Évaluer l’effet des tolérances sur les exigences fonctionnelles des assemblages

Dans la maquette numérique (DMU), la simulation des déformations mécaniques et la prédiction des besoins fonctionnels représentent deux principales phases de la conception du produit. Les résultats de ces calculs dépendent du modèle CAO adoptée. Dans la DMU, la pièce et l’assemblage sont représentés par leurs dimensions nominales. En fait, les tolérances sont définies comme annotations et sont négligées dans le modèle géométrique. Par conséquent, en négligeant les effets de tolérance, cela provoque un dysfonctionnement du système au processus de montage et pendant la phase d’exploitation. Dans la gestion du cycle de vie des produits en cours (PLM), la détection des impacts des tolérances se fait uniquement après la réalisation du produit [1]. La correction de ces erreurs à ce stade nécessite des coûts supplémentaires [2]. Par conséquent, la prédiction de ces erreurs représente un besoin industriel et un gain financier, et peut être effectuée en utilisant un modèle réaliste (Fig. 1).

La suite de ce RPI sera présentée dans sa langue originale de rédaction.

Fig. 1. Correction of design defects caused by tolerances in an early phase of product life cycle management.

Thus, the main objective of this paper is to integrate the tolerancing in the CAD model by determining worst case configurations of components and assemblies: realistic models. A model taking into account dimensional tolerances has been developed in our previous work [3]. In this paper, worst case configurations are modelled using the geometrical tolerances.

Algorithm of realistic assemblies modelling

The integration of tolerances in the geometric model allows obtaining realistic assemblies. These assemblies enable the possibility to evaluate tolerance impacts on assembly requirements and to predict assembly deformations. To obtain realistic assembly an algorithm is proposed (Fig. 2).

Fig. 2. Modelling of the realistic assembly.

The methodology consists in determining worst case configurations of an assembly that are required and imposed by tolerances. These models are given by applying the worst case tolerancing [4,5] to all toleranced faces of the components. In our study, the principle of tolerance independency according to ISO 8015 is considered. Mathematical formulations of tolerance zones are obtained by the domains method. Indeed, the SDTs are used to model the geometrical deviations. Then, the assumption of neglecting form defects relative to orientation and position defects is adopted. Hence, allowed extreme positions of faces are calculated. Thereafter, displacements parameters of faces, which are required by the tolerance, are deduced. Thus, realistic assemblies, which take into account the tolerances in CAD model, are obtained by rebuilding assemblies.

A model taking into account tolerances in CAD model

The consideration of the tolerances in CAD model is obtained by face displacements. Therefore, subalgorithms were developed to realize the desired displacements of faces. Parameters of each displacement are calculated by using the domains method and a worst case approach. The deviation between nominal and realistic feature is determined by the SDT tool. Then, form deviations are neglected relative to those of orientation and position. The model is integrated under the CAD software “SolidWorks”. The methodology to determine worst case faces depends on the type of the tolerance face. The face displacements and the identification of the tolerance type and the toleranced feature are automated.

Displacements of a planar face

In the case of planar surface, components of variations between the real coordinate system (associated with real surface) and the nominal coordinate system (associated with nominal surface) are one translation and two rotations (Fig. 3).

Fig. 3. Definition of the deviation torsor of a planar surface.

In order to illustrate the possible displacements of a planar face, a prismatic part is taken as an example (Fig. 4a). The determination of all configurations with worst case tolerancing for this surface requires the calculation of displacements of points I, J, K and L [4].

Fig. 4. (a) Planar face subjected to a positional tolerance. (b) Rotation of a planar face.

A sub-algorithm is developed to determine the worst case configurations of the Planar Face with Quadratic Loop which is subjected to Positional Tolerance (Sub-algorithm PFQLPT) (Fig. 5). Figure 4b shows the rotation method of a planar face about the x axis (small median of rectangle) by an angle equal to Tl/b. The rebuilding of the geometrical model requires the updating of the faces influenced by the displacement of that toleranced one.

