Shape control of smart composite plate with non-rectangular piezoelectric actuators

Quasi-static shape control of a smart structure may be achieved through optimizing the applied electric fields, loci, shapes and sizes of piezoelectric actuators attached to the structure. In this paper, a finite element analysis (FEA) software has been developed for analyzing static deformation of smart composite plate structures with non-rectangular shaped PZT patches as actuators. The mechanical deformation of the smart composite plate is modeled using a 3rd order plate theory, while the electric field is simulated based on a layer-wise theory. The finite element formulation is verified by comparing with experimentally measured deformation. Numerical results are obtained for the optimum values of the electric field in the PZT actuators to achieve the desired shape using the linear least square (LLS) method. The numerical results demonstrate the influence of the shapes of actuators.     

A FEM model for the active control of curved FGM shells using piezoelectric sensor/actuator layers

In this paper, a generic finite element formulation is developed for the static and dynamic control of FGM (functionally graded material) shells with piezoelectric sensor and actuator layers. The properties of the FGM shell are graded in the thickness direction according to a volume fraction power‐law distribution. The proposed finite element model is based on variational principle and linear piezoelectricity theory. A constant displacement and velocity feedback control algorithm coupling the direct and inverse piezoelectric effects is applied in a closed‐loop system to provide feedback control of the integrated FGM shell structure. Both static and dynamic control of FGM shells are simulated to demonstrate the effectiveness of the proposed active control scheme within a framework of finite element discretization and piezoelectric integration. Copyright © 2002 John Wiley & Sons, Ltd.     

Feedback control of vibrating plates using piezoelectric patch sensors and actuators

An analysis of the solutions for various feedback control laws applied to vibrating simply supported plates is evaluated. The control is carried out via a piezoelectric patch sensor and patch actuator. By considering an integral equation formulation, which is equivalent to the differential equation formulation, the analytical results are investigated. The conversion is accomplished by introducing an explicit Green’s function. The feedback controls implemented include displacement, velocity, and a combination of these. A numerical comparison of eigenvalues is presented to illustrate the efficacy of the method and to contrast the effects of the controls. The results presented in the study can be used for benchmarking solutions based in numerical or approximation approaches.     

Geometrically non-linear analysis of composite structures with integrated piezoelectric sensors and actuators

This paper deals with the geometrically non-linear analysis of thin plate/shell laminated structures with embedded integrated piezoelectric actuators or sensors layers and/or patches. The motivation for the present developments is the lack of studies in the behavior of adaptive structures using geometrically non-linear models, where only very few published works were found in the open literature.

The model is based on the Kirchhoff classical laminated theory and can be applied to plate and shell adaptive structures with arbitrary shape, general mechanical and electrical loadings.

The finite element model is a non-conforming single layer triangular plate/shell element with 18 degrees of freedom for the generalized displacements and one electrical potential degree of freedom for each piezoelectric layer or patch.

An updated Lagrangian formulation associated to Newton–Raphson technique is used to solve incrementally and iteratively the equilibrium equations.

The model is applied in the solution of four illustrative cases, and the results are compared and discussed with alternative solutions when available.     

Optimal shape control of functionally graded smart plates using genetic algorithms

This paper deals with optimal shape control of functionally graded smart plate containing patches of piezoelectric sensors and actuators. The genetic algorithm (GA) is designed to search for optimal actuator voltage and displacement control gains for the shape control of the functionally graded material (FGM) plates. The work extends the earlier finite element formulations of the two leading authors, so that it can be readily treated using genetic algorithms. Numerical results have been obtained to study the effect of the shape control of the FGM plates under a temperature gradient by optimising (i) the voltage distribution for the open loop control, and (ii) the displacement control gain values for the closed loop feedback control. The effect of the constituent volume fractions of zirconia, through varying the volume fraction exponent n, on the optimal voltages and gain values has also been examined.     

