On the special properties of graphic and co-graphic bondgraphs

Abstract

The special combinatorial properties of graphic and co-graphic bondgraphs are analysed. A graphic bondgraph is one which has an associated graph, meaning that the two represent the same combinatorial information, expressed in the junctions of the bondgraph or the cycles of the graph. Dually a co-graphic bondgraph is one whose dual bondgraph has an associated graph. The combinatorial basis for constructing a systematic bondgraph model is explained, leading to the definitions of vertex and mesh bondgraphs and their duals. For each part of a vertex or mesh bondgraph one junction, called an auxilliary junction, and all its internal bonds may be eliminated. This procedure, called grounding a node in the bondgraph literature, is a purely combinatorial procedure and has no physical interpretation, nor does the choice of the auxilliary junction affect the meaning and interpretation of the effort and flow variables of the elements of a bondgraph model. Inverting this construction of a systematic bondgraph motivates the definition of nodal and mesh bondgraphs, with special properties parallel to those expressed in the vertex structure of a graph. It is shown how to construct a nodal (mesh) bondgraph from any graphic (resp. co-graphic) bondgraph and extend these by adding new external bonds called nodal (resp. mesh) bonds. This procedure may be applied to a bondgraph model to yield alternative formulation methods, pseudo-nodal (resp. pseudo-mesh) formulations, with precisely the same advantages as the nodal (resp. mesh) techniques of network analysis.