Moffatt (Nature 404, 833 - 834) discusses the ever-intriguing motions of a rolling disk, as it whirrs and shudders to a horizontal stop

Regarding generality, the paper does not apply to the two examples mentioned, coins and a toy (Fig. 2 therein) by lack of the conformal contact that the theory demands (Fig. 1 therein): a coin by its raised images and generally rough edges, and the toy by its concave support bowl (Fig. 2 of the paper). One can roll almost any disk, including the toy and a coin, on a variety of nominally flat surfaces and note that the settling time differs by factors of two or so (although the disk parameters and air viscosity vary by far less than 1% between observations). Thus the viscous mechanism can't generally be the primary damping.