Modified Method of the Unified Jacobian-Torsor Model for Tolerance Analysis and Allocation

다국어 초록 (Multilingual Abstract)

The unified Jacobian-Torsor model uses the torsor model for tolerance representation and the Jacobian matrix for tolerance propagation.
The torsor model is composed of six components, i.e., three translational vectors and three rotational vectors. However, previous studies about this model have only considered the constraint of individual component. In fact, this constraint is the scope of a single component.
It is called variation. Relations between these components are constraints which reflect the interaction between them in a tolerance zone.
Integrating all limited values of components of torsors into the unified Jacobian-Torsor model may lead to an inaccurate result.
Meanwhile, the variations and constraints of torsor for a feature specified by more than one tolerance have been not illustrated clearly.
In this paper, a modified method of the unified Jacobian-Torsor model considering constraints between components of torsor is presented.
The variations and constraints of torsors for cylindrical and planar features are proposed. These constraints are calculated by means of a modified Monte Carlo method based on the previous work. Moreover, tolerance allocation of this modified method in a statistical way is also introduced. Two case studies have been performed to demonstrate the modified method.

참고문헌 (Reference)

1 Ameta, G., "Using Tolerance-maps to Generate Frequency Distributions of Clearance and Allocate Tolerances for Pin-hole Assemblies" 7 (7): 347-359, 2007

2 Rivest, L., "Tolerancing A Solid Model with A Kinematic Formulation" 26 (26): 465-476, 1994

3 Ameta, G., "Tolerance-Maps Applied to a Point-line Cluster of Features" 129 (129): 782-792, 2007

4 Clément, A., "Theory and Practice of 3D Tolerancing for Assembly" 25-55, 1991

5 Ameta, G., "The Effects of Different Specifications on the Tolerance-Maps for an Angled Face" 303-311, 2004

6 Lin, C. H., "Study of an Assembly Tolerance Allocation Model based on Monte Carlo Simulation" 70 (70): 9-16, 1997

7 Laperrière, L., "Statistical and Deterministic Tolerance Analysis and Synthesis using a Unified Jacobian-Torsor Model" 51 (51): 417-420, 2002

8 Ghie, W., "Statistical Tolerance Analysis using the Unified Jacobian-Torsor Model" 48 (48): 4609-4630, 2010

9 Jin, J., "State Space Modeling of Sheet Metal Assembly for Dimensional Control" 121 (121): 756-762, 1999

10 Bourdet, P., "Series on Advances in Mathematics for Applied Sciences Advanced Mathematical Tools in Metrology, Vol. 40" World Scientific Publishing 110-122, 1996

11 Nigam, S. D., "Review of Statistical Approaches to Tolerance Analysis" 27 (27): 6-15, 1995

12 Whitney, D. E., "Representation of Geometric Variations using Matrix Transforms for Statistical Tolerance Analysis in Assemblies" 6 (6): 191-210, 1994

13 Roy, U., "Representation and Interpretation of Geometric Tolerances for Polyhedral Objects. II. Size, Orientation and Position Tolerances" 31 (31): 273-285, 1999

14 Bhide, S., "Models for Computer Aided Tolerancing in Design and Manufacturing" Springer 45-54, 2007

15 Ghie, W., "Models for Computer Aided Tolerancing in Design and Manufacturing" Springer 95-104, 2007

16 Desrochers, A., "Modeling Three Dimensional Tolerance Zones using Screw Parameters" 895-903, 1999

17 Laperrière, L., "Modeling Assembly Quality Requirements using Jacobian or Screw Transforms: A Comparison" 330-336, 2001

18 Cai, M., "Mathematical Model of Form Tolerance for Cylindrical Feature based on Mathematical Definition" 39 (39): 86-90, 2003

19 Lafond, P., "Jacobian-based Modeling of Dispersions Affecting Predefined Functional Requirements of Mechanical Assemblies" 20-25, 1999

20 Ameta, G., "Influence of Form on Tolerance-Map-generated Frequency Distributions for 1D Clearance in Design" 34 (34): 22-27, 2010

21 Chase, K. W., "Including Geometric Feature Variations in Tolerance Analysis of Mechanical Assembly" 28 (28): 795-807, 1996

22 Anselmetti, B., "ISO Manufacturing Tolerancing: Three-Dimensional Transfer with Analysis Line Method" 61 (61): 1085-1099, 2012

23 Davidson, J. K., "Geometric Tolerances: A New Application for Line Geometry and Screws" 216 (216): 95-103, 2002

24 Fleming, A., "Geometric Relationships between Toleranced Features" 37 (37): 403-412, 1988

25 Gao, J., "General 3-D Tolerance Analysis of Mechanical Assemblies with Small Kinematic Adjustments" 30 (30): 367-377, 1998

26 Chase, K. W., "General 2-D Tolerance Analysis of Mechanical Assemblies with Small Kinematic Adjustments" 5 (5): 263-274, 1995

27 Ameta, G., "Effects of Size, Orientation, and Form Tolerance on The Frequency Distributions of Clearance between Two Planar Faces" 11 (11): 011002-, 2011

28 The American Society of Mechanical Engineers, "Dimensioning and Tolerancing"

29 Portman, V. T., "Computer-Aided Tolerancing" Chapman & Hall 71-85, 1995

30 Salomons, O. W., "Computer Support in The Design of Mechanical Products" University of Twente 1995

31 Ameta, G., "Comparison of Spatial Math Models for Tolerance Analysis: Tolerance-Maps, Deviation, Domain, and TTRS" 11 (11): 012004-, 2011

32 Desrochers, A., "Application of a Unified Jacobian-Torsor Model for Tolerance Analysis" 3 (3): 2-14, 2003

33 Cai, W., "A Variational Method of Robust Fixture Configuration Design for 3D Workpieces" 119 (119): 593-602, 1997

34 Glancy, C. G., "A Second Order Method for Assembly Tolerance Analysis" 1999

35 Marziale, M., "A Review of Two Models for Tolerance Analysis of an Assembly: Jacobian and Torsor" 24 (24): 74-86, 2011

36 Davidson, J. K., "A New Mathematical Model for Geometric Tolerances as Applied to Round Faces" 124 (124): 609-622, 2002

37 Mujezinovic´, A., "A New Mathematical Model for Geometric Tolerances as Applied to Polygonal Faces" 126 (126): 504-518, 2004

38 Bhide, S., "A New Mathematical Model for Geometric Tolerances as Applied to Axes" 329-337, 2003

39 Desrochers, A., "A Matrix Approach to the Representation of Tolerance Zones and Clearances" 13 (13): 630-636, 1997

40 Salomons, O. W., "A Computer Aided Tolerancing Tool II: Tolerance Analysis" 31 (31): 175-186, 1996

41 Teissandier, D., "A Computer Aided Tolerancing Model: Proportioned Assembly Clearance Volume" 31 (31): 805-817, 1999

42 Chen, H., "A Comprehensive Study of Three Dimensional Tolerance Analysis Methods" 53 : 1-13, 2014

43 Hong, Y. S., "A Comprehensive Review of Tolerancing Research" 40 (40): 2425-2459, 2002

44 Shen, Z., "A Comparative Study of Tolerance Analysis Method" 5 (5): 247-256, 2005