At the end of a moving string

TL;DR

This paper presents a new exact solution for the dynamics of a flexible string with a free end, revealing a time-dependent range and a singularity in acceleration, with implications for slender structures and fluid-structure interactions.

Contribution

It introduces a novel exact solution to the string equations with a free boundary, advancing understanding of nonlinear dynamics in flexible structures.

Findings

Derived a new exact solution valid for distances greater than t^{4/3} from the free end.

Identified a t^{-2/3} acceleration singularity indicating whipping motion.

Combined solutions for curved and straight segments to model string dynamics.

Abstract

One cannot pull an open, curved string along itself. This fact is clearly reflected in the unwrapping motion of a string or chain as it is dragged around an object, and implies strong consequences for slender structures in passive locomotion, whether industrial cables and sheets or the ribbons of rhythmic gymnastics. We address a basic problem in the dynamics of flexible bodies, namely the solution of the string equations with a free boundary. This system is the backbone of many fluid-structure interactions, and also a model problem for thin structures where geometric nonlinearities cannot be ignored. We consider planar dynamics under the restriction that the spatially-dependent stress profile in the string is time-independent, which results in a conservation law form for the equations. We find a new exact solution whose range of validity is time-dependent, limited to greater than a distance scaling as $t^{\frac4/3}$ from the free end. The remainder of the distance is covered by splicing to another exact solution for a straight string, which introduces an error into the combined solution. The splicing also implies a $t^{-\frac2/3}$ singularity in acceleration, apparently corresponding to a whipping motion of the vanishing straight segment.