Dynamics of taut strings undergoing large changes of tension caused by a force-driven traveling mass

Abstract

The nonlinear dynamics of a horizontal, initially taut, elastic string, subjected to a traveling point-mass driven by an assigned time-dependent force, is studied. A kinematically exact nonlinear model for the free boundary problem, previously derived by the authors through a variational procedure, is resumed. The model is developed in the framework of a mixed formulations, ruling the time-evolution of the configuration variables and the reactive contact forces. The model is then consistently simplified, via a static condensation of the horizontal displacements of the string. Differently from most of papers in literature, here the dynamic tension is accounted in the equations of motion, as a nonlinear effect. The simplified equations are discretized by the Galerkin method, which leads to a set of ordinary differential equations in the amplitudes of the trial functions and in the unknown abscissa of the point-mass. As a further simplified model, the mass of the string is neglected, and a set of two ordinary differential equations in the coordinates of the point-mass is derived. Numerical results are obtained, aimed at exploring the influence of the mass of the string and of the dynamic tension. The focus is on analyzing peculiar ‘back and forth’ motions of the point-mass, when this is driven by small or even zero forces.