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Multi-pulse jumping orbits and homoclinic trees in motion of a simply supported rectangular metallic plate

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Abstract

The global bifurcations in mode interaction of a simply supported rectangular metallic plate subjected to a transverse harmonic excitation are investigated with the case of the 1:1 internal resonance, the modulation equations representing the evolution of the amplitudes and phases of the interacting normal modes exhibit complex dynamics. The energy-phase method proposed by Haller and Wiggins is employed to analyze the global bifurcations for the rectangular metallic plate. The results obtained here indicate that there exist the Silnikov-type multi-pulse orbits homoclinic to certain invariant sets for the resonant case in both Hamiltonian and dissipative perturbations, which imply that chaotic motions may occur for this class of systems. Homoclinic trees which describe the repeated bifurcations of multi-pulse solutions are found. To illustrate the theoretical predictions, we present visualizations of these complicated structures and numerical evidence of chaotic motions.

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References

  1. Sathayamorthy M.: Nonlinear vibration analysis of plates: a review and survey of current developments. Appl. Mech. Rev. 40, 1553–1561 (1987)

    Article  Google Scholar 

  2. Yeh F.H., Liu W.H.: Nonlinear analysis of rectangular orthotropic plates. Int. J. Mech. Sci. 33, 563–578 (1991)

    Article  MATH  Google Scholar 

  3. Sridhar S., Mook D.T., Nayfeh A.H.: Non-linear resonances in the forced responses of plate, part I: symmetric responses of circular plates. J. Sound Vib. 41, 359–373 (1975)

    Article  MATH  Google Scholar 

  4. Sridhar S., Mook D.T., Nayfeh A.H.: Non-linear resonances in the forced responses of plate, part II: asymmetric responses of circular plates. J. Sound Vib. 59, 159–170 (1978)

    Article  MATH  Google Scholar 

  5. Hadian J., Nayfeh A.H.: Modal interaction in circular plates. J. Sound Vib. 142, 279–292 (1990)

    Article  MathSciNet  Google Scholar 

  6. Yang X.L., Sethna P.R.: Local and global bifurcations in the parametrically excited vibrations of nearly square plates. Int. J. Non-Linear Mech. 26, 199–220 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  7. Chin, C., Nayfeh, A.H.: Bifurcation and chaos in externally excited circular cylindrical shells. In: Proceedings of the 36th Structures, Structural Dynamics, and Materials Conference, pp. 1–11. New Orleans, Louisiana (1995)

  8. Elbeyli O., Anlas G.: The nonlinear response of a simply supported rectangular metallic plate to transverse harmonic excitation. J. Appl. Mech. 67, 621–626 (2000)

    Article  MATH  Google Scholar 

  9. Anlas G., Elbeyli O.: Nonlinear vibrations of a simply supported rectangular metallic plate subjected to transverse harmonic excitation in the presence of a one-to-one internal resonance. Nonlinear Dyn. 30, 1–28 (2002)

    Article  MATH  Google Scholar 

  10. Touzé C., Thomas O., Chaigne A.: Asymmetric non-linear forced vibrations of free-edge circular plates. Part I: theory. J. Sound Vib. 258, 649–676 (2002)

    Article  Google Scholar 

  11. Thomas O., Touzé C., Chaigne A.: Asymmetric non-linear forced vibrations of free-edge circular plates. Part II: experiments. J. Sound Vib. 265, 1075–1101 (2003)

    Article  Google Scholar 

  12. Yeo M.H., Lee W.K.: Evidence of global bifurcations of an imperfect circular plate. J. Sound Vib. 293, 138–155 (2006)

    Article  Google Scholar 

  13. Samoylenko S.B., Lee W.K.: Global bifurcations and chaos in a harmonically excited and undamped circular plate. Nonlinear Dyn. 47, 405–419 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  14. Tong X., Tabarrok B.: Melnikov’s method for rigid bodies subject to small perturbation torques. Arch. Appl. Mech. 66, 215–230 (1996)

