Explorations into the Nonlinear Dynamics of a Single DOF System Coupled to a Wideband Auto-Parametric Vibration Absorber

  • Anil K. Bajaj
  • Ashwin Vyas
  • Arvind Raman
Conference paper
Part of the Solid Mechanics and its Applications book series (SMIA, volume 122)


The nonlinear dynamics of a resonantly excited linear oscillator coupled to an array of weakly coupled nonlinear pendulums is investigated under 1:1:…:1:2 internal resonance between the pendulums and the linear oscillator. In the first part of the work, periodic solutions and bifurcations under harmonic excitation of the linear oscillator are investigated. In the second part of the work, numerical simulations of the unperturbed Hamiltonian are presented to demonstrate the complex dynamics of the system even in the absence of external excitation.

Key words

Autoparametric system vibration absorber internal resonances Hamiltonian 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    A.H. Nayfeh and D.T. Mook, Nonlinear Oscillations. Wiley Interscience, New York, 1979.zbMATHGoogle Scholar
  2. [2]
    P.R. Sethna, “Vibrations of dynamical systems with quadratic nonlinearities,” Journal of Applied Mechanics, 32:576–582, 1965.MathSciNetGoogle Scholar
  3. [3]
    R.S. Haxton and A.D.S. Barr, “The autoparametric vibration absorber,” Journal of Engineering for Industry, 94:119–225, 1972.Google Scholar
  4. [4]
    A. Vyas and A.K. Bajaj, “Dynamics of autoparametric vibration absorbers using multiple pendulums,” Journal of Sound and Vibration, 246:115–135, 2001.MathSciNetCrossRefGoogle Scholar
  5. [5]
    E. Doedel, Auto: Software for continuation and bifurcation problems in ordinary differential equations. Report, Department of Applied Mathematics, California Institute of Technology, Pasadena, CA, 1986.Google Scholar
  6. [6]
    A. Vyas, A.K. Bajaj, and A. Raman, “Dynamics of flexible structures with wideband autoparametric vibration absorbers — theory,” Proceedings of the Royal Society of London, 460(2046):1547–1581, 2004.MathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos. Springer Verlag, New York, 1990.zbMATHGoogle Scholar
  8. [8]
    A. Vakakis, T. Nayfeh, and M. King, “A multiple-scales analysis of nonlinear, localized modes in a cyclic periodic system,” Journal of Applied Mechanics, 60:388–397, 1993.MathSciNetzbMATHGoogle Scholar
  9. [9]
    B. Banerjee and A.K. Bajaj, “Amplitude modulated chaos in two degree-of-freedom systems with quadratic nonlinearities,” Acta Mechanica, 124:131–154, 1997.MathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    F.A. McRobie, A.A. Popov, and J.M.T. Thompson, “Auto-parametric resonance in cylindrical shells using geometric averaging,” Journal of Sound and Vibration, 227(1):65–84, 1999.CrossRefGoogle Scholar
  11. [11]
    P. Holmes, “Chaotic motions in a weakly nonlinear model for surface waves,” Journal of Fluid Mechanics, 162:365–388, 1986.zbMATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer 2005

Authors and Affiliations

  • Anil K. Bajaj
    • 1
  • Ashwin Vyas
    • 1
  • Arvind Raman
    • 1
  1. 1.School of Mechanical EngineeringPurdue University West LafayetteUSA

Personalised recommendations