Free vibration of axially functionally graded beams using the asymptotic development method

Axially functionally graded (AFG) beams, with variable coefficients in the governing equation, are a novel class of composites structures that have continuous variations in material properties from one component to another. In this paper, the asymptotic development method (ADM) is utilized to investigate the free vibration of uniform AFG beams with different boundary conditions. By decomposing the variable flexural stiffness and mass per unit length into reference invariant parts and variant parts, perturbation theory is introduced to obtain an approximate formula of the natural frequencies of the uniform AFG beams. The numerical results of the proposed method are confirmed by comparing the obtained results with those obtained via finite element analysis and the published literature results, the comparison reveals the proposed method yields an accurate estimate of the first three order natural frequencies of the AFG beam. Moreover, the influences of the gradient parameter and support conditions on the first three natural frequencies are discussed. The proposed analytical method is simple and efficient and can be used to conveniently analyze uniform AFG beams with arbitrary changes in the material properties along the axial direction.