Mathematical Modelling of Mechanical Structures and Assembly Processes of Complex Technical Systems

The paper discusses a hypergraph model of the mechanical structure of a technical system. The model describes the coordination of parts during assembly of a product, obtained by basing on design bases. This model correctly describes the operations and assembly processes of products that have the properties of sequentiality and coherence. The sequential and coherent assembly operations are prevalent in the assembly of modern technical systems: machines and mechanical devices. In terms of this model, the assembly operation is represented as a normal contraction of an edge. The sequence of contractions that transforms the hypergraph into a point is a mathematical description of the assembly process. A theorem on the necessary conditions for contractibility of hypergraphs is presented. It is shown that the necessary conditions are not sufficient. An important theorem on sufficient conditions of contractibility is proved. The concept of an ns-hypergraph is introduced. Ns-hypergraph is a mathematical model of the mechanical structure that cannot be assembled due to structural defects. Computational experiment was carried out to enumerate the ns-hypergraphs of various orders. The proposed apparatus can be used in computer-aided design systems for structural analysis of complex projects and computer-aided planning of assembly processes.