: analysis, identification and control using the Bouc-Wen model. Wiley, Chichester

    Book  Google Scholar 

  14. Iwan WD (1966) A distributed-element model for hysteresis and its steady-state dynamic response. ASME J Appl Mech 33:893–900

    Article  Google Scholar 

  15. Janaideh MA, Naldi R, Marconi L, Krejčí P (2013) A hybrid model for the play hysteresis operator. Phys B 430:95–98

    Article  Google Scholar 

  16. Klein O, Krejčí P (2003) Outwards pointing hysteresis operators and asymptotic behaviour of evolution equations. Nonlinear Anal Real World Appl 4:755–785

    Article  MathSciNet  Google Scholar 

  17. Kottaria AK, Charalampakis AE, Koumousi VK (2014) A consistent degrading Bouc-Wen model. Eng Struct 60:235–240

    Article  Google Scholar 

  18. Krasnosel’skii MA, Darinskii VM, Emelin IV, Zabreiko PP, Lifshitz EA (1970) Operator-hysteron. Doklady AN SSSR 190:29–33 (in Russian)

    Google Scholar 

  19. Krasnosel’skii MA, Pokrovskii AV (1989) Systems with hysteresis. Springer, Berlin

    Book  Google Scholar 

  20. Kuehn C, Münch C (2017) Generalized play hysteresis operators in limits of fast-slow systems. SIAM J Appl Dyn Syst 16:1650–1685

    Article  MathSciNet  Google Scholar 

  21. Lacarbonara W, Bernardini D, Vestroni F (2004) Nonlinear thermomechanical oscillations of shape-memory devices. Int J Solids Struct 41:1209–1234

    Article  Google Scholar 

  22. Lacarbonara W, Talò M, Carboni B, Lanzara G (2018) Nonlinear damping: from viscous to hysteretic dampers. In: Belhaq M (ed) Recent trends in applied nonlinear mechanics and physics, vol 199. Springer, New York, pp 227–250 (Proceedings in Physics)

    Chapter  Google Scholar 

  23. Lacarbonara W, Vestroni F (2003) Nonclassical responses of oscillators with hysteresis. Nonlinear Dyn 32:235–258

    Article  Google Scholar 

  24. Lin CJ, Lin PT (2012) Tracking control of a biaxial piezo-actuated positioning stage using generalized Duhem model. Comput Math Appl 64:766–787

    Article  Google Scholar 

  25. Masri SF, Ghanem R, Arrate F, Caffrey J (2006) Stochastic nonparametric models of uncertain hysteretic oscillators. AIAA J 44:2319–2330

    Article  Google Scholar 

  26. Mayergoyz ID (1986) Mathematical models of hysteresis. Phys Rev Lett 56:1518–1521

    Article  Google Scholar 

  27. Mayergoyz ID, Bertotti G (eds) (2005) The science of hysteresis, vol 3. Academic Press, New York

    Google Scholar 

  28. Mayergoyz ID, Dimian M (2005) Stochastic aspects of hysteresis. J Phys Conf Ser 22:139–147

    Article  Google Scholar 

  29. Naser MFM, Ikhouane F (2013) Consistency of the Duhem model with hysteresis. Math Probl Eng 586130:1–16

    Article  MathSciNet  Google Scholar 

  30. Padthe AK, Drincic B, Oh J, Rizos DD, Fassois SD, Bernstein DS (2008) Duhem modeling of friction-induced hysteresis. IEEE Control Syst Mag 28:90–107

    MathSciNet  MATH  Google Scholar 

  31. Rachinskii D (2016) Realization of arbitrary hysteresis by a low-dimensional gradient flow. Discret Contin Dyn Syst B 21:227–243

    Article  MathSciNet  Google Scholar 

  32. Rios LA, Rachinskii D, Cross R (2017) A model of hysteresis arising from social interaction within a firm. J Phys Conf Ser 811(1–12):012011

