On aspects of boundary damping for cables and vertical beams

Abstract

Elastic structures are susceptible to wind- and earthquake-induced vibrations. These vibrations can damage a structure or cause human discomfort. To suppress structural vibrations, various types of damping mechanisms, active or passive, can be applied. In this thesis the model of a weakly damped, standing Euler-Bernoulli beam in a (turbulent) wind-field and the model of a standing Timoshenko beam will be used as a simple model of a tall building. These models will be used to study the stabilizing effect of dampers which are installed at the top of the beam (the so-called boundary dampers), the self-weight effect of a beam on its stability, and the possibly destabilizing effect due to galloping (a dynamic wind response). In this thesis two passive control methods will be applied to the Euler-Bernoulli beam. Moreover, the string-equation will be used to study the dynamics of a cable with an end-mass and subjected to boundary damping. The model of a tensioned beam will be used to examine the damping behavior of a tensioned cable with small bending stiffness and an attached tuned mass damper. The vibrations of these beam and string models can be described by (stochastic) initial-boundary value problems. The problem will be stochastic if a beam in a turbulent wind-field is studied. It is assumed that the damping effect, the self-weight effect, and the wind-force in these problems are small but not negligible. The multiple-timescales perturbation method, the method of separation of variables, and a combination of the Galerkin truncation method and a numerical scheme, will be used to construct (explicit) approximations of the beam-like problems. The Laplace transform method will be applied tothe string-like problem. In this way a so-called characteristic equation has been obtained. This equation have been solved by using (adapted) classical perturbation methods. For both control methods, the uniform stability of an Euler-Bernoulli beam subjected to boundary damping has been established and it has been concluded that these strategies can be used effectively to damp the wind-induced vibrations of a standing Euler-Bernoulli beam. Furthermore, it has been found that the self-weight effect on the frequencies and damping rates of an Euler-Bernoulli and Timoshenko beam is small. For the string problem approximations of the damping rates have been constructed. These have been used to conclude that a string with an end-mass can be damped uniformly by applying boundary damping. Lastly, for the tensioned beam with attached damper it has been shown that small bending stiffness only slightly influences the damping rates of the cable.