# Coupled Dynamic Analysis of a Bottom-Fixed Elastic Platform with Wave Energy Converters in Random Waves

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Analysis Methods

#### 2.1. Equation of Motion of a Constrained Rigid Body System

#### 2.1.1. Equation of Motion

#### 2.1.2. External Forces Acting on a Floating Body

#### 2.2. Equation of Motion of a Bottom-Fixed Structure

#### 2.3. Irregular Wave Analysis

## 3. Numerical Simulations

#### 3.1. Verification of Numerical Results

#### 3.1.1. Heave RAO of a Spherical Float

^{3}. The float was modeled with 7200 quadrilateral elements. The float has one degree of freedom, with the motion constrained except for vertical (heave) motion. Frequency-domain hydrodynamic parameters (added mass, radiation-damping coefficient, and diffraction force) were obtained using WAMIT. Among the external forces acting on the float, the diffraction and radiation forces were applied as linear forces.

#### 3.1.2. Pitch RAO of a Spherical Float Constrained to a Hinge Point

^{3}. The float was modeled with 7200 quadrilateral elements. The float was constrained in motion except for the rotational motion in the y-axis by a hinge joint. A linear rotational damper was applied to the hinge point to describe the PTO energy extraction (damping) system. The moment acting on the hinge point (${M}_{PTO}$) can be expressed as

^{6}N·m/(rad/s) was applied for comparison. Frequency-domain hydrodynamic parameters were also obtained using WAMIT. The diffraction and radiation forces were applied as linear forces among the external forces.

#### 3.2. Simulation Results and Analysis of the Proposed System

^{3}. Each float was fixed to a massless rigid rod, the other end of which was hinge-connected to the foundation structure. This connection allowed rotational movement with respect to the joint at the foundation structure as the center of rotation. The rotational damping proportional to the rotational velocity was applied to the hinge point to describe the PTO system, as shown in Equation (23). This PTO moment resists the rotational motion. The extraction coefficient of 100 kN·m/(rad/s) was applied through a parametric study for a regular wave [19]. Figure 8 and Figure 9 compare their pitch RAOs, showing the body-nonlinear results. The pitch RAOs of the present body were larger than those of the sphere when the wave frequency was larger than 0.5 rad/s, i.e., in practical sea states, and the present body was superior to the sphere. In this regard, the present body shape was chosen for the subsequent section.

^{3}.

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Conflicts of Interest

## References

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**Figure 4.**Comparison of heave RAOs of a spherical float (Lin: Linear analysis; B-N: Body-nonlinear analysis) [41].

**Figure 7.**Description of a hemispherical floating body with a vertical circular cylinder on top: (

**a**) Numerical model; (

**b**) Panel distribution.

**Figure 8.**Description of single and dual floating bodies: (

**a**) Single sphere: front; (

**b**) Single present body: front; (

**c**) Top view of single sphere/present body: front; (

**d**) Single sphere: rear; (

**e**) Single present body: rear; (

**f**) Top view of a single sphere/present body: rear; (

**g**) Side view of a dual present body; (

**h**) Top view of a dual present body.

**Figure 12.**Description of the bottom-fixed platform with 2 WECs: (

**a**) 3D view; (

**b**) Top view; (

**c**) Side view.

**Figure 13.**Time histories of dynamic responses of Leg 1: (

**a**) x-axis displacement at the top; (

**b**) Stress at the bottom.

**Figure 15.**Time histories of the constraint forces acting on the hinge points: (

**a**) x-axis; (

**b**) z-axis.

**Table 1.**Five-year average winter wave data in Buan measured by the KMA [44].

Year | Average Significant Wave Height [m] | Average Peak Period [s] | Overall Availability [%] |
---|---|---|---|

2017 | 1.387 | 5.65 | 99.41 |

2018 | 1.211 | 5.28 | 98.75 |

2019 | 1.150 | 5.24 | 93.96 |

2020 | 1.182 | 5.33 | 89.93 |

2021 | 1.446 | 5.80 | 99.34 |

Component | Description | Value | Unit |
---|---|---|---|

Platform | $\mathrm{Deck}\mathrm{size}(\mathrm{x}\times \mathrm{y}\times \mathrm{z})$ | $3\times 2.4\times 2$ | m |

