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Research Papers

# A Perturbation Approach for Predicting Wave Propagation in One-Dimensional Nonlinear Periodic Structures

[+] Author and Article Information
Raj K. Narisetti, Massimo Ruzzene

School of Aerospace Engineering, Georgia Institute of Technology, Atlanta, GA 30332

Michael J. Leamy1

School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, GA 30332michael.leamy@me.gatech.edu

Strictly speaking, each atom in the unit cell may have its own unique neighbor spacing, where the spacings are not all equal. The function $s(j,n)$ would be evaluated as a float instead of an integer.

1

Corresponding author.

J. Vib. Acoust 132(3), 031001 (Apr 14, 2010) (11 pages) doi:10.1115/1.4000775 History: Received January 13, 2009; Revised June 16, 2009; Published April 14, 2010; Online April 14, 2010

## Abstract

Wave propagation in one-dimensional nonlinear periodic structures is investigated through a novel perturbation analysis and accompanying numerical simulations. Several chain unit cells are considered featuring a sequence of masses connected by linear and cubic springs. Approximate closed-form, first-order dispersion relations capture the effect of nonlinearities on harmonic wave propagation. These relationships document amplitude-dependent behavior to include tunable dispersion curves and cutoff frequencies, which shift with wave amplitude. Numerical simulations verify the dispersion relations obtained from the perturbation analysis. The simulation of an infinite domain is accomplished by employing viscous-based perfectly matched layers appended to the chain ends. Numerically estimated wavenumbers show good agreement with the perturbation predictions. Several example chain unit cells demonstrate the manner in which nonlinearities in periodic systems may be exploited to achieve amplitude-dependent dispersion properties for the design of tunable acoustic devices.

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## Figures

Figure 1

Chain of unit cells, each cell consisting of N elements

Figure 2

Schematic of PML in a 100 mass (monoatomic) chain

Figure 3

Monoatomic chain

Figure 4

Dispersion behavior of nonlinear monoatomic chain with increasing amplitude A1(0) predicted by perturbation analysis

Figure 5

Numerical estimation of dispersion in nonlinear monoatomic chain with increasing amplitude A1(0), as shown in limited frequency range

Figure 6

Real part of wavenumber versus frequency of the wave with respect to amplitude A1(0)

Figure 7

Imaginary part of wavenumber versus frequency of the wave with respect to amplitude A1(0)

Figure 8

Monochain with an attached nonlinear base

Figure 9

Dispersion behavior of nonlinear monoatomic chain with attached nonlinear base with increasing amplitude A1(0) predicted by perturbation analysis

Figure 10

Numerical estimation of nonlinear monoatomic chain with attached nonlinear base with increasing amplitude A1(0), as shown in limited frequency range

Figure 11

Real part of wavenumber versus frequency of the wave with respect to amplitude A1(0)

Figure 12

Imaginary part of wavenumber versus frequency of the wave with respect to amplitude A1(0)

Figure 13

Nonlinear diatomic chain with m2≥m1

Figure 14

Dispersion trend with respect to amplitude in nonlinear diatomic chain as predicted by perturbation analysis

Figure 15

Numerical estimation of dispersion in optical mode of nonlinear diatomic chain with respect to amplitude in certain frequency range

Figure 16

Numerical estimation of dispersion in acoustic mode of nonlinear diatomic chain with respect to amplitude in certain frequency range

Figure 17

Schematic of an amplitude dependent frequency isolator. Input signal is transformed by nonlinear diatomic chain. Depending on the amplitude of input signal the output has different frequency content.

Figure 18

Input signal containing frequencies 1.50 rad/s and 2.46 rad/s at A0=0.5; FFT of output signal is shown on the right, which depicts the existence of two input frequencies

Figure 19

Input signal containing frequencies 1.50 rad/s and 2.46 rad/s at A0=0.75; FFT of output signal is shown on the right, which depicts the elimination of high frequency content from the input signal

Figure 20

Input signal containing frequencies 1.75 rad/s and 2.5 rad/s at A0=0.7; FFT of output signal is shown on the right, which depicts the existence of two input frequencies as well as a third one generated due to nonlinearity

Figure 21

Input signal containing frequencies 1.75 rad/s and 2.5 rad/s at A0=0.1; FFT of output signal is shown on the right, which depicts the elimination of lower frequency content from the actual input signal

Figure 22

Schematic of narrow band-pass filter. Input signal is first filtered by a nonlinear monoatomic hard chain and then filtered again in series by a nonlinear chain with attached base. The resulting output signal contains narrow bandwidth and the bandwidth can be tuned by changing amplitude at gain 1 and gain 2.

Figure 23

Schematic describing the tunable narrow band output with designed configuration

Figure 24

Input and output signal frequency spectrum corresponding to input amplitude A0=1.0; output signal is normalized

Figure 25

Input and output signal frequency spectrum corresponding to input amplitude A0=11.0; output signal is normalized

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