This paper presents a comparison of degree of freedom (DOF) based math models, viz., tolerance-maps, deviation domain, and TTRS, which have shown most potential for retrofitting the nuances of the ASME/ISO tolerance standards. Tolerances specify allowable uncertainty in dimensions and geometry of manufactured products. Due to these characteristics and application of tolerances, it is necessary to create a math model of tolerances in order to build a computer application to assist a designer in performing full 3D tolerance analysis. Many of the current efforts in modeling tolerances are lacking, as they either do not completely model all the aspects of the ASME/ISO tolerance standards or are lacking the requisite full 3-D tolerance analysis. Some tolerance math models were developed to suit CAD applications used by the designers while others were developed to retrofit the ASME/ISO tolerance standard. Three math models developed to retrofit the ASME/ISO standard, tolerance-maps, deviation domain, and TTRS are the main focus of this paper. Basic concepts of these math models are summarized in this paper, followed by their advantages and future issues. Although these three math models represent all aspects of the ASME/ISO tolerance standard, they are still lacking in one or two minor aspects.

References

1.
ASME Y14.5–2009 Dimensioning and Tolerancing
, 2009, ASME, New York, NY.
2.
ISO 1101:2004 Geometrical Product Specifications (GPS) — Geometrical tolerancing — Tolerancing of form, orientation, location and run-out.
3.
Shah
,
J. J.
,
Ameta
,
G.
,
Shen
,
Z.
, and
Davidson
,
J.
, 2007, “
Navigating the Tolerance Analysis Maze
,”
Comput.-Aid. Des. Appl.
,
4
(
5
), pp.
705
714
. See http://www.cadanda.com/CAD_4_5__13.PDFhttp://www.cadanda.com/CAD_4_5__13.PDF
4.
Shen
,
Z.
,
Ameta
,
G.
,
Shah
,
J. J.
, and
Davidson
,
J. K.
, 2005, “
A Comparative Study of Tolerance Analysis Methods
,”
ASME J. Comput. Inf. Sci. Eng.
,
5
(
3
), pp.
247
256
.
5.
Chase
,
K. W.
, and
Parkinson
,
A. R.
, 1991, “
A Survey of Research in the Application of Tolerance Analysis to the Design of Mechanical Assemblies
,”
Res. Eng. Des.
,
3
(
1
), pp.
23
37
.
6.
Hong
,
Y. S.
, and
Chang
,
T. C.
, 2002, “
A Comprehensive Review of Tolerancing Research
,”
Int. J. Prod. Res.
,
40
(
11
), pp.
2425
2459
.
7.
Nigam
,
S. D.
, and
Turner
,
J. U.
, 1995, “
Review of Statistical Approaches to Tolerance Analysis
,”
Comput.-Aid. Des.
,
27
(
1
), pp.
6
15
.
8.
Prisco
,
U.
, and
Giorleo
,
G.
, 2002, “
Overview of Current CAT Systems: Review Article
,”
Integ. Comput.-Aid. Eng.
,
9
(
4
), pp.
373
387
.
9.
Davidson
,
J. K.
,
Mujiezinovic
,
A.
, and
Shah
,
J. J.
, 2002, “
A New Mathematical Model for Geometric Tolerances as Applied to Round Faces
,”
ASME J. Mech. Des.
,
124
(
4
), pp.
609
622
.
10.
Marziale
,
M.
, and
Polini
,
W.
, 2011, “
A Review of Two Models for Tolerance Analysis of an Assembly: Jacobian and Torsor
,”
Int. J. Comput. Integ. Manuf.
,
24
(
1
), pp.
74
86
.
11.
Marziale
,
M.
, and
Polini
,
W.
, 2009, “
A Review of Two Models for Tolerance Analysis of an Assembly: Vector Loop and Matrix
,”
Int. J. Adv. Manuf. Technol.
,
43
(
11
), pp.
1106
1123
.
12.
Hillyard
,
R. C.
, and
Braid
,
I. C.
, 1978, “
Analysis of Dimensions and Tolerances in Computer-Aided Mechanical Design
,”
Comput.-Aid. Des.
,
10
(
3
), pp.
161
166
.
13.
Gossard
,
D. C.
,
Zuffante
,
R. P.
, and
Sakurai
,
H.
, 1988, “
Representing Dimensions, Tolerances, and Features in MCAE Systems
,”
IEEE Comput. Graphic. Appl.
