During ‘float-out’ installation of export cables, the cable is pulled to shore while floating, supported by inflatable floaters. Understanding the hydrodynamic behaviour of these cable-floater-systems (CFS’s) is important, to be able to relate the hydraulic environment to the cable stresses and determine the maximum allowed hydraulic conditions for these procedures. The software package OrcaFlex has appeared to be insufficient and gives untrustworthy results, namely really high peak axial tension and compression forces in the cable. In this research, the CFS is studied with three methods to explain these extreme forces and improve the CFS design. With the first method, the CFS is modelled as an Euler Bernoulli Beam (EBB) on a continuous elastic foundation. In this analytical model, straight, accessible formulas showed that the CFS is a really stiff dynamic system, dominated by the buoyancy stiffness of the floaters. As a consequence, the natural frequencies are relatively high and in the same range as the wave frequency spectrum. This lead to this research’ hypothesis that the extreme, internal stresses in the cable-floater-system are caused by resonance taking place between the external wave forcing and the CFS itself. With the second method, the CFS is modelled with the finite-element-method. In the vertical direction, the natural frequencies are again in the range of wave frequencies and are barely increasing for higher modes. The explanation for both of these observations is given by the local oscillations in the modal shapes occurring in between floaters. A parametric study on floater spacing shows that the length of these oscillations is determined by the floater spacing. Since the original floater spacing is small, it causes small wave lengths and high frequencies. With the third method, dynamic analyses are done in OrcaFlex with cases varying in wave frequency and presence/absence of a current in order to test the hypothesis of resonance. It can be concluded that the CFS is really sensitive to Stokes drift when an other current is absent. This causes a different positioning of the CFS and a mean axial tension in the cable decreasing in wave direction. Consequently, the hydrodynamic behaviour of the CFS is changing throughout its length. Near the cable end most closest from where waves originate, the Stokes drift induced axial tension causes the CFS to be unable to move vertically in waves. Consequently, no interaction between wave and CFS, necessary for resonance, is taking place. Near the other cable end, the mean axial tension is close to zero and the most extreme axial forces occur for resonance conditions with wave frequency equal to natural frequency. The results do not fully confirm the hypothesis, but do give answer to the first part of the research question. Regarding the second part on the optimisation of a CFS design, one could focus on eliminating the chance on the occurrence of resonance, by increasing the floater spacing or decreasing the floater dimensions, or focus on decreasing the impact of resonance by increasing the floater’s drag area in vertical direction.

This thesis is about homological algebra and singular (co)homology. In the first chapter the notions of complexes of abelian groups, (co)homology of these complexes and injective resolutions will be introduced. Then Ext-groups will be defined and various properties dervied. A particularly interesting group, Ext(Q,Z), will be calculated which involves the p-adic integers. Lastly we will prove the universal coefficient theorem for complexes of free abelian groups. In the second chapter we will use the tools provided by the previous chapter to calculate the singular homology groups of topological spaces. First we will explicitely describe the zero'th and first singular homology groups for any topological space. For the spheres S^n and real projective n space P^n(R) we will calculate all singular homology and cohomology groups. For this we will use the universal coefficient and properties about Ext-groups which have been proven in chapter 1. We will also prove and use the long exact sequence of Mayer-Vietoris. This theorem proposes a way to calculate the singular homology groups of a space by using the singular homology groups of two subspaces.

This thesis is about Van Kampen's theorem and fundamental groupoids. Van Kampen's Theorem is a classical result in algebraic topology, which proposes a way of calculating the fundamental group of a topological spaces using the fundamental groups of certain subspaces. In this thesis we will construct the fundamental group, which intuitively counts "holes" in a topological space and is mainly used to distinguish topological spaces. Van Kampen's theorem is then proven for a arbitrary large cover using covering spaces. In the proof we glue topological spaces together and an example of this gluing is given for the Klein bottle. Van Kampen's theorem does not work for every topological space, so we take a look at a generalization of the fundamental group, the so-called fundamental groupoid. Van Kampen's theorem can be upgraded to calculate fundamental groupoid and this theorem is proven in this thesis as well.