Probability distribution functions for a single-particle vibrating in one dimension: experimental study and theoretical analysis

We consider the form of the rebound velocity, ν0, particle velocity, ν, 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 (vibration frequency ⪢ collision rate). Experimentally, we find that these PDFs are well-approximated by Pν0(ν0) ∞ ν0 exp(− αν02), a Gaussian Pν(ν) ∞ exp(− αν2) and a Boltzmann-type function Ph(h) ∞ exp(− 2αgh), where α is a constant and g is the acceleration due to gravity. We develop an analytical model which accurately predicts the general form for the rebound velocity PDF; the other two PDFs are then analytically shown to follow as a consequence. Scaling laws for the particle granular temperature with peak base velocity and particle-base restitution coefficient, determined from previous work, can 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 vibrating base. Good agreement between scaling laws from the theory and simulation is found but insufficient data is obtainable to derive accuracy exponents experimentally.