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Research Papers

Wave Motion in Periodic Flexural Beams and Characterization of the Transition Between Bragg Scattering and Local Resonance

[+] Author and Article Information
Liao Liu

Department of Aerospace Engineering Sciences,  University of Colorado Boulder, Boulder, CO 80309

Mahmoud I. Hussein1

Department of Aerospace Engineering Sciences,  University of Colorado Boulder, Boulder, CO 80309mih@colorado.edu

These specific material and geometric properties are chosen arbitrarily; it should be noted that the main conclusions we arrive at in this section and in the rest of the paper hold for other choices including beams much smaller in dimension, limited only to scales that can be adequately described by the continuum hypothesis.

We refer to this ratio as the relative band-gap size.

If any of the resonator frequencies lie above the acoustic branch, a corresponding locally resonant band gap that might emerge will not be a subwavelength band gap.

1

Corresponding author.

J. Appl. Mech 79(1), 011003 (Nov 14, 2011) (17 pages) doi:10.1115/1.4004592 History: Received July 07, 2010; Revised June 21, 2011; Posted July 13, 2011; Published November 14, 2011

Band gaps appear in the frequency spectra of periodic materials and structures. In this work we examine flexural wave propagation in beams and investigate the effects of the various types and properties of periodicity on the frequency band structure, especially the location and width of band gaps. We consider periodicities involving the repeated spatial variation of material, geometry, boundary and/or suspended mass along the span of a beam. In our formulation, we implement Bloch’s theorem for elastic wave propagation and utilize Timoshenko beam theory for the kinematical description of the underlying flexural motion. For the calculation of the frequency band structure we use the transfer matrix method, derived here in generalized form to enable separate or combined consideration of the different types of periodicity. Our results provide band-gap maps as a function of the type and properties of periodicity, and as a prime focus we identify and mathematically characterize the condition for the transition between Bragg scattering and local resonance, each being a unique wave propagation mechanism, and show the effects of this transition on the lowest band gap. The analysis presented can be extended to multi-dimensional phononic crystals and acoustic metamaterials.

Copyright © 2012 by American Society of Mechanical Engineers
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Figures

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Figure 1

Types of periodicity: (a) material, (b) geometric, (c) boundary (discrete elastic foundation) and (d) elastically suspended mass. Dimensions for the case of a uniform beam with a square cross-section are shown in panel (a).

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Figure 2

Material periodicity: Frequency band structure for thin and thick Timoshenko beams. For the homogenous cases, the TM results are verified using a direct analytical expression for the dispersion relation [30].

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Figure 3

Material periodicity: Bloch mode shapes for Points P1 and P2 marked in Fig. 2

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Figure 4

Material periodicity: Bloch mode shapes for Points S1 and S2 marked in Fig. 2

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Figure 5

Material periodicity: Timoshenko beam theory versus Euler-Bernoulli beam theory

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Figure 6

Material periodicity: Effect of unit cell aspect ratio on relative band gap size

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Figure 7

Geometric periodicity: Frequency band structure for thick Timoshenko beam

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Figure 8

Boundary periodicity: Frequency band structure for thick Timoshenko beam for different values of elastic support spring constant

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Figure 9

Suspended mass periodicity: Frequency band structure for thick Timoshenko beam for different values of spring constant and suspended object mass

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Figure 10

Bragg scattering to local resonance transition for the suspended mass periodicity model. Evolution of (a)(b) frequency band structure and (c) group velocity with decrease in spring stiffness, (d) effect of spring stiffness on width of first band gap, width of first optical branch, and minimum frequency of first optical branch, (e)–(g) evolution of real and imaginary band structure before, at and after the transition.

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Figure 11

Bragg scattering to local resonance transition for the suspended mass periodicity model. Evolution of (a)(b) frequency band structure and (c) group velocity with increase in mass of suspended object, (d) effect of mass of suspended object on width of first band gap, width of first optical branch, and minimum frequency of first optical branch, (e)–(g) evolution of real and imaginary band structure before, at and after the transition.

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Figure 12

Effects of periodicity properties on the size of the first relative band gap

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Figure 13

Effects of periodicity properties on first band gap location and width. Band gap region is shaded; solid lines represent ωlower and ωupper, and dashed line represents ωc.

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Figure 14

Combination of periodicity types ((a) material/boundary, (b) geometry/boundary, and (c) material/geometry): Frequency band structure for thick Timoshenko beams

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Figure 15

Effects of material and/or geometric periodicity properties on band gap characteristics

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Figure 16

Combination of periodicity types (material/geometry/boundary): Frequency band structure for thick Timoshenko beam

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