SCHRIFTENREIHE SCHIFFBAU

Cheung Hun Kim

Über den Einfluß nichtlinearer Effekte auf hydrodynamische Kräfte bei erzwungenen Tauchbewegungen prismatischer Körper

158 | Juli 1965

INSTITUT' FUR SCHIFFBAU DER UNIVERSITÄT HAMBURG

Uber den Einflu~ nichtlinearer Effekte auf hydro-

dynamische Kräfte bei erzwungenen Tauchbewegungen

prismatischer Körper.

Der Fakultät für Maschinenwesen

der Technischen Hochschule Hannover

vorgelegte und genehmigte Dissertation des

Cheung Hun K I M

Hauptreferent Prof. Dr.-Ing. otto G r i m

Korreferent Prof. Dr.-Ing. Kurt Wen deI

Juli 1965

'.

:

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I~Uj'1Jf ('\1'

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.. ...1 C/\.. . I ..." UL' . 1 . rl""I n"", '1

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b. ,'cl, 1'\1.1,'lrr. t

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;...

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. ."p I ," r. I 11 I.L I

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- I Be I . -""r;... ~ .,.." t .. ~. I . UI'\~ i I :t t - . .(1:t

.. - I -I ." ,. I I' .' ~ ~ 1

.. 1 .. .n ...

· I' i., . 1 .. I '-, II 'r.' Ir ,l111" '1 I I I' 1 L 01 J 1.. -," ...--

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,..,'" I ITt'P:J PI .. ,. ' n111 In 1Ll~lri-,. 'r p'. 't. 1)' ,i f"'tJi, ' '1 df'r ,It -, '_

I - . ; . (t ,n f' 1n i - .' 1\' - h Ir", -" 1 'l j, I n 11C . I" I .' ..'L . 11'~n .., _

LI"}"p, ,.,

I' r f LI] I . I., LI' 1 , _'1 ,.'1''''

, r ,. nOI I

"rIT .,. 1 """"(' 1 . "I'~ ( .. 1Z"1 l~'l" Ir'fU/'rl? ' "''''''', )(' 't"

PI' . ., t c\J ."

n, I '), I i 1 '.. r1.

-·

Cl.· I I' I I

... 1 L '., l r (",L" r

I _

I , ." l.'n "","" ' "

(I, elH' T \;

I"" I'" ." n '1 t I "11 "

. I r

'1 \ "J In. llht'n, I", U l{!', ,''''"'''' r,',,' 1.1

,;", , f' I I "11 I'" b...;t 1t: It . 1 '(1.I.' ., ('('b 'OfetltliJ~ .,," 711 d.~,

I \ ,.. I "1'

__.sl

. ' 1r . _

. ..

.. [1]. (J [~]. (1

.., [] I. 1 .', , 1 r 1"rt . . . ..~, I.. '. 1 I .. \ 1

1'1. . 1 I )- '. , ... t .. , · f ", I, 1 1 ., ,1n. l 011-

, , .. ...,. I " M

'- - -

1

,-

.. ., .

,

~y

-52-X

l' r, i r" I '

00

'.' iwt

[ f

i h ( )( + iy ) + i -qr == Ue AO lim e . dl-<

p+O k- V+- I}J

o co

[""

J

Z{n-I) i k (x+ iy)

]+ An k (I- I)hpr J:C' f 1 .t 1 P.1 ,lnrJ }

.'"

cp == Re[

.. ,( .11

1

\.1 I' c:P I 0/

,~ Ir" 1

.,(

cp = u r {[ (AnJr

(cp n ), - ( A.)j (Ibn) i ]cos '"t

n=o .

- [(An)j(Ib.J,+ (A.),"((bn)j ]sin wt),

,lr=U fj[

(A ) (\V ) - (An). ( 'V).!

cos wt'I' L I1r nr I 111n=O

-[

( A ) (rp ) + (A ) ( cp ).J

sin wt]

Jnj n r n r nl

1 I 1 t .

J.1; ..

(CPJr (CPr)i ('/1Jr(~n)i t

. GO

cp = (cp); i (CP). = II'm

/e-kYcos(kx)

dko .. r 0 I )J. +0 k- v + 1)-<

..

co

'/1 :- (0/) t ('(0/ ).= ll~ Ie-kYsin.(k)d

dko 0 r .. I ;U + 0 k - V + ,}-' I

o

""

[{2(n-l) ik(>C+lyJ

1cfJh= (cpfl)r+i.O- Re k (k+v)e Qk

..

[1 iv

1

= Re (x+ l'y)2"-

(x+ iy)zn-'(Zn-t)

co

r{2(".1) ik()(+IY)

11fIn=(I/1,,)r+l'O=Jm k (k+Ll)e dk

..

=- J tn[

I2n - ( . '~n-I

]()( + i

Y

) )( + I Y) (Zn- I )

h ... J1 = J, .!, . .. .

00( .

I-ky ik)(

rp t i If ::: Im e e d ko 0 H..O k ; M,~ - J)+ 1/

"

- L. I r

..

1.

-

VIX2+yZ'

(4).), = e-'Ycos (v X). {- In (1.781. v Vx' + y')

Oll "_ \ (vYx'"+ y"') . ces (nS)

)L n.n!nol

+ e-IIY.

sin (vx).{

_ [CO

(I) Vx2t y2 )"sin (ne) _ arctg ~]n' n ! Y

""I

-1Jy

(m ). = - lTe . ces ( \1x )Ta I )

co n

('I'J =e-IlY.

cos( \JX)'{ [

(VVX2+ yZ)sin (n8)+ arctg~

)r n . n ! Ynol

+ e-'Y sin vx{

- In ( 1.78 I v 11x'+ y'. ) -

f (v VX 2 + y 2 )n . cos ( n 8)]~ n . n !

}

n=1

( )-1JY

t . = -1fe . sin ( VX )o I I

Xe = arctgT

.

v J X2+ y2

co

("' ) -_.1.. [ (n-f)~r

(Y+iX)"+(Y-iX(

)

-Ily.( )

.( 11)'f/ - 2 n 2 Z n -lTe. slqn X . sm \) X J°r 11 C y +x)

nst

-"y(~J.=-Tre 'COS(VX)I

I

(111) i [CD

{n-t)1

(

(Y+iX)"-(Y_ix)h

j

_-vy .T. =

_2 n z 2" + 11e . 51gn

°r \I (y+x)"zt

-lly

(% )i=-- lre . si n (v x) .

~ ,

I , :;. (CPn+ i \jJ n ) . n = . I. ) ) r. .

-i2nlD . -i(2n-l)lD. e ""1 1ve ""1

-r \1 0

--- - ..

