Studies of the wave resistance to the rolling of ships, Part VI: Effects of motion ahead on wave resistance to rolling

TUtIE. Ci' T1F

VE RESISTkN

TO THE }L

OF SHIPS

(Part II - EffFcts c' ?-oticr

aheae on Wave Re

.stacce to Rdllingt

y To-n

.i.hica

original

aper wa

puhiishe

in the journal of th Eocity

c

t'aia1

.

httct

f .iap. Vol.

7.

This is th

t.raysiat.i.'r

:f the criinal paper, reviied

anì en1ared by the author foi'

cirf.fl if th çoirS isussd

by

orNn1r

ft with a

sist.: parer "flolkr

riments

with e1i-Prope1ir.g

odei

htp.

Thi-

blihi in the journal of th

Kan

a;

Arcflitects cf Japar.. Vol.

72).

CJ!TENTS

Intcictior.

...

'ave-Makir.g 1esi'tace cf ar. ¿llipsoid

ol' PjcliC.

3.

Disctssions of the above results.

.

Effcts

f Lirited width of Water.

.

Disc

sior.

he aov

Result-s.

olli.ng

xperthentS witl. self-Propelling Model

h2ps.

7.

_Suinuary.

.ppenuix.

IrtrdUctiO.

[t i

well knowr, fact that wave resistance

to sh

;.Luing

increases '.i.ih ìncreang her forwaro

speed.

This p

nomencn has

bn examined both thec.retìally an

x:erimentally by ïny

investiLat:rs

among t.hn, W. White

(1), A.W. Johns (2), E. Brtin

3),

.F. Fayne (4)

lì. Legendre (5), R. 5ar.tis and N. Russe (b,.

.

S. Baker (7), N. Kato (8) and so on.

riowever, thorough

explara-tion .of the rrchanis cf augmentaexplara-tion of resistance to rolling

still recainS to he done.

The author disuseS Ìre the resLlts öf

rofling exj.erimers

'sing sef-propelling model ships with and without bilge kc'is

as well as the theoretical

crnpitatio

cf wave resitr.ce t.

rolling of an ellipsoid of re.îc1ution in

notion ahead.

2.

Ware-t'aKing t&istance of an ELlipsd

tieV?lution.

e tae the x-, y- and z-axes as in F'i. 1. Ari ellipsoid

of revolutin wt.h a major axis 2a and ai equatorial radtus h

rcves thrcu

water carI1eì to the fr

surface with a

con-tant:peed U in the drtr of x posìtve.

If

he ellipsoid

rolls about. a longitudinal axis

above

the

ajor axis, the

trans-vere velocity V may be expressed by Eq. (2), provided

that

tw rolling angle is defined by

8

t9 sinrt,

= r

9o- coscrt,

& is the amp1tde of rclling, t tii

and r0 the height

th roing axi

above the major axis.

The ve.ccity potential ' for the fluid motion is gi7en by

ti

real part cf (rfer to Appendix T)

ae

_2ezh2\hJ

-

D

x os 9dB dk

tJ(1h2)dhJLkk(f)(b)9

.fYSiì.GJjifl8d9dk

(3)

00

aZe _JZ

_)dh

_k?(k,e

+Y5ifl8] CO5bdBdk

(a1

-h1 )dh

_j

kG(k)8 )exp-k(f-z )+ik t(x-h)co&

wr1er

-ac

+Ysin&] sir&d8dk

i

r&0a-'

e

i

l.ie'

.,

1+e

i-2e

'

'°gT

icg1-

2e12

(22ufrt+v22

_n

aU1

-F(k,)=

!+(g/U')sec6+i(P/U)sec

k-(g/i, )iec+i(/L/U)sec6

'

o -. .

g:4(cos6)-ucos4-ifr)

gk-p--Ukcos)(r Jk-)

_cos

(1)

f2.

\ I v1

/4.

and G are functions to be dterniind by the condition of the

free surÇce given belo.

--IT.-;=O,

ULL5o.O.

}

5)

(4-or

There fore

In Ihese eqationS

is the elevation o.f the free

surface and

p- Rayleigh's internal resistance.

It is evident that the first

and third teriis or the right-hand side of (3) havé no effect or.

wave resistance to rolling nor

do the second and f ot3rth terris

on speed ahead.

Taking therefore only the second and fourth

terrs we write again

-utbtng z=O in the real part of

-*fco

,.(

F1 cos(kcosO)sin(rsin&)4F25in(C0)5in(uh18)

kddk.

Similarly

JJicos(

xcos

in(kysin8)+G2sinUxcos)sin(kysi48) kd6dk,

-co

where -h

_{)dhÇ°os(khcos6)}

=

i_ae e -h1 )dh

keSinO,.

