Free and forced localization in a nonlinear periodic lattice

Research output: Chapter in Book/Report/Conference proceedingConference contribution


Free and forced localized periodic motions in an infinite nonlinear periodic lattice are analytically investigated. The lattice consists of weakly coupled identical masses, each connected to the ground by a nonlinear stiffness. In order to study the localized motions of the discrete system a continuous approximation is assumed, and the ordinary differential equations of motion are replaced by a single nonlinear partial differential equation. The time-periodic solutions of this equation are then obtained by an averaging method, and their stability is examined using an analytic linearized method. It is shown that localized periodic motions of the lattice correspond to standing solitary solutions of the partial differential equation of the continuous approximation. For the free lattice, localized free motions occur when the coupling stiffness forces are much smaller than the nonlinear effects of the grounding stiffnesses. Moreover, these free localized motions are detected in the perfectly periodic nonlinear lattice, i.e., even in the absence of structural disorder (a feature which is an essential prerequisite for linear mode localization). When harmonic forcing is applied to the chain, localized, non-localized, and chaotic motions occur, depending on the spatial distribution and the magnitude of the applied loads. A variety of spatially distributed harmonic loads and analytic expressions for the resulting localized motions of the chain are derived.

Original languageEnglish (US)
Title of host publicationStructural Dynamics of Large Scale and Complex Systems
EditorsMo Shahinpoor, H.S. Tzou
PublisherPubl by ASME
Number of pages8
ISBN (Print)079181176X
StatePublished - 1993
Event14th Biennial Conference on Mechanical Vibration and Noise - Albuquerque, NM, USA
Duration: Sep 19 1993Sep 22 1993

Publication series

NameAmerican Society of Mechanical Engineers, Design Engineering Division (Publication) DE


Other14th Biennial Conference on Mechanical Vibration and Noise
CityAlbuquerque, NM, USA

ASJC Scopus subject areas

  • General Engineering


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