基于二维声子晶体波分离理论的周期结构振动带隙特性研究
|
摘要
机械振动是物理学及工程技术中广泛存在的现象,抑制有害振动一直是物理学和工程技术中迫切需要解决的问题。在工程巾,当作周期结构处理的结构具有机械振动带隙特性,即当弹性波在周期结构中传播时,某些频率范围内,振动不能通过。
周期结构的研究在物理领域中已广泛开展,有电子晶体、光子晶体声子晶体。声子晶体是指具有声波或弹性波禁带的周期性结构材料。在带隙频率范围内,声波和振动不允许通过。声子晶体在带隙理论和带隙算法方面已取得重要进展,但应用探索才刚开始。
本文以此为背景,利用散度和旋度算符对弹性波巾的压缩波和剪切波进行分离,提出利用声子晶体波分离理论,对周期结构的压缩波和剪切波振动带隙特性进行深入研究。
首先从周期结构、光子晶体引入声子晶体的概念、属性、禁带机理、非完整周期性、能带结构计算方法、并阐述了声子晶体的研究现状、研究意义。
接着,我们从理想弹性介质中的弹性波波动方程出发,利用散度和旋度算符分离出纵波和横波。介绍了声子晶体常见的排列,推导了固-固系统和液-液(气-气)系统的平而波展开方法的本征值方程,并给出了几种常见单胞的结构函数和旋转结构函数。且详细介绍了一维声子晶体的传输矩阵法。
第三,我们研究了二维固液和固固系统的弧状声子晶体。对于钢和水组成的系统,研究了低频能带结构；模态和第一带隙上边沿的关系：钢层的厚度对第一声波带隙的影响。对于钢和环氧树脂组成的系统,分别研究纵波和横波低频复能带结构,发现存在一些局域平带,其局域频率对应于材料弹性参数；研究钢层厚度对局域共振频率的影响。结果表明低模态下第一声波带隙从零赫兹开始。在高阶旋转对称下,可以获得局域共振带隙,局域的频率对应于声波在材料中的速度。这些特性可以有效应用于低频禁带结构中,或窄带通器件设计中。
第四,我们研究了在环氧树脂基体中嵌入正方形晶格钢圆柱体和正方形晶格矩形横截面柱体组成的二维声子晶体系统压缩波低频能带结构。发现能带结构中存在一此局域化平带，分析了最低几条能带Γ点的局域模在不可约Brillion区产生的压力分布图。比较发现,第二个系统产生局域模的数目减少,并且动幅度增加,局域化现象更为明显。振动在多通道的局域模式得到加强,波将沿着通道传播。结果表明散射体对称性降低是增强局域的有效方法。这种特性对设计波导有潜在应用。
第五,我们研究了磁流变弹性体基体中嵌入铅柱的正方形格的二维声子晶体剪切波的能带结构。发现剪切波带隙对柱体横截面长度是敏感的,剪切波带隙仅在很小的长度范围内打开,迅速增加,并最终消失；带隙宽度及中间频率随着外磁场强度的增加呈线性增加,并出现旧的带隙向新的带隙切换现象；当30°<θ<45°时,剪切波带隙与外部磁场的旋转角度几乎是线性关系。结果表明,可以通过外部磁场旋转或磁场强度变化进行非接触式调谐选频滤波。这些特点潜在应用于剪切波带隙可调装置的设计。
最后,研究了二维声子晶体点群对称对不可约布里渊区的影响。我们在单柱体二维声子晶体单胞中增加一个柱体,设计成不同点群对称操作数的对称结构,计算了最低5条能带的本征频率值在第一布里渊区的分布图,结果表明：在计算二维声子晶体的能带结构时,如果声子晶体的点群对称操作数为n,所取的不可约布里渊区应为整个第一布里渊区的1/n。
本文将声子晶体理论和算法引入到周期结构的振动带隙特性研究中,应用理论分析和仿真相结合的研究方法,进一步深化了周期结构的振动带隙理论,特别是通过弹性波分离的方法,为周期结构的低频减振降噪提供了新的思路和技术途径。这些研究对推动声了晶体理论在减振降噪领域巾的工程应用具有重要的理论意义和工程参考价值。
Mechanical vibration is a widespread phenomenon in physics and engineering, inhibition of harmful vibration has been the urgent need to address the problems in the physics and engineering. In engineering, the structures which is treated as periodic structures have mechanical vibration band gaps, i.e.. when the elastic waves propagate in the periodic structures, the vibration can not be within a certain frequency range.
Periodic structure has been widely carried out in the physical realm, electronic crystals, photonic crystals, phononic crystals (PCs). PCs are the periodic structural materials with acoustic or elastic band gaps. Acoustic waves and vibrations in the frequency range of the phononic band gaps (PBGs) are not allowed through. The PBGs theory and algorithm have made important progress, but the application of exploration has just begun.
In this background, we separated compression and shear waves in the elastic waves by divergence and curl operators. Based on the elastic waves separation theory of PCs, we conduct in-depth research of compression and shear waves vibrating bandgaps characteristics of the periodic structure.
First, from periodic structures and photonic crystals, we introduced the PCs concept, properties, mechanism of PBGs, incomplete periodic PCs, band structures calculation method, applications test and research significance.
Next, we proceed from the elastic wave equation of ideal elastic medium and separate of longitudinal and transverse waves using the divergence operator and curl operator. We have given the PCs common arrangement, the derivation of the eigenvalue equation of the plane wave expansion method of solid-solid and liquid-liquid (gas-gas) system, and several common unit structure functions and rotating structure functions. Details of the transfer matrix method in the one-dimensional PCs are described.
Third, we study the two-dimensional solid-liquid and solid-solid systems arc-shaped PCs. For steel and water systems, we have studied the low-frequency band structure; the relationship of the upper edge in the first band gap and the modes; the thickness of the steel layer to affect the first PBGs. For steel and epoxy systems, we have studied the longitudinal and transverse waves low frequency complex band structures, respectively; found the localized flat bands with local frequency corresponds to the elastic parameters; the thickness of the steel layer to affect the locally resonant frequency. The results showed that the first PBGs start with zero Hz with low modes. The locally resonant gaps are obtained with higher-order rotation symmetry, for locally resonant frequencies corresponding to the speeds of acoustic waves in the background materials. These properties can be efficiently used in a structure for low frequencies that are forbidden, or in a device that permits the propagation of signals within a narrow window of frequencies.
Fourth, by the plane wave expansion method based on the decomposition of elastic waves, a study on P-wave band structures in2D solid-solid PCs revealed multiple flat bands at low frequency. We considered the systems which comprising a circular scatterer in the square lattice and a rectangular scatterer in the square lattice. An analysis of the pressure field distribution of the localised modes at Γ points in the lowest bands of the systems shows that a lower scatterer symmetry are more effective in strengthening localisation. The number of localisations is reduced, and the vibration amplitudes increase. The localisation phenomena are more evident. Vibrations in the multi-channels of the localised modes are stronger. The proposed property has potential applications in the design of waveguides.
Fifth, we study the shear wave band structures of a square lattice2D PCs comprising the lead cylinders embedded in the Magneto-rheological elastomers matrix. Shear wave band gaps (SBGs) are sensitive to the length of the cylinders, such that the SBGs opened up only within small ranges of crystal length, increasing rapidly and finally disappearing; the widths of the SBG frequency and the middle frequency of gaps increased linearly along with the external magnetic magnitude. And the old gaps shifted toward new gaps; when30°<θ<45°the SBGs are almost linearly related to the external rotational angle of the magnetic field. The results show that by introduction of an external contactless magnetic field which can be rotated or has a variable magnitude, the SBGs can be obtained and tuned as frequency-selective filters. These methods can be potential applications in the design of SBG-tunable devices.
Finally, we study the point group symmetry of two-dimensional PCs effecting on the irreducible Brillouin zone. Additional atom in the unit cell, designing it in kinds of structures with different operation number of symmetry, we calculated eigen-frequencies of the five lowest bands in the Brillouin zone, the results show that:in the calculation of2D PCs band structure, if the PCs point group symmetry operation number is n, the irreducible Brillouin zone should be1/n of the whole first Brillouin zone.
In this paper, the PCs theory and algorithms introduced to the research of periodic structure vibration bandgap characteristics, with the application of theoretical analysis and simulation combining research methods, further deepen the vibration bandgap theory of the periodic structure, especially through the elastic wave separation method, it provide new ideas and technical approaches to the low-frequency noise and vibration reduction for the periodic structure. These studies have important theoretical significance and value of engineering, promoting the engineering application of the theory of PCs in the field of noise and vibration reduction.
