On the transverse vibrations of a revolving tether

We analyse the transverse vibrations of a tether, modelled as an inextensible cable, and revolving at an average rate equal to the orbital rate. The reference motion is a revolving rigid tether. During this motion the force in the tether (time and location dependent) remains, in a first approximation, aligned with the tether axis. Separation of variables for the vibrations about this motion gives a Legendre equation for the spatial dependency of the deformations and Hill's equations for time dependency of the in- and out-of-plane deformations. The boundary conditions on the Legendre equation generate a series of admissible values of the separation constant that become equidistant. The two Hill's equations generate a series of intervals, contracting to equidistant critical values, where the solutions are unbounded. The admissible values of the separation constant must avoid these intervals. Asymptotic expressions for the separation constant and the critical values are given. The first and second in-plane deformation mode arc unstable for zero end masses. By increasing the ratio of the concentrated over the distributed mass the deformation modes can be stabilised and the values of the separation constant can be made a multiple of the distribution of the critical points. Introducing unequal tip masses does not affect this result.