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Analytical Assessment of the Mooring System Stiffness

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This presentations reviews The definition of the mooring systems is one of the most important stages on the design any offshore unit. Some effects associated with it, on the responses of the floating body, are still under investigation. In this context, the full nonlinear high-order hierarchical modeling and the numerical integration of the resulting equations of motion might not be the most cost effective approach for the evaluation of those effects during the early design process. Thus, an expedite analytical formulation to assess the mooring system stiffness, a tool that could help the initial design and analysis. Using classic approaches from Analytical Mechanics, the nonlinear generalized restoring forces associated with the mooring acting on the vessel due to the mooring lines are formulated. The six-degree-of-freedom (DoF) problem is herein addressed. The stiffness matrix is obtained from the linearization of the generalized forces around a generic position. Mooring line characteristic tension function is an input of the method. The closed formulation does not requires a specific line model, although the formulation for a multi-segment mooring line is also derived. The methodology is applied taking the OC4-DeepCwind Floating Wind Turbine semi-submersible platform as a case study. Two Spread Mooring Systems arrangements are studied, in order to demonstrate the use of the presented formulation as a design tool. The calculated mooring system stiffness matrix, evaluated at the trivial equilibrium position, exhibits good agreement with numerical results found in the literature by high hierarchy models. Additionally, the stiffness coefficients are evaluated for other positions than the trivial equilibrium one in the form of colored maps. The natural periods of the motions on the horizontal plane are also mapped. These maps help to understand the effects of the static vessel mean position on the mooring system stiffness and, consequently, on the natural periods associated with the motions on the horizontal plane. Considering the original OC4 mooring system, the effects of the mooring line pre-tensioning are also investigated. Some conclusions on the axial stiffness of catenary cables are also made. The main contributions of the present master dissertation are: (i) the stiffness matrix analytical closed formulation and (ii) the use of colored maps to evaluate the stiffness and the natural periods as functions of the mean offset position. The present master dissertation brings then an innovative closed-form formulation with important practical applications.
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... As discussed in Section 3, a second numerical model was constructed, considering a heeled mesh, as presented in Figure 11, and with a similar correction of the ballast mass made in the physical model. In addition, the linearized external mooring stiffness was taken into account considering the six dof, with the stiffness matrix computed by means of the analytical formulation proposed in [27]. As shown in Section 3, the mooring system stiffness dependency on both the floater mean position and the mean attitude of the platform (trim/heel) had to be considered for a proper modeling of the motions. ...
... Due to the mean thrust load, the FOWT platform drifts to a mean point far from the trivial unloaded position. As discussed by [27,30], the new equilibrium position changes the mooring system stiffness, mainly for horizontal motions, as they are the most affected by mooring restoring forces. Figure 13 brings the stiffness-offset relation, considering different tilt angles of the floater, for surge, pitch and heave motions only (possible couplings included). ...
... Figure 13 brings the stiffness-offset relation, considering different tilt angles of the floater, for surge, pitch and heave motions only (possible couplings included). Notice that the plots in Figure 13 represent the coefficients of the 3 × 3 symmetric mooring system stiffness matrix ( [27]). As a consequence of this change, the natural periods also change. ...
Article
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The present work highlights some of the dynamic couplings observed in a series of tests performed in a wave basin with a scaled-model of a Floating OffshoreWind Turbine (FOWT) with semi-submersible substructure. The model was moored by means of a conventional chain catenary system and an actively controlled fan was used for emulating the thrust loads during the tests. A set of wave tests was performed for concomitant effects of not aligned wave and wind. The experimental measurements illustrate the main coupling effects involved and how they affect the FOWT motions in waves, especially when the floater presents a non-negligible tilt angle. In addition, a frequency domain numerical analysis was performed in order to evaluate its ability to capture these effects properly. The influence of different modes of fan response, floater trim angles (changeable with ballast compensation) and variations in the mooring stiffness with the offsets were investigated in the analysis. Results attest that significant changes in the FOWT responses may indeed arise from coupling effects, thus indicating that caution must be taken when simplifying the hydrodynamic frequency-domain models often used as a basis for the simulation of FOWTs in waves and in optimization procedures for the design of the floater and mooring lines.
... which requires the values of K M and F M to be known beforehand, using, for example, the formulation proposed by Pesce, Amaral, and Franzini (2018) and Amaral (2020). Although the linear model is considered to be sufficient for this work, as the focus is on the hydrodynamic model, improving the modeling of the mooring system is one of the main steps for the future development of the software, not only for a more accurate calculation of the motions of the FOWT, but also because the analysis of the mooring dynamics itself is a particularly important application of a numerical tool such as the one developed here. ...
