Algebraic Models of Product Assembly Process

Assembly process engineering is one of the biggest challenges in modern preproduction engineering. The quality of the process largely depends on the sequence of machine or device assembly. With increasing number of parts in the engineering system the number of allowable assembly sequences is rapidly increasing. It is impossible to analyse this copious combinatorial space without using the cutting-edge methods of mathematical modelling. The paper offers the algebraic models that can be used to select the rational design solutions at the preproduction engineering stage of the assembly operation. When assembling any engineering system the coherence and sequence conditions should be met. It is shown that an adequate mathematical description of sequential and coherent assembly operation is shrinkage of edges of the hyper-graph, which describes a mechanical structure of the product. For the product and its parts must be provided a property of the independent assembly. It is shown that this property can be represented as an action of the closure operator on a set of the product parts. The representations of this operator are parts to be assembled independently (the s-sets). Arranged by inclusion, an aggregate of all the sets is a lattice. The lattice is an algebraic structure where are specified two stable operations, namely: lattice intersection and lattice jog. It turned out that it is possible to use this structure, as a universal generating medium to a diversity of design options for the assembly conversion. The lattice terms are used to describe the sequences of product assembly and disassembly, the sequences of assembly and disassembly of the assembly units, the multi-level diagrams of the assembly decomposition, etc. The properties of the independent assembly are required not only to provide the assembly. A lot of design and technology operations can be performed, provided that a set of the parts has a stable and coordinated configuration within the product, i.e. an s-set of the lattice. For example, those are adjustment operation, various types of tests, fitting and trial assembly, etc. One of the most important operations when designing the engineering system is the synthesis of a rational system of design dimension chains. It is shown that the lattice operations can be used to find the minimum length design chains. The lattice structure allows us to state and solve a problem of minimizing the number of tests for geometric solvability (geometric access) while assembling the products of complicated Science & Education of the Bauman MSTU 231 configuration. It is shown that the set of all solvable configurations is the sub-lattice within the lattice of all the assembled configurations. This allows us to solve the problem of minimizing the number of geometrical tests through the algebraic lattice restoration methods