Nonlinear resonance of a subsatellite on a short constant tether

The nonlinear resonant behavior of a subsatellite on a short constant tether during station-keeping phase is investigated in this paper. The nonlinear dynamic equations of in-plane motion of the system are derived based on Kane's method first. Then an approach of multiple scales expressed in matrix form is employed in solving the simplified nonlinear system of cubic nonlinearity near its local equilibrium position. Analysis shows that there exists a three-to-one resonance in such a nonlinear system with two degrees of freedom. Afterward, the approximate solution up to third order determined analytically by the Weierstrass elliptic function is obtained and the comparison between the approximate and numerical solutions presented as well. The results show that the approximate solution is coincide well with the numerical solution of original system. The nonlinear resonance of the subsatellite on short tether exhibits coexistent quasiperiodic motions or a quasiperiodic oscillation near local equilibrium position.