Bending and torsion vibrations of a beam excited by a moving load

Abstract

The linear dynamic bending- and torsion vibrations of a simply supported Bernoulli-Euler beam under a moving load with constant velocity are analysed. The main target is to derive an analytical solution of the governing partial differential equations. The deformations of the beam are computed based on the eigenmode expansion. For the computation of the solution first the method of generalized finite integral transformation is used followed by a transformation using the Laplace-Carson integral transformation. The resulting algebraic equation then is rearranged considering the boundary and initial conditions. The inverse transformation has to consider the position of the poles and after a further transformation the solution in the time domain results. The convergence and the necessary number of eigenmodes of this procedure is analysed. For the pure bending load the analysis is performed and the beam vibrations are calculated for various longitudinal speeds. For the special case of static deflections the solution is also given. The resulting deformations are plotted over time and space and the occurring phenomena are discussed. For the pure torsion deformation the Saint-Venant torsion theory is used. The described methods for solving the equations are used again. Based on the boundary and initial conditions the solution shows a similar structure like the pure bending solution. Finally the analytic solutions are compared to a numerical calculation of a finite element model which shows good agreement of the two solutions.