(_XR+SY)cosrt-(SX+RY)Sir3f t X

°

_{_(_xR+SY)sinrt-(SX+RY)cosrt;} R= gk_+2Urkcos8-U1k1coS & S=,u.(r-UkcosB), X (r-Ukcos8)2 ,

Energy dissipated per unit time by wave-making3s given by the formula P-fl

l.í

E

lirnj

J J

(9g)

dxdydt', (1.0) .

where T is a rolling perio&

However, since

jfrx,Y)

47jd9jCF,G1

4FG2)kdk.

(11)

ws now have

F1 G1 4F2

G=

x

cos(khcosû)cos'(krncos6)dhdxn.

00 x

F=ìin4flNlgf

d8.

f

dk( dhÇ dm(a2 e1_hl

)(a2!n2)

3 i ).-ee

j-3e

(13) -L

Sifl26

_{h )} (iou a) R1i-S2 ijae f ae -211f

(a2

e2 -h2 )(a e2

2)k e

siI?8 'X

j-aej

ae

B *5

Transforming by a contour integratthn (refer to

_{Appendix II),}

we have

Cr3 LN'.

Çk,))).

4

1 2

(1+2)cos8+/1449CPBß

v'l4cos&

_42jCo!l

)

xexp(- 2f .l+2+4)cOs8)+il

r

23, .ose

,'. il+2'CosD-/1+vcOs6 )exp(-

2f i+2Pcos-.Ji+4'cos

4)COS

1

2Xcosô

sìn2

X

cos&5

with

91Z for

_2E-cos11

_for

_i..

(17)

L in (14) to (16) approxivteìv represente _{the length of}

th ellipsoid if it is long andslender 1ike _{a ship.}

_{¿Vis a}

rtio of th

speed of ahead motion to the propagaticn _velocity of a wave ripple and

_X

_{a Fróude 's number.}

Although is the function of 'and)., it _{may be considered} as that o',\ and k0L, for

-¡' _/g.

And kL/2means the ratio of L

to the

length of tw

_dimensional

waves of the period T. _{cor.puted by the numerical integration}

i _{shown in Fig. 2 in which 2f/L is assumed to be 1/50.}

3. Discussions of the above Results.

It appears in Fig. 2 that wave-.aaking reiistanee incrèäses

witn speed, very

gradually at first

but rapidly at Froude's

numbers over 0.1. _{There is an inf1ction point at V= 1/4, at}

which 9, passeu from 7C to ic . _{As to k0L, the smaller it is,}

the greater the resistance.

When an ellipsoid has no speed, wave-making resistance is due to

progressive waves.

_{The energy per unit tire}

transported

- by these waves is of an order c, A1 L, provided the mean hèight

of- the waves is 2A _{and e, a proportional constant.}

When an

ellipsoid is in motion ahead, it moves through undisturbed _ter constantly iv-ing

it energy.

_{Some of the energy is left behind}

the e11ipojd, and some reriains _{in a conser'r-tive system of} staniiñg waves around the ellipsoid. _{Let us suppose the}

o

-

2ee2)' ;

À=, t=2ae.

'

(16)

-' a Bessel function. The

upper limit of the

jiwan htLght of the standing waves to be 2A nd the energy left behind per unit ti is c2Â U.

Then tn

rate

of increase in wave resistance is c1A U/c1 A L. The faster is U or the shorter L the larger is the rate.

Furthermore, A5 varies with U. If the ellipsoid bas no speed, the height of

standing waves is

comparatively

great on

both sides of the ellipsoid and it is small in the fore- and aft- body. But when the ellipsoid r'ioves ahead, the ditribution

of wave heights relative to it

changes:

i.e.,

dense at the bow .;hile thin at the stern, for

elementary waves,

created

6y all

parts of the ellipsoid which compose a standing wave, are crowded at the bow.

I

Therefore, the increint of lost energr changes gradually up to a certain Speed because the height of the standing

waves

left behind the ellipsoid is coiarative1y low, although its length increases with U.

When speed reaches the

magnitude corresponding to V = 1/4, the whole system of standing waves is left behind the ellisoid. A

then becomes constant.

The

incrernt of the resistance is linear'y proportional to U.

The physical meaning of y= 1/4 may be

explained

as follows: For small U, although the ellipsoid is in

a wave systeii

generated

by itself; with incréasing U, the wave shifts

back relative to

t.

At )=U/g=l/4, wave speed becomes

equal to U.

gain, when

the stem of the ellipsoid is at A in

Fig. 4, the principal part

of the waves generate.d by the stem

is expressed as

w-re r is the distance from A to the

princia1 part. of wave

ripples. Mter the period 2/-, when the stem comes to S, ir.

order that the principal part may hev reached B, r must be

U./r.

