An energy-based approach to computing resonant nonlinear normal modes

M. E. King, A. F. Vakakis

Research output: Contribution to journalArticlepeer-review


A formulation for computing resonant nonlinear normal modes (NNMs) is developed for discrete and continuous systems. In a canonical framework, internal resonance conditions are immediately recognized by identifying commensurable linearized natural frequencies of these systems. Additionally, a canonical formulation allows for a single (linearized modal) coordinate to parameterize all other coordinates during a resonant NNM response. Energy-based NNM methodologies are applied to a canoni- cal set of equations and asymptotic solutions are sought. In order to account for the resonant modal interactions, it will be shown that high-order terms in the 0(1) solutions must be considered (in the absence of internal resonances, a linear expansion at 0(1) is sufficient). Two applications (*3:1’ resonances in a two-degree-of-freedom system and ‘3:1’ resonance in a hinged-clamped beam) are then considered by which to demonstrate the resonant NNM methodology. It is shown that for some responses, nonlinear modal relations do not exist in the context of physical coordi- nates and thus a transformation to a canonical framework is necessary in order to appropriately define NNM relations.

Original languageEnglish (US)
Pages (from-to)810-819
Number of pages10
JournalJournal of Applied Mechanics, Transactions ASME
Issue number3
StatePublished - Sep 1996

ASJC Scopus subject areas

  • Condensed Matter Physics
  • Mechanics of Materials
  • Mechanical Engineering


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