Relative Positioning of Planar Parts in Toleranced Assemblies

Accounting for geometric variability in mechanical assemblies is a key component of modern design methodologies. This paper presents a framework for worst case analysis of the relative position variation of toleranced parts. The framework is based on our general parametric tolerancing model for planar parts whose boundaries consist of line and arc segments and whose vertices are described by standard elementary functions of part dimensions, which vary within tolerance intervals. Here, we present six types of relative position constraints designed to model all types of contact and clearance specifications between features of two parts. For these pairwise constraints, we describe an algorithm to compute the sensitivity matrix of each vertex. This matrix describes the vertex position variation satisfying the constraints as a linear function of the two part parameters. To model the relative part position variation in the entire assembly, we introduce the assembly graph, a generalization of Latombe’s relation graph that includes cycles, toleranced parts, and three degrees of freedom. We show how to compute the sensitivity matrices of each vertex from the pairwise relative position constraints and the assembly graph. These matrices serve to compute the tolerance envelopes bounding the areas occupied by the parts under all possible assembly instances. The envelopes are an accurate characterization of geometric uncertainty for assembly planning and mechanism design.