Geometric tolerancing of products

Geometric Tolerancing of Products Geometric Tolerancing of Products Edited by François Villeneuve Luc Mathieu First published 2010 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc Adapted and updated from Tolérancement géométrique des produits published 2007 in France by Hermes Science/Lavoisier © LAVOISIER 2007 Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address: ISTE Ltd 27-37 St George’s Road London SW19 4EU UK John Wiley & Sons, Inc 111 River Street Hoboken, NJ 07030 USA www.iste.co.uk www.wiley.com © ISTE Ltd 2010 The rights of François Villeneuve and Luc Mathieu to be identified as the authors of this work have been asserted by them in accordance with the Copyright, Designs and Patents Act 1988 Library of Congress Cataloging-in-Publication Data Geometric tolerancing of products / edited by Francois Villeneuve, Luc Mathieu p cm Includes bibliographical references and index ISBN 978-1-84821-118-6 Tolerance (Engineering) Geometry, Descriptive I Villeneuve, Francois, 1960- II Mathieu, Luc, 1954TS172.G467 2010 620'.0045 dc22 2010003707 British Library Cataloguing-in-Publication Data A CIP record for this book is available from the British Library ISBN 978-1-84821-118-6 Printed and bound in Great Britain by CPI Antony Rowe, Chippenham and Eastbourne Table of Contents PART I GEOMETRIC TOLERANCING ISSUES Chapter Current and Future Issues in Tolerancing: the GD&T French Research Group (TRG) Contribution Luc MATHIEU and François VILLENEUVE 1.1 Introduction 1.2 Presentation of the Tolerancing Resarch Group: objectives and function 1.3 Synthesis of the approach and contributions of the group 1.3.1 Languages for geometric specification 1.3.2 Dimension chains in 3D 1.3.3 Methods and tools 1.3.4 Manufacturing dimensioning and tolerancing 1.3.5 Uncertainties and metrology 1.4 Research perspectives 1.5 Media examples: “centering” and “connecting rod-crank” 1.6 Conclusion 1.7 Bibliography 10 11 12 13 15 17 19 PART II GEOMETRIC TOLERANCING LANGUAGES 21 Chapter Language of Tolerancing: GeoSpelling Alex BALLU, Jean-Yves DANTAN and Luc MATHIEU 23 2.1 Introduction 2.2 Concept of the GeoSpelling language 2.3 Geometric features 2.3.1 Ideal features 2.3.2 Non-ideal features 2.3.3 Limited features 23 24 26 26 29 29 vi Geometric Tolerancing of Products 2.4 Characteristic 2.4.1 Intrinsic characteristic 2.4.2 Situation characteristic 2.4.3 Situation characteristic between ideal features 2.4.4 Situation characteristic between limited and ideal features 2.4.5 Situation characteristic between non-ideal and ideal features 2.4.6 Situation characteristic between non-ideal features 2.5 Operations 2.5.1 Operations to identify the geometric features 2.5.2 Evaluation operation 2.6 Conditions 2.7 Specifications on assemblies – quantifiers 2.8 Applications to part specification 2.9 Applications to product specifications 2.10 Conclusion 2.11 Bibliography 29 29 30 33 36 36 37 38 39 42 43 44 45 48 51 52 Chapter Product Model for Tolerancing Denis TEISSANDIER and Jérôme DUFAURE 55 3.1 Introduction 3.2 Objectives and stakes 3.2.1 Cover the design cycle of the product 3.2.2 Propose an environment of collaborative work 3.2.3 Ensure the traceability of geometric specifications 3.3 Proposal for a product model 3.3.1 History 3.3.2 General description of the IPPOP product model 3.3.3 Basic entities definition of the product model 3.3.4 Description of the connection links between basic entities 3.3.5 Description of the decomposition and aggregation of basic entities 3.3.6 Correspondence between tolerancing data and product model data 3.4 Benefits of the IPPOP product model 3.4.1 Description of the transfer principle 3.4.2 Formalization of the geometric condition transfer activity 3.4.3 Traceability of specifications 3.5 Application on the centering device 3.5.1 Description of the case studied 3.5.2 Functional analysis of the centering device 3.5.3 Transfer in preliminary design (stage 1) 3.5.4 Transfers in embodiment design (stages and 3) 55 56 56 57 57 58 58 58 59 65 67 68 68 69 70 73 73 73 74 76 77 Table of Contents 3.5.5 Transfer in detailed design (stage 4) 3.5.6 Traceability of specifications of axis 3.6 Conclusion 3.7 Bibliography 80 82 84 84 Chapter Representation of Mechanical Assemblies and Specifications by Graphs Alex BALLU, Luc MATHIEU and Olivier LEGOFF 87 4.1 Introduction 4.2 Components and joints 4.2.1 Components, surfaces and datum features 4.2.2 Joints 4.2.3 Models of joints 4.2.4 Models of contacts 4.3 The requirements, technical conditions and specifications 4.3.1 The requirements 4.3.2 Technical conditions 4.3.3 The specifications 4.4 Manufacturing set-ups 4.5 Displacements between situation features and associated loops 4.5.1 Relative displacements 4.5.2 The loops 4.5.3 Loops with or without a coordinate system on the components 4.6 The key elements 4.6.1 The key