Non-linear dynamics of a system of coupled oscillators with essential stiffness non-linearities

Alexander F. Vakakis, Richard H. Rand

Research output: Contribution to journalArticlepeer-review


We study the resonant dynamics of a two-degree-of-freedom system composed of a linear oscillator weakly coupled to a strongly non-linear one, with an essential (non-linearizable) cubic stiffness non-linearity. For the undamped system this leads to a series of internal resonances, depending on the level of (conserved) total energy of oscillation. We study in detail the 1:1 internal resonance, and show that the undamped system possesses stable and unstable synchronous periodic motions (non-linear normal modes - NNMs), as well as, asynchronous periodic motions (elliptic orbits - EOs). Furthermore, we show that when damping is introduced certain NNMs produce resonance capture phenomena, where a trajectory of the damped dynamics gets 'captured' in the neighborhood of a damped NNM before 'escaping' and becoming an oscillation with exponentially decaying amplitude. In turn, these resonance captures may lead to passive non-linear energy pumping phenomena from the linear to the non-linear oscillator. Thus, sustained resonance capture appears to provide a dynamical mechanism for passively transferring energy from one part of the system to another, in a one-way, irreversible fashion. Numerical integrations confirm the analytical predictions.

Original languageEnglish (US)
Pages (from-to)1079-1091
Number of pages13
JournalInternational Journal of Non-Linear Mechanics
Issue number7
StatePublished - Sep 2004


  • Capture
  • Coupled oscillators
  • Non-linear
  • Resonance

ASJC Scopus subject areas

  • Mechanics of Materials
  • Mechanical Engineering
  • Applied Mathematics


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