Input-Output Equation for Planar Four-Bar Linkages

Abstract

In this paper the generalised input-output (I-O) equation for planar 4R function generators is derived in a new way, leading to the algebraic form of the well known Freudenstein equation. The long term goal is to develop a generalised method to derive constraint based algebraic I-O equations that can be used for continuous approximate synthesis, where the synthesis equations are integrated between minimum and maximum input angle values resulting in a linkage whose objective function has been optimised over every output angle. In this paper we use a planar projection of Study’s soma and the Cartesian displacement constraints for the dyads. These are mapped to the image space leading to four constraint equations in terms of the image space coordinates and the sines and cosines of the input and output angles. Using the tangent of the half angle substitution the trigonometric equations are converted to algebraic ones. Algebraic methods are used to eliminate the image space coordinates, then the polynomial resultants are found to obtain common roots leading to the desired equations.