Vertical exciting forces on a restrained cylinder from waves in shallow water

CHALMERS UNIVERSITY OF TECHNOLOGY

DEPARTMENT OF NAVAL ARCHITECTURE

AND MARINE ENGINEERING

GOTHENBURG - SWEDEN

VERTICAL EXCITING FORCES ON A

RESTRAINED CYLINDER FROM WAVES

IN SHALLOW WATER

by

CHEUNG H. KIM

DIVISION OF SHIP HYDROMECHANICS REPORT NO. 37

Gothenburg, May 1967

CONTENTS

.t'.

Part I: Vertical Exciting Forces due to Transverse Incident Wave

1. Intröductiön . s s s s

i

s . s s s s s

i

s

i

i

2

Potential

and boundary

condition

. .

.

. s

.

.

1

3. Exciting forces s s. e s s s s s s s

.

s s e s s e e

'.

Numerical calculation, and discussion . . . .

Apliication - Heave öf freely floating cylinders 5

Discussion .. . . .. .. . .

.

. .

.

. . . 6

Part II: Vertical Exciting Forces due to Longitudinal Ináident

Waves.,

1. Introduction .. s. s s s. 's. s

2. Potential and bôundary condition . . .

s .

. 7

3.Excitingforces.

.. ... .... .-. ...

8

t.Numerical calculation and discussion

.. . . .

9

5 . App ii. cat-i on . . . - . . . . - . . . s s s s s s iO

Acknowledgement ...

..

.. .

...

. . s e s s

s es es

References

.

.

.. .. -. .

s_11

N orne nc lature . . . s s s s e s s e e s s s s _. .12

Vertical Exciting Forces na.Restràined GXlinder from Waves

ihallow' Water.

Part I.

1., Introduction.

Suppose that a cylinder is fixed on a calm water surface and that laterally oncoming waves then are passing the cylinder in a directio

perpendicular to its axis. As a resul.t of the, wave motion the

pressure distribution yields an oscillating force and montent. Our aim is to calculate the vertical component: of the exciting force

in .shallòw water. Our basic assumptions are:

ideal flUid water,

linearized boundary conditions.

.

Grim [i] has computed the exciting forces in water of infinite

depth. The author exténds his theory to shallow water.

The forces as well as the heaving motions and phase 'lags' of a freely

floating cylinder in beam seas are also calculated ànd represented

in figures and discussed.

The calculations show that the heaving' amplitude and phase lags in Shallow water are generally higher than those of, deep water waves.

The computation has been carried out on the SAAB D21 - computer of

the University in Gothenburg. . .

2. Potential and boundary condition.

The origin of the coordinate system is at the cross point of the waterline in calm water and the centrline'of the profile. The.

x-axis lies horizontally to the eight an4 the y-x-axis vertically

pointing downward. .

Grim

fi]

assumed that the potential of fluid motion dùe to a beam wave passing a fixed cylinder is composed of the potential of the laterally passing wave and the potential, which describes the

For this latèr coTnonent the potential for the forced. heaving

motion in calm water is assumed [1], wheré the amplitude of the hèavé velocity 'TJ=1,

Generally the potential of the incident wave in shallow water is

-

+ i

=.gE

w

.w w

amplitude of the wave gravitational constant

2

shallow water wave number; = v0.tanh

vh

circular frequency of the wave

water depth.

complex coordinate x+iy

'For the calculation of vertical forces one needs only the even

potep'tial function about the y-axis. Therefore our potential and

stream function of the wave should be cosh v_(h-y) cosh v0h s Inh ' (h-y) cosh v0h cos(-v0z

+ iv0h -- wt)

cosh v0h

COS \)0XCOS wt

sin

VX.COS

(A)t

Thê potential for the disturbed motion of the fluid in shallow water [3] is

+

iwt

{A r_!

cosh K(hy)'

h ' h e L

'iRh

_K

2ir.cosh

1.

20h+sinh 21i

cosh

v0(h-y

(-1 +

ZA

n1

n (2n-. s s s s s s s (l.l) s s s s s s s (1.1'.)

eot]+

2rr'sv2.èosh y

(ji_y)e0

(2v0h+sinh 2h)scosh v0h

where g = = = w = h z =

$ o

3

where is a component stram function for forced heave in calm

water of finite, depth (see []').

3. Exciting forces.

By satisfying the boundary condition.(L3) One obtains the

Intensi-ties of sources A0 and A and. therefore the potential of the fluid

motion which satisfies all the boundary conditions is determined. The exciting force is then obtained by integrating the hydrodynamic

pressures along the surface of the cylinder:

dx =

w[zJ

Re(A4,.)dx].sin wt

--

coshv0(h-y)

cas vx.dx].sin wt

-cosh

v0h

-

_{PwrEfImcA,dx].còs}

wt ; '.

