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Research Papers

An Intrinsic Geometric Framework for the Building Block Synthesis of Single Point Compliant Mechanisms

[+] Author and Article Information
Girish Krishnan

Department of Mechanical Engineering, University of Michigan, Ann Arbor, MI 48105gikrishn@umich.edu

Charles Kim

Department of Mechanical Engineering, Bucknell University, Lewisburg, PA 17837charles.kim@bucknell.edu

Sridhar Kota

Department of Mechanical Engineering, University of Michigan, Ann Arbor, MI 48105kota@umich.edu

J. Mechanisms Robotics 3(1), 011001 (Nov 23, 2010) (9 pages) doi:10.1115/1.4002513 History: Received July 03, 2009; Revised August 20, 2010; Published November 23, 2010; Online November 23, 2010

In this paper, we implement a characterization based on eigentwists and eigenwrenches for the synthesis of a compliant mechanism at a given point. For 2D mechanisms, this involves characterizing the compliance matrix at a unique point called the center of elasticity, where translational and rotational compliances are decoupled. Furthermore, the translational compliance may be represented graphically as an ellipse and the coupling between the translational and rotational components as vectors. These representations facilitate geometric insight into the operations of serial and parallel concatenations. Parametric trends are ascertained for the compliant dyad building block and are utilized in example problems involving serial concatenation of building blocks. The synthesis technique is also extended to combination of series and parallel concatenation to achieve any compliance requirements.

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Copyright © 2011 by American Society of Mechanical Engineers
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Figures

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Figure 1

Eigentwist and eigenwrench parameters for a particular building block geometry

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Figure 2

Compliance ellipse and compliance coupling vector (cv)

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Figure 3

Stiffness ellipse and stiffness coupling vector (sc)

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Figure 4

Two building blocks BB1 and BB2 in series. The final coupling vector is the vector addition of the modified coupling vector of BB1(rI/kg1) and the coupling vector of BB2(rE2/kg2)

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Figure 5

The compliance ellipse of BB2 is augmented by a degenerate shift ellipse rm2/(kg1+kg2)

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Figure 6

Addition of building blocks in parallel involves addition of the individual stiffness ellipses and the coupling vectors

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Figure 8

Guidelines with an example: (a) problem specification in terms of compliance ellipse and coupling vector; (b) choose E1, E2, Ipm, and evaluate shift ellipse; (c) net ellipse evaluation and subdivision into smaller building block ellipses; and (d) design geometry of the two building blocks and their orientation

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Figure 9

Design for ((a) and (b)) circular compliance ellipse and (c) zero coupling vector

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Figure 7

(a) A plot of np, (b) a plot of the normalized value of af1, and (c) normalized value of rE, (d) angle β, and (e) normalized stress factor σn with respect to the dyad angle and dyad length ratios l2norm

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Figure 10

Parallel combination: (a) two symmetric halves, (b) addition of stiffness coupling vectors, (c) addition of stiffness ellipses, and (d) final mechanism with a rigid probe

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Figure 11

The center of elasticity of any mechanism due to a series combination of building blocks will always lie within its footprint: (a) entire mechanism, (b) mechanism divided into a number of beams of length l, and (c) curve traced by the coupling vectors, which define the position of the center of elasticity

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