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RESEARCH PAPERS

On Higher Order Point-Line and the Associated Rigid Body Motions

[+] Author and Article Information
Yi Zhang

Center for Manufacturing Research, Tennessee Technological University, Cookeville, TN 38505

Kwun-Lon Ting

Center for Manufacturing Research, Tennessee Technological University, Cookeville, TN 38505kting@tntech.edu

J. Mech. Des 129(2), 166-172 (Jan 17, 2006) (7 pages) doi:10.1115/1.2406086 History: Received February 15, 2005; Revised January 17, 2006

This paper presents a study on the higher-order motion of point-lines embedded on rigid bodies. The mathematic treatment of the paper is based on dual quaternion algebra and differential geometry of line trajectories, which facilitate a concise and unified description of the material in this paper. Due to the unified treatment, the results are directly applicable to line motion as well. The transformation of a point-line between positions is expressed as a unit dual quaternion referred to as the point-line displacement operator depicting a pure translation along the point-line followed by a screw displacement about their common normal. The derivatives of the point-line displacement operator characterize the point-line motion to various orders with a set of characteristic numbers. A set of associated rigid body motions is obtained by applying an instantaneous rotation about the point-line. It shows that the ISA trihedrons of the associated rigid motions can be simply depicted with a set of 2 cylindroids. It also presents for a rigid body motion, the locus of lines and point-lines with common rotation or translation characteristics about the line axes. Lines embedded in a rigid body with uniform screw motion are presented. For a general rigid body motion, one may find lines generating up to the third order uniform screw motion about these lines.

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Copyright © 2007 by American Society of Mechanical Engineers
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Figures

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Figure 1

Two finitely separated point-line positions

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Figure 2

The dual geodesic trihedron of the l-surface

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Figure 3

The dual geodesic trihedron of the ISA surface

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Figure 4

The ISA and ISA trihedron of an associated rigid body motion

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