Abstract
In this paper, a modified Jeffcott model is proposed and studied in order to shed light into the dynamics of a complex system, the Short Electrodynamic Tether (SET), which is similar to an unbalanced rotor. Due to the internal damping, a geometrically linear SET model appears to be unstable as predicted by the linear rotordynamics theory. Some studies in the field of rotordynamics suggest that this instability caused by internal damping do not appear if geometric nonlinearities are taken into account in the system equations of motion. Stability and bifurcation analysis have been carried out on the modified Jeffcott model, which accounts for geometric nonlinearities, orthotropy in the shaft's cross section, and a viscous damping-based internal damping model. The stability results analytically obtained have been compared with a nonlinear multibody model by means of time simulations and good agreement has been found.
Similar content being viewed by others
References
Valverde, J., Escalona, J. L., Mayo, J., and Domínguez, J., ‘Dynamic analysis of a light structure in outer space: Short Electrodynamic Tether’, Multibody System Dynamics 10(1), 2003, 125–146.
Cosmo, M. L. and Lorenzini, E. C., Tethers in Space Handbook, 3rd edn., prepared for NASA/MSCFC by Smithsonian Astrophysical Observatory, NASA Marshall Space Flight Center, Cambridge, MA, 1997.
Ahedo, E. and Sanmartín, J. R., ‘Analysis of bare-tether systems for deorbiting low-Earth-orbit satellites’, Journal of Spacecrafts and Rockets 39(2), 2002, 198–205.
Steiner, W., Zemann, J., Steindl, A., and Troger, H., ‘Numerical study of large amplitude oscillations of a two-satellite continuous tether system with varying length’, Acta Astronautica 35(9–11), 1995, 607–621.
Childs, D. W., Turbomachinery Rotordynamics, Wiley Interscience, New York, 1993.
Genin, J. and Maybee, J. S., ‘Stability in the three dimensional whirling problem’, International Journal of Non-Linear Mechanics 4, 1969, 205–215.
Bolotin, V. V., Dynamic Stability of Elastic Systems, Holden Day, San Francisco, 1964.
Shaw, J. and Shaw, S. W., ‘Instabilities and bifurcations in a rotating shaft’, Journal of Sound and Vibration 132(2), 1989, 227–244.
Chang, C. O. and Cheng, J. W., ‘Non-linear dynamics and instability of a rotating shaft-disk system’, Journal of Sound and Vibration 160(3), 1993, 433–454.
Valverde, J., Escalona, J. L., Domínguez, J., and Freire, E., ‘Stability and dynamic analysis of the SET (Short Electrodynamic Tether)’, in Proceedings of the ASME DETC & CIE Conference (ISBN: 0-7918-3698-3), Chicago, IL, Sept. 2–6, 2003.
Novozhilov, V. V., Foundations of the Nonlinear Theory of Elasticity, Dover, New York, 1953.
Snowdon, J. C., Vibration and Shock in Damped Mechanical Systems, Wiley, New York, 1968.
Meirovitz, L., Methods of Analytical Dynamics, McGraw-Hill, 1970.
Roseau, M., Vibrations in Mechanical Systems, Springer-Verlag, Paris, 1984.
Strogatz, S. H., Nonlinear Dynamics and Chaos, Perseus Books, Cambridge, MA, 1994.
García de Jalón, J. and Bayo, E., Kinematic and Dynamic Simulation of Multibody Systems – the Real-Time Challenge, Springer-Verlag, New York, 1993.
Wu, S. and Haug, E. J., ‘Geometric non-linear substructuring for dynamics of flexible mechanical systems’, International Journal for Numerical Methods in Engineering 26, 1988, 2211–2226.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Valverde, J., Escalona, J.L., Freire, E. et al. Stability and Bifurcation Analysis of a Modified Geometrically Nonlinear Orthotropic Jeffcott Model with Internal Damping. Nonlinear Dyn 42, 137–163 (2005). https://doi.org/10.1007/s11071-005-2365-3
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/s11071-005-2365-3