Hydrodynamic forces and moments for heaving, swaying and rolling cylinders on water of finite depth

CHALMERS UNIVERSITY OF TECHNOLOGY DEPARTMENT OF NAVAL ARCHITECTURE

AND MARINE ENGINEERING

GOTHENBURG - SWEDEN

HYDRODYNAMIC FORCES AND MOMENTS FOR

HEAVING SWAYING AND ROLLING CYLINDERS

ON WATER OF FINITE DEPTH.

by

CHEUNG H. KIM

DIVISION OF SHIP HYDROMECHANICS REPORT NO. 43

Gôteborg, April 1968

CONTENTS

Pae

Abstract

NOmene1ature Introduction Ettén±ón óf poténtials 2

HödynamiÔ föröéà àrid moments

...

9 Da ping and. excitrg forces and Fórnents

Numerica1 oa1ct1atión and discussIon

_..,..

_n.

Pigu.res

Acknowledgments.

Reférenoes

ABSTRACT

The potentials for forced heaving swaying and rolling cyiirders

on the càlm water surface. of deep vater used by Grim and Tamura

were extendedto

b.àppiiedfor the oòi11tiofi o1aetjn

shallow water by Thorne'smethòd!

P-irstly the hydrodynamic forces and moments on oscillating

Lewis cylinders were calculated by Grim's method; secondly the exciting forces and moments on a fixed cylinder in a transverse incident wave system were calculated according to the H.skird newman method; and finally the influencè of the

shallow

watèr effect on the forces were illustraded in tiures and d.iscusseda

NOMENCLATURE

a amplitude of oscillation

A wave aplitude ratio

A source intensity in complex number

B beam

C äddes mass coefficient E éxciting force and moment

F force

gravity constant

water depth wave amplitude

half

beam draft ratio or subscript designating heave

LITT

or subscript designating imaginary part

Im imaginary part of

I" added moment of inertia

1 moment

arm

added mass M moment n integer N damping coefficient I r subscript designating real part ori/x

Re real part of

subscript designating roll

S subscript designating sway or sectional area

t time.

T draft

U velocity amplitude of sway

V velocity amplitude of heave

a phase 1ag

or

lead

w

polar cbordinate tan1

(L)or

uliness cylinder

damping parameter wave elevation

2.

deep water wave number (ci, ¡g)

shallow water wave number density of water

veloeity potential stream fu.nction

cirkular frequency

¿giiiar ëlocity amplitude of roll subscript designating infinite depth

5 T)

INTRODUCTION

We consider two problems: (a) the problem of forced oscilla-tions have sway and roll of a cylinder on the calm water sur-face of shallow water and(b) the problem of a fixed cylinder in a transverse incident wave system of shallow wier. Our

problems are. to determine the hydrodynamic forces and moments

on the cylinder in both cases and to determine the influenbe of shallow water efíeòt on the forces and moments. There are

two reports on the first problem b31 Yu - Ursell [5] and Wang [8]

and two reports on the second problem by Haskind

_16]

and New-man [7].

Firstly the potentials of the forced oscillations on the calm water surface of deep water used by Grim and Tamura [1] [2]

are extended tó those of shallow water by Thorne's method [3]. There are two reports on such extensions [4] [5]

Secondly by making use of the potential of shallow water

oscillation, the hydrodynamic förces and moments are calculated.

Thirdly the exciting fòrces and moments on a fixed cylinder in beam waves are calculated by using the preceding calculations

for force.d oscillations äccording to the Haskind - Newman method. [6] [7]; alternatively the exciting forces and moments can be

calculated by Grim's method [1] [2] and a comparison of the re-sults o,f both methods shows that they are in good agreement.

Finally some calcúlated results are represented in conventional dimensionless forms including the depth parameter as functions of

EXTENSION OF POTENTIALS

First of all we define our rectangular coordinate system O - xy: x - axIs, on the calm water level, is pointing to the

right and y - .xis, in the centerline of the cylinder at rest, is. pointing, downward. In the following discussion the

irrotatio-nal motion of incompressible inviscid fluid and the linearized boundary conditions and linear hydrodynamic pressure are assumed.

To begin with, the potentials for oscillations in deep water

used by the authors [i] and [2] mu.st be reproduced. They are

ge-nerally represented in the following form

rv

+ ioo = U

et

A (p00 '+ 1'floo . . . (i)

., where V,U,Q: linear velocity amplitude of heave and sway. and

angular velocity amplitude of roll; A : complex cource 'intensi.''

ties to be dètermiied by the bounda'y condition on the cylinder

surface; cp cp

+ i

CP _: partial velocity potential which

satisf.ie,s the laW.of continuity everywhere in the liquid, the

free surface condition at y o, the condition of rest of water,

at y co _{and radiation condition at bcH co.}

. +

floe _{flroe ruco:} partial stream function.

