Non-periodical forces and moments on a ship in waves

EPORT TO THE SEAKEEPING CCN4ITTEE OF THE

.-oer

ai fcrce

wave.

kud nov

ngr a.: :.r

Institute.

;9()

ARCHIEF

Lab.

_{v. Scheepsbouwkunde}

Tecnsche

Hogeschool

Deift

REPOPT to the Seakeeping Committee of the 12th :.T.T.c.

Non-periodical forces and moments on a ship in waves

by Dr. N.K. Ankudinow, U. ... Leningrad, Leningrad Shipbuilding

Insti tate.

SU MM A P Y

A new method to determine the average non periodic hydrodynamic forces and moments exerted on a ship in waves, is advanced. The calculating formulae are derved for unmoving and moving ships among waves in deep and shallow water. The results of Kochin,

1asknd, Haveock, Maruc, ::ewman are obtained from author's

analysis as particular cases.

A very simple formula is proposed for practical evaluation _{of the} added resistance of a ship in obiiue waves. The results can be applied to ships in irregular waves.

-1-Non-periodical fosces and moments on a ship is waves by Dr. ..K. .-inkuiinow

U.O.2.., Leningrad. Leningrad Shipbuilding _Thstitute.

REPORT _{to the Seakeeping Committee of the 12th I.T.T.C.}

Par. 1. Unmoving 3hip.

At first we consiier a floating body with no forward sreed, on the surface of an ideal incompressible fluid _{of finite depth}

h.

and is the presence of an incoming _{wave system, which is incident} plane and progressive. The fluid motion is _{assumed to be harmonic} in time with the frequency of oscillating

_=2/Th

Besides the conventional unsteady harmonic _{forces which are due} to the oscillatious of the ship in _{waves there are also} higher-order steady forces due to various _{non-linear effects. These}

latter forces are generaly too small to

influence

_{the oscillatory}

motions of the ship but they _{can nevertheless be isrortant in} certain practical cases (for instance in _{evaluating the power of} ships in a seaway and in maneuvering). There _{is not very much} theoretical work available (1) - (6; in this subject.

Chiefly the longitudional and lateral _{osciilatiofls have been}

investigated.

The main reason for this is that the _{problem is} nonlinear in nature and rather complicated. _{Below we develop the} general method for evaluating these forces _{for a ship with six} degrees of freedom. All the abovementioned _{results can be obtained} from it as rartcular cases. Our analysis makes use of the usuul

techniques of bear wave theory.

Let (x1

y,

_{z) be Cartesian coordinates with Z = 0 the plane}

of the undisturbed free surface and the Z _{- axis positive} upwards. The forces and moments are determined _{by integrating} the pressure on the body's wetted surface S.

T

= -

JJp-p0)dS

_{=-fJ1/o-pO)Izn)ds,}

(1)

where [p-p0) s expressed by _{the pressure equation}

p

= - p

°

ad82

pz

(2)

and

_{°(X,Y,Z,t)°X,Y,2)eXfi(L6ut)}

_{represent the full velocity} otential.

The formula (1) can be written in an other form making use of the following relations

pJJas +pjjf

(gza.d

Y_(

QJldS * Fits

And by ana.ogy with the :orces we find for the moments

f1J

JfO(xs

+pfJ[f

Z

2((

°)Jds *

_J7hs

where

_F

, express the hydrostatic forces and are considered to be known.

If we take the average value of the above equations during one period, the first periodical terms vanish and we obtain the time independent mean hydrodynrimic forces and moments, which are the square of tne first-order quantities.

If we put ,where designates the function

wnicn has a source singularity on S and is the reguiar

function in the whole half space1 we obtain (for brevity only for the forces)

,=jFct

(d°gwd 1D)a% ojdJdt

Or taking into account the hrmonio motion

=pRejJ[

a4p2° ccL?o_

fl°2ad

o_

where O _{here and below is the conjugate complex of} °.

.'e note (7) keeps its value over any surface S., enclosing

_[].

Next we can ut

.,00=ow*o

,

(3)

where

-K,(XCOS *yWv&)

()

e

is the (linearized) potential of the incident wave system and is the disturbance due to the presence of the body.

