Free vibration of two-directional functionally graded beams

This paper presents a theoretical investigation in free vibration of a functionally graded beam which has variable material properties along the beam length and thickness. It is assumed that material properties vary through the length according to a simple power law distribution with an arbitrary power index and have an exponential gradation along the beam thickness. The characteristic equations are derived in closed form. The governing equation can analytically reduce to the classical forms of Euler–Bernoulli beams if the gradient index disappears. Analytical solutions of the natural frequencies are obtained for graded beams with clamped-free and hinged–hinged end supports. Results show that the variations of material properties in the beam length and thickness have a strong influence on the natural frequencies. It is also shown that there exists a critical frequency depending on the gradient parameter. The natural frequencies have an abrupt jump when across its critical frequencies. The derived results can be useful for designing non-homogeneous beams which may be required to vibrate with a particular frequency.