Semi-inverse method for axially functionally graded beams with an anti-symmetric vibration mode

In this paper, we use the semi-inverse method to find the solutions to the dynamic equation of inhomogeneous, functionally graded simply supported beams. For the anti-symmetric mode and the material density (or the Young's modulus) that are both polynomial functions and and have been pre-specified, we find the Young's modulus (or the density) in polynomial functions and determine the closed from expression for the natural frequency.