The secon,d term represents the .hydrodynamic force due to

Froude-Krylov assumption i.e. the force from theundeformed surface wave,

while the rest terms do the c'ornponeùts of hydrodynainic forces Çaused. by the disturbance of the fluid owing to the presence of.

the body.

where A A = unknown intensities of sources

V =

u2/g

= shallow water wave number

h water depth

w

circular fequençy

The boundary çondition on the 'surface of the cylinder is thus written as follows: sinh

v(h-y)

Re z (A _{= -} L_.. _{sin, y x} fl .

coshlvh

O (1 3)

Im z(A,) = O

The exciting force obtained consists of two components with a

phase difference of 90°. One is in phase with the vertical

dis-placements of the water particles and the other is in phase with their vertical velocities at xo, y=y. Each of them is defined

by the real and imaginary parts of the force and represented by

r ç cosh v0(h-y) = p w E j Re dx + p gj

---.- - cos

V

noS

s

coshvh

F..

= pwE

jIm(Adx

or in dimensionless form F r r pgBn F. E. = ____ i pgBn i j .

Thus Er and E are the dimensionless components of exciting forces

due to the lateral incident wave of unit amplitude on a cylinder

with unit width of beam.

4, Numerical calculation and discussion.

The method of calculation is the same as in [3]. The number of

the terms of the Fourier expansion is five as in ref [3].

Calculations are carried out for several Lewis cylinders and

represented in Figurés l-5.

For the cylinder of H0,8 and=0,8

the forces in infinitely deep water are calculated according to

Grim and plotted in Figure 1. The exciting forces for depth

para-meter h/T10 are almost identical with those of deep water, except

in the very low frequency range. This fact is due to the

hydro-dynamic character of heave in shallow water as discussed in [3].

It is generally stated that the influence of limited water depth on the exciting forces is remarcable, see Figures l5. The effects

of the form of the cylinders are also illustrated. The deep,.

narrow cylinders are less influenced by water depth,. than the shallow draft fuller ones.

As a comparison the Froude-Kryiov fbrces were calculated and drawn

5

5. Application - heave of freely floating cylinders due to beam

waves.

The space and body coordinate systems are taken. as shown in the Figure A.

The equation of the heaving motion of the freely flóátlng cylinder

due t laterally oncoming waves n

sin(v0x+wt)

(m+m") + .N + pgB = F

where n

= mass of the cylinder

= hydrodynamic mass

N = damping coefficient.

B = breadth of the cylinde.r

p water density

= ç0e1Wt, _{is a complex number}

F =

F0et,

F0 = (E+iEj)pgB

=

circular frequency of encounter

The equation is reduced to

(.R+iR.)

_{= (E+iE1).1T} r

_{+ n)}

_N where Rr = i pgB W and R =

The amplitude ratio of heave and wave is Er + iE1

- R + iR

the magnitude and the argument of which _{are respectively}

o_

Er2+E2

li.

R2+R2

.

E.R- ER.

s =

tn

flhr

+

ER)

where s is the phase lag i.e. the lag of maximum heave motion

behind the maximum of the wave.

6. Discussion.

6

The amplitude ratios of heave and wave and the phase lags

in degrees are calculated for several cylinders as funçtions of

frequency and depth parameter , see Figures 6-9.

From the Figures and the results of [] it is seen, that the

heaving motIon and phase lag in shallow water are generally larger

than those in deep water. For depth parameter 10 and they are.

almost identical.

The form effects of the cylinders on the mötions are also illustrated

in the Figures i.e. the deep narrower cylinders are less influenced.

by the change of depths, while the shallöw fuller ones are largely

Part II

1. Introduction.

As an extention of the work by Grim [i] and Abels [s] let a three-dimensional ship be fixed in calm water of limited depth and

suppose that longitudinal waves are passing the ship in a direction

parallel to its longitudinal axis. Our attention is confined to

the plane motion of the fluid around. a section between two

trans-verse control planes of the ship. It is then required to determine

the

hydrodynamic

force

on the section.

.

Grim's assumption is also applied to this problem and numerical

calculation are carried out in order

_tO

find the inftuence of

water depth. on the hycirodynamic force. The results are given in

figúres and discussed. Assumed:

ideal fluid,.

linöarized boundary conditions.

2. Potêntial and boundary condition on the cylinder.

The coordinates are the same as in 1. (Part I) except for the.

z-axis, which is taken as the longitudinal axis of a ship, see

Figure B.

