A practical method of estimating ship motions and wave loads in large amplitude waves

j.

Tech&sche HogeschaQJ

Deift

A PRACTICAL METHOD OF ESTIMATING SHIP MOTIONS AND WAVE LOADS

IN LARGE AMPLITUDE WAVES by

MFujino* and B.S. YOon**

1. Introduction

The prediction of ship motions and wave loads acting on the ship structure has been a principal theme

in the research of seakeeping qualities of ships. The

Ordinary Strip Method (hereafter called O.S.M.),

which has been developed under the assumption that both amplitudes of incident waves and ship motions are infinitesimal, has proved to be sufficiently useful

for predicting ship motions and wave loads [1, 2). The so-called 'total system of prediction of seakeeping qualities of ships' based on the stnp method has been developed in shipyards, classification societies, etc.

and is widely used at design stage in shipyards, for

in-stance, in order to provide excellent seakeeping qual-ities as well as sufficient strength of ship structure. In such a system, the principle of linear superposition is applied to perform the short-term prediction of ship responses in irregular seas using the response am plitude operators of ship responses defined in regular seas, and then the long-term prediction is obtained by the use of the long-term wave statistics in the service route of a particular ship concerned. However, when discussing the long-term prediction of a particular response, of which the nonlinearity in large amplitude

waves seems significant, it must be reminded that the

long-term prediction based on the linear superposition might not exactly correspond to the reality.

So far many papers have been devoted to solve the

nonlinear radiation and/or diffractiOn problems which are deeply related to prediction of ship motions in large amplitude waves. To our knowledge, however,

the practical methods have not been fully developed to

predict the wave induced motions and wave loads of ship-like three-dimensional body advancing in large

amplitude waves except several attempts [3, 4].

In this paper, a practical method is advoôated for

predicting nonlineaE ship responses in waves by extend-ing the strip method synthesis. The nonlinearity of hydrodynamic forces included in the present method stems from the time-variation of ship's submerged portion; the sectional hydrodynamic forces are

cal-culated for the time-varying submerged portion at each time step, and then integrated in the longitudinal

direction of ship's hull to obtain the total

hydro-dynamic forces and moments acting on a ship.

')University of Tokyo, Japan. **) University of Ulsan, Korea.

The resultant equations ofmotions are coupled in five-degree of freedom, that is to say, heave, pitch,

sway, yaw, roll and hydrodynamic coefficients

includ-ed in the equations vary with time. When _estimating

the wave loads acting on _{a transverse section of ship's}

hull by the present method, therefore, _{not only the} effect of time-varying vertical displacement but also the effect of nonlinear coupling between vertical _and lateral hydrodynamic forces are taken into consider-ation. The sectional hydrodynamjc forces by radiation and wave diffraction are evaluated upon the same theoretical basis as in O.S.M., and such hydrodynamic coefficients as the added mass and damping forces_are

calculated by Frank close-fit method [5]. The

hydro-static and Froude-Krylov forces are calculated exactly

by integrating the water pressure over the instantan-eous submerged Portion of ship's hull. The ship motions and wave loads predicted by the present method are compared with experimental

measure-ments with emphasis on the nonlinear characteristics

in wave loads.

2. Equations of ship motion andwave loads

As shown in Figure 1, _{the Cartesian coordinate}

systems o-xyz and o'-x'y'z' are moving with ship's average velocity U, and O-XYZ fixed in space. The' X-Y and x'-y' planes coincide with the undisturbed

water surface. The origin o' is located at the midship and o denotes the connecting point of towingcarriage

and ship model when the model is towed in waves on the occasion of experiments.

2.1. Sectional force components

Forces acting on a transverse section of ship's hull can be described as the sum of the following

compo-nents:

dF. _dFM. dF11. dF.

_L__J...__ +._!

dx dx dx dx

where the subscript I means I = I : vertical force 2 : horizontal force 3 : roll moment dFMI

- :

inertia force dx (1)

z

Figure 1. Coordinate system.

dFHI dx

dF.

static force. and dt at ax

s' : f-mode displacement of the point o',

m11 _{i-mode added mass by /-mode motion,}

i-mode damping coefficient by f-mode motion. The definition of i- and /-mode motion are as

fol-lows:

= 1 : vertical displacement,

i,j = 2 : horizontal displacement, = 3 : rotatiOn about o'x'.

dw..

