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Research Papers

# Geometrically Exact Kirchhoff Beam Theory: Application to Cable Dynamics

[+] Author and Article Information
Frédéric Boyer

Institut de Recherche en Communications et Cybernétique de Nantes (IRCCyN), Ecole des Mines de Nantes, La Chantrerie, 4, rue Alfred Kastler, B. P. 20722, F-44307 Nantes, Cedex 3, Francefrederic.boyer@emn.fr

Guillaume De Nayer

Fakultät für Maschinenbau, Helmut-Schmidt Universität, Holstenhofweg 85, Hamburg, Germanydenayer@hsu-hh.de

Alban Leroyer

Laboratoire de Mécanique des Fluides, UMR CNRS 6598, Ecole Centrale de Nantes, 1 rue de la Noë, B.P. 92101, Nantes 44321, Cedex 3, Francealban.leroyer@ec-nantes.fr

Michel Visonneau

Laboratoire de Mécanique des Fluides, UMR CNRS 6598, Ecole Centrale de Nantes, 1 rue de la Noë, B.P. 92101, Nantes 44321, Cedex 3, Francemichel.visonneau@ec-nantes.fr

Referring to Riemannian geometry, any $Λu(v)$ defines on the unit sphere $S2$, the action of the parallel transport of the Levi–Civita connection along the geodesic (great circle) linking $u∊S2$ with $v∊S2$.

Given in the case of $SO(3)$ by $exp(v̂)=cos(‖v‖)I3+(1−cos(‖v‖))(v/‖v‖)⊗(v/‖v‖)+sin(‖v‖)(v/‖v‖)̂$.

$d/dϵ∣ϵ=0$ denotes the directional (Gateau) derivative.

In fact, one of the two integrations by parts is done; the second will be “replaced” by the polynomial interpolation (Hermit) of the finite element method.

J. Comput. Nonlinear Dynam 6(4), 041004 (Apr 05, 2011) (14 pages) doi:10.1115/1.4003625 History: Received July 14, 2010; Revised January 19, 2011; Published April 05, 2011; Online April 05, 2011

## Abstract

In this article, the finite element simulation of cables is investigated for future applications to robotics and hydrodynamics. The solution is based on the geometrically exact approach of Cosserat beams in finite transformations, as initiated by Simo in the 1980s. However, the internal basic kinematics of the beam theory is not those of Reissner–Timoshenko but rather those of Kirchhoff. Based on these kinematics, the dynamic model adopted is a nonlinear extension of the so-called linear model of twisted and stretched Euler–Bernoulli beams. In agreement with the investigated applications, one or both of the ends of the cable are submitted to predefined motions. This model is also implemented into a computational fluid dynamics code, which solves the Reynolds-averaged Navier–Stokes equations. Regarding this last point, an implicit/iterative algorithm including a conservative load transfer for the variable hydrodynamic forces exerted all along the beam length has been used to reach a stable coupling. The relevance of the approach is tested through three advanced examples. The first is related to the prediction of cable motion in robotics. Then, the two last illustrations deal with fluid-structure interaction (FSI). A 2D classical benchmark in FSI is first investigated, and, at last, a computation illustrates the procedure in a 3D case.

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## Figures

Figure 8

Models of the structure and fluid loads

Figure 9

Transfer of fluid contact forces to external wrenches compatible with Kirchhoff beam kinematics

Figure 10

Hübner test-case results with the Bossak method (Δt=10−3 s) on the coarse mesh

Figure 11

Pressure field around the filament, which deforms itself during a half period T/2≈0.16 s (calculation on the fine mesh) : (a) t=0, (b) t≈T/12, (c) t≈T/6, (d) t≈T/4, (e) t≈T/3, and (f) t≈T/2

Figure 12

Deformable cable partly in water

Figure 6

Diagram of FSI coupling

Figure 7

Algorithm of the FSI coupling

Figure 13

Cable with inflow piercing the free surface: (a) initial configuration and (b) deformed cable at t=15 s

Figure 14

Towed cable into a multifluid domain (snapshots every 0.5 s)

Figure 15

Deformed mesh at t=15 s: (a) global view and (b) zoom at the free cable extremity

Figure 1

Basic kinematics of a beam

Figure 2

Parameterization of the rotation field

Figure 3

Initial configuration of the six-axis “Puma robot”

Figure 4

A cable fixed at the end of a moving Puma type manipulator: (a) planar projection in (e1,e2), (b) planar projection in (e3,e1), (c) planar projection in (e2,e3), and (d) 3D visualization

Figure 5

Elastic filament fixed to a square rigid body in an incompressible flow

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