Dynamic Analysis of Isotropic Homogeneous Beams Using the Method of Initial Functions and Comparison with Classical Beam Theories and FEM

Abstract

The main aim of this study is the dynamic analysis of isotropic homogeneous beams using the method of initial functions (MIFs) and comparison with classical beam theories and FEM. Also, this research employs the state space methodology, with a special emphasis on isotropy, to analyse simply supported beam systems. A mathematical model for the dynamic response of beams is constructed using the method of initial functions. The novelty of this study lies in its approach to dynamic analysis, where isotropic homogeneous beams are explored without making assumptions, thus ensuring increased precision using the method of initial functions. Importantly, the approach remains free from restrictive assumptions and relies solely on mathematical formulations, yielding results superior to classical beam theories such as Euler–Bernoulli, Timoshenko, and Rayleigh beam theories. In this work, the application of MIFs of various orders (4th, 6th, 8th, and 10th) to calculate natural frequencies is explored, enabling a thorough examination of the beam’s dynamic characteristics. In addition, parameters such as normal stresses, shear stresses, and deflections in different directions are considered to provide a comprehensive understanding of beam behaviour. To validate the findings, a detailed comparison with a finite element method (FEM) is conducted, achieving excellent agreement between the analytical results and FEM solutions. Furthermore, the influence of Poisson’s ratio (μ) on natural frequencies is investigated by varying its value from 0.18 to 0.30. The research also explores the deviation of plane stress values of the beam section from those estimated using the FEM for the corresponding components.