Lazhar Homri

Tolerance analysis by polytopes

To determine the relative position of any two surfaces in a system, one approach is to use operations (Minkowski sum and intersection) on sets of constraints. These constraints are made compliant with half-spaces of R n where each set of half-spaces defines an operand polyhedron. These operands are generally unbounded due to the inclusion of degrees of invariance for surfaces and degrees of freedom for joints defining theoretically unlimited displacements. To solve operations on operands, Minkowski sums in particular, cap half-spaces are added to each polyhedron to make it compliant with a polytope which is by definition a bounded polyhedron. The difficulty of this method lies in controlling the influence of these additional half-spaces on the topology of polytopes calculated by sum or intersection. This is necessary to validate the geometric tolerances that ensure the compliance of a mechanical system in terms of functional requirements. In tolerance analysis, sets of constraints can be compliant with operand polyhedra.These operands are generally unbounded due to the inclusion of degrees of freedom.Cap half-spaces are added to each polyhedron to make it compliant with a polytope.The influence of the cap half-spaces on the topology of polytopes must be controlled.It is necessary to ensure the compliance of a mechanism in terms of requirements.

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

Abstract 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.

Tolerancing Analysis by Operations on Polytopes

Geometric tolerancing analysis consists of simulating the behavior of a mechanical system according to geometric defects in the constituent parts. The aim is to verify system compliance in terms of the functional requirements for its expected operation. When carrying out the simulation the geometric specifications of the constituent parts and specifications of parts potentially in contact must be taken into account. One approach using polytopes consists of characterizing the specifications of the parts, the specifications of the contacts and the functional requirements of the mechanical system using sets of geometric constraints. This article describes modeling different sets of constraints manipulated by polytopes. We introduce the operations that are applied (Minkowski sum and intersection) to determine the relative position of any two surfaces of a mechanical system. Finally, tolerancing analysis of a simple mechanical system is described.