Nonlinear Forced Vibration of Functionally Graded Magneto-Electro-Elastic Rectangular Plate based on the Third Order Shear Deformation Theory

In this paper, nonlinear vibration of the functionally graded rectangular plates made of piezoelectric BaTiO3 and magnetostrictive CoFe2O4, whit simply supported boundary condition has been investigated. It is assumed that the composition is varied from the bottom surfaces to top surface, i.e., the top surface of the plate is piezoelectric-rich, whereas the bottom surface is magnetostrictive-rich. In addition, material properties are graded along the thickness according to volume fraction power-law distribution. Based on the Reddy’s third-order share deformation plate theory, the governing equations of motion, whereas Maxwell equations for electrostatics and magnetostatics are used to model the electric and magnetic behavior. 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 which include nonlinear inertia and stiffness terms with quadratic and cubic nonlinear terms. A perturbation method is used to solve the equation of motion analytically. The results for natural frequency are compared with the available results for isotropic, laminated and piezo-laminated plates and good agreement is found between the results of present study with the results of previously published papers. In the forced vibration, primary, super-harmonic resonances are studied and the frequency response equation has been obtained. Because of the importance of the primary resonance, the stability of the steady-state motion is investigated for the primary resonance, The applied external force is assumed to be harmonic in time with a constant amplitude.