Dimensional Synthesis of Watt II and Stephenson III Six-Bar Slider-Crank Function Generators for Nine Prescribed Positions

Abstract

This paper proposes an efficient exact dimensional synthesis method for finding all the link lengths of the Watt II and Stephenson III six-bar slider-crank function generators, satisfying nine prescribed precision points using the homotopy continuation method. The synthesis equations of each mechanism are initially constructed as a system of 56 quadratic polynomials whose Bézout number, which represents the maximum number of solutions, is 256 ≅ 7.21 × 1016. In order to reduce the size of the system, multi-homogeneous formulation is applied to transform the system into 12 equations in 12 unknowns, and the multi-homogeneous Bézout number of the system is 286,720. The Bertini solver, based on the homotopy continuation method, is used to solve the synthesis equations to obtain the dimensions of the two mechanisms. For the arbitrarily given nine precision points, the proposed method yields 37 and 31 defect-free solutions of Watt II and Stephenson III six-bar slider-crank mechanisms, respectively, and it is confirmed that they pass through the prescribed positions.