Analysis of the Axial Vibration of Non-Uniform and Functionally Graded Rods via an Analytical-Based Numerical Approach

In this study, an analytical-based numerical approach was proposed for the analysis of the free axial vibration of homogeneous and functionally graded rods with varying cross-sectional areas. The proposed approach is based on analytical approximation techniques, such as the Adomian decomposition method, variational iteration method, and homotopy perturbation method. However, the governing equations of the problems solved in this study were variable coefficient differential equations. These equations provide analytical solutions for strictly limited cases. Analytical approximation methods easily handle problems with uniform material properties and constant cross-sections, whereas with varying cross-sectional areas, the analytical integration process becomes a difficult task for the software. If the rod's material is functionally graded with varying cross-sectional areas, the analytical integration process becomes a cumbersome task. The proposed approach eliminates all difficulties and requires computation within several seconds. The application of this method is straightforward, and the results obtained in this study are in excellent agreement with the solutions provided in the literature.