The fluid motion of the undeformed longitudinally oncoming waves

is represented by the potential

+

"w = % cosh

v0h

cos iv0(h_y)_(v0z+wt)] . . . . (2.1

From the above our rêquired potential is written as equation (1.1'):

. cosh y. (h-y)

=.g . wt

W W

coshy0h

O

which gives the oscillating vertical velocity of a water particle

at a point around the sectiOn at z: sinh y (h-y)

V=w

..-

CO5Wt

The factor cos is not included, for the problem is confined to

a sectiOn. It must however be used in the appiicati.on of the

re-sults to a three dimensional ship, see 5. Application.

Consequently there exists oscillating flujd flow with the velocity

y through .the section.

On the other hand the boundary condition on the profile requires that no fluid is penetrating the surfäce of the cylinder. To ful-fi.1 this condition one needs therefore to add another stream. The

potential of forced heaving motion in calm water equation (1.2) with unit amplitude o th heaving velocity U is used for this

purpose.. The potential represents the motion of h? djsturbed

water of the longitudinally passing wave (2.1) in the lateral plane. The boundary condition is thus written as follOws

ç sinh y (h-y)

JO

dx+1phO

. . . .

. ...

..(2.2)

"s cosh

where is the stream function of the forced heave in shallow

water (see [3]).

3. Exciting fOrce..

By solving the boundary condition (2.2) one obtains the required

potential of the. deformed motion of the longitudinally oncoming

wave (.2.1) in the lateral plane. The hydrodynamic pressure on the

surface of the section thus consists of two components: 1, Pressure due to the deformed fluid motion.

2. Pressure due to the undeformed fluid motion,

The force is therefore obtained by

-8-ç ç cosh y (h-y) I _h dx - pgh J ° dx (2 3)

j5

s cash v0h

The second integral is the exciting force according to Froude-Krylov

in shallow water. The force is composed of two components with a

phase diff. ir/2 in a manner similar to the equat±ons (1.3). We define the real component as:

Ísn0nr

the imaginary component as

C) E

I

_(A nr 9 - A . .)dx + ni nra r cosh \ (h-y)

+ pg

J ° dx , s _{eosh v0h} + A .ó. .)dx ni nnj

In dimensiönless form they are given as

F F.

t.

-pgBh i pgBh

where B breadth of the cylinder

E

amplitude of the incident wave,

4, Numerical calculation and discussion,

Numerical öalculation are done _{as in Part I, The dimensionless}

exciting fOrces as functionC öf frequency _{for depth parameter}

hIT were computed. and some of them are plotted in the Figures

10-13. .

From the Figure 10 and 12 it is seen that the _{òurves of Grim at}

infinite depth and of hIT 10 are almost identical. _{As a omparison}

the Fraude-Krylov forces _{were also calculated ad plotted in the}

figures.

It is generally stated _{that the influence of shallow water on the}

force are larger for fu1let sections.. than fOr _{fines ones.}

j \

s e (2.4)

5.Applicätion - Exciting forces on a fixed ship in head waves.

Consider.

a ship fixed in a longitudinally oncoming surface wave

n

sin(vz + wt)

to calculate the exciting .force and moment on the hull (see Fig.C).

The exciting force for an arbitrary section at thiökness dz is

i(v z+wt)

pgBsdz(E

+ iE1.)e

Taking the origiorl of the còord.inate system at the midship section,

the mornnt of the àbov

förce about

he pitching axis

X

is

approxi-mately written as

i(v z+wt)

gBiszsdz(E

+ iE1)e

°

Integrating the elementary forces and moments along the total ship

length, one obtains the heaving force and pitching moment aböut the

axis x at the midship section:

lo

-L

-

i(v z+wt)

pgh J

Bz(Er+iEi)e

°

.

dz

. e e s -. s s e i

(2.6)

2

iCy z+ot)

pgh J

Bz(Er+iEi)e

°

z.dz,

where

is the beam of cylinders.

I

The author expresses sincerely his gratitude

_{to Prof..Falkemo, who}

has supported this work

He takes this opportunity to acknowledge Dipi.-Ing.

_{P.Claussen for}

L T

L .

Fe1y f1Oating

1 nde

in a bam Waver.

H

r r

Fjg

4 4

.L'HH;

I j r

/1lI/1r(II/I(IIrffrflf,//I1rfffIJlff///fff11l1f1fr1/f«fIIffI/fff///

Sie*atib viewr.ò

à

e.ction fixed ir. .spce.

,i .

'. .

siri.(v0z+irt)

z"'

4

LJ2

:

Fig. A, B and C

CTH-SH

Report 37

h

1/1/11(1 f/f/If//f/f

i71/If/f/ff

-

of ¿_fixed S}li.p

alongitudinal wave.