\.1--0 - I\.1+0

v

«p + I \{J = UeiL.>t

{A

° [- In(rv V x t iy) - i'Ir

I+ t A n (X t /Y)' n ]n-I

t ,J. 11 . \ 'IL

I \ I

. '

~+ i 'Ir · U{

eiot

Ao. [- In(rv V x t iy )- i'lrJte "0' Ao. [-tn(4 roVx+iy)-ilrOll

+ ~ Fn(t). An (xt i1y),n )

AOI A02 Fn(t) An . I .I

( 1)

{

iut

[ ]

i2wt~cf> = URe e AOI (CPOI)r t l (CPol)i te Aozl(CPoz)rt i (CPoz)i

.+ L Fn (t)'An'(x+~y)2n)

}

)

n"'lr

{

iwt

fi l

i 2u t

[ ],Ir

== URe e A ()Ir ) + i (V ). + e A 2 (0/ ) + i ( 11r ) .T 01 'f 01 r 01 l 0 oZ r T 02 l

+ t F" (t). A" ( (K+ljy)'" )jJ

I(z)

n=1

1

(cl» = -ln(yvr)01 r

(cp ). = -Il01 l

(rp 02) = - l n ( 4 t v r )r(

-

I

Y = -axt b + c. cos(C.Jt)} (3) BI /~__~ 1: I: /

f T' ---I / XI 1 / II' / 1I 1 1I 1 1I

/

~I/ /1

I 11 1I

'/ 1I 1I 1I /

in /u1__ 1 .

V

v- = - U . X . sin (wt)Ist

r = b + C . CQS (lJ t)P si n e + Q cos e

o/- t=-Ur -c05(8). sin(IJt)

15 p

'VA

o/::r 0/. (5)H.t

_ Tr

a 8 - T II

. r

4 .L

( T( ljJ01 \ \fIOi!)r 2.. I

_ 1..._

(~) ==; - e } ( tp ) = TI - e01 r 02 r 2. EI =0 · I .

1 ,. 11~. . ,

01 0,

A -~i A - C'01 - Ta } 02 - lra l }

... r I 1,..1 e I I I I00

[. Zn F (t )A . (sm e + Q . cas e) . sin (2 n 8) nn (b + c. C05(wt))~n

n=1

={

.cos e L ( j-

1T

2 8)]

.{( b+ C'CDS wt).sin (wtJ

)sm EI t Q.cas e a

I -2nt 1 .

Fn (t) = ( b + c. cas (wt)) . sm (wt)

(;()

[lntl

fAn (sin Elt a.cas 8) . :>in(2n e)=co58-Q(I-~e)(sine+Qcose)

".18=0

'irS'" -2

- 1 T . ., ..

Ilr' N _

.I

An

si n (2 m 8) A tJ

'. NZn+1

[(s,-n 8+ a-c05) sin (2n8): . Bmn - .sin (2m 8)m=1

N

C05 e - f- ( 1- ~ e) ( sin 8 + a . cos 6) = [ C m sin ( 2 m 8)111= I

11

4- fz

Zn+1

Bmn =-;r (sin6+a-cos6) 5in(2n8)- sin(2m8)- d8

o !!.

C m · ~ r [C05 e - ~ ( 1- ~ 6 J(sin 6 + Q. cos 6 ) 1 si n (2 m 6) d 6o

N N N

[[ An Bmn - [ Cmm~1 n=1 m=1

An

z nt 1 n

fB =- ( J-.. 1 'm.n' (az+1 ) ~ ) - \ ( '2nit ) .mn 1i 2 Zn - 5 L k

k-O

[

ITI-k

[ f]

. (-1) . C05[(2n+1-2k)arctg~Jtsin (2n+1-2k)arctgaJ .

[

(2ntl-2k)

1].

((4n-2kt02- 4mZ)((2k-I)~- 4m2)

C8

[(

mTI t 4 m]

m"'1fz(-1)

-a)(4m2_1)Z

"'., I" .'1'

( .

cp = u{ ;~ ln(r\)r).sih(

z.. (sin e + a. C05 8 )

1) p~~ = pUW{ : ln(rl1rp).cos{wt)+: (~)ln(4rllrp)'cos(2wt)Ranct

- B 51'n(wt) - B (~

) . sin ( 2 wt)

+ AI cos (2 e)( sin e + a. GOS8/ [-3 c, sin(wt)t ( b+ c .cos Qt). COS(tJt)]

+ Al. cos (4 eH sin 8 + a . cos e)~

[- 5 C . sin{wtH ( b+ c. ws wt). cos(wt)]

+ A3 cos (6 e)( sin e +0 . cos e)' [- 7c. sin 2(wf)+ ( bt c. cos wt). cos(lJt)]. S

[z.

]+ AI, cos(8B)(smGta..cos6) -9c's,n(w-l)+(b+c.coswt).c.os(wt)

+,pp,

]

2) _ 1.. pf( H-)

Z

+ (1- ~)Z]

=2 a r r a eRand

Z

[

z . 2t

z . .) 2 Z . 2 t)

]_ _1.. U B sm (w) z + ~ E. sm (L.lt)'sln(2wt +.!..!. ~ sin (u- Z P [? (b1"c.coswt) lTZ(b) (btc.c:os wt)Z 4 1f~( b) (btc.cos &.>t)2

.~I. ...

. ~

P' P~ i -~ p ( '1 ~)' · l' :: - -}p \ ( ~ ~ )' + ( T ~: q

[

. 2_ 4- A ! 5'" (ut)

1 11" (btc.tOsl.Jt,t .!.. ~ (.E..) sin{wt).sin(Zwt)

],c05(2 e).(sin eta cose)

4Z 1i b ( b tc . c.os wt)

" 1.cos(~ e). (sin et Q C05e)'11 J. cos( 6 e).( sin S+ 0 cos e)8

1

1011 'Cos(Be),(sIne+Q cose)

.

-8 Az [

- 12 A3 [

-16 A" [

~ 1

k' . - = P/ ~ dsinstatlonar a tRand

I2-

K . .. '" - ~ ( \J

q ~Q"d 0

11"

Z

- ..!..P / (\l cf>/ ds = - p I (\l tp/ b :c . cos (w t) Z cl e2 !land (sin e+a'Cose)Rand 0

K :inst

K;n"~ f' U0 tr ~ ln [ r~ (bH' COS 01)] COS (wt). (btC. cos "I)

+ z: (~

) . In [4 rv (b+ c . cos wt)] cOs (2 wt). ( b+ c. cos wf)

- 2 B si n (w t) . ( b + c . cos wt )iLI

- 2 B(~ ). Sin ( 2 wt) ( b +c . cos wt)]. fde

(sin 9 ta. cose)z

I

i

I

I.,

I

o

- [2 B

. cos ( wt) ( b+ c . cos wt )11

IrZ

+2B. (E.. )'COS(2wt)(b+COCoswt)}

' ftn.

+I

(a+~-~ )1+ a2. Q 2

d8

o

=Q

ln [V1+QZ", sin (arctg ~ )}

+I

')' ln [v 1 + Ol . cas (arctg ~ )]

'Irl

+ A, ( b + c . cos wt) [- 3 c. sin' ..,1+( b+ c. cos I).

I I Kinstr

.r I ~"" tierI I K.

t 2 3 4.

5 InstI r

. . " · r I ~ "" tle r,

l'

W 4

.. .. I"

!

