Analytical free vibration solutions of Euler-Bernoulli beams resting on Winkler foundation using the Elzaki transform method

Abstract

The natural frequency analysis of a freely vibrating thin beam on Winkler foundation is a significant part of the analysis for design against resonance failures which occur when the natural frequency coincides with the excitation frequency due to loads. This article explores the natural frequency determinations of an Euler-Bernoulli beam on a Winkler foundation using Elzaki transform method (ETM). The main objective is to develop a new analytical solution procedure using the ETM and demonstrate its efficiency for beam foundation vibration problem. The ETM is adopted because of its effectiveness in solving both linear and partial differential equations problems, computational simplicity and versatility. For harmonic vibrations, the governing partial differential equation (GPDE) of motion is simplified to an ordinary differential equation (ODE) in terms of the space variable. The ETM further simplifies the ODE to an algebraic problem whose inverse yields the solution for the modal displacement in the physical domain. The application of boundary conditions gives the characteristic frequency equation as a transcendental equation. The roots give the eigenvalues from which the natural frequencies are found. The boundary conditions considered are the classical boundary conditions (BCs). The natural frequencies for all the classical BCs are found as exact solutions within the assumptions and theoretical framework of Euler Bernoulli theory, Winkler foundation model, linear elastic behavior, small displacement elasticity theory and classical boundary conditions. They are identical with previous solutions using Generalized Integral Transform Methods (GITM). The present solutions are close to the previous solutions based on semi-analytical methods of Differential Transform Method (DTM). Analytical solutions for the natural frequencies are obtained for the classical BCs considered and for the foundation parameters considered. The effectiveness of the ETM in simplifying the natural frequency analysis of thin beam on Winkler foundation to an algebraic formulation is demonstrated.