Seaworthiness tests with three geometrical similar ship models

SEAWORTHINESS TESTS WITH THREE OEOMETRIOAL SIMILAR

SHIPM0DEL3

P u b l i c a t i o n n r

. - 7

^ P

by

i r J . GERRITSMA

D e l f t S h i p b u i l d i n g Laboratory^

2, P r o f . Mekelweg, D e l f t ( H o l l a n d )

To i n v e s t i g a t e the i n f l u e n c e of model s c a l e on the r e s u l t s o f

s e a w o r t h i n e s s t e s t s , t h r e e models of the Todd s e r i e s

6o,

0

,6o

b l o c k c o e f f i c i e n t were t e s t e d i n r e g u l a r waves.

Assuming a s h i p l e n g t h of 400', t h e s c a l e f a c t o r s were

40,50

and

65,

and t h e corresponding model l e n g t h s

10

f t , 8 f t and

6 . 1 5

' f t .

The wave l e n g t h v a r i e d from O

.75

t o I

. 7 5

model l e n g t h s and i n

a l l c a s e s a wave h e i g h t of I

/ 4 8

t h of the model l e n g t h was

cho-s e n .

Mean speed, r e s i s t a n c e , h e a v i n g , p i t c h i n g and phase l a g between

t h e s e motions were measured| some of the t e s t r e s u l t s a r e

com-pared w i t h those of s i m i l a r t e s t s , c a r r i e d out a t t h e T a y l o r

Model Basin«

A f a i r l y good agreement between the measured motions i s found5

the d i f f e r e n c e s a r e of t h e same order a s t h e i n a c c u r a c i e s of

the measurements*

T h i s does not h o l d f o r the added r e s i s t a n c e and t h e l o s s of

speed i u v/aves.

I t i s thought t h a t w a l l i n f l u e n c e might p l a y an important r o l e

i n t h i s k i n d of t e s t s ,

I . IATR0DÏÏCTI01-].

The s u b j e c t o f t h i s paper a r e t h e c o n c l u s i o n s of the i n t e r

-n a t i o -n a l comparative s e a w o r t h i -n e s s t e s t s ( I ) . I -n p a r t i c u l a r ,

g r e a t a t t e n t i o n i s p a i d t o t h e a c c u r a c y o f t h e measurements

and t o t h e p r e v a i l i n g c o n d i t i o n s .

'fhree geosims of the Todd

6 o

- s e r i e 3 of O

.60

b l o c k c o e f f i

-c i e n t were t e s t e d i n r e g u l a r waves i n t h e D e l f t tank. A

comparison w i t h t h e r e s u l t s of the other t a n k s was made

and t h e i n f l u e n c e of tank b r e a d t h was s t u d i e d .

The l a r g e d i f f e r e n c e s between the r e s u l t s of v a r i o u s t a n k s

a s shown i n V e d e l e r ' s r e p o r t ( I ) a r e caused by a number of

f a c t o r s of which we s h a l l c o n s i d e r the f o l l o w i n g s

a ) Inaoouraoy of the measuremtnta»

C o n s i d e r i n g f o r i n s t a n c e the measurement of p i t c h

am-p l i t u d e , a s c a t t e r of a t l e a s t 0

, 1 °

seems

u n a v o i d a b l e .

T h i s l e a d s t o a

3^

e r r o r in. the c a s e of a

5 °

p i t c h

a m p l i t u d e .

wave s l o p e ) i n c l u d e s the i n a c c u r a c i e s of the wave h e i g h and wave l e n g t h measurements a s w e l l .

Assuming a ^fi. e r r o r f o r the f i r s t and 2 ^ f o r the l a t t e r a ponsiiMM e r r o r of 3';' > the d i m e ; o ^ } g;-!?; -..v-ü.^l^ ^.-^.-^-^ meter w i l l he the r e s u l t .

E r r o r s of the same o r d e r of magnitude may a f f e c t the d i m e n s i o n l e s s heave parameter and the added r e s i s t a n c e c o e f f i c i e n t .

b ) S c a l e e f f e c t s .

I t seems improbable t h a t model motions a r e much a f f e c -ted by s c a l e e f f e c t s i the damping c u r v e s of the motions i m p l y s u c h a h i g h damping c o e f f i c i e n t t h a t the g r e a t e r p a r t o f the damping m s t be caused by energy d i s s i p a -t i o n of -the waves g e n e r a -t e d by -the mo-tions.

The added r e s i s t a n c e i n waves may be p a r t l y due to v l s -oOLia (iritg,, 'Mt ilo : 'uh b^ou to^ i/lrit'e:;tö''(t;: e'y:rh<y,i

the v i s c o u s p a r t w i l l a l s o be dependent on wave dimen-s i o n dimen-s and the phadimen-se l a g between wave and raotiona,

e ) Wall e f f e c t s .

A p i t c h i n g and h e a v i n g shipmodel g e n e r a t e s a number of wave s y s t e m s . Ihe v e l o c i t y of t h e s e wave systems depends on the p e r i o d of e n c o u n t e r .

üepending on the model speed and the v e l o c i t y o f the wave systeme, the waves r e f l e c t e d by the tank w a l l s w i l l d i s t u r b more or l e s s the f l o w p a t t e r n around the model, and t h i s w i l l a f f e c t the behavioiir of the model.

The above mentioned p o i n t s should be kept i n mind when a n a l y s i n g s e a w o r t h i n e s s t e s t s .

Ij i t h i s r e p o r t t h e methods of measuring, the t e s t p r o -gram, the r e s u l t s and a few remarks on the r e s u l t s a r e g i v e n .

I Ï I t ^Ix.G MBTHODS.

I n f i g . 1 the arrangement of the model g u i d e s and the meaau-r i n g system i s shown.

G u i d i n g of the model i s done by two p a i r s of l i g h t h o r i z o n t a l t u b u l a r r o l l e r s connected to the model. Together w i t h two v e r t i c a l r o l l e r s connected t o the c a r r i a g e , e a c h of which s l i d e s between one p a i r of the h o r i z o n t a l r o l l e r s , freedom i n h e a v i n g p i t c h i n g and s u r g i n g motions of the model i s ob-t a i n e d , whereas yawing i s noob-t p o s s i b l e .

p i t c h i n g and h e a v i n g motions were measured w i t h p r e c i s i o n low

f r i c t i o n potentiometersp the p i t c h a n g l e was m a g n i f i e d

15

times by a gear.

