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Research Papers

A Generalized Constraint Model for Two-Dimensional Beam Flexures: Nonlinear Strain Energy Formulation

[+] Author and Article Information
Shorya Awtar1

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

Shiladitya Sen

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

1

Corresponding author.

J. Mech. Des 132(8), 081009 (Aug 18, 2010) (11 pages) doi:10.1115/1.4002006 History: Received October 16, 2009; Revised May 16, 2010; Published August 18, 2010; Online August 18, 2010

The beam constraint model (BCM), presented previously, captures pertinent nonlinearities to predict the constraint characteristics of a generalized beam flexure in terms of its stiffness and error motions. In this paper, a nonlinear strain energy formulation for the beam flexure, consistent with the transverse-direction load-displacement and axial-direction geometric constraint relations in the BCM, is presented. An explicit strain energy expression, in terms of beam end displacements, that accommodates generalized loading conditions, boundary conditions, initial curvature, and beam shape, is derived. Using energy-based arguments, new insight into the BCM is elucidated by fundamental relations among its stiffness, constraint, and energy coefficients. The presence of axial load in the geometric constraint and strain energy expressions—a unique attribute of distributed compliance flexures that leads to the elastokinematic effect—is highlighted. Using the principle of virtual work, this strain energy expression for a generalized beam is employed in determining the load-displacement relations, and therefore constraint characteristics, of a flexure mechanism comprising multiple beams. The benefit of this approach is evident in its mathematical efficiency and succinctness, which is to be expected with the use of energy methods. All analytical results are validated to a high degree of accuracy via nonlinear finite element analysis.

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Copyright © 2010 by American Society of Mechanical Engineers
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Figures

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Figure 1

Simple beam flexure

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Figure 3

Multibeam parallelogram flexure

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Figure 4

Parasitic axial displacement ux (DoC) versus transverse displacement uy (DoF)

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Figure 2

Undeformed and deformed beam geometries

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Figure 6

Axial stiffness (DoC) versus transverse displacement uy (DoF)

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Figure 7

Initially slanted and curved beams

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Figure 5

Parasitic stage rotation θz (DoC) versus transverse displacement uy (DoF)

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