Equilibria of Multibody Chain in Orbit Plane

Abstract

We study equilibria of a multibody system in the orbit plane within the framework of a model of n + 1 material points connected by n light rods into an n-link chain. The junctions are spherical hinges. The center of mass of the system moves along a circular orbit. The equilibrium equations are reduced to a fairly simple form that enables their analysis.We find all the equilibria of an n-link chain in the orbit plane and prove that each rod can occupy one of the following three positions: it can be directed along the tangent to the orbit of the center of mass of the chain; it can be a member of a group of adjacent vertical rods, being the center of mass of this group situated on the tangent to the orbit; and, finally, an oblique orientation is possible if the rod joins either two vertical groups of rods or the end of a vertical group with the tangent to the orbit. It is shown that the number of equilibria does not exceed 2^(2n). We include the analysis of two examples (three- and four-link chains) and represent the schemes of all the realizable equilibria in these cases.