周期结构的研究在物理领域中已广泛开展,有电子晶体、光子晶体声子晶体。声子晶体是指具有声波或弹性波禁带的周期性结构材料。在带隙频率范围内,声波和振动不允许通过。声子晶体在带隙理论和带隙算法方面已取得重要进展,但应用探索才刚开始。
本文以此为背景,利用散度和旋度算符对弹性波巾的压缩波和剪切波进行分离,提出利用声子晶体波分离理论,对周期结构的压缩波和剪切波振动带隙特性进行深入研究。
首先从周期结构、光子晶体引入声子晶体的概念、属性、禁带机理、非完整周期性、能带结构计算方法、并阐述了声子晶体的研究现状、研究意义。
接着,我们从理想弹性介质中的弹性波波动方程出发,利用散度和旋度算符分离出纵波和横波。介绍了声子晶体常见的排列,推导了固-固系统和液-液(气-气)系统的平而波展开方法的本征值方程,并给出了几种常见单胞的结构函数和旋转结构函数。且详细介绍了一维声子晶体的传输矩阵法。
第三,我们研究了二维固液和固固系统的弧状声子晶体。对于钢和水组成的系统,研究了低频能带结构；模态和第一带隙上边沿的关系：钢层的厚度对第一声波带隙的影响。对于钢和环氧树脂组成的系统,分别研究纵波和横波低频复能带结构,发现存在一些局域平带,其局域频率对应于材料弹性参数；研究钢层厚度对局域共振频率的影响。结果表明低模态下第一声波带隙从零赫兹开始。在高阶旋转对称下,可以获得局域共振带隙,局域的频率对应于声波在材料中的速度。这些特性可以有效应用于低频禁带结构中,或窄带通器件设计中。
第四,我们研究了在环氧树脂基体中嵌入正方形晶格钢圆柱体和正方形晶格矩形横截面柱体组成的二维声子晶体系统压缩波低频能带结构。发现能带结构中存在一此局域化平带，分析了最低几条能带Γ点的局域模在不可约Brillion区产生的压力分布图。比较发现,第二个系统产生局域模的数目减少,并且动幅度增加,局域化现象更为明显。振动在多通道的局域模式得到加强,波将沿着通道传播。结果表明散射体对称性降低是增强局域的有效方法。这种特性对设计波导有潜在应用。
第五,我们研究了磁流变弹性体基体中嵌入铅柱的正方形格的二维声子晶体剪切波的能带结构。发现剪切波带隙对柱体横截面长度是敏感的,剪切波带隙仅在很小的长度范围内打开,迅速增加,并最终消失；带隙宽度及中间频率随着外磁场强度的增加呈线性增加,并出现旧的带隙向新的带隙切换现象；当30°<θ<45°时,剪切波带隙与外部磁场的旋转角度几乎是线性关系。结果表明,可以通过外部磁场旋转或磁场强度变化进行非接触式调谐选频滤波。这些特点潜在应用于剪切波带隙可调装置的设计。
最后,研究了二维声子晶体点群对称对不可约布里渊区的影响。我们在单柱体二维声子晶体单胞中增加一个柱体,设计成不同点群对称操作数的对称结构,计算了最低5条能带的本征频率值在第一布里渊区的分布图,结果表明：在计算二维声子晶体的能带结构时,如果声子晶体的点群对称操作数为n,所取的不可约布里渊区应为整个第一布里渊区的1/n。
本文将声子晶体理论和算法引入到周期结构的振动带隙特性研究中,应用理论分析和仿真相结合的研究方法,进一步深化了周期结构的振动带隙理论,特别是通过弹性波分离的方法,为周期结构的低频减振降噪提供了新的思路和技术途径。这些研究对推动声了晶体理论在减振降噪领域巾的工程应用具有重要的理论意义和工程参考价值。
Mechanical vibration is a widespread phenomenon in physics and engineering, inhibition of harmful vibration has been the urgent need to address the problems in the physics and engineering. In engineering, the structures which is treated as periodic structures have mechanical vibration band gaps, i.e.. when the elastic waves propagate in the periodic structures, the vibration can not be within a certain frequency range.
Periodic structure has been widely carried out in the physical realm, electronic crystals, photonic crystals, phononic crystals (PCs). PCs are the periodic structural materials with acoustic or elastic band gaps. Acoustic waves and vibrations in the frequency range of the phononic band gaps (PBGs) are not allowed through. The PBGs theory and algorithm have made important progress, but the application of exploration has just begun.
In this background, we separated compression and shear waves in the elastic waves by divergence and curl operators. Based on the elastic waves separation theory of PCs, we conduct in-depth research of compression and shear waves vibrating bandgaps characteristics of the periodic structure.
First, from periodic structures and photonic crystals, we introduced the PCs concept, properties, mechanism of PBGs, incomplete periodic PCs, band structures calculation method, applications test and research significance.
Next, we proceed from the elastic wave equation of ideal elastic medium and separate of longitudinal and transverse waves using the divergence operator and curl operator. We have given the PCs common arrangement, the derivation of the eigenvalue equation of the plane wave expansion method of solid-solid and liquid-liquid (gas-gas) system, and several common unit structure functions and rotating structure functions. Details of the transfer matrix method in the one-dimensional PCs are described.
Third, we study the two-dimensional solid-liquid and solid-solid systems arc-shaped PCs. For steel and water systems, we have studied the low-frequency band structure; the relationship of the upper edge in the first band gap and the modes; the thickness of the steel layer to affect the first PBGs. For steel and epoxy systems, we have studied the longitudinal and transverse waves low frequency complex band structures, respectively; found the localized flat bands with local frequency corresponds to the elastic parameters; the thickness of the steel layer to affect the locally resonant frequency. The results showed that the first PBGs start with zero Hz with low modes. The locally resonant gaps are obtained with higher-order rotation symmetry, for locally resonant frequencies corresponding to the speeds of acoustic waves in the background materials. These properties can be efficiently used in a structure for low frequencies that are forbidden, or in a device that permits the propagation of signals within a narrow window of frequencies.
Fourth, by the plane wave expansion method based on the decomposition of elastic waves, a study on P-wave band structures in2D solid-solid PCs revealed multiple flat bands at low frequency. We considered the systems which comprising a circular scatterer in the square lattice and a rectangular scatterer in the square lattice. An analysis of the pressure field distribution of the localised modes at Γ points in the lowest bands of the systems shows that a lower scatterer symmetry are more effective in strengthening localisation. The number of localisations is reduced, and the vibration amplitudes increase. The localisation phenomena are more evident. Vibrations in the multi-channels of the localised modes are stronger. The proposed property has potential applications in the design of waveguides.
Fifth, we study the shear wave band structures of a square lattice2D PCs comprising the lead cylinders embedded in the Magneto-rheological elastomers matrix. Shear wave band gaps (SBGs) are sensitive to the length of the cylinders, such that the SBGs opened up only within small ranges of crystal length, increasing rapidly and finally disappearing; the widths of the SBG frequency and the middle frequency of gaps increased linearly along with the external magnetic magnitude. And the old gaps shifted toward new gaps; when30°<θ<45°the SBGs are almost linearly related to the external rotational angle of the magnetic field. The results show that by introduction of an external contactless magnetic field which can be rotated or has a variable magnitude, the SBGs can be obtained and tuned as frequency-selective filters. These methods can be potential applications in the design of SBG-tunable devices.
Finally, we study the point group symmetry of two-dimensional PCs effecting on the irreducible Brillouin zone. Additional atom in the unit cell, designing it in kinds of structures with different operation number of symmetry, we calculated eigen-frequencies of the five lowest bands in the Brillouin zone, the results show that:in the calculation of2D PCs band structure, if the PCs point group symmetry operation number is n, the irreducible Brillouin zone should be1/n of the whole first Brillouin zone.
In this paper, the PCs theory and algorithms introduced to the research of periodic structure vibration bandgap characteristics, with the application of theoretical analysis and simulation combining research methods, further deepen the vibration bandgap theory of the periodic structure, especially through the elastic wave separation method, it provide new ideas and technical approaches to the low-frequency noise and vibration reduction for the periodic structure. These studies have important theoretical significance and value of engineering, promoting the engineering application of the theory of PCs in the field of noise and vibration reduction.
引文
[1]丁文镜.减振理论[M].北京：清华大学出版社,1988.
[2]郁殿龙.基于声子晶体理论的梁板类周期结构振动带隙特性研究[D].长沙：国防科技大学,2006.
[3]M I Hussein. Dynamics of banded materials and structures:analysis, design and computation in multiple scales[D]. The University of Michigan,2004.
[4]L Brillouin. Wave propagation in periodic structures.2nd edition[M]. New York:Dover Publications,1953.
[5]J W Mile. Vibrations of beams on many supports[C]. Proceedings of the American Society of Civil of Kngineering 82. EMI,1956:1.
[6]M Heckl. Investigations on the vibrations of grillages and other simple beam structures[J]. J. Acoust. Soc. Am.,1964,36(7):1335-1343.
[7]D J Mead. Wave propagation in continuous periodic structures:Research contributions from Southampton,1964-1995[J]. J. Sound Vib.,1996, 180(3):495-524.
[8]S Gupta. Dynamics of periodically stiffened structures using a wave approach[D]. University of Southampton,1970.
[9]S Gupta. Natural frequencies of periodic skin-stringer structures using a wave approach[J]. J. Sound Vib.,1971,16(4):567-580.
[10]S Gupta. Natural flexural waves and the normal modes of periodically-supported beams and plates[J]. J. Sound Vib.,1970,13(1):89-101.
[11]D J Mead. Free wave propagation in periodically supported, infinite beams[J]. J. Sound Vib.,1970,11(2):181-197.
