Research Papers

Nonlinear Strain Energy Formulation of a Generalized Bisymmetric Spatial Beam for Flexure Mechanism Analysis

[+] Author and Article Information
Shorya Awtar

e-mail: awtar@umich.edu
Precision Systems Design Lab,
Mechanical Engineering,
University of Michigan,
2350 Hayward Street,
Ann Arbor, MI 48109

A slender beam is defined as a beam, the thickness and width of which are at least 1/20th of its length.

1Corresponding author.

Contributed by the Mechanisms and Robotics Committee of ASME for publication in the JOURNAL OF MECHANICAL DESIGN. Manuscript received October 5, 2012; final manuscript received September 27, 2013; published online November 26, 2013. Assoc. Editor: Oscar Altuzarra.

J. Mech. Des 136(2), 021002 (Nov 26, 2013) (13 pages) Paper No: MD-12-1499; doi: 10.1115/1.4025705 History: Received October 05, 2012; Revised September 27, 2013

Analytical load–displacement relations for flexure mechanisms, formulated by integrating the individual analytical models of their building-blocks (i.e., flexure elements), help in understanding the constraint characteristics of the whole mechanism. In deriving such analytical relations for flexure mechanisms, energy based approaches generally offer lower mathematical complexity, compared to Newtonian methods, by reducing the number of unknowns—specifically, the internal loads. To facilitate such energy based approaches, a closed-form nonlinear strain energy expression for a generalized bisymmetric spatial beam flexure is presented in this paper. The strain energy, expressed in terms of the end-displacement of the beam, considers geometric nonlinearities for intermediate deformations, enabling the analysis of flexure mechanisms over a finite range of motion. The generalizations include changes in the initial orientation and shape of the beam flexure due to potential misalignment or manufacturing. The effectiveness of this approach is illustrated via the analysis of a multilegged table flexure mechanism. The resulting analytical model is shown to be accurate using nonlinear finite elements analysis, within a load and displacement range of interest.

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Fig. 1

Spatial beam flexure, undeformed, and deformed configurations

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Fig. 2

A 3-DOF spatial flexure mechanism

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Fig. 3

(a) tilted spatial beam deformation (b) relating the orientation of the XT-YT-ZT co-ordinate frame and Xd-Yd-Zd co-ordinate frame

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Fig. 4

(i) normalized force fy1 versus normalized displacement uy1 for varying tilt angle β and γ, (ii) normalized (mz1 − 4θz1)/mxd1 versus normalized moment mx1 at θy1 = 0 and θz1 = 0.02 rad for varying tilt angle β, (iii) Normalized displacement ux1 versus normalized displacement uy1 for varying β and γ, (iv) rotational displacement θxd1 versus rotational displacement θy1 for varying γ

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Fig. 5

(i) load stiffening during in-plane translation along y, (ii) in-plane translation along y due to in-plane rotation in the presence of mys, (iii) in-plane translation along y due to in-plane rotation in the presence of mzs, (iv) load stiffening during in-plane rotation due to x

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Fig. 6

Parasitic error motion in uxs due to uys and θxs




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