Guide to computational procedure for analytical evaluation of ship bending-moments in regular waves

(1)

Lab.

_{y. Scheepsbouwkunde}

Techñische Hogeschoo

D1fL

DAVIDSON

LABORATORY

GUIDE TO COMPUTATIONAL PROCEDURE FOR

ANALYTICAL EVALUATION OF

SHIP BENDING-MOMENTS

IN REGULAR WAVES

by

W.. R. Jacobs, J.

DaZze ¿Z,

and P. Lalangas

(2)

l3eport No. 791

October 1960

DAVIDSON LABORATORY

STEVENS INSTITUTE OF TECHNOLOGY CAS1LE POINT STATION

HOBOKEN. NEW JERSEY

GUIDE TO COMPUTATIONAL PROCEDURE. FOR

ANALYTICAL EVALUATION OF

SHIP BENDING-MOMENTS

IN REGULAR WAVES

by

W.. R.. Jacobs, J. Dälzell, and P. Lalangas

Wilbur Marks

HEAD. SHIP HYDRODYNAMICS DvISIoN

(3)

tUT4 G.RiM'5

i'17

TABLE

A-1

OF kDtD

iV3Ç

Pr,Vc 124,y%P/AJ MODEL CHARACTERISTICS

coPFIC l,JT.S

Fss AA-!

,A4-LI,

Model f?!? 2_..

CSE1,es1

ößLocK')

Length L.B.P., ft.

Maximum Beam,ft û-f Gyradlus. k01ft

1,19 7

Draft ,ft O. 2Z.

L. C. G.

Position

With

Respect

to

Station iO+OiIZ'(fodeI

Displacement , A ,Ib 4

34 mà/g

J mk0' STATION NUMBER I 3 5 9 II 13 15 17 19

(D ¿,ft. from LCG

/.s.' ,.o"

O.1i

Ò,/2..

-O.3&'. '-Afl' -1.32. -/. tCi -ìì2

® ¿2,ft.2

q-4

4q.3

j/7

o.3C

O,OK ò,L3 O.7/

/.7/

.3.24' -J

® ?

1ft.

43 yf ,b,yr.sys'

6/i- .5. 3o0

H ft. ,

® s.ft.2

,ò'/

/t7 ./Kq

.,q

,'

/3

.17..i..

//?Ô3J

Wt.Ift. lb/ft. ///t

/2ji.

/2(02. J/éi... //.) /t2,t

4.

33 2.7/

®

YaS/B,tt.1®+()

ì2-ì-,U3 ;TV .2-fi

'2.17 -1

.ig'2_

.2..(.S

.117

®

B'H

®X®.

,/ /7i ITf ,/Ç .J(( ;/rf

,,gj

,,

//

C9 S / B'H

®

+ ® 77f

f 9

99Ç

.

?f?

. '733 .4 7

B°/H a +

/

'2.4/ l.2.4

2-iJ.

24/,

272& /.J/)V5

rAL-4-2_

3 B2 a

)31

V/3

'2o

e'/2.Ò I

'/2.

'w

"/2e 3,4

ò,

®

L 0

c' ® ®

ek2

-S ¿

Y1'3

p

IVM®-jK,- a/

f+1 8m

®/

lb-sec2/ft2

.

//e ij3/

.

j-.3.7e

.37

.

3/

.3j

.3/a

. /,9 ,

'37

6J(8m)C2a®x® ,Ib-sec2

r9/J

/&o,Y .)Y7 i2t)

'3J.37

pgBOpgX®,Ib/f?2

9o.io

0y2.

6g

,'ì- 3o1/;7 33i i7i-.z

®

pgB(e.îjx®,Ib/f,

%,)(,

l'f242f

ØÍ

-if 33-i.S-L36'

'-24.23

-/'

() pgBC" ® X

Ib

;97

71i

V7.n-.57 5zf ?J3

I/3.$7

'72-t i7 L73.O

(4)

TABLE

A-2

COMPUTATION OF HYDRODYNAM IC COEFFICI ENTS OF EQUATIONS OF MOTION FOR

HEAVE AND

PITCH Model Model Speed

, V ,ft/sec

-, 2/9

Wave Length , A

, ft

;2o

Wove

Celerity ,c, ft/sec 4 L

Frequency of Encounter, w,

2/2g

J.'-/3.) 2-STATION NUMBER I 3 5 7 9 II 3 15 I? 19 (13

¿,ft. from LCG

2y /Y

/JZ'

'IO

o,p-tY-L32 -/.

