Study on lateral motions of a ship in waves by forced oscillation tests

Study on Lateral Motions of a Ship in Waves

by Forced Oscillation Tests

Hitoshi Fujii*

Takeshj Takahashi**

The coefficients of equations of motion were determined by the forced oscillation technique, asa step of improvement of the calculation of lateral motion sway yaw and roll of a ship in waves. The effects of frequency advance speed and bilge keels on ship motions were clarified experimentally.

The experimental values were compared with the calculated ones by the strip method. For the main terms of sway, yaw and roll, except roll damping terms, the both show fairly good agreement. For the coupling terms of sway-yaw, yaw-roll and roll-sway, the

both also show comparatively good agreement.

-It is pointed out that some proper method should be developed for the approximation of the effects of viscous damping advance speed bilge keels and frequency of motion to obtain better prediction of the roll damping term

1. Introduction

It is considered that the strip method has been confirmed to be applicable to the prediction of the vertical motions of a ship in waves; heave and pitch, and the computing. program based on it has been developed for the practical ship design. Despite of still remaining amounts to be refined and to be more precisely evaluated, it may be allowed to say that the

strip method has come to the extent of practical use. Now that

the efforts are focused upon establishment or application of the calculation method based on the strip method again for

lateral motions: sway, yaw and roll(1 )(2) However, there

could not be found so ample experimental data of lateral

motions as of

vertical

ones and so the study on the

applicability of the method has been delayed.

In order to apply the theoretical calculation to the practical use, it is necessary to certify empirically or modify

theoretical-ly each item of the equations of motions; (a) coefficients of the equation of motions, (b) terms of wave exciting force, (C)

amplitudes of motions, respectively. It can be summarized as shown below.

Experimental Tank Towing tank

Item Experiment

Check coefficients Forced oscillation of equations of. model tests in calm

motions water

Check terms of Restrained model Seakeeping basin

wave exciting tests in waves

force

Check amplitudes Free model tests in Do.

of motion waves

Although there have been some experimental data Of (C)

amplitudes of motions which are connected with the equations

of motions made of (a) and (b), concerning (a) and (b), however, there are only few experimental data useful for

checking the calculation method on the coupled motions of

sway, yaw and roll.

In general, to perform the experiments on (b) and (c)

terms, the seakeeping basin equipped with wave maker in

which a model can run obliquely to the waves comes in need.

TherefOre, it is considered as a proper step to evaluate the coefficients of equations of motions at first then to get their

values experimentally by the forced Oscillation tests and lastly

to compare the experimental values with the calculated ones

and improve the method of theoretical calculation. In

connec-tion with the study on ship manoeuvrability, a number of lateral forced oscillation tests on sway and yaw are being

carried out. However, these data can not be used directly to

the ship motions in waves because of the frequencies of their

motion being very low. Leeuwen3 conducted experiments

including high frequencies of motion with the Series 60 model

(C6 = 0.7), but they were on sway and yaw only without any

concern with roll and its coupled motion, sway and yaw. Furthermore, as a forced oscillation test for the study on

lateral motions in waves, we can barely find basic experiments

by Vugts(4) on two-dimensional body. In order to establish

the calculation method for lateral motions under these

circumstances, we explain the following items in this report;

A forced oscillator which can give a model into sway yaw and roll motions with given amplitude and frequency was

designed and constructed.

Using this apparatus, some forced oscillation tests were

carried out with a container ship mOdel, a tanker model and

others, and the coefficients of equations of coupled

motions of sway, yaw and roll were obtained

experimental-* Dr. Engr. Nagasaki Technical Institute. Technical Headquarters

MTB87 August 1973

(3) Comparing the calculated values based on the strip method with the experimental ones, and the effects of

advance speed, three-dimension and viscous damping of

roll, which can not be amended properly by the

two-dimensional strip method, were investigated.

2. Method of lateral forced oscillation tests

2.1 Equations of motion and method ofanalysis

Linearized equations of coupled lateral motions sway yaw

and r011 are expressed as equations (2.1), neglecting the restoring terms except roll for brief analyses' sake. (Restoring terms due to forward speed appear in the equations of motion

obtained by the momentum theory using the strip method.

