Force calculation of a rectangular plate moving obliquely in water channels

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Force Calculation of a Rectangular Plate Moving

Obliquely in Water Channels

By Shosuke INOUE

and Katsuro KIJIMA

1bliotheek van .Sbouw)cunde

Ondetad1fl

-- nische HogeschOO

'

3.9

DOCUMENTATIE D AT U M: 39

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zu iu

453

Reprinted from

JOURNAL OF SEIBU Z5SEN KAI

(THE S0CETY OF NAVAL ARCHITECTS OF WEST JAPAN)

No. 39 Merch 1970

Lab. v Scheepsbouwkunde

Technische Hogeschool

Deift

¡2&

z2

&)

1, NOV. 1972

ARCHIEF

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Force Calculation of a Rectangular Plate Moving

Obliquely in Water Channelä

By Shosu1e INOUE and Katsurö KIJIMA

Abstract

Growing more and more t he size of the Oil-tanker, restricted water effect on rnanoeuvrabilitv, which has not been so seriois problems in the days before, is

becoming more important problems. In the previous paper, S. INOUE and K. MURAYAMA proposed how to calculate the force acting on a turning ship in shallow water by the analogy of Bollay's wing theory. In this paper, by their

method, the authors dalculated the derivatives of the force acting on a rectangular plate moving obiquely in finite water depth and in finite water channel width using

infinite images. According to this result, it showed in Fig. 31Fig. 34, it can be said that the ratio of derivatives in infinite water channel width and in finite water channel width coincided with the experimental result in model ship. Accordingly, we think that this method is a foundatinn of the study on manoeuvrability of a

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{2(2i)x-2sine}irxø+ (xE)

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i/T22n-1) ?-2E + (x_e)sineTz+ (x )2®+k2 {2(n2+1)hb-1}2

k{2(m+1)hb+1)cosi _{2(2nl)X-2sin8± (x)si}2+kZ{2(m±1)hb±1}2 (X ) cosO ₊₁

k{2(m+1)hblcos'

(2(2fliP2Sifl0± (x_)sin1Ei}2+k2{2(m+i)hb_i}2

(x)cos

x Lv{22n_nx_e+

(2(2nl)?-2sinø+ (xE)sini}2+k2 I /

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112 E (xC)cosS ₊₁ L1/2(2n_1)x_2sin8+(x_C)sth:}2+(x_C)2cosze+k2 +

2(xC)

{4nX+ (xC)sinø}2-i- (xC)2cos2® 4nXsin+ (xe)

i/{4nX+ (xC)Sin}2+ (x)2cos2e+k2

+ 2(xC)cose {4nx+(_C)sirie}2+k2 E

(x)ccse

L1/{4nx+ (x_C)siue}2+ (x_C)2cos+k2 + 2(xC) {2 (2n 1) X+2sinø - (xC) sinO} 2+ (E C)2cds29 {2(2nÏ)X+2sn®}cine_(x_C)

V{2(2n-1»+2sin8 (xC)sir9)2+ (xC)2cos&+k2

2(x=-C)cos@

{2(2ñ i) X+2sin& (xC) sine} 2ik2

< L(xC)cos®

i/ {2(2m-1)X+2ine (x_C)sine}2+ (xC)2cos2'9+k2

2(xC)

X m1 {4nX (xC) sin®)2+ (x_C)2cos2 4nXsinI (xC) (xC)sin9}2+ (x_C)2os2®+k2 + 2(xC)cos& {4nX(xC)sin8)2+k2 (xC)cos®

-X Lv'{4nx_(x_C)sine)2+ (x_C)2cos28+k2 T/(x_C)2±k2{2(m+1)hb±1)2 /(xC)2+k2{2(m+1)flb-1}2 + {2(m+1)hòr--1)(xC)cos® 2(m+1)kb-1}(xE)cose (±_C)sin2+k2 {2(m+1)hb+1}2 (x_C)2i2+k2{2(m+1)hD_1)2 + {2(m+1)h+1} (xC)2cos2® (xC)2siri2®+ k2 {2 (m+ 1) hb+ 1)2 1 (xC)2 +k2 {2 (m + 1) hb + 12 )2(m+1) kb-1) (x_C)2cos2 (x_C)23in2+kZ{2(m+1)kb_1}2 i V (xC)2 + k2 (2 (m + 1) kb _1)2

r

2mhb-1 2mkb+1 Ll/(x_C)2+k2(2mhb_1)2 v'xC2+k2(2mhb+12 X

'i (xC)cos'