-2,2-á-® ¿,tt.2

4j' Z/3 ¿/7 O3

¿/1 ¿,,3 £.7/ ,73,2y'

® Bw212g

5 .9-i ?3O .93t)

,'7-93ô 7 r/

'4, C

(i'c. P;\-')

, 57 / ¿ 1.0

/

¿ ¿

3

57

7

®

/3 .3'

,33

33)

.33Ò

33o 33) .2y o7/

® pSk2k4, Ib-sec2/ft2:9),

o$

,2-Y.

3O 33ò .3?)

33ò 3ib '2-7.3

/y/

)3j3

137 1./Ti

pSk2iI4ClDx®Ibrca

«

'1

,9

,tL/Ò -.i'?

-J.77 -.3t)

2ry

.t43

® pSk2k4CC 2x

,Ibsec2 ).-YI .5Yò

.I/

.OÔ.(

;63

:2-3g .Ç 1Y7

172- 2.712- /3

® d(pSkskj/dC,Ib.seca/f13

-. a3 -,V&3 -.\3/

O O £ 'OCO

,,;;7

2e- ./j3

Vd(pSk2k4)/dC,1bSec/f12

-,t?7t' -J7f .-?,V

1 û

/«fr' .72-.

.L7

.

® ¡

(FIC, FiA-2.

.'/L.

,73 .fío

.5Z 1S-û

5 .f;io '77

,jy)

Q

¡Z

2-YIL-,y)3

'31.3 .31)

t"

(j)

N(C)AZpg2,w5,Ib_sec,ft2

553 /.l.3

:7ô7 .7ó7

.7e? .7o7

.?í

/5$

.7 S/J7 3.'/4

Q (c)x?

i/

2..Y

'í-V .i-1Y

.ex

,ò7-

/7/2-/Ç77

Q N() -v dpS 112 "4) / dC j,

3j À 7e /

/

,

7 7 .7 L) 7

. .

/

q '.57/

'7i_.

Q

N(C)-Vd pSk2k4 ¡dEC

ib5j

Z

Yi-/ .i9'f -.26V

-.s'J -7/-i 3

.2éJ

/J7j'

Coefficients of Equations of Motion a m + AC (pS112k4)

;? S?1

B LE [N(e)]

2. z 7"

C c AC egB*C2) -VE 2-:3 A J + AC (,psk2k4e2)

/1-E = Le [NC -v d(pSkek4)/dC]C. /

J,<í

G AC (pgBC) g

-/ 2-.

d D AC !(pSI1k4C) o E -2v[Ae !(psk2k4)] '-3,9y'_3 g

a G -Vb:

-/, b AC N(e) s c AC

' (pgB)

)12'.3

(5)

TABLE

A-3

CALCULATION OF EXCITING FORCES

AND MOMENTS

F\/Ç2 cos[wt-orctanF2/F,]

i1

-M \,41,2+ M22 C0S[ wt- arctofl M2IIJ

[

r

t-. i F,, lb F2,.lb

M,. ft-lb

M2, ft-lb STATION NUMBER I 3 5 7 9 II 13 15 17 19 ¿

ft

from L.C.G. Z.Ó

/.- i'

t;,

/z -3 -o,X7 /3. --/6 -2,21'

® 2irC/X ,

rod.

y/#?f .7i- .1i7 -5'J -Im -i. ;i

î-1f/

® sIn(2'yr/X)

í.ÇY

/

767

'-v

)ri

® cos(2,r/X)

.7g/ _p/

[7

iy

.$'/ 17'fr'

./)Z

:ì -,

2'ny/X

,

.'f/

.3)i- .37._

.3'71-,37j

I.3

yy

®

7i//

74

[Ç'7 ¿,ÇQ

( E,r/

767 7i,.

(

pgB'h

(/

¿û2/

¿,/

® psktk4(-hw:)

73

.-,E5'j _if3',3 ,$5

®

+®

/1cc)7

J3»

134 ¡.I' /,yVá' /V52

ç [N()-vpdSk2k4/de] (hwo) 3'? ,JTÇi 1-J'3 ,

vi)'

i,-4

,

-IJÇ;:-i.'1jj--i.c',-.i3(

,

I')

i t'

-o 7!

D

.tI:c1

I'i

"i

-.L3

.2

©

X®

Th35'

-O

/k l'14

©

IO1." ì,û

1.3?!

1./o!

.993 .-39

-4o93 -I.'&"7 -'

(íj

®x®

.