23 32C:: )I P J' 634 I.

ILgA

' -F I r 4 --

sx+t)

Exciting forces in transverse

as functions of frequency

9

depth parameter hIT.

Pig.1

CT14-SH

732.501.

c 1634

Excitin2 force in

transverse

.

u

functions of frequency

dspTh partér hIT.

.

waves

for

Fig.2

jjj

. 37 :1 _ I

::::;

i___ L J. ' i .-4L JL l;:r t u L4_i ';»

-4 --

.__L t O _ _ :i_ - i'14 .14.:-' ri!t1_. I .i

1:T;

t:::iH:T::

.:. ' .4fl. 4::::

.--

-:i:::::..::ij.;j. :.::;::

::::::.. .::. t : . T

I1_,

:

Th LH L_ !HH .::: ..:1

812

.:::;:

.:i;::.

-I

! ..

:;I:;

i-Eii

I.tJ-. m- tj I I ::f;.: :::;. t:.! :4 .::

:;:.

.::..t-_ J_;: .:; LLL. IL

.: :

:;:

_;.::

:: .

-

.:::::::-L ...

..::

.. L .',:-

_I:.

.::: :1:::

i:-. !i:Ifi

:. ±IiIiE

-fl:.

- :::- . -:

i:.

14

:!!::1

:::

::

;.; :. .L:i :1.1:1 tiF.t1 i

::::H

:

.L:.:

:; .

:

itfl

:::j...:;:

:i!r

t t I

_r

! _{t J} I 2_t ii t_J r

tit

j}j

! i i

I1I

._.

:L;:

i r ;.ti g:

LL

L T _li

_Lf

- ì

I .i...' ::: :.hI I :

j

i il

jJj

:i »- 1j'- L. _t L t4 _it --_ J-! I i ! I I

-;llh

-1

41I

i i -. I r-'- _

!:-

fri13 , 1 I I

JI

ìI:-...

I :

'.::::L'

:: I

jH

I 1 P-' Ìr_l. i r L I I Jr I I L_ .4j.11

HTFL..t1

4LE..:., r1 r Tr-r--r:--'--

r::.::

.. I '

.f .,± ...

i.:::i.::,1_;::;:

_-:.,--7...--I ...

...,

i:

1i;

I;.rj.:-T

:;:: :.,

Y, Jr . ::1I ;h: : ::r r: . ....

-

I-:. . r: ,. : : :. . -r : . ... ... I . r

1ft1F

t. L I ''I

t;r

-I -E:r.;:T

i:f:j

i4i:.. r-'T-

I ' H.:. ::J?'L1

'tt

kL"

. i

:Li

. :Ir:

lt:r::

-H

4 .J

.i--':

.: i.:Ir;: ;.I I

.:

: r .i:::r:,:: .

;:

r:: .1.4-: .1.4-:.1.4-:.1.4-:1 .1.4-: :: .: ::

-'

-.. NO : :: -

j

: . ..

:.;i:

I 1 : :... I -: I -

i:!

r I:: . : . _j : I ji .i ±_ 4 pr 4R t

:L.

.. H:. LII frJ u tL '

*J:

r I -' I I I iti IL ;:T;7:1.. .1t r 1

;i1,

I .. -

_T

-wr.;::HH.

I 4 1 1r1 H

:H :..;

J r

..

:::;...2

r' _J O L 1f2

h

I r _H

-

-- r r

:;:;r;::rLH;rHJ' r'.

i .r.:.:.:.:J:::;'::r

$ A4

732501

-lip,I'

Nr 1634

Exciting force8 in traneverse waves

¿a functions of frequency

%i

f

,

depth paraeter hIT.

Fig. 3

31

--Report

i:

:

j:::

î :

r11;

l.-l---I Ji 1.4 r : 4L-

:tr

: :- t--:: Y :.:_;::.. L . .::. -. .:. .;::L:: .: ; . r_1 :: :;-v r , .. ,. I _;-. , .

,,

I t1 , I i ,, :: :L i I I

-:::.

L :1:î _{I - '} '

.ft

'H--:1 _; t I

:

--t::l

IiI

,

i

.:t -h;fi

i4fit

L';U

T

td'

1r -::' . 19h

«

I T r E r t u t I i

L'

I t L Ï L

ii

i_L

J I {1 I

I1

-:1l t' i 4 ' .- IL _{4_.0} ' Li_.-Ij ' I T I _ , -j-!I I - _fI I ; r I I j I I : _{flJi1 :1} +J I !

ii

'

4-

t

I 1ii

llI:!llqIllI Lt

" I Hh ' I L i I I I I +r.

