# Time-Domain Implementation and Analyses of Multi-Motion Modes of Floating Structures

^{1}

^{2}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. TALOS WEC

## 3. Frequency-Domain Governing Equation and Responses

#### 3.1. Frequency-Domain Governing Equation

#### 3.2. Added Mass

^{6}). Therefore, the weak coupling terms can be fully omitted in the numerical modeling since these terms would not have any influence on the overall performance of the structure motions and responses, although such a weak coupling presented between the heave and pitch of TALOS may be an indication that the heave and pitch could be strongly coupled if a different structure, for instance, a floating structure with no symmetry for the y axis was applied (here, TALOS has 2 symmetries for the x and y axes, respectively).

#### 3.3. Radiation-Damping Coefficients

^{6}). Therefore, the weak coupling terms can be omitted from the numerical modeling. In fact, in the corresponding time-domain analysis, such weak coupling terms must be dropped; otherwise, it may cause divergence problems in the numerical modeling due to the possibly corresponding divergent-impulse functions (the details are discussed later in the research).

#### 3.4. Wave Excitation

#### 3.5. Response Amplitude Operators (RAOs)

^{6}N/m and ${C}_{66}$ = 5 × 10

^{6}Nm (and these added restoring coefficients mean a relatively stiff mooring system). It should be noted that these linear restoring coefficients can be obtained by linearizing the mooring forces, and the linearization of the mooring forces can be valid if the structure motions are small in magnitude.

## 4. Time-Domain Dynamic Equation and Analysis

#### 4.1. Dynamic Equation

^{E}’ are the parameters/forces externally added to the dynamic system, such as ${f}_{j}^{E}\left(t\right)$ (j = 1, 2, …, 6) the external force, for instance, the force from power take-off (PTO), or the control force, or the force from the mooring system, and these forces can be both linear and nonlinear.

#### 4.2. Memory Effect and the Impulse Functions

#### 4.3. Impulse Function

#### 4.4. Added Mass at Infinite Frequency

_{15}, with an error of about 2.5%. Hence, we can conclude that all the errors should be within the acceptable range in the engineering calculations.

## 5. Approximations of Impulse Function and Memory Effect

#### 5.1. Approximation of Impulse Function

#### 5.2. Calculation of the Memory Effect

_{0}). Subsequently, two different ways to solve ${I}_{k}\left(t\right)$ would be introduced.

- (1)
**Method ‘1’: solving additional differential equations [31]**

- (2)
**Method ‘2’: a recursive method for calculating I**_{k}[32]

## 6. Implementation and Validation of Time-Domain Analysis

#### 6.1. Implementation of Time-Domain Analysis

#### 6.2. Validations of Time-Domain Modeling

- The transformation from the radiation-damping coefficients to impulse functions, as well as how to efficiently approximate the impulse functions;
- The transformation of the frequency-dependent added mass to the added mass at the infinite frequency;
- The transformation from the radiation-damping effects to the memory effects, together with the method for how we can reliably and rapidly calculate the memory effect;
- The inclusions of the coupling terms between different motion modes, especially the calculation of the coupled memory-effect terms;
- The transformation of the excitation responses to the forces in the time domain for a given wave spectrum.

- (1)
- The time-domain analysis can be obtained by solving Equation (22), in which only the linear forces are applied. For the purpose of comparison, all these linear forces must be presented in the frequency-domain analysis, too, and the corresponding motion responses in the frequency domain can be obtained under same conditions as those in the time-domain analyses.
- (2)
- For a comparison, a time history can be generated directly based on the RAOs in the frequency-domain analysis, in which a transformation is made in a formula as

_{6}), the results from the time-domain and frequency-domain analyses are very different. However, the yaw motions are so small in magnitude (~10

^{−9}), and they can be fully ignored in the analysis.

