Free vibration of hybrid FG-CNT/GPL spherical–conical shells on localized elastic foundations via GDQ method

Abstract

This study investigates the free vibration characteristics of spherical–conical joined shells reinforced with functionally graded carbon nanotubes (CNTs) and graphene platelets (GPLs), resting on localized elastic foundations. Addressing the limitation of idealized boundary conditions in previous studies, this research simulates practical environments, such as partial substrate contact, by incorporating localized elastic supports. The governing equations are derived using Hamilton’s principle within the framework of the first-order shear deformation theory (FSDT), ensuring the inclusion of shear deformation effects. For numerical solution, the generalized differential quadrature (GDQ) method is employed to discretize the domain efficiently. Comprehensive parametric studies analyze the effects of foundation stiffness, reinforcement distribution patterns, and geometric curvature on the fundamental frequencies. The results demonstrate that localized foundations significantly shift the dynamic response compared to full foundations, while the hybrid CNT/GPL reinforcement offers superior stiffness-to-weight ratios and tunable vibration resistance. This work provides a robust analytical framework for the design of advanced multi-phase composite structures in complex mechanical environments