Dynamics of an unbalanced ring spinning on a rough horizontal surface
An interesting stability property, as fascinating as that of spinning tops and gyroscopes, is observable in the motion of an unbalanced ring spinning on a rough horizontal surface. An analytical and numerical study is performed to investigate the general motion of an unbalanced ring modeled as a thin ring with a particle attached to its rim. The translational motion is represented by the rectangular coordinates of the ring geometric center. The rotational motion is represented by a 1-2-3 set of Euler angles. The kinetic motion equations are derived with the use of Newton's second law and Euler's rotational motion equations.
The types of motion considered are the pure-rolling and rolling-with-slipping motions. Given favorable initial conditions, ring properties, and a sufficiently large constraint force in the form of friction, the ring undergoes a pure-rolling motion. For other conditions, however, limitations on the magnitude of the friction force render the pure the mathematical model to allow switching from pure-rolling to rolling-with-slipping motion and vice versa.
The general motions of the unbalanced ring, obtained by numerically integrating the governing equations with the use of the seventh-eighth order Runge-Kutta method, are in very good qualitative agreement to those observed during an experiment performed with the use of a high-speed video camera.