Extension of Freudenstein’s Equation to Geared Linkages

Abstract

Freudenstein’s equation for planar four-bar function generators correlates input and output crank positions implicitly in a scalar expression, with coefficients that are functions of link proportions. Applying this approach to planar geared function generator linkages leads to nonlinear systems of algebraic equations. By the principle of superposition taken from the matrix theory of linear systems and by Sylvester’s dyalitic elimination, closed form solutions are obtained. When the geared linkages are changed into the planar four-bar by setting certain link lengths equal to zero, the generalized equations derived here specialize to Freudenstein’s well-known equation. Results of computer programs for synthesis and analysis based on this theory are tabulated.