# A Coupled Hydrodynamic–Structural Model for Flexible Interconnected Multiple Floating Bodies

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Frequency-Domain Model

_{b}, the motions of the action point between the flexible substructures and barges are defined as master DOF (degrees of freedom), and the nodal motions on the substructures are deemed as slave DOF, which are designated as X

_{m}and X

_{s}, respectively. The establishment of this dynamic model can help to condense the flexible substructures to the barges through master nodes in the form of a condensed mass and stiffness matrix using static condensation; therefore, the wave-induced dynamics of the interconnected system can be conveniently obtained.

**F**

_{m}(t) at the connections, the motion equation for the substructure can be written in the following form [52,53]:

**I**denotes the identity matrix and

**T**

_{G}is the transpose matrix.

**M**, ${A}_{H}(\omega )$ and ${B}_{H}(\omega )$ are the (6N) × (6N) structural mass, hydrodynamic added mass, and added damping matrices, respectively; ${C}_{bb}$ is the hydrostatic stiffness matrix assembled by each 6 × 6 sub-hydrostatic stiffness of individual structure along the diagonal; ${\widehat{F}}_{}^{E}(i\omega )$ is the 6N × 1 wave excitation force at frequency ω.

#### 2.2. Coupled Time-Domain Model

**K**(t) within the time range from 0 to t is calculated at each time step, therefore the integrating process is time-consuming and easy to accumulate and amplify errors. Since the convolution term is a linear time-invariant system, one can replace it with either a state–space model or a transfer function to improve its accuracy and efficiency [55]. A single-input–single-output (SISO) state–space model can be easily established to be equivalent to each convolution term [42]. Once the SISO state–space models for all the impulse response functions are obtained, the multi-input–multi-output (MIMO) state–space model replaces the matrix of the convolution terms [43]:

**z**(t) represents the state vector;

**P**,

**Q**, and

**S**are the parametric matrices of the MIMO state–space model, which have the following forms [4]:

^{2}is applied [56]:

_{l}for the input being the Dirac delta function, and ${\overline{\mathrm{K}}}_{i,j}$ is the mean value of ${\hat{\mathrm{K}}}_{i,j}(t)$.

## 3. Discussion of the Development of CPHSTDM and Parametric Study on Connector Parameters

#### 3.1. Particulars of the Analyzed Model

_{m}, while the motion at the joint is defined as X

_{s}.

#### 3.2. Frequency-Domain Simulations of the Interconnected Three-Module System

^{10}N and 10 kg/m, respectively, and the kinematic characteristics of the system with different bending stiffness EI are to be discussed. There is no restoring force in the surge direction considered in CPHSTDM by now, whether from mooring force or hydrostatic force. To remain consistent with the following time-domain analysis, axial stiffness EA is set to be a constant. As for the mass of the connector, it should be noted that the mass of the connected component, consisting of four Euler–Bernoulli beams, accounts for a very small proportion of the total mass of the system, therefore the response to the change can also be ignored. Moreover, the stiffness of the connection joint C is selected as the same value as bending stiffness.

^{9}Nm

^{2}. With the increase in the bending stiffness, the differences between different modules decreased, because the overall system is more rigid due to the strengthening of the constraint effect. In addition, it should be noted that both the heave and pitch motions of Module 2 showed an obvious trend of weakening around 0.5 rad/s, while this phenomenon did not occur in Module 1 and Module 3. This is because it was in the center of the system and received the connection constraints from both the front and back sides. However, when the bending stiffness EI changes from 1.0 × 10

^{6}Nm

^{2}to 1.0 × 10

^{7}Nm

^{2}, it has almost no effect on the heave motion of the floating body, and only has a slight change in the pitch motion at some frequencies.

#### 3.3. Discussion of the Development of CPHSTDM and Parametric Study on Connector Parameters

^{2}defined in Equation (31), where the value of 0.97 for R

^{2}is proved by Duarte et al. [52] to be sufficient to satisfy the time-domain accuracy in the single floating system.

^{2}of the SSMs for the coupled terms between different modules in the selected motions are summarized in Table 4. The R

^{2}values of the system shown in Figure 11 and Table 4 are greater than 0.99, surpassing the suggested value of 0.97 by Duarte et al. [56].