    Article  MATH  Google Scholar 

  15. Ravindra B., Zhu W.D.: Low-dimensional chaotic response of axially accelerating continuum in the supercritical regime. Arch. Appl. Mech. 68, 195–205 (1998)

    Article  MATH  Google Scholar 

  16. Wang, Y.G., Song, H.F., Li, D., Wang, J.: Bifurcations and chaos in a periodic time-varying temperature-excited bimetallic shallow shell of revolution. Arch. Appl. Mech. (2009). doi:10.1007/s00419-009-0341-y

  17. Kovačič G., Wiggins S.: Orbits homoclinic to resonances, with an application to chaos in a model of the forced and damped sine-Gordon equation. Phys. D 57, 185–225 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  18. Haller G., Wiggins S.: N-pulse homoclinic orbits in perturbations of resonant Hamiltonian systems. Arch. Ration. Mech. Anal. 130, 25–101 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  19. Haller G.: Chaos near resonance. Springer, New York (1999)

    MATH  Google Scholar 

  20. Feng Z.C., Wiggins S.: On the existence of chaos in a class of two-degree-of-freedom, damped parametrically forced mechanical systems with broken O(2) symmetry. ZAMP 44, 201–248 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  21. Feng Z.C., Sethna P.R.: Global bifurcation in motions of parametrically excited thin plates. Nonlinear Dyn. 4, 398–408 (1993)

    Article  Google Scholar 

  22. Haller G., Wiggins S.: Multi-pulse jumping orbits and homoclinic trees in a modal truncation of the damped-forces nonlinear Schrödinger equation. Phys. D 85, 311–347 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  23. Malhotra N., Namachchivaya N.S.: Chaotic motion of shallow arch structures under 1:1 resonance. ASCE J. Eng. Mech. 123, 620–627 (1997)

    Article  Google Scholar 

  24. Feng Z.C., Liew K.M.: Global bifurcations in parametrically excited systems with zero-to-one internal resonance. Nonlinear Dyn. 21, 249–263 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  25. Samaranayake G., Samaranayake S., Bajaj A.K.: Non-resonant and resonant chaotic dynamics in externally excited cyclic system. Acta Mech. 150, 139–160 (2001)

    Article  MATH  Google Scholar 

  26. Wang M.N., Xu Z.Y.: On the homoclinic orbits in a class of two-degree-of-freedom systems under the resonance conditions. Appl. Math. Mech. 22, 340–352 (2001)

    Article  MATH  Google Scholar 

  27. Zhang W., Wang F.X., Yao M.H.: Global bifurcations and chaotic dynamics in nonlinear nonplanar oscillations of a parametrically excited cantilever beam. Nonlinear Dyn. 40, 251–279 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  28. McDonald R.J., Sri Namachchivaya N.: Pipes conveying pulsating fluid near a 0:1 resonance: global bifurcations. J. Fluids Struct. 21, 665–687 (2005)

    Google Scholar 

  29. Zhang W., Yao M.H.: Multi-pulse and chaotic dynamics in motion of parametrically excited viscoelastic moving belt. Chaos Solitons Fractals 28, 42–66 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  30. Wiggins S.: Global bifurcations and chaos. Springer, New York (1988)

    MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mechanics, Nanjing University of Aeronautics and Astronautics, Nanjing, 210016, People’s Republic of China

    Weiqin Yu

  2. Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing, 210016, People’s Republic of China

    Fangqi Chen

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  1. Weiqin Yu
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  2. Fangqi Chen
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Correspondence to Weiqin Yu.

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Yu, W., Chen, F. Multi-pulse jumping orbits and homoclinic trees in motion of a simply supported rectangular metallic plate. Arch Appl Mech 80, 1103–1123 (2010). https://doi.org/10.1007/s00419-010-0431-x

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