Calculation of heaving and pitching motions of a ship by the Strip Method

Introduction

Ship motions in waves are generally calculated by the so-called strip method from the practical point of

view, and it has been found that this theory can describe those motions with fairly good accuracy. As the

numeri-cal numeri-calculation is still rather complex and needs much

time, the authors tried to get the numerical values using a high speed computer IBM 7044, limiting the case to

the motions of a ship travelling in two-dimensional regular waves and in a direction normal to the wave

crests.

We used the

equations of motion proposed by Watanabew, and obtained added mass and damping

coefficient according to Tasai's method2. Even with the high speed computer the calculations of added mass and damping. coefficient require much time, so that we

calculated them to the parameters for a wide range in

advance, from which we picked up the required values

by interpolation or extrapolation. By this procedure

the computing time has been reduced very much.

Comparing the calculating results with the experi-mental values, it is noticed that there are some differ-ences between them owing to the three-dimensional

effect and to the effect of forward speed To investigate these effects, we used Gerritsma's experimental values,

and corrected the calculating manner to the

three-dimensional effect for the heave damping coefficient.

Now we can calculate the ship motions in regular waves with good accuracy, and put our calculating method into practical use. This may be extended to the motions in irregular waves, and be useful in calculating the increase in the resistance of a ship in waves.

Equations of Motion

We used the equations of motion for heave and pitch in waves presented by Fukuda18> who summarized the

Watanabe's idea1. Assumptions and the coordinate

systems are as follows:

of a Ship by the Strip Method

Hitoshi Fujii* and Yoichi ogawara**

Abstraèt

In general, it is very difficult and needs much time to calculate theoretically heaving and pitching motions of a ship in waves. The authors tried to calculate these motions in regular waves by the well known strip method using a high speed computer IBM 7044. They compared their calculating results with Gerritsma's experimental values on Series Sixty models, and

con-sidered the three-dimensional correction for the heave damping coefficient to their original

calcu-lating manner. The test results at Mitsubishi experimental tank were found to fit their modified

values very well.

In this way, we succeeded in putting the calculation of heaving and pitching motions of a ship into practical use. By employing this method and a high speed computer, it is possible to calculate these motions on 60-70 conditions of one ship only in 1-2 minutes with pretty good accuracy.

Assumptions

The oscillatory surging motion appears to be of no

direct interest in defining the seagoing qualities of a ship. This is to be an extension of the practice

of neglecting the effect of surging on coupled heave-pitch motion.

According to the strip theory, a three-dimensional effect is not to be considered.

A ship's hull form is wall-sided or nearly wall-sided

in the neighbourhood of the water line, and her

draft is to be considered equal throughout her whole length.

A ship is travelling in two-dimensional regular

waves and in the direction normal to wave crests.

Among wave exciting forces, the Froude-Kriloff force is to be considered as the pressure of waves

acting on the bottom of the rectangular section with

the same breadth and the same sectional area as the

ship section. Coordinate systems

As shown in Fig. 1, the coordinate systems o0-x0y0z0

fixed to the space and o-xyz fixed to the ship are

employed.

Ooxo.. ye. Z5 Space axes

ox. Il

Z Body axes

Dr; Eng. Hydrodynamics Laboratory, Kobe Technical Institute, Technical Headquarters Hydrodynamics Laboratory, Kobe Technical Institute. Technical Headquarters

Fig. 1.

A ship is assumed to go straight under a constant

speed V in the direction of positive x, and when one of the wave crests is at midship at t=0, the surface eleva-tion of the waves may be written as

cos (kx+wt) (1)

By these assumptions the equations of motion about the heaving at C. G. z0 and the pitching can be writ-ten as follows.

azoc+ bz0c +cz0G+dO + eO + gO =F

These coefficients are explained in detail in Ref. (8)

The exciting force F and exciting moment M in the right hand of the equations (2) are given, resulting in

the form of

F=F'0 cos (Wet+EF) 1 ₍₃₎

M=MOcos(wOt+CM) J

and the solutions of the equations (2) can be obtained in the form of

ZQGZ0GACO5(Wet+EZ) 00A cos (wt+eg)

3. Approximation of a Ship Section

with the Lewis Form Section

According to Ref. (2), the added mass coefficient C0K4 and the damping ratio A for the heaving motion of the

two-dimensional cylinder in the surface of a fluid are

given to the Lewis form section for the parameter, half

breadth-draft ratio a, the sectional area coefficient j3

and the nondimensional circular frequency Ea. The

co-efficients of the Lewis transformation, a1 and a3, are related to a and as follows.

a-i

a1=

_a+l

(a3+l)

I(_B'+B'2_A'C'), when A'O

(5)

a3= I. 2B' when A'=O where +4+3ora =A'

3,ra

=A'

-4ra

One of the results of the Lewis transformation is

shown in Fig. 2 in regard to the cargo liner, Nissei Maru. It is seen that the forward part of a ship can be approximated very well, but the afterboby is

trans-formed into a slightly diiTerent shape, and especially it

is striking at the square stations 1/2-4½. The results

about some other ships like an oil tanker, a cargo liner

L.W.L

(2)

Fig. 4. Lewis Sections including limit values of

and a passenger boat, not presented here, showed the same tendency and approximation as the preceding

example.

