Parametric Resonance of a charged pendulum with suspension point oscillating between two vertical charged lines

TL;DR

This paper investigates the dynamics and parametric resonance phenomena of a charged pendulum with an oscillating support point between charged wires, analyzing stability regions and extending classical Mathieu equation results.

Contribution

It introduces a mathematical model of a charged pendulum with a oscillating support between charged wires, analyzing stability and resonance phenomena with new parameter space insights.

Findings

Identified stability and instability regions in the parameter space.

Derived boundary curves for the zero-charge case matching Mathieu equation.

Analyzed parametric resonance effects in a charged pendulum system.

Abstract

In this work, we study a mathematical planar pendulum whose support point is positioned equidistant between two vertical and uniformly electrically charged wires. Its bob carries an electric charge and, its support point oscillates vertically, following a harmonic law of motion. We study the dynamics of such phenomenon and the parametric resonances of the equilibria. Moreover, we obtain the surface in the parameter space (since such system presents three parameters) which separates the region of stability from the region of instability. On the particular case of zero charge, we obtain the boundary curves of the stability/instability of Matheiu equation.