U

Cf3C

k

¡fr

/'1'

.?:24) -1.ûz

-'Sl

®

o7

i

',,

ò

i,

77t7

O3i f ¿

-.iV

/.7

-.117 -, I,-I1

-ij,3 -/b3

-.öi/

'-7Y5

®

-©

© (JJX(

dF,/d

.7t1

"731 7' 2'Y-ì,7

._

21J'-/3û'i'/uf.-/77

-.3t)

«ìS-I

(j) (î)x(,)

dF2/di

1)

-3s'

.?

$'" -ôí''-4'5

cI

dM,/d

t'Ç

J.oû(

.i'3'S' O31

-tC'9'

l.}(,e!

1.ti'

(i1

'33'1" c(' dM2/dz

,.j.3 ,..3,

.3z../ ./Ç -.'3i'

'.13 ..'"

ÌII'(

I(S'3 .-./iS" -.OI

Model Model Speed . V ft /sec Wave Length ,

A , ft.

76

Wove Amplitude , h ,

ft

Wave

CelerIty. c,

. ft/sec

4. ?$

2,r/X )3c?

Wave Frequency w0

2vc/X

:

(6)

TABLE A-4

SOLUTION OF

EQUATIONS

OF MOTION FOR

ÑEAVE AND PITCH

Model

Speed

, V ,ft /sec

l 3

Phase

log with respect to wove node at CG.

Wove Length

Ì ft

Wave

AmplItude , h, ft

Frequency

of

Encounter w

9. -w2

92_,J

F0

JF +

s J3

PS

- /q7. c2rS

'1'-arc ton F2/F1

2h?!

f rom Table A-3

OR _

______

M0 =./M+ M

33Ç(

PS-OR

-arc tan M2/M1s-34

PS-QR

Z-jC?7 -r j/öi

F(COS0-Isincr) =

- jì -./Uj4,

(Ps_QR)(Ps_QR)

3r S73 ,k'

0 M0(cosTIslnr)

j

rLi L

/i- - /:)1.

p

-ow2 + c +Ibw

(I.5t

S 0 ¿ i

02-+4j .C7

S = Aw2 +

+iBw

icL4

from

(sQ)(Ps_QR)

1-®

Q

-dw2+g +Iew

Table A-2

p

2-lÇ

-ìi'Ça_,

RDw2+G+iEw/r/j'/.

R

.1_t

®

-R)(PS -OR) - /oi,

/fí

-i

'5J,

''

--.-i

*-,OÒJ,

Z0Z+Z

ft

-arcton (z2/z1)

Is

°I2 ûòq

-0= rad

*

arcton (;io,)

(7)

TABLE

A-5

CALCULATION OF SHEAR AND BENDING MOMENT AT MIDSECTION Model

¡(f,

Model Speed V

ft /sec

L1 3? ¿/

Wove Length / Model

Length AIL Wove Amplitude h

ft

Frequency of Encounter w 2 J w

z0,ft

sin B cos 8

.?/

rod.0'OY

sin «

/976

cos

t

F'C/A

Tftr3

E

'

Shear ,lb cos(wt - ardan f2/f1) l(

Cd-(tT_52.

BM., ft-Ibm cos(wt-orctan m2/rn1)

1 + f

i. v)

f1 , lb m1, ft-lb lb lt - lb STATION NUMBER I 3 5 7 9 II 13 15

li

19

Ii 9 91

_{[( -)J}

o e.ft

from L.C.G. i.oY û.LÒ

Jz--O)

¿,,24/ /.y.i

FO] o

ft from Sta. IO

ZI6

¿.7L

g.,.ì.'

i7j.. J ¿)

_j,--.

[O]©)

i'L

II

î .3

3j/ .3

jjg

[O]O)

[O]LOO

l)ri'

'

I 3) i);?b

33)

. '.-?3

I V/

.L 7cL

.7)3 .b5/

¿

S

J.3

[O]

41)

'/'

/s/

[O][OO

[O][OZOIO)

____ _

Z33 /,7I

7/

7ô

,y S,

'-"

A- t'i?

só

9toL L

iry

-J2)

.

7/

.

[]

[1IG28

](2wV90sInlr)x

[J(.zcos8)

(w z cosS) x x

-!.fJ-..j

i.

,

105c))c

O.o1

(Ii-,CL.L. t3 ./SJ

jc

./L.O .ù70

OJl .ìc.

