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About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as PREDICTION OF NONLINEAR MOTIONS OF HIGH-SPEED VESSELS IN OBLIQUE WAVES 161 (24) into equation (23), the rotation related 3 elements of symmetrical matrix can be expressed in terms of the elements of as (25) The sectional velocity and angular velocity in Fb frame is described as Moreover, the sectional fluid momentum described in Fb frame can be expressed in terms of which described in frame as (26) and then, (27) or (28) can be obtained. Where (29) and (30) Substituting equation (29) and (30) into equation (28), the relationships between the 6 elements of symmetrical matrix and those of can be obtained as follows. (31) described in Fb Finally, substituting equation (24) and (25) into equation (31), the sectional added mass matrix frame can be transformed from the sectional added mass matrix described in frame. In a manner similar to the above stated derivation of the transformation of sectional added mass matrix, denoting the sectional damping coefficient frame, [N] in the Fb frame, then the equation (24), (25) and (31) are matrix with in the frame, in the also effective for the transformation of sectional damping coefficient matrix. Sectional Force Compone nts Since u, v, w and p, q, r are denoted as the translational velocity and angular velocity of the ship described in FB the authoritative version for attribution. frame, then the average relative velocity and relative angular velocity to the water at section x described in FB frame, denoting with and can be given as (32) where the velocity of the point y=z=0 is defined as the average velocity at section x, and denote the sectional average of the orbital

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About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as PREDICTION OF NONLINEAR MOTIONS OF HIGH-SPEED VESSELS IN OBLIQUE WAVES 162 velocity of wave particles given by equation (20). i.e. (33) where ∫c means the integration along the hull section contour. denotes the equivalent average angular velocity of the orbital motion of wave particles relative to the point y=z=0 at section x, and is described by (34) where In the subsequent formulation of the equations of motions for a high-speed vessel, which are derived by following the Ordinary Strip Method synthesis, and like Fujino & Chiu (1983), the state of steady running in calm water is considered as the initial reference state from which the ship motions are reckoned. Therefore, both the relative velocity and the hydrodynamic coefficients are decomposed into the oscillatory motion related component and the steady forward motion related component. By using equation (8)~(11), the equation (32) becomes (35) where the term of sin on the right-hand side of equation (9) is neglected by considering that it is much smaller . Moreover, V, W are defined as than the term of cos and respectively. The effects of surge to the other motion modes are assumed to be negligible, and the surge mode is decoupled in the subsequent formulation of the equations of motions. Sectional force F m and moment Mm due to the change of fluid momentum—The sectional hydrodynamic force and moment due to the time variation of fluid moment can be described as (36) where “*” denotes the sectional added mass for steady running in calm water, and those at infinite frequency are used under the assumption of high speed running condition. Subscript “0” denotes the sectional added mass which is evaluated for the submerged portion under the undisturbed water surface while steady running in still water. Due to the symmetry of hull section, can be substituted into equation (36). Sectional damping force F r and moment Mr—Similar to Fm and Mm, the sectional damping force and moment are decomposed into the oscillatory motion related component and the steady forward motion related component, and is described as (37) where “*” and “0” have the same meaning mentioned above. Then the second term of right-hand side has no the authoritative version for attribution. contribution therefore. Sectional restoring and Froude-Krylov force Fs and moment Ms—Denoting the submerged portion under the undisturbed water surface while steady running in still water with c0, pressure acting

