Band gaps of elastic waves in three-dimensional piezoelectric phononic crystals with initial stress

In this paper, the stop band properties of elastic waves in three-dimensional piezoelectric phononic crystals with initial stress are studied taking the mechanical and electrical coupling into account. The band gap characteristics for three kinds of lattice arrangements (i.e. sc, bcc and fcc) are investigated by the plane wave expansion (PWE) method. Regarding the variables of mechanical and electrical fields as the elements of the generalized state vector, the expression of the generalized eigenvalue equation for three-dimensional piezoelectric periodic structures is derived. Numerical calculations are performed for the PZT-2/polymer and ZnO/polymer phononic crystals. It can be observed from the results that the fcc lattice is more favorable to create the stop band than the sc and bcc lattices for the piezoelectric phononic crystals, which has also been proved for the pure elastic periodic structures. Compared with the PZT-2/polymer systems, the band gap of the sc lattice for the ZnO/polymer structures is narrower. However, the widths of the bcc and fcc lattices for the ZnO/polymer phononic crystals are much larger than those for the PZT-2/polymer structures. The lattice arrangements and the piezoelectricity have remarkable influences on the stop band behaviors.     

压电超构材料及其波动控制研究:现状与展望

弹性波超构材料是一种人为设计的周期结构材料, 因其独特的力学性能而受到广泛的关注, 在军用和民用领域都展现出重要而独特的应用前景. 根据需求主动或被动地调控弹性波超构材料的力学特性, 能够赋予其更强的适用性能. 其调控的方式多种多样, 其中运用压电材料进行调控是一种方便、速度快、精度高、体积小且价格低的调控方式. 文章中首先简要地介绍弹性波超构材料、可调超构材料、压电材料和几种常用的分流电路的基本特性. 然后依据压电材料在弹性波超构材料中应用形式的不同, 将其分为两大类: 第一类中, 压电材料作为主体结构材料或主体结构的一部分组成材料; 第二类中, 压电材料主要以压电弹簧或压电片的形式贴附于主体结构的表面或内嵌在结构中, 作为激励器或/和传感器. 文章主要介绍两种类型弹性波超构材料的研究内容和发展历史, 涉及带隙调控、波导、负折射、超传输、拓扑态、隐身以及外接分流电路等. 最后总结压电弹性波超构材料研究的不足之处并给出相应的未来研究展望.      

弯曲波彩虹捕获效应及其在能量俘获中的应用

弹性波在色散关系经过设计的梯度结构中传播时会产生空间分频现象和波场能量增强现象,即不同频率的弹性波会在结构的不同位置停止向前传播并发生能量聚集,这就是弹性波彩虹捕获效应.其相关研究成果可以促进结构健康监测、振动控制以及能量俘获等领域的发展.本文通过所设计的梯度结构梁,系统地研究了弯曲波彩虹捕获效应及其在压电能量俘获中的应用.首先,利用传递矩阵法获得了梯度结构梁元胞能带结构的解析解,进而分析了弯曲波彩虹捕获效应的产生机理:不同频率的弯曲波会在不同元胞附近群速度减小到零,从而停止向前传播并发生反射;入射波和反射波的叠加,以及群速度减小带来的能量聚集,会显著增强反射处的波场能量.其次,通过有限元仿真和实验验证了弯曲波彩虹捕获效应的空间分频现象和波场能量增强现象.最后,通过有限元多物理场耦合仿真和实验,研究了粘贴PVDF压电薄膜的梯度结构梁相对于均匀梁的弯曲波能量俘获效果及其随入射波频率的变化规律.结果表明,在弯曲波彩虹捕获效应发生频带内,粘贴在梯度结构梁上的PVDF压电薄膜的输出电压约为粘贴在均匀梁相应位置处的PVDF压电薄膜的输出电压的2倍.     

