Counterweight Balancing for Vibration Reduction of Elastically Mounted Machine Frames: A Second-Order Cone Programming Approach

A moving linkage exerts fluctuating forces and moments on its supporting frame. One strategy to suppress the resulting frame vibration is to reduce the exciting forces and moments by adding counterweights to the linkage links. This paper develops a generic methodology to design such counterweights for planar linkages, based on formulating counterweight design as a second-order cone program. Second-order cone programs are convex, which implies that these nonlinear optimization problems have a global optimum that is guaranteed to be found in a numerically efficient manner. Two optimization criteria are considered: the frame vibration itself and the dynamic force transmitted to the machine floor. While the methodology is valid regardless of the complexity of the considered linkage, it is developed here for a literature benchmark consisting of a crank-rocker four-bar linkage supported by a rigid, elastically mounted frame with three degrees of freedom. For this particular benchmark, the second-order cone program slightly improves the previously known optimum. Moreover, numerical comparison with current state-of-the-art algorithms for nonlinear optimization shows that our approach results in a substantial reduction of the required computational time.