A Unique Approach to Investigate the Free Vibrations of Non-uniform and Functionally Graded Euler-Bernoulli Beams

In this article, the free vibrations of non-uniform and non-homogeneous beams with the general non-classical boundary conditions including end objects with mass moment of inertia, eccentricity, and rotational and translational flexible constraints are investigated. A review of related beam sources shows that unlike many studies and various methods used to investigate the vibrations of non-uniform and non-homogeneous beams, the presented method often does not lead to a unique solution for the beam with the general boundary conditions. Hence presented solution cannot be used for the other special cases. Therefore, the purpose of this thesis is to present a new method based on the application of the Cauchy’s formula for iterated integrals which according to the best of author’s knowledge has not been considered so far. By using this method and its related equations, it is possible to analyze the vibrations of different types of homogeneous uniform, homogeneous non-uniform, functionally graded uniform, and functionally graded none-uniform beams. The validation of the obtained results for some of the studied configurations have been compared with the available data reported in related studies and good agreement observed. Also parametric study has been presented in order to investigate the effect of aspect ratio for wedge and tapered beams. To carry out convergence test and other calculations throughout this article a code in Mathematica software has been written.