Thereadersmay refer to sectiOn 4.)

auy+a2yai4cb±ai5cb+ai70+ai= Y

a+aa24+a+av0+a&N

"(2.1)

a±a±a+a+a+a3sØ+a0 =L

The first equation descibes sway, the second yaw and the

third roll motions. Let a

be coefficients of equations of motion, y, and be displacement of sway, yaw and roil

respectively, and let y, N and L denote sway force, yaw

moment and roll moment respectively.

In forced oscIllation tests, coefficients of equations of

motion, , are determined experimentally by measuring

hydrodynamic reactions Y, N and L at the time when the model moves in the manner determined by y , 0 and

Here the given motions are chosen sinusoidal ones which are

considered to be components of actual motion of ship.

Analyzing the phases of measured hydrodynamic reactions,

fOrces and moments are separated into in-phase and out-phase

terms: inertia term proportional to acceleration or angular acceleration and damping term proportional to velocity or

angular velocity, and the coefficients of equations of motion

are obtained.

Fundamental modes of forced oscillation and method of

analysis are described below.

(1) Pure sway test

Keeping the center line of a model in the fOrward direction,

the center of gravity of a model is forced to move in the direction of axis only in the manner; y = YASiflcot. In

this case, if the hydrodynamic reactions are represented as Y= YAsin(Wt+ey)

N=NAsin(Wt+ CN) L=LAsin(oit+CL)

considering =cb=0

and 0=0=0

, equation _(2.1)

re-duces to the fo!lowing.

a

a24=

aujj+ai,jj=

Y

a2iy+ayN

ay+ay=L

From (13), coefficients au , a31 , of main terms of sway

motion and coefficients a , a22 , a31 , a of coupled

terms of sway-yaw and sway-roll are given as follows. YACOSEY a31 - - -YASInCY a31 -YA W LACOSCL a31 -YA C') NASinCN

a24

2 cbA LACOSEL ₊ a OACO2 W2 LA sin CL,

virtual mass of sway

damping force coefficient of sway

NASinCN a31 = YAW LA sine a31 -YAW (2.r3) coefficients of coupled terms of sway-yaw coefficients of coupled terms of sway-roll (2.4) Pureyawtest

A model is forced to move horizontally as its center of

gravity makes sinusoidal cUrvC With its center line being always in the tangential direction Of this curve. Putting the

motion of a model 0=coswt and noting

jy=0 and

6= 0 = Ô

the coefficients of main terms of yaw and

coupled terms of yaw-sway and yawroll are given as follows.

virtual mass moment of inertia of yaw NACOS £N _{: damping moment coefficient of yaw}

OA t'

__YAsinCv

a YACOSCY: coefficientsof coupled

-

ct'A U)2 OA Co

terms of yaw-sway

-

LA Sin CL

_{a =}

LAcos CL: _{coefficients of cOUpled}

CO2 ' OA Co

terms of yaw-roll

(2.5)

Pure roll test

Restraining the center line of a model in the forward

direction, the model is forced to roll about water plane axis (0

point) sihusoidally Assuming _{that 0 = OA siñwt} , and

= = and ç = = 0, coefficients of main terms of roll

and coupled terms of roll-sway and roll-yaw are given as f011ows.

(2.6)

virtual mass moment of inertia

of roll

damping moment coefficient

of roll

YACOS y Y SIflCY

WACO2

a12 coefficients of coupled.

terms of r011sway NA COS Cs NA SIflCN _{coefficients of coupled}

--

a31

U)

terms of roll-yaw

a NACOSCN

II,

YAWING GEAR

YAWING

SUDE UN1(

FRD: Forced Rolling Dynanrometer PI-LASE SHIFTER

2.2 Forced oscillator

Forced oscillation tests require an apparatus which has an

oscillating mechanism and a measuring System, where the

former makes a model in sway, yaw, roll or their coupled

motions with prescribed period and amplitude and the latter

measures corresponding hydrodynamic reactions.

This apparatus is called "Forced rolling dynamometer" from the viewpoint that roll is the most important among

lateral motions in waves and that this apparatus is used for the

study on lateral motions, especially roll. The outline of this

dynamometer is shown in Fig. 1.

It has a swaying frame which slides transversely in the outer

frame fixed to the towing carriage, while a driving motbr and

three gears for sway, yaw and roll are mounted on that

swaying frame. Each gear system has a scotch yoke device which enable yaw or roll to be superposed on the motion of

the swaying frame. The phase between sway and yaw or sway and roll are adjustable by every 15 degrees.

