Nonlinear free vibration of magneto-electro-elastic rectangular plates

Nonlinear free vibration of symmetric magneto-electro-elastic laminated rectangular plates with simply supported boundary condition is studied for the first time. The first order shear deformation theory considering the von Karman’s nonlinear strains is used to obtain the equations of motion, whereas Maxwell equations for electrostatics and magnetostatics are used to model the electric and magnetic behavior. Closed circuit electro-magnetic boundary condition at top and bottom surfaces of the plate is considered. Then, the nonlinear partial differential equations of motion are transformed into five coupled nonlinear ordinary differential equations by using the Galerkin method. Afterward, the obtained coupled ordinary differential equations are reduced to a single nonlinear differential equation with quadratic and cubic nonlinear terms. A perturbation method is used to solve the equation of motion analytically and a closed-form solution is obtained for the nonlinear frequency ratio. The results for natural frequency and nonlinear frequency ratio are compared with the available results for isotropic, laminated and piezo-laminated, and laminated magneto-electro-elastic plates and good agreement is found between the results of present study with the results of previously published papers. Several numerical examples are carried out to show the effects of different parameters on the nonlinear behavior of these hybrid plates.