Bifurcation of nonlinear normal modes in a class of two degree of freedom systems

  • R. H. Rand
  • C. H. Pak
  • A. F. Vakakis
Part of the Acta Mechanica book series (ACTA MECH.SUPP., volume 3)


This work concerns the nonlinear vibrations of a class of two-degree-of-freedom autonomous conservative systems consisting of two coupled nonlinear oscillators with cubic coupling forces. For such a system, nonlinear normal modes (NNM’s) have long been studied as “vibrations in unison”, i.e. periodic motions in which x and y simultaneously achieve zero velocity. We investigate the stability and bifurcation of NNM’s by using a perturbation method together with the computer algebra system MACSYMA.

The results of our study include the characterization of those systems which are maximally degenerate in the sense of changing the number of elementary periodic motions which they exhibit in response to a small change in parameters. By looking in the neighborhood of such a degenerate system, we obtain universal unfoldings. In particular we show how NNM’s, which project onto the configuration plane as (curved) line segments, are related to a family of periodic orbits which look like ellipses on the configuration plane. Our analytic results are shown to compare favorably with the results of numerical integration.


Periodic Orbit Singular Point Phase Portrait Perturbation Method Periodic Motion 
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Copyright information

© Springer-Verlag Wien 1992

Authors and Affiliations

  • R. H. Rand
    • 1
  • C. H. Pak
    • 2
  • A. F. Vakakis
    • 3
  1. 1.Department of Theoretical and Applied MechanicsCornell UniversityIthacaUSA
  2. 2.Department of Mechanical EngineeringInha UniversityInchon, 160Korea
  3. 3.Department of Mechanical EngineeringUniversity of IllinoisUrbanaUSA

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