FORMAL DEFINITION OF TOLERANCING IN CAD AND METROLOGY

Transcription

1 P. SERRÉ, A. CLÉENT, A. RIVIÈRE FORAL DEFINITION OF TOLERANCING IN CAD AND ETROLOGY Abstract: Our aim is to unify mathematically the secification and the metrological verification of a given geometrical object. The observed differences result from the fact that strictly geometrical arameters are used in CAD systems while in mechanical engineering distances are aramount. A new mathematical concet - Near Surfaces - will be defined. This concet enables secifications and metrological determination to be incororated within a unified mathematical theory. 1. INTRODUCTION The need to unify the secification of a given geometrical object from a mathematical standoint is based on the following 2 observations. First, The variation observed between the secification of a geometrical object by means of Euclidean geometry and its execution in a CAD software database is erfectly similar as regards its rincile to the dimensional variation observed between the CAD model, taken as the secification by a NC machine tool, and its mechanical execution. A NC machine tool is actually a coying machine, which coies the CAD model errors onto the art, adding in its own uncertainties. However, these errors are currently of the same magnitude as a result of the convergence of the effect of imrovements to NC machines tools and the increased comlexity of the surfaces used. This variation must, therefore, be considered on a global basis, or, at the very least, its relations determined. Second, In the domain of CAD, we maniulate secifications whereas in the domain of mechanical engineering, we maniulate distances. Therefore, we need to be able to exress the one in terms of the other, at least on a local basis. For examle, there are 2 methods for defining the accetable variation for a geometrical object in relation to its secification; one consists of limiting secification variation and the other of limiting the Euclidean distance from a real oint to the theoretical object. The first is basically used in CAD and the second in the metrology of mechanical arts and in tolerancing. We roose to define a new mathematical concet Near Surfaces which enables the relation between secifications and verification of geometrical objects to be amalgamated. 2. OTIVATION athematicians from time immemorial have had to contend with insufficient rational numbers for modelling the continuity roerties of Euclidean geometry E 3 ;

2 P. SERRÉ, A. CLÉENT, A. RIVIÈRE however, it was not until the 19 th century that they were able to formally define the mathematical set of real numbers R. They were robably given this name sarcastically since these numbers cannot be hysically realised and can only be reresented by symbols such as 3 or π! The best known examle is the calculation of the number π by Archimedes who used 2 convergent series constituted by a series of 2 olygons, one inscribed and the other circumscribed to a circle with a radius of 1, with an indefinitely increasing number of sides. It will be noted that the rocedure can be stoed at any moment to obtain a given recision value. The series of mathematical tools used in CAD/CA and engineering science is develoed from the toology of the numerical straight line R thus created. However, this sort of model never exists, neither in comuter science nor on a machine! In ractice, in comuter science, the set of floating numbers F is the best we have and, in mechanical engineering, we only have decimal numbers of limited accuracy. This has the following disastrous consequence: from a mathematical oint of view, numerous equivalent definitions exist for the same E 3 geometric object. Unfortunately, the same is not true from a hysical or numerical standoint. Widesread belief in the ractical equivalence of all mathematical definitions results in numerous trials and tribulations. In some circumstances, certain definitions are unusable but, on the other hand, they are ideal in others. It is said that the roblem is or is not well conditioned for the values of the arameters under consideration. We will call secification this mathematical definition: A secification is a certain mathematical definition of a geometrical object. It is always evidenced by one or several equations (imlicit functions) between certain of the object s arameters. By definition, the secification variation ε is the value of the secification at measurement oint. i mes ( ) Equation ( ) = ε Equation 0 nom = i mes A oint of the real object (hysical or numerical) always resents a variation in relation to its secification that must be limited by a tolerance called the secification tolerance IT sec : ε < ITsec. The standardised method consists of limiting the Euclidean distance variation between measurement oint mes and the surface secified by a tolerance that we will call the standardised tolerance - IT standard. For the same geometrical variation, we have a comletely different secification variation, deending on the lace where the oint is located. We roose to establish the relationshi between the geometrical variation (Euclidean distance) and the secification variation.