## 7. Conclusions

- A discussion of how the transformation from the frequency domain to the time domain can be made, and a direct transformation is possible, but not very useful due to its inherent limitations;
- The method for calculating the impulse function and the added mass at infinite frequency based on the frequency-domain prediction was presented, and a MATLAB function was presented for reference;
- The approximation of the impulse function using the Prony approximation method was introduced, and a comparison with the results from WAMIT F2T were made to ensure that the calculation method was reliable;
- A simple recursive method for calculating the memory-effect based on the Prony approximation was introduced and the results were validated for its accuracy;
- A validation method for the time-domain implementation was explained, which can be used for ensuring the correctness of the time-domain analysis;

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

## Appendix B

## References

- Brommundt, M.; Krause, L.; Merz, K.; Muskulus, M. Mooring System Optimization for Floating Wind Turbines using Frequency Domain Analysis. Energy Procedia
**2012**, 24, 289–296. [Google Scholar] [CrossRef][Green Version] - Nagata, S.; Toyota, K.; Imai, Y.; Setoguchi, T.; Mamun, M.A.H.; Nakagawa, H. Frequency domain analysis on primary conversion efficiency of a floating OWC-type wave energy converter ‘Backward bent Duct Buoy’. In Proceedings of the 9th European Wave and Tidal Energy Conference, Southampton, UK, 5–9 September 2011. [Google Scholar]
- Fitzgerald, J.; Bergdahl, L. Including moorings in the assessment of a generic offshore wave energy converter: A frequency domain approach. Mar. Struct.
**2008**, 21, 23–46. [Google Scholar] [CrossRef] - Liu, Y. HAMS: A Frequency-Domain Preprocessor for Wave-Structure Interactions—Theory, Development, and Application. J. Mar. Sci. Eng.
**2019**, 7, 81. [Google Scholar] [CrossRef][Green Version] - Perez, T.; Fossen, T.I. Practical aspects of frequency-domain identification of dynamic models of marine structures from hydrodynamic data. Ocean Eng.
**2010**, 38, 426–435. [Google Scholar] [CrossRef] - Kim, D.; Chen, L.; Blaszkowski, Z. Linear frequency domain hydroelastic analysis for McDermott’s mobile offshore base using WAMIT. In Proceedings of the 3rd International Workshopon Very Large Floating Structures (VLFS ‘99), Honolulu, HI, USA, 22–24 September 1999. [Google Scholar]
- Michele, S.; Zheng, S.; Greaves, D. Wave energy extraction from a floating flexible circular plate. Ocean Eng.
**2022**, 245, 110275. [Google Scholar] [CrossRef] - Michele, S.; Buriani, F.; Renzi, E.; Van Rooij, M.; Jayawardhana, B.; Vakis, A.I. Wave Energy Extraction by Flexible Floaters. Energies
**2020**, 13, 6167. [Google Scholar] [CrossRef] - Sheng, W.; Lewis, A. Assessment of Wave Energy Extraction from Seas: Numerical Validation. J. Energy Resour. Technol.
**2012**, 134, 041701. [Google Scholar] [CrossRef] - Budal, K.; Falnes, J. A resonant point absorber of ocean-wave power. Nature
**1975**, 256, 478–479. [Google Scholar] - Falnes, J. Wave-Energy Conversion through Relative Motion between Two Single-Mode Oscillating Bodies. Trans. ASME
**1999**, 121, 32–38. [Google Scholar] [CrossRef] - Sheng, W.; Lewis, A. Power Takeoff Optimization for Maximizing Energy Conversion of Wave-Activated Bodies. IEEE J. Ocean. Eng.