(xC)2sin2e+k2 (2mhb-1)2 + (2?1ZI2bl) (xC)2cos2e (xC)2sin2e+k2(2mhb-1)2 2('z+1)hb-1 i V(xC)2+k2(2mhbi)2 (2mhb+1) (xC)cos8 (EC) 2sin29+k2 (2mhb ±1)2

(6)

Î31-Íi)

<jJ® 113

(2mhb+1)(xE)2cos2®

(xE)2sin28+k2 (2mhb+ 1)2

(x E) 2+ k2 (2mhb+ 1)2

-

(xE) L{2(2n-1)X-2sin®}sinø+ (xE) m0 {2(2n-1)X-2sin9+ (xE)sin)2+ (xE)2cosZ®

2(m+1)hb+1

X

L-v(2(2n-1)x-2sine+ (x)Sjfl9}2+ (x_E)2cos2®+k2{2(m+1)hb+1)2 2(m+1)hb-1

V{2(2n-1)X-2sin9± (x)Sjne}2+ (x.E)2cos2+k2 {2(m+1)hb-1}2

I

{2(m+1)hb+1}(xE)cos'

m=O Lt2(2n1)?2sifl8+ (xE)sin®}2+k2{2(m+1)hb+1}2

><

t(xE)cose

_V {2(2n1)X-2sine+ (xE)sifl®}2+ (E)2cos2+k2{2(m+1)hb+1}2

{2(m±1)hb-1} (xE)cosø

t2(2n-1)X-2sin®+ (xE)sin}2+k2 {2(m±1)hb-1)2

><

f

(xE)cos'

li/ {2(2n-1)X-2s1n8+(xE)sin®}2±(xE)2cos2+k2{2(2ñ+1)hb 1)2

(xE)[{2(2n-1)X-2sin1E}sin+(xE)1

m (2(2n_Ï)X2sin)+ (x_E)sine)2+ (xE)2cos2

E 2mhb+1 X

Ly{2(2n_1_2sine+_esine2+x_2cos2ek22mhb+1)2

2mh-1

v'{2(2n-1)X-2sine+ (xE)sin')2+(xE)2cos2+k2(2mhb-1)2 E (2mhb+1) (xE)cos8 L{2(2ñ-1)X-2sin®+ (xE)sin®2} +k2(2m/zb+1)2 (xE)cos® Vt2(2n-1)?-2sin®+(xE)sin}2+(xE)2cos2+k2(2mhb±1)2+1 (2mhb-1) (ìE)cos8 {2(2n-1)X-2sin®+ (x_E)sin}2+k2 (2mhb _1)2 (xE)cos

X

_L

_{2(2n-1)-2sine+ (EE)si}2+ (E)2cOs2+k2(2nthb-1)2}

+ 4nXsin8+ (xE)

inO{4n) + (xE) sin®)2 + (xE) 2cos2®

X 2(m+1)hb+1

LV(4n+

(x_E)Sin®)2+ (xE)2cbs2®+k22(m+1)hb+1)2 2(m+1)hb-1 -/4nX+(xE)sin®}2+(xE)2cos2®+k2{2(m+1)hb-1}2

-

E 2(m+1)hb-1}(xE)cos® m0 L{4nX+ (xE)sin®}2+k2{2(m+1)hb-1}2

E

m1 x

(7)

114 ± 2Th O < ç (xC)cosø ₊₁ 11/4nX+. (xE)sine}2+ (xC)2cos2e+k2(2(m+i)h-1}2 2(m+1)hb±1}(xE)cosø 4nx+(x_C)sin2}2+k2 {2(m+1)hb +i}2

x i

_{V{4n+ (xC)snø}2+(xC)2cos2e+k2{2(m+1)h+1}2}

(x)cosø

±1 + (xE){4nxsine+(xC)} m=1 {4nX+ (xE)siri9}2+ (x_E)Oc28. E 2mhb+1

Lv'{4ñ) + (EC) sin9}2 + (x _C)2cos20 +k2 (2mhb +1)2

2mhb 1 i/{4nX+ (xE)sine}2+(xC)2cos2e+k2(2mhb-1)2

r

(2mhb-1)(xC)cos 7ThT L{4flX+(x_C)Sifle}2+k2(2mhb_1)2 x (xE)cos9 li/4nX+ (x_C)sine}:2+(x_E)26i2e+k2(2mhb _1)2 +1 (2mhb±1) (xe) cosø 4n± (xC) sin®}2+k2(2mhD+1)Z

