Resonance of Beams Due to Cyclic Moving Loads

An analysis is presented for determining the conditions of resonance and the dynamic response of a simply supported beam subjected to a moving point load of variable magnitude P cos ωt oscillating longitudinally along the beam about a fixed point. In addition to ordinary resonance as produced by a variable magnitude load applied at a fixed point on an elastic body, it is shown that other interesting resonance conditions occur because of the oscillation of the force along the surface of the elastic body. The time rate of increase of the vibrational amplitude of the beam is determined numerically for two sets of initial conditions corresponding to: (1) a load initially at rest on the beam and (2) an oscillating load dropped from zero height on an initially undeformed beam. It is shown that the time rate of buildup of vibrational amplitude increases rapidly as the amplitude of longitudinal oscillation is increased for the typical examples presented.