Free vibration analysis of stepped FGM nanobeams using nonlocal dynamic stiffness model

A nonlocal Dynamic Stiffness Model (DSM) for free vibration analysis of Functionally Graded Material (FGM) nanobeams on a Winkler elastic foundation based on the Nonlocal Elastic Theory (NET) is proposed. The NET model considers the length scale parameter, which can capture the small scale effect of nanostructures considering the interactions of non-adjacent atoms and molecules. Material characteristics of FGM nanobeams are considered nonlinearly varying throughout the thickness. The nanobeams are 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. This nonlocal DSM proposed has overcome the stiffening phenomena of the cantilever beam fundamental frequency and validated by comparing the obtained results with published results. Afterwards the proposed model is applied to investigate free vibrations of stepped FGM nanobeams. Numerical results are presented to show the influence of the material distribution profile, geometry, nonlocal, elastic foundation and boundary conditions on the free vibration of stepped FGM nanobeams. It is shown that the proposed nonlocal DSM can be applied to more complex stepped nanostructures.