The pendulum under vibrations revisited

Abstract

A simple intuitive physical explanation is offered, of the stability of the inverted pendulum under fast violent vibrations. The direct description allows to analyze, both intuitively and rigorously, the effect of vibrations in similar, and in more general, situations. The rigorous derivations in the paper follow a singular perturbations model of mixed slow and fast dynamics. The approach allows applications beyond the classical inverted pendulum model.

The Spherical Kapitza-Whitney Pendulum

“…It contradicts the property of the considered periodic solution to remain in W c,δ for all t. Finally, we conclude that the periodic solution obtained for the modified system remains a solution for the original system. Below we present some numerical results concerning periodic solutions of system (5). Our main goal is to show that periodic solutions without falling, the existence of which is guaranteed by Theorem 1, can be asymptotically stable.…”

“…We have already proved that there always exists a periodic non-falling solution in the presence of a periodic horizontal force (in case of the Kapitza pendulum, this solution also always exists and it is the vertical equilibrium). The conditions for the asymptotic stability for the Kapitza pendulum are known and it might be useful to find similar conditions for system (5). Another open problem related to the considered systems is the problem of precise estimation of values of ε for which there exist a periodic solution.…”

“…The simple planar pendulum with a vibrating pivot point has been one of the most well-studied nonlinear systems in the 20th century [1][2][3][4] and still remains an object of mathematical research (see, for instance, the most recent papers [5][6][7][8]). A comprehensive overview of the works related to the topic and the history of the problem can be found in [9,10].…”

Vibrational Stabilization of the Kapitza Pendulum Using Model Predictive Control with Constrained Base Displacement

“…Vibrational stabilization of the Kapitza pendulum (KP) has been widely studied within the framework of the Mathieu equation [3], [4], [5], [6], [7], [8]. A summary of various techniques applied to this problem along with intuitive explanations are given in [9].…”