# The Effect of Mooring Line Parameters in Inducing Parametric Resonance on the Spar-Buoy Oscillating Water Column Wave Energy Converter

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## Abstract

**:**

## 1. Introduction

## 2. Device and Mathematical Model

#### 2.1. Equation of Motion

#### 2.2. Nonlinear Froude–Krylov Force Model

#### 2.3. Viscous Drag Force Model

#### 2.4. Mooring System Model

## 3. Results

#### 3.1. Configuration ${m}_{3}$

#### 3.2. Analysis of Mooring Configurations

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Cross-section view of the floating oscillating water column geometry with full scale dimensions.

**Figure 2.**Inertial frame $\left(\right)$, with the origin at still water level (SWL), and body-fixed (non-inertial) frame $\left(\right)$, after an arbitrary displacement. At rest the two frames coincide. The velocities are according to the inertial frame $\left(\right)$ and the body-fixed frame $\left(\right)$.

**Figure 3.**Mooring system layout with three lines 120${}^{\circ}$ apart. Each line is divided in three segments of length ${L}_{1}$, ${L}_{2}$, and ${L}_{3}$. ${F}_{C}$ is the net clump-weight force and ${F}_{J}$ is the net jumper force (negative in the figure). The quasi-static model solves for the tension at the vessel (${T}_{v}$), the tension at the anchor (${T}_{0}$), and the angles of the three lines (${\mathsf{\Phi}}_{1}$, ${\mathsf{\Phi}}_{1}$, and ${\mathsf{\Phi}}_{3}$).

**Figure 4.**Map of jumper mass (${M}_{J}$) and clump-weight mass (${M}_{C}$) to the resulting draft of the top cylindrical section of the floater. Five different configurations are studied (${m}_{1}$ to ${m}_{5}$), as tabulated in Table 4, all resulting in a draft of 7.91 m.

**Figure 5.**Mean displacement in surge $\overline{x}$ due to drift effects for configuration ${m}_{3}$ as a function of ${T}_{w}$ and ${H}_{w}$. The dashed and dash-dotted red lines correspond to ${T}_{w}=\frac{1}{2}{T}_{n,5}$ and ${T}_{w}={T}_{n,5}$, respectively.

**Figure 6.**Amplitude of the response for configuration ${m}_{3}$ as a function of ${T}_{w}$ and ${H}_{w}$. The dashed and dash-dotted red lines correspond to ${T}_{w}=\frac{1}{2}{T}_{n,5}$ and ${T}_{w}={T}_{n,5}$, respectively. The green dotted line refers to ${T}_{w}=\frac{1}{2}{T}_{n,6}$.

**Figure 7.**Response amplitude operator for configuration ${m}_{3}$ as a function of ${T}_{w}$, for different ${H}_{w}$, as shown in the color bar. The dashed and dash-dotted red lines correspond to ${T}_{w}=\frac{1}{2}{T}_{n,5}$ and ${T}_{w}={T}_{n,5}$, respectively. The green dotted line refers to ${T}_{w}=\frac{1}{2}{T}_{n,6}$.

**Figure 8.**Peak tension of the front mooring at the fairlead as a function of ${T}_{w}$, for different ${H}_{w}$. The dashed and dash-dotted red lines correspond to ${T}_{w}=\frac{1}{2}{T}_{n,5}$ and ${T}_{w}={T}_{n,5}$, respectively.

**Figure 9.**Mean displacement in surge $\overline{x}$ due to drift effects as a function of ${T}_{w}$ and ${H}_{w}$, for different mooring configurations. The dashed and dash-dotted red lines correspond to ${T}_{w}=\frac{1}{2}{T}_{n,5}$ and ${T}_{w}={T}_{n,5}$, respectively.

**Figure 10.**Peak tension of the front mooring at the fairlead as a function of ${T}_{w}$, for different ${H}_{w}$ and for different mooring configurations, as tabulated in Table 3. The black line represents the pre-tension. The dashed and dash-dotted red lines correspond to ${T}_{w}=\frac{1}{2}{T}_{n,5}$ and ${T}_{w}={T}_{n,5}$, respectively.

**Figure 11.**Amplitude of the heave response for different mooring configurations, as tabulated in Table 3. For a fair comparison, the same color bar applies to all plots. The dashed and dash-dotted red lines correspond to ${T}_{w}=\frac{1}{2}{T}_{n,5}$ and ${T}_{w}={T}_{n,5}$, respectively.

**Figure 12.**Amplitude of the roll response for different mooring configurations, as tabulated in Table 3. For a fair comparison, the same color bar applies to all plots. The dashed and dash-dotted red lines correspond to ${T}_{w}=\frac{1}{2}{T}_{n,5}$ and ${T}_{w}={T}_{n,5}$, respectively.

**Figure 13.**Amplitude of the yaw response as a function of ${T}_{w}$ and ${H}_{w}$, for different mooring configurations, as tabulated in Table 3. For a fair comparison, the same color bar applies to all plots. The dashed and dash-dotted red lines correspond to ${T}_{w}=\frac{1}{2}{T}_{n,5}$ and ${T}_{w}={T}_{n,5}$, respectively. The green dotted line refers to ${T}_{w}=\frac{1}{2}{T}_{n,6}$.

