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Nonlinear oscillations of an elastic two-degrees-of-freedom pendulum

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Abstract

Nonlinear oscillations of the vertical plane swinging spring pendulum in the resonance case are studied (frequencies ratio regarding horizontal and vertical directions is equal to 1:2). Square and cubic terms of the Hamiltonian are taken into account. Novel normal form method, i.e., the so called invariant normalization is applied to solve the stated problem.

Full system of integrals exhibits equations of the normal form, and solution for the pendulum coordinates is expressed via elementary functions. Frequencies of modes of oscillations are proportional to the first power of amplitude, and not to the second power as it is exhibited by one dimensional Duffing oscillator. Amplitudes of the modes are changed periodically, and energy from one mode is transited to energy of the second one, whereas the period of oscillations depends on the initial conditions. It is illustrated that asymptotic solution with small amplitudes approximates well numerical solution of the governing equations. In addition, an example of a periodic stable solution with constant amplitudes of the oscillation modes is given. Stability of this solution is proved.

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References

  1. Poincaré, A.: Collected Works in Three Volumes, vol. II. Nauka, Moscow (1972), in Russian

    Google Scholar 

  2. Birkhoff, G.D.: Dynamical Systems. American Mathematical Society, New York (1927)

    MATH  Google Scholar 

  3. Bruno, A.D.: The Restricted Three Body Three Problem. Walter de Gruyter, Berlin (1994)

    MATH  Google Scholar 

  4. Arnold, V.I.: Mathematical Methods of Classical Mechanics. Academic Press, Boston (1978)

    MATH  Google Scholar 

  5. Arnold, V.I.: Geometrical Methods in the Theory of Ordinary Differential Equations. Springer, New York (1983)

    MATH  Google Scholar 

  6. Arnold, V.I., Kozlov, A.I., Neishtadt, A.I.: Mathematical Aspects of Classical and Celestial Mechanics Dynamical Systems III. In: Encyclopaedia of Mathematical Sciences, vol. 3. Springer, Berlin (1993)

    Google Scholar 

  7. Verhulst, F.: Differential Equations and Dynamical Systems, 2nd edn. Springer, Berlin (2006)

    Google Scholar 

  8. Nayfeh, A.H.: Perturbation Methods. Wiley, New York (1973)

    MATH  Google Scholar 

  9. Awrejcewicz, J., Krysko, V.A.: Introduction to Asymptotic Methods. Chapmann and Hall/CRC Press, Boca Raton/London/New York (2006)

    MATH  Google Scholar 

  10. Zhuravlev, V.F.: Introduction to Theoretical Mechanics. Nauka Fizmatlit, Moscow (1997)

    Google Scholar 

  11. Zhuravlev, V.F.: Invariant normalization of non-autonomus Hamiltonian systems. J. Appl. Math. Mech. 66(3), 347–355 (2002)

    Article  MathSciNet  Google Scholar 

  12. Petrov, A.G.: A parametric method of construcing Poincaré mapping in hydrodynamic systems. J. Appl. Math. Mech. 66(6), 905–921 (2002)

    Article  MathSciNet  Google Scholar 

  13. Petrov, A.G.: Modification of invariant normalization method of Hamiltonians with help parametrization of the canonical transformations. Dokl. Phys. 47(10), 742–744 (2002)

    Article  MathSciNet  Google Scholar 

  14. Petrov, A.G.: Invariant normalization of non-autonomus Hamiltonian systems. J. Appl. Math. Mech. 68(3), 357–367 (2004)

    Article  MathSciNet  Google Scholar 

  15. Awrejcewicz, J., Petrov, A.G.: On the normal forms of Hamiltonian systems. Nonlinear Dyn. 48, 185–197 (2007)

    Article  MathSciNet  Google Scholar 

  16. Zaripov, M.N., Petrov, A.G.: Nonlinear oscillations of a swinging spring. Dokl. Phys. 49(11), 691–696 (2004)

    Article  MathSciNet  Google Scholar 

  17. Petrov, A.G.: Nonlinear oscillations of a swinging spring for resonance. Mech. Solids 40(5) (2006)

  18. Bruno, A.D., Petrov, A.G.: On computation of the Hamiltonian normal form. Dokl. Phys. 51(10), 555–559 (2006)

    Article  Google Scholar 

  19. Bruno, A.D.: Analytical form of differential equations. Trans. Mosk. Math. Soc. 26, 199–239 (1972)

    MATH  Google Scholar 

  20. Cherry, T.M.: On the solution of Hamiltonian systems of differential equations in the neighbourhood of a singular point. Proc. Lond. Math. Soc. Ser. 2 27, 151–170 (1928)

    Article  Google Scholar 

  21. Gustavson, F.G.: On constructing formal integrals of a Hamiltonian system near an equilibrium point. Astron. J. 71, 670–686 (1966)

    Article  Google Scholar 

  22. Belitskii, G.R.: Normal Forms, Invariants and Local Mapping. Naukova Dumka, Kiev (1979)

    Google Scholar 

  23. Hori, G.I.: Theory of general perturbations with unspecified canonical variables. Astron. Soc. Jpn. 18(4), 287–296 (1966)

    Google Scholar 

  24. Deprit, A.: Canonical transformations depending on a small parameter. Celest. Mech. 1(1), 12–30 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  25. Vitt, A., Gorelik, G.: Elastic pendulum oscillation as an example of parametrically coupled linear systems. J. Tech. Phys. 3, 294–307 (1933)

    Google Scholar 

  26. Bogaevskii, V.N., Povzner, A.Y.: Algebraic Methods in Nonlinear Theory of Perturbation. Nauka, Moscow (1987), in Russian

    Google Scholar 

  27. Starzhinskiy, V.M.: Applied Methods of Nonlinear Oscillations. Nauka, Moscow (1977), in Russian

    Google Scholar 

  28. Mersman, W.A.: A new algorithm for the Lie transformation. Celest. Mech. 3(1), 81–89 (1970)

    Article  MathSciNet  MATH  Google Scholar 

  29. Markeev, A.P., Sokolskiy, A.G.: Algorithms of Hamiltonian Systems Normalization. Institut of Applied Mathematics of AN SSSR, Moscow (1976)

    Google Scholar 

  30. Awrejcewicz, J., Holicke, M.M.: Smooth and Nonsmooth High Dimensional Chaos and the Melnikov-Type Methods. World Scientific (2007)

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Authors and Affiliations

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

    Jan Awrejcewicz

  2. Institute for Problems in Mechanics, Russian Academy of Sciences, Moscow, Russia

    Alexander G. Petrov

Authors
  1. Jan Awrejcewicz
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  2. Alexander G. Petrov
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Correspondence to Jan Awrejcewicz.

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Awrejcewicz, J., Petrov, A.G. Nonlinear oscillations of an elastic two-degrees-of-freedom pendulum. Nonlinear Dyn 53, 19–30 (2008). https://doi.org/10.1007/s11071-007-9292-4

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