A New Mathematical Model for Geometric Tolerances as Applied to Axes

Abstract

A new mathematical model for representing the geometric variations of lines is extended to include form and accumulation (stackup) of tolerances in an assembly. The model is compatible with the ASME/ANSI/ISO Standards for geometric tolerances. Central to the new model is a Tolerance-Map©, a hypothetical volume of points which corresponds to all possible locations and variations of a segment of a line (the axis) which can arise from tolerances on size, position, orientation, and form. Every Tolerance-Map is a convex set in a metric space. The new model makes stackup relations apparent in an assembly, and these can be used to allocate size and orientational tolerances; the same relations also can be used to identify sensitivities for these tolerances. All stackup relations can be met for 100% interchangeability or for a specified probability. Much of the detail in this paper would probably reside internally to software for designers, yet would not be included in the interface; its workings should be invisible to the user.