.The partial potentials are for heave

° -ky.

e _-v

oroc + = COS kx. dic - lire ' ces. vx

o k-v

and for sway and roll:

+ lcD_'fl]oo

ooky

± sin ß + e sth kx dk - sin vx

r

ok-y

vi

U

e1t

nfcos2n.

y Cos (2n_1)ß 2fl i 2fl-1 r r

-3-nr

+

r-2n s:in (2n+

)1

₊ 2n r21

r2'1

where A

_nh

₊ loO

"

()

y = w2/g, _r y2 =

tan1

(x/y) , and

ik(x+iy)

the wave source integrai dk expanded by G-rim may

o k-v be seen in [9].

The aböve potentials for infinitely deep water aise extended

to those of limited water depth by making use of Thorne's method

[]

[4]

_[5].

We set the complete potential for the osc±llat±onC

in shallow water, wbi,eh are to be extended, in the following

form.

)

, Where. the subsöript. h designates the potential for limited depth

of water and cp. _{= c} + 1w mu.st satisfy the following

condi-nh

nrh

Tflih

nr ad,

4)

( ₊

Vonh )

, x \Ix2,

y = y0 tanh v0i,

ôx

: shallow water number.

Since the pötential cpdoes iot satisfy the bottöm conditiOn 3

it is to. be extended, by introducing an adjusting potential _{nr aci,,} to the corresponding potential oÍ' finite water depth, i.e.

nh =

+.

ad) +

'nih'

The adjusting potentaials

_{nr ad.}

fOr

heave and sway and roll

are asumed in the

following

forms respectively.' POr' heave:

o

for sway and roil:

Pnr ad. =

f[y(k) sinh cy + ô(k) cash k(h_y)] sin kx dk

where h: depth

of

wa.ter;'(k),

(k), y(k), 5(Ïc): un.lrnown

oitat .

-4-.

evereywherè in the liquid:,

at y = o;

aty='h;

(k) sinh ky-- i1(k)

cosh k.(h-y)] cos kx dk

A _{n h}

'nh +

= o

ÖPnh

Firstly the unkriknown constants (k.) and (k) are

elimina-ted by satisfying the free surface condition 2) by the

poten-tials dalone. Secondly the unknown constants (k) and.

are determined by fulfilling the bottom condition 3) by the potentials (p

nr

_{+ p} ). T}nis we determine the required

nrad.

adjusting pOtential

nr ad, and the

ötenal

h in he fol1oying f:orws. Namely,

for heave:

y sinh_k

_{- kcbshk}

cos icx dic

= o 1E- y cosh ki -

k

sinh ich

-ki'l VSIflLI _{ky-kcosh icy}

voosh kh-icsiith ich or a orh = nr ad.. co (p = (2n-1.) o -5-co

(-1)

_{k2(1_1) (v2-k2}

cash k(h-y) dk o V öosh kh - k sinh kh i

(k+v)e vsinh. ky-iccosh icy sin kx dic (2n)i

vcosh kh-ksjnh ich

- k cosh k(h-y).

sin kx dk o y cash kh - k sinh ici.

co k (h.--y)

cas kx dk vcosh ich - kinh ich

for sway and röll:

co

p '

=- e

y sinh ky - k cosh icy

kx dk orad, _{o k-v}

y cash ich - k sixth kh

t 2 coshk(h-y)

/

(2n) o y ÒOS1'i kh - k sinh kìi

sin kx ilc X dic rn' ad.

::zt)-''

.. n-i) (k±v o

for heave:

nrh

= Re['ti Res(v0)]

and for sway and roll:

nrh

=Im [7cl Res(v0)]

By limitting process, the residues of the function

F(k)e1

at k = y0

i.e. Res

(v0) are calculated and thus the

asymp-totic expressions for X± are

for heave:

- 2it

coh2

y h

_.0

cosh y (h-y)

. O

sin y

2v .h + smb 2v h

cosh 'j

h

o... -o o

n

2c cosh2v h

cosh y (h_y)

y

2(-tanh2y h,)

° 9 _-

Sin v

°

_{2v h+sinh 2v h}

_{cosh y h}

'o

o. o

J

orh

-6-(2n-i)1

°

Finally the potential

_Piih

is determined by satisfri..ng

the radiation condition 4). Before fuÍilhing the con4tion

We rieed the asymptotiò expression of the potential cpflrhfor X--+oo

Por this purpose we consider the äomplex contour integra

of the function of the form

p(k)1

,

which may generally

re±esent the potentials

Pnrhin k-complex pihne.

