Influence of form on Tolerance-Map-generated frequency distributions for 1D clearance in design

Tolerance-maps are used to generate the frequency distribution for 1D clearance between two planar features, one of which is manufactured on a part to given size and form tolerances. A Tolerance-Map (T-Map) [Davidson JK, Shah JJ, Mujezinović A. Method and apparatus for geometric variations to integrate parametric computer-aided design with tolerance analysis and optimization, US Patent 6,963,824; November 8, 2005] is the range of points resulting from a one-to-one mapping from all the variational possibilities of a perfect-form feature, within its tolerance-zone, to a specially designed Euclidean point-space. A functional T-Map represents both the acceptable range of 1D clearance and the acceptable limits to the 3D variational possibilities of the target face consistent with it. The geometric shapes of the perfect-form feature T-Map, together with subsets of certain perfect-form variations within it, represent all variational possibilities of manufacture of the part. This array of possibilities is combined with the functional T-Map to compute a measure of all variational possibilities that will give each value of clearance. The measures are then arranged as a probability density function over the acceptable range of clearance. In this paper form variations are incorporated as subsets in order to model the coupling between form variations and the allowable range of actual-mating-size, as required by Rule #1 of the ASME Standard. Two distributions for flatness are used as examples to illustrate the influence of form on the frequency distribution for 1D clearance: the uniform distribution and the Gaussian distribution that has been truncated symmetrically to a range of six standard deviations.