HydrOdynamic force is measured as transverse force by

three gauge springs y1 , y2 and , and these ones yield

sway force, yaw moment and roll moment as followings,

referring Fig. 2.

Fig. 2 Forces and levers SWAYING GEAR SWAYING SCOTCH YOKE DRIVING MOTOR SWAYING FRAME EXTERNAl. FRAME (flood to carriage) ROUJNG GEAR ROLLING GEOTCH YOKE

Fig. 1 Scheme of mechanism of forced rolling dynamometer

AACOSE

2

AA sIne-i

MTB87 August 1973

Sway Force Y = Y1± Y2+ 1'3

Yaw Moment N = Y111 Y212 (2.7)

RollMoment

LY013

The model is supported by a towing rod

longitudinally, and a gauge spring in the rod can indicate the total resistance. The specifications

of this forced rolling dynamometer are shown

in Table 1. Standard size of ship models for the

tests is 3.0 meters in length. The photographs of the dynamometer and its experimental condition during fOrced rolling tests with a

run-ning model are given in Fig. 3 and Fig. 4,

respectively.

2.3 Data processing

Data obtained by tank tests are recorded and treated analytically in the way shown below.

(1) The amplitudes of Y ,N and L, reduced

from the output of gauge springs Y1 , Y2. and Y3 by an analogue computer, are monitored.

The outputs of gauge springs Y , Y2 and Y are also

recorded by a data recorder with the timing signal marks

showing the origin of the phase of the motion.

Y5 , Y2 and Y3 on magnetic tape are processed

through the spectrum computer JRA-5 with sampling

time At = 0.01 second.

By use of IBM 360 computer program compositiOn

of Y , L

and N is made from the processed paper tape according to equations (2.7), and analysis is made for ai

from Y

, N and L based on the phase analysis method by way of sin and cos component, i.e.;

fTAi(t+)i o.t dt

(2.8)

JTA.(+)td

where A denotes

Y , N

and L

3. Forced oscillation tests 3.1 Tested models

Four models were chosen for test; two container ships with

a single and a twin screw type as typical fine ships, 120000 DWt ore carrier and 210000 DWt tanker as typical full ships. Principal particulars and test conditions of the models are shown in Table 2. Taking into consideration that bilge keels play an important part in the resisting moment against rolling and that the effects of those should be investigated, model

tests for a single screw container ship and a tanker were

performed each for the two case of with and without bilge

keels.

MTB87 August1973

Fig 3 Forced rolling dynamometer (FRD)

Table 1 Principal particulars of FRD

3.2 Test conditions

In forced oscillation tests,

(1) mode of motion,

(2)

frequency of motion, (3) amplitude of motion and (4) advance

speed are considered as parameters.

(1) Mode of motion

As basic mode of motions, pure sway, pure yaw and pure roll motion tests were conducted, and supplementary test

was carried out for combined yaw motion. It is very.

convenient for analysis to let rolling axis lie through the

center of gravity because restoring force comes from

hydro-dynamic one alone. In addition, it is more practical for the comparison of the experimented values and the

-Fig. 4 Forced rolling test of

a container ship model

(ØA=15°,

F=0.2O)

calculated ones to assume the rolling axis lie through the

point 0 on the still water plane.

According to these consideration, thetests were carried out

to roll a container ship model about the point 0, puffing the center of gravity in it. In case of a tanker model,

however, it was rolled about G adopting the reasonable GM

as GM is too small to put CG. on the point 0. When pure

roll motion is aumed to be roll about the point 0, roll

about the other point except 0 should be considered to be a combined motion of sway and roll.

(2) Frequency of motion

Frequency range of tests was determined as o. = cv

Model length

L9 =

3.0 m (standard) Swaying amplitude

Yawing amplitude Rolling amplitude Sway. yaw phase Sway. roll phase

Period

Circular frequency

Driving motor

Force and moments capacity Restriction for longitudinal motion

y A = 0

35 mm (variable, continuous)

ØA = 0

5° ( 0

= 0

30° ey -CA = 0 - ± 180: (variable, every 150) = 0 ± 180 T

= O.8'-'6sec

cv = 1 - 7.8 rad/sec

0.75 kW DC motor with speed control by

-static Leonard method.