... However, as the simulations include second-order loads, it is important to properly model the first-order motions of the body in the WAMIT analysis, meaning that mooring and viscous effects need to be accounted for. The action of the mooring system was modeled by the following external linear stiffness matrix, which was evaluated using the analytical formulation proposed by Pesce, Amaral, and Franzini (2018) and Amaral (2020) around the mean body position (taken as the initial body position, since mean drift was small for the cases without wind): ...
Thesis
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This thesis describes the development of METiS, a numerical tool for the seakeeping analysis of floating offshore wind turbines (FOWT). It is based on a slender-body approximation for evaluating the first- and second-order wave loads acting on a floating structure comprised of slender cylinders that combines Rainey's equation, which can be seen as an extension of the inertial part of Morison's equation to include nonlinear terms, with Pinkster's formulation for the low-frequency second-order loads on floating bodies. This combination is followed to allow the evaluation of the forces considering the mean body position, in such a way that an Inverse Fast Fourier Transform algorithm can be used to efficiently compute the time series of wave kinematics and second-order wave loads in a real sea condition directly in time domain, as opposed to the most common procedure of solving the second-order radiation/diffraction problem in frequency domain and importing the results to time-domain solvers. As a drawback of the approximation, end effects due to the extremities of the cylinder and effects due to wave scattering and radiation are lost, which is acceptable as long as the diameters of the cylinders that compose the structure are small in face of their draft and in face of the length of the incoming waves. These conditions may be too restrictive to modern oil & gas spars and semi-submersibles, which have large diameter columns, but it is satisfied by FOWTs in many significant wave conditions. The slender-body approximation is first verified by analyzing the simple case of a single surface piercing cylinder under the action of bichromatic waves, both bottom mounted and floating, for different combinations of the dimensionless parameters that describe the problem. The results are compared with the ones obtained with diffraction theory and Newman's approximation, evidencing an interesting complementarity between the two approximations and the conditions in which each of them performs the better. The relevance of the second-order terms from Rainey's formulation is shown, demonstrating that the common practice of analyzing second-order loads by simply applying Morison's equation with second-order wave kinematics is not strictly correct. The method is then applied to the analysis of a semi-submersible FOWT model, moored by three caternary lines, that was tested at the wave basin of the Numerical Offshore Tank of the University of São Paulo. Three sets of tests are presented: free decays of the moored model; forced oscillations of the hull; and motions under the action of waves (bichromatic, JONSWAP and white-noise) and wind. The results obtained with METiS are compared with the experiments and with WAMIT and OpenFAST, illustrating the capabilities and limitations of each software.
... Fig. 10 brings the panels limits adopted for the WAMIT® mesh. In addition, the linearized external mooring stiffness was taken into account considering the 6 dof, with the stiffness matrix computed by means of the analytical formulation proposed by [18]. As it will be shown in Section 3, the mooring system stiffness dependency on both the floater mean position and the mean attitude of the platform (trim/heel) had to be considered for a proper modeling of the motions. ...
... Due to the mean thrust load, the FOWT platform drifts to a mean point far from the trivial unloaded position. As discussed by [21] and [18], the new equilibrium position changes the mooring system stiffness, mainly for horizontal motions, as they are the most affected by mooring restoring forces. Fig. 13 brings the stiffness-offset relation, considering different tilt angles of the floater. ...
Preprint
Full-text available
The present work highlights some of the dynamic couplings observed in a series of tests performed in a wave basin with a scaled-model of a FOWT with semi-submersible substructure. The model was moored by means of a conventional chain catenary system and an actively controlled fan was used for emulating the thrust loads during the tests. A set of wave tests comprising regular and irregular waves was done for different wave angles and wind velocities. The experimental records illustrate the main coupling effects involved and how they affect the FOWT motions in waves, especially when the floater presents a non-negligible tilt angle.. In addition, an analysis of the frequency-domain dynamic model was made in order to evaluate its ability to capture these effects properly. The influence of different modes of fan response, floater trim angles (changeable with ballast compensation) and variations of the mooring stiffness with the offsets were investigated in the analysis. Results attest that significant changes in the FOWT responses may indeed arise from coupling effects, thus indicating that caution must be taken when simplifying the hydrodynamic frequency-domain models often used as a basis for the simulation of FOWTs in waves and in in optimization procedures for the design of the floater and mooring lines.
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