Vibrational transportation of deformable axisymmetric particles

A particle on a substrate supporting a surface acoustic wave can experience horizontal drift excited by the dry friction force. The effect is referred to as vibrational transportation, or as a surface acoustic wave motor, and is used in a number of industrial applications. A traditional theory of vibrational transportation considers a particle as a material point moving on a rigid substrate. A more realistic representation is a contact model based on Cattaneo-Mindlin (also called Hertz-Mindlin) mechanics applicable to an axisymmetric deformable particle. The contact zone in this case is not a point but rather a circle that generally contains a smaller circle of stick and a surrounding annulus of slip. A recent semianalytical extension of the Cattaneo-Mindlin solution called the method of memory diagrams allows one to compute the hysteretic friction force for an arbitrary loading history in terms of contact displacements and, subsequently, to numerically solve the equations of motion. Depending on the materials' and excitation parameters, the particle can stay in permanent contact with the substrate or experience multiple jumps. In the former case, the particle can drift in a horizontal direction due to asymmetric sliding condition created by oscillating normal and tangential contact forces. In other words, during each wave period, the particle advances and recedes with different efficiencies, which finally results in a drift. The drift can occur in the wave propagation direction and against it and require a specific choice of system's parameters. In the regime of multiple jumps, directed horizontal motion is also possible. However, it is governed by a completely different mechanism based on synchronization between the wave period and rebounding events. There exist cases where the rebound occurs once per period and consistently at the same phase. During each rebound, the particle gets horizontal momentum of the same sign (against the wave propagation direction). The value of this momentum depends on the horizontal velocity mismatch between the particle and the substrate; therefore, at the beginning of the process, the particle moves with an acceleration that decreases and finally disappears. Exactly the same type of motion against the wave has been observed in our preliminary experiments. In other cases, the time of flight of the particle and the wave period are uncorrelated, thus resulting in a chaotic motion. We also demonstrate that a point mass in the same situation behaves differently. In particular, in a regime of permanent contact, negative and positive sliding are equilibrated, which produces no drift. In addition, multiple rebounds of a point mass are always chaotic, at least for fully conservative collisions. In conclusion, the deformable particle model can be a better guide for various applications, such as particle micropositioning or acoustic dust cleaning.