Local resonance phononic band gaps in modified two-dimensional lattice materials

In this paper,modified two-dimensional periodic lattice materials with local resonance phononic bandgaps are designed and investigated.The design concept isto introduce some auxiliary structures into conventional periodic lattice materials.Elastic wave propagation in this kindof modified two-dimensional lattice materials is studied using a combination of Bloch’s theorem with finite elementmethod.The calculated frequency band structures of illustrative modified square lattice materials reveal the existenceof frequency band gaps in the low frequency region due tothe introduction of the auxiliary structures.The mechanismunderlying the occurrence of these frequency band gaps isthoroughly discussed and natural resonances of the auxiliarystructures are validated to be the origin.The effect of geometric parameters of the auxiliary structures on the width ofthe local resonance phononic band gaps is explored.Finally,a conceptual broadband vibration-insulating structure basedon the modified lattice materials is designed and its capability is demonstrated.The present work is anticipated to beuseful in designing structures which can insulate mechanicalvibrations within desired frequency ranges.     

INFLUENCES OF ANISOTROPY ON BAND GAPS OF 2D PHONONIC CRYSTAL

Band gaps of 2D phononic crystal with orthotropic cylindrical fillers embedded in the isotropic host are studied in this paper. Two kinds of periodic structures, namely, the square lattice and the triangle lattice, are considered. For anisotropic phononic crystal, band gaps not only depend on the periodic lattice but also the angle between the symmetry axis of orthotropic material and that of the periodic structure. Rotating these cylindrical fillers makes the angle changing continuously; as a result, pass bands and forbidden bands of the phononic crystal are changed. The plane wave expansion method is used to reduce the band gap problem to an eigenvalue problem. The numerical example is given for YBCO/Epoxy composites. The location and the width of band gaps are estimated for different rotating angles. The influence of anisotropy on band gaps is discussed based on numerical results.     

PHONONIC BAND GAPS IN TWO-DIMENSIONAL HYBRID TRIANGULAR LATTICE

<正>Absolute phononic band gaps can be substantially improved in two-dimensional lattices by using a symmetry reduction approach.In this paper,the propagation of elastic waves in a two-dimensional hybrid triangular lattice structure consisting of stainless steel cylinders in air is investigated theoretically.The band structure is calculated with the plane wave expansion (PWE)method.The hybrid triangular Bravais lattice is formed by two kinds of triangular lattices. Different from ordinary triangular lattices,the band gap opens at low frequency(between the first and the second bands)regime because of lifting the bands degeneracy at high symmetry points of the Brillouin zone.The location and width of the band gaps can be tuned by the position of the additional rods.     

一维格子结构动力分析的传递矩阵法

本文将传递矩阵法推广应用于分析一维格子结构的波传播和动力响应特性。一个格子结构的元件可分为主元件和次元件,传递矩阵沿主元件形成并考虑次元件的作用。文中通过例子说明形成一个周期单元传递矩阵的方法,指出利用传递矩阵计算无限或半无限长格子结构波传播的传播常数及有限长格子结构固有频率和频率响应函数的方法。作为数例,文中计算了一维平面格子结构的传播常数和频率响应函数。     

圆管型局域共振声子晶体三维构型振动带隙研究

采用多重多级子结构方法计算具有一定刚度的圆管型局域共振声子晶体三维构型振动带隙特性。考察包裹方向对带隙特性的影响,并对第一带隙上下边界点的单胞振动形式进行分析。结果表明,两种包裹形式都可以得到较低较宽的第一带隙,并且带隙特性相似,因而其周期结构都可以大幅减弱带隙范围内弹性波的传播。但两种构型带隙上下边界点对应振动形式不同,此外带隙特性还受单胞尺寸的影响。通过给定评价指标得到相关材料参数与带隙特性关系的相图,由此分析包裹层材料属性对带隙特性的影响。     

Tuning of band gaps for a two-dimensional piezoelectric phononic crystal with a rectangular lattice

In this paper, the elastic wave propagation in a two-dimensional piezoelectric phononic crystal is studied by considering the mechanic-electric coupling. The generalized eigenvalue equation is obtained by the relation of the mechanic and electric fields as well as the Bloch-Floquet theorem. The band structures of both the in-plane and anti-plane modes are calculated for a rectangular lattice by the planewave expansion method. The effects of the lattice constant ratio and the piezoelectricity with different filling fractions are analyzed. The results show that the largest gap width is not always obtained for a square lattice. In some situations, a rectangular lattice may generate larger gaps. The band gap characteristics are influenced obviously by the piezoelectricity with the larger lattice constant ratios and the filling fractions.     

