Gradient Descent-Based Optimization Method of a Four-Bar Mechanism Using Fully Cartesian Coordinates

Machine vibrations often occur due to dynamic unbalance inducing wear, fatigue, and noise that limit the potential of many machines. Dynamic balancing is a main concern in mechanism and machine theory as it allows designers to limit the transmission of vibrations to the frames and base of machines. This work introduces a novel method for representing a four-bar mechanism with the use of Fully Cartesian coordinates and a simple definition of the shaking force (ShF) and the shaking moment (ShM) equations. A simplified version of Projected Gradient Descent is used to minimize the ShF and ShM functions with the aim of balancing the system. The multi-objective optimization problem was solved using a linear combination of the objectives. A comprehensive analysis of the partial derivatives, volumes, and relations between area and thickness of the counterweights is used to define whether the allowed optimization boundaries should be changed in case the mechanical conditions of the mechanism permit it. A comparison between Pareto fronts is used to determine the impact that each counterweight has on the mechanism's balancing. In this way, it is possible to determine which counterweights can be eliminated according to the importance of the static balance (ShF), dynamic balance (ShM), or both. The results of this methodology when using three counterweights reduces the ShF and ShM by 99.70% and 28.69%, respectively when importance is given to the static balancing and by 83.99% and 8.47%, respectively, when importance is focused on dynamic balancing. Even when further reducing the number of counterweights, the ShF and ShM can be decreased satisfactorily.