A g r a v i t y type dynamometer w i t h a r e d u c t i o n of

1

s

15

waa

f i t t e d t o p r o v i d e a c o n s t a n t towing f o r c e t o the model by

means of a sub c a r r i a g e .

G R A V I T Y D Y N A M O M E T E R .

PITCH POTENTIOMETER.

The a u b c a r r i a g e i s guided by a p o l i s h e d r o d , u s i n g b a l l

b e a r i n g s t o minimize f r i c t i o n .

The wave h e i ^ t meter ( r e s i s t a n c e t y p e ) was connected t o the

c a r r i a g e a t a d i s t a n c e o f about

15

f e e t i n f r o n t of t h e

mod e l . A c a l i b r a t i o n i s c o n t i n u o u s l y a v a i l a b l e u s i n g two s l l

-vwr c o n t a c t s on two e x t r a non conducting w i r e s .

V.hen i n c o n t a c t w i t h t h e wave s u r f a c e , e a c h of t h e s e

con-t a c con-t s g i v e s a p i p on con-the wave h e i g h con-t r e g i s con-t r a con-t i o n . Wicon-th a

known v e r t i c a l d i s t a n c e between the c o n t a c t s i t i s p o s s i b l e

t o c a l i b r a t e the r e g i s t r a t i o n of each wave.

The r e l a t i o n between wave p e r i o d and wave l e n g t h was taken

a s T - ( T » period? X « wave l e n g t h ) .

I t was p o s s i b l e t o r e g u l a t e the c a r r i a g e speed i n s u c h a way

t h a t mean model speed was equal t o meeui c a x r i a g e speed w i t h

-i n

Vfo*

T h i s was v e r i f i e d by u s i n g a surge p o t e n t i o m e t e r

con-n e c t e d t o t h e dycon-namometer w h e e l .

An e l e c t r o n i c counter i n c o n n e c t i o n w i t h a p u l s e g e n e r a t o r p r o v i d e s f o r the speed measurement e v e r y one o r t e n seconds. The l o n g i t u d i n a l moment of i n e r t i a was measured w i t h a so üBile.i i,.ert:la table..

The p e r i o d of o s c i l l a t i o n o f t h e systemi t a b l e and model, i s r e t f f t e a to tlj.e t - a r t t t i ov Lhe model o

T h i s t a b l e was a l s o u s e d t o check t h e l o n g i t u d i n a l p o s i t i o n ot -bhG ,uj .i;v-e o f ; ? , ï ' a v l t y o

III«TEST PROGRAM.

As mentioned i n t h e i n t r o d u c t i o n , t h e program c o n s i s t e d o f t e s t s w i t h t h r e e g e o m e t r i c a l l y s i m i l a r shipmodels i n r e g u l a r waves. The s h i p f o r m was o f t h e Todd a e r i e s 6o, p a r e n t form,

.;Xoekceef:dLcie-t O060 v i t h s t a-dai- d r;beer a-ti mea-. f nr,-.^^ The model l e n g t h s were r e s p e c t i v e l y 1 10 f t , 8 f t and 6 . I 5 f t c o r r e s p o n d i n g w i t h s c a l e f a c t o r s 40, 50 and 65 (aasumintr a s h i p l e n g t h of 4 0 0 ' ) .

The r a d i u s of t h e l o n g i t u d i n a l moment o f i n e r t i a amounted t o

0,25 L f o r each model.

The Froude number v a r i e d from gero tO 0.30 and the wave dimensions wares

wave h e i g h t » 1/48 L

^ 1 i f " ^ * ^ ' 1.00, 1*25, 1.50, 1.75 L . ihe a c t u a l wave dimensions a r e summariaed i n t a b l e 1.

Vz

10'

A C>n model Vz

0,74

1

5,1

0,78

1,00 305 6 4 5tö 1,00

1,23

r

3,0

:a,2o

452

2,5

i.:v-;-1,75

'3 a t 1

2,1

.

1,74

'a,.aa 1 uave diuio^ifjio- 3 8» model X C>v> a . i -6.15' model Vz •ipü

_0,75

l;Vi

5,0

:a.O 1 , 0 1 _{189 , Q}

_3»7

3,0

2m '

_{3 , 0}

2,4 1,50

281 2,d> 2 , 2

_1,75

328 2 , 1 The a c t u a l wave l e n g t h s d i f f e r somewhat from t h e nominal v a l u e s g i v e n b e f o r e but t h e d e v i a t i o n s a r e s m a l l e r t h a n

RESULTS.

!• P i t c h i n g and h e a v i n g .

F l g u r e a

2, } , 4

and

5

g i v e the r e s u l t s of the motion

t e s t s .

The d i m e n s i o n l e s s p i t c h parameter i s deflïied a s

'^L p i t c h amplitude , , , ^,

" mxlmum wave s l o p e ^""^ s i m i l a r l y the heave

parameter

heave a m p l i t u d e

The ploase l£g (heave a f t e r p i t c h ) i s g i v e n i n

de-g r e e s . ,

5

/1- aiid ^ a r e p l o t t e d on a base of

t h e Froude nriraber F r , a s functloi'^s of the wave l e n g t h

parameter yl ,

F o r Froude numbers l e a s tlion O.lO no p o i n t s a r e showü,

on account of the l a r g e s c a t t e r i n t h i s r e g i o n . T h i s

p o i n t w i l l be d i s c u s s e d l a t e r .

Reaistaxioe and apeed l o s s .

The r e s u l t s of tha r e s i s t a n c e t e s t s a r e shown i n f i g

. 6 ,

7 and 80 I n t h i s f i g u r e s , the measured r e s i s t a n c e s were

c o r r e c t e d f o r d i f f e r e n c e s between nominal and a c t u a l

wave h e i g h t , s i n c e the l a t t e r v a r i e d somewhat f o r each

r u n .

According t o

(2)

the c o r r e c t e d r e s i s t a n c e i s taken a s

where t

Tfrt.- added r e s i s t a n c e a t the nominal wave h e i g h t 2 r

t o t a l r e s i s t a n c e measured i n waves of h e i g h t 2 r

R < - s t i l l / w a t e r r e s i s t a n c e .

The r e l a t i o n ( 1 ) was not v e r i f i e d e x p e r i m e n t a l l y , but the

c o r r e c t i o n s proved t o be v e r y s m a l l .

