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