Non-linear free vibration of a spinning tether

This paper considers a non-linear free vibration of a spinning tether. The tether is assumed to be a part of a spinning tethered satellite system in the station-keeping phase so that it has a constant nominal length, and it is forced to spin at a constant rate about its nominal axis. Using the Hamilton's principle, the governing equations of motion are derived retaining the non-linear terms up to the third order as well as the gyroscopic terms. Then, these equations are discretized by the assumed-modes method, followed by the application of the method of averaging. When the tether has moderately high nominal tension, averaging with two variables results in a closed form solution, which shows the dependence of the frequency content on the initial amplitude parameters. In the case of very low nominal tension, averaging with a single variable is necessary to obtain accurate solutions. One can obtain the steady state and the limit steady state solutions, both of which represent circular whirling motion like a skip-rope. In the absence of damping, a general transverse motion appears to be quasi-periodic, but it can be periodic depending on the initial conditions. Numerical investigations reveal that the material damping through the longitudinal mode drives the steady state to the limit steady state.