Finite element piezothermoelasticity analysis and the active control of FGM plates with integrated piezoelectric sensors and actuators

 An efficient finite element model is presented for the static and dynamic piezothermoelastic analysis and control of FGM plates under temperature gradient environments using integrated piezoelectric sensor/actuator layers. The properties of an FGM plate are functionally graded in the thickness direction according to a volume fraction power law distribution. A constant displacement-cum-velocity feedback control algorithm that couples the direct and inverse piezoelectric effects is applied to provide active feedback control of the integrated FGM plate in a closed loop system. Numerical results for the static and dynamic control are presented for the FGM plate, which consists of zirconia and aluminum. The effects of the constituent volume fractions and the influence of feedback control gain on the static and dynamic responses of the FGM plates are examined. Received: 13 March 2002 / Accepted: 5 March 2003 The work described in this paper was supported by a grant awarded by the Research Grants Council of the Hong Kong Special Administrative Region, China (Project No. CityU 1024/01E).     

Optimal location of piezoelectric actuators for active vibration control of thin axially functionally graded beams

Up to now, optimal location for active control studies concern principally multilayers or homogeneous structures. In the case of functionally graded materials, very few papers exist and they only concern cross section variations. In this way, this paper deals with the optimization of piezoelectric actuators locations on axially functionally graded beams for active vibration control. For this kind of structures, the free vibration problem is more complicated as the governing equations have variable coefficients. Here, the eigenproblem is solved using Fredholm integral equations. The optimal locations of actuators are determined using an optimization criterion, ensuring good controllability of each eigenmode of the structure. The linear quadratic regulator, including a state observer, is used for active control simulations. Two numerical examples are presented for two kinds of boundary conditions.     

Finite element modelling and vibration control study of active plate with debonded piezoelectric actuators

A finite element model for a piezoelectric plate with edge debonded actuators is presented. This model is employed to investigate the effect of edge debonding on actuation authority, natural frequencies and vibration control performance. The regions of the plate with the piezoelectric patches are modelled such that each layer undergoes rotation due to shear deformation independently. The necessary constraints for continuity of displacements at the interfaces of the layers are imposed. The plate with edge debonded actuators is idealized by dividing it into debonded regions and healthy regions. A finite element procedure for imposing the constraints regarding continuity of displacements at the interfaces of the adjacent regions is developed and is implemented using MATLAB. Experiments are conducted for finding the actuation authority and natural frequencies of the plate with debonded actuators. It has been found that the developed model has predicted the mechanics of actuator debonding properly. The investigations have revealed the fact that the edge debonding of actuators will result in considerable degradation in actuation authority and vibration control performance.     

Finite element analysis on deflection control of plates with piezoelectric actuators

A finite element model for the deflection control of plates with piezoelectric actuators is presented. This model contains an actuator element, an adhesive interface element and an eight-node isoparametric plate element. The actuator element developed here is based on first-order shear deformation theory. An analytical solution is also derived in comparison with results using the finite element model. The analyses articulate separate response of the plate; actuators and the adhesive give the flexible meshing advantage of solving the- smart structure problem with any type of boundary conditions and geometry configuration.     

Response and control of functionally graded laminated piezoelectric shells under thermal shock and moving loadings

Based on the first-order shear deformation theory (FSDT), approximate solution for FG (functionally graded) laminated piezoelectric cylindrical shells under thermal shock and moving mechanical loads is given utilizing Hamilton’s principle. The thin piezoelectric layers embedded on inner and outer surfaces of the functionally graded layer are acted as distributed sensor and actuator to control dynamic characteristics of the FG laminated cylindrical shells. Here, the modal analysis technique and Newmark’s integration method are used to calculate the dynamic response of FG laminated cylindrical shells. Constant-gain negative velocity feedback approach is used for active vibration control. The active vibration control to a single moving concentrated loading, thermal shock loading and a continuous stream of moving concentrated loadings is, respectively, investigated. Results indicate that the control gain and velocity of moving loadings have significant effects on the dynamic response and resonance of the system.     