And we get

=2*:

¿r tWç

07g

_{= 1/4.}

This value of 's"

is independent of the dimension and form

of

the moving obstacle. For speeds U.over

g/4a, the obstacle

outruns ripples, as shown

ifl

Fig. 5.

We see these properties of wave resistance at U in the experimental results on "Conte di Savoia". Fig. 6 shows this

n comparison with an

ellipsoid, where resiStance to the rolling

o

"Conte di Savoia II

includes slight resistance due

tó very

srmll bU-ge keels.

The same tendency nay.be found in the

experiments on A-model ("Royal Sovereign without bulges nor

bilge keels) after

Payne (4), although we can

not make conparative

study because the particulars of the ship are not

known.

-5--

h.

Effect of Limited Width of Wter.

A motion ahead always increases wave

_{eSitèLnce to the}

rolling as discussed above.

ut there. are soi

reports showing

contrary results made by- Legendre (5), Kato (8), etc.

The author has guessed tt such cofltrary resultsy

possibly $ a.ttributed to sidewall effects 1 tbe expériintal

tanks. Accordingly he examines the sìdeÑ11'effects ònan ellipsoid of revolution.

Approximately, we consider an aigebx'iÍ supe osition of

an infinite number of velocity potentials f'or each of iirga

ellipsoids which áre assumed to be in an infinite number and

separated one from the other by a distance q representing the

tank width.

-The disturbance on the surface of thè ellj.ps

oid coming from the images, will be small if q is modertely

large conared

with the transverse sectional length of the. 'ellipsoid and

we

may explain general. tendencies (réfer to Appeñdix III).

Velocity potential for each image is

y replaced by y+i

(rn=±l, *2,. ...) on theight-hand side of Eq. (3) and

multi-piled by (-l)'..

Rewriting similarly in .(7).and (8) and suflning

with respect to m (including rn=O), we have

rn J

)

F1cos(coS8)sin(ksi(4s):).pF2sjri(.}ocOs&)

rn0

_\

(J,)"

xsin(ksins(Y+mq))jkd&di(I

r°

{).

)frt

cos(kxcos8)sin(1cysin8)+F2s](1Keos)sin(kysjfle)j

Ofl

x

1+2

(_lfLcos(nqksin8) kdôdkl

,

(n

1,2,3 . .

_.

_{). (19)}

t

ffl'Io

À similar forti way be had for-..

Eqs. (12) and l3) change therefore to :those

with

integrands

multiplied by [1+2

(_l)tcos(knqsjie)j2

_{and made tcnd to n-o°}

ni

As this multiplier inc1ues.only circular

functions, if we

consider it together with cos(khcbs&)(csknIós8) in (13), it is

transformed Into a form

consisting only of circular

îunctìcns

like the righthand

member of Eq (x) in Appendix II _Accordingly with respect to the integration of.E.,1

also, the first and second

groups in (xvi) in Appendix II must be multiplied by the same multiplier

within which k is replaced by k1

_{or k2 . Consequently} we have . corresponding tó of Eq. (25):

{2n:ltt

(9)jls2$1(1)tcos(k,nqsiz1e)3

e

(_l)1tco(k1nqsin6)*JJ]

(20)

fl]

where f, and f are the first and second terms

01

the irite.grar.d in

Eq. (15) respectively.

Ising a well-known surcation forrala of a trigonometrical

series,

l2

(l)'cos(k1nqsjn5)

_{cos(n+)k,qsine}

rl

I

-2n+1

_J

L(2nf1)c0s2 k,qsine

This is very s11 for every value of 9

, excepting

ir and

â11-which make cok,qsin&/2)zero 1ik kjqsin4= (2r+1)-7L

and

k1qsin 8,',.

2rt (r..O,l,2,.,.) wider liiitations

_{¿9, >6,,.>o -a-nd}

b

> o

When k2 replaces k, taking

or

we have

f

( I. i -C

)ri,

ì

,

(&,r ).

)

2n-i-I 8sr 2fr.,)

-Z2(2) (,

2+l

1+2

(-1)cos(k2nqsiri

ne].

)

)h10.

2t-Putting

_{'qsin& =o<.}

2 j n

1+2

_{(-1)"cos(k, nqsin8)}2d6}

ri=l

t-+1)

(-1)"cos(n)

t

2n+1T

L'2r

t

l+2.'co

k1+4Vco

_.5mG.

L

2cos'8

For such r, that for whjh

&,,. or

_{coincides with either}

integration i.mit

or ò, the values under

_{in the above}

equation i-iust be ha1ve.

n

-(1+4

.

'

n1

_2C

2n + ,1 2

I

[

(kqsin)1

r, (

'r

) +

¡(k2qsin)18

f2(

2,j

b,,.

