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Effects produced by oscillations applied to nonlinear dynamic systems: a general approach and examples

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Abstract

A general approach to study effects produced by oscillations applied to nonlinear dynamic systems is developed. It implies a transition from initial governing equations of motion to much more simple equations describing only the main slow component of motions (the vibro-transformed dynamics equations). The approach is named as the oscillatory strobodynamics, since motions are perceived as under a stroboscopic light. The vibro-transformed dynamics equations comprise terms that capture the averaged effect of oscillations. The method of direct separation of motions appears to be an efficient and simple tool to derive these equations. A modification of the method applicable to study problems that do not imply restrictions on the spectrum of excitation frequencies is proposed. It allows also to abandon other restrictions usually introduced when employing the classical asymptotic methods, e.g., the requirement for the involved nonlinearities to be weak. The approach is illustrated by several relevant examples from various fields of science, e.g., mechanics, physics, chemistry and biophysics.

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References

  1. Blekhman, I.I.: Vibrational Mechanics. Nonlinear Dynamic Effects, General Approach, Applications. World Scientific, Singapore (2000), 509 p

  2. Blekhman, I.I.: Theory of Vibrational Processes and Devices: Vibrational Mechanics and Vibrational Rheology (in Russian). “Ruda I Metalli”, St. Petersburg (2013), 640 p

  3. Kapitsa, P.L.: Pendulum with vibrating axis of suspension. Uspekhi fizicheskich nauk 44(1), 6–11 (1951). (in Russian)

    Google Scholar 

  4. Thomsen, J.J.: Vibrations and Stability: Advanced Theory, Analysis and Tools. Springer, Berlin (2003)

    Book  MATH  Google Scholar 

  5. Landa, P.S.: Nonlinear Oscillations and Waves in Dynamical Systems, p. 564. Kluwer, Dordrecht (1996)

    Book  MATH  Google Scholar 

  6. Neimark, YuI, Landa, P.S.: Stochastic and Chaotic Oscillations, p. 424. URSS, Moscow (2008) (in Russian)

  7. Blekhman, I.I.: A general approach to study effects of high-frequency actions on dynamical systems. Vestnik Nignegorogskogo Universiteta 2(4), 65–66 (2011). (in Russian)

    Google Scholar 

  8. Blekhman, I.I.: Oscillatory strobodynamics—a new area in nonlinear oscillations theory, nonlinear dynamics and cybernetical physics. Cybern. Phys. 1(1), 5–10 (2012)

    MathSciNet  Google Scholar 

  9. Nayfeh, A.H., Mook, D.T.: Nonlinear Oscillations, p. 720. Wiley-Interscience, New York (1979)

    MATH  Google Scholar 

  10. Bogoliubov, N.N., Mitropolskii, JuA: Asymptotic Methods in the Theory of Non-linear Oscillations. Gordon and Breach, New York (1961)

    Google Scholar 

  11. Sanders, J.A., Verhulst, F.: Averaging Methods in Nonlinear Dynamical Systems. Springer-Verlag, Berlin (1985)

    Book  MATH  Google Scholar 

  12. Sorokin, V.S.: Analysis of motion of inverted pendulum with vibrating suspension axis at low-frequency excitation as an illustration of a new approach for solving equations without explicit small parameter. Int. J. Non Linear Mech. 63, 1–9 (2014)

    Article  Google Scholar 

  13. Sorokin, V.S.: On the unlimited gain of a nonlinear parametric amplifier. Mech. Res. Commun. 62, 111–116 (2014)

    Article  Google Scholar 

  14. Magnus, K.: Vibrations. Blackie, London (1965)

    Google Scholar 

  15. Selected Topics in Vibrational Mechanics (edited by I.I. Blekhman). World Scientific, New Jersey (2002), 409 p

  16. Bogoliubov, N.N.: Selected Works in 3 Volumes. Naukova Dumka, Kiev (1969). (in Russian)

    Google Scholar 

  17. Volosov, V.M., Morgunov, B.I.: The Method of Averaging in the Theory of Nonlinear Oscillatory Systems. Moscow State University, Moscow (1971). (in Russian)

    MATH  Google Scholar 

  18. Blekhman, I.I., Myshkis, A.D., Panovko, Ya.G.: Applied Mathematics: Subject, Logic, Specific Approaches 3rd edn. Comkniga/URSS, Moscow (2005) 376 p (in Russian)

