TY - GEN

T1 - Free and forced localization in a nonlinear periodic lattice

AU - Vakakis, Alexander F.

AU - King, Melvin E.

AU - Pearlstein, Arne J.

N1 - Publisher Copyright:
© 1993 American Society of Mechanical Engineers (ASME). All rights reserved.

PY - 1993

Y1 - 1993

N2 - 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 continuoum 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 stiffnesses 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, nonlocalized, 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.

AB - 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 continuoum 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 stiffnesses 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, nonlocalized, 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.

UR - http://www.scopus.com/inward/record.url?scp=85104197884&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85104197884&partnerID=8YFLogxK

U2 - 10.1115/DETC1993-0164

DO - 10.1115/DETC1993-0164

M3 - Conference contribution

AN - SCOPUS:85104197884

T3 - Proceedings of the ASME Design Engineering Technical Conference

SP - 27

EP - 34

BT - 14th Biennial Conference on Mechanical Vibration and Noise

PB - American Society of Mechanical Engineers (ASME)

T2 - ASME 1993 Design Technical Conferences, DETC 1993

Y2 - 19 September 1993 through 22 September 1993

ER -