Modeling and analysis of laminate composite plates with embedded active-passive piezoelectric networks

The objective of this work is to present the finite element modeling of laminate composite plates with embedded piezoelectric patches or layers that are then connected to active-passive resonant shunt circuits, composed of resistance, inductance and voltage source. Applications to passive vibration control and active control authority enhancement are also presented and discussed. The finite element model is based on an equivalent single layer theory combined with a third-order shear deformation theory. A stress-voltage electromechanical model is considered for the piezoelectric materials fully coupled to the electrical circuits. To this end, the electrical circuit equations are also included in the variational formulation. Hence, conservation of charge and full electromechanical coupling are guaranteed. The formulation results in a coupled finite element model with mechanical (displacements) and electrical (charges at electrodes) degrees of freedom. For a Graphite-Epoxy (Carbon-Fibre Reinforced) laminate composite plate, a parametric analysis is performed to evaluate optimal locations along the plate plane (xy) and thickness (z) that maximize the effective modal electromechanical coupling coefficient. Then, the passive vibration control performance is evaluated for a network of optimally located shunted piezoelectric patches embedded in the plate, through the design of resistance and inductance values of each circuit, to reduce the vibration amplitude of the first four vibration modes. A vibration amplitude reduction of at least 10 dB for all vibration modes was observed. Then, an analysis of the control authority enhancement due to the resonant shunt circuit, when the piezoelectric patches are used as actuators, is performed. It is shown that the control authority can indeed be improved near a selected resonance even with multiple pairs of piezoelectric patches and active-passive circuits acting simultaneously.     

Thickness vibration of piezoelectric plates of 6mm crystals with tilted six-fold axis and two-layered thick electrodes

We perform a theoretical analysis of thickness vibrations in piezoelectric plates of crystals with 6 mm symmetry. The six-fold axis is tilted with respect to the plate surfaces. The major surfaces of the plate are covered with two layers of electrodes of different metals. The equations of linear piezoelectricity are used for the crystal plate. The electrodes are modeled by the equations of elasticity. Thickness vibrations frequencies and modes as well as impedance are calculated and examined.     

Lamb wave dispersion in a PZT/metal/PZT sandwich plate with imperfect interface

Abstract

The Lamb wave dispersion in a PZT/Metal/PZT sandwich plate is investigated by employing the exact linear equations of electro-elastic waves in piezoelectric materials within the scope of the plane-strain state. It is assumed that at the interfaces between the piezoelectric face layers and metal core layer, shear-spring and normal-spring type imperfect conditions are satisfied. The degree of this imperfectness is estimated through the corresponding shear-spring and normal-spring type parameters which appear in the contact condition characterizing the transverse and normal displacements’ discontinuity. The corresponding dispersion equation is derived, and as a result of the numerical solution to this equation, the dispersion curves are constructed for the first and second lowest modes in the cases where the material of the face layers is PZT and the material of the middle layer is Steel (St). Consequently, for the PZT/St/PZT sandwich plate, the study of the influence of the problem parameters such as the piezoelectric and dielectric constants, layer thickness ratios, non-dimensional shear-spring, and normal-spring type parameters, is carried out. In particular, it is established that the imperfectness of the contact between the layers of the plate causes a decrease in the values of the wave propagation velocity.     

Finite element modelling of piezolaminated smart structures for active vibration control with distributed sensors and actuators

Finite element modelling of laminated structures with distributed piezoelectric sensor and actuator layers and control electronics is considered in this paper. Beam, plate and shell type elements have been developed incorporating the stiffness, mass and electromechanical coupling effects of the piezoelectric laminates. The effects of temperature on the electrical and mechanical properties and the coupling between them are also taken into consideration in the finite element formulation. The piezoelectric beam element is based on Timoshenko beam theory. The plate/shell element is a nine-noded field-consistent element based on first order shear deformation theory. Constant-gain negative velocity feedback, Lyapunov feedback as well as a linear quadratic regulator (LQR) approach have been used for active vibration control with the structures subjected to impact, harmonic and random excitations. The influence of the pyroelectric effects on the vibration control performance is also investigated. The LQR approach is found to be more effective in vibration control with lesser peak voltages applied in the piezo actuator layers as in this case the control gains are obtained by minimizing a performance index. The application of these elements in high-performance, light-weight structural systems is highlighted.     

A single plate ultrasonic gyroscope in plane motion

We have proposed a vibration gyroscope made of a single piezoelectric plate in plane motion. The gyroscope makes use of the degenerated modes associated with a square plate. In the present paper, the experimental verification using an enlarged piezoelectric ceramic plate is presented, whose results are compared with the numerical prediction based on two-dimensional and three-dimensional finite element modelling.     