    Article  MathSciNet  Google Scholar 

  33. Semenov ME, Shevlyakova DV, Meleshenko PA (2014) Inverted pendulum under hysteretic control: stability zones and periodic solutions. Nonlinear Dyn 75:247–256

    Article  MathSciNet  Google Scholar 

  34. Semenov ME, Solovyov AM, Meleshenko PA, Balthazar JM (2018) Nonlinear damping: from viscous to hysteretic dampers. In: Belhaq M (ed) Recent trends in applied nonlinear mechanics and physics, vol 199. Springer, New York, pp 259–275 (Proceedings in Physics)

    Chapter  Google Scholar 

  35. Semenov ME, Solovyov AM, Popov MA, Meleshenko PA (2018) Coupled inverted pendulums: stabilization problem. Arch Appl Mech 88:517–524

    Article  Google Scholar 

  36. Shiryaev AN (2016) Probability-1. Springer, New York

    Book  Google Scholar 

  37. Solovyov AM, Semenov ME, Meleshenko PA, Reshetova OO, Popov MA, Kabulova EG (2017) Hysteretic nonlinearity and unbounded solutions in oscillating systems. Proc Eng 201:578–583

    Article  Google Scholar 

  38. Wen YK (1976) Method for random vibration of hysteretic systems. ASCE J Eng Mech 102:249–263

    Google Scholar 

  39. Zorich VA (2015) Mathematical analysis I, 2nd edn. Springer, Berlin (Universitext)

    Book  Google Scholar 

Download references

Funding

This study was funded by the RFBR (Grants 18-08-00053-a and 19-08-00158-a). The contributions by M.E. Semenov and P.A. Meleshenko (“Generalized stochastic play-operator” and “Definition of the output on piece-wise monotonic inputs”) were supported by the RSF Grant no. 19-11-00197.

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Author notes

  1. Mikhail E. Semenov

    Present address: Digital Technologies Department, Voronezh State University, Universitetskaya sq. 1, 394006, Voronezh, Russia

Authors and Affiliations

  1. Digital Technologies Department, Voronezh State University, Universitetskaya sq. 1, 394006, Voronezh, Russia

    Sergei V. Borzunov & Peter A. Meleshenko

  2. Geophysical Survey of Russia Academy of Sciences, Lenina av. 189, 249035, Obninsk, Russia

    Mikhail E. Semenov

  3. Meteorology Department, Zhukovsky–Gagarin Air Force Academy, Starykh Bolshevikov st. 54 “A”, 394064, Voronezh, Russia

    Mikhail E. Semenov

  4. Mathematics Department, Voronezh State Technical University, XX-letiya Oktyabrya st. 84, 394006, Voronezh, Russia

    Mikhail E. Semenov

  5. State Research Institute of Aviation Systems, Viktorenko st. 7, 125319, Moscow, Russia

    Nikolay I. Sel’vesyuk

  6. Target Search Lab of Groundbreaking Radio Communication Technologies of Advanced Research Foundation, Plekhanovskaya st. 14, 394018, Voronezh, Russia

    Peter A. Meleshenko

Authors
  1. Sergei V. Borzunov
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  3. Nikolay I. Sel’vesyuk
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Correspondence to Mikhail E. Semenov.

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Sergei V. Borzunov declares that he has no conflict of interest. Mikhail E. Semenov declares that he has no conflict of interest. Nikolay I. Sel’vesyuk declares that he has no conflict of interest. Peter A. Meleshenko declares that he has no conflict of interest.

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This paper is a revised and expanded version of a paper entitled “The stochastic play-operator: definition and properties” presented at The 15th International Conference on VIBRATION ENGINEERING AND TECHNOLOGY OF MACHINERY (VETOMAC 2019), 10–15 November 2019, Curitiba, Brazil.

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Borzunov, S.V., Semenov, M.E., Sel’vesyuk, N.I. et al. Generalized Play-Operator Under Stochastic Perturbations: An Analytic Approach. J. Vib. Eng. Technol. 9, 355–365 (2021). https://doi.org/10.1007/s42417-020-00234-1

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