0
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.

FIGURES IN THIS ARTICLE
<>
Copyright © 2014 by ASME
Your Session has timed out. Please sign back in to continue.

References

Smith, S. T., 2000, Flexures: Elements of Elastic Mechanisms, Gordon and Breach Science Publishers, New York, NY.
Jones, R. V., 1988, Instruments and Experiences: Papers on Measurement and Instrument Design, John Wiley & Sons, New York, NY.
Sen, S., and Awtar, S., 2013, “A Closed-Form Non-Linear Model for the Constraint Characteristics of Symmetric Spatial Beams,” ASME J. Mech. Des., 135(3), p. 031003. [CrossRef]
Rasmussen, N. O., Wittwer, J. W., Todd, R. H., Howell, L. L., and Magleby, S. P., 2006, “A 3D Pseudo-Rigid-Body Model for Large Spatial Deflections of Rectangular Cantilever Beams,” International Design Engineering Technical Conferences & Computers and Information in Engineering Conference, Philadelphia, PA.
Ramirez, I. A., and Lusk, C., 2011, “Spatial-Beam Large-Deflection Equations and Pseudo-Rigid Body Model for Axisymmetric Cantilever Beams,” Proceedings IDETC/CIE 2011, Washington D. C., Paper # 47389.
Tauchert, T. R.,1974, Energy Principles in Structural Mechanics, McGraw-Hill, New York, NY.
Chen, K. S., Trumper, D. L., and Smith, S. T., 2002, “Design and Control for an Electromagnetically Driven X–Y–θ Stage,” Precis. Eng., 26, pp. 355–369. [CrossRef]
Samuel, H. D., and Sergio, N. S., 1979, “Compliant Assembly System,” U.S. Patent No. 4155169 A.
Ding, X. L., and Dai, J. S., 2006, “Characteristic Equation-Based Dynamics Analysis of Vibratory Bowl Feeders With Three Spatial Compliant Legs,” IEEE Trans. Rob. Autom., 5(1), pp. 164–175 [CrossRef].
Awtar, S. T., Trutna, T., Nielsen, J. M., Abani, R., and Geiger, J. D., 2010, “FlexDex: A Minimally Invasive Surgical Tool With Enhanced Dexterity and Intuitive Actuation,” ASME J. Med. Devices, 4(3), p. 035003. [CrossRef]
Hao, G., and Kong, X., 2012, “A Novel Large-Range XY Compliant Parallel Manipulator With Enhanced Out-of-Plane Stiffness,” ASME J. Mech. Des.134(6), p. 061009. [CrossRef]
Awtar, S., and AlexanderH, S., 2007, “Constraint-Based Design of Parallel Kinematic XY Flexure Mechanisms,” ASME J. Med. Devices, 129(8), pp. 816–830 [CrossRef].
Kim, C. J., Moon, Y. M., and Kota, S., 2008, “A Building Block Approach to the Conceptual Synthesis of Compliant Mechanisms Utilizing Compliance and Stiffness Ellipsoids,” ASME J. Mech. Des., 130(2), p. 022308 [CrossRef].
Hao, G., Kong, X., and Reuben, R. L., 2011, “A Nonlinear Analysis of Spatial Compliant Parallel Modules: Multi-Beam Modules,” Mech. Mach. Theory, 46, pp. 680–706. [CrossRef]
Timoshenko, S., and Goodier, J. N., 1969, Theory of Elastisity, McGraw-Hill, New York, NY.
DaSilva, M. R. M. C., 1988, “Non-Linear Flexural-Flexural-Torsional-Extensional Dynamics of Beams-I. Formulation,” Int. J. Solids Struct., 24, pp. 1225–1234. [CrossRef]
Awtar, S., Slocum, A. H., and Sevincer, E., 2006, “Characteristics of Beam-Based Flexure Modules,” ASME J. Mech. Des., 129(6), pp. 625–639. [CrossRef]

Figures

Grahic Jump Location
Fig. 1

Spatial beam flexure, undeformed, and deformed configurations

Grahic Jump Location
Fig. 2

A 3-DOF spatial flexure mechanism

Grahic Jump Location
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

Grahic Jump Location
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

Grahic Jump Location
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

Grahic Jump Location
Fig. 6

Parasitic error motion in uxs due to uys and θxs

Tables

Errata

Discussions

Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In