Skip to main content
SpringerLink
Log in

Orbits homoclinic to resonances in a harmonically excited and undamped circular plate

  • Published:
Meccanica Aims and scope Submit manuscript

Abstract

Orbits homoclinic to resonances in mode interactions of an imperfect circular plate with 1:1 internal resonance are investigated. The case of primary resonance is considered. The damping force is not included in the analysis. The energy-phase criterion is used to give a fairly complete picture of the complex dynamics associated with orbits homoclinic to the resonances. A saddle-node bifurcation of homoclinic orbits occurs. The existence of homoclinic orbits in the unperturbed system may lead to chaos in the sense of Smale horseshoes under perturbation.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Efstathiades GJ (1971) A new approach to the large-deflection vibrations of imperfect circular disks using Galerkin’s procedure. J Sound Vib 16:231–253

    Article  MATH  ADS  Google Scholar 

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

    Article  MATH  ADS  Google Scholar 

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

    Article  MATH  ADS  Google Scholar 

  4. Yeo MH, Lee WK (2002) Corrected solvability conditions for non-linear asymmetric vibrations of a circular plate. J Sound Vib 257:653–665

    Article  ADS  Google Scholar 

  5. Lee WK, Yeo MH (2003) Non-linear interactions in asymmetric vibrations of a circular plate. J Sound Vib 263:1017–1030

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

  8. Zenkour AM, Mashat DS (2009) Exact solutions for variable-thickness inhomogeneous elastic plates under various boundary conditions. Meccanica 44:433–447

    Article  MATH  MathSciNet  Google Scholar 

  9. Gupta AK, Kumar L (2009) Effect of thermal gradient on vibration of non-homogeneous visco-elastic elliptic plate of variable thickness. Meccanica 44:507–518

    Article  MathSciNet  Google Scholar 

  10. Yang XL, Sethna PR (1991) Local and global bifurcations in the parametrically excited vibrations of nearly square plates. Int J Non-linear Mech 26:199–220

    Article  MATH  MathSciNet  Google Scholar 

  11. Samoylenko SB, Lee WK (2007) Global bifurcations and chaos in a harmonically excited and undamped circular plate. Nonlinear Dyn 47:405–419

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  ADS  Google Scholar 

  13. Feng ZC, Wiggins S (1993) 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

    Article  MATH  MathSciNet  ADS  Google Scholar 

  14. Feng ZC, Sethna PR (1993) Global bifurcation in motions of parametrically excited thin plates. Nonlinear Dyn 4:398–408

    Article  Google Scholar 

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

    Article  Google Scholar 

  16. Feng ZC, Liew KM (2000) Global bifurcations in parametrically excited systems with zero-to-one internal resonance. Nonlinear Dyn 21:249–263

    Article  MATH  MathSciNet  Google Scholar 

  17. Zhang W, Wang FX, Yao MH (2005) Global bifurcations and chaotic dynamics in nonlinear nonplanar oscillations of a parametrically excited cantilever beam. Nonlinear Dyn 40:251–279

    Article  MATH  MathSciNet  Google Scholar 

  18. Yeo MH, Lee WK (2006) Evidence of global bifurcations of an imperfect circular plate. J Sound Vib 293:138–155

    Article  ADS  Google Scholar 

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

    Article  MATH  Google Scholar 

  20. Avramov KV, Awrejcewicz J (2006) Frictional oscillations under the action of almost periodic excitation. Meccanica 41:119–142

    Article  MATH  MathSciNet  Google Scholar 

  21. Mikhlin YV, Manucharyan GV (2006) Determination of the chaos onset in mechanical systems with several equilibrium positions. Meccanica 41:253–267

    Article  MATH  MathSciNet  Google Scholar 

  22. Haller G, Wiggins S (1993) Orbits homoclinic to resonances: the Hamiltonian case. Physica D 66:298–346

    Article  MATH  MathSciNet  ADS  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mechanics, Nanjing University of Aeronautics and Astronautics, Nanjing, 210016, China

    Weiqin Yu

  2. Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing, 210016, China

    Weiqin Yu & Fangqi Chen

Authors
  1. Weiqin Yu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Fangqi Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Weiqin Yu.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Yu, W., Chen, F. Orbits homoclinic to resonances in a harmonically excited and undamped circular plate. Meccanica 45, 567–575 (2010). https://doi.org/10.1007/s11012-009-9273-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11012-009-9273-4

Search

Navigation