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Nonlinear Mode Localization in Systems Governed by Partial Differential Equations

[+] Author and Article Information
Alexander F. Vakakis

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

Appl. Mech. Rev 49(2), 87-99 (Feb 01, 1996) (13 pages) doi:10.1115/1.3101890 History: Online April 20, 2009

Abstract

The concept of nonlinear normal mode (NNM) is used to study localized oscillations in certain classes of oscillators governed by nonlinear partial differential equations. NNMs are synchronous free oscillations during which all positional coordinates of the system reach their extreme values or pass through the equilibrium position at the same instant of time. Although such motions can be regarded as nonlinear analogs of the linear normal modes of classical vibration theory, not all NNMs are analytic continuations of linear ones. Continuous systems of finite and infinite spatial extent are considered. For periodic assemblies consisting of a finite number of nonlinear structural members, the NNMs are computed asymptotically by solving nonlinear sets of equations possessing regular singular points. Some of the computed NNMs are spatially localized to only a limited number of components of the assembly. The bifurcations giving rise to nonlinear mode localization are examined using the perturbation method of multiple-scales. The implications of nonlinear mode localization on the vibration and shock isolation of periodic flexible structures are discussed. In particular, localized NNMs lead to passive motion confinement of disturbances generated by impulsive loads. Finally, the concept of NNMs is extended to analytically study standing waves with spatially localized envelopes in a class of nonlinear partial differential equations defined over infinite domains. It is shown that NNM-based methodologies can be an effective tool for analyzing such motions.

Copyright © 1996 by American Society of Mechanical Engineers
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