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The Mapping Dynamics of a Three-Piecewise Linear System under a Periodic Excitation

  • Albert C.J. Luo
Conference paper
  • 1.2k Downloads
Part of the Solid Mechanics and its Applications book series (SMIA, volume 122)

Abstract

The mapping dynamics of non-smooth dynamical systems is presented through a three-piecewise linear system with a periodic excitation. The mapping structures for periodic motions are developed and a transition from a periodic motion to another one is qualitatively discussed through the mapping structures. From such mapping structures, the stable and unstable periodic motions can be uniquely determined, and generic mapping series in chaotic motion can be certainly found. This methodology is extendable to any non-smooth dynamical system.

Key words

Piecewise linearity mapping dynamics grazing non-smooth systems 

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References

  1. [1]
    J.P.D. Hartog and S.J. Mikina, “Forced vibrations with non-linear spring constants,” ASME Journal of Applied Mechanics, 58, pp.157–164, 1932.Google Scholar
  2. [2]
    S.W. Shaw and P.J. Holmes, “A periodically forced piecewise linear oscillator,” Journal of Sound and Vibration, 90(1), pp.121–155, 1983.MathSciNetCrossRefGoogle Scholar
  3. [3]
    M. Kleczka, E. Kreuzer, and W. Schiehlen, “Local and global stability of a piecewise linear oscillator,” Philosophical Transactions: Physical Sciences and Engineering, Nonlinear Dynamics of Engineering Systems, 338(1651), pp.533–546, 1992.MathSciNetGoogle Scholar
  4. [4]
    S. Natsiavas, ”Periodic response and stability of oscillators with symmetric trilinear restoring force,” Journal of Sound and Vibration, 134(2), pp.315–331, 1989.MathSciNetCrossRefGoogle Scholar
  5. [5]
    S. Theodossiades and S. Natsiavas, “Non-linear dynamics of gear-pair systems with periodic stiffness and backlash,” Journal of Sound and Vibration, 229(2), pp. 287–310, 2000.CrossRefGoogle Scholar
  6. [6]
    S.J. Hogan and M.E. Homer, “Graph theory and piecewise smooth dynamical systems of arbitrary dimension,” Chaos, Solitons and Fractals, 10, pp.1869–1880, 1999.MathSciNetCrossRefGoogle Scholar
  7. [7]
    S. Lenci and G. Rega, “Periodic solutions and bifurcations in an impact inverted pendulum under impulsive excitation,” Chaos, Solitons and Fractals, 11, pp. 2453–2472, 2000.MathSciNetCrossRefGoogle Scholar
  8. [8]
    R.P.S. Han, A.C.J. Luo, and W. Deng, “Chaotic motion of a horizontal impact pair,” Journal of Sound and Vibration, 181, pp.231–250, 1995.MathSciNetCrossRefGoogle Scholar
  9. [9]
    A.C.J. Luo, “An unsymmetrical motion in a horizontal impact oscillator,” ASME Journal of Vibrations and Acoustics, 124, pp. 420–426, 2002.CrossRefGoogle Scholar
  10. [10]
    G.X. Li, R.H. Rand, and F.C. Moon, “Bifurcation and Chaos in a forced zero-stiffness impact oscillator,” International Journal of Nonlinear Mechanics, 25, pp.414–432, 1990.MathSciNetCrossRefGoogle Scholar
  11. [11]
    S. Menon and A.C.J. Luo, “An analytical prediction of the global period-1 motion in a periodically forced, piecewise linear system,” International Journal of Bifurcation and Chaos, 2003, in press.Google Scholar
  12. [12]
    A.C.J. Luo, and S. Monen, “Global chaos in a periodically forced, linear system with a dead-zone restoring force”, Chaos, Solitons and Fractals, 19, pp.1189–1199, 2004.MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer 2005

Authors and Affiliations

  • Albert C.J. Luo
    • 1
  1. 1.Department of Mechanical and Industrial EngineeringSouthern Illinois University EdwardsvilleEdwardsvilleUSA

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