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.
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January 2006
Research Papers
Extension of Graph Theory to the Duality Between Static Systems and Mechanisms
Offer Shai,
Offer Shai
Department of Mechanics, Materials and Systems,
Tel Aviv University
, Ramat Aviv 69978, Israel
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Gordon R. Pennock
Gordon R. Pennock
School of Mechanical Engineering,
Purdue University
, West Lafayette, IN 47907, USA
Search for other works by this author on:
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, USAJ. Mech. Des. Jan 2006, 128(1): 179-191 (13 pages)
Published Online: July 4, 2005
Article history
Received:
May 3, 2005
Revised:
July 4, 2005
Citation
Shai, O., and Pennock, G. R. (July 4, 2005). "Extension of Graph Theory to the Duality Between Static Systems and Mechanisms." ASME. J. Mech. Des. January 2006; 128(1): 179–191. https://doi.org/10.1115/1.2120827
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