Nonlinear Free Vibration Analysis of Micro-beams Resting on Viscoelastic Foundation Based on the Modified Couple Stress Theory

In this paper, nonlinear free vibration analysis of micro-beams resting on the viscoelastic foundation is investigated by the use of the modified couple stress theory, which is able to capture the size effects for structures in micron and sub-micron scales. To this aim, the governing equation of motion and the boundary conditions are derived using the Euler–Bernoulli beam and the Hamilton’s principle. The Galerkin method is employed to solve the governing nonlinear differential equation and obtain the frequency-amplitude algebraic equation. Finally, the effects of different parameters, such as the mode number, aspect ratio of length to height, the normalized length scale parameter and foundation parameters on the natural frequency-amplitude curves of doubly simply supported beams are studied.