Generalized multiple scale approach to the problem of a taut string traveled by a single force

The strongly nonlinear dynamics of taut strings, traveled by a force moving with uniform velocity, is analyzed. A change of variable is performed, which recasts the equations of motion in terms of a linearized dynamic displacement, measured from the nonlinear quasi-static response. Under the hypothesis the load velocity is far enough from the celerity of the string, the system appears in the form of linear PDEs whose coefficients are slowly variable in time. Since the classic perturbation methods are not applicable to such kind of equations, the Generalized Method of Multiple Scales is developed, by directly attacking the PDEs, to derive asymptotic solutions. The validity of the analytical predictions is assessed by comparisons with numerical simulations, aimed to prove the accuracy of (1) linearization, and (2) the asymptotic approach.