.L in the equation (4)

denotes the average velocity of wave particle defined in a transverse section of submerged ship's hull:

/ = 1 : vertical velocity, 1 = 2 : horizontal velocity,

/ = 3 : rate of change of wave slope. dF

The static force..- can be written as follows:

0

hydrodynamic force

c

C,

Figure 2. Configuration of a ship's transverse section.

dF1

w - t p dy'

dx C

dF2

! p dz' B(Y

,Z)

dF

f3fPiYdY' zdz') +w(z' z) sinØ

(7) where

W _{sectional weight of ship,}

p: water pressure,

C' _{instantaneous Submerged surface of a section,}

(see Figure 2)

Z,, _{z'coordinate of point o,}

z'-coordinate of sectional center of gravity. Z2. Evaluation of sectional forces

The above mentioned _{N,7, dw1/dt and p are}

evaluated as follows. As stated in the Introduction, the hydrodynamic coefficients _{and are defined for}

the instantaneous submerged portion ofa section, and

are computed by Frank close-fit method at each time

step. When the instantaneous values of sectional hydro-dynamic coefficients are evaluated, the _submerged portion is approximately substituted by the _sectional form under the dotted line shown in Figure _{2. The} equations of motion which. will be derived in 2.3 des-cribe the ship motion in waves in terms of translations of the point o as well as rotatiOn about the same

point. Therefore, the hydrodynamic coefficients. for r011 motion, which are defined with respect to the point o', must be transformed into those defined with

respect to

the point o. Denoting the difference

between the reference points o and o' with the sub-script o and o' respectively, the. transformation of the

added mass and added moment of inertia, for instance, as follows:

y,

Furthermore, the nto the radiation

Hi,D force hydrodynamic dFH..R force dFH,D

-dF.

force _{is divided} dx

and the diffraction

dFH, dFHIR (2) dx dx dx where

dF.

_Hz,R

= --1

d'3

E m. _L. )

ds\

\j=i '/ dr

/ dt

-. ds

Z N..!-

(3) j=1 dt

dF10 d13

;;

3 = _____ ..-,-i

m.. -Li +

" dt /

1=1N.. ._.L.dt (4) dx dt

where

.Ozcos0

,

O=z,,sin,

(9)

and the subscripts S. H and R denote the_{sway, heave}

and roll, respectively. The same transformation is true for damping coeffiëients.

The sectional average wave surface , and the

sec-tional average velocities of wave partiôles at each time step are obtained as follows (see Figure 2): wave surface:

=

acos(Kcosx' Ksinxy'

-

₍₁₀₎

wa'e subsurface:

e

ae

cos(Kcosxx Ksinyy

_et)

(11)

FrOm equation (1Q),is

=a(sinQ2 sinQ)/Ksinx(y -;4)

and similarly,

dw _Kd

'(cosQB _c05QA)/'o5X(YB yd) (12)

where S is the area of instantaneous submerged sec-tion.

The water pressure p in equations (5-7) can be ex-pressed as follows:

180w 11

_[_..j

_.p([S7Ø]2

[rq]2)

₍₁₈₎

where _{is the incident wave potential given by}

= -!

eT _{sin(Kcosxx' - Ksinxy' _c)et) (19)}

2.3.Equations_{of motion}

Integrating equation (1) from A.P.

in the following equations of motion:

F. dF

heave:

_f

dx=0

A.P. dx F.F.,dF1 pitch:

_I

A.P. dx FP dF2 sway:

A.dX

F.F. dF2 yaw:

fxdx=0

AJ. dx

F? dF

roll: AR.dx

Arranging equation (20), the resultant heave-pitch-sway-yaw-roll coupld equations of motion become as

follows:

[A] (R}+ [B] {R}+.[C] {R}= {E} (21)

{R}=[Z0,e0,Y0,,0,0JT

(22)

The elements of coefficient. matricesA, Band c, and the wave exciting force vector E are described in the

Appendix.

