On the Stability Analysis of a Double-Link Inverted Pendulum Subject to an Oscillatory Tilted Excitation

Abstract

This paper presents the study of the stability characteristics of a two-link (double) inverted pendulum system subject to a tilted and oscillatory excitation at the bottom pivot. Each of the two links are assumed to have a concentrated point mass at their ends. The excitation force acts at the bottom of the first link and its amplitude, frequency and tilt angle can be varied. The system is also assumed to have very light damping between the two links. Euler-Lagrange formulation is used to develop the equations of motion of the double-link inverted pendulum which results in a non-autonomous dynamic system. The method of multiple timescales is implemented to convert it to an autonomous system which allows for determination of the equilibrium points. Bifurcation analysis of the damped and undamped system with respect to frequency and amplitude of the input excitation is performed using a continuation analysis software. Further, the evolution of the equilibrium points and their stability properties of the system using extensive Monte-Carlo simulation of the nonlinear system as several parameters such as damping coefficient, excitation frequency and tilt angle are varied is presented.