B[

( I CZ! c4 ) BK. = p Uw - 5 c.ln(bp)+c+-c {- ) +-c{-) --.5.cInst TI 1 8 b 6 b ".. ~

-~ bc'(3AzS3+5A3S4+7A4SS)1

+[ ~St(2b'ln (bpl)+c(~)ln(4rvb)+: c.(~)+: c(~/)

-: S;t (2b+ c.( ~)) + b2 (AZS3T A3S4 +A-tSs)I z

1

- 2C (AZS3 + 2A3S~ + 3 A4SS) . C05 (wt)

-[B5i(2btC(~})J'5in (wt)

+ [ ~ 51 (c.ln(brv) + 2 c.ln (4 brv)+c+ +c (~)\ ~ c( ~)4)

-~ Sz(3c)+i-bc(7AzS3+9A354+11A"Ss)]'COS (2wt)

- [B51 (3d]' sin (2Qt)

+[~S (c.(E-) In (4brv) +-~c(~)+.!.. c (~)3 )1(" 1 b ~ b 12 b

- ~ 5l (C .( ~) ) + ~ cZ ( 3 A

Z 53 + 4 A354 + 5 A455 )J

cos(3(.)t)

-[ BS, (c.(~))] sin (31)]

K .=_ U2B [[l ( ~_2.. ( C3 i... (~ )

4 ~ ( .E.E )G

)9LtQ5\

f '�iO 2 4 b) + 4 b + t28 b

+[

J ( 1-(~ ) t ~ ( ~)3

+ ~ ( f. )5

) - ~ ( ~ ) ] COS ( wt )1ia 4 b 32 b 64 b TI" b

[

1 ( tj C 6 2S

J+ va -2+32 ("b»)+ 1f COS ( 2 wt)

+[

J... (_J... ( ~ ) t ( c5 S c

J ]lia 4 b - 64 b)) + 'Ir ( b) cos ( 3 wt)

AiJI+2AzJ2+3A3J3+4A~J4.

J1 ) J Z 1 J 3 und J 41 1"

ii

I

fZ TI Z

J1 = _COS ( 2 e). (sin e +a . cos e) d e = I; (a -1)

o ~

1

2 4J

2.::: COS (4 8). (5 in e + O. cos e) cl 8

o

1 3 ii" Z 4 )= - 3" (Q + Q ) + 32( I - 6 Q + 0

iI

J3 =J

icos ( 68). ( sin 8 t Q. cos 8)' cl e

o3 5 11 Z 4 6 )= -(0-0)+-(-1+150-150+010 ~8 J

J

~ 8J4 ::: COS ( 8 e) . ( sin e + Q . cos e) cl e

K .'t~QSI

.

., . I

o

1 ( 3 5 23 (7 TL 2 4 6 8

= "5 Q + a ) - 105 a + 0 ) + 512( 1- 28 a + 70a - 280 + 0 )

Q H ß0,57735 1,73205 0,5

1, 0000 1,0000 0,5

1,73205 0,57735 0,5

)(

y

.I.

~/

..

H 1,73205 t / 0000 0, 5 7735

AOIB .

/1-=- I J/Il

Aozc.H B .-,-1 11 /J

b ii'

AI -0,2912 0 - 0,0390

Az - 0, 18 77 - 0,0184 0,0084

A3 - 0, t0 g 1 0 - 0,0011

Alt - 0,0605 - 0,0021 0, 0° 03

St 1,73205 1,0000 0, 5 7 7 3 5

S2 - 0,1604 0,2146 0, 2 ~38

S3 - 0/ 19 2 ~-0,3333 - 0, 57 7 3

S4 0,0769 0 - 0, 69 2 8

S5 - 0,064 t - 0, 1429 - '.7320L 0, 031 b 0,0 ° b ~-0,0046----J, - 0,5235 0 1,570 B

Jz - 0,3437 - ',0592 - 3,0946

J3 ° 2119 0- 5, 724 7,

J4 - 0, 13 68 - 0, 3b 74 - f1,0877

5 0,2449 0,04 \ 6 - 0,1076I.uL.

Jn

~

,

r

22 B SQ 2.K=pUwB ~ln(2Qr\l)-~ +~ L}cOs(wt)-pUwBSrn(wt)

nV.B

~

2

22

[

1 1 SQ a

]

2.K=pUwB lrlnF+Tiln(ay)-

~ +"4L COS(LJt)-pUwBsrn(wt)

, . I- Usin (wt) - Uw cos(ut)

UNm ,

2M 2

[

1 t SQ a]

m = -p B - In F + - ln(a y) - -L.. t - L'Ir 1t' Tl 4 }

2N =pwB

. C T (" . .I ., J I . '.

( I i ".

'"a .. . I I I .

I .. ,I 'L, . ... .',

1.10

'r" . f- I

[

2

1

8 8 I 52 Q Q- - . ln F - - - Inp.r)- - + - L1[2 . 11' 'Ir 1t" 4-

B'\J=2F

- . ~- .

I<T .

p.g'C'B

1, 73 205 0,57735 "

- 0,0I 0, 04

H (P-g~C-B ) F = 0, 041, 73 205 - 0,0 78 7 cos (wt) - 0,08 5in (wt)

0,57735 - 0, 065 j cos(wt) - 0,08 sin (wt)

H ( f-9"~" B ) F .. 0,01t, 73 205 - 0,0285 cos (wt) - 0/ 02 sin (wt)

0,57735 - 0/ 0 2 51 C05 (wt) - 0,02 sin (wt)

k' -qL.tQSI= ')( + ~ cos (wt)

f" 9 . C - Bot ,+ K3 cos (2wt)

}

+ x 5 cos (3 wt)

~ 'I

= (C\C + de ) t (dt +- K ) Cos (w t) + K si n (w t )r-'J"c"B 0 0 1 I 2

»

+ (~3+ )(3) cos(2wt) + x4sin(2wf)

+ ()fs +)t;) cos (3 wt) -+ d{ (; sln (3wt)

u ~ dt

I

. . . . "

I .

I..

... I )~] .. . .

f{-InstB

=)( + x cos (wt) + x. sin (wt)P-'J"c- 0 1 Z

+ ')( 3 cos (2wt) + )(4 sin (2wt)

+ KSCOS(3wt)+x{; sin (3wt)

F H (lto ()CII ~2 I ~3 I X" MS ".)t,

I, 73 205 -0,0038 - 0,0290 -0,0211 .0,0105 -0,0103 - 0,0006 - 0,00120,01 1,0000 - 0,0020 - 0,0271 - ° 0203 -00056 - 0,0060 - 0,0001 - 0,0003

0,57735 - 0,0010 - ° 02~5 - 0,0201 -0,0029 - 0,0034 - 0,0000 - 0,00011,73205 - 0,0092 - 0,0 786 -0, 08~7 -0,0239 - 0,0415 - 0,0006 - 0,001.7

0,04 1,0000 -0, 0047 - 0,0724 - 0,0815 -~ 0120 - 0,0240 - 0,0000 - 0,00150.57735 -0,0022 -0,0650 -0,0805 - 0, 0058 - 0,0138 -0, 0000 - 0,0005

F H ~o ~t ~z M3 ~4 I ()Cs I ~6

"73205 -0,0075 -0,0304- -0,0247 - 0,0207 - 0,0207 -0,002.7 -0,0047

0, 01 1,0000 -0,0041 - 0,0275 - 0, 0215 -0,0112 - 0,0120 - 0,0008 - 0,001S0,57735 - 0,0020 -O,02~b - 0,0205 -0,0058 - o,oob9 -0,0002 - 0,0005I, 73205 -0,0178 -0,0777 - 0,0991 - 0,0~b2 -0,0831 - 0,0024 - 0,° 1S1