24. Wave loads acting on a transverse sectioA of

ship's hull

If we define positive directions of wave lOads acting

on a section (x x) as shown in. Figure 3, they may be calculated in the following manner. Let E,' and be defined as f011ows:

Figure 3. Positive directions ofwave loads.

x1

=f

e.dx A.?. x1 G

f g.dx

A.?.

where e and g1 express the sectional forces due to the diffraction of incident waves and the ship motion, respectively. Then, wave loads at the section x nx1

may be obtained by the following equations. dw (13)

sinQ)/K(y y)

_ac,.,eM2(sinQB di (14)

__ae12(sinQB

et

sinQA)/(y y)

(IS) where A =Kcosxx'Ksjny (16) QB =

Kcosx' - Ksinxy

-The sectional draft, d1 and

d1 =S/(y y)+.

d2, are defined by

d2S/2(yy)+

(17) m13 m31 =mHRO=mHRO. +QmHH OmSH m23 m32 ₌mSRO m0, + OSmHS +

_Om

(8)

m33 _{=mRRO 1RRO' + °sHRO + mHRO. )}

+Q(mSRO+mSRO,)

dt

=a,e

to F.F. results

xl

F1,(x1) =f(e3g3)dx.

M(x1)=

_I

(xx1)(e3 g3)dx

M.(x1) =

g5)dx

where F, M,, F, M,, M, denote vertical shear

force, vertical bending moment horizontal shear

force, horizontal bending moment, torsional moment, respectively. From the telations e2 = Xe1 , e = Xe3,

g2 xg1 and g4 = xg3, equation (25) can be

ex-pressed as follows: -(25)

F,(x1) =EG

M,(x.1) = x1F,(x1)+E G

F,(x1) =E.G

M(x1)= x1F,(x1)+EG

M(x1) = E G

Finally in order to describe the wave loads in

ship-fixed coordinate system, the following transformations are to be made.

Fv(x1) = F,,(x1)cos0 F sinØ0

M(x1) = M,(X1)cÔsØ M1 sinØ0

F(x1) = F1,(x1)cosçb0 +F. sinØ0 x1

+ f (wpgS)dxsinçb

A.P. MH(xl) M1(x1)cosçt0 +M,sin0 xi

+1 (x x)(wpgS)dxsind

A!. 0 0 Mr(Xi) M,(x1)

The additive terms in the horizontal shear force and horizontal bending moment M11 in equation (27) account for the roll-induced lateral components of vertical shear forces in still water, and S, denotes the

sectional area.in still water.

3. Experiments and numerical prediction

3.1. Model tests

The principal particulars of the ship model used in the experiment and its weight distribution arô shown

(26) (27) Displacement (W) 70.604 kgf Trim = 0.0 KM = 0.1495m KG = 0.1209m GM 0.0286m

Radius of gyration in roll = 0.30MB

RadiuS of gyration in yaw (about ) = 0.2216 L Rolling period = 1.45 sec.

LcG 0.0373maft x

= 0.292m

z _{= 0.0135m(KO=0.1222m)}

WH (heave rod and balancing weight) 5.27 kgf W (weight of sway subcaniage) 15.63 kgf

K0 (yaw balancing spring constant) 3.352 kgf.m/rad

ZB (block gage axis above ox axis) 0.0144 m

60 W(X) 40 20 heave rod balancing Weight towing carriage - I I I I 2 3 4 6 7 8 9 FP

Figure 4. Weight distribution of the model.

sway subcarriage

yav' baj1ancing spring

block gage

- ₁₎

-

I W.L

npnjg ,flfl4(a,a tflS,n -direc. gyro Z8 _vert.gyro 71/2

Figure 5. Outline of the model and towing carriage.

TabiC 2 Wave heights used in

the experimehts

c/L7j

HW/L 2.0 1/20.8

2.0-26

1/31.2 2.6 1/41.7 x1 _{Table I}

F,(x1) = f(e1 g1)dx

Principal partiulars of the model

Length L) = 2.500 m

M'(x )

f (x x)(e g1)dx

Breadth B) 0.3629m

AJ'. Draft(d) = O.1357m

in Table 1 and Figure 4, respectively. The ship model is divided into three parts which are connected at the

midship with a 5-component force block gage (F

_{, M,}

1,

M11, MT) and the suare station 7 1/2 with a 2-component force block gage (F ,M).

The configuration of the towing carriage and the ship model is shown in Figure 5. As shown there, the weight of heave rod is counterbalanced with a balan-cing weight, and a yaw balanbalan-cing spring is added to

prevent a yaw drift of the ship model. At experiments,

only surge motion of the model was constrained, but

sway, heave, roll, pitch and yaw were free. The height of waves generated at the experiments were as large as possible within the limit Of capacity of the wave maker, but was adjusted to avoid the occurrence of water shipping on the deck and/or the bottom slam-ming. The wave heights used in the experiments are shown in Table 2. Experiments were conducted only in regular large amplitude waves under the following condition: Froude number (F) = O0, 0.1 and encoun-ter angle (x) 0, 30, 60, 90, 120,, 150, 180 deg. The picked-up values in the experiments were ship motions at the point o, and wave loads at the mid-ship as well as the square station 7 1/2.