(21)

-Resúlts of nwr.erical coutatior1

_{for k0L=4 are shown above}

the abscissa J=2qk in Figs. 8

_{for J}

11, in

hich

represents

O,

and

_{for ti}

_o,

O Y

_

-.

When the speed U is-ciiparativèly

low, wave-mald.ng

resistance Is very great if the tank width q is about odd :ies

of hail' the length 'ji

a two dimensional.waé corresponding tc

the rolling pericd, ihile wave-making resistance. decreases

rapidly as q cones off these widths decrease to less than

or there.

-The widths correspond to wave lengths for free transverse

oscillations oí' the

tank water. For snail U, wave creats are

accumulated.

The lower is U th narrower

are the widths.

It

is con-sidered from

this

that reflected

waves

sent by the ellipsoid's

entrance in the direction of its sides, superpose. themélves

on the waves chót in the sa

direction by the run, and. the

effective widths. of water increases to. some extent.

When the tank dLdth is any multiple of the wave length of

the characteristic osrillation, resistance takes the least values.

esistance vanishes if

the

length of the ellipsoid. is infinite,

except where the tank width equals odd times of half the wave

length. if the ellipsoid ! of

finite length, however, portionS

of the eleruitary waves it creates propagate obliquely toward

side wal1

reulting in resistance to some degre.

At the

satie time the energy giver. b

an ellipsOid

t!

motion is absorbed

to create standing ives

in the water that comes in contact

wit-h the eJpsoid as the latter advances through it.

For a constant speed U, with an increase of th width,

t he curves of / repeate s mular Thrrr. The. hus ab out synchror.ous width are slackened and the hollows gradually

approach the

value for an infinite width. Wé can hardly expect,

however, a tank width being close to infinite.

50 to conduct

rolling experiments at rather low speeds of advance is practi-cally impossible as long as suitable wave breakers are not

ai1able along the tánk'! walls.

The above discussions are on the rolling that has continued

uniformly for a long period of time. In the càse.of experiments, .

where rolling ceases after betweenseveral and oie fifteen

swings, some nodificat ions

are required.

.

If the period, between the ti.-te when the first waves are

shot by a ship and th time when the reflectéd- waves come back

to her, equals or exceeds the tire required

for one rollig.--t

coiet.e., the oonditioñ is ecuivalent to

that on an unbohded

water surface.. This condition is

.fq

group y ocity

)riuverf rolls (N)xrolling period(T).

owe ver.,

half wave length

1

5. Discussions of the Above kesults.

T

k0L.

If k0L 5. ¿

This is hardly possible

to prattce in experiment.

n the other hand if assumed q/L 1.5 is a mean, is

nearly unity, that is,

only the first two swings are of use.

Further, since the uirst. swing contains transitional

extra-ordinary phenomena (9), only

the record for half the

period

uccecdir3g the first swing is

reliable.

Th ?igs. 2 - II, the range of qjk in existing experimental tanks is sho hatched. The range lies in the width of water where wave-making resistance i very sxrall. This makes the

record of

wave-mking resistance

ccnservatiVe.

-Again, when the speed of

advance exce&ls U

corresponding

to V

1/4, the form of resistance curves

suddenly changeS.

Thotgh the first 'hump

correpond'thg to q/

/4 for U0 ramainS

for larger values of q a curve

gets considerably gradual

and

approach that representing the value for ininite width. This

of course'

iS de tQ the fact that

the major part of

wave

resistance consists of waves

other than progressive waves. Consequently, it is difficult

to avoid side-wall effects unìes

a speed is made to exceed V 1/4

or speed-length

ratio 0.8.

In Fig. 12 is plotted the comparison between wave-making resistance affected by walls 1.5L.apart and that'in the water with an unbounded surface.

The relation in (22) holds'

approxintély true for the

finite depth h of 'water.

Wave-length J

of progrSSiVe waves

produced by the rolling in the same period is given by the equation

2CL'

2V

2.

cotn--7-

T

g _L

L' is shorter than. j for infinite depth, while

the group velocity of.waves increases by the ratio

Sfly

Hence, instead' of (22),we have for k0L

4

tL).

However, the riht-hand side

is nearly

/2'N for h up to

approxi-mately-L/l5.

It is clear from the above that special

conSidera-tions need to be rrade with

respect to side wall effects whñ

rolling experiments are carried out with actual ships.

6.

Rolling experiments with

self-propelling model ships.

In order to learn resistance

to

the ro1lLg of ships in

notiofl, it is necessary to examine

the

variations of component resistances, due to bilge keels, wave-making, etc., together

with the speed of advance.

t

-9-A.W. Johns (2) stated that the resistances .-dtce to surface

friction and bilge keels would be ir1creaec becave' these vere

attacked by a vater flow with a resultant velocity of both

rolling and advance.