Optimal 2D auxetic micro-structures with band gap

A systematic investigation is presented that explores band gap properties of periodic micro-structures architected for maximum auxeticity. The design of two-dimensional auxetic cells is addressed using inverse homogenization. A non-convex optimization problem is formulated that is solved through mathematical programming. Different starting guesses are used to explore local minima when distributing material and void or two materials and void. The same numerical tool succeeds in capturing re-entrant, chiral and anti-chiral layouts with negative Poisson’s ratio, retrieving solutions originally found through other approaches as well as generating variations. A Floquet–Bloch approach is then applied to the achieved periodic cells to investigate possible band gaps characterizing the in-plane wave propagation. Directional and full band-pass filters are found in the case of micro-structures whose auxetic behavior comes from the arising of a rotational deformation of the periodic cell. Such kind of topologies could be exploited to design tunable wave guides and tunable phononic crystals, respectively.

     

Wave propagation in two-dimensional anisotropic acoustic metamaterials of K4 topology

An acoustic metamaterial is envisaged as a synthesised phononic material the mechanical behaviour of which is determined by its unit cell. The present study investigates one aspect of mechanical behaviour, namely the band structure, in two-dimensional (2D) anisotropic acoustic metamaterials encompassing locally resonant mass-in-mass units connected by massless springs in a K4 topology. The 2D lattice problem is formulated in the direct space (r-space) and the equations of motion are derived using the principle of least action (Hamilton’s principle). Only proportional anisotropy and attenuation-free shock wave propagation have been considered. Floquet–Bloch’s principle is applied, therefore a generic unit cell is studied. The unit cell can represent the entire lattice regardless of its position. It is transformed from the direct lattice in r-space onto its reciprocal lattice conjugate in Fourier space (k-space) and point symmetry operations are applied to Wigner–Seitz primitive cell to derive the first irreducible Brillouin Zone (BZ). The edges of the first irreducible Brillouin Zone in the k-space have then been traversed to generate the full band structure. It was found that the phenomenon of frequency filtering exists and the pass and stop bands are extracted. A follow-up parametric study appreciated the degree and direction of influence of each parameter on the band structure.     

Multi-resonator coupled metamaterials for broadband vibration suppression

In this study,multi-resonator coupled metamaterials(MRCMs)with local resonators are proposed to obtain the multiple and wide band gaps.Kinetic models of the MRCMs are established,and the boundary conditions of the unit cell are obtained with Bloch's theorem.The effects of structural parameters,including the mass of the resonator and the spring stiffness,on the distributions of the band gaps are studied.Furthermore,the frequency domain responses and the time domain responses are calculated for analyzing the structural vibration characteristics and the effects of damping on structural vibration.The results show that the frequency domain response can accurately express the distributions of the band gaps of the MRCMs,and we can increase the number and the width of the band gaps by using the MRCMs for the superior vibration suppression capability.     

Propagation of bending waves in phononic crystal thin plates with a point defect

In this paper, the band structures of bending waves in a phononic crystal thin plate with a point defect are studied using an improved plane wave expansion method combined with the supercell technique. In particular, a phononic crystal thin plate composed of an array of circular crystalline Al₂O₃ cylinders embedded in an epoxy matrix with a square lattice is considered in detail. Full band gaps are shown. When a point defect is introduced, the bending waves are highly localized at or near the defect, resulting in defect modes. The frequency and number of the defect modes are strongly dependent on the filling fraction of the system and the size of the point defect. The defect bands appear from the upper edge of the gap and move to the middle of the gap as the defect size is reduced. For a given defect, the frequencies of the defect bands increase as the filling fraction increases.     

蜂窝材料的弹性波传播特性

通过研究蜂窝材料的弹性波频散关系,分析了其弹性波传播特性. 采用基于小波理论的分析方法,将材料参数和位移均展开为双正交周期小波基函数的形式,利用Bloch定理将波动方程转化为特征值方程,求解该方程得到3种典型结构------正方、三角与六角排列的铝(Al)和聚丙烯(PP)蜂窝材料的频散关系,并进行了比较分析. 结果显示:结构形式的不同显著地影响其波动特性,而制作材料的不同则没有影响;3种结构形式都不存在完全带隙,但正方和三角形结构在一定的传播方向范围内存在方向带隙,而六角形结构则在任何方向都不存在方向带隙;与正方结构相比,三角结构在相同孔隙率下,在更广的传播方向内和更低的频率下,能产生较宽的方向带隙.      