To compare the r e s u l t s of the motion t e s t s , c r o s s c u r v e s

a r e c o n s t r u c t e d from the c u r v e s c o n t a i n i n g the b a s i c

d a t a . Thus f i g .

9

g i v e s the motion parameters ^y< , S/h

and the phase l a g s on a base of Vl f o r the Froude

The w e l l krxown speed l o s s c u r v e s a r e shown i n f i g u r e 10.

They a r e t h e c r o s s c u r v e s of t h e r e s i s t a n c e - s p e e d c u r v e s Assuming t h a t the added r e s i s t a n c e i s n o t o f a v i s c o u s o r i g i n t h e s p e e d - l o s s c u r v e s a r e s u i t a b l e f o r a compa-r i s o n o f d i f f e compa-r e n t model l e n g t h s .

T h i s i s v a l i d a l s o f o r t h e added resishiUiot; uortffiolosroi: ''a " 1^ f - ^ * ^ ^ " wetted s u r f a c e ^ p l o t t e d on a base of Froude number ( f i g u r e 11)

DISCUSSION OF THE RESULTS.

The o r i g i n a l t e s t d a t a o f t h e motion t e s t s show a mean d e v i a t i o n fgom t h e curves' of +_ 0,1 f o r p i t c h , 1 mm f o r heave and 5 f o r t h e phase l a g s .

The c r o s s c u r v e s p r e s e n t e d I n f i g u r e 9 show a somewhat i n -c r e a s e d s o a t t e r i n g when -comparing the 10' f t ahd 6,15 f t models e s p e c i a l l y a t Vz - l,50and 1,75 su^^d i t a p p e a r s t h a t t h e p i t c h i n g motion a t t h e s e wave l e n g t h s i s somewhat l a r g e r f o r tho 10' model and a m a l l e s t f o r th-^ ^,15' model. But t h e d i f f e r e n c e s a r e too s i n a l l (+, 0 , 2 ° 5. +. 0,3 ) i n

: •;) '\i, 'l^l,a >:'hr.'iU.:o'.: iu;iu;vre;,ov -bo'ijJloU any c o n c l u s i o n s t o be drawn*

A comparison w i t h t h e T.i.!.3. r e s u l t s ( t a k e n from r e f e r e n c e

( 2 ) ) i s made f o r a Froude number o f 0,25 i n t a b l e 2 .

TABLE 2 Froude number 0,25 i

1

,^0

3

,8

1

,25

3,3

1

,50

2,5

0,78

1,20 1

,40

1,42

_1,24

0,58

1.1T+

l,27++

A + 10 f t model •M- 5 f t model 0,88

1

,63

(1,33+

4

6+4- '1,16++

The d l f f s r e n o s s I n fk w i t h t h e T a y l o r Model B a s i n v a l u e u a r e of t h e same o r d e r a s the s c a t t e r o f t h e p o i n t s i n t h e c r o s s - c u r v e s o f f i g u r a «nd t h e motion c u r v e s i n ( 2 ) .

The i / h v a l u e s o f t h e TMB a r e somewhat h i g h e r , whereas t h e i r phase l a g s a r e about 10 - I 5 d e g r e e s l o w e r .

T h i s same p i c t u r e h o l d s f o r t h e o t h e r F r a i d e numbers. As w i l l be seen I n t h e f i g u r e s 2, 5 and 4 no e x p o r i m e n t a l p o i n t s ivr& g i v e n below a F r a u d s number of 0,10 « t h e l a r g e

s c a t t e r o f t h e p o i n t s i n t h i s r s g i o n would n i s t i f y t h s p i o

-t u r s .

I h s r s a s o n of t h i s s c a t t e r i s due t o t h s waves g s n s r a t e d by

the model motions. Vflisn the speed o f t h e s e waves i s auch

t h a t the a r e a ahead o f t h e model i a covered w i t h t h s s s d i s

t u r b i i n c e s j Lnöip2.rur motisjia m y bo expected which a r e d i f

-f i c u l t t o a n a l y s e . :arard

(5)

showed t h a t t h i s t a k e s p l a o s

a t a c e r t a i n speed d e f i n e d b y j

where V » model spaed

Te - p e r i o d o f encomiter

F o r ^ < 'A the d i o t u r b a n o s s t r a v e l f a s t e r than the modsl

whereas f o r ^ >'// t h e y c o v e r an a r e a enclos»3d i n a V

(opening a n g l s

3

) H k e ote ave system of a s h i p i n s t i l l

w a t e r .

:"tei'rftaö,a';,-.''.g;]

. , . . t . ^

0 - A / i ^ y - x / ^ - - ^ ^

U s i n g f o r m u l a s (^3) w i t h y » t h e o r i t i o a l s p s s d s s s a

ftanotion of t h e wave l e n g t h parameter oan be o a l o u l a t s d

( e e a t a i sio t ] :

T A B L E 3

0,75

1

,00

1

,25

1

,50

1

,75

j f G,07 0,03 0

,09

0,10 0,11

The i n f l u s n o s o f t h s tank w a l l s on motions and r s s l s t a n o s

w i i l be due, f o r the g r e a t e r p a r t , t o t h s r s f l e e t e d wavs

s y s t e m s g s n s r a t e d by t h e model.

Tank b r e a d t h B and model l e n g t h L, i n oonneotloa w i t h modsl

speed V, groitp v e l o c i t y o f t h e wavss g e n e r a t e d by t h s modsl

ajid t h s a n g l e ^/^6 i n which the l a t t e r a r e enoloaed d e t e r

-mine whether t h e r e f l s o t s d wavs systems w i l l i n f l u s n o s the

motion and r e a i a t a n c e , o r n o t .

Ths f o l l o w i n g parameters t h e r e f o r e may be u s e d f o r t h a p l o t

-t i n g of r e s u l -t s w i -t h r e g a r d -t o w a l l e f f e c -t s j

( s e e a l s o form 3)

» B / L , -^Z

, F r . ( b e i n g d e f i n e d by

F r

I t aeeiua d i f f i c u l t t o c a l c u l a t e t h e o r e t i c a l l y the c o n d i t i o n s

g o v e r n i n g the e f f e c t which the r e f l e c t i o n s may have on t h e

motions and r e s i s t a n c e o f the model. I n a n y c a s e a t i n c r e a

-s i n g v a l u e -s o f ^ ( d e c r e a -s i n g ) the e f f e c t mu-st become

s m a l l e r .