.o ,&-ò

SiI 9/ (,I

;Püf .9e'

.'(('l

,4'17

[J (-B0cos)

x .13 i

,1k

,(.l,L1

I .0

C'-II

-.it

-.!z-f

[1

(-wzosIn8+v90cose)xJ

c

,coV

c/

col

oc ,ôô i 07f, 1.O/Ò _{7 L. 7}

j

_J1

.IL.L 7JJ' -ro 12 ,

1 Où

-.

[1[1O

df1 ide

ot,c'*

I C

mIJJtJ_;ûlL-itq

]

_[]

(20&ra) x

-.Oj'

fl-cs

PI

Ei(2wvo,cosr)x

J(-zsln3)

(w8enc) x

'x

Ù

(.oö

3' .'Y0

.ci .. o

cq3

c'3

.&q

:/)

0o9

û'

.

'o'

[

(-9sine) x

, -1. isi

1

J. J4 (.7'10 Iq

_ot

-û?'I'L..Ci7

[]

[(w8cosc) x

i

CìOL' , 0b7

'JJ

dFidx flrwJ4E

.'o'? ö

i

ITh 1 ¿'i

df/di

:1,3-

lFi1J!LIL9ifl-3 ')c .1',

LWiII

t i

o df2/de

33

u1

Th

tI

I! .klO

.DI"

(8)

CotWhR

oc Ex

w

su

,

ir

4'D T44EOTICL

LCULntJ

öp

oflûj M)C)

EMDI!JÇ

o1

4S-For

ôO

L

_{L9I9.2 (SiE £7) T ¿.394 Ft/SEc.}

)9OC#L(

I94-CJLC

PoV6LEApL7LJD

OF44EI

,

'OULE#FVPL,iTUD

oF Pic,

&EftJD!AY-rnoJrRMJE; ¡M-LB.

4.40 O,7

4.3 aiÇ

O.34

USi N

Po M

S _û1J4-

IIJ

7/* CoEPFc i

_{w a}

URLL.$

F-CT4,Ç

.EET

uS

S3

rfP,Jc. CPF/C/.'r;

fMfi&J

4i/.S (-PC!&rAJr.s

(9)

ADDENDUM

The numerical example given in the appendix was computed before

*

publication of Grim's recalculations of the hydrodynarnic mass and

damping coefficients of two-dimensional, floating, F.M. Lewis forms.

The new results were calculated on a high speed digital computer and re

more extensive and considered to be more accurate than his 1953 desk

calculator results. _{Both sets of results are based on the same theoretical}

considerations and assumptions.

Figures 13 through 18 of Dr. Grim's 1959 report, which give the

hydrodynamic forces in heaving motions, are reproduced here as Figures

Ak-i and Ak-2. _{The first figure presents the hydrodynamic mass coefficients}

C in heaving motion, where C k2k4/(ÍTB*2/8S) in the notation of this

report. The second figure presents the amplitude ratio in heaving motion0

2*

These coefficients are plotted versus c B ¡2g for a range of section

coefficients (C S/B*H) from 0.5 to 1.0, and for different beam-draft

ratios (B ¡H) from 0.4 to 4.4.

Fig. AA-2 should be substituted for Fig. A-3 Which gives the 1953

computations of . Fig. AA-1 takes the place of both Figs. A-1 and Â-2.

Because of these changes, rows 11, 13 and 14 of Table A-1 are not computed.

Instead, there will be a computation òf ptrB*2/8 . Row 4 In Table A-2

is replaced by values of C _{interpolated from the curves on Fig. M-l.}

ydrodynamic. mass pSk2k4 , row 6 of Table A-2, Will be obtained by

*2

multiplying C by pnB _/8

*TtDje Schwingungen von schwimmenden, zweidimensionalen Korpern", _{Harúburgische}

Schiffbau-Versuchsanstalt Gesellschaft Report No. 1171, September. 1959.

R-791

(10)

'.4 L2 I.0 0.4 0.2 .4 1.2 I.0 0.8 0.6 0.4 0.2

C8:O.5

00 ₀₂

₀₄

w2 ß*/2g

FiGURE 4 A

I a GRIM1S (1959) COMPUTATIONS OF HYDRODYNAMIC MASS

COEFFICIENTS C FOR TWODIMENSIONAL FLOATING

BODIES VN HEAVING MOTION

C:O.6

IO 12 14 16 0.8 C 0.6 02

04 W2B/2g

Io

- 12 14 1.6

(11)

