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Research Papers

Tolerance-Maps to Model Composite Positional Tolerancing for Patterns of Features

[+] Author and Article Information
Gaurav Ameta, Gagandeep Singh

Department of Mechanical
and Aerospace Engineering,
Arizona State University,
Tempe, AZ 85287

Joseph K. Davidson

Design Automation Laboratory,
Department of Mechanical
and Aerospace Engineering,
Arizona State University,
Tempe, AZ 85287
e-mail: j.davidson@asu.edu

Jami J. Shah

Honda Professor of Engineering Design
Department of Mechanical
and Aerospace Engineering,
The Ohio State University,
Columbus, OH 43210

1 Present address: Dakota Consulting, Inc., Silver Spring, MD 20899.

2 Present address: SalesForce.com, Inc., San Francisco, CA 94105.

Contributed by the Computers and Information Division of ASME for publication in the JOURNAL OF COMPUTING AND INFORMATION SCIENCE IN ENGINEERING. Manuscript received October 13, 2017; final manuscript received February 5, 2018; published online June 12, 2018. Assoc. Editor: Jitesh H. Panchal. This work is in part a work of the U.S. Government. ASME disclaims all interest in the U.S. Government's contributions.

J. Comput. Inf. Sci. Eng 18(3), 031003 (Jun 12, 2018) (9 pages) Paper No: JCISE-17-1221; doi: 10.1115/1.4039473 History: Received October 13, 2017; Revised February 05, 2018

For the first time, Tolerance-Maps (T-Maps) are constructed to model composite positional tolerancing applied to patterns (arrays) of features. The T-Map for a feature is a range (codomain) of points obtained by mapping all the variational possibilities (domain) of a feature within its tolerance zone to a hypothetical Euclidean point space. T-Maps have already been developed for tolerances applied to single features, such as to a simple axis (line), a plane, and a cylinder, but not for the special methods available for tolerancing patterns of features. In this paper, the different pattern tolerancing methods listed in the standards produce distinctions in geometric shape, proportions, and/or dimensions of a T-Map. The T-Map geometry is different when tolerances are specified with composite position tolerancing rather than with two-single-segment control frames. Additional changes to geometry occur when material modifiers are also specified. Two levels of T-Maps are proposed for a pattern of features. One is at the assembly level to ensure the assembly of an engaging pattern of pins and holes, such as the array of pins on an integrated circuit, which are to be inserted into a base. The second is at the part level to model the variations between the two parts that contain the engaging patterns. The assembly-level T-Maps apply to any number of engaging pin/hole features arranged in any pattern: linear, circular, rectangular, or irregular. In this paper, the part-level T-Map is restricted to linear patterns. The different specifications are also compared with a statistical analysis of misalignment for an assembly with a pattern of pins and holes.

