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

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 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.

Highlights:

• The dynamics of a flexible pendulum with a pivoting tip-mass is studied.
• Its resonant frequency is one order of magnitude higher than that of a single string.
• A technique of remote magnetic parametric forcing is experimentally used.
• Parametric resonances, such as harmonics and period doubling, are evidenced.
• The resonant tongues are theoretically modeled using a Hill equation.