Stability, bifurcation and chaos of a pendulum on a rotating arm

Abstract

This paper presents an analytical and numerical study of the dynamical behavior of a pendulum on a rotating arm. There are four control parameters, which influences the behaviors of mechanical system, namely: the length ratio of the arm to pendulum, the damping, the frequency and the gravity force. The sufficient conditions for the stability are obtained using damped Mathieu equation theory and the Lyapunov direct method. It is found that stable equilibrium positions depend on four control parameters, the gravity force that appears as as an oscillating exciting torque and the length ratio in system causes the existence of a chaotic attractor or regular, several different types of transitions to chaos are studied. The Melnikov method is used to show the existence of chaotic motion. The treatment of the Melnikov integral is different for different length ratio. The magnitude effects of the length ratio parameter is important to acknowledge the existence of a chaotic attractor or regular motion. Moreover, numerical simulations including bifurcation diagrams are constructed to investigate dynamical behavior of the system. The transitions to chaos are confirmed by calculation of Lyapunov exponents. More detailed numerical investigations appear in phase trajectories and Poincare maps.