transverse vibration for non-uniform timoshenko nano-beams

in this paper, eringen’s nonlocal elasticity and timoshenko beam theories are implemented to analyze the bending vibration for non-uniform nano-beams.  the governing equations and the boundary conditions are derived using hamilton’s principle. a generalized differential quadrature method (gdqm) is utilized for solving the governing equations of non-uniform timoshenko nano-beam for pinned-pinned, clamped–clamped, clamped–pinned, clamped–free, clamped–slide, and pinned-slide boundary conditions. the non-dimensional natural frequencies and the normalized mode shapes are obtained for short and stubby nano-beams where influences varying cross-section area, small scale, shear deformation, rotational moment of inertia, acceleration gravity and the self-weight of the non-uniform timoshenko nano-beam are discussed. the present study illus-trates that the small scale effects are more significant for smaller size of nano-beam, larger nonlocal parameter and higher vibration modes. further, the compression forces due to gravity and the self-weight of the nano-beam also like the small scale effect are reduced the magnitude of the fre-quencies of the nano-beam.