**Table 1.**Physical properties of the spar-buoy oscillating water column (OWC) device shown in Figure 1, in full-scale.

Parameter | Symbol | Value | Units |
---|---|---|---|

Water depth | h | 80.00 | [m] |

Diameter of the top cylinder | ${d}_{c}$ | 16.00 | [m] |

Draft of top cylinder | ${L}_{c}$ | 7.91 | [m] |

Total submerged length | ${L}_{t}$ | 50.91 | [m] |

Vertical coordinate of Centre of Gravity | ${z}_{\mathrm{CoG}}$ | −31.96 | [m] |

Vertical coordinate of Centre of Buoyancy | ${z}_{\mathrm{CoB}}$ | −22.14 | [m] |

Mass | M | 2.86$\xb7{10}^{6}$ | [kg] |

Perpendicular moment of inertia | ${I}_{x}={I}_{y}$ | 1.53$\xb7{10}^{9}$ | [kg m${}^{2}$] |

Axial moment of inertia | ${I}_{z}$ | 1.12$\xb7{10}^{8}$ | [kg m${}^{2}$] |

Orifice diameter | ${d}_{o}$ | 0.864 | [m] |

${\mathit{C}}_{\mathit{d},\mathit{c}}$ | ${\mathit{C}}_{\mathit{d},3}$ | ${\mathit{C}}_{\mathit{d},37}$ | ${\mathit{C}}_{\mathit{d},6}$ |
---|---|---|---|

1.75 | 0.175 | 0.2 | 0.125 |

**Table 3.**Parameters of the full-scale mooring system, based on the experimental tests in [31].

Parameter | Symbol | Value | Units |
---|---|---|---|

Line diameter | ${d}_{l}$ | 32 | [mm] |

Net line density | ${\rho}_{L}^{*}$ | 3.55 | [kgm${}^{-3}$] |

Jumper mass | See Table 4 | ||

Jumper density | ${\rho}_{J}$ | 123.00 | [kgm${}^{-3}$] |

Clump-weight mass | See Table 4 | ||

Clump-weight density | ${\rho}_{C}$ | 8097.50 | [kgm${}^{-3}$] |

Length from anchor to jumper | ${L}_{1}$ | 143.28 | [m] |

Length from jumper to clump-weight | ${L}_{2}$ | 37.01 | [m] |

Length from clump-weight to buoy | ${L}_{3}$ | 50.40 | [m] |

Radius at the anchor | ${R}_{a}$ | 211.2 | [m] |

Depth at the anchor | h | 80 | [m] |

Radius of attachment at the buoy | ${R}_{b}$ | −9.28 | [m] |

Depth of attachment at the buoy | ${h}_{b}$ | −2.58 | [m] |

**Table 4.**Jumper and clump weight masses for five different mooring configurations that ensure the same draft of the top cylinder of the floater ${L}_{c}$ of 7.91 m, as shown in Figure 4. The consequent pre-tension is also reported.

Configuration | Jumper Mass [kg] | Clump-Weight Mass [kg] | Pre-Tension [N] |
---|---|---|---|

${m}_{1}$ | 2015.2 | 27,737 | 2.34$\xb7{10}^{5}$ |

${m}_{2}$ | 3022.8 | 32,050 | 2.71$\xb7{10}^{5}$ |

${m}_{3}$ | 4030.5 | 36,044 | 3.14$\xb7{10}^{5}$ |

${m}_{4}$ | 5038.1 | 39,460 | 3.63$\xb7{10}^{5}$ |

${m}_{5}$ | 6045.7 | 42,742 | 4.18$\xb7{10}^{5}$ |

**Table 5.**Natural periods from free decay tests for different mooring configurations, as in Table 4.

${\mathit{T}}_{\mathit{n}}\left[\mathit{s}\right]$ | ||||||
---|---|---|---|---|---|---|

${\mathit{m}}_{\mathbf{1}}$ | ${\mathit{m}}_{\mathbf{2}}$ | ${\mathit{m}}_{\mathbf{3}}$ | ${\mathit{m}}_{\mathbf{4}}$ | ${\mathit{m}}_{\mathbf{5}}$ | ||

Surge & Sway | ${T}_{n,1}$ | 187 | 167 | 150 | 132 | 117 |

Heave | ${T}_{n,3}$ | 10.2 | 10.1 | 10.1 | 10.1 | 10.1 |

Roll & Pitch | ${T}_{n,5}$ | 19.3 | 19.2 | 19.1 | 19.1 | 18.9 |

Yaw | ${T}_{n,6}$ | 30.6 | 26.8 | 24.0 | 21.8 | 20.0 |

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## Share and Cite

**MDPI and ACS Style**

Giorgi, G.; Gomes, R.P.F.; Bracco, G.; Mattiazzo, G.
The Effect of Mooring Line Parameters in Inducing Parametric Resonance on the Spar-Buoy Oscillating Water Column Wave Energy Converter. *J. Mar. Sci. Eng.* **2020**, *8*, 29.
https://doi.org/10.3390/jmse8010029

**AMA Style**

Giorgi G, Gomes RPF, Bracco G, Mattiazzo G.
The Effect of Mooring Line Parameters in Inducing Parametric Resonance on the Spar-Buoy Oscillating Water Column Wave Energy Converter. *Journal of Marine Science and Engineering*. 2020; 8(1):29.
https://doi.org/10.3390/jmse8010029

**Chicago/Turabian Style**

Giorgi, Giuseppe, Rui P. F. Gomes, Giovanni Bracco, and Giuliana Mattiazzo.
2020. "The Effect of Mooring Line Parameters in Inducing Parametric Resonance on the Spar-Buoy Oscillating Water Column Wave Energy Converter" *Journal of Marine Science and Engineering* 8, no. 1: 29.
https://doi.org/10.3390/jmse8010029