3.2. Numerical prediction of ship response

At numerical prediction of ship response, linear roll damping moment, B, which was derived by the strip method was substituted by the, nonlinear roll damping moment, which was obtained by the free

rolling of, the model. According to the suggestion of the .16th ITTC Seakeeping Committee [6], sec-tional mass moment of inertia of roll and secsec-tional height of the center of gravity are assumed as

fol-lows:

sectional roll damping coefficient b55(x): sectional mass moment of inertia in roll, i55 (x): sectional height of the center of gravity kg(x):

The last assumption says that the sectional height of

the center of gravity is constant along the whole length

of ship, and coincides the height of center of gravity

of the whole ship.

For the purpose of comparison, two kinds 'of

non-linear calculations were performed in addition to thern linear calculation. The first one (ôalled NONi) is a nonlinear calculation in which only the time-variatiOn

of sectiOnal draft is taken into account, and the other one (NON2) includes the hydrodynamic coupling ef-fect due to the asymmetry of the submerged portion

of a ship section by the presence of ship's heel as well.

Needless to say, in infinitesimal waves, the results of NONI and NON2 coincide with those of the linear calculation. The difference between the results of NON 1 and NON2, which manifests itself in large

am-plitude waves, represents the effects of hydrodynamic

coupling between vertical and horizontal ship

respons-es, while the difference between the results of NON 1 and linear calculation denotes the effect of time-varying vertical displacement of the ship.

The time interval used at the numerical integration of equations of motion is 7/32(7: encounter period),

and such hydrodynamic coefficients as sectional added mass and sectional damping coefficients are inter-polated from the table of hydrodynamic coefficients prepared in advance for various sectional draft and heel angle.

4. Comparison of experimental results and predicted

ship responses

4.1. Ship motions

As mentioned earlier, the ship motions were not measured at the center of gravity, but at the point o

(see Figure 1). Comparison between the measured and

predicted motions is made for positive and negative

peak values separately in case of heave, pitch and. roll,

while the mean value of positive and negative peak values is compared in case of sway and yaw. As an example of comparison, only one case of x = 120° and = 0.0 'is shown in Figure 6; where the circle and triangle of heave, pitch and roll responses denote the positive and negative peak values respectively. As shown there, the discrepancies between, the ship motions obtained at experiments, linear prediction and nonlinear prediction are not so large except for roll. In other words, it can be said' from the practical point of view that the nonlinearity of ship motions in

waves is negligibly small

unless such an impact

phenomena as slamming does occur. This holds true for any other tncOunter angles at zero-forward

speed, and furthermore in cases where the' ship travels at a speed equal to 1 = 0.10, although the compari-sons between measured and predicted ship motions in

those cases are not presented in this paper. 4.2. Wave loads

The wave loads are nondimensionalized in the

fol-lowing manner: = F,/pgLBa = M/pgL2Ba

= F/pgLBa

(29) = M/pgL2Ba b55 (x)xS0(x) i55 (x)w(x)2 (28) kg(x) KG

0.8 0.4 +PEAK 'S X:120' F:0.0 o EXP(..-PEAK) -- -- LINEAR CAL. --NON1

NON2

4 Figure 6(a). = M7/pgL2Ba

In this Section, wave loads are presented only for

x = 1200 and F, = 00 As seen in Figures 7(a) and (b),

large discrepancies manifest themselves between positive and negative peak values of vertical wave loads in particular, and the predicted peak values by the present method (NON2) are in good agreement with the experimental results. From tiie fact that the

difference between the predicted 'values by NON I and

NON2 is not large except for the' wave length

corres-ponding to the roll resonance, it can be said that the

nonlinearities of F and M are mainly due to the

draft variation with time.

To the contrary, the nonlinearities of Fj and _AIH are small compared with those of, F;,, and

clearly understood from the Figures 7(c) and (d).

