Algorithm to Find and Analyze All Configurations of Four-Bar Linkages with Different Geometric Loci Degenerate Forms

A general algorithm to determine the coupler link geometric loci, such as centrodes, inflection and return circles, as well as circling-point and centering-point curves, is formulated to analyze any type of four-bar linkages with the main target to find all mechanism configurations, in which at least one of the above-mentioned loci degenerates. Thus, different types of four-bar linkages, such as crank-rocker, double-crank, double-rocker and triple-rocker, are classified according to Grashof’s law, in order to distinguish and analyze their corresponding geometric loci. In particular, the proposed algorithm is based on four diagrams of the angular velocity ratios versus the mechanism driving angle, which consider the links pairs of input/output, input/coupler, and output/coupler, along with those of coupler/input and coupler/output for their relative motion. These diagrams allow the determination of all mechanism configurations according to Freudenstein’s theorems, where the aforementioned geometric loci degenerate into straight lines, including the line at infinity, ϕ-curves, and/or equilateral hyperbolas. This algorithm has been implemented in Matlab in order to run several examples regarding different four-bar linkages, according to Grashof’s law, and analyzing the degenerate forms of their inflection and return circles, as well as the circling-point and centering-point curves, that are also validated by using the collineation axis.