This paper is devoted to obtain the vibrational behavior of doubly curved shells with paraboloidal and hyperboloidal geometries using an efficient semi-analytical solution method. To obtain the governing differential equations of the shells, the First-order Shear Deformation Theory (FSDT) is used. In addition, the shell structure is composed of a new hybrid three phases nanocomposite material. The material includes three parts: (1) polymer matrix, (2) carbon macroscale fiber, and (3) Graphene NanoPlatelets (GNP) nanoscale filler. Three different multiscale approaches including (1) bridging model, (2) Mori-Tanaka scheme, and (3) generalized self-consistent model are employed for homogenization of hybrid matrix and macroscale carbon fibers. The governing equation of motions are obtained using Hamilton\\\'s principles and Green-Gauss theory. Afterwards, an efficient semi-analytical solution procedure entitled Generalized Differential Quadrature Method (GDQM) is used for solving the governing differential equations of the structure. Finally, the frequency parameter of the structure is achieved for different states of boundary conditions and geometric properties. Moreover, the results are verified with the responses of other references obtained for a paraboloidal (Cap) shell structure. It is worth mentioning that the effect of homogenization approach, boundary conditions, volume fraction of carbon fibers and distribution pattern of GNP within the polymer matrix are studied numerically. It is concluded that the solution method is accurate, efficient and capable to obtain the natural frequency of doubly curved shells.

Keywords