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.     

Numerical and experimental analysis of uncertainty on modal parameters estimated with the stochastic subspace method

Modal parameters of structures are often used as inputs for finite element model updating, vibration control, structural design or structural health monitoring (SHM). In order to test the robustness of these methods, it is a common practice to introduce uncertainty on the eigenfrequencies and modal damping coefficients under the form of a Gaussian perturbation, while the uncertainty on the mode shapes is modeled in the form of independent Gaussian noise at each measured location. A more rigorous approach consists however in adding uncorrelated noise on the time domain responses at each sensor before proceeding to an operational modal analysis. In this paper, we study in detail the resulting uncertainty when modal analysis is performed using the stochastic subspace identification method. A Monte-Carlo simulation is performed on a simply supported beam, and the uncertainty on a set of 5000 modal parameters identified with the stochastic subspace identification method is discussed. Next, 4000 experimental modal identifications of a small clamped–free steel plate equipped with 8 piezoelectric patches are performed in order to confirm the conclusions drawn in the numerical case study. In particular, the results point out that the uncertainty on eigenfrequencies and modal damping coefficients may exhibit a non-normal distribution, and that there is a non-negligible spatial correlation between the uncertainty on mode shapes at sensors of different locations.     

Multimode shunt damping of piezoelectric smart panel for noise reduction

Multimode shunt damping of piezoelectric smart panel is studied for noise reduction. Piezoelectric smart panel is a plate structure on which a piezoelectric patch is attached with an electrical shunt circuit. When an incidence sound is impinged on the panel structure, the structure vibrates and the attached piezoelectric patch produces an electrical energy, which can be effectively dissipated as heat via the electrical shunt circuit. Since the energy dissipation strongly depends on the vibration mode of the panel structure, many patches are required for multiple vibration modes. Instead of using multiple piezoelectric patches, a single piezoelectric patch is used in conjunction with a blocked shunt circuit for multimode shunt damping. Modeling, shunt parameter tuning, and implementation of the blocked shunt circuit along with an acoustic test of the panel are explained. A remarkable reduction of the transmitted noise was achieved for multiple modes of the panel. Since this technology has many merits in terms of compactness, low cost, robustness, and ease of installation, practical applications in many noise problems can be anticipated.     

A new method for accurately determining the modal equivalent stiffness ratio of bonded piezoelectric structures

This paper describes new methods for measuring the modal equivalent stiffness ratios and modal electromechanical coupling coefficients of piezoelectric elements attached to a host structure such as a beam. Modal equivalent stiffness ratios and modal electromechanical coupling coefficients are essential for estimating the performance and determining an optimum design of active vibration control and passive vibration suppression systems that use piezoelectric elements. Accurate determination of these modal parameters is also useful for other systems including piezoelectric sensors and energy generators. This paper not only describes the measurement methods but also presents the theoretical formulations derived by taking into account the effect of adhesive bonds. The formulations in this paper demonstrate the necessity of experimental measurements and the accuracy enhancements that the theoretical estimations can provide. Conventional methods for obtaining the modal equivalent stiffness ratios are sensitive to measurement errors, which result in the loss of accuracy, rendering these methods unreliable for many practical applications. The proposed methods use an inductor instead of an open circuit to address the abovementioned issue and, thereby, provide significant improvement in the accuracy. Because the loss factors of the experimental apparatus tend to compromise the accuracy of the proposed methods, a method using a negative resistor is proposed, theoretically analyzed, and confirmed to eliminate some of the errors introduced by loss factors. The advantages of the proposed methods and the effectiveness of theoretical analysis, considering the effect of adhesive bonds, are verified experimentally.     

Electro-mechanical response of functionally graded beams with imperfectly integrated surface piezoelectric layers

The time-dependent behavior of a simply-supported functionally graded beam bonded with piezoelectric sensors and actuators is studied using the state-space method. The creep behavior of bonding adhesives between piezoelectric layers and beam is characterized by a Kelvin-Voigt viscoelastic model, which is practical in a high temperature circumstance. Both the host elastic functionally graded beam and the piezoelectric layers are orthotropic and in a state of plane stress, with the former being inhomogeneous along the thickness direction. A laminate model is employed to approximate the host beam. Moreover, the coupling effect between the elastic deformation and electric field in piezoelectric layers is considered. Results indicate that the viscoelastic property of interfacial adhesives has a significant effect on the function of bonded actuators and sensors with time elapsing.     