含电学边界的压电层合梁的非线性弯曲波

非线性科学己成为近代科学发展的一个重要标志, 特别是非线性动力学和非线性波的研究对于解决自然科学各领域中遇到的复杂现象和问题有着极其重要的意义. 本文研究了含电学边界条件的压电层合梁的非线性弯曲波传播特性.首先, 考虑几何非线性效应和压电耦合效应, 利用哈密顿原理建立了一维无限长矩形压电层合梁弯曲波的非线性方程.其次, 采用Jacobi椭圆函数展开法对非线性弯曲波方程进行求解, 得到了非线性弯曲波动方程在近似情况下对应的冲击波解和孤波解.最后, 利用约化摄动法得到了非线性薛定谔方程, 进一步得到了亮孤子和暗孤子解.基于两种方法具体研究了外加电压、压电层厚度等参数对冲击波和孤立波以及亮孤子和暗孤子特性的影响. 研究结果表明, 在波速较小时, 外加电压对冲击波的影响较大, 波速较大时, 外加电压对孤立波影响减弱.通过调整作用在压电层合梁上的电压发现了存在亮孤子和暗孤子, 分析结果表明随着外加电压值的增大, 亮孤子和暗孤子的振幅都增大.      

Electro-mechanical coupling wave propagating in a locally resonant piezoelectric/elastic phononic crystal nanobeam with surface effects

The model of a "spring-mass" resonator periodically attached to a piezoelectric/elastic phononic crystal(PC) nanobeam with surface effects is proposed, and the corresponding calculation method of the band structures is formulized and displayed by introducing the Euler beam theory and the surface piezoelectricity theory to the plane wave expansion(PWE) method. In order to reveal the unique wave propagation characteristics of such a model, the band structures of locally resonant(LR) elastic PC Euler nanobeams with and without resonators, the band structures of LR piezoelectric PC Euler nanobeams with and without resonators, as well as the band structures of LR elastic/piezoelectric PC Euler nanobeams with resonators attached on PZT-4, with resonators attached on epoxy, and without resonators are compared. The results demonstrate that adding resonators indeed plays an active role in opening and widening band gaps. Moreover, the influence rules of different parameters on the band gaps of LR elastic/piezoelectric PC Euler nanobeams with resonators attached on epoxy are discussed, which will play an active role in the further realization of active control of wave propagations.     

Vibration control of piezoelectric beam-type plates with geometrically nonlinear deformation

This paper presents a wavelet-based approach of deformation identification and vibration control of beam-type plates with geometrically nonlinear deflection using piezoelectric sensors and actuators. The identification is performed by transferring the nonlinear equations of identifying deflection into a system of solvable nonlinear algebraic equations in terms of the measurable electric charges and currents on piezoelectric sensors. After that, a control law of negative feedback of the identified signals of deflection and velocity is employed, and the weighted residual method is chosen to determine control voltages applied on the piezoelectric actuators. Due to that the scaling function transform is like a low-pass filter which can automatically filter out high-order signals of vibration or disturbance from the measurement and the controller employed here, this control approach does not lead to the undesired phenomenon of control instability which is generated by the spilling over of high-order signals. Finally, some numerical simulations are carried out to show the efficiency of the proposed approach.     

磁耦合式双稳态宽频压电俘能器的设计和俘能特性

为了同时提高振动能量俘获系统的效率和实用性, 俘能器主结构的振动特性与环境振动特性的匹配度显得尤为重要. 非线性系统复杂的动力学行为为设计高效的俘能器奠定了基础, 但结构一旦被设计、生成出来, 其工作频率往往是固定的, 无法根据环境中的振动而发生相应的改变. 本文利用可移动铰支座和非线性磁力设计了一种具有双稳态特性的宽频压电俘能器, 通过拓宽压电俘能器的工作频带, 来匹配环境中较宽的振动频率. 为了保证系统低频宽带的俘能效果, 详细分析了结构的长度比、磁间距、负载阻抗、外激励频率和幅值等对系统线性刚度、非线性刚度以及动力学行为的影响, 并进行了实验验证. 首先将系统简化为欧拉-伯努力梁, 利用拉格朗日方程建立系统的非线性动力学方程, 并利用谐波平衡法进行求解. 针对理论分析给出的不同外激励频率下的最优长度比, 搭建了实验平台进行验证. 理论和实验的结果表明: 非线性磁力的引入使系统呈现负刚度特性, 使俘能器能够在单稳态和双稳态之间的变换, 实现低频俘能效果; 通过调节可移动铰支座的位置, 改变系统的长细比, 能够实现从0到16 Hz的宽频俘能效果.      

