DYNAMIC BALANCING OF FOUR-BAR LINKAGES: A CONVEX OPTIMIZATION FRAMEWORK FOR EFFICIENTLY OBTAINING GLOBALLY OPTIMAL COUNTERWEIGHTS

This paper focusses on reducing the dynamic reactions (shaking force, shaking moment and driving torque) of plane, crank-rocker four-bars through counterweight addition. Determining the mass parameters of the counterweights constitutes an optimization problem, which is classically considered to be nonlinear and hence difficult to solve. A first contribution of this paper is the proof that this optimization problem can be reformulated as a convex program, that is, a nonlinear optimization problem that still has a unique (and hence guaranteed global) optimum, which can be found with great efficiency. Because of the unique features of this formulation, it becomes possible to investigate (and by the guarantee of obtaining a global optimum, in fact prove) the ultimate limits of dynamic balancing, in a reasonable amount of time. When applied to a particular example, this results in design charts, which clearly illustrate (i) the tradeoff between minimizing the different dynamic reactions, and (ii) the fact that adding counterweights is effective, but at the cost of a significant amount of added mass. These design charts constitute a second contribution of the present work