deviations, surfaces, joints and components 4.6.2 The loops and key sub-graphs 4.7 Conclusion 4.8 Bibliography 87 89 90 91 92 95 97 97 99 99 100 103 103 104 106 107 107 107 109 110 Chapter Correspondence between Data Handled by the Graphs and Data Product Denis TEISSANDIER and Jérôme DUFAURE 111 5.1 Introduction 5.2 Correspondence between tolerancing graphs and the product data 5.2.1 Kinematic graphs 5.2.2 Graph of the elementary joints 5.2.3 Closings of influential loops and traceability of specifications 5.3 Correspondence between manufacturing set-ups and the data product 5.3.1 Manufacturing graph of body 5.3.2 Manufacturing set-up 10 of the body 5.4 Conclusion vii 111 112 112 114 116 118 118 120 121 viii Geometric Tolerancing of Products PART III 3D TOLERANCE STACK-UP 123 Chapter Writing the 3D Chain of Dimensions (Tolerance Stack-Up) in Symbolic Expressions Pierre BOURDET, François THIÉBAUT and Grégory CID 125 6.1 Introduction 6.2 A reminder of the establishment of the unidirectional chain of dimensions by the Δl method 6.2.1 Definition and properties 6.2.2 The Δl model 6.2.3 A reminder of the Δl method 6.3 Establishment in writing of a chain of dimensions in 3D by the method of indeterminates in the case of a rigid body 6.3.1 General points 6.3.2 Model of the indeterminates 6.3.3 Laws of geometric behavior of a mechanism with gaps and defects 6.3.4 An example 6.4 Consideration of the contact between parts in the mechanisms 6.4.1 General theory 6.4.2 Calculation of the distance between a point and a surface 6.4.3 Utilization of the distance function expressed in the symbolic calculation 6.5 Mechanisms composed of flexible parts, joints without gap (or imposed contact) and imposed effort 6.5.1 General theory 6.5.2 Utilization of a coordinate system on the parts 6.5.3 Modeling of form defects and deformations 6.5.4 Integration of flexibility of the parts 6.5.5 The principle of writing an equation(s) for a mechanism composed of a single flexible part 6.6 Conclusion 6.7 Bibliography 125 126 126 130 132 135 135 136 138 140 142 142 143 144 144 144 144 145 146 146 147 148 Chapter Tolerance Analysis and Synthesis, Method of Domains Max GIORDANO, Eric PAIREL and Serge SAMPER 151 7.1 Introduction 7.2 Deviation torsor and joint torsor 7.2.1 Cartesian frame linked to a surface 7.2.2 Deviation torsor 7.2.3 Relative deviation torsor and absolute deviation torsor 7.2.4 Joint torsor, kinematic torsor and clearance torsor 151 152 152 153 154 155 Table of Contents 7.3 Equations of loops 7.3.1 Mechanism without clearance or deviation 7.3.2 Taking into account the clearances and deviations 7.4 Deviation and clearance domains 7.4.1 Deviation domain 7.4.2 Clearance domain 7.5 Representation and properties of the domains 7.5.1 Change of Cartesian frame 7.5.2 Symmetry with regard to the origin 7.5.3 Representation by polytopes 7.5.4 Stacking of tolerances and sum of Minkowski 7.5.5 Resulting clearance domain 7.5.6 Zone corresponding to a domain 7.5.7 Cases of axisymmetric systems 7.6 Application to the analysis of simple chains 7.6.1 Condition of assembly for one loop 7.6.2 Application to a chain of dimension taking angular defects into account 7.6.3 Application to a connecting rod-crank system 7.6.4 Application to the synthesis of tolerances 7.6.5 Condition of assembly, virtual state and domain 7.7 Case of assemblies with parallel joints 7.7.1 Notion of residual clearance domain and inaccuracy domain 7.7.2 Condition of assembly for joints in parallel 7.8 Taking elastic displacements into account 7.8.1 Elastic deviation and joint torsor definition 7.8.2 Elastic deviation torsors 7.8.3 Elastic joint torsors 7.8.4 Use rate and elastic domains 7.8.5 Elastic clearance domain 7.8.6 Elastic deviation domains 7.8.7 Elastic domain duality 7.8.8 Application to a simple assembly 7.8.9 Assembly without clearances 7.8.10 Assembly with clearances in joints 7.9 Conclusion 7.10 Bibliography ix 155 155 156 158 158 161 162 162 163 164 165 167 167 167 168 168 169 171 171 172 173 173 174 176 176 176 176 177 177 178 178 178 179 179 180 180 Chapter Parametric Specification of Mechanisms Philippe SERRÉ, Alain RIVIÈRE and André CLÉMENT 183 8.1 Introduction 183 x Geometric Tolerancing of Products 8.2 Problem of the parametric specification of complete and consistent dimensioning 8.2.1 Model of dimensioning 8.2.2 Case study 8.2.3 Analysis of the coherence and completeness of dimensioning 8.3 Generation of parametric tolerancing by the differential variation of the specification of dimensioning 8.3.1 Generation of implicit equations of a parametric tolerancing 8.3.2 Case study (continuation) 8.3.3 Analysis and resolution of compatibility relations 8.4 Problem of the specification transfer 8.5 Expression of parametric tolerancing 8.5.1 Relation between the variation intervals of specification parameters 8.5.2 Interchangeability