As the

eon-tour we take the quadrant Re(k)>o

,

I(k)>o

Which pases

over the singular point k=v0 (v>o),v

being the only positive

root of the equation y cosh v0h - y

sinh. y h = o.

Taking the radius of the quadrant

infinitely large and the identation

radius infinitely small, we find

for sway and roll:

2itoosh2v h cash v(h-y)

(p -s ' o . cos V lxi

orh . - 0

2v h + sjnh 2v h cosh y h

o o o

2it cosh2v h cosh y (h-y)

cp +

v2cth2v h1)

0 065 _{V lx}

nrh

_-

_(2n)j

2v h+sjnh 2v h Cc)sh y h °

o o o

We now return to the radiation condition 4) and substitute the above potentials in the condition and thus obtain the

potential

_h1

the following foms,

for heave:

oih

=

-and. for sway and roll:

27c coshv h cash y (h-y)

y

. . 0

.0

sinvjxt oih -+

02v h

+ sinh 2v h

coshv h

o o o

27t cosh h cosh y (h-y)

= -

v21+l(tanh2v _h.i) 0 O

_{sj v!}

Tnih + O . O

2v h+sinh 2v h cash y h

7..

27t cosh2vh cosh v0(h-y)

vh+sjnh2vh

coshvh

o. . o o

cas vc

" _i\

2fl ₂ cosh2v h co.h y (h-y)

nih - '- I. y (1-tanh y

h)-'--. O 0.

cas y

(2n-1)i

°

2v h+sinh 2v h cosh y h

o o o

-8-, where

corresponds to x±

The extension is now conp1ete. The two sets oÍ' the equations (6) and 11) and (8) and (12) are the extended potentials p for heave and sway or roll respectively. Since the adjusting potential p ) are obtained, in

theform..öf..Qauchy-nr ad.

rLr ad.

integral, they are expanded in series to be used for numerical

The condition of

_{zero relative normal véióoity to be}

satis-fied on the cylindeÌ surface

_{for heave sway and xoIl}

_{my be}

wr.tten ii the following form

X

=- ow

HYDRODYNÀMIC FORCES AND MOMENTS

where. the stream

_{ftntiônsEAflnh}

tion. This condition determines the unknown source intensities

and conseuentiy the velocity potential

_h

desigiiated in

advance in theequation(4).

Having obtained the velocity

_potential

_,

_{one. may integrate}

r

i

the hydrodynamic pressure

_{distributioñ Re}

rp-J

on the

sub-Lät

i

merd cylinder surface and determine

_{the hydrodynamic forces}

and moments:

vetkicai forces

in heaving

horizontal forces in swaying.

_{and rolling}

longitudinal moments in swaying

_{and rolling}

J

_v_

clx

-T,Q

fre[

E A f

h dy

]sin wt +

u; 2 n=o S

xdx+ydy

cos wt

_'(13)

depend on the mode Of

mo-dx

dy Ç dx+ydy 7

]cos wt

Re.,

31WtA

₁

_y

00 -

nh

-lo-where the constants A depend on each tode of motion. The förces (moment) consist of' inertial and damping part, both of which are in phase with the ¿cceleratiòn and velocity Of' öscillation.

They are called respectively hydrodynamic inertial fo'ces

(mo-ents) and hydrodynamic damping forces (mom(mo-ents).

From the inèrtial and damping forces (moments) we define the added mass and added moment of Inertia and damping force (moment) coefficient and they are represented in dimensionless forms.

In case of sway and roll, it is necessary, in addition to the above, to define added moment arm and damping moment arm, which

relate the moment and force of inertial an.d damping part

respec-tively. For our convenience the above mentioned may be

-11-heave sway roll

inertial force

Rr

damping force _FHI .

inertial mment _Me..,

M

__

Ri added mass

m}= r

Hr m,, Sr = added moment of inertia . I -damping force coefficient _{NH =} N = -damping moment coefficient . . MRI 'R = addèd mass coefficient 0S

_pT

CH

pB

adde:mom:of

inertia .coeffjcjen-. . .. . c R T4 damping ferce

parmeer

o

- ptB2

pwT2

damping moment parameter . . R

pwT4

added moment arm

-i

Sr_

MSr 1

Rr

M

_Hr

T

- Fc'..

T T - T

damping moment arm M

Ps±T

1

T T F5.T

, where B: beam T: draft.

