Nonlinear Vibrations of Functionally Graded Graphene Reinforced Composite Cylindrical Shells Using the Multiple Scale Method

In this research, the nonlinear dynamic behavior of a cylindrical shell reinforced with graphene platelets, examining both uniform and non-uniform distributions of the platelets, is explored. The classical shell theory, complemented by von Karman's nonlinear model for strain-displacement relationships, facilitates the derivation of the governing equations of motion. The mechanical properties of the reinforced shell are ascertained through the Halpin-Tsai micromechanical model and the rule of mixtures. For the transition from partial differential equations to nonlinear ordinary differential equations, the Galerkin method is employed. These resultant equations are subsequently addressed using a multiple scale method. Initial validation of the proposed approach is achieved by comparing the results with the existing literature, and by employing the Runge-Kutta numerical method. Following this validation, the study investigates the impact of various parameters associated with the functionally graded material, such as the number of layers, the distribution patterns of the graphene platelets (including uniform distribution, FG-X, FG-O, and FG-V), and the proportion of these platelets in the composition, all of which significantly influence the nonlinear dynamic behavior of the shell as represented by the response curves for the primary resonance frequency.