New experimental results of forced yaw tests in shallow water

NEW EXPERIMENTAL RESULTS OF FORCED YAW TESTS IN SHALLOW WATER

by Masataka FUJINO

October 1972

DEPARTMENT OF NAVAL ARCHITECTURE

THE FACULTY OF ENGINEERING

UNIVERSITY OF TOKYO

BUNKYO-KU, TOKYO. JAPAN

Yours sincerely,

Masataka Fujino

14th November, 1972

Submission for Correction

Dear Sir,

After I sent you the report, of which the title was

"New Experimental Results of Forced Yaw Tests in Shallow Water", some errors were found in it.

Then I enclosed an errata and two revised pages.

Please substitute those new pages for the old ones and correct the errors listed in the errata.

ERRATA

Page Line Original form Correct form

3 16 may mix in may be mixed in

5 16 Fig. 2 showes Fig. 2 shows

7 16 Fig. 6 showes Fig. 6 shows

7 22 which are defined by which are defined as

Pages 15 and 29 must be substituted by the revised ones enclosed with this errata.

New Experimental Results of Forced Yaw Tests in Shallow Water

Masataka Fujino

Introduction

Some years ago, the author published the _{papers on the} manoeuvrability of ships in restricted waters

_{11], [2].}

In

those papers, the manoeuvrability of ships, in shallow water

and in narrow channel was discussed with the experimental results of stability derivatives obtained by the planar motion mechanism. Recently, the author had an opportunity of makthg an experi-ment in shallow water, and on that occasion the stability deriva-tives were measured again by the planar motion mechanism.

The main aim of this experiment was to clarify the speed effect on the stability derivatives in shallow water.

In this report, the experimental results obtained at this time are put together.

Outline of the experiment

The model used

at 'this time is the same one as used at the last experiment, and its prototype is a full-bodied oil tanker. The particular dimensions of the model are shown in Table 1.

The advance speedSof the model were the scaled-down speeds of 5, 7, 9, 11 and 13 knots of the actual ship, while at the last experiment they were only.two speeds corresponding to 7 and .12 knots. The propeller revolution was adjusted for the propeller thrust to cancel the total resistance of the model. In other. words, they were theso-called self-propelling revolutions of

the model.

The ratios of the water depth H to the draft d of the model, to say H/d, were 1.2, 1.3, 1.5, 1.8, 2.3, 3.0 and 29.0. The depth of water with the H/d ratio of 29.0 can be regarded as infinitively deep.

With regard to the method to determine the stability deriva-tives by analyzing the hydrodynamic force and moment acting on the model, there is a great difference between the present system

and.'the old one. At the last time, the stability derivatives were determined by such method as described, in the following.

Namely provided that the force and moment necessary to make the model move in the prescribed manner pure swaying motion, pure yawing motion, etc. have respectively

V0 añdN

as

their amplitudes, and have

E1

and

E'2

as their phase lags behind the prescribed sinusoidal motion, we can determine the stability derivatives from the next foUr values, to say

YocosEl,

sin

El',

N00O5E2 andisinE2.

In order to obtain these four values, the electric signals equivalent to the force and moment are fed to the sine-cosine potentiometer syncronizing with the ship motion. In consequence,

we can get Vcos(u.t-E1)coswt, Tocos(cAIt-E1)sincl)t,

cos

(c&t-E2)coswt and icos(wt-E2)sinWt,

as the outputs of the potentiometer, and these four outputs are soon fed 'to an analogue integrator. The integrated functions consist of the two kinds of timé-functions, one of which represents such one as increases with constant rate, for instance (V,cosE1)/2 in case of

fvcos

(wt- 1)costdt,

and the other is a sinusoidal function with

'the frequency 2W

. The relevant coefficient, (Vcos61)/2, is

which is recorded continuously on a pen-recorder. Other coef-ficients are also determined in the same manner.

