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RESEARCH PAPERS

Finding Dead-Point Positions of Planar Pin-Connected Linkages Through Graph Theoretical Duality Principle

[+] Author and Article Information
O. Shai

Department of Mechanics Materials and Systems, Faculty of Engineering, Tel-Aviv University, Tel-Aviv, 69978 Israelshai@eng.tau.ac.il

I. Polansky

Department of Mechanics Materials and Systems, Faculty of Engineering, Tel-Aviv University, Tel-Aviv, 69978 Israel

J. Mech. Des 128(3), 599-609 (Jun 04, 2005) (11 pages) doi:10.1115/1.2179461 History: Received August 31, 2004; Revised June 04, 2005

The paper brings another view on detecting the dead-point positions of an arbitrary planar pin-connected linkage by employing the duality principle of graph theory. It is first shown how the dead-point positions are derived through the interplay between the linkage and its dual determinate truss—the relation developed in the previous works by means of graph theory. At the next stage, the process is shown to be performed solely upon the linkage by employing a new variable, the dual of potential, termed face force. Since the mathematical foundation of the presented method is discrete mathematics, the paper points to possible computerization of the method.

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Copyright © 2006 by American Society of Mechanical Engineers
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Figures

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Figure 1

Schematic illustration of the essence of the suggested methodology.

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Figure 2

(a) A simple truss with the dual linkage superimposed and (b) the dual linkage

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Figure 3

(a) Linkage in a dead-point position and (b) the corresponding deformation configuration of the truss with the dual topology

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Figure 4

(a) Arbitrary displacement vectors associated with truss joints, (b) yield feasible sets of rod deformations

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Figure 5

(a) Double-butterfly linkage; (b) the structures of the dual truss superimposed upon the linkage; (c) the truss possessing the dual topology

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Figure 6

(a) Arbitrary displacements assigned to all the joints of the truss and the corresponding deformation vectors. (b) The double-butterfly linkage redrawn in accordance to the angles of the deformations.

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Figure 7

Additional dead-point configurations of the double-butterfly linkage obtained by means of the suggested method (Step 5). (a) The dead-point configuration obtained, when two of the dual truss joints, D and F, are assigned zero displacement, (b) the dead-point configuration obtained, when two of the dual truss joints, A and C, are assigned the same displacement vector.

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Figure 8

The face forces in the linkage in a dead-point position

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Figure 9

Example for employing face forces in detecting dead-point positions of the Stephenson II linkage. (a) Stephenson II linkage, (b) randomly chosen face-force vectors, (c) the resultant dead-point position.

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Figure 10

Stephenson III mechanism, for which all the possible dead-point positions are to be found

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