Anomalous curvature evolution and geometric regularization of energy focusing in the snapping dynamics of a flexible body

TL;DR

This paper investigates the violent snapping motion of flexible structures, revealing how geometric effects and initial conditions lead to energy focusing and singular behaviors, with implications for predicting large energy amplifications.

Contribution

It introduces a numerical scheme validated against experiments and uncovers the role of geometry in regularizing finite-time singularities in flexible body dynamics.

Findings

Identification of geometric regularization effects.

Observation of anomalously slow curvature scaling.

Development of a simple predictive model for energy amplification.

Abstract

We examine the focusing of kinetic energy and the amplification of various quantities during the snapping motion of the free end of a flexible structure. This brief but violent event appears to be a regularized finite-time singularity, with remarkably large spikes in velocity, acceleration, and tension easily induced by generic initial and boundary conditions. A numerical scheme for the inextensible string equations is validated against available experimental data for a falling chain and further employed to explore the phenomenon. We determine that the discretization of the equations, equivalent to the physically discrete problem of a chain, does not provide the regularizing length scale, which in the absence of other physical effects must then arise from the geometry of the problem. An analytical solution for a geometrically singular limit, a falling perfectly-folded string, accounts surprisingly well for the scalings of several quantities in the numerics, but can only indirectly suggest a behavior for the curvature, one which seems to explain prior experimental data but does not correspond to the evolution of the curvature peak in our system, which instead displays a newly observed anomalously slow scaling. A simple model, incorporating only knowledge of the initial conditions along with the anomalous and singular-limit scalings, provides reasonable estimates for the amplifications of relevant quantities. This is a first step to predict and harness arbitrarily large energy focusing in structures, with a practical limit set only by length scales present in the discrete mechanical system or the initial conditions.