声子晶体板复能带计算方法
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摘要
声子晶体具有弹性波带隙,可以用于结构振动与噪声控制。声子晶体传统能带算法一般给定波矢k在不可约布里渊区边界取值,然后求解特征频率ω,得到ω-k曲线。因而,传统方法中波矢k只取实数,只能求解实能带。为了求解复能带,一般需要给定频率ω,求解特征波矢k,从而得到k-ω曲线。提出了一种参数变换方法,解决了特征波矢求解中复杂的非线性特征值问题,实现了复能带的快速求解。最后,采用两个算例对文中算法进行了验证,包括布拉格声子晶体板和局域共振声子晶体板,研究了带隙内衰减常数随波传播方向的变化和阻尼对带隙的影响。
Phononic crystals possess elastic wave band-gaps,which can be used to control the vibration and noise of structures.To obtain the band structures of the phononic crystals,the conventional procedures are as follows:given the wave vector k,whose value sweep the boundary of Brillouin zone,the eigenfrequencyωcan be evaluated,resulting in theω-kcurve.However,this method can only yield the real band structure.In order to get the complex band structure,the frequencyωusually is given and then the eigenvalue of wave vector kis calculated,resulting in the k-ωcurve.This work proposes a parameter transformation method,which can resolve the complicated non-linear eigenvalue problem and achieve the rapid solution of complex band structure.Finally,two examples,i.e.,a Bragg phononic plate and a locally resonant phononic plate,are adopted to validate the proposed method.The variation of the attenuation constant along with the wave direction in the band gap and the influence of damping on the band gap are investigated in detail.
Phononic crystals possess elastic wave band-gaps,which can be used to control the vibration and noise of structures.To obtain the band structures of the phononic crystals,the conventional procedures are as follows:given the wave vector k,whose value sweep the boundary of Brillouin zone,the eigenfrequencyωcan be evaluated,resulting in theω-kcurve.However,this method can only yield the real band structure.In order to get the complex band structure,the frequencyωusually is given and then the eigenvalue of wave vector kis calculated,resulting in the k-ωcurve.This work proposes a parameter transformation method,which can resolve the complicated non-linear eigenvalue problem and achieve the rapid solution of complex band structure.Finally,two examples,i.e.,a Bragg phononic plate and a locally resonant phononic plate,are adopted to validate the proposed method.The variation of the attenuation constant along with the wave direction in the band gap and the influence of damping on the band gap are investigated in detail.
引文
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[5] Kushwaha M S,Halevi P,Dobrzynski L,et al.Acoustic band structure of periodic elastic composites[J].Physics Review Letter,1993,71(13):2022-2025.
[6] Martinez-Salar R,Sancho J,Sanchez J V,et al.Sound attenuation by sculpture[J].Nature,1995,378(6554):241.
[7] Liu Z Y,Zhang X,Mao Y,et al.Locally resonant sonic materials[J].Science,2000,289(5485):1734-1736.
[8] Li J,Chan C T.Double-negative acoustic metamaterial[J].Physics Review E,2004,70(5):055602
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[10]Milton G W.New metamaterials with macroscopic behavior outside that of continuum elastodynamics[J].New Journal of Physics,2007,9(39):46-50.
[11]Li Y F,Meng F,Li S,et al.Designing broad phononic band gaps for in-plane modes[J].Physics Letters A,2018,382(10):679-684.
[12]Srivastava A,Lu Y.Variational methods for phononic calculations[J].Wave Motion,2016,60:46-61.
[13]Coffy E,Euphrasie S,Addouche M,et al.Evidence of a broadband gap in a phononic crystal strip[J].Ultrasonics,2017,78:51-56.
[14]Chen S B,Wen J H,Wang G,et al.Tunable band gaps in acoustic metamaterials with periodic arrays of resonant shunted piezos[J].Chinese Physics B,2013,22(7):074301.
[15]Yu D L,Wang G,Liu Y Z,et al.Flexural vibration band gaps in thin plates with two-dimensional binary locally resonant structures[J].Chinese Physics,2006,15(2):266-271.
[16]Xiao Y,Wen J,Wen X.Flexural wave band gaps in locally resonant thin plates with periodically attached spring-mass resonators[J].Journal of Physics D:Applied Physics,2012,45(19):195401.
[2] Mead D J,Markus S.Coupled flexural-longitudinal wave motion in a periodic beam[J].Journal of Sound Vibration,1983,90(1):1-24.
[3] Mead D J.A new method of analyzing wave propagation in periodic structures:Applications to periodic Timoshenko beams and stiffened plates[J].Journal of Sound Vibration,1986,104(1):9-27.
[4] Sigalas M M,Economou E N.Elastic and acoustic wave band structure[J].Journal of Sound Vibration,1992,158(2):377-382.
[5] Kushwaha M S,Halevi P,Dobrzynski L,et al.Acoustic band structure of periodic elastic composites[J].Physics Review Letter,1993,71(13):2022-2025.
[6] Martinez-Salar R,Sancho J,Sanchez J V,et al.Sound attenuation by sculpture[J].Nature,1995,378(6554):241.
[7] Liu Z Y,Zhang X,Mao Y,et al.Locally resonant sonic materials[J].Science,2000,289(5485):1734-1736.
[8] Li J,Chan C T.Double-negative acoustic metamaterial[J].Physics Review E,2004,70(5):055602
[9] Fang N,Xi D,Xu J,et al.Ultrasonic metamaterials with negative modulus[J].Nature Materials,2006,5(6):452-456.
[10]Milton G W.New metamaterials with macroscopic behavior outside that of continuum elastodynamics[J].New Journal of Physics,2007,9(39):46-50.
[11]Li Y F,Meng F,Li S,et al.Designing broad phononic band gaps for in-plane modes[J].Physics Letters A,2018,382(10):679-684.
[12]Srivastava A,Lu Y.Variational methods for phononic calculations[J].Wave Motion,2016,60:46-61.
[13]Coffy E,Euphrasie S,Addouche M,et al.Evidence of a broadband gap in a phononic crystal strip[J].Ultrasonics,2017,78:51-56.
[14]Chen S B,Wen J H,Wang G,et al.Tunable band gaps in acoustic metamaterials with periodic arrays of resonant shunted piezos[J].Chinese Physics B,2013,22(7):074301.
[15]Yu D L,Wang G,Liu Y Z,et al.Flexural vibration band gaps in thin plates with two-dimensional binary locally resonant structures[J].Chinese Physics,2006,15(2):266-271.
[16]Xiao Y,Wen J,Wen X.Flexural wave band gaps in locally resonant thin plates with periodically attached spring-mass resonators[J].Journal of Physics D:Applied Physics,2012,45(19):195401.