As s t a t e d b e f o r e , the r e s u l t s of the motion t e s t s f o r d i f f e

-r e n t model l e n g t h s show d i f f e -r e n c e s , which a -r e of t h e same

o r d e r a s the e x p e r i m e n t a l i n a c c u r a c y . T h e r e f o r e no attempt

oan been made t o r e v e a l w a l l e f f e c t from t h e s e r e s u l t s .

The r e s i s t a n c e c u r v e s ( f i g . 6, 7, 8 ) show a marked hump a t

about F r => 0,12 f o r W/ m 1,00.

As c a n be seen t h i s hump i s i n c r e a s i n g r a p i d l y w i t h

model-langthó _

The added r e s i s t a n c e c o e f f i c i e n t G = '^A/^' show t h s same

tendency ( s e e f i g , 1 1 ) a v

A n o t a b l e w a l l e f f e c t seems t o be p r e s e n t h e r e and t h i s h a s

a g r e a t i n f l u e n c e on the speed r e d u c t i o n c u r v e s ( s e e f i g u r e

1 0 ) a s t h e minimum o f t h i s c u r v e s I s m o s t l y I n t h s v i c i n i t y

of Vi- ° 1. The 10 f t and 8 f t speed r e d u c t i o n c u r v e s show

t h i s i n f l u e n c e a t ^/i. - 1 a s a v e r y l a r g e speed r e d u c t i o n

f o r medium speeds and a s m a l l i n c r e a s e f o r t h e lower speeds.

I n g e n e r a l t h e r e i s l i t t l e c o n f o r m i t y between t h e t h r e e s e t s

of c u r v e s .

From r e f e r e n c e s ( 1 ) and ( 2 ) speed r e d u c t i o n c u r v e s could be

c o n s t r u c t e d f o r t h e TMB and MIT t e s t s .

Together w i t h our own v a l u e s t h e speeds i n waves a t c o n s t a n t

towing f o r c e s t a r t i n g from t h e s t i l l w a t e r speeds, F r a 0,280

0,255 s.nd 0,187, a r e g i v e n i n t a b l e 4. The v a l u e o f 1

ta. .Ir ..readth i s alelo in.oluded«

Spetd I n v a r a s a t o o n s t a n t towing f o r o e

- Vz

₀,28 Fr a t

0,235

X -

0

0,187 L

_BA

Tank

1,00

0,23

0,13

0,09

2

1,25

0,21

0,16

0,12

10'

5,1

1

,50 0,25

•

0,15

1,00

0,23

0 , l 6

0,13

1,25

0,24 0,18

_0,15

_6,15'

2,28 D e l f t

1.50 0,27

0,21

0,17

1,00

0,22

0,13

0,11

1

,25 0,23

0,18

_0,14

8'

_1,75

D e l f t

1,50

0,27

0,21

0,16

i , e ö 0,22

0,11

0,10

1,25

0,21

0,16

0,13

10'

1,40

D e l f t

1,50

0,25

0,19

0,15

1,00

0,22

0,14

0,10

1,25

0,22 0,17

0,14

5'

1,60

MIT

1

,50

0 , 2 4

1 0,20

0,17

I t oan be concluded from t a b l e 4 t h a t the d e v i a t i o n s a r e f a l t l y l a r g e i n some c a s e s .

The I'JIIT v a l u e s and the D e l f t 1 0 ' and 8' r e s u l t s ( w i t h B / L r e s p . l, 6 0 j 1 , 4 0 and 1 , 7 5 ) show the b e s t agreement, which should em-p h a s i z e the i n f l u e n c e of the B / L parameter. However, a s the MIT v a l u e s a r e d e r i v e d from a s m a l l s c a l e r e p r o d u c t i o n i u ( l ) t h e s e may be not e x a c t .

The added r e s i t a n c e c u r v e s ( f i g u r e 1 1 ) show d i f f e r e n c e s of about p l u s or mixius 1 0 ^ w i t h r e g a r d to the t o t a l r e s i s t a n c e c o e f f i c i e n t s f o r tho F r . numbers > 0 , 1 5 ( t h e s t i l l water r e s i s -t a n c e c o e f f i c i e n -t b e i n g a p p r o x i m a -t e l y 0 , 0 0 5 f o r speeds up t o F r - 0, 2 8 )

Again, the h i g h e r v a l u e s a r e o b t a i n e d w i t h the

10

f t model and

t h e lower w i t h the

6 , 1 5

f t model.

I a I t u n l i k e l y t h a t t h e s e d i f f e r e n c e s a r e due t o a x p a r l m s n t a l

e r r o r s , but an e x p l a n a t i o n of the d e v i a t i o n s i s not a t hand.

S o a l e s f f e c t

i t

p r s s s n t a t a l l should have the o p p o s i t e e f f s o t t

h i g h e r addsd r s s l s t a n o s o o s f f l o l s n t s f o r the s m a l l e r models

assuming t u r b u l e n t f l o w *

A oomparlson w i t h the C c u r v e s o f t h s TMB I s not g i v e n , beosuss

o f t h s s m a l l s c a l e of t h e i r f i g u r e s i n ( 2 ) .

The

B / L

v a l u e s of the t h r e e models cover a range of

1 , 4 0

t o

2,26 s h l o h a p p s a r s t o be too s m a l l to p l o t t h s C «alu#s on a

b a s s o f

B/L.

A d d i t i o n a l v a l u e s a t

B/L • 3 , 4

and

3

* r s n s s d s d f o r t h i s purposs

COHCLUDIivG RE^IARKS.

The f o l l o w i n g c o n c l u s i o n s may be drawn from t h s t e s t s .

1)

Within e x p e r i m e n t a l aoouraoy, the motion parameters of the

t h r e s modols a r s the same* There i s a s l i g h t i n d i c a t i o n t h a t

t h s p l t o h paramotsr i s somswhat l a r g e r f o r the

10

f t modsl.

2 )

Ths s p s s d l o s s o u r v s s a r s g r s a t l y I n f l u s n o s d by w a l l e f f seta;;

F r o u d s numbsrs*

At t h s h l f l h s r Frou.de numbsrs the added r e s l s t a n o s o o s j ^ l

-o l e n t s sh-ow d l f f s r e n c e s f -o r t h e t h r e s m-odslsi

t h s h l g h s s t v a l u s s a r e shown by t h s I s r g s s t modsl.

