The free vibration analysis of a cantilever trapezoidal plate with non-uniform thickness reinforced with non-uniformly distributed nanofillers

Abstract

The presented work is devoted to studying the dynamics of cantilever trapezoidal plates with non-uniform thickness and enriched with either carbon nanotubes (CNTs) or graphene nanoplatelets (GNPs). It is assumed that the volume fractions of nanofillers (CNTs or GNPs) and thickness of the plate vary in one direction from the clamped edge of the plate to the outer free edge. The mathematical modeling of the plate is performed according to the first-order shear deformation theory (FSDT), and the density and the effective values of elastic constants of the plate are calculated utilizing the rule of mixture (ROM), the Halpin-Tsai model, and the Eshelby-Mori-Tanaka scheme. Hamilton’s principle is employed to derive the governing equations and boundary conditions at a clamped edge and three free edges of the plate. An approximate solution is provided via the differential quadrature method (DQM) to estimate the natural frequencies of the plate and achieve the corresponding mode shapes. It is concluded that to attain higher natural frequencies in the low vibrational modes, it is more effective to distribute most of the nanofillers as close as to the clamped edge and reduce their volume fraction from the clamped edge to the outer free edge.