Denis Teissandier

A computer aided tolerancing model: proportioned assembly clearance volume

Abstract The object of this article is to present a tolerancing model, the “Proportioned Assembly Clearance Volume: PACV”. The finality of the PACV is to create a three-dimensional (3D) tolerancing analysis tool that takes into account only standardized specifications. The PACV is based on the Small Displacements Torsor (SDT) concept. SDT is used to express the relative position between two ideal surfaces. The rotations between two geometric features are linearized, i.e. displacements are transformed into small displacements. An ideal surface is a surface, which can be characterized by a finite number of geometric features: point, centerline, part face, etc. A nominal surface is an ideal surface by definition. By modeling fabricated surfaces in ideal surfaces, it is possible to compute the limits of small displacements of a fabricated surface inside a tolerance zone. The values of these limits define a PACV. With a similar method, the limits of small displacements between two surfaces of two distinct parts, e.g. clearance in a joint, can be determined: they define a PACV. Using a graph, we illustrate how PACV (edges) could be associated in series or in parallel between two any surfaces (vertices) in an assembly, in order to create 3D dimension-chains. The governing rule of PACV in series is introduced. In addition, one example of computation of 3D dimension-chain (result of an association of PACV in series) is presented.

Operations on polytopes: application to tolerance analysis

This article presents numerical methods in order to solve problems of tolerance analysis. A geometric specification, a contact specification and a functional requirement can be respectively characterized by a fmite set of geometric constraints, a finite set of contact constraints and a finite set of functional constraints. Mathematically each constraint formalises a n-face (hyperplan of dimension n) of a n-polytope (1 ≤ n ≤ 6). Thus the relative position between two any surfaces of a mechanism can be calculated with two operations on polytopes: the Minkowski sum and the Intersection. The result is a new polytope: the calculated polytope. The inclusion of the calculated polytope inside the functional polytope indicates if the functional requirement is satisfied or not satisfied. Examples illustrate these numerical methods.

Algorithm to calculate the Minkowski sums of 3-polytopes based on normal fans

Prompted by the development of algorithms for analysing geometric tolerancing, this article describes a method to determine the Minkowski sum for 3-dimensional polytopes. This method is based exclusively on intersection operations on normal cones, using the properties of the normal fan of a Minkowski sum obtained by common refinement of the normal fans of the operands. It can be used to determine from which vertices of the operands the vertices of the Minkowski sum derive. It is also possible to determine to which facets of the operands each facet of the Minkowski sum is oriented. The basic properties of the algorithms can be applied to n-polytopes. First, the main properties of the duality of normal cones and primal cones associated with the vertices of a polytope are described. Next, the properties of normal fans are applied to define the vertices and facets of the Minkowski sum of two polytopes. An algorithm is proposed, which generalises the method. Lastly, there is a discussion of the features of this algorithm, developed using the OpenCascade environment.

Tolerance analysis with polytopes in HV-description

This article proposes the use of polytopes in

Tolerancing Analysis by Operations on Polytopes

\mathcal{HV}

Minkowski Sum of Polytopes Defined by Their Vertices

-description to solve tolerance analysis problems. Polytopes are defined by a finite set of half-spaces representing geometric, contact or functional specifications. However, the list of the vertices of the polytopes are useful for computing other operations as Minkowski sums. Then, this paper proposes a truncation algorithm to obtain the

Minkowski sum of HV-polytopes in Rn

\mathcal{V}

An Adaptive Tolerance Model For Collaborative Design

-description of polytopes in

How to trace the significant information in tolerance analysis with polytopes

\mathbb{R}^n

Three-dimensional Functional Tolerancing with Proportioned Assemblies Clearance Volume (U.P.E.L : Unions Pondérées d’Espaces de Liberté) : application to setup planning

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