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RESEARCH PAPERS

Extension of Graph Theory to the Duality Between Static Systems and Mechanisms

[+] Author and Article Information
Offer Shai

Department of Mechanics, Materials and Systems, Tel Aviv University, Ramat Aviv 69978, Israel

Gordon R. Pennock

School of Mechanical Engineering, Purdue University, West Lafayette, IN 47907, USA

J. Mech. Des 128(1), 179-191 (Jul 04, 2005) (13 pages) doi:10.1115/1.2120827 History: Received May 03, 2005; Revised July 04, 2005

This paper is a study of the duality between the statics of a variety of structures and the kinematics of mechanisms. To provide insight into this duality, two new graph representations are introduced; namely, the flow line graph representation and the potential line graph representation. The paper also discusses the duality that exists between these two representations. Then the duality between a static pillar system and a planar linkage is investigated by using the flow line graph representation for the pillar system and the potential line graph representation for the linkage. A compound planetary gear train is shown to be dual to the special case of a statically determinate beam and the duality between a serial robot and a platform-type robot, such as the Stewart platform, is explained. To show that the approach presented here can also be applied to more general robotic manipulators, the paper includes a two-platform robot and the dual spatial linkage. The dual transformation is then used to check the stability of a static system and the stationary, or locked, positions of a linkage. The paper shows that two novel platform systems, comprised of concentric spherical platforms inter-connected by rigid rods, are dual to a spherical six-bar linkage. The dual transformation, as presented in this paper, does not require the formulation and solution of the governing equations of the system under investigation. This is an original contribution to the literature and provides an alternative technique to the synthesis of structures and mechanisms. To simplify the design process, the synthesis problem can be transformed from the given system to the dual system in a straightforward manner.

Copyright © 2006 by American Society of Mechanical Engineers
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Figures

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Figure 1

The static pillar system

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Figure 2

The FLGR corresponding to the static pillar system

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Figure 3

The PLGR of the six-bar linkage

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Figure 4

The two-dimensional static pillar system and the dual linkage

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Figure 5

The double flier eight-bar linkage and the corresponding static pillar System

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Figure 6

A determinate beam and the dual planetary gear train

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Figure 7

A multi-dimensional pillar system and the dual serial robot

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Figure 8

A two-platform robot and the dual spatial linkage

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Figure 9

Static pillar system and dual linkage

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Figure 10

The mobility of the serial robot

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Figure 11

Two topologically identical concentric platform systems

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Figure 12

The FLGR and the dual PLGR of the platform systems

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Figure 13

The linkages dual to the two platform systems

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Figure 14

A static beam system to amplify the input force

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Figure 15

The planetary gear train in an electrical screwdriver

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Figure 16

The graph representing the planetary gear train

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Figure 17

The dual graph representation of the planetary gear train

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Figure 18

The original gear train and the resulting static beam system

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Figure 19

(a). The 3-2-1 platform. (b). A six-link serial robot manipulator.

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