[(xC)cos®

+i}

i/4nX+ (XC)Sifle}2+ (xC)2cos2e+k2(2mhb+1)2 (xC) [2(2n-1) X+2sinE»sinø (xC)]

?(2ñ1) ?+2sin® - (xC) sine) 2+ (xC)2cos2O

2(m+1)/ob+1

Lj/{2(2n-1)x+2sin® (x_E)sine2+ (xC)2cos2e+k2{2(m-i-1)hb+112

2(in+1)hb-1

-V(2n-1)x+2sine (xE)sin'9}2± (x_C)2i2ø+k2{2m+1)hb_1p

r

{2(m+1)hb+1}(xE)cos9

=o L2(2ni)X±2sine (xC)sin'}2±k2{2(m±1)ñb+1}2

> Ç (xE) cosi8 +

lj/{2(2nfl?+2sin® (x_)sine}2 (x...C)2co52e+k2{2(m+1)flb+1}2

2(m±1)hb-1} (x-E)cosø

{2(2n-1)X+2sin9 (xE)sin9}2+k22(m+1)/zb-1}2

Ç

(xe)cose

V{2(2n_i)x+2sine_(E)sin9}2±(x_E)2cos2e+k2{2(m+1)hb_1}2+1

(xE) [2(2n-1)X+2sine}sine (xE)]

7l1 2(2n-1)X±2sine (x_E)sinI}2+ (xE)2cos2e

E 2mhb+1

(8)

E

1Th=l 2mhb-1 y2(2n_1)x+2sine_(x_)sine}2+(x_)2Cos2e+k2(2mhb_1)2 E (2mkb+1) (xe)cose L{2(2n-1)x+2sine--(x-r-E)sine)2+k2(2mflb±1)2 X Ç

(xe)cose

±1 Li/ {2n-1)X+2sin9(x)sine}2+(x)2COs28+k2(2mhb+1)2 (2mho-1) (x)cosø {2(2n-1)X+2sin®(xE)sin8}2+k2(2mhs-1)2 Ç (xe)coscE X l/(2(2n_1)x±2sin9_(x_)sin2}2+(x_)2cos2e+k2(2mhb_1)2 +

-

(xe){4n?.sine(xe)}

(4ñ? - (X - ) sine)2 + (ïe) 2cos2®

2 (m + 1) hb + i i/ {4n? (xe)sine)2+ (x)2cos2'+k2{2(m+1)hb+1}2 2(m+1)hb-1 /{4n?_(x_e)sin8)2+ (xE)2cos2e+k2(2(rn+1)h-1)2

-

{2(m+l)hb-1}(x)cos9

7Th0 (4n?(±)sin'E)2+k2{2m+1)/zs-1))2 X

(x)cose

lV{4nX_(x.E)sin8}2+ (x_E)29±k2{2(2±1)hb_1}2+1 {2(m+1)hb+l} (xe) cose {4nX (xE)sin9}2+k2{2(m+1)hb+1}2

t(xC)cose

V{4(_e)sine}z+(xC)2cos2e+k2{2(m+1)nb+1}2+11

-

(xC){4nxsinI(xC)}

n {4nX - (x C) sine) 2± (x E) 2cos29 E 2mhb+1 X Lv' {4nx (x_E)sine}2+ (xE)2cos2'+k2(2mhb+1)2

2mhbi

{4n? (x_E)in®}2+ (xE)2CO529+k2(2mhs-1)2

-

r

(2mhb-1)(xC)cose L{4nX(xE)sin9}2+k2(2mhbi)2 Ç (xC)cos8 X /{4x (±_E)sinI}2+(x_E)2cos2+k2(2m/zb1)2+ (2mhb.+l) (xC)cos' {4nX (xC)sifle}?±k2(2mhb+1)2 ç (xC)cos® X 7j(

4t<Jjøtt

115

(9)

116 a,

(12o),t 5

Vsina= K(x,C)dC (2) 3. K Kutta OL) 7CC) =Va(C)i/ 1L à(C) ) Fourier L 7, (lo) )