However, the author cannot irudiately

agree with him.

So far as the author is thÍoriid tbre are no

other invastigaions treating the relatiân betwe n the. oiponente

of resistaice to rólling an

the sed of advance,.

Almost all works have dealt with total resiatane,

regard-less whether a ship has bilge keels or uot. Ts makes it ail

the more difficult to grasp the general properties of ship's

resistance.

With this

vtew the author has carr'iéd out experilTnts

usirg ship models with and wit1out bilge keels and under the

same conditions throughout.

Table 1.

smalipassenger ship, G of a cargo ship, B of a tz'aiÉ1x and I,

t ae Sijyijiar' models of 1oenge.ahape.

The particnlrs of

these' models are shown in Tablè 1, amI the lines 'of hull toxins 0f 1

re indicated in Figs. 13,

»».

Sh.iç

F

H.I,

3cale cf niodel

1/25

1/50

1/25

1 2

L (m)

1.60

1.86

1,52

100

2.00

!

(m)

.300'

.274

.288

.30

6OQ

ci (cm)

).0

10.0

11.6

rrn (cm)

o

o

4

0

-' by stern

0

Dsplacernent (kg)2.9

37.3

2L$

25.0

200.<>.

3M (cri)

3.48

2.00,.:

1,'.O

_2,45

CG (cm).

..

-1.72 O.O

-252

5..2

10.44

(tckX1eth)51

5x54

,-

5

10x60.

Bar keel

»

(thick.x1eight

"

-

.2.SX6.4.

. -. .

T (sec,)

1.20

140

1.10

122

,l.74

.42

,2

48

k,L

¿.5

3.8

5.0

2.7.

2.7

The experinents were performed ï twopon;, for t.h use

of an experinntaI tank is not prcper for the euttor's purpose as is clear from the above discusions. The an depths of water

are

2 n and 1.8 m respective1y while th shortést

die-. tarces frOErn

the ship's courses to thé 1orders re 30 rn and 15 n

repective1y The model ships are prope1ed byscrew propellers driven bj small electric motors As electric sources

batterte

,are used.

'4 Rolling is starte4 by the rieans ás shown in Fig. .

Weights are attached to the arms of frames using

electrô-magnets.

After à model ship i .startéd at an-estimated time

when its speed has become constant,

a

eléct'ic current in. a

magnet on one side is cut off b7 a chÑnometer switch and a

weight on the same, side falls down along a guide wire to a shock

absorber situated at the

módel's center.

Plling

velocity of

the weight is absorbed by the friction Of

the wire ans ¿ spring.

The ship coiences to heel. When the

bèeling angle is above

a ce"tain amount, a small weight. in-an electric

relay tti1es, thereby making a switch off for another magnet. The heeling mônent 7anishes, but rolling remains. At the time, when rolling damps out, the

chronometer switch' cut.s

of? all current. Then the &hip comes to a stop.

To record rolling, a gyro-recorder is used. This is adjusted before thé model is started. The

speed of advance is

measured by the same method as in a speed tria1., except that it

is done on ].and,in the period ftï'required to

run out the

co e which is indicated by buOys placed 5 n apart.. The

st _{Ang point is at a distance about ¿ m tò the first buoy.} Some 'items in the process hd to be modified for the model

H becaise of damage to te

chronometer switch. Ro]Jing was

started by pushing

the frame when the ship was alose to the

first buoy. The recorder was set in action at the

ame time

when the ship was started. -And another boat was sent tO stop

the ship. _{The rolling curves, bòwever. are in good coincidere} with those ot the rolling automatically start.ed.

'5

Fig. and _{show a ciibration curve' Of the} gyro-recorder and a curve of extinction of the rolling of a model ship obtained

'from the recorder in çomparion wth those

_{of an}

optical recording. _{Fig.'729 is an exale of records made by the}

gyrc-reorder, which are affected by the

precession of the gyro.

A middle line

between th

both aplitudé curves, theréfore,

_needs

to be taken as a neutral liÀe-.

¡8 2Z

The experimental results are shown in Figs, - . They

are the data measured on a perfectly smooth water .'ón midsummer

days.

23

In Figs.

6

9, the data are reaÑriged

_{on abscissae of}

the. speed of advance, -taking the roIling ¿litude as a.

parameter.

.

-It is found from these 'figures that the differene in

resistance between ships with and without . bilge keels is

approxi-mately constant, alt'hough some of such diiTerenòes. decreases

slightly in degree at higher speeds. Te rame end ay is found

an opinion that a bilge keel is affected by a 1ft because a streir atta* tts faces with an ancle of incidence and the

rcr'

ispor.ent of the lift cojtributes to anti-rolling. rt

is supposed, however, that the effect of a. lift seen on such a long and slender plate as a bilge keel wld be localized

within a rather short part abtft the leading edge. This tendency is endorsed by the fact that bilge fins öf a comb form give

strong dátTlping to a ship in motiofl head.

