Research Papers

Analysis of Bloch’s Method and the Propagation Technique in Periodic Structures

[+] 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(3), 031010 (Mar 29, 2011) (7 pages) doi:10.1115/1.4003202 History: Received March 03, 2010; Revised August 24, 2010; Published March 29, 2011; Online March 29, 2011

Bloch analysis was originally developed by Bloch to study the electron behavior in crystalline solids. His method has been adapted to study the elastic wave propagation in periodic structures. The absence of a rigorous mathematical analysis of the approach, as applied to periodic structures, has resulted in mistreatment of internal forces and misapplication to nonlinear media. In a previous article (Farzbod and Leamy, 2009, “The Treatment of Forces in Bloch Analysis,” J. Sound Vib., 325(3), pp. 545–551), we clarified the treatment of internal forces. In this article, we borrow the insight from the previous work to detail a mathematical basis for Bloch analysis and thereby shed important light on the proper application of the technique. For example, we conclusively show that translational invariance is not a proper justification for invoking the existence of a “propagation constant,” and that in nonlinear media, this results in a flawed analysis. We also provide a simple, two-dimensional example, illustrating what the role stiffness symmetry has on the search for a band gap behavior along the edges of the irreducible Brillouin zone. This complements other treatments that have recently appeared addressing the same issue.

Copyright © 2011 by American Society of Mechanical Engineers
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Figure 1

The envelope function, unit cell vibration, and their product are depicted here for a 1D case and some ω and k. Graph a shows the external excitation exerted on the structure. In graph b, the vertical lines are the cell boundary. The bottom graph c shows the vibration of the lattice as the product of the envelope function (graph a) and the unit cell vibration (graph b).

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Figure 2

A honeycomb lattice, with the unit vectors a1 and a2. The minimum set of displacements in a unit cell are qi=[q1] and q̃=[q2 q3]T. The rest of the displacements can be defined by pushing these minimal sets in the a1 and/or a2 direction: [q4]=Tx(q̃)=q̃x, [q5]=Ty(q̃)=q̃y, and [q6 q7]T=Txy(q̃)=q̃xy.

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Figure 3

A schematic view of the unit cell with its neighboring cells. In the cell (n1,n2), q̃ represents the coordinates of the lower left shaded region. T−x is an operator, which pulls forces and displacements back in the x direction. The operators T−y and T−xy would do the same in their respective directions.

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Figure 4

The force exerted on the shaded region by the neighboring cells is depicted in the right picture. The sum of all forces on this imaginary region should be zero.

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Figure 5

A unit cell of a simple lattice structure with the out of plane displacements for the masses

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Figure 6

Contour graph of ω as a function of μx and μy. As it can be seen, the maximum ω occurs inside of the Brillouin zone.

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Figure 7

ω as a function of μx and μy for the example in Fig. 5

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Figure 8

An infinite mass-spring system with nonlinear spring stiffness. The unit cell boundaries are marked by dashed lines.



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