Extension/Compression-Controlled Complete Band Gaps in 2D Chiral Square-Lattice-Like Structures

Achieving tunable band gaps in a structure by external stimuli is of great importance in acoustic applications. This paper aims to investigate the tunability of band gaps in square-lattice-like elastic periodic structures that are usually not featured with notable band gaps.Endowed with chirality, the periodic structures here are able to undergo imperfection-insensitive large deformation under extension or compression. The influences of geometric parameters on band gaps are discussed via the nonlinear finite element method. It is shown that the band gaps in such structures with curved beams can be very rich and, more importantly, can be efficiently and robustly tuned by applying appropriate mechanical loadings without inducing buckling. As expected, geometry plays a more significant role than material nonlinearity does in the evolution of band gaps. The dynamic tunability of band gaps through mechanical loading is further studied. Results show that closing, opening, and shifting of band gaps can be realized by exerting real-time global extension or compression on the structure. The proposed periodic structure with well-designed chiral symmetry can be useful in the design of particular acoustic devices.     

弹性波超材料板中拓扑交界态与缺陷保护的主动控制研究

在波动周期结构中,六边形单胞是常见的具有动态拓扑性质的重复单元。本文设计了具有主动控制特性的六边形单胞,通过粘贴压电片并连接负电容电路对材料参数进行调控,实现了弯曲波拓扑交界态传输与缺陷保护特性的主动控制,并发现了该保护行为具有方向选择特性。上述工作将物理学与声学中拓扑态概念引入到弹性波超材料板中,从而实现了其中弯曲波拓扑传输的主动控制功能。     

Folding beam-type piezoelectric phononic crystal with low-frequency and broad band gap

A folding beam-type piezoelectric phononic crystal model is proposed to isolate vibration. Two piezoelectric bimorphs are joined by two masses as a folding structure to comprise each unit cell of the piezoelectric phononic crystal. Each bimorph is connected independently by a resistive-inductive resonant shunting circuit. The folding structure extends the propagation path of elastic waves, while its structure size remains quite small. Propagation of coupled extension-flexural elastic waves is studied by the classical laminated beam theory and transfer matrix method. The theoretical model is further verified with the finite element method(FEM). The effects of geometrical and circuit parameters on the band gaps are analyzed. With only 4 unit cells, the folding beam-type piezoelectric phononic crystal generates two Bragg band gaps of 369 Hz to1 687 Hz and 2 127 Hz to 4 000 Hz. In addition, between these two Bragg band gaps, a locally resonant band gap is induced by resonant shunting circuits. Appropriate circuit parameters are used to join these two Bragg band gaps by the locally resonant band gap.Thus, a low-frequency and broad band gap of 369 Hz to 4 000 Hz is obtained.     

Complete band gaps in two-dimensional piezoelectric phononic crystals with {1–3} connectivity family

In this paper, we present results of full band structures for two-dimensional piezoelectric phononic crystals with {1–3} connectivity family. The plane-wave-expansion (PWE) method is applied to the theoretical derivation of secular equations of the two polarization modes: a transverse polarization mode and a mixed (longitudinal-transverse) polarization mode. And the band structures of the two modes for both the case of piezoelectric rods embedded in a polymer matrix and the case of polymer rods embedded in a piezoelectric matrix are calculated for two different cross-sections of the rods, i.e., circular and square, considering the practical fabrication of phononic crystals. We reveal the existence of several very large complete band gaps in a material of practical interest such as PZT rods reinforced polythene composite. The effects of shapes and filling fraction of the rods on band gaps are discussed in detail. The existence of these gaps in relation to the physical parameters of the constituent materials involved is studied. Understanding the band structures of piezoelectric phononic crystals can give some information for improvements in the design of acoustic transducers.     

Multiple Crack Detection of Concrete Structures Using Impedance-based Structural Health Monitoring Techniques

This paper presents a feasibility study for practical applications of an impedance-based real-time health monitoring technique applying PZT (Lead–Zirconate–Titanate) patches to concrete structures. First, comparison between experimental and analytical studies for damage detection on a plain concrete beam is made. In the experimental study, progressive surface damage inflicted artificially on the plain concrete beam is assessed by using both lateral and thickness modes of the PZT patches. Then, an analytical study based on finite element (FE) models is carried out to verify the validity of the experimental result. Secondly, multiple (shear and flexural) cracks incurred in a reinforced concrete (RC) beam under a third point bending test are monitored continuously by using a sensor array system composed of the PZT patches. In this study, a root mean square deviation (RMSD) in the impedance signatures of the PZT patches is used as a damage indicator.     

