A Study of the Duality Between Planar Kinematics and Statics

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).