Friction-Induced Lines of Attraction and Repulsion for Parts Sliding on an Oscillated Plate

We show that the frictional forces arising from simultaneous small amplitude periodic translation and rotation of a rigid plate cause parts on the plate to converge to or diverge from a line coinciding with the rotation axis. The relative phase between the translation and rotation determines whether the parts are attracted to or repelled from the rotation axis. Assuming that both the translational and rotational accelerations of the plate are ldquobang-bangrdquo and have identical frequencies, we derive the resultant velocity fields for point parts on the plate. For many choices of phase the speed of the part is approximately proportional to its distance from the rotation axis. The strength of the velocity field can be controlled by modulating the amplitude of the translational acceleration, or modulating the relative phase between the translational and rotational acceleration profiles. We also determine the phases that maximize part speed towards and away from the rotation axis. These optimal phases not only maximize part speed but also generate velocity fields that are nearly independent of the coefficient of friction.