The paper investigates the elastic, thermal, and electrical coupling effects on the dynamics and libration stability of flexible bare electrodynamic tethers in the end-of-mission satellite deorbit. A high-fidelity model is developed by considering transverse and longitudinal dynamics and libration dynamics of the tether and environmental effects with the latest models for atmospheric and plasma density, as well as Earth gravity and magnetic fields. The long-term orbital and libration dynamics of the tether are analyzed by a nodal position finite element method and symplectic time integration. Orbital motion limited theory is used to model the electron collection by bare electrodynamic tethers, whereas a Fowler-Nordheim equation is used for a Spindt array cathode. The thermal effect and its coupling with the dynamics of electrodynamic tethers are investigated parametrically. It is found that the thermal effect significantly affects the stability of electrodynamic tethers, which must be considered in the stability control. Two practical and effective electrical current on/off control strategies are developed based on the libration energy and libration angles. Although the libration energy control is found to be more stable and efficient than the libration angle control, the latter is more appealing for practical applications due to its simplicity and low computational effort.
NomenclatureA t , A s = cross section or stress area of tether and projected drag area of satellite, m 2 B g = vector of Earth magnetic field strength in the global frame, T c m = specific heat of tether material, J · kg · K −1 d = tether diameter, m e E = eccentricity of electrodynamic tether orbital plane H 0 = Hamiltonian of tethered system in equilibrium configuration, J I t = electrical current at electron emitter device, A L, L = total and characteristic length of tether, m L e0 L e = unstretched length (instant length) of tether element, m M yaw , M in , M out = components of electrodynamic torque in system orbital frame, N · m n orb = orbital mean motion, rad · s −1 Q = thermal flux, W · m 2 R e = electrical resistance in tether, Ω R, S, W = unit vector of orbital frame of tethered system T, T 0 = instant and initial temperature, K t = time t, n, b = unit vectors of the local frame of tether element V cc = potential bias at the cathodic end C with respect to the ambient plasma, V X, Y, Z = components of position vector in global geocentric inertial frame, m Z T = impedance of emitter device, Ω α, _ α = pitch angle and pitch angular velocity of virtual tether, rad α p , _ α p = periodic solution of pitch angle, rad; and angular velocity of virtual tether, rad · s −1 α s ∕ε = ratio of absorptivity to emissivity of the tether β, _ β = roll angle, rad; and angular velocity of virtual tether, rad · s −1 θ, θ 0 = north latitude and colatitude, rad λ = east longitude, rad μ g = gravitational constant of Earth, m 3 s −2 ρ a ,ρ t = atmospheric density and material density of tether, kg · m −3 σ = electrical conductivity of tether material, Ω −1...
