On Aspects of Asymptotics for Axially Moving Continua

Abstract

In this thesis two models for axially moving continua have been studied: a string-like model and a beam-like model. Mathematically, a string-like model is described by a wave equation and a beam-like model is usually described by the Euler-Bernoulli beam equation. A string-like model has been used to describe the transversal vibrations of a vertically moving elevator cable system with time-varying length and a beam-like model has been used to describe the transversal vibrations of a constant length conveyor belt system between two supports. For the string model, a rigid body is attached to the lower end of the string, and the suspension of this body against the guide rails is assumed to be rigid. For the string model, it is assumed that the length changes linearly in time or that the length changes harmonically about a constant mean length. For the linear length-variations it is assumed that the axial velocity of the string is small compared to nominal wave velocity and the string mass is small compared to the mass of the rigid body and, for the harmonically length variations small oscillation amplitudes are assumed. The case with boundary excitations has also been investigated in detail, and interesting resonance conditions have been found. For the beam model, the axial velocity is assumed to be constant and relatively small compared to the wave speed. The case with boundary damping has been investigated in detail for the beam equation and interesting damping properties have been obtained. The corresponding initial boundary value problems have been formulated, and in all cases formal asymptotic approximations of the analytic solutions have been constructed by using the multiple timescales perturbation methods. For the string-like problem, it has been shown that Galerkin's truncation method can not be applied in order to obtain asymptotic approximations valid on long timescales. For boundary excitations the interesting phenomenon of autoresonance occurs when there is passage through (dynamic) resonance. The maximal amplitude of the autoresonant solution and the time of autoresonant growth of the amplitude of the modes of fast oscillations have been determined. Interior layer analysis has been provided systematically and it has been shown that there exists an unexpected timescale of order (1/?(?). For this reason three timescales have been introduced when constructing asymptotic results. For the beam-like problem, by using the energy integral, it has been shown that the solutions are bounded for times t of order 1/?. It has also been analytically and numerically shown that all solutions (up to order ?) are uniformly damped.