It is customary to represent the damping force and moment

coefficients _{N11, N}

_{and NR}

by wave amplitade ratios AH,

s and

.where

-12-DA1VJPING AND EXCITIN PORCESÂNID MOMENS

The aymptotic expressions of the velocity potentials for heave sway and roll are easily obtained by

combiniig the equations (9) and (ii) and. (i0) arid (12). They are, omitting the subscript h,

for heave:

00 _{cosh y (h-y)}

iVE

°

ei(_wt)

X4+co _fl=o

'

_{cash y h}

o H.

(_1)fl

v (i - tanh2v0h.)fl ri (2n-1).

for Sa3r and roll.:

,: eos.h v(h-y) H --

cosh vh

H'=v

2iEcoshv h -o o o___. 2v h + sirth 2v h o o i 2fl (tanh2v h -(2n) ° o o ± : corresponds to X-?+oo

source 1.ntensïtios, depending on sway and roll.

(15) + E

flo

2it cosh2y h , where o 2v h-i-sinh 2v h

0

o + wt) (16)

The gei.erated wavès by the oscillations progressing at far

dstänOe from the body r = -

fi

at y = o are obtained from the. above asymptotic expressions in complex number, i.e.

for heave:

= - vaFI

flEO

A HeH?

wt) fOr sway and roll:.

(17') i v.a 00 E A

fin

IF n=o 00 = ;

iva/B/2

E A H

AH

/B/2

13-+ wt)

xI+ wt)

whoro depend on the mcde o motion.

, where _a : linear amplitude of heave and sway, B

: correspond to

x.-Consequently the wave amplitude 'atios for } 'r qwav and roll

are written in the following form.

00

=.v EAH

_{n n}

00

-14-Ii addition, bt equating the mean power dissipated by the damping force in the forced oscillation and the mean power used in forming the progressive wave system in both positive and negative direc-tion of x- axis, we obtain the reladirec-tion between damping coeffi-cient N and wave amplitude rtio A in the following form:

Ti

2(13

Next we consider the vertical and horizontal exciting _forces and longitudinal exciting moment _{on a fixed cylinder in a} trans-verse incident Wave system. Haskind's theory [6] sta

_dthat

the exciting forces and moments depend only on the asymptotic

behaviöur of the potential for forced oscillations of the body

with unit veloCity amplitude.. Newman[7] extended tI'e theory and further established the formula:e relating between exciting

forces änd moments and wave amplitude ratios _{in two dim.ensirinal} case. The exc±ting force (moment) in our case is represented by

the following óontour integral [7],

, where : potential of transverse wave system

p : asymptotic potential of unit velocity amplitude.

We take two incident waves o± the _{ollöwing form, each of which}

is oncoming from the positive and negative end of x-axis;

pg +

2vh

o

(1g)

wv

sinh2vh

h

- cosh

y (h-y)

e- o

CP

-cosh vh

Each asymptotic potential p is obtained from the potentials

and by taking unit velocity amplitude (seo eq. _(15),(lG)),:

Substituting the above defined potentials in the equation (20)

and executing the integral we obtain the heave exciting and

sway exciting forces. _EH , E5 and roll exciting moment ER:

JJ

pgv0ST

T : draft. -15-r' -V 1+ ° ₎ smb 2v h o O 00 + i E A H'

- n=onn

where _± correspond to the waves which are oncoming from the

positive and negative end of x - àxis A : depend on the moae

of motion. Making use of the relation

_{y = y}

_tanh and the formula of the wave amplitude ratios [eq. ₍₁₈₎₁ we obtaii the

magnitude of E in conventional dimensionless forms:

IEHÌ AH (J )

pgB

v0B smb 2v0h lE_S

=(1+

2vh

_°

)

pgv0S

vS

smb 2v0ri (21 )

(23)

.,

where S : submerged section area; H : half beam draft ratio;

EH IJs --iwt = pghe E A H n n 00

+ i E A H'

-y h + sinh 2v h o o

(22)

2 cosh2v h

-1

6-The phase diffcr-ence. bctween th _{ct,ng forces (momonts)}

and both oncoming waves from right and left are obtained by comparing the phase angles of thewave elevations

Re[_. pie1t1

with those of exciting forces (moments) Re [EH,ES,E1, j.

They are for heave always

_H,

while for sway and roll

and R±

, where

s co

= arg( E A H ), = arg(E A H').

