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Higher-Order Dispersion, Stability, and Waveform Invariance in Nonlinear Monoatomic and Diatomic Systems

[+] Author and Article Information
Michael Leamy

George W. Woodruff School of Mechanical Engineering Georgia Institute of Technology Atlanta, GA 30332-0405
michael.leamy@me.gatech.edu

Matthew Fronk

George W. Woodruff School of Mechanical Engineering Georgia Institute of Technology Atlanta, GA 30332-0405
mfronk3@gatech.edu

1Corresponding author.

ASME doi:10.1115/1.4036501 History: Received May 24, 2016; Revised April 11, 2017

Abstract

Recent studies have presented first-order multiple time scale approaches for exploring amplitude-dependent plane-wave dispersion in weakly nonlinear chains and lattices characterized by cubic stiffness. These analyses have yet to assess solution stability, which requires an analysis incorporating damping. Furthermore, due to their first-order dependence, they make an implicit assumption that cubic stiffness influences dispersion shifts to a greater degree than quadratic stiffness, and they thus ignore quadratic shifts. This paper addresses these limitations by carrying-out higher-order, multiple scales perturbation analyses of linearly-damped nonlinear monoatomic and diatomic chains. The study derives higher-order dispersion corrections informed by both quadratic and cubic stiffness, and quantifies plane wave stability using evolution equations resulting from the multiple scales analysis and numerical experiments. Additionally, by reconstructing plane waves using both homogeneous and particular solutions at multiple orders, the study introduces a new interpretation of multiple scales results in which predicted waveforms are seen to exist over all space and time, constituting an invariant, multi-harmonic wave of infinite extent analogous to cnoidal waves in continuous systems. Using example chains characterized by dimensionless parameters, numerical studies confirm that the spectral content of the predicted waveforms exhibits less growth/decay over time as higher-order approximations are used in defining the simulations’ initial conditions. Thus the study results suggest that higher-order multiple scales perturbation analysis captures long-term, non-localized invariant plane waves, which have the potential for propagating coherent information over long distances.

Copyright (c) 2017 by ASME
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