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Research Papers

Analysis of Bloch’s Method in Structures with Energy Dissipation

[+] Author and Article Information
Farhad Farzbod

George W. Woodruff School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, GA 30332-0405

Michael J. Leamy

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

J. Vib. Acoust 133(5), 051010 (Sep 20, 2011) (8 pages) doi:10.1115/1.4003943 History: Received July 22, 2010; Revised February 06, 2011; Published August 31, 2011; Online September 20, 2011

Bloch analysis was originally developed to solve Schrödinger’s equation for the electron wave function in a periodic potential field, such as found in a pristine crystalline solid. In the context of Schrödinger’s equation, damping is absent and energy is conserved. More recently, Bloch analysis has found application in periodic macroscale materials, such as photonic and phononic crystals. In the vibration analysis of phononic crystals, structural damping is present together with energy dissipation. As a result, application of Bloch analysis is not straightforward and requires additional considerations in order to obtain valid results. It is the intent of this paper to propose a general framework for applying Bloch analysis in such systems. Results are presented in which the approach is applied to example phononic crystals. These results reveal the manner in which damping affects dispersion and the presence of band gaps in periodic systems.

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Copyright © 2011 by American Society of Mechanical Engineers
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Figures

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Figure 1

Square honeycomb with out of plane motion. For this geometry, qi=[q1], q̃=[q2], q̃x=[q3], q̃y=[q4], and q̃xy=[q5].

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Figure 2

A simple mass-spring-damper structure where a is the lattice constant, and K and C are the stiffness and damping coefficients, respectively

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Figure 3

Plot of z(y) for varying values of C using a fixed 2πkR=2.4, mass m=10, and stiffness K=5

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Figure 4

Dispersion curves for the mass-spring-damper structure of Fig. 2 for various damping coefficients C and stiffness K=5, lattice vector a=0.1, and mass m=5. The real part of the wavevector is depicted here with the simulation results marked by open circles. As is evident, the real part of the wavevector does not cover the interval [0,0.5]. Due to symmetry, only the positive part of the graph is depicted.

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Figure 5

Dispersion curves for the mass-spring-damper structure of Fig. 2 for various damping coefficients C and stiffness of K=5, lattice vector a=0.1, and mass m=5. The imaginary part of the wavevector is depicted here with the simulation results marked by open circles. Due to symmetry, only the positive part of the graph is depicted.

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Figure 6

A model of a bimaterial chain which includes linear damping

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Figure 7

Real part of the wavevector for the bimaterial chain

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Figure 8

Imaginary part of the wavevector for the bimaterial chain with damping

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Figure 9

Unit cell of a simple two-dimensional lattice with out of plane motion

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Figure 10

Dispersion surfaces for the structure depicted in Fig. 9 with K1=1, K2=2, C1=0.1, C2=0.2, and m=1. For k2I=0, (k1R,k2R,ω) and (k1I,k2R,ω) are plotted in (a) and (b), while (c) and (d) are the dispersion surfaces for k2I=3/2π.

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Figure 11

Symmetry lines Γ-X, X-M and M-Γ and contour graph of the dispersion surface depicted in Fig. 1. It is evident that these symmetry lines do not capture the extreme values of ω.

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