On the other hand, this transformation is not

appli-cable to all form sections, but the range of.a and $, that is, a1 and a3 should be limited in order to get the figure transformed as the section shape. This condition can be derived in the following form from the characteristics of the mapping function.

Ia1I+3a3-1O, when a,O

a02-3a,-l<0, when a,<O This is shown in Fig. 3.

The shapes corresponding to the limit values of S

for each a are shown as the most inner and outer curves

in Fig. 4. It is seen that the Lewis transformation can represent the section profile of a ship below the water

line with rather good accuracy, and that it can also re-present the bulge at the bow to a certain degree.

4. Added Mass Coefficient and Damping Ratio

Added mass coefficient C0K4 and amplitude ratio A

of a cylinder with the Lewis form section heaving in

the free surface of a fluid are theoretically obtained by Tasai02 and Grimdll). Here these values are calculated by Tasai's method using a high speed computer IBM 7044. Considering both the grade of approximation of a ship

section with the Lewis form section and the results of

Tasai's experiments9 (10), the added mass and the

damp-ing coefficient of the cylinder with a ship-like section

may be estimated by this method with good accuracy.

In order to investigate the motions of a ship by, the strip method, the length of the ship must be divided

into 20 sections or so, and the preceding hydrodynamic characteristics must be calculated for. each section with

Correspond to a-3a3J=Q Transformable range i3=3or(2-1/á)/32 O=3or(2a)/32 1.0 2.0 a

Fig. 3. Limits of a and 9 for Lewis Section

} (6)

- Original

Transformed

Fig. 2. Transformation to Lewis Section (Nissei Maru)

2.0 1.0 0 (4) fl=0.5596 0.8247 1.0897 1.3548 1.6179

about 60 conditions respectively. And it was found that these computations required about 450 minutes per one ship with IBM 7044. As this was not practical, the au-thors adopted the following method to reduce the com-puting time, that is, they calculated these coefficients for wide ranges of parameters a, jS and in advance, put them into the computer, and obtained the necessary value by interpolation or extrapolation, the method of curve fitting. The ranges of a, $ and were decided as fol-lows, considering both the transformable range of a and $ (Fig. 3) and for various ship speeds.

a =O.2--.0.6

e3=0.5'l.l = 0.l-2.6

In the computation of CO3 K4 and A, the velocity potential was approximated with the first four terms,

and the boundary condition on the surface of the cylin-der was satisfied at 19 points. By the consicylin-deration on

the energy balance, the accuracy of the results was

found to be higher than 0.1% in all ranges. One of the

results is shown in Fig. 5 compared with Grim's

values">. They coincide very well, except C0K4 for the small value of . Grim's values were read from the

figures in his copied paper, so that there may be some errors in copying or reading them.

The curve fitting is carried out by the procedure of interpolation to the four points with the fitting of the

Lagrange's interpolation formula on a cubic polynomial expression, or by the procedure of extrapolation to the

three points near the end with the fitting of the quad-ratic polynomial expression. But in the extrapolation

of K4 to the very small value of , the fitting of the

logarithmic curve is assumed according to Ursell(12) who found that when an elliptic cylinder heaves very slowly and the free surface condition can be approximated to

=0 (y=O) K4 is to the first order

0.1862-1.8664 [log +log (1+a)J (7)

Under the assumption of this relation to the cylinder with the Lewis form section, the formula (7) is applied

for and for the formula of the

fol-lowing form is fitted. K4=a±b log

5. Calculation of Heave and Pitch

According to the preceding calculating manner, the

computation of the ship motions of heave and pitch is carried out by IBM 7044. The input data are the length

of a ship; the breadth, the draft and the sectional area

at midship respectively; the displacement, the position of C.G., the radius of gyration for pitch, the fluid densi-ty; the breadth, the draft and the sectional area for each section divided along the length of a ship, the forward

speed, the wave length and the wave height. As an

example of the output data shown in Table 1, there are the main dimensions of a ship, C0K4, A2 for each section,

coefficients of the equations of motion, amplitudes of heave and pitch, their phase lags to the wave and the data for the estimation of natural periods. It requires only about 1.5 minutes to compute 6---7 conditions on waves and 8.-9 conditions on ship speeds per one ship.

