Synthesis of Six-link, Slider-crank and Four-link Mechanisms for Function, Path and Motion Generation Using Homotopy with m-homogenization

Abstract

This paper presents solutions to the function, motion and path generation problems of Watt’s and Stephenson six-link, slider-crank and four-link mechanisms using homotopy methods with m-homogenization. It is shown that using the matrix method for synthesis, applying m-homogeneous group theory, and by defining auxiliary equations in addition to the synthesis equations, the number of homotopy paths to be tracked is drastically reduced. To synthesize a Watt’s six-link function generator for 6 through 11 precision positions, the number of homotopy paths to be tracked to obtain all possible solutions range from 640 to 55,050,240. For Stephenson-II and -III mechanisms these numbers vary from 640 to 412,876,800. It is shown that slider-crank path generation problems with 6, 7 and 8 prescribed positions require 320, 3840 and 17,920 paths to be tracked, respectively, whereas for four-link path generators with 6 through 8 specified positions, these numbers range from 640 to 71, 680. The number of homotopy paths to be tracked to body guidance problems of slider-crank and four-link mechanisms is exactly the same as the maximum number of possible solutions given by Burmester-Ball theories. Numerical examples dealing with the synthesis of slider-crank path generators for 8 precision positions, and six-link Watt and Stephenson-III function generators for 9 prescribed positions are also presented.