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PREDICTION OF NONLINEAR MOTIONS OF HIGH-SPEED VESSELS IN OBLIQUE WAVES 157
Prediction of Nonlinear Motions of High-speed Vessels in Oblique
Waves
F-C Chiu,1 Y-H Lin,1 C-C Fang,2 S-K Chou2
(1National Taiwan University, 2United Ship Design and Development Center, Taiwan)
ABSTRACT
A practical method, based on a nonlinear strip synthesis scheme, to calculate the nonlinear motions of a high-speed
vessel in oblique waves is presented in this paper. In this method, the equations of motions are described by the body-fixed
coordinate, rather than the conventionally used ship-carried vertical coordinate. Moreover, the time-varying submerged
hull surface and the coupling effect between transverse and vertical motions are considered. By using the momentum
theory, the flare impact and dynamic lift are also taken into account. In the time domain simulation, to prevent the
numerical divergence due to the drift of sway and/or yaw motions, artificial springs in sway and yaw modes are introduced.
In order to clarify the validity of the proposed prediction method, a series of seakeeping tests in oblique waves have
been carried out in SSPA with a model of 90-meter patrol vessel, which is designed by USDDC (United Ship Design &
Development Center, Taiwan), The experimental results are compared with the calculation by the present method and some
of the selected results of comparison study are shown in this paper. As a practical tool for predicting the nonlinear motions
of high-speed vessels in oblique waves, the validity of the present method is verified.
INTRODUCTION
In this two decades, the expanding demand of large-sized high-speed ocean-going vessels urged the necessity of
developing analytic tools to evaluate their nonlinear behavior in rough sea. Up to the present, several more sophisticated
methods have been proposed to predict nonlinear motions and wave loads of a ship at forward speed in head sea. For
example, a three-dimensional Rankine Panel Method (Kring et al 1996) and a three-dimensional transient free-surface
Green function source distribution method (Lin & Yue 1990) have proven to be sufficiently useful. On the other hand, a
practical technique basing on a nonlinear strip synthesis (Chiu & Fujino 1989; Chou, Chiu & Lee 1990) has also proven to
be accurate enough for practical use.
Several years ago, one of the authors Chiu and Liaw (1993), following the same nonlinear strip synthesis scheme,
developed a practical method for predicting the nonlinear motions of a high-speed vessel in oblique waves. Based on the
numerical investigation, an existing 60-feet planing boat had been shown its fundamental characteristics of vertical and
transverse motions in bow/beam sea. In this method, the equations of motions are described by the body-fixed coordinate,
rather than the conventionally used ship-carried vertical frame. Moreover, the time-varying submerged hull surface and the
coupling effect between transverse and vertical motions are considered. Besides, using the momentum theory, the flare
impact and dynamic lift are also taken into account. However, in the time domain simulation, artificial springs in sway and
yaw modes are introduced to prevent the numerical divergence due to the drift of sway and/or yaw motions.
In this paper, the dynamic responses of a 90-meter high-speed patrol vessel RD-200, which is designed by USDDC,
travelling in oblique waves are predicted by the present method and compared with the result of experiments carried out by
Lundgren (1997) at SSPA, in order to confirm the validity of the present method. The detailed formulation of the present
method was fully described in Chiu & Liaw (1993). For convenience sake, however, the basic concept of the method will
be described briefly.
THEORETICAL FORMULATION
Coordinate System
The right hand Cartesian coordinate systems and sign convention used for following theoretical formulation are shown
in Figure 1.
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PREDICTION OF NONLINEAR MOTIONS OF HIGH-SPEED VESSELS IN OBLIQUE WAVES 158
Figure 1 Coordinate system
The space-fixed coordinate system O–XYZ (hereafter FO frame) is defined so that the X–Y plane coincides with the
undisturbed water surface and the Z-axis is vertically downward. The incident waves propagate toward positive X-
direction. Ship-carried vertical coordinate system o′ –x′y′z′ (hereafter FV frame) is moving at ship speed U in x′-direction and
keeping z′-axis vertically downward. x′-axis is laid on the undisturbed water surface. The angle between x′-direction and
X-axis is denoted with χ. Another coordinate system o–xyz is ship-fixed (hereafter FB frame) with origin located at the
center of gravity of the ship and x-axis parallels to the base line of the ship. The ship is assumed to be steady running in calm
water and let the moment just encountering with the wave as starting time t=0. At this moment, the center of the waterline
at the midship is set to be o′ and coincides with O. And the coordinates of o in F V frame are τs and ζs are the
increments of trim and sinkage due to steady running in calm water respectively, while ξ, η, ζ and denote surge,
sway, heave displacement and roll, pitch, yaw angle of FB frame relative to F V frame respectively due to waves. If we
define τ ≡τs+θ then the Euler angles of FB frame relative to FV frame for a ship running in waves can be described as
and the relationships between the FV coordinates and FB coordinates are
(1)
where the transformation matrix LVB is defined as
(2)
Moreover, the relationships between the FO coordinates and FV coordinates are
(3)
where the transformation matrix LOV can be defined as equation (2) but just substituting the Euler angles by (χ,
0, 0).