₍₂₅₎

n=o

It should be noted that the minus sign of phase difference. means

the phase lag and vice versa. The wave force on a fixed cylinder consists of vertical and horizontal component and the former con-tributes to the heave exciting, while the latter does to bOth sway and roll exciting at the same instant. Thus the identity

a0 _{is valid and therefore we need to calculate heave and}

sway exciting force together with the roll exciting moment arm

nh

(a,b,G)

= b0 (

-

G)

and in odd sine series for sway and roll:

4

nh

(a,b,@) =

E csin (2m.-1)e

-17-NUMERICAL CALCULATION AND DISCUSSION

By the transformation x-4-iy = e-ae©+be'

the stream ftinlctions in the boundary condition [the equation (13)]

are represented as finitions of three variables

,

a and b. The

functions are expanded in finite trigonometric series on the

cylinder contour, namely in even sine series for heave:

4,

E b

sin2m8

nia

The boùndary conditions are thus reduced to 10 and 8 simultaneous

-

linear equation systems respectively, from which the source

in-tensities are determined.

The adjusting potentials or stream functions are given in

in-finite trigonometric series (Appendix) and in our calculation the

first 9 terms were taken. In the numerical integration of the

sin-gular functions G(h) andF251(vh)the.limits were takeii at

and 30 and the increment of (2p) was o.5 (see Pig. 2). The

cal-culated forces and moments were represented as the functions of

dimensionless frequency in Figures. The numerical calculations

were carried out by the computer IBM 360 at Chalmers University.

Our discussion will be primarily concerned with the influence

of finite depth effect upon the hydrodynaniic behaviours of forces

-18-WAVE LENGTH: To begin with we consider the lengths of the _waves generated by the oscillations on the surface of limited Water depth. The formula for shallow water wave number v= _{tanh y h}

o o

reveals that at low frequency rango 'the relation y>y is valid. This explains that at _{the sathe frequency of oscillation} the generatd wave length of shallow water _{is shorter than that}

of deep water.. It s'höud be noted that the _{approximate relation} is valid at

_voo

SOURCE INTENSITIES: In our discussion we are obliged to depend on the calculated source strengths. Por vanishing frequency

_v-o

it is needed for us to consider only the significant _source in-tensity A. From Fig. 3 we see that firstly _{at y the} nondimen-sionài source intensities for heave A _{7/B and A .,t/B approachto}

or

unit and zero respectively for any depth of water; secondly in the vicinity of

_vo

_{the intensity A0r/B for finite depth becomes} higher than that of infinite depth as the depth decreases, while

for finite depth decreases first and then _{increases as y} increases. Although Fig _{3 hows the calculated result of a}

oir-oflar cylinder, the above statements _{are valid for any form of}

cylinder. As to the swaying oscillation it is known _from seve-rai calculations that the intensities A0 for _{finite depth are} generally higher than those of infinite depth when

ADDED MASS COEFICIENTS: One may obtain the approximate formula of the added mass coeff±cient for heavthg _{oscillation of 'a}

oir-iar. cylinder in the following form

+ 92S+1 (vh)

h=h h-oo v-o

o

2L_i

h-oo 0'H h=h h4od + 1ÇT\2 2'h1 -19-- 0.577

From the above formulae we see that the influence of shallow water effect on the behaviour of added mass coefficients at

may be explained by the behaviour of the singtlar func-tion G251 (vh) See Fig. 4, 5, 6, 7 in connection with 2. It is stated that CH for finite depth is higher than that of infinite depth of water at v-o and tl4s is valid fOr

any form of cylinder. For a moment we observe the addes mass

coeffic.ent at high frequency. Pig. 5 and 7 illustrate the

added mass coefficients C at wide frequency range. We se.e

frOm them that at high frequency the values o± CH for finite

depth are generally higher than those of deep water and from.

Pig. 5 that the values of CH tend to approach the known values

of CH for v->o by Havelock

_tu]

i.e.

Fig, 7 also illustrates the tendency of CH at very high frequen-cy range, which, seems reasonable. Now we come to the discussion

on the added mass coefficient _{C in swaying oscillation. For}

infinite depth.of water at v-e-o the exact formula. of C is

(1 - a)2 + 3b2

(i - a + _b)2

This value is also obtaiied numeridally _{by applying the potential} of infinite water depth at the _{vanishing frequency i.e.}

-

4m As

h-boo B

and for sway:

2 2 or oi. v.-o li =(;; h=h VB

2'

-20-Por the finite depth of water at , on the other hand,

the velocity poteiltial is approximately written by

A[ si:

