In this paper, free vibration analysis of sandwich composite joined conical-cylindrical-conical shells is implemented using First-order Shear Deformation Theory (FSDT). The sandwich structure is made of three layers including two face sheets and one core. These layers are reinforced with functionally graded carbon nanotubes (FG-CNTs) and functionally graded graphene nanoplatelets (FG-GNPs). Eight patterns are considered for distribution of CNTs and GNPs throughout the thickness of layers. To obtain the equivalent mechanical properties of each layer, the well-known rule of mixture is utilized. The coefficients of material matrix is calculated using the Equivalent Single Layer (ESL) theory. Moreover, the Donnell\\\'s approach is employed for obtaining the equations of motion associated to the conical and cylindrical shells. The principle of Hamilton is used for achieve the governing system of partial differential equation (PDE) of joined shells. Afterwards, this system of PDE is solved by using an efficient semi-analytical solution method, namely Generalized Differential Quadrature Method (GDQM). Different cases of boundary conditions are investigated in this study. By comparing the obtained results with the reference solutions, the correctness and efficiency of the proposed formulation are proved. Furthermore, several new, complicated and applicable examples are considered to study the effect of different parameters of geometric, material and boundary conditions on the natural frequency of the joined shells as well.

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