**2016**, 41, 529–540. [Google Scholar] [CrossRef] - Lasa, J.; Antolin, J.C.; Angulo, C.; Estensoro, P.; Santos, M.; Ricci, P. Design, Construction and Testing of a Hydraulic Power Take-off for Wave Energy Converters. Energies
**2012**, 5, 2030–2052. [Google Scholar] [CrossRef][Green Version] - Falcão, A.F.D.O. Modelling and control of oscillating-body wave energy converters with hydraulic power take-off and gas accumulator. Ocean Eng.
**2007**, 34, 2021–2032. [Google Scholar] [CrossRef] - Henderson, R. Design, simulation, and testing of a novel hydraulic power take-off system for the Pelamis wave energy converter. Renew. Energy
**2006**, 31, 271–283. [Google Scholar] [CrossRef] - Aggidis, G.; Taylor, C.J. Overview of wave energy converter devices and the development of a new multi-axis laboratory prototype. IFAC-PapersOnLine
**2017**, 50, 15651–15656. [Google Scholar] [CrossRef] - Kelly, J.F.; Wright, W.M.D.; Sheng, W.; O’Sullivan, K. Implementation and Verification of a Wave-to-Wire Model of an Oscillating Water Column with Impulse Turbine. IEEE Trans. Sustain. Energy
**2016**, 7, 546–553. [Google Scholar] [CrossRef] - Natanzi, S.; Teixeira, J.A.; Laird, G. A novel high-efficiency impulse turbine for use in oscilalting water column device. In Proceedings of the 9th European Wave and Tidal Energy Conference, Southampton, UK, 5–9 September 2011. [Google Scholar]
- Pereiras, B.; Castro, F.; El Marjani, A.; Rodríguez, M.A. An improved radial impulse turbine for OWC. Renew. Energy
**2010**, 36, 1477–1484. [Google Scholar] [CrossRef] - Henriques, J.; Portillo, J.; Sheng, W.; Gato, L.; Falcão, A. Dynamics and control of air turbines in oscillating-water-column wave energy converters: Analyses and case study. Renew. Sustain. Energy Rev.
**2019**, 112, 571–589. [Google Scholar] [CrossRef] - Sheng, W.; Alcorn, R.; Lewis, A. On improving wave energy conversion, part I: Optimal and control technologies. Renew. Energy
**2015**, 75, 922–934. [Google Scholar] [CrossRef] - Todalshaug, J.H.; Ásgeirsson, G.S.; Hjálmarsson, E.; Maillet, J.; Möller, P.; Pires, P.; Guérinel, M.; Lopes, M. Tank testing of an inherently phase controlled wave energy converter. In Proceedings of the 11th European Wave and Tidal Energy Conference, Nantes, France, 6–11 September 2015. [Google Scholar]
- Falcão, A.F.O.; Henriques, J. Effect of non-ideal power take-off efficiency on performance of single- and two-body reactively controlled wave energy converters. J. Ocean Eng. Mar. Energy
**2015**, 1, 273–286. [Google Scholar] [CrossRef][Green Version] - Li, G.; Belmont, M.R. Model predictive control of sea wave energy converters—Part I: A convex approach for the case of a single device. Renew. Energy
**2014**, 69, 453–463. [Google Scholar] [CrossRef] - Taghipour, R.; Perez, T.; Moan, T. Hybrid frequency–time domain models for dynamic response analysis of marine structures. Ocean Eng.
**2008**, 35, 685–705. [Google Scholar] [CrossRef] - Rahmati, M.; Aggidis, G. Numerical and experimental analysis of the power output of a point absorber wave energy converter in irregular waves. Ocean Eng.
