Portal de Periódicos da CAPES

Infinitesimal motions of a bipartite framework

1984; Mathematical Sciences Publishers; Volume: 110; Issue: 1 Linguagem: Inglês

10.2140/pjm.1984.110.233

1945-5844

Walter Whiteley,

Advanced Materials and Mechanics

A recently established criterion for stresses on any bar and joint framework with a complete bipartite graph is converted into explicit criteria for infinitesimal motions of the framework.These criteria, based on quadric surfaces through the joints, include and complete those developed in recent years in the study of geodesy with range satellite networks.The infinitesimal motions are displayed in a simple geometric form, appropriate to any dimension.This geometric description is used to establish necessary and sufficient conditions, based on ruled quadric surfaces, for which bars may be added to a bipartite framework without removing a prior infinitesimal motion.These criteria are applied to the behaviour of structural engineering space frames. Introduction.In a recent paper, Bolker and Roth presented an analysis of static stresses in any bar and joint framework with an underlying bipartite graph [1].In spite of their title ("When is a bipartite graph a rigid framework?"), the paper gave explicit criteria only for the number of stresses in the framework, and left implicit the number of infinitesimal (or finite) motions which resulted.Our work began with an observation that their static criterion yielded explicit criteria (both necessary and sufficient) for the presence of nontrivial infinitesimal motions: essentially that all joints of the framework lie on a quadric surface ( §2) [7].This simple geometric conclusion suggested that the motions must flow in some natural way from the quadric surface and we present two descriptions of the motion drawn from the quadric surface ( §3).One description is informal and geometric, the other is more detailed and analytic though still geometric.We feel that the general geometric approach gives a stronger intuition for the basic connections, but the analytic approach is easier to condense, and easier to use in detailed form later in the paper.Once we "see" the source of the motions we can pose, and answer, an interesting question: What bars can be added to a bipartite framework without blocking the infinitesimal motion?( §4).When additional bars are added to a bipartite framework, the essential condition for an infinitesimal motion is a ruled quadric surface through all the joints, containing the lines of all added bars (Theorem 3).Ironically, while Bolker and Roth 233