An Energy-Based General Method for the Optimum Synthesis of Mechanisms

The aim of the present paper is to set forth a method for the optimum synthesis of planar mechanisms. The method in question can be applied in the case of any mechanism and kinematic synthesis (function, path generation, rigid-body guidance, or combination of these). The mechanisms are discretized with bar-finite elements that facilitate the computation of the geometric matrix, which is a stiffness matrix. The error function is based on the elastic energy accumulated by the mechanism when it is compelled to fulfill exactly the synthesis data. Thus during the iterative process the elements of the mechanism may be considered deformable. The energy computation for each synthesis datum requires the solution of the nonlinear equilibrium position. This problem is solved with the help of the geometric matrix and the force vector of the deformed system. The minimization of the error function is based on Newton’s method, with a semianalytic approach. Analytic and finite-difference concepts are used together in the calculation of the gradient vector and of the second-derivative matrix. The method has proved very stable for a wide range of step sizes. There is convergence to a minimum even when the start mechanism is far from a solution. Examples with different mechanisms and syntheses are also provided.