Vibro-levitation and inverted pendulum: parametric resonance in vibrating droplets and soft materials

Abstract

The phenomenon of liquid droplets “levitating” or bouncing off a liquid vibrating surface has attracted attention of scientists due to its possible application in microfluidics and novel nanostructured superhydrophobic materials. Several models have been suggested in the literature, and the effect is usually attributed to non-linear viscosity. Here we suggest a simple model relating the effect to the parametric resonance as described by the Mathieu equation, which explains stabilization of an inverted pendulum with vibration foundation. Small fast vibrations can be substituted by an effective “levitation” force. We present modeling and experimental results for oil droplets and discuss how the mathematical separation of the slow and fast motion provides insights on the relation of vibro-levitation of oil droplets and soft materials with the vibro-stabilization of an inverted pendulum, and the “Indian rope” and “Cornstarch monster” tricks.