A New Mathematical Model for Geometric Tolerances as Applied to Round Faces

A new mathematical model for representing geometric tolerances is applied to polygonal faces and is extended to show its sensitivity to the precedence (ordering) of datum reference frames. The model is compatible with the ASME/ISO Standards for geometric tolerances. Central to the new model is a Tolerance-Map® 2 , a hypothetical volume of points that corresponds to all possible locations and variations of a segment of a plane which can arise from tolerances on size, position, 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. 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.

[1]  Joseph K. Davidson,et al.  Geometric tolerances: A new application for line geometry and screws , 2002 .

[2]  R. Light,et al.  Modification of geometric models through variational geometry , 1982 .

[3]  Alain Riviere,et al.  A matrix approach to the representation of tolerance zones and clearances , 1997 .

[4]  Jami J. Shah,et al.  A new mathematical model for geometric tolerances as applied to rectangular faces , 2001 .

[5]  M. Giordano,et al.  Mathematical representation of Tolerance Zones , 1999 .

[6]  Serge Samper,et al.  Taking into account elastic displacements in 3D tolerancing , 1998 .

[7]  Aristides A. G. Requicha,et al.  Toward a Theory of Geometric Tolerancing , 1983 .

[8]  D. M. Y. Sommerville,et al.  An Introduction to The Geometry of N Dimensions , 2022 .

[9]  Aristides A. G. Requicha,et al.  Representation of geometric features, tolerances, and attributes in solid modelers based on constructive geometry , 1986, IEEE J. Robotics Autom..

[10]  Daniel E. Whitney,et al.  Representation of geometric variations using matrix transforms for statistical tolerance analysis in assemblies , 1994 .

[11]  R. C. Hillyard,et al.  Analysis of dimensions and tolerances in computer-aided mechanical design , 1978 .

[12]  H. Coxeter,et al.  Introduction to Geometry. , 1961 .

[13]  K. H. Hunt,et al.  Geometry—The key to Mechanical movements , 1976 .

[14]  P. Zsombor-Murray,et al.  Elementary Mathematics from an Advanced Standpoint , 1940, Nature.

[15]  Louis Rivest,et al.  Tolerancing a solid model with a kinematic formulation , 1994, Comput. Aided Des..

[16]  Jami J. Shah,et al.  Attributed graph model for geometric tolerancing , 1992, DAC 1992.

[17]  H. Hadwiger.  Vorlesungen über Inhalt, Oberfläche und Isoperimetrie , 1957 .

[18]  Herbert B. Voelcker,et al.  The current state of affairs in dimensional tolerancing: 1997 , 1998 .

[19]  Vijay Srinivasan,et al.  Towards an ISO Standard for Statistical Tolerancing , 1996 .

[20]  Utpal Roy,et al.  Representation and interpretation of geometric tolerances for polyhedral objects. II.: Size, orientation and position tolerances , 1999, Comput. Aided Des..

[21]  Jami J. Shah,et al.  Dimension and tolerance modeling and transformations in feature based design and manufacturing , 1998, J. Intell. Manuf..

[22]  Spencer P. Magleby,et al.  Including Geometric Feature Variations in Tolerance Analysis of Mechanical Assemblies , 1996 .

[23]  David C. Gossard,et al.  Representing dimensions, tolerances, and features in MCAE systems , 1988, IEEE Computer Graphics and Applications.

[24]  Luc Laperrière,et al.  Tolerance Analysis And Synthesis Using Virtual Joints , 1999 .

[25]  Jami J. Shah,et al.  Structure for supporting geometric tolerances in product definition systems for CIM , 1990 .

[26]  David C. Gossard,et al.  Variational geometry in computer-aided design , 1981, SIGGRAPH '81.

[27]  Utpal Roy,et al.  Feature-based representational scheme of a solid modeler for providing dimensioning and tolerancing information , 1988 .

[28]  Glenn A. Kramer.  Solving geometric constraint systems a case study in kinematics , 1992, Comput. Aided Des..