Piezoelectric Shell Research Articles

Broadband multimode baffled piezoelectric cylindrical shell transducers

Hollow piezoelectric cylindrical shell transducers may be made directional for underwater acoustic applications by the use of suitable acoustical baffles and the operational bandwidth may be extended by using multiple resonant modes. A theoretical and experimental investigation was performed for circumferentially baffled piezoelectric cylindrical shell transducers operating in the zero and one modes of extensional vibration. The frequency responses and directivity patterns were analyzed under various conditions of energizing separate halves of electrodes. It was found that the broadest frequency response with nearly constant beamwidth can be obtained when the two halves of the piezoelectric ring are electromechanically excited 90 deg out-of-phase. The experimental results obtained with a proof-of-concept transducer were in good agreement with the theoretical predictions.

An analysis of a cylindrical thin shell as a piezoelectric transformer

The vibration of a circular cylindrical piezoelectric ceramic shell with tangential poling is investigated for transformer application and an analytical solution is obtained. Numerical calculations of the output voltage, input admittance and the efficiency of the transformer are presented. The results indicate that the structure analyzed can be used as a piezoelectric transformer with a certain harmonic driving frequency and matching load impedance.

Piezoelectric generator based on torsional modes for power harvesting from angular vibrations

Torsional vibration of a circular piezoelectric shell of polarized ceramics mounted on a rotationally vibrating base is analyzed. The shell is properly electroded and connected to a circuit such that an electric output is generated. The structure analyzed represents a piezoelectric generator for converting mechanical energy from angular vibrations to electrical energy. Analytical expressions and numerical results for the output voltage, current, power, efficiency and power density are given.

Exact solutions of functionally graded piezoelectric shells under cylindrical bending

Within a framework of the three-dimensional (3D) piezoelectricity, we present asymptotic formulations of functionally graded (FG) piezoelectric cylindrical shells under cylindrical bending type of electromechanical loads using the method of perturbation. Without loss of generality, the material properties are regarded to be heterogeneous through the thickness coordinate. Afterwards, they are further specified to be constants in single-layer homogeneous shells and to obey an identical exponent-law in FG shells. The transverse normal load and normal electric displacement (or electric potential) are, respectively, applied on the lateral surfaces of the shells. The cylindrical shells are considered to be fully simple supports at the edges in the circumferential direction and with a large value of length in the axial direction. The present asymptotic formulations are applied to several benchmark problems. The coupled electro-elastic effect on the structural behavior of FG piezoelectric shells is evaluated. The influence of the material property gradient index on the variables of electric and mechanical fields is studied.

Wave propagation in Reissner–Mindlin piezoelectric coupled cylinder with non-constant electric field through the thickness

In a recent paper by Berg et al. [Berg, M., Hagedorn, P., Gutschmidt, S., 2004. On the dynamics of piezoelectric cylindrical shell, Journal of Sound and Vibration 274, 91–109], equations for the piezo-ceramic cylindrical shell are derived. These equations differ from known equations because the electric field is not assumed to be constant over the thickness, but is obtained by solving the Maxwell’s equation. The considered shell model is the so-called Flugge’s shell theory where the transverse shear deformations is not considered, and the Maxwell’s equation is reduced in form. In this paper the analysis of wave propagation in Reissner–Mindlin piezoelectric coupled cylinder is studied. The shear strain is modelled and the Maxwell’s equation is fully considered.

Vibration of axisymmetric composite piezoelectric shells coupled with internal fluid

AbstractThis paper presents the theoretical and finite element formulations of piezoelectric composite shells of revolution filled with compressible fluid. The originality of this work lies (i) in the development of a variational formulation for the fully coupled fluid/piezoelectric structure system, and (ii) in the finite element implementation of an inexpensive and accurate axisymmetric adaptive laminated conical shell element. Various modal results are presented in order to validate and illustrate the efficiency of the proposed fluid–structure finite element formulation. Copyright © 2007 John Wiley & Sons, Ltd.

