On transversal oscillations of a vertically translating string with small time-harmonic length variations

In this paper, the free transverse vibrations of a vertically moving string with a harmonically time-varying length are studied. The string length variations are assumed to be small. By using the multiple‐timescales perturbation method in conjunction with a Fourier series approach, we determine the resonance frequencies and derive the non-secularity conditions in the form of an infinite dimensional system of coupled ordinary differential equations. This system describes the long time behavior of the amplitudes of the oscillations. Then, the eigenvalues of the obtained system are studied by the Galerkin truncation method, and applicability of this method is discussed. Apart from this, the dynamic stability of the solution is investigated by an energy analysis. Additionally, resonance detuning is considered.