Nonlinear dynamics of a hanging string with a freely pivoting attached mass

TL;DR

This study investigates how attaching a freely pivoting mass to a hanging string drastically alters its natural frequency and induces complex nonlinear behaviors, including oscillations and instabilities, modeled analytically and validated experimentally.

Contribution

It reveals the significant impact of a pivoting mass on the nonlinear dynamics of a hanging string and provides an analytical model for the observed behaviors.

Findings

Resonant frequency is increased by an order of magnitude due to the pivoting mass.

Experimental observation of harmonic and period-doubling instabilities.

Analytical modeling with Hill equation accurately predicts instability curves.

Abstract

We show that the natural resonant frequency of a suspended flexible string is significantly modified (by one order of magnitude) by adding a freely pivoting attached mass at its lower end. This articulated system then exhibits complex nonlinear dynamics such as bending oscillations, similar to those of a swing becoming slack, thereby strongly modifying the system resonance that is found to be controlled by the length of the pivoting mass. The dynamics is experimentally studied using a remote and noninvasive magnetic parametric forcing. To do so, a permanent magnet is suspended by a flexible string above a vertically oscillating conductive plate. Harmonic and period-doubling instabilities are experimentally reported and are modeled using the Hill equation, leading to analytical solutions that accurately describe the experimentally observed tonguelike instability curves.