Analytical analysis of free vibration of non-uniform and non-homogenous beams: Asymptotic perturbation approach

•Perturbation method is proposed to solve differential equation with variable coefficients.•Analytical formula is derived for frequencies of non-uniform and non-homogeneity beams.•Laminated beam and axially functional graded beam are analyzed.•Different boundary conditions are considered.•Solution is validated by the FEM and the published references.