Sway force : 50 kg

Yaw moment : 25 kg m

Roll moment :

4 kg m

Heave, Pitch free for small displacement

=0.2- 1.1 . Considering the encountered period in beam sea,

this range corresponds to AlL = G.5-5.O. Amplitude of motion

Pure sway and pure yaw tests were conducted with the

small oscillating amplitude as that the assumption of

linearity of hydrodynamic forces is satisfied as well as in

heave and pitch motions. However, pure roll test was conducted with several oscillating amplitudes because the

non-linearity of hydrodynamic moment is considerable

amount by the viscous damping effect:

Advance speed

Several advance speeds were selected to investigate its

effect on hydrodynamic forces including F = 0 at which

the strip method is most reliable.

4. Coefficients of equatiOns of motions based on the strip.

method

Coefficients of equations of motion

aj (i1-3, j1-9)

based on the strip method by Takagi ) are given as follows.

aii=M+fmedx

aj'Nsdx

a= 0

a= f(x_± ) dx+

fN5 dx

aiszrJ'Ns(x_xc) dx- U(mydx

_{a= 0}

Table 2 Characteristics of the tested models

MT887 August 1973

Ship

Single screw

container ship Tanker container shipTwin screw Ore carrier

Scale 1/58.333 11103.333 1170 1/82.333 Lpp (m) 3.000 3.000 3.500 3.000 B (in) 0.4464 0.4719 0.4606 0.49312 d (in) 0.1632 0.1828 0.1575 0.19433 4 (kg) 121.8 220.6 147.11 23702 L.C.B. (in) -0.0385 0.0104 -0.0648 0.08868 KM (m) 0.1851 0.1917 0.2134 0.19931 GM (in) 0.0219 0.0597 0.0559 0.05016 K/Lp, 0.2190 0.2323 0.2508 0.2362 K/B. 0.3062 0.3108 0.3546 - 0.2228 Appendages Bilge keels (with, without)

Bilge keels. Rudder

(with, without)

Bilge keels. Rudder, Bowing

Bilge keels, Rudder

(Self ProPelll atmodelpoin! Rolling axis _00G_ M M

OOG-

I

-

-

.---

-

G

O-=-=-=

O=0.O508m

-

=-=--

---ã=0

-G_ OG=0.0452m

-aj7fmy(zc- iy)dx

alsfNY(zG

- I

dx

a = 0

a21=fmY(xxG) dx

ainfNe(x_x)dx+ Ufmy dx

=UJN5 dx

ainI+fmY(xxG )2 dx+ -fNe(x-xG)dz

a=fNe(x_xc)2 dx+ 2f N5 dx

a UJNS(x-xG)dx- U2j'mvdx

az7fms(zc_I y)(x-xG)dx

asfNs(zG

Iw)(x_xc)dx+Ufme(zo Iy)dx

awUfNs(zG

i)dx

asi=fmv(zc- Iy)dx

MTBB7 August 1973

af.ms(zc1s)(xxc)dx+JNs(zc_ 1,,,)dx

a=O

1,,)2 dx

aM,g

GM where M, : mass of a ship

mass moment of inertia about z axis mass moment of inertia about x axis x-coordinate of the center of gravity of ship 23 : z-coordjnate of the center of gravity of ship sectional added mass to the y-axis direction N9 : sectional damping force coefficient to the y-axis

direction

lever of sectional added mass inertia force due to

rolling motion with respect to 0' which is the pro-jéction of 0 on the transverse section

191 = i/msl

lever of sectional force due to rolling motion with respect to 0'

sectional added mass moment of inertia with

re-spectto 0'

circular frequency of the motion

U advance speed

5. Results and considerations

The coefficients of equations of coupled motions: svvay, yaw and roll, of a tanker model obtained by forced oscillation

tests are shown in below Fig. 5, for instance, as compared with the calculated values. For the main terms of roll, the results of

other ship forms are also shown. And, the non-dimensiOnal

representations of the coefficients are shown in Table 3. 5.1 Main terms of sway, yaw and roll

Except roll damping term, the experimental values show fairly good agreement with the calculated ones by the strip

method.