Electroelasticity Relations and Fracture Mechanics of Piezoelectric Structures

Three-dimensional (3D) constitutive equations of piezoelectric (PZ) plates and shells are considered for inverse linear and electrostrictive (quadratic) piezoeffects. Prestressed multilayer PZ shells reinforced with metal including the case of uneven thickness polarization are studied. Asymptotic and variational methods to solve the governing differential equations of PZ shells are considered. Concentrations of electrical and mechanical fields near structure imperfections and external local loading are investigated. The electrothermoviscoelastic heating of PZ shells is considered at harmonic excitation. From numerical analysis and the experimental data of energy dissipation and the temperature behavior of PZ shell the conditions of optimal transformation of electric energy into mechanical deformations are defined. Thus, the geometrical parameters and working frequencies are determined with due account of dielectric relaxation processes. The following nonlinear phenomena are studied: acoustoelectronic wave amplification; electron injection into metalized polar dielectric; resonance growth by 5–20 times of internal electrical field strength in the PZ shells and plates; and autothermostabilization of ferroelectric resonators. For a better understanding of R.D. Mindlin’s gradient theory of polarization in view of electron processes in thin metal-dielectric-metal structures, use was made of solid state physics interpretations as well as experimental data. High concentration of mechanical stresses and temperature and electrical fields near structure defects (first of all, near boundary between various materials) defines the main properties of polar dielectrics. An unknown domain of electrode rough surface influence was estimated, and as result an uneven polarization distribution was found. A theory of nonlinear autowave systems with energy dissipation was used in a physical model of the electrothermal fracture of dielectrics (contacting with metal electrodes), and as a result a nondestructive testing method to study the microstructure defect formation has been suggested.

Analysis of a piezoelectric ceramic shell in thickness-shear vibration as a power harvester

We study mechanical-to-electrical power conversion in a piezoelectric ceramic shell mechanically forced into thickness-shear vibrations. A theoretical analysis is performed. The efficiency of energy conversion, the power density of the device and their dependence on various physical and geometric parameters are examined. The structure may be used as an electric power source for wireless devices.

A Four‐node Finite Element for Piezoelectric Shell Structures in Convective Co‐ordinates

AbstractIn this contribution, an isoparametric piezoelectric shell element is presented which is based on convective coordinates and which allows for the analysis of arbitrary shell geometries. A two‐field variation formulation [1, 2] is used in which the displacements and the electric potentials serve as independent variables. Especially, for thin‐walled structures under certain boundary conditions and load cases, the displacement based element tend to shear and membrane locking. In order to avoid this poor behaviour, the Assumed Natural Strain (ANS) method [3] is introduced into the piezoelectric shell element. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Wave propagation in laminated piezoelectric cylindrical shells in hydrothermal environment

This paper reports the result of an investigation into wave propagation in orthotropic laminated piezoelectric cylindrical shells in hydrothermal environment. A dynamic model of laminated piezoelectric cylindrical shell is derived based on Cooper-Naghdi shell theory considering the effects of transverse shear and rotary inertia. The wave characteristics curves are obtained by solving an eigenvalue problem. The effects of layer numbers, thickness of piezoelectric layers, thermal loads and humid loads on the wave characteristics curves are discussed through numerical results. The solving method presented in the paper is validated by the solution of a classical elastic shell non-containing the effects of transverse shear and rotary inertia. The new features of the wave propagation in laminated piezoelectric cylindrical shells with various laminated material, layer numbers and thickness in hydrothermal environment and some meaningful and interesting results in this paper are helpful for the application and the design of the ultrasonic inspection techniques and structural health monitoring.

Experimental investigation of coupled vibrations in piezoelectric cylindrical shells

An experimental investigation of the electromechanical properties of radially polarized thin-walled piezoelectric ceramic cylinders as a function of the height-to-diameter aspect ratio is presented. The focus of this investigation is on the lowest order axially symmetric modes of vibration for cylinders with free ends having aspect ratio less than 3 with emphasis on the region of strongest coupling around aspect ratio π∕2. The resonance frequencies and coupling coefficients are presented vs aspect ratio and quantify the lower, upper, and an intermediate resonance branch. It is shown that the intermediate branch is flexural in origin and clearly crosses the so-called “dead zone,” which is an inadequacy of the membrane theory applied to the vibration of thin shells [Junger and Rosato, J. Acoust. Soc. Am., 26, 709–713 (1954)]. The coupling coefficient for the lower branch is shown to drop to zero at h∕2a=π∕2, whereas the coupling coefficient associated with the upper branch reaches a maximum at an aspect ratio near this region. As an example of the practical application of this study, the performance of two underwater electroacoustic transducers comprised of cylindrical elements having different aspect ratios is reported.

