Allocating tolerances statistically with tolerance-maps and beta distributions: The target a planar face

A new math model for geometric tolerances is used to build the frequency distribution for clearance in an assembly of parts, each of which is manufactured to a given set of size and orientation tolerances. The central element of the new math model is the Tolerance-Map® (T-Map®); it is the range of points resulting from a one-to-one mapping from all the variational possibilities of a feature, within its tolerance-zone, to a specially designed Euclidean point-space. A functional T-Map represents both the acceptable range of 1-D clearance and the acceptable limits to the 3-D variational possibilities of the target face consistent with it. An accumulation T-Map represents all the accumulated 3-D variational possibilities of the target which arise from allowable manufacturing variations on the individual parts in the assembly. The geometric shapes of the accumulation and functional maps are used to compute a measure of all variational possibilities of manufacture of the parts which will give each value of clearance. The measures are then arranged as a probability density function over the acceptable range of clearance, and a beta distribution is fitted to it. The method is applied to two examples.