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Research Papers

A Numerical Method to Model Dynamic Behavior of Thin Inextensible Elastic Rods in Three Dimensions

[+] Author and Article Information
Stephen Montgomery-Smith

Department of Mathematics,
University of Missouri,
Columbia, MO 65211
e-mail: stephen@missouri.edu

Weijun Huang

Mechanical and Aerospace Engineering,
University of Missouri,
Columbia, MO 65211
e-mail: whfy6@mail.missouri.edu

Some people might argue that a vector of vectors should really be called a tensor.

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received October 25, 2012; final manuscript received October 1, 2013; published online November 7, 2013. Assoc. Editor: Johannes Gerstmayr.

J. Comput. Nonlinear Dynam 9(1), 011015 (Nov 07, 2013) (10 pages) Paper No: CND-12-1182; doi: 10.1115/1.4025627 History: Received October 25, 2012; Revised October 01, 2013

Static equations for thin inextensible elastic rods, or elastica as they are sometimes called, have been studied since before the time of Euler. In this paper, we examine how to model the dynamic behavior of elastica. We present a fairly high speed, robust numerical scheme that uses (i) a space discretization that uses cubic splines, and (ii) a time discretization that preserves a discrete version of the Hamiltonian. A good choice of numerical scheme is important because these equations are very stiff; that is, most explicit numerical schemes will become unstable very quickly. The authors conducted this research anticipating describing the dynamic Kirchhoff problem, that is, the behavior of general springs that have natural curvature, and for which the equations take into account torsion of the rod.

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Figures

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Fig. 1

A heavily damped, coiled, elastic rod unraveling

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Fig. 2

An elastic rod rotating about a point

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Fig. 3

An elastic rod fixed at two ends, one of which is oscillating

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Fig. 4

An elastic rod, clamped at one end, free at the other end, under the influence of gravity

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Fig. 5

A steady state solution

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Fig. 6

Plot of 4rErrr,r+1 for r = 0,1,2

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Fig. 7

Plots of Errr,r+1/Errr+1,r+2 for r = 0,1

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