Research Papers

Tolerance-Maps for Line-Profiles Formed by Intersecting Kinematically Transformed Primitive Tolerance-Map Elements

[+] Author and Article Information
Y. He

Siemens PLM Software, Inc.,
2000 Eastman Drive,
Milford, OH 45150
e-mail: yifei.he@siemens.com

J. K. Davidson

Design Automation Laboratory,
Department of Mechanical and
Aerospace Engineering,
Arizona State University,
Tempe, AZ 85287-6106
e-mail: j.davidson@asu.edu

N. J. Kalish

Design Automation Laboratory,
Department of Mechanical and
Aerospace Engineering,
Arizona State University,
Tempe, AZ 85287-6106
e-mail: njkalish@asu.edu

Jami J. Shah

Design Automation Laboratory,
Department of Mechanical and
Aerospace Engineering,
Arizona State University,
Tempe, AZ 85287-6106
e-mail: jami.shah@asu.edu

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTING AND INFORMATION SCIENCE IN ENGINEERING. Manuscript received June 18, 2015; final manuscript received March 23, 2016; published online April 29, 2016. Assoc. Editor: Vijay Srinivasan.

J. Comput. Inf. Sci. Eng 16(2), 021005 (Apr 29, 2016) (14 pages) Paper No: JCISE-15-1201; doi: 10.1115/1.4033236 History: Received June 18, 2015; Revised March 23, 2016

For the purposes of automating the assignment of tolerances during design, a math model, called the Tolerance-Map (T-Map), has been produced for most of the tolerance classes that are used by designers. Each T-Map is a hypothetical point-space that represents the geometric variations of a feature in its tolerance-zone. Of the six tolerance classes defined in the ASME/ANSI/ISO Standards, profile tolerances have received the least attention for representation in computer models. The objective of this paper is to provide a comprehensive treatment of T-Map construction for any line-profile by using primitive T-Map elements and their Boolean intersection. The method requires (a) decomposing a profile into segments, each of constant curvature; (b) creating a solid-model T-Map primitive for each in a common global reference frame; and (c) combining these by Boolean intersection to generate the T-Map for a complete line-profile of any shape. Freeform portions of a profile are modeled as a series of closely spaced points and subsequent formation of short circular arc-segments, each formed from the circle that osculates to three adjacent points.

