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TECHNICAL PAPERS

A New Mathematical Model for Geometric Tolerances as Applied to Round Faces

[+] Author and Article Information
J. K. Davidson

Department of Mechanical and Aerospace Engineering, Arizona State University, Tempe, AZ 85287-6106 e-mail: J.Davidson@asu.edu

A. Mujezinović

Power Systems Division, General Electric Co., Schenectady, NY

J. J. Shah

Arizona State University, Tempe, AZ 85287-6106

J. Mech. Des 124(4), 609-622 (Nov 26, 2002) (14 pages) doi:10.1115/1.1497362 History: Received June 01, 2000; Online November 26, 2002
Copyright © 2002 by ASME
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References

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Figures

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(a) A diametral cross-section of a round bar with size tolerance t; the vertical scale in the tolerance-zone is exaggerated. (b) A point-map that represents half the planes in one diametral section of the tolerance-zone of Fig. 1(a).
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The Tolerance-Map®, or T-Map® (three dimensional range of points), for the tolerance-zone shown in Fig. 1(a)
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(a) A diametral section of the accumulation tolerance-zone for an assembly of two of the parts in Fig. 1(a). (b) A half-section of its T-Map.
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A round bar with size, form, and orientation tolerances specified
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(a) A diametral half-section of the T-Map in Fig. 2 and its two diconical sub-sets of dimensions t and (t−t) that lie within when size and form tolerances t and t are specified. (b) A diametral half-section of the dicone of dimension t and a vertical line of dimension (t−t) as sub-sets within a truncated diconical T-Map that results when size and orientation tolerances, t and t, are specified.
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The tradeoff between the internal sub-sets for form and range of positions which lie within the T-Map for one end-face of a round bar (This figure prepared by Mr. S. Ramiswami, Design Automation Laboratory, Arizona State University.)
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A coaxial assembly of two round parts 1 and 2 with their individual tolerance-zones shown with t1=t2; the functional feature is the outer face of Part 2.
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(a) and (b) Diametral half-sections of the T-Maps for each of the parts in Fig. 7, the aspect ratio of the one for Part 2 being unity. (c) A half-section of the accumulation map, point σ3a corresponding to the CW-most plane. (d) A half-section σ1fσ2 fσ3f of the functional T-Map and, inscribed in it, the accumulation T-Map (heavy solid line), the dashed line showing the result of the effective (bounty) orientational tolerance tfb.
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Diametral half-section of the functional T-Map with aspect ratio unity
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The assembly of the same round parts shown in Fig. 7, now with the reference and target faces interchanged; the functional feature is the outer face of Part 1
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(a) and (b) Diametral half-sections of the T-Maps for each of the parts in Fig. 10, the aspect ratio of the one for Part 1 being unity. The dashed line in (a) shows the effect of an orientational tolerance t2. (c) A half-section of the accumulation map (heavy line) with central axis length t1+t2 and of the effective accumulation map (isosceles triangle bounded in part by narrow lines) with central axis length t1+d1t2/d2.
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The magnification of tolerance t2 in any diametral cross-section of the assembly in Fig. 10 because the functional surface has a larger diameter than d2
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Three superimposed diametral half-sections of T-Maps for the assembly in Fig. 10, now with an orientational tolerance t2: the accumulation map (heavy line) with central-axis length t1+t2, the effective accumulation map (trapezoidal) in which the length along its central axis is given by Eq. (6), and the circumscribed functional map (triangular) with aspect ratio unity and height given by Eq. (6).
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Two stacked round parts 1 and 2, both frustums of a cone, with their individual tolerance-zones shown with tb=ts; the functional feature is the face of diameter db on Part 2
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An assembly of a round part and an offset link, the target on each being its upper face
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Tolerance-zones and associated coordinate frames for the assembly in Fig. 15
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A modified T-Map for Part 1 in Fig. 15 which accounts for the amplification of orientational variations arising from an offset to the functional target face (This figure prepared by Mr. S. Ramiswami, Design Automation Laboratory, Arizona State University.)
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Cross-section of the functional map circumscribing the octagonal qs-section of the accumulation map for the parts in Fig. 15 when b>b0. Drawn for t1=t2.
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The qs-section of the two operand dicones from Fig. 18, now including an orientational tolerance t1. The oblique dicone is truncated by a cylinder, removing the dotted portions.
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Three basis points K,L, and M that form a reference triangle for areal coordinates in a plane
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(a) A reference tetrahedron KLMN in which the four basis points are chosen to lie on the three axes of a Cartesian frame. (b) The ordering of points in matrix (A2) which is required to get a positive volume.
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A plane in the Cartesian xyz-frame of a tolerance-zone

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