Research Papers

Equations of Motion for an Inextensible Beam Undergoing Large Deflections

[+] Author and Article Information
Earl Dowell

Department of Mechanical Engineering and
Materials Science,
Duke University,
Durham, NC 27708
e-mail: earl.dowell@duke.edu

Kevin McHugh

Department of Mechanical Engineering and
Materials Science,
Duke University,
Durham, NC 27708
e-mail: kevin.mchugh@duke.edu

1Corresponding author.

Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received January 14, 2016; final manuscript received February 15, 2016; published online March 10, 2016. Editor: Yonggang Huang.

J. Appl. Mech 83(5), 051007 (Mar 10, 2016) (7 pages) Paper No: JAM-16-1024; doi: 10.1115/1.4032795 History: Received January 14, 2016; Revised February 15, 2016

The Euler–Lagrange equations and the associated boundary conditions have been derived for an inextensible beam undergoing large deflections. The inextensibility constraint between axial and transverse deflection is considered via two alternative approaches based upon Hamilton's principle, which have been proved to yield equivalent results. In one approach, the constraint has been appended to the system Lagrangian via a Lagrange multiplier, while in the other approach the axial deflection has been expressed in terms of the transverse deflection, and the equation of motion for the transverse deflection has been determined directly. Boundary conditions for a cantilevered beam and a free–free beam have been considered and allow for explicit results for each system's equations of motion. Finally, the Lagrange multiplier approach has been extended to equations of motion of cantilevered and free–free plates.

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Tang, D. , Gibbs, S. , and Dowell, E. , 2015, “ Nonlinear Aeroelastic Analysis With Inextensible Plate Theory Including Correlation With Experiment,” AIAA J., 53(5), pp. 1299–1308. [CrossRef]
Tang, D. , Zhao, M. , and Dowell, E. , 2014, “ Inextensible Beam and Plate Theory: Computational Analysis and Comparison With Experiment,” ASME J. Appl. Mech., 81(6), p. 061009.
Simmonds, J. , and Libai, A. , 1979, “ Exact Equations for the Inextensional Deformation of Cantilevered Plates,” ASME J. Appl. Mech., 46(3), pp. 631–636. [CrossRef]
Simmonds, J. , and Libai, A. , 1979, “ Alternate Exact Equations for the Deformation of Arbitrary Quadrilateral and Triangular Plates,” ASME J. Appl. Mech., 46(4), pp. 895–900. [CrossRef]
Simmonds, J. , and Libai, A. , 1981, “ Exact Equations for the Large Inextensional Motion of Plates,” ASME J. Appl. Mech., 48(1), pp. 109–112. [CrossRef]
Darmon, P. , and Benson, R. , 1986, “ Numerical Solution to an Inextensible Plate Theory With Experimental Results,” ASME J. Appl. Mech., 53(4), pp. 886–890. [CrossRef]
Paidoussis, M. , 2004, Fluid-Structure Interactions: Slender Structures and Axial Flow, Vol. 2, Academic Press, London.
Tang, D. , Yamamoto, H. , and Dowell, E. , 2003, “ Flutter and Limit Cycle Oscillations of Two-Dimensional Panels in Three-Dimensional Flow,” J. Fluids Struct., 17(2), pp. 225–242. [CrossRef]
Novozhilov, V. , 1953, Foundations of the Nonlinear Theory of Elasticity, Graylock Press, Mineola, NY.
Hamdan, M. , and Dado, M. , 1997, “ Large Amplitude Free Vibrations of a Uniform Cantilever Beam Carrying an Intermediate Lumped Mass and Rotary Inertia,” J. Sound Vib., 206(2), pp. 151–168. [CrossRef]
Villanueva, L. , Karabalin, R. B. , Matheny, M. H. , Chi, D. , Sader, J. , and Roukes, M. , 2013, “ Nonlinearity in Nanomechanical Cantilevers,” Phys. Rev. B, 87(2), p. 024304. [CrossRef]
Lacarbonara, W. , and Yabuno, H. , 2006, “ Refined Models of Elastic Beams Undergoing Large In-Plane Motions: Theory and Experiment,” Int. J. Solids Struct., 43(17), pp. 5066–5084. [CrossRef]
McHugh, K. , and Dowell, E. , 2016, “ Modal Formulations of Equations of Motion for an Inextensible Beam Undergoing Large Deflections,” (unpublished).
Goldstein, H. , Poole, C. , and Safko, J. , 2002, Classical Mechanics, 3rd ed., Addison Wesley, San Francisco, CA.


Grahic Jump Location
Fig. 1

(a) Cantilevered and (b) free–free beam configuration sketches to illustrate the evaluated system



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