:"r

' L

L.L.)J!I

I ' ' I '

t

-t_.4 I ' I :

tj-4

t- _VtT- I' + ' i r I i ±T L

VI_

I ¡ ff- I I '

r-j11j-.:LT i) I

Pft iii ji:

;i:

T I-

'[:

I ;1:-

-!-:

» ' I

:,.,

--4II4

L;

:fT

H_Ipr'

I-: 1-ti-I --r: t

.

:ftr

tt:: t::ijjl,r j4I-,;,..:.i.:.

j4j:

; -,;-r:.

r:.-:::, N'

\,-.:;-

- -: : .". -i :L -L _f i-..-.-. ':!- -«- ::-1T-i i 41±11 -I fj _I

-'

I 'II& -;

-I

[L ... ., ... \._ - - I .. -f 1, .4 T

_t

1, t;_{- H} t, -11a4! r4

rt iii

-- .i-I .i _i. rn:- :i4 '----I I J2Ihj

:

;:t . T :.z: -r-jr4 i:. I L L: ' L

rt

-:; tra,_.. I :t ;-: . -: I I t; . t . s. -; '-I : wi4 ¡d I i:j!

t": lJ

I

'

-. -- I .I. :

,... ...

_-...

-

-. ." ...

i..\ ;iI

:i.;Ie

I , -I I . -- I I 4-LT t1

rr

ch . .._{!! Ttk:tti:...}

if- ..

I' I :, .-Lh1! H

M1.k Ii

T

LlL

tr

,j i4 4;ïi :T:1-; : f I ::I:-r ::TI:.i: :

..._:ìk

:---

--L -:. =F2 I 1 :

._ty

:.

' . . -: H '. I , -: i1-2 rfE EI1j

ffIIO!.i

!j-it

'i- JÇL1 I

- A:r-..;t--r ';: : I T... I L ... I

J

I I-._I._ --:-±:.iL I I I

::..

-I I :-: I Il I

:i:i: .

II I I H _{i .} I : I f 1

fL:4rI.1

, I1 ;-:

-.-

ISL..!'/

lJt_f

I r i.

- I

.

_"h

L HL »111 _ I I j -$ :;1i-1 I

-::.I

--::: II I :LI,.:

L-H

, -L I I --- - ,.; -1 I t ¡13 .I ' O81.i I 1 &I1 'I I :.

:\i;.

i - I

Hj

-r J I r

:L,;.-,:i:

tÊ

t'I

I I L

-732501 Nr 1634

E*dti

forces

aø fwiction6 of

depth pareter

in tr*never

wavs

frsqusncy ti for

hiT.

Fig.

. _{.-.j:_} ..:-: ;; -: rtt.; I _{i_d:} . t l-i .I :_j . 1_J_ : :'t; :4 i.:: . - - ::r .L. :1 . 1 ._:.:- .. : : . L

..;_::;:

L -I*T_. L -} :

Ij

i

-ji1 J1 r _h r q rLi4j J d H

U1

: ' i-.rc1e -_-. :i:h_ I ii H j ; _ii FL f n : S:f .j T

:::

;:i::

I [fii t J 1

-

i 1z i I L tr L . Lr.4..i t j l _:::. : ,t:::;L-:

jj-

.:ii:t. jI

''-J,

j _A "

'j...

' : I

li

;

:!!IItt

t :I

if.

1t

H4 -

i

1i:NI I4J.r1 ; lrf[ -I L I » . I i

j

II rl 'I JjJff I r1 iI - .1-i .:rI _L L r ' F j j i .- It i i :: _rì3 i _{« .t} i-I

!IJÍ i-

Ft!I!.II

. i1t.- .1 j ii I

iI4

j14

:

lL L

- :

r- r1 :1 r T' :EFt

:jj!j

. :i: ;. H :j:: _________

!:

_¡!OO1!?! 't-r L r ' I L il I i i4gaJ 4

jjj

.: ' Tt i _IFIL i ' I . i

ih iiiii

_

llM

S

i r i , i- rT i T

i

+

ï1IIIiI I

r 1 i I r

r-t

jTi1t

i r i -i.i , I

.T:1

t I i_:_4i r i I i 4;.: I ..J.. iL: ti e-r. t-j ' ;Ì i

i .j&l

I i :1

:i: i_tL

:

i.::;

_iL-

_tit_tt

zrt1

14_r ;rF :,_t:-.;

:- i

. . :; 1_ . :::' j

'pr -rj

r ,: L. _'i ...,' ,.: ... . . .::H: . H ' :. .:

-:: ::r::,:

::i:r. :1.: -.::

J.LTLl1 -I 6 L ir L _:i- h: r -.. r:r i-'-'i '.r';,

:::

.:i-. ... -;;

,,:jr

+1

,: ...

