Solution to the Problem of a Mass Traveling on a Taut String via Integral Equation

Abstract

The problem of a massive taut string, traveled by a heavy point mass, moving with an assigned law, is formulated in a linear context. Displacements are assumed to be transverse, and the dynamic tension is neglected. The equations governing the moving boundary problem are derived via a variational principle, in which the geometric compatibility between the point mass and the string is enforced via a Lagrange multiplier, having the meaning of transverse reactive force. The equations are rearranged in the form of a unique Volterra integral equation in the reactive force, which is solved numerically. A classical Galerkin solution is implemented for comparison. Numerical results throw light on the physics of the phenomenon and confirm the effectiveness of the algorithm.