WAVE PROPAGATION IN TWO-DIMENSIONAL DISORDERED PIEZOELECTRIC PHONONIC CRYSTALS

The wave propagation is studied in two-dimensional disordered piezoelectric phononic crystals using the finite-difference time-domain （FDTD） method. For different cases of disorder, the transmission coefficients are calculated. The influences of disorders on band gaps are investigated. The results show that the disorder in the piezoelectric phononic crystals has more significant influences on the band gap in the low frequency regions than in the high frequency ones. The relation between the width of band gap and the direction of position disorder is also discussed. When the position disorder is along the direction perpendicular to the wave transmission, the piezoelectric phononic crystals have wider band gaps at low frequency regions than the case of position disorder being along the wave transmission direction. It can also be found that the effect of. size disorder on band gaps is analogous to that of location disorder. When the perturbation coefficient is big, it has more pronounced effects on the pass bands in the piezoelectric phononic crystals with both size and location disorders than in the piezoelectric phononic crystals with single disorder. In higher frequency regions the piezoelectric effect reduces the transmission coefficients. But for larger disorder degree, the effects of the piezoelectricity will be reduced.     

Application of piezoelectric actuation to regularize the chaotic response of an electrostatically actuated micro-beam

The impetus of this study is to investigate the nonlinear chaotic dynamics of a clamped–clamped micro-beam exposed to simultaneous electrostatic and piezoelectric actuation. The micro-beam is sandwiched with piezoelectric layers throughout its length. The combined DC and AC electrostatic actuation is imposed on the micro-beam through two upper and lower electrodes. The piezoelectric layers are actuated via a DC electric voltage applied in the direction of the height of the piezoelectric layers, which produces an axial force proportional to the applied DC voltage. The governing differential equation of the motion is derived using Hamiltonian principle and discretized to a nonlinear Duffing type ODE using Galerkin method. The governing ODE is numerically integrated to get the response of the system in terms of the governing parameters. The results show that the response of the system is greatly affected by the amounts of DC and AC electrostatic voltages applied to the upper and lower electrodes. The results show that the response of the system can be highly nonlinear and in some regions chaotic. Evaluating the K–S entropy of the system, based on several initial conditions given to the system, the chaotic response is distinguished from the periodic or quasiperiodic ones. The main objective is to passively control the chaotic response by applying an appropriate DC voltage to the piezoelectric layers.     

Nonlinear active control of damaged piezoelectric smart laminated plates and damage detection

Considering mass and stiffness of piezoelectric layers and damage effects of composite layers, nonlinear dynamic equations of damaged piezoelectric smart laminated plates are derived. The derivation is based on the Hamilton＇s principle, the higher- order shear deformation plate theory, von Karman type geometrically nonlinear straindisplacement relations, and the strain energy equivalence theory. A negative velocity feedback control algorithm coupling the direct and converse piezoelectric effects is used to realize the active control and damage detection with a closed control loop. Simply supported rectangular laminated plates with immovable edges are used in numerical computation. Influence of the piezoelectric layers＇ location on the vibration control is in- vestigated. In addition, effects of the degree and location of damage on the sensor output voltage are discussed. A method for damage detection is introduced.     

Active vibration control of thin constrained composite damping plates with double piezoelectric layers

Vibrations and the damping behaviour of thin constrained composite plates with double piezoelectric layers are analytically explored by using Fourier transformation and classical laminated plate theory. Electric potential equations in the double piezoelectric layers are solved with respect to closed and open circuit boundary conditions, an exterior dielectric slab and active control. The natural frequencies and loss factors of the constrained smart composite plates with passive control methods are not notably changed in comparison with those of the constrained composite plates without piezoelectric effects since vibrational energy does not efficiently convert to electrical energy. The loss factors of the composite plates with active constrained damping increase and the natural frequencies have significant variations as the proportional derivative gains increase. Transverse displacement power spectra of the piezoelectric composite plates with active control are compared with those of the piezoelectric composite plates with passive control showing that active control has the best suppression performance of vibrations for the constrained laminated plates with double piezoelectric layers. Radial power spectral density, phase angles and cylindrical-wave power spectral density are calculated. Interesting patterns of wave propagation are explained when plane wave expansion is used to obtain Bessel cylindrical waves.     

