Absolute versus relative formulations of the moving oscillator problem

In the general framework of the bridge-vehicle dynamic interaction, the so-called "moving oscillator" problem is revisited in order to provide a deeper insight into some analytical and physical aspects not specifically analyzed in previous investigations. Without lack of generality, the case of a stream of moving oscillators crossing a simply supported beam with arbitrary time law is considered. The formulations in terms of both absolute and relative displacements of the moving oscillators are critically reviewed and compared, and alternative sets of differential equations with time-dependent coefficients are derived. The study enlightens, both theoretically and numerically, that impulsive contributions to the dynamic response appear in the relative displacement formulation at the time instants in which each vehicle enters or exits the bridge. It is demonstrated that such contributions, somehow "hidden" in the absolute displacement formulation, may have a significant influence on the vibration of the moving oscillators, and thus cannot be a priori neglected in the analysis. It is also shown that the analytical and computational difficulties associated with these additional impulses make preferable the use of the absolute displacement formulation. Far from being restricted to the case of simply supported beams, these findings are valid for any type of bridge structure which induces a discontinuity in the slope of the road profile experienced by the vehicles.