Vibration analysis of a Timoshenko non-uniform nanobeam based on nonlocal theory: An analytical solution

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In this article free vibration of a Timoshenko nanobeam with variable cross-section is investigated using nonlocal elasticity theory within the scope of continuum mechanics. Small scale effects are modelled after Eringen’s nonlocal elasticity theory while the non-uniformity is presented by exponentially varying width through the beam length with constant thickness. Analytical solution is achieved for both Timoshenko beams and nanobeams with different boundary conditions including both ends being simply-supported (S-S), both ends being clamped (C-C) and one end clamped other free (C-F). It is shown that section variation accompanying small scale effects has a noticeable effect on natural frequencies of non-uniform Timoshenko beams at nano-scale. In order to illustrate these effects, Natural frequencies of single-layered graphene nano-ribbons (GNRs) with various boundary conditions are obtained for different nonlocal and non-uniform parameter which shows a great sensitivity to non-uniformity in different shape modes.