C

.4

1.2 I.0 0.8 as 0.4 0.2 I2 o.é C 06 04. 0.2 0,0 Q,

/

e0

e

C9: 0.7

w2 8/ 2g

0.2 04

'Io

2 ':4 C

Q8

FIGURE Â A - Ib

GRIMS (1959) COMPUTATIONS OF HYDRODYNAMIC MASS'

COEFFICIENTS C FOR TWO - DIMENSIONAL FLOATING

BODIES IN HEAVING MOTION'

I.8

(12)

C '4 2 '.0 02 o

2 FIGURE A A

Ic

GRIM'S (1959) COMPUTATIONS OF HYDRODYNAMIC MASS

COEFFICIENTS C FOR TWO - DIMENSIONAL FLOATING

BODIES IN HEAVING MOTION

B* / 2g l.Ó 12

'4

16

Qe C

(13)

A

0.2

04 w2*/2g

I 0

12 4 16

FIGURE AA - 2o GRIM'S (1959) COMPUTATIONSOF AMPLITUDE RATIOS

LFOR TWO DIMENSIONAL FLOATING BODIES IN

HEAVING MOTION

0.

0.2 0.4 w2 1.0 2

¡4

1.6

(14)

14 1.2 1.0 0.8 0.6 0.4 02 o 1.4 I .2

Io

0.8 0. 0. 0.

FIGURE A A - 2b GRIIAS (1959) COMPUTATIONS OF AMPLITUDE RATIOS

A FOR

TWO - DIMENSIONAL FLOATING BODIES IN HEAVING MOTION

I

C6:O.7

B'/Hz 4.4

3.6 2.6 2.4

A

p

0.4

C8:O.8

B*/H: I 4 .

I6

1.2

-:

T08

- . . o

02

04

(LI2

B'/2

1.0. I 2 1.4 1.6 0.2 04 WE_B4/2 _1.0 _1.2 1.4 1.6

(15)

1.4 1.2 1.0 0.8 0.6 0.4 0.2 2 1.0 0.8

A

os 0.4 0.2 0 0. 1.4 0.2 0.4

w2 8*/29

I I

0.9

4.4 3.6

- 2.8

2.4 2.0 1.6 1.2 0.8 0.4 B*T

24 H:

02

94 W2 8*/29

IO 12 1.4 1.6

FIGURE 4 A 2c GRIM'S (1959) COMPUTATIONS 0F AMPLITUDE

_RATIOS

FOR TWO .- DIMENSIONAL FLOATING BODIES IN

HEAVING MOTION

(16)

TABLE OF CONTENTS

P age

Introduction i

Statement of Problem 3

Supporting Material on Hydrodynamic Coefficients 9

References ii

Appendix: Numerical Example A-i

LIST OF TABLES

Table Page

A-1 Model Characteristics A-3

Â-2 Computation of Hydrodynamic Coefficients of Equations of

Motion for Heave and Pitch A-5

A-3 Calculation of Exciting Forces and Moments A-7

A-4 Solution of Equations of Motion for Heave and Pitch

A-9

A-5 Calculation of Shear and Bending Moment at Midsection A-11

(17)

LIST OF ILLUSTRATIONS

Figure

A-1 Séctional inertia coefficients C as functions of the B/H ratio

and section coefficient C From Prohaska

s

A-2 Ursell's

k4

for circular cylinder (Free surface effect on virtual

mass.)

A-3a Grim's A vs section coefficient, C

A-3b Grim's A vs section coefficient, C

A-4 Virtual mass

ØSkk

vs

(18)

INTRODUCTION

This step-by-step guide to coriputationa1 procedure is presented to

encoUrage the use of a theoretical method, developed by Körvin-Kroukovsky,

to calculate the forces that.äct on a hip tri regular head-seas and result

in heaving and pitching motions and longitudinal bending-moments. References

1 and 2 p-esent the complete development of the theoretical derivation.

Reference 3' is a concise summary and presents certain simplifications of the

calculations.

A numerical example of a complete motion and bending-moment

cal-culation is given to show the setup for desk calcal-culation. Blank tabular

forms that 'divide thè work into five major parts are enclosed and can be

reproduced. Definitions and notes accompany the tablés to clarify each

step. Charts of the supplementary hydrodynamic coefficients are also

included. '

The tables are devised to implement the linearized theory based on

the assumption that wave heights and motions are small enough so that the

'coefficients of the equations of motion may be considered as constant in

time. All dimensions and hyd,rodynmic'çoefficiçnts are for the part of the

hull below the still-water level. The analytically calculated motions and.

bending-moments therefore vary linearly with wave height.