,
8
(
2
), pp.
51
59
.
14.
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. Robot. Automat.
,
2
(
3
), pp.
156
166
.
15.
Requicha
,
A. A. G.
, 1983, “
Toward a Theory of Geometric Tolerancing
,”
Int. J. Robot. Res.
,
2
(
4
), pp.
45
.
16.
Roy
,
U.
, and
Liu
,
C. R.
, 1988, “
Feature-Based Representational Scheme of a Solid Modeler for Providing Dimensioning and Tolerancing Information
,”
Robot. Comput.- Integr. Manufact.
,
4
(
3–4
), pp.
335
345
.
17.
Pasupathy
,
T. M. K.
,
Morse
,
E. P.
, and
Wilhelm
,
R. G.
, 2003, “
A Survey of Mathematical Methods for the Construction of Geometric Tolerance Zones
,”
ASME J. Comput. Inf. Sci. Eng.
,
3
(
1
), pp.
64
76
.
18.
Martinsen
,
K.
, 1993, “
Vectorial Tolerancing for all Types of Surfaces
,”
ASME Adv. Design Autom.
,
2
, pp.
187
198
.
19.
Turner
,
J. U.
, and
Wozny
,
M. J.
, 1990, “
The M-Space Theory of Tolerances
,”
Proceedings of the ASME 16th Design Automation Conference
,
Chicago, IL
, pp.
217
225
.
20.
Chase
,
K. W.
,
Gao
,
J.
,
Magleby
,
S. P.
, and
Sorenson
C. D.
, 1996, “
Including Geometric Feature Variations in Tolerance Analysis of Mechanical Assemblies
,”
IIE Trans.
,
28
(
10
), pp.
795
808
.
21.
Rivest
,
L.
,
Fortin
,
C.
, and
Morel
,
C.
, 1994, “
Tolerancing a Solid Model with a Kinematic Formulation
,”
Comput.-Aid. Design
,
26
(
6
), pp.
465
476
.
22.
Chase
,
K. W.
,
Gao
,
J.
, and
Magleby
,
S. P.
, 1995, “
General 2-D Tolerance Analysis of Mechanical Assemblies With Small Kinematic Adjustments
,”
J. Design Manuf.
,
5
(
4
), pp.
263
274
.
23.
Gao
,
J.
,
Chase
,
K. W.
, and
Magleby
,
S. P.
, 1998, “
Generalized 3-D Tolerance Analysis of Mechanical Assemblies With Small Kinematic Adjustments
,”
IIE Trans.
,
30
(
4
), pp.
367
377
.
24.
Kramer
,
G. A.
, 1992,
Solving Geometric Constraint Systems: A Case Study in Kinematics
,
The MIT Press
,
Cambridge, MA
.
25.
Coxeter
,
H. S. M.
, 1961,
Introduction to Geometry
,
Wiley
,
New York
.
26.
Bhide
,
S.
,
Davidson
,
J. K.
, and
Shah
,
J. J.
, 2003, “
A New Mathematical Model for Geometric Tolerances as Applied to axes
,”
CD Proceedings of ASME Design Engineering Technical Division, Design Automation Conference
, Paper No. 48736.
27.
Davidson
,
J. K.
, and
Shah
,
J. J.
, 2002, “
Geometric Tolerances: A New Application for Line Geometry and Screws
,”
Proc. Inst. Mech. Eng., C
,
216
(
1
) , pp.
95
103
.
28.
Ameta
,
G.
,
Davidson
,
J. K.
, and
Shah
,
J. J.
, 2004, “
The Effects of Different Specifications on the Tolerance-Maps for an Angled Face
,”
Proceedings of ASME Design Engineering Technical Conferences – Design Automation Conference
,
Salt Lake City, Utah
, September–October 2004, Paper No. 57199.
29.
Jian
,
A.
,
Ameta
,
G.
,
Davidson
,
J.
, and
Shah
,
J.
, 2007, “
Tolerance Analysis and Allocation Using Tolerance-Maps for a Power Saw Assembly
,”
Models for Computer Aided Tolerancing in Design and Manufacturing
,
J. K.
Davidson
, ed., pp.
267
276
,
Springer
,
Dordrecht, The Netherlands
.
30.
Ameta
,
G.
