Dynamic analysis of a beam under a moving force: A double Laplace transform solution

Abstract

The response problem of a simply supported and damped Bernoulli-Euler uniform beam of finite length traversed by a constant force moving at a uniform speed is solved by applying the double Laplace transformation with respect both to time and to the length co-ordinate along the beam. This leads to obtaining the sum of the Fourier series which represents the forced vibration part of the transient response in closed form. The solution thus obtained is effective for computing beam stresses. It is also shown that the forced vibration part can be expanded in a double power series, and that the coefficients of the series at the point of application of the moving force can be readily obtained by making use of Bernoulli polynomials. As a numerical example, simple approximate formulae obtained from the series are used to compute the forced vibration parts of the deflection and the beam stresses at the mid-span of the beam when a moving load is exactly at the mid-point of the beam, and their truncation errors are calculated.