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Research Papers

Complete Bandgap in Three-Dimensional Holey Phononic Crystals With Resonators

[+] Author and Article Information
Yue-Sheng Wang

Institute of Engineering Mechanics,
Beijing Jiaotong University,
Beijing 100044, China
e-mail: yswang@bjtu.edu.cn

1Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF VIBRATION AND ACOUSTICS. Manuscript received April 29, 2012; final manuscript received October 1, 2012; published online June 6, 2013. Assoc. Editor: Massimo Ruzzene.

J. Vib. Acoust 135(4), 041009 (Jun 06, 2013) (10 pages) Paper No: VIB-12-1123; doi: 10.1115/1.4023823 History: Received April 29, 2012; Revised October 01, 2012

In this paper, the bandgap properties of three-dimensional holey phononic crystals with resonators are investigated by using the finite element method. The resonators are periodically arranged cubic lumps in the cubic holes connected to the matrix by narrow connectors. The influence of the geometry parameters of the resonators on the bandgap is discussed. In contrast to a system with cubic or spherical holes, which has no bandgaps, systems with resonators can exhibit complete bandgaps. The bandgaps are significantly dependent upon the geometry of the resonators. By the careful design of the shape and size of the resonator, a bandgap that is lower by an order of magnitude than the Bragg bandgap can be obtained. The vibration modes at the band edges of the lowest bandgaps are analyzed in order to understand the mechanism of the bandgap generation. It is found that the emergence of the bandgap is due to the local resonance of the resonators. Spring-mass models or spring-pendulum models are developed in order to evaluate the frequencies of the bandgap edges. The study in this paper is relevant to the optimal design of the bandgaps in light porous materials.

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Figures

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Fig. 1

The unit cells of two kinds of 3D PCs and their finite element models as well as their associated Brillouin zones

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Fig. 2

Band structures of 3D holey phononic crystal in a simple cubic lattice with (a) the six-necked resonators, (b) the one-necked resonators, (c) cubic holes, and (d) spherical holes. The insets show the cross section of the corresponding unit cell. The solid and dashed lines represent the band structure along R-M-Γ-X-R and R-M'-Γ-X'-R, respectively.

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Fig. 3

Vibration modes of the six-necked resonator at the band edges corresponding to Fig. 2(a). Panels (a)–(c) show the lower-edge modes at different cross sections; panels (d)–(f) show the upper-edge modes at different cross sections.

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Fig. 4

Vibration modes of the one-necked resonator at (a) point C and (c) point D corresponding to Fig. 2(b), and (b) upper edge mode at point D for b/a = 0.9, c/a = 0.7 and d/a = 0.05. Only vibration modes at some typical cross-sections are shown for simplicity.

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Fig. 5

Configuration of the unit cell of phononic crystals with (a) cubic holes and (b) spherical holes corresponding to Figs. 2(c) and 2(d), respectively

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Fig. 6

Frequencies at the edges of the lowest bandgap for the six-necked resonator with the geometric parameter (a) d/a (b/a = 0.9 and c/a = 0.8) or (b) c/a (b/a = 0.9 and d/a = 0.1). The hollow/solid dots represent the lower/upper bandgap edges calculated by FEM. The thin dash-dotted line represents the bandgap width obtained by FEM, and the other lines represent the bandgap edges evaluated by the equivalent models.

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Fig. 7

Frequencies at the edges of the lowest bandgap for the one-neck resonator with the geometric parameter (a) d/a (b/a = 0.9 and c/a = 0.8) or (b) c/a (b/a = 0.9 and d/a = 0.05). The hollow/solid dots represent the lower/upper bandgap edge calculated by FEM. The thin dash-dotted line represents the bandgap width obtained by FEM, and the other lines represent the bandgap edges evaluated by various equivalent models.

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