Tliess d l f f s r e n o s s a r s b s l l s v s d t o be l a r g e r t h a n t h s s x p s

-r l m e n t s e -r -r o -r s *

3 )

A d d i t i o n a l t e s t s a t

B / L - 3» 4

and

5

m i ( ^ t r s v s a l soms of t h s

w a l l e f f e c t s .

REFERKi:CESi

(1)

Seventh I n t e r n a t i o n a l C o n f s r s n o s on Ship Hy&rodynaraloa

S u b j e c t 6s Ssagoln4 ^ l i t l e s of S h i p s

1934*

( 2 ) Ur, V.G. S z s b s l t e l y , M.B. B&sdsos, G.P. S t e f u n .

S e a l s s f f o o t s i n S s a w o r t h i n s s s TMB Rsp. no.

( 3 )

P r o f . R. ^ a r d i I n t r o d u c t i o n a l*<tude t h i o r i q u e du tangage

tu

'o

O!

(

'

1

u

I

i ,

•H

- P

O

0)

§

I

S

Ti +»

.

Q

•

•rl

fin

,1

+»

'II

•V

F i g . 8 R s s i s t a n o e

6 . I 5

f t model

« M E A N V A L U E S

0 . 7 5 I.OO 1.25 L 5 0 1.75

^

V

L

1.5

H E A V E

1.0

25

T H A N N I V E R S A R Y O F T H E N E T H E R L A N D S S H I P M O D E L B A S I N W A G E N I N G E N R e l^ ' c e r - - ^ - P l a b . V.

Sc'neepsboijwkunrj

Technische Hogeschool

P R O C E E D I N G S

S Y M P O S I U M O N T H E B E H A V I O U R

OF SHIPS

I N A SEAWAY

V O L U M E I S E P T E M B E R 7 T H - 1 0 T H 1 9 5 7 1959 H. V E E N M A N & Z O N E N N V . — W A G E N I N G E N

( . ' T I . : H 27

S E A W O R T H I N E S S T E S T S W I T H T H R E E G E O M E T R I C A L S I M I L A R S H I P M O D E L S

hy]. G B R R I T S M A

Doll'l Shipbuilding Labonilory, Nelherlands

S Y N O P S I S :

To i n v e s t i g a t e the i n f l u e n c e of model soale on the r e s u l t s of seaworthiness t e s t s , t h r e e models of the Todd s e r i e s 60, O.60 b l o c k c o e f f i c i e n t were t e s t e d i n r e g u l a r waves.

Assuming a ship l e n g t h o f 400', the scale f a c t o r s were 40,50 and 65, and the corresponding model lengths 10 f t , 8 f t and 6 . 1 5' f t . The wave l e n g t h v a r i e d from 0.75 t o I. 7 5 model lengths and i n a l l oases a wave height of I/ 4 8 t h of the model l e n g t h was chosen.

Mean speed, r e s i s t a n c e , heaving, p i t c h i n g and phase l a g be-tween these motions were measured; some of t h e t e s t r e s u l t s are compared w i t h those of s i m i l a r t e s t s , c a r r i e d out a t the Taylor Model 3asin.

A f a i r l y good agreement between the measured motions i s found; the d i f f e r e n c e s are o f the same order as the i n a c c u r a c i e s of the measurements.

This does not hold f o r the added r e s i s t a n c e and the loss of speed i n waves.

I t i s thought t a t w a l l i n f l u e n c e might p l a y an important r o l e i n t h i s k i n d of t e s t s .

I . DJTROIXJCTIOM.

The subject of t h i s paper are the conclusions o f the i n -t e r n a -t i o n a l compara-tive seawor-thiness -t e s -t s ( 1 ) . I n p a r -t i c u l a r , grea-t a -t -t e n -t i o n i s paid -t o -the accuracy of -the measurements and t o the p r e v a i l i n g c o n d i t i o n s .

Three geosims of t h e Todd 60-series o f O.60 block c o e f f i -c i e n t were t e s t e d i n r e g u l a r waves i n t h e B e l f t tank. A comparison w i t h the r e s u l t s of the other tanks was made and tne i n f l u e n c e of tank breadth was s t u d i e d .

The l a r g e d i f f e r e n c e s between the r e s u l t s of v a r i o u s tanks as shown i n Vedeler's r e p o r t (1) are caused by a number of f a c t o r s o f which we s h a l l consider the f o l l o -wing:

a) Inaccuracy of the measurements.

Considering f o r i n s t a n c e the measurement o f p i t c h am-p l i t u d e y , a s c a t t e r of a t l e a s t 0 , 1 ° seems unavoidable.

This leads t o a J,fo e r r o r i n the case o f a 3° p i t c h amplitude.

A dimensionless p r e s e n t a t i o n such as Ot = maximum

4 6 8

-J. GERKITSM A

w^ve slope) i n c l u d e s the i n a c c u r a c i e s of the wave h e i g h t and wave l e n g t h measurements as w e l l .

Assuming a e r r o r f o r t h e f i r s t and 2'fo f o r the l a t t e r , a p o s s i b l e e r r o r of B-fo i n t h e dimensionless p i t c h p a r a -meter w i l l be t h e r e s u l t .

E r r o r s of the same order o f magnitude may e f f e c t t h e dimensionless heave parameter and the added r e s i s t a n c e c o e f f i c i e n t .

b) Scale e f f e c t s .

I t seems improbable t h a t model motions are much e f f e c -ted by scale e f f e c t s : the damping curves o f the motions i m p l y such a h i g h damping c o e f f i c i e n t t h a t the g r e a t e r p a r t o f the damping mast be caused by energy d i s s i p a -t i o n o f -the waves genera-ted by -the mo-tions.

The added r e s i s t a n c e i n waves may be p a r t l y due t o v i s -cous drag, but we do n o t know t o what e x t e n t ; anyhow the viscous p a r t w i l l a l s o be dependent on wave dimen-s i o n dimen-s and the phadimen-se l a g between wave and motiondimen-s. c) \7all e f f e c t s .

A p i t c h i n g and heaving shipmodel generates a number o f wave systems. The v e l o c i t y o f these wave systems depends on t h e p e r i o d of encounter.

Depending on the model speed and the v e l o c i t y of t h e wave systems, the waves r e f l e c t e d by t h e tank w a l l s w i l l d i s t u r b more or l e s s t h e f l o w p a t t e r n around the model, and t h i s w i l l a f f e c t the behaviour o f t h e model.

