Weakly nonlinear dynamics of taut strings traveled by a single moving force

The dynamical response of a taut string traveled by a single moving force is here studied in the nonlinear regime. The equations of motion of the system, accounting for geometric nonlinearities and external damping, are discussed and then studied through perturbation and numerical methods. In particular, the Multiple Scale Method and the Straightforward Expansion are successfully applied to obtain semi-analytical results, and direct numerical integrations are performed on the equations of motion discretized via a Galerkin approach; a solution through the finite-difference method is also developed. Particular attention is devoted to the dynamic increment of tension, which is the main nonlinear effect induced by the traveling force. Using values of model parameters deducted from the literature, the agreement of semi-analytical results with numerical ones is discussed, showing the good behavior of the Straightforward Expansion and pointing out the importance of the geometric nonlinearity for certain combinations of the parameters involved.