Inelastic impact of a sphere on a massive plane: Nonmonotonic velocity-dependence of the restitution coefficient

Abstract

We have studied the coefficient of restitution, η, in normal collisions of a non-rotating sphere on a massive plate for a range of material parameters, impact velocity and sphere size. The measured coefficient of restitution does not monotonically vary with velocity. This effect is due to dynamics that occur during the finite duration of impact: the contact time varies as a function of velocity is comparable to the time-scales of the vibrational modes of the plate. The measured effect is robust and is expected to be ubiquitous in fluidized granular media. We also find that η is a decreasing function of particle size, a dependence that is not captured by existing models of impact.

Stochastic behavior of the coefficient of normal restitution

“…In virtually all publications on granular gases and rapid granular flows, it is assumed that the COR is either a constant or a function of the impact velocity. This is in contrast to several experimental results, e.g., [13][14][15][16][17][18], where it was found that even for almost spherical particles, the COR reveals significant scatter, which cannot be explained by the imperfections of the experiment but must be attributed to tiny imperfections of the surfaces in contact. Performing large-scale bouncing ball experiments using a robot, Montaine et al [19] analyzed the fluctuations of the COR of more than 10 5 single impacts and found that besides the known dependence on impact velocity, the COR may be described as a fluctuating quantity whose probability distribution is a combination of two exponentials,…”

“…In virtually all publications so far, it is assumed that the COR is either a material constant or a deterministic function of the material and system characteristics and the impact velocity. This assumption is in contrast to experimental results, e.g., [13][14][15][16][17][18], which show that even tiny surface textures, that is, even weak roughness, cause significant scatter of the COR, which suggests that the COR is a fluctuating quantity. These fluctuations have been investigated in large-scale experiments [19] to obtain the probability density for the COR of almost smooth particles, which was reported to be of asymmetric Laplacian shape.…”

Coefficient of restitution of aspherical particles

“…This can be seen in many experimental results published in the literature, e.g., [12][13][14][15][16][17], where the strong scatter of the measured coefficient of restitutions cannot be explained by the imperfections of the experiment [10]. This scatter can be attributed to microscopic imperfections of the surface of the particles [11] in very good agreement with the experiment [10].…”

“…Therefore, the deviation from the Laplacian for very small velocity may be attributed to correlations of the impacts. We believe that for most bouncing-ball experiments reported in the literature, this regime is not relevant; however, there are experiments reported by King et al [14] where impact velocities as low as 0.001 m/s are considered such that the described motion may become relevant. Analyzing the probability density of the ratio E rot / (E rot + E tran ) in the same way (Fig.…”

“…Namely, in the present paper, we consider the coefficient of restitution for the repeated bounce of smooth but slightly eccentric particles of dumbbell type. This is directly related to experiments where the coefficient of restitution was obtained from the time lap between consecutive impacts of a particle bouncing on a flat plate [12][13][14][15][16][17].…”