The above mentioned p o i n t s should be k e p t i n mind when a n a l y s i n g seaworthiness t e s t s . •

I n t h i s r e p o r t the methods of measuring, the t e s t p r o -gram, the r e s u l t s and a few remarks on t h e r e s u l t s a r e g i v e n .

I I . MEASÜRIKG yiETHOJS.

I n f i g . 1 the arrangement o f the model guides and the measuring system i s shown. ^ ^ . ^ • O a i d i n g of the model i s done by two p a i r s of l i g h t h o r i -z o n t a l t u b u l a r r o l l e r s connected t o t h e model. Together w i t h two v e r t i c a l r o l l e r s connected t o t h e c a r r i a g e , each of which s l i d e s between one p a i r of the h o r i z o n t a l r o l l e r s , freedom i n heaving p i t c h i n g and s u r g i n g motions of t h e model i s obtained, whereas yawing i s not p o s s i b l e .

-PART 4

P i t c h i n g and heaving n o t i o n s were measured w i t h p r e c i s i o n low f r i c t i o n p o t e n t i o m e t e r s ; t h e p i t c h angle was m a g n i f i e d 15 t i m e s by a gear.

A g:ravity type dynamometer w i t h a r e d u c t i o n of 1 : 5 was f i t t e d t o provide a constant t o w i n g f o r c e t o t h e model by means o f a sub c a r r i a g e .

GRAVITY DYNAMOMETER .

/PITCH POTENTIOMETER.

F i g . 1 Towing arrangement

The suboarriage i s guided by a p o l i s h e d r o d , u s i n g b a l l b e a r i n g s t o minimize f r i c t i o n .

The wave h e i g h t meter ( r e s i s t a n c e t y p e ) was connected t o t h e c a r r i a g e a t a d i s t a n c e o f about I J f e e t i n f r o n t of the mod e l . A c a l i b r a t i o n i s c o n t i n u o u s l y a v a i l a b l e u s i n g two s i l -v e r c o n t a c t s on two e x t r a non conducting w i r e s ,

'.'ftien i n contact w i t h t h e wave s u r f a c e , each o f these con-t a c con-t s gives a p i p on con-t h e wave h e i g h con-t r e g i s con-t r a con-t i o n . Wicon-th a known v e r t i c a l d i s t a n c e between t h e c o n t a c t s i t i s p o s s i b l e t o c a l i b r a t e the r e g i s t r a t i o n o f each wave.

The r e l a t i o n between wave p e r i o d and wave l e n g t h was taken as T - 1/2'TA (T = p e r i o d ; A =. wave l e n g t h ) .

I t was p o s s i b l e t o r e g u l a t e t h e c a r r i a g e speed i n such a way t h a t mean model speed was equal t o mean c a r r i a g e speed w i t h -i n l<fa. This was v e r i f i e d by u s i n g a surge p o t e n t i o m e t e r con-nected t o the dynamometer wheel.

4 7 0

-J. GERRITSMA

An e l e c t r o n i c counter i n conneoti,on w i t h a pulse genera-t o r provides f o r genera-the speed measuremengenera-ts every one or genera-t e n seconds. The l o n g i t u d i n a l moment of i n e r t i a was measured w i t h a 30 c a l l e d i n e r t i a t a b l e .

The p e r i o d of o s c i l l a t i o n of the system: t a b l e and model, i s r e l a t e d t o the i n e r t i a of the model.

This t a b l e was a l s o used t o check the l o n g i t u d i n a l p o s i -t i o n of -the cen-tre o f g r a v i -t y .

III.TSST PROGRAJjl.

As mentioned i n the i n t r o d u c t i o n , the program consisted of t e s t s w i t h t h r e e g e o m e t r i c a l l y s i m i l a r shipmodels i n r e g u l a r waves. The s h i p f o r m was o f the Todd s e r i e s 6o, parent form, b l o c k c o e f f i c i e n t 0.60 w i t h standard sheer and mean f l a r e .

The model l e n g t h s were r e s p e c t i v e l y : 10 f t , 8 f t and 6.15 f t corresponding w i t h soale f a c t o r s 40,50 and 65

(assuming a s h i p l e n g t h of 400').

The r a d i u s o f t h e l o n g i t u d i n a l moment o f i n e r t i a amoun-ted t o 0,25 L f o r each model.

The Froude number v a r i e d form zero t o 0.50 and the wave dimensions were:

wave h e i g h t = l/48 L

wave l e n g t h = 0.75, 1-00, 1.25, 1-50, 1-75 L. The actij.al wave dimensions are summarized i n t a b l e 1 .

TABLE 1 Wave dimensions

10' model 8 ' model

i

.15' model

V/.

_Cim A d e j r e ö oc

Vz.

_cm A zr cm A cm ar _{cm ol€3i-ees} oc 0,74 1,00 1,23 1,48 1,75 225 505 354 452 534 6,4 5,1 3,8 3,0 2,5 2,1 0,78 1,00 1,26 1,54 1,74 190 244 515 376 425 5,1 4,8 3,8 5,0 2,4 2,2 0,75 1,01 1,26 1,50 1,75 141 189 236 281 328 3,9 5,0 5,7 5,0 2,5 2,1 The a c t u a l wave l e n g t h s d i f f e r somewhat from the nominal values given before b u t the d e v i a t i o n s are smaller than

-IV. RESULTS.

P i t c h i n g and heaving.

Figures 2, 3, 4 and 5 g i v e the r e s u l t s o f the motion t e s t s .

The dimensionless p i t c h parameter V/d i s d e f i n e d as

parameter

f - heave a m p l i t u d e _{irave amplitude}

The phase l a g (heave a f t e r p i t c h ) i s g i v e n i n de-grees, ^/a , and t?" are p l o t t e d on a hase o f the Froude number F r , as f u n c t i o n s of the wave l e n g t h parameter ^/L. .

For Froude numbers l e s s thaui 0.10 no p o i n t s axe shown, on account of the l a r g e s c a t t e r i n t h i s r e g i o n . This p o i n t w i l l be discussed l a t e r .

Resistance and speed l o s s .

The r e s u l t s of the r e s i s t a n c e t e s t s are shown i n f i g ,6 , 7 and 8. I n t h i s f i g u r e s , the measured r e s i s t a n c e s were c o r r e c t e d f o r d i f f e r e n c e s between nominal and a c t u a l wave h e i g h t , since the l a t t e r v a r i e d somewhat f o r each r u n .

