Abstract
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.
Highlights
A complete mechanical balance or dynamic balance of a mechanism consists of eliminating the dynamic reactions at the base of a mechanism produced by the movement of its structure.These dynamic reactions are the shaking force (ShF) and the shaking moment (ShM)
Three solutions of the Pareto front are taken; the first one is the best result when optimizing the index corresponding to the ShM (β ShM = 0.1600587, β ShF = 0.9152829), the second one is the best result when optimizing the index corresponding to the ShF
By using fully Cartesian coordinates to represent a mechanism, the equations that define the reactions are less complex than those obtained with other methods, the use of this kind of coordinates is suitable for complete balancing, minimization of reactions, and calculation of the ShF
Summary
A complete mechanical balance or dynamic balance of a mechanism consists of eliminating the dynamic reactions at the base of a mechanism produced by the movement of its structure. The balancing conditions are commonly obtained using methods that involve Cartesian coordinates and the use of angles; this implies the use of trigonometric functions that derive into complex expressions [3,4,5,6] In relation to this point, this work exploits the use of fully Cartesian coordinates ( called natural coordinates [7]). Dynamic balancing optimization of a four-bar mechanism was achieved through the sole addition of counterweights [23,24]. Pareto Fronts are used to present a sensitivity analysis of the response of the whole mechanism when using three, two, or one counterweights This type of analysis allows to define the importance of each counterweight and determine which ones can be dispensed, obtaining optimum results when trying to optimize the ShF, the ShM, or both.
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