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

•Perturbation method is introduced to solve the differential equation with variable coefficients.•Analytical formula is derived for frequencies of axially functionally graded(AFG) beams.•Finite segment model is proposed to obtained solutions of the finite element simulation.•Analytical solutions are validated by the finite element method(FEM) and the published references.•Various gradient parameters and different boundary conditions are discussed.