0,04 1,0000 - o,oo!J4 -O,071S -0,08'3 -0,0239 -o,o~80 - 0,0003 - 0,00630,57735 - 0,0044 - 0,0647 - 0,0821 -0,0115 -0,0277 - 0,0000 - 0,0021

I

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,I

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( p; c ) :=< V [ ~ ln (b rv) - ~ ln ( sin e + a cos e)Rand

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4+ Abo cos ( 4 e) 0 (sin e + Q' cos a )2

(. I;

+ A3 b . cos 6 a) . ( Sin e + a. cos e)

+ A". b . cos ( 8 e) 0( si n e t Q 0cos e) 8] COS( wt)

+ v. B0sion( wt)

-- - - - - - -- --- -

P ins!pqc

t3_

w" TU

e 0 il Ti:; 1~1 L;"' 3d

1/73205

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( p; c ) :; v [ ~ ln (b t v)- : ln ( sin e + a cos e)Rand

2

+ A1 b. COS ( 2 e) . (51'n e + a . C05e )

4+ A z b

. COS( 4 e). ( sin e + a. COSa )

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+ A~. b . cos ( Be). ( 5 in eta. cos e) 8] cos ( wt)

t 1.1.6- sin (wt)

P ins!pgc

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p' q' c' B

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-'-'-- -...

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( p; c) .:= y [ ~. ln (b rv) - ~ ln (sin e + a cos e)Rand

2

+ Ai b 0 C05 ( 2 6) . (slOn 6 + a . C05 e)

4+ A2 b

0 COS ( 4 e). (sin e + Q' COSa )

(0

(;

+ A3 b . cos 6 a) . ( Sin e + a 0 cos e)

+ A~. b 0cos ( 8 e) . ( 5 in e t Q 0cos e )

8

J

cos ( wt)

tvoß,sIOn(wt)

..

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

P ins!I

pgc

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t 3 - 7. W=yll -'Tr._ _ 4 _6=0

il ..!.!-..!! .J!....16'4'3 Z.. d

.~ ~. . .

K q..as;!''l.c.8

IK;nst

,og' c, 8

K q\.lQsi

p' q' c' B

. .~~ " t \-

r, .

3 7wt::: -"Ir -'Ir2 4-

. ...... .. .. ..

1',CI ..J _.lJ...-.

c Ä4

..'---

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ß=o.5

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. 00.i 2 ut

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co .i 3wt

[ li k (Je +ly)

+ e A lim e . cl k03 fJ.-+O k- 911 + If'N-I 0

1

+ Afn(t} _ i9vGn3(t)[ n3 ( ()(+ iy)2n (x+ iy)zn-I(Zn-1) )]

11"'1

fn(t) Gns (0

( C. ]2n+1

fnet)= 1+b"sm(wt)11

2i I fn(t)Gns(t);: fn(t)- sc..> fn

(t) - SZWZ) 5 = 1,2 J 3

AOl J An! , A02 J An2 ...U

c

b

w

1J

.

tJ (CP ).-s,'n(swt). (cp )

]. I ns I ns r(0

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t (Ans)' , [ -cos (swt)('o/ ).- sin (sut).("o/ )]ns l ns r

(2)

\.J

N-1

epns + i 1j1ns =

(4J+i'lr)

(CP -. 9 cp ) ;:: 0 für y = 0tI YI .

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6 x N ( A ns)r ( A ns ) l ..

tp = tlJist (4)

f

.

, .

3 N-'

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f(A )

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+(A ).(

sin (swt)(€/J ). -c.os(sut)(CP )])ns I ns I hS r

1 )z

-Zp('Vq>

3 rHa cP { [ [ (

i $wt 0 rbns

]a x :: URe' e. Ans '0 X )5= 1 n= 1

() cp= URe [ [

3

[NOI

( eiswt . A . a cbns ) ja y ns 0 y

5=1 li-IOos _ _ x _ 52.,) ,Iro X

- X 2 t- Y 2 't' oS

CJcp oS _ _ y _52\) rha y - x2 + y 2 't' OS

()ePns:: _ 2n 'COS [(2ni-1)'fl.fn(t) + 52\1' Sin(Zhlf').6ns(t)Ox rzn+1 r2n

a cb 2n .sin[(2nt1)'fJ.fn(t) 52V' cos(Zn

· tJI. = UXc.os wtISt

r,.. b+c'sin (wt) (5)P sin e + a. cos e

b

C a

x

..Q

(J

Y. .

4

~ N-I

~ k {( A"s)r [C05 ( 5 wl) -(o/"s) r - sin (s wt) ( 'Ir.,) i ]+CA"s)

d- cos (5wl) - ('lrJ

i- ,in ( 5wl)-(tj1"s)

r J)

= x. cos ( wt ) (4 )

.I.

b x N (A) (A ).ns r ns lbxN

'. wt.O iL 311 "iT5 iI 3» 4- 4-- -4- 2

~ ~ -2

. . .1 .""';11 . i. )

I

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t 0 ii 3i1 Sii 3iT(..):= - - 11 -'4 4 I 4 ""2

.- ~ ~

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Schematische Dar.stellu?'Jd

des Keils in ver.schiede77e~

Positionen

- - ----

"t 4. I I.

K .

B= ;}f + dC. cos (wt) + i)C sm (wt)p.g.c. 0 2

+ de3 cos (2wt) + ~4 sin (2wt)

+ dCSCOS(3wt) + ~,sin (3wt)

~ --- __ ;---;-- - - J.~

?Jl'" := : (F2 + F3 + Fs + F7)?(

t = ~(

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dC2. = ~

(- F, t F 2 t F3 - F4 + F5 - 2 F, + F7 )

+ 5, \. 5' ( Fz + F3 - F5 - F7 )

~ =.!..'(2F -F -F t2F -F -F)34 123 4 57

I~-j

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(I)

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(1)

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I

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(lI

[

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-911Y.

+ ge (-(A0 3 ),'cos(9I1X-3wt)+(A 3) 'sl'n(9vx-3LJt)'

l 0 r ,

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+ 81 ( (A ) + (A 3 ). )03r

0 I

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J01 r I 02 r 01 l

+

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). ) . cos ( 8 V)( - 2 (.J t )01 r 0 r 01 I 0 I

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]03 r 01 I

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+ ( (A ) . (A ). - CA) (A ). ) , 5 in (5 v X - w t )]

J

( 6)02 r 03 I 03 r oz I

2

{

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-(( A ) . (A ). - (A ). (A >.) cos (3 v x - wt > ]01 r 02 1 02 r 01 I

t36 [( (A ) ,(A 3) +(A ),'(A

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- ((A ). (A ). - (A3

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+72 [( (A ). (A ) + CA ).'(A ). )5in (Svx-wt)02 r 03 r 02 I 03 I

( (A ) . (A 3 ). - ( A ). (A ) . ) C05 (5 vx -wt)l}02 r 0 I 03 r 02 I I .I~

(7 )

((11 I (1)

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I

I -..1 .. t ..

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ror 2 3I .I jI .

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, I . ~ . · 1"

I (1., I

I .

