Using Fully Cartesian Coordinates to Calculate the Support Reactions of Multi-Scale Mechanisms

It is well known that machine vibrations often occur due to dynamic unbalance. They induce noise, wear, and fatigue problems that prevent many machines from attaining their full potential. Dynamic balancing of mechanisms allows to limit the transmission of vibrations to the base, an advantage that could be exploited in hand tools, machinery, vehicles moving in free space and robots. This work presents a method for representing a four-bar mechanism with the use of Fully Cartesian coordinates and its equations for calculating the shaking force (ShF) and the shaking moment (ShM). Such method can be applied and it is useful for the analysis of multi-scale (meso/micro/nano) mechanisms. The projected Gradient Descent is used to minimize the ShF and ShM functions with the aim of balancing the system. The Taylor's theorem was used for calculating the step length. The multiobjective optimization problem was solved using a linear combination of the objectives. The result of this methodology reduces successfully the ShF and the ShM by 31% and 65%, respectively.