Tunable low frequency band gaps and sound transmission loss of a lever-type metamaterial plate

A novel metamaterial plate with subwavelength lever-type resonators is proposed to obtain low frequency broadband band gaps and good sound insulation performance. The band structure is theoretically derived, and the validity of the theoretical method is verified by the finite element method. The formation mechanisms of the band gaps are illustrated by the analysis of the effective dynamic mass density and group velocity. The effect of the lever ratio on the band gaps is analyzed. The results indicate that as the lever ratio increases, the first band gap shifts to lower frequencies, while the bandwidth is widened. Moreover, the sound insulation performance of the proposed metamaterial plate is evaluated via examining the sound transmission loss (STL). Compared with the metamaterial plates without lever accessories, the proposed metamaterial plates with a suitable lever ratio have better sound insulation performance at low frequencies.     

Broadband elastic metamaterial with single negativity by mimicking lattice systems

The narrow bandwidth is a significant limitation of elastic metamaterials for practical engineering applications. In this paper, a broadband elastic metamaterial with single negativity (negative mass density or Young's modulus) is proposed by mimicking lattice systems. It has two stop bands and the bandwidth of the second one is infinite theoretically. The effect of the relevant parameters on band gaps is discussed. A continuum model is proposed and the selection of materials is discussed in detail. It shows that continuum metamaterials can be described accurately by using the lattice model, and the second stopband can be ultra-broad but not infinite. This discrepancy is investigated and a method is provided to calculate the upper limit of the second stopband for a continuum metamaterial. As a verification, the proposed metamaterial is used for wave mitigation over broadband frequency ranges. Moreover, the present method is extended to design 2D anisotropic elastic metamaterials, and a device to control the direction of elastic wave transmission is proposed as an example.     

低频弹性波超材料的若干进展

超材料是一类新兴的具有超常物理性质的人造周期/拟周期材料, 能够改变电磁波、声波以及弹性波等在介质中的传播特性. 因在航天、国防以及民用科学等方面的巨大应用潜力, 超材料自被提出后便受到极大的关注并引发研究热潮. 弹性波超材料是超材料的一种, 能够基于弹性波与超材料结构的相互耦合作用实现对弹性波的操控. 带隙是评估弹性波超材料实现弹性波操控的重要工具, 其性质与超材料的材料参数、晶格常数以及局域振子的固有频率相关. 受制于超材料的承载能力、外观尺寸以及局域振子结构等因素, 利用传统超材料开启低频(约100 Hz)弹性波带隙依然存在较大困难. 文章首先简要介绍超材料开启弹性波带隙的基本原理, 然后从低频弹性波超材料基本结构与低频带隙实现方法、低频带隙优化与调控策略、低频带隙潜在应用等三个方面详细总结低频弹性波超材料的研究工作. 其中, 低频带隙超材料的基本结构主要包括布拉格散射型超材料、传统局域共振型超材料以及准零刚度局域共振超材料. 文章通过总结低频弹性波超材料的研究进展, 分析了目前研究中的不足并对未来低频弹性波的研究方向进行了展望.      

Discrete bright–dark soliton solutions and parameters controlling for the coupled Ablowitz–Ladik equation

Analysis of piecewise-linear nonlinear dynamical systems is critical for a variety of civil, mechanical, and aerospace structures that contain gaps or prestress that are caused by cracks, delamination, joints or interfaces among components. Recently, a technique referred to as bilinear amplitude approximation (BAA) was developed to estimate the response of bilinear systems that have no gap or prestress. The method is based on an idea that the dynamics of a bilinear system can be treated as a combination of linear responses in two time intervals both of which the system behaves as a distinct linear system: (1) the open state and (2) the closed or sliding state. Both geometric and momentum constraints are then applied as compatibility conditions between the states to couple the linear vibrational response for each time interval. In order to estimate the response for more general cases where there are either gaps or prestress in the system, a generalized BAA method is proposed in this paper. The new method requires inclusion of contact stiffness and damping to model contact behavior in the sliding state, and new equilibrium positions for each state to establish proper coordinates. The new method also finds the bilinear frequency of the system, which cannot be computed using the bilinear frequency approximation method previously developed since that method is only accurate for the zero gap and no prestress case. The generalized BAA method is demonstrated on a single degree of freedom system, a three degree of freedom system, and a cracked cantilever beam model for various gap sizes and prestress levels.     

Piezoelectric-sensitive mode of lamb wave in one-dimensional piezoelectric phononic crystal plate

We investigate the transmission modes in one-dimensional piezoelectric phononic crystal plate, which is consisted of piezoelectric ceramics placed periodically in epoxy. The influences of piezoelectricity and different electrical boundary conditions on the Lamb waves in the composite plate are studied in detail, and an important elastic wave mode named as piezoelectric-sensitive mode (PSM) is defined. Furthermore, the influences of the piezoelectric constants, the filling fraction and the ratio of thickness to lattice pitch on PSM are given, and the physical mechanism of PSM is also discussed. The numerical results show that PSM has a relatively larger frequency shift and significant change in the displacement distribution when piezoelectricity is considered; the electromechanical coupling coefficient of PSM is much larger than the other modes. It reveals that PSM is an important and useful mode in piezoelectric periodic structures. This investigation is helpful in controlling the band gaps and also gives a potential application in design of sensing system and different piezoelectric devices.     

