Fundamental and subharmonic resonances in a system with a '1-1' internal resonance

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The fundamental and subharmonic resonances of a two degree-of-freedom oscillator with cubic stiffness nonlinearities and linear viscous damping are examined using a multiple-seales averaging analysis. The system is in a '1-1' internal resonance, i.e., it has two equal linearized eigenfrequencies, and it possesses 'nonlinear normal modes.' For weak coupling stiffnesses the internal resonance gives rise to a Hamiltonian Pitchfork bifurcation of normal modes which in turn affects the topology of the fundamental and subharmonic resonance curves. It is shown that the number of resonance branches differs before and after the mode bifurcation, and that jump phenomena are possible between forced modes. Some of the steady state solutions were found to be very sensitive to damping: a whole branch of fundamental resonances was eliminated even for small amounts of viscous damping, and subharmonic steady state solutions were shifted by damping to higher frequencies. The analytical results are verified by a numerical integration of the equations of motion, and a discussion of the effects of the mode bifurcation on the dynamics of the system is given.

Original languageEnglish (US)
Pages (from-to)123-143
Number of pages21
JournalNonlinear Dynamics
Issue number2
StatePublished - Mar 1992


  • Nonlinear normal modes
  • nonlinear resonances
  • perturbation methods

ASJC Scopus subject areas

  • Control and Systems Engineering
  • Aerospace Engineering
  • Ocean Engineering
  • Mechanical Engineering
  • Applied Mathematics
  • Electrical and Electronic Engineering


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