Fig. 5. Sub-Algorithm PFQLPT.

Case of a cylindrical tolerance zone

In the case of positional tolerance (t) assigned to an axis AB of a cylindrical face (or a conical face), the tolerance zone is a cylinder as illustrated (Fig. 6a). The three data (A, B and C) are supposed ideal faces.

Fig. 6. (a) Positional tolerance allotted to cylinder axis. (b) Calculation of the deviation torsor of a cylindrical face.

Geometric deviations are represented by SDT. Then, form defects are neglected relative to orientation and position defects (Fig. 6b). To determine the realistic configurations of the axis, the discretization of the tolerance zone (virtual zone) is carried out (Fig. 7).

Fig. 7. A discretization of the tolerance zone (with n = 8) to obtain worst case configurations of the axis.

In the CAD model, the discretization is carried out by using the polar coordinate system [6]. The displacement of the toleranced face implies the change of the adjacent faces geometry (Fig. 8).

Fig. 8. Displacement of a face: case of the cylindrical tolerance zone.

Worst case configurations of the part (Fig. 6a) are calculated and modelled in CAD model according to the method detailed previously. Two cases are illustrated in Figure 9: a rotation of the cylindrical face and a translation of the same face along the axis.

Fig. 9. Realistic modeling of a cylindrical face.

Displacement of a planar face with non quadratic loop

In the case of toleranced planar surface with non quadratic loop, the associated surface is the bounding polygonal one [7]. In fact, the surface loop is discretized by vertices. The n vertices obtained by this discretization must remain inside the tolerance zone. Then, worst case displacements of the n vertices are defined by n inequalities. Indeed, the number of worst case configurations of the face is proportional to the number of discretization vertices. The number of configurations becomes very large, especially if the contour contains a circular portion.

The oriented bounding box (OBB) [8] allows enveloping the surface. The extremes OBB displacements along its eigenvectors are the extremes displacements of the corresponding surface. In the case of planar surface, the OBB is rectangular and planar. Then, the OBB is associated to toleranced surface. The identification of worst cases of toleranced surface is realized by using a method based on OBB (Fig. 10). In fact, worst case displacements of the toleranced face are deduced from worst case displacements of the corresponding OBB.

Fig. 10. Methodology of determination of face displacement parameters: case of planar face with complex loop.

Initially, the OBB of the toleranced face is calculated. The obtained OBB is a flattened parallelepiped (the box height is zero) and it has a four directing vertices (a rectangular loop). Hence, extreme displacements of driving vertices generate those of the face. Displacement settings of vertices are calculated by the previous algorithm used in the case of planar face with rectangular loop (sub-Algorithm PFQLPT in Fig. 5). Figure 11 illustrates an example of OBB.

Fig. 11. Example of OBB.

A demonstration of the complete method is given with an example in the author’s article cited below.

Conclusion

In this paper, a model is presented in order to obtain realistic assemblies. The model enables the tolerance analysis. The approach is based on tolerancing by the domains method and on the approach of worst cases. The realistic model is obtained by displacements of tolerance features to worst case configurations. Sub-algorithms are developed to manipulate some particular cases. In fact, the tool of oriented bounding box allows circumventing the displacement problem of planar faces with complex loop. In addition, the MMC (or LMC) requirement and datum priority order are respected in the proposed model. The difference between the numerical model and the real product is reduced by the proposed model. Then, in the digital Mock-up, the control of functional requirements and the optimization of assembly deformations are performed with realistic model.

For a more comprehensive discussion about “Evaluating the effect of tolerances on the functional requirements of assemblies”, we invite you to read the following Research Paper:

Mehdi Tlija , Borhen Louhichi and Abdelmajid BenAmara. Evaluating the effect of tolerances on the functional requirements of assemblies Mechanics & Industry 14, 191–206 (2013).