and “clearance effect” 8.6 Case study 8.6.1 Representation of parts 8.6.2 Assembly representation 8.6.3 Generation of the equation system associated with the mechanism 8.6.4 Generation of compatibility relations 8.6.5 “Clearance effect” calculation 8.7 Conclusion 8.8 Bibliography 184 185 185 187 188 188 189 192 192 193 194 196 198 199 200 201 201 202 204 205 PART IV METHODS AND TOOLS 207 Chapter CLIC: A Method for Geometrical Specification of Products Bernard ANSELMETTI 209 9.1 Introduction 9.2 Input of a tolerancing problem 9.2.1 Definition of nominal model 9.2.2 External requirements 9.3 Part positioning 9.3.1 Setting up of parts 9.3.2 Positioning tables 9.3.3 Selection of positioning surfaces 9.3.4 Virtual part assembly 9.4 Tolerancing of positioning surfaces 9.4.1 Generation of positioning requirements 9.4.2 Generation of positioning tolerancing 209 210 210 211 212 212 213 215 216 217 217 218 Table of Contents 9.5 Generation of functional requirements 9.5.1 Generation of proximity requirements 9.6 Specification synthesis 9.6.1 Principle 9.6.2 Simple requirement 9.6.3 Decomposition of complex requirements 9.6.4 Tolerancing of the support 9.7 Tolerance chain result 9.7.1 Analysis lines method 9.7.2 Application 9.7.3 Statistical result 9.7.4 Representation in Excel ranges 9.8 Tolerance synthesis 9.8.1 Variation of nominal models 9.8.2 Quality optimization 9.8.3 Effective method for maximizing tolerances 9.9 Conclusion 9.10 Bibliography 221 221 222 222 222 223 225 227 227 229 232 232 234 234 234 235 238 238 Chapter 10 MECAmaster: a Tool for Assembly Simulation from Early Design, Industrial Approach Paul CLOZEL and Pierre-Alain RANCE 241 10.1 Introduction 10.2 General principle, 3D tolerance calculation 10.2.1 Kinematic definition of the contact 10.2.2 Calculation principle 10.2.3 “3D chains of dimension” results 10.2.4 Tolerance definition 10.3 Application to assembly calculation 10.3.1 Preamble: definition of surfaces playing a part in the model 10.3.2 Model definition 10.3.3 Hyperstatism calculation and analysis 10.3.4 Possible assembly configurations 10.3.5 Quantification of functional conditions, choice of system architecture 10.4 From model to parts tolerancing 10.4.1 Choice of reference system 10.4.2 Connections graph 10.4.3 Identification of specifications: example 10.4.4 Identification of numerical values: example 10.5 Statistical tolerancing 10.6 Industrial examples xi 241 242 242 243 244 245 245 246 248 251 253 255 263 263 264 265 267 268 269 Uncertainties 365 geometric specification (ISO1101) and to check it The sequence of processing operations propagates acquisition uncertainty [BAC 03] The transfer function enables the transition from acquisition uncertainty to distance uncertainty This function must be minimal in order not to propagate the same uncertainty several times and amplify it unjustifiably With the proposed method these two conditions will be respected To illustrate the effect of the measurement planning process, we are going to take the example of coaxiality, defined in Figure 13.21 First of all, we will demonstrate the effect of constructions and the number of points defining the surfaces Then, we will explain propagation mechanisms graphically with the concept of the limit envelope On the reference surface, two circles in 12 points have been acquired On the specified surface, three circles in 12 points have been probed To check this specification, two measurement ranges can be created The first one starts with the best-fitting of two circles on the reference surface Through the two derived centers, a line is drawn For each circle taken on the specified surface, a best-fit with a circle is carried out To finish, distances between the reference line and the three circumcenters on the specified surface are calculated The second one is more straightforward The probed points belonging to the reference element are grouped together in the same file The best-fitting of a cylinder to this file will allow us to obtain the specified reference Next, the three distances between the reference and three circumcenters on the specified surface are calculated Procedure 05 A Second A Top c ircle Bottom circle Specified circlei First Line Distance Circle number Mean Value di Standard deviation 0.0135 8,1E-03 0.0148 7,2E-03 0.0132 6,7E-03 0.0135 4,9E-03 0.0148 4,6E-03 0.0132 4,8E-03 Cylinder Distance Specified circle i Distance Distance Figure 13.21 Impact of the measurement planning process 366 Geometric Tolerancing of Products The propagation method enables the calculation of the mean value of the distance and its uncertainty (k = 1) In Figure 13.21, the mean values of the distances and their uncertainties are carried forward We observe that the mean values of the distances are identical The uncertainties of the results provided by the range based on two circles acquired on the