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 functionally graded laminated cylindrical shells

An analytical method on active vibration control of smart FG laminated cylindrical shells with thin piezoelectric layers is presented based on Hamilton’s principle. The thin piezoelectric layers embedded on inner and outer surfaces of the smart FG laminated cylindrical shell act as distributed sensor and actuator, which are used to control vibration of the smart FG laminated cylindrical shell under thermal and mechanical loads. Here, the modal analysis technique and Newmark’s integration method are used to calculate the dynamic response of the smart FG laminated cylindrical shell with thin piezoelectric layers. Constant-gain negative velocity feedback approach is used for active vibration control with the structures subjected to impact, step and harmonic excitations. The influences of different piezoelectric materials (PZT-4, BaTiO₃ and PZT-5A) and various loading forms on the active vibration control are described in the numerical results.     

Variational asymptotic modeling of the multilayer functionally graded cylindrical shells

An efficient high-fidelity shell model is developed for heterogeneous multilayer cylindrical shells made of functionally graded material by using the variational asymptotic method (VAM). Taking advantage of the smallness parameters inherent in the shell structure, the VAM is applied to rigorously decouple the 3-D, anisotropic elasticity problem into a 1-D through-the-thickness analysis and a 2-D shell analysis. The through-the-thickness analysis servers as a link between the original 3-D analysis and the shell analysis by providing a constitutive model for the shell analysis and recovering the 3-D field variables in terms of global responses calculated by the shell analysis. The present model is valid for large displacements and global rotations and can capture all the geometric nonlinearity of a shell when the strains are small. A cylindrical bending example of a homogeneous substrate with a thin SiC-Al functionally graded coating under sinusoidal pressure on the top surface is used to validate this model.     

Time-harmonic crack problems in functionally graded piezoelectric solids via BIEM

In-plane crack analysis of functionally graded piezoelectric solids under time-harmonic loading is performed by using a non-hypersingular traction based boundary integral equation method (BIEM). The material parameters are assumed to vary quadratically with both spatial variables. A frequency dependent fundamental solution, as well as its derivatives and asymptotic expressions, is derived in closed-form by using an appropriate algebraic transformation for the displacement vector and the Radon transform. Numerical results for the stress intensity factors (SIFs) are discussed for different examples. The accuracy of the presented method is checked by comparison with available results from the literature. Investigated are the effects of the inhomogeneity parameters, the frequency of the applied electromechanical load and the geometry of the crack scenario on the K-factors.     

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.     

Cylindrical bending vibration of functionally graded piezoelectric shells using the method of perturbation

Based on the three-dimensional (3D) piezoelectricity, two asymptotic formulations for the cylindrical bending vibration of simply supported, functionally graded (FG) piezoelectric cylindrical shells with open-circuit and closed-circuit surface conditions are presented. The normal electric displacement and electric potential are prescribed to be zero on the lateral surfaces. In the present asymptotic formulations the material properties are regarded to be heterogeneous through the thickness coordinate. Afterwards, they are further specified to be constant in single-layer shells, to be layerwise constant in multilayered shells and to obey an identical exponent-law distribution in FG shells. The method of multiple time scales is used to eliminate the secular terms arising from the regular asymptotic expansion. The orthonormality and solvability conditions for various orders are derived. The recursive property among the motion equations of various order problems is shown. The present asymptotic formulations are applied to several illustrative examples. The accuracy and the rate of convergence of the present asymptotic solutions are evaluated. The coupled electro–elastic effect and the influence of the material-property gradient index on the free-vibration behavior of FG piezoelectric shells are studied.     

Time and frequency domain analysis of piezoelectric energy harvesters by monolithic finite element modeling

The successful design of piezoelectric energy harvesting devices relies upon the identification of optimal geometrical and material configurations to maximize the power output for a specific band of excitation frequencies. Extendable predictive models and associated approximate solution methods are essential for analysis of a wide variety of future advanced energy harvesting devices involving more complex geometries and material distributions. Based on a holistic continuum mechanics modeling approach to the multi‐physics energy harvesting problem, this article proposes a monolithic numerical solution scheme using a mixed‐hybrid 3‐dimensional finite element formulation of the coupled governing equations for analysis in time and frequency domain. The weak form of the electromechanical/circuit system uses velocities and potential rate within the piezoelectric structure, free boundary charge on the electrodes, and potential at the level of the generic electric circuit as global degrees of freedom. The approximation of stress and dielectric displacement follows the work by Pian, Sze, and Pan. Results obtained with the proposed model are compared with analytical results for the reduced‐order model of a cantilevered bimorph harvester with tip mass reported in the literature. The flexibility of the method is demonstrated by studying the influence of partial electrode coverage on the generated power output.     