(1) Main terms of sway an : virtual mass of sway

The experimental values roughly coincide with the càlculat-ed ones as shown in Fig. 5. The decreasing tendency of the

virtual mass is observed with the increase of the advance

speed, however, it is considered that the modification is not

necessary to the estimation method as the difference

between them is small in comparison with the total value

of an

a12: damping force coefficient of sway

The experimental values agree with the calculated ones as

shown in Fig. 5, and it seems that the damping force of

Table 3 N on-dimensional: quantity

(1) Coefficients of equations of motion

a'2 a11 a12.

a53 ----,i

a37 a,7 VB2 a58 a28

pVB2j.B

a28,a,5

_{- pVLB}

2g 2g

a3, - a29 ,a35

a3,

_pgV(2B)

a29 ,a36

Circular frequency of motion

0= CL) Advance speed U

F,,

a, Where L: ship length CL99) B: breadth V: displacement volume a14 ,a23 a14,a22

_pVL

a55 ,a22 a22 -

_pVL

_a16,a23 B a16,a23

_{- pg V}

a24 a24 pV/L2

a25 a19 , a32

a25 pVL2 2g a18,a32 pVB a57 ,a31 a27 ,a31 - _pVB - _a37,a34 a57 ,a34 pVLB 2g p : density of fluid g : acceleration of gravity U: advane speed

swayat F,, = 0 can be accounted for by the effect due to wavemaking. As the effect of advance speed on a12 is

slightly observed,

it would be better to make some

modificatiOns to the estimation method. The application of the non-steady wing theory, must be one of the possible method for the modification.

(2) Main terms of yaw

a24 : virtual mass moment of inertia of yaw

The experimental values of a24a24/w' roughly coincide

with the calculated ones as shown in 'Fig. 6 The reason why

this coefficient does not agree so well as in the case

of a12 may. be attributed to the fact that the end effects are emphasized due to the multiplication (x x3 )2; such

as three dimensional effect and discrepancy of section form at after body with Lewis form. The effect of advance speed on a,4 is larger than that on .ali and the value of a5, at

F,, = 0.15 decreases by about 20% from that at F,, = 0 in

the neighbourhood of &i = 0.7.

damping moment coefficient of yaw

calculat-0.15 0.10 0.05 0 I ! I 0.2 0.4 06 0.8 1.0

virtual mass of sway

a

: virtual mass moment of

a24

0) 2 _{inertia of yaw}

M0 : mass of the ship 0

I ,: mass moment of inertia of yaw

0.2 0.4 0.6

0)

Fig. 5 Coefficients of main terms of sway

0.8 1.0

ed ones except for the low frequency range, as shown in

Fig. 6. It seems that the damping moment of yaw is almost attributed to the wave-making damping as the same in the

case of aa , but some modifications should be made on

am because the effect of advance speed on it is con-siderable.

(3) Main terms of roll

am virtual ma moment of inertia of roll

The experimental values are in the same order with the calculated ones, as shown in Fig. 7. Though some

dis-crepancies between them are observed, it is considered that

this difference does not affect the estimation of ship

motions as it is small compared with the total value of.

0.04

0.02

damping force coeff. of sway

damping moment coeff. of yaw

Fig. 6 Coefficients of main terms of yaw

0 /

/

F

/

0 ' 0 0.2 0.4 0.6 C,) F,,-0.15 MTB87 August 1973. F,, =0

damping moment coefficient of roll

The values of the coefficient am , asshown in Figs.8 and 9 on a single screw container ship model and in Fig. 10 on a tanker model, show complicated variation according to the

advance speed, with and without the bilge keels and also circular frequency of forced oscillation. The consideration

on am is discribed in the following section.

5.2 Roll damping moment

The roll damping momenti

calculated by the linear

potential theory as the wave-making damping moment.

How-ever, the roll angle is generally overestimated in case of using

this moment only. Therefore,

it

is

urged to find more

reasonable roll dampin moment, including the effect of 7 Calculated Experimental

-+-

---0--F,,= 0 0.15 Swaying amplitude YA1B00424 Calculated Experimental

-+-

--0--F,,0

0.15 Yawing amplitude 1.0 0.5 Tanker ao 1.0 0 _0.8 0.2 0.4 0.6 0) 1.0 0.08 Tanker 0.8 1.0

MTB87 August 1973

0.010

0.005

0

0 Single screw container ship

0 0

I

mass moment of inertia of roll

0

-cv

Fig. 7 Coefficients of main terms of roIl (1)

Single screw container ship

Rolling

amplitude

A7.5

: damping moment coeff. of roll

0.15 0.10

of

0.010 0.005

/

0

Fig. 8 Coefficients of main terms of roll (2)

advance speed and taking- account of the viscous damping

aas,+a3a,,±a

(5.1)

efféct Considering the abovementioned situation, the co- where

efficient aw is assumed to be composed of three terms, that : coefficient corresponding to the wave-making

Is, damping calculated by usual potential theory

I

: mass moment of inertia of roll

F 02 Tanker 0 0.2 0.4 0.6 0.8 1.0 cv Rolling amplitude -= 15

-: damping moment coeff. of roll

F =0. Calculated - Experimental .

