Deflection of beams by the operational method

Abstract

Laplace transform operator only accepts initial values at zero in evaluating derivative functions. This limits its use in vibration analysis of beams as a tool for merely finding analytical solutions and coefficients. By extending Laplace transform’s capacity to take non-zero initial conditions, a novel operational method for developing dynamic beam elements is proposed. The method avoids integration and differentiation, which are traditionally used in developing dynamic vibration elements. The ease of application of this operational method is demonstrated by developing dynamic stiffness elements of Euler-Bernoulli beams under various boundary conditions. The proposed technique also allows for easy handling of coupled differential equations, such as the ones applied to bending-torsion vibration of beams, as well as discontinuous functions which appear in elastic support modelling. MATLAB® software has been used as graphing tool and for the mathematical calculations in this research.