Myriam Verschuure

Ultimate limits for counterweight balancing of crank-rocker four-bar linkages

This paper focuses on reducing the dynamic reactions (shaking force, shaking moment, and driving torque) of planar crank-rocker four-bars through counterweight addition. Determining the counterweight mass parameters constitutes a nonlinear optimization problem, which suffers from local optima. This paper, however, proves that it can be reformulated as a convex program, that is, a nonlinear optimization problem of which any local optimum is also globally optimal. Because of this unique property, it is possible to investigate (and by virtue of the guaranteed global optimum, in fact prove) the ultimate limits of counterweight balancing. In a first example a design procedure is presented that is based on graphically representing the ultimate limits in design charts. A second example illustrates the versatility and power of the convex optimization framework by reformulating an earlier counterweight balancing method as a convex program and providing improved numerical results for it.

A General and Numerically Efficient Framework to Design Sector-Type and Cylindrical Counterweights for Balancing of Planar Linkages

This paper extends previous work concerning convex reformulations of counterweight balancing by developing a general and numerically efficient design framework for counterweight balancing of arbitrarily complex planar linkages. At the numerical core of the framework is an iterative procedure, in which successively solving three convex optimization problems yields practical counterweight shapes in typically less than 1 CPU s. Several types of counterweights can be handled. The iterative procedure allows minimizing and/or constraining shaking force, shaking moment, driving torque, and bearing forces. Numerical experiments demonstrate the numerical superiority (in terms of computation time and balancing result) of the presented framework compared to existing approaches.

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.

Counterweight Balancing for Reducing Machine Frame Vibration: A Convex Optimization Framework

This paper focusses on reducing, through counterweight addition, the vibration of an elastically mounted, rigid machine frame that supports a linkage. In order to determine the counterweights that yield a maximal reduction in frame vibration, a non-linear optimization problem is formulated with the frame kinetic energy as objective function and such that a convex optimization problem is obtained. Convex optimization problems are nonlinear optimization problems that have a unique (global) optimum, which can be found with great efficiency. The proposed methodology is successfully applied to improve the results of the benchmark four-bar problem, first considered by Kochev and Gurdev. For this example, the balancing is shown to be very robust for drive speed variations and to benefit only marginally from using a coupler counterweight.Copyright