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A research paper introduction is a blog article presenting a research paper done by researchers from École de technologie supérieure (ÉTS) de Montréal.
In the Digital Mock-Up (DMU), the simulation of the mechanical deformations and the prediction of the functional requirements represent two main phases of the product design. The results of those calculations depend on adopted CAD model. In the DMU, the part and the assembly are represented by their nominal dimensions. In fact, tolerances are defined as annotations and are neglected in the geometric model. Consequently, neglecting tolerance impacts causes system malfunctions at mountain process and during the operating phase. In the current Product Life cycle Management (PLM), the detection of those tolerance impacts is made only after the product manufacturing . The correction of these errors at this stage requires additional costs . Hence, the prediction of these errors represents an industrial need and a financial gain, and can be performed by using a realistic model (Fig. 1).
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 . 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).
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).
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 .
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.
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.
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).
In the CAD model, the discretization is carried out by using the polar coordinate system . The displacement of the toleranced face implies the change of the adjacent faces geometry (Fig. 8).
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.
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 . 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)  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.
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.
A demonstration of the complete method is given with an example in the author’s article cited below.
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).
Mehdi Ttlijalija is an assistant at the Institute of Applied Sciences and Technology of Sousse and a Ph.D. student in the National Engineering School of Monastir, Tunisia. His research interests are computer aided design and tolerancing.