[12]D J Mead. A general theory of harmonic wave propagation in linear periodic systems with multiple coupling[J]. J. Sound Vib.,1973,27(2): 235-260.
[13]D J Mead. Wave propagation and natural modes in periodic systems:Ⅰ. Mono-coupled systems[J]. J. Sound Vib.,1975,40(1):1-18.
[14]D J Mead. Wave propagation and natural modes in periodic systems:Ⅱ. Multi-coupled systems, with and without damping[J]. J. Sound Vib. 1975,40(1):19-39.
[15]D J Mead, S Parthan. Free wave propagation in two-dimensional periodic plates[J]. J. Sound Vib.,1979,64(3):325-346.
[16]D J Mead, S Markus. Coupled flexural-longitudinal wave motion in a periodic beam[J]. J. Sound Vib.,1983,90(1):1-24.
[17]D J Mead, N S Bardell. Free vibration of a thin cylindrical shell with discrete axial stiffeners[J]. J. Sound Vib.,1986,111(2):229-250.
[18]D J Mead, N S Bardell. Free vibration of a thin cylindrical shell with periodic circumferential stiffeners[J]. J. Sound Vib.,1987,115(3): 499-520.
[19]M Toso. Wave propagation in rods, shells, and rotating shaft with non-uniform geometry[D]. University of Maryland.2004.
[20]S M Jeong. Analysis of vibration of 2-D periodic cellular structures[D]. Georgia Institute of Technology,2005.
[21]J S Jensen, O Sigmund. J J Thomsen, M P Bendsoc. Design of multi-phase structures with optimized vibrational and wave-transmitting properties[C].15th Nordic Semillar on Computational Mechanics,2002: 63-66.
[22]V Mangaraju, V R Sonti. Wave attenuation in periodic three-layered beams:Analytical and FEM study [J]. J. Sound Vib.,2004,276(3-5): 541-570.
[23]D Richards, D J Pines. Passive reduction of gear mesh vibration using a periodic drive shaft[J]. J. Sound Vib..2003,264(2):317-342.
[24]M Tawfik, J Chung, A Baz. Wave attenuation in periodic helicopter blade[C].5th Jordan International Mechanical Engineering Conference, Amman, Jordan,2004.
[25]S Asiri. Isolation of helicopter gearbox vibration using periodic support struts[D]. University of Maryland,2003.
[26]夏建白,朱邦芬.半导体超晶格物理[M].上海：上海科学技术出版社,1995.
[27]E Yablonovitch. Inhibited spontaneous mission in solid-state physics and electronics[J]. Phys. Rev. Lett.,1987,58(20):2059-2062.
[28]S John. Strong localization of photons in certain disordered dielectric superlattices[J]. Phys. Rev. Lett.,1987,58(20):2486-2489.
[29]M M Sigalas, E N Economou. Elastic and asoustic wave band structure[J]. J. Sound Vib.,1992,158(2):377-382.
[30]M S Kushwaha, P Halevi, L Dobrzynski, B Djafari-Rouhani. Acoustic band structure of periodic elastic composites[J]. Phys. Rev. Lett.,1993, 71(13):2022-2025.
[31]R Martinez-Sala, J Sancho, J V Sanchez, V Gomez, J Llinares, F Meseguer. Sound attenuation by sculpture[J]. Nature,1995,378(6554):241-1.
[32]W M Kuang, Z L Hou, Y Y Liu. The effects of shapes and symmetries of scatterers on the phononic band gap in 2D phononic crystals[J]. Phys. Lett. A,2004,332(5-6):481-490.
[33]A N Guz, N A Shulga. Dynamics of laminated and fibrous composites[J]. Appl. Mechanics Rev.,1992,45(2):35-60.
[34]Z Y Liu, X X Zhang, Y W Mao, Z Z Zhu, Z Y Yang, C T Chan, P Sheng. Locally resonant sonic materials[J]. Science,2000,289(5485): 1734-1736.
[35]Z Liu, C T Chan, P Sheng. Analytic model of phononic crystals with local resonances[J]. Phys. Rev. B. 2005, 71(1): 014103-8.
[36]C Goffaux, J Sanchez-Dehesa. Two-dimensional phononic crystals studied using a variational method: Application to lattices of locally resonant materials[J]. Phys. Rev. B. 2003. 67(14): 144301-10.
[37]C Goffaux, J Sanchez-Dehesa. A L Yeyati, P Lambin. A Khelif, J O Vasseur, B Djafari-Rouhani. Evidence of fano-like interference phenomena in locally resonant materials[J]. Phys. Rev. Lett.. 2002,88(22)：225502-4.
[38]M H Kin, C K Cheng, Z Yang, X X Zhang, P Sheng. Broadband locally resonant sonic shiclds[J]. Appl. Phys. Lett., 2003, 83(26): 5566-5568.
[39]G Wang. X S Wen, J H Wen. L H Shao, Z Y Liu. Two-dimensional locally resonant phononic crystals with binary structures[J]. Phys. Rev. Lett., 2004, 93(15): 154302-4.
[40]R E Vines, J P Wolfe. Scanning phononic lattices with surface acoustic waves[J]. Phys. B: Conden. Matt.. 1999, 263-264: 567-570.
[41]M Torres, F R Montero de Espinosa, D Garcia-Pablos, N Garcia. Sonic band gaps in finite elastic media: Surface states and localization phenomena in linear and point defectsfJ]. Phys. Rev. Lett., 1999, 82(15): 3054-3057.
[42]J Gao, X Y Zou, J C Cheng, B W Li. Band gaps of lower-order Lamb wave in thin plate with one-dimensional phononic crystal layer:Effect of substrate[J]. Appl. Phys. Lett.,2008,92(2):023510-3.
[43]J C Hsu, T T Wu. Lamb waves in binary locally resonant phononic plates with two-dimensional lattices[J]. Appl. Phys. Lett.,2007,90(20): 201904-3.
[44]B Bonello, C Charles, F Ganot. Lamb waves in plates covered by a two-dimensional phononic film[J]. Appl. Phys. Lett.,2007,90(2): 021909-3.
[45]D G Zhao, Z Y Liu, C Y Qiu, Z J He, F Y Cai, M Z Ke. Surface acoustic waves in two-dimensional phononic crystals:Dispersion relation and the eigenfield distribution of surface modes[J]. Phys. Rev. B,2007,76(14): 144301-6.
[46]R Sainidou, B Djafari-Rouhani, J O Vasseur. Surface acoustic waves in finite slabs of three-dimensional phononic crystals[J]. Phys. Rev. B, 2008,77(9):094304-9.
[47]M Z Ke, Z J He, S S Peng, Z Y Liu, J Shi. Surface resonant-states-enhanced acoustic wave tunneling in two-dimensional phononic crystals[J]. Phys. Rev. Lett.,2007,99(4):044301-4.
[48]T T Wu, Z G Huang, S Lin. Surface and bulk acoustic waves in two-dimensional phononic crystal consisting of materials with general anisotropy[J]. Phys. Rev. B,2004,69(9):094301-10.
[49]V Laude, M Wilm, S Benchabane, A Khelif. Full band gap for surface acoustic waves in a piezoelectric phononic crystal[J]. Phys. Rev. E,2005, 71(3):036607-7.
[50]R James, S M Woodley, C M Dyer, F H Victor. Sonic bands, band gaps, and defect states in layered structures:Theory and experiment[J]. J. Acoust. Soc. Am.,1995,97(4):2041-2047.
[51]J N Munday. C B Bennett, W M Robertson. Band gaps and defect modes in periodically structured waveguides[J]. J. Acoust. Soc. Am.,2002, 112(4):1353-1358.
[52]W M Robertson, C Baker. C B Bennett. Slow group velocity propagation of sound via defect coupling in a one-dimensional acoustic band gap array[J]. Am. J. Phys.,2004,72(2):255-257.
[53]M M Sigalas. Elastic wave band gaps and defect states in two-dimensional composites[J]. J. Acoust. Soc. Am.,1997.101(3): 1256-1261.
[54]M M Sigalas. Defect states of acoustic waves in a two-dimensional lattice of solid cylinders[J]. J. Appl. Phys.,1998,84(6):3026-3030.
[55]M Kafesaki, M M Sigalas, N Garcia. Frequency modulation in the transmitivity of wave guides in elastic-wave band-gap materials[J]. Phys. Rev, Lett.,2000,85(19):4044-4047.
[56]M Kafesaki, M M Sigalas, N Garcia. Wave guides in two-dimensional elastic wave band gap materials[J]. Phys. B:Conden. Matt.,2001, 296(1-3):190-194.
[57]M Torres, F R Montero de Espinosa, D Garcia-Pablos, N Garcia. Sonic band gaps in finite elastic media:Surface states and localization phenomena in linear and point defects[J]. Phys. Rev. Lett.,1999,82(15): 3054-3057.
[58]F G Wu, Z L Hou, Z Y Liu, Y Y Liu. Point defect states in two-dimensional phononic crystals[J]. Phys. Lett. A,2001,292(3): 198-202.
