In this paper, we study the free transverse vibrations of an axially moving (gyroscopic) material represented by a perfectly flexible string. The problem can be used as a simple model to describe the low frequency oscillations of elastic structures such as conveyor belts. In order to suppress these oscillations, a spring–mass–dashpot system is attached at the nonfixed end of the string. In this paper, it is assumed that the damping in the dashpot is small and that the axial velocity of the string is small compared to the wave speed of the string. This paper has two main objectives. The first aim is to give explicit approximations of the solution on long timescales by using a multiple-timescales perturbation method. The other goal is to construct accurate approximations of the lower eigenvalues of the problem, which describe the oscillation and the damping properties of the problem. The eigenvalues follow from a so-called characteristic equation obtained by the direct application of the Laplace transform method to the initial-boundary value problem. Both approaches give a complete and accurate picture of the damping and the low frequency oscillatory behavior of the traveling string.

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