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Generalized Krylov-Bogoliubov Method for Solving Strong Nonlinear Vibration

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Lectures on Nonlinear Dynamics

Part of the book series: Understanding Complex Systems ((UCS))

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Abstract

In this paper an analytic procedure for solving strong nonlinear differential equation of the vibration is developed. The Krylov–Bogoliubov method, introduced for weak nonlinear equations, is generalized for the strong nonlinear ones with power order nonlinearity. Based on the exact solution of the truly nonlinear differential equation (without linear term), given in the form of the Ateb function, the approximate solution of the perturbed version of vibration equation is obtained. In addition, the method is applied for strong nonlinear oscillator with time variable parameter. As the special case, motion of the strong nonlinear oscillator with slow variable mass is investigated. In the paper the steady state motion of the forced strong nonlinear one-degree-of-freedom oscillator is also analysed. Using the Melnikov criteria the condition for existence of deterministic chaos in excited strong nonlinear oscillator with linear damping is determined. For suppressing of the chaos the delayed feedback control, i.e., the ‘Pyragas method’ based on the idea of the stabilization of unstable periodic orbits embedded within a strange attractor is suggested.

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References

  1. N.N. Bogoliubov and J.A. Mitropolski. Asimproticheskie methodi v teorii nelinejnih kole-banij. Nauka, Moscow, 1974.

    Google Scholar 

  2. W.-H. Chen and R. F. Gibson. Property distribution determination for nonuniform composite beams from vibration response measurements and Galerkin’s method. Journal of Applied Mechanics, 65(1):127–133, mar 1998.

    Article  Google Scholar 

  3. L. Cveticanin. Dynamics of machines with variable mass. Gordon and Breach Science Publishers, London, 1998.

    MATH  Google Scholar 

  4. L. Cveticanin. The approximate solving methods for the cubic duffing equation based on the Jacobi elliptic functions. International Journal of Nonlinear Sciences and Numerical Simulation, 10:1491–1516, 2009.

    Article  Google Scholar 

  5. L. Cveticanin. Dynamics of the mass variable body. Chapter 3, in “Dynamics of mechanical systems with variable mass”. Series CISM International Centre for Mechanical Sciences, 2014.

    Google Scholar 

  6. L. Cveticanin. Strongly nonlinear oscillators – analytical solutions. Springer, Berlin, 2014.

    Book  MATH  Google Scholar 

  7. L. Cveticanin. Strong nonlinear oscillators – analytical solutions. Springer, Berlin, 2018.

    Book  MATH  Google Scholar 

  8. L. Cveticanin, S. Vujkov, and D. Cveticanin. Application of Ateb and generalized trigonometric functions for nonlinear oscillators. Archive of Applied Mechanics, 90(11):2579–2587, 2020.

    Article  MATH  Google Scholar 

  9. L. Cveticanin and M. Zukovic. Melnikov’s criteria and chaos in systems with fractional order deflection. Journal of Sound and Vibration, 326(3-5):768–779, oct 2009.

    Article  Google Scholar 

  10. L. Cveticanin, M. Zukovic, and D. Cveticanin. Exact steady states of periodically forced and essentially nonlinear and damped oscillator. Communications in Nonlinear Science and Numerical Simulation, 78:104895, nov 2019.

    Article  MathSciNet  MATH  Google Scholar 

  11. L. Cveticanin, M. Zukowic, and D. Cveticanin. Steady state vibration of the periodically forced and damped pure nonlinear two-degrees-of-freedom oscillator. Journal of Theoretical and Applied Mechanics, 57(2):445–460, apr 2019.

    Article  Google Scholar 

  12. I.S. Gradstein and I.M. Rjizhik. Tablici integralov, summ, rjadov I proizvedenij. Nauka, Moscow, 1971.

    Google Scholar 

  13. J. Guckenheimer and P. Holmes. Nonlinear oscillations, dynamical systems, and bifurcations of vector fields. Springer, New York, 1983.

    Book  MATH  Google Scholar 

  14. H. W. Haslach. Post-buckling behavior of columns with non-linear constitutive equations. International Journal of Non-Linear Mechanics, 20(1):53–67, jan 1985.