Leg height | 52.6 | m | |

Diameter of leg member | 0.45 | m | |

Thickness of leg member | 12 | mm | |

Inertia coefficient | 2.0 | - | |

Drag coefficient | 1.0 | - | |

Deck weight | 980 | kN | |

Structural weight per unit volume | 78.6 | kN/m^{3} | |

Modulus of elasticity | $2.1\times $ 10^{8} | kN/m^{2} | |

Modulus of rigidity | $8.33\times $ 10^{7} | kN/m^{2} | |

Number of elements | 124 | - | |

Number of nodal points | 122 | - | |

Float | Diameter | 2.0 | m |

Draft | 1.0 | m | |

Mass | 2147 | kg | |

Damping coefficient of a rotational damper | 100 | kN·m/(rad/s) | |

Number of elements | 8520 | - |

1st Mode | 2nd Mode | 3rd Mode | 4th Mode | 5th Mode | |
---|---|---|---|---|---|

Natural frequency [rad/s] | 0.4026 | 0.4059 | 3.1132 | 6.3066 | 6.3739 |

Natural period [s] | 15.60 | 15.48 | 2.02 | 1.00 | 0.99 |

Parameter | Case | Maximum | Minimum | RMS | Standard Deviation |
---|---|---|---|---|---|

x-axis displacement at the top [m] | 2 floats | 0.635 | −0.546 | 0.122 | 0.121 |

No floats | 0.059 | −0.060 | 0.014 | 0.014 | |

Stress at the bottom [MPa] | 2 floats—Leg 1,2 | −40.31 | −102.11 | 50.99 | 8.02 |

2 floats—Leg 3,4 | −39.54 | −110.74 | 51.17 | 8.23 | |

No floats—Leg 1,2 | −40.61 | −46.39 | 41.85 | 0.78 | |

No floats—Leg 3,4 | −40.53 | −46.23 | 41.82 | 0.73 |

Parameter | Float | Maximum | Minimum | RMS | Standard Deviation |
---|---|---|---|---|---|

x-axis constraint force [kN] | Float 1 | 45.05 | −9.97 | 2.90 | 2.78 |

Float 2 | 4.33 | −71.79 | 3.49 | 3.29 | |

z-axis constraint force [kN] | Float 1 | 22.48 | −47.53 | 6.70 | 6.66 |

Float 2 | 21.18 | −28.50 | 6.14 | 6.07 |

Case | Float | Maximum | Minimum | RMS | Standard Deviation |
---|---|---|---|---|---|

Coupled analysis | Float 1 | 0.528 | −0.525 | 0.126 | 0.126 |

Float 2 | 0.673 | −0.574 | 0.134 | 0.134 | |

Fixed platform analysis | Float 1 | 0.460 | −0.515 | 0.106 | 0.106 |

Float 2 | 0.679 | −0.365 | 0.118 | 0.118 |

Case | Float | Average | Maximum | RMS | Standard Deviation |
---|---|---|---|---|---|

Coupled analysis | Float 1 | 1.847 | 58.62 | 3.34 | 2.78 |

Float 2 | 2.206 | 74.98 | 4.14 | 3.50 | |

Fixed platform analysis | Float 1 | 1.862 | 92.32 | 3.81 | 3.32 |

Float 2 | 2.169 | 74.53 | 4.10 | 3.48 |

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**MDPI and ACS Style**

Heo, S.; Koo, W.; Kim, M.-H.
Coupled Dynamic Analysis of a Bottom-Fixed Elastic Platform with Wave Energy Converters in Random Waves. *Appl. Sci.* **2022**, *12*, 7915.
https://doi.org/10.3390/app12157915

**AMA Style**

Heo S, Koo W, Kim M-H.
Coupled Dynamic Analysis of a Bottom-Fixed Elastic Platform with Wave Energy Converters in Random Waves. *Applied Sciences*. 2022; 12(15):7915.
https://doi.org/10.3390/app12157915

**Chicago/Turabian Style**

Heo, Sanghwan, Weoncheol Koo, and Moo-Hyun Kim.
2022. "Coupled Dynamic Analysis of a Bottom-Fixed Elastic Platform with Wave Energy Converters in Random Waves" *Applied Sciences* 12, no. 15: 7915.
https://doi.org/10.3390/app12157915