Analysis of a rotating pendulum with a mass free to move radially

Gravity combined with a rotating pendulum shaft causes a parametric excitation which can create a surprisingly steady-state motion. Conventional intuition would suggests that the pendulum will swing briskly away from the axis of the shaft and the mass will slide off the rod. While this motion is likely to happen, the analysis shows that the pendulum can take on this motion: the pendulum oscillates near the axis of the shaft and the mass approximately maintains a constant radius.