reference surface are roughly twice as great as in the other process The number of probed points has a beneficial effect on the value of uncertainty Figure 13.22 Impact of the number of points on uncertainty Figure 13.22 depicts the results of a trial measurement using a gauge ring The abscissa shows the number of points acquired on the gauge Along the ordinate, values of the diameter estimated by the software program and the calibrated diameter values of the gauge, i.e 39.998 mm, are entered We note that the calibrated value of the gauge is always included in the bar representing uncertainty (U=2.u, i.e a risk α = 5%) We also observe that the reduction of the error bar is related to the number of probed points Reduction is achieved in on the square root of N (where N is the number of probed points) If we return to our example of coaxiality, the ratio of uncertainty in procedure by the second uncertainty of the first distance is 1.65 Remember that in procedure 1, the number of points taken for the best-fit was 12 (two circles of 12 points) and 24 in the second procedure (one cylinder) With the hypothesis that σcylinder is close to σcircle then the effect of the number of points EN on this global uncertainty can be approached as follows: Uncertainties 367 σ cylinder EN = 24 σ circle = 1.41 12 By simple deduction, we can estimate the effect of propagation EP at 0.24 for this example To explain the weak impact of propagation on the final result, we reported in Figure 13.23 the two variance covariance matrices of the line and of the cylinder The values of the second moments of the reference points of the line and cylinder are nearly identical However, the propagation of procedure brings about an increase in the variances covariances on the vector of the line Procedure Procedure e3 e3 IOAOB O OA V e2 O e1 O e1 = OA2 VOAOB e2 VOAOB V 8,7E-08 -8,3E-11 -1,3E-10 1,8E-10 -1,7E-10 -7,5E-10 2,7E-07 -2,0E-10 -3,9E-10 1,1E-07 1,5E-10 -1,6E-10 -8,3E-11 8,7E-08 2,4E-10 -1,8E-10 -1,8E-10 -2,1E-09 -2,0E-10 2,7E-07 7,3E-10 1,5E-10 1,1E-07 9,7E-11 -1,3E-10 2,4E-10 8,2E-13 -7,5E-13 -2,3E-13 -4,5E-12 -3,9E-10 7,3E-10 2,5E-12 -1,6E-10 3,0E-10 4,9E-13 1,8E-10 -1,8E-10 -7,5E-13 2,6E-06 -2,9E-09 -3,8E-09 1,1E-07 1,5E-10 -1,6E-10 2,0E-06 -1,5E-09 -3,3E-09 -1,7E-10 -1,8E-10 -2,3E-13 -2,9E-09 2,6E-06 7,1E-09 1,5E-10 1,1E-07 3,0E-10 -1,5E-09 2,0E-06 6,1E-09 -7,5E-10 -2,1E-09 -4,5E-12 -3,8E-09 7,1E-09 8,0E-11 -1,6E-10 9,7E-11 4,9E-13 -3,3E-09 6,1E-09 2,4E-11 O ≈ IOAOB Figure 13.23 Variance covariance matrices for the two procedures 13.4 Application to tolerance analysis In the conclusion of the preceding section, the choices made by the designer in the tolerancing phase of a mechanical system have an influence on the uncertainty of specification verifications If we wish to remain in the logic of a concurrent engineering approach to the industrial phase of a product, we must integrate professional constraints as soon as possible Moreover, the ISO has introduced the notion of generalized uncertainty It consists of: 368 Geometric Tolerancing of Products – correlation uncertainty: it represents the adequacy that should exist between the functional needs expressed by the customer and the functionality of the ultimate design; – specification uncertainty: it defines the gap between the functions requested and the toleranced functions of the design; – measurement uncertainty: it corresponds to the geometric zone in which the true value of the measured magnitude is located The vector approach, as shown in section 13.1.2.1, is capable of describing the location of geometric elements In this section, we are going to use the notion of random vector in tolerance analysis in order to highlight the contribution of this type of modeling to the representation of specification uncertainty [LIN 03] This approach can help the designer in the choices he or she makes in conceptual product design as well as to help him or her reduce uncertainty In metrology, we are striving to minimize the difference between the model and the real entity In spite of all the precautions taken and corrections made, uncertainty persists in the announced result In design, the same parallel can be made between the conceptual model present in the mind of the designer and the technical model displayed or proposed by a software program Our approach is capable of modeling ignorance and propagating it in the different steps of modeling within the digital framework of the software program This process is close to the one presented in metrology Correlation uncertainty could thus be approached In this section, we will extend the scientific basis of metrology to the tolerancing approach ELS Mi tA A n uMi e3 ELS A V O uA e2 e1 u C< d > = i u2 M i + u2A Figure 13.24 Review of the principle Uncertainties 369 13.4.1 Review of the principle of modeling Uncertainty ucdi, see