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Research Papers

Multicell Homogenization of One-Dimensional Periodic Structures

[+] Author and Article Information
Stefano Gonella

School of Aerospace Engineering, Georgia Institute of Technology, 270 Ferst Drive, Atlanta, GA 30332

Massimo Ruzzene2

School of Aerospace Engineering, Georgia Institute of Technology, 270 Ferst Drive, Atlanta, GA 30332massimo.ruzzene@ae.gatech.edu

2

Corresponding author.

J. Vib. Acoust 132(1), 011003 (Jan 08, 2010) (11 pages) doi:10.1115/1.4000439 History: Received January 24, 2007; Revised January 23, 2008; Published January 08, 2010

Much attention has been recently devoted to the application of homogenization methods for the prediction of the dynamic behavior of periodic domains. One of the most popular techniques employs the Fourier transform in space in conjunction with Taylor series expansions to approximate the behavior of structures in the low frequency/long wavelength regime. The technique is quite effective, but suffers from two major drawbacks. First, the order of the Taylor expansion, and the corresponding frequency range of approximation, is limited by the resulting order of the continuum equations and by the number of boundary conditions, which may be imposed in accordance with the physical constraints on the system. Second, the approximation at low frequencies does not allow capturing bandgap characteristics of the periodic domain. An attempt at overcoming the latter can be made by applying the Fourier series expansion to a macrocell spanning two (or more) irreducible unit cells of the periodic medium. This multicell approach allows the simultaneous approximation of low frequency and high frequency dynamic behavior and provides the capability of analyzing the structural response in the vicinity of the lowest bandgap. The method is illustrated through examples on simple one-dimensional structures to demonstrate its effectiveness and its potentials for application to complex one-dimensional and two-dimensional configurations.

Copyright © 2010 by American Society of Mechanical Engineers
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References

Figures

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Figure 1

Single-cell and multicell (two-cell) approaches to the analysis of a generic 1D periodic medium

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Figure 2

Periodic spring-mass system and considered macrocell

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Figure 3

Comparison between exact (solid line) and approximate dispersion relations (first branch “o” and second branch “+”) for the spring-mass system

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Figure 4

Schematic of unit cell and boundary degrees of freedom and generalized forces

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Figure 8

Exact and two-cell approximations of the dispersion relations of the 1D truss (exact: solid line, first branch approximation “o,” and second branch approximation “+”)

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Figure 9

Two-cell approximation of the truss dispersion relations for increasing Taylor series order (exact: solid line, first branch approximation “o,” and second branch approximation “+”)

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Figure 10

Four-cell approximation of the truss dispersion relations for increasing Taylor series order (exact: solid line and approximated “o”)

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Figure 11

Unit cell and properties of the 1D frame

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Figure 12

Two-cell approximation of the frame dispersion relations for increasing Taylor series order (exact: solid line, first branch approximation “o,” and second branch approximation “+”)

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Figure 13

Four-cell approximation of the frame dispersion relations for increasing Taylor series order (exact: solid line and approximated “o”)

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Figure 14

Considered 1D periodic structures and homogenized continuous rod

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Figure 16

Comparison between exact and approximate harmonic responses for the truss (exact “o” and approximate: solid lines)

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Figure 5

Macrocell for two-field approximation

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Figure 6

One-dimensional truss lattice with highlighted elementary unit cell

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Figure 7

Unit cell and degrees of freedom for the 1D truss

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Figure 15

Comparison between exact and approximate harmonic responses for the spring-mass system (exact “o” and approximate: solid lines)

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