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Research Papers

A Perturbation Approach for Analyzing Dispersion and Group Velocities in Two-Dimensional Nonlinear Periodic Lattices

[+] Author and Article Information
R. K. Narisetti1

School of Aerospace Engineering,  GA Institute of Technology, Atlanta, GA 30332rajknarisetti@gatech.edu

M. Ruzzene

School of Aerospace Engineering, GA Institute of Technology, Atlanta, GA 30332; School of Mechanical Engineering,  GA Institute of Technology, Atlanta, GA 30332

M. J. Leamy

School of Mechanical Engineering,  GA Institute of Technology, Atlanta, GA 30332

1

Corresponding author.

J. Vib. Acoust 133(6), 061020 (Dec 01, 2011) (12 pages) doi:10.1115/1.4004661 History: Received November 24, 2009; Revised March 22, 2011; Published December 01, 2011; Online December 01, 2011

The paper investigates wave dispersion in two-dimensional, weakly nonlinear periodic lattices. A perturbation approach, originally developed for one-dimensional systems and extended herein, allows for closed-form determination of the effects nonlinearities have on dispersion and group velocity. These expressions are used to identify amplitude-dependent bandgaps, and wave directivity in the anisotropic setting. The predictions from the perturbation technique are verified by numerically integrating the lattice equations of motion. For small amplitude waves, excellent agreement is documented for dispersion relationships and directivity patterns. Further, numerical simulations demonstrate that the response in anisotropic nonlinear lattices is characterized by amplitude-dependent “dead zones.”

Copyright © 2011 by American Society of Mechanical Engineers
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References

Figures

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Figure 1

Schematic of a generalized two-dimensional periodic structure and considered system of reference

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Figure 2

Nine-cell assembly and associated numbering scheme

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Figure 3

Monoatomic lattice of identical masses connected by nonlinear springs

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Figure 7

Dispersion isofrequency contours for a nonlinear lattice with nonlinearity in the a1 direction (m1=1.0kg, k1=1.0Nm-1, k2=1.0Nm-1, Γ1=-3.0Nm-3, Γ2=0.0Nm-3), show noticeable stretching in one direction as amplitude increases (—: A1  = 0.1,···: A0  = 1.0, —: A0  = 2.0)

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Figure 4

Band structure of the nonlinear monoatomic lattice (m=1.0kg, k1=1.0Nm-1, k2=1.5Nm-1, A0=2.0,---:Γ1=Γ2=1.0,-:linear(Γ1=Γ2=0),•••:Γ1=Γ2=-1.0)

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Figure 5

Dispersion isofrequency contour plot of an anisotropic monoatomic linear lattice m=1kg, k1=1Nm-1, k2=1.5Nm-1

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Figure 6

The group velocity plot corresponding to isofrequency contour (Fig. 5) (—: ω = 1.65 rad s−1 , ···: ω = 2.10 rad s−1 )

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Figure 18

Point harmonic response for excitation amplitude A0=3.0 (m1=1.0kg,K1=1.0Nm-1, k2=1.0Nm-1, Γ1=-3.0Nm-3, Γ2=0.0Nm-3)

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Figure 19

Point harmonic response for excitation amplitude A0=4.5 (m1=1.0kg,k1=1.0Nm-1, k2=1.0Nm-1, Γ1=-3.0Nm-3,Γ2=0.0Nm-3)

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Figure 20

Point harmonic response for excitation amplitude A0=4.75 (m1=1.0kg,k1=1.0Nm-1, k2=1.0Nm-1, Γ1=-3.0Nm-3, Γ2=0.0Nm-3)

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Figure 8

Group velocity for the nonlinear mono-atomic lattice (parameters m=1kg, k1=1.0Nm-1, k2=1.0Nm-1, and Γ1=−3.0Nm−3, Γ2=0.0Nm-3) at frequency 1.75 rad s−1 and varying amplitude. (—: A0  = 0.1,— —A0  = 1.0, o: A0  = 2.0)

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Figure 9

Dispersion isofrequency contours for a nonlinear lattice with nonlinearity in the a2 direction (m1=1.0kg and Stiffness parameters k1=1.0Nm-1, k2=1.0Nm-1, Γ1=0.0Nm-3,Γ2=-3.0Nm-3) (- - - -: A0  = 0.1,···: A0  = 1.0, —: A0  = 2.0)

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Figure 10

Group velocity for the nonlinear mono-atomic lattice (parameters m=1kg, k1=1.0Nm-1, k2=1.0Nm-1, and Γ1=0.0Nm−3, Γ2=-3.0Nm-3) at frequency 1.75 rad s−1 and varying amplitude. (—: A0  = 0.1, — —A0  = 1.0, o: A0  = 2.0)

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Figure 11

Nonlinear diatomic lattice

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Figure 12

Amplitude dependent band diagram of the nonlinear diatomic lattice (m1=2.0kg, m2=1.0kgk1=1.0Nm-1, k2=1.5Nm-1), A0=2.0, −−−:Γ1=Γ2=1.0(hard),−:linear(Γ1=Γ2=0),···:Γ1−Γ2=−1.0(soft)

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Figure 13

Lattice with inclusion with unit cell schematic located on top right corner

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Figure 14

Band diagram for nonlinear lattices with inclusion (mi=4.0kg, m=1.0kg, k1=1.0Nm-1, k2=1.5Nm-1), A0=2.0, ---:Γ1=Γ2=2.0(hard),-linear(Γ1=Γ2=0),···:Γ1-Γ2=-2.0(soft)

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Figure 15

Schematic of a finite monoatomic lattice and incident wave at angle α

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Figure 16

Comparison of dispersion is-frequency contours and numerically estimated wavenumbers at ω0=1.60rads-1 (m1=1.0kg, k1=1.5Nm-1, k2=1.0Nm-1, Γ1=1.0Nm-3, Γ2=-1.0Nm-3) and two values of amplitude. -:A0=0.1(perturbationanalysis),•:A0=0.1(numericalestimation),···:A0=2.0(perturbationanalysis),▪:A0=2.0(numericalestimation)

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Figure 17

Comparison of dispersion isofrequency contours and numerically estimated wavenumbers at ω0=1.90rads-1 (m1=1.0kg,k1=1.5Nm-1,k2=1.0Nm-1, Γ1=1.0Nm-3, Γ2=−1.0Nm−3) and two different amplitudes. Circled outliers in high amplitude curve indicate evanescent waves in forbidden propagation direction. -A0=0.1(perturbationanalysis),•:A0=0.1(numericalestimation),···:A0=2.0(perturbationanalysis),▪:A0=2.0(numericalestimation)

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