^{9}Nm

^{2}, and a resonance phenomenon occurs, making it impossible to obtain stable time-domain results. This phenomenon may have something to do with the hydrostatic restoring force in pitch–pitch mode, with the pitch–pitch stiffness being 4.1732 × 10

^{10}Nm/rad. Therefore, for the time-domain comparison, the value of EI equal to 1.0 × 10

^{9}Nm

^{2}, is not considered.

^{−3}. However, it is crucial to note that the motion amplitude of each module is not necessarily the same. On the other hand, it can also be seen that the heave response amplitude of the floating system under various connection stiffness is different, but there is no significant difference between the results when the stiffness EI is 1.0 × 10

^{6}Nm

^{2}and 1.0 × 10

^{7}Nm

^{2}, which remain consistent with the conclusion drawn in the frequency-domain model.

^{6}Nm

^{2}to 1.0 × 10

^{7}Nm

^{2}, the phases of the responses of all three modules are hardly affected. However, when the EI is set to 1.0 × 10

^{8}Nm

^{2}, the response amplitudes of Module 2 and Module 3 exhibit differing extents of increase compared to the former two cases. Notably, the response amplitude of Module 1 experiences a slight decrease. No phase shift occurs on the responses of both Module 2 and Module 3, while Module 1 seems to have a positive phase shift when the value of EI is set to 1.0 × 10

^{8}Nm

^{2}, which indicates that the heave response of Module 1 lags more compared to the previous cases.

^{8}Nm

^{2}, the relative motion between Module 2 and Module 3 is significantly greater than those when the value of EI is lower.

## 4. Concluding Remarks

- The effects of hydrodynamic interactions in the multi-module floating system play a crucial role in the response of the system and directly affect the feasibility of the numerical model calculation, especially when transforming the frequency-domain results into the time domain. When the distorted hydrodynamic interaction results in potential theory are corrected, it would lead to stable time-domain results.
- Connector parameters such as bending stiffness have a significant impact on overall system performance. Both the response of the floating body and the internal load such as forces and moments obtained through the connections between adjacent modules are affected to varying degrees. For the semi-rigid connection system analyzed here, since the tension–compression characteristics are mainly determined by the axial stiffness EA, the force along the connector layout shows signs of insensitivity to different bending stiffness EI. With the exponential increase of the bending stiffness, loads in the surge direction seem to be insensitive for both connectors and the joint, while the loads in the heave and pitch direction of the connectors exhibit first exponential growth and then a slow-down at EI = 1.0 × 10
^{9}Nm^{2}. - The development of CPHSTDM makes it possible not only to analyze the system behaviors such as the specific motion state at a certain time and the relative motion between adjacent modules under the connection constraints, but also to judge the influence of stiffness selection on the whole system so that the phenomenon of global resonance caused by inappropriate stiffness selection can be avoided in the analysis of practical problems.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 11.**Fitting quality of the MDOF SSM for K(t) based on the frequency-domain results with damping lid factor α = 0.1.

**Figure 12.**Comparisons of heave and pitch motions of Module 3 between the RAO-based response and CPHSTDM-based response under the regular wave of ω = 1.26 rad/s and wave amplitude of 2 m in the head sea for different connector stiffness.

**Figure 13.**The system’s instantaneous motion state under different connector stiffness when the windward module’s heave motion is at the peak at ω = 1.26 rad/s and wave amplitude of 2 m in a head sea.

**Figure 14.**Relative motion between adjacent modules in a heave direction with different connector stiffness.

Module Characteristic | Value |
---|---|

Length, L(m) | 100 |

Breadth, B(m) | 50 |

Depth (m) | 5 |

Draught, D(m) | 2 |

Center of gravity above base, KG (m) | 2.5 |

Radius of roll gyration, Rxx (m) | 14.5 |

Radius of pitch gyration, Ryy (m) | 28.9 |

Type | Bending Stiffness of the Connector | On Module 1 | On Module 2 | On Module 3 |
---|---|---|---|---|

F_{x} | EI = 1.0 × 10^{6} Nm^{2} | 1.74 × 10^{6} N | 2.80 × 10^{6} N | 1.80 × 10^{6} N |

