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Research Papers: Design Automation

An Analytical Formulation for the Lateral Support Stiffness of a Spatial Flexure Strip

[+] Author and Article Information
Marijn Nijenhuis

Precision Engineering,
University of Twente,
Enschede 7522 NB, The Netherlands
e-mail: m.nijenhuis@utwente.nl

J. P. Meijaard

Precision Engineering,
University of Twente,
Enschede 7522 NB, The Netherlands;
Olton Engineering Consultancy,
Enschede 7522 NB, The Netherlands
e-mail: j.p.meijaard@utwente.nl

Dhanushkodi Mariappan

Precision Systems Design Lab,
Mechanical Engineering,
University of Michigan,
Ann Arbor, MI 48109
e-mail: dhanush@umich.edu

Just L. Herder

Precision Engineering,
University of Twente,
Enschede 7522 NB, The Netherlands
e-mail: j.l.herder@utwente.nl

Dannis M. Brouwer

Precision Engineering,
University of Twente,
Enschede 7522 NB, The Netherlands
e-mail: d.m.brouwer@utwente.nl

Shorya Awtar

Precision Systems Design Lab,
Mechanical Engineering,
University of Michigan,
Ann Arbor, MI 48109
e-mail: awtar@umich.edu

1Corresponding author.

Contributed by the Design Automation Committee of ASME for publication in the JOURNAL OF MECHANICAL DESIGN. Manuscript received July 1, 2016; final manuscript received January 11, 2017; published online March 20, 2017. Assoc. Editor: Nam H. Kim.

J. Mech. Des 139(5), 051401 (Mar 20, 2017) (11 pages) Paper No: MD-16-1478; doi: 10.1115/1.4035861 History: Received July 01, 2016; Revised January 11, 2017

A flexure strip has constraint characteristics, such as stiffness properties and error motions, that govern its performance as a basic constituent of flexure mechanisms. This paper presents a new modeling approach for obtaining insight into the deformation and stiffness characteristics of general three-dimensional flexure strips that exhibit bending, shear, and torsion deformation. The approach is based on the use of a discretized version of a finite (i.e., nonlinear) strain spatial beam formulation for extracting analytical expressions that describe deformation and stiffness characteristics of a flexure strip in a parametric format. This particular way of closed-form modeling exploits the inherent finite-element assumptions on interpolation and also lends itself for numeric implementation. As a validating case study, a closed-form parametric expression is derived for the lateral support stiffness of a flexure strip and a parallelogram flexure mechanism. This captures a combined torsion–bending dictated geometrically nonlinear effect that undermines the support bearing stiffness when the mechanism moves in the intended degree of freedom (DoF). The analytical result is verified by simulations and experimental measurements.

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Figures

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Fig. 1

Deformed fixed–free flexure strip with general end-load of forces Fx, Fy, Fz,  and moments Mx, My, Mz

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Fig. 2

(a) Deformed configuration of a flexure strip modeled as a beam (thickness exaggerated), (b) orientation of a single cross section, and (c) position of a point on the elastic line

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Fig. 3

Interpretation of average bending moment M̃y using an imaginary lumped torsion spring

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Fig. 4

Interpretation of average bending moment M̃y and shear force Ñz using imaginary lumped torsional and linear springs

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Fig. 5

Discrete model of a flexure strip

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Fig. 6

Effect of decreasing element size (the use of multiple elements per flexure strip means that the element size reduces). (a) Lateral support stiffness and (b) cumulative angle of twist.

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Fig. 7

Contributions to torsion moment Mx¯(ξ), directed along the elastic line tangent

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Fig. 8

Parallelogram flexure mechanism. Internal bending moment Mz(s) is shown on the right.

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Fig. 9

Experimental setup for measuring the lateral support stiffness: (a) full setup and (b) parallelogram flexure mechanism

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Fig. 10

Experimental and theoretical results

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Fig. 11

Nonideal flexure strip geometry, governed by uyi

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