Vsina=J

ai/1

K(x,C)dC

4. x=cos9, C=cosû'

a(C)K(x, C) =a(9')K(6,6') ') ¿+1 1+1

=- E 69a(09)K(D, 6) E

cosqe cosqo'

+J. p=o qo P»a=O X ¡+1 2

(o)«ht"

8e', _q=l

x e=o.=S/(1+1), 6'=6=P=r/(l+1)

o t. ¿+1 8

sina= E a(6)K(O3,O)C

p IBL

_-

C8---

I

1(-1)P'

1+cosO +1. p 1+1 I. 2 cosO,cosO3 2

c8

e,,

°11+

_i 2(1+1) ' o 2 4(1+1)

(4)O)j2 ¡=15 L17LK

_o

o)t )itM- :'

-o)-E) FACOM23O-60

a(9,,) :*.: 1(6')

) 7(6') 8,,Va(6,,) EqcOsqOpCOsqO'

*lo

N

PLJ

_{(VcOs+We)7(C)d(C) M} pbL2

L

(Vcosa+we)C7(C)dC (5)

w P#o)x u

7CC) we -i-(.3) (4)

(10)

-j&< }3øtE

k

f'

r

2sin WE

= ---J

_,

7 (C) L_e22®+kz

2(xC)sincosø

_d

+(xC)2sin2e+k2)T/(xe722

C

+ k

f'

r

{2(m+1))b+1}sin® J L(x_C)2sin2e+kz{2(,n+1)hb+1}2 {2(m+1)hd}sin9 (xC)2sin2I+k2(2(m+1)hb-1}2 + {2(m+1)hb+1} (xC)sinecos' [(xC)2sin2'+k2{2m±1)h+1}2]

i

f I

(C) 47r _{n=t J} -1 {2(m+1)hb-1} (xC)sin4a'cose [(xE)2siñ2+k2 {2(m+i)hbi} 2]

XV(C)2+k22+1)fl1}2

dC + k 1' _(C)

r

(2mhb+1)sinY m1_J ' L(x_C)2sin2±k2(2*nhb±1j2 (2mho 1) sin9 X (xC)2sin29+k2(2mh- 1)2 + (2mhb+1) (xE)sinecose_{(xC)2sin2'+k2(2mhb+1)2)} i /(xC)2+k2(2mhb+1)2 (2mhb-1) (xe)sin®cos0 {(xE)2sin2®+k2(2rnhb _1)2}

XV(C)2+2(2fll)2

dC 2

{2 (2ñ 1) X2sint9+ (xC) sin9}2+ (xC) 2cos2® {2(2n-1)X-2sin®}cos® V{2(2n-1)X-2sin9+(xC)sin®}2+(xC)2cos29+k2

2iiø

{2(2 1) ?-2sinø+ (xC) sine}2+k2 <

r

_{L2(2n_x_2sipe+x_sine}2+x_2cos2e+k2.}

(xE)cbs0 +

i

2 {4ñX+ (xC)sin'3}2+ (xC)2cos9 117

(11)

118

k39

+_m1 2sinø + {4nx+(xe)sirie}2k2 r-

(x)cosO

X

_{Li/4nx+x_E)sine)2 (x_e5262}

2 {2(2n-1)X+2in0} cosO

i/{2(2n-1)X+2sin - (x_e)sn2+(_y22e+k2

2sinø + 4nXcosø X_{1/{4X + (x ) sinO)2 + (z e5229 +k2} {2(2n-1)X+2sinø (x)sisø}2±k2 r-

(x)cosO

L2(2n_1)X+2sinO(._)sin0}2+(xE)2cos20+k2

+ 2 {4nX (x_)Sin0)2+ (xE)2cos2' 4nXcosø X _{{4nX (x.e)sino)2+ (x_522ø+k2} +_(4nX_{(z_e)sine}2+k2}2sin9 r- (xE)cosø X

_{LV{4nx_x_sine2+ (522O + k2}

± -I-k2 +

il]

