Dynamical elastic contact of a rope with the ground

TL;DR

This paper investigates the complex elastic contact dynamics of a rope on the ground under time-dependent forcing, revealing resonance behaviors, bifurcations, and bistability through analytical, numerical, and experimental methods.

Contribution

It provides a comprehensive analysis of elastic contact dynamics of a rope, including resonance phenomena, bifurcation analysis, and the effects of bending stiffness and nonlinearity.

Findings

Fundamental frequency is independent of tension in weakly bent ropes.

Transition from non-radiating to radiating states occurs via grazing bifurcations.

Bistability observed in ropes with large bending stiffness and significant nonlinearity.

Abstract

A rope laid on the ground with one end subjected to time-dependent forcing is proposed as a prototypical elastic dynamical contact problem, which we study analytically, numerically, and experimentally. The dynamics is governed by an infinite set of linear and nonlinear resonances. In the limit of weak bending stiffness, the fundamental frequency is found to be independent of the rope tension. A transition between a radiation-less and a wave radiating state occurs via a series of grazing bifurcations, whereby new contacts between the rope and the ground are formed. The grazing bifurcations form overlapping Arnold tongues in the frequency-amplitude parameter space. Finally, for ropes with large bending stiffness and when the geometric nonlinearity is important, bistability is observed between several wave-making regimes.