Smart Materials Research

Volume 2015, Article ID 748459, 15 pages

http://dx.doi.org/10.1155/2015/748459

## Thermal Effects on Vibration and Control of Piezocomposite Kirchhoff Plate Modeled by Finite Elements Method

^{1}Equipe Sciences et Technologies Avancées, Ecole Nationale des Sciences Appliquées, Université Abdelmalek Essaadi, 93030 Tétouan, Morocco^{2}Laboratoire d’Etude des Matériaux Avancés et Applications, Faculté des Sciences et Ecole Supérieure de Technologie, Université Moulay Ismail, 50040 Meknès, Morocco

Received 5 December 2014; Revised 2 April 2015; Accepted 3 April 2015

Academic Editor: Weihua Li

Copyright © 2015 M. Sanbi et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Theoretical and numerical results of the modeling of a smart plate are presented for optimal active vibration control. The smart plate consists of a rectangular aluminum piezocomposite plate modeled in cantilever configuration with surface bonded thermopiezoelectric patches. The patches are symmetrically bonded on top and bottom surfaces. A generic thermopiezoelastic theory for piezocomposite plate is derived, using linear thermopiezoelastic theory and Kirchhoff assumptions. Finite element equations for the thermopiezoelastic medium are obtained by using the linear constitutive equations in Hamilton’s principle together with the finite element approximations. The structure is modelled analytically and then numerically and the results of simulations are presented in order to visualize the states of their dynamics and the state of control. The optimal control LQG-Kalman filter is applied. By using this model, the study first gives the influences of the actuator/sensor pair placement and size on the response of the smart plate. Second, the effects of thermoelastic and pyroelectric couplings on the dynamics of the structure and on the control procedure are studied and discussed. It is shown that the effectiveness of the control is not affected by the applied thermal gradient and can be applied with or without this gradient at any time of plate vibrations.

#### 1. Introduction

In the piezoelectric sensors applications, mechanically or thermally induced deformations can be determined from measurement of the induced electrical potential, whereas in piezoelectric actuator applications deformation of strain can be controlled through the introduction of appropriate electric potential. By integrating distributed piezoelectric sensors/actuators and advanced composites, the potential exists for forming high-strength, high stiffness, lightweight structures capable of self-monitoring and self-controlling. Typical applications of such structures are envisioned in the thermal distortion management of propulsion components and space structures. Before they can be utilized in these applications, the performance of piezoelectric structures in thermal environment must be quantified. Thermal effects become important when the piezoelectric structure has to operate in either extremely hot or cold temperature environments. These extreme conditions may severely affect the response of piezoelectric elements by induction of thermal stresses resulting from thermoelastic and pyroelectric coefficients.

Recently, there have been various mathematical models developed to describe the behavior of the piezocomposite or laminated plates actuated and sensed by piezoelectric materials. Görnandt and Gabbert [1] have analysed the finite element method of thermopiezoelectric smart structures. Vel and Batra [2] presented a generalized plane strain thermopiezoelectric analysis of multilayered plates. de Abreu et al. [3] have implemented finite element modeling of a plate with localized piezoelectric sensors and actuators. Deü and Benjeddou [4] have studied free-vibration analysis of laminated plates with embedded shear-mode piezoceramic layers. Tanaka and Sanada [5] have detailed modal control of a rectangular plate using smart sensors and smart actuators. Trindade and Benjeddou [6] have discussed the critical evaluation and optimization of the effective electromechanical coupling coefficients of piezoelectric adaptive structures. Sanbi et al. [7] have analyzed thermoelastic and pyroelectric couplings effects on dynamics and active control of piezolaminated beam. A general solution for piezothermoelasticity of transversely isotropic piezoelectric materials and its applications have been deduced by Ding et al. [8].

In this work, fundamental equations governing the behavior of smart structures are at first given, and, based on the work of de Abreu et al. [3], the generalized linear finite element formulation of the problem is described. Thermopiezoelastic characteristics of aluminium rectangular plate and the theory of active structures in sensing and optimal control are established. A generic thermopiezoelastic theory for piezocomposite plate is derived, using linear thermopiezoelasticity and* Kirchhoff* assumptions. Generalized finite element equations for the thermopiezoelasticity are obtained by using the linear constitutive equations in Hamilton’s principle together with the finite element approximations. A* Kirchhoff* four-node rectangular element with one electrical, one temperature, and three mechanical degrees of freedom is considered. The structure consists of a modeling of cantilevered piezocomposite plate with perfectly surface bonded thermopiezoelectric elements. The structure is modelled analytically and then numerically and the results of simulations are presented in order to visualize the states of their dynamics and the state of control. The optimal control* LQG-Kalman* accompanied is applied. The effects of thermoelastic and pyroelectric couplings coefficients on the dynamics of the structure and on the control effectiveness are discussed. We show that the control procedure cannot be perturbed by applying a thermal gradient and the control can be applied at any time during the period of vibration of the plate.

#### 2. Basic Equations and FEM Implementation

The physical and dimensional characteristics of the plate material are given in Table 1. Properties for the two piezoelectric materials used in the simulation can be found in the literature as in [9, 10]. The piezoelectric and elastic properties of the sensor are chosen smaller compared to those of the actuator. This is justified by the fact that the sensor must have high sensitivity to capture the deformation of the structure, while the actuator is required to act as a secondary source on the plate to deaden its vibrations. In order to model the structure by finite element method, we consider an element with four-node quadrilateral flexible element type “plate.” Each node has three mechanical degrees of freedom (dofs), the displacement in -direction and two rotations in - and -direction, one electric dof , and one temperature dof (Figure 1).