; · TI' . I . , r 1'" l r n A _(1)

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, t, (... . .. t I' 2w ' (.,.1 {7,

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01 r °3 1 03 r 03 1

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(9 )

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><

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it (AoI)

i. CAol)

i ] JA: =36f-[CAol)r'(A03\ - (Ao3)r' (Aol}iJ}

B: = 36 [f( Aal) r

. (A 03) r t (A 01 \ . (A 03 \ ]1

A. .. 72{- [ (A ). ( A ). - (A ) . (A ) .

113 02 r 03 I 03 r 02 I

S$ = 72 {[ CA ) . (A o,.) + (A ),'CA 3 ). ]J3 02 r ,. r oz( 0 I .

,j

1

I

r

. "(f - 9 lf) = A cos (rx + st)! B sin (rx + st)tt y0y"

[ '".

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sin (rx) . [A. sin (st) + s.cos (st)]

z5 - gr + 0

> un<

lf.

CI. blf= lf + f

.. · ryi'Q = -

A COS (st)~ B sin (st) . Te-

COs (rx)1r(S -gr)

[

GO-ky

-f

A.COS(st)-S"Sin(st)l

.D e ~?S{.kx) dkk--tl;U

oq

· ..

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I

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co

(w

b _ g tD b]

_ lsin (st) + ß*cos (st)

(1

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Icos [x ( u -t r) J cl u

y=o U Llx f 0 r -r

co co

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r -rOJ 00

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r -r

= sin (rx) .f!~- Si(rx):: ~ + Si (rx)]

I

b b

I

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A Si" (st) + B cos(st} 2f e cos (kx)

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o

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°

cp(2)

(Z)U

z z

(w \J 11 '" .. . ii -3 "ycl> =- g [Alcos(wt)-Btstn(wtJ].0),e cos(3vx)

[ . .. ] il -811Y+ Azcos(Zwt)- Bzsin (2wt) '(-4)' e cos (B vx)

[· · '

J

'Ir - 5 vy+ A3cos(wt)- 8zsm(wt) '(-4)' e cos(5vx)

ja>

-ky

j"" -ky

+ [A'*sin(wt)tB'*cOS(wt) ).[!.. e cos(kJ()dkt.!. e cos(kx)(1 1 4 k+ 3 v z k- 3vo

~ ~ 0

+ [A. .

(2 t) B- (2 t)ll.[1.je-klCOSCkX)dk .L/

e'klcoslkx)d k2

sm ("J t zcos w1

12 k+ 8 v t 4 k - 8 11

jom

-ky0

J

QI

-ky

+[A.sin(wt)+B*COs(wt)

].[.!. e CosCkx)dk+!. e c.os(k)()dk

3 3 b k t 511 4 k- 5 v

° -

I

o

( 14-)

2 Z

{

3(2)W U v '[.. ,1f . 11 - 111

1Jr = -9

[AI cos Cwt) - BI sm (Wt)]. (-2)' e sin (3 vx)

;.

[

.., ..

]

ff -8vy

+ A2cos (2wt)- 8zsin(2wt) '(-4)' e sin (Bvx)

[

.. .]

TL-5vy

+ A3 cos(wt)- B3sinCwt) .~.e sin(5vx)(-4)

00 Co

T[A" sin (wt)+ s.COS(wtJ

].[!. j e-k'sl'O CklC)dk+ .!.../ e

-kysi" (kx) dk

]I I 4- k+3v 2 11'-3\.1

o 0

CI> GO

+[

A. sin (2 wt)+ B. cos (2wt)J

.[

.!.

Je

-kysi" (k)() dkt l..

J

e-ky

si" (kx) dkJ

2 2 12 k + 8 v 4 11'- 8 vo 0

[

. .] [ I

'"' -1

rvxZ+ yZ

j"" -ky ikx

-ry +- irx

\J [

00

[( ') 1" x

)

e e dk=e -ln [yr/xZtyZ _ r ~-\)( +iardg-k-r 11 n. y"=\

o

=e-'Y';" {-t"[r"/ x' + y'] -

t.

(rv'x';y: (-e-;..mq ~

t iQrctq~]

r V XZ t yZ.

j

'"' k '"

x-ky I J(

[il"lorctq y -r1 irx

e edk=- e

J"+ iTre sign (X) ek-r n! (rv'yz+xZ'

1"1-1o

,

...

Je -

kye

ik)(

d kk + r

o

n/xZ+('

00

J

k . k

{

- ~1

(13) \.I vxz + yz'UZ w rrlv

(Z) 9cp 11Vx2 + yZ

... ~

. -ry. irJ(l1Te slgn()().e

I"" -ky

I"" -ky -hy

~ e cos(kx)dkt-.L e cos(kJ()dk~_!e sign(x).sin(3vx)4 k+3r 2 k-3r 2

o 0

00 ~

/-ky

I-ky - -8vy

.!.. e cos(kx)dk+..!.. e COS(kX)dk~_~e .sign(x)-sin(Bvx)12 k t 8 v 4 k - 8 LI 4

o 0

J

"" -ky

J

"" -ky -SIIY

J e cosCkx)dk+.!.. e COS(kx)dk~_ile .sign(x).sin(5vx)6 k+5v 4 k-5v 4

o 0

Cl)I ~

(2) Z Z

(

_ 311 Ycp =-

UWV'fr[-lcos(wt)+ S:sin(wt»

),e COS (3vx)

2. q t

[

. .]

f-811y

+ - Azcos(2wt) + Bzsin (2wt) . '2 e cos (8 vx)

[

· ·J

1 -511Y

+ -A3cosewt)+B3sin(wt) 'Ze COS(SVX)

[· · ] -

3 .,y .t -A, 5in (wt)- 8, cos(wt) . e . sign (x), sm (3vx)

(

· ·1

t -8vy, . ( )T -Azsin(2wt)-BzcoS(2wt).z.e ,slgn(x),sln 8vx

[

" ·J

f -5 vy .]

+ -A3~in(wt)-B3cos(wt) '2.e .sl'gn(>CZ)

'lF>

I.I

- ~-----. .

;:I

.

t+.

(Z) . (Z)I

'Y +lo/ .

".1

, I(1) k

(l)

lJI. = UXCOS wt- tV (16)I&t

(1)

cp 1 I . -(2)

. j. \V

k k~

kZ

P = f ~ - ~ p ('iJ ~ ) + T f (v

~ ,

H = I} 73 205 , ~::o q5 wt =-L 'if .

4 }-cB/ 2 = 0,05} 0, 15

1 ,. F= 0,4 } 0, 6 0,8.

-,-

.

... . .L l

~Pmax. pgc

H= 1,0 1,73205 -c

B/2 F = 0,2 } 0,3 } 0,4, 0,5 , q6,0,7, 0,f.

. .

. " "'., -,. .

. .I

Pmax~/Z UZ . - ." -

C J. I. It..., . . '} f)

~ ) Pmax

P/2 Uz-; v

. ,.

h

..!to ,

_ oe [ J. L T)

- -]

rgilt für den Bereich OC zwischen 25° und 50°. Wagner

hat allerdings für die konstante Geschwindigkeit des

Keils ~egen ruhiges Wasser gerechnet, während unsere

~eschwindigkeit als harmonische Funktion U cos (wt )

ge~eben und dafür der maximale Druck in dem Zeitpunkt

wt = -t-n' errechnet is t.

10 (;.(

8!'

