Equilibrium states of an elastic conducting rod in a magnetic field

Seidman, Thomas I.; Wolfe, Peter

doi:10.1007/bf00251533

Cited by 15 publications

(8 citation statements)

References 6 publications

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“…In this paper we have tried to show how the techniques of modern nonlinear analysis may be brought to bear on a classical problem. Of course these techniques can be applied to more complicated models in which the rod is assumed to be nonlinearly elastic and can undergo stretch, twist and shear, as well as being subject to a variety of forces [6,16,18]. These problems can be three dimensional as well as planar.…”

Section: Discussionmentioning

confidence: 99%

Buckling of a Nonlinearly Elastic Column: Variational Principles, Bifurcation and Asymptotics

Wolfe¹

1995

Rocky Mountain J. Math.

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We study the equilibrium states of an elastic column subject to an end thrust. The column is modeled as an inextensible rod with a nonlinear moment-curvature equation. We bring various mathematical theories to bear on this problem; the calculus of variations, bifurcation theory, phase plane analysis, and singular perturbation theory for ordinary differential equations. We study the connections between the various approaches.Many of the results presented here are not new. Our object is to show how this problem can serve to illustrate various abstract theories and to show how the various approaches compliment each other.

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Section: Discussionmentioning

confidence: 99%

Buckling of a Nonlinearly Elastic Column: Variational Principles, Bifurcation and Asymptotics

Wolfe¹

1995

Rocky Mountain J. Math.

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“…The existence and regularity of solutions to this problem have been shown by Seidman and Wolfe [17]. The main difficulty is the condition (9) on the preservation of orientation, since it is a strict inequality and hence the admissible set is open.…”

Section: Forces and Momentsmentioning

confidence: 97%

Geodesic finite elements for Cosserat rods

Sander¹

2009

Numerical Meth Engineering

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SUMMARYWe introduce geodesic finite elements as a new way to discretize the non-linear configuration space of a geometrically exact Cosserat rod. These geodesic finite elements naturally generalize standard onedimensional finite elements to spaces of functions with values in a Riemannian manifold. For the special orthogonal group, our approach reproduces the interpolation formulas of Crisfield and Jelenić. Geodesic finite elements are conforming and lead to objective and path-independent problem formulations. We introduce geodesic finite elements for general Riemannian manifolds, discuss the relationship between geodesic finite elements and coefficient vectors, and estimate the interpolation error. Then we use them to find static equilibria of hyperelastic Cosserat rods. Using the Riemannian trust-region algorithm of Absil et al. we show numerically that the discretization error depends optimally on the mesh size.

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“…Eqs (3.19) have been considered by Wolfe in bifurcation studies from the trivial straight solution [4,5,6]. In the director frame the governing equation is a non-canonical Hamiltonian system of the form  20) where the structure matrix J = J (m, n, B) is given by…”

Section: The Conducting Rod In a Uniform Magnetic Field -Three-field mentioning

confidence: 99%

Integrability of a conducting elastic rod in a magnetic field

Sinden¹,

Heijden²

2008

J. Phys. A: Math. Theor.

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Abstract. We consider the equilibrium equations for a conducting elastic rod placed in a uniform magnetic field, motivated by the problem of electrodynamic space tethers. When expressed in body coordinates the equations are found to sit in a hierarchy of noncanonical Hamiltonian systems involving an increasing number of vector fields. These systems, which include the classical Euler and Kirchhoff rods, are shown to be completely integrable in the case of a transversely isotropic rod; they are in fact generated by a Lax pair. For the magnetic rod this gives a physical interpretation to a previously proposed abstract nine-dimensional integrable system. We use the conserved quantities to reduce the equations to a four-dimensional canonical Hamiltonian system, allowing the geometry of the phase space to be investigated through Poincaré sections. In the special case where the force in the rod is aligned with the magnetic field the system turns out to be superintegrable, meaning that the phase space breaks down completely into periodic orbits, corresponding to straight twisted rods.

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Equilibrium states of an elastic conducting rod in a magnetic field

Cited by 15 publications

References 6 publications

Buckling of a Nonlinearly Elastic Column: Variational Principles, Bifurcation and Asymptotics

Buckling of a Nonlinearly Elastic Column: Variational Principles, Bifurcation and Asymptotics

Geodesic finite elements for Cosserat rods

Integrability of a conducting elastic rod in a magnetic field

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