:-

t"

-r-.:...; : h:-L L.' tr

r

" 1

1íj

i'

9 '1

'[i

r'

i T ' i L I i , i ']1:iL j L : : i i : i 4j 4 :-: r.J .Fdi9T L ._-1 _{:jiî _ t-_:} t-- _'r i ir

r

i i _j i j j i

:ii4

i 4 1 i

IVs

i_, 4_ 1 .E ' L ' i

I..

i i I i r i I f i f

111'k._

i I r -

H

TI H

_L

r i -t--t

-

-t--- -4 F-+- 1 -'--4 1"

.:::

-t -_

L-:.::

- - - .- -.

.:.j:j.:i.

- iii

::I.:.

I :.

.:,

i I . .

::,.

Jï

F1 i ri I

J2 r

_..L I T L L I

:.i:::;:

' ...

_'j

11ff

..nh.S*.a.s IflEbp.uI...uflÌ b

1S1

J!i

-1ri1

!1

qi1IL

-, - +-F , i-,44-r

4i'.T;riF

,

.!IJ

IFIllhIll

piIqiRit:. 4dt

IdE LIJllØdIllÎll

iù jiL

llllllllllffpr L pqi

' j

-InØflqL

illi

iib.

pJ

i!

krnpkL:arwki::ppwÎk

w':hI

]WIJIII

hIPII

hhMP

fi

uL:hL

. .

q jq

i.JWjJj: j q,r1ijj

h.:qu

j]Eqg

rÇ

ffjhg

dill

$ill' J?JOII

llLJ!!hthfiE1ll

iriiiW!E!llJC8fi3

h ::nn.

! .

kwib

LP

ÇL

LI

jE

j

qq: -. -+ - 4.-

-

-

-:

-

-:

:

2

:

..

-

.E I 2 -. 1222::.2JJ222 r. . . -. -

21!!!.!!!!!!!!!!!!!!..!-.--.--....

. .:.'1Ijju;zq2a: -

---

-

:

.:. .::.22-:. 222... -

rj3EE

-

-

222F -..

.q::::!:::2-

EE5ELr.1;::

z:::.;:2r.

--.---

_

IIt1I!

r.E..

222C.

. -- J -. !IE:

ii

hp

:-

::

-

!-

-

-

#..1L _2LT - I _

-

-- -

-

._._:::::::::2___:._:...

I

2jr..

I

-

-.

--

-

..

2

-

-

-

--

-

-

-

-.

-

J

..ErELI

P-EbPE4hJ

-

-

-

..

-

-

.f

-

I2

-

IlIdIL

_

b.

i!Ì:z EEl !E2

r'

2E!2

-

::::III:

::2:::r:82

::L.::: Irn2::IL::2-.:r---:r.::

:

2

Ldz

::E:IE

;:-.

zth-2

-

-

-

-

2jjIH::L:::2::J::;;:I

¡gJ

-

. -

-

--

.-..

-,

-:

::

Lj1E

IIEd

.

..

-

-

-2r22t1..

:

i

::

- qj

23-1Er:

JEjEj:

&Er

iIh

-- -. '

I.2.

E4EEEEEE

-

-

2...

:...::::

;..

IIJL

fI

2:2 E2Ej - - :- ;::::J2: ,:::::::: 2

--&

E!EErF !EE

L:

-

iH

-fffiì

jji

hI

2E

-

;

h%

r2r1

hi

--E

.L

jE-

:-

-

r

:uIIL! :::: -.

r.:::::

E'EEEEE

!ffEWEELE! !EEEh.. .:L. -. -

U2.l.

:222E:2:

.ii2.I

::2:-

r

2

-

U 2

Ij

1UI

l

-

-

--

-

.

-

_rL

-#:

_r- . -

.

-

:p:l:p::::::-...2.#.1.rn

m... U I U EWL.I h 2E

ci

U:-rv.r

-

E'

IJWh!J

I iEEE I:

;'

veibaaj

3°

T'3

a#

aò5átd

dsp

¿i

i

tw

.Ie

¡e

!sq

SAV

'P%$1T4IV

I-'

---2....

222

i L' _{.L L}

':

and

r.

Amplitude

phase

aefunctiona

ratio of heave and

.

leg e fore depth

of frequency

...

wave

parameter

t

I

o

: i: I

---...

