PROBABILITY-DISTRIBUTION FUNCTIONS FOR A SINGLE-PARTICLE VIBRATING INONE-DIMENSION - EXPERIMENTAL-STUDY AND THEORETICAL-ANALYSIS

We consider the form of the rebound velocity, nu(o), particle velocity , nu, and height, h, probability density functions (PDFs) for the one- dimensional motion of a single particle on a sinusoidally oscillating base. The motion is considered in the limit of high excitation (vibrat ion frequency much greater than collision rate). Experimentally, we fi nd that these PDFs are well-approximated by P-nu o((nu o)) proportiona l to (nu o) exp(- alpha nu(o)(2)), a Gaussian P-nu(nu) proportional to exp(- alpha nu(2)) and a Boltzmann-type function P-h(h) proportional to exp(- 2 alpha gh), where alpha is a constant and g is the accelerat ion due to gravity. We develop an analytical model which accurately pr edicts the general form for the rebound velocity PDF; the other two PD Fs are then analytically shown to follow as a consequence. Scaling law s for the particle granular temperature with peak base velocity and pa rticle-base restitution coefficient, determined from previous work, ca n also be predicted from the PDF. A fine scale ''spiky'' structure in the rebound velocity PDF is found, using numerical simulations, to be a consequence of resonance phenomena between the particle and vibratin g base. Good agreement between scaling laws from the theory and simula tion is found but insufficient data is obtainable to derive accuracy e xponents experimentally.