Nonlinear Dynamics of a Space Tethered System in the Elliptic Earth-Moon Restricted Three-Body System

Abstract

This paper focuses on nonlinear oscillation of a space tether system connected to the Moon’s surface, elongating along the Earth-Moon line at L1 and L2 sides respectively. The full nonlinear elongation dynamics of the space tether system were established for a superlong viscoelastic massless tether with a large tip mass. The equilibria and their stabilities were investigated by removing the external force in the established dynamics. The equilibria and the stabilities depend on the elasticity and natural length of the tether, and the dependence is the key to determining the parameters for successful operation of the tether system. The equilibria were also used as reference positions to facilitate dynamic analysis. The method of multiple scales was utilized to obtain the analytical asymptotic solutions to the dynamics. The analytical results agree well with the numerical ones quantitatively and qualitatively in terms of the steady-state magnitudes of the length and the length rate of the tether, as well as the power harvested for a large range of parameters. The detailed investigation reveals that the dynamic responses were affected by three important parameters, namely, the damping, the elasticity, and the original length of the tether. In particular, the increasing damping stabilizes the motion around the expected equilibrium and makes the quasi-periodic motion periodic. It was also indicated that one should design a tether with small rigidity for efficient power generation, but the largest steady-state length should be restricted within prescribed ranges when selecting the rigidity. There is also a compromise between the power output and elastic tension when designing the tether system, i.e., the tether will experience a larger elastic tension for greater harvested power.