0-3型压电复合材料覆盖层水下 吸声性能的理论研究

目前压电分流阻尼技术在振动和噪声领域的应用得到了广泛的关注. 本文尝试将压电分流阻尼技术应用于水下吸声领域, 以提高覆盖层的吸声性能. 将压电覆盖层厚度模态的机电方程和声波传播的传递矩阵相结合, 建立一维电声模型. 该模型可以用于分析多层压电和非压电水下吸声覆盖层的吸声性能. 采用该模型分析了0-3型压电复合材料覆盖层的水下吸声性能. 压电复合材料的参数是采用Furukawa的模型计算的. 研究结果表明, 采用合适的分流电阻, 负电容分流电路可以在较宽的频率范围显著提高覆盖层的吸声性能. 其原理可以从阻抗匹配的角度解释, 负电容分流电路可以调整压电覆盖层的表面声阻抗, 使之与水的特性声阻抗相匹配.     

Dynamics modelling of beams with shunted piezoelectric elements

General modelling of a resonant shunting damper has been made from piezoelectric sensor/actuator equations. It is found that an additional damping, which is augmented to a system, is generated by the shunt damping effect. The transfer function of the tuned electrical absorber is derived for both series and parallel shunt circuit. The governing equations and associated boundary conditions are derived using Hamilton's principle. The shunt voltage equation is also derived from the charge generated in PZT due to beam vibration. The frequency response function of the obtained mathematical model is compared with that of the tuned electrical absorber and experimental work. The vibration amplitude is reduced about 15 dB at targeted second mode frequency.     

LCR分流电路下压电声子晶体智能材料的带隙

将带有LCR分流电路的压电陶瓷片对贴在铝和环氧树脂组成的声子晶体结构中.使智能材料的机械振动与压电陶瓷的压电效应耦合起来,推导出机械振动在压电陶瓷片上的等效附加应力;使LCR分流电路中的电磁振荡效应和声子晶体的能带特性有机结合,计算了在分流电路作用下智能材料扭转和弯曲振动的带隙特性,研究了电阻、电感、电容元件的改变对压电声子晶体智能材料带隙的影响.研究结果表明:在合理尺寸下,随着分流电路中电阻值的增大,带隙的频率范围变宽,但衰减幅值有所降低;电感和电容值的增大都可以使带隙向低频移动,带隙的衰减幅值随着电感值的增大而升高,但随着电容值的增大而降低.从而给压电声子晶体智能材料减震降噪的控制提供了一种新思路.     

Design of shaped piezoelectric modal sensor for beam with arbitrary boundary conditions by using Adomian decomposition method

The theory for designing distributed piezoelectric modal sensors is well established for beam structures. However, the current modal sensor theory is limited in scope in that it can only be applied in the case of classical boundary conditions (i.e., either clamped, free, simply supported or sliding). In this paper a solution to the problem of finding the shape of piezoelectric modal sensors for a beam with arbitrary boundary conditions is proposed, using the Adomian decomposition method (ADM). A general expression for designing the shape of a piezoelectric modal sensor is presented, in which the output signal of the designed sensor is proportional to the response of the target mode. Other modes are filtered out. The modal sensor shape is expressed as a function of the second spatial derivative of the structural mode shape function. Based on the ADM and employing some simple mathematical operations, the closed-form series solution of the second spatial derivative of the mode shapes can be determined. Then the shapes of the designed modal sensors are obtained. Finally, some numerical examples are given to demonstrate the feasibility of the proposed modal sensors. It is shown that, for classical boundary conditions, the shapes of the modal sensors based on the ADM agree well with analytical and numerical results given in the literature. For general boundary conditions it is found that the shape of the modal sensors is influenced by the number of modes of interest because the second spatial derivatives of the mode shapes are not orthogonal to one another. The modal sensors for general boundary conditions can be considered as modal filters within a limited frequency band.     

Modal analysis of the electromechanical conversion in piezoelectric ceramic spherical shells

The current study considers piezoelectric ceramic electromechanical transducers utilizing axisymmetric vibrations of complete and incomplete spherical shells. Analysis is focused on generating the modes of vibration that can be employed in the design of multimode unidirectional electroacoustic transducers for underwater applications and on determining the electrode configurations that achieve optimal electromechanical coupling for the different modes of vibration considered. Analytical expressions are presented for the modal and intermodal equivalent parameters characterizing the energy state of the shell vibration. Results of calculation and experimental verification of the resonance frequencies and effective coupling coefficients for different modes of vibration of the complete and incomplete spherical shells are in good agreement.     

Modal characteristics of a flexible beam with multiple distributed actuators

A flexible structure with surface-bonded piezoceramic patches is modelled using Timoshenko beam theory. Exact mode shapes and natural frequencies associated with the flexural motion are computed for various piezoceramic distributed actuator arrangements. The effects of patch placement and of shear on the modal characteristics are demonstrated using a cantilevered beam as an example. Perfect bonding of the piezoceramic to the beam substructure is assumed, and for the purposes of this paper only passive piezoceramic properties are considered. The modelling technique and results obtained in a closed form are intended to assist investigations into the modelling and control of active structures with surface-bonded piezoceramic actuators.     