13

G251(vh)

s=1

_{r sin}

I

and since the source intensity Aor for finite water depth is in general relatively higher than that of infinite depthand since we see the behaviour of the singular function

at v-o (see Pig. 2), we come to the conclusion that for

finite depth is higher than

tha.t

of infinite depth when

v-o

(see Fig. 8) and the behaviour of C in the vicinity of is caused by the contribution of the singular function

G25+1

(vh)1

WAVE AIVifLITUDE RATIO: The influence of the shallow Water effect

on the WaVe amplitude ratio Ä11 and

at vo

may be easily

discussed by observing the asymptotic formula for y _o ,

namely for heave:

(vB

v-.o

=

l/or

A0 V -.O (VB '2

The slope of the curves A. aiad at the origin are

respec--i

hh

tively 2 and , the value of Which is infinitely large

(Pig, 1). It Is obvious therefore that the amplitude ratio for shallow water is higher than that of deep water at y. (Pig. 10,

ii, 12). Further it is stated that theamplitude ratio for Challow water is higher

than

that of deep water at the whóle

Ì-I ) ;

-21-frequency range (Pig. ii). This is due to the cont-ribution of

the higher order potentials to forming the progressive wave

system [eq. (18)]

As to the amplitude ratio for sway it is easily observed from. the above formula that at vo A is a 2nd order

para-h

bola with zero slope ,while is a s±raight line with finite

hh

positive slope and tIs both curves must cross each other

(Pig. 13, 14). It should be. noted that the coefficients for roll and must have similar characters to those of sway,

for both oscillations are essen±ially similar from the point of view of antisymmetry about the y - axis. (Fig. 9, 14).

ADDED MOMENT AND DA]VIPING MMENT ARM: The added moment arms 'Sr i

and Rr for shallow water are generally shorter than those T

of deep water at v-o but at the other low frequency range in the vicinity of v-o the Opposite phenomena appear (Fig. 15, 16). The damping rnoment arms l l and the exciting moment arm

T T

coincide with each other and decrease as the depth of water

decreases (Pig. 17).

EXCITING FORCES: In the preceding section the discussion was confined to the Haskind - Newman method. According to Grim's

method, we can also easily calculate. the exciting forces and

moments. By applying the latter method the.same calculations were carried out and both results were compared. It was found that they were in good agreement. In this report the _results by the Haskind - Newman method were representd in Pig. 18, 19 and 20. It is obvious from the wave amplitude ratios

H Fig.

14) that the- heave éxciting forces of shallow water are

generally higher than those of deep water. Similary the expia-natiön o± the behaviöur pf the sway exciting force may be

ob-tained from the behaviour of the wave amplitude ratio for _sway

-22-MINIMUM DTH: To

find the mininum allowable depth parameter for which our solution converges the calculation bas een. carried oi.t for several widely varying forms of cylinders.

From the comparison of the results it was conclued that:

or deep draft cylinders the minimu parameters e very low0

For shallow draft cylinders the values of minimum parameter

are relatively high.

The influence of fullness coefficient on the minimum

allow-able depth is also signifïcent when the shallow draft

cylin-ders are considered.

The minithum parameters for sway and roll are lower than those of heave.

CONCLUSION: In general the influence of shállow water effect on the hydrodynamic forces and moments are remarkable. The curve of added mass coefficient C(CH, C, CR) for finite depth gener1-ly örosses the curve of C for infinite depthtwLce añdthus:the

two crossing points divide the frequency range into three parts. The values of are first higher and then lôwer and higher

again than those of Ch at the ist and 2nd and 3rd frequency range respectively. As to the wave damping or wave amplitude

ratio Ä, is generally highe:r than at whole frequency

hzho.. -. 1'ioo

-range1while (As or ÄR is higher and hen lower than (A5 or A,) =h

CHALMERS

TEKNISKA HOGSKOLA

Shallow water wave number y0

as

functïon of frequency y and

depth b

I

I_

I

VA

_I'

k!.

-ru

dVA

r.

vb

v0h.tanh.

V11 I

Pig.

i

'CTE-SR

Report 43

Vb

0,00.2 0.4 0.60.8 1.0 1.21.4 1.6 1.8 2.O

3O

3.0

$

2.0

les

1. 6

1.4

1.2

.1.0

0.8

0.6

0.4

0.0

C HA LM ER s

Ovesof the

inguiar. integrai

TEKN5KA GSKOLA

G21 (vii)

15

I

O 5 G

O5

1.5

2.0

o

0 0) E

'AA

CHALME RS

TEKNISKA HOGSKOLA À_o1

1.0

0.9

008

0.7

0.6

005

094

0. 3

0,.2

091

0.0

0.0

0,2

Source intensities for Heave

0.4

or

0.6

0.8.

