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

TL;DR

This paper analyzes the non-stationary oscillations of an infinite string on a Winkler foundation influenced by a non-uniformly moving discrete mass-spring system at sub-critical speeds, revealing trapped modes and energy propagation behaviors.

Contribution

It introduces an analytic approach to describe free and forced oscillations with a slowly varying moving oscillator, including trapped modes and stability analysis with destabilizing springs.

Findings

Identification of trapped modes during initial acceleration stage

Analytic solutions for non-vanishing free oscillations

Numerical verification and discovery of unexpected energy propagation results

Abstract

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).