阶梯压电层合梁的波动动力学特性

采用行波理论系统地研究了压电阶梯梁的自由振动分析以及强迫响应的分析方法. 基于分布 参数理论研究了压电阶梯梁的波传播特性，忽略柔性梁横向剪切和转动惯量的影响，给出了 梁的轴向和横向的简谐波解. 将压电阶梯梁离散化为单元，考虑压电片的刚度和质量的影响， 建立了节点散射模型. 应用位移连续和力平衡条件，推导了节点的波反射和波传递矩阵，在 此基础上，引入波循环矩阵的概念，给出波循环矩阵、波传递系数矩阵的确定方法. 应用波 循环矩阵可以有效地计算结构的固有频率. 另外，应用波传递系数研究了压电陶瓷作动器位 置对其驱动能力的影响. 得出两个主要结论：1)作动器靠近悬臂梁固定端将有较强的驱动 能力，悬臂梁边界反射行波产生弯曲消失波有利于增大压电波的模态传递系数；2)模态传 递系数与固有频率的灵敏度密切相关，波传递系数越大, 对应该处固有频率变化灵敏度越大. 另外，数值算例表明了行波方法比有限元方法具有更高的计算精度.     

Vibration band gap analysis of a new periodic beam model using GDQR method

In this study, a new periodic beam model is introduced. This beam consists of the concentrated rigid masses and tapered beam elements with linearly variable width. The theoretical equations are derived by employing the Euler-Bernoulli beam and the Bloch–Floquet theorem and then solved using the generalized differential quadrature rule method to calculate the first two band gaps. The effects of the mass, mass moment of inertia and taper ratio on the widths and central frequencies of the first two band gaps are investigated. Results show that the wide band gaps at low frequency ranges can be obtained by changing the geometrical parameters. This is of interest for different applications of the band gap phenomenon such as broadband piezoelectric energy harvesting. Finally, the finite element simulation (ANSYS software) is used to validate the analytical method and good agreement is found.     

Band structure properties of elastic waves propagating in the nanoscaled nearly periodic layered phononic crystals

The localization factor is used to describe the band structures for P wave propagating normally in the nanoscaled nearly periodic layered phononic crystals. The localization factor is calculated by the transfer matrix method based on the nonlocal elastic continuum theory.Three kinds of nearly periodic arrangements are concerned, i.e., random disorder, quasiperiodicity and defects. The influences of randomly disordered degree of the sub-layer's thickness and mass density, the arrangement of quasi-periodicity and the location of defect on the band structures and cut-off frequency are analyzed in detail.     

基于渐近均匀化方法的准周期梁结构等效刚度数值实现方法研究

相比周期梁结构,准周期梁结构沿轴向梯度变化,具有更大的设计自由度,能够获得更好的结构性能。由于其非均质性,一般将其均匀化为具有等效性质的均质梁结构,但现有工作很少涉及准周期梁结构等效性质的计算。本文针对由周期梁结构映射而成的准周期梁结构,通过引入雅可比矩阵,基于渐近均匀化方法推导的单胞方程及其等效性质计算列式,并建立了其单胞方程及等效刚度的有限元求解列式。该方法可以处理沿轴向变形的任意微单胞构型,数值算例验证了其正确性和有效性。     

一维磁电弹准周期结构带隙特性的研究

采用传递矩阵方法,研究了横波(SV波)垂直入射时压电/（弹性/压磁）和（压电/弹性）/压磁两种Fibonacci准周期结构的频带特性,通过计算局部化因子和位移透射系数,数值揭示了此两种Fibonacci准周期结构频带特性的差异以及与相应理想周期结构频带特性的不同,而且表明（压电/弹性）/压磁Fibonacci准周期结构的频带特性与纯弹性材料Fibonacci准周期结构的频带特性是相似的。     

Modeling and analysis of piezoelectric beam with periodically variable cross-sections for vibration energy harvesting

A bimorph piezoelectric beam with periodically variable cross-sections is used for the vibration energy harvesting. The effects of two geometrical parameters on the first band gap of this periodic beam are investigated by the generalized differential quadrature rule (GDQR) method. The GDQR method is also used to calculate the forced vibration response of the beam and voltage of each piezoelectric layer when the beam is subject to a sinusoidal base excitation. Results obtained from the analytical method are compared with those obtained from the finite element simulation with ANSYS, and good agreement is found. The voltage output of this periodic beam over its first band gap is calculated and compared with the voltage output of the uniform piezoelectric beam. It is concluded that this periodic beam has three advantages over the uniform piezoelectric beam, i.e., generating more voltage outputs over a wide frequency range, absorbing vibration, and being less weight.     

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。     

存在间隙的多自由度系统的周期运动及Robust稳定性

研究一类存在间隙的多自由度振动系统的动态响应．系统由线性元件构成，但其中一个元件的最大位移不能超过由刚性平面约束所确定的阀值．应用模态矩阵方法将系统解耦，并根据碰撞条件和由碰撞规律所确定的衔接条件求得系统的周期运动及其稳定条件．将Ｌｙａｐｕｎｏｖ方法应用于周期运动的扰动差分方程，导出了含不确定参数的碰撞振动系统周期运动的鲁棒（Ｒｏｂｕｓｔ）稳定性条件．文末用一个二自由度系统阐明了方法的有效性     

Asymptotic Analysis of High-Contrast Phononic Crystals and a Criterion for the Band-Gap Opening