Then, substituting equation (1) into the right hand side of equation (3), the relationships between the FO coordinates
and FB coordinates can be obtained as
(4)
u, v, w and p, q, r are the translational velocity and angular velocity of the ship described by FB frame in x–, y–, z–
direction respectively. The relationships between these velocity and those described by FV frame, such as and
are
(5)
(6)
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PREDICTION OF NONLINEAR MOTIONS OF HIGH-SPEED VESSELS IN OBLIQUE WAVES 159
where the transformation matrix R−1 is defined as
(7)
Using the reverse transformation, equation (5) can be expressed as
(8)
where LBV is the transverse matrix of LVB, i.e. The first term of the right-hand side of equation (8)
(9)
denotes the velocity component related to the steady forward speed, while the second term denotes the velocity
component related to the oscillatory motion speed and can be defined as
(10)
τ=τs into equation (9) and define the velocity component of a ship running
Furthermore, we can substitute
in calm water as
(11)
Incident Waves
The incident wave ζw and the subsurface of incident wave ζε are described as follows in the space-fixed FO frame.
(12)
(13)
where ζa is the wave amplitude, κ the wave number, ω the wave frequency, X and Z can be expressed by equation (4).
By defining equation (12) and (13) become
(14)
(15)
where ωe is the encounter wave frequency defined by ωε=ω−κU cos χ. Moreover, the velocity potential of the
incident wave is expressed by
(16)
where g is the acceleration of gravity.
The instantaneous immersed depth at any point of hull surface Zd is expressed by
(17)
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The instantaneous submerged portion of a hull section can be obtained from equation (17), by letting Z d=0
The orbital velocity of wave particles described in FO frame can be divided into two components, VX and VZ, which
are parallel to X and Z—axes, respectively.
(18)
(19)
Moreover, they are described in FB frame by

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PREDICTION OF NONLINEAR MOTIONS OF HIGH-SPEED VESSELS IN OBLIQUE WAVES 160
(20)
where LVO is the transverse matrix of LOV, i.e.
The pressure of incident wave is given by Bernoulli's equation.
(21)
where ρ is the density of water.
Transformation of Hydrodynamic Coefficient Matrix
The right hand Cartesian coordinate systems and sign convention used for the transformation of sectional
hydrodynamic coefficient matrix are shown in Figure 2. The -axis of the vertical coordinate system (hereafter
frame) is laid on the still water surface. Point b is the intersection of the x-axis of F B frame and the hull section.
frame. Coordinate system b–yz
Another vertical coordinate system (hereafter frame) is parallel to
(hereafter Fb frame) is fixed in the hull section. We denote the sectional added mass matrix for oscillatory motion with
frame, [m] in the Fb frame. i.e
in the frame, in the
Figure 2 Coordinate system for defining sectional hydrodynamic coefficients
(22)
in -axis, and with p for
Denoting the translational velocity of a hull section in frame with in -axis,
angular velocity, the sectional velocity and angular velocity in frame is described as and
p respectively. Furthermore, the sectional fluid momentum described in frame can be expressed in terms of which
described in frame as
(23)
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where
is the coordinates of b in frame.
Considering the symmetry of and substituting the equation

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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
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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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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
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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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PREDICTION OF NONLINEAR MOTIONS OF HIGH-SPEED VESSELS IN OBLIQUE WAVES 163
on it with P0, and the submerged portion under the undisturbed incident wave surface while running in wave with c,
pressure acting on it with P, the sectional restoring and Froude-Krylov force and moment are described as
(38)
where ∫c means the integration along the contour of submerged portion from port to starboard. denote the
coordinates of sectional center of gravity in FV frame. m is sectional mass. P 0 can be obtained from equation (4) and is
expressed as
(39)
while P is given as equation (21).
Sectional inertia force F i and moment Mi—The sectional inertia force and moment are expressed as
(40)
where ixx denotes the sectional moment of inertia relative to o−x axis.
Equations of Motions
Taking the summation of all the sectional force and moment stated above, then integrating the total sectional force and
moment from the aftmost water-hull intersection A to the foremost water-hull intersection F yields the equations of motions
in FB frame as follows. In which the state of steady running in calm water is considered as the initial reference state from
which the ship motions are reckoned
(41)
Substituting equation (35) into equation (36)~(38), (40) and then substituting these equations into equation (41), the
resultant equations of 5-D coupled motions are expressed in a matrix form as
(42)
where the detailed expressions of the various elements included in the coefficient matrices and the force vectors can be
referred in Chiu & Liaw (1993)
NUMERICAL ALGORITHM
Sectional Hydrodynamic Coefficients
The procedure to calculate the instantaneous sectional hydrodynamic coefficient matrices [m], [N] and [∂ m/∂t] in Fb
frame during numerical integration of equations of motions can be described briefly as follows.
