Bifurcation Theory of an Elastic Conducting Rod in a Magnetic Field

Wolfe, Peter

doi:10.1093/qjmam/41.2.265

Cited by 7 publications

(17 citation statements)

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“…Wolfe also considered a conducting rod in a uniform magnetic field [23]. In addition to extension a rod can undergo flexure, torsion and shear, and for the case of welded boundary conditions it was found that in certain cases bifurcation occurs, with the usual infinity of non-trivial equilibrium states.…”

Section: Introductionmentioning

confidence: 99%

Magnetically-Induced Buckling of a Whirling Conducting Rod with Applications to Electrodynamic Space Tethers

Valverde

Heijden

2010

J Nonlinear Sci

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We study the effect of a magnetic field on the behaviour of a slender conducting elastic structure, motivated by stability problems of electrodynamic space tethers. Both statical (buckling) and dynamical (whirling) instability are considered and we also compute post-buckling configurations.The equations used are the geometrically exact Kirchhoff equations. Magnetic buckling of a welded rod is found to be described by a surprisingly degenerate bifurcation, which is unfolded when both transverse anisotropy of the rod and angular velocity are considered. By solving the linearised equations about the (quasi-) stationary solutions we find various secondary instabilities. Our results are relevant for current designs of electrodynamic space tethers and potentially for future applications in nano-and molecular wires.

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Section: Introductionmentioning

confidence: 99%

Magnetically-Induced Buckling of a Whirling Conducting Rod with Applications to Electrodynamic Space Tethers

Valverde

Heijden

2010

J Nonlinear Sci

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“…This is because the equations are invariant under rotation about the axis (e 3 axis) of the supports. This symmetry property complicated Wolfe's analysis (for non-rotating states of isotropic rods), which had to take account of the variational nature of the problem to prove existence of non-trivial bifurcating states [5]. By contrast, the coat hanger boundary conditions here introduced break the S 1 symmetry down to Z 2 symmetry (reflection symmetry along v 0 and v 1 ), and no problems in the application of standard results from bifurcation theory should arise.…”

Section: Discussionmentioning

confidence: 99%

“…Wolfe also considered a conducting rod in a uniform magnetic field [5]. In addition to extension a rod can undergo flexure, torsion and shear, and for the case of welded boundary conditions it was found that in certain cases bifurcation occurs, with the usual infinity of non-trivial equilibrium states.…”

Section: Introductionmentioning

confidence: 99%

“…In this paper we consider the post-buckling behaviour of a conducting rod. Wolfe considered welded boundary conditions in [5] and encountered degeneracies (even-dimensional eigenspaces) because of rotational symmetry of the problem. In previous work [6] we showed that further complications occur and that magnetic buckling of a welded transversely isotropic rod (i.e., a rod with unequal bending stiffnesses about the two principal axes of its cross-section) is described by a remarkably degenerate pitchfork bifurcation.…”

Section: Introductionmentioning

confidence: 99%

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Helical buckling of a whirling conducting rod in a uniform magnetic field

Valverde

Heijden

2012

International Journal of Non-Linear Mechanics

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We study the effect of a magnetic field on the behaviour of a conducting elastic rod subject to a novel set of boundary conditions that, in the case of a transversely isotropic rod, give rise to exact helical post-buckling solutions. The equations used are the geometrically exact Kirchhoff equations and both static (buckling) and dynamic (whirling) instability are considered. Critical loads are obtained explicitly and are given by a surprisingly simple formula. By solving the linearised equations about the (quasi-)stationary solutions we also find secondary instabilities described by (Hamiltonian-)Hopf bifurcations, the usual signature of incipient 'breathing' modes. The boundary conditions can also be used to generate and study helical solutions through traditional non-magnetic buckling due to compression, twist or whirl.

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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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Bifurcation Theory of an Elastic Conducting Rod in a Magnetic Field

Cited by 7 publications

References 0 publications

Magnetically-Induced Buckling of a Whirling Conducting Rod with Applications to Electrodynamic Space Tethers

Magnetically-Induced Buckling of a Whirling Conducting Rod with Applications to Electrodynamic Space Tethers

Helical buckling of a whirling conducting rod in a uniform magnetic field

Integrability of a conducting elastic rod in a magnetic field

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