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RESEARCH PAPERS

# Ultimate Limits for Counterweight Balancing of Crank-Rocker Four-Bar Linkages

[+] Author and Article Information
Bram Demeulenaere1

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

Erwin Aertbeliën, Myriam Verschuure, Jan Swevers, Joris De Schutter

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

Tepper and Lowen optimize minimum inertia counterweights attached to the crank and rocker of a crank-rocker four-bar, so as to minimize the root mean square of the shaking force, subject to an upper limit on the root mean square (rms) of the ground bearing forces.

For instance, if other constraints, like upper limits on the root mean square driving torque, are considered.

This is also the case when the radius, thickness and, COG coordinates of circular counterweights are used as optimization variables.

This is due to the aforementioned presence of optima in which the actually obtained $αfsh$ or $αdrv$ are lower than the imposed upper limits $αfshM$ or $αdrvM$.

Note that the vertical line $αfsh=0.66$ in Fig. 3 and the vertical line $αdrv=1.20$ in Fig. 3 intersect corresponding trade-off lines at equal $αmsh$ values.

Not shown in Fig. 4 is that, for $η→∞$, $αmsh→0.07$, illustrating that for a general crank-rocker four-bar as the one considered here, shaking moment balance cannot be obtained by only adding counterweights. If shaking force balance is imposed $(αfshM=0)$, while dropping the upper limit 15 on $αdrv$, $η→∞$ results in $αmsh→0.16$.

In fact, the counterweight shape does not matter for the crank, since it revolves at constant speed, resulting in no influence of the counterweight inertia on the balancing effect indices.

Sadler and Mayne chose the reference point for the shaking moment calculation halfway between the fixed pivots $p$ and $s$, as opposed to the example of Sec. 5, in which the reference point coincides with $p$.

This has to be checked always in the obtained optimum: if $Ji*=0,∀i$, relaxing Eq. 17 to Eq. 18 has had no resulting effect on the obtained optimum. If, on the other hand, $Ji*>0$ for some $i$, the obtained optimal value provides a lower bound for the “true” optimal value.

Each of the unconstrained optimization problems is solved using the Davidon-Fletcher-Powell method, a so-called quasi-Newton method for unconstrained optimization.

1

Corresponding author.

J. Mech. Des 128(6), 1272-1284 (Jan 09, 2006) (13 pages) doi:10.1115/1.2337313 History: Received July 30, 2005; Revised January 09, 2006

## Abstract

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.

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## Figures

Figure 1

(a) Kinematic scheme of a planar crank-rocker four-bar mechanism: link 1 (crank) =pq; link 2 (coupler) =qr; link 3 (rocker) =sr. (b) Corresponding free-body diagram.

Figure 2

Contour plots of αmsh [-] as a function of αfsh [-] and αdrv [-] for η equal to {0.50(a),0.75(b),1.00(c),2.00(d)}. The white region marked I is the infeasible region. The white region marked F denotes the part of the feasible region for which data are not available. In each figure, the value of αmsh increases from the top to the bottom of the figure.

Figure 3

(a)αmsh as a function of αfsh for a fixed value αdrv=1.20. The four lines mark four different values of η={0.50,0.75,1.00,2.00}. (b)αmsh as a function of αdrv for a fixed value αfsh=0.66.

Figure 4

Ultimate balancing limits as a function of η for (αfsh=0.66,αdrv=1.20). (a)αmsh; (b) mass of crank counterweight; (c) mass of coupler counterweight; (d) mass of rocker counterweight.

Figure 5

Counterweighted crank-rocker four-bar, after transforming the point-mass counterweights for (η=0.80,αfsh=0.66,αdrv=1.20) to circular, minimum inertia counterweights. The counterweight radiuses equal: Rc,1*=26.45 and Rc,3*=26.37mm. The counterweight thicknesses equal tc,1*=2.83 and tc,3*=18.20mm.

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