~Oll

.;t- ~() $!()' C$ ,..:-

"H . ..

\j Il.. ::t Oi

+h=:

0.'

~~~\8

~~l;o;

IIÖ I"I/N

."S ..t-::.tI\:;:: .

~to.'.()c:) k.. .......i

to.-

'" 'il ---- ca. ~~'".::t '-~i .....1j 118 Oll-E

........äGI0-

~~l;:.....

§ .!:- 3 +'-ca. ......1:

",~\;:.~c::

.~~~~'""'" Inl..t-~'--

)c ;..

~co~SI c:\j

t'II

;;t-;;t-~

sr~-c::i

~It . ..v

~~"'-I.,..

~u'a~t

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~~~t::::i-- ~.....

§ "~~~~~......t:"- ::s ..~.......'3...

~~cu N~5i~5-

2 ~~~li:: "'I~

~~"-c5--.

eh '-~§~..... I

ca.

.'5

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\6::

81$

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:3

§.........

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e.a

"

..

~:)~......~13 iä~:"'- t .~8ii ~1

~""'= ~~~:;:.

~t ~~\0...'C ~C)(

~;:--.

.....

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.......~... I11

-!I

......

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.~~ §:-.;. ~

"b i l\c:::: ...

t.&ca:. !I! t:

~ &.ES ~

'-t; cu

"ij ~-~ 'Qj

§ 1:="" '1

~ I-g

:

~ IKJ \

-----......-.-,

l-._--

II\lt'I

~~:()

~~IOj.

~.C1I

ii

~.Q~\Q

"ts~§

1:) a...... t:....Ctl :::s......

''!o.,

&:~~",1..-~::.

~CoJ

~Qc.oC\J....

~11

~§

~~.

~"

q;.....

"8

""10 ~

Q- ~- ""--

"N 11

(" Ic... ~

~ co ~~

cu

'S~~C) "tlt\f

~~c§

.i~~~I""

~'b

§..Q

~~~l~"'". .iJ~....

1s~-~.....-t;qa:::s

~l~~I'P\/"

"b

2':::s (0

~~C\J..

""8 ~~I'":...0 I....ii~Kj

q10

qos

- aas'I

- qlO

Kfog"CoB

Kra ft koe ff izien t

F = q 04ß=o,sH= ',73205

-----------

o0,1 0,4

2cB

L.,,~Q.rer Fo.Ll- - - -- -- -----------

0,10

0,05

- 005I

- 0,10

Kp"g"c"8

Bild 4.8 b

Kra ft koeffizient

F zr 0,04

~-----

o

ß=o,sH :: 'i

()

x.

2cB

.

q25

q15

0,10

Q05

- 005I

-010I

- q15

Bild 4.9 aKraft koeffizien t

F:: 0,2 {3=D,5H= f r~K

1'°9'c. B

oq 1

li""e~II".r Fall----

K0,25 p.g.C-S

q05

- qos

- 0,10

- ql5

- Q20

Bild 4. 'I b

Kraft koeffizient

F=O,2

0,15

qtO

o

li7learer F41f ft = 0,SH = 1,0

0,1 0,2

Xz.

0,3--

2c--B

o,s

0,3

- 0 tI

- qz

- 0,3

Bild 4.10 a

Kp-g"c-8

Kra ft koeffizient

F=D,4

ß = 0,5 -H= t 73205I

I i "ea re r F4 11

oq3 2c 0,+8-

I ; ." e a re r FCi IJ ----------

-qt

-0,2

-03I

Bild 4. 10 b

qs. K

p"'J"c.8

Kraftkoeffizient

F-O,4

0,3-'

0,2

ql

oqf 0,2

li?1earer F.1l

ß=D,SH :::1,0

2c- -----.B

- 0,5

- 1,0

Bild 4.11 Q

Kra (t koeff"zientFe 0,6

0,5

o

-- -Li1tta rer FaU,

0,25

ß=D,SH = 1,73 205

0,3 2 c 0,+---.

B

1,0

Bild 4.11 b

Kra ft koeffizien t

I<

p.g.c.8

qs )tj

o

-qs

-1,0

- -- ------------

ql

- -- ---

ß = 0,5H= 1,0

Jt".

2c8

-0,5

- 1,0

Bild 4.12 a

Kraftkoeffizient

F c 0,8

Kp.g.c.8

13= 0,5H= t 73205,

I,D

qs

Zc--8

oo,Z 0,3

-0,5

- 1,0

0,10

0,5

o

Bild 4.12 b

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F = 0,8

1

"?

i'~ ,,' ""

"

," " ,".--....

..°1LN'o.

,.-

",

, ,.

.....

~s::s--.-

",'\,,' \

"\

üe,b /\ 'Ol~',' \ ',"

. \

,.'\

\

, ,. \

"

\

,,'. \\

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,"-\\,

\ \. \\ '. ,.,~ \\ \ ~

. .0<6

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"~

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~: \.,. \ A, ..."

,

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,,,' . ).\ ,~ ".

". \ \.',., ... "'\\ ,... "'"

\

, :'\ //

,. \,),.'

. . \\, \,\ \.,,"., "

,./

l.t)

,

C),I

eS"I

ItI

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L'

810

.I

I

: ,, ,r:;i

-hf\/'

.1.__

-

.....

~ ao 1::- '-0 Lr, .:::f- ("rj~ ci ci' 0'"

n ßQl:t:

-

2

15

fO

20

15

{O

B ; I J 1f../4Q

N.xim_!e, "1'TOtLfn. O,,,,cK

ü.be, A.,p/it",le"ve,hältnis8/2

PmllX-fU'

f3:= 0, 5

H:= 1.73205

0.2c

B/20,4-

ß = 0,5

H ==1,0

5 cB/2a

..

~.. pMaxima.ler hydrodyn. DrucK +uz-

ü.ber" WinKe I J.s k~;ls"

G(

5

(1 + 4-~~ ) von Wagner

2

15

f

o20 30

_ :-)7

'I. lusammenfassung, und AusblIck.

Ila .'rohlpm oer rauC'hs('h"'ingungen mit endllchpr .\mpI1tuoe

... i np s I~'! I n cl"

~;{ ,1 s t 111) c h n ic' h t \ rc} Ist an d 1 2 g

p1 (1s t. I" eH'

,q Ps gf'lungf'nt fur klf>inp f'rpquenzen und\mplltllc1l?tI his

C't") X halhp!, Hr4'1t.' ausrpl('h~ndp 10...lln~pn zu flndl-'nt 1-"111'

sntd:., lall.. slnrl Hechnungen durchgefllhrt uflcf ,fl~ Ergf'bn1SSf'

d fi nz. e s teilt \\ 0 r d p n. TIi (' s.' Er J;!;e h ni s s f' Z f' i I!:e n. da 1.1 d i 1:> n i (, h t -linpnff'Ti j l'l'l\lt, /"1 t t. n. "e Ich e bel g t "1.',,, r f' n F r (~q tH' n -

