Terminal retrograde turn of rolling rings

We report an unexpected reverse spiral turn in the final stage of the motion of rolling rings. It is well known that spinning disks rotate in the same direction of their initial spin until they stop. While a spinning ring starts its motion with a kinematics similar to disks, i.e., moving along a cycloidal path prograde with the direction of its rigid body rotation, the mean trajectory of its center of mass later develops an inflection point so that the ring makes a spiral turn and revolves in a retrograde direction around a new center. Using high speed imaging and numerical simulations of models featuring a rolling rigid body, we show that the hollow geometry of a ring tunes the rotational air drag resistance so that the frictional force at the contact point with the ground changes its direction at the inflection point and puts the ring on a retrograde spiral trajectory. Our findings have potential applications in designing topologically new surface-effect flying objects capable of performing complex reorientation and translational maneuvers.

Figure 1

(a) Geometry of a ring spun on a horizontal table. The dashed line is perpendicular to the surface of the table. The origin of the coordinate frame defined by $({\mathbf{e}}_1,{\mathbf{e}}_2,{\mathbf{e}}_3)$ coincides with the center of mass of the ring, point $C$. The point $A$ is the center of the upper circular edge. (b) The quasiperiodic trajectory of ${\mathbf{r}}_A\left(t\right)$ projected on the surface of the table, and in the absence of dissipative effects. We have set the initial conditions to ${\theta }_0=0.55$ rad, ${\dot{\varphi }}_0=4.5$, and ${\dot{\psi }}_0=0$. All other conditions have been set to zero.

Figure 2

(a) First cycloidal turn of the ring. The four marks for motion tracking have been labeled by numbers 1, 2, 3, and 4. (b) The full trajectory of the center of the top circular edge, point $A$, of the ring. The ring undergoes a directional walk along the arrow. (c) The first few prograde turns of the rolling disk. (d) The trajectory of the center of mass of the disk until it reaches a small inclination angle.

Figure 3

Experimentally computed velocity of the ring parallel to the surface of the table. The magnitude of ${\mathbf{v}}_p$ in units of pixel/frame vs the frame number $n$. The imaging speed has been 300 frames per s, and approximately $135.6$ pixels correspond to $40.3$ mm.

Figure 4

(a) Projection of the simulated trajectory of ${\mathbf{r}}_A\left(t\right)$ on the surface for the same ring of Figs. 1 and 1 and with $(C_1,C_2,C_3)=(0.03,0.063,0.085)$. We have also reproduced the experimental trajectory of Fig. 2 for comparison. The motion between the initial prograde and final retrograde turning phases is directional along the arrow. (b) Same as panel (a) but for $(C_1,C_2,C_3)=(0.2,0.063,0.2)$. All lengths and position vectors have been normalized to the mean radius $R-w/2$ of the ring. (c) The variation of the friction force $F_f$ vs time for the models of panels (a) and (b). Variables are in dimensionless units.