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Theoretical study on the effect of parametric rolling on added resistance in regular head seas

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Abstract

A formula based on Maruo’s theory is presented for the added resistance in regular head seas when parametric rolling occurs. Firstly, the velocity potential of the radiation waves due to parametric rolling, sway and yaw motions should be superposed on the conventional velocity potential. Secondly, the added resistance is averaged within the time duration that is double the encounter period. Thirdly, the stationary-phase method which is used in Maruo’s method is also used to obtain the formula of added resistance in waves with parametric rolling taken into account as well as viscous roll damping. Fourthly, source distribution based on Maruo and Ishii's and Maruo and Iwase’s works is used to calculate the added resistance. Finally, the effect of parametric rolling on added resistance in regular head seas mainly results from viscous roll damping and it becomes larger as the amplitude of parametric rolling becomes larger.

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Acknowledgments

J. Lu's visit to Osaka University was supported by China Scholarship Council (No. 2008606031) and the research described here was supported by a Grant In Aid for Scientific Research of the Japan Society for Promotion of Science (No. 21360427). Prof. M. Kashiwagi from Osaka University provided the authors with his useful advice on the calculation of added resistance in waves. The authors sincerely thank the above organisations and individual.

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Authors and Affiliations

  1. Department of Naval Architecture and Ocean Engineering, State Key Laboratory of Structural Analysis for Industrial Equipment, Dalian University of Technology, No. 2 Linggong Road, Ganjingzi District, Dalian, 116024, Liaoning, China

    Jiang Lu

  2. Department of Naval Architecture and Ocean Engineering, Osaka University, 2-1 Yamadaoka, Suita, Osaka, 565-0871, Japan

    Jiang Lu & Naoya Umeda

  3. Ship CAD Engineering Centre, Dalian University of Technology, No. 2 Linggong Road, Ganjingzi District, Dalian, 116024, Liaoning, China

    Kun Ma

Authors
  1. Jiang Lu
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  2. Naoya Umeda
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  3. Kun Ma
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Correspondence to Jiang Lu.

Appendix: Validation of the calculation of added resistance in waves various methods for the source distribution

Appendix: Validation of the calculation of added resistance in waves various methods for the source distribution

The authors calculated added resistance with six different methods of source distribution [18] for the modified Wigley model [19] in head waves. Here no lateral motions are allowed. And then the calculated results are compared with the model experiments published by Kashiwagi et al. [19] and their results of enhanced unified theory (EUT) [20] are shown in Figs. 13 and 14. As reported by Kashiwagi et al. [19], EUT shows reasonably good agreement in added resistance but its agreement in heave and pitch motion is not so good as a strip theory. Since parametric roll prediction requires both accurate added resistance and the heave and pitch motions, it is worthwhile to examine the applicability of strip theory-based approach to added resistance. It is impressive here that the line source distribution by Maruo and Ishii’s method [13] based on strip theory provides relatively good agreement with the model experiments particularly in the region of the wavelength to ship length ratio of around one, where parametric rolling could be significant. This is because only Maruo and Ishii’s method takes all the effects of radiation, diffraction, steady motion and 2D wavy free surface conditions on the line source distribution into account among the six methods. Therefore, the author used Maruo and Ishii's method for source distribution. It is noteworthy here that even Maruo and Ishii’s method slightly overestimates the experiment.

Fig. 13
figure 13

Added resistance of the modified Wigley ship model with difference methods and experimental (EXP) and theoretical (EUT) results by Kashiwagi et al. [19] at Fn = 0.1

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Fig. 14
figure 14

Added resistance of modified Wigley ship model with difference methods and experimental (EXP) and theoretical (EUT) results by Kashiwagi et al. [19] at Fn = 0.15

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Lu, J., Umeda, N. & Ma, K. Theoretical study on the effect of parametric rolling on added resistance in regular head seas. J Mar Sci Technol 16, 283–293 (2011). https://doi.org/10.1007/s00773-011-0127-0

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  • DOI: https://doi.org/10.1007/s00773-011-0127-0

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