Time-Domain Implementation and Analyses of Multi-Motion Modes of Floating Structures
Abstract
:1. Introduction
2. TALOS WEC
3. Frequency-Domain Governing Equation and Responses
3.1. Frequency-Domain Governing Equation
3.2. Added Mass
3.3. Radiation-Damping Coefficients
3.4. Wave Excitation
3.5. Response Amplitude Operators (RAOs)
4. Time-Domain Dynamic Equation and Analysis
4.1. Dynamic Equation
4.2. Memory Effect and the Impulse Functions
4.3. Impulse Function
4.4. Added Mass at Infinite Frequency
5. Approximations of Impulse Function and Memory Effect
5.1. Approximation of Impulse Function
5.2. Calculation of the Memory Effect
- (1)
- Method ‘1’: solving additional differential equations [31]
- (2)
- Method ‘2’: a recursive method for calculating Ik [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
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
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Added Mass | WAMIT | Calculation | Error (%) |
---|---|---|---|
(kg) | 2.047 × 106 | 2.040 × 106 | −0.338 |
(Ns2) | 1.582 × 106 | 1.620 × 106 | 2.430 |
(kg) | 2.583 × 106 | 2.580 × 106 | −0.116 |
(Nms2) | 1.353 × 108 | 1.360 × 108 | 0.517 |
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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
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 StyleSheng, 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
APA StyleSheng, W., Tapoglou, E., Ma, X., Taylor, C. J., Dorrell, R., Parsons, D. R., & Aggidis, G. (2022). Time-Domain Implementation and Analyses of Multi-Motion Modes of Floating Structures. Journal of Marine Science and Engineering, 10(5), 662. https://doi.org/10.3390/jmse10050662