Comparison of analysis line and polytopes methods to determine the result of a tolerance chain

Abstract : Functional tolerancing must ensure the assembly and the functioning of a mechanism. There are several types of functional requirements: mainly the assembly requirements which are expressed in terms of maximum material conditions and accuracy requirements that impose minimum material conditions. The approach consists of five steps: -Definition of each functional requirement by geometrical characteristics between some functional surfaces of mechanism and determination of the limit values of this characteristic allowing insuring the requirement with an acceptable risk, -Determination of the annotations of each part with the standards ISO of dimensioning and tolerancing, -Choice of the tolerances of each specification of each part, -Verification that the accumulation of the tolerances guarantees the respect for all the requirements, -Optimization of the allocation of tolerances to minimize cost or to maximize the capability of manufacturing processes. In this approach of functional dimensioning, the tolerance analysis is a very delicate stage to take into account the dimensional and angular effects. The problem is to ensure that the studied functional requirement will be well respected considering the tolerances on influential parts. The aim of this paper is to compare two methods of tolerance analysis of a mechanical system: the method of "analysis lines" and the method of "polytopes". The first method needs a discretization of the ending functional surface according to various analysis lines placed on the outer-bound of the face and oriented along the normal of the surface. For each analysis line, the result is the sum of the influences of each junction defects on the search characteristic using pre-established relations for classical types of junctions. The calculation is very fast and it can express the result as a formula according to the tolerances, if necessary with a statistical approach. The second method uses polytopes (finite sets of point in n). The polytopes are defined from the acceptable limits of the geometric deviations of parts and possible displacements between two parts. Minkowski sums and intersections polytopes are then carried out to take into account all geometric variations of a mechanism. Tools like OpenCASCADE have been developed to quickly realize intersections and Minkowski sums. This comparison is performed on a simple example specified with the method CLIC. The requirement is the assembly condition of a shaft in two bores of two different parts which involves straightness in common zone of the bores. Comparison of these results shows that solutions are identical. Assumptions and model behaviour bonds are also identical.
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Laurent Pierre, Bernard Anselmetti. Comparison of analysis line and polytopes methods to determine the result of a tolerance chain. 13th CIRP International Seminar on Computer Aided Tolerancing, May 2014, Hangzhou, China. ⟨hal-01094151⟩

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