Abstract :
The work reported here was derived as a part of a research program aimed at investigating various discrete mathematical representations for a wide variety of engineering problems. For the results in this paper, the starting point was to represent trusses and mechanisms using graph theory and matroid theory, then using knowledge and algorithms embedded in these theories, to develop analysis methods and understanding of those two kinds of engineering systems.
The research program was approached by searching for mathematical representations which may be useful for representing engineering problems. For each chosen representation, the attributes of the representation were mapped, or matched, to attributes of the engineering system considered. Properties and algorithms embedded within the representation were then applied to the corresponding attributes of the engineering system. In this way, new properties of the engineering system were found, together with new techniques to derive known properties. This paper deals exclusively with the representations applied to trusses and mechanisms.
The paper first shows how plane axial force trusses and mechanisms can be represented as graphs and then analyzed using the language and algorithms of graph theory. These sections are based upon material, which has already been reported in the literature, but also include many new details in the solution methods.
Having defined the dual of a planar graph, the paper shows that, in general, the graph representations of mechanisms and trusses are mathematically dual. From the perspective of this duality a number of classes of trusses and mechanisms can be defined. The duality is demonstrated for plane statically determinate trusses, and for the corresponding class of mechanisms.
This direction of investigation enhances understanding of the connection between static and kinematic systems, and enables utilization of knowledge from one field to solve problems in the other.
