This paper evaluates the vibrational behavior of the complex types of nanocomposites paired shells with arbitrary support conditions. The paired shell comprises two segments, including a paraboloid and a cylinder. In addition, arbitrary support conditions are incorporated and investigated by thirty-six different cases of boundary conditions. The influence of using agglomerated Graphene Nano-Platelets (GNP) for reinforcing the polymer matrix on the natural frequency of the nanocomposite shells is studied by conducting the Eshelby-Mori-Tanaka scheme as an efficient homogenization procedure. Furthermore, the impact of the GNP accumulation throughout the polymer matrix is also considered in formulation by defining agglomeration factors. To attain the displacement fields of the shell structure, the three-dimensional (3D) shell approach and First-order Shear Deformation Theory (FSDT) are engaged. Afterward, Hamilton\\\'s principle determines the structure\\\'s total strain and kinetic energies. By applying the variational calculus, the governing differential equations of the structural model are available. In this step, a robust and reputable approximated numerical solution method, namely Generalized Differential Quadrature (GDQ), is used to discretize the shell structure by defining several grid points. Finally, the established eigenvalue problem is solved numerically to capture the structure\\\'s natural frequency. The authors validate their formulation by comparing the results obtained by the present procedure and the outputs of the Finite Element Method (FEM). Some more functional, intricate, and new nanocomposite Paired Paraboloidal-Cylindrical Shells (PPCS) are considered to be analyzed for achieving the associated natural frequency.

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