# Initial Design Phase and Tender Designs of a Jacket Structure Converted into a Retrofitted Offshore Wind Turbine

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## Abstract

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## 1. Introduction

## 2. Theoretical Background

**M**is assumed to be a diagonal matrix with elements ${m}_{i}>0$, $i=1,2,\dots ,N$, in which the element’s subscript is its associated node point. It has been said that each generalized coordinate represents either the displacement or the rotation of a portion of the structure’s mass at a specific node point. The mass lumping method is probably the most popular method of discretizing the supporting framework and the rigid body portions of an offshore structure. In this research, the mass lumping method will be applied where all the offshore mass structure is modeled as lumped at the top of a cantilever beam. The stiffness matrix

**K**for an N degree of freedom structure is a symmetric matrix of $N\times N$ elements. The stiffness is defined as the force applied to the structure in order to produce a unitary displacement. The constant ${k}_{ij}$ in other terms is that force that is required at node i to counteract a unit elastic displacement ${\xi}_{sj}=1$ imposed at node j, under the condition that all displacements ${\xi}_{si}=0$ for $i\ne j$. If a displacement condition is applied sequentially to each node, then the net force at each node j can be obtained by superposition. For a structure with N degrees of freedom, the damping matrix

**C**is defined as a symmetric array of $N\times N$ constants ${c}_{ij}$. Damping is an influence within or upon an oscillatory system that has the effect of reducing, restricting, or preventing its oscillations. In physical systems, damping is the capacity of the system to dissipate the energy stored in the oscillation within itself without damage. In this analysis, the damping force ${q}_{Di}$ for the structural mode coordinate $\xi $ is assumed to be a linear combination of the generalized coordinate velocities ${\dot{\xi}}_{i}$, $i=1,2,\dots ,N$:

**C**is proportional to the system’s mass and also the system’s stiffness:

## 3. Reference Model

- Structural and environmental parameters
- Undamped motion of the structure
- Damped frequencies’ definition
- Response to environmental loads
- Maximum load pile applied
- Time domain solution
- Transfer function definition
- Spectral density function of the generic load
- Response spectrum
- $3\sigma $ approach

**M**of the 5DOF model expressed in kg developed in MATLAB:

**K**has been determined along the direction x. Once having applied the definition of stiffness to all the system, it is possible to define (using MATLAB) the matrix

**K**in N/m as:

**X**is also identified for each natural structural frequency ${\omega}_{n}$. Each modal vector is identified by ${\widehat{\mathbf{\xi}}}_{\mathit{n}}={[{\widehat{\xi}}_{1n},{\widehat{\xi}}_{2n},\dots ,{\widehat{\xi}}_{Nn}]}^{T}={[1,{\widehat{\xi}}_{2n},\dots ,{\widehat{\xi}}_{Nn}]}^{T}$. Those vectors are normalized with respect to the mass matrix to form the new modal vectors ${\mathit{x}}_{\mathit{n}}={\widehat{\xi}}_{\mathit{n}}/{e}_{n}$, where ${e}_{n}$ is a set of positive and real constants computed from ${\widehat{\mathbf{\xi}}}_{\mathit{n}}^{\mathit{T}}\mathit{M}{\widehat{\mathbf{\xi}}}_{\mathit{n}}={e}_{n}^{2}$. It follows that the modal shape matrix

**X**of the 5DOF model is defined as the assembly of the normalized modal vectors ${\mathit{x}}_{\mathit{n}}$:

#### Spectral Density Function of the Generalized Force Component in Modal Coordinates

## 4. Jacket Supporting the Wind Turbine

**M**of the 6DOF model in kg is:

**K**has been determined along the x direction using the proper definition. Thus, matrix

**K**in N/m is:

**X**of the 6DOF model is defined as the assembly of the normalized modal vectors ${\mathit{x}}_{\mathit{n}}$ as:

**X**of the 6DOF model. It follows that the equations are the same as the 5DOF configuration, but the functions of the five wave transfer functions are slightly different.

## 5. Improved Retrofitting Models

#### 5.1. Crown Pile Configuration

**M**, where the mass ${m}_{6}=$ 19,202 kg. The six undamped natural structural frequencies for free vibration in rad/s are: ${\omega}_{1}=3.3879$, ${\omega}_{2}=8.5391$, ${\omega}_{3}=35.7613$, ${\omega}_{4}=70.5186$, ${\omega}_{5}=89.1430$, ${\omega}_{6}=111.4539$. For the nth frequency ${\omega}_{n}$ exists a modal vector ${\xi}^{n}$, which can be computed solving the linear eigenvalue problem (see [30] p. 226).

#### 5.2. Mooring Line Configuration

**M**in kg:

**K**has been determined over the x direction. Once having applied the definition of stiffness to all the system, it is possible to define the matrix

**K**in N/m as:

#### 5.3. The 2-MW Wind Turbine Configuration

**M**of the 6DOF model in kg is:

**K**has been determined over the x direction. Once having applied the definition of stiffness to all the system, it is possible to define the matrix

**K**in N/m as:

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 6.**Spectral density functions of the generalized force component in modal coordinates for the 5DOF model.

**Figure 12.**Spectral density of the generic environmental load in modal coordinates of the 6DOF model.

**Figure 13.**Response spectra for the displacement in physical coordinates of the offshore wind turbine plot.

**Figure 17.**Response spectra for the displacements in physical coordinates of the offshore wind turbine crown pile configuration.

**Figure 21.**Response spectra for the displacements in physical coordinates of the offshore wind turbine mooring line configuration.

**Figure 24.**The 2-MW configuration spectral density of the generic environmental load in modal coordinates.

**Figure 25.**Response spectra for the displacements in physical coordinates of the 2-MW offshore wind turbine configuration.

MATLAB | FE | Ratio | |
---|---|---|---|

${\xi}_{1}$ | 0.0281 | 0.0384 | 0.73 |

${\xi}_{2}$ | 0.0272 | 0.0348 | 0.78 |

${\xi}_{3}$ | 0.0236 | 0.0278 | 0.85 |

${\xi}_{4}$ | 0.0177 | 0.0187 | 0.95 |

${\xi}_{5}$ | 0.0107 | 0.0101 | 1.06 |

MATLAB | FE | Ratio | |
---|---|---|---|

${\omega}_{1}$ | 3.38 | 3.39 | ≈1 |

${\omega}_{2}$ | 8.53 | 8.54 | ≈1 |

${\omega}_{3}$ | 36.02 | 36.00 | ≈1 |

${\omega}_{4}$ | 71.90 | 71.88 | ≈1 |

${\omega}_{5}$ | 90.62 | 90.60 | ≈1 |

${\omega}_{6}$ | 111.50 | 111.53 | ≈1 |

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**MDPI and ACS Style**

Alessi, L.; Correia, J.A.F.O.; Fantuzzi, N.
Initial Design Phase and Tender Designs of a Jacket Structure Converted into a Retrofitted Offshore Wind Turbine. *Energies* **2019**, *12*, 659.
https://doi.org/10.3390/en12040659

**AMA Style**

Alessi L, Correia JAFO, Fantuzzi N.
Initial Design Phase and Tender Designs of a Jacket Structure Converted into a Retrofitted Offshore Wind Turbine. *Energies*. 2019; 12(4):659.
https://doi.org/10.3390/en12040659

**Chicago/Turabian Style**

Alessi, Lorenzo, José A. F. O. Correia, and Nicholas Fantuzzi.
2019. "Initial Design Phase and Tender Designs of a Jacket Structure Converted into a Retrofitted Offshore Wind Turbine" *Energies* 12, no. 4: 659.
https://doi.org/10.3390/en12040659