The geometry of 3D statistical tolerancing

Radial errors and non-commuting manufacturing processes

Abstract

Normally distributed Cartesian errors centered at nominal produce a radial probability density concentrated away from nominal, becoming increasingly shell-like in higher dimensions. Position and runout measurements are examples of this in two dimensions. Relations are given for the variance of a gap in a three-dimensional stack path and for radial error when Cartesian component variances are equal and for when they are unequal, and examples are given of each. The effect of physically ordered, non-commuting manufacturing operations on covariance-based error propagation is examined. Higher-order moment methods are also developed for nonlinear error propagation and compared with Monte Carlo simulations.