Here a = is the wave amplitude, K

_=---

is the wave

number connected with the frequency 6a by the relationship

=-=

Koth.Koh

E is

the angle of wave incidence relative

to the -axis ( = 1300 for head waves). The potential

satisfying Laplace's equation and all boundary conditions, can be written as

4-7Zao -K/LKW I f

JJ

(K*p)CIi,ic'(2+1i)ê

2=

8L2f Je

--y o

_-o(L)

z2= (X_)2(y_)2(2_C)2,

w0=XCL7SO+i$LI7..8

Here

H(KJ8) and

H(K,S)

are

Kochin Hfunctions[H(X,B)H(K,8)J

h-Hu(,8) jJ(__

cft K(C

'e

K(cos8 +Lit8)d

On

CKh

and L

lays in the upper half of the K-Diane near the pole K

Now

'e substitute

(3)

into

(7)

='JOReJJf[zod

r

-

+

+ [a

-

*

(12)

The members _[

]O

and[.. ]D

Lecause , and 02

are

harmonic inside S and therefore tne surface S may be reduced to a

point. The bracket [.. ]- exrresses the forces due to oscillation

of the body in still water. The members [..] correspond to the

interaction of the oscilations of a body and waves. Finally e

derive

i=fPRe[-JJ e2H(-8)O1-G)kKd&

--

2fJ

(

Kh(D)C

_K

-Ch(,9) =-1s

+

sid7* )IK,8).

t

is easy to obtain Koch:n' s results

{ 1 3 assuming ft = U.

Separating the real and imaginary part of the above

exrression,

the projections of re given by

=

sf!Hh(Ko,e)I2c:d8 +

c

_Re[H-Ka,E)J

2m

I52

e/2K9 *

C=- '

L0f(1*

R/HKa,E)J.

Suppostng h - we get aruo' s and Newman' s seoul ts for

the

forces X and Y . If we put

m

6 _./t

°m Um

HK,8)=Y hç

(K,8) + H7 {xç8)

/7tl

in still water; O7 _{is the scattering potential, we find}

nr RX

(to)

(lLf)

(15)

(i6

R'=Rt *R *Rt #RC3

[Um=um6Dt)

D* jO9K0 6

I\t

_fflrf

/R(-,8)u/2cos8d9,

(17)

R=

I /Hh(#,9)/2cosOde

coRe[H(-k0,)],

R-

Re{

L K(-k0,D)H

(-K9,8)

(cosf-cos8)8),

_{(1 7c)} D (flf -IL

Rscos/c.[XFm]mea

=Le0afJ(W

)s

(17d)

Here is R - from oscillation n still water, R - from

St _Sc

scattering, R*.t is the interfering component, R is the component from the interaction of the incident waves and the oscillation of a body.

The expression (17) has been derived by Haskind using the momentum

theorem

[5].

For the small wave amplitudes

a,

generated by the

disturbances of a body

aH2O()

, (m=1,2...7)

R(2)

Rt(2),

R3 ()

and for practical purposes we can assume

Rz

= CO

f_p9aetH

(ko,Em

]mett fz pezLa

The nathema:icai anaiys:s for t4 .s anaogica to

The fin expression is given by

2k

m{/J j e

KtftKkK,9,8}

w n crc 5h

_{8)-{t[H (K,8)}

hb(K.8)]+j[tcosB

_{- H(K,a)]}

_{[i sn B H(K,5)}

-H(kB)=JJn(h)ceC0S8+Ts

/

For exampe, the first projec:on

t.l,

becomes equal

efJ je

(-KO)[N (-K,8)+

tSJZ9h2 (-,k9

(_K,

/H,e-L

--

_{2 jH(-,e)LH(k.}

e-l the above resUe-lts can

_be

applied to deep water if we take into

account that for h= ri'

and also-for the oscillation in still water if a3.

Par. 2 ship ith the constant Thrward speed U

Keepin, the early approximations and connect:ng the coordinate system with the moving ship, for and 11 we can use the

cz

formulae of type (6) substituting only

o

p0

=

-e

_(=-uK0coE)

(21)

Here _{is the potential corresponding to motion of the ship in} stil water with speed

U

As a result we derive from (6).

