This paper analyzes the dynamical response of taut strings crossed by systems of traveling forces at constant velocity. Starting from the classic solution for the single moving load, the effect of trains of forces having a step equal to the string length is dealt with. The response is formulated in terms of a linear map, whose reiteration furnishes the discrete-time response, and enables the investigation of the asymptotic behavior of the system. The analytical solution highlights the presence of many critical velocities, for which an instability phenomenon by response accretion may occur. The presence of damping inhibits the onset of instability but also allows to attain large displacements, especially in correspondence of the first critical velocities of the undamped string. Finite-difference numerical solutions confirm the full validity of the proposed analytical solutions. A simple procedure to deduce an improved solution for the problem of the single moving force is outlined in the Appendix.
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