A model for a 3D spinning rigid electrodynamic tether

For the stability analysis of an electrodynamic tethered satellite system a mechanical model was proposed for a 3D rigid and spinning tether. The model took into account both the spinning speed and the electrodynamic force generated by the earth's magnetic field. Inertial geocentric, orbital, satellite-centred and tether body-fixed co-ordinate frames were introduced and the usual Euler angles for the description of the orientation of moving bodies are used to relate the various frames to each other. The tether was taken as a rigid rod with mass and moment of inertia included for the satellite and the contactors. Based on the non-tilted dipole geomagnetic field model and an assumption that the magnetic field does not vary along the length of the tether, an expression was derived for the induced electromotive force (EMF) between the ends of the tether using Faraday's law of induction when the tether moves through the earth's magnetic field. This EMF then causes the current to flow and an electrodynamic (Lorentz) drag or thrust force to be generated. Lagrang's method was employed to derive the equations of motion for the system when the satellite was restricted to an orbit plane (such as the polar plane) and moved at a uniform angular velocity. In special cases (e.g., planar tether motion, no magnetic field, no spin) the equations reduce to known models in the literature. The stability of the simplest orbiting motions (fixed points of the differential equations) was analytically determined and changes of stability were found as tether length or axial spinning speed varied. Increase of length is generally found to have a destabilising effect, while increase of spin speed is found to have a stabilising effect. These stability results were also confirmed by direct numerical integration of the equations. The effect of the electrodynamic force on a simple orbital motion was also investigated.