Rolling and slipping motion of Euler’s disk

Abstract

We present an experimental study of the motion of a circular disk spun onto a table. With the help of a high speed video system, the temporal evolutions of (i) the inclination angle α, (ii) the angular velocity ω and (iii) the precession rate Ω are studied. The influence of the mass of the disk and the friction between the disk and the supporting surface are considered. The inclination angle α and the angular velocity are observed to decrease according to a power law. We also show that the precession rate Ω diverges as the disk stops. Exponents are measured very near the collapse as well as on long range times. Collapsing times have been also measured. The results are compared with previous theoretical and experimental works. The major source of energy dissipation is found to be the slipping of the disk on the plane.

Experimental and theoretical investigation of the energy dissipation of a rolling disk during its final stage of motion

“…The experimental results agree with values of n between 1 2 and 2 3 . Measurements on a torus are believed in [5] to confirm the supposition of van den Engh et al [23] that air drag is only of minor importance.…”

“…Caps et al [5] present a rather detailed experimental study of various rolling disks using a high-speed video camera (125-500 fps) and a laser beam during about 10 s. The inclination angle, precession rate and angular velocity around the axis of symmetry of the disk are each measured with a different experimental setup during a different run and, therefore, have not been obtained simultaneously. The experimental results agree with values of n between 1 2 and 2 3 .…”

“…There exists a tremendous amount of literature on the dynamics of the rolling disk, e.g. [1][2][3][4][5][6][7][8][11][12][13][17][18][19][20][21][22][23][24] but this list is far from complete. Here, we will only give an overview of the literature on dissipation mechanisms which explain the finite-time singularity and of the literature reporting measurements of this phenomenon.…”

Rolling friction and energy dissipation in a spinning disc

“…Since then, results on the dynamics of a rigid body have been reported in a tremendous body of literature [2][3][4][5]. Owing to Moffatt's recent work [6], this classical problem has recently revived interest for studying the energy dissipation in the motion of a 'Euler's disc', namely a circular (homogeneous) metal disc spinning on a flat surface [7][8][9][10][11][12][13][14]. In this system, while its mechanical energy is dissipating, paradoxically, the speed of the disc's rolling increases rapidly.…”

“…Easwar et al [9] reported an exponent n = 3 through experiments using a single high-speed camera to measure the precession rate. Caps [8] performed a series of experiments to separately measure the inclination angle, precession rate and angular velocity around the symmetric axis of the disc, and presented the exponent with a value varying between n = 3 and 4. Leine [10] adopted a high-speed camera to synchronically measure the inclination angle and the precession rate, and suggested that the exponent is either n = 4 or 3.…”