Net trajectory equation of the particle in vibrated granular material

Abstract

Inspired by results from close observations of vibrated granular material, we considered the constituent particles as projectiles. We derived an equation from the recursive form of the projectile motion equation to quantify the conditions for a particle to rise, sink or remain stationary in the granular bed. Our formulation implies that characterizing vibrated granular materials is about determining the boundary conditions or constitutive relations for cascaded projectile motions. Details of segregation or mixing mechanisms will be contained in the transfer function in the constitutive relation for the initial velocities and the drag coefficient.