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research-article

# 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
Nathan Kalish

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

Joseph K. Davidson

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

Satchit Ramnath

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

Payam Haghighi

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

Jami S. Shah

Honda Professor of Engineering Design, Department of Mechanical & Aerospace Engineering, The Ohio State University, Columbus, OH 43210
shah.493@osu.edu

1Corresponding author.

ASME 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 5D T-Map for controlling the minimum wall thickness between two concentric cylinders with a least-material-condition 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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