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Research Papers

Synthesis of Planar Rigid-Body Mechanisms Approximating Shape Changes Defined by Closed Curves

[+] Author and Article Information
Justin A. Persinger

 Babcock and Wilcox, Barberton, OH 44203japersinger@babcock.com

James P. Schmiedeler

Aerospace and Mechanical Engineering, University of Notre Dame, Notre Dame, IN 46556schmiedeler.4@nd.edu

Andrew P. Murray

Mechanical and Aerospace Engineering, University of Dayton, Dayton, OH 45469murray@udayton.edu

J. Mech. Des 131(7), 071006 (Jun 24, 2009) (7 pages) doi:10.1115/1.3149843 History: Received January 01, 2008; Revised April 19, 2009; Published June 24, 2009

This paper presents a procedure to synthesize planar linkages, composed of rigid links and revolute joints, that are capable of approximating a shape change defined by a set of closed curves possessing similar arc lengths. The synthesis approach is more rigorous and more broadly applicable to dramatic changes between larger numbers of shapes than existing techniques that employ graphical methods. It specifically addresses the challenges of approximating closed curves, but the methodology is equally applicable to open curves. Link geometry is determined through an existing procedure, and those links are then joined together in a chain using numerical optimization to minimize the error in approximating the shape change. Binary links are added to this chain via a search of the design space, forming a single-degree-of-freedom mechanism in which an actuated link can be driven monotonically to exact the shape change. The procedure is applied to synthesize an example mechanism that changes between circular, elliptical, and teardrop shapes as inspired by an aerodynamic flow field modification application.

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Copyright © 2009 by American Society of Mechanical Engineers
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Figure 1

(a) Three closed-curve design profiles that define a shape change from a circle to an ellipse to a teardrop. (b) Chain of six rigid segments jointed together shown relative to the elliptical design profile. (c) Solution single-DOF mechanism synthesized from the chain, shown in its circle-approximating position.

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Figure 2

Example of a design profile (dashed line) and a target profile (solid line segments) formed by an exaggeratedly small number of points on the design profile

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Figure 3

Segmentation result—six segments (solid line segments) shown in their error-minimizing positions relative to an elliptical design profile (dashed line), one of the three design profiles used to determine the segment geometries

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Figure 4

Variable definitions for optimization of the positions of the mean segments as an assembled closed chain. Six segments are shown, but the definitions apply to any m number of segments.

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Figure 5

Mechanization Step 1 result—six segments (solid line segments) shown as a closed chain in the error-minimizing position relative to an elliptical design profile (dashed line) as determined using optimization

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Figure 6

Definition of the variables, associated with one five-bar sublinkage, that appears in the kinematic equations of motion for the mechanism

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Figure 7

Single-DOF mechanism changing between three design profiles. (a) and (h) indicate the range of motion of the mechanism. (b), (e), and (g) show matching of the circular, elliptical, and teardrop shapes. (c), (d), and (f) show intermediate positions.

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