    Article  MATH  Google Scholar 

  15. H. W. Haslach. Influence of adsorbed moisture on the elastic post-buckling behavior of columns made of non-linear hydrophilic polymers. International Journal of Non-Linear Mechanics, 27(4):527–546, jul 1992.

    Article  MATH  Google Scholar 

  16. I. Kovacic, L. Cveticanin, M. Zlokolica, and Z. Rakaric. Jacobi elliptic functions: A review of nonlinear oscillatory application problems. Journal of Sound and Vibration, 380:1–36, 2016.

    Article  Google Scholar 

  17. G. Lewis and F. Monasa. Large deflections of cantilever beams of non-linear materials of the ludwick type subjected to an end moment. International Journal of Non-Linear Mechanics, 17(1):1–6, jan 1982.

    Article  MATH  Google Scholar 

  18. V.K. Melnikov. On the stability of the center for time periodic perturbations. Transactions of the Moscow Mathematical Society, 12:1–57, 1963.

    MathSciNet  Google Scholar 

  19. R.E. Mickens. Truly nonlinear oscillations. World Scientific, 2010.

    Google Scholar 

  20. A.H. Nayfeh and D.T. Mook. Nonlinear oscillations. Wiley, New York, 1979.

    MATH  Google Scholar 

  21. G. Parthap and T.K. Varadan. The inelastic large deformation of beams. Journal of Applied Mechanics, 45:689–690, 1976.

    Article  Google Scholar 

  22. W.N. Patten, S. Sha, and C. Mo. A vibrational model of open celled polyurethane foam automative seat cushion. Journal of Sound and Vibration, 217(1):145–161, oct 1998.

    Article  Google Scholar 

  23. R.M. Rosenberg. The Ateb(h) functions and their properties. Quaterly of Applied Mathematics, 21:37–47, 1963.

    Article  MathSciNet  MATH  Google Scholar 

  24. D. Russel and T. Rossing. Testing the nonlinearity of piano hammers using residual shock spectra. Acoustica-Acta Acustica, 84:967–975, 1998.

    Google Scholar 

  25. A. F. Vakakis and A. Blanchard. Exact steady states of the periodically forced and damped duffing oscillator. Journal of Sound and Vibration, 413:57–65, jan 2018.

    Article  Google Scholar 

  26. B. Van der Pol. On relaxation-oscillations. The London, Edinburgh and Dublin Philosoph-ical Magazine, 2(7):901–912, 1926.

    Google Scholar 

  27. Q Zhu and M Ishitobi. Chaos and bifurcations in a nonlinear vehicle model. Journal of Sound and Vibration, 275(3-5):1136–1146, aug 2004.

    Article  Google Scholar 

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Authors and Affiliations

  1. Obuda University, 1034 Budapest, Becsi ut 96/B, Hungary

    Livija Cveticanin

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  1. Livija Cveticanin
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Correspondence to Livija Cveticanin .

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Editors and Affiliations

  1. Escola Politécnica da Universidade de São Paulo, São Paulo, Brazil

    José Roberto Castilho Piqueira

  2. Escola Politécnica da Universidade de São Paulo, São Paulo, Brazil

    Carlos Eduardo Nigro Mazzilli

  3. Escola Politécnica da Universidade de São Paulo, São Paulo, Brazil

    Celso Pupo Pesce

  4. Escola Politécnica da Universidade de São Paulo, São Paulo, Brazil

    Guilherme Rosa Franzini

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Cveticanin, L. (2024). Generalized Krylov-Bogoliubov Method for Solving Strong Nonlinear Vibration. In: Castilho Piqueira, J.R., Nigro Mazzilli, C.E., Pesce, C.P., Franzini, G.R. (eds) Lectures on Nonlinear Dynamics. Understanding Complex Systems. Springer, Cham. https://doi.org/10.1007/978-3-031-45101-0_9

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  • DOI: https://doi.org/10.1007/978-3-031-45101-0_9

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-031-45100-3

  • Online ISBN: 978-3-031-45101-0

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