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RESEARCH PAPERS

A Study of the Duality Between Planar Kinematics and Statics

[+] 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

J. Mech. Des 128(3), 587-598 (Oct 12, 2005) (12 pages) doi:10.1115/1.2181600 History: Received December 14, 2004; Revised October 12, 2005

Abstract

This paper provides geometric insight into the correlation between basic concepts underlying the kinematics of planar mechanisms and the statics of simple trusses. The implication of this correlation, referred to here as duality, is that the science of kinematics can be utilized in a systematic manner to yield insight into statics, and vice versa. The paper begins by introducing a unique line, referred to as the equimomental line, which exists for two arbitrary coplanar forces. This line, where the moments caused by the two forces at each point on the line are equal, is used to define the direction of a face force which is a force variable acting in a face of a truss. The dual concept of an equimomental line in kinematics is the instantaneous center of zero velocity (or instant center) and the paper presents two theorems based on the duality between equimomental lines and instant centers. The first theorem, referred to as the equimomental line theorem, states that the three equimomental lines defined by three coplanar forces must intersect at a unique point. The second theorem states that the equimomental line for two coplanar forces acting on a truss, with two degrees of indeterminacy, must pass through a unique point. The paper then presents the dual Kennedy theorem for statics which is analogous to the well-known Aronhold-Kennedy theorem in kinematics. This theorem is believed to be an original contribution and provides a general perspective of the importance of the duality between the kinematics of mechanisms and the statics of trusses. Finally, the paper presents examples to demonstrate how this duality provides geometric insight into a simple truss and a planar linkage. The concepts are used to identify special configurations where the truss is not stable and where the linkage loses mobility (i.e., dead-center positions).

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Figures

Figure 1

Forces F⃗1 and F⃗2 acting along lines of action l1 and l2

Figure 2

The equimomental lines for the three coplanar forces

Figure 3

Point Eij which lies on the equimomental line of two forces F⃗i and F⃗j

Figure 4

Example of the truss-linkage duality. (a) A simple truss (the dual linkage is superimposed). (b) The dual linkage.

Figure 5

(a) A simple truss. (b) The faces of the simple truss.

Figure 6

The dual Kennedy circle (a) The primary equimomental lines. (b) The secondary equimomental line mPG. (c) The secondary equimomental line mPC. (d) The secondary equimomental line mCO.

Figure 7

Six of the equimomental lines

Figure 8

(a) The perpendicular distances to the equimomental lines. (b) The direction of the face forces F⃗C and F⃗G.

Figure 9

Kinematic synthesis of a four-bar linkage. (a) Original geometry of the four-bar linkage. (b) Modified geometry of the linkage.

Figure 10

The synthesis technique applied to a simple truss. (a) Original truss and the new position of rod 4. (b) Equimomental lines of the original truss. (c) Geometry of the modified truss.

Figure 11

The rigidity of the truss. (a) The truss. (b) The equimomental line mBE.

Figure 12

The dual linkage in a dead-center position. (a) The nonrigid truss. (b) The dual linkage. (c) Instant center I13 is coincident with instant center I23.

Figure 13

(a) Double flier eight-bar linkage in an arbitrary configuration. (b) The equimomental line mBF for a singular configuration.

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