## Abstract

A diffraction-radiation analysis is usually required when the hydrodynamic interactions between structural members occur in short waves. For bracings or small cylindrical members, which play important roles in the vicinity of the natural frequency of a floating platform, special care should be taken into account for the effect of viscous damping. Two hybrid panel-stick models are, therefore, developed, through the combination of the standard diffraction-radiation method and the Morison’s formulae, considering the effect of small members differently. The fluid velocity is obtained directly by the panel model. The viscous fluid force is calculated for individual members by the stick model. A semi-submersible type platform with a number of fine cylindrical structures, which is designed as a floating foundation for multiple wind turbines, is analyzed as a numerical example. The results show that viscous force has significant influence on the hydrodynamic behavior of the floating body and can successfully be considered by the proposed hybrid models.

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## Acknowledgments

This research is supported in part by Grants-in-Aid for Scientific Research (B), MEXT (no. 15H04215). We would like to thank ClassNK (Nippon Kaiji Kyoukai), Oshima Shipbuilding Co. Ltd., Shin Kurushima Dockyard Co. Ltd., and Tsuneishi Holdings Corp. for funding this study and for permission to publish this paper. The first author gratefully acknowledges the financial support provided by the MEXT Scholarship (Grant no. 123471) from Japanese Government during the three-year Ph.D. research.

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## Additional information

An erratum to this article is available at http://dx.doi.org/10.1007/s00773-016-0427-5.

## Appendix

### Appendix

A procedure for calculating natural periods of a semi-submersible is given here. Since in the neighborhood region of the natural periods, the wave damping is negligible. From Eq. 6 we know the free motion equation of a semi-submersible without damping is

In these matrices, many of the elements are zeroes, only leaves the diagonal terms and some coupling terms are nonzero. We can thus further deduce characteristic equations for each mode from Eq. 22 as the following: for heave motion, we have

For yaw motion, we have

For surge and pitch motion, we have

where

and

For sway and roll motion, we have

where

and

It should be noted that, for Eqs. 25 or 29, the two modes are coupled and thus should be solved simultaneously. The two positive solutions of Eq. 25 correspond to the surge (the smaller) and the pitch (the larger) natural angular frequencies, respectively; similarly, the two positive solutions of Eq. 29 correspond to the sway (the smaller) and the roll (the larger) natural angular frequencies, respectively. In addition, since the added mass coefficients which depend on the wave frequency are contained, all the above equations need to be solved through iteration processes to find the exact values.

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Liu, Y., Hu, C., Sueyoshi, M. *et al.* Motion response prediction by hybrid panel-stick models for a semi-submersible with bracings.
*J Mar Sci Technol* **21, **742–757 (2016). https://doi.org/10.1007/s00773-016-0390-1

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### Keywords

- Floating offshore wind turbine
- Semi-submersible
- Potential theory