Elastic guided wave propagation in a periodic array of multi-layered piezoelectric plates with finite cross-sections

In this study, we present a model study of guided wave dispersion and resonance behavior of an array of piezoelectric plates with arbitrary cross-sections. The objective of this work is to analyze the influence of the geometry of an element of a 1D-array ultrasound transducer on generating multi-resonance frequency so as to increase the frequency bandwidth of the transducers. A semi-analytical finite-element (SAFE) method is used to model guided wave propagation in multi-layered 1D-array ultrasound transducers. Each element of the array is composed of LiNbO₃ piezoelectric material with rectangular or subdiced cross-section. Four-node bilinear finite-elements have been used to discretize the cross-section of the transducer. Dispersion curves showing the dependence of phase and group velocities on the frequency, and mode shapes of propagating modes were obtained for different geometry consurations. A parametric analysis was carried out to determine the effect of the aspect ratio, subdicing, inversion layer and matching layers on the vibrational behavior of 1D-array ultrasound transducers. It was found that the geometry with subdiced cross-section causes more vibration modes compared with the rectangular section. Modal analysis showed that the additional modes correspond to lateral modes of the piezoelectric subdiced section. In addition, some modes have strong normal displacements, which may influence the bandwidth and the pressure field in front of the transducer. In addition, the dispersion curves reveal strong coupling between waveguide modes due to the anisotropy of the piezoelectric crystal. The effect of the matching layers was to cluster extensional and flexural modes within a certain frequency range. Finally, inversion layer is found to have a minor effect on the dispersion curves. This analysis may provide a means to analyze and understand the dynamic response of 1D-array ultrasound transducers.     

On the propagation of long thickness-stretch waves in piezoelectric plates

We study the propagation of thickness-stretch waves in a piezoelectric plate of polarized ceramics with thickness poling or crystals of class 6 mm whose sixfold axis is along the plate thickness. For device applications we consider long waves with wavelengths much longer than the plate thickness. A system of two-dimensional equations in the literature governing thickness-stretch, extensional, and symmetric thickness-shear motions of the plate is further simplified. The equations obtained can be used to analyze piezoelectric plate acoustic wave devices operating with thickness-stretch modes.     

Design and modeling of inversion layer ultrasonic transducers using LiNbO3 single crystal

Using inversion domain engineering controlled by heating temperature, the LiNbO(3) (LNO) piezoelectric plate with both odd and even-order thickness-extensional modes can be excited simultaneously. Therefore, the inversion layer ultrasound transducer is expected to be capable of operating over a wider frequency range. In this paper, the electrical impedance and the acoustic characteristics of LiNbO(3) (LNO) inversion layer transducer have been studied by finite element modeling (FEM). The transducer designed for this study uses a 36 degrees rotated Y-cut LiNbO(3) thin plate with an active element thickness of approximately 100 microm. First the electrical and elastic properties of the 36 degrees rotated Y-cut LNO were obtained by transforming a basic piezoelectric matrix for Z-cut LNO. In order to validate the FEM using the transformed properties several pieces of pure and 50% inversion layer LNO were tested on the electrical impedance analyzer. The modeled impedance characteristics were consistent with the measured data. Next the model was used to design 50-60 MHz transducers using pure and 30% inversion LNO. Two lambda/4 matching layers and a Tungsten loaded epoxy backing were used in these designs. The modeled results show that an over 90% bandwidth transducer can be made with proper matching and 30% inversion layer.     

Closed-form solution for free vibration of piezoelectric coupled annular plates using Levinson plate theory

Free vibration analysis of annular moderately thick plates integrated with piezoelectric layers is investigated in this study for different combinations of soft simply supported, hard simply supported and clamped boundary conditions at the inner and outer edges of the annular plate on the basis of the Levinson plate theory (LPT). The distribution of electric potential along the thickness direction in the piezoelectric layer is assumed as a sinusoidal function so that the Maxwell static electricity equation is approximately satisfied. The differential equations of motion are solved analytically for various boundary conditions of the plate. In this study the closed-form solution for characteristic equations, displacement components of the plate and electric potential are derived for the first time in the literature. To demonstrate the accuracy of the present solution, comparison studies is first carried out with the available data in the literature and then natural frequencies of the piezoelectric coupled annular plate are presented for different thickness-radius ratios, inner-outer radius ratios, thickness of piezoelectric, material of piezoelectric and boundary conditions. Present analytical model provides design reference for piezoelectric material application, such as sensors, actuators and ultrasonic motors.     