Comparison with experiments at Mitsubishi experi-mental tank on the oil tanker and the cargo liner are

shown in Fig. 6. The main particulars of these models are shown in Table 2. In spite of the linearized, strip

theory, the computed values can be said to coincide with

the experiments very well as a whole. But near the natural period, the calculated heaving amplitude be-comes considerably large, and the calculated pitching

amplitude becomes conversely small. On the other hand, the difference between the calculated and experimental values becomes larger with the increase in the advance speed of the ship. It seems that there are some effects of the three-dimension and those of the advance speed which are not considered in the strip method. The phase lags of the motions to the wave are not shown here, but they denote the same degree of agreement as in the case

of amplitudes. 1.5 1.0 0.5 3 9=0.8

-Authors Grim

\

_s. \d' Authors

-

Gr 02 06 10 0 02 06 10

Fig. 5. Comparison of calculated values of A and C0K4 with Grim's results for $=O.8 1.0

Table 1. Example of output

SECTION (0 DENOTES THE VALUE AT MIDSHIP)

BETWEEN PP (MEAN DRAFT (DM) (M)=0.12438E-00)

1-1

SHIP SPEED KT) =0.18990E 01 PERIOD OF ENCOUNTER (5) = 0.83276E 00

(M/S)=O.97693E 00 CIRCULAR FREQUENCY OF ENCOUNTER (1/S)= 0.75450E 01

FROUDE NO. =0.19980E-00

SECTJONAL ADDED MASS AND DAMPING COEFFICIENTS

COEFFICIENTS OF THE EQUATIONS OF MOTION

D= -0.12467E-00

E= 0.31184E 01

G = -0.34359E 02

STATION HEAVING

X (M) AMP(M) 2AMP/HW AMP(DEG)

C.G. 0.18652E-OI 0.73447E 00 2.558

(0.16047E-0I) (O.63189E 00) (2.520)

ESTIMATION FOR THE NATURAL PERIOD

THO (SEC)= 0.95498E 00 TPO (SEC)=0.91957E 00 (0.95947E 00)

Table .2. Main particulars of models

EPSI M= 98.571

IN ( ) DENOTE THE CORRECTED VALUES) PHASE LAG PITCHING _{TO WAVE (DEG)} AMP/MAX. WAVE SLOPE HEAVING PITCHING

0.68208E 00 -111.351 -18.424 )0.67188E 00) (-104.165) (-20.290)

Models L55 mx B mx d m C, L9/B d/B d/L5 ton Trim m

Tanker A 4.20 x 0.6027 x 0.220 0.7973 6.969 0.3652 0.0524 0.4445 0.2581 0

Nissei Maru 4.20 x 0.5758 x 0.2634 0.7272 7.295 0.4575 0.0627 0.4636 0.2795 0.0269 A

SerIeS 60 _{2.438x0.348x0.139 0.7000}

7.005 0.3994 0.0570 0.0829 0.250 0

HEAVING PITCHING

A= O.15233E 02 FC = 0.10034E 01 A= O.473I3E 01 MC = -0.53838E 00 B = 0.20842E 02 FS = -0.11518E-00 B = 0.82631E 01 MS = 0.35720E 01 C = 0.66653E 03 FO = O.1OIOOE 01 C = 0.22450E 03 MO = 0.36123E 01

2X/LPP 2YB/BO T/TO S/SO ALPHA BETA

-1.00000 0.08900 0.10000 0.00500 1.11410 0.55399

-0.90000 0.36800 1.00000 0.14000 0.46066 0.375 15

1.00000 0. 0. 0. 0. 0.

2X/LPP ALPHA BETA XID COK4 ABAR**2

-1.00000 1.1 1410 0.55399 0.80689E-OI 0.15197E 01 0.20249E-01 -0.90000 0.46066 0.37515 0.80689E-00 0.58417E 00 0.22480E-00

1.00000 0. 0. 0. 0. 0.

SHIP

LENGTH BLOCK COEFFICIENT FLUID DENSITY (KG. S2. M-4)

LPP )M) =0.24380E 01 (CBPP) =0.70361 (RHO)=0.I0188E 03

LWL (M) =0:24788E 01 (CBWL) = 0.69202

BREADTH (M) =0.34800E-00 LONG. RADIUS OF GYRATION (K/LPP) =0.25000

DRAFT (M) =0.13900E-00 LONG. MOMENT OF INERTIA (KG. S2. M)=0.31404E 01

MIDSHIP AREA (M2) =0.47700E-OI WATER PLANE

DISPLACEMENT (TON)=0.82900E-01 AREA )M2)= 0.667I6E 00

VOLUME (M3) =0.82978E-01 CENTER OF FLOATATION (M) = -O.39349E-0I

MIDSHIP TO C.G. (M) =0.12200E-01 MOMENT OF INERTIA ABOUT C.G. (M4)= 0.22598E-00

LPP/BO= 7.00575 TO/LPP=0.05701 BML )M) = 0.27234E 01

BO/TO =2.50360

RESULTS OF CALCULATION

CASE I HEAD SEA

WAVE LENGTH/LPP =1.00000 WAVE PERIOD (5) = 0.I2498E 01 WAVE LENGTH/WAVE HEIGHT=0.48000E 02 WAVE CIRCULAR FREQUENCY (IfS) = 0.50273E 01

