Theoretical velocity of an object frictionally coupled to a two-mode vibrating plate

Net velocity has been demonstrated for objects frictionally coupled to a flat plate that oscillates periodically in-plane with two frequencies, provided plate displacement is nonantiperiodic: the ratio of frequencies γ cannot be the ratio of two odd integers. We give a mathematical derivation of the experimentally determined dependence of mean velocity on the relative amplitudes of the two frequency modes, and the phase lag between the modes, when γ=2, and when the magnitude of plate acceleration is much larger than the magnitude of acceleration by static friction. The approach uses an analysis of the symmetry properties of the roots of trigonometric polynomials, without explicit determination of those roots. The behavior when γ=1/2, and specific phase lags that inhibit net velocity for general γ, are also determined.