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About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as PREDICTION OF NONLINEAR MOTIONS OF HIGH-SPEED VESSELS IN OBLIQUE WAVES 165 The comparisons between experimental results and present predictions for the cases in regular bow sea (χ=135 deg) with various wavelengths at the forward speed corresponding to the Froude number in ship length of 0.416 are selected to be shown in this paper. The wavelength to ship length ratio (λ/L) and wave steepness (Hw/λ) are shown in Table 2, varied from 0.476 to 4.885 and 1/29.9 to 1/136.5 respectively. The corresponding data used in prediction for comparison are also shown in Table 2. The Heave, roll and pitch motions as well as vertical accelerations at main deck of FP, LCG stations and at helicopter platform were measured and compared. The steady running trim and CG rise measured at the above mentioned forward speed in calm water are approximately 0.39 degree and 0.33 m respectively. The roll damping factor αe and natural period T4 obtained from the roll decay test at the forward speed corresponding to 24 knots of the full-scale ship are 0.18 and 8.1 second respectively. Table 2 Wave length and steepness Experiments Prediction λ/L Hw/λ λ/L Hw/λ 0.476 1/29.9 0.623 1/39.0 0.6 1/30 0.687 1/30.0 0.848 1/29.9 0.8 1/30 1.073 1/37.9 1.0 1/40 1.25 1/40 1.401 1/39.1 1.5 1/40 1.75 1/60 1.909 1/53.3 2.0 1/60 2.271 1/63.4 2.747 1/76.7 2.5 1/90 3.0 1/90 3.393 1/94.8 3.5 1/120 4.292 1/119.9 4.0 1/120 4.885 1/136.5 4.50 1/120 Figures 4 through 9 illustrate the wavelength dependence of responses of RD-200 travelling in bow seas at the forward speed corresponding to the Froude number of 0.416. In these figures, the nondimensionalized amplitudes of 1st order and the phase angle, which is related to when the wave trough is at the ship's CG, of heave ζ /ζa, roll and pitch θ/κζa as well as vertical accelerations at main deck of FP station of LCG station and at helicopter platform are plotted together with predicted responses obtained by the present computation. The abscissa of the figures denotes the wavelength to ship length ratios λ/L. Figure 5 Roll response in bow sea at Fn 0.416 Figure 4 Heave response in bow sea at Fn 0.416 the authoritative version for attribution.

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About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as PREDICTION OF NONLINEAR MOTIONS OF HIGH-SPEED VESSELS IN OBLIQUE WAVES 167 and heave motions are nondimensionalized by wave amplitude, while roll, pitch and yaw motions are nondimensionalized by the amplitude of wave slop. Furthermore, the calculated heave motion is kept in phase with the measured heave motion, and it can be found that the relative phase angles between motions obtained by present calculation agree well with that of experiment results. Figure 11 Comparison of time histories of motions in bow Figure 10 Comparison of time histories of motions in bow sea with λ/L 1.909 at Fn 0.416 sea with λ/L 1.073 at Fn 0.416 CONCLUSION A prediction method, basing on a nonlinear strip synthesis scheme, to calculate the nonlinear motions of a high-speed vessel in oblique waves is presented and applied to a high-speed patrol vessel RD-200 travelling in bow sea. The present results of ship motions and vertical accelerations at three different positions, have been validated by a proper comparison with experimental data. And the following conclusion may be drawn. Through the comparison between the dynamic responses predicted by the present nonlinear calculation and experimental results, it is confirmed that the present method can be applied to estimate the ship motions and vertical accelerations along ship length of a high-speed vessel in oblique waves with accuracy enough for practical use. Furthermore, it can be expected that other dynamic responses—for instance, wave loads and pressure on hull panel—can be predicted by extending the present method. the authoritative version for attribution.

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About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as PREDICTION OF NONLINEAR MOTIONS OF HIGH-SPEED VESSELS IN OBLIQUE WAVES 168 ACKNOWLEDGEMENTS This research was funded by the National Science Council Taiwan under Grant Numbers NSC80–0403-E-002–01, NSC88–2611-E-002–01, and the Ministry of Economic Affairs Taiwan under Grant Number S891030. REFERENCES F.C.Chiu & M .Fujino, ‘Nonlinear prediction of vertical motions and wave loads of high-speed crafts in head sea', International Shipbuilding Progress, Vol. 36, No. 406, 1989 F.C.Chiu & Y.C.Liaw, ‘A practical method for estimating ship motions of high-speed crafts in oblique waves', Journal of the Society of Naval Architects of Japan, Vol. 174, 1993 S.K.Chou, F.C.Chiu, Y.J.Lee, ‘Nonlinear motions and whipping loads of high-speed crafts in head sea', 18th ONR Symposium on Naval Hydrodynamics, Ann Arbor, 1990 W.Frank & N.Salvesen, ‘The Frank close-fit ship motion computer program', NSRDC Report No. 3289, Bethesda, Md., 1970 M.Fujino & F.C.Chiu, ‘Vertical motions of high-speed boats in head sea and wave loads', Journal of the Society of Naval Architects of Japan, Vol. 154, 1983 D.Kring, Y.-F.Huang, P.Sclavounos, T.Vada, A. Braathen, ‘Nonlinear ship motions and wave-induced loads by a Rankine method', 21st ONR Symposium on Naval Hydrodynamics, Trondheim, 1996 W.-M.Lin & D.Yue, ‘Numerical solutions for large-amplitude ship motions in the time domain', 18th ONR Symposium on Naval Hydrodynamics, Ann Arbor, 1990 J.Lundgren, ‘USDDC OPV Seakeeping tests in regular and irregular waves', SSPA Report 97 4256–1, 1997 the authoritative version for attribution.

