Linear and nonlinear free vibration of a multilayered magneto-electro-elastic doubly-curved shell on elastic foundation

Nonlinear and linear free vibration of symmetrically laminated magneto-electro-elastic doubly-curved thin shell resting on an elastic foundation is studied analytically. The shell is considered to be simply-supported on all edges and the magneto-electro-elastic body is poled along the z direction and subjected to electric and magnetic potentials between the upper and lower surfaces. To obtain the equations of motion, the Donnell shell theory in the presence of rotary inertia effect is used. Moreover, Gauss' laws for electrostatics and magnetostatics are used to model the electric and magnetic behavior. The nonlinear partial differential equations of motion are reduced to a single nonlinear ordinary differential equation by introducing a force function and using the single-term Galerkin method. The resulting equation is solved analytically by Lindstedt-Poincare perturbation method. After validation of the present study, several numerical studies are done to investigate the effects of foundation parameters, geometrical properties of the shell, and electric and magnetic potentials on the linear and nonlinear behavior of these smart shells.