Prediction and Control of Slip-Free Rotation States in Sphere Assemblies

We study fixed assemblies of touching spheres that can individually rotate. From any initial state, sliding friction drives an assembly toward a slip-free rotation state. For bipartite assemblies, which have only even loops, this state has at least four degrees of freedom. For exactly four degrees of freedom, we analytically predict the final state, which we prove to be independent of the strength of sliding friction, from an arbitrary initial one. With a tabletop experiment, we show how to impose any slip-free rotation state by only controlling two spheres, regardless of the total number.

Figure 1

Operation of an assembly of spheres in a slip-free state. (a) Three pairs of plastic spheres in contact were stacked onto each other with an alternating stacking angle of 35° and their positions were fixed using ball rollers. When forcing the bottom pair of spheres to rotate according to the scheme shown in (b) using external driving wheels, the other spheres adjust their rotation due to friction. In the stationary state the top pair of spheres rotates more than 3 times faster than the bottom pair.

Figure 2

Relaxation of a single pair of spheres toward the slip-free state. Rotating spheres with sliding forces at their contact (left) and in the final slip-free state (right), in which the contact moves along circles (dashed). Because of Newton’s third law of motion ${\overrightarrow{F}}_2=-{\overrightarrow{F}}_1$, where ${\overrightarrow{F}}_1$ and ${\overrightarrow{F}}_2$ are the sliding forces acting on the first and second sphere, respectively.

Figure 3

Parameters describing the slip-free state. The $c_{ij}$ parameters that uniquely relate the angular velocities of contacting spheres $i$ and $j$ in the slip-free state are independent for an open chain (a) and are all equal to one single $c$ for a noncoplanar-4 loop (b), which is a 4DOF assembly. For $c_{ij}=0$, the angular velocities of spheres $i$ and $j$ are antiparallel.

Figure 4

Description of different slip-free states of a 4DOF assembly. For $\overrightarrow{A}\ne \overrightarrow{0}$ and $B=0$ all axes of rotation are parallel to $\overrightarrow{A}$. For $B\ne 0$ all axes of rotation of the spheres meet at the position $\overrightarrow{x}=I\overrightarrow{A}/(BM)$, and, in particular, for $\overrightarrow{A}=\overrightarrow{0}$ they meet at the center of mass.

Figure 5

Tabletop experiment to compare with the model. (a) Two pairs of contacting TPU spheres were stacked onto each other and their positions were fixed using ball rollers. The two lower spheres were forced to rotate with equal absolute angular velocity using driving wheels attached to motors. (b) Ratio of the angular velocities of an upper and a lower sphere [experiment (symbols) in comparison with theory (lines)] versus different stacking angles $\alpha$ [compare (c)], for two different modes of rotation indicated in (c) (constant mode) and (d) (amplifying mode).