An application of the Lyapunov stability theorem: a bead on a rotating hoop

The motion of a bead on a rotating hoop is a classical problem in mechanics. The problem describes that a bead is placed on the track of a hoop rotating around a vertical axis. Although this issue is researched by lots of scholars, this system is usually analyzed without considering the friction. However, when considering friction, it becomes difficult to analyze the stability. The purpose of this study is to investigate the stability of the system with friction more easily. We derive the equation of motion of the bead and use the Lyapunov stability theorem to analyze the stability. The stability is determined by the angular velocity of the hoop. If the angular velocity is less than or equal to the critical value the bead is asymptotically stable at origin (the bottom of the hoop) otherwise it is asymptotically stable at a non-zero position. In addition, we study the difference between mass point bead and rigid body bead and find that both have identical bifurcation diagram and stability, whereas the trajectories have a difference on the phase plane. In the end, the numerical simulation method is used to verify the conclusion above.