Next, the ratios of resistance to the rolling of a bare ship Ln notion ahead and that of a ship in tand-still are shown in Figs. 12in coarison with the theoretical

curves f ro Fig. 2 above.

These experimantal ratios containing

fríctto:al resistances of sa1Ï anount, ágree rather well with

those represented by the theoretical curves of an ellipsoid

of rvo1ution. That L to say, they agree with the residuary resi,'ance (the

niaor part of which is wave-iaking resistance)

ratío, arid probably these ratios roughly equal the residuary

resistance cf an eflipoid regardless of ship fs. In. this

connection the experintal values of the model ship R8(a)

after

3aker (7) are found to agree approximately with those

represented by the theoretical curve Shown in Ftg 94.

33

It ry be said that an Increase of resistance to rolling

with

speed of advance is mainly due to wave-making resistance.

Summary.

Wave.reaistance to the

rclUngòf aneliipsoid of

revñution in motion ahead has been calculated. A motion ahead always causes an increase of resistance. The rate f the increase i coiaratjvely snail at rather low speed, while it shows a

rìpid increa8e at a speed about =. Ujg =. 1/4.. Añd the sn1ler k01, the larger is the rate of

resistance increase in. the sana

speed-length ratio.

rapidincease in the vicinity of ))= 1/4 is observed also in the experintal results in foreign countries..

i= 1/4

corresponds to

the Speed of advance which equals the propaga tier.

velecity of

ve group.

With this, speed as a boundary, the

ooxçonents of 'ave resistance cbaneu: for at lawer

eeda

resistancE is of progressive waves, ìiil

at higher abeeds

resistane is conssiciicais

du

t.

.

-.

,

water

as it contacted by a ship advang. The larger k0L, the

løwer-is a boundary line in the speidlength ratio.

The ratio between the

cononenta o1 wave-.making

resist-anc (which may contain fricttona.l resistance, if it i? sr1l in quar.tity) of a model ship is app xitely- euqal to tht of wave-making resistance of an el poi.d which has the Same length, speed and rolling period a the former.

The wave re.sistamce to rolling on a re5tricted water surface with a constant Width q varies frequently with q or

the

rolling period T.

it greatly increaeee if q is near odd times

of ha1

the length of a progressive. waVe with the periód T

being in two dr.sios. Fut if q .les between such lengths,

resistance js smaller than that on n unrestricted surace.

o

/

In the seed-ength ratio apprcite1y above O., however,

the said variation is considrab1y mitinated arid resistance

ztproaches th ' zn an unrest icted surface because the ship

i

lit.t)e affected by r'

ed waves corfling

froi

sidewalls.

At a low

speed of advance, the record in

only the

second

ewing

of roll is attributable, if the tank has an

ordinary

width. And the

succeeding swing record will

give

sinal].er

res5tane than in a water with

an infinite width.

It is not awisable to conduct an

experinrit

on

the

r2J-ing of ships unless it is done at roderately

high

speeda of

.dvance.

It is necessary to choose a sea surface

as

wide

as

possible for rolling exeriints on an actual ship,

too.

The resistance due to bilge keels is affected little.

by the speed of advance.

It may be said, therefore, that an

increase in resistance th

rolling

due to an increase in speed risists

rin1y

of wave-a1cing resistance.

(p7)

_Forn

the ahoye, resistance to the rolling of a

in

notiûn

ahead is approximately qual to the sun of the two

parts;

the resist.ance due .o bilge keels and a bar kee. of

the

shi: in

a standstiU and the resistance of the bare hull in

a standstil) irultiplied by the wave reistanoe ratio

of

an

e1lipsc6 which hà.s the arrie length, speed and. rolling period

asthe.'ip.

AP?ENIX

T.

The velocity potential due to doublets in an unbounded

fluid, whi±

t.and in a row between x= -ae and x= as

on

the

x-axis and bave axes parallel to the sarte x-axis, is given by the

ì?wirg

',,

if the strngth per unit length is proprtionai

to

a2ezh

n the vicinity of x=h.

=

(x-h)(a't-h )dh

.

1(x_h)hs;7+zJ4a '

The inter..ticn turns easily to

-

/(ae.:x

2+y2+z_J(ae+x)2

i_)

(ii)

Puiting x=ae,iÇ , y1sz* a1e

=

2aelogaP

_-2aei.

-ae-aeì+ae(ç+.)

If

M=.

e

1

_14e

log113

-(H: a C3nSt.).

(i)

Eq. (iii) represents evidently the velocity potential (10) of

the fluid in which an ellipsoid of revolution -d.th the focal

length 2ae advances with a constant velocity U in the direction

of the major axis.

