Free vibration response of micromorphic Timoshenko beams

In this paper the authors investigate the free vibration of a two-length-scale nonlocal micromorphic Timoshenko beam, which is shown to overlap with the nonlocal strain gradient Timoshenko beam under certain conditions. Hamilton's principle is utilized to obtain a system of two coupled fourth-order equations of motion governing the eigen-deflection and the eigen-rotation of the beam. Uncoupling both equations leads to two eight-order differential equations. Using Ferrari's method, exact solutions are derived for the eigenfrequencies for various boundary conditions, including simply supported, clamped-clamped, clamped-free, and clamped-hinged boundary conditions. The obtained results are compared with those published in the literature using similar nonlocal strain gradient cases. A detailed parametric study is then performed to emphasize the role of the variationally-derived higher-order boundary conditions (natural higher-order boundary conditions or mixed higher-order boundary conditions). It is noted that when the difference in length-scales is large, the effect of the slenderness of the beam on the frequencies is amplified. Finally, the hardening or the softening effect of the beam model can be achieved through a choice of the ratio between the two length-scales.