Keywords: actuators, forced vibration, laminated composite plates, piezoelectric materials, sensors.

A numerical method is presented in this paper that investigates the linear and geometrically non-linear transient vibrations of laminated composite piezoelectric plates subject to mechanical or electrical loading. A displacement based p-type method, which is a generalized form of finite element method (FEM) is formulated where the geometry is represented by a very coarse mesh and the displacement fields are defined by polynomials. Convergence of the results is achieved by increasing the degrees of the displacement polynomials, while the geometry is kept the same and defined by relatively low order polynomials depending upon the complexity of the shape under consideration. Equations are based on the Reissner-Mindlin plate theory with five mechanical degrees of freedom per node, three translational and two rotational components. The non-linearity is retained with the in-plane strain components only and the transverse shear strains are kept linear. A layer-wise approximation is used for the electric potentials across the thickness of the piezoelectric layers. Each piezoelectric layer is divided into sub-layers and the potential is linearly interpolated for each sub-layer in the thickness direction. The in-plane potentials at the top and bottom of each piezoelectric sub-layer is defined by the same shape functions used for the displacement fields. The equation of motion is obtained by Hamilton's principle wherein the potential energy functional contains strain and kinetic energies and energy from the piezoelectric part. Then it is solved by Newmark's method. A Newton-Raphson iterative technique is applied simultaneously at each time step to obtain the converged solution. Plate problems presented in this study are analyzed by using only single domain, which does not require any meshing as required in the FEM. A single domain solution represents a true continuous system; it drastically reduces the sizes of the matrices involved in the solution; and yields fast and improved convergence of the results.

For the validation of the present numerical approach, results from the present method are successfully compared with those from the published papers by other researchers. Additional results include the linear and non-linear forced vibrations of sandwich plates with piezoelectric facings and laminated composite material core. The plate responses are studied first under mechanical load and then electric load. They are presented and discussed in the form of the displacement and electric potential time histories. The variation of electric potential along the thickness direction of the plate is also presented. The non-linearity due to large deformations is seen to produce stiffening effects, reduces the amplitude of vibrations and increases frequency. The authors believe that the method used in this study is efficient and accurate and can very effectively solve complicated geometry with only a single domain.

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