According t o (2) the c o r r e c t e d r e s i s t a n c e i s taken as

in where:

- added r e s i s t a n c e a t the nominal wave h e i g h t 2r t o t a l r e s i s t a n c e measured i n waves of h e i g h t 2r s t i l l s - w a t e r r e s i s t a n c e .

The r e l a t i o n ( I ) was not v e r i f i e d e x p e r i m e n t a l l y , b u t the c o r r e c t i o n s proved t o be v e r y s m a l l .

To compare the r e s u l t s of the motion t e s t s , cross curves are c o n s t r u c t e d f r o m the curves c o n t a i n i n g the basio d a t a . Thus f i g , 9 g i v e s the motion parameters 9^/o(. ,

and the phase l a g s on a base of -^JL f o r the Froude numbers Tr.'U/^/^ ; o . l 5 , 0.20, 0.25, 0 . 5 0 .

4 7 2

-J. CERRITSM/

The w e l l known speed l o s s curves are shown i n f i g u r e 10. They are the cross curves of the resistance—speed curves Assuming t h a t the added r e s i s t a n c e i s not of a viscous o r i g i n the speed—loss curves are s u i t a b l e f o r a compa-r i s o n of d i f f e compa-r e n t model l e n g t h s .

This i s v a l i d a l s o f o r the added r e s i s t a n c e c o e f f i c i e n t s C ^p<\/ii (where S - wetted s u r f ace_)plotted on a base of Froude number ( f i g u r e l l )

DISCUSSION OF THE RESULTS.

The o r i g i n a l t e s t data of the motion t e s t s show a mean d e v i a t i o n from t h e curves of + 0,1 f o r p i t c h , + 1 mm f o r heave and 5° f o r the phase l a g s .

The cross curves presented i n f i g u r e 9 show a somewhat i n -creased s c a t t e r i n g when comparing the 10' f t ahd 6,15 f t models e s p e c i a l l y a t V L - l,50and 1,75 and i t appears t h a t the p i t c h i n g motion a t these wave lengths i s somewhat l a r g e r f o r the 10* model and smallest f o r the 6,1^' model. But the d i f f e r e n c e s are ±00 small ( + 0 , 2 5. + 0,5 ) i n comparison w i t h the estimated absolute accuracy t o a l l o w any conclusions t o be drawn,

A comparison w i t h the T.M.B. r e s u l t s ( t a k e n from r e f e r e n c e (2)) i s made f o r a Froude number of 0,25 i n t a b l e 2.

TABLE 2 Froude number 0,25

VL

D e l f t + 10_{++ S f t model} f t model

VL

i/t S/t-1,00 1,25 1,50 5,6 5,5 2,5 0,64 1,20 1,42 0,78 1,40 1,24 0,58 i l, 1 7 + 11,27++ J l, 5 5 + U,46++ 0,88 1,63 ƒ1,55+ l l, l 6 + +

The d i f f e r e n c e s i n f/oL w i t h the Taylor Model Basin values are of the same order as the s c a t t e r of the p o i n t s i n the cross—curves of f i g u r e S and the motion curves i n ( 2 ) . The Sir values of the TMB are somewhat h i g h e r , whereas t h e i r phase l a g s are about 10 - 15 degrees lower.

This same p i c t u r e h o l d s f o r the other Froude numbers. As w i l l be seen i n ^ t h e f i g u r e s 2, 3 and 4 no experimental p o i n t s are g i v e n below a Froude number of 0,10 : the l a r g e

-PART 4

s c a t t e r o f the p o i n t s i n t h i s r e g i o n would m i s t i f y the p i c -t u r e .

The reason of t h i s s c a t t e r i s due t o the waves generated by the model motions. When the speed of these waves i s auch t h a t t h e area ahead o f t h e model i s covered w i t h these d i s t u r b a n c e s , i r r e g u l a r motions may be expected which are d i f -f i c u l t t o analyse. Brard (5) showed t h a t t h i s takes place a t a c e r t a i n speed d e f i n e d by:

y- 2TT V_ ^ / A 2

-where V - model speed

- p e r i o d o f encounter

For y•^'//f the d i s t u r b a n c e s t r a v e l f a s t e r than the model whereas f o r Y> '/^ t h e y cover an area enclosed i n a V

(opening angle p, ) l i k e t h e wave system of a s h i p i n s t i l l w a t e r .

S u b s t i t u t i n g :

U s i n g formulae 5 w i t h ^ - ^ t h e c r i t i c a l speeds as a f u n c t i o n of t h e wave l e n g t h parameter oan be c a l c u l a t e d (see t a b l e J )

TABLE 5

VL

0,75 1,00 1,25 1,50 1,75 FT 0,07 0,08 0,09 0,10 0 , 1 1

The i n f l u e n c e of the t a n k w a l l a on motions and r e s i s t a n c e w i j . ! be due, f o r the g r e a t e r p a r t , t o the r e f l e c t e d wave systems generated by t h e model.

Tank b r e a d t h B and model l e n g t h L, i n connection w i t h model speed V, group v e l o c i t y of the waves generated by the model and the angle / 3 i n which the l a t t e r are enclosed d e t e r -mine whether the r e f l e c t e d wave systems w i l l i n f l u e n c e the m o t i o n and r e s i s t a n c e , o r n o t .

The f o l l o w i n g parameters t h e r e f o r e may be used f o r t h e p l o t -t i n g of r e s u l -t s w i -t h r e g a r d -t o w a l l e f f e c -t s : 4 7 4 -J. GERRITSMA (see a l s o form J ) : B / L , , Fr. ^ ( b e i n g d e f i n e d by FT and ^ A ) -I t seems d i f f i c u l t t o c a l c u l a t e t h e o r e t i c a l l y the c o n d i t i o n s governing t h e e f f e c t which the r e f l e c t i o n s may have on the motions and r e s i s t a n c e o f the model. I n any case a t i n c r e a ^ s i n g values of (decreasing /2, ) the e f f e c t must become s m a l l e r .

Aa s t a t e d b e f o r e , t h e r e s u l t s of t h e motion t e s t s f o r d i f f e -r e n t model l e n g t h s show d i f f e -r e n c e s , which a-re of the same

order as t h e experimental inaccuracy. Therefore no attempt can been made t o r e v e a l w a l l e f f e c t from these r e s u l t s . The r e s i s t a n c e curves ( f i g . 6, 7, 8) show a marked hump a t about Fr - 0,12 f o r A / ^ . 1,00.