[59]F G Wu, Z Y Liu, Y Y Liu. Splitting and tuning characteristics of the point defect modes in two-dimensional phononic crystals[J]. Phys. Rev. E,2004,69(6):066609-4.
[60]J H Sun, T T Wu. Analyses of mode coupling in joined parallel phononic crystal waveguides[J]. Phys. Rev. B,2005,71(17):174303-8.
[61]J H Sun, T T Wu. Propagation of acoustic waves in phononic crystal plates and waveguides using a finite-difference time domain method[J]. Phys. Rev. B, 2007. 6(10): 104304-8.
[62]I E Psarobas, N Stefanou, A Modinos. Phononic crystals with planar defects[J]. Phys. Rev. B, 2000. 62(9): 5536-5540.
[63]H Chandra, P A Deymier, J O Vasseur. Elastic wave propagation along waveguides in three-dimensional phononic crystals[J]. Phys. Rev. B, 2004. 70(5): 054302-6.
[64]J O Vasseur, P A Deymier, B Djafari-Rouhani, Y Pennec, A C Hladky-Hennion. Absoult forbidden bands and waveguiding in two-dimensional phononic crystal plates[J]. Phys. Rev. B, 2008, 77(8): 085415-15.
[65]J O Vasseur, A C Hladky-Hennion, B Djafari-Rouhani, F Duval, B Dubus, Y Pennec, P A Deymier. Waveguiding in two-dimensional piezoelectric phononic crystal plates[J]. J. Appl. Phys., 2007, 101(11): 1 14904-6.
[66]C Y Qiu, Z Y Liu, J Shi, C T Chan. Directional acoustic source based on the resonant cavity of two-dimensional phononic crystalsfJ]. Appl. Phys. Lett., 2005, 86(22): 224105-3.
[67]X F Wang, M S Kushwaha, P Vasilopoulos. Tunability of acoustic spectral gaps and transmission in periodically stubbed wavcguidcs[J]. Phys. Rev. B, 2001, 65(3): 035107-10.
[68]A C Hladky-Hennion, J O Vasseur, B Djafari-Rouhani, M Billy. Sonic band gaps in one-dimensional phononic crystals with a symmetry stub[J]. Phys. Rev. B,2008,77(10):104304-7.
[69]Z L Hou, F G Wu, Y Y Liu. Acoustic wave propagating in one-dimensional Fibonacci binary composite systems[J]. Phys. B,2004, 344(1-4):391-397.
[70]M M Sigalas, E N Economou. Attenuation of multiple-scattered sound[J]. Europhys. Lett.,1996,36(4):241-246.
[71]F G Wu, Z Y Liu, Y Y Liu. Acoustic band gaps created by rotating square rods in two-dimensional lattice[J]. Phys. Rev. E,2002,66(4): 046628-5.
[72]S J Mizuno. Eigen-frequency and decay factor of the localized phonon in a super lattice with a defect layer[J]. Phys. Rev. B,2002,65(19): 193302-4.
[73]C Goffaux, J P Vigneron. Theoretical study of a tunable phononic band gap system[J]. Phys. Rev. B,2001,64(7):075118-5.
[74]王刚,温激鸿,韩小云,赵宏刚.二维声子晶体带隙计算中的时域有限差分方法[J].物理学报,2003,52(8)：1943-1947.
[75]M M Sigalas, N Garcia. Theoretical study of three dimensional elastic band gaps with the finite-different time domain method[J]. J. Appl. Phys. 2000,87(6):3122-3125.
[76]M Kafesaki, E N Economou. Multiple-scattering theory for three-dimensional periodic acoustic composites[J]. Phys. Rev. B,1999, 60(17):11993-12001.
[77]Z Y Liu, C T Chan, P Sheng, A L Goertzen, J H Page. Elastic wave scattering by periodic structures of spherical objects:Theory and experiment[J]. Phys. Rev. B,2000,62(4):2446-2457.
[78]J Mei, Z Y Liu, J Shi, D Tian. Theory for elastic wave scattering by a two-dimensional periodical array of cylinders:An ideal approach for band-structure calculations[J]. Phys. Rev. B,2003.67(24):245107-7.
[79]A Khelif, P A Deymier, B Djafari-Rouhani, J O Vasseur, L Dobrzynski. Two-dimensional phononic crystal with tunable narrow pass band: Application to a waveguide with selective frequency [J]. J. Appl. Phys. 2003,94(3):1308-1311.
[80]G Wang, J H Wen, Y Z Liu, X S Wen. One-dimensional phononic crystals with locally resonant structures[J]. Phys. Lett. A,2004,327(5-6): 512-521.
[81]王刚,温激鸿, 刘耀宗,郁殿龙,温熙森. 大弹性常数差二维声子晶体带隙计算中的集中质量法[J]. 物理学报,2005,54(3)：1247-1252.
[82]G Wang, J H Wen, Y Z Liu. Lumped-mass method for the study of band structure in two-dimensional phononic crystals[J]. Phys. Rev. B,2004, 69(18):184302-6.
[83]温熙森等.光子／声子晶体理论与技术[M].北京：科学技术出版社,2006.
[84]温熙森等.声子晶体[M].北京：国防工业出版社,2009.
[85]王刚.声子晶体局域共振带隙机理及减振特性研究[D].长沙：国防科技大学,2005.
[86]温激鸿.声子晶体振动带隙及减振特性研究[D].长沙：国防科技大学,2005.
[87]孟浩.基于声子晶体理论的潜艇推进轴系纵向减振技术研究[D].长沙：国防科技大学,2007.
[88]钱斯文.带隙原位可调型固固二维声了晶体研究[D].长沙：国防科技大学,2004.
[89]赵宏刚.基于声子晶体理论的水声吸声材料吸声特性研究[D].长沙国防科技大学,2008.
[90]籍夫建.外场对声子晶体梁弯曲振动带隙特性影响研究[D].长沙：国防科技大学,2007.
[91]刘竞文.周期结构振动传播特性研究[D].长沙：国防科技大学,2007.
[[92]沈惠杰.基于带隙理论的管路系统振动特性研究[D]. 长沙：国防科技大学,2009.
[93]花拓.一维声子晶体振动禁带的理论分析及其实验研究[D].兰州：兰州大学,2009.
[[94]周梦平.声子晶体理论在建筑隔声板材中的应用性研究[D].哈尔滨：哈尔滨工业大学,2008.
[95]张三强.基于周期性阻尼结构的的车内振动与噪声控制研究[D].武汉：湖北工业大学,2009.
[96]周明刚,周小强,陈源, 黄涛, 阢晓辉. 周期阻尼结构低频带隙的研究与应用[J].拖拉机与农用运输车,2012,39(1)：26-30.
[97]左曙光,魏欢,何宇漾,文歧华.声子晶体在车身顶棚减振中的应用研究[J].材料导报,2011,25(12)：124-127.
[98]马琮淦, 左曙光,何昌昌, 魏欢.声子晶体与轮边驱动电动汽车振动噪声控制[J].材料导报,2011,25(8)：4-8.
[99]沈礼,吴九汇, 陈花玲.声子晶体结构在汽车制动降噪声中的理论研究及应用[J].应用力学学报,2010,27(2)：294-298.
[100]钟会林.声子晶体带隙结构优化研究[D].广州：广东工业大学,2005.
[101]钟兰花.二维声子晶体对称性声波带隙的影响[D]. 广州：广东工业大学,2006.
[102]黄丹.二维声子晶体声波及弹性波带隙的研究[D].广州：广东工业大学,2010.
[103]卢明辉. 声子晶体及其物理效应的研究[D]. 南京：南京大学,2008.
[104]邱春印.二维声子晶体的层间多重散射理论及声子晶体相关的应用设计[D].武汉：武汉大学,2005.
[105]梅军.二维声子晶体带结构的多重散射理论[D].武汉：武汉大学,2002.
[106]张欣.声子晶体中声波或弹性波带隙的研究[D].广州：华南理工大学,2004.
[107]曹永军.弹性波在二维声子晶体中传播特性的研究[D].广州：华南理工大学,2005.
[108]Z L Hou, X J Fu, Y Y Liu. Singularity of the Bloch theorem in the fluid/solid phononic crystal[J]. Phys. Rev. B,2006,73(2):024304-5.
[109]Z L Hou, X J Fu, Y Y Liu. Calculational method to study the transmission properties of phononic crystals[J]. Phys. Rev. B,2004, 70(1):014304-7.
[110]王毅泽.磁电弹性声带隙材料结构巾的弹性波传播与局部化研究[D].哈尔滨：哈尔滨工业大学,2009.
[111]沈耀辉.二维三组元局域共振型声了晶体稳态响应研究[D].哈尔 滨：哈尔滨工业大学,2010.
[112]陈阿丽. 随机失谐和准周期声子晶体中弹性波传播特性研究[D].北京：北京交通大学,2008.
[113]门志忠.基于小波理论的二维声子晶体带隙结构分析[D]. 北京：北京交通大学,2007.
[114]J J Chen, K W Zhang, J Gao, J C Cheng. Stopbands for lower-order Lamb waves in one-dimensional composite thin plates[J]. Phys. Rev. B, 2006, 73(9): 094307-5.