.4

1.0

Fig. 3

OTFI-SH

Report 43

H=1

= 0.785

1.2

h

_{= 1.5}

o C3A4

e

C H AtM E R

_{Added Miss Coefficient. for Heave}

0.0

0.0

0.4

0.6

0.8

1,0

E1.

p

0.755

1,2 I 4

1epo143

TEKNtSKA HGSKOLA

P3 M A

Ij_

-ar1

I-Ij

CHALMERS

TEKNISKA HOGSKOIA p.

- tJrse].1

= p

H=1

= 0.785

h

_{- 1.5}

2,0

4.0

lo

2 B

.1,6

1.4

1.2

1.0

0.8

0.6

0.4

0.2

0.0

0.0.

1,0

2.0

340

40

5.0

60

Pig5_-CTHSH

Report 43

Added Mass Ooeficient Íor Heave

PiA

CHALMERS

TEKNISKA HOGSKOLA

Added

ass Coefficient for Heave

1.75

ri=i

CHALMERS

TEKNtSKA H$GSKÓLA

Added Mass Coefficient for:, Heave

H=1

p= i

Pig. 7

0TH-SN

Report 43

0.0

- '1

0.0

1.0

2.0

3,0

4.0

r

5.0

*4

CHALMERS

TEKNISKA .HOGSKOLA

,.

5

0.0

0.2

Ade

Mase

Coefficient for Sway

0.4

0.6

0.8

1.0

K=1

i

1.2

Pig.8

CTE - SE

Report 43

yB

Added Moment of Inertia for.. Rbll

CHALME RS

TEKNISKA HCGSKOLA

0.3

Ursell

theory

CHALME RS

TEKNSKA HOGSKOLA

Fig0 10

CTR Sii

Report 43

H=1

= 0.785

J

CHALMERS

TEKNISKA HOGSKOLA

Wave Amplitude Ratio Íor Heave

1.0

0.8

0.6

0,4

0.2

.0

0,0

1.0

2.0

B=1

p

0.785

Yu - Ursefl

I.

_{= 2.0}

'1 .5

10.0

9

vB

3.0

4.0

5.0

6.0

CHALMERS

CHALMERS

TEKNSKA HOSKOLA

2.0

4.0

6.0

Pig.. 13

CTE-SH

TEKNISKA HDGSKOLA

CHALMERS

Wave Amplitude Ratio for R11

0.0

0.2

_0.4

_0.6

_0.8

_1.0

_1.2

Fig. 14

CTH - SU

Report 43

C HALME R

TEKNISKA HOGSKOLA

Added Moment Arm for Sway

0.8

1.0

1.2

Pig, 15

CTH- SR

Report 43

CHALME RS

TEKNISKA HDGSKOtA

Added Moment Arm

Or Roll

H=1

= i

Pig. 16

0TH

-Report 4

.. . . . R R i . I i i

0.0.

0..2

0.4

0.6

0.8

1.0

1.2

CHALME RS

TEKNISKA HOGSKOLA 203A4

'Si

1Ri

'E

T

r

0,35

4.Q

0.30

0.25

0.20

0. 10

0.0

0.2

Damp ihg Moment Arm l'or Sway and

Roll and Exciting Moment Arm

= ,10.0.,6.0

2.0

1.5

I I

-O4

0.6

0.8

1.0

1.2

Fig. 17

CTII-SH

_j

Resort 43

C.H A 1M E RS

TEKNISKA HOGSKOLA

90

80

70

1 .Ç:::l5.

60

.2

50

40 0.2

3:

20 0.1.

He

Exciting Force Coefficient

Pig. 18

Report 43

Oj0,

. .

I i

j

i

CHALME RS

TEKNISKA HOGSKOLA

IfiL

pgB

1,0

Q.9

0.8

0.7

0.6

go

50 O4

70

60 0.3

50

40 0.2

30

20 04

10

0 0.0

0,5

HeaiExcitin Porce Coefficient

= 1 .75

11=1

Pig, 19

0TH SN

0.0

0.4.

0.6

0.8

1.0

1.2

1.4

Rort 4.3

CHALMERS

TEKNISKA HOGSKOLA

lEsi

p g y0

S s

Sway Exciting

PoThe

Coefficient

H=1

i

Pig. 20

CTHSH

Report 43

ACKNO1NIEDGEMENT S

The author expresses sincerely his gradlt'u.de to the head

f the institute, Prof. Palkerno, who has supported this work,

Aböve ¿il, thê author is deeply indebted to

G'im

furnihing his inpublished computer progt'am for heave and pitch

of ships in regular waves and also Prof. porter, Whö allowed

e

to copt the work of u and Ursell from his own library and

wil-lingly hiped me by

ending the rescent report on a heave theory

of a sphere in shallow water by Dr. Wang.