Figure 13.24, corresponds to uncertainty created from the coaxiality specification The principle in calculating measurement uncertainty can be extended to the phase of geometric dimensioning and tolerancing This indicator can represent the variability of technical solutions chosen This feature is a function of a great number of variables The variance covariance matrix in design is a function of: – the quality of topological algorithms (technological choices made and quality of the geometric modeling software); – the variability of the substitute model as a function of the constraints imposed by the real model (surface condition, variability in the computer system, variability in the manufacturing process, etc.); – the surface extent (designer’s choice); – geometric constructions created by the designer; – the encumbrance of the mechanism (lever arm effect); and – the quality of the propagation in the geometric constructions In the following sections, we will explain some of these variables n e3 tA A Mi uMi O V L1 e1 uA(L1) e2 O n Mi A uMi e3 tA O V L2 e1 O Figure 13.25 The effect of surface extent uA(L2) e2 370 Geometric Tolerancing of Products 13.4.2 Effect of the reference surface extent In section 13.1.2.1 we showed that by using vector modeling, the interpretation of a coaxiality specification was independent from the reference surface extent If we use the concept of a statistical limit envelope to represent the geometric elements participating in the coaxiality specification, we obtain the two diagrams in Figure 13.25 We note that the reduction in the length of the reference surface results in an amplification of the curve of the statistical limit envelope This increase entails greater specification uncertainty We will return to the variance covariance matrix in the example of the line in the plane (below) The sum of squares of xk is reduced when the length of the reference is decreased With a constant variability, this leads to the increase in variance of a1 Parameter a1 is the slope of the line ⎛ σ p2 ⎜ ⎜ N Cov a = ⎜ ⎜ ⎜ ⎝ σ p2 ∑x k ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ [13.5] The variance covariance of the direction vector V of the axis of reference has the same behavior when the length of the surface decreases to a constant diameter Figure 13.26 depicts the impact of this decrease on the statistical limit envelope tA A L = 15mm L = 5mm L = mm tA A L = 15 mm Figure 13.26 Limit envelopes versus the length Uncertainties 371 13.4.3 Effect of surface spacing In section 13.1.2.1 we showed that the interpretation of coaxiality specification was independent from the spacing of participating surfaces Figure 13.27, based on the concept of a statistical limit envelope highlights the impact of surface spacing on specification uncertainty In order to study the behavior of the statistical limit envelope as a function of the remoteness of the estimation point, we will take up the case of point/line distance again n e3 t A A Mi uMi V O uA(S1) E1 E2 A e2 O e1 n e3 t A Mi uMi O V uA(S2) e1 O e2 Figure 13.27 Effect of surface spacing The frame of reference is positioned at feature point C of the line thus: Var(d) = Var(Y ) + ⎡⎣Var(a0 ) + X Var(a1 )⎤⎦ Variable X locates point M in relation to the feature point of the line The more remote the specified surface, the more the value of X increases Consequently, the impact of this variable is a function of the square of its value 13.4.4 Effect of shape defect on reference surfaces Form defect is a geometric specification that is often forgotten in the tolerance analysis of mechanical parts The classic vector approach does not take this parameter into account If we take up the variance covariance matrix of the line in the plane again and make a model of the shape defect with a variance, we note that 372 Geometric Tolerancing of Products the variance of points around the mean element σ2 has an influence on the position (a0) and orientation (a1) of the derived element (see equation [13.5]) Figure 13.28 depicts the effect of the shape defect on the statistical limit envelope In conclusion, it is not realistic to tolerance a surface in relation to a reference surface if the latter is not specified by the form tolerance 0.05 A A tA 0.02 tA Form specification: 0.02 mm Form specification : 0.05 mm Figure 13.28 Effect of form defect A B A B X X t A-B t A-B Points Mi Cylinder (A + B) Distance di Circle A Circle B Points Mi Line A-B Distance di Coaxiality e3 O e1 O Coaxiality e3 Mi V Mi OA e2 e1 O Figure 13.29 Common reference OB e2 Uncertainties 373 13.4.5 Effect of the choice of a reference system In the ISO1101 standard without modifiers, the specified reference can be a point, a line or a plane These geometric elements can be obtained directly or through a set of constructions In Figure 13.29, two cases of common references are proposed In the