R-791

(19)

-1-II. STATEMENT OF PROBLEM

The analytical calculation of the longitudinal bending-moment at

any seçtion o a ship in regular head-seas requires a solution of the coupled

equations of motion in heave (z) and: pitch (e), given in references i and 2as

a'+ b

+ cz +.d? + eÓ,+ ge Pe

A + BO + ce + D'

+ E± +Gz

e1 (i)

The coefficients of .quation i are defined and evaluated in Tables A-1 and

A-2 in the Appendix.,

-ic,t _i()t

-'

Only the real parts of Fe and Me are to be taken. F = Foe

and M Moe , where' F arid M are amplitudes of heaving force and pitching

moment caused by waves and body-wave interference; a and are phase angles

with respect to the wave. These can be obtained by a íummation of the unit

exciting-forces and moments over the length of the hull for different

posi-tioris of the ship relative to the waves--that is, different ct . ' The unit

exciting-force acting on a ship section is given

ence 3 as

and the' unit

dF dx * pgB q exciting moment

-X,

r

S

24

d is dF/dx, where

the

lòngitudinai

center of gravity (LcG)

The symbols of equation 2 are defined in the notes for Tables A-1

through A-3. -The first term of' -this equation is the changé in buoyancy that

results from the wave pattern on an assumed wall-sided part of the section,

the second term is the dissipative damping-effect,, the third the dynamic

effect due to fore-and-aft asythmetry, and the last the inertial effect of the

in

simplified form

in

refer-r

,?

'-2ny

+ p(sk2k)i e

X

is station distance from

(2)

R-791

(20)

-3-R-791

-4-water flow in vaves. The exponential, exp

(-2uy/x),

where X is the wave

length and -y the mean draft measured down from the'designed water-line DWL,

takes into account the effect of the pressure gradient in the wave.

As in references 1 through 3, two set,s of còordinat.e axes are 'used

to orient the wave and ship motions. One system is fixed in the ship with

its origin at the LCG; the longitudinal coordinate is designated by and is

pósitive in the forward direçtion.. This coordinate system moves with the

ship. The othr coordinate system has its origin at the wave nodal-point

r1 h sin

where x = + Vt + ct andh=wave amplitude,

aridV=forward velocity of the ship. 'At t O, when the wave

nodalpoint

is

at the LCG, X ' ' ' ' ' '

COORDINATE SYSTEM AND NOTATION TO 'ORIENT

AXES

OF

SHIP IN

WAVES :

The vertical velocity çf the water particles in the wave is

dt at

adt

at

dv dX cih

(____ = - - = O

'dt dt dt

= wave length, C= wave cèler:ity,,

preceding the wave crest, the abscissa (x) positive toward this crest.

(21)

Therefore,

and acceleration

where the frequency of encounter c =

dF dx F + Sin ú)t 2.nhc -

w[N()

2 nhc w

.,2n.

cos--2 2

4thc

2

sin( +

t) Vd( psk2k)1 d

J

t) + c w

Equation 2 can be divided into cos t. and sin ct terms:

2.2

V

4nhc

-p Sk2k4 sin 2Tthc

N()

Vd(pSk2k4) co e k

r

22

. V I . ''

41hò

IhpgB' 2 p Sk k

L

4.

The integrated force and moment are then of the form

cos ()t + . 1F i 2 1M F 2 2 ii X -2ny

eX

n,

{a}

= arc tan (phase lags after wave node at the c.g.)

si.n ()t (3) R-791

-5-F

cos(t

o - o) F = M. cos(ot o, - -r) V M F o

fF2

i lj and = l

M

'O

IMJ

i

"V

2 n sin X

(22)

R-791

-6-Equation 3 is evaluated in Table A-3. F , M , o , and 'r are

entered in Table A-4.

The particular (steady-state)solution of equation 1 is

-z = Ze

-e = -e-e

where ₍₌

ze6)

and

(= ee) are complex amplitudes; again, only

the real parts of z and e are to be taken.

z = z cos (cat - ô) and e = e cos (út - c)

The zero subscript signifies amplitude and ô and c are phase lags of ship

motion after the waves. and are obtained in Table A-4.

After the solutions are obtained in Table A-4 for heave (z) and

pitch (e), the longitudinal distributions of the total force that acts on

the ship can be determined. The load equation (reference 3) is

= - 5m( +

)-p Skk

( + - 2Vó) [N d(Sk k )1

241

* dF - _{- Vp} d j

(i + ô ve) - pgB

(z +

e)+ -

(6)

where dF/dx is the total of loads due to waves as defined in equation 2; the

other terms are loads similar to the constituent loads of equation 2 but

result from the pitching and heaving displacements, velocities, and

accel-erations.