This'work has been financially supported

_y

Swêdib.

REFERENCES

[i]. Grim, 0.: "Eine Methöde für eine genauere Berechnung

der Tauch.- und Stamph-bewegungen in glattem Wasser." H.S.V.A. Bericht No. 1217, 1960.

[21.

Tamura, K.: "The öalculation of hydrodynamical forces and moments acting on the two dimensional body:- accor

ding to the Grim's theory -." Jour. SZK, No. 26, Sept.

1963.

-Thöre, R,C.: "Multiple expansions in theory of surface

waves." Proc. Camb. Philos. Soc. 69,

_1953.

Porter, W.R,: "Pressure Distributions, Added Mass, and Damping CoeffIcient for Crlinders Oscillating in a Free Surface." Report 8216, Inst. Eng. Res., University of

CalifOrnia, 1960.

Yu, Y.S. and Ursell, F.: "Surface waves generated by an

oscillating circular cylinder on water of finite depth: theory and experiment". J. Fluid Mech. 11 (1961), 529-551.

Haskind, MiD.:" The exciting forces and wetting of ships

in waves." DTME Translation

₃₀₇

by J.N. Newrnan Nov. 1962.

Newman, J.N.: "The exciting foròes on fixed bodies in

[8]. Wang, S.: "The hydrodynamic forces and pressure

distributions for a oscillating sphere in a fluid of 'irite depth0t' Dissertation, M.14. Çambridge, June 1

966.

Kim, C.H.: "Uber den Einfluss njcht linearer Ef ekte auf hydrodynamische Kräfte bei erzwungenen

Tauch-bewegangen prismatischer Krper.' Shiffstecbiìik 73. Heft, Sept.

_1967.

[io _{Wehausen, J.V.: "Surface Wàves." Handbuch der Physik,,} Band IX, Springer Verlag 1960.

[ii]. Havelock, T.H..: "Ship Vibrations the Virttal Inertia of' a Spheroid in Shallow Water." INA, Jan, 1953.

Haskind, M.D0: "Wayes arising from osc-illation of bodies in shallow water", SNAME T.R,B. No..

1961.

Appenlix

1. Adjusting potentials and. stream functions, for heve.

G (vh)

-vE

2S+1 2S+1 br ad. - s=Ç2s+1) h2S S1fl

(2s1)-.2(s+1)+1

r22

_{sin (2s-i-2).}

(-i

)fl

i

'2fl+2S-1 (vh) 2S

2s+

d..

(2n-i)Ls=o(2z»

.1 _{r25 Cos (2s+1)p,} T2n-1)L"s=o(2s+1)t _h2fl+2S î, = ______

i

2fl+2S-1 (vh) 25+1

nr ad, (21)t s=o(2s+1) h2fl+2S

r

sin (2s-i-1) +

(_1)h1

_i

(2n-F)Ls=o(2s+2), sin (2z2)j3.

L

¶

ad. so(T

G 2S+ (V h) _r2S cos 2s -i G251(vh) 2S+1 Cos (2s+1).

2. Adjusting potentials and stream functions for sway and roll.

G (v'h) = V E 25+3

r22sin (2s+2)

+ or ad.

_{(2s+2)L h22}

G .(vh) +

So

(2+1)Vh22:

r2S+t _{sin (2s+1)ß.} r ad. ad. 4,

nr

ad

G251 (y)

P2S+1 = = E

G251

(vh) (2s)L h2:8 G

(vb)

+

₈₀

E .28 3

(2s+i)

h2Sl2 ?n) r2S.cos

_?s.+

28+1 r (vb 2fl+28+1 (2s-i-2)L

h2+2S+2

(vh) i 2X1+25+1 4(2n)1 °(2s+1)L h2hl+2S+2 ø _{P (vh)} V

2n+2S-1

= .(2ñJT 80 (2z}1 'b228 e S+1

.(ii)(v.

Oosh u-u

slnhu

ju11

y coshu-usjnhu

cos (2s+1)3, sin (.2s+2) + sin (2s+1). cos 2s + F.. + E 2X1+28+1 .. 8+1

Ç2n).

_{So(2+1yh2n+2s+2}

i,2

cos (2s+U.

du

The epansion of the :ábove. Caucy integrals for numerical calculations are obtained from the reference [

_43.