Advertisement

Applied Mathematics and Mechanics

, Volume 24, Issue 3, pp 261–273 | Cite as

Coexisting periodic orbits in vibro-impacting dynamical systems

  • Li Qun-hong
  • Lu Qi-shao
Article

Abstract

A method is presented to seek for coexisting periodic orbits which may be stable or unstable in piecewise-linear vibro-impacting systems. The conditions for coexistence of single impact periodic orbits are derived, and in particular, it is investigated in details how to assure that no other impacts will happen in an evolution period of a single impact periodic motion. Furthermore, some criteria for nonexistence of single impact periodic orbits with specific periods are also established. Finally, the stability of coexisting periodic orbits is discussed, and the corresponding computation formula is given. Examples of numerical simulation are in good agreement with the theoretic analysis.

Key words

vibro-impact system periodic orbit existence stability 

Chinese Library Classification

O175 O322 

2000 MR Subject Classification

70K50 70K20 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    Guckenheimer J, Holmes P.Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields[M]. New York: Springer-Verlag, 1986.Google Scholar
  2. [2]
    Wigins S.Introduction to Applied Nonlinear Dynamical Systems and Chaos[M]. (Reprinted) New York: Springer-Verlag, 1991.Google Scholar
  3. [3]
    Wiggins S.Global Bifurcations and Chaos, Analytical Methods[M]. New York: Springer-Verlag, 1988.Google Scholar
  4. [4]
    Bazejczyk-Okolewska B, Kapitaniak T. Co-existing attractors of impact oscillator[J].Chaos, Solitons & Fractals, 1998,9(8):1439–1443.CrossRefGoogle Scholar
  5. [5]
    Feudel U, Grebogi C, Poon L, Yorke J A. Dynamical properties of a simple mechanical system with a large number of coexisting periodic attractors[J].Chaos, Solitons & Fractals, 1998,9(1/2):171–180.zbMATHMathSciNetCrossRefGoogle Scholar
  6. [6]
    Whiston G S. Global dynamics of a vibro-impacting linear oscillator[J].J Sound Vib, 1987,118 (3):395–424.MathSciNetCrossRefGoogle Scholar
  7. [7]
    Shaw S W, Holmes P J. A periodically forced piecewise linear oscillator[J].J Sound Vib, 1983,90 (1):129–155.zbMATHMathSciNetCrossRefGoogle Scholar
  8. [8]
    Ivanov A P. Stabilization of an impact oscillator near grazing incidence owing to resonance[J].J Sound Vib, 1993,162(3):562–565.zbMATHCrossRefGoogle Scholar
  9. [9]
    Whiston G S. Impacting under harmonic excitation[J].J Sound Vib, 1979,67(2):179–186.zbMATHMathSciNetCrossRefGoogle Scholar
  10. [10]
    Whiston G S. The vibro-impact response of a harmonically excited and preloaded one-dimensional linear oscillator[J].J Sound Vib, 1987,115(2):303–319.MathSciNetCrossRefGoogle Scholar
  11. [11]
    Nordmark A B. Non-periodic motion caused by grazing incidence in an impact oscillator[J].J Sound Vib, 1991,145(2):279–297.CrossRefGoogle Scholar
  12. [12]
    Foale S, Bishop S R. Dynamical complexities of forced impacting systems[J].Phil Trans Royal Soc London A, 1992,338(4):547–556.zbMATHMathSciNetGoogle Scholar
  13. [13]
    Nordmark A B. Effects due to low velocity in mechanical oscillators[J].Int J Bifurcation and Chaos, 1992,2(3):597–605.zbMATHMathSciNetCrossRefGoogle Scholar
  14. [14]
    Shaw S W, Rand R H. The transition to chaos in a simple mechanical system[J].Int J Nonlinear Mech, 1989,24(1):41–56.zbMATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Editorial Committee of Applied Mathematics and Mechanics 2003

Authors and Affiliations

  • Li Qun-hong
    • 1
  • Lu Qi-shao
    • 1
  1. 1.School of ScienceBeijing University of Aeronautics and AstronauticsBeijingPR China

Personalised recommendations