Absolute band gaps and waveguiding in free standing and supported phononic crystal slabs

Using the finite element method (FEM), we investigate the existence of absolute band gaps and localized modes associated with a guide in thin films of phononic crystals. Two different structures based on two-dimensional (2D) phononic crystals are considered, namely a free standing plate and a plate deposited on a silicon substrate. The 2D phononic crystal is constituted by a square array of cylindrical holes drilled in an active piezoelectric PZT5A matrix. We demonstrate the existence of absolute band gap in the band structure of the phononic crystal plate and, then, the possibility of guided modes inside a linear defect created by removing one row of air holes. In the case of the supported plate, we show the existence of an absolute forbidden band in the plate modes when the thickness of the substrate significantly exceeds the plate thickness.     

Importance of the near Lamb mode imaging of multilayered composite plates.

In recent years Lamb waves are being used for internal defect detection in multilayered composite plates. Different Lamb modes generate various stress levels in different layers. As a result, all Lamb modes are not equally sensitive to internal defects of various layers. A number of studies have been carried out to identify which Lamb mode is most effective for detecting defects in a specific layer. However, one shortcoming of the Lamb wave inspection technique is that in a symmetrically layered composite plate stress and displacement magnitudes and energy distribution profiles for all Lamb modes are symmetric about the central plane of the plate. As a result, the ability of a Lamb mode to detect defects in a specific layer of the plate is identical to its ability to detect defects in the corresponding layer of mirror symmetry. Hence, from the Lamb wave generated image one cannot distinguish between the defects in these two layers of mirror symmetry. In this paper it is investigated how by fine-tuning the frequency and the striking angle of the incident beam in the neighborhood of a Lamb mode one can separately detect internal defects in layers of mirror symmetry in the upper and lower halves of a plate.     

Guided waves in a stratified half-space

The dispersion and excitation mechanisms and the energy distribution of guided waves in a stratified half-space are studied. All possible guided waves excited by a symmetric point source in two or three-layer medium models and their relation to the medium parameters are analyzed in detail. The excitation and propagation characteristics, as well as the energy distribution along the depth direction, of all modes of the surface waves and trapped waves are numerically investigated and analyzed thoroughly not only in the case when the shear wave velocity increases from up to down layers but also when a low-velocity layer is contained in halfspace, especially when the shear wave velocity decreases from up to down layers. It is found that there exist many guided wave modes in the case where the shear wave velocity of each layer increases from up to down layers. However, there is less than one guided wave mode in the case where the shear wave velocity of each layer decreases from up to down layers. The trapped waves exist and propagate along the low-velocity structure in the stratified half-space. It is also found that the characteristic of a mode is related to the source frequency. It is possible that a surface wave at one value of frequency is like a trapped wave at another value of frequency. Finally, the relation of the characteristics of all guided waves (surface waves and trapped waves) to the parameters of media is studied.     

Sound absorption characteristics of a thin plate with PZT and shunt circuitⅠ.Theoretical analysis

The feasibility of sound absorbing by a vibrating plate with piezoelectric material and shunt circuits is theoretically investigated.Based on an equivalent compliance of a piezoelectric wafer shunted with RL circuits,the governing equations for the flexural vibration of the plate with the piezoelectric wafer are derived using Lagrange's approach.The equations take into account not only the mass,stiffness and structural damping of the plate and the wafer,but also the electrical resistance and electrical i...     

Focusing and waveguiding of Lamb waves in micro-fabricated piezoelectric phononic plates

This paper presents results on the numerical and experimental studies of focusing and waveguiding of the lowest anti-symmetric Lamb wave in micro-fabricated piezoelectric phononic plates. The phononic structure was based on an AT-cut quartz plate and consisted of a gradient-index phononic crystal (GRIN PC) lens and a linear phononic plate waveguide. The band structures of the square-latticed AT-cut quartz phononic crystal plates with different filling ratios were analyzed using the finite element method. The design of a GRIN PC plate lens which is attached with a linear phononic plate waveguide is proposed. In designing the waveguide, propagation modes in square-latticed PC plates with different waveguide widths were studied and the results were served for the experimental design. In the micro-fabrication, deep reactive ion etching (Deep-RIE) process with a laboratory-made etcher was utilized to fabricate both the GRIN PC plate lens and the linear phononic waveguide on an 80 μm thick AT-cut quartz plate. Interdigital transducers were fabricated directly on the quartz plate to generate the lowest anti-symmetric Lamb waves. A vibro-meter was used to detect the wave fields and the measured results on the focusing and waveguiding of the piezoelectric GRIN PC lens and waveguide are in good accordance with the numerical predictions. The results of this study may serve as a basis for developing an active micro plate lens and related devices.     