Finite element analysis for geometrically nonlinear deformations of smart functionally graded plates using vertically reinforced 1-3 piezoelectric composite

In this paper, a finite element model has been developed for the geometrically nonlinear static analysis of simply supported functionally graded (FG) plates integrated with a patch of vertically reinforced 1-3 piezoelectric composite material acting as a distributed actuator. The material properties of the functionally graded substrate plate are assumed to be graded only in the thickness direction according to the power-law distribution in terms of the volume fractions of the constituents. The analysis of the electro-elastic coupled problem includes the transverse deformations of the overall plate to utilize the transverse normal actuation by the distributed actuator for counteracting the nonlinear deformations of smart functionally graded plates. The nonlinear governing equations of equilibrium are solved by using direct iteration method with under-relaxation. The numerical illustrations suggest the potential use of the distributed actuator made of vertically reinforced 1-3 piezoelectric composite material for active control of nonlinear deformations of smart functionally graded plates. The effect of variation of piezoelectric fiber orientation in the distributed actuator on its control authority for counteracting the nonlinear deformations of smart functionally graded plates has also been investigated.     

Analysis of laminated composite beams and plates with piezoelectric patches using the element-free Galerkin method

 An efficient meshfree formulation based on the first-order shear deformation theory (FSDT) is presented for the static analysis of laminated composite beams and plates with integrated piezoelectric layers. This meshfree model is constructed based on the element-free Galerkin (EFG) method. The formulation is derived from the variational principle and the piezoelectric stiffness is taken into account in the model. In numerical test problems, bending control of piezoelectric bimorph beams was shown to have the efficiency and accuracy of the present EFG formulation for this class of problems. It is demonstrated that the different boundary conditions and applied actuate voltages affects the shape control of piezolaminated composite beams. The meshfree model is further extended to study the shape control of piezo-laminated composite plates. From the investigation, it is found that actuator patches bonded on high strain regions are significant in deflection control of laminated composite plates. Received: 23 October 2001 / Accepted: 29 July 2002     

Finite element formulation of smart piezoelectric composite plates coupled with acoustic fluid

In the context of noise and vibration reduction by passive piezoelectric devices, this work presents the theoretical formulation and the finite element (FE) implementation of vibroacoustic problems with piezoelectric composite structures connected to electric shunt circuits. The originalities of this work concern (i) the formulation of the electro-mechanical-acoustic coupled system, (ii) the implementation of an accurate and inexpensive laminated composite plate FE with embedded piezoelectric layers connected to resonant shunt circuits, and (iii) the development of an efficient fluid-structure interface element. Various results are presented in order to validate and illustrate the performance of the proposed fully coupled numerical approach.     

Adhesive element modelling and weighted static shape control of composite plates with piezoelectric actuator patches

A novel finite element model is presented for static and dynamic analysis of composite plates integrated with a laminated piezoelectric layer, a host laminated composite plate and an adhesive layer between them. A new adhesive element is developed which includes both peel and shear effects in the adhesive layer based on first‐order shear deformation plate theory. The thin adhesive layer between the piezoelectric layer and the host plate is modelled by assuming that it carries constant shear and peel strains throughout its thickness. In addition, a weighted static shape control scheme for finding the optimal voltage distribution for static shape control is given. By selecting different weighting matrices, a variety of items such as displacements, slopes, curvatures, strains and even generalized forces, can be included in finding the optimal actuating voltage for static shape control. The present model is validated by comparing with those results available in the literature. The numerical results show that the weighted linear least method can give a satisfactory voltage distribution to best match the desired shape. Copyright © 2004 John Wiley & Sons, Ltd.     

