# Modelling of Parametric Resonance for Heaving Buoys with Position-Varying Waterplane Area

^{*}

## Abstract

**:**

## 1. Introduction

#### 1.1. Parametric Resonance in Heaving Wave Energy Converters

#### 1.2. Modelling and Analysis of Parametric Resonance

#### 1.3. Outline and Objectives of Paper

## 2. Mathematical Model

#### 2.1. Conventional Linear Hydrodynamic Model

#### 2.1.1. The Hydrostatic Restoring Force

#### 2.1.2. The Wave Radiation Force

#### 2.1.3. The Wave Excitation Force

#### 2.1.4. The Linear Hydrodynamic Model

#### 2.2. Analytical Model for Parametric Resonance

#### 2.2.1. Nonlinear Hydrostatic Restoring Force

#### 2.2.2. Parametric Excitation

#### 2.2.3. Mathieu Type Equation

## 3. Test Case

#### 3.1. Test Case Geometry

#### 3.2. The Analytical Model

#### 3.2.1. Hydrostatic Restoring Force

#### 3.2.2. Parametric Excitation

#### 3.2.3. Wave Radiation Force

#### 3.3. The Nonlinear Hydrostatic Restoring Force Model

#### 3.4. The Nonlinear Froude–Krylov Force Model

## 4. Results

#### 4.1. Comparison against Nonlinear Hydrostatic Restoring Force Model

#### 4.2. Code-to-Code Verification against a NLFK Force Model

#### 4.3. Two-Parameter Analysis

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 4.**The nonlinear hydrostatic force ${F}_{s}$ as a function of the buoy displacement z for the test case geometry.

**Figure 5.**The mesh for NEMOH, which considers the buoy fixed at one position and only discretises the surface area of the buoy beneath the still water level ($z\le 0$) into panels.

**Figure 6.**The wave excitation force (a) amplitude ${f}_{e}(z,\omega )$ and (b) phase ${\theta}_{e}(z,\omega )$, for the test case geometry.

**Figure 7.**The excitation force polynomial coefficients for the amplitude, (

**a**) ${a}_{0}\left(\omega \right)$, (

**b**) ${a}_{1}\left(\omega \right)$, (

**c**) ${a}_{2}\left(\omega \right)$, and phase (

**d**) ${b}_{0}\left(\omega \right)$, (

**e**) ${b}_{1}\left(\omega \right)$, (

**f**) ${b}_{2}\left(\omega \right)$, as function of the wave frequency.

**Figure 8.**The excitation force (

**a**) amplitude ${f}_{e}(z,\omega )$ and (

**b**) phase ${\theta}_{e}(z,\omega )$ as a function of the buoy position z for a wave frequency of $\omega =1$ [rad/s], and the excitation force (

**c**) amplitude ${f}_{e}(z,\omega )$ and (

**d**) phase ${\theta}_{e}(z,\omega )$ as a function of the buoy position z for a wave frequency of $\omega =2$ [rad/s].

**Figure 9.**The hydrodynamic coefficients for the (

**a**) added mass ${m}_{a}\left(\omega \right)$ and (

**b**) damping ${c}_{a}\left(\omega \right)$ coefficients.

**Figure 10.**Steady-state oscillations for various integration step sizes (wave parameters: $H=2.2$ [m], $\omega =1.87$ [m].

**Figure 11.**The buoy geometry visualised in the NLFK4ALL toolbox, which analytically recalculates the Froude–Krylov force at each time step from the evolving submerged surface area of the buoy based on the instantaneous buoy position and wave elevation.

**Figure 15.**(

**a**) The transient and (

**b**) the steady-state oscillations due to the parametric resonance. Wave amplitude $H=2.2$ [m] and frequency $\omega =1.87$ [rad/s].

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

Opening angle of the conical part | $\alpha $ | 11.31 | ${}^{\circ}$ |

Radius of the conical part at equilibrium state at still water level | ${r}_{0}$ | 3 | $\mathrm{m}$ |

Half-length of the conical part | ${h}_{0}$ | 5 | $\mathrm{m}$ |

Radius of the cylindrical extension | ${r}_{1}$ | 2 | $\mathrm{m}$ |

Length of the cylindrical extension | ${h}_{1}$ | 15 | $\mathrm{m}$ |

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## Share and Cite

**MDPI and ACS Style**

Lelkes, J.; Davidson, J.; Kalmár-Nagy, T.
Modelling of Parametric Resonance for Heaving Buoys with Position-Varying Waterplane Area. *J. Mar. Sci. Eng.* **2021**, *9*, 1162.
https://doi.org/10.3390/jmse9111162

**AMA Style**

Lelkes J, Davidson J, Kalmár-Nagy T.
Modelling of Parametric Resonance for Heaving Buoys with Position-Varying Waterplane Area. *Journal of Marine Science and Engineering*. 2021; 9(11):1162.
https://doi.org/10.3390/jmse9111162

**Chicago/Turabian Style**

Lelkes, János, Josh Davidson, and Tamás Kalmár-Nagy.
2021. "Modelling of Parametric Resonance for Heaving Buoys with Position-Varying Waterplane Area" *Journal of Marine Science and Engineering* 9, no. 11: 1162.
https://doi.org/10.3390/jmse9111162