0.4 kQ 0.2 (102) 0 2 Figure 6(e). Figure 6. Motion responses

However, the difference between positive and nega tive peak values of 2cç. _{is not neglible in relatively}

short waves although the roll amplitude is small. On the other hand, the agreement between _{the results} of 'prediction by NON I and &periments is not good while the agreement between NON2 and _experiments is satisfactory. From this fact, it can be 'said that the

hydrodynamic coupling due to vertical and lateral ship motions has a siguificant. effect

on the torsional

moment Al?..

2 3 _L1/E7

7(a) vertical shear force -PEAI< HEAVE

4

2

0

2

.3

_LJJE7

7(b) vertical bending moment

2 3

7(c) horizontal shear force

3

4

Figure 7. Wave loads. 0.3 In Figure 8, the time histOries of I,,,iCl and M.

are shown together with their two components, of which one is due to wave diffraction and the other due to the ship motion, and they are discriminated from each other by the subscripts E and R, respec-tively. In each component Of wave loads, there exist large discrepancies in phases between the linear and nonlinear calculations. However, in wave loads them selves expressed as the sum of the two components, the significant difference in phases are not fOund between the linear and nonlinear calculations. The time histones of wave loads predicted by the non-linear calculation NON2 are in good agreement with

those of measured wave loads.

Similar comparisons between predicted wave loads

and experimental results were made in the case of (ld2)

non-zero forward speed, that is to say. F,, = 0.10. As a result, it was confirmed that the above-mentioned facts found in case of zero fOrward speed hold true

even at non-zero forward speed [7].

0 -0.3 0.3 0 -0.3 - X120° 0.0 2 3

wJt74

7(d) horizontal bending momCñt

at 8(a) vertical shear force

PEAI< - - - ₀ EXP - - -. LINEAR CAL. - - NON1 NON2 X:12Q,Fn:0.0 W4E7:2.3, Hw/X1/38 M1 (-1ci3) w Jt7 2 3

1.2 MIE (1O2,) 0 3

/

o EXR

'LINEAR

-- NON1 - NON 2 o EXP. ----LINEAR --NON1 NON2 MV-MVEMVR at

fl

8(b) vertical bending moment

Te/ 2 X:12d Fn:O.O wjE72.3, H/X:1 /38 X:12d' .Fn:O.O WJt7i:2.3. Hw/X1/36

-'S 06o,. 's 0 0 0 \ 0 " 0 o

/

/

0

_/

00

/'

OL'

,//

MTMTE+MTR 8(c) torsiOnal moment Figure 8. Time histories of wave loads. 4.3. Wave height dependence of wave loads

Figure 9 illustrates the wave height dependence of peak values of wave loads at the midship by

com-paring predicted values with experimental results in

the case of w'JL7j=

3.0, x = 600 and _{= 0.1. As} discussed earlier, F and M proved to be significantly nonlinear, and in particular,. positive peak of F and

negative peak of.M are more pronounced than those of the other peaks in opposite directions The pre-dicted values by NON1 and NON2 show qualitatively

similar trends, but the experimental results are closer

to the prediction by NON2 as the wave height increas' es.

In and shown in Figures 9(c) and (d), non-linearities are relatively small compared with vertical

wave loads, and and the agreement of meas-ured values with the prediction by NON2 is fairly good.

In case of .A., NON I seems to have a trend to

over-estimate the wave height dependence of peak value. From, the fact that the prediction by NON 1 does. not agree well With the measurement While the latter shoWs a very good agreement with NON2, it tUrned out that the nonlinearity of M is mainly due to the

9(a) vertical shear fOrce

effect of the hydrodynamic coupling _{as mentioned} earlier. Such a fact is much clearly recognized in the Eange of wave length shorter than that shOwn in

Fi-gUre 9(e).

According to additional calculations, of Which the results are not presented in this paper, it can be said that the above-mentioned discussion on the wave

valid for the other transverse section of the ship hull [7]. 2

(x12)

M 0

0

X 60°

- F:Q1

2/100 _4/100

wjC7:ao

at

9(b) vertical ben4ing moment

X:60° F:0.1

w at

9(c)horizontal shear force

LINEAR

NON1

N

NON2

Hw/X

Figure 9. Wave height dependence of peak values of wave loads.

(X = wave length)

0

N0N1

JEXP.

9(d) horizontal bending moment

9(e) torsional mothent

NON1 N0N2

(_.._ __...._Q._...

