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Research Papers

# Mathematical Tools for Automating Digital Fixture Setups: Constructing T-Maps and Relating Metrological Data to Coordinates for T-Maps and Deviation Spaces

[+] Author and Article Information
N. J. Kalish

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

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

S. Ramnath

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

P. Haghighi

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

Jami J. Shah

Engineering Design,
Department of Mechanical and
Aerospace Engineering,
The Ohio State University,
Columbus, OH 43210
e-mail: shah.493@osu.edu

Jiten V. Shah

Product Development & Analysis, LLC,
Naperville, IL 60563
e-mail: info@pda-llc.com

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTING AND INFORMATION SCIENCE IN ENGINEERING. Manuscript received August 4, 2017; final manuscript received November 22, 2017; published online July 30, 2018. Editor: Satyandra K. Gupta.

J. Comput. Inf. Sci. Eng 18(4), 041009 (Jul 30, 2018) (12 pages) Paper No: JCISE-17-1157; doi: 10.1115/1.4038821 History: Received August 04, 2017; Revised November 22, 2017

## Abstract

Mathematical tools underlie a method that has strong potential to lower the cost of fixture-setup when finishing large castings that have machined surfaces where other components are attached. One math tool, the kinematic transformation, is used for the first time to construct Tolerance-Map® (T-Map)® models of geometric and size tolerances that are applied to planar faces and to the axes of round shapes, such as pins or holes. For any polygonal planar shape, a generic T-Map primitive is constructed at each vertex of its convex hull, and each is sheared uniquely with the kinematic transformation. All are then intersected to form the T-Map of the given shape in a single frame of reference. For an axis, the generic T-Map primitive represents each circular limit to its tolerance-zone. Both are transformed to a central frame of reference and are intersected to form the T-Map. The paper also contains the construction for the first five-dimensional (5D) T-Map for controlling the minimum wall thickness between two concentric cylinders with a least-material-condition (LMC) tolerance specification on position. It is formed by adding the dimension of size to the T-Map for an axis. The T-Maps described are consistent with geometric dimensioning and tolerancing standards and industry practice. Finally, transformations are presented to translate between small displacement torsor (SDT) coordinates and the classical coordinates for lines and planes used in T-Maps.

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## References

Haghighi, P. , Ramnath, S. , Kalish, N. , Shah, J. V. , Shah, J. J. , and Davidson, J. K. , 2016, “ Method for Automating Digital Fixture Set-Ups That are Optimal for Machining Castings to Minimize Scrap,” J. Manuf. Syst., 40(2), pp. 15–22.
Kalish, N. J. , 2016, “ The Theory Behind Setup Maps: A Computational Tool to Position Parts for Machining,” M.Sc. thesis, Arizona State University, Tempe, AZ.
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## Figures

Fig. 1

AutoFix® work flow. Modified from [1]

Fig. 2

CAD model of the valve body: (a) datum targets A, B, and C and (b) round openings Ci, the hexagonal plane F, and simulated fitted features for Ci, Co, F, and targets A

Fig. 3

(a) The tolerance zone for the planar end face of a rectangular bar and (b) its corresponding T-Map

Fig. 4

The 4D T-Map H4 and line coordinates (L,M,P,Q) for position of the axis for a hole or pin. (a) Tolerance specification for a pin of length d. (b) The corresponding tolerance zone. (c) Line coordinates L and M for $= AB as shown in (b). (d)–(g) Four 3D hypersections of the T-Map H4: M′ = 0, P = 0, L′ = 0, and Q = 0, respectively [8,10]. Fig. 5 The fixture for the valve body showing the adjustable locators at datum points A1, A2, A3, B1, and B2, and a rigid stop at C Fig. 6 The tolerance-zone (dashed lines) for a hexagonal planar segment F, the six vertices V1V6 of its convex hull, the two knot-points (heavy dots) at each vertex, local coordinate frame Oxyz, and parallel vertex-frames at Vi, all used in constructing the T-Map for the segment Fig. 7 A trimmed portion of the point-space of the generic T-Map that models constraints from two opposite knot-points of the tolerance-zone boundaries at any vertex of a plane-segment (Fig. 6). Coordinates correspond to the vertex (V) coordinate frame. Fig. 8 Sample T-Map primitives derived by transforming the T-Map in Fig. 7 with Eq. (7). All correspond to the local () coordinate frame Oxyz in Fig. 6 after shearing (a) for vertex V3, (b) for vertex V4, and (c) for vertex V5. Fig. 9 The local T-Map for the profile tolerance on the hexagonal face, F, and for the translational SDT displacement Δz measured in the local Oxyz-frame (Fig. 6). Labeled faces correspond to vertices in Fig. 6. Faces for V2 appear as lines. Fig. 10 (a) T-Map for the pentagonal face obtained by removing vertex V2 from F in Fig. 6. (b) T-Map for the rectangular face obtained by removing both V2 and V5 from F. Fig. 11 (a) Simplified tolerance-zone for an axis and local () coordinate frame Oxy, from Fig. 4(b). (b) Freedom and constraint of lines provided by any point A on one constraint circle C in (a). (c) The 4D point-space of the generic T-Map that models constraint from either of the circles C or C¯ in (a). Translational coordinates correspond to a circle (C) coordinate frame, e.g., OCxCyC shown in (b). Fig. 12 Notional representation of the 5D T-Map for an engaged pin-and-hole assembly, here drawn for minimum radial clearance cmin = 0. The shaded region is the hypersurface Se representing locations for the pin (left) which, when engaged with an upright hole of VC size, give a radial clearance c′. The scale is exaggerated in the direction of envelope size. Fig. 13 Notional representation of the 5D T-Map that controls minimum wall thickness wmin for a tube. The shaded region is the hypersurface Si, all points of which give the same value of wall thickness with an outer wall Co that has position and envelope size corresponding to$6 (▼) in the T-Map. Three partial sections of the tube show variations in position and size of the inner wall Ci which correspond to points on Si. The scale is exaggerated in the direction of envelope size.

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