CTH-SH

.Report 37

:--

---'li

--:-i f.-' i -i:i: ... _.l I L '

.J :;:It::::.j:::t;

_ i __ _ __ I ':7 i j -1 -_ :. -1 -.-1 i -..-,-1-:-'.-: :: :

--

:, : . J J '-I i f

j:

i

i B

2-'

r

-;-'H.

_J -_ - -: -i - i r J

-=

r _ i : 1-r r-1LL1 i _

H'i

L

-

-

-- t -: i _ _ - - -t

-:

.-.--...

i!;í:.

: t-. -.--.I::

IDrti

I -I--- _: Ir ' r _{-t-' -i--- - .-II} L :

r

r: r: . :., !: J_i:i : ;:: ._ ..!:::. i::i.,',: I;t

' L ir

f

t

r . -rÍH1 Li: I J II i- _{i 't} r r _L

r__r

I -i.. ___ ___ _: ¿__

tir'

r ,-.- Il I

j.

.t -r -

j-r r ' -

h ti

L

j I i i _.t,___»I

---

-r'

.:;t:r;I:-,

-

--r r.: 4

i'i

'

II

:. r_ ---

_---

. I

1:2ì

-n r il. r--i:i.: .r ;__ _:;;1; I1-a -r t - I , _-i - -, -I J

-r

-L... ___.t _{_l_} .i:: _ ---i,

-.

ï--'_1 r-t r

;.':

' r !'::TT I , --':t 4ri11 : . L _____i__ L , ' i - .-t f_J ________ _______f_-.---_

:.:

t - -- --- - - -'-i T r ;t:- ...

r11ii

I I

-J r

r-r '

i!-i

-i-J--t !

:ï

:;i rl r 4... -

-iiIIk

i -f- : I

1hT-i 2

t 110

: r

L

,i.4.

--'

-_

I I Jr

I

T

.j4.4'j

-A'

_-J,

i

L-L

H-H

I f _L

1H IL..

I I _j

;ri

i

Otß 10 If2 14 i r ii) I ' LI Ii i I J r I I 523 A4 732.501 -.fIppI "Jr 1634

7325W

Nr 1634

Exciting fotc

in longitudinal

.

vavee as functions of frsqusncy

.

i.!

2

for depth pareeter' hIT.

Fi

10

Rsport 37

I _-LL i L1::t.: 1r .::

:4i::

. .--__L I ; I;: L : ..;:: ! :;:

':;:i.

. ::; .::;::

__,

I

4L

:::..

: -:!:;1 . .:- .;.. , ;:. :: I ' j:j I

: ';:.

t ... : .: . :.: ... . . ::

::

;. :..: . :-II:: :-: I 1:: ._:; .: r :V -: ::

"iI

1 j .1 -. I

t11J1

I II

_iI_

I ili I I

t.tiJ-i

I1 :tt

T

1:I"

Ifrj1[

I _1r} I

h:;_.

I

r-_i_ .-.'-- -t.

,

-

.

14 II4 !t Ti fjl.t1 ;;i

J_I.I

I 4__j

_.-h-

I T

.j:tt;l.1

' ;;i:111t T;-F1

r

-.:. --i:

--:::L::I'

4.-..

...:i:g

:. ...

-. l [ } + ri I I -I-I lIt h1.._1t4 ;-r-i ill J! Ti t .T ...J i _. _ I ' I i i t

-

n

_4IIII

_:L

I .L- L... _ _ .L --. r I

irr

WP2

_... i

...'

..i:;;::

: . I L I

r- -

_iI r' - H . i1

.:;iL

:

fi:;

..

..-

-.-It_

::... :

.:

J -i ₁

1rdt:t

I : _{H1 -} -I rLn

HlI

_{i :iJ}

:t:;!:.!;

rT "

: ,:'

,.. L Y:I i .L ::: 1 -. "1

1jJ'1

I .f :i ;: :.:, :. i-i ,. . :; :... -

. I.:.L:

»:

r

,LiH:

.

...

::i:

I H1...f i ... .L j.

r 1:fJ

_{iI ..t. LL}

::j.

:1 I,. ._1

111I

J T Li-it 4fi

j

T t i i . -.4 J 'q 1 'i -k_ii H .9 i

-'i---t--

I

-1 -

-I - -I

.1

I ... .. I. i .1F

j!'

L1

Lt1

T-I2- :1 i- ii I I [II' t

: L

., j r _:r:i

-iI

£I:4_. .1 : -i-' '

- "hk1.

I L i 1 ,_ ._ . ' _I t. .1

'::

_H-.,' f --H- 'ft

_:!i1::.-

k

T r ... L.._.

:.;1:

i -.-.

...

i i._ ..:

...

