Gravitational Stability of Frictionless Assemblies

Abstract

The stability of an assemblage of contacting rigid bodies without friction is investigated. A method is presented for finding an orientation of the assembly so that the assembly remains motionless under gravity. If no stable orientation exists for an assembly, the method finds the "least" unstable orientation. The metric used to measure stability is based on the second time-rate of change of the gravitational potential energy, and the desired orientation for an assembly is expressed in terms of an optimization problem involving changes in potential energy. The problem of finding stable or maximally-stable orientations is formulated as a constrained maximin problem. The maximin problem is shown to be a variant of standard zero-sum matrix games, and can be solved using linear programming. The method is the first general method for automatically determining stable orientations. Example assemblies are presented.