The method of averaging for the Kapitza-Whitney pendulum

TL;DR

This paper generalizes the classical Kapitza pendulum by analyzing an inverted pendulum with a vibrating pivot and horizontal force, demonstrating conditions for stable non-falling oscillations and multiple periodic solutions.

Contribution

It introduces a new method of averaging for analyzing a generalized Kapitza-Whitney pendulum with time-periodic forcing and vibration, establishing existence and stability of non-falling solutions.

Findings

Existence of a periodic solution where the pendulum never falls.

Sufficient conditions for multiple non-falling periodic solutions.

Numerical evidence of stable non-falling oscillations.

Abstract

A generalization of the classical Kapitza pendulum is considered: an inverted planar mathematical pendulum with a vertically vibrating pivot point in a time-periodic horizontal force field. We study the existence of forced oscillations in the system. It is shown that there always exists a periodic solution along which the rod of the pendulum never becomes horizontal, i.e. the pendulum never falls, provided the period of vibration and the period of horizontal force are commensurable. We also present a sufficient condition for the existence of at least two different periodic solutions without falling. We show numerically that there exist stable periodic solutions without falling.