WAVE LENGTH (M)=0.24380E 01 MAX. WAVE SLOPE (DEG)= 0.37500E 01

WAVE HEIGHT (M)=0.50792E-01 WAVE CELERITY (M/S) = -O.19507E 01

HEAVING AND PITCHING MOTIONS OF A SHIP IN HEAD OR FOLLOWING SEA DATE 1965-06-01 NOTE SERIES 60 CB=0.7 (B CORRECTED)

D= -0.12467E-00 EPSI F=-6.584

E=-0.10126E 02

G=-0.54721E 02

B (CORRECTED)=0.28457E 02

Tanker A Nissei Maru Series 60 C=0.7 1.5 1.0 0.5 0 Calculated values

Calculated values (b corrected)

Fig. 6. Comparison of calculated and measured motions

Pitch Exp. values

+ A/L=0.75 A _1.00 o _1.25 o 1.50 1.75 2.00 0.75 Heave , / Exp. values + X/L=0.75 A _1.00 a _1.25 0 1.50 1.75 2.00

,iJ

i,;i _L

".1

/

lo

"

\ ' 1.75

20'

1.5

,//°,

,' /

5- _o,,I

,.

/ / i -- / I A0 - n/-1.25

/

A / 10 _0.75' o.\ ,. + + * * Pitch 1.25°

/

1.0

\A

+ Heave

'II

i!iI I

-'

2.0/','

\

1.75 _, I 15

_/

/ I 1.25' / 1.0/ A a 0.75 + Pitch

1.75f

0.7

5/"\

+ Heave

tf

1.75j

1.5

/

_{/ I} .1

/

_/ / _I / ' 1.25 _,/'

':

0.5 1.5 05 15 05 10 15 ilz 05 10 1.5 lie 05 10 15 lie 05 10 15 lie 1.5 1.0 0.5 0 1.5 1.0 0.5 0 1.5 1.0 0.5 0 1.5 1.0 0.5 0 1.5 1.0 0.5 0

6 1.5 0.5 0 Cal. Value \ * Relative error(%)= (1.0 Exp. Value) x 100 The natural periods of heave and pitch at zero speed, as shown in Table 3, coincide very well with each other.

6. Gerritsma's Experiment on the Series

Sixty Models

As has been stated, the calculating results show

some disagreement with the experimental values, due

to the effect of the three-dimension and that of the

forward speed. Now comparing these results with the

Gerritsma's experiments on the Series Sixty models, it was tried to investigate these effects, and to modify the calculating values carried out by the strip method.

Heave

--S

-9-0.75 Phase lag 1.00 1.25

Table 3. Natural periods of heave and pitch for F0=0

1.50 1.75

Fig. 7. Comparison of calculated and measured motions

for Cb=O.7, F=0

Gerritsma used three models with block coefficients C=0.6, 0.7 and 0.8, and experimented with the conditions

F0=0.15'-.0.30, 2/L5=0.75-1.75 (head sea) and h/L=

1/48. He measured the amplitudes of heave and pitch, the phase lag between heaving and pitching, coefficients of the equations of motion, and finally the forcing func-tions of the models in waves.

6.1 Motions in waves

An example of the comparison of our calculated and Gerritsma's measured motions is shown in Fig. 6. Ger-ritsma also tried to solve the equations of motion using the coefficients obtained from his own experiment. One of these results is shown in Fig. 7 compared with ours.

From these comparisons we can conclude as follows: The calculated amplitudes of both heaving and

pitch-ing show a good agreement with the measured results so far as A/L5=0.75-.1.Oo. However, the calculated

values of the amplitude of heaving become slightly larger than experimental values, and the calculated values of the amplitude of pitching become slightly

smaller, when 2/L5 is larger than the above-mentioned. And though there are shown no results, they are found to be in better agreement as C5 increases from 0.6 to 0.8.

On the other hand, the estimation of natural periods for heave and pitch has a good accuracy, as shown in

Table 3.

6.2 _{Comparison of the coefficients of the equations}

of motion

Comparison and its investigation were carried out

under the following assumptions.

The difference between the calculated and measured

values is due to the three-dimensional effect when

the speed effect is not recognized, because the mea-sured added mass and damping coefficient of the two-dimensional cylinder with a ship-like section showed

a good agreement with the calculated values of the

cylinder with the Lewis form section.

According to Gerritsma, in a first approximation, the coefficients a-g and A-G are the same for the

motion in waves and in calm water.

