Analytical Solution for Free Vibration of a Variable Cross-section Nonlocal Nanobeam

In this article, small scale effects on free vibration analysis of non-uniform nanobeams is discussed. Small scale effects are modelled after Eringen's nonlocal elasticity theory while the non-uniformity is presented by exponentially varying width among the beams length with constant thickness. Analytical solution is achieved for free vibration with different boundary conditions. It is shown that section variation accompanying small scale effects has a noticeable effect on natural frequencies of non-uniform beams at nano scale. First, five natural frequencies of single-layered graphene nanoribbons (GNRs) with various boundary conditions are obtained for different nonlocal and non-uniform parameters which shows a great sensitivity to non-uniformity in different shape modes.