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Research Papers

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

[+] Author and Article Information
B. Demeulenaere, M. Verschuure, J. Swevers

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

J. De Schutter

Department of Mechanical Engineering, Katholieke Universiteit Leuven, Leuven B-3001, Belgiumjoris.deschutter@mech.kuleuven.be

The number of iterations is fixed to three: doing more iterations generally results in only marginal an improvement.

This CPU-time is necessary to have a higher probability to obtain the global optimum and is therefore used as comparison.

J. Mech. Des 132(1), 011002 (Dec 09, 2009) (10 pages) doi:10.1115/1.4000532 History: Received March 05, 2009; Revised October 10, 2009; Published December 09, 2009; Online December 09, 2009

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.

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Copyright © 2010 by American Society of Mechanical Engineers
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References

Figures

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Figure 1

Sector-type counterweight (left) compared to minimum-inertia counterweight (right)

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Figure 2

Poor choice of LCS origin q. The sector-type counterweight with sector center in q′ is the truly optimal coupler counterweight.

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Figure 3

Two typical cases for iterative procedure to optimally choose LCS origin: (i) closest point to point-mass counterweight on the binary link is one of its end-points (a−b); (ii) closest point to point-mass counterweight on the binary link is the point v lying between the end-points (c−d−e).

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Figure 4

Kinematic scheme of the considered ten-bar.

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Figure 5

Bounds on the COG coordinates of counterweight for a (a) binary link and (b) ternary link

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Figure 6

Trade-off curves αmsh as a function of αfsh with ηm=1.5, ηb=1, ηR=1, and tM=0.03 (a) with αbearing=1.5 for αdrvM={1.15,1.2,1.25,1.3}, respectively, (b) with αdrvM=1.25 for αbearingM={1.5,2,10}, respectively

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Figure 7

Trade-off curves αmsh as a function of αfsh with αdrvM=1.25, αbearingM=1.5, ηm=1, and ηb=1 for (a) ηR=0.06 for point-mass (dashed-dotted line), minimum-inertia counterweights (dashed-line) and 90 deg-sector counterweights (solid line), (b) ηR={0.4,0.5,0.6}, respectively

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Figure 8

Configuration of sector-type counterweights for partially balanced ten-bar of Table 2 columns 2–4.

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Figure 9

Comparison of the (a) shaking force, (b) bearing force in A, (c) shaking moment, and (d) driving torque of the original linkage (dashed-line) and the partially balanced ten-bar (solid line)

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Figure 10

Histograms of the results for the Mstart-SQP for the considered ten-bar (a) exitflag, (b) optimal value αmsh, and (c) computation time

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