We investigate the band-gap structure of the frequency spectrum for elastic waves in a high-contrast, two-component periodic elastic medium. We consider two-dimensional phononic crystals consisting of a background medium which is perforated by an array of holes periodic along each of the two orthogonal coordinate axes. In this paper we establish a full asymptotic formula for dispersion relations of phononic band structures as the contrast of the shear modulus and that of the density become large. The main ingredients are integral equation formulations of the solutions to the harmonic oscillatory linear elastic equation and several theorems concerning the characteristic values of meromorphic operator-valued functions in the complex plane, such as the generalized Rouché’s theorem. We establish a connection between the band structures and the Dirichlet eigenvalue problem on the elementary hole. We also provide a criterion for exhibiting gaps in the band structure which shows that smaller the density of the matrix is, the wider the band-gap is, provided that the criterion is fulfilled. This phenomenon was reported by Economou and Sigalas (J Acoust Soc Am 95:1734–1740, 1994) who observed that periodic elastic composites whose matrix has lower density and higher shear modulus compared to those of inclusions yield better open gaps. Our analysis in this paper agrees with this experimental finding.     

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.     

Optical properties of two-dimensional polymer photonic crystals after deformation-induced pattern transformations

The photonic band structure and optical transmittance of two-dimensional periodic elastomeric photonic crystals are studied computationally to understand the effects of large strains on optical properties of the structures. The large compressive deformation patterns of the two-dimensional periodic structure studied by Mullin and coworkers [Mullin, T., Deschanel, S., Bertoldi, K., Boyce, M.C., 2007. Pattern transformation triggered by deformation. Physical Review Letters 99(8), 084301] are first reproduced using hyperelastic material models for the elastomer SU-8. Finite element analysis is then used to solve Maxwell's equations to obtain light transmittance through both the undeformed and deformed structures; simultaneously the wave equation resulting from the appropriate two-dimensional form of Maxwell's equations is solved as an eigenvalue problem to obtain the band structure. The deformation-induced shift in transmission spectrum valleys for different bands is calculated, and the changes in the width of these reflectance peaks are also obtained. The band structure calculation shows that there are no complete photonic band gaps as expected for the low dielectric contrast system. However, the effect of the observed reversible, symmetry-breaking deformation pattern is to uncouple many of the photonic bands in all three high symmetry directions, i.e. Γ–X, X–M, and Γ–M. New non-degenerate deformation-induced optical modes appear in both the real space transmittance spectra and the band structure with lower reflectance values. Analyses of the deformation pattern, the optical mode shapes, and the photonic band structure reveal that localized regions of large rotation are responsible for the significant changes in optical transmittance. The results have practical importance for the design of strain-tunable optomechanical materials for sensing and actuation.     

桁架材料弹性波带隙特性分析

研究了弹性波在周期性桁架材料中的传播特性,并根据桁架材料的周期性特点和杆纵向振动模态,给出了基于单胞的桁架材料弹性波色散(dispersion)方程。分析了1维和2维问题的色散特性,研究了相应的弹性波带隙性质;以CAE分析软件为工具平台对桁架材料的带隙特性进行了数值仿真实验,给出了基于谐响应和特征频率变化特征的仿真实验方法。仿真实验确认了所分析的桁架材料的带隙特性,同时说明所用的仿真实验方法是可行的。     

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.     

Bandgap characteristics of the two-dimensional missing rib lattice structure

In this paper, the bandgap characteristics of a missing rib lattice structure composed of beam elements are investigated by using the Floquet-Bloch theorem. The tuning of the width and position of the bandgap is achieved by changing the local structural parameters, i.e., the rotation angle, the short beam length, and the beam thickness. In order to expand the regulation of the bandgap, the influence of the material parameters of the crossed long beams inside the structure on the bandgap is analyzed. The results show that the mass density and stiffness of the structure have significant effects on the bandgap, while Poisson's ratio has no effect on the bandgap. By analyzing the first ten bands of the reference unit cell, it can be found that the missing rib lattice structure generates multiple local resonance bandgaps for vibration reduction, and these bandgap widths are wider. The modal analysis reveals that the formation of the bandgap is due to the dipole resonance of the lattice structure, and this dipole resonance originates from the coupling of the bending deformation of the beam elements. In the band structure, the vibrational mode of the 9th band with a negative slope corresponds to a rotational resonance, which is different from that with the conventional negative slope formed by the coupling of two resonance modes. This study can provide a theoretical reference for the design of simple and lightweight elastic metamaterials, as well as for the regulation of bandgaps and the suppression of elastic waves.     

Shear wave propagation in periodic phononic/photonic piezoelectric medium

Coupled electro-elastic SH waves propagating oblique to the lamination of a one dimensional piezoelectric periodic structure are considered in the framework of the full system of Maxwell’s electrodynamic equations. The dispersion equation has been obtained and numerical analyses carried out for two kinds of composites both consisting of two different piezoelectric materials. The results demonstrate the significant effect of piezoelectricity on the widths of band gaps at acoustic frequencies and confirm that it does not affect the band gaps at optical frequencies.