# Insight into Hydrodynamic Damping of a Segmented Barge Using Numerical Free-Decay Tests

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## Abstract

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## 1. Introduction

## 2. Numerical Methods

#### 2.1. Modal Solver for the Structure

#### 2.2. Rayleigh Damping

#### 2.3. Damping Determination

#### 2.4. Flow Solver

## 3. Modal-Based Coupling

Algorithm 1 Reduced order modal approach to weakly coupled FSI. |

1. Calculate dry vibration natural frequencies ${\omega}_{1},{\omega}_{2},\cdots ,{\omega}_{n}$ and their mass-normalised mode shapes 2. Setup simulation initial conditions (e.g., initial structure displacement and fluid boundary conditions) 3. Build RBF connections fluid face/structural node 4. Simulation loop, for each time-step $\Delta t$: (a) Transfer fluid loads to structural nodes (b) Solve set of Equation (10) for all mode shapes (c) Obtain new shape of the structure via Equation (6) (d) Perform RBF mesh interpolation from the old to the new structure shape (e) Solve flow for time $t+\Delta t$, influenced by the deformation of the structure (f) If residuals of fluid solver are too high, go to step 4.(a) |

## 4. Vertical Vibrations of a Flexible Barge

#### 4.1. Barge Characteristics

#### 4.2. Extraction of Mode Shapes

#### 4.3. Hydroelastic Simulations

#### 4.4. Discussion

## 5. Conclusions

## References

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**Finite element model of the barge (red— beam elements, blue—plate elements, black—mass elements, white—rigid connections).

**Figure 4.**Modal acceleration of first (

**top**) and second (

**bottom**) vibration modes for the segmented model (blue) and monohull (red), compared to the decay based on experimental modal damping ratio (dashed line).

**Figure 5.**Top view of free surface elevation contours for the first mode of vertical vibration (segmented hull—top image, monohull—bottom image).

**Figure 6.**Top view of free surface elevation contours for the second mode of vertical vibration (segmented hull—top image, monohull—bottom image).

**Figure 7.**Profile view of free-surface elevation along the barge’s length for the first mode of vertical vibration.

**Figure 8.**Profile view of free surface elevation along the barge’s length for the second mode of vertical vibration.

**Figure 9.**Section cut at the centreline, showing streamlines and the dynamic pressure field for the first mode of vertical vibration of the segmented barge.

**Table 1.**Wet natural frequencies and damping ratios for the first two vibration modes, obtained using the modal analysis and experimental data.

Mode | Natural Frequency ${\mathit{\omega}}_{\mathit{i}}$ | Average Damping Ratio ${\mathit{\zeta}}_{\mathit{i}}$ |
---|---|---|

1 | 1.24 Hz | 7.3% |

2 | 2.40 Hz | 6.2% |

**Table 2.**Dry natural frequencies obtained analytically and numerically, in Hz; relative differences compared with the analytical solution.

Mode | Timoshenko | FEM, Mass Above Deck | Relative Difference | FEM, Figure 2 | Relative Difference |
---|---|---|---|---|---|

1 | 1.367 Hz | 1.389 Hz | +1.6% | 1.295 Hz | −5.3% |

2 | 3.769 Hz | 3.823 Hz | +1.4% | 3.346 Hz | −11.2% |

3 | 7.389 Hz | 7.487 Hz | +1.3% | 5.900 Hz | −20.1% |

**Table 3.**Wet natural frequencies for the first two vibration modes, in Hz, obtained with different methods and models.

Mode | Experiment | CFD, Segmented | Relative Difference | CFD, Monohull | Relative Difference | FEM, Monohull | Relative Difference |
---|---|---|---|---|---|---|---|

1 | 1.24 Hz | 1.26 Hz | +1.6% | 1.12 Hz | −9.7% | 1.02 Hz | −17.7% |

2 | 2.40 Hz | 2.54 Hz | +5.8% | 2.33 Hz | −2.9% | 2.48 Hz | +3.3% |

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**MDPI and ACS Style**

Bašić, J.; Degiuli, N.; Malenica, Š.
Insight into Hydrodynamic Damping of a Segmented Barge Using Numerical Free-Decay Tests. *J. Mar. Sci. Eng.* **2023**, *11*, 581.
https://doi.org/10.3390/jmse11030581

**AMA Style**

Bašić J, Degiuli N, Malenica Š.
Insight into Hydrodynamic Damping of a Segmented Barge Using Numerical Free-Decay Tests. *Journal of Marine Science and Engineering*. 2023; 11(3):581.
https://doi.org/10.3390/jmse11030581

**Chicago/Turabian Style**

Bašić, Josip, Nastia Degiuli, and Šime Malenica.
2023. "Insight into Hydrodynamic Damping of a Segmented Barge Using Numerical Free-Decay Tests" *Journal of Marine Science and Engineering* 11, no. 3: 581.
https://doi.org/10.3390/jmse11030581