On the terminal motion of sliding spinning disks with uniform Coulomb friction

We review previous investigations concerning the terminal motion of disks sliding and spinning with uniform dry friction across a horizontal plane. Previous analyses show that a thin circular ring or uniform circular disk of radius $R$ always stops sliding and spinning at the same instant. Moreover, under arbitrary nonzero initial values of translational speed $v$ and angular rotation rate $\omega$, the terminal value of the speed ratio $\epsilon_0=v/R\omega$ is always 1.0 for the ring and 0.653 for the uniform disk. In the current study we show that an annular disk of radius ratio $\eta=R_2/R_1$ stops sliding and spinning at the same time, but with a terminal speed ratio dependent on $\eta$. For a twotier disk with lower tier of thickness $H_1$ and radius $R_1$ and upper tier of thickness $Р_2$ and radius $R_2$, the motion depends on both $\eta$ and the thickness ratio $\lambda=H_1/H_2$. While translation and rotation stop simultaneously, their terminal ratio $\epsilon_0$ either vanishes when $k>\sqrt{2/3}$, is a nonzero constant when $1/2<k<\sqrt{2/3}$, or diverges when $k<1/2$, where k is the normalized radius of gyration. These three regimes are in agreement with those found by Goyal et al. [S. Goyal, A. Ruina, J. Papadopoulos, Wear 143 (1991) 331] for generic axisymmetric bodies with varying radii of gyration using geometric methods. New experiments with PVC disks sliding on a nylon fabric stretched over a plexiglass plate only partially corroborate the three different types of terminal motions, suggesting more complexity in the description of friction.