**2016**, 111, 483–492. [Google Scholar] [CrossRef][Green Version] - McCabe, A.; Aggidis, G.; Stallard, T. A time-varying parameter model of a body oscillating in pitch. Appl. Ocean Res.
**2006**, 28, 359–370. [Google Scholar] [CrossRef] - Cummins, W.E. The Impulse Response Function and Ship Motions; Department of the Navy: Monterey, CA, USA, 1962.
- Ogilvie, T.F. Recent Progress Toward the Understanding and Prediction of Ship Motions. In Proceedings of the 5th Symposium on Naval Hydrodynamics, Bergen, Norway, 10–12 September 1964. [Google Scholar]
- Duarte, T.; Sarmento, A.; Alves, M.; Jonkman, J. State-space realization of the wave-radiation force within FAST. In Proceedings of the ASME 2013 32nd International Conference on Ocean, Offshore and Arctic Engineering, Nantes, France, 8–13 June 2013. [Google Scholar]
- Duclos, G.; Clement, A.H.; Chatry, G. Absorption of outgoing waves in a numerical wave tank using a self-adaptive boundary condition. Int. J. Offshore Polar Eng.
**2001**, 11, ISOPE-01-11-3-168. [Google Scholar] - Sheng, W.; Alcorn, R.; Lewis, A. A new method for radiation forces for floating platforms in waves. Ocean Eng.
**2015**, 105, 43–53. [Google Scholar] [CrossRef] - UKRI. Projects to Unlock the Potential of Marine Wave Energy. 2021. Available online: https://www.ukri.org/news/projects-to-unlock-the-potential-of-marine-wave-energy/ (accessed on 1 November 2021).
- SmartWave. High Resolution Sea State Simulation with SmartWave. 2021. Available online: https://www.offshorewindlibrary.com/smartwave/ (accessed on 15 December 2021).
- Mei, C.C.; Stiassnie, M.; Yue, D.K. Theory and Applications of Ocean Surface Waves: Linear Aspects and Nonlinear Aspects; World Scientific: Singapore, 2005. [Google Scholar]
- Newman, J.N. Marine Hydrodynamics, 40th anniversary ed.; The MIT Press: Cambridge, MA, USA, 2017. [Google Scholar]
- Faltinsen, O.M. Sea Loads on Ships and Offshore Structures; Cambridge University Press: Cambridge, UK, 1990. [Google Scholar]
- Liu, Y. HAMS: An Open-Source Computer Program for the Analysis of Wave Diffraction and Radiation of Three-Dimensional Floating or Submerged Structures. 2020. Available online: https://github.com/YingyiLiu/HAMS (accessed on 19 October 2021).
- WAMIT. User Manual (v73). Available online: https://www.wamit.com/manualupdate/v73_manual.pdf (accessed on 1 November 2021).
- Sheng, W.; Tapoglou, E.; Ma, X.; Taylor, C.; Dorrell, R.; Parsons, D.; Aggidis, G. Hydrodynamic studies of floating structures: Comparison of wave-structure interaction modelling. Ocean Eng.
**2022**, 249, 110878. [Google Scholar] [CrossRef] - Babarit, A.; Hals, J.; Muliawan, M.; Kurniawan, A.; Moan, T.; Krokstad, J. Numerical benchmarking study of a selection of wave energy converters. Renew. Energy
**2012**, 41, 44–63. [Google Scholar] [CrossRef] - Holthuijsen, L.H. Waves in Oceanic and Coastal Waters; Cambridge University Press (CUP): Cambridge, UK, 2007. [Google Scholar]
- Duclos, G.; Babarit, A.; Clément, A.H. Optimizing the Power Take off of a Wave Energy Converter with Regard to the Wave Climate. J. Offshore Mech. Arct. Eng.
**2006**, 128, 56–64. [Google Scholar] [CrossRef]

