Optimum Balancing of the Four-Bar Linkage Using Fully Cartesian Coordinates

This article presents the design optimization of the four-bar linkage for the reduction of the Shaking Force and the Shaking Moment. It is followed a novel procedure based on fully Cartesian Coordinates (Natural Coordinates) and the use of counterweights. These counterweights with as little restrictive constraints on their geometric parameters as possible. No particular counterweight shape in advance is assumed. The analytical equations for the linear momentum and for the angular momentum are obtained, resulting in equations expressed in terms on the mass and the coordinates of the center of mass of the counterweights. These expressions are then used to obtain the Shaking Force and the Shaking Moment of the system in a very direct way. To solve the inverse dynamics problem the inversión of the transpose of the Jacobian matrix, associated to the kinematic constraints, is not required. The links’ masses are restricted only by imposing positive mass values. As a novelty, the most influencing optimization variables are identified using a global sensitivity analysis method, leading to a reduction on the number of optimization variables. The minimization is done using Evolutionary Computation. The results obtained are validated by simulations, and compared to those presented in previous representative works.