含共振单元的二维多孔声子晶体带隙特性研究
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摘要
声子晶体因具有带隙特性而在减震降噪方面有潜在的应用,近年来吸引了诸多关注。基于声带隙的产生机理可知,局域共振带隙比布拉格散射带隙频率更低,且局域共振机理突破了对结构周期性的要求限制,更有利于声子晶体在实际中的应用。本文即是基于局域共振机理,设计了一种含共振单元的二维多孔声子晶体。并利用有限元方法,研究了该体系的带隙特征,主要工作如下：
1.计算了该体系的能带结构、响应谱及能带上特殊点的振动模态,结果表明：单元的局域共振是引起带隙产生的原因。
2.研究了结构的几何参数对能带结构的影响,分析了带隙边界随单元几何参数的变化,讨论了使带隙宽度最大、中心频率最低的几何参数取值。
3.通过研究带隙边界点的振动模态,提出了等效弹簧-质量模型和弹簧-钟摆模型,并给出了带隙边界的近似公式。
Phononic crystals have attracted extensive attentions for noise reduction and vibration isolation due to their properties of bandgaps. It is known that the locally resonant bandgaps are lower than those of the Bragg type, and therefore are more significant in practical applications. Based on the locally resonance, we design a type of two-dimensional holey phononic crystal with unit cells of resonators in this thesis. The bandgap properties are studied by using the finite element method. The main contributions of the thesis include:
1. Band structures and vibration modes of the proposed phononic crystal are studied. It is found that bandgaps appear in the low frequency region and that the generation is due to the local resonance of the resonant units.
2. The influences of the geometric parameters on the bandgaps are discussed, and the variations of the bandgap edges versus the geometric parameters are studied. The optimal values of the geometric parameters for the widest and lowest bandgap are suggested.
3. Equivalent spring-mass/pendulum models are developed to predict the frequencies of the bandgap edges by studying the vibration modes at the bandgap edges.
1.计算了该体系的能带结构、响应谱及能带上特殊点的振动模态,结果表明：单元的局域共振是引起带隙产生的原因。
2.研究了结构的几何参数对能带结构的影响,分析了带隙边界随单元几何参数的变化,讨论了使带隙宽度最大、中心频率最低的几何参数取值。
3.通过研究带隙边界点的振动模态,提出了等效弹簧-质量模型和弹簧-钟摆模型,并给出了带隙边界的近似公式。
Phononic crystals have attracted extensive attentions for noise reduction and vibration isolation due to their properties of bandgaps. It is known that the locally resonant bandgaps are lower than those of the Bragg type, and therefore are more significant in practical applications. Based on the locally resonance, we design a type of two-dimensional holey phononic crystal with unit cells of resonators in this thesis. The bandgap properties are studied by using the finite element method. The main contributions of the thesis include:
1. Band structures and vibration modes of the proposed phononic crystal are studied. It is found that bandgaps appear in the low frequency region and that the generation is due to the local resonance of the resonant units.
2. The influences of the geometric parameters on the bandgaps are discussed, and the variations of the bandgap edges versus the geometric parameters are studied. The optimal values of the geometric parameters for the widest and lowest bandgap are suggested.
3. Equivalent spring-mass/pendulum models are developed to predict the frequencies of the bandgap edges by studying the vibration modes at the bandgap edges.
引文
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[22]吴福根.声子晶体中的带隙及缺陷态研究[D].广州：华南理工大学,2001.
[23]Benchabane S, Khelif A, Choujaa A, Djafari-Rouhani B, Laude V. Interaction of waveguide and localized modes in phononic crystal [J]. Europhysics Letters, 2005,71:570-575.
[24]Khelif A, Choujaa A, Benchabane S, Djafari-Rouhani B, Laude V. Guiding and bending of acoustic waves in highly confined phononic crystal waveguides [J]. Applied Physics Letters, 2004,84:4400-4402.
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[26]Torres M, Montero de Espinosa F R. Ultrasonic bandgaps and negative refraction [J]. Ultrasonics,2004,42:787-790.
