Polyhedral-based Modeling and Algorithms for Tolerancing Analysis

Abstract

The cumulative stack-up of geometric variations in mechanical systems can be modeled summing and intersecting sets of constraints. These constraints derive from tolerance zones or from contact restrictions between parts. The advantage of this approach is its robustness for treating any kind of mechanisms, including the over-constrained ones. However, the sum of constraints, which must be computed when simulating the accumulation of defects in serial joints, is a very time-consuming operation. In previous papers, we proposed to virtually limit the degrees of freedom of the toleranced features and joints turning the polyhedra into polytopes to avoid manipulating unbounded objects. Even though this approach enables to process the whole mechanism, it also introduces bounding or cap facets which increase the complexity of the operand sets after each operation until becoming far too significant. In this work, we introduce algorithms summing, intersecting and testing inclusions. As they operate on sets of constraints using unbounded polyhedral objects, we identify the smaller sub-space in which the projection of these operands are bounded sets. Calculating the sum in this sub-space allows reducing the operands complexity significantly and consequently the computational time. Then, checking the final inclusion informs us not only about the compliance of the mechanism tolerances with respect to the functional specification but also to quantify how far we are from this target. Finally prismatic polyhedra integrate ISO and contacts specifications in a very natural way and are able to perform a full kinematic analysis of the mechanism. After presenting the geometric properties on which this approach rely, we demonstrate it on an industrial case. Then we compare the computation times, prove the robustness of the new method and show how to quantify the functional condition compliance with respect to a given set of tolerances.