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Many mathematical models, which describe oscillations in elastic structures such as suspension bridges, conveyor belts and elevator cables, can be formulated as initial-boundary value problems for string (wave) equations, or for beam equations. In order to build more durable, elegant and lighter mechanical structures, the undesired vibrations can be suppressed by using dampers. In this thesis, the effect of boundary damping on elastic structures is studied. In Chapter 2, as a simple model of oscillations of a cable, a semi-infinite string-like problem is modelled by an initial boundary value problem with (non)-classical boundary conditions. We apply the classical method of D'Alembert to obtain the exact solution which provides information about the efficiency of the damper at the boundary.In Chapter 3, initial-boundary value problems for a beam equation on a semi-infinite interval and on a finite interval have been studied. The method of Laplace transforms is applied to obtain the Greens function for a transversally vibrating homogeneous semi-infinite beam, and the exact solution for various boundary conditions are examined. The analytical results confirm earlier obtained results, and are validated by explicit numerical approximations of the damping and oscillating rates. The study shows that the numerical results approximate the exact results for sufficiently large domain lengths and for a sufficiently high number of modes. Moreover, the study provides an understanding of how the Greens functions for a semi-infinite beam can be computed analytically for (non)-classical boundary conditions.Finally, in Chapter 4 the studies as presented in Chapter 2 and in Chapter 3 are extended to inclined structures. A model is derived to describe the rain-wind induced oscillations of an inclined cable. For a linearly formulated initial-boundary value problem for a tensioned beam equation describing the in-plane transversal oscillations of the cable, the effectiveness of a boundary damper is determined by using a two timescales perturbation method. Not only the influence of boundary damping but also the influence of the bending stiffness on the stability properties of the solution have been studied. |
