Vibration analysis of FG nanobeams on the basis of fractional nonlocal model: a variational approach

Abstract

The nonlocal vibrations of Euler–Bernoulli nanobeams are studied in this paper within the framework of fractional calculus. It is assumed that the material properties are functionally graded in the thickness direction and are estimated using the power-law function. Hamilton’s principle is applied to drive the fractional equation of motion which is then solved based on a new numerical approach named as variational finite difference method (VFDM). VFDM is formulated by the finite difference method (FDM) and matrix differential/integral operators. Since the method is directly applied to the variational form of governing equation, it is advantageous over existing approaches used for the fractional nonlocal models. The effects of nonlocality, fractional parameters and gradient of material on the fundamental frequencies of nanobeams subject to fully clamped, fully simply supported and clamped-simply supported boundary conditions are analyzed through illustrative examples.