Free Vibration Analysis of a Functionally Graded Micro-Beam with Tapered Cross Section

In this study, the free vibration analysis of a functionally graded (FG) microbeam with tapered cross-section was carried out theoretically. The beam has material distribution according to the power law throughout the thickness. The governing equation was reduced to an ordinary differential equation for a tapered beam with cross-sectional geometry whose width varies exponentially. Vibrations of a FG tapered microbeam was analyzed analytically in the elastic variation range based on the modified stress couple theory. Motion equations and boundary conditions were derived from the Hamilton principle. Analytical results of natural frequencies were calculated for cantilever exponential FG beams. Solutions for natural frequencies were obtained as the ratio of the beam's characteristic size to the material internal length parameter and according to the FGM distribution function characteristics