--O'---F=O

0.15 Rolling amplitude 95A=1O -- Calculated Experimental

+

--0--F = 02 0 Rolling amplitude QA7.5 Bilge Keels Calculated Experimental

F=0

F=0.2 Without

+

---0--With

----

---s--

----virtual mass moment of inertia of roll virtual mass moment of inertia of roll

02 0.4 0.6 0.8 1.0

02 0.4 0.6 0.8 1.0

0.10

0.05

0.010

0.005

0.010

damping moment coeff. of roll

(non-dimensional)

0

0 02 0.4- , 0.6

C,)

Twiri screw container ship

damping moment coeff. of roll

Fig. 10 Coefficients of main terms of roll (4).

Tanker

0.010

0.005

Fig. 9 Coefficients of main terms of roll (3)

damping moment coeff. of roll

(non-dimensional)

0 I I

0 0.2 - 0.4 0.6

C,)

aUU : coefficient representing the effect of advance

speed

The percentage of the wave-making damping and the

viscous damping is not yet clarified, however, it is supposed that the same modification factor can be applied to both of them, considering the result of comparison of the calculated

values with the experimehted ones. Coefficient N, usually

used for roll damping coefficient, is in itself effective around the point of synchronism of roll and, therefore, the effect of frequency of motion should be introduced into it. Moreover,

as the increment of damping moment by the bilge keels can be

almost attributed to the viscosity, the increment of coefficient N due to the bilge keels- should be added. When a ship has

advance speed, the roll damping moment increases in general,

but the reason and the amount of increment can not yet be

clarified correctly. Here, supposing that the roll damping

moment is increased simply by rolling which -alters the

direction of flow around the-ship hull, only -the moment due

to the term of Yfi are used. So each term of a39 is expressed as follows.

-a3&fNv(zc-1 ,,)2dx

F,, =0 MTB87 AugUst 1973 4-9 Bilge keels -Calculated Experimental F,=0 F,=0.15 Without

--i--With -

--

-.--Calculated Experimental

±

--0--F,=0 0275 Rolling amplitude ftA=10'

a3,= ( Nio+ Npic ) ---a37 & (5.- 2)

coefficient corresponding to the viscous damping,

which is a part of N coefficient obtained experi- -

_a()jmvdx

U d 2!'

mentally 0.2 0.4 0.6 1.0 U 0.8 1.0 0.8 1-.0 Rolling amplitude = 5 Rolling amplitude A=1O'

MTB87 Au9ust 1973 where

N10 : coefficientN at dA = 10 degrees

NBK coefficientNdue to the bilge keels

,, : encountered circular frequency of the motion The ratio of these components is shown in Fig. 11.

The calculated values shown in Figs. 8 and 9 are ones adopting the formulae (5.1) and (5.2). and the whole feature

of experimental results is röpresented

fairly well by the

calculated ones. The results on a twin screw and single rudder

container ship model as shown in Fig. 11 and the value

of a38 amounts to twice that of the single screw container ship model ifl the case with advance speed. In the estimation formulae, the effect of propeller bossing and filet of the twin

ship on roll damping moment are not considered, so the calculated values show smaller ones than those from the experiments. The effect of forced rolling angle on a3s is

shown in Fig. 12 on an ore carrier model, where the forced

rolling angles are changed 50, 10° and 15°. In each case

damping moment coeff. of roll

0.010

0.008

0.006

0.004

0.002

Fig. 11 Components of roll damping moment coefficient

Ore carrier

0.010

0.2 0.4 0.6 0.8 1.0

C')

damping moment coeff. of roll

Effect of advance speed

Effect of BILGE KEELS Viscous damping

Wave damping

Freg. Calculated Experimental

F.=0 F,=0.15

0.6

0

-:-.--

S A

the calculated values stay in the same order as the

experi-mental ones, with some agreeable and some discrepant. 5.3 Coupling terms of sway, yaw and roll