EI = 1.0 × 10^{7} Nm^{2} | 1.67 × 10^{6} N | 2.80 × 10^{6} N | 1.74 × 10^{6} N | |

EI = 1.0 × 10^{8} Nm^{2} | 1.56 × 10^{6} N | 2.80 × 10^{6} N | 1.80 × 10^{6} N | |

EI = 1.0 × 10^{9} Nm^{2} | 1.53 × 10^{6} N | 2.77 × 10^{6} N | 1.80 × 10^{6} N | |

F_{z} | EI = 1.0 × 10^{6} Nm^{2} | 2.33 × 10^{5} N | 4.03 × 10^{5} N | 2.34 × 10^{5} N |

EI = 1.0 × 10^{7} Nm^{2} | 1.97 × 10^{6} N | 3.42 × 10^{6} N | 1.97 × 10^{6} N | |

EI = 1.0 × 10^{8} Nm^{2} | 7.80 × 10^{6} N | 1.35 × 10^{7} N | 7.70 × 10^{6} N | |

EI = 1.0 × 10^{9} Nm^{2} | 1.11 × 10^{7} N | 1.94 × 10^{7} N | 1.11 × 10^{7} N | |

M_{ry} | EI = 1.0 × 10^{6} Nm^{2} | 1.22 × 10^{7} Nm | 1.69 × 10^{7} Nm | 1.22 × 10^{7} Nm |

EI = 1.0 × 10^{7} Nm^{2} | 1.03 × 10^{8} Nm | 1.39 × 10^{8} Nm | 1.04 × 10^{8} Nm | |

EI = 1.0 × 10^{8} Nm^{2} | 4.09 × 10^{8} Nm | 5.02 × 10^{8} Nm | 4.04 × 10^{8} Nm | |

EI = 1.0 × 10^{9} Nm^{2} | 5.82 × 10^{8} Nm | 6.80 × 10^{8} Nm | 5.82 × 10^{8} Nm |

Type | Bending Stiffness of the Connector | At Joint 1 | At Joint 2 |
---|---|---|---|

Fx | 1.0 × 10^{6} Nm^{2} | 36.9 N | 36.9 N |

1.0 × 10^{7} Nm^{2} | 36.9 N | 36.9 N | |

1.0 × 10^{8} Nm^{2} | 36.8 N | 36.8 N | |

1.0 × 10^{9} Nm^{2} | 36.5 N | 36.5 N | |

Fz | 1.0 × 10^{6} Nm^{2} | 1.20 × 10^{4} N | 1.23 × 10^{4} N |

1.0 × 10^{7} Nm^{2} | 1.24 × 10^{5} N | 1.27 × 10^{5} N | |

1.0 × 10^{8} Nm^{2} | 1.31 × 10^{6} N | 1.24 × 10^{6} N | |

1.0 × 10^{9} Nm^{2} | 1.32 × 10^{7} N | 1.16 × 10^{7} N |

**Table 4.**R

^{2}values for SSMs identified based on the frequency-domain results for the coupling term between different modules.

k-SSM | R^{2} Defined by Equation (31) | ||
---|---|---|---|

40 | K_{1,13}(t) | K_{3,15}(t) | K_{5,17}(t) |

0.99568 | 0.99957 | 0.99286 | |

K_{1,7}(t) | K_{3,9}(t) | K_{5}_{,11}(t) | |

0.9986 | 0.99905 | 0.99862 | |

K_{7,13}(t) | K_{9,15}(t) | K_{11,17}(t) | |

0.9986 | 0.99903 | 0.9984 |

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**MDPI and ACS Style**

Chen, M.; Ouyang, M.; Guo, H.; Zou, M.; Zhang, C.
A Coupled Hydrodynamic–Structural Model for Flexible Interconnected Multiple Floating Bodies. *J. Mar. Sci. Eng.* **2023**, *11*, 813.
https://doi.org/10.3390/jmse11040813

**AMA Style**

Chen M, Ouyang M, Guo H, Zou M, Zhang C.
A Coupled Hydrodynamic–Structural Model for Flexible Interconnected Multiple Floating Bodies. *Journal of Marine Science and Engineering*. 2023; 11(4):813.
https://doi.org/10.3390/jmse11040813

**Chicago/Turabian Style**

Chen, Mingsheng, Mingjun Ouyang, Hongrui Guo, Meiyan Zou, and Chi Zhang.
2023. "A Coupled Hydrodynamic–Structural Model for Flexible Interconnected Multiple Floating Bodies" *Journal of Marine Science and Engineering* 11, no. 4: 813.
https://doi.org/10.3390/jmse11040813