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Realistic Assembly Modeling for Computer Aided Tolerancing -

# Realistic Assembly Modeling for Computer Aided Tolerancing

## Introduction

In CAD models, the tolerances are represented as annotations. In fact, the numerical model is used in the nominal configuration. The values of the dimensional and geometrical tolerances do not have any impact on CAD models or the results of numerical simulations. Research asserts that the tolerance stack-up has an impact on the Finite Element (FE) results [1] and the assembly process [2], as is the case with sheet metal assembly [3]. Thus, a realistic modeling of the assembly, which considers the geometrical and dimensional tolerances on CAD models, is a very real industrial need. In this paper, a method of modelling realistic assembly (variational assembly) is presented. This new method allows for tolerance analysis while considering assembly process planning, contact types and assembly motion.

In our previous work, a method taking into account the tolerances on CAD models was established [4]. This model is based on the assumption of obtaining the worst case assemblies by using components in the worst case configurations. The worst case configurations of components are computed using the worst case tolerancing. The randomness of obtaining mechanical parts and the assembly complexity cast doubt on this assumption. Thus, in this work, the assumption of obtaining worst case assemblies by random components is adopted. New algorithms to compute the possible configurations of components and redefine the assembly mating constraints will be established.

## Research done

In this paper, the components with dimensional and geometrical tolerances are modeled in CAD models by using two algorithms to respect the principle of independence (ISO 8015):

1. The first one leads to obtaining a component with dimensional defects. This algorithm is founded on a numerical perturbation method which determines the relationships between dimensions. The tolerance interval is discretized to define the realistic configuration of components;
2. The second algorithm allows the determination of components with geometrical tolerances. The tolerance zone is discretized according to the parameters specified by Small Displacement Torsors (SDT) [5]. The face displacements are performed to obtain components with geometrical defects.

In nominal assembly, the ideal components are linked by mating constraints. The replacement of those components by the realistic ones requires the updating of mating constraints. A tolerance analysis is performed by using the proposed method and showed through the crankshaft system (fig. 1). The proposed analysis method takes into account not only part deviations but also assembly process planning and contact types between parts. In addition, the impact of the tolerance stack-up on the functional requirement in a dynamic environment can be determined.

Fig. 1. (a) Nominal assembly; (a) Realistic assembly configuration Cm (α= 50°); (b) Detection of interferences in the realistic configuration Cm (α= 129°).

## Literature review synthesis

The literature review relates two aspects. The first aspect consists of tolerance analysis and synthesis [5, 7 -24]. The second aspect consists of the variational geometry and the redefinition of assembly mating constraints [30, 33-38].
In most of the references cited, the tolerance analysis and syntheses are based on the spatial math representations (polyhedral and Technical Manual Application System (T-Maps)) or mathematical model. The solutions for the tolerance analysis and syntheses, which are presented in mathematical form (equations), are modeled in a geometrical solution. The spatial math models (polyhedral and T-Maps) are difficult tools to use in industry as this model does not predict the impact of dimensional and geometrical deviation, permitted by tolerances, on assembly deformations. The proposed model in this paper is considered a tool for visualization and tolerance impact analysis (interference detection) in assembly simulation cases. The second aspect addressed in the literature appears to be closest to this work [30, 33-38]. Indeed, we can learn a great deal from the methods used in this second aspect. The later can be applied to solve some issues of the problem as contemplated: The parametric variational features is used to generate realistic assembly.

## Research approach

The approach presented in this paper is an extension of these works. The realistic modeling of components allows us to take into account the tolerances in CAD models. The redefinition of mating constraints leads us to consider the relationships between realistic components and the assembly process planning. The realistic assembly modeling in a CAD environment allows for direct use of computation applications which already exist in CAD software as assembly motion simulations.

## Modeling of a realistic component with dimensional defects

In CAD model, obtaining realistic configurations allowed by dimensional tolerances requires modifications of the component dimensions. CAD software defines two dimension types. The driving dimensions are used to build the CAD model. The driven dimensions are controlled by driving ones. In fact, the driven dimension values are deduced from driving dimension ones. The toleranced dimension can correspond to a driving dimension: the driven dimension is controlled by one driving dimension in Fig. 2(a). Therefore, the modification of the driving dimension value ( 40 mm ) allows us to directly change the driven dimension’s value. In the other case, obtaining a target value of the driven dimension requires the determination and the modification of the corresponding values for each driving dimensions (25 mm,35 mm and 100 mm in Fig. 2 (b)).

Fig. 2. Driven and driving dimensions: (a) driven dimension controlled by one driving dimension, (b) driven dimension controlled by three driving dimensions.

In the general case, the designer defines the component model by using driving dimensions. In the following steps, he assigns tolerances to the driven dimensions. The determination of the components with dimensional defects requires the identification of the relationships between thedriving and driven dimensions.

