Tolerance analysis by polytopes: Taking into account degrees of freedom with cap half-spaces

Homri, Lazhar; Teissandier, Denis; Ballu, Alex

doi:10.1016/j.cad.2014.11.005

Cited by 42 publications

(26 citation statements)

References 24 publications

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“…In comparison with the strategy based on 6-polytopes [8,9], the method proposed in this paper allows decreasing the computation time of Minkowski sums of polytopes by taking information from the tolerance analysis problem to simplify the operands and to perform the computation in the subspace of smallest possible dimension.…”

Section: Discussionmentioning

confidence: 99%

Adapting Polytopes Dimension for Managing Degrees of Freedom in Tolerancing Analysis

Arroyave-Tobón¹,

Teissandier²,

Delos³

2016

Procedia CIRP

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In tolerancing analysis, geometrical or contact specifications can be represented by polytopes. Due to the degrees of invariance of surfaces and that of freedom of joints, these operand polytopes are originally unbounded in most of the cases (i.e. polyhedra). Homri et al. proposed the introduction of virtual boundaries (called cap half-spaces) over the unbounded displacements of each polyhedron to turn them into 6-polytopes. This decision was motivated by the complexity that operating on polyhedra in R 6 supposes. However, that strategy has to face the multiplication of the number of cap half-spaces during the computation of Minkowski sums. In general, the time for computing cap facets is greater than for computing facets representing real limits of bounded displacements. In order to deal with that, this paper proposes the use of the theory of screws to determine the set of displacements that defines the positioning of one surface in relation to another. This set of displacements defines the subspace of R 6 in which the polytopes of the respective surfaces have to be projected and operated to avoid calculating facets and vertices along the directions of unbounded displacements. With this new strategy it is possible to decrease the complexity of the Minkowski sums by reducing the dimension of the operands and consequently reducing the computation time. An example illustrates the method and shows the time reduction during the computations.

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Section: Discussionmentioning

confidence: 99%

Adapting Polytopes Dimension for Managing Degrees of Freedom in Tolerancing Analysis

Arroyave-Tobón¹,

Teissandier²,

Delos³

2016

Procedia CIRP

Self Cite

View full text Add to dashboard Cite

show abstract

“…Here, the equations define the relations of displacements in the different loops of the joint graph, the relations of displacements represent the linear compatibility constraints between deviations and gaps in different loops, while the inequalities and equations define the interface constraints that characterize the non-interferences between surfaces that are nominally in contact with each other [32][33][34][35][36]. Nevertheless, considering positional deviations in a 3-dimensional context could lead to highly nonlinear functions which then have to be linearized piecewise [7,37].…”

Section: Geometrical Behavior Modelingmentioning

confidence: 99%

“…The model enabled a 3D simulation of all variations of features (i.e., size, orientation and form) [44,45] and links [35]. Homri et al [7] also developed polytopes method to simulate variations in the overconstrained mechanisms. These mathematical representations of tolerances allowed computing accumulation of the tolerances by Minkowski sums or intersections and the operations depended on the kinematic chains.…”

Section: Technical Solutions and Analysis Approachesmentioning

confidence: 99%

“…Pertaining to technical solutions, a rich overview of tolerance analysis models of over-constrained mechanisms is also given in [3,6,7,34,38,41,[51][52][53][54][55][56][57]. Globally, the main shortcomings of the tolerance analysis approaches are their insufficient consideration of parts' form defects in the respective systems.…”

Section: Technical Solutions and Analysis Approachesmentioning

confidence: 99%

“…Moreover, tolerancing activities, when simulating geometric deviations, are generally based on the hypothesis that parts are ideals and surfaces have no form defects [4][5][6][7]. The consideration of parts' form defects is integral to the tolerancing activities and hence, make it significant to highlight its impacts on the assembly probability estimation in the product quality and cost assessment [8,9].…”

Section: Introductionmentioning

confidence: 99%

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Tolerance analysis — Form defects modeling and simulation by modal decomposition and optimization

Homri

Goka

Levasseur

et al. 2017

Computer-Aided Design

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is an open access repository that collects the work of Arts et Métiers ParisTech researchers and makes it freely available over the web where possible. Tolerance analysis aims on checking whether specified tolerances enable functional and assembly requirements. The tolerance analysis approaches discussed in literature are generally assumed without the consideration of parts' form defects. This paper presents a new model to consider the form defects in an assembly simulation. A Metric Modal Decomposition (MMD) method is henceforth, developed to model the form defects of various parts in a mechanism. The assemblies including form defects are further assessed using mathematical optimization. The optimization involves two models of surfaces: real model and difference surface-base method, and introduces the concept of signed distance. The optimization algorithms are then compared in terms of time consumption and accuracy. To illustrate the methods and their respective applications, a simplified over-constrained industrial mechanism in three dimensions is also used as a case study.

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How to trace the significant information in tolerance analysis with polytopes

Delos¹,

Teissandier²,

Arroyave-Tobón³

2016

Lecture Notes in Mechanical Engineering

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International audienceManufacturing a mechanical system is tightly related to geometric tolerancing. In this field, verifying the respect of the functional conditions relies on a way to model both the contacts and the manufacturing defects. Polytopes have proven to be a powerful tool to fulfill such objectives. However they've been used so far as restrictions of underlying polyhedra, with the aim of handling bounded bodies to fit with algorithmic requirements. This approach has reached its limit in terms of complexity. The method presented here allows us to treat polytopes, as if they were polyhedra, separating the facets holding the relevant physical information from the bounding facets. Separating the facets not only saves computation time but also memory space in a very significant way. The second part of the paper is dedicated to running the algorithm on a mechanical example to prove its efficiency

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Tolerance analysis by polytopes: Taking into account degrees of freedom with cap half-spaces

Cited by 42 publications

References 24 publications

Adapting Polytopes Dimension for Managing Degrees of Freedom in Tolerancing Analysis

Adapting Polytopes Dimension for Managing Degrees of Freedom in Tolerancing Analysis

Tolerance analysis — Form defects modeling and simulation by modal decomposition and optimization

How to trace the significant information in tolerance analysis with polytopes

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