Herve the tstribution cf the doubletts

strength fcr the ellipsoid is to be a2e2.h'.

If E. (i) is rewritten by the reiation

-

J

i

cos8d&fk

-.

(y)

'e bave the first term in the righthand

mber of (3). in the

text.

5mi1arly the second term in the &are equation is.

obtained whex doutlets with axes in the y-direction are

distri-buted.

ssurTIing the third and fourth terw.s in the sara equatioi

to satisfy the free surface coMition,

..

(3) is cox1eted.

II.

Extracting the ínt1egration I with. respect to k ar.d 9

from (13)

-:

j4eJ

k3 2kf

28.X

s('

)s()dk

(vi)

Rf

= (gk- o +2iJkcos6-12k2 soso)z +fr-Ukcos

6)zuZ _.

_cvii)

If th

roots of a hiqadrtic 'quaion with respect to k, the

righthand side of (vii) = O, are k,,k,, k3 and k,,

X

cr-Ukcos8

_{1 1 1}

r+s2

-2piU2cose

J T

k3 -k

$ìn.e

i

cc(khcosO)coa(ktIlcos8)=.rLe

df

k

g+2tJrcos8 +i$.Uco58+J(g+i Tos9)2 34Uagcosß

I.

2U2cs

k3 =expression with.-

instead of 1zi. in the above,

k2, k

_{expressio with -f- insteaã of J}

in k,, 1(3 respectively.

)I

d

- r

(r

_{(0= ±(h*os8.}

a compLex constarrt,

(Ç=k.iu,

(fl; poeitive integr,

(5=1,2,3,4,).

14

-_k(h-m)o58

th

integra tian with reapect to k is eontatned. in the citur

integration

.

Co!ing respectiv

contours a

shown in Fig. 3+ according to

the sin of the rea]. or the imaginary par

oX r5 and that Q.f

wehave

J,.

K e

dkJ, +i2i

exp(-2r;+ir,)

,

( w > O)

J2,

,

(w<O)

-

(i>O)

(xi)

=

-i2xr

exp (-ar; +1i"x)

,

(w

o)

JI3,4,

(u40)

(w< O)

In the above equations

I.

)t ...2fU.k)&L

.,.

(iu) e

_j

_.(iu

_e

idi.

(xii)

j

in-r5

s

'j

We trnsforr (ix) in order to examine the sgn of the real

and the iirary parts of k1

k. Name1y

j (g±iutJcos9Jz +h.Ugcoe

+6iLzUzgcos2J4

I

i

-,

2gUcosB

x exp±

_{ga_}

_l2tOS

+4UrgçosT

J.

I

UcoaB

H

(g+4Ugcosa)2

_.

eXP±

_e+4UosoJ

Therefore, when g44Ua-cos B > O,

!, ,k2 ,k3 ,k4

1k1 ,k2

_{jJos.(l±g//g2 +4Urgc'osej/2U2co2O}

_,

1k3 ,k

=

_;

5.f y+4tYucos

O,

_(xiv)

2k, ,k, ,k1 ,k

n., ,k3

=

/jz_t7o.gcoso

_/2Uc8

Ik

,k4

=

In the above eqtatiçns, take pIus for k1

_{an k3}

_{from the double}

signs and R, I specify the real and the iginarj

_{parts respec-.}

tivel.

The crresponIence between k, -w k and

_{is, therefore, as fo1iows}

15

-in 4hiCh

9= 7L.-cûs1 (g/Ur),

t-COS1 (g/2U)

iid only the range of

> ô

is cnsidered because the integration

iitI

respect to 6

iS

1. 1

Thefefire, since k1 -

k3 arid k2 -

k4 or 4:

4her.

O,

arid J,

vanish in ail cases and the term t2xir,"

.exp(-2fr, -i'l'r

al -ne re-iains.

Further, this term appears only for the

root

with a plus real part and the

double sigri

of the term are

chosen in accordance with the

sign ofthe irrginary part o

the

root.

These combinations are shown in

F1.g.

and Table 2.

35

Table 2.

O V

f

gfItJc.S8 > O,

g+zUrcoS&K O,

0=0 -,

z

I,

1e knoi fror the above

are indep

sent of the

sign for

, if

integration is redaced

k1

= r1

r2

= r1

= r3

k,

r1

r,

r.. r4

k -

_{3 2} r1 r1

r3

o

o.

