Geometry-adaptive nonlinear free vibration modeling of FG-CNTRC plates: a VDQ-transformed TSDT framework

Abstract

This study presents a comprehensive nonlinear free vibration analysis of functionally graded carbon-nanotube-reinforced composite (FG-CNTRC) plates of arbitrary geometries. The proposed numerical formulation integrates the third-order shear deformation theory (TSDT) with a variational differential quadrature-transformed (VDQ-T) technique, enabling precise modeling of complex plate shapes such as rectangular, skew, quadrilateral, and annular sectors without re-derivation of the governing equations for each geometry. The governing equations of motion are obtained from Hamilton’s principle, incorporating geometric nonlinearity of the von Kármán type. The VDQ-transformation facilitates direct discretization of the system energy functionals and leads to a compact matrix formulation suited for high-accuracy vibration calculations. Validation against published results demonstrates excellent agreement of the developed approach. A detailed parametric investigation reveals the pronounced influence of CNT dispersion pattern, plate aspect ratio, and boundary conditions on the nonlinear vibration characteristics. The amplitude-dependent natural frequencies highlight strong coupling between material gradation and geometric nonlinearity. The proposed formulation provides a unified computational framework for nonlinear free vibration analysis of FG-CNTRC plates with various geometrical configurations.