Averaging method and asymptotic solutions in some mechanical problems

TL;DR

This paper investigates systems under oscillating forces using averaging methods, providing conditions for forced oscillations and analyzing asymptotic behaviors, including a stable solution for an inverted pendulum in oscillating gravity.

Contribution

It introduces sufficient conditions for forced oscillations and explores asymptotic solutions, including a novel stable solution for an inverted pendulum in oscillating gravity.

Findings

Existence of forced oscillations under certain conditions

Asymptotic behavior of solutions analyzed

Stable solution for inverted pendulum in oscillating gravity

Abstract

In the paper we consider systems in oscillating force fields such that the classical method of averaging can be applied. We present sufficient conditions for the existence of forced oscillations in such systems and study the asymptotic behaviour of some solutions. In particular, we show that for an inverted pendulum with a horizontally moving pivot point in an oscillating gravity field there exists a solution along which the pendulum never falls.