The results of calculations and experiments> are presented in Fig. 8. Due to wall effects the measure-ments for frequency w < 4 are not very reliable. Fig. 8 indicates that except the coefficients of the cross

coupling terms there are no three-dimensional and ad-vance speed effects in the mass a, the mass moment of

inertia A, the damping coefficient for pitch B and the coefficient of the restoring function for heave c, and these calculated values coincide with the experiments

very well. Models

Heave _Pitch

Exp. values Cal. values Relative_error* Exp. values Cal. values Relative_error* Original Corrected Tanker A 1.27 s 1.265s 1.270S

0 %

1.27 s 1.207s 4.36 % Nissei Maru 1.28 1.261 1.269 0.86 1.26 1.274 -1.11 Series 60 C5=0.7

-

0.951 0.960

-

-

0.915

-Pitch 9

/

OExp.values . Cal. values (Gerritsma) 0.75 1.00 1.25 1.50 1.75 0.75 1.00 1.25 1.50 1.75 AlL Calculated values

Calculated values (b corrected) go 'U L 'U 0 1.0 0.5 0

2.0

_.

1.0 0.95 0.9 0 Exp. values -O---Cb =0.6 ---a-- 0.7

-°--

0.8 6 (Gerritsma) I-I 2.0 1.0 0.20 0.15 0.2 0 -0.2 1.0 0.95 0.9 Exp. values -A/L= 0.75 1.00 1.25 1.50 1.75 0.20 0.30 5 7

L!L

-D 2 4 Exp. F 6 values = 0.15 0.20 0.25 0.30 Mean Cal. values L/g o A o

- --0 0X x

\

- >-c - --values Cal. value 2 - ' I' w/ CbSO 6 F=0.2 Exp. valued d D 2

4w7

Cal.

,Ii:1iuxp.

6 values V values(d= D)

---'

--Cal. 0.25 0.30

---6 (Thi 6 2 4 wIL/g

hal. values (for each Gb)

'

c)

F

0.10 0.20 0.30 0 0.10

F

Fig. 8. Coefficients of equations of motion (Series Sixty C5=O.7) 0 g -50 -100 0.04 0.02 0 -0.02 -0.04

On the other hand, the damping coefficient for heave

b has a considerable three-dimensional effect, and its

effect increases with the decrease in the block coefficient.

On the coefficient of the restoring function for pitch C, it is the same tendency for both the calculated and the measured results that they decrease with advance speed, but considering the fact that the measurements were carried out for the case w = 0 and corresponded

to the calculations for the case 2/L, =

, it seems

that there is a considerable difference between them. Though the cross coupling coefficients

e, E and g

are nearly equal between the calculated and the

mea-sured results, they show. considerably different

tenden-p P

cies to frequency and ship speed, and these variatiOns

cannot be estimated here clearly. The coupling terms

seem to play no significant part on a ship motion,

considering the necessary forces or moments for unit dis-placement of heave or pitch. But according to Gerritsnia these terms cannot be negligible, because the solutions of the uncoupled equations are very different from those of the coupled equations.

Besides these experiments, Gerritsma(6) carried

out forced heaving experiments with a seven-section model of the Series Sixty with C = 0.7 in still water,

to investigate the distribution of added mass and

damp-ing along the length of a ship model for the case of

w=6rad/s w=l2rad/s

-w =6 rad/s = iiiii (1) = 1 0rad/s _ ___ _{- w ra} /_,s w=6rad/s

,/

F1!I.w 8rad/s Iw=l0rad/s

F1"

w=6rad/s _.p-.

L::T

717 .n'w p F Experimental values Calculated values

Fig. 9. Comparison of calculated distribution of a11, b', d' and e' with experimental values

forC5= 0.70, F5 =0.20 4 2 0 4 2 0 4 2 0 20 10 0 20 10 . 20 10 0 20 10 0 1.0 0

1.0

1.0 E 0 1.0 0

1.0

1.0 0

1.0

10 0

10

10 0 10 0

10

10 0

10

A

= 0.15-.0.30 and w = 4'--14 rad/s. Fig. 9 indicates a

comparison of the calculated distributions of sectional coefficients, a11, b', d' and e', with experimental values for Froude number 0.20. Computations are carried out in the following formulae.

a1' = pjrB2CoK4 d (82C0K4) e'= b'(x-x0)-2Va,' where g: Acceleration of gravity

It is remarked that mutual interference, the

influ-ence of the fore section against the aft section, appears obviously in b' and e'. But these influences disappear when a model is considered as one body and the

calcu-lated results become nearly equal to the measured

values, when sectional computed values are integrated in ship's lengthwise direction.

On the other hand, Gerritsma obtained the coeffi-cients of the equations of motion except those of the restoring functions, regarding a model as one body at the seven-section model experiment7. The followings

are found for each coefficient by the investigation of the results:

"a" and "A" coincide with calculated values even in the low-frequency band.

A three-dimensional effect for "b" is smaller than

the values in Ref. (5).

"B" differs slightly from calculated values, and an advance speed effect is noticed a little.

Coupling coefficients d. e, D and E coincide with calculated values very well both qualitatively and

quantitatively.

Inthis way according to the values in Ref. (7), there is

noproblem about the coupling terms; but the correction of "B" should be considered.

