Density of states in granular media in the presence of damping.

We consider the density of states of granular media in which each grain-grain contact is damped with a damping force proportional to the relative velocity of the two grains, in addition to the usual spring constant. Under the assumption that the so-called criterion of proportional damping is only weakly violated we are able to deduce the density of states for undamped frequencies from the measured complex-valued frequencies of damped oscillations. We deduce a quantitative estimate of the deviation from the proportional criterion. We consider, specifically, numerical simulations of cases in which the grains are frictionless spheres that interact via Hertz central forces and all the nonzero contacts are damped with the same damping constant. We show how these ideas can be applied to data on real granular systems.