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About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as PREDICTION OF NONLINEAR MOTIONS OF HIGH-SPEED VESSELS IN OBLIQUE WAVES 169 DISCUSSION T.Fukasawa Kanazawa Institute of Technology, Japan In the time domain simulation of ship motion in oblique waves, it is most important to secure the numerical stability in simulation, because there is no restoring force and moment in sway and yaw motions. The absence of the restoring force and moment causes the numerical drifting and diverging of ship motions in the simulation. The authors adopted the artificial springs to remove the numerical drifting in sway and yaw motions. However, it is not easy to determine the adequate spring constant, with which the drifting of motions can be controlled, so that the motion amplitude and phase angle are not affected by the springs. The discusser, on the other hand, has proposed a procedure to remove the numerical drifting of ship motions with the use of a numerical filter.[1] In this procedure, there is no messy problem like the determination of the spring constant, and the ship motion amplitude is not affected at all. In the Figures 10 and 11, the predicted swaying and yawing amplitude has not enough accuracy comparing with the other motions. Does this mean that the artificial spring constant using in the paper is not adequate? And, if it is difficult to choose the adequate spring constant, isn't it better to use such a procedure as the numerical filter to remove the numerical drifting? On the other hand, the actual drifting in sway and yaw motions is inevitable in the experiments in the case where a free-running model is used. The drifting in yawing motion, in particular, causes the shift of attack angle of ship to wave, and the mean encounter angle between ship and wave differs from the expected one. I would like to hear the authors' comment on the comparison of the drifting in sway and yaw motions in the experiments and in the simulations. How can we predict the actual drifting in sway and yaw motions, avoiding the numerical drifting in these motions? And also, in case the drifting in the simulation is removed, how do we consider the encounter angle shift in the experiments? 1. Fukasawa, T., “On the Numerical Time Integration Method of Nonlinear Equations for Ship Motions and Wave Loads in Oblique Waves,” Journal of the Society of Naval Architects of Japan, Vol. 167, June 1990, pp. 69–79. (in Japanese) AUTHOR'S REPLY The predicted lateral motions are relatively sensitive to the values of artificial spring constants, and the predicted sway and yaw amplitude is not satisfactory. The authors agree with the discussor's opinion that it's better to use a numerical filter to avoid the numerical instability. It requires at least N times of the computer time, where N denotes the order of the numerical filter, which values might be, say, 50 or 60. Basing on results shown in this paper, the future study on the employing of a numerical filter into the present model is undergoing. The physical drifting in sway and yaw motions are not considered in the present method. As shown in the experimental records of Figures 10 and 11, the overall yaw drift are no more than 3 degrees. The effect of the drifts on the encounter angle seems to be little. the authoritative version for attribution.

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About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as PREDICTION OF NONLINEAR MOTIONS OF HIGH-SPEED VESSELS IN OBLIQUE WAVES 170 DISCUSSION D.Yue Massachusetts Institute of Technology, USA In the present model for the prediction of nonlinear motions of high-speed vessels, artificial springs are employed to suppress the numerical instability associated with the drift of sway and yaw motions. How sensitive is the overall solution to the choice of the spring constants? Since the drift of sway and yaw motions is physical and should be considered as part of the solution, it is probably more reasonable to include physical damping (such as the wave-drift damping) rather than artificial restoring forces to retain the stability of the scheme. Could the authors comment on this? AUTHOR'S REPLY The following is a typical example, Figure A shows the sensitivity of ship motions to the spring constants. The vertical motions seem not be affected significantly, while the transverse motions are quite sensitive to the spring constant. This sensitivity may be considered as an important factor that results in unsatisfactory lateral motion predictions. The authors would suggest that a numerical treatment may be needed to obtain a stable solution. The authors also agree to the discussor's viewpoint that it may offer a more reasonable solution to take into account the physically existing drifting force which was not considered in the present model. Figure A Effects of artificial Spring Constants on ship motions (λ/L=1.0 at Fn=0.416) DISCUSSION J.Xia The University of Western Australia, Australia Could the authors please comment on the influence of neglecting memory effects on their modeling of hydrodynamic forces and vessel motions. AUTHOR'S REPLY Since we just take into account the effects of noncirculatory part of dynamic lift on the ship motions, so there is no need to consider the memory effects. However, the reduced frequency of a planning vessel running in head sea seems not to be small enough to neglect the memory effects if the circulatory part of dynamic lift is taken into account. The authors think that it needs further study to clarify the influence of neglecting the circulatory part of dynamic lift on ship motions. the authoritative version for attribution.