{2(2n-1)A-2srnO} cosO l2(2n_1)X_2sinO+(x)sifl®)2+ (t_C)cos2o r- k{2(m+1)hb±1}

LvT2(2n_1_2sinO±_esinO2+ (x_522I+k2{2(m+1)hb+1}2

k{2(m +1)hb 1) /{2(2n_1)X_2sinO+(x_C)siflO}2+(XE)2C0S2®+kZ {2(m+1)hb-1)2 k {2 Cm + 1) h ± 1) sine (2(2ñ-1) )-2siñO+ (xC)sinO)2+k2{2(m+1)ho+1)2 r- (xC)cosO + k{2(m+1)hb-1}sinø {2(2n-1)X-2sinø+ (xe)sino}2+k2 {2m+1)hb-1)2 X r-_{Lv'{2(2n_1)x_2sinO+(x)sinø}2+(x)2cos28+k2 {2(ñ±1)hb-1)2}

(x)cosø

+ {2 (2n 1) X-2sinø) cosO

(2(2n_1)X_2O+C_E)sin)2+(x)2osc2O

dC i:i de

(12)

74%th < )3O)

r

k(2mhb+1) X Li/2(2n_1)x_2sine+(x_)sine}2+(x_e)2osc2e+k2(2mhb1)2 k(2mhb-1) k(2mhb+1)sin (2(2n-1)?_2sinO±(x_)sin9}2+k2(2mhb+1)2

r

(x)cose

_{+ i}

L/ {2(2ni)-2siri9+ ()SiflI}2+ (x)2cos2W+k2(2nhb±1)2

k(2mho-1)sin'

{2(2ni)X-2spi®+ (x)sine}2±k2(2mhb-1)2

r

(x)cosø

₊

Li" {2(2n-1)X-2sin8+ (x_)sin9}2(x_)2cos2e+k2(2mh_1)2.

{4nX+ (x)siñ®}2+k2{2(m-4-i)hb-l-1}2 < L

(x)cos9

+ k{2 m+ 1 hb - 1} ® {4nX+ (x)sine) 2+k2 {2(m+1)hb-1}2

r

(x)cos

+ X Lv'{4ñ+(È_&sinø}2+(x_C)2cos29±k2{2(m+1)hb--,1}

-

rl

₍₎

-

4nxcosø ml_J _{4nX+(x_)Sin)2+(x_)2Cos2®

f

k(2mhb+i)

Lv' {4nX+(±) sin} 2 +(X_i:)2cos2ø+k2 (2mh0+ 172

k(2mhb-1)

X

y'{4nX+ (xe)sine}2+ (xe)2eos2+k2(2mnbi)2

+ k(2mhb+1)sine {4ñ+ (x)sinø}2+k2(2mhb+i)2 k(2mhö-1)sinø {4nx+(x_)sine}2+k2(2mhb_1)2 11 (xC)cose +1 }2±(xC)2cos2e+k2(2mhb+1)2 de 119 X

r

_L'

k{2(m+i)hb±i} {4n?.+ (x)sine}2+ (x)2cos29±k2{2(rn+1)hb-j-1}2 k{2(m+1)hb-1} y' {4nx+(x_)sin}2+(x_)2cos2e+kZ(2(rn+1)hb_1}2 k(2(m+1)hb+jjsine +_iiiD)

I

4nxcose -1

(13)

120 f13 ì3 fG

-

_I

7(E) m1 _{J -1} +

i

?n=o J_i i-.

(±E)cbs

_i -Lv (42A.+ (x) )2±(xE)2cos20+k2(2mhb-1)2+

{2(2z-1) ?+2sin9 (xE) sine) 2+(xE)2cos28{2(2n-1)±2sine)cos

< r-k{2(m+i)hb+Ï} Li/{2(2ni)x+2sin® (x_E)sine)2+(x_E)2cosze+k2{2(m+i)hb+1)2 k{2(m+l5hb-1} k{2Çm+1)hb-1}sin' {2(2nI)X±2sin9 (X E)sin8}2+k2 )2(m +1)hb- 1} X

(xl:)cose

4

Lv {2(2ni)X+2sine (xl:)sin®)2+ (x_E)2i2®+k2{2(m+i)hb_1)2

+ k{2(m+1)hb+i}sin9

{2(2n-=i)x+2sine ()ine}2k2{2(flZ+i)hb.i)2

(x-l:Jcos ₊₁ LVl2(2n-.1x+2sinO(x_l:)sine}2+ -l:)2cbs2®±{2(m± 1)]2b+1) dl: {2(2ni)X+2sin®cos' {2t2n-1)X+2sine(xE)slne}2+(xl:)2cos29 k(2mhb+Ï)

