Analysis of tolerances and thermo-mechanical strains by operations on polytopes for hyperstatic mechanism architecture

Abstract

simulate system compliance with functional requirements (stator rotor clearance, flush, etc.). The tolerance optimization is a critical point to address since the mechanisms most of time corresponds to hyperstatic architectures. In such cases, the contact between surfaces of parts induced by these architectures considerably increases the complexity of the problems to be solved. More to this, the parts are generally assumed to have an infinitely rigid behaviour. This limitation should be overcome in case of flexible parts of the mechanisms subject to thermal expansion and external mechanical loads.Studies introduced by Fleming in 1988 provided the foundation for a variational approach to tolerance analysis, based on operations by sets of geometric constraints. 1 A set of geometric constraints defines all the possible positions of a surface within a tolerance zone 2 which can be generated by offsets of the nominal model of the part. 3 In this way the geometric variations of a part that are compliant with ISO specifications for orientation or position tolerances can be characterised. [4][5][6][7] In the same way, a set of geometric constraints can also be used to characterise all relative positions between two distinct surfaces that are potentially in contact. 8 Fleming established the correlation between cumulative defect limits on parts in contact and the Minkowski sum of finite sets of geometric constraints. 1 A detailed synthesis of this is given in. 9 Algorithms of Minkowski sums applied to the problem of tolerance analysis have also been developed. 10,11 Giordano showed that modelling the relative positions of two parts resulting from several potential contacts can be formalised by an operation involving the intersection of sets of geometric constraints. 12 More generally, the variational approach to tolerancing consists of characterising the relative position of two surfaces from any two parts of a system by intersections and Minkowski sums of sets of geometric constraints derived from ISO specifications for the parts and specifications formulated specifically for two parts potentially in contact. 13 The variational approach to geometric tolerancing differs from parametric approaches. 14 Parametric approaches, especially those used in the various commercial tools, formalise the relative position of any two surfaces of a mechanism at a specific point by a simple relation (linear or non-linear) between parameters of position (translation and/or rotation). This relation is obtained using either an analytical method 15-18 or a Monte Carlo method. 19 This type of approach does not support the redundancy of suppressing degrees of freedom between two parts. In addition, it is generally necessary to generate several equations to simulate the relative position of two surfaces. Historically the procedures for tolerance analysis using a variational or parametric approach are based on the following physical hypotheses: no defect in the shape of the real surfaces, no local strain on surfaces in contact, and no flexible pa...