Spectral Estimates for Two-Dimensional Schrödinger Operators with Application to Quantum Layers

A logarithmic type Lieb-Thirring inequality for two-dimensional Schrödinger operators is established. The result is applied to prove spectral estimates on trapped modes in quantum layers.     

Transfer constant of a multilayer acoustic transducer at the direct and inverse transformation

The algorithms for calculating the direct and inverse transfer constants of an acoustic transducer with an arbitrary number of intermediate layers between the piezoelectric layer, the acoustic duct, and the rear acoustic load are described. The results of a numerical analysis are presented and discussed. As an illustration, a 100-MHz transducer formed by a (Y+36°)-cut LiNbO₃ plate fixed on a fused-quartz acoustic duct with the help of five metal layers is considered. The other side of the plate carries two metal layers and a rear load. The phase-frequency characteristics and the transformation loss as a function of frequency are analyzed for the cases of direct and inverse transformation under the assumption that the signal is supplied and retrieved by a two-wire line.     

应用长度极化压电陶瓷33模态的 半主动减振技术研究*

本文提出利用长度方向极化的压电材料的33模态来实现半主动减振。论文以横梁为例,通过理论分析和有限元仿真,对比研究了当压电材料分别连接31, 33两种模态对应的最佳分流电路时,压电材料两种模态在横梁的共振频率附近的减振效果。结果表明33模态比31模态具有更高的减振效率。此外,鉴于33模态存在极化长度有限的问题,仿真分析了压电材料的尺寸和位置对减振效果的影响。在此基础之上,提出了一个利用压电材料33模态的多模态减振的组合设计,对横梁的前三个模态起到了很好地减振作用。相对31模态而言,横梁的每个振动模态均有约15dB的减振提升。     

Active modal control simulation of vibro-acoustic response of a fluid-loaded plate

Active modal control simulation of vibro-acoustic response of a fluid-loaded plate is presented. The active modal control of the vibro-acoustic response is implemented using piezoelectric actuators/sensors. The active modal damping is added to the coupled system via negative velocity feedback. The feedback gain between the piezoelectric actuators/sensors for the modal control is obtained using the in-vacuo modal matrix and the incompressible fluid-loaded modal matrix. The modal control performance of structural vibration and acoustic radiation of a baffled plate is numerically studied. It is shown that the proposed method increases the modal damping ratio and achieves reduction in the mean square velocity and the sound power for given modes of the fluid-loaded plate.     

采用单个压电传感器的单模式兰姆波激发频率的选择

考虑了压电传感器(PZT)与板之间的耦合作用,从理论上研究了单个压电传感器激发时产生单模式兰姆波的频率调节方法,实验给出了模式选择在兰姆波结构健康监测中的应用. 在板材、板厚以及PZT尺寸一定的情况下,从理论上能够预测到作为频率函数的各兰姆波模式的幅值变化. 根据某特定兰姆波模式的幅值最大而其他模式幅值相对最小时所对应的频率,即可识别出该兰姆波模式优化的激发频率. 数值仿真验证了理论的有效性以及单模式兰姆波选择的可能性. 在不同的激发频率下,分别激发了优化的A0 模式,优化的S0模式以及共存的A0和S0模 关键词： 兰姆波 压电传感器 激发频率 结构健康监测     

Analytical study of an active piezoelectric absorber on vibration attenuation of a plate

The attenuation of the transverse vibration of a plate, subjected to a harmonic force, is studied. This goal can be achieved by using an active dynamic absorber. The active absorber is made of a pair of piezoelectric sheets, attached to both sides of the plate, and closed electric circuits. One piece of the piezoelectric material provides a sensor for detecting the motion of the plate. Another piece serves as an active dynamic absorber. The equations of motion of the composite plate, including the plate and the piezoelectric material, and the circuit equations of the sensor and the absorber are derived. The displacements of the plate and the currents in the circuits are calculated. The active absorber can successfully attenuate the vibration. The numerical results show that the proposed active absorber can offer more reduction than that using a passive absorber while the absorber is designed to suppress the resonance of a particular vibration mode. Moreover, the active absorber can also reduce the displacements corresponding to other uncontrolled modes. The effects of altering various parameters of the active absorber are studied and discussed.