first one on the left, the specified reference is made up of the line common to two parts of cylinders A and B In the second case, the specified reference is the line passing though the circumcenters of A and B taken on the portions of cylinder x mm from respective shoulders Case: Cylinder n uMi e3 O Mi V Case: Line (IAB,VOAOB) n e3 OA uA e1 O uMi M i e2 e1 O uA OB IOAOB VOAOB e2 Figure 13.30 Effect of the construction In the two cases, the specified surface will be defined by the set of circumcenters taken in the right sections of the toleranced cylinders We are in the presence of a case that is identical to the one studied in metrology in section 13.2.6 As we have shown in metrology, the constructions required in order to obtain the line passing through the two circumcenters are going to amplify initial uncertainties The reference thus obtained will not be as robust as in the other case (see Figure 13.30) 13.5 Conclusion In this chapter, we have presented a new approach to geometric modeling The notion of the random vector has been proposed First and second central moments of these random vectors provide additional indications to geometric users This modeling accounts for the surface extent and their shape defects Second moments represented graphically have allowed us to implement the concept of a statistical limit envelope for ordinary geometric elements (point, line and plane) The propagation of variance covariance matrices enabled us to include the effect of the position and orientation of estimated magnitude in the determination of its uncertainty Models made with this new approach have met with success in practical 374 Geometric Tolerancing of Products and industrial cases 3D metrology has been the main field of experimentation The statistical approach to measurement issues is an alternative to the strictly geometric modeling of best-fit surfaces The thoughts and reflections of Professors Estler and Cox, reinforce our choice [COX 01, EST 99] 13.6 Bibliography [BAC 03] BACHMANN J., Contribution la propagation des incertitudes dans les gammes de mesure des machines mesurer par coordonnées, PhD thesis, University of the Mediterranean, October 2003 [COX 01] COX M G., “Measurement uncertainty and the propagation of distributions”, Proceedings of the 10th International Metrology Congress, Saint Louis, France, CDRom, 2001 [EST 99] ESTLER W T., “Measurement as inference: fundamental ideas”, Annals of the CIRP, vol.47, no 2, pp 611-632, 1999 [LIN 03] LINARES J.M., BACHMANN J., SPRAUEL J.M., P BOURDET, “Propagation of specification uncertainties in tolerancing”, Proceedings of the 8th CIRP Seminar on Computer Aided Tolerancing, Charlotte, pp.301-310, April 28-29, 2003 [SPR 03] SPRAUEL J.M., LINARES J.M., BOURDET P., “Contribution of non linear optimization to the determination of measurement uncertainties”, in: PIERRE BOURDET and LUC MATHIEU (eds), Computer Aided Tolerancing CIRP, Geometric Specification and Verification: Integration of Functionality, Kluwer Academic Publisher, pp.237-244, 2003 [WEC 95] WECKENMANN A., EITZERT H., GARMER M., WEBER H., “Functionalityoriented evaluation and sampling strategy in coordinate metrology”, Precision Engineering, vol.17, pp 244-252, 1995 List of Authors Bernard ANSELMETTI LURPA ENS Cachan France Alex BALLU LMP Bordeaux University France Pierre BOURDET LURPA ENS Cachan France Grégory CID LURPA ENS Cachan France André CLÉMENT LISMMA Supméca Saint-Ouen France Paul CLOZEL Ecole Centrale de Lyon France Jean-Yves DANTAN LGIPM ENSAM Metz France Jérôme DUFAURE LMP Bordeaux University France Max GIORDANO Lméca SYMME University of Savoy Annecy France Olivier LEGOFF IRCCyN Ecole Centrale de Nantes France Jean-Marc LINARES EA(MS)2 University of the Mediterranean Aix Marseille France Geometric Tolerancing of Products Edited by François Villeneuve and Luc Mathieu © 2010 ISTE Ltd Published 2010 by ISTE Ltd 376 Geometric Tolerancing of Products Luc MATHIEU LURPA ENS Cachan France Denis TEISSANDIER LMP Bordeaux University France Eric PAIREL Lméca SYMME University of Savoy Annecy France François THIÉBAUT LURPA ENS Cachan France Pierre-Alain RANCE Ecole Centrale de Lyon France Alain RIVIÈRE LISMMA Supméca Saint-Ouen France Serge SAMPER Lméca SYMME University of Savoy Annecy France Philippe SERRÉ LISMMA Supméca Saint-Ouen France Jean-Michel SPRAUEL EA(MS)2 University of the Mediterranean Aix Marseille France Stéphane TICHADOU IRCCyN Ecole Centrale de Nantes France Frédéric VIGNAT G-SCOP University of Grenoble France François VILLENEUVE G-SCOP University of Grenoble France Index A active surface, 100, 284, 306-307, 315, 329, 333 actual surface, 137, 142 assembly requirement, 16, 50, 172, 221, 249, 254-255, 260, 263 associated surface, 12, 153, 318, 325 C characteristic, 9-10, 25-26, 29-30, 3337, 42-50, 63, 141, 279 clearance domain, 158, 161-162, 167-171, 173-174, 