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References

ASME, 2009, “Dimensioning and Tolerancing,” American Society of Mechanical Engineers, New York, Standard No. ASME Y14.5.
ISO, 2017, “Geometrical Product Specifications (GPS)—Geometrical Tolerancing—Tolerances of Form, Orientation, Location and Run-Out,” International Organization for Standardization, Geneva, Switzerland, Standard No. ISO 1101:2017. https://www.iso.org/standard/66777.html
Giordano, M. , Pairel, E. , and Samper, S. , 1999, “Mathematical Representation of Tolerance Zones,” Sixth CIRP International Seminar on CAT, Enschede, The Netherlands, Mar. 22–24, pp. 177–186. [CrossRef]
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. [CrossRef]
Clément, A. , Rivière, A. , Serré, P. , and Valade, C. , 1997, “The TTRS: 13 Constraints for Dimensioning and Tolerancing,” Geometric Design Tolerancing: Theories, Standards and Applications, H. A. ElMaraghy , ed., Springer, Boston, MA, pp. 28–29.
Zou, Z. , and Morse, E. P. , 2003, “Applications of the GapSpace Model for Multidimensional Mechanical Assemblies,” ASME J. Comput. Inf. Sci. Eng., 3(1), pp. 22–30. [CrossRef]
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(4), pp. 191–210. [CrossRef]
Pasupathy, T. M. , 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. 64–75. [CrossRef]
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. [CrossRef]
Shah, J. J. , Ameta, G. , Shen, Z. , and Davidson, J. , 2007, “Navigating the Tolerance Analysis Maze,” Comput.-Aided Des. Appl., 4(5), pp. 705–718.
Ameta, G. , Samper, S. , and Giordano, M. , 2011, “Comparison of Spatial Math Models for Tolerance Analysis: Tolerance-Maps, Deviation Domain, and TTRS,” ASME J. Comput. Inf. Sci. Eng., 11(2), p. 021004.
Polini, W. , 2012, “Taxonomy of Models for Tolerance Analysis in Assembling,” Int. J. Prod. Res., 50(7), pp. 2014–2029. [CrossRef]
Mansuy, M. , Giordano, M. , and Davidson, J. K. , 2013, “Comparison of Two Similar Mathematical Models for Tolerance Analysis: T-Map and Deviation Domain,” ASME J. Mech. Des., 135(10), p. 101008. [CrossRef]
Marziale, M. , and Polini, W. , 2011, “Review of Variational Models for Tolerance Analysis of an Assembly,” Proc. Inst. Mech. Eng. Part B: J. Eng. Manuf., 225(3), pp. 305–318. [CrossRef]
Marziale, M. , and Polini, W. , 2011, “A Review of Two Models for Tolerance Analysis of an Assembly: Jacobian and Torsor,” Int. J. Comput. Integr. Manuf., 24(1), pp. 74–86. [CrossRef]
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. [CrossRef]
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. [CrossRef]
Nigam, S. D. , and Turner, J. U. , 1995, “Review of Statistical Approaches to Tolerance Analysis* 1,” Comput.-Aided Des., 27(1), pp. 6–15. [CrossRef]
Hong, Y. S. , and Chang, T. C. , 2002, “A Comprehensive Review of Tolerancing Research,” Int. J. Prod. Res., 40(11), pp. 2425–2459. [CrossRef]
Davidson, J. K. , Mujezinovič, 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. [CrossRef]
Mujezinović, A. , Davidson, J. K. , and Shah, J. J. , 2004, “A New Mathematical Model for Geometric Tolerances as Applied to Polygonal Faces,” ASME J. Mech. Des., 126(3), pp. 504–518. [CrossRef]
Bhide, S. , Ameta, G. , Davidson, J. K. , and Shah, J. J. , 2007, “Tolerance-Maps Applied to the Straightness and Orientation of an Axis,” In Models for Computer Aided Tolerancing in Design and Manufacturing, J. K. Davidson , ed., Springer, Dordrecht, The Netherlands, pp. 45–54. [CrossRef]
Davidson, J. K. , and Shah, J. J. , 2002, “Geometric Tolerances: A New Application for Line Geometry and Screws,” Proc. Inst. Mech. Eng., Part C, 216(1), pp. 95–103. [CrossRef]
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.. [CrossRef]
Bhide, S. , 2002, “A New Mathematical Model for Geometric Tolerances Applied to Cylindrical Features,” M.S. thesis, Arizona State University, Tempe, AZ.
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.
Singh, G. , Ameta, G. , Davidson, J. K. , and Shah, J. J. , 2013, “Tolerance Analysis and Allocation for Design of a Self-Aligning Coupling Assembly Using Tolerance-Maps,” ASME Mech. Des., 135(3), p. 031005.
Ameta, G. , 2004, “Tolerance-Maps Applied to Angled Faces and Two Clusters of Features,” M.S. thesis, Arizona State University, Tempe, AZ.
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(8), pp. 782–792. [CrossRef]
Clasen, P. J. , Davidson, J. K. , and Shah, J. J. , 2009, “Modeling of Geometric Variations Within a Tolerance-Zone for Circular Runout,” ASME Paper No. DETC2009-86283.
Davidson, J. K. , and Shah, J. J. , 2012, “Modeling of Geometric Variations for Line-Profiles,” ASME J. Comput. Inf. Sci. Eng., 12(4), p. 041004. [CrossRef]
He, Y. , Kalish, N. , Davidson, J. K. , and Shah, J. J. , 2016, “Tolerance-Maps for Line-Profiles Formed by Intersecting Kinematically Transformed Primitive T-Map Elements,” ASME J. Comput. Inf. Sci. Eng., 16(2), p. 021005.
Ameta, G. , 2006, “Statistical Tolerance Analysis and Allocation for Assemblies Using Tolerance-Maps,” Ph.D. thesis, Arizona State University, Tempe, AZ.
Jiang, K. , Davidson, J. K. , Liu, J. , and Shah, J. J. , 2014, “Using Tolerance Maps to Validate Machining Tolerances for Transfer of Cylindrical Datum in Manufacturing Process,” Int. J. Adv. Manuf. Technol., 73(1–4), pp. 465–478. [CrossRef]
Mohan, P. , Haghighi, P. , Shah, J. J. , and Davidson, J. K. , 2015, “Development of a Library of Feature Fitting Algorithms for CMMs,” Int. J. Precis. Eng. Manuf., 16(10), pp. 2101–2113.
Gunasena, N. U. , Lehtihet, E.-A. , and HAM, I. , 1991, “The Verification of Composite Position Tolerance,” IIE Trans., 23(3), pp. 290–299. [CrossRef]
Ranade, S. , Lehtihet, E. A. , and Cavalier, T. M. , 2000, “Comparative Evaluation of Composite Position Tolerance Specifications for Patterns of Holes,” 33rd International MATADOR Conference, Manchester, UK, July, pp. 533–538.
Xi, M. , Lehtihet, E. A. , and Cavalier, T. M. , 2004, “Numerical Approximation Approach to the Producibility of Composite Position Tolerance Specifications for Pattern of Holes,” Int. J. Prod. Res., 42(2), pp. 243–266. [CrossRef]
Jiang, Y. , 2016, “Evaluation on the Accuracy of Multiple Machines Using Composite Position Tolerances,” M.S. thesis, Pennsylvania State University, State College, PA. https://etda.libraries.psu.edu/catalog/9c67wm80s
He, G. , Guo, L. , Zhang, M. , and Liu, P. , 2016, “Evaluation of Composite Positional Error Based on Superposition and Containment Model and Geometrical Approximation Algorithm,” Measurement, 94, pp. 441–450. [CrossRef]
Saravanan, A. , Balamurugan, C. , Sivakumar, K. , and Ramabalan, S. , 2014, “Optimal Geometric Tolerance Design Framework for Rigid Parts With Assembly Function Requirements Using Evolutionary Algorithms,” Int. J. Adv. Manuf. Technol., 73(9–12), pp. 1219–1236. [CrossRef]
Ameta, G. , Singh, G. , Davidson, J. K. , and Shah, J. J. , 2017, “Application of T-Maps for Composite Position Tolerance for Patterns of Features,” ASME Paper No. DETC2017-68391.
Coxeter, H. S. M. , 1961, Introduction to Geometry, Wiley, New York.
Giordano, M. , Samper, S. , and Petit, J. , 2007, “Tolerance Analysis and Synthesis by Means of Deviation Domains, Axi-Symmetric Cases,” Ninth CIRP International Seminar on CAT, Tempe, AZ, Apr. 10–12, pp. 85–94.