At this time, however, these four coefficients, to say

(V0cosE1)/2, (Y,0sinE1)/2, (icosE2)/2

and (sinE2)/2, were

obtained directly by integrating those four outputs of the sine.-cosine potentiometer for the prescribed number of periods on the analogue, integrator. Namely, integration was started and stopped by controlling signals generated from a microswitch which was connected to the planar motion mechanism.

As soon as the integration for the prescribed number of periods 'finished, the integrated values were held on the output terminals of the analogue integrator, and in turn were measured by a digital voltmeter. ' Therefore, if the calibration curve of

the force gauge is prepared in advance, .the stability derivatives can be obtained with ease from the integrated values. By adopt-ing.this analyzing method, the error wtii.ch maymixin the expert-mental results of the stability derivatives mainly at. the time when the rates of increase are determined on' an. oscillopaper, can be diminished remarkably.

As another improvement of the planar motion mechanism, the driving motor was exchanged for a 500W servo-motor,' of which the number of revolution was controlled by a SCR controller. Because the old one was not controlled and was also short of the power.

As a' result of such exchange, the condition that the prescribed motions. of the planar motion mechanism must be purely sinusoidal,

was improved. . ' .

Through the experiments, the driving frequency of the planar motion mechanism was unchanged regardless of the advance speed of

reduced frequency w ' in order .to avoid the resonance frequency of tank.

The amplitudes of drift angle and yaw rate at the pure sway-i.ng motion and the pure yawing motion, to say

and T',

were 0.087266 and 0.156 respectively regardless of the advance speed of the model.

Results of the experiment and discussion

The experimental results of the stability derivatives are shown in Table 2 - Table 7 in the non-dimensional forms. The

lack of data at high Speeds in the shallowest waterway was due to difficulty in execution. Because the sinkage at high speed was so, large that the model might touch the .bottom of the water-way.

The. shallow water effect on each derivative is represented by the ratio of the relevant derivative at the finite water depth

H to that in infinitively deep water. However since it is impossible to make an experiment in infinitively deep water, the latter one is substituted by that at the waterway29.0 times as deep as the draft of the model. Namely, thesubscripts H and

co of the derivatives represent a finite depth of water H and infinitely deep water respectively.

The shallow water effects on the six, derivatives that play the important role to decide the manoeuvrability of a ship, are.

shown in Fig. 1 Fig. 6. .

In the following, the shallow water effecton each

deriva-tive, will be discussed in turn. . .

added mass along the lateral axis, my'

From Table 2', it can be said that the advance speed of the model have an influence on the added mass along the lateral. axis.

Namely with a few exceptions, the lateral added mass,

becomes larger with increase of the advance speed. However, the speed effect on the shallow water effect on the lateral added mass is negligibly small as shown in Fig. 1, where the ratios of the added mass at a finite depth to that in infinitely deep water are plotted with Froude number being as the parameter.

added 'mass moment of inertia about the vertical axis,

J7'

Being not so evident as in case of the lateral added mass, there' seems to be speed effect on the added mass moment of inertia.

In relatively shallow water speaking concretely when the Hid ratio is larger than about 1.8 the added mass moment of inertia diminishes as the advance speed increases. On the other hand, in relatively deep water, to say in case of the ' H/d

smaller than about 1.5, the added mass moment of inertia increases with' increase of .the advance speed.

Fig. 2.'.showØs the speed effect on the, shallow water effect on the added mass moment of inertia. Although the speed effec.t on the' added mass moment of inertia makes a striking contrast according to the depth of water as stated right, above, the shallow water'effect on the added mass moment of inertia in relatively shallow water, to say

_'zH/'z

is not so dependent of the advance

speed. However, when the H/d ratio becomes smaller than about 2.0, the speed effect on the shallOw water effect becomes remarkably

large. Narnély, the J'ZH/J'ZØ.ratio becomes larger with' increase of Froude number.

partial derivative of the lateral force with respect to drift

angle,

Y.

' ,

Similarly to the case of the. added mass moment of inértia, the speed effect on the partial derivative ' in deep. water makes a

striking contrast. with that in shallow water.

