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RESEARCH PAPERS

An Energy-Based Approach to Computing Resonant Nonlinear Normal Modes

[+] Author and Article Information
M. E. King

Department of Aerospace and Mechanical Engineering, Boston University, Boston, MA 02215

A. F. Vakakis

Department of Mechanical and Industrial Engineering, University of Illinois at Urbana-Champaign, Urbana, IL 61801

J. Appl. Mech 63(3), 810-819 (Sep 01, 1996) (10 pages) doi:10.1115/1.2823367 History: Received March 10, 1995; Revised October 22, 1995; Online December 04, 2007

Abstract

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 canonical 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 O(1) solutions must be considered (in the absence of internal resonances, a linear expansion at O(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 coordinates and thus a transformation to a canonical framework is necessary in order to appropriately define NNM relations.

Copyright © 1996 by The American Society of Mechanical Engineers
Topics: Resonance , Equations
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