-.

o o

o

taule that (a.) the integrated values

sign of w., (b) the values change..the

they contain c, and (o) the rang

of

to 9- O-6,

I. 4

b .-'t,

.,0

o

-O

>0

o

'O Q

o

o

r,

r

.2

r,

r4

Comeq'ueìt1y I beeces

, -2 _51T1& ¡A

₎

_+/4ThrgcoS8 x

s(k,hco8)co5(k,rC3S&)6

_,(_1Jk2â)k exp(-2fk1)

C

x

ccs(k2cs&)coS(kiO8)1J

k,

g+2'Jico5

±/g2+4Tir3S

2L13co5'& K2

next., the

with respect to h or m turns to

ae

J -ae

a3e3 J

(k, aecos

ing;.\and ?

we have (14) and (15).

III.

The velocity potent4al when an

infinite

er of

1lip!jdS 3tand abrvast, with

distance

betweon bath othei.

N

((a2eh2)dh

h)

z-jr

(xix)

velcity on the surface of

he ellipsoid (m=O)

to -.

?.,67at xeO, z=-f and y=O.

(

ae

a2e_h%)dh(_1)"Çh_2t

(rrt' = ,2,..)'.

)

-ac

If q is at least as large as 2ae, it is reduced ap:oxite1y to

LÍ

ae3h1)dh=

(_l)m<2 L3

(tq)3)

3

q3

'

m' 3

'

'raking the ratio of this to tte

aiopitudè of V in

2)

2

i

The amount of the (

) is very eI1 sine e

' 1 for such an

e1lipsid which is as slender as a snip.

The disturbance,

t.herefoÑ,S very sa11 unless Lfq is very

large.

T!EL

disturbance

is nearly equal

e

- 17

(xvit:

(xz

s ç

REFENCES

WHITE,. W.:

_{Notes on further ex eriHr1t8}

_{with first-class}

battle-shjp, T.I.N.A., 1896.

JOHNS, A..:

_{The effects of rotiot ahead on the rolling}

of ships, T.I.N.A,, 1905.

BERTIN, L.:

_{The azrlitude of rolling}

_{on a nonsynchro.}

wave, T.I.N.A., 1896.

(4)

PAYflE, 14.?. :

iesi1ts cf sáe roilitg

_{experiments on ship}

rwdels, T.I.NA., 1924.

LE3ENflR., R.:

_{The influence of speed on ro11ng.}

Shtpbi1der, April, 1933.

6)

SANTIS, R. AND RUSSO,

_{Ro.1ingof the S.S. "Conte di}

Saroia

_{in tank experimer.t and}

_{at sea, T.LJ;.A.LE., 1936.}

13AKLR, (LS.:

_{Roliing of ships under}

_way.

_{Shipb. and}

shipping record, 1ov. 9, 1939, Nov. 23, 1939.

KtTO, H.:

_{On the stability' ofsmall}

_{vessels,(jn Japanese).}

J..K., Vol. '79, 1948.

ISHIDA, T.;

_{TranSitional phenomena in}

_{wave resstanc}

_to

the roll ing of. s hips.

_Jour

_{of Kansai Soc. N.A., Vol. 67,}

150.

(lo)

LAMB, i-L:

_{"Hydrodynarni:s", 6th edition, p. 142.}

18

-(7)

5-/

Fig. ¡

Fg.Z

I I 1,2

/4

1.6 .4

6

(.6

.6

.8

1.0

Sp..d MngM ra'o

F

9:11

16

20

9 (deg.)

..."

Io

o....

-

X

/0.

20

U(kno)

OI I 6 2

i,

o

o.

F19.8

Fig.1/

.2E,

k0L=1

t'-L0L

&j.

t)ocrL

L=L'---X

7_= 4 3-2 / o

Syuh1Q +, COt with reipect te m-seçtoyi

I

q

I

_{6 s.}

F.I2

0/

DZ 03

F9.I3

I1. ModeL SCm

20

FQ. 14

Fig.

7,ostren9th..

e

8 volti.

o

.I0

_vo1hged motor

x--

Bvolts u /OVolt 100 /4-0 200

/2

16 ₂₀

60(deq.)

With iI5e )eeIs

e

Without bilge

keeLs

22

Fi9.

23

v -.

bilge kee1

/

Wi$1ouf

'F9. 24

6

"o

-o

Fia.

25

D I I 2

t

Fi9.

26

'o

---- wiTh

b*r keeL

- -

Wliho"t q

o'

F

Fig. 27

.1

_/

32 -

/

/

V

(

iup.soid

-.4 ,..volw#

.'

S?eed U,,g rio

e

(IiJ_= 3.9)

Fag. 2'?

.6

Speed ¡Qn9M ra*tO

Speed ¡esiØ& rgto.

o. 4s°

V

/5°

(0L = 2.7)

Steed

Fig. 32

o

I-a .4 .8

io

(L!

_{I.ngM raii.o}

Fi9. 33

T

IJ k)I

a! k

cos>o

gj

>0 >0 5,

83

'<o(;

:,

Fu .

34

Fig. 3S