These results are found only for the case of C1=0.7,

AlL 1.751.501.25 1.00 0.75

0.5

AlL 1.75 1.50 1.25 1.00 0.75

Fig. 10. Wave exciting force and moment (Series sixty C1=O.7)

and the other cases like Gb = 0.6 or 0.8 are not obtained

here, so that the authors did not dare to consider them

in the following, from a point of view that they aimed to obtain the effect to the change of C1.

6.3 Wave exciting force and moment

Fig. 10 indicates a comparison of calculated values with experimental values on the nondimensionalized amplitude of the exciting force C and that of the ex-citing moment CM and their phase lags a and 8 with

regard to the wave, the crest of which is at the position of C.G. of a ship. Coefficients, CF and CM, are defined as follows.

Cf= Fo/TAWhA

CM = Mo/iK 0WA

It seems that there is no effect of three-dimension and that of ship speed, and calculated results agree satisfactorily with experimental values. Here we

re-versed the sign for a measured from what was presented in Ref. (5). Considering its physical meaning and refer-ing to other results (13) (14), we estimated that, on the sign

for a, either a part or all results in Ref. (5) was by

mistake put conversely.

6.4 Correction of the calculating manner

In the foregoing, we compared the coefficients of the equations of motion obtained by Gerritsma by his

experiments with our calculated values. This compari-son fully explains the contrast of the motions shown in Fig. 6, that is, owing to the underestimate of the damp-ing coefficient for heave, the heavdamp-ing amplitude becomes too large, and as the coefficients of the inertia terms and

those of the restoring functions for zero speed can be computed with good accuracy, the estimated natural

periods show a very good agreement with the measured values.

Among the coefficients of the equations of motion

the following cannot be calculated with good accuracy, and their influences on the motion must be investigated.

9 F Exp. Cal.

0.150

-0.20 A 0.25 0 F Exp. Cal.

0.150

-0.20 £ 0.25o 0.30x 0 ifti

i/il

AlL 1.75., 1.2/.1.O0 0.75 150' 100' 50' 0 0 0.5 LI). 1.0 1.5 AlL 1.751.501.25 1.00 0.75 0 LI).

05i1.0

1 5 0 0.5 LI). 10 15 05 LI). 1.0 15 0.5 L) -150' -100'

-50'

w

Fig. 11. Three-dimensional effect of heave damping

coefficient

The three-dimensional effect of the damping coeffi-cient for heave.

The influences of frequency and ship speed to the

coefficient of the restoring function for pitch. The effects of the coupling terms to motion.

For the second item, any suitable estimation can

hardly be found for the moment, but the difference be-tween their calculated and measured value seems to be

small in general.

And similarly the third item can

hardly be analized for any factor, due to their compli-cated variances. So that it is tried here to obtain the

three-dimensional effect of the coefficient b which seems to affect ship's motion exceedingly, and to modify the calculating manner. The dotted lines in Fig. 7 represent the results calculated by the use of the mean measured values shown in Fig. 8. As a whole, the heaving ampli-tude becomes small, and the pitching ampliampli-tude increases slightly. In this way, they are able to agree with the

experimental values considerably. Then it was decided to modify the calculated values of the damping coeffi-cient for heave b, introducing the ratio of its measured value to the calculated one as the correction coefficient

Kb.

This three-dimensional effect of the coefficient b is

shown in Fig. 11.

This effect has been obtained theoretically by

Have-lock', Vossers(16), Kaplanu7) and experimentally by Tasai(18). They are also shown in Fig. 11 as Kb.

It is

seen that the result obtained by the authors is

consider-ably different from the values given by Havelock or Kaplan for a submerged spheroid or by Vossers for a

Michell ship. It seems that the value of the coefficient of three-dimensional effect is larger than any obtained up to date within the range of wJL) generally employed for the calculation of ship motion.

7. Results of the Modified Calculation

One example of motions calculated on Series Sixty

Table 4. Relative values of the modified part of the heaving amplitude*

Corrected value

) <

* Relative value (°M= (i.o

Original älue

models by the use of the correction coefficient shown in Fig. 11 is presented in Fig. 6 with full lines. It is noticed

that in general a good agreement exists between the

calculated and experimental values, except some parts in the pitching motion.

Though no figures are shown here about the

com-parison of phase differences between heave and pitch, it cannot be recognized how far they are improved, owing to the scattering of the measured data.

The natural periods for heave at zero speed are

nearly equal to the values before correction, but they indicate rather a better agreement with experimental

values as shown in Table 3.

This modified calculating manner is applied to the

following, nine models tested at Mitsubishi experimental tank, in order to investigate its general effect.

Oil tanker C=0.740 L/B=7.397 ® C,=0.767 L/B=6.268

® C=0.797

L/B=6.969 ® C,=0.826 L/B=6.756 Cargo Liner ® C=O.727 L/B=7.295

® C=0.733

L/B=7.156 Passenger Boat

® C=0.533

L/B=5.874 ® C,=0.547 L/B=7.230 Cable Layer Cb=0.631 L/B= 4.989 The results are as follows:

When CD lies in between 0.6 and 0.8, calculated values are improved as far as the case of Series Sixty models, as shown in Fig. 6 with full lines. As an example

show-ing this correction effect, the relative values of the

modified part of the heaving amplitude are shown in

Table 4 in regard to oil tankers and cargo liners. On the other hand, as the correction coefficient K,, is

obtained from the experiments carried out about the --Cb=0.61

..

