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

Abstract

• 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. This paper applies the asymptotic perturbation approach (APA) to obtain a simple analytical expression for the free vibration analysis of non-uniform and non-homogenous beams with different boundary conditions. A linear governing equation of non-uniform and non-homogeneous beams is obtained based on the Euler–Bernoulli beam theory. The perturbative theory is employed to derive an asymptotic solution of the natural frequency of the beam. Finally, numerical solutions based on the analytical method are illustrated, where the effect of a variable width ratio on the natural frequency is analyzed. To verify the accuracy of the present method, two examples, piezoelectric laminated trapezoidal beam and axially functionally graded tapered beam, are presented. The results are compared with those results obtained from the finite element method (FEM) simulation and the published literature, respectively, and a good agreement is observed for lower-order beam frequencies.