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Technical Briefs

# Assessing Position Order in Rigid Body Guidance: An Intuitive Approach to Fixed Pivot Selection

[+] Author and Article Information
David H. Myszka

Department of Mechanical and Aerospace Engineering, University of Dayton, 300 College Park, Dayton, OH 45469dmyszka@udayton.edu

Andrew P. Murray

Department of Mechanical and Aerospace Engineering, University of Dayton, 300 College Park, Dayton, OH 45469murray@udayton.edu

James P. Schmiedeler

Department of Mechanical Engineering, The Ohio State University, 201 W. 19th Avenue, Columbus, OH 43210schmiedeler.2@osu.edu

J. Mech. Des 131(1), 014502 (Dec 16, 2008) (5 pages) doi:10.1115/1.3013851 History: Received October 17, 2007; Revised September 18, 2008; Published December 16, 2008

## Abstract

Several established methods determine if an RR dyad will pass through a set of finitely separated positions in order. The new method presented herein utilizes only the displacement poles in the fixed frame to assess whether a selected fixed pivot location will yield an ordered dyad solution. A line passing through the selected fixed pivot is rotated one-half revolution about the fixed pivot, in a manner similar to a propeller with infinitely long blades, to sweep the entire plane. Order is established by tracking the sequence of displacement poles intersected. With four or five positions, fixed pivot locations corresponding to dyads having any specified order are readily found. Five-position problems can be directly evaluated to determine if any ordered solutions exist. Additionally, degenerate four-position cases for which the set of fixed pivots corresponding to ordered dyads that collapse to a single point on the center point curve can be identified.

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## Figures

Figure 2

An alternative relationship between the fixed pivot of two positions, the pole, and the crank angle

Figure 5

Equations 14,15 ensure the order of all N positions

Figure 9

Synthesized dyad for Example 2 with a crank rotation of zero

Figure 10

An ordered dyad solution for Example 2

Figure 11

An unordered dyad solution for Example 2

Figure 12

The five-position application of the propeller method from Example 3, with ordered regions crosshatched

Figure 13

An ordered dyad solution for Example 3

Figure 14

The five-position problem from Example 4, without any ordered regions

Figure 1

The relationship between the fixed pivot of two positions, the pole, and the crank angle

Figure 3

The relationship between the fixed pivot, the poles, and three positions

Figure 4

The angle between line GPij¯ and GPjk¯ is βik∕2

Figure 6

The four-position application of the propeller method in Example 1, with the regions that satisfy the theorem crosshatched

Figure 7

The synthesized dyad for Example 1

Figure 8

The special four-position application of the propeller method in Example 2

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