[115]肖伟.场子晶体型周期复合结构禁带特性研究[D].武汉：华中科技大学，2007.
[116]王勇.基于声子晶体的多层周期性振系统理论与方法研究[D].武汉：华中科技大学，2006.
[117]X C Li, Z Y Liu. Coupling of cavity modes and guiding modes in two-dimensional phononic crystals[J]. Solid State Commun., 2005, 133(6): 397-402.
[118]李晓春,易秀英,肖清武, 梁宏宇.三组元声子晶体中的缺陷态[J].物理学报,2006,55(5)：2300-2306.
[119]许震宇,卢强,李斌.三维阵列亥姆霍兹共鸣管的声带隙性能[J].同济大学学报.2011,39(6)：926-929.
[120]靳晓雄,邵建旺,彭为.基于声子晶体的车内噪声研究[J].振动 与冲击,2009,28(12)：107-109.
[121]卢天健,高国钦, 马守林,金峰,金东范.二维四方排列半圆铝管/空气声子晶体的禁带特性[J]. 中国科学：技术科学,2009,28(1)：57-64.
[122]B Wu, R J Wei, H Y Zhao, C F He. Phononic band gaps in two-dimensional hybrid triangular lattice[J]. Acta Mechanica Solida Sinica,2010,23(3):255-259.
[123]C F He, H Y Zhao, R J Wei, B Wu. Existence of complete band gaps in 2D steel-water phononic crystal with square lattice[J]. Frontiers of Mechanical Engineering in China,2010,5(4):450-454.
[124]程祖依.弹性动力学基础[M].武汉：中国地质大学出版社,1993.
[125]J L Rose.固体中的超声波[M].何存富.吴斌.王秀彦译.北京：科学出版社,2004.
[126]吴福根.声子晶体中的带隙及缺陷态的研究[D].广州：华南理工大学,2002.
[127]P G Luan, Z Ye. Acoustic wave propagation in a one-dimensional layered system[J]. Phys. Rev. E,2001,63(6):066611-8.
[128]杨燕明.经典波在一维结构中带隙优化的研究[D].广州：广东工业大学,2007.
[129]K Bertoldi, M C Boyce. Mechanically triggered transformations of phononic band gaps in periodic elastomeric structures[J]. Phys. Rev. B. 2008,77(5):052105-4.
[130]R Sainidou, B Djafari-Rouhani, Y Pennec. J O Vasseur. Locally resonant phononic crystals made of hollow spheres or cylinders[J]. Phys. Rev. B,2006,73(2):024302-7.
[131]H Larabi, Y Pennec, B Djafari-Rouhani, J O Vasseur. Multicoaxial cylindrical inclusions in locally resonant phononic crystals| J J. Phys. Rev. E,2007,75(6):066601-8.
[132]N Horiuchi, Y Segawa, T Nozokido, K. Mizuno,H Miyazaki. Isotropic photonic gaps in a circular photonic crystal[J]. Opt. Lett.,2004,29(10): 1084-1086.
[133]W Zhong, X Zhang. Localized modes in defect-free two-dimensional circular photonic crystals[J]. Phys. Rev. A,2010,81(1):013805-6.
[134]D Torrent, J Sanchez-Dehesa. Radial wave crystals:Radially periodic structures from anisotropic metamaterials for engineering acoustic or electromagnetic waves[J]. Phys. Rev. Lett.,2009,103(6):064301-4.
[135]J M Bendickson, J P Dowling, M Scalora. Analytic expressions for the electromagnetic mode density in finite, one-dimensional, photonic band-gap structures[J]. Phys. Rev. E,1996,53(4):4107-4121.
[136]R E Camley, B Djafari-Rouhani, L Dobrzynski, A A Maradudin. Transverse elastic waves in periodically layered infinite and semi-infinite media[J]. Phys. Rev. B,1983,27(12):7318-7329.
[137]Y W Gu, X D Luo, H R Ma. Low frequency elastic wave propagation in 2D locally resonant phononic crystal with asymmetric resonator[J]. J. Appl. Phys.,2009,105(4):011903-7.
[138]D Torrent, J Sanchez-Dehesa. Acoustic resonances in two-dimensional radial sonic crystal shells[J]. New J. Phys.,2010,12(7):073034-18.
[139]M S Kushwaha, P Halevi. Giant acoustic stop bands in two-dimensional periodic arrays of liquid cylinders[J]. Appl. Phys. Lett.,1996,69(1): 31-33.
[140]J J Chen, Q Wang, X Han. Lamb wave transmission through one-dimensional three-component fibonacci composite plates[J]. Mod. Phys. Lett. B,2010,24(2):161-167.
[141]H Chandra, P A Deymier, J O Vasseur. Elastic wave propagation along waveguides in three-dimensional phononic crystals[J]. Phys. Rev. B, 2004,70(5):054302-6.
[142]A Khelif, A Choujaa, B Djafari-Rouhani, M Wilm, S Ballandras, V Laude. Trapping and guiding of acoustic waves by defect modes in a full-band-gap ultrasonic crystal[J]. Phys. Rev. B,2003,68(21): 214301-4.
[143]J Dellinger, J Etgen. Wave-field separation in two-dimensional anisotropic media[J]. Geophysics,1990,55(7):914-919.
[144]J Yan, P Sava. Improving the efficiency of elastic wave-mode separation for heterogeneous tilted transverse isotropic media[J]. Geophysics,2011,76(4):65.
[145]J Baumgartl, M Zvyagolskaya, C Bechinger. Tailoring of phononic band structures in colloidal crystals[J]. Phys. Rev. Lett.,2007.99(20): 205503-4.
[146]J Y Yeh. Control analysis of the tunable phononic crystal with eleclro-rheological material[J]. Physica B:Condensed Matter,2007,400(1-2): 137-144.
147] Z G Huang, T T Wu. Temperature effect on the bandgaps of surface and bulk acoustic waves in two-dimensional phononic crystals[J]. IEEE Ultrasonics, Ferroelectrics. and Frequency Control Society,2005,52(3): 365-370.
[148]J O Vasseur, O B Matar, J F Robillard, A C Hladky-Hennion, P A Deymier. Band structures tunability of bulk 2D phononic crystals made of magneto-elastic materials[J]. AIP Advances,2011,1(4):041904-12.
[149]J F Robillard, O B Matar, J O Vasseur, P A Deymier, M Stippinger, A C Hladky-Hennion, Y Pennec, B Djafari-Rouhani. Tunable magnetoclastic phononic crystals[J]. Appl. Phys. Lett.,2009,95(12):124104-3.
[150]M R Jolly, J D Carlson, B C Munoz, T A Bullions. The magnetoviscoelastic response of elstomer composites consisting of ferrous particles embedded in polymer matrix[J]. J. Intel. Mater. Syst. Struct.,1996,7(6):613-622.
[151]L C Davis. Model of magnetorheological elastomers[J]. J Appl. Phys., 1999,85(6):3348-3351.
[152]B Wu, C F He, R J Wei, H Y Zhao. Research on two-dimensional phononic crystal with magnetorheological material[J]. IEEE Ultrasonics Symposium,2008,2:1484-1486.
[153]F G Wu, Z Y Liu, Y Y Liu. Stop gaps and single defect states of acoustic waves in two-dimensional lattice of liquid cylinders[J]. Chin. Phys. Lett.,2001,18(6):785.
[154]X C Li, Z Y Liu, H Y Liang, Q W Xiao. Band gap and waveguide states in two-dimensioanl disordered phononic crystals[J]. Chin. Phys. Lett. 2006,23:1830-1833.
[155]L H Zhong, F G Wu, X Zhang. Effects of orientation and symmetry of rods on the complete acoustic band gap in two-dimensional periodic solid/gas systems[J]. Phy. Lett. A,2005,339(1-2):164-170.
[156]赵芳,苑立波.二维复式格了声子晶体带隙结构特性[J].物理学报, 2005,54(10):4511-4516.
[157]H F Dong. F G Wu, H L Zhong. X Zhang, Y W Yao. Effects of asymmetrical rotated rectangular basis on the acoustic band gap in two-dimensional acoustic crystals:The bands are twisted[J]. J. Phys. D: Appl. Phys.,2010,43(10):105404-5.
[158]董华锋,吴福根,牟中飞,钟会林.二维复式声子晶体中基元配置对声学能带结构的影响[J].物理学报,2010,59(2)：754-758.
[2]郁殿龙.基于声子晶体理论的梁板类周期结构振动带隙特性研究[D].长沙：国防科技大学,2006.
[3]M I Hussein. Dynamics of banded materials and structures:analysis, design and computation in multiple scales[D]. The University of Michigan,2004.
[4]L Brillouin. Wave propagation in periodic structures.2nd edition[M]. New York:Dover Publications,1953.
[5]J W Mile. Vibrations of beams on many supports[C]. Proceedings of the American Society of Civil of Kngineering 82. EMI,1956:1.
[6]M Heckl. Investigations on the vibrations of grillages and other simple beam structures[J]. J. Acoust. Soc. Am.,1964,36(7):1335-1343.
[7]D J Mead. Wave propagation in continuous periodic structures:Research contributions from Southampton,1964-1995[J]. J. Sound Vib.,1996, 180(3):495-524.
