Research Papers

A Closed-Form Nonlinear Model for the Constraint Characteristics of Symmetric Spatial Beams

[+] 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

Uniform implies a non-varying cross- section along the beam length. Symmetric implies equal moments of area of the beam cross-section about the Y and Z axes.

Slender generally implies a length to thickness ratio greater than 20 [12,13].

1Corresponding author.

Contributed by the Mechanisms and Robotics Committee of ASME for publication in the Journal of Mechanical Design. Manuscript received March 1, 2012; final manuscript received October 25, 2012; published online January 24, 2013. Assoc. Editor: Ashitava Ghosal.

J. Mech. Des 135(3), 031003 (Jan 24, 2013) (11 pages) Paper No: MD-12-1139; doi: 10.1115/1.4023157 History: Received March 01, 2012; Revised October 25, 2012

The constraint-based design of flexure mechanisms requires a qualitative and quantitative understanding of the constraint characteristics of flexure elements that serve as constraints. This paper presents the constraint characterization of a uniform and symmetric cross-section, slender, spatial beam—a basic flexure element commonly used in three-dimensional flexure mechanisms. The constraint characteristics of interest, namely stiffness and error motions, are determined from the nonlinear load–displacement relations at the beam end. Appropriate assumptions are made while formulating the strain and strain energy expressions for the spatial beam to retain relevant geometric nonlinearities. Using the principle of virtual work, nonlinear beam governing equations are derived and subsequently solved for general end loads. The resulting nonlinear load–displacement relations capture the constraint characteristics of the spatial beam in a compact, closed-form, and parametric manner. This constraint model is shown to be accurate using nonlinear finite element analysis, within a load and displacement range of practical interest. The utility of this model lies in the physical and analytical insight that it offers into the constraint behavior of a spatial beam flexure, its use in design and optimization of 3D flexure mechanism geometries, and its elucidation of fundamental performance tradeoffs in flexure mechanism design.

Copyright © 2013 by ASME
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Jones, R. V., 1988, Instruments and Experiences: Papers on Measurement and Instrument Design, Wiley, New York.
Smith, S. T., 2000, Flexures: Elements of Elastic Mechanisms, Gordon and Breach Science Publishers, New York.
Blanding, D. K., 1999, Exact Constraint: Machine Design Using Kinematic Principles, ASME Press, New York.
Awtar, S., 2004, “Analysis and Synthesis of Planer Kinematic XY Mechanisms,” Sc. D., Massachusetts Institute of Technology, Cambridge, MA.
Hopkins, J. B., and Culpepper, M. L., 2010, “Synthesis of Multi-Degree of Freedom, Parallel Flexure System Concepts Via Freedom and Constraint Topology (FACT)—Part I: Principles,” Precis. Eng., 34, pp. 259–270. [CrossRef]
Su, H. J., and Tari, H., 2010, “Realizing Orthogonal Motions With Wire Flexures Connected in Parallel,” ASME J. Mech. Des., 132, p. 121002. [CrossRef]
Sen, S., and Awtar, S., 2010, “Nonlinear Constraint Model for Symmetric Three-Dimensional Beams,” Proceedings of IDETC/CIE 2010, Montreal, Canada, pp. 607–618.
Awtar, S., Slocum, A. H., and Sevincer, E., 2006, “Characteristics of Beam-based Flexure Modules,” J. Mech. Des., 129, pp. 625–639. [CrossRef]
Zelenika, S., and DeBona, F., 2002, “Analytical and Experimental Characterization of High Precision Flexural Pivots Subjected to Lateral Loads,” Precis. Eng., 26, pp. 381–388. [CrossRef]
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]
Hodges, D. H., and Dowell, E. H., 1974, “Nonlinear Equations of Motion for the Elastic Bending and Torsion of Twisted Non-Uniform Rotor Blades,” NASA Technical Note D-7818.
Timoshenko, S., and Goodier, J. N., 1969, Theory of Elastisity, McGraw-Hill, New York.
Crandall, S. H., Dahl, N. C., and Lardner, T. J., 1972, An Introduction to the Mechanics of Solids, McGraw-Hill Book Company, New York.
Euler, L., 1744, Methodus Inveniendi Lineas Curvas Maximi Minimive Proprietate Gaudentes, “Lausannæ, Genevæ, apud Marcum-Michaelem Bousquet & socios”. Available at http://archive.org/details/methodusinvenie00eulegoog
Awtar, S., and Sen, S., 2010, “A Generalized Constraint Model for Two-Dimensional Beam Flexures: Non-Linear Load-Displacement Formulation,” ASME J. Mech. Des., 132, p. 0810091.
Frisch-Fay, R., 1962, Flexible Bars, Butterworth & Co. Ltd., London.
Przemieniecki, J. S., 1968, Theory of Matrix Structural Analysis, McGraw-Hill, New York.
Bisshop, K. E., and Drucker, D. C., 1945, “Large Deflections of Cantilever Beams,” Q. Appl. Math., 3(3), pp. 272–275.
Ramirez, I. A., and Lusk, C., 2011, “Spatial-Beam Large-Deflection Equations and Pseudo-Rigid Body Model for Axisymmetric Cantilever Beams,” Proceedings of the IDETC/CIE 2011, Washington D. C., pp. 43–49.
Shames, I. H., and Dym, C. L., 1985, Energy and Finite Element Methods in Structural Mechanics, Taylor and Francis, New York, NY.
Rubin, M. B., 2000, Cosserat Theories: Shells, Rods, and Points, Springer, New York.
Kirchhoff, G., 1859, “Uber des Gleichgewicht und die Bewegung eines unendlich dunner elasticschen Slabes,” J. Reine Agnew. Math., 56, pp. 285–313. [CrossRef]
Chouaie, B. N., and Maddocks, J. H., 2004, “Kirchhoff's Problem of Helical Equilibria of Uniform Rods,” J. Elasticity, 77, pp. 221–247. [CrossRef]
Whitman, A. B. and DeSilva, C. N., 1974, “An Exact Solution in a Nonlinear Theory of Rods,” J. Elasticity, 4(4), pp. 265–280. [CrossRef]
Antman, S. S., 1974, “Kirchhoff's Problem for Nonlinearly Elastic Rods,” Q. Appl. Math., 32(3), pp. 221–240.
Krylov, A. N., 1931, Calculation of Beams on Elastic Foundation, Russian Academy of Sciences, St. Petersburg.
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]
Popescu, B., and Hodges, D. H., 1999, “Asymptotic Treatment of the Trapeze Effect in Finite Element Cross-Sectional Analysis of Composite Beams,” Int. J. Non-Linear Mech., 34, pp. 709–721. [CrossRef]
Sen, S., and Awtar, S., 2011, “Nonlinear Strain Energy Formulation to Capture the Constraint Characteristics of a Spatial Symmetric Beam,” Proceedings of the IDETC/CIE 2011, Washington, pp. 127–135.


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Fig. 1

Spatial beam flexure—undeformed and deformed

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Fig. 2

A 3 DOF patial flexure mechanism

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Fig. 3

Spatial kinematics of beam deformation

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Fig. 5

Kinematic matrices [H2], [H3], and [H7]

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Fig. 4

Elastic matrix [H1]

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Fig. 6

Elastokinematic matrix [H4]

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Elastokinematic matrix [H5]



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