Abstract
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 static (buckling) and dynamic (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.
Similar content being viewed by others
References
Ahedo, E., Sanmartín, J.R.: Analysis of bare-tether systems for deorbiting low-earth-orbit satellites. J. Spacecraft Rockets 39(2), 198–205 (2002)
Antman, S.S.: Nonlinear Problems of Elasticity. Springer, Berlin (1995)
Beletsky, V.V., Levin, E.M.: Dynamics of Space Tether Systems. Advances in the Astronautical Sciences, vol. 83. American Astronautical Society, San Diego (1993)
Coleman, B.D., Dill, E.H., Lembo, M., Lu, Z., Tobias, I.: On the dynamics of rods in the theory of Kirchhoff and Clebsch. Arch. Ration. Mech. Anal. 121, 339–359 (1993)
Cosmo, M.L., Lorenzini, E.C.: Tethers in Space Handbook, 3rd edn. Smithsonian Astrophisical Observatory (NASA Marshall Space Flight Center, Huntsville, Alabama) (1997)
Doedel, E.J., Champneys, A.R., Fairgrieve, T.R., Kuznetsov, Yu.A., Sandstede, B., Wang, X.J.,: AUTO97: Continuation and bifurcation software for ordinary differential equations (1997). Available from http://indy.cs.concordia.ca/auto
Fraser, W.B., Stump, D.M.: Yarn twist in the ring-spinning balloon. Proc. R. Soc. Lond. A 454, 707–723 (1998)
Goriely, A., Tabor, M.: Nonlinear dynamics of filaments I. Dynamical instabilities. Physica D 105, 20–44 (1997)
Healey, T.J.: Large rotating states of a conducting elastic wire in a magnetic field: subtle symmetry and multiparameter bifurcation. J. Elast. 24, 211–227 (1990)
Hughes, P.C.: Spacecraft Attitude Dynamics. Dover, New York (1986)
Jackson, J.D.: Classical Electrodynamics, 2nd edn. Wiley, New York (1975)
Krupa, M., Poth, W., Schagerl, M., Steindl, A., Steiner, W., Troger, H., Wiedermann, G.: Modelling, dynamics and control of tethered satellite systems. Nonlinear Dyn. 43, 73–96 (2006)
Maiya, B.G., Ramasarma, T.: DNA, a molecular wire or not—the debate continues. Curr. Sci. 80(12), 1523–1530 (2001)
Scheibel, T., Parthasarathy, R., Sawicki, G., Lin, X.M., Jaeger, H., Lindquist, S.L.: Conducting nanowires built by controlled self-assembly of amyloid fibers and selective metal deposition. Proc. Natl. Acad. Sci. USA 100, 4527–4532 (2003)
Sinden, D., van der Heijden, G.H.M.: Integrability of a conducting elastic rod in a magnetic field. J. Phys. A: Math. Theor. 41, 045207 (2008) (16 pp)
Valverde, J., Escalona, J.L., Mayo, J., Domínguez, J.: Dynamic analysis of a light structure in outer space: short electrodynamic tether. Multibody Syst. Dyn. 10(1), 125–146 (2003)
Valverde, J., Escalona, J.L., Freire, E., Domínguez, J.: Stability and bifurcation analysis of a geometrically nonlinear orthotropic Jeffcott model with internal damping. Nonlinear Dyn. 42(2), 137–163 (2005)
Valverde, J., Escalona, J.L., Domínguez, J., Champneys, A.R.: Stability and bifurcation analysis of a spinning space tether. J. Nonlinear Sci. 16(5), 507–542 (2006)
Wolfe, P.: Equilibrium states of an elastic conductor in a magnetic field: a paradigm of bifurcation theory. Trans. Am. Math. Soc. 278, 377–387 (1983)
Wolfe, P.: Rotating states of an elastic conductor. In: Lightbourne, J., Rankin, S. (eds.) Physical Mathematics and Nonlinear Partial Differential Equations. Dekker, New York (1985)
Wolfe, P.: Bifurcation theory of an elastic conducting rod in a magnetic field. Quart. J. Mech. Appl. Math. 41(2), 265–279 (1988)
Woodson, H.H., Melcher, J.R.: Electromechanical Dynamics, Part II: Fields, Forces, and Motion. Wiley, New York (1968)
Ziegler, S.W., Cartmell, M.P.: Using motorized tethers for payload orbital transfer. J. Spacecraft Rockets 38, 904–913 (2001)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by A.R. Champneys.
Rights and permissions
About this article
Cite this article
Valverde, J., van der Heijden, G.H.M. Magnetically-Induced Buckling of a Whirling Conducting Rod with Applications to Electrodynamic Space Tethers. J Nonlinear Sci 20, 309–339 (2010). https://doi.org/10.1007/s00332-010-9059-9
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00332-010-9059-9
Keywords
- Rod mechanics
- Kirchhoff equations
- Magnetic buckling
- Degenerate pitchfork bifurcations
- Hopf bifurcation
- Spinning electrodynamic tether