For a simply supported Bernoulli‐Euler beam under a moving load with constant velocity the bending and torsion vibrations are analysed. An analytical solution of the decoupled partial differential equations is derived based on the eigenmode expansion considering the boundary and initial conditions. For the pure bending and the pure torsion the analysis is performed and the beam vibrations are calculated. For various longitudinal speeds the results are plotted. The coupled bending and torsion deformations are computed using a modal transformation. The modal decoupling yields two separated ordinary differential equations in time which are computed using the derived closed form solutions. The solutions of the coupled equations are plotted and discussed. A comparison with a numerical calculation of a finite element model which shows good agreement.

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.

Frequently slender structures are excited by moving loads and show the corresponding transient vibrations. A model is derived using a simply supported Bernoulli-Euler beam and the coupled bending and torsion vibration equations are derived. An analytical solution of the decoupled partial differential equations is derived based on the eigenmode expansion considering the boundary and initial conditions. For the pure bending and the pure Saint-Venant torsion the analysis is performed and the beam vibrations are calculated. The coupled bending and torsion deformations are computed using a modal transformation, where it is essential that the eigenmodes are the same for both kinds of vibrations. The resulting modal decoupling yields two separated ordinary differential equations in time with the assumed constant velocity of the axially moving excitation load. Closed form solutions are used which show a very good convergence. A transformation into the original coordinates results in the solutions of the coupled equations. The results are plotted in a dimensionless form. The evaluation and comparison with a suitable numerical calculation using a finite element model shows a good agreement of the results. It turns out that a fine discretization is necessary and the computation effort for computing the transient dynamic solution is very much higher.