L =

-[qta0

-°

±

*

-

_g

]dsJdt22

The members in ( give the forces F st.w. _{exerted on a ship}

during motion in still water. (.. )= 0 and (

i=

0

because of periodicity in each of them. Finally

d(wz)rd(

*)_

₊

.c. _{the mean hydrodynamic forces in waves don't depend on the} forces from motion in quiet water.

The subseauent transformations (23) are practically the same as

for J=0 except as regards tse neaning of which must satisfy

more _{oralex boundary conditions. Jsing the potentia. , obtained}

by the author, we find for

_m

_w

-=-p

Re{-f Je2

I i(J

K2 H&[K,) CK&

K

r c'

Q

- Kca 5)2J

2c fleLL L

represented in projections

with

the relp of (13a).

Expressions for fri differ fro by _{5 and S instead}

of C and . in the expressions (2') the contour

(i,) and (L) are defined

2=VKtKh

Y=_Vt

foz cos8>0

czndi<'foz co8<O

0

where roots

k

_{d;erir,±ated from equation}

or graphically from the foJowing figure

ç L

2

(23)

-6

8

is determined by the condition

₍₂₅₎

/ VKB LhK9JL ±6

[C08]cDniptexCoS8oJ

_2K0U

_/

z oo t.s

where K5 is the real root of the equation

Sit KI1ChKIt

6

_(25a)

2dL2Kit/xthKk

j

For h.-oo

from (25) ar.d (25a)

_cos8

_'

For the most interesting force, i.e. the added resistance

R(XmXtw)

we obtain the more simple expression

X-80 _/2 k',.

/Hh(K,L.,5)Co38a.8

+

R=(J

(k2)

_J(K1)±J(g))

2K228

0 ₈ Ch-2K,,jt

-Re[Hk0,a)]cos.

(2/)

(26)

Using (16))we derive R , , and . It is easy to

st SC In WS

prove that _{is the leading term in the series expansion to}

ws

respect j'3 and for practical application we recommend tuse

6

0056

[

where the exciting forces and moments must include the influence of speed. For example, for longitudional waves we have

L(6ti-Ep)

c'os=±1

,

U,O

,

-z.e

_,

e=F0

_exp(L6t*E)

_{=1 exp(t6t}

RfOeeowX[1oJtFaCL1

0

-.ssuming

?=O and

_FFW

, we derive Havelock's formula (2].

Figure 1 _{(a, b) shows the cilculated and experimental added resistance}

of cargo ship ( - =

7,3)

for = 3,2,

The experiment was carried out in the model Basin af Leningrad Shipbuilding Institute, the calculated results have been obtained with computer programs.

In conclusion we indicate that the above results can be applied for evaluating the mean forces in irregular waves if we ise the possibility of linear superposition and assume these forces to be proportional to the squared amplitude of the wave. For a ship moving into two-dimensional irregular waves with the spectrum E(6)

(corrected for the speed) we can assume

where

4(5) is the mean force

in a plane regular wave of

v=/,Z.G)

(30)

REFERENCES

Kochin, N.E., "The Theory of Waves Generated by

Oscillations

of a Body under the Free Surfice of a Heavy Incorrpressible

Fluid",

Moscow, 1938.

Havelock, T.H., "Drifting Force on

a Ship among Waves",

Philosophical Magazine,

v.33, Nr. 224, 191+2.

Maruo, H., "The Drift of a Body Floating on Waves", Journal of Ship Research, v. , Nr. ,

1960.

. The Society of Naval Architects of Japan 60th Anniversary

Series V.

2, 1957, v.

8

1963.

Haskind, M.D.,

"The Theory

of

the Resistance of Ships Moving

Through 'Naves", Jzv. Akad. Nauk. SSSP. Otd. Tekhn. Nauk.,

Mek.., Mashinostr.,

Nr. 2, 1959.

6.

Newman, J.N., "The

Drift

Forces

and Moments or. Ships i

Waves",

journal of Ship

Pea. v.11

Nr. I

1967

-A

1ThT°

-6' fl 'q

0'

0-I

' 0=1 / to

1j) uiu:

9'0

'0 =

=

i/y

I---

-o--o--o-L

,7

ro

05924 V

=

-.1 j

ot '0

0