Namely in relatively shallow water,

- when the

Hid ratio

is larger than about 1.8the derivative

diminishes against increase of the advance speed. HoWever, in relatively deep water,

- when the

/d ratio is smaller than about 1 .8 - the deriva-tive increases with increase of the speed.

This tendency seems to be exaggerated by the essentialdecrease of the water depth which is brought by the remarkable sinkage of the mode] running with high speed.

The shallow water effect on the derivative Yfi which is as-sessed with the ratio Y'a H/Yeoo, is not affected so remarkably by the advance speed. However at a small Hid ratio, the speed effect on. the shallow water effect becomes remarkably larger with

increase of 'the advance speed.

partial derivative of, the yaw moment, with respect to drift angle,

As shown in Table 4, the derivative becomes larger with increase of the advance speed. This tendency is striking especial-ly in shallower water

The speed effect on the shallow water effect on this derivative is negligibly small in case that the H/d ratio is larger than abo.ut 1.8. On the other hand, when the water depth becomes smaller than that, the advance speed affects the shallow water effect on the derivative N .

partial derivative of the latèra.l force with respect to yaw rate,

'r

It may be'said that the speed effect on the shallow water effect on the derivative

'r' is very 'small. However, it must be noted that the shallow water effect on the partial derivative

1'

of

the lateral force with respect to yaw rate has to be discussed

by the shallow water effect on the pure. hydrodynamic component contained in the derivative yr'" because the derivative _'r.1

consists of the centrifugal force component and the hydrodynamic

one, to say Yr* as defined in the next equation.*

= - m

Fig. 5 showes the shallow water effect on the derivative However, it is difficult to make a definite statement as to the speed effect on the derivative Yr*I _{because of scattering of the.} plotted points. Therefore,, the curve described on the figure represents the mean tendency of the derivative Yr*I _{versus the} depth of Water.

partial derivative of the .yaw moffient with respect to yaw rate, Ny.'

The speed effect on the derivative Nr' is negligibly srnall. Fig. 6 showfs the shallow, water effect on the derivative Nr'.

From this figure, it may be said that there is no definite speed effect on the shallow water effect on the derivative Nr'.

Now, we shall examine the course stability of the model Iy using the experimental results stated above. Course stability of a ship in shallow water is decided by the sign of the difference

between the. two levers, to say ir' and lp' which are defined

*) In the papers [1] and [2], the author' defined the hydrodynamic component Yr*I by the following equation, in which meant the added mass along the longitudinal axis. Yr=Yr*l_(mI+mxi)

Since the added mass m'

is also affected by the finite depth of water, the shallow water effect on the added mass mx'

must be taken into consideration in order to discuss the shallow wate.r effect on the derivative Yy.*'., _{However, it is very} difficult to do so only by the forced yaw test. Hence, the author decided to make the derivative Yr*I contai.n the added mass component mr'.

:the ratios _Nr'/Yr' and Np'/Yp' respectively. Namely, if

r' is larger than the ship is stable in course stability

and vice versa.

These two levers are calculated with the four partial deriva.-tives, to say Np's Yp' _Nr' and and are shown in, Table 8 and 9. As the result, it may be said that both the shallow water effect and the speed effect on the value _ip' have definite tenden-cies; Namely as far as the water depth is same, the

_lp'

value becomes larger with increase ofthe advance speed. This fact means that the course stability of the model has. a tendency to become worse as the advance speed increases. As another

dis-tinctive feature, the _1p' value has the so-called convex character at any Froude number. The H/d ratio at which the

1p' value

has its maxium value., is between about 1.5 and about 1.8.

On the other hand, however, the speed effect on the 1r' value is not so remarkable. .

In Fig. 7 - 'Fig. 11, these two levers at each Froude number are shown in the Same figures. From these figures, it can be deduced that in any case of Froude number, thecourse stability of the model becomes unstabTe at the H/d ratio ranging between about 1.6 and about2.6, in which the lever l.r' becomes smaller than the lever

_1p'

The upper limit of this unstable region becomes largerslightly with increase of the advance speed, while the lower limit remains almost unchanged. Therefore, the unstable region of the Hid ratio becomes broader with increase of the advance speed. However. fromthe practical point of view, it can be said that the.. unstable region remains a1most unchanged

Besides the forced yaw tests,the so-called oblique tow tests' were executed only at Froude number equal to 0.0868, in order to compare the partial derivatives

Y'

and N'

obtained by these tests with those from the forced yaw tests. The lateral force and the yaw moment are plotted in Figs. 12 and 13 in the non-dimensional forms.

