Imperfections, impacts, and the singularity of Euler's disk

Abstract

The motion of a rigid, spinning disk on a flat surface ends with a dissipation-induced finite-time singularity. The problem of finding the dominant energy absorption mechanism during the last phase of the motion generated a lively debate during the past two decades. Various candidates including air drag and various types of friction have been considered, nevertheless impacts have not been examined until now. We investigate the effect of impacts caused by geometric imperfections of the disk and of the underlying flat surface, through analysing the dynamics of polygonal disks with unilateral point contacts. Similarly to earlier works, we determine the rate of energy absorption under the assumption of a regular pattern of motion analogous to precession-free motion of a rolling disk. In addition, we demonstrate that the asymptotic stability of this motion depends on parameters of the impact model. In the case of instability, the emerging irregular motion is investigated numerically. We conclude that there exists a range of model parameters (small radii of gyration or small restitution coefficients) in which absorption by impacts dominates all preiously investigated mechanisms during the last phase of motion. Nevertheless the parameter values associated with a homogenous disk on a hard surface are typically not in this range, hence the effect of impacts is in that case not dominant.

Effects of rolling friction on a spinning coin or disk

“…The noise generated by a hard object rolling on a hard surface is the result of multiple collisions of this type [17]. The effect of such collisions on a rolling disk has recently been examined theoretically by Baranyai and Varkonyi [19].…”

The abrupt ending of a spinning disk

“…As a result, the disk emits an audible sound that increases in pitch over time. Most authors [1][2][3][4][5][6][7][8][9][10][11][12][13] have concluded that the disk rolls rather than slides arounds its circumference, but there is no consensus on the mechanism that causes it to come to a sudden stop. Some authors have suggested [2], and others have disproved [3,8], that air resistance is responsible.…”

“…More likely, rolling is responsible for the gradual loss of energy, although the sudden stop remains a mystery. It has been suggested [6,13] that vibrations of the disk and/ or the horizontal surface might cause the disk to contact the surface at several different points simultaneously when the angle of inclination is very small, but there has been no direct experimental evidence of that effect.…”

Comments on ‘The abrupt ending of a spinning disk’

“…The deformations will influence significantly the role the (high) spots play in an abrupt stopping of Euler's disk. We noticed that for investigation of a similar spinning disk's rolling friction, the important role of deformations could be traced in [3][4][5][6][7] with respect to steelʼs deformations (thermal-induced surface damage at contacts possibly due to hightemperature friction in-between) [8,9]. Furthermore, the deformations will possibly change those results as Cross claimed in [1]: possible collisions between the high spots of the disk act to enhance the normal reaction force at those spots, with the result that the offset distance increases as the angle of inclination decreases.…”

“…The latter (role of deformation [3]) will influence the changes of the friction coefficient which was not mentioned in [1]. Meanwhile it was suggested in [4] that a detailed analysis of energy dissipation via microscopic impacts in a model including (elastic) deformations should be a possible direction of future research. Moreover, considering the loss of microscopic contact of the (Euler's) disk (rolling on a horizontal surface) from the horizontal surface before its sudden stop [5], Borisov et al proposed that the micro-losses of contact were most likely due to micro-deformations of the surface and possibly of the disk itself, because they observed a steel sheet was deformed by the disk moving on it, resulting in a decrease in the duration of the loss of contact of the disk with the steel sheet [5].…”