Lv' {2(2n - Dx ±2sinø (xE) sin8) 2+ . l:)2cos2e+k2(2ñthb+ 1)

k(2mhbi) 7{2(2ñiP±2sin8 (x-.=l:)sin®}2+(x=l:)2cos29+k2(2mhb--1)2 k(2mfrzb-1)sin®

(2(2ni)x2sine (xl:)fnei2+k2(2mhb-1)2

r

(xl:)cose Li/{22ni+2iii® x_l:)sine)2±(±_E)22e+k2(2mhb_i)z

+ _{(2(2nDx+2sine (xl:)sin9}2+k2(2mh+l)2}k(2mh ± 1) sine

(xl:)xs® LV{22ix+2siex_sie}2±(x-E)2cos2e±k2(2mhb+1)2 4ñxcose {4n).- (xE)sn }2+(x_l:)2ços2e

r

k{2(m+1)hb+1) Lv' {4nx(xl:)sin9)2+ (x_E)2cos2e+k2{2(m+i)hb+1)2

+ _{4nX_{(xE)sine} 2+k2{2(m+}k{2(m±1)hb±1)sin _1)hb+i)2

dl: dl: k{2(m+ 1)hbi}

il

i

+1

(14)

t

o F

(ì)cos

₊₁ Lv' {4nX (xe)sine}2± (x)2cos2+k2{2(m±1)hbF1}2 k{2(m±1)h0-1}sine {4nX (x_)sin®}2+k2 {2(m+1)ho _1)2

X [

_V{4X (

e)ie}2±(e)2C2e±k2{2(+1)h

1}2 + +

I

m1 J 1

-

4nxcos9 {4ñX (±e)sine 2+ (x ) 2cos2®

r

k(2mhb+1)

L1' {4nX (xe) Sifle)2+ (xe) 2Cos29+k2(2mhb ±1)2 k(2mh0l) T" {4nX (x_)sjne}2+ (x)2cos2'+k2(2rnhbl)2 ± _{{4n (xe)sin®} 2±k2(2ffhb + 1)2}k(2mhb+1)sinø >

r

(x°)cose

LT'(4nX (x)siri) 2+ (x)2cos2+ h2 (2mhb +l) + k(2mhb-1)sin' (4nX (x)sinO}2+k2(2mh-1)2

r

(x)cos8

-LT' (4nX (x )sinø}2 ± (X ) 2cos2ik22rnhb_ 1)2 ± 0.10, 50, H/a=1.5 Vcòs

U7cosa

=0. 99619 ¿U w/V0. 00358

w ø/J\0

N4Vcosaj

7()d,

M pbL2 Vcosaj

7()d

N M M -pV2bL2

Ui)3;

¿+1

C =

cosa Eqa(Gp) (1+Ços9)

±J. q=O ¿+1 CM = E 8a(9) {1+2cos9+cos28} 4(1+1) =o X, ' i- CML

Ml-*43-Írn < 3a)ftr

121 ¿+1 4(1+1)

E

a(O2,) (1cos29) f, 7cosa L'MT. p=o II d

(15)

122

4.

k=o. 05,0.10,0.15 0)3

l-l-'-Th, XIdL,

Ø.

W/Lo.5,U.7,i.0

H/d=1.3, 1.5,2.0 0)

DIt

DV'o

/JGMk CN _-*5Z, n

0)0)0) Cv

CN(n) {Cjq(n)C(n-1))/Cv(n><íJ.001 DtB LZCN

C, CML

t0)

Fig.4, Fig.5 Z, Xøll0) dCN/dxP

Fig6 ¿tO

C, C

(C

Ot--' i') o)3nhj a= 0)0)

ø0t 5 lty0

(dCzqwH\ (dClqH \ (dCvswÊ\

\

da

_/=

k da 1= k da J k da 1=

{.L CNWJ%

Wl2i, H0 íEt C4VWH

l7I(7iÇ

CN

ttU, CNff Jjlj,

CN CMSWH, CM,H 0)

*0)f

( (dC;wH/da)/(dCrç.H/da), (dCMØWff/dv)/(dCM,,H/da)

l,

Jk

W/L, Hid

Lt:øô Fig.7Fig.30

O X,

lGOèlLè-Fig.31Fig.34 lt0 3JZ *0)tilll±,

0J ,

bJt I0)±I

*r

11 V' . X, V'0Y ;ò,

0)-

Fig.31Fig.34 IEALCV'O

5-

7it

e

_n 3r*Zc

K)i Y -'

ll4t-DtV'0 øy

0)l2t V'0 X,

7J(1 0)

_'0

Ji lE :

124- Vu43$

5jIl,

115- flu39

±IE,

W. Bo11a,r, A non-linear wing thenry and its applccation to rectangulâr

wings of small aspect ratio

ZAMM, 1939.

