Explicit Equations of Motion of Discrete System of Disks in Two Dimensions

Abstract

A granular system comprising interconnected disks is treated as a multibody system the motion of which is governed by the algebraic‐differential equations. In computer simulations of the flow of granular‐type materials, every time the topology of the system changes, a new set of equations must be generated; thus the efficiency of this procedure becomes very important. The algebraic equations are associated with the geometrical constraints imposed by the loops in the system graph and are easily generated in this case. The differential equations, describing the system of disks arranged in a topological tree, require special consideration. The use of a path matrix in describing the topological tree helped derive explicit expressions for the differential equations. The derivations are based on Lagrange's equations of the second type, so that the minimum set of equations is obtained. Also, some terms in Lagrange's equations cancel each other out analytically, thus eliminating redundant calculations if numerical generation of equations was used. These two results make the computer simulation of motion of granular‐type materials more efficient and accurate. An illustrative example of generating the differential equations of motion of a three‐disk system moving down a ramp is given.