A new mathematical model is presented for representing the tolerances of planar surfaces. The model is compatible with the ASME Standard for geometric tolerances. Central to the new model is a Tolerance-Map®,1 a hypothetical volume of points which corresponds to all possible locations and variations of a segment of a plane which can arise from tolerances on size, form, and orientation. Every Tolerance-Map is a convex set. This model is one part of a bi-level model that we are developing for geometric tolerances. The new model makes stackup relations apparent in an assembly, and these can be used to allocate size and orientational tolerances; the same relations also can be used to identify sensitivities for these tolerances. Stackup relations are developed for parts where the centers of faces are offset laterally. All stackup relations can be met for 100% interchangeability or for a specified probability. Methods are introduced whereby designers can identify trade-offs and optimize the allocation of tolerances. Examples are presented that illustrate important features of the new model.
Issue Section:
Technical Papers
1.
American National Standard ASME Y14.5M, 1994, Dimensioning and Tolerancing, The American Society of Mechanical Engineers, NY.
2.
Gossard
, D. C.
, Zuffante
, R. P.
, and Sakurai
, H.
, 1988
, “Representing Dimensions, Tolerances, and Features in MCAE Systems
,” IEEE Comput. Graphics Appl.
, 8
(2
), pp. 51
–59
.3.
Hillyard
, R. C.
, and Braid
, I. C.
, 1978
, “Analysis of Dimensions and Tolerances in Computer-Aided Mechanical Design
,” Comput.-Aided Des.
, 10
(3
), pp. 161
–166
.4.
Light
, R. A.
, and Gossard
, D. C.
, 1982
, “Modification of Geometric Models through Variational Geometry
,” Comput.-Aided Des.
, 14
(4
), pp. 209
–214
.5.
Lin
, V. C.
, Gossard
, D. C.
, and Light
, R. A.
, 1981
, “Variational Geometry in Computer-Aided Design
,” Proc. of Siggraph ’81, Comput. Graph.
, 15
(3
), pp. 171
–177
.6.
Turner, J. U., and Wozny, M., 1988, “A Mathematical Theory of Tolerances,” In Geometric Modeling for CAD Applications, Wozny, McLaughlin, Encarnacao, eds., Elsevier.
7.
Turner, J. U., and Wozny, M. J., 1990, “The M-space Theory of Tolerances,” In Proc. of 16th ASME Design Automation Conf., B. Ravani, ed., ASME Press, pp. 217–225.
8.
Requicha
, A. A. G.
, 1983
, “Toward a Theory of Geometric Tolerances
,” Int. J. Robot. Res.
, 2
(4
), pp. 45
–60
.9.
Requicha
, A. A. G.
, and Chan
, S. C.
, 1986
, “Representation of Geometric Features, Tolerances, and Attributes in Solid Modelers Based on Constructive Geometry
,” IEEE J. Rob. Autom.
, RA-2
(3
), pp. 156
–166
.10.
Roy
, U.
, and Liu
, C. R.
, 1988
, “Feature-Based Representational Scheme of a Solid Modeler for Providing Dimensioning and Tolerancing Information
,” Rob. Comput.-Integr. Manufact.
, 4
(3/4
), pp. 335
–345
.11.
Martinsen, K., 1993, “Vectorial Tolerancing for All Types of Surfaces,” Proc. of 19th ASME Design Automation Conf., Albuquerque, Vol. 2, ASME Press.
12.
Turner, J. U., 1990, “Exploiting Solid Models for Tolerance Computations,” In Geometric Modeling for Product Engineering, M. J. Wozny, J. U. Turner, and K. Preiss, eds., pp. 237–258, North-Holland.
13.
Rivest
, L.
, Fortin
, C.
, and Morel
, C.
, 1994
, “Tolerancing a Solid Model with a Kinematic Formulation
,” Comput.-Aided Des.
, 26
, pp. 465
–476
.14.
Chase
, K.
, Gao
, J.
, Magelby
, S.
, and Sorensen
, C.
, 1998
, “Including Geometric Feature Variations in Tolerance Analysis of Mechanical Assemblies
,” IIE Transactions
, 28
, pp. 795
–807
.15.
Bernstein, N., and Preiss, K., 1989, “Representation of Tolerance Information in Solid Models,” DE-Vol. 19–1, Proc. of 15th ASME Design Automation Conf., Montreal, September 17–21, ASME Press, pp. 37–48.
16.
Zhang, B. C., 1992, “Geometric Modeling of Dimensioning and Tolerancing,” PhD Thesis, Arizona State University.
17.
Kramer, G. A., 1992, Solving Geometric Constraint Systems: A Case Study in Kinematics, MIT Press.
18.
Cle´ment, A., Desrochers, A., Rivie`re, A., 1991, “Theory and Practice of 3D Tolerancing for Assembly,” Second CIRP Seminar on Comp. aided Tolerancing, PennState, May.
19.
Cle´ment, A., Rivie`re, A., and Temmerman, M., 1992, Cotation Tridimensionelle des Systemes Mecaniques, Theorie et Pratique, in French, Yvry-sur-Siene. (English translation in progress by Addison-Wesley, Boston).
20.
