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TECHNICAL PAPERS

Modeling of Flexural Beams Subjected to Arbitrary End Loads

[+] Author and Article Information
Chris Kimball

Department of Mechanical Engineering and Institute for Systems Research, University of Maryland, College Park, MD 20742e-mail: ckimball@isr.umd.edu

Lung-Wen Tsai

Department of Mechanical Engineering, University of California, Riverside, CA 92521le-mail: lwtsai@engr.ucr.edu

J. Mech. Des 124(2), 223-235 (May 16, 2002) (13 pages) doi:10.1115/1.1455031 History: Received February 01, 2001; Online May 16, 2002
Copyright © 2002 by ASME
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References

Midha,  A., Norton,  T. W., and Howell,  L. L., 1994, “On the Nomenclature, Classification, and Abstractions of Compliant Mechanisms,” ASME J. Mech. Des. 116, No. 1, pp. 270–279.
Howell,  L. L., and Midha,  A., 1993, “A Method for the Design of Compliant Mechanisms with Small-Length Flexural Pivots,” ASME J. Mech. Des. 116, No. 1, pp. 280–290.
Howell,  L. L., and Midha,  A., 1995, “Parametric Deflection Approximations for End-Loaded, Large Deflection Beams in Compliant Mechanisms,” ASME J. Mech. Des. 117, No. 1, pp. 156–165.
Saxena,  A., and Kramer,  S. N., 1998, “A Simple and Accurate Method for Determining Large Deflections in Compliant Mechanisms Subjected to End Forces and Moments,” ASME J. Mech. Des. 120, No. 3, pp. 392–400.
Saxena,  A., and Kramer,  S. N., 1999, “A Simple and Accurate Method for Determining Large Deflections in Compliant Mechanisms Subjected to End Forces and Moments,” ASME J. Mech. Des. 121, No. 2, pp. 194.
Lyon, S. M., Howell, L. L., and Roach, G. M., 2000, “Modeling Flexible Segments with Force and Moment End Loads via the Pseudo-Rigid-Body Model,” Proceedings of the ASME Dynamic Systems and Control Division-2000. Vol. 69-2, pp. 883–890.
Derderian, J. M., and Howell, L. L., et al., 1996, “Compliant Parallel-Guiding Mechanisms,” Proceedings of the 1996 ASME Design Engineering Technical Conferences, Irvine, CA, MECH:1208.
Bisshop,  K. E., and Drucker,  D. C., 1945, “Large Deflection of Cantilever Beams,” Q. Appl. Math. 3, No. 3, pp. 272–275.
Frisch-Fay,  R., 1963, “Applications of Approximate Expressions for Complete Elliptic Integrals,” Int. J. Mech. Sci. 5, No. 3, pp. 231–235.
Mattiasson,  K., 1981, “Numerical Results from Large Deflection Beam and Frame Problems Analyzed by Means of Elliptic Integrals,” Int. J. Numer. Methods Eng. 17, pp. 145–153.
Byrd, P. F., and Friedman, M. D., 1954, Handbook of Elliptic Integrals for Engineers and Physicists, Springer-Verlag, Berlin.
Howell,  L. L., Midha,  A., and Norton,  T. W., 1996, “Evaluation of Equivalent Spring Stiffness for Use in a Pseudo-Rigid-Body Model of Large-Deflection Compliant Mechanisms,” ASME J. Mech. Des. 118, No. 1, pp. 126–131.
Kimball, C., and Tsai, L. W., 2000, “Modeling and Batch Fabrication of Spatial Micro-Manipulators,” Proceedings of the 2000 ASME Design Engineering Technical Conferences, Baltimore, MD, MECH:14116.

Figures

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Beam with external bending moment, Mo, vertical force, P, and horizontal force, nP, causing an inflection point, where n is the ratio of horizontal force to vertical force and (xo,yo) is the position of the free end of the beam.
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Pseudo rigid body model which can accurately model a beam with an inflection point.
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Free body diagram of the moving links of the pseudo rigid body model. See Fig. 2 for dimensions.
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The moment index for joint 1 (β1) plotted as a function of Θ for n=1 and n=−1
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Pivot locations, γ1 and γ2, plotted as a function of loading conditions, n and κn. (a), γ1(n,κn) and (c), γ2(n,κn) were determined using the elliptic integral solutions and optimization; (b), γ̃1(n,κn) and (d), γ̃2(n,κn) are polynomial approximations of γ1(n,κn) and γ2(n,κn).
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Ratio of the two joint displacements, Kθ, plotted as a function of loading conditions, n and κn. (a), Kθ(n,κn), was determined using the elliptic integral solutions and optimization; (b), K̃θ(n,κn) is a polynomial approximation of Kθ(n,κn).
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Coefficients of the load/deflection polynomial, Eq. (80), as a function of loading conditions, n and κn. (a) C11(n,κn); (b) C12(n,κn); (c) C21(n,κn); (d) C22(n,κn).
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Polynomial approximation of Cij(n,κn) as a function of loading conditions, n and κn. (a) C̃11(n,κn); (b) C̃12(n,κn); (c) C̃21(n,κn); (d) C̃22(n,κn).
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Beam deflection as κ is increased for fixed values of n and α. In cases (a) and (e) no inflection point exists; in cases (b) through (d) an inflection point exits. The discontinuity in Kθ occurs at the transition from (c) to (d). Hence the discontinuity occurs close to the value of κn at which the beam no longer possesses an inflection point, but not necessarily at exactly that value of κn.

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