Euler's disk and the rocking can : a study of three-dimensional problems in rotational dynamics

• Main Author: Collins, Ben W. B
• Format: Dissertation
• Language: English
• Published: University of Bristol 2023

When tilted and released, a can falls towards a flat, upright state. Instead of falling flat, the can experiences a bouncing motion before rising up and falling over. During the bounce, the can rotates through ±∆ψ, an angle greater than π and observable in experiments. This asymmetric behaviour is known as the rocking can phenomenon. A special case of Euler's disk, the equations of motion describing the rocking can contain two conserved quantities, for which we provide a physical justification of their existence. The dynamics are reduced to a singularly perturbed, second order ODE, with a small parameter corresponding to a combination of angular momenta. Using matched asymptotics, we split the dynamics into an outer region, which describes the initial rocking motion, and an inner region, which describes the bounce. The solution to the inner problem yields the same angle of turn found by Srinivasan and Ruina. We gain new information about the sign of the angle of turn, with good agreement to the full non-linear equations. We find that large angles of turn require prohibitively large coefficients of friction. The related problem of Euler's disk arises when dissipation is introduced to the system. When spun on a hard, smooth surface, Euler's disk rolls with a whirring noise that increases infrequency until an abrupt halt. This halting behaviour is investigated in two experiments. In the first experiment, we spin the disk on base-plates of different materials. We find that the energy closely follows the well-known power law, but that the energy exponent n, commonly used to characterise the motion, varies with the material characteristics of the base-plate. In the second experiment, we spin the disk and film it from two angles, tracking both its angle of inclination and the rotation of a strip drawn on the disk. We show that the disk continues to rotate after falling flat. The duration of the continued rotation can be increased by lubricating the surface and decreased by spinning the disk on a concave base-plate. It is suggested that the disk loses contact with the plane and is briefly supported by a layer of fluid between the disk and the base-plate. To analyse Euler's disk, we derive the equations of motion subject to Coulomb friction and another unspecified dissipation mechanism. We check for Painlevé paradoxes, a mechanism for loss of contact, and find them to be impossible. In a relaxation of the rigid body assumption, we find equivalent definitions of classical rolling friction and contour friction. Taking contour friction, we numerically solve the equations of motion with initial conditions informed by experiments. We find that the disk undergoes alternating periods of slipping and rolling motion, ending with slipping motion. Furthermore, the disk encounters the discontinuity set of contour friction, providing a mechanism for the continued rotation observed in experiments. Finally, we analyse the post-falling flat motion, consisting of a lubrication theory-informed, contact-less phase followed by a Coulomb-governed slip to a halt. The predicted duration of the post-falling flat motion is in agreement with experiments. We posit that, on the concave base-plate, contour friction is the dominant dissipation mechanism. However, on flat, hard base-plates, we conjecture the presence of an air resistance based mechanism.