Stochastic dynamics of a rod bouncing upon a vibrating surface

Abstract

We describe the behavior of a rod bouncing upon a horizontal surface which is undergoing sinusoidal vertical vibration. The predictions of computer simulations are compared with experiments in which a stainless-steel rod bounces upon a metal-coated glass surface. We find that, as the dimensionless acceleration parameter Gamma is increased appreciably above unity, the motion of a long rod passes from periodic or near-periodic motion into stochastic dynamics. Within this stochastic regime the statistics of the times between impacts follow distributions with tails of approximately Gaussian form while the probability distributions of the angles at impact have tails that are close to exponential. We determine the dependence of each distribution upon the length of the rod, upon frequency, and on Gamma. The statistics of the total energy and of the translational and rotational components each approximately follow a Boltzmann distribution in their tails, the translational and rotational energy components being strongly correlated. The time-averaged mean vertical translational energy is significantly larger than the mean rotational energy, and both are considerably larger than the energy associated with horizontal motion.