As can be seen t h i s hump i s i n c r e a s i n g r a p i d l y w i t h model-l e n g t h .

The added r e s i s t a n c e c o e f f i c i e n t C - —7TJ> shows the same tendency (see f i g . 11) ^ JiPV-S

A n o t a b l e w a l l e f f e c t seems t o be present here and t h i s has a g r e a t i n f l u e n c e on t h e speed r e d u c t i o n curves (see f i g u r e lO) as the minimum of t h i s curves i s mostly i n the v i c i n i t y of ^/L - 1 . The 10 f t and 8 f t speed r e d u c t i o n curves show t h i s i n f l u e n c e a t ^/l, = 1 as a v e r y l a r g e speed r e d u c t i o n f o r medium speeds and a small increase f o r t h e lower speeds. In g e n e r a l t h e r e i s l i t t l e c o n f o r m i t y between the t h r e e sets of curves.

From r e f e r e n c e s ( l ) and (2) speed r e d u c t i o n curves could be c o n s t r u c t e d f o r t h e TMB and MIT t e s t s .

Together w i t h our own values t h e speeds i n waves a t constant t o w i n g f o r c e s t a r t i n g from the s t i l l water speeds, Fr - 0,280 0,255 and 0,187, are g i v e n i n t a b l e 4 , The value of

tank b r e a d t h ^ / i s a l s o i n c l u d e d , model l e n g t h " '

I t can be concluded from t a b l e 4 t h a t the d e v i a t i o n s are f a i r l y l a r g e i n some cases.

The MIT values and the D e l f t 10' and 8' r e s u l t s ( w i t h B / L r e s j ) ,

1,60; 1,40 and 1,75) show t h e best agreement, which should emphasize the i n f l u e n c e of the B / L parameter. However, as the MIT values are d e r i v e d from a small scale r e p r o d u c t i o n i n ( l ) these may be n o t e x a c t .

-TABLE 4

Speed i n waves a t c o n s t a n t towing f o r c e

Er a t A - 0 L B / L Tank 0 , 2 8 _0,235 0 , 1 8 7 B / L Tank 1 , 0 0 0,25 0,13 0 , 0 9 5 ' 2 and TMB 1,25 0 , 2 1 0,16 0 , 1 2 10' 5 , 1 1 , 5 0 0,25 0,19 0,15 1 , 0 0 0,23 0,16 0,13 1,25 0 , 2 4 0 , 1 8 _{0,15 6,15'} 2 , 2 8 D e l f t 1 , 5 0 0,27 0 , 2 1 0,17 1 , 0 0 0 , 2 2 0,'13 0 , 1 1 1,25 0,23 0 , 1 8 0 , 1 4 8 ' 1 , 7 5 D e l f t 1 , 5 0 0 , 2 7 0 , 2 1 0,16 1 , 0 0 0 , 2 2 0 , 1 1 0,10 1,25 0 , 2 1 0,16 0,13 10' 1 , 4 0 D e l f t 1 , 5 0 0,25 0,19 0,15 1 , 0 0 0 , 2 2 0 , 1 4 0,10 1,25 0 , 2 2 0,17 0,14 5 ' 1,60 MIT 1 , 5 0 0 , 2 4 0 , 2 0 0,17

The added r e s i s t a n c e curves ( f i g u r e 11) show d i f f e r e n c e s o f ahout p l u s o r minus 10^ w i t h r e g a r d t o the t o t a l r e s i s -t a n c e c o e f f i c i e n -t s f o r -the F r numbers &g-t; 0,15 ("the s t i l l w a t e r r e s i s t a n c e c o e f f i c i e n t b e i n g approximately 0,005 f o r speeds up t o Pr =- 0,28).

A g a i n , the h i g h e r values are o b t a i n e d w i t h the 1 0 f t model and t h e lower w i t h the 6 . 1 5 f t model.

I s i t u n l i k e l y t h a t these d i f f e r e n c e s are due t o experimen-t a l e r r o r s , buexperimen-t an e x p l a n a experimen-t i o n o f experimen-the d e v i a experimen-t i o n s i s noexperimen-t a experimen-t hand.

Scale e f f e c t i f present a t a l l should have the opposite e f f e c t

4 7 6

-J. CERRITSM.^

h i g h e r added r e s i s t a n c e c o e f f i c i e n t s f o r the smaller models assuming t u r b u l e n t f l o w ,

A comparison w i t h the 0 curves o f the TMB i s not given, be-cause o f the small scale of t h e i r f i g u r e s i n ( 2 ) .

The B / L values o f the three models cover a range o f 1 , 4 0 t o

2 , 2 8 which appears t o be too small t o p l o t the C values on a

base of B / L . ^

A d d i t i o n a l values a t B / L - 3, 4 and 5 are needed f o r t h i s pur-pose.

COHCLÜDIMS REMABgS.

The f o l l o w i n g conclusions may be drawn from the t e s t s .

1 ) W i t h i n experimental accuracy, the motion parameters of the three models are the same. There i s a s l i g h t i n d i c a t i o n t h a t the p i t c h parameter i s somewhat l a r g e r f o r the 10ft model.

2) The speed l o s s curves are g r e a t l y i n f l u e n c e d by w a l l e f f e c t s ; the same holds f o r the added r e s i s t a n c e c o e f f i -c i e n t s a t low Froude numbers.

At the h i g h e r Froude numbers the added r e s i s t a n c e c o e f f i -c i e n t s show d i f f e r e n -c e s f o r the t h r e e models:

the h i g h e s t values are shown by the l a r g e s t model.

These d i f f e r e n c e s are b e l i e v e d t o be l a r g e r than the expe-riments e r r o r s ,

3) A d d i t i o n a l t e s t s a t B / L « 3, 4 and 5 might r e v e a l some of the w a l l e f f e c t s ,

REFERENCES:

(1) Seventh I n t e r n a t i o n a l Conference on Ship Hydrodynamics Subject 6: Seagoing Q i a l i t i e s o f Ships 1954,

( 2 ) Dr. Y.G. Szebekely, M.D. Bledsoe, G.P. Stefun. Scale e f f e c t s i n Seaworthiness TMB Hep. no.

(3) P r o f . R. Brardi I n t r o d u c t i o n a l'étude t h ^ o r i q u e du tangage en marche. ATSI 1 9 4 8 .