In this research, in order to investigate the effect of the piezoelectric patch which is used as a sensor or actuator in rotating flexible structures such as a helicopter blade, the free vibrations of the rotating rectangular sheet with and without the piezoelectric patch have been presented. First-order shear deformation theory is considered for plate displacement and piezoelectric field. Considering the effect of Coriolis acceleration, centrifugal acceleration and centrifugal in-plane forces, the equations of motion are derived from Hamilton's principle and the electromechanical couple equation is obtained from Maxwell's equation. For piezoelectric, two electrical conditions, open circuit and closed circuit, which are used in sensors and actuators, respectively, have been considered. The equations are discretized with the help of the numerical method of generalized differential squares and the matrices of inertia mass, eccentricity, Coriolis and stiffness matrix are obtained. Natural frequency values for beam and rotating plate have been compared in Abaqus software. Also, the values obtained from the numerical solution in MATLAB have been verified with articles and ABAQUS, which have high accuracy. The effect of parameters such as hub radius, rotation speed, sheet thickness, aspect ratio, piezoelectric patch thickness and applied voltage on the natural frequency of the system has also been investigated.

Based on the desirable properties of smart materials, many studies on the application of these materials in the field of vibration control, damage detection, shape control and buckling control have been undertaken. For the design and analysis of smart beams, plates or shells, several considerations, such as the type and number of sensors and actuators and their locations, the method of modelling, and different analyses and the types of control system, are considered.In this paper, vibration control of a laminated composite plate, with piezoelectric actuators and sensors, is investigated.Governing equations of the laminated composite plate were derived using piezoelectric constitutive equations and Hamilton' s principle. For modelling of the laminated plate, one of the numerical mesh-less methods; Element Free Galerkin (EFG), is used. In such methods, for discretization of a continuous field, a scattered set of particles instead of elements are distributed in the field, and the method of moving least squares (MLS) is used to construct the shape functions. In order to model the displacement and strain field of the laminated plate, the first-order shear deformation theory (FSDT) is used to consider its shear deformation effects.To verify the results, the natural frequency of isotropic and orthotropic plates with different boundary conditions was calculated and compared with other studies.The accuracy of the piezoelectric modelling is verified by calculating the displacement in different points of a bimorph piezoelectric beam in response to applying the electrical potential. After verification of the proposed method, the Newmark method is used to solve the dynamic equations of the laminated composite plate. For active vibration control of the laminated plate, a feedback control algorithm is used. The model is applied for the solution of two illustrative cases and the results are presented and discussed.

Analytical investigation of the vibration behavior of thin circular functionally graded (FG) plates integrated with two uniformly distributed piezoelectric actuator layers based on the classical plate theory (CPT) is presented. The material properties of the FG substrate plate are assumed to be graded in the thickness direction according to the power-law distribution. The differential equations of motion are solved, analytically for clamped edge boundary condition of the plate. The detailed mathematical derivations are presented and numerical investigations are performed while the emphasis is placed on investigating the effect of varying the gradient index of FG plate on the vibration characteristics of the structure.

A plate periodically bonded with piezoelectric patches on its surfaces is considered. The differential quadrature element method is used to solve the wave motion equation for the two-dimensional periodic structure. The method is very simple and easy to implement. Based on the method, band structures for in-plane wave propagating in the periodic piezoelectric plate are studied, from which the frequency band gap is then obtained. Parametric studies are also performed to highlight geometrical and physical parameters on the band gaps. It is found that the thickness of the piezoelectric patches have no effect on the upper bound frequency of the band gap. Physical mechanism is explained for the phenomena. Dynamic simulations are finally conducted to show how the band gap works for a finite quasi-periodic plate. Numerical results show that the vibration in periodic plates can be dramatically attenuated when the exciting frequency falls into the band gap.

In this paper, an analytical solution for bending analysis of functionally-graded piezoelectric plate with two simply-supported parallel edges and two other arbitrary boundary conditions under uniformly-distributed transverse loading is presented. The five-variable refined plate theory is employed for describing the displacement field. This theory, despite the few numbers of unknown variables, predicts a parabolic distribution for transverse shear stresses across the thickness and also considers the thickness-stretching effect. The governing equations are obtained using Hamilton’s principle and Maxwell's equation. The Levy-type solution in conjunction with the state-space approach is used to solve them. Comparing the results with those obtained by the higher-order shear theories and Abaqus finite element simulation confirms the accuracy and efficiency of the proposed method. It can be seen that for the length-to-thickness ratio of 10 and the power-law index of 0.5, the value of non-dimensional deflection of the plate with the clamped boundary condition is 0.3327, which has the largest amount of stiffness, while the value of the non-dimensional deflection of the plate with two parallel free boundary condition edges having the lowest amount of stiffness is 2.2036. In addition, for the plate with a clamped boundary condition and length-to-thickness ratio of 10, with the increase of the power index from 0.5 to 10, the value of displacement changes from 0.3327 to 0.3545, which means an increase of about 6%.

Active control vibrations of an arbitrary point of a simply supported rectangular plate is presented in this paper. The governing differential equation of plate with attached piezoelectric sensor/actuator patches, whose effects of mass and stiffness are included by the local functions, is solved at first. The response of the plate is computed using the first nine natural modes. Then, piezoelectric patches are attached about the point based on a special arrangement and possibility of vibration reduction of the local area of the plate without reducing the global vibration of the plate is discussed. Numerical examples with four and nine patches are discussed to contrast the present method. The results show the superiority of the present approach.