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Research Papers: Mechanisms and Robotics

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

[+] Author and Article Information
M. Verschuure1

Department of Mechanical Engineering, Katholieke Universiteit Leuven, Celestijnenlaan 300B, B-3001 Heverlee, Belgiummyriam.verschuure@mech.kuleuven.be

B. Demeulenaere, J. Swevers, J. De Schutter

Department of Mechanical Engineering, Katholieke Universiteit Leuven, Celestijnenlaan 300B, B-3001 Heverlee, Belgium

For the more general case of $N$-bar linkages with possibly multiple degrees of freedom, a wider range of techniques exists for achieving full shaking force and shaking moment balance. These techniques are comprehensively reviewed in Ref. 6.

In this respect, our methodology deviates slightly from Ref. 15, in which a slightly different mass matrix $M$ is used and the damping and stiffness matrices are referred to $c¯$, the average COG of the ensemble of the frame and the linkage. Both modeling approaches are approximate, since they are based on the aforementioned decoupling between the frame vibration and the shaking force and shaking moment.

For the benchmark problem of Sec. 3, all numerical experiments with Eqs. 8,8 resulted in point-mass counterweights.

See Table 2 of this reference for the numerical values of the four-bar parameters.

It would be more accurate to use the bounds chosen in Ref. 15, but these bounds are unavailable.

www.lmsvirtuallab.com

The relation between the optimization variable $x$ and the resulting frame vibration is much more complicated in this case, given the coupling between the frame vibration and the shaking force/moment that cause it.

Due to the nonconvex nature of the problem, “solving” implies “determining a local optimum:” no claims concerning global optimality can be made.

There is no way of proving that the $pglob$ found here is the global optimum, since the considered optimization problem is nonconvex. It can only be guessed from the fact that a large number (128) of initial guesses $xk$ have been considered and that 21 of them give a similar low value.

As discussed in Sec. 3, it is assumed that the frame is at rest, implying that the acceleration of the frame coordinates system ${b}$ with respect to the world coordinate system ${a}$ is zero.

1

Corresponding author.

J. Mech. Des 130(2), 022302 (Dec 27, 2007) (11 pages) doi:10.1115/1.2812420 History: Received July 17, 2006; Revised March 01, 2007; Published December 27, 2007

## Abstract

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.

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## Figures

Figure 1

Planar crank-rocker four-bar linkage supported by an elastically mounted, rigid frame with three degrees of freedom (qx,qy,qt). (qx,qy) are the coordinates of the frame COG c4 with respect to {a}. The shaded boxes indicate spring-damper systems.

Figure 2

Uncoupled model: average frame kinetic energy versus crank speed for the unbalanced linkage (dashed line) and the balanced linkage (solid line) with the three counterweights of Table 2, column 6

Figure 3

Uncoupled model: one period (Ω=20rad∕s) of steady-state frame acceleration components q̈x (a), q̈y (b), and q̈t (c) for the unbalanced linkage (dashed line), full force balanced linkage (dash-dotted line; Table 2, column 8), and E¯kin-balanced linkage (solid line; Table 2, column 6)

Figure 4

One period (Ω=20rad∕s) of steady-state frame kinetic energy ((a) and (c)) and mounting potential energy ((b) and (d)) for the unbalanced linkage ((a) and (b)) and E¯kin-balanced linkage ((c) and (d)): comparison of coupled model results (solid line) and uncoupled model results (dashed line)

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