Research Papers

Elastic Averaging in Flexure Mechanisms: A Three-Beam Parallelogram Flexure Case Study

[+] Author and Article Information
Shorya Awtar1

Mechanical Engineering, University of Michigan, 2350 Hayward Street, Ann Arbor, MI 48109awtar@umich.edu

Kevin Shimotsu, Shiladitya Sen

Mechanical Engineering, University of Michigan, 2350 Hayward Street, Ann Arbor, MI 48109

This might appear counterintuitive since the axial beam stiffness increases linearly with beam thickness t. However, k33 represents the axial stiffness normalized with respect to the bending stiffness, which leads to an inverse quadratic dependence on t.


Corresponding author.

J. Mechanisms Robotics 2(4), 041006 (Sep 30, 2010) (12 pages) doi:10.1115/1.4002204 History: Received November 12, 2009; Revised July 06, 2010; Published September 30, 2010; Online September 30, 2010

Redundant constraints are generally avoided in mechanism design because they can lead to binding or loss in expected mobility. However, in certain distributed-compliance flexure mechanism geometries, this problem is mitigated by the phenomenon of elastic averaging. Elastic averaging is a design paradigm that, in contrast with exact constraint design principles, makes deliberate and effective use of redundant constraints to improve performance and robustness. The principle of elastic averaging and its advantages are illustrated in this paper by means of a three-beam parallelogram flexure mechanism, which represents an overconstrained geometry. In a lumped-compliance configuration, this mechanism is prone to binding in the presence of nominal manufacturing and assembly errors. However, with an increasing degree of distributed-compliance, the mechanism is shown to become more tolerant to such geometric imperfections. The nonlinear elastokinematic effect in the constituent beams is shown to play an important role in analytically predicting the consequences of overconstraint and provides a mathematical basis for elastic averaging. A generalized beam constraint model is used for these predictions so that varying degrees of distributed compliance are captured using a single geometric parameter. The closed-form analytical results are validated against finite element analysis, as well as experimental measurements.

Copyright © 2010 by American Society of Mechanical Engineers
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Figure 1

Parallelogram mechanism: (a) traditional linkage, (b) lumped-compliance flexure, and (c) distributed-compliance flexure

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Figure 2

Generalized beam flexure

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Figure 3

Beam characteristic coefficients versus beam shape

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Figure 4

Three-beam parallelogram flexure with a geometric error

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Figure 5

Analytically predicted Y-direction stiffness of the three-beam parallelogram flexure

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Figure 6

Elastic averaging metric versus beam shape

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Figure 7

Reconfigurable three-beam parallelogram flexure experimental set-up

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Figure 8

Alignment and assembly of the ground frame and motion stage

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Figure 9

Normalized Y DoF force versus normalized Y DoF displacement: BCM (line), FEA (○), and Exp (×)

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Figure 10

Normalized X displacement: BCM (line), FEA (○), and Exp (×)

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Figure 11

Motion stage rotation: BCM (line), FEA (○), and Exp (×)



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