,
Davidson
,
J. K.
, and
Shah
,
J. J.
, 2007, “
Tolerance-Maps Applied to a Point-Line Cluster of Features
,”
ASME J. Mech. Des.
,
129
, pp.
782
792
.
31.
Mujezinović
,
A.
,
Davidson
,
J. K.
, and
Shah
,
J. J.
, 2004, “
A New Mathematical Model for Geometric Tolerances a Applied to Polygonal Faces
,”
ASME J. Mech. Des.
,
126
(
3
), pp.
504
518
.
32.
Giordano
,
M.
,
Samper
,
S.
, and
Petit
,
J.
, 2007, “
Tolerance Analysis and Synthesis by Means of Deviation Domains, Axi-Symmetric Cases
,”
Models for Computer Aided Tolerancing in Design and Manufacturing
,
J. K.
Davidson
, ed.
Springer
,
Dordrecht, The Netherlands
, pp.
85
94
.
33.
Samper
,
S.
,
Petit
,
J. P.
, and
Giordano
,
M.
, 2007, “
Elastic Clearance Domain and Use Rate Concept Applications to Ball Bearings and Gears
,”
Models for Computer Aided Tolerancing in Design and Manufacturing
,
J. K.
Davidson
, ed.,
Springer
,
Dordrecht, The Netherlands
, pp.
331
340
.
34.
Samper
,
S.
,
Petit
,
J. P.
, and
Giordano
,
M.
, 2006, “
Computer Aided Tolerancing-Solver and Post Processor Analysis
,”
Advances in Design
,
Hoda
ElMaraghy
and
Waguih
ElMaraghy
, eds.,
Springer
,
London
, pp.
487
497
.
35.
Adragna
,
P. A.
,
Samper
,
S.
, and
Pillet
,
M.
, 2010, “
A Proposition of 3D Inertial Tolerancing to Consider the Statistical Combination of the Location and Orientation Deviations
,”
Int. J. Prod. Develop.
,
10
(
1
), pp.
26
45
.
36.
Adragna
,
P. A.
,
Samper
,
S.
, and
Favreliere
,
H.
, 2010, “
How Form Errors Impact on 2D Precision Assembly with Clearance?
,”
Precision Assembly Technologies and Systems
,
Svetan
Ratchev
, ed.,
Springer
,
Berlin
pp.
50
59
(Proceedings of 5th IFIP WG 5.5 International Precision Assembly Seminar, IPAS 2010, Chamonix, France, February 14–17, 2010).
37.
Fukuda
,
K.
,
Petit
,
J. P.
, and
Annecy
,
F.
, 2003, “
Optimal Tolerancing in Mechanical Design Using Polyhedral Computational Tools
,”
Proceedings of the 19th European Workshop on Computational Geometry
, pp.
117
120
38.
Germain
,
F.
,
Giordano
,
M.
, and
Adragna
,
P. A.
, 2006, “
Taking Manufacturing Dispersions into Account for Assembly: Modelling And Simulation
,”
J. Mach. Eng.
,
6
(
1
), pp.
83
94
.
39.
Reuleaux
,
F.
, 1875, Theoretische Kinematik: Grundzüge einer Theorie des Maschinenwesens, F. Vieweg und Sohn.
40.
Hunt
,
K. H.
, 1976, “
Geometry–The Key to Mechanical Movements
,”
Mech. Mach. Theory
,
11
(
2
), pp.
79
89
.
41.
Desrochers
,
A.
, and
Clement
,
A.
, 1994, “
A Dimensioning and Tolerancing Assistance Model for CAD/CAM Systems
,”
Int. J. Adv. Manuf. Technol.
,
9
(
6
), pp.
352
361
.
42.
Desrochers
,
A.
, 2003, “
A CAD/CAM Representation Model Applied to Tolerance Transfer Methods
,”
ASME J. Mech. Des.
,
125
, pp.
14
22
.
43.
Clemént
,
A.
,
Rivière
,
A.
,
Serré
,
P.
, and
Valade
,
C.
, 1997, “
The TTRS: 13 Constraints for Dimensioning and Tolerancing
,”
Proceedings of the 5th CIRP International Seminar on Computer-Aided Tolerancing
.
44.
Desrochers
,
A.
, 1999, “
Modeling Three Dimensional Tolerance Zones Using Screw Parameters
,”
CD-ROM Proceedings of ASME 25th Design Automation Conference
, DAC–8587,
Las-Vegas, NY
.
