On rotational solutions for elliptically excited pendulum

TL;DR

This paper investigates the rotational behavior of an elliptically excited pendulum, deriving conditions for stable solutions and comparing approximate analytical results with numerical simulations.

Contribution

It introduces new approximate solutions for the elliptically excited pendulum with small ellipticity, extending previous circular pivot analyses.

Findings

Stable rotational solutions are identified and their existence conditions are derived.

Approximate solutions match well with numerical results for various damping levels.

The analysis covers both high and low damping scenarios.

Abstract

The author considers the planar rotational motion of the mathematical pendulum with its pivot oscillating both vertically and horizontally, so the trajectory of the pivot is an ellipse close to a circle. The analysis is based on the exact rotational solutions in the case of circular pivot trajectory and zero gravity. The conditions for existence and stability of such solutions are derived. Assuming that the amplitudes of excitations are not small while the pivot trajectory has small ellipticity the approximate solutions are found both for high and small linear damping. Comparison between approximate and numerical solutions is made for different values of the damping parameter.