Piezoelectric actuators and sensors location for active control offlexible structures

A method for optimal positioning of piezoelectric actuators and sensors on a flexible structure is presented. First, a two-dimensional (2-D) model of a piezoelectric actuator bonded to a plate is obtained. Then, a Ritz formulation is used to find a state model of the system in view of its control. To define an optimal positioning strategy, an energy based approach is developed. This leads quite naturally to the study of controllability and observability properties of the overall dynamical model. A new criterion based on energy assessment is proposed to locate actuators and sensors     

Solid freeform fabrication of piezoelectric sensors and actuators

The last two decades have witnessed the proliferation piezoelectric composite transducers for an array of sensor and actuator applications. In this article, a concise summary of the major methods used in composite making, with special emphasis on Solid Freeform Fabrication (SFF), is provided. Fused Deposition of Ceramics (FDC) and Sanders Prototyping (SP) are two SFF techniques that have been utilized to make a variety of novel piezocomposites with connectivity patterns including (1-3), (3-2), (3-1), (2-2) and (3-3). The FDC technique has also been used to prototype a number of actuators such as tube arrays, spiral, oval, telescoping, and monomorph multi-material bending actuators. It has been demonstrated that SFF technology is a viable option for fabricating piezocomposite sensors and actuators with intricate geometry, unorthodox internal architecture, and complex symmetry. The salient aspects of processing of such composite sensors and actuators are summarized, and structure-processing-property relations are elaborated on.     

压电集成碳纳米管CNT增强功能梯度结构非线性建模与仿真

将碳纳米管(carbon nanotube, CNT)以梯度形式分布与基体材料结合,形成功能梯度(functionally graded,FG)结构.为了实现FG-CNT增强复合板在发生大变形时的准确计算,考虑四种典型的CNT分布形式,均匀分布、V型分布、O型分布和X型分布,建立基于Reissner-Mindlin板壳...     

Application of triangular element invariants for geometrically nonlinear analysis of functionally graded shells

This paper is an attempt to construct a computationally effective curved triangular finite element for geometrically nonlinear analysis of elastic shear deformable shells fabricated from functionally graded materials. The focus is on the concise finite-element formulation under the demand of accuracy-simplicity trade-off. To this end, a nonconventional approach based on the invariants of the natural strains of fibers parallel to the element edges is used. The approach allows one to obtain algorithmic formulas for computing the stiffness matrix, gradient, and Hessian of the total strain energy of the finite element. Transverse shear deformation effects are taken into account using the first order shear deformation theory with the shear correction factor dependent on the material property distribution across the shell thickness. The performance of the proposed finite element is demonstrated using problems of functionally graded plates and shells under mechanical and thermal loads.     

Network formalism for modeling functionally gradient piezoelectric plates and stacks and simulations of RAINBOW ceramic actuators

A simple network representation is given for a stack of thin, homogeneous piezoelectric plates, executing a single thickness mode of motion. All plates may differ in thickness and material properties, including dielectric loss, ohmic conductivity, and viscous loss. Each plate is driven by a thickness-directed electric field, and all stack elements are connected electrically in series. Functionally gradient single plates and composites are readily modeled by the network, to a desired precision, using a sequence of circuit elements representing stepwise variations in material properties and layer thicknesses. Simulations of RAINBOW (reduced and internally biased oxide wafer) ceramics are given     

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.     

Finite element evaluation of mixed mode stress intensity factors in functionally graded materials

This paper is directed towards finite element computation of fracture parameters in functionally graded material (FGM) assemblages of arbitrary geometry with stationary cracks. Graded finite elements are developed where the elastic moduli are smooth functions of spatial co‐ordinates which are integrated into the element stiffness matrix. In particular, stress intensity factors for mode I and mixed‐mode two‐dimensional problems are evaluated and compared through three different approaches tailored for FGMs: path‐independent J^*_k‐integral, modified crack‐closure integral method, and displacement correlation technique. The accuracy of these methods is discussed based on comparison with available theoretical, experimental or numerical solutions. Copyright © 2001 John Wiley & Sons, Ltd.