3 FORAL DEFINITION OF TOLERANCING IN CAD AND ETROLOGY 3. SPECIFICATION AND VARIATION OF SPECIFICATION 3.1. Definitions A surface secification is a set of constraints between the co-ordinates of the running oint and its RGEs (inimum Reference Geometrical Element) by the intermediary of a certain number of osition, orientation and dimension arameters (Clément, Rivière & Serré, 1999). List of Constraints F(, ) between RGEs (Point, Straight Line, Plane) This is the skeleton The Running Point (X, Y, Z COORDINATES) This is the skin The arameters - of osition, orientation and/or dimension There are also secifications which are the relative osition constraints between 2 RGEs. List of Position Constraints G() between RGEs (Point, Straight Line, Plane) This is the skeleton The arameters - of osition, orientation and/or dimension Finally, there can be secifications which are engineering constraints resulting from the laws of hysics or technology. List of Engineering Constraints between The Running Point (X, Y, Z, dx, dy and dz COORDINATES) This is the skin The arameters - of osition, orientation and/or dimension 3.2. Relative osition of 2 surfaces at a finite distance The standardised, classic method for secifying the relative ositon of surface A in relation to surface B consists of using situation elements (a cylinder axis, Bézier s surface olygon, for examle) which have been defined as RGEs (inimum Reference Geometrical Elements) in TTRS theory (Clément, Rivière, Serré & Valade, 1997) and (Srinivasan, 1999). An examle of RGE (Fig. 1): The arabola below can be fully defined by using either of the following as RGEs: the directrix P P 1 2 and the axis of symmetry OF which constitute the axes of a Cartesian reresentation; the 2 tangents S S and S S at 2 oints secific oints and 2 which constitute the olygon of a Bézier reresentation. To osition this arabola in relation to another object, we can use the relative ositioning of any of either of these 2 RGEs and, generally seaking, it always comes down to ositioning oints, straight lines and lanes in relation to oints, straight lines and lanes. Justification for this method is immediate: the distance of any 2 surfaces using Hausdorff s distance, for examle, is totally unusable in ractice. The current method is unavoidable for the moment: the RGEs must be used to osition 2 surfaces at a finite distance.

4 P. SERRÉ, A. CLÉENT, A. RIVIÈRE P 1 1=S 0 t 1 S 1 O X F P 2 2=S 2 t2 Figure 1: Declaration of a arabola with RGE However, the situation is different when the 2 surfaces are infinitely close to one another. We roose to show that it is ossible to directly define the distance of 2 surfaces in this situation without using their RGEs. The advantageous results of this concet will be seen in the tolerancing secifications and their metrological verification Relative osition of 2 surfaces at a very small finite distance We immediately notice (in the figure below for examle) that not only is the secified noiminal surface ositioned by means of its RGEs but all the accetable surfaces by this standardised tolerancing are also defined by the same method. B d± d O 1 A B a I b A Figure 2: Standardised tolerancing

5 FORAL DEFINITION OF TOLERANCING IN CAD AND ETROLOGY For a fully secified theoretical surface, we will not only consider one surface measured as being near but also all the surfaces secified by standardised tolerancing. We will model the actual nearness of 2 surfaces at a finite distance, albeit small, by mathematical nearness at an infinitely small distance. 4. THE CONCEPT OF NEAR SURFACES We first of all start by defining what an objet is. A function ( x y) considered in 2 different ways: either as an alication of the domain of x to the domain of y, f, can be or as all the coules ( x, y) of the Cartesian-roduct of these 2 domains. This is what is termed entity-relation duality. In comuter science, this duality, known as object-oriented rogramming, has had considerable success for the ast twenty years or so. We say that we rogramme a function in a rocedural manner when it is considered as a relation and that we rogramme a function in an objectoriented manner when it is considered as an entity. A solution to a set of relations is called an instance Definition A surface secification can be reresented by a vectorial function of vectorial variables: F G (, ) ( ) = 0 = 0 with the arameters denoted by and the surface running oint by. We consider the ossible dislacements d of a oint obtained by a variation d of the arameters of the initial surface, such that oint + d belongs to the disturbed surface: F G ( + d, + d ) ( + d ) = 0 We seek the relations between the initial and the disturbed surfaces. For this, we carry out a Taylor s series develoment, limited to the first order (we assume here that none of the artial derivatives are equal to zero). F G (, ) + grad F(, ) ( ) + grad G d = 0 = 0 d + grad F d = 0

