The Duality Between Planar Kinematics and Statics

Abstract

This paper shows that there is a correlation between basic concepts underlying the kinematics of mechanisms and the statics of 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 the statics of mechanical systems. The paper begins by proving the existence of a unique line (referred to as the equimomental line) where the moments, at each point on this line, caused by two arbitrary co-planar forces are equal. The dual concept in kinematics is the instantaneous center of zero velocity and two theorems are presented based on the duality between equimomental lines and instantaneous centers. The first theorem states that the three equimomental lines defined by three co-planar forces must intersect at a unique point. The second theorem states that the equimomental line for two co-planar forces acting in a trusss with two degrees of indeterminacy must pass through a unique point. The paper presents several practical examples to demonstrate how the duality between kinematics and statics provides a better understanding of planar linkages and trusses. The new concepts are used to identify the singular configurations of linkages and the configurations of determinate trusses where they are not rigid. Finally, the paper takes advantage of some important relationships between linkages and trusses to provide a general perspective of the duality between the kinematics of mechanisms and the statics of trusses.