Non-stationary oscillation of a string on the Winkler foundation subjected to a discrete mass-spring system non-uniformly moving at a sub-critical speed.

We consider non-stationary free and forced transverse oscillation of an infinite taut string on the Winkler foundation subjected to a discrete mass-spring system non-uniformly moving at a given sub-critical speed. The speed of the mass-spring system is assumed to be a slowly time-varying function less than the critical speed. To describe the non-vanishing free oscillation we use an analytic approach based on the method of stationary phase and the method of multiple scales first time suggested in Gavrilov, Indeitsev (J. Appl. Math. Mech. 66(5), 2002) for simpler problem concerning a moving point mass, but now we significantly simplify the calculations using some mathematical trick. This allows us to obtain the analytic solution of the more complicated problem in an easier way and to discover an error in that previous paper. The obtained solution is valid under certain conditions in the absence of resonances if a trapped mode initially exists in the system. We also take into consideration the forced oscillation caused by a force being a superposition of harmonics with time-varying parameters (the amplitude and the frequency). We demonstrate that the analytic solution is in a very good agreement with the numerical one.