Directed transport of a particle on a horizontal surface under asymmetric vibrations

Abstract

Asymmetric unbiased fluctuations coupled with the effect of nonlinear dry (Coulombic) friction are observed to generate a directed transport even in the absence of any net external force or gradient. This phenomenon, called the ratchet effect, occurs when the driving force does not overcome the static friction threshold symmetrically. An elementary model is used to generate directed and unidirectional motion of a particle placed on top of a horizontal rigid surface which is given a zero mean spatially asymmetric motion through a multi-frequency in-plane excitation. We obtain steady-state periodic solutions of the nondimensional transport equations analytically to assess the distinct single/multiple sticking and sliding phases of particle motion and various parametric conditions for phase transitions. These analytical results are then validated by solutions obtained semi-analytically using Galerkin projection and numerically using the fourth-order Runge–Kutta method (RK4) and Newton–Raphson method with a fixed arc-length based continuation scheme. Also, a multibody dynamics model of this particle-on-plate system is built in MSC Adams to help us visualize the actual particle behavior under real dimensional system parameters and compare it with the foregoing analyses. A highly counter-intuitive phenomenon is encountered as the particle drift velocity increases non-monotonically with the increase in friction whereas a very high or a very low amount of friction impedes motion. Maximum transport velocity is found to be independent of the amplitude ratio of the vibratory input for low values of kinetic friction and is observed to manifest at a critical static friction threshold beyond which the particle motion becomes unidirectional. The effect of viscous drag of the ambient fluid on transport is assessed to be quite insignificant. Optimal vibratory input parameters to achieve the best transport current under varying friction conditions are determined. At low kinetic friction, a fixed set of system parameters is observed to generate distinct solution branches that coexisted just beyond the static friction threshold corresponding to a motion transition from “slip” to “stick–slip”.