On resonances in transversally vibrating strings induced by an external force and a time-dependent coefficient in a Robin boundary condition

In this paper an initial–boundary value problem on a bounded, fixed interval is considered for a one-dimensional and forced string equation subjected to a Dirichlet boundary condition at one end of the string and a Robin boundary condition with a slowly varying time-dependent coefficient at the other end of the string. This problem may serve as a simplified model describing transverse or longitudinal vibrations as well as resonances in axially moving cables for which the length changes in time. By introducing an adapted version of the method of separation of variables, by using averaging and singular perturbation techniques, and by finally using a three time-scales perturbation method, resonances in the problem are detected and accurate, analytical approximations of the solutions of the problem are constructed. It will turn out that small order ɛ excitations can lead to order ɛ responses when the frequency of the external force satisfies certain conditions. Finally, numerical simulations are presented, which are in full agreement with the obtained analytical results.