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American National Standard, 2009, “ Dimensioning and Tolerancing,” American Society of Mechanical Engineers, New York, Standard No. ASME Y14.5M.
ISO, 1983, “ Geometric Tolerancing—Tolerancing of Form, Orientation, Location, and Run-Out—Generalities, Definitions, Symbols, and Indications on Drawings,” International Organization for Standardization, Geneva, Switzerland, Standard No. ISO 1101.
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]
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. 64–75. [CrossRef]
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. [CrossRef]
Shen, Z. , Ameta, G. , Shah, J. J. , and Davidson, J. K. , 2007, “ Navigating the Tolerance-Analysis Maze,” Comput. Aided Des. Appl., 4(5), pp. 705–718.
Giordano, M. , Pairel, E. , and Samper, S. , 1999, “ Mathematical Representation of Tolerance Zones,” Global Consistency of Tolerances, F. vanHouten and H. Kals , eds., Kluwer, Amsterdam, The Netherlands, pp. 177–186.
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(4), pp. 273–285. [CrossRef]
Giordano, M. , Duret, D. , Tichadou, S. , and Arrieux, R. , 1992, “ Clearance Space in Volumic Dimensioning,” Ann. CIRP, 41(1), pp. 565–568. [CrossRef]
Homri, L. , Teissandier, D. , and Ballu, A. , 2015, “ Tolerance Analysis by Polytopes: Taking Into Account Degrees of Freedom With Cap Half-Spaces,” Comput. Aided Des., 62, pp. 112–130. [CrossRef]
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]
Davidson, J. K. , Savaliya, S. B. , He, Y. , and Shah Jami, J. , 2012, “ Methods of Robotics and the Pseudoinverse to Obtain the Least-Squares Fit of Measured Points on Line-Profiles,” ASME Paper No. DETC2012-70203.
Savaliya, S. B. , Davidson, J. K. , and Shah, J. J. , 2013, “ Using Planar Kinematics to Construct the Full 4D Tolerance-Map for a Line-Profile,” ASME Paper No. DETC2013-12682.
He, Y. , Davidson, J. K. , and Shah, J. J. , 2015, “ Tolerance-Maps for Line-Profiles Constructed From Boolean Intersection of T-Map Primitives for Arc-Segments,” J. Zhejiang Univ. Sci. A, 16(5), pp. 341–352. [CrossRef]
He, Y. , 2013, “ Generation of Tolerance-Maps for Line-Profiles by Primitive T-Map Elements,” M.S. thesis, Arizona State University, Tempe, AZ.
Spatial Corp., 2012, “ 3D ACISModeler,” accessed April 26, 2016, https://doc.spatial.com/get_doc_page/articles/a/c/i/Portal~ACIS_e5cc.html
Teissandier, D. , 2012, “ Contribution à l'analyse des tolerances géométriques d'un système mécanique par des polytopes,” Ph.D. Habilitation, Université Bordeaux 1, Bordeaux, France.
Davidson, J. K. , Shah, J. J. , and Mujezinović, A. , 2005, “ Method and Apparatus for Geometric Variations to Integrate Parametric Computer-Aided Design With Tolerance Analysis and Optimization,” U.S. Patent No. 6,963,824.
Coxeter, H. S. M. , 1969, Introduction to Geometry, 2nd ed., Wiley, Toronto, ON, Canada.
Lawrence, J. D. , 1972, A Catalog of Special Plane Curves, Dover, New York.
Von Seggern, D. H. , 1993, CRC Standard Curves and Surfaces, CRC Press, Boca Raton, FL.
Hunt, K. H. , 1979, Kinematic Geometry of Mechanisms, Clarendon Press, Oxford, UK.
Hain, K. , 1967, Applied Kinematics, 2nd ed., McGraw-Hill, New York.
Mohan, P. , Haghighi, P. , Vemulapalli, P. , Kalish, N. , Shah, J. J. , and Davidson, J. K. , 2014, “ Toward Automated Tolerancing of Mechanical Assemblies: Assembly Analyses,” ASME J. Comput. Inf. Sci. Eng., 14(4), p. 041009. [CrossRef]
Uicker, J. J. , Pennock, G. R. , and Shigley, J. E. , 2013, Theory of Machines and Mechanisms, Oxford University Press, Oxford, UK.
Davidson, J. K. , and Hunt, K. H. , 2004, Robots and Screw Theory: Applications of Kinematics and Statics to Robotics, Oxford University Press, Oxford, UK.
Brannan, D. A. , Esplen, M. F. , and Gray, J. J. , 2012, Geometry, 2nd ed., Cambridge University Press, Cambridge, UK.
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. [CrossRef]
Ameta, G. , Davidson, J. K. , and Shah, J. J. , 2011, “ Effects of Size, Orientation, and Form Tolerances on the Frequency Distributions of Clearance Between Two Planar Faces,” ASME J. Comput. Inf. Sci. Eng., 11(1), p. 011002. [CrossRef]
“Equation of Circle Passing through 3 Given Points,” accessed April 26, 2016, www.qc.edu.hk/math/Advanced Level/circle given 3 points.htm


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Fig. 1

Triangular boss with its shape controlled by the profile tolerance ŧ = 0.2 mm relative to Datums A, B, and C

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Fig. 2

The exaggerated tolerance-zone (between two solid-lined triangles) for the triangular line-profile in Fig. 1

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Fig. 3

The eight points representing one rounded vertex of the line-profile in Fig. 2, the spiral curve used in their genesis, and the centers of the osculating circles for the six short overlapping arcs

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Fig. 4

The MSP and the true inner and outer boundaries to the tolerance-zone for side #1 from Fig. 2, which is joined with continuity C0 to adjacent segments; the true and approximated limits to a rotation of the MSP

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Fig. 5

(a) Approximated tolerance-zone for one independent line-segment of length i = 1, its MSP (line AB), and three perfect-form displaced variations C, D, and E (dotted lines). (b) Its 2D and (c) its 3D T-Map primitives.