At F',, = 0 , the experimental values correspond fairly well

with the calculated ones, but the effect of advance speed on

these terms should be investigated still mOre. Sway-yaw coupling terms

a14 coupling force coefficient of yaw into sway

am_am/a2 :. coupling moment coefficient of sway into

yaw

The experimental values and the calculated ones show

comparatively good agreement, as shown in Fig. 13. The effect

of advance speed on these terms is mod ifièd b:y the term

UJN5dx , but it seems that this term only is not enough.

a,0 : coupling force coefficient of yaw into sway am : coupling moment coefficient of sway into yaw In regard to the sway force, the term due to centrifugal

force should be added as in the case of the equation of motion on manoeuvrability, then am becomesas follows.

aiourfNs(xxc)dxUfrnsdx±M0U

(573)

Some different effect of advance speed on these terms are observed between the experimental values and the calculated ones, especially on a38 , and it is considered that investigation should be made still more into this point.

Yaw-roll coupling terms

coupling moment coefficient of roll into y

a34 : coupling moment coefficient of yaw intd roll a38 : coupling moment coefficient of roll into yaw

a38 : coupling moment coefficient of yaw into roll As shown in Fig. 14, the effect of advance speed on these terms are comparatively large and the. feature of variation of

experimental values due to circular frequency of forced

oscillation differs from the calculated ones, especially on a38. Sway-roll coupling terms

a,7 : coupling force coefficient of roll into sway

a38 : coupling moment coefficient of sway into roll

The experimental values and the calculated ones of a38 Ofl a tanker model are shown in Fig. 15. The reason why the both show fairly good agreement may be based on the fact that the

ratio of

zofmsdx occupying .in this coefficient is large comparing with 1sfmydx . Contrary to this, the absolute

value of a1, on a container ship model is small as the

experiments were conducted under the conditiOn of ZG' = 0, and agreement of the both is not good. In the Calculation.

formulae, the value of am is equal to that of a17,, but in the experiments this relation is not always satisfied. In order to

investigate the relation of a38 and a17 , let the depth of the

draftwise center of added mass distribution denote as follows.

ZG ly-

fms(zc

ly)dx

fmsdx

- or -

Cu a

a17 am

And the values of 18/d are as shOwn in Fig. 16. Though the

0 10 15

Rolling amplitude tlA

Fig. 12 Effect of rolling angle on damping moment

0.05

coupling force coeff. of yaw into sway

a21

a

coupling moment coeff. of

2

sway intd yaw

0.2 0.4 0.6 0.8 1.0 0.05

/

/

/

0.10

_'

/

p. F=0.15 F,=0

_/

/

I

/

values of 1, obtained from pure swaying test are smaller than that from pure rolling test, the both coincide with each calculated values and it can be regarded as a17 nearly equal

to am.

a1 : coupling force coefficient of roll into sway am : coupling moment coefficient of sway into roll

Similarly to the case of a and am, let the depth of the

draf-twise center of damping force distribution denote as

follows.

Tanker

Tanker

all coupling force coeff. of yaw into sway

Fig. 13 Coefficients of sway-yaw coupling terms

coupling moment coeff. of sway into yaw F= 0. 15 MTB87 August 1973 11 Calculated Experimental

--F.

--0--F0

0.15 Yawing amplitude.

9A2

Calculated Experimental -

+--- F =0

Swaying amplitude

--0--

0.15 YA/BQ.0424

[Ny(z 1)dx

018 am I.

-

or

-J

Nydx am am

And the value of I .,/d are as shown in Fig. 16. The values

of 1w obtained from pure swaying test are smaller than that

from pure rolling test and the both differ each other at low

frequency range of forced oscillation. The calculated values nearly equal to the results of pure swaying test. As the effect of viscosity are contained in the results of pure rolling test,

MTB87 August 1973

C

0.01

F, =0

coupling moment coeft. of yaw into roll

F, =0.1-5

F, =0

such difference is found and it may be appropriate, in general,

to consider that a is not equal to 0m

-6. Conclusion

The coefficients of equations of motiOn were determined by the forced oscillation technique, as a step of improvement of the calculation of lateral motion; sway, yaw and roll, of a ship in waves. The values of coefficient, extracted from the experimental results of a tanker model and container ship

models, were presented in comparison with the calculated ohes by the strip method. The experimental values show fairly good

agreement with the calculated ones and the both show the

- Tanker

Tanker

coupling moment coeff. of roll into yaw

0.005

,

F, =0.15

- 0.005

Fig. 14 Coefficients of yaw-roll coupling terms

F, = 0.15

F, 0

coupling moment coeff. of yaw into roll

02 0.4 0.6 0.8 1.0

same feature also in the case of other ship models, which are not presented in this report.