Since z = z (cos cüt cos 5+

sin ot sin o)

j =

-z (sin

t cos o -

cos ct

sin

ô)

= -cz

(cos t cos 5+ sin ()t sin 5)

(23)

6

.-e

(sin ct cos e - cos ct sin e)

-oe

(cos ct cos + sin ct sin e)

equation 6 can be expressed as

- -

COS ()t + sin ()t df df1 df dx dx dx (7) where = (5m +

pSkk)(2z

cos 6 +

¿ecos

_e) +

(psk2k4)(2Vce0

sin e) d(Sk k )1

-

[N()

-

Vp

d] (z sin 6 +

e sin £ -

ve

cos e)

dF (pgB*)(z _cos _{5 +} _e _{cos e) +} o o dx and 2 = (6m +

pSk2k)(2z

sin 6 + c e sin, e) - (pSk2k4)(2Ve cos e)

+ E

d(Sk2k )1

N() -

vp

d

I (coz cos 6 + e cos

£ + ve

sin ₎

J

o o o

(pgB*)(z _{sin 6 +}

_{9sin e) +}

Equation 7 and a(df/dx), where a is the moment arm about the

mid-section, are evaluated in Table A-5 for each station. The integration of

df/dx over the forebody or afterbody length gives the shear at midsection;

the integration of a(df/dx) over the forebody or afterbody length gives

the longitudinal bending-moment at midsection.

R-791

(24)

-7--79l

-8-and and Letting

[df

f ₌

J __!x

i dx 1/2.8

[df

f

I

dx

2

.'

dx

1/2

the amplitude of shear at the midsectionis equal to

i m1

J

df

a - dx

dx 1/2 £ m2 =

J

df 2 a -s-- dx 1/2

the amplitude of bending moment at the midsection + m22 . Phase

lags of shear and bending moment at the midsection, with respect to the time

that the wave node preceding the crest is at the cg are computed as:

f m

arc tan _ for shear and arc tan for moment.

f1 m1

These represent lags of maximum sagging-moment and positive shear after wave

nodal-point at the cg.

2 2

(25)

III.

SUPPORTING MATERIAL ON. HYDRODYNAMIC COEFFICIENTS

The virtual mass and dissipative damping-coefficients used :th. the

present calculation are given in Figures A-1 to A-3 of the appendix. If

other virtual mass iand damping coefficients, theoretical or experimental,

become available and are judged more suitable for given section forms, those

may be. substituted.

Figure

A-1,

the coefficient of accession to inertia C' for

analyt-ical forms that resemble ship sections, wastaken from Prohaska (reference 4).

Prohaska derived the sections by conformal transformation of the

known flow

about an elliptical cylinder. The transformation used was

-m -n

zaz +bz

with (m,n) '. (1,5), (1,7), and (3,7). F. M. Lewi.s (reference 5)had invest

igated this transformation

Ñith (m,n)

= (1,3).

His

results were. included in

Prohaska's work. The sections are tangent at the bottoii to' the baseline and

at the LWL to the vertical. Thê. k coefficient of accession to iner.tia in

the z-plane is k2 Ct(TtB*2/8S) The figure shows contours of C' for

vary-ing

beam-draft ratio (B*/H) and section coefficient (C S/B*H).

Figure A-2 is a plot, of the k4 factor to correct.k2 for

free-sur-face effects. It is actually' Ursell 's computation of the virtual-mass

coef-ficient k2k4 for the submerged half of: a circulär cylinder (for which k2 = 1)

in the presence of a free surface (reference 6 and discussion of reference i),

The curve of k4 versusdimensionless frequency (B*c2/2g) approaches thfihity

as either beam (B*) or frequencyof encounter () approaches O. Hòwever,

according to 'Haskind's theory for a thin ship, k2k4 is .about equal to 3.0

for B /2g = O to 0.1 (reference 7). Model tests reported by Grim confirm

Haskind's theoretical computations in this region.