The curves connecting the plotted points in these figures rep-resent the next cubic algebraic equations, of which 'the' coefficients, to say V01. Yp'

Ya ', N0, N' and N,

' were determined by

the method of least squares.

+ Y1g'fl'

+ Yef3'3

N' = N0'

+N',S'

+ Nti?Iô'3

The good agreement of these curves with the plotted points means that the cubic algebraic equations represent the hydrodynamic force and moment correctly. In Table 10, the partial derivatives Vp' and Np' àbtained by this method are compared with those from the forced yaw tests. The derivatives from 'these different methods agree well with each other.

For reference, the sinkage and the trim of the model advancing in shallow water are shoWn in Table 11, 12 and also in Fig. 14.

Reference

FuJino, M. :"Studies on Manoeuvrability of Ships in Restricted Waters", Journal of Society of Naval Architects of Japan, Vol.124, 1968 (in Japanese)

Fujino,M-

: "Experimental Studies on Ship Manoeuvrabil'ity in

Restricted Waters - Part I", International Ship-building Progress, 15, No.168, 1968 (in English)

-9-VI - 1

'0

m

m

x,,

List of Symbols

Syüibol . Non-dimensional form Definition

L length ofthe ship

d .. draft of the ship

H depth of water

U . . advance speed of the ship

g . . . acceleration of gravity

F

F= 1-L

. Froude number

CA) _CU'

. circular frequency

fi' =p

drift angle

= _{amplitude of drift angle}

yaw angular velocity

amplitude of yaw angular velocity

density of water sinkage of the ship

trim of the ship, trim by the stem is positive

lateral force

yaw moment

mass of the ship

mx,y. . added mass along the

longi-tudinal axis and the lateral axis respectively

-,Iz mass moment of inertia of

1'Ld. the ship

added mass moment of inertia

ffL'cL

about the vertical axis

10 -r

t

m

Symbol Non-dimensional, form F

_-

_kJ'LLU2

Np

p

*PLcLU2

J)L2ctU

fL2cLU

N'-

_r Nr

*2LctU

'r -Definition

partial derivative of the lateral force with respect to drift angle

partial derivative of the yaw moment with respect to. drift angle

partial derivative of the lateral force with respect to yaw rate

hydrodynamic component of _r

partial derivative of the yaw moment with respect to yaw rate

point of application of sway damping force

point of application of yaw damping force,

lp

iF '=.

Np

12

-

I-Table 1. Particular dimensions of the model

Length between..perpendiculars,

Breadth, m

mm 2,000.0 327.6 Draft, mm 110.3 Displacement, kg 58.44 Block coefficient, _Cb 0.8054 Prismatic coefficient, C 0.8102

Water plane area coefficient, Midship section coeificient, .

Ci,,

.

0.8653 0.9939

Radius of gyration, kL 0.2687L

LCB from midship, mm 50.8 fore

KG, m

. 84.2

Rudder area, mm2 3390.9

Propeller diameter, m

: . 53.8

Pitch, m

39.8

Expanded area ratio 0.619

Projected area ratio 0.554

Boss ratio. . . 0.1821

Rake angle . . . 70 01'

Blade thickness ratio 0.0572

Number of blades . 5

Table 2 : Added mass along the lateral axis, F 1.2 ' 1.3 ' 1.5 1.8 : 2.3 3.0 ' 29.0 0.0482 0.700 0.576 0.440 0.359 0.280 0.244 0.193 0.0675 0.659 0.542 0.419 0.339 0.277 0.243 0.192, 0.0868 0.731 0.560 0.440 0.339 0.279 '0.243 0.198 0.1061 0.613 0.447 0.371 0.290. 0.247 0.207 0.1254 0.639 0.461 0.353 0.294 0.254 0.210

