Second-order solutions for the dynamics of a semi-infinite cable on a unilateral substrate

We present an asymptotic solution of a moving-boundary problem which describes the nonlinear oscillations of semi-infinite cables resting on an elastic substrate reacting in compression only, and subjected to a constant distributed load and to a small harmonic displacement applied to the finite boundary. Our solution is correct through the second-order terms in a smallness parameter, which we identify with the amplitude of the harmonic oscillation at the boundary, and it complements the first-order solution presented in an earlier work. The second-order analysis confirms the existence of two different regimes in the behaviour of the system, one below (called subcritical) and one above (called supercritical) a certain critical (cutoff) excitation frequency. In the latter, energy is lost by radiation at infinity, while in the former this phenomenon does not occur and various resonances are observed instead. We show that these two regimes exist at all orders in the expansion parameter, and that the cutoff frequency decreases at each order. We also perform a limited comparison of our asymptotic results with a numerical solution. The two approaches show very good agreement.