Watt I Indirect Cognates of Four-Bar Linkages

Abstract

AbstractThis paper describes a method for synthesizing a Watt I linkage that exactly traces the coupler curve of a given four-bar linkage. The six-bar coupler curve contains the original four-bar curve and an additional branch. Because the Watt I linkage is a different linkage type than the four-bar, it is termed an indirect cognate. The method is derived from a study of the symmetry of linkage skeletons and their graphs. In its simplest form, the algorithm consists of taking a copy of a given four-bar, rotating the copy 180°, and then pinning the midpoint of each link to its opposite, e.g. the original’s ground link to the copy’s coupler link, the original’s input side link to the copy’s output side link, etc. This results in a redundantly constrained linkage with eight links and twelve joints and five loops. The five vector-loop equations have rank one when expressed in complex number form. Editing the redundant linkage to remove five joints, two links and three loops can result in a Watt I mechanism (six links, seven joints, and two loops) with the specified coupler curve. A principal use of cognate mechanisms is to reposition the ground link with respect to the coupler curve; a generalization of the midpoint indirect cognate algorithm allows one joint of the ground link of the original four-bar to be moved to any point on the pencil of lines passing the four-bar’s other ground joint. The method is illustrated by synthesizing and prototyping two different indirect cognates of a Roberts’ straight-line mechanism.KeywordsMechanism synthesisCognate mechanisms