The effect of non-spherical aspect of a dimer on the dynamic behaviors

Shape-anisotropic particles on vibrating plates exert diverse behaviors of interest. But there is very limited work that devoted to the effect of particle shape, let along the dynamic reasons behind. In this paper, we exhibit that a non-spherical particle called a dimer encounters double impacts with friction, superstatic problems and the Painlevé paradox, which are different from those of a spherical particle, when it is put on a harmonically vibrating plate. The dimer system was firstly presented by Dorbolo et al. (Phys Rev Lett 95:044101, 2005), who found experimentally that the bouncing dimer exerted a horizontal drift on the vertically vibrating plate, forwards or backwards. We reveal that the drifting direction of the dimer is closely related to a dual singularity, a superstatic singularity during a double impact when the tangential velocities vanish simultaneously at the two impacting tips. A correlative coefficient of friction (CCoF) is defined on the basis of the Coulomb frictional law to determine stick or slip at the tips at the moment of the dual singularity. The CCoF correlates the static coefficient of friction between the dimer and the plate to its aspect ratio. It decides the bifurcation of the drifting directions. Interestingly, the CCoF is independent of the characteristics of the plate’s vibration, which is also verified by the experiment of Dorbolo. The non-spherical aspect of the dimer also leads to the Painlevé paradox where the solution is inconsistent or indeterminate. A parametric study is carried out in this paper to identify the Painlevé paradox in the dimer system.