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Research Papers

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

[+] Author and Article Information
Matthew D. Fronk

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

Michael J. Leamy

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

1Corresponding author.

Contributed by the Technical Committee on Vibration and Sound of ASME for publication in the JOURNAL OF VIBRATION AND ACOUSTICS. Manuscript received May 24, 2016; final manuscript received April 11, 2017; published online June 12, 2017. Assoc. Editor: Matthew S. Allen.

J. Vib. Acoust 139(5), 051003 (Jun 12, 2017) (13 pages) Paper No: VIB-16-1265; doi: 10.1115/1.4036501 History: Received May 24, 2016; Revised April 11, 2017

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.

Copyright © 2017 by ASME
Topics: Stability , Waves , Chain
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Figures

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Fig. 1

Cnoidal waves captured by the U.S. Army bombers over the Panama coast in 1933

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Fig. 2

Schematic of the unit cell for the monoatomic chain

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Fig. 3

Schematic of the unit cell for the diatomic chain

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Fig. 4

Zeroth-order dispersion relationships for the monoatomic and diatomic chains

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Fig. 5

First- and second-order dispersion shifts in the monoatomic (left) and diatomic (right) chains. Monoatomic: Π1=0,Π2=0.04  (top), Π2=0 (bottom), Π3=0.04, Diatomic: Π1=0,Π2=0.04 (top), Π2=0 (bottom), Π3=0.04, Π4=1.5.

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Fig. 6

Fixed points and basins of attraction for the monoatomic chain

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Fig. 7

An example chain exhibiting a breakdown in spectral content: Π1=0.01,Π2=0, Π3=3.7, second-order initial conditions (ICs)

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Fig. 8

An example chain exhibiting a growth in amplitude: Π1=0.01, Π2=1, Π3=0, second-order ICs

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Fig. 9

Monoatomic chain stability study: second-order ICs, Π1=0.01, μ=(π/4)

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Fig. 10

Diatomic chain stability study: xa, second-order ICs, Π1=0.01, Π4=1.5, μ=(π/4), acoustic (a), and optical (b)

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Fig. 11

Reduction of variance of the (a) second and (b) third harmonics, monoatomic chain, μ=(π/4), Π1=0

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Fig. 12

Reduction of variance in xa, second (a, c) and third (b,d) harmonics, diatomic chain. Π4=2, μ=(π/4),Π1=0 acoustic (a and b), and optical (c and d).

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Fig. 13

Diatomic chain stability study: xb, second-order ICs, Π1=0.01, Π4=1.5,μ=(π/4),  acoustic (a), and optical (b)

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Fig. 14

Reduction of variance in xb, second (a) and third (b) harmonics, acoustic branch, Π4=2, μ=(π/4),Π1=0

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Fig. 15

Reduction of variance in xb, second (a) and third (b) harmonics, optical branch, Π4=2, μ=(π/4), Π1=0

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