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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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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