Duality Transformation Research Based On Graph Representation And Its Application In Engineering

The thesis introduces a general approach for solving engineering problems by transforming them to problems in other engineering fields, through general discrete mathematical models called graph representations. Through the transformation between engineering structures, the thesis shows some new methods and theories for complicated engineering problems. Firstly, the thesis introduces briefly appliance in engineering structures of graph and reviews some frequently-used methods of mechanical kinematics. At the same time, several concepts and correlation theories are presented in this thesis. Secondary, different kinds of engineering structures are represented by graph through the mathematic relations between different kinds of graph representation (GR) and transformation thinking. By analyzing the conjunction of various graph representation, transformation relations between various engineering structures are established. Then, the thesis views the mechanisms and trusses as main research subject and represents mechanisms as potential graph (PGR), trusses as flow graph (FGR). Based on the duality between potential graph and flow graph, the same relation and transformation are established. Through the duality transformation between mechanisms and trusses, the thesis uses some special methods from structural analysis in machine theory, such as Henneberg's method, and also employs methods from machine theory in structural analysis, such as the vector resolution method.Then, the thesis regards multi-engineering structures as subject investigated. Through graph representation, the transformation relations between these engineering structures are established. At the same time, the loop method from structures mechanics is described by Kirchhoff's current law (KCL) and Kirchhoff's voltage law (KVL). Based on the duality between mechanisms and trusses and the loop method, which is described by KVL, the thesis shows a new method called loop method to analyze multi-loop mechanism.Finally, the thesis shows the deficiencies of duality transformation and the goals in the future.