Re-analysis procedure for laminated plates using FSDT finite element model

 A new method is proposed for effective analysis of laminated plates incorporating accurate through-the-thickness distribution of displacements, strains and stresses in the finite element formulation. It is a two-step analysis procedure. In the first step, displacements are obtained using a post-processing procedure based on the three-dimensional stress equilibrium equations and the thermoelasticity equations, from the results of FSDT finite element analysis. In the second step, the higher-order through-the-thickness distribution of displacements are reflected on the subsequent finite element analysis. The effectiveness of the present approach for the analysis of laminated plates is shown by numerical examples. Received: 13 September 2001 / Accepted: 23 May 2002     

Integration of active vibration control methods with finite element models of smart laminated composite structures

Vibration control problems can be directly and systematically solved in a single analysis stage using commercial finite element programs. Integration of control methods into the finite element solutions (ICFES) can be achieved in ANSYS. In this work, first, the direct velocity feedback (DVF) control is tested on a 3-DOF mechanical system under a step input. The simulation results obtained by the ICFES are compared with the analytical results obtained by the Laplace transform method. Then, active control of free and forced vibrations in a smart laminated composite structure (SLCS) with two different lay-ups is studied numerically and experimentally. The SLCS consists of a symmetric laminated glass–epoxy composite beam with [0/90]_s and [45/−45]_s lay-ups and a piezoelectric actuator. For the vibration suppression, the DVF control tested on a mechanical system is applied to the SLCS. In addition, displacement feedback (DF) control is studied. Experiments are conducted to verify the natural frequencies and the closed loop time responses. Analytical results for the mechanical system and experimental results for the SLCS match well to the corresponding results obtained using the ICFES technique.     

Analysis of active damping in composite laminate cylindrical shells of revolution with skewed PVDF sensors/actuators

Active damping in a FRP composite cylindrical shell with collocated piezoelectric sensors/actuators is studied. The electrode on the sensors/actuators are spatially shaped to reduce spillover between circumferential modes. A three noded, isoparametric, semianalytical finite element is developed and used to model the cylindrical shell. The element is based on a mixed piezoelectric shell theory which makes a single layer assumption for the displacements and a layerwise assumption for the electric potential. The effects of location of patch of collocated piezoelectric sensors/actuators, percentage length of the shell covered with these patches, fiber angle of the laminae in the composite laminate, stacking sequence of laminae in a laminate and skew angle of the sensor/actuator piezoelectric material, on the system damping for various modes is studied.     

A point interpolation mesh free method for static and frequency analysis of two-dimensional piezoelectric structures

 A mesh free method called point interpolation method (PIM) is presented for static and mode-frequency analysis of two-dimensional piezoelectric structures. In the present method, the problem domain and its boundaries are represented by a set of properly scattered nodes. The displacements and the electric potential of a point are interpolated by the values of nodes in its local support domain using shape functions derived based on a point interpolation scheme. Techniques are discussed to surmount the singularity of the moment matrix. Variational principle together with linear constitutive piezoelectric equations is used to establish a set of system equations for arbitrary-shaped piezoelectric structures. These equations are assembled for all quadrature points and solved for displacements and electric potentials. A polynomial PIM program has been developed in MATLAB with matrix triangularization algorithm (MTA), which automatically performs a proper node enclosure and a proper basis selection. Examples are also presented to demonstrate the accuracy and stability of the present method and their results are compared with the conventional FEM results from ABAQUS as well as the analytical or experimental ones. Received: 6 February 2002 / Accepted: 5 August 2002     

A finite element model for the analysis of 3D axisymmetric laminated shells with piezoelectric sensors and actuators

This paper presents the development of a semi-analytical axisymmetric shell finite element model with piezoelectric layers using the 3D linear elasticity theory. The piezoelectric effect of the material could be used as sensors and/or actuators in way to control shell deformation. In the present 3D axisymmetric model, the equations of motion are expressed by expanding the displacement field using Fourier series in the circumferential direction. Thus, the 3D elasticity equations of motion are reduced to 2D equations involving circumferential harmonics. In the finite element formulation the dependent variables, electric potential and loading are expanded in truncated Fourier series. Special emphasis is given to the coupling between symmetric and anti-symmetric terms for laminated materials with piezoelectric rings. Numerical results obtained with the present model are found to be in good agreement with other finite element solutions.     

