Semi-analytical approaches for the nonlinear dynamics of a taut string subject to a moving load

The nonlinear behavior induced by a moving force on the dynamics of a nonlinear taut string is investigated. In these cases, classic perturbation solutions are only applicable to the weakly nonlinear problem and pertain to small values of dimensionless parameters governing the system. A change of coordinate that is dependent on the nonlinear quasi-static response of the string enables the formulation of a new set of governing equations for an incremental dynamic variable. These new equations are obtained using a closed-form expression for the nonlinear quasi-static displacement of the string. The solution to the linearized form of the governing equations allows us to compute the effective dynamic response even in the presence of strong nonlinearities, for which string dynamic tensions can reach values over twice the static ones. The proposed procedure also shows good reduction characteristics and permits a description of the dynamic response using a small number of eigenfunctions. Applications of the procedure show that, unless the speed of the moving load is sufficiently large, the nonlinear behavior of the string is captured by the quasi-static solution, whereas the dynamics are captured by the linearized equations of motion about this state.