Reuleaux, F., 1875, Theoretische Kinematik: Grundzu¨ge einer Theorie des Maschinenwesens, Vieweg, Braunschweig. Trans. Kennedy, A. B. W. (1876) as The Kinematics of Machinery, MacMillan, London. Reprinted (1963) with a new introduction by E. S. Ferguson, Dover, NY.
21.
Hunt
, K. H.
, 1976
, “Geometry—The Key to Mechanical Movements
,” Mech. Mach. Theory
, 11
, pp. 79
–89
.22.
Desrochers, A., 1999, “Modeling Three Dimensional Tolerance Zones Using Screw Parameters,” CD-rom Proceedings, ASME Design Engr. Technical Conf’s. (25th Design Automation Conf.), Las Vegas, NV, Sept. 12–15, 1999, Paper #DETC99/DAC-8587, 9 pp.
23.
Salomons, O., 1995, “Constraint Specification and Satisfaction in Feature Based Design for Manufacturing,” Ph.D. thesis, Univ. of Twente, Enschede, Netherlands.
24.
Srinivasan, V., 1996, “Towards an ISO Standard for Statistical Tolerancing,” In Computer-Aided Tolerancing Techniques, Proc., 4th CIRP Int’l Seminar on Computer-Aided Tolerancing, Chapman-Hall, pp. 159–172.
25.
Voelcker
, H.
, 1998
, “The Current State of Affairs in Dimensional Tolerancing: 1997
,” Integrated Manufacturing Systems
, 9
(4
), pp. 205
–217
.26.
American National Standard ASME Y14.5.1M, 1994, Mathematical Definition of Dimensioning and Tolerancing Principles, The American Society of Mechanical Engineers, NY.
27.
Shah, J. J., and Zhang, B., 1992, “Attributed Graph Model for Geometric Tolerancing,” Proc. of 18th ASME Design Automation Conf., Scottsdale, AZ, Sept. 13–16, Vol. H0770B, pp. 133–139.
28.
Shah
, J. J.
, Yan
, Y.
, and Zhang
, B.-C.
, 1998
, “Dimension and Tolerance Modeling and Transformations in Feature Based Design and Manufacturing
,” J. Integrated Manufacturing
, 9
, N5
N5
.29.
Yan, Y., and Shah, J., 1996, “Representation and Mapping of Geometric Dimensions from Design to Manufacturing,” ASME Computers in Engineering Conference, CD-ROM Proc., Aug., Irvine, CA, Paper #96-DETC/MECH-1481.
30.
Giordano, M., Pairel, E., and Samper, S., 1999, “Mathematical Representation of Tolerance Zones,” In Global Consistency of Tolerances, Proc., 6th CIRP Int’l Seminar on Computer-Aided Tolerancing, Univ. of Twente, Enschede, Netherlands, March 22–24, F. vanHouten and H. Kals, eds., pp. 177–86.
31.
Teissandier, D., Delos, V., and Couetard, Y., 1999, “Operations on Polytopes: Application to Tolerance Analysis,” In Global Consistency of Tolerances, Proc., 6th CIRP Int’l Seminar on Computer-Aided Tolerancing, Univ. of Twente, Enschede, Netherlands, March 22–24, F. vanHouten and H. Kals, eds., pp. 425–434.
32.
Coxeter, H. S. M., 1969, Introduction to Geometry, 2nd ed. Wiley.
33.
Klein, F., 1939, Elementary Mathematics from an Advanced Standpoint, Geometry. MacMillan. (Trans. from the 3rd German ed. (1925) by E. R. Hedrick and C. A. Noble).
34.
Cromwell, P. R., 1997, Polyhedra, Cambridge U. Press.
35.
Valentine, F. A., 1964, Convex Sets, McGraw-Hill, NY.
36.
Mujezinovic´, A., 1999, “A New Mathematical Model for Representing Geometric Tolerances,” MSc thesis, Arizona State University, Tempe, AZ, May.
37.
Roy
, U.
, and Li
, B.
, 1999
, “Representation and Interpretation of Geometric Tolerances for Polyhedral Objects: II. Size, Orientation and Position Tolerances
,” Comput.-Aided Des.
, 31
, pp. 273
–285
.38.
Whitney
, D. E.
, Gilbert
, O. L.
, and Jastrzebski
, M.
, 1994
, “Representation of Geometric Variations Using Matrix Transforms for Statistical Tolerance Analysis in Assemblies
,” Res. Eng. Des.
, 6
, pp. 191
–210
.39.
Hadwiger, H., 1957, Vorlesungen u¨ber Inhalt, Oberfla¨che und Isoperimetrie, Springer.
40.
Jensen, C., 1993, Geometric Dimensioning & Tolerancing, Delmar Pubs.
41.
Laperrie`re, L., and Lafond, P., 1999, “Tolerance Analysis and Synthesis Using Virtual Joints,” In Global Consistency of Tolerances, Proc., 6th CIRP Int’l Seminar on Computer-Aided Tolerancing, Univ. of Twente, Enschede, Netherlands, March 22–24, F. vanHouten and H. Kals, ed., pp. 405–414.
42.
Sommerville, D. M. Y., 1929. An Introduction to the Geometry of n Dimensions, Methuen, London.
Copyright © 2002
by ASME
You do not currently have access to this content.