177, 179 effect, 192, 196-197, 202-205 torsor, 155-158, 161-164 CLIC, 10, 70, 83, 209, 210-212, 216223, 227, 237-238 compatibility equation, 141-143, 321 relation, 10, 125, 184, 188-193, 195, 201, 204 component, 9, 58-60, 65-75, 80-84, 90, 97, 99, 107, 112-113, 129, 135139, 165, 170, 177, 179, 196, 213216, 221, 243, 252, 280-283 connection link, 75, 84 contact, 57, 59, 61-62, 66-68, 71, 7580, 87-96, 99, 107, 110-115, 119, 133, 141-147, 151, 161, 170, 177, 213-222, 229-231, 241-245, 249251, 254-257, 260, 263, 266, 284, 316, 319, 321-322, 325-329, 334 D design cycle, 56-58, 62-63, 68-69, 71, 74, 82-84, 111, 118, 121 deviation, 10, 13, 29, 43, 97, 107, 128-132, 136-142, 144-147, 151179, 197, 227-235, 280, 296, 300, 315-319, 332-344, 350, 357 domain, 10, 158-174, 178, 228 torsor, 10, 136-140, 142, 144, 151159, 165, 168, 173-176, 315 dimensioning, 10, 11, 58, 111, 126, 179, 184-188, 241, 277, 298, 300302, 369 displacement, 3-13, 17, 26, 96-97, 103-104, 107, 135-139, 146, 148, 154-155, 163, 176, 183-184, 229231, 242-244, 260, 278, 288-289, 292, 302, 315, 322, 324, 338 domain, 23, 27, 152, 158-159, 160180 Geometric Tolerancing of Products Edited by François Villeneuve and Luc Mathieu © 2010 ISTE Ltd Published 2010 by ISTE Ltd 378 Geometric Tolerancing of Products E K, L elastic deviation domain, 179 torsor, 176 external requirement, 211-212 key deviation, 107 element, 107, 110 loop, 107-108 surface, 107-108 kinematic graph, 87, 113-116, 121 link torsor, 322, 325 F feature, 25-42, 44, 58, 59, 61-68, 71, 75-76, 79-86, 89, 137, 158, 167, 172, 213-214, 219, 360-361, 369, 371 flexible part, 9, 126, 144, 146 function, 4, 11-12, 28, 30, 35-38, 4143, 48, 93-94, 97, 112-114, 139, 144-147, 192, 204, 224, 227, 229, 235, 268, 270, 301-302, 313, 316, 321-322, 325-331, 333-338, 352, 356-358, 363, 365, 369, 371 functional requirement, 6-9, 14-15, 55-58, 62, 69, 88-89, 104, 107, 152, 165, 167, 177, 209, 221, 223, 249, 268, 278, 289, 301 tolerancing, 10, 211, 238 G geometric condition transfer, 68-73 feature, 24-29, 38, 58, 99, 126-127 graph of the elementary joints, 114 I ideal feature, 26, 29, 33-41 interface, 24, 58-61, 64-71, 77, 82, 84, 113, 214, 245, 250, 256, 257, 282, 318, 323 inter-parts specification parameter, 192 intrinsic characteristic, 26, 29, 30, 35, 61, 64, 280, 315 M machined surface, 100, 119, 130, 290-291, 296, 306-309, 316, 333 manufactured surface, 100, 143, 287, 315, 316, 329, 344 manufacturing set-up, 100, 103, 118, 120, 279 tolerance, 11-12, 301-302, 305306, 310-314, 323, 329-338 measured deviation, 126, 127 method of indeterminates, 9, 126, 135 Minimum Geometric Reference Element (MGRE), 27-28 Model of Manufactured Part (MMP), 12, 306, 314-317, 323-330, 334, 338 modeling parameter, 185, 187, 190, 201 N, P nominal surface, 138, 142-143, 234 non-ideal feature, 31, 36 parameter, 29, 145, 180, 194, 195, 201, 234, 295-296, 321, 330-334, 358, 371 positioning requirement, 97, 217-218 surface, 140, 213, 215, 217, 224, 227, 253-254, 282, 286, 324325, 330, 332-333 table, 213, 215-219, 222 Index process signature, 350 product model, 8-9, 56-69, 71-84, 111-114, 118, 120-121 propagation of uncertainties, 12 R random vector, 12, 350-353, 356358, 368, 373 real surface, 153, 343-348, 354-355 requirement, 11, 16, 44-51, 55-57, 61-63, 69-73, 76-78, 97, 101, 105108, 112, 116, 152, 165, 204, 209, 217-229, 232-238, 242-243, 266268, 313 rigidity matrix, 126, 146, 147 S Situation characteristic, 26, 30-38 feature, 27-28, 33-35, 88-90, 99, 103-104, 109, 136-137, 143145 skin model, 8, 39-40, 46-50 small displacements torsor, 151 specification, 6-8, 12-14, 23-26, 29, 39, 43-51, 55-57, 61-72, 77, 79, 81, 83, 97, 101, 118-119, 183204, 211, 219, 223-225, 227, 230, 234, 266-268, 299, 330, 333, 336337, 343, 346-347, 364-365, 368371 model, 12 parameter, 184-186, 189-196, 201204 synthesis, 223 statistical, 12-13, 43, 55, 128, 134, 218, 232-233, 245, 269, 338, 343, 350-351, 354-355, 360-363, 370373 limit envelop, 12, 351, 360-361, 363, 370-373 379 substituted surface, 60 surface, 10-11, 26-32, 40-50, 60-61, 65, 75-83, 90-94, 99, 104-107, 126, 130-133, 136-139, 142-145, 152155, 163, 167, 171-175, 183, 193, 197, 213, 217, 219, 221-224, 227229, 234-235, 241, 245, 248, 255257, 263-268, 282-283, 286-288, 291, 306-309, 314-334, 343-351, 354-356, 360-366, 369-373 synthesis, 4-6, 9-13, 17, 147, 151152, 171-172, 222, 234, 238, 263, 288, 292, 302-306, 313-317, 329, 338 T, U, W technologically and topologically related surfaces (TTRS), 13, 135, 183, 224 tolerance, 4-14, 23-26, 43-49, 51, 63, 83, 110, 126-135, 142, 152, 158160, 163-167, 170-172, 175, 180, 184, 195-197, 202, 209, 217-218, 221, 223, 227-238, 241-242, 244245, 248, 252, 255-257, 263, 267272, 295, 306-307, 313-317, 322338, 343-347, 350, 364, 368, 371 analysis, 6, 9-11, 110, 234, 271, 315, 317, 323, 329, 343, 345, 350, 368, 371 synthesis, 6, 329, 330 torsor, 9-11, 104, 135-139, 142-143, 148, 151-165, 168, 174, 176-179, 184, 229, 278, 289-292, 302, 316320, 323, 325 traceability of the tolerances, uncertainty, 12, 44-45, 350-351, 358373 worst case, 55, 134, 165, 171, 197, 229, 231, 233, 260, 269, 309-310, 317, 328-329, 335, 338 [...]