Analysis on vibration characteristics of large-size rectangular piezoelectric composite plate based on quasi-periodic phononic crystal structure

Based on the theory of composite materials and phononic crystals (PCs), a large-size rectangular piezoelectric composite plate with the quasi-periodic PC structure composed of PZT-4 and epoxy is proposed in this paper. This PC structure can suppress the transverse vibration of the piezoelectric composite plate so that the thickness mode is purer and the thickness vibration amplitude is more uniform. Firstly, the vibration of the model is analyzed theoretically, the electromechanical equivalent circuit diagram of three-dimensional coupled vibration is established, and the resonance frequency equation is derived. The effects of the length, width, and thickness of the piezoelectric composite plate at the resonant frequency are obtained by the analytical method and the finite element method, the effective electromechanical coupling coefficient is also analyzed. The results show that the resonant frequency can be changed regularly and the electromechanical conversion can be improved by adjusting the size of the rectangular piezoelectric plate. The effect of the volume fraction of the scatterer on the resonant frequency in the thickness direction is studied by the finite element method. The band gap in X and Y directions of large-size rectangular piezoelectric plate with quasi-periodic PC structures are calculated. The results show that the theoretical results are in good agreement with the simulation results. When the resonance frequency is in the band gap, the decoupling phenomenon occurs, and then the vibration mode in the thickness direction is purer.     

A tri-modal directional transducer

Acoustic transducers made from piezoelectric ceramic cylinders usually exploit the breathing or omnidirectional (omni) mode of vibration. However, with suitable voltage distribution, higher order extensional modes of the cylinder can be excited which produce directional radiation patterns. These modal radiation patterns can then be combined to synthesize desired beam patterns which may be steered by incrementing the excitation. This paper describes a model for the combined acoustic response of the extensional modes of vibration of a piezoelectric ceramic cylinder, a method of synthesizing a desired radiation pattern, and an experimental implementation of a directional transducer that uses these techniques. This tri-modal transducer is broadband and directional with a frequency independent beam pattern yet simple, small, and lightweight.     

Design of shaped piezoelectric modal sensors for cantilever beams with intermediate support by using differential transformation method

In this paper a solution to the problem of finding the shape of piezoelectric modal sensors for a cantilever beam with intermediate support is proposed by using the differential transformation method (DTM). A general expression for designing the shape of a piezoelectric modal sensor is presented, in which the output signal of the designed sensor is proportional to the response of the target mode. Other modes are filtered out. The modal sensor shape is expressed as a linear function of the second spatial derivative of the structural mode shape function. By using boundary condition and continuity condition equations at intermediate support, the closed-form series solution of the second spatial derivative of the mode shapes can be determined based on DTM. Then the shapes of the designed modal sensors are obtained. Finally, numerical examples are given to demonstrate the feasibility of the proposed modal sensors for the cantilever beam with intermediate support.     

MODELLING AND VIBRATION CONTROL OF BEAMS WITH PARTIALLY DEBONDED ACTIVE CONSTRAINED LAYER DAMPING PATCH

A detailed model for the beams with partially debonded active constraining damping (ACLD) treatment is presented. In this model, the transverse displacement of the constraining layer is considered to be non-identical to that of the host structure. In the perfect bonding region, the viscoelastic core is modelled to carry both peel and shear stresses, while in the debonding area, it is assumed that no peel and shear stresses be transferred between the host beam and the constraining layer. The adhesive layer between the piezoelectric sensor and the host beam is also considered in this model. In active control, the positive position feedback control is employed to control the first mode of the beam. Based on this model, the incompatibility of the transverse displacements of the active constraining layer and the host beam is investigated. The passive and active damping behaviors of the ACLD patch with different thicknesses, locations and lengths are examined. Moreover, the effects of debonding of the damping layer on both passive and active control are examined via a simulation example. The results show that the incompatibility of the transverse displacements is remarkable in the regions near the ends of the ACLD patch especially for the high order vibration modes. It is found that a thinner damping layer may lead to larger shear strain and consequently results in a larger passive and active damping. In addition to the thickness of the damping layer, its length and location are also key factors to the hybrid control. The numerical results unveil that edge debonding can lead to a reduction of both passive and active damping, and the hybrid damping may be more sensitive to the debonding of the damping layer than the passive damping.     

