Vibrations Analysis of Functionally Graded Beams at Different Boundary Conditions

Functionally graded materials (FGM) are very important and modern applications in engineering due to the unique properties of these materials. In this work, the vibrations of FGM beams are studied at eight different boundary conditions such as F=F, SS=SS, C=C, C=SS, C=F, F=SS, C=SL, and F-SL. The exact equations of natural frequencies of FGM beams are derived for that different boundary conditions. A good agreement between present results and the results of previous literature. At all the different boundary conditions, the same behavior observed for natural frequencies at all modes when the physical properties of the beam are changing such as the thickness, the length, the density of material or ceramic, the elasticity of material or ceramic, and the ratio of volume fraction. When the ratio of volume fraction increases, the natural frequencies decrease. As the dimensionless ratio (L/h) increases, the natural frequencies will be decreased. When the dimensionless ratio (Em/Ec) increases, the natural frequencies increase too, but that frequencies decrease when the dimensionless ratio (ρm/ρc) increases.