LINEAR MAMD - _- u,u1_ - LINEAR 2 (x1Ø2j X:600 Fn:0.1 2/100 41100 Hw/X

w J[7:3O

at

4.4. Ship speed dependence of wave loads

Let wave loads in large. amplitude waves, which are

obtained by the nonlinear caiëulation NON2, be.ex-pressed by the sum of the linear part and the residual. Then, the magnitude of the latter component repre-sents the extent of nonlinear characteristics of wave loads.. For instance, the entire vertical shear force Fv

is described as follows:

Pv = + (30)

where

VL : linear component of vertical shear force

F.

: residual component.

In the above equatiOn, Ft,,L is obtained by O.S.M., and

then is defined by the difference between the predicted wave loads by NON2 and

Comparisons are made on the basis of calculated results at the square station 7 1/2 for shear forces and at the midship fOr bending moments. As a typical example, the ship speed dependence of wave loads only in a case for x = 120°, = 1/25 is presented in this paper as shown in Figure 10 where X denotes

wave length.

As seen in Figure 10(a), ñonlinearities that is

to say, F', reduces at frequencies which are close to roll resonance as the ship speed increases. Because, with increment of ship speed, the roll amplitude de-creases and the resonance frequency of roll moves to a range of wave length where the vertical shear force

diminishes. However, in relatively short waves an op-posite tendency is clearly found: nonlinearities become more significant in case of x = 60°, which is not presented in this article. Almost the same featurecan be found in_M as easily understood from Figure 10(b).

Generally speaking, nonlinearities of vertical wave

loads J. and become significant with increment of ship speed, and therefore, the peak values of wave loads in large amplitude waves _{may exceed by far the} peak values predicted by linear calculation.

The trend that nonlinearities of_{.id and fH become} less significant as the ship speed increases is detected at frequencies close to roll resonance.. This is due to

the same reasOn for the vertical wave lOads, and holds

true in almost the whole frequency range dipicted in the figures when the ship advances in head waves. That is to say, the predicted peak values of I _andMH by nonlinear calculation are closer to those by the linear calculation as the ship speed increases. On the contrary, in following waves, the increase of ship speed results in enhancing the. nonlinearities so that the wave loads themselves become smaller than those obtained by the linear calculation.

The ship speed dependence of torsional moment has qualitatively the same trend as that of F, and Mi,, but it is relatively small compared with that of and.

M as shown in Figure 10(e).

MVL

4

0

4

10(a) vertical shear force

10(b) vertial bending moment

Xl20

atm.

F

-0-0.0

_6_p.1.

/

/

. . . -w17

PEAK

-PEAK 1 Tp ,,_L,

/

,/ 7

D---\

FHL

4

10(c) horizontal shear force

LL) 1i7 X:120 at Fn -o-- 0.0 -'a- 0.1

10(d) horizontal bending moment.

5. Conclusions

From the investigation into ship motion and wave loads in regular waves of large amplitude, in which neither water shipping on the deck nor bottom

slam-ming were assumed to occur, the following conclusions may be drawn:

(1) Ship motions are less sensitive to the. wave am-plitude and retain their linearities within the

above-.3

2 MTL 0 2

+PEAK

-PEAK .

-10(e) torsional moment

Figure 10.' Ship speed dependence of nonlinear wave loads in

laige"amplitude waves.

mentioned limit of wave height. Therefore, except the roll at frequencies close to roll resonance, the

am-plitude of ship motions can be estimated by the

linear calculation with 'an accuracy sufficient from the practical point of view.

Vertical shear force, vertical bending moment and torsional moment are likely to manifest. their sig-nificant nonlinearities as the wave height increases. The nonlinearities of vertical shear force and vertical bending moment are mainly due to the time-variation of submerged sectiOnal area of the ship's hull while the, nOnlinearity of torsional momerit is attributed to the hydrodynanüc coupling between vertical and lateral ship motions, which becomes dominant in relatively short waves. '

As the ship speed increases,roughlyspeaking, the nonlinearities of vertical shear force and vertical

bend-ing moment become so significant that vertical wave

loads in large amplitude waves may be underestimated by the linear prediction method. Qualitatively the same trend is found in the nonlinearity of torsional moment, but its ship speed dependence is small

com-pared with the ship speed dependence of vertical wave loads.

The wave height dependence of horizontal shear force and horizontal bending moment is small

corn-pared with that of vertical wave loads. Increase Of the

ship speed results in reducing nonlinearities of

horizon-tal Wave loads in head waves, but enhancing them in

following, waves.