FINITE ELEMENT FORMULATION AND ACTIVE VIBRATION CONTROL STUDY ON BEAMS USING SMART CONSTRAINED LAYER DAMPING (SCLD) TREATMENT

This work deals with the active vibration control of beams with smart constrained layer damping (SCLD) treatment. SCLD design consists of viscoelastic shear layer sandwiched between two layers of piezoelectric sensors and actuator. This composite SCLD when bonded to a vibrating structure acts as a smart treatment. The sensor piezoelectric layer measures the vibration response of the structure and a feedback controller is provided which regulates the axial deformation of the piezoelectric actuator (constraining layer), thereby providing adjustable and significant damping in the structure. The damping offered by SCLD treatment has two components, active action and passive action. The active action is transmitted from the piezoelectric actuator to the host structure through the viscoelastic layer. The passive action is through the shear deformation in the viscoelastic layer. The active action apart from providing direct active control also adjusts the passive action by regulating the shear deformation in the structure. The passive damping component of this design eliminates spillover, reduces power consumption, improves robustness and reliability of the system, and reduces vibration response at high-frequency ranges where active damping is difficult to implement. A beam finite element model has been developed based on Timoshenko's beam theory with partially covered SCLD. The Golla-Hughes-McTavish (GHM) method has been used to model the viscoelastic layer. The dissipation co-ordinates, defined using GHM approach, describe the frequency-dependent viscoelastic material properties. Models of PCLD and purely active systems could be obtained as a special case of SCLD. Using linear quadratic regulator (LQR) optimal control, the effects of the SCLD on vibration suppression performance and control effort requirements are investigated. The effects of the viscoelastic layer thickness and material properties on the vibration control performance are investigated.     

Wave propagation in a periodic elastic-piezoelectric axial-bending coupled beam

The wave propagation in a periodic elastic-piezoelectric axial-bending coupled beam is investigated in this paper by considering the mechanical–electrical coupling behavior. The strain energy and kinetic energy of each sub-cell are first formulated to extract the dynamic stiffness matrices, and then the compatibility and continuity conditions at the interface between the adjacent cells are utilized to derive the transfer matrix that governs the propagation of the wave along the periodic piezoelectric beam. By employing the Lyapunov exponent method, the dynamic behaviors of the periodic beam structure are evaluated with different base beam materials, dimension ratios, piezoelectric constants and elastic stiffness. The results indicate that regardless of the length ratio, there exist certain frequency intervals, where the width and magnitude of the prominent stop band of the aluminum beam with periodic piezoelectric patches are always broader and larger than those of the steel base system. In addition, as the thickness ratio decreases, the location of the stop band tends to move toward a higher frequency. Numerical studies also demonstrate that different piezoelectric constants and elastic stiffness affect the characteristics of wave propagation in completely different fashions. The investigation in the present study provides basic guidelines to design periodic elastic-piezoelectric laminate structures in order to achieve desired filtering characteristics.     

A generic double-curvature piezoelectric shell energy harvester: Linear/nonlinear theory and applications

Converting vibration energy to useful electric energy has attracted much attention in recent years. Based on the electromechanical coupling of piezoelectricity, distributed piezoelectric zero-curvature type (e.g., beams and plates) energy harvesters have been proposed and evaluated. The objective of this study is to develop a generic linear and nonlinear piezoelectric shell energy harvesting theory based on a double-curvature shell. The generic piezoelectric shell energy harvester consists of an elastic double-curvature shell and piezoelectric patches laminated on its surface(s). With a current model in the closed-circuit condition, output voltages and energies across a resistive load are evaluated when the shell is subjected to harmonic excitations. Steady-state voltage and power outputs across the resistive load are calculated at resonance for each shell mode. The piezoelectric shell energy harvesting mechanism can be simplified to shell (e.g., cylindrical, conical, spherical, paraboloidal, etc.) and non-shell (beam, plate, ring, arch, etc.) distributed harvesters using two Lamé parameters and two curvature radii of the selected harvester geometry. To demonstrate the utility and simplification procedures, the generic linear/nonlinear shell energy harvester mechanism is simplified to three specific structures, i.e., a cantilever beam case, a circular ring case and a conical shell case. Results show the versatility of the generic linear/nonlinear shell energy harvesting mechanism and the validity of the simplification procedures.     

Damping as a result of piezoelectric energy harvesting

Systems that harvest or scavenge energy from their environments are of considerable interest for use in remote power supplies. A class of such systems exploits the motion or deformation associated with vibration, converting the mechanical energy to electrical, and storing it for later use; some of these systems use piezoelectric materials for the direct conversion of strain energy to electrical energy. The removal of mechanical energy from a vibrating structure necessarily results in damping. This research addresses the damping associated with a piezoelectric energy harvesting system that consists of a full-bridge rectifier, a filter capacitor, a switching DC–DC step-down converter, and a battery. Under conditions of harmonic forcing, the effective modal loss factor depends on: (1) the electromechanical coupling coefficient of the piezoelectric system; and (2) the ratio of the rectifier output voltage during operation to its maximum open-circuit value. When the DC–DC converter is maximizing power flow to the battery, this voltage ratio is very nearly 1/2, and the loss factor depends only on the coupling coefficient. Experiments on a base-driven piezoelectric cantilever, having a system coupling coefficient of 26%, yielded an effective loss factor for the fundamental vibration mode of 2.2%, in excellent agreement with theory.     

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。