45.
Laperrière
,
L.
, and
Desrochers
,
A.
, 2002, “
Modeling Assembly Quality Requirements Using Jacobian or Screw Transforms: A Comparison
,”
Proceedings of the IEEE International Symposium on Assembly and Task Planning
, pp.
330
336
.
46.
Clément
,
A.
,
Rivière
,
A.
, and
Serré
,
P.
, 1996, “
The TTRS: A Common Declarative Model for Relative Positioning, Tolerancing and Assembly,” MICAD proceedings
,”
Revue de CAD-CAM et d’informatique graphique
,
1
(
1–2
), pp.
149
164
.
47.
Liu
,
J.
, and
Wilhem
,
R.
, 2001, “
Genetic Algorithms for TTRS Tolerance Analysis
,”
Proceedings of the 7th CIRP International Seminar on Computer Aided Tolerancing
,
Paris, France
, pp.
55
64
.
48.
Ameta
,
G.
,
Davidson
,
J. K.
, and
Shah
,
J. J.
, 2007, “
Using Tolerance-Maps to Generate Frequency Distributions of Clearance and Allocate Tolerances for Pin-Hole Assemblies
,”
ASME J. Comput. Inf. Sci. Eng.
,
7
(
4
), pp.
347
359
.
49.
Ameta
,
G.
, 2006, “
Statistical Tolerance Analysis and Allocation for Assemblies Using Tolerance-Maps
,” Ph.D. dissertation, Arizona State University, Arizona.
50.
Ameta
,
G.
,
Davidson
,
J. K.
, and
Shah
,
J.
, 2010, “
Influence of Form on Tolerance-Map-Generated Frequency Distributions for 1D Clearance in Design
,”
Precis. Eng.
,
34
(
1
), pp.
22
27
.
51.
Ameta
,
G.
,
Davidson
,
J. K.
, and
Shah
,
J. J.
, 2010, “
Statistical Tolerance Allocation for Tab-Slot Assemblies Utilizing Tolerance-Maps
,”
ASME J. Comput. Inf. Sci. Eng.
,
10
(
1
), p.
011005
.
52.
Singh
,
G.
,
Ameta
,
G.
,
Davidson
,
J. K.
, and
Shah
,
J. J.
, 2009, “
Worst-Case Tolerance Analysis of a Self-Aligning Coupling Assembly Using Tolerance-Maps
,”
Proceedings 11th CIRP International Seminar, Annecy, France
.
53.
Nejad
,
M. K.
,
Vignat
,
F.
,
Desrochers
,
A.
, and
Villeneuve
,
F.
, 2010, “
3D Simulation of Manufacturing Defects for Tolerance Analysis
,”
ASME J. Comput. Inf. Sci. Eng.
,
10
(
2
), p.
011005
.
54.
Villeneuve
,
F.
, and
Vignat
,
F.
, 2005, “
Manufacturing Process Simulation for Tolerance Analysis and Synthesis
,”
Advances in Integrated Design and Manufacturing in Mechanical Engineering
,
Alan Bramley
,
Daniel
Brissaud
,
Daniel
Coutellier
, and
Chris
McMahon
, eds.,
Springer
,
Dordrecht, The Netherlands
, pp.
189
200
.
55.
Jayaprakash
,
G.
,
Sivakumar
,
K.
, and
Thilak
,
M.
, 2010, “
Parametric Tolerance Analysis of Mechanical Assembly by Developing Direct Constraint Model in CAD and Cost Competent Tolerance Synthesis
,”
Intell. Control Autom.
,
1
(
1
), pp.
1
14
.
56.
Jaballi
,
K.
,
Bellacicco
,
A.
,
Louati
,
J.
,
Riviere
,
A.
, and
Haddar
,
M.
, 2011, “
Rational Method for 3D Manufacturing Tolerancing Synthesis Based on the TTRS Approach
,”
Comput. Ind.
,
62
(
5
), pp.
541
554
.
57.
Stuppy
,
J.
,
and Meerkamm
,
H.
, 2009, “
Tolerance Analysis of Mechanisms Taking into Account Joints With Clearance and Elastic Deformations
,”
Proceedings of the 17th International Conference on Engineering Design (ICED’09)
, Vol.
5
, pp.
489
500
.
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