[27]Li J, Liu Z, Qiu C. Negative refraction imaging of acoustic waves by a two-dimensional three-component [J]. Physics Review B,2006,73:054302.
[28]Hu X, Shen Y, Liu X, Fu R, Zi J. Superlensing effect in liquid surface waves [J]. Physics Review E, 2004, 69:030201.
[29]Qiu C Y, Liu Z Y. Acoustic directional radiation and enhancement caused by band-edge states of two-dimensional phononic crystals [J]. Application Physics Letters, 2006, 89:063106.
[30]Ivansson S. Sound absorption by viscoelastic coatings with periodically distributed cavities [J]. Journal of the Acoustical Society of America, 2006, 119: 3558-3567.
[31]张荣英,姜根山,王璋奇,吕亚东.声子晶体的研究进展及应用前景[J].声学技术,2006,25：35-42.
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[36]Liu Z, Zhang X, Mao Y. Locally resonant sonic materials [J]. Science, 2000,289:1734-1736.
[37]Dowling J P. Sonic band structure in fluids exhibiting periodic density variations [J]. Journal of the Acoustical Society of America, 1992,91:2539-2543.
[38]Esquivel-Sirvent R, Cocoletzi G H. Band structure for the propagation of elastic waves in superlatticcs [J]. Journal of the Acoustical Society of America, 1994, 95:86-90.
[39]James R, Woodley S M, Dyer C M. Sonic bands, bandgaps and defect states in layered structures-theory and experiment [J]. Acoustical Society of America, 1995,97:2041-2047.
[40]Sigalas M M, Soukoulis C M. Elastic-wave propagation through disordered and/or absorptive layered systems [J]. Physics Review B,1995,51:2780-2789.
[41]Hou Z, Wu F, Liu Y. Phononic crystals containing piezoelectric material [J]. Solid State Communications,2004,130: 745-749.
[42]Wilim M, Khelif A, Ballandras S. Out-of-plane propagation of elastic waves in two-dimensional phononic band-gap materials [J]. Physics Review E, 2003,67:065602.
[43]Zhang X, Liu Z, Mei J. Acoustic bandgaps for a two-dimensional periodic array of solid cylinders in viscous liquid [J]. Journal of Physicsl-Condensed Matter, 2003,15:8207-8212.
[44]Platts S B, Movchan N V, McPhedran R C. Bandgaps and elastic waves in disordered stacks: normal incidence [J]. Proceeding of the Royal Society of London A, 2003,459:221-240.
[45]Psarobas I E, Sigalas M M. Elastic bandgaps in a fee lattice of mercury spheres in aluminum [J]. Physics Review B,2002,66:052302.
[46]温维佳,沈平局.局域共振的光子、声子功能材料[J].物理,2004,33：106-110.
[47]Liu Z Y, Zhang X X, Mao Y W, Zhu Z Z, Yang Z Y, Chan C T, Sheng P. Locally resonant sonic materials [J]. Science,2000,289:1734-1736.
[48]Wang G, Wen X S, Wen J H. Two-dimensional locally resonant phononic crystals with binary structures [J]. Physics Review B,2004,69:184302.
[49]Chen J J, Bonello B, Hou Z L. Plate-mode waves in phononic crystal thin slabs:mode conversion [J]. Physical Review E, 2008, 78:036609.
[50]Brunet T, Vasseur J, Bonello B, Djafari-Rouhani B, Hladky-Hennion A C. Lamb waves in phononic crystal slabs with square or rectangular symmetries [J]. Journal of Applied Physics, 2008,104:043506.
[51]Zhao Y C, Wu Y B, Yuan L B. Characteristics of the localized modes in 2D phononic crystal with heterostructure crossover region defect [J]. Physica Scripta, 2009,80:065401.
[52]Robillard J F, Matar O B, Vasscur J O, Dcymier P A, Stippinger M, Hladky-Hennion A C, Pennec Y, Djafari-Rouhani B. Tunable magnctoclastic phononic crystals [J]. Applied Physics Letters, 2009, 95:124104.
[53]Checoury X, Lourtioz J M. Wavelet method for computing band diagrams of 2D photonic crystals [J]. Optics Communications, 2006, 259:360-365.
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