For the main terms of sway,yaw and rol!, except roll

damping term, the coefficient can be approximately esti mated by the strip method with the satisfactory order of

agreement.

To obtain the reasonable prediction of the roll damping term, it is pointed out that some proper method should be. developed for the approximation of the effect of viscous

damping, advance speed, bilge keels and frequency of

motion.

For the coupling terms of sway-yaw, yawroll and roll

Calculated Experimental

+

--0--F,=0 0.15 Rolling amplitude =10' Calculated Experimental

+

---0--F,=0

0.15 Yawing amplitude 029

coupling moment coeff.

of roll into yaw

-coupling moment coeff. of sway into roll

..-sway, the experimental values correspond fairly well with

the calculated ones at F = 0. but the effect of advance

speed on these terms should be investigated still more.

Further investigation should be made on the effect of the

discrepancies of the coupling terms on the amplitudes on ship

motions in waves. As for the effect of advance speed, it is considered to be necessary to investigate the effects of the

wave profile, the mean sinkage and the trim on the roll

damping term which are not included in the present calcula-Tanker

Tanker 0.06

0.04

0.02

Fig. 15 Coefficients of roll-sway coupling terms

/

,

0.2 0.4 0.6 0.8 1.0

coupling moment coeff. of sway into roll

a)

0.2

Q0

0.6 0.8 1.0

0.4 C,)

MTB87 AUgust 1973

tion method. In general, it is necessary to modify or develop the present estimation method of the important terms in the equations of motion, to improve the accuracy of the

predic-tion of the ship mopredic-tion in waves. Acknowledgement

The authors wish to express their gratitude to Dr. J.

Fukuda, professor of Kyushu University and Dr. K. Watanabe, vice director of Nagasaki Technical Institute, Mitsubishi Heavy

13 Calculated Experimental Rolling

F=0

amplitude

--0---

0.15

_A10

Calculated Experimental

---0---F=0

0.15 Swaying amplitude YA/B=Q.0424

coupling moment coeff. of roll into sway coupling moment coeff. of roll into sway

0.2 0.4 0.6 0.8 1.0

0.10

0.05

MTB87 August 1973

Industries Ltd. fOr their continuing guidance and encourage rnent. The authors also wish to express their appreciation to

Mr.. K. Hatakenaka and Mr.. N. Matsunaga who cooperated

in carrying out this investigation.

References

F. Tasai; "On the Swaying, Yawing and Rolling Motions

of Ships in Oblique Waves" Journal of the Society of Naval Architects of West Japan, No. 32(1966)

J. Fukuda and others; "Theoretical Calculations on the

MotiOns, Hull Surface Pressures and Transverse Strength of 02 Mo Gop 0

02

0.4

0.6

0.8

1.0

K Mi

Go

Single screw container ship

F=02

0

02

0.4

0.6

0.8

i/d: draftwise center of added mass

distribution

draftwise center of damping force distribution

from Swaying test - - - frOm Rolling test

O2. 0.4 .8 1.0

K

Fig. 16 Hydrodynamical levers

0

0.2

0.4

0.6

0.8

1.0

Tanker F =0.15

1/d: draftvise center of added mass

distribution

0.2 0.4 0.6 0.8 1.0

i/d:

draftwise center of damping

force distribution

02 0.4 0.6 0.8 1.0

a Ship in Waves" Journal of the Soiiety of Naval Architects of Japan, Vol. 129 (1971)

G. Van Leeuwen; "The Lateral Damping and Added Mass

of HorizOntal Oscillating Ship Model" TNO Report No. 65s

(1964)

J. H. Vugts;"The Hydrodynamic Coefficient for Swaying, Heaving and Rolling Cylinders in a Free Surface" ISP Vol.

15 No. 167 (1968)

F. Tasai; 'On the Sway, Yaw and Roll Motions of a Ship in Short Crested Waves" Journal of Naval Architects of

West Japan. No. 42 (1971)

1..0

02 0.4 _0.6

0.8

Expennwntal Calculated