Time-harmonic P-waves engulfing a rectangular limited-permeable crack in piezoelectric medium: Energy density and energy release

The dynamic behavior of a limited-permeable rectangular crack in a transversely isotropic piezoelectric material is impinged by to a P-wave. The generalized Almansi theorem and the Schmidt method are used to determine the stress intensity factor and energy density factor as the primary fracture criterion of failure. The mixed boundary value problem entails the evaluation of the appropriate crack edge stress singularities that are characteristics of the fundamental functions. The stress and electric displacement intensity factors are also used to find the energy release rate that can be computed numerically and compared with the results corresponding to those of the stress intensity factor, and energy density factor. Graphical presentation shows that the energy release rate is always negative for the boundary conditions considered while the energy density factors always remain positive. Under certain conditions, the stress and electric displacement intensity factors can be negative and subject to physical limitations. Piezoelectric material boundary value problem solutions should therefore be qualified by the application of failure criteria by fracture of otherwise, particularly when the mechanical and electrical energy can release by creating free surface at the macroscopic and microscopic scales. Negative energy release rate found for the piezoelectric medium in this work can be a case in point.Positive definiteness of the energy density factor can be applied to mutliscale fracture. This is not true for the stress intensity factor nor the energy release rate. Hence, crack initiation behavior for the permittivity of a rectangular crack due to the wave propagation effects may be studied. In particular, the initiation of micro-cracks may be identified with certain critical stress wave frequency band. Negative stress intensity factor may not enhance macrocracking but it does not exclude microcrack initiation.     

WAVE LOCALIZATION IN RANDOMLY DISORDERED PERIODIC PIEZOELECTRIC RODS WITH INITIAL STRESS

The elastic wave localization in disordered periodic piezoelectric rods with initial stress is studied using the transfer matrix and Lyapunov exponent method. The electric field is approximated as quasi-static. The effects of the initial stress on the band gap characteristics are investigated. The numerical calculations of localization factors and localization lengths are performed. It can be observed from the results that the band structures can be tuned by exerting the suitable initial stress. For different values of the piezoelectric rod length and the elastic constant, the band structures and the localization phenomena are very different. Larger disorder degree can lead to more obvious localization phenomenon.     

运动柔性梁振动主动控制的有限段法分析

基于柔性梁的多刚性－弹簧系统模型，采用分段线性化的思想，由线性二次优化理论导出了有闭环反馈控制的以分段压电片为执行器的平面运动柔性杆梁结构非线性振动的主动控制的分析方法。两个算例验证了所提方法的有效性。     

不同类型水平弹簧组合刚度非线性吸振器的性能分析及稳定性研究

动力吸振器作为一种振动控制单元被广泛运用于各种工程场合,但传统的线性吸振器只能实现窄带振动控制.文章在线性吸振器的基础上引入对称水平弹簧构建线性刚度与非线性刚度相结合的组合刚度非线性吸振器,以提升吸振器的吸振性能.考虑实际工程中可能的安装方式,分别建立水平弹簧接地安装和不接地安装的组合刚度非线性吸振器模型,利用谐波平衡法结合弧长延拓法解析求解动力学响应,并与数值结果相互验证,证明了求解结果的准确性.随后分析比较两种组合刚度非线性吸振器与线性吸振器以及非线性能量阱之间的吸振性能,发现水平弹簧接地安装类型的组合刚度非线性吸振器在保留线性吸振器优势的同时又改善其吸振频带窄的缺点,且与非线性能量阱相比在主共振频率附近的较宽频内吸振性能更优.在此基础上,讨论了水平弹簧参数以及吸振器阻尼对主结构振动幅频响应和稳定性的影响,最后观察分析主结构幅频响应曲线不稳定区内的复杂动力学行为.研究结果表明合适的设计参数能够使得主结构振动峰值较低的同时,频响曲线不稳定运动区域的范围也较小.     