6 P. SERRÉ, A. CLÉENT, A. RIVIÈRE Note: We note the derivative of the vectorial function F in relation to the variable, as follows: grad F F, = 0 And, since ( ) disturbance G, = 0, we can deduce a relation between and ( ) d and variation d : (, ) grad F d + grad grad G d = 0 F d = Geometrical interretation A geometrical interretation is made starting with a scalar function ( ) the following notation: grad f = grad f u (, ) (, ) with the unit vector carried by f ( ) carried by f ( ) f, using grad f = grad f v and (, ) (, ) grad, reresented by u and the unit vector grad, reresented by v. For a given disturbance d, the dislaced oint ' = + d is located on a lane arallel to the lane tangent to the surface under consideration at oint, at a distance equal to ξ the value of which we will now determine. grad / f, v d + grad / f, u d = 0 ( ) ( ) by substituting e = v d and ξ = u d, we obtain: grad f, ξ = K e with K = grad f, ( ) ( ) Thus, when arameters are subject to random variation d such that: v d < e, then the dislacement of oint extends into a zone ξ wide on either side of oint. The factor K should be considered as an amlification factor of

7 FORAL DEFINITION OF TOLERANCING IN CAD AND ETROLOGY the disturbance of arameters on the shae of the curve at oint. oreover, it can be seen that this factor can be extremely large (for examle, if the first derivatives are cancelled at a dual oint) or may be zero at a secific oint. Generally seaking, as we were already aware, we observe that the mathematical form of the secification can affect accuracy (Farouki & Rajan 1988). Figure 3: Geometrical interretation of a near surface When the surface f is comletely regular, a continuous family of vectors ξ u erendicular to this surface can be traced for a given value of d. The ends of these vectors form a continuous surface termed the generalised offset of this surface. By definition, the air of surfaces constituted by the surface and its generalised offset surface will be called near surfaces. It should be noted that, although the arameters adoted are confined to osition and orientation arameters, this method is the equivalent of methods using small dislacements; to a certain extent, it is a sort of generalisation of the small dislacements method (Bourdet & Clément 1988) and (Bourdet, athieu, Lartigue & Ballu, 1995) Proerties Since we know F (, ) = 0, the near surfaces family is therefore a family with a vectorial arameter: d. For a given value of this arameter, we will be able to determine: F ( + d, ) ; F (, + d ); grad F( d, ) grad F(, + d ) and the relations between these different values. + ; The geometrical significance of these various arameters is illustrated below by the examle of a lane curve, deendent uon 2 arameters 1 and 2, in 2 different lanes. One is lane ( x, y) of the curve and the other the arametric lane (, 1 2 ). We have also reresented the gradients and numerical variation.

8 P. SERRÉ, A. CLÉENT, A. RIVIÈRE Figure 4: Illustration of the different arameters in lane ( x, y ) and in lane (, ) CONCLUSION We have shown that a near surface of the same nature can be made to corresond to any surface, differing only with resect to the second order of the initial reference. The symmetry of the near surfaces relation allows these 2 surfaces to be fully defined if we know the analytical arameters of one or the other or any combination of arameters and oints belonging to one or the other. This roerty gives considerable flexibility to identification of the air of near surfaces, erceived as a unique entity. This oens u the way to the resolution of extremely comlex identification roblems by global otimisation, which is known to be more efficient than a series of artial otimisations. 6. REFERENCES Srinivasan V. (1999). A geometrical roduct secification language based on a classification of symmetry grous. Comuter Aided Design, 31, Bourdet, P., & Clément, A. (1988). A study of Otima-criteria identification based on the small dislacement screw model. CIRP Annals, 37(1), Bourdet, P., athieu, L., Lartigue, C., & Ballu, A. (1995). The concet of the small dislacement torsor in metrology. In P. Ciarlini,. G. Cox, F. Pavese & D. Richter (Eds), Advanced athematical Tools in etrology II ( ). Oxford, United Kingdom. Clément, A., Rivière, A., & Serré, P. (1999). Keynote: Global Consistency of Dimensioning and Tolerancing. In F. van Houten & H. Kals (Eds), Global Consistency of Tolerances (. 1-26). Enschede, The Nederlands. Clément, A., Rivière, A., Serré, P., & Valade, C. (1997). The TTRSs: 13 Constraints for Dimensioning and Tolerancing, In H. A. Elaraghy (Eds), Geometric Design Tolerancing: Theories, Standards and Alications ( ). Toronto, Ontario, Canada. Farouki, R.T., & Rajan, V.T. (1988). On the numerical condition of algebraic curves and surfaces. 1. Imlicit equations. Comuter Aided Geometric Design, 5,