A Nonlocal Dynamic Stiffness Model for Free Vibration of Functionally Graded Nanobeams

This paper presents a nonlocal dynamic stiffness model (DSM) for free vibration analysis of Functionally Graded (FG) nanobeams. The nanobeam is investigated on the basis of the Nonlocal Elastic Theory (NET). The NET nanobeam model considers the length scale parameter, which can capture the small scale effect of nano structures considering the interactions of non-adjacent atoms and molecules. Material characteristics of FG nanobeams are considered nonlinearly varying throughout the thickness of the beam. The nanobeam is modelled according to the Timoshenko beam theory and its equations of motion are derived using Hamilton’s principle. The DSM is used to obtain an exact solution of the equation of motion taking into account the neutral axis position with different boundary conditions. The DSM is validated by comparing the obtained results with published results. Numerical results are presented to show the significance of the material distribution profile, nonlocal effect, and boundary conditions on the free vibration of nanobeams. It is shown that the study can be applied to other FG nanobeams as well as more complex of framed nanostructures.