Figures

Grahic Jump Location
Fig. 1

(a) A hole in a plate of thickness h. (b) A cylindrical tolerance zone, for the axis of a hole, of diameter equal to position tolerance t.

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Fig. 2

Four 3D hypersections of the T-Map and its basis four-simplex for the tolerance zone shown in Fig. 1(b); all circles are of diameter t

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Fig. 3

Two three-dimensional hypersections of the T-Map of the axis when position tolerance t and perpendicularity tolerance t″ are specified

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Fig. 4

Five-dimensional tolerance maps for an engaging pin-hole pair

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Fig. 5

Tolerance specifications for a pattern of holes, using (1) a composite feature control frame with datum A repeated in the feature-relating control frame, (2) a composite feature control frame with datum A and B repeated in the feature-relating control frame, and (3) two single-segment feature control frames

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Fig. 6

One possible position of the FRTZF relative to PLTZF when (1) datum A is repeated in the feature-relating control frame of the composite tolerance specification, (2) both data A and B are repeated in the feature-relating control frame of the composite specification, and (3) the specifications are made with two-single-segment feature control frames

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Fig. 7

One hypersection of the T-Map for the axis of a hole when the tolerances are specified by two single-segment frames, as shown in Fig. 5 control frame (3)

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Fig. 8

The assembly-level T-Map of a pin-hole assembly. Tolerance symbols with subscript p refer to the pin and tolerance symbols with subscript h refer to the hole. ΔFmin is the minimum diametral clearance between the pin and the hole when both are at their MMC sizes.

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Fig. 9

(a) Plate with pattern, of holes, oriented at the maximum possible angle. (b) Point-Line cluster, with the point at the midpoint of line, to represent the location of the pattern; the coordinate system with the y-axis aligned with the line. (c) Maximum misalignment between a hole in the pattern and a gage pin.

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Fig. 10

Effect of different composite tolerancing schemes on the allowable variations of the point-line cluster. Possible position of point-line cluster: (a) with one datum repeated, (b) with two datum repeated, and (c) with two single-segment frames forming the pattern-locating (effective) and feature-relating zones for shown in Fig. 5.

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Fig. 11

The part-level T-Map for pattern of holes when (a) only datum A is also repeated in the lower segment of the composite feature control frame, (b) datum A and datum B are repeated in the lower segment of the composite feature control frame, and (c) two single-segment feature control frames are used

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Fig. 12

Comparison of frequency distributions of only radial (labeled cylindrical) and combined radial and angular (labeled concentric) misalignment of engaging pattern of holes with gage pattern of pins when one datum is repeated in the composite feature control frame

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Fig. 13

Comparison of frequency distributions of combined radial and angular misalignment of engaging pattern of holes with gage pattern of pins for the three composite tolerancing schemes shown in Fig. 5

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Fig. 14

(a) Convolved T-Map, (b) convolved T-Map (shown transparent) with weights corresponding to the shades (light gray = 0.1, gray = 0.25, dark gray = 0.5, and black = 0.75), (c), (d), and (e) Intersection of cylinder and the convolved T-Map at different values of the cylinder diameter. The intersection surface includes the weights from the convolved T-Map.

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Fig. 15

Comparison of only radial (cylindrical) and combined radial and angular (concentric) misalignment for engaging pattern of pins and holes with composite feature control frame of one repeated datum

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