. . . -., . ri . 1' i

...

-r i I - i L

AA

I i i -

I-"rl...

L I 1Î

rfti

-'

a

__i1tr

_{_} :. . . I

''I.

:1:r

I iIt II t I I i i I i i __j__ t 1 _J t '

L/'t ;sw,xd qdsp puQ &ou.nb.zj io.. suoi 3uflJ iv s*Ai* t,UTPU$U0t UÇ S03403 2UTIT33 r.: ;:;::: i, .,'

:

:

-:

--. 4' .

L

r

4

L

L:L IF '- 1-:TFT:.t. Ft -:Ik1 L:..t4' F4

i

ÌJI-'

'1

r

L

i

r

illJI

'

I

j

F tJ

ii

AlIT ir IilV 1Î1

r

r

b1

.1v! Ld1 IJ-4rJWJjj L%]r

d

:!f :z:.: iL: -. .:i:m 1J3

. . -u.0 : . -: . j i: i:

-..a:

LhUft '

m.u'iii .kfiirj'

ì!5L. fW: . p.

r::cj;E,.;:'

'J:

n!ft!uJr:flEj.H::n: 1

h' ..

rfiqmj i

. :

JLE Ç.

!ff!EffPV.jj

..J:.J:..:::. : ;

dI!]!hw

mi mk. u _Z: _ .r:

-; I iu;h.0 :h:!, :

e i

&jfr

, i

Irn!1

rPL]E .4 I :

.i

E ¡ qq :

:9

"

z

r

. iE ..zr k.

z:

.. rnzz.ij

-g. :zuiad

::r

z

"i r

r

i: lh:lIIIzz&4:d1z

,J

:,zj EIh.I

z . :uiriq qu: u: U rr ?zilr:ui Huuzk z: i 11 EEl 2! U i g U IW:Wr Fzr1:uzij:!r'

i

11 UI!

E '.F h U1,EE1'IÑU 13% 1 í,,J ,!

'!T'

E E': E

r

-z u&..q j3

r i: i

LIIhik

'!h1fuzEt.z 1

z..

r

,

r'''u::I zlhiizz_z i_.. I

11..J11 u:Jz.ihl.._: 1!! ff1 ff1

tf

rdb4ll

Ili!IIIH

FfflJhEiÌ

LjII!JI.

44 .ttT 'i ¼ E hE2!ØW u:

Reférences..

[i]

Grim,O.:

"Eine Methode für eine genauere Berechnung der

Tauch-und Stampfbewegungen in glattém Wasser Tauch-und in

Wellene"

HSVA - BerichtNr 1217.

Juni 1960.

M.ilne-Thornson,L.M.

"Theoretical Hydrodynamics."

Third Edition.

1955.

Kirn,C.H.:

"Calculation öf Hydrodynamic Forces for CylinderS

Oscillating in Shallow Water,"

Division of Ship Hydromechanics Report No 36,

Cha.lmers UniverSity of Technology.

Feb. 1967.

Grirn,O.:

"Die durdh. eine Oberfläcbenwelle erregte Tauchbewegun

Schifftechnik Bd.!4_1957. Heft 20.

Abels,Fa:

"Die Druckverteilung an einem festgehaltenen

Schiffsmödell

im regelrn.ssigen Seegang."

Jahrbuch der STG, Vol.53,, 1959.

[6]

Grim,O. u Kirsch,i1.:

"TR-4 Prograsrun zur Berechnung, der Thuch- und

Starnpf-schwingungen nach der Strei.fenmethode."

Inst. für Schiffbau, Hamburg, Jan.1966.

[2]

[31

[4].

12

-Nomenclature.

A source intensity

Ann source intensities (real, jmgináry)

B beam of cylinders

E E1 dimensionless exciting force (rea.]-, imaginary)

dimensionless Foude-Krylov exciting force

F exciting force

Fri F1 excititig force (real, imaginary)

depth of water òr sbscript fOr finjte depth

wavè amplitude

half-beam draft ratio

in

imaginary part

L length betieen perpendicular

m mass of a section

in" hydrodynamic mass of a sect-ion

N darnpiñg coeff. of a section

real part

-t draft of ä cylinder

X,

y body coordinate system

x0,

y0

space coordinate system

z an axis of the coordinate system xyz

section coefficient

E -phase lag

13

-shallow water wave number

density of wäter velocity potential

velocity potential for shallow water

component potential for shallow wate.r (real, imaginary) complex velocity potential for limjted water depth

stream function

stream function for shallow water

component stream function for shallÒw water (real,

imaginary) complex stream. function for shallow water

circular frequency

complex heave, amplitude ;uhr, ''hni