High-Performance 4-Node Shell Element with Piezoelectric Coupling

The paper focuses on the topic of smart and adaptive structures using embedded piezoelectric sensors and actuators. A new high-performance degenerated continuum based solid piezoelectric shell element is developed herein. It is an isoparametric 4-noded quadrilateral layered element based on the Reissner-Mindlin theory of plates with piezoelectric coupling. It assumes linear elasticity and small displacements, rotations and strains theories. The element is free of both membrane and transverse shear locking effects, and it has fully ranked stiffness matrix. This is ensured by a carefully selected combination of the Enhanced Assumed Strain, Discrete Shear Gap, and Drilling Rotations approaches. The element was implemented into ANSYS 7.1 and named SHELL10X. The pure-mechanical and static behavior is tested, e.g., using the standard set of test problems—patch tests—proposed by Harder and MacNeal. Furthermore, an experimental measurement has been carried out on a steel beam with applied piezoelectric patches. The harmonic response of the beam was studied to test the piezoelectric behavior and to show the reliability of the element. Good agreement between the measured and calculated data was achieved.

Dynamic analysis of thick laminated shell panel with piezoelectric layer based on three dimensional elasticity solution

Elasticity solution is presented for infinitely long, simply-supported, orthotropic, piezoelectric shell panel under dynamic pressure excitation. The direct and inverse piezoelectric effects are considered. The highly coupled partial differential equations (p.d.e.) are reduced to ordinary differential equations (o.d.e.) with variable coefficients by means of trigonometric function expansion in circumferential direction. The resulting ordinary differential equations are solved by the finite element method. Numerical examples are presented for [0/90/ P] lamination, where P indicates the piezoelectric layer. Finally the results are compared with the published results.

Modelling and optimization of laminated adaptive shells of revolution

In this paper two shell finite element models are presented for the structural analysis of composite laminated piezoelectric shells. One is an axisymmetric conical frustum with two nodal rings and the other is a conic shell panel with eight nodes. Both models are based in a mixed laminated theory that combines a higher order shear deformation theory for the mechanical displacement field with a layerwise representation with linear functions for the electric potential through each piezoelectric layer. In order to obtain the optimal design sensitivities analysis and optimization techniques based in the nonlinear mathematical programming are used. The design objectives can be the minimization of the deformed structure or the maximization of the natural fundamental frequency and the design variables are the electric potential difference applied to the actuators or the ply thicknesses among others.

Closed-form solutions for the free-vibration problem of multilayered piezoelectric shells

This paper addresses the free-vibration problem of multilayered shells with embedded piezoelectric materials. A series of hierarchic, two-dimensional axiomatic shell theories are presented within the “Unified Formulation” introduced by the last author. Shells of constant curvature are considered, and no simplifying assumptions on the curvature terms are made in the geometric relations. Closed-form solutions are given for the free-vibration problem of simply supported, orthotropic piezoelectric laminates. The formulations are applied to study the influence of the electro-mechanical coupling on the resonant frequencies. It is demonstrated that the slenderness of plates with through-thickness polarized piezoelectric layers increases the electro-mechanical coupling. For comparison purposes, the fundamental axisymmetric mode of hollow cylinders has been exemplarily considered: with respect to flat plates, the thickness and the curvature of the shells have a less important effect on the piezoelectric coupling.