[8]S Gupta. Dynamics of periodically stiffened structures using a wave approach[D]. University of Southampton,1970.
[9]S Gupta. Natural frequencies of periodic skin-stringer structures using a wave approach[J]. J. Sound Vib.,1971,16(4):567-580.
[10]S Gupta. Natural flexural waves and the normal modes of periodically-supported beams and plates[J]. J. Sound Vib.,1970,13(1):89-101.
[11]D J Mead. Free wave propagation in periodically supported, infinite beams[J]. J. Sound Vib.,1970,11(2):181-197.
[12]D J Mead. A general theory of harmonic wave propagation in linear periodic systems with multiple coupling[J]. J. Sound Vib.,1973,27(2): 235-260.
[13]D J Mead. Wave propagation and natural modes in periodic systems:Ⅰ. Mono-coupled systems[J]. J. Sound Vib.,1975,40(1):1-18.
[14]D J Mead. Wave propagation and natural modes in periodic systems:Ⅱ. Multi-coupled systems, with and without damping[J]. J. Sound Vib. 1975,40(1):19-39.
[15]D J Mead, S Parthan. Free wave propagation in two-dimensional periodic plates[J]. J. Sound Vib.,1979,64(3):325-346.
[16]D J Mead, S Markus. Coupled flexural-longitudinal wave motion in a periodic beam[J]. J. Sound Vib.,1983,90(1):1-24.
[17]D J Mead, N S Bardell. Free vibration of a thin cylindrical shell with discrete axial stiffeners[J]. J. Sound Vib.,1986,111(2):229-250.
[18]D J Mead, N S Bardell. Free vibration of a thin cylindrical shell with periodic circumferential stiffeners[J]. J. Sound Vib.,1987,115(3): 499-520.
[19]M Toso. Wave propagation in rods, shells, and rotating shaft with non-uniform geometry[D]. University of Maryland.2004.
[20]S M Jeong. Analysis of vibration of 2-D periodic cellular structures[D]. Georgia Institute of Technology,2005.
[21]J S Jensen, O Sigmund. J J Thomsen, M P Bendsoc. Design of multi-phase structures with optimized vibrational and wave-transmitting properties[C].15th Nordic Semillar on Computational Mechanics,2002: 63-66.
[22]V Mangaraju, V R Sonti. Wave attenuation in periodic three-layered beams:Analytical and FEM study [J]. J. Sound Vib.,2004,276(3-5): 541-570.
[23]D Richards, D J Pines. Passive reduction of gear mesh vibration using a periodic drive shaft[J]. J. Sound Vib..2003,264(2):317-342.
[24]M Tawfik, J Chung, A Baz. Wave attenuation in periodic helicopter blade[C].5th Jordan International Mechanical Engineering Conference, Amman, Jordan,2004.
[25]S Asiri. Isolation of helicopter gearbox vibration using periodic support struts[D]. University of Maryland,2003.
[26]夏建白,朱邦芬.半导体超晶格物理[M].上海：上海科学技术出版社,1995.
[27]E Yablonovitch. Inhibited spontaneous mission in solid-state physics and electronics[J]. Phys. Rev. Lett.,1987,58(20):2059-2062.
[28]S John. Strong localization of photons in certain disordered dielectric superlattices[J]. Phys. Rev. Lett.,1987,58(20):2486-2489.
[29]M M Sigalas, E N Economou. Elastic and asoustic wave band structure[J]. J. Sound Vib.,1992,158(2):377-382.
[30]M S Kushwaha, P Halevi, L Dobrzynski, B Djafari-Rouhani. Acoustic band structure of periodic elastic composites[J]. Phys. Rev. Lett.,1993, 71(13):2022-2025.
[31]R Martinez-Sala, J Sancho, J V Sanchez, V Gomez, J Llinares, F Meseguer. Sound attenuation by sculpture[J]. Nature,1995,378(6554):241-1.
[32]W M Kuang, Z L Hou, Y Y Liu. The effects of shapes and symmetries of scatterers on the phononic band gap in 2D phononic crystals[J]. Phys. Lett. A,2004,332(5-6):481-490.
[33]A N Guz, N A Shulga. Dynamics of laminated and fibrous composites[J]. Appl. Mechanics Rev.,1992,45(2):35-60.
[34]Z Y Liu, X X Zhang, Y W Mao, Z Z Zhu, Z Y Yang, C T Chan, P Sheng. Locally resonant sonic materials[J]. Science,2000,289(5485): 1734-1736.
[35]Z Liu, C T Chan, P Sheng. Analytic model of phononic crystals with local resonances[J]. Phys. Rev. B. 2005, 71(1): 014103-8.
[36]C Goffaux, J Sanchez-Dehesa. Two-dimensional phononic crystals studied using a variational method: Application to lattices of locally resonant materials[J]. Phys. Rev. B. 2003. 67(14): 144301-10.
[37]C Goffaux, J Sanchez-Dehesa. A L Yeyati, P Lambin. A Khelif, J O Vasseur, B Djafari-Rouhani. Evidence of fano-like interference phenomena in locally resonant materials[J]. Phys. Rev. Lett.. 2002,88(22)：225502-4.
[38]M H Kin, C K Cheng, Z Yang, X X Zhang, P Sheng. Broadband locally resonant sonic shiclds[J]. Appl. Phys. Lett., 2003, 83(26): 5566-5568.
[39]G Wang. X S Wen, J H Wen. L H Shao, Z Y Liu. Two-dimensional locally resonant phononic crystals with binary structures[J]. Phys. Rev. Lett., 2004, 93(15): 154302-4.
[40]R E Vines, J P Wolfe. Scanning phononic lattices with surface acoustic waves[J]. Phys. B: Conden. Matt.. 1999, 263-264: 567-570.
[41]M Torres, F R Montero de Espinosa, D Garcia-Pablos, N Garcia. Sonic band gaps in finite elastic media: Surface states and localization phenomena in linear and point defectsfJ]. Phys. Rev. Lett., 1999, 82(15): 3054-3057.
[42]J Gao, X Y Zou, J C Cheng, B W Li. Band gaps of lower-order Lamb wave in thin plate with one-dimensional phononic crystal layer:Effect of substrate[J]. Appl. Phys. Lett.,2008,92(2):023510-3.
[43]J C Hsu, T T Wu. Lamb waves in binary locally resonant phononic plates with two-dimensional lattices[J]. Appl. Phys. Lett.,2007,90(20): 201904-3.
[44]B Bonello, C Charles, F Ganot. Lamb waves in plates covered by a two-dimensional phononic film[J]. Appl. Phys. Lett.,2007,90(2): 021909-3.
[45]D G Zhao, Z Y Liu, C Y Qiu, Z J He, F Y Cai, M Z Ke. Surface acoustic waves in two-dimensional phononic crystals:Dispersion relation and the eigenfield distribution of surface modes[J]. Phys. Rev. B,2007,76(14): 144301-6.
[46]R Sainidou, B Djafari-Rouhani, J O Vasseur. Surface acoustic waves in finite slabs of three-dimensional phononic crystals[J]. Phys. Rev. B, 2008,77(9):094304-9.
[47]M Z Ke, Z J He, S S Peng, Z Y Liu, J Shi. Surface resonant-states-enhanced acoustic wave tunneling in two-dimensional phononic crystals[J]. Phys. Rev. Lett.,2007,99(4):044301-4.
[48]T T Wu, Z G Huang, S Lin. Surface and bulk acoustic waves in two-dimensional phononic crystal consisting of materials with general anisotropy[J]. Phys. Rev. B,2004,69(9):094301-10.
[49]V Laude, M Wilm, S Benchabane, A Khelif. Full band gap for surface acoustic waves in a piezoelectric phononic crystal[J]. Phys. Rev. E,2005, 71(3):036607-7.
[50]R James, S M Woodley, C M Dyer, F H Victor. Sonic bands, band gaps, and defect states in layered structures:Theory and experiment[J]. J. Acoust. Soc. Am.,1995,97(4):2041-2047.
[51]J N Munday. C B Bennett, W M Robertson. Band gaps and defect modes in periodically structured waveguides[J]. J. Acoust. Soc. Am.,2002, 112(4):1353-1358.
[52]W M Robertson, C Baker. C B Bennett. Slow group velocity propagation of sound via defect coupling in a one-dimensional acoustic band gap array[J]. Am. J. Phys.,2004,72(2):255-257.
[53]M M Sigalas. Elastic wave band gaps and defect states in two-dimensional composites[J]. J. Acoust. Soc. Am.,1997.101(3): 1256-1261.
[54]M M Sigalas. Defect states of acoustic waves in a two-dimensional lattice of solid cylinders[J]. J. Appl. Phys.,1998,84(6):3026-3030.
[55]M Kafesaki, M M Sigalas, N Garcia. Frequency modulation in the transmitivity of wave guides in elastic-wave band-gap materials[J]. Phys. Rev, Lett.,2000,85(19):4044-4047.
[56]M Kafesaki, M M Sigalas, N Garcia. Wave guides in two-dimensional elastic wave band gap materials[J]. Phys. B:Conden. Matt.,2001, 296(1-3):190-194.
