On the Lateral Vibrations of a Vertically Moving String With a Harmonically Varying Length

In this paper, the free lateral vibrations of a vertically translating continuum modeled as a taut string with variable length are studied. The time-varying length of the cable is described by a harmonically varying function about a constant mean length. This model can be used to describe the lateral vibrations of an elevator cable with changing length. A Fourier series approach is used to predict resonant frequencies arising because of the length fluctuation. The amplitudes of vibrations are represented by the infinitely dimensional system of coupled ordinary differential equations. This system is studied by Galerkin’s truncation method for the lowest resonant frequency. The obtained results are in agreement with the energy analysis, but the truncation technique leads to inaccurate approximations on long timescales. Hence, an alternative analytical method is proposed promising an accurate approximation of the response on long timescales for the general resonant frequency.