Active control of flexural waves in a phononic crystal beam with staggered periodic properties

The band gaps of a phononic crystal beam with staggered periodic structure are investigated. The periodic system consists of a pure elastic (i.e. PMMA) matrix beam and some piezoelectric (i.e. PZT) patches with coupling between the mechanical–electrical components. The PZT patches connected by negative capacitance circuits are applied to function as the active control system. Based on the condition at the interface between adjacent unit cells, the transfer matrix and localization factor are derived. The influence of the degree of interlacing and negative capacitance circuits are discussed. The numerical results show that another band gap can be generated by the staggered periodic structure of PZT patches. The widths and locations of the band gaps can be changed by the degree of interlacing.     

偏压电场对压电板中Lamb波相速度的影响

本文研究了偏压电场作用下,Lamb波在压电板中的传播行为，首先给出了偏压电场作用时压电板中的应力场及电位移场，然后通过求解含初应力及初电位移的小幅波动问题的耦合方程，分别给出了Lamb波的对称模态和反对称模态的相速度方程，以典型的PZT－5H压电陶瓷板为例进行了数值计算,并讨论了偏压电场对Lamb波相速度及频散曲线的影响，结果表明，偏压电场可以显著地改变Lamb波的传播速度，借此可使声波器件获得延时效果。     

Love waves in a two-layered piezoelectric/elastic composite plate with an imperfect interface

Based on the shear spring model, the propagation of Love wave in two-layered piezoelectric/elastic composite plates under the influence of interfacial defect is investigated. The piezoelectric layer is electrically shorted at both top and bottom surfaces. The wave form solutions of the piezoelectric and elastic layers are obtained, and the dispersion equation is derived by subjecting the boundary conditions and the continuity conditions to the obtained wave form solutions. Numerical results are performed for PZT4/aluminum composite plate. The phase velocities and the mode shapes of mechanical displacement and electric potential are illustrated graphically. The results show that both the interfacial defect and the thickness ratio between the piezoelectric and elastic layers have significant effect on the propagation characteristics of Love wave. One important feature is observed that the interfacial defect always decreases the phase velocities.     

Controllable wave propagation in a weakly nonlinear monoatomic lattice chain with nonlocal interaction and active control

The one-dimensional monoatomic lattice chain connected by nonlinear springs is investigated, and the asymptotic solution is obtained through the Lindstedt-Poincar′e perturbation method. The dispersion relation is derived with the consideration of both the nonlocal and the active control effects. The numerical results show that the nonlocal effect can effectively enhance the frequency in the middle part of the dispersion curve.When the nonlocal effect is strong enough, zero and negative group velocities will be evoked at different points along the dispersion curve, which will provide different ways of transporting energy including the forward-propagation, localization, and backwardpropagation of wavepackets related to the phase velocity. Both the nonlinear effect and the active control can enhance the frequency, but neither of them is able to produce zero or negative group velocities. Specifically, the active control enhances the frequency of the dispersion curve including the point at which the reduced wave number equals zero, and therefore gives birth to a nonzero cutoff frequency and a band gap in the low frequency range. With a combinational adjustment of all these effects, the wave propagation behaviors can be comprehensively controlled, and energy transferring can be readily manipulated in various ways.     

COMBINED DAMAGE FRACTURE CRITERIA FOR PIEZOELECTRIC CERAMICS

Mechanical and electrical damages are introduced to study the fracture mechanics of piezoelectric ceramics in this paper. Two kinds of piezoelectric fracture criteria are established using the method of least squares combined with a damage analysis of the well-known piezoelectric fracture experiments of Park and Sun‘s. One is based on a linear combination of the mechanical and electrical damages and the other on their nonlinear combination. When the combined damage D is up to its critical value De, piezoelectric fracture occurs. It is found from the qualitative comparison of the numerical results with the experimental data that the nonlinearly combined damage fracture criterion can give a better prediction of piezoelectric fracture. And it is concluded from the nonlinearly combined damage fracture criterion that a negative electric field impedes fracture whereas the effect of a positive electric field on fracture depends on its magnitude.