Table Partial derivative of the lateral force with respect to drift angle, '(p'

F 1.2 1.3 1.5 1.8 2.3 3.0 29.0 ô.o482. 1.44' 1.05

0681

0.543 0.475 0.454 0.417 0.0675 1.45 1.0? 0.650 0.516 0.460 0.436 0.379 0.0868

1.62:

1.04 0.650 0.529 .0.464 0.408 0.368 0.1061 1.25 0.716 0.544 0.451 0.424 0.358 0.1254 1.27 0.756 0.540 0.429 0.383 0.333

'Table 4 Partial' derivative of the yaw moment with respect to drift angle, Np'

1.2 1.3: _{1.5 1.8 2.3} 3.0 29.0 0.0482 0.404 0.333 0.254 0.200 0.1.43 0.108 0.0887 0 .0675 0.438 0.347 0.253 ' 0.199 0.146' 0.108 0.0893 0 .0868 0.498 . 0.359 0.256 ' 0.200 0.152 0.117 0.0901 0.1061 0.433 0.284 0.212 0.155 ' 0.120 . 0.0897 0.1254' 0!459 0.302 0.214 0.150 0.116 0.0868 axis, 1.2 1.3 1.5

1.8'

2.3 3.0 29.0 0.0482 0.0238 0.0206 0.0175 0. 0141 0.0121 0.0116 0.0106 0.0675 0. 0272 0.0241 0.0191 0.0142 0.0126 0.0115 0.0107 0.0868' 0.0298 0.0254 0.0195 0.0142 0.0114 '0.0117 0.0105 .0.1061 0.0282 0.0198 0.0157 0.0116 0.00975 .0.0101 0.1254 0.0268 0.0176 0.0124 0.00861 0.008,48 0.O083l

-13

o. 0482

0.0675 0.0868 0.1061 0.1254

Table 6 : Partial derivative of the lateral force with

14

-Table 7 : Partial derivative of the yaw moment with.

respect to yaw rate, Nr'

1.2 1.3 1.5 1.8 2.3 3.0 29.0 0.0482 -0.107 -0.0832. -0.0646 -0.0586 -0.0542 -0.0581 -0.0536 0.0675 -0.0928 -0.0797 -0.0622 -0.0555 -0.0540 -0.0536 -0.0522 0.0868 -0.0978 -0.0830 -0.0728 -0.0648 -0.0567 -0.0541 -0.0532 0.1061

-

. -0.0931 -0.0701 -0.0681 -0.0624 -0.0605 -0.0584 0.1254

- -0.0943

-0.0716 -0.0669 -0.0615 -0.0585 -0.057.7

Table 8 : Point of application of sway damping force, _lp'

1.2 1.3 1.5 1.8 . 2.3 3.0 29.0.. .0.0482 0.281. 0.317 0.373 0.368 0.301 0.239' 0.211 0.0675 0.302 0.341 .0.390 0.385 0.317 ,

0249

0.236 0.0868 0.307 0.345 0.394 0.377 0.327 0.?87 0.245 0.1061

0346

0.397 0.390 0.343 0.282 0.251 0. 1254

-

0.363 0.400 0.396 0.349 0.303 0.261

Table 9 : Point of application of yaw damping force, 1

r 1.2 1.3 1.5 1.8 2.3 3.0 29.0 0.0482 3.45 0.634 0.412 0.330 0.288 0.298 0.285 0.0675 3.09 .' 0.674 0.427 0.346 0.298 0.295 0.283 0.0868 2.61 0.705 0.445 ' 0.372 0.314 0.301 0.285 0.1061

-

0788

0.429 0.363 0.326 0.300. 0.290 0. 1254 0.644 0.424 0.365 0.337 0.307 . 0.280

respect to yaw rate,

1.2 1.3 1.5 r' 1.8 2.3 3.0 29.0 -0.0311 -0.131 -0.156 -0.178 -0.189 -0.195 -0.188 -0.0300 '-0.118 -0.146 -0.161 -0.181 -0.181 -0.185