E I1 i :

(16)

123

'J'tLìtÌ

_"4O

j J On the turning ships

(17)

124

k39

0.06 0.05 0.04 0.03 0.02 0.01 0.005 0.06 0.05 0.04 0.03 0.02 0.01 * 0.1 O 3 500 - 1000 NO.OF IMAOE Fig. 4 t1.3 0010 -500 1000 - NO. OF IMAGE Fig. 5 3 0.5 3. 0.5 20 40 60 1. thord t,n, I.adI,g èd9. Fig. G trIIng 1d5

(18)

dGwH/ =1.3 áCNH dOc 125 0.05 0.10 0.15 - Aspect ratio Fig. 7

=1.5

0.05 0.10 0.15 Aspect ratio Fig. 8

(19)

126 d 2 O 1.8 1.6 1.4 1.2 1.0 Fig. 9 Fig, 10

2.0

1.3 0.05

010

0.15 Aspect ratio I. 0.05 0.10 0.15 Aspect ratio

(20)

dGwit

I, f

jcjS ç ;bøt

127 Fig, 12 'kd = 1.5 0.05 0.10 0.15 Aspect ratio Fig. 11

0.05

0.10 0.15 Aspect ratio

(21)

128 dC4wH, d d

Aspect ratio

0.05

FigS 14

=1.0

1.0

1.5

2.0

Fig 13 dC.lwH

Aspect ratio

0.1

1.0

1.5

2.0

(22)

dCNwM

d4

dt

1.0 QH

do

1.0 *r

C4Ît

3IÑ&Ith (3O)t

Aspect ratio

0.15

1.5

Fig. 15

2.0 Aspect ratio

0.05

1.5

Fig. 16

2.0

129

(23)

130 dGwaii7

dfrQi

do

Aspect ratio

0.1 = 0.5

Fig. 17

2.0

= 0. 5 H

.4

1.0

1.5

Fig. 18

1.0

1.5

2.0 dGwH,

(24)

CNWI

d4

'L. = 0 .5

1.0

dGlwH/ doc J3o)tE 131

1.5

Fig. 19 Fig. 20

= 0.7

0.15

2.0

H

4

o5

= 0.15

I I

1.0

1.5

2.0

H

(25)

132 dCNwH

dQH

1.0

Fig. 22 o

= 0.15

1.0

1.5

2.0

Fix.21

(26)

2.0

18

1.6

1.4

1.2

1.0 *C

i-i&«J< MO 133 FiÉ. 2

't=1.0

Fig 24

0.7

1.0

1.5

2.0

1.0

15

2.0 H,

(27)

134

2.0

1.2 1.0 dCNWH dyA'CNWH d

05 -

0.75 Fig. 25 d0rwH/

dC

d

0.5 = 1.3

1.5:

0.75 Fig. 26 1.0

_w

/L

-1.8

16

1.4 1.2

1;0

2.0

1.8 1.6 1.4 1.0 L

(28)

dCNw dCM

d.

_CMH dc -JJ K 135

= 2.0

0.o5

0.10 = 0.15

0.5

0.75

FigS 27

0.5

Fig. 28 = 1.3 0.75

1.0

(29)

136

dGwi,

Ø4GCH

,

0.,5

=1.5

0.75

Fig. 29 H,

_2.0

Fig. Q I

-1.0

dCt dt( CMH do G. 5

0.75

1.0

(30)

2.0 1.8 16 1.4 1.2 10 o 0.5 0.75 1.0

z

< & . o _d=1.3 o _k=15 2.0 .8 1.6 I .4 1.2 1.0 dC ° _tj 1.5 u: 137 ° = 1 .5 I: Fig.31 Fig32

_it

Fig. 33 _-

-flG

Fig.34 3Z

it

0.5 0.75 1.0 0. 5 0.75 I.0 0.5 0.75 1,0