A solution method based on Lagrange multipliers and Legendre polynomial series for free vibration analysis of laminated plates sandwiched by two MFC layers

In this paper, a new solution approach based on Lagrange multipliers is developed to investigate the free vibrations of rectangular composite plates equipped with piezoelectric layers such as Macro Fiber Composites (MFCs). The piezo-composite plate has general stacking sequences and is subjected to the elastic edge restraints. Using the first-order shear deformation theory (FSDT) and Hamilton's principle, the boundary conditions of the problem are deduced. To solve the problem, the generalized displacements and electric potentials are expanded using the Legendre polynomial series as the base functions. Afterward, the strain and kinetic energies of the problem are achieved. Then, all the boundary conditions have been added to the energy expression by means of Lagrange multipliers to form the functional. This functional is extremised to give the natural frequencies and mode shapes of the problem through the generalized eigenvalue problem. Capability and credibility of the proposed approach are confirmed by comparing the results with those achieved by the experimental, finite element method (FEM) and 3D elasticity. The method finally is used to investigate the effects of MFCs orientations on the vibrational behavior of the smart plates.