Nonlinear free vibration modeling of anisogrid lattice sandwich plates based on a weak formulation analysis

Abstract

This research examines the geometrically nonlinear free vibrations of a type of sandwich plate. This plate consists of an anisogrid lattice core and two nanocomposite face sheets. The lattice core is made up of a set of straight and oblique ribs with rectangular cross-sections that meet at nodes that are not all in the same place (anisogrid nodes). The nanocomposite skins are reinforced with carbon nanotubes (CNTs) based on the various functionally graded models. The displacement field of the sandwich plate is estimated based on the third-order shear deformation plate theory. Also, the nonlinear von-Kármán strain–displacement relations are used. Based on the Hooke and Hamilton laws, the motion equations of the system for nonlinear free vibrations are derived in a weak form. Then, the nodal weak formulation method based on the Lagrange interpolation functions and Gauss–Lobatto–Chebyshev node distribution, for the first time, is employed to calculate the time-dependent equations. The residuals of the obtained equations are minimized by the Galerkin method. Next, a displacement control iterative technique is applied to compute the nonlinear frequency of the sandwich plate. The effect of anisogrid lattice core characteristics and nanocomposite face sheets properties on the linear and nonlinear frequencies and also mode shapes are examined in the results.