The components with dimensional defects are obtained through a numerical perturbation method. In fact, the influence coefficients between driven and driving dimensions are identified. Therefore the values of the new driving dimensions are computed according to those coefficients and tolerance values. After the allocation of those new dimensions to the CAD model, the components with dimensional defects were deduced.

## Modeling of a realistic component with geometrical defects

An algorithm to determine all components with geometrical defects is developed. Those realistic configurations are obtained by modeling the geometrical tolerances by the displacements of the corresponding faces. This approach depends on the shape of the tolerance zone, the toleranced feature type and the tolerance type. The overview table in Tab. 1 comprises the sub-algorithms used to model components with geometrical defects.

Tab. 1. Sub-algorithms used to model components with geometrical defects

The deviation between the nominal and the realistic feature is determined by the Small Displacement Torsors (SDT) tool. Then, the assumption of neglecting the form defects relative to the position and orientation defects is used. The deviation torsor, which is based on a SDT tool, defines the Degrees of Freedom (DoFs) of the toleranced feature. In the proposed model, the deviation parameters between the nominal and realistic features are determined by an analogy to the parameters defined by the SDT. Sub-algorithms are developed to obtain realistic configurations of a planar face or a cylindrical axis subjected to a positional and/or orientation tolerance, as the case of planar face subjected to a positional tolerance (Fig. 3). In a CAD model, the realistic part configurations are generated with face displacements: Rotation and/or translation using the parameters deduced from the discretization. In the proposed methodology, the original shape of the feature is conserved as the planar faces remain planar. The algorithm starts by identifying toleranced features, tolerance types and datum references, which are specified by ISO standards.

Fig. 3. (a) The part drawing, (b) The sub-algorithm PCCPTP, (c) The discretization of the tolerance zone, (d) The setting of the tolerance zone.

## Redefinition of the assembly mating constraints

In the realistic assembly, these mating constraints must be redefined. These realistic mating constraints are obtained by modifications of the relations between Minimum Geometrical Reference Elements (MGREs). This method depends on the Objective Function of the Assembly (OFA) specified by the designer. In CAD software, the OFA is automatically deduced from the nominal model:

• The mating constraint order, specified in the feature manager design tree of the software, defines the mounting order of the assembly and the joint order priority. Thus, the planning of the assembly process is identified;
• The kinematic state of the nominal assembly defines the DoFs which are to be conserved in the realistic model. The DoFs are identified by a method based on the graphs of primitive kinematic joints: Each assembly or sub-assembly is defined by a graph. In this graph, a node represents a part and an edge represents a primitive joint;
• The contact between the features is conserved;
• The joint type between each couple of components is respected.

In the context of this article, three primitive mating constraints are considered: A coincident between two planar faces, a coaxiality between two axes and a distance between two planar faces.

## Conclusion

In this paper, a model for the integration of dimensional and geometrical tolerances in CAD models is presented: the components with defects are automatically obtained according to the tolerances and a new method to redefine mating constraints according to the OFA is established.

The developed model is a tool for tolerance analysis while considering the assembly process planning and the various types of contact between parts. The tolerance integration allows us to use the DMU in a realistic configuration. The tolerance impacts on the results of the F.E. calculation or the dynamic computation can be performed.

Future research will focus on the consideration of form defects on CAD model through another type of tolerancing approach. In addition, a modeling of the realistic assemblies according to statistical tolerancing can be obtained.

## Research paper

For a more comprehensive discussion about “An algorithm for CAD Tolerancing Integration: Generation of assembly configurations according to dimensional and geometrical tolerances”, we invite you to read the following Research Paper:
Borhen Louhichi, Mehdi Tlija, Abdelmajid Benamara, Antoine Tahan. An algorithm for CAD Tolerancing Integration: Generation of assembly configurations according to dimensional and geometrical tolerances. The International Journal of Advanced Manufacturing Technology, July 2014, DOI: 10.1016/j.cad.2014.07.002.

#### Abdelmajid BenAmara

Profil de l'auteur(e)

Mehdi Tlija est assistant à l’ISSAT. Il est actuellement candidat au doctorat à l’ENIM. Ses principaux intérêts de recherche sont la conception assistée par ordinateur et le tolérancement.

#### Bohren Louhichi

Profil de l'auteur(e)

#### Souheil-Antoine Tahan

Profil de l'auteur(e)

Antoine Tahan est professeur de génie mécanique de l’ÉTS.

Programme : Génie mécanique

Laboratoires de recherche : LIPPS – Laboratoire d'ingénierie des produits, procédés et systèmes

#### Mehdi Tlija

Profil de l'auteur(e)

Mehdi Ttlijalija est assistant de recherche à l’ISSAT de Sousse et doctorant à l’ENIM de Monastir en Tunisie. Ses intérêts de recherche sont la conception assistée par ordinateur et le tolérancement.

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