Zt'n und Amplituden auftreten, sind \,'a~l,",,~L":~,lii h ".tdul'ch

Vpnp""d.!:t. '!d' .i',. jloIllkontur zu nahe an die tH'i XC",)

und y CI IIPf!pnd(,t1 -..i';E:11:i11"itKfpn d",r h..nllt~t.'n !'['fPntlaJ.>

ht>lanh ,mmt. j)f>r DnlCk\'prlaut' Ulill d i" Ll'fullung dpr j/andt".,~

tjllJ~;ln!.' mr! t" !UI !.I'''V 1" 11lf.Pl'..~ '1,uhp r~"f (,!),'rsti>n j'osl-

ti on.' dp... l'n)t1J... "lng,'hpndpr unf':.rs!leht ,,'prd..n und ;,.''\\,11'

f1!r Pl!ip e.r'c;/\pt.p lahl \'on I'ttnkt..n an .lpI' 1'l'ldl!hl)l~f1lr.nar811s

h;\ 11n t I' fj dan n""

h I' S eh,' I nil (' h Sc h j ti s s , .!!f>zog (' n \,'" f' d .' n f 11r .' i ri I'

.\h,lnl!t'runQ dp1' '!pthndf't '\\'p)('h(> dann \,',thl's('h.'!nl i(~h 7U aus-

I ,. 1eh (>n d (' n Erg'. b n IsS (' n a 11I' h f 11r l!.roß e re Fr e q n e n Z f' nun d .\m _

~,; i i!:.:' l f IJh r-e n "

11r d .' n .

/llt' \!e11l1l(1.' 1st nIcht nur 1"111' dip DI'eiecksproflJ., hrauch-

11a f. S 1 (' j... t p h f' n :.,0 gut f 11r h p I j .>h i I!p P 1"1)r 1 ] p h rau I~h h a f, 0 1 p

dann ifl rlas H(>('hpnprol!ramm durch di:-.kr':.1.> !"n,hf.' Plngegebt'tl

werden können. Rechnungen sInd auch fur andere Profile aus-

L",'fÜhrt worden. Die Ergebnisse führen ,jedoch z. 7. /i. i"';I,""t' . . I. . 1. '..::~~t t '.

,\ 1 .. d .' r' e.i.ih p v (>r /. i (' h t e t'"

0 nl e n .

[61 G. Vossers Resistance, Propulsion and Steering

of Ships.Behaviour of Ships in Waves.

(The Technica1 Pub1ishing Company H. Stam

N.V., Haar1em, Nether1ands, 1962)

- -

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{

0170180190210211023 3

02027028

02903003103203303403503637

038039040041042043044045046047048 .049

l050051053054

-STARTUEBERSETlERSB516 KIM

'BEGIN''COMMENT' ZEITABLEITUNG DER DRUCKVERTEILUNG .,

'REAL' NUE, BA, BB, BC, BD, BE, BF, BG, GA, GB, GC, GD, GE,AM. UE,

GF, C, S, A, B, BH, GG, RHO, RHO2, RHOJ, PHI . ,

'INTEGER' M, N, T, TT, K, I, J, L, Q, E, EI, Tl, R,NN,Z,MM,D .,'PROCEDURE' TRANSP ., 'CODE' ., 005

'PROCEDURE' ADIVB., 'CODE' .~ 006'PROCEDURE' MAMUBT., 'CÖDE' ., 007

NIN..READ (N) ., 008READ (UE) .. 010EI = 0 .. Oll'BEGIN' 013

'ARRAY' X,Y,XX,AT(1..N),Bl,B2,B3,B4,Gl,G2,G3(1..N,I..N),VX,VY,POT,VTX,VTY,VY2,VTTY(1..7*N,I..6*N) ,

VXX, VYY, PT, VTXR, VTYR, VY2R, VTTYR, Cl ( 1..7*N),C2(l..7*N),GES( 1..1 . 1..6*N) ,

S1(1..3) .,

'PROCEDURE' UPKIM .,'BEGIN'

J= 0 ..SJ.. J=J+l..

LABLl..I = 1 ..SI.. XII) = XX(I) .'

B2(I,J) = -3.141592652*EXP(-NUE*Y(I»*SIN(NUE*X(I» .,LA3..G2(I,J) = -3.141592652*EXP(-NUE*Y(I»*COS(NUE*X(I» .,

BA = BB = NUE*SQRT(X(I)*X(I)+Y(I)*Y(I» .,'1Ft Y(I) tLESS' 0.1$-3 'THENt BC =1.570S*SIGN(X(I» 'ELSE'BC = ARCSI.HNUE*X( I )/BA) .,BD = BE = 0 .,'1Ft BA 'L~SSt 6 'THEN' 'BEGINtMM = 1 .,

LA4..BD = BD+BB*COS(MM*BC) ..BE = BE-BB*SIN(MM*BC) ..BB = BB*BA*MM/«MM+U*(MM+l» .,MM = MM+1 ..

tlF'BB'GREATER' 0.1$-5*BA 'THEN"GOTO'LA4..BD = (-BD-LN(1.781*BA»*EXP(-NUE*Y(I» .'BE = (-BE+BC)*EXP(-NUE*Y(I» .,GA = BD*COS(NUE*X(I) )-BE*SIN(NUE*X(I» .'GB = BE*COS(NUE*X(I»+BD*SIN(NUE*X(I»'END' BA LESS'ELSE' 'BEGIN''FOR' MM = 1 'STEP' 1 'UNTJL' 5 'DO' 'BEGIN'BD = BD-COS(MM*BC)/BB ..BE = BE-SIN(MM*BC)/BB ..BB = BA*BB /Mt-1

'END' MM .,GA = BD-3.141592652*EXP(-NUE*Y( I»*SIGN (X( I»*(SIN(NUE*X( 11» .,GB = BE+3.141592652*EXP(-NUE*Y(I) )*SIGN(X(I»*COS(NUE*X(I»'END' BA GREATER .,BI (I ,J) = +GB .,

LA5..G1(I,J)= +GA.,B3(I.J) = -(NUE*Y(!)/(BA*BA)+ G1(I,J»*NUE .,G3(I.J) = -(NUE*X(I)/(BA*BA) + Bl(I,J»*NUE .,B4(I,J)=-NUE*(B3(I,J)+NUE*NUE*NUE*(X(I)*X(I)-Y(I)*Y(I))/(BA'POWER'4) ) .,

1=1+1.. 'IF' I 'LESSt N+l 'THEN' 'GOTO' SI.,'FOR' 1=1 'STEP' 1 'UNTIL' N 'DO''BEGIN' RHO= SQRT(XX(I)*XX(IJ+ Y(I)*Y(I» .,

.,

------_.--------

VTTY(K+T1*N,L+(2*TT-l)*N)=TT*TT*(-C*NUE*G2(K,L)+S*B3(K,L» .,'END' .. 'END' .,Tl = Tl + 1 ..

'IF' Tl 'LESS' 7 'THEN' 'GOTO' F3 ..NUE = 9*NU~/4 ..TT = TT+l . ,I1F' TT 'L'::SS' 4.0 'THEN' 'GOTO' KI1 ..