Surface effect on band structure of magneto-elastic phononic crystal nanoplates subject to magnetic and stress loadings

This paper presents a theoretical model for the size-dependent band structure of magneto-elastic phononic crystal(PC) nanoplates according to the Kirchhoff plate theory and Gurtin-Murdoch theory, in which the surface effect and magneto-elastic coupling are considered. By introducing the nonlinear coupling constitutive relation of magnetostrictive materials, Terfenol-D/epoxy PC nanoplates are carried out as an example to investigate the dependence of the band structure on the surface effect, magn...     

低频振动隔离和能量采集双功能超材料

振动隔离和能量采集一体化是一种能够将有害振动隔离并转化为电能收集利用的动力学机制. 本文从局域共振超材料存在低频带隙特性出发, 研究了振动隔离和能量采集双功能超材料的动力学行为. 通过在球型磁腔内放置固接了感应线圈的球摆构成具有能量采集功能的球摆型谐振器, 并将其周期性的放置在基体梁中, 可以将带隙频率范围内的振动聚集在谐振器内, 以实现振动隔离和能量采集双功能. 建立了横向激励下双功能超材料梁的动力学方程, 应用Bloch's定理得到超材料的能带结构, 通过有限元仿真验证了理论模型和研究方法. 研究了不同参数下超材料梁的带隙特性. 进一步将一维拓展到二维, 研究了二维双功能超材料板的振动隔离和能量采集性能. 最后, 设计并建造了振动隔离和能量采集一体化双功能超材料动力学实验平台, 解析、数值和实验结果表明, 在局域共振带隙的频率范围内, 超材料梁主体的振动明显被抑制, 与此同时, 振动被局限在谐振器中, 使采集到的电压达到了最大值. 通过对附加谐振器和没有附加谐振器的能带结构和幅频响应的对比, 发现球摆型谐振器的加入可以在低频范围内形成了一个局域共振带隙, 有效提高了超材料梁在低频处的振动隔离和能量采集性能.      

Topology design and optimization of nonlinear periodic materials

This paper explores optimal topologies yielding large band gap shifts in one- and two-dimensional nonlinear periodic materials. The presence of a nonlinearity in a periodic material system results in amplitude-dependent dispersion behavior, leading to novel wave-based devices such as tunable filters, resonators, and waveguides. The performance of these devices over a broad frequency range requires large, tunable band gaps, motivating the present study. Consideration of a one-dimensional bilayer system composed of alternating linear and nonlinear layers shows that optimal designs consist of thin, compliant nonlinear layers. This is at first surprising considering the source of the shift originates from only the nonlinear layer; however, thin layers lead to localized stresses that activate the nonlinear character of the system. This trend persists in two-dimensional materials where optimization studies are performed on plane-stress models discretized using bilinear Lagrange elements. A fast algorithm is introduced for computing the dispersion shifts, enabling efficient parametric analyses of two-dimensional inclusion systems. Analogous to the one-dimensional system, it is shown that thin ligaments of nonlinear material lead to large dispersion shifts and group velocity variations. Optimal topologies of the two-dimensional system are also explored using genetic algorithms aimed at producing large increases in complete band gap width and shift, or group velocity variation, without presupposing the topology. The optimal topologies that result resemble the two-dimensional inclusion systems, but with small corner features that tend to enhance the production of dispersion shift further. Finally, the study concludes with a discussion on Bloch wave modes and their important role in the production of amplitude-dependent dispersion behavior. The results of the study provide insight and guidance on selecting topologies and materials which can yield large amplitude-dependent band gap shifts and group velocity variations, thus enabling sensitive nonlinear devices.     

Investigation of flexural wave band gaps in a locally resonant metamaterial with plate-like resonators

This paper presents an investigation of flexural wave band gaps in locally resonant metamaterials (LRMs). An LRM is a periodic structure consisting of repeated unit cells containing a local resonator. Due to the local resonance occurring in the unit cell, the LRM induces a band gap (a frequency band in which no waves propagate). Discrete-like or beam-like resonators have generally been used to realise LRMs in previous research. By extending the beam-like resonator configuration, this paper studies LRMs with a plate-like resonator to exploit its advantages with respect to large design freedom. In order to understand flexural wave band gaps in an LRM with plate-like resonators, parametric studies are conducted with the development of a finite element model. Further, the influences of the plate-like resonator design parameters on flexural wave band gaps are investigated. Based on the parametric studies, the rules governing band gap properties are determined. Finally, tailoring flexural wave band gaps by adjusting the parameters is discussed.