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TECHNICAL PAPERS

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

[+] Author and Article Information
A. Mujezinović

Power Systems Division, General Electric Co., Schenectady, New York

J. K. Davidson, J. J. Shah

Ira A. Fulton School of Engineering, Department of Mechanical and Aerospace Engineering, Arizona State University, Tempe, Arizona 85287-6106

J. Mech. Des 126(3), 504-518 (Oct 01, 2003) (15 pages) doi:10.1115/1.1701881 History: Received May 01, 2002; Revised October 01, 2003
Copyright © 2004 by ASME
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References

American National Standard ASME Y14.5M, 1994, Dimensioning and Tolerancing, The American Society of Mechanical Engineers, NY.
ISO 1101, 1983, Geometric Tolerancing—Tolerancing of Form, Orientation, Location, and Run-Out—Generalities, Definitions, Symbols, and Indications on Drawings, International Organization for Standardization.
Davidson,  J. K., Mujezinović,  A., and Shah,  J. J., 2002, “A New Mathematical Model for Geometric Tolerances as Applied to Round Faces,” ASME J. Mech. Des., 124, pp. 609–622.
Shah, J. J., and Mäntylä, M., 1995, Parametric and Feature-Based CAD/CAM, Wiley, New York.
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American National Standard ASME Y14.5.1M, 1994, Mathematical Definition of Dimensioning and Tolerancing Principles. The American Society of Mechanical Engineers, NY.
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Figures

Grahic Jump Location
The reference tetrahedron σ1σ2σ3σ4 in which the four basis-points are chosen to lie on the three axes of a Cartesian frame. Dimensions are σ1σ2=Oσ3=Oσ4=t and σ3σ4=t2.
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The end of a rectangular bar with size tolerance t; the vertical scale in the tolerance-zone is exaggerated. Symbols B□ and C□ designate the front sides that are used as datums in Section 4. Datum A is the lower end of the bar (not shown). Drawn with dy>dx.
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(a) One view of the bar in Fig. 2 and its tolerance-zone (looking along the x-axis). (b) A point-map that represents half of the planes that are parallel to the x-axis and lie in the tolerance-zone of Fig. 2.
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The Tolerance-Map®, or T-Map® (three dimensional range of points), for the tolerance-zone on the rectangular bar shown in Fig. 2. The dotted lines form three edges of the reference tetrahedron in Fig. 1.
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A triangular bar formed by removing the front half of Fig. 2. Origin O and the z-axis lie in that face of the bar having the greatest width.
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The Tolerance-Map for the triangular bar in Fig. 5, obtained by translating the faces σ1σ3σ4 and σ2σ7σ8 in Fig. 4 outward until they become points σ9 and σ10
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A central assembly (no offset) of two rectangular 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) Half of the conformable qs-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 σ1fσ2fσ3f 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 tfby.
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(a) Half of the qs-section of the orientational T-Map (dipyramid σ1Cσ3Cσ2Cσ7Cσ4Cσ8C) and half (hatched triangle with base and height tA−tC) of the rhombic T-Map that represents its displacement when orientational tolerances tA and tC are specified for a rectangular face as in feature control frames (7). (b) Half of the ps-section of the same two T-Maps, the rhombus now appearing on edge (heavy line).
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The T-Map for only the orientational variations of a rectangular face (Fig. 2), as specified in feature control frames (7). Length σ1Aσ2A=Oσ3A=tA.
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The entire T-Map for the rectangular face in Fig. 2 on which orientational tolerances tA and tC are specified with frames (7). Length σ1σ2=t, just as in Fig. 4.
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A solid model of the entire T-Map for the rectangular face in Fig. 2 on which orientational tolerances tB and tC are specified with frames (8)4. Length σ1σ2=t.
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The two-part assembly for Section 5
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The solid model of the Minkowski sum formed from a right circular dicone (see Section 5) and the solid shown in Fig. 104
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Portions of two superimposed pq-sections of Fig. 10 (heavy lines) and of the functional map less the T-Map for Part 2 (full dicone). The larger circle corresponds to contact only at the antipodal point (point 1), the smaller circle to contact at both the antipodal point (point 2) and at the q-axis (point 3).
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The pq-section of Fig. 12 and of the functional map less the T-Map for Part 2 (truncated dicone)
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A plane in the Cartesian xyz-frame of a tolerance-zone
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Symmetrical polygons inscribed in a circle. (a) A hexagon. (b) An octagon.
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Normalized values of the coordinates for points R and S in Fig. 18(b)

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