An asymptotic theory for dynamic response of laminated piezoelectric shells

A three-dimensional (3D) asymptotic theory for dynamic analysis of doubly curved laminated piezoelectric shells is formulated on the basis of 3D piezoelectricity. By using the direct elimination, we reduce the twenty-two basic equations of 3D piezoelectricity to eight differential equations in terms of eight primary variables of elastic and electric fields. In the formulation, multiple time scales are introduced to eliminate the secular terms so that the asymptotic expansion is uniform and feasible. By means of nondimensionalization, asymptotic expansion and successive integration, we finally can obtain recurrent sets of governing equations for various order problems. The classical laminated piezoelectric shell theory (CST) is derived as a first-order approximation to the 3D piezoelectricity. Higher-order corrections can be determined by considering the solvability and orthonormality conditions in a systematic and consistent way. Several benchmark solutions for various piezoelectric laminates are given to demonstrate the performance of the theory.

Active control of a piezoelectric funnel-shaped structure based on subspace identification

This paper presents the results of active control design and its experimental implementation for the vibration suppression of a piezoelectric funnel-shaped shell structure. The controller design is a model-based one: optimal LQ control modified by the introduction of additional dynamics in order to face the periodic sinusoidal excitations/disturbances. Design of the controller, in terms of determining the controller parameters as well as the parameters of the Kalman filter used for the estimation of the unmeasurable state variables, is based on the state space model obtained through a subspace-based identification approach. The results of the single-input single-output (SISO) and the multiple-input multiple-output (MIMO) control are presented in the time domain, together with the frequency response functions obtained from the measurements and based on the identified models for selected actuator/sensor pairs. Implementation of the active control results in suppression of the vibration magnitudes. Copyright © 2005 John Wiley & Sons, Ltd.

General micromechanical modeling of smart composite shells with application to smart honeycomb sandwich structures

Analytical solutions for the static deformations of piezoelectric composite shells with a periodic structure are developed and effective elastic and piezoelectric coefficients are determined. The derived model also allows the determination of the displacement, strain and stress fields. The underlying mathematical framework is that of asymptotic homogenization. It is shown that the original boundary value problem decouples into a set of four simpler problems called local or unit cell problems. It is precisely these unit cell problems that determine the effective elastic and piezoelectric coefficients. These coefficients are universal in nature and can be used to study a wide variety of boundary value problems. The model is applied to the practically important case of a hexagonal honeycomb-type sandwich structure composed of an isotropic core and orthotropic carrier faces that also exhibit piezoelectric behavior.

Coupled mechanics and finite element for non‐linear laminated piezoelectric shallow shells undergoing large displacements and rotations

AbstractA theoretical framework is presented for analysing the coupled non‐linear response of shallow doubly curved adaptive laminated piezoelectric shells undergoing large displacements and rotations. The formulated mechanics incorporate coupling between in‐plane and flexural stiffness terms due to geometric curvature, coupling between mechanical and electric fields, and encompass geometric non‐linearity effects due to large displacements and rotations. The governing equations are formulated explicitly in orthogonal curvilinear co‐ordinates and are combined with the kinematic assumptions of a mixed‐field shear‐layerwise shell laminate theory. Based on the above formulation, a finite element methodology together with an incremental‐iterative technique, based on Newton–Raphson method is formulated. An eight‐node coupled non‐linear shell element is also developed. Various evaluation cases on laminated curved beams and cylindrical panels illustrate the capability of the shell finite element to predict the complex non‐linear behaviour of active shell structures including buckling, which is not captured by linear shell models. The numerical results also show the inherent capability of piezoelectric shell structures to actively induce large displacements through piezoelectric actuators, by jumping between multiple equilibrium states. Copyright © 2005 John Wiley & Sons, Ltd.

Transfer Function and the State Space Model Identification of a Funnel Shaped Piezoelectric Structure for the Controller Design Purposes

AbstractFor the controller design purposes in order to suppress vibration magnitudes of a funnel shaped piezoelectric shell structure the mathematical model of the structure was identified in the form of the transfer function and the state space representation. The excitation of the structure with different signal types as well as the measurement of the responses were achieved using the piezoelectric actuator and sensor patches attached to the funnel. From the frequency responses obtained as a ratio of the Fast Fourier Transforms between the appropriate sensor and the actuator signals, discrete‐time transfer functions were determined by best curve fitting of the model‐based frequency response. For the state space model identification the subspace based identification approach was used. Obtained models were used for the optimal LQ controller design. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)