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Fig. 6

(a) MSP for the entire triangular profile (Fig. 2), the tolerance-zone for side #3, and the superimposed limits of angular location of the MSPs for sides #1 and #3. (b) Superimposed 2D T-Map primitives for all the three sides.

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Fig. 7

The tolerance-zone of Fig. 2, now with an undisplaced perfect-form triangular manufacturing variation (long-and-short dashed line) of size ΔF larger than the MSP

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Fig. 8

(a) A profile variation A′B′ in the tolerance-zone of Fig. 5(a), which is larger than the MSP by ΔF; one displaced perfect-form variation (dotted line) and (b) its 2D and (c) 3D T-Map primitives for +ΔF

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Fig. 9

(a) The MSP for one generic arc-segment (dashed-lined arc) in the (exaggerated) tolerance-zone that is specified with the profile tolerance ŧ, two perfect-form variations C and D (dotted lines); (b) the approximated 2D T-Map; (c) the continuous 2D T-Map boundary; and (d) the 3D T-Map primitive. Adapted from Ref. [14].

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Fig. 10

(a) A profile variation A′B′ for the generic arc-segment which is larger than the MSP by ΔF, two variations C′ and D′ (dotted arcs) and (b) its 2D T-Map primitive. Adapted from Ref. [14].

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Fig. 11

The MSP of the triangular line-profile, five local frames of reference, global frame Gx′y′, and notionally shown frame Px′y′

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Fig. 12

Displaced and undisplaced MSP for one generic line-segment AB, pole P, fixed local frames Oixiyi and Oix′iy′i, and fixed global frame Gx′y′

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Fig. 13

T-Map primitive for the MSP of side #1 of the triangular profile in Figs. 1 and 2. (a) From Fig. 5(c) and with coordinates (ex ey θ′). (b) After transformation to coordinates (e′x e′y θ′) for the Gx′y′-frame.

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Fig. 14

The three T-Map primitives for the MSP of the triangular profile after transformation to coordinates e′x e′y θ′ for the Gx′y′-frame. The unsheared 2D eyθ′-sections (dotted lines) are from Fig. 6(b).

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Fig. 15

Boolean intersection of the three T-Map primitives which forms, for global frame Gx′y′, the 3D T-Map for the MSP of the triangular profile in Fig. 1. The two points with coordinates represent the limits to rotation of the MSP.

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Fig. 16

The canonical 3D T-Map for the MSP of the triangular profile in Fig. 1, represented in the Px′y′-frame

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Fig. 17

The 4D T-Map, referred to frame Px′y′, for the triangular line-profile specified in Fig. 1; its five basis-points ψ1,…,ψ5; and the changes of its morphology as a function of size. For clarity of the graphics, the linear scale in the direction of size (ψ1ψ2) has been shown nonlinear. Values are in millimeter.

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Fig. 18

Form tolerance-zone and subset T-Map for the minimum-zone that captures 35 points measured on one triangular profile. (a) Its tolerance-zone that bounds the points. (b) A portion of Fig. 17 and the subset T-Map for form within it (lengths of dotted lines are exaggerated).

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Fig. 19

The triangular profile of Fig. 1 with one rounded corner: (a) profile and (b) 3D T-Map for its MSP

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Fig. 20

A square line-profile with tolerance ŧ/2 on one side: (a) specification and (b) tolerance-zone and five basis profiles

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Fig. 21

Three-dimensional T-Maps for a square line-profile: (a) MSP with tolerance ŧ/2 on the top side, (b) MSP with tolerance ŧ everywhere, and (c) skeleton e′xe′yΔF-hypersection showing the shift of e′xe′yθ′-centers

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Fig. 22

(a) Profile of a buttress thread to be made on a cylinder of OD 200 mm. (b) T-Map of the MSP for the buttress thread. (c) Unnecessary segment (dotted line EF) to complete the convex hull of the arc-slot profile from Ref. [14].



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