Active control of FGM shells subjected to a temperature gradient via piezoelectric sensor/actuator patches

A generic static and dynamic finite element formulation is derived for the modelling and control of piezoelectric shell laminates under coupled displacement, temperature and electric potential fields. The base shell is of functionally graded material (FGM) type, which consists of combined ceramic–metal materials with different mixing ratios of the ceramic and metal constituents. A multi‐input–multi‐output (MIMO) system is applied to provide active feedback control of the laminated shell using self‐monitoring sensors and self‐controlling actuators through a close loop. Numerical studies clearly show the influence of the positional configurations of sensor/actuator pairs on the effectiveness of static and dynamic control for the shell laminates. The effects of the constituent volume fractions on the static and dynamic responses of the shell laminate are also elucidated. Copyright © 2002 John Wiley & Sons, Ltd.     

Modeling of 3D transversely piezoelectric and elastic bimaterials using the boundary element method

 This paper presents a numerical model for three-dimensional transversely isotropic bimaterials based on the boundary element formulation. The point force solutions expressed in a united-form for distinct eigenvalues are studied for transversely isotropic piezoelectricity and pure elasticity. A boundary integral formulation is implemented for the modeling of two-phase materials. In this study, the stress distributions are computed for a near interface flaw. The influences of the shape and location of the flaw on the the stress concentration are examined. The accuracy of the numerical procedures is validated through selected example problems and comparison studies. Received 3 October 2001 / Accepted 9 April 2002     

Finite element analysis of cracks in piezoelectric materials taking into account the permittivity of the crack medium

In this paper, the influence of the electric boundary conditions on cracks in piezoelectric components shall be studied. Several electric boundary conditions have been proposed in the literature. Here, influence of the permeability of the crack on electric and mechanical fields near the crack tip is considered. Cracks of lower permeability lead to stronger electric singularities. Furthermore, the influence on the stress intensity factors and energy release rate will be discussed. Finally, an experiment with piezoceramic CT specimens, which was performed by Park and Sun, will be evaluated taking into account the permeability of the crack.     

On material forces and finite element discretizations

 The idea of using material forces also termed configurational forces in a computational setting is presented. The theory of material forces is briefly recast in the terms of a non-linear elastic solid. It is shown, how in a computational setting with finite elements (FE) the discrete configurational forces are calculated once the classical field equations are solved. This post-process calculation is performed in a way, which is consistent with the approximation of the classical field equations. Possible physical meanings of this configurational forces are discussed. A purely computational aspect of material forces is pointed out, where material forces act as an indicator to obtain softer discretizations. Received 12 December 2001 / Accepted 18 March 2002     

Finite element analysis of piezoelectric corner configurations and cracks accounting for different electrical permeabilities

Based on eigenfunctions of asymptotic singular electro-elastic fields obtained from a kind of ad hoc finite element method [Chen MC, Zhu JJ, Sze KY. Finite element analysis of piezoelectric elasticity with singular inplane electroelastic fields. Engng Fract Mech 2006;73(7):855-68], a super corner-tip element model is established from the generalized Hellinger-Reissner variational functional and then incorporated into the regular hybrid-stress finite element to determine the coefficients of asymptotic singular electro-elastic fields near a corner-tip. The focus of this paper is not to discuss the well-known behavior of electrically impermeable and permeable (usually it means fully permeable, hereinafter the same) cracks but analyze the limited permeable crack-like corner configurations embedded in the piezoelectric materials, i.e., study the influence of a dielectric medium inside the corner on the singular electro-elastic fields near the corner-tip. The boundary conditions of the impermeable or permeable corner can be considered as simple approximations representing upper and lower bounds for the electrical energy penetrating the corner. Benchmark examples on the piezoelectric crack problems show that present method yields satisfactory results with fewer elements than existing finite element methods do. As application, a piezoelectric corner configuration accounting for the limited permeable boundary condition is investigated, and it is found that the limited permeable assumption is necessary for corners with very small notch angles.