Free vibration analysis of FGM framed nanostructures using variational-consistent boundary conditions

Abstract

This paper analyses free vibrations of framed nanostructures made of Functionally Graded Material (FGM) based on the Nonlocal Elastic Theory (NET) and the Dynamic Stiffness Method (DSM). FGM characteristics vary nonlinearly throughout the height of the beam element. The NET considers the nonlocal parameter that perfectly captured the size effect of nanostructures. However, the NET makes nonlocal paradoxes in the bending and vibration behaviour of framed nanostructures with the free ends. To overcome these phenomena, the nanostructure is modelled according to the Euler–Bernoulli beam theory and the variational-consistent nonlocal boundary conditions have been derived. The exact solutions of differential equations of motion and variational-consistent nonlocal boundary conditions are found using the DSM. The influences of the nonlocal, material, geometry parameters and Pasternak’s foundation on the free vibration are then analyzed. It is shown that the study can be applied to other FGMs as well as more complicated framed structures.