  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. To describe a non-vanishing free oscillation we use an analytic
approach based on the method of stationary phase and the method of multiple
scales. The moving oscillator is characterized by a partial frequency, which
can be greater or less than the cut-off frequency. Accordingly, a sub-critical
uniformly accelerated motion generally has two stages. At the first stage there
exists a trapped mode, and, therefore, a part of the wave energy is localized
near the moving oscillator and does not propagate away. For this stage we
obtain the analytic solution in a simple form describing non-vanishing free
oscillation and verify it numerically. For the second stage there is no trapped
mode, and all the wave energy propagate away. This stage is investigated
numerically, and some unexpected results are obtained. Additionally, we
consider the case of the oscillator with a destabilizing spring. The dynamics
of the system in the latter case is quite different from the case of commonly
used stabilizing spring, since the system loses the stability during an
accelerated motion. We also take into consideration the forced oscillation
caused by an external load being a superposition of harmonics with time-varying
parameters (the amplitude and the frequency).