(v) _{The predicted results o ship motions and wave} loads by the present method show good agreement with the experimental restilts. The main advantage of the present method; which is principally based on the conventional Ordinary Strip Method, is that the pre-diction of nonlinear wave loads in large amplitude waves can be easily performed with Sufficient accur-acy without solving three dimensional higher order

potential problems. 6. Acknowledgement

The computation was carried out by HITAC M 200H/280H in the Computer Center, the University of Tokyo. This research is partly supported by the Grant-th-Aid for Scientific.Research of Ministry of

Education of Japan.

The authors wish to express their cordial thanks to Mr. M. Koyanagi, Mr. T. Wada and Mr. T. Kawainura for their cOoperation in carrying out the experiments. References

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Wahab, R. and Vink, 1.11, 'Wave induced motion and wave

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Progress, VoL 122, No. 249, 1975.

Leibowitz. R.C., 'Computation of theoiyand experiment for slamming of a Dutch destroyer', NSRDC Report No. 1511,

1962.

Yajnamoto, Y., Fujino, M. and Fukasawa, T., 'Motion and longitudinal, strength of a ship in head sea and effects of

nonlinearities' (1st, 2nd, 3rd Report), Jour. Society of Naval Architects of Japan, Vol. 143, 144,1978; Vol. 145,1979. Frank, W. and Salveseñ, N., 'The Frank close-fit ship motion computer program', NSRDC Report No. 3289, 1970. Seakeeping Committee, 'Comparison of results obtained with computer programs to predict ship motion in six.degree-of freedom and associated respOnses',Proc. 16thITrC,'1981. Fujino, M. and Yoon, B.S., 'A study on wave loads acting on a ship in large amplitude waves' (3rd Report), Jour. Society of Naval Architects of Japan. Vol. 158, 1985.

Appendix

The elements of coefficient matrices, A,,, B,, and c of equation (21) and vectorE _{are as follows [7]}

A11 ₌

f

_{dx + I mHHdx +}

12 =fxdrfm

xdx g A13 _{=J m11.dx} A14 _{I mJf,xdx}

A'=

_{øo.fg(Zo_gc+fmHROdX}

A21

=_fxdx_fmJfHxdx=Al2

fx2dx+fmHHx2dx

g A23 ₌f mflxdx A1, A24 ₌

f mHSxdx

A, ='ø

f(z-

xdx

-

JmHROxdx A31

=fm,dx

=A13 A =

f m,xdx

=A23 A33

=fdx+fmdx+m5

A4=f!xdx+finjcdx

dx A41 =f m11xdx=A14 A42 ₌

mx2dx

=A24

A43 =fxdx+fmxdx=AM

444

fx2dx+fmx2th

A45

=f(z z)xdx(l

A51

=øof(zz)dr+fmHRQdx=AlS

A52

øf(z _z)xdx_fmHROxcjJ=

_A

A53 =f(zzdx+fm0dx

A54 =f(z z)xdx+fm0xdx

A55 =fidx+fW_(z_zp2dx+fmodx

= dx + (z, - z, )2 f dx m0 dx B11 =fNHffdx_U[mHffJ B12 _{= _fNffHxdx+U[mHHx]}

+Ufmgdx

B13 fNHSdx -U[mfl$J B14 _{=fNHSxd.é_U[mffSxI}

Uf

_mHSdx B15

_{nfNHROdx_U[mHRO]+fw_(z,_.z;)dx}

B21

= -

I Nxdx +

U

_{[mHHxJ -}

_U

_f

mHfldx B =fNHHx2dx_u[mHHx2] B _{=-fN,jrxdx+U[msx] UfmHSdr} B24

= -

_{f NHSx2dx + U[mHSx2]}

-

C14 E3 = =

-

_{I N110xdx +U} [InhfRox I - 4,f! -- zg)xdx

_UfmHROdx

B3, =1

NdrU[pn1

B13 B32

fNxdx+U[mx1

+Ufmdx=-F B33

fNdxU[mJ

B

=fNxdx -

_{U[mx] - UI rndx} B35

fNdx(J[m0J

B4

f4xthU[tnxJ +UfmSh,dx= B23.