MAMUBT(GES,VX,VXX).,MAMUBT(GES, VY,VYY ) .,MAMUBT( GES, POT, PT) .,MAMUBT( GES, VTX..,"VTXR)

. . ,MAMUBT(GES, VTY, VTYR) .,MAMUBT(GES,VY2, VY2R) .,MAMUBT(GES, VTTY, VTTYR) .,

'FOR' K=l 'STEP' 1 'UNTIL' 7*N 'DO''BEGIN'Cl( K) = 2* (VXX(K)*VTXR(K) + VYY(K)*VTYR(K»

PT(K)*(VY2R(K)-U~*VTTYR(K» .,C2(K)=- PT(K)*(VY2R(K)-UE*VTTYR(K» .,'END' ..OUTPUT< 1, Cl, (2) .,

'GOTO' 9 .,9.. 'GOTO' NIN .,'END' ..'END' .,

152

* 5. 0.2. 7, 7.4, 7.8, 8.2, 8.6, 0, 0, 0, 0, 0,4.825700549881- 1,-2.57287563358$- 2, 2.52454711974$- 2.

-0.93341622600$- 2, 0.97602740850$- 2, 1.38838304636$- 1,-0.59171597890$- 2, 0.59928481298$- 2,-2.84836993606$- 3,

2.86408431136$- 3, 4.86863961944$- 2, 3.60450192292$- 3,1.64641383030$- 3, 0.94920110815$- 4, 1.66111376784$- 4,

-3.07018488438$- 2. 0.85877085628$- 2,-1.45814156112$- 2,0.56866812001$- 2,-0.77073192860$- 2,-2.40072603057$- 2,

-0.55244781684$- 3,-0.90894135551$- 2, 3.69252726699$- 3,-0.66862290739$- 2.-2.01747962319$- 2,-0.42052509431$- 2,

0.40115448730$- 2,-1.86186262975$- 3, 2.57668624421$- 3,

[

RHOJ = 1 .,'IF' XX(I) 'EQUAL' 0 'THEN' PHI=l.5708 'ELSE'

PHI = ARCSIN( Y(J)/RHO) .,'FOR' J=Z 'STEP' 1 'UNTIL' N '00''BEGIN' RHOJ = RHOJ*RHOZ .,

Bl(I,J)= (-SIN(2*(J-l)*PHI)-NUE*COS«Z*J-3)*PHII*RHO/(2*J-3»/RHOJ .,

GI(I,J)=(COS(Z*(J-l)*PHII-NUE*SIN«Z*J-3)*PHI)*RHO/C2*J-3»/RHOJ .,

B3(I,J)=-(2*(J-II*SIN«2*J-I)*PHI)/RHO+NUE*COS(2*(J-l)*PHI»/RHOJ.,G3(I,J)=-(2*(J-I)*COS( (Z*~-1)*PHI')/RHO-NUE*SIN(2*(J-l)*PHI) )/RHOJ.,B4CI,J) = (-2*(J-l)*(2*J-I)* COS(Z*J*PHI)/RHO*RHO + 2*CJ-l)*NUE*SIN«Z*J-l)*PHI)/RHO)/RHOJ .,BZCI,J)= GZ(I,J) = 0 .,

'END' J .,'END' I ..

'END' UPKIM .,FI.. READC XX, Y) ., NUE = UE., READ( GES) .,Tl = 0 .,

UPKIM .,F2 .. '�F' Tl 'G~EATER' 1 'THEN' EI = I 'ELSE' EI = 0 .,

C = COS«Tl+El)*O.785398) .,S = SIN«Tl+El)*O.785398) .,'FOR' K = 1 'STEP' I 'UNTIL' N '00' 'BEGIN''FOR' L = 1 'STEP' I 'UNTIL' N '00' 'BEGIN'

VX(K+Tl*N,L) = C*G3CK,L) + S*NUE*B2CK,L) .,VX(K+TI*N,L+~ ) = -S*G3(K,L) + C*NUE*B2(K,L) .,VY(K+Tl*N,L)= C*B3(K,L) + S*NUE*GZ(K,L) .,VY(K+TI*N,L+N )=-S*B3(K,L)+ C*NUE*G2(K,L) .,

POT (K+Tl*~,L) = -S*GI(K,L)-C*GZ(K,L) .,POT(K+Tl*N,L+N )=S*G2(K,L)-C*Gl(K,L).,

VTX( K+Tl*N, LI = -S*G3(K,LI + C*NUE*B2(K,L) .'VTX( K+Tl*N, L+N I = -S*NUE*BZ(K,L) - C*G3(K,LI .,VTY(K+Tl*N, LI = -S*B3(K,LI + C*NUE*G2( K, LI.,VTY( K+Tl*N, L+N I = -S*NUE*G2(K,LI - C*B3( K, L) .,VY2( K+Tl*N, LI = C*B4(K, L) - S*NUE*NUE*G2( K,L) .,VY2( K+Tl*N, L+N ) = -C*NUE*NUE*G2(K,LI - S*B4( K,LI .,VTTY(K+Tl*N,~1 = -C*B3(K,L) -S*NUE*G2(K,LI .,VTTYCK+Tl*N,L+NI = -C*NUE*G2(K,LI + S*B3(K,L) .,'END' ., 'END' .,Tl = Tl + I .,

'�F' Tl 'LESS' 7 'THEN' 'GOTO' F2 .,NUE = 4*NUE .,TT = 2 .,KIl .. UPKIM .,Tl = 0 .,

F3 .. 'IF' Tl 'GREATER' I 'THEN' EI = I 'ELSE' El=O .,C = COS(TT*(Tl+EI)*O.7853981 .,S = SIN(TT*(Tl+EI)*O.785398) .''FOR' K = I 'STEP' I 'UNTIL' N '00' 'BEGIN''FOR' L = 1 'STEP' I 'UNTIL' N '00' 'BEGIN'

VX(K+Tl*N,L+Z*(TT-II*N ) = C*G3(K,LI + S*NUE*B2(K,LI .,VX(K+TI*N,L+(2*TT-ll*N ) = -S*G3(K,LI+C*NUE*BZ(K,L) .,VYCK+Tl*N,L+Z*(TT-l)*N )=C*B3(K,L) + S*NUE*G2CK,LI .,VYCK+Tl*N,L+CZ*TT-ll*N ) = -S*B3(K,L)+ C*NUE*G2CK,LI .,POT(K+Tl*N,L+2*(TT-I)*N ) =TT*(-S*GICK,LI-C*G2CK,L) I .,POT(K+Tl*N,L+(2*TT-I)*N )=TT*(S*G2(K,L)-C*Gl(K,LII.,VTX( K+Tl*N, L+2*(TT-l)*N I=TT*(-S*G3(K,LI + C*NUE*B2(K,LI).,VTXCK+Tl*N, L+C2*TT-II*N ) =TT*(-S*NUE*B2(K,LI -C*G3(K,L) I .,VTY(K+Tl*N, L+2*(TT-l)*N )=TT*(-S*B3(K,L) + C*NUE*G2(K,LII .,VTY(K+Tl*N, L+(2*TT-ll*N )=TT*(-S*NUE*GZ(K,L) -C*B3(K,LI) .,VY2(K+Tl*N, L+2*(TT-l)*N ) = C*B4( K,L) -S*NUE*NUE*G2(K,LI .,VY2(K+Tl*N, L+(2*TT-II*N ) = -C*NUE*NUE*G2(K,L) -S*B4(K,LI .,VTTY(K+T1*N.L+?*CTT-1 \*N\=TT*TT*I-r~-R~(I(.I \-c;*I\IIII:"*(;?(\(.1\ \

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