**Figure 1.**TALOS I structure/panel (

**a**); TALOS II and the multi-axis power take-off (PTO) (

**b**); and TALOS PTO test rig (

**c**).

**Figure 2.**The conventional added-mass dampening of the TALOS structure. (

**a**) ${A}_{11}$; (

**b**) ${A}_{22}$; (

**c**) ${A}_{33}$; (

**d**) ${A}_{44}$; (

**e**) ${A}_{55}$ (

**f**) ${A}_{66}$ for motions of surge, sway, heave, roll, pitch, and yaw, respectively.

**Figure 3.**The coupled added mass for the TALOS structure: (

**a**) for surge-pitch coupling (${A}_{15}$); (

**b**) sway-roll coupling (${A}_{24}$); (

**c**) for heave-pitch coupling (${A}_{35}$).

**Figure 4.**The conventional radiation-damping coefficients of the TALOS structure. (

**a**) ${B}_{11};\left(b\right){B}_{22};\left(c\right){B}_{33};\left(d\right){B}_{44};$ (

**e**) ${B}_{55};\left(f\right){B}_{66}$ for motions of surge, sway, heave, roll, pitch, and yaw, respectively.

**Figure 5.**The coupled radiation-damping coefficients for the TALOS structure: (

**a**) for surge-pitch coupling (${B}_{15}$); (

**b**) sway-roll coupling (${B}_{24}$); (

**c**) heave-pitch coupling (${B}_{35}$).

**Figure 6.**The wave-excitation forces on the TALOS structure (incoming wave angle = 45°). (

**a**) Surge; (

**b**) sway; (

**c**) heave; (

**d**) roll; (

**e**) pitch; and (

**f**) yaw.

**Figure 7.**The responses of the TALOS motions in waves (wave angle = 45°).

**(a**) Surge; (

**b**) sway; (

**c**) heave; (

**d**) roll; (

**e**) pitch; and (

**f**) yaw.

**Figure 8.**The conventional impulse functions: (

**a**) ${K}_{11}$; (

**b**); ${K}_{22}$; (

**c**) ${K}_{33}$; (

**d**) ${K}_{44}$; (

**e**) ${K}_{55}$; (

**f**) ${K}_{66}$.

**Figure 10.**Conventional impulse functions (${N}_{0}$ = 10): in the figure, ‘IRF’ means the impulse response function, calculated based on the results from the panel method, while ‘approx’ is the Prony approximation, calculated from Equation (14).

**Figure 13.**Approximation of impulse function vs. order of Prony function. (

**a**) ${N}_{0}$ = 5; (

**b**) ${N}_{0}$ = 6; (

**c**) ${N}_{0}$ = 8; (

**d**) ${N}_{0}$ = 10.

**Figure 15.**The comparisons of the time histories from the time-domain solution and from the transformation of frequency-domain responses. (

**a**) surge; (

**b**) sway; (

**c**) heave; (

**d**) roll; (

**e**) pitch; (

**f**) yaw.

Added Mass | WAMIT | Calculation | Error (%) |
---|---|---|---|

${A}_{11}$(kg) | 2.047 × 10^{6} | 2.040 × 10^{6} | −0.338 |

${A}_{15}$(Ns^{2}) | 1.582 × 10^{6} | 1.620 × 10^{6} | 2.430 |

${A}_{33}$(kg) | 2.583 × 10^{6} | 2.580 × 10^{6} | −0.116 |

${A}_{55}$(Nms^{2}) | 1.353 × 10^{8} | 1.360 × 10^{8} | 0.517 |

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**MDPI and ACS Style**

Sheng, W.; Tapoglou, E.; Ma, X.; Taylor, C.J.; Dorrell, R.; Parsons, D.R.; Aggidis, G.
Time-Domain Implementation and Analyses of Multi-Motion Modes of Floating Structures. *J. Mar. Sci. Eng.* **2022**, *10*, 662.
https://doi.org/10.3390/jmse10050662

**AMA Style**

Sheng W, Tapoglou E, Ma X, Taylor CJ, Dorrell R, Parsons DR, Aggidis G.
Time-Domain Implementation and Analyses of Multi-Motion Modes of Floating Structures. *Journal of Marine Science and Engineering*. 2022; 10(5):662.
https://doi.org/10.3390/jmse10050662

**Chicago/Turabian Style**

Sheng, Wanan, Evdokia Tapoglou, Xiandong Ma, C. James Taylor, Robert Dorrell, Daniel R. Parsons, and George Aggidis.
2022. "Time-Domain Implementation and Analyses of Multi-Motion Modes of Floating Structures" *Journal of Marine Science and Engineering* 10, no. 5: 662.
https://doi.org/10.3390/jmse10050662