B42

fNx2dx+U[mx2]

=B24 B43 fN5gxdxUErngx) +Ufmdx B

=1 Nx2dx -

_U[mx2J B4

fN0xdxU[m0x1 +Ufm0dx

= _{øo I} (z - z) dx₊INHR0dx - U

[rn0]

B52 =

-4, f(z z)xdxfNHRoxdx

U[mHROx] +UImHROdX B53

fN0dxU[m0]

=B35 B fN0xdx - U[m0xJ - UI

rn,dx

B55 =1 N0dx - U[mRRO] _AJ4,I +B,A _and

B are determined from free rolling of a model

Cl'

=0

=UfNHHdX - U2[mflHl c'3

=0

Cl4

= -

Uf

N,dx +

U2 _s1 C15

=0

Cl, =0

C2i =UfNHHxdx+U2[rnHHxJ

-

U2f mHHdX

c23

=0

Cl4 = UI N,xdx U2[mHSx) + U2f C25

=0

C31 =0

C3 =UfNdx -

U2[rn,1 C33 =0

= -

Uf

Ndx

+.U2:[m] C35. =

rndx

dw 1 +m_HRO

Id2

dt

pgff.[z' a(1 +e' -

e'cos(Kcos,x'

-

Ksinxy'

- et

dz'dx-If

a;;,

-(_x)]

_UfLmHH_'+mHS_

dw3 mHRO dw2

C4' =0

C42

=UfNxdx_U2[mthx] +U2fmdx=C

C43 =0 C44

= -

UI Nxdx + U2

[m..x] -

U2f mdx + K0

:_0f!!(Z_.zpxdx

Cs' c52 _{=UJNHROdx_U2[mHRO]} C53

=0

C54

UIN,dx

+U2[rn0] C55 0 =

fpgS0dx+pgff. [z'+ a(1+ez'

- e_") cos(Kcosxx' - Ksinxy'

_et)j dydx+.pa2c..,2ffc,[e_212'

-

dy'dx 1 {mHH

-(')

+mff

__)

+m_-)] dx

af3\T

I

dw dw

dwl

+f1HH 7' NHs.-- + NHRO._-3_:I dx

r

dw1 '2 dw3 -LmHH

HSHRO

dt E2 = fpgS0 (x)dx + pgff. [z' + a( 1 + e

-.

et)] (-x)

dy'dx +.fpa2,2ffci[e_2KE'

-

erJ (.t)dy'dx +f [

mHHA--+ mHS +

3)i±_x

dx

2Kz'

-

_{eI dz'dx +fLrn,,}

a fdw2\ div3

+m__)+rn

_(-)dx

I

dw dw dw 1

+fIN 1+N +jV Idx

_{i' di}

_di SRO _dt

dw3i

Tj

E4 =

_{Pff.[' -a(1 +e'_e_Kr)cos(Kcosxx!}

- Ksin" - Wet)] xdz'dx

-

fpa2w2ff[e21_. e2']

xdz'dx

r

a fdw1\

\

1 cIw2

1Lm"

af\1

I

ã +rnSO

jxdx +f

dw _dw3l

+N+N0

----jxdx

If

dw dw2 _dw3\

)xj

I

dw dw

_dwl

Uf[ms!+mrn0 -Jdr

E5 ₌

[-z'

_{+ a( I} -FeCZ'- e )cos(Kcosx' - Ksinxy' _(&)t)l :(ydy'+zdz') dx

+.pa2o.,2ff, [e'.- er2lCj

_(ydy'

+zdz')dx-f-fw(z' -z;)dX.SflQ

r

a fdw1\ a fdw2

Lm'°

j J)sRO

T

fdw\1

'I

dw -I-rn _N

'°at\

dtJ

HROdt

dw 1 1 dw1

+N0--+N

_iJdr

U[rnHRO-dw2 dw

+rn

_dt RRO

_dt

where

m11 : mass of heave rod and heave balancing weight, m5 : mass of sway subcarriage,

K, : spring constant of yaw balancing spring

and A.P. and F.P., which are the lower and upper bOund of the integrations along the ship length respec-tively, are omitted for simplicity of descnption

The and

_'G

included in A55 denote sectional

moment Of inertia of roll about axis through g and G

which are the centers, of gravity of a section and of

the whOle ship, respectively In the present calculation,

they are assumed to be equivalent from the equation

(28).

The parentheses included in B,,, C, and E1, so-called

end terms, mean the differences between the values

in themselves at F.P. and A'.P

The equations of motion include nonhnear

coup-ling terms, for instance, A15, A, B and etc. How-ever, in this time step analysis, they can be easily treated by introducing an iterative solution scheme at

each time step.