The Stability of Relative Equilibrium Positions of a Pendulum on a Mobile Platform

Abstract

The motion of a pendulum mounted on a platform that rotates around a vertical line with a constant velocity and simultaneously makes specified harmonic oscillations along this vertical line is considered. The platform angular velocity is assumed to be precisely equal to the frequency of small oscillations of the pendulum in the case of an immobile platform. There are relative equilibrium positions of the pendulum when it is oriented along the platform rotation axis (in the hanging or inverted state). The nonlinear problem of stability of these relative equilibria is considered.