I

4\!

p.

_{.-.--- .}

b_O.7ITaSai Vossers16

/

-x-

Cb= 0.7

-- C= 0.8

Authors Model AlL 0 0.05 0.10 0.15 0.20 1.25 5.8 10.1 13.3 15.3 14.4 Tanker 1.50 5.9 7.9 9.9 12.9 16.0 Cb=0.767 1.75 4.5 6.5 8.3 9.3 11.4 2.00 3.6 52 6.1 8.0 9.3 1.25 7.5 13.5 15.9 20.7 18.9 Tanker 1.50 5.8 7.8 9.8 14.4 18.5 Cb=0.797 1.75 5.3 7.5 9.1 11.7 14.1 2.00 4.3 6.3 82 10.1 102 1.25 3.9 6.7 102 18.9 22.6 Cargo 1.50 3.6 5.0 6.6 10.1 14.5 Cb=0.727 1.75 3.8 4.6 5.4 7.1 10.5 2.00 2.0 3.0 4.4 5.7 7.7 1.25 4.1 10.4 18.4 25.6 27.7 Cargo 1.50 5.1 8.8 10.6 17.0 22.4 Cb=0.733 1.75 4.4 6.0 7.5 10:3 15.9 2.00 3.7 4.2 5.7 7.5 10.4 2 4 6 1.5 1.0 0.5 0

models with Cb=0.6, 0.7 and 0.8, it will be a question to

apply this result by the extrapolation to a ship with a

larger or a smaller block coefficient than 0.8 or 0.6

re-spectively. However, so far as the authors investigate

on the oil tanker model ® and the passenger boat model ® and ®, heaving and pitching motions can be estimated

pretty well by the original strip method. For this

rea-son, original results before correction are also shown in the output data of a computer as shown in Table 1.

But, there is still a problem that, although this

cor-rection coefficient is obtained for the parameters Gb and (4L/g using Gerritsma's work, the parameters B/L and T/B, etc. must be also considered which, we think,

con-tribute to a three-dimensional effect more than Gb or

w.JL/1. In fact, on the cable layer model, the

experi-mental values coincide with the original calculated values more than those corrected, if any thing. The value of L/B of this model is very small, and it may be assumed that this L/B-effect is one of the reasons why the modified calculating manner cannot be applied suc-cessfully to this model. These effects should be studied

in the future.

In this way, this calculating manner is useful and must be limited to the ship having an ordinary hull

form.

8. Conclusion

Results obtained by this study are as follows:

The underwater section of a general floating ship is represented by the Lewis form section very well,

and its added mass and damping coefficient as a

two-dimensional body can be calculated with good

accuracy.

The computing time of ship motion can be reduced

very much by the method of the curve fitting in the

calculation of added mass coefficient C0K4 and that of amplitude ratio A.

Among the coefficients of the equations of motion

and those of the forcing functions, the calculated values by the strip method agree satisfactorily with

the experimental values, except the damping coeffi-cient for heave, the coefficoeffi-cient of the restoring

func-tion for pitch and the coefficients of the coupling

terms. a-g, A-G A a1, a3 B Co F, F0, F h, hA, h 'U,

Coefficients of the equations of motion Ratio of amplitude of dissipated wave

to that of the motion

Water plane area

Coefficients of the Lewis transformation Breadth of section

Added mass coefficient in a infinite fluid Exciting force, its amplitude and phase angle

Froude number

Elevation, amplitude and double ampli-tudes of wave

Added mass moment of inertia

Nomenclature

K K,, K4 k=2or/2 1W, M, M In'

When the damping coefficient for heave is corrected

to the three-dimensional effect, the calculated

re-suits by the strip method show a sufficient accuracy for practical use. The correction coefficient is larger

than what has been obtained up to date, and varies

with block coefficient complicatedly.

The mutual interference with each section appears

to be canceled favourably when a ship is taken into account as one body. This seems to be one of the

reasons why the calculation by the strip method

shows a better agreement with the experiment than

expected.

Now the calculations of the heaving and pitching

motions of a ship in regular waves can be put into prac-tical use by utilizing the high speed computer, but there

are still some problems left behind. The

three-dimen-sional effect of the damping coefficient for heave must

be investigated not only for the block coefficient and

the non-dimensional circular frequency, but also for the parameters, B/L, TIB, etc. These parameters seem to be significant factors to this effect. Furthermore, the block coefficient of a full ship built recently or that of a passenger boat will protrude beyond the range of Gb obtained here as the parameter of the correction coeffi-cient.