-0.0374-0.118

-0.164 -0.175 -0.181 -0.180 -0.187 -0.163 -0.188 -0.192 -0.202 -0.201. -0.118 -0.169 -0.184 -0.183 -0.190 -0.206 -0.147

Table 11 : Sinkage of the model in shallow water, - x 1000

290

0.05 Q.11 0.26 0.43 0.67

Table 12 : Trim of the model in shallow water, --x 10000

F'd

1.2 .1.3 1.5 1.8 2.3 3.O 29.0 0.0482 . 0.7 0.5 0.2 0.2 0.8 1.2 0.0675 . . . 3.3

33

-2.3 2.5 2.5 2.5 0.0868 6.3 5.3 4.6 : 4;O 0.1061 11.4 9.7 7.8 7.8, 6.8 6.4

01254

199

148

123

127

96

100

15

-12

13

15

8

23

0.0482 0.33 0.30 .0.25 0.19 0.16. 0.0675 0.75 0.59 0.50 0.39 0.34 0.0868 1.35. 1.07 0.89 0.72 0.55 0.1061 2.18 . 1.61 1.38 1.09 0.82 0.1254 3.13 2.42 1.98 1.56 1.16

Table 10 : Comparison of the experimental results obtained by

the forced yaw tests and oblique tow tests

Hid 1.5 1.8 2..3 3.0 29.0

Forced yaw tests 0.650 0.529 0.464 0.408: 0.368 Oblique tow tests 0.689 0.530 0.464 0.408 0.324 Forced yaw tests 0.256 0.200 0.152. 0.117 0.0901 Oblique tow tests 0.258 0.197 0.137 0.113 0.0921

4.0

3.0

2.0

I.0

1.0

myH/

,myø

0 Fn0.0482

0.0675

0.0868

*

_0.1061

A

0.1254

I

1.5

I

2.0

2.5

tYd

3.0

Fig. 1. Shallow water effect on added mass, my'

3.0

2.0

JH

i.c0.

1.5

2.0

°Fn: 0.0482

.---

0.0675

0.0868

0.1061

--

_..-

_0.1254

2.5

Fig. 2. Shallow water effect on added mass moment of inertia,

J'

5.0

4.O

3.0

2.0

Ypi

Fig. 3. Shallow water effect on static derivative Vp'

Fn 0.048 2

0.06 75

0.0868

0.1061

0.1254

H'

3.0

/d

6.0

.5.0

4.0

3.0

2.0

I.0'

1.0.

.1.5

Fig. 4. Shallow water effecton static derivative Np' I.

2.0

°---.--- Fn: 0.0482

0.0675

0.0868

0.1061

---0--$

.2.5

0.1254

3.0

o

_{Fn: OO4.82}

o

0.0675

0.0868

0.1061

£

_0.1254

2.0

2.5

_3.0

1.0

1.5

Fn 0.0482

0.0675

0.0868

£

_0.1061

£

0.1254

2.0

2.5

3.0

0.8

1Qrip

0.6

.0.4

0.2

l.a..

_'.5

_2.0

_2.5

1Yd

3.0

Ô.8

0.2'

.1.0

21Qp'

1.5

____1

I

2.0

_2.5

_H/d

_3.0

Fig. 9. _{Variation of the points of application of sway and yaw damping forces. F=O.O868}

0.6

/

Fig. 10. _{Variation of the points of application}

0.8

lop'

0.6

04

0.2

I.0

1.5

2.0

2.5

_H/d

3.0

27

-Fig. 12.

0.04

0.03

0.02

0.01

-0.0I

.1 I I &

28

-2.4

6:8

10

H

(deg)

Fig. 13.

Results of.oblique tow tests in shallow

3.0

I

o Fn0°482

0.0675

00868

£

0.106l

0.1254

SINKAGE

;TRIM

Fig. 14., Sinkage and trim of the model advancing in shallow water

ExiOOC

A

3.0

2.0

.2.0

w

z

U)

,I.Q