Transient response of supported beams to moving forces with sinusoidal time variation

Moving forces are a common loading pattern for flexible beams, which is found in many applications in both Civil and Mechanical Engineering. These give rise to a transient response, whose nature depends on the time variation of the amplitude of the force and of its position along the beam. Besides the possibility of the numerical evaluation, closed form solutions of the beam response are beneficial for their simplicity of use, and for the fact that they allow an understanding of the system behaviour. On the other hand, these prove to be rather complicated in most of the cases, and only a limited number of them are available in the literature. In this paper the simple but common case of a supported beam loaded by a force with sinusoidal time variation moving at a constant speed is dealt with. Simple equations for the approximated response at and away from resonance are presented, and their accuracy is discussed. Transient Frequency Response Functions are also shown. Finally, as an example, the results are applied to the evaluation of the response of beam footbridge to the action of a walker, and compared to Code specifications.