Mathematical Problems in Engineering 
Volume 2006 (2006), Article ID 10847, 15 pages
doi:10.1155/MPE/2006/10847

Nonlinear normal modes and their application in structural dynamics

Christophe Pierre,1 Dongying Jiang,2 and Steven Shaw3

 1Department of Mechanical Engineering, University of Michigan, Ann Arbor 48109, MI, USA
2MKP Structural Design Associates, Inc., Ann Arbor 48104, MI, USA
3Department of Mechanical Engineering, Michigan State University, East Lansing 4882, MI, USA
Received 12 February 2005; Revised 13 June 2005; Accepted 12 July 2005

Abstract

Recent progress in the area of nonlinear modal analysis for structural systems is reported. Systematic methods are developed for generating minimally sized reduced-order models that accurately describe the vibrations of large-scale nonlinear engineering structures. The general approach makes use of nonlinear normal modes that are defined in terms of invariant manifolds in the phase space of the system model. An efficient Galerkin projection method is developed, which allows for the construction of nonlinear modes that are accurate out to large amplitudes of vibration. This approach is successfully extended to the generation of nonlinear modes for systems that are internally resonant and for systems subject to external excitation. The effectiveness of the Galerkin-based construction of the nonlinear normal modes is also demonstrated for a realistic model of a rotating beam.