[57]M Torres, F R Montero de Espinosa, D Garcia-Pablos, N Garcia. Sonic band gaps in finite elastic media:Surface states and localization phenomena in linear and point defects[J]. Phys. Rev. Lett.,1999,82(15): 3054-3057.
[58]F G Wu, Z L Hou, Z Y Liu, Y Y Liu. Point defect states in two-dimensional phononic crystals[J]. Phys. Lett. A,2001,292(3): 198-202.
[59]F G Wu, Z Y Liu, Y Y Liu. Splitting and tuning characteristics of the point defect modes in two-dimensional phononic crystals[J]. Phys. Rev. E,2004,69(6):066609-4.
[60]J H Sun, T T Wu. Analyses of mode coupling in joined parallel phononic crystal waveguides[J]. Phys. Rev. B,2005,71(17):174303-8.
[61]J H Sun, T T Wu. Propagation of acoustic waves in phononic crystal plates and waveguides using a finite-difference time domain method[J]. Phys. Rev. B, 2007. 6(10): 104304-8.
[62]I E Psarobas, N Stefanou, A Modinos. Phononic crystals with planar defects[J]. Phys. Rev. B, 2000. 62(9): 5536-5540.
[63]H Chandra, P A Deymier, J O Vasseur. Elastic wave propagation along waveguides in three-dimensional phononic crystals[J]. Phys. Rev. B, 2004. 70(5): 054302-6.
[64]J O Vasseur, P A Deymier, B Djafari-Rouhani, Y Pennec, A C Hladky-Hennion. Absoult forbidden bands and waveguiding in two-dimensional phononic crystal plates[J]. Phys. Rev. B, 2008, 77(8): 085415-15.
[65]J O Vasseur, A C Hladky-Hennion, B Djafari-Rouhani, F Duval, B Dubus, Y Pennec, P A Deymier. Waveguiding in two-dimensional piezoelectric phononic crystal plates[J]. J. Appl. Phys., 2007, 101(11): 1 14904-6.
[66]C Y Qiu, Z Y Liu, J Shi, C T Chan. Directional acoustic source based on the resonant cavity of two-dimensional phononic crystalsfJ]. Appl. Phys. Lett., 2005, 86(22): 224105-3.
[67]X F Wang, M S Kushwaha, P Vasilopoulos. Tunability of acoustic spectral gaps and transmission in periodically stubbed wavcguidcs[J]. Phys. Rev. B, 2001, 65(3): 035107-10.
[68]A C Hladky-Hennion, J O Vasseur, B Djafari-Rouhani, M Billy. Sonic band gaps in one-dimensional phononic crystals with a symmetry stub[J]. Phys. Rev. B,2008,77(10):104304-7.
[69]Z L Hou, F G Wu, Y Y Liu. Acoustic wave propagating in one-dimensional Fibonacci binary composite systems[J]. Phys. B,2004, 344(1-4):391-397.
[70]M M Sigalas, E N Economou. Attenuation of multiple-scattered sound[J]. Europhys. Lett.,1996,36(4):241-246.
[71]F G Wu, Z Y Liu, Y Y Liu. Acoustic band gaps created by rotating square rods in two-dimensional lattice[J]. Phys. Rev. E,2002,66(4): 046628-5.
[72]S J Mizuno. Eigen-frequency and decay factor of the localized phonon in a super lattice with a defect layer[J]. Phys. Rev. B,2002,65(19): 193302-4.
[73]C Goffaux, J P Vigneron. Theoretical study of a tunable phononic band gap system[J]. Phys. Rev. B,2001,64(7):075118-5.
[74]王刚,温激鸿,韩小云,赵宏刚.二维声子晶体带隙计算中的时域有限差分方法[J].物理学报,2003,52(8)：1943-1947.
[75]M M Sigalas, N Garcia. Theoretical study of three dimensional elastic band gaps with the finite-different time domain method[J]. J. Appl. Phys. 2000,87(6):3122-3125.
[76]M Kafesaki, E N Economou. Multiple-scattering theory for three-dimensional periodic acoustic composites[J]. Phys. Rev. B,1999, 60(17):11993-12001.
[77]Z Y Liu, C T Chan, P Sheng, A L Goertzen, J H Page. Elastic wave scattering by periodic structures of spherical objects:Theory and experiment[J]. Phys. Rev. B,2000,62(4):2446-2457.
[78]J Mei, Z Y Liu, J Shi, D Tian. Theory for elastic wave scattering by a two-dimensional periodical array of cylinders:An ideal approach for band-structure calculations[J]. Phys. Rev. B,2003.67(24):245107-7.
[79]A Khelif, P A Deymier, B Djafari-Rouhani, J O Vasseur, L Dobrzynski. Two-dimensional phononic crystal with tunable narrow pass band: Application to a waveguide with selective frequency [J]. J. Appl. Phys. 2003,94(3):1308-1311.
[80]G Wang, J H Wen, Y Z Liu, X S Wen. One-dimensional phononic crystals with locally resonant structures[J]. Phys. Lett. A,2004,327(5-6): 512-521.
[81]王刚,温激鸿, 刘耀宗,郁殿龙,温熙森. 大弹性常数差二维声子晶体带隙计算中的集中质量法[J]. 物理学报,2005,54(3)：1247-1252.
[82]G Wang, J H Wen, Y Z Liu. Lumped-mass method for the study of band structure in two-dimensional phononic crystals[J]. Phys. Rev. B,2004, 69(18):184302-6.
[83]温熙森等.光子／声子晶体理论与技术[M].北京：科学技术出版社,2006.
[84]温熙森等.声子晶体[M].北京：国防工业出版社,2009.
[85]王刚.声子晶体局域共振带隙机理及减振特性研究[D].长沙：国防科技大学,2005.
[86]温激鸿.声子晶体振动带隙及减振特性研究[D].长沙：国防科技大学,2005.
[87]孟浩.基于声子晶体理论的潜艇推进轴系纵向减振技术研究[D].长沙：国防科技大学,2007.
[88]钱斯文.带隙原位可调型固固二维声了晶体研究[D].长沙：国防科技大学,2004.
[89]赵宏刚.基于声子晶体理论的水声吸声材料吸声特性研究[D].长沙国防科技大学,2008.
[90]籍夫建.外场对声子晶体梁弯曲振动带隙特性影响研究[D].长沙：国防科技大学,2007.
[91]刘竞文.周期结构振动传播特性研究[D].长沙：国防科技大学,2007.
[[92]沈惠杰.基于带隙理论的管路系统振动特性研究[D]. 长沙：国防科技大学,2009.
[93]花拓.一维声子晶体振动禁带的理论分析及其实验研究[D].兰州：兰州大学,2009.
[[94]周梦平.声子晶体理论在建筑隔声板材中的应用性研究[D].哈尔滨：哈尔滨工业大学,2008.
[95]张三强.基于周期性阻尼结构的的车内振动与噪声控制研究[D].武汉：湖北工业大学,2009.
[96]周明刚,周小强,陈源, 黄涛, 阢晓辉. 周期阻尼结构低频带隙的研究与应用[J].拖拉机与农用运输车,2012,39(1)：26-30.
[97]左曙光,魏欢,何宇漾,文歧华.声子晶体在车身顶棚减振中的应用研究[J].材料导报,2011,25(12)：124-127.
[98]马琮淦, 左曙光,何昌昌, 魏欢.声子晶体与轮边驱动电动汽车振动噪声控制[J].材料导报,2011,25(8)：4-8.
[99]沈礼,吴九汇, 陈花玲.声子晶体结构在汽车制动降噪声中的理论研究及应用[J].应用力学学报,2010,27(2)：294-298.
[100]钟会林.声子晶体带隙结构优化研究[D].广州：广东工业大学,2005.
[101]钟兰花.二维声子晶体对称性声波带隙的影响[D]. 广州：广东工业大学,2006.
[102]黄丹.二维声子晶体声波及弹性波带隙的研究[D].广州：广东工业大学,2010.
[103]卢明辉. 声子晶体及其物理效应的研究[D]. 南京：南京大学,2008.
[104]邱春印.二维声子晶体的层间多重散射理论及声子晶体相关的应用设计[D].武汉：武汉大学,2005.
[105]梅军.二维声子晶体带结构的多重散射理论[D].武汉：武汉大学,2002.
[106]张欣.声子晶体中声波或弹性波带隙的研究[D].广州：华南理工大学,2004.
[107]曹永军.弹性波在二维声子晶体中传播特性的研究[D].广州：华南理工大学,2005.
[108]Z L Hou, X J Fu, Y Y Liu. Singularity of the Bloch theorem in the fluid/solid phononic crystal[J]. Phys. Rev. B,2006,73(2):024304-5.
[109]Z L Hou, X J Fu, Y Y Liu. Calculational method to study the transmission properties of phononic crystals[J]. Phys. Rev. B,2004, 70(1):014304-7.
[110]王毅泽.磁电弹性声带隙材料结构巾的弹性波传播与局部化研究[D].哈尔滨：哈尔滨工业大学,2009.
[111]沈耀辉.二维三组元局域共振型声了晶体稳态响应研究[D].哈尔 滨：哈尔滨工业大学,2010.