  19. Bolotin, V.V.: The Dynamic Stability of Elastic Systems. Holden-Day, San Francisco (1964)

    MATH  Google Scholar 

  20. Troger, H., Steindl, A.: Nonlinear Stability and Bifurcation Theory. Springer, Wien (1991)

    Book  MATH  Google Scholar 

  21. Shishkina, E.V., Blekhman, I.I., Cartmell, M.P., Gavrilov, S.N.: Application of the method of direct separation of motions to the parametric stabilization of an elastic wire. Nonlinear Dyn. 54, 313–331 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  22. Vasilkov, V.B.: Influence of Vibration on Nonlinear Phenomena in Mechanical Systems. Thesis Doct. Techn. Nauk, St. Petersburg (2008) (in Russian)

  23. Chelomey, V.N.: Mechanical paradoxes caused by vibrations. Sov. Phys. Dokl. 28, 387–392 (1983)

    Google Scholar 

  24. Chelomey, S.V.: Dynamic stability upon high-frequency parametric excitation. Sov. Phys. Dokl. 26, 390–392 (1981)

    Google Scholar 

  25. Paidoussis, M.P., Sundararajan, C.: Parametric and combination resonances of a pipe conveying pulsating fluid. J. Appl. Mech. Trans. ASME 42, 780–784 (1975)

    Article  MATH  Google Scholar 

  26. Blekhman, I.I.: Vibrational dynamic materials and composites. J. Sound Vib. 317(3–5), 657–663 (2008)

    Article  Google Scholar 

  27. Andrievsky, B.R., Fradkov, A.L.: Selected Chapters of Control Theory. Nauka, St. Petersburg (1999). (in Russian)

    MATH  Google Scholar 

  28. Kuznetsov, A.P., Kuznetsov, S.P., Riskin, N.M.: Nonlinear Oscillations. Fizmatlit, Moscow (2005). (in Russian)

    Google Scholar 

  29. Efimov, D.V.: Robust and Adaptive Control of Nonlinear Oscillations. Nauka, St. Petersburg (2005). (in Russian)

    Google Scholar 

  30. Scott, A.: Nonlinear Science, Emergence and Dynamics of Coherent Structures, 2nd edn. Oxford University Press, Oxford (2003)

    MATH  Google Scholar 

  31. Zeldovich, Y.B., Barenblatt, G.I.: Theory of flame propagation. Combust Flame 3, 61–74 (1959)

    Article  Google Scholar 

  32. Offner, F., Weiberg, A., Young, C.: Nerve conduction theory: some mathematical consequences of Bernstein’s model. Bull. Math. Biophys. 2, 89–103 (1940)

    Article  MathSciNet  Google Scholar 

  33. Brillouin, L.: Wave Propagation in Periodic Structures, 2nd edn. Dover Publications, New York (1953)

    MATH  Google Scholar 

  34. Hayashi, C.: Nonlinear Oscillations in Physical Systems. McGraw-Hill, New York (1964)

    MATH  Google Scholar 

  35. Postma, H.W.C., Kozinsky, I., Husain, A., Roukes, M.L.: Dynamic range of nanotube- and nanowire-based electromechanical systems. Appl. Phys. Lett. 86, 223105 (2005)

    Article  Google Scholar 

  36. Rhoads, J.F., Shaw, S.W.: The impact of nonlinearity on degenerate parametric amplifiers. Appl. Phys. Lett. 96, 234101 (2010)

    Article  Google Scholar 

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Acknowledgments

The work is carried out with financial support from the Russian Science Foundation, Grant 14-19-01190.

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Authors and Affiliations

  1. Institute for Problems in Mechanical Engineering, V.O., Bolshoj pr. 61, St. Petersburg, Russia, 199178

    I. I. Blekhman & V. S. Sorokin

  2. Department of Mechanical Engineering, Technical University of Denmark, Nils Koppels Allé, Building 404, 2800, Kgs. Lyngby, Denmark

    V. S. Sorokin

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  1. I. I. Blekhman
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  2. V. S. Sorokin
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Correspondence to V. S. Sorokin.

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Blekhman, I.I., Sorokin, V.S. Effects produced by oscillations applied to nonlinear dynamic systems: a general approach and examples. Nonlinear Dyn 83, 2125–2141 (2016). https://doi.org/10.1007/s11071-015-2470-x

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