The sectional hydrodynamic coefficients in frame at several different prescribed drafts and heel angles of a
section at an encounter frequency, i.e. [mb], [N b], and those at infinite frequency, i.e. are computed in advance by
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Frank close-fit method (Frank & Salvesen, 1970) for each transverse section. Moreover,
can be calculated then. Where dr and
and denote the sectional draft and heel angle of a
section relative to undisturbed wave surface respectively. The obtained results are saved as

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PREDICTION OF NONLINEAR MOTIONS OF HIGH-SPEED VESSELS IN OBLIQUE WAVES 164
a database. Making use of such a database, the sectional hydrodynamic coefficients are evaluated by interpolation at each
time step during numerical integration of equations of motions. as well as can be calculated by
(43)
Then the instantaneous sectional hydrodynamic coefficient matrices [m], [N] and [∂ m/∂t] in F b frame can be obtained
following the transformation described in the previous section.
Time Integration
For the numerical integration of the equations of motions described in ship-fixed FB frame, i.e. equation (42), the
Newmark-β method with β=1/4 is used from the viewpoint of the stability and the accuracy of numerical integration. The
discrete time interval ∆t adopted for time integration is 1/60 of the encounter period.
Viscous Roll Damping
In order to take into account the viscous effect in roll motion, the equivalent roll damping coefficient is
, where m44 is the virtual moment of inertia in roll, and αe denotes the roll damping factor
expressed as
which is evaluated from roll decay test results according to
(44)
denotes the initial roll amplitude used, n the number of swings, the roll amplitude of nth swing, and T4
where
denotes the natural roll period.
Artificial Spring
In general, to solve the 5 degrees coupled motions in time domain, the stability of solution will be affected by sway
and/or yaw motions due to no restoring force and moment in these two modes. Therefore, to prevent the numerical
divergence due to the numerical drift of sway and/or yaw motions, artificial springs in sway and yaw modes are introduced.
Other methods such as introducing rudder force with auto-pilot or introducing a numerical filter may be considered,
however making use of artificial springs seems to be the simplest way to meet this purpose. The strength of the artificial
springs is decided by a trade-off, that is to say it has to be strong enough to keep the drifts small, and weak enough not to
affect the motions significantly.
In this paper, the artificial spring constant is given by K · (ωe /2)2 · M, where M denotes the mass for sway mode, and
the longitudinal moment of inertia for yaw mode respectively. while K is a factor for tuning, K =10.0 is adopted to carry out
the numerical computation.
COMPARISON OF PREDICTION AND EXPERIMENTAL RESULTS
In order to verify the validity of the present nonlinear prediction method, the model test results of a free-running 1:25
scale hull model carried out in SSPA Maritime Dynamics Laboratory are used for comparison. The model is self-propelled,
autopilot steered and free to move in all six degrees of freedom. There are over 120 runs performed in this test project. They
covered investigation of the seakeeping performance of the vessel, which is designed by USDDC, in both regular and
irregular waves. This paper describes some of the selected results of the comparison study. Solid model of the RD-200 is
shown in Figure 3. Some of its principal particulars are shown in Table 1.
Figure 3 Solid model of the RD-200
Table 1 Principal particulars of the RD-200
Length between perpendiculars, L 90.0 m
Breadth of Water Line, B 12.2 m
Draft 3.55 m
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Displacement 1840 tonnes
Vertical position of CG 5.59 m abv B.L.
Longitudinal position of CG 46.92 m aft F.P.
Radius of gyration in roll 4.43 m
Radius of gyration in pitch 22.5 m

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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
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PREDICTION OF NONLINEAR MOTIONS OF HIGH-SPEED VESSELS IN OBLIQUE WAVES 166
As seen in these figures the present calculations tend to somewhat underestimate the pitch and vertical acceleration
responses in the range of longer wavelength, while the agreement between the responses predicted by the present
calculation and the experimental results seems satisfactory. It is especially evident that the peak location by the calculation
agrees remarkably well with the experimental results.
Figure 7 Response of vertical acceleration at main deck of
Figure 6 Pitch response in bow sea at Fn 0.416
FP station in bow sea at Fn 0.416
Figure 8 Response of vertical acceleration at main deck of Figure 9 Response of vertical acceleration at helicopter
LCG station in bow sea at Fn 0.416 platform in bow sea at Fn 0.416
The recorded time histories of motions together with calculated results of two test runs of λ/L 1.073 and 1.909 are
shown in Figure 10 and Figure 11 respectively. In these figures wave as well as sway
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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.
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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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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.
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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.