[112]陈阿丽. 随机失谐和准周期声子晶体中弹性波传播特性研究[D].北京：北京交通大学,2008.
[113]门志忠.基于小波理论的二维声子晶体带隙结构分析[D]. 北京：北京交通大学,2007.
[114]J J Chen, K W Zhang, J Gao, J C Cheng. Stopbands for lower-order Lamb waves in one-dimensional composite thin plates[J]. Phys. Rev. B, 2006, 73(9): 094307-5.
[115]肖伟.场子晶体型周期复合结构禁带特性研究[D].武汉：华中科技大学，2007.
[116]王勇.基于声子晶体的多层周期性振系统理论与方法研究[D].武汉：华中科技大学，2006.
[117]X C Li, Z Y Liu. Coupling of cavity modes and guiding modes in two-dimensional phononic crystals[J]. Solid State Commun., 2005, 133(6): 397-402.
[118]李晓春,易秀英,肖清武, 梁宏宇.三组元声子晶体中的缺陷态[J].物理学报,2006,55(5)：2300-2306.
[119]许震宇,卢强,李斌.三维阵列亥姆霍兹共鸣管的声带隙性能[J].同济大学学报.2011,39(6)：926-929.
[120]靳晓雄,邵建旺,彭为.基于声子晶体的车内噪声研究[J].振动 与冲击,2009,28(12)：107-109.
[121]卢天健,高国钦, 马守林,金峰,金东范.二维四方排列半圆铝管/空气声子晶体的禁带特性[J]. 中国科学：技术科学,2009,28(1)：57-64.
[122]B Wu, R J Wei, H Y Zhao, C F He. Phononic band gaps in two-dimensional hybrid triangular lattice[J]. Acta Mechanica Solida Sinica,2010,23(3):255-259.
[123]C F He, H Y Zhao, R J Wei, B Wu. Existence of complete band gaps in 2D steel-water phononic crystal with square lattice[J]. Frontiers of Mechanical Engineering in China,2010,5(4):450-454.
[124]程祖依.弹性动力学基础[M].武汉：中国地质大学出版社,1993.
[125]J L Rose.固体中的超声波[M].何存富.吴斌.王秀彦译.北京：科学出版社,2004.
[126]吴福根.声子晶体中的带隙及缺陷态的研究[D].广州：华南理工大学,2002.
[127]P G Luan, Z Ye. Acoustic wave propagation in a one-dimensional layered system[J]. Phys. Rev. E,2001,63(6):066611-8.
[128]杨燕明.经典波在一维结构中带隙优化的研究[D].广州：广东工业大学,2007.
[129]K Bertoldi, M C Boyce. Mechanically triggered transformations of phononic band gaps in periodic elastomeric structures[J]. Phys. Rev. B. 2008,77(5):052105-4.
[130]R Sainidou, B Djafari-Rouhani, Y Pennec. J O Vasseur. Locally resonant phononic crystals made of hollow spheres or cylinders[J]. Phys. Rev. B,2006,73(2):024302-7.
[131]H Larabi, Y Pennec, B Djafari-Rouhani, J O Vasseur. Multicoaxial cylindrical inclusions in locally resonant phononic crystals| J J. Phys. Rev. E,2007,75(6):066601-8.
[132]N Horiuchi, Y Segawa, T Nozokido, K. Mizuno,H Miyazaki. Isotropic photonic gaps in a circular photonic crystal[J]. Opt. Lett.,2004,29(10): 1084-1086.
[133]W Zhong, X Zhang. Localized modes in defect-free two-dimensional circular photonic crystals[J]. Phys. Rev. A,2010,81(1):013805-6.
[134]D Torrent, J Sanchez-Dehesa. Radial wave crystals:Radially periodic structures from anisotropic metamaterials for engineering acoustic or electromagnetic waves[J]. Phys. Rev. Lett.,2009,103(6):064301-4.
[135]J M Bendickson, J P Dowling, M Scalora. Analytic expressions for the electromagnetic mode density in finite, one-dimensional, photonic band-gap structures[J]. Phys. Rev. E,1996,53(4):4107-4121.
[136]R E Camley, B Djafari-Rouhani, L Dobrzynski, A A Maradudin. Transverse elastic waves in periodically layered infinite and semi-infinite media[J]. Phys. Rev. B,1983,27(12):7318-7329.
[137]Y W Gu, X D Luo, H R Ma. Low frequency elastic wave propagation in 2D locally resonant phononic crystal with asymmetric resonator[J]. J. Appl. Phys.,2009,105(4):011903-7.
[138]D Torrent, J Sanchez-Dehesa. Acoustic resonances in two-dimensional radial sonic crystal shells[J]. New J. Phys.,2010,12(7):073034-18.
[139]M S Kushwaha, P Halevi. Giant acoustic stop bands in two-dimensional periodic arrays of liquid cylinders[J]. Appl. Phys. Lett.,1996,69(1): 31-33.
[140]J J Chen, Q Wang, X Han. Lamb wave transmission through one-dimensional three-component fibonacci composite plates[J]. Mod. Phys. Lett. B,2010,24(2):161-167.
[141]H Chandra, P A Deymier, J O Vasseur. Elastic wave propagation along waveguides in three-dimensional phononic crystals[J]. Phys. Rev. B, 2004,70(5):054302-6.
[142]A Khelif, A Choujaa, B Djafari-Rouhani, M Wilm, S Ballandras, V Laude. Trapping and guiding of acoustic waves by defect modes in a full-band-gap ultrasonic crystal[J]. Phys. Rev. B,2003,68(21): 214301-4.
[143]J Dellinger, J Etgen. Wave-field separation in two-dimensional anisotropic media[J]. Geophysics,1990,55(7):914-919.
[144]J Yan, P Sava. Improving the efficiency of elastic wave-mode separation for heterogeneous tilted transverse isotropic media[J]. Geophysics,2011,76(4):65.
[145]J Baumgartl, M Zvyagolskaya, C Bechinger. Tailoring of phononic band structures in colloidal crystals[J]. Phys. Rev. Lett.,2007.99(20): 205503-4.
[146]J Y Yeh. Control analysis of the tunable phononic crystal with eleclro-rheological material[J]. Physica B:Condensed Matter,2007,400(1-2): 137-144.
147] Z G Huang, T T Wu. Temperature effect on the bandgaps of surface and bulk acoustic waves in two-dimensional phononic crystals[J]. IEEE Ultrasonics, Ferroelectrics. and Frequency Control Society,2005,52(3): 365-370.
[148]J O Vasseur, O B Matar, J F Robillard, A C Hladky-Hennion, P A Deymier. Band structures tunability of bulk 2D phononic crystals made of magneto-elastic materials[J]. AIP Advances,2011,1(4):041904-12.
[149]J F Robillard, O B Matar, J O Vasseur, P A Deymier, M Stippinger, A C Hladky-Hennion, Y Pennec, B Djafari-Rouhani. Tunable magnetoclastic phononic crystals[J]. Appl. Phys. Lett.,2009,95(12):124104-3.
[150]M R Jolly, J D Carlson, B C Munoz, T A Bullions. The magnetoviscoelastic response of elstomer composites consisting of ferrous particles embedded in polymer matrix[J]. J. Intel. Mater. Syst. Struct.,1996,7(6):613-622.
[151]L C Davis. Model of magnetorheological elastomers[J]. J Appl. Phys., 1999,85(6):3348-3351.
[152]B Wu, C F He, R J Wei, H Y Zhao. Research on two-dimensional phononic crystal with magnetorheological material[J]. IEEE Ultrasonics Symposium,2008,2:1484-1486.
[153]F G Wu, Z Y Liu, Y Y Liu. Stop gaps and single defect states of acoustic waves in two-dimensional lattice of liquid cylinders[J]. Chin. Phys. Lett.,2001,18(6):785.
[154]X C Li, Z Y Liu, H Y Liang, Q W Xiao. Band gap and waveguide states in two-dimensioanl disordered phononic crystals[J]. Chin. Phys. Lett. 2006,23:1830-1833.
[155]L H Zhong, F G Wu, X Zhang. Effects of orientation and symmetry of rods on the complete acoustic band gap in two-dimensional periodic solid/gas systems[J]. Phy. Lett. A,2005,339(1-2):164-170.
[156]赵芳,苑立波.二维复式格了声子晶体带隙结构特性[J].物理学报, 2005,54(10):4511-4516.
[157]H F Dong. F G Wu, H L Zhong. X Zhang, Y W Yao. Effects of asymmetrical rotated rectangular basis on the acoustic band gap in two-dimensional acoustic crystals:The bands are twisted[J]. J. Phys. D: Appl. Phys.,2010,43(10):105404-5.
[158]董华锋,吴福根,牟中飞,钟会林.二维复式声子晶体中基元配置对声学能带结构的影响[J].物理学报,2010,59(2)：754-758.
© 2004-2018 中国地质图书馆版权所有 京ICP备05064691号 京公网安备11010802017129号 地址：北京市海淀区学院路29号 邮编：100083 电话：办公室：(+86 10)66554848；文献借阅、咨询服务、科技查新：66554700 |