Multiple degree of freedom inverted pendulum dynamics: Modeling, computation, and experimentation

A pendulum is statically unstable in its upright inverted state due to the Earth's gravitational attraction which points downward. However, with proper forcing, the pendulum can be stabilized in its upright inverted state. Special interest is on periodic vertical forcing applied to the pendulum's base to stabilize it around the upright inverted equilibrium.;Many researchers have studied how to stabilize the system by varying various parameters, in particular its amplitude and frequency. Most have focused on the single degree of freedom inverted pendulum case, which with linear assumption can be described via Mathieu's equation. The system stability can then be characterized by Floquet theory. Our focus is on searching for the periodic solutions inside the linearly stable region of the pendulum's inverted state when the pendulum is under proper periodic forcing. Our research shows that under appropriate excitation by controlling the forcing amplitude and frequency, the pendulum can maintain certain periodic orbits around its inverted state which we characterize in a systematic way.;In this thesis, we applied four different kinds of geometric realizations of the system response: system time traces, system phase portraits, three dimensional views of the system phase portrait as a function of input forcing, and the system's power spectral density diagram. By analyzing these four diagrams simultaneously, we characterize different kinds of multi-frequency periodic behavior around the pendulum's inverted state. To further discuss the effect of the nonlinearity, we applied perturbation techniques using the normalized forcing amplitude as a perturbation parameter to carry out the approximate periodic solutions on a single degree of freedom inverted pendulum nonlinear model.;We also discuss the multiple degree of freedom inverted pendulum system. Both numerical simulation and experiments were performed and detailed comparisons are discussed. Our numerical simulations show close qualitative agreement with experiments.