... Synthesis of the approach and contributions of the group Without trying to be exhaustive, the chapters of this book reflect the present state of knowledge and research in tolerancing in France The domains of activity and research in tolerancing can be resumed thus (see Figures 1.1 and 1.2): 6 Geometric Tolerancing of Products – The specification of products This domain tries to define some geometric. .. DANTAN and Luc MATHIEU Geometric Tolerancing of Products Edited by François Villeneuve and Luc Mathieu © 2010 ISTE Ltd Published 2010 by ISTE Ltd 24 Geometric Tolerancing of Products minimal and maximal tolerance This language corresponds to the concept of tolerancing by dimension, and was the direct reflection of the possibilities of the measuring equipment for dimensions of this epoch A great advance... Figure 1.2 Research branches into tolerancing and metrology domains 8 Geometric Tolerancing of Products The different branches of research in the domains of tolerancing are shown in Figure 1.2 This graph is inspired by the work of François Villeneuve and Frédéric Vignat in the PhD thesis of the latter [VIG 05] It succinctly presents the contributions of the authors of this book, where the numbers in... P3 Figure 1.6 Parameter set-up of the connecting rod-crank example Issues in Tolerancing 19 This work is evidently not yet finished Section 1.4 of this chapter offers a range of new research areas aimed at economic mastery of the geometric variations all along the product life cycle The next objective of the TRG resides in the framework of a national research project in tolerancing and metrology, mobilizing... important role in the quality and cost of products Mastering these geometric variations throughout the product life cycle remains an undeniable performance Chapter written by Luc MATHIEU and François VILLENEUVE Geometric Tolerancing of Products Edited by François Villeneuve and Luc Mathieu © 2010 ISTE Ltd Published 2010 by ISTE Ltd 4 Geometric Tolerancing of Products factor for companies Moreover, in... measuring equipment for dimensions of this epoch A great advance was made by the arrival of measuring equipment using coordinates and by the international introduction of the concept of geometric tolerancing This concept, which consists of defining a zone of variation of the geometric features, was the subject in the 1980s of international standards that are still being used for the most part today Even though... Figure 2.2) Figure 2.2 Definition of a specification In the following sections, the principle concepts of GeoSpelling are explained, i.e the geometric features, characteristics, operations and conditions Section 2.8 illustrates these concepts in the expression of specifications on individual parts and on assemblies 26 Geometric Tolerancing of Products 2.3 Geometric features Geometric features are distinguished... domain of displacements allowed by the gap Relating the gap domains and deviation domains enables the analysis and synthesis of tolerances Chapter 8 covers the notion of tolerance transfer from a parametric point of view, i.e with a vectorial parametric transformation of the surfaces and links of a mechanism The deviations of the mechanism are seen as a variation of the characteristic parameters of each... of the different French research teams today, but also to offer a shared vision of examples in common resulting from a regular exchange of views that have animated meetings of the Tolerancing Research Group (TRG) since 2001 1.2 Presentation of the Tolerancing Resarch Group: objectives and function The first discussions about the creation of the Tolerancing Research Group (TRG) go back to April 2001 at... 373 374 List of Authors 375 Index 377 PART I Geometric Tolerancing Issues Chapter 1 Current and Future Issues in Tolerancing: the GD&T French Research Group (TRG) Contribution 1.1 Introduction This book, entitled Geometric Tolerancing of Products, shows that especially in France a wealth of research ... international introduction of the concept of geometric tolerancing This concept, which consists of defining a zone of variation of the geometric features, was the subject in the 1980s of international... uncertainties Figure 1.2 Research branches into tolerancing and metrology domains Geometric Tolerancing of Products The different branches of research in the domains of tolerancing are shown in Figure 1.2... DANTAN and Luc MATHIEU Geometric Tolerancing of Products Edited by François Villeneuve and Luc Mathieu © 2010 ISTE Ltd Published 2010 by ISTE Ltd 24 Geometric Tolerancing of Products minimal and