A Global Analysis of an Harmonically Excited Spring-Pendulum System With Internal Resonance

Abstract

A non-linear system can often have multiple attractors. In this case, it is important to study the global behavior of the system by determining the global domain of attraction of each attractor. This paper is concerned with the global behavior of an harmonically excited spring-pendulum system with internal resonance. Before studying the global behavior, the local behaviors of the system around its steady state periodic solutions are examined. Two non-linearly coupled second order non-autonomous ordinary differential equations governing the spring-pendulum system are transformed into a four-dimensional autonomous system in amplitude and phase variables. Next the stability of equilibrium states of the autonomous system, corresponding to steady state periodic solutions of the non-autonomous system, is examined. When certain resonance conditions are satisfied, the spring-pendulum system has a very complex behavior including jump phenomena and Hopf bifurcation. Global domains of attraction for the spring-pendulum system are investigated when two stable steady state periodic solutions exist. The interpolated mapping method, a numerical method for determining global domains of attraction, is presented so as to be applied to systems with dimension higher than two. Then the domains of attraction for a specific set of parameter values are obtained by the method. It is shown that the interpolated mapping method can be an efficient tool for analyzing the global behaviors of higher-dimensional systems. Also examined is how the global domains of attraction evolve as the forcing frequency is varied across critical values at which jump phenomena occur. It is discovered that there exists a special plane in the four-dimensional state space which contains all the attractors. By using this plane, called the principal plane, the global domains of attraction can be discussed more effectively. Results show that knowledge of this evolution is essential to better understand the mechanism of the jump phenomena.