Higher-Order Dispersion, Stability, and Waveform Invariance in Nonlinear Monoatomic and Diatomic SystemsSource: Journal of Vibration and Acoustics:;2017:;volume( 139 ):;issue: 005::page 51003DOI: 10.1115/1.4036501Publisher: The American Society of Mechanical Engineers (ASME)
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 the cubic stiffness influences dispersion shifts to a greater degree than the 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, multiharmonic 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 the higher-order multiple scales perturbation analysis captures long-term, nonlocalized invariant plane waves, which have the potential for propagating coherent information over long distances.
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contributor author | Fronk, Matthew D. | |
contributor author | Leamy, Michael J. | |
date accessioned | 2017-11-25T07:20:12Z | |
date available | 2017-11-25T07:20:12Z | |
date copyright | 2017/12/6 | |
date issued | 2017 | |
identifier issn | 1048-9002 | |
identifier other | vib_139_05_051003.pdf | |
identifier uri | http://138.201.223.254:8080/yetl1/handle/yetl/4236271 | |
description 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 the cubic stiffness influences dispersion shifts to a greater degree than the 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, multiharmonic 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 the higher-order multiple scales perturbation analysis captures long-term, nonlocalized invariant plane waves, which have the potential for propagating coherent information over long distances. | |
publisher | The American Society of Mechanical Engineers (ASME) | |
title | Higher-Order Dispersion, Stability, and Waveform Invariance in Nonlinear Monoatomic and Diatomic Systems | |
type | Journal Paper | |
journal volume | 139 | |
journal issue | 5 | |
journal title | Journal of Vibration and Acoustics | |
identifier doi | 10.1115/1.4036501 | |
journal fristpage | 51003 | |
journal lastpage | 051003-13 | |
tree | Journal of Vibration and Acoustics:;2017:;volume( 139 ):;issue: 005 | |
contenttype | Fulltext |