Besides, the differences between the calculated and the measured results about the coefficient C and the

coefficients of the coupling terms must be also studied, which are not considered here. But as has been stated, the results described in Ref. (7) are considerably differ-ent from those in Ref., (5) which are refered here. This problem must be studied with the improvement of the

experimental technique.

If so, the accuracy of the strip method is hoped to

become higher.

Acknowledgement

The authors greatly appreciate many suggestions and instructions given to them by Dr. F. Tàsai, the professor of Kyushu University, Dr. J. Fukuda, the assistant professor of Kyushu University and Dr. S.

Nakamura, the assistant professor of Osaka University.

Longitudinal moment of inertia of water plane

Correction coefficient of heave damping coefficient to the three dimensional effect.

Correction coefficient of added mass to the free surface effect

Wave number

Ship's length between perpendiculars

Exciting moment, its amplitude and

phase angle Added mass

o-xyz S T V COG, Z5GA, e : cx=B/2T 19=S/BT r

Co-ordinates fixed to the space Co-ordinates fixed to the ship

Sectional Area Draft of section Ship's speed

Distance from midship to C.G. Heaving displacement at C.G., its am-plitude and phase angle.

Half breadth-draft ratio of section

Sectional area coefficient of section Specific gravity of fluid

Ship's displacement

Reference

0WA /10 2 =w2T/g p Ot (L)g

Pitching angle, its amplitude and phase angle

Max. wave slope

Tuning factor of heaving Tuning factor of pitching Wave length

Nondimensional circular frequency Nondimensional circular frequency Fluid density

Circular frequency

Circular frequency of encounter of the ship to waves

Surface of a Fluid", J. Soc. Naval Architects of West Japan,

No. 21 (1960) P. 101.

0. Grim. "A Method for a More Precise Computation of Heaving and Pitching Motions both in Smooth Water and in

Waves", Third Symposium of Naval Hydrodynamics, Scheven-ingen (1960).

F. Ursell, "On the Roiling Motion of Cylinders in the Surface of a Fluid", Quart. J. of Mechanics and Applied Mathematics, Vol. 2, part 3 (1949) P. 335.

A. Gersten, "A Comparison of Experimental and Theoretical Forces and Moments Acting on a Restrained 5urface Ship in

Regular Waves ", J. of Ship Research, Vol. 6, No. 4 (1963)

P. 47.

T. Jinnaka, "Some Experiments on the Exciting Forces of

Waves Acting on the Fixed Ship Models ", J. Soc. Naval Ar-chitects of Japan, No. 103 (1958) P. 47.

T.H. Havelock, ' The Damping of Heave and Pitch: A Com-parison of Two-dimensional and Three-dimensional Calcula-tions", T.I.N.A. (1956) P. 464.

G, Vossers. Discussion in (15).

P. Kaplan and P.N. Ku, "Three-dimensional Stripwise Damp-ing Coefficients for Heave and Pitch of a Submerged Slender Spheroid", Journal of Ship Research, Vol.4. No. 1 (1960) P. 1. F. Tasai, "Damping Force and Added Mass of Ships Heaving

and Pitching (Continued)", Journal of the Society of Naval

Architects of West Japan, No. 21 (1960) P. 109.

Reprinting or reproduction without written permission prohibited.

We would appreciate receiving technical literature published by you.

Y. Watanabe, "On the Theory of Heaving and Pitching Motions of a Ship". Technology Report of the Faculty of Engineering, Kyushu University. Vol. 31, No. 1 (1958) P. 26.

F. Tasai, "On the Damping Force and Added Mass of Ships Heaving and Pitching ", J. Soc. Naval Architects of Japan,

No. 105 (1959) P. 47.

(8) J. Gerritsuna, "Experimental Determination of Damping Added Mass and Added Mass Moment of Inertia of Shipmodel" Isst. Shipb. Pr., Vol, 4, No. 38 (1957) P. 505.

J. Gerritsma, "An Experimental Analysis of Shipmotions in

Longituthnal Regular Waves ", mt. Shipb. Pr., Vol. 5. No. 52 (1958) P. 533.

J. Gerritsma, "Shipmotions in Longitudinal Waves", tnt. Shipb. Pr., Vol. 7, No. 66 (1960) P. 49.

J. Gerritsma. "Distribution of Damping and Added Mass along the Length of a Shipmodel", tnt. Shipb. Pr., Vol. 10, No. 103 (1968) P. 73.

J. Gerritama. "The Distribution of the Hydrodynamic Forces

on a Heaving and Pitched Shipmodel in Still Water ", tnt.

Shipi.. Pr., Vol. 11, No. 123 (1964) P. 506.

S. Fukuda, "On the Bending Moments of a Ship in Regular

Waves (Continued) ", J. Soc. Naval Architects of Japan. No. 111 (1962) P. 204.

F. Tasai. "Measurement of the Wave Height Produced by Forced Heaving of the Cylinders ", J. Soc. Naval Architects

of Japan, No. 107 (1960) P. 38.