Multi-dimensional tolerance analysis of mechanical assemblies

Mechanical engineers use tolerances to define allowable manufactured uncertainties to insure interchangeable assemblies with proper functionality. It is necessary for a Computer-Aided Design system to help designers to validate the functionality and interchangeability. The development of a new method for multi-dimensional tolerance analysis of mechanical assemblies, called “GapSpace model”, is described. In this model a gap describes the relation between a pair of features of different components, and is used as an arc in a graph representation of an assembly. Necessary and sufficient conditions (called fitting conditions) for testing if the assembly can be assembled together are formulated by searching the graph based on the spatial relations of related features. An algorithm is designed to search the assembly graph to find all fitting conditions. Two equivalent representations for such conditions are induced mathematically. One of them is a function of the dimensions and tolerances of related components. Based on the two representations, (1) the assembleability of the assembly can be easily checked for nominal dimensions, in worst or statistical case; (2) the quality of an assembly (usually the range of a gap) can be transformed into a linear optimization problem; (3) the sensitivity of a design variable with respect to a dimension or a tolerance is gotten from an explicit function; (4) the relative motion range between components can be calculated using Minkowski sum. These problems can be considered in the assembly space, which is a vector space generated by all fitting conditions and a subspace of the GapSpace spanned by all gaps. Comparing assembly regions defined by the functionality of the assembly, with the tolerance regions constrained by designers, tolerance analysis problems can be analyzed graphically and mathematically. Over-constrained assemblies can be detected by the model. The model is particularly useful for non-orthogonal and over-constrained multidimensional assemblies.