The dynamical evolution of realistic Euler's Disks – experiment and theory

The "Euler's Disk", a rigid, axisymmetric disk spinning and rolling on a horizontal surface, is a well-known physical toy for its power law increasing precession rate and sound frequency approaching to a finite-time singularity, as its inclination angle decreases. While the final few seconds before this singularity have received much attention in literature, exploring the disk dynamics and dissipation mechanism, The full dynamics of the disk (over 120 seconds), where no single power law can describe the energy dissipation and a small angle approximation for the inclination is no longer valid, have received less focus.

In this work, we extended the perspective to the full dynamical evolution of the Euler’s Disk, from initial release to final collapse. We experimentally (video and sound) track the disk trajectory to reveal its epicyclic path, chased by oscillations in its inclination angle, evolving throughout its stages of motion.  We will model the motion of the disk, for the dissipation-free case, using Lagrangian mechanics with a formal extension to the regime of semi-holonomic constraint, which naturally predicts the inclination angle oscillations. Additionally, we report, for the first time, the appearance of a second oscillatory mode in the disk’s motion. We conclude with a discussion on dissipation mechanisms and present a phenomenological function that can fit the observed dissipation behavior over time.