Tolerance analysis in mechanical design

Products cannot be manufactured precisely. Some deviations from nominal design will always be present, no matter how good manufacturing processes become. The precision with which a product is manufactured affects its functionality. Higher precision results in higher manufacturing cost. Tolerances describe allowed deviations from nominal design to ensure that a product is functional. Unfortunately tolerances interact in complicated ways which are difficult and often impossible to analyze. The reason for it is that tolerances can describe a large variety of allowed geometric variations from nominal design. While individual variations can be parameterized and modeled easily, combinations of even a few variations may result in extremely complex multi-dimensional spaces.; In this thesis we develop a mathematical theory and a representation to express engineering tolerance constraints. This representation is then used to formulate problems of tolerance analysis and synthesis. The approach is illustrated with examples. Mathematical solutions to tolerance analysis and synthesis problems are discussed.; The representation is developed to satisfy the following requirements: (1) representation is independent of the underlying geometric modeler; (2) both representation and tolerance analysis and synthesis techniques are uniform for individual components and for assemblies; (3) representation is fully symbolic; (4) representation unifies current engineering standards, geometric modeling techniques and tolerance analysis techniques; (5) representation can be used to model errors in manufacturing processes.; The representation uses a graph of symbolic geometric entities and symbolic geometric constraints to model designs of constrained parts and assemblies. Geometric information which is not included explicitly in the model, but required for tolerance analysis, is computed symbolically by MAPLE, a symbolic manipulation package. The translation of ANSI constraints into symbolic geometric and algebraic constraints is fully automatic and is performed in such a way as to: (i) minimize the symbolic complexity of the resulting tolerance representation, and (ii) preserve the geometric intent of high level ANSI constraints on the low levels of representation.; One of the natural applications of our representation is modeling errors in manufacturing and inspection processes. We demonstrate the technique by modeling errors and estimating error parameters for the calibration of a coordinate measuring machine.