R-791

(26)

-9-R -791

-lo-FiguresA.r3a and A-3b give Grim's (reference 7) values of Ä for the

F. M. Lewis forms, A is the ratio of the amplitude of waves radiated by an

oscillating body to the amplitude of body oscillation and therefore is a unit

of the dissipative damping force. The Ígures presented here are cross-plots

of Grim's original curves of Ä versus B*c,2/2g. Linear interpolation is

per-*2

*

-missible between the given values of B c ¡2g and B ¡H and also between A = O

(27)

IV. REFERENCES

Korvin-Kroukovsky, B. V.: "Investigation of Ship Motions in Regular Waves," Trans. SNAME, 1955.

Korvin-Kroukovsky, B. V., and Jacobs, W. R.: "Pitching and Heaving

Motions of a Ship in Regular Waves," Trans. SNAME, 1957.

Jacobs, W. R.: 'The Analytical Calculation of Ship Bending Moments in

Regular Waves," Journal of Ship Research, June 1958.

Prohaska, C. W.: "The Vertical Vibration of Ships,"The Shipbuilder and

Marine Engine-Builder, Oct-Nov. 1947.

Lewis, F. M.: "The Inertia of the Water Surrounding a Vibrating Ship,"

Trans. SNAÌVIE, 1929.

-Ursell, F.: "Water Waves Generated by Oscillating Bodies," Quarterly

Journal of Mechanics and Applied Mathematics, Vol. 7, 1954.

Grim, O.: 'TBerechnung der durch Schwingungen eines Schiff skorper

erzeugten hydrodynamischen Kraf te," Jahrbuch der Schiffbautechnischen Gessellschaft, 1953.

Jacobs, W. R., and Dalzell, J.: "Theory and Experiment in the Evaluation

of Bending Moments in Regular Waves," International Shipbuilding Progress oport, Qc-tober 1960.

R-791

(28)

-11-APPENDIX: NUMERICAL EXAMPLE

INTRODUCTION

The calculation demostrated in the appended tables is for a 4.8 ft

model of the Series 57 parent hull of 0.8 block coefficient (D.L.Model No.

1919.2), running at 2.394 ft/sec in waves of model length L and height L/48.

The hull lines of the model, a tanker type, and complete calculated and

experimentì. data are given in reference 8.

Although this is a typical sample of the computational procedure,

the results pertain only to the given vessel, speed, and wave length. This

warning is especially apropos when considering phase angles of wave, heaving.

force, and pitching moment. Theoretically, for a doubly symmetric

canoe-form stationary in waves, the maximum heaving-force due to waves should occur

when the wave crest is at midship (90° lag); the maximum pitching-moment

should occur when the wave node is at midship (0° lag). For hulls that are

asymmetric fore and aft, maxi.ma will occur at positions other than midship.

Also, at high speeds the phase lags of the exciting force and momeñt, after

the wave, would be larger than at zero speed, as would the phase lag of: force

after moment,

All tabulated dimensions and coefficients aré for the hull in still

water. These are defined at evenly spaced stations.

Extensive computations have shown that (for ordinary ship forms)

it is sufficient to determine force and moment distributions at 10 or 11

stations. It has been also found that a straight summation of the únit

forces and moments at the 10 odd stations, Of the 20 into which the model is

usually divided, yields results very little different from those obtained by

a Simpson numerical integration of the values at the 11 even stations. The

work has therefore been simplified by using the straight summation method.

R-791 A-1

(29)

TABLE A-1

DEFINITIONS AND NOTES

distance from LCG, positive forward distance between stations listed

B* section beam at DWL H section draft to DWL S section area to DWL WT./FT. =

average loading over interval

-to + y mean draft C' =

coefficient of accession to inertia (from Figure A-l)

z

k2

k coefficient of accession to inertia; k2

p

=

mass density of water; p = 1.937 for f.w. at

70°F p = 1.990 for s.w. at 59°F 6 m = mass loading g =

acceleration due to gravity

Integration of row 15, should give m =

/g, where L is the displacement in pounds

integration of row 16

should give a moment of inertia of J = (/g) k

2

If the weight distribution is not given, motions may

be calculated by substituting

/g for row 15 and (/g) k2 for row 16, assuming 0.2L < k

< O.3L.

(30)

TABLE

A-1

MODEL

CHARACTERISTICS

_{Model !?I.2}

Length L.B. P

,

ft.

4.O

Maximum Beam. ft

Gyradius .k0 . ft

1.147

Draft .ft

O.2it.

L. C. G.

Position.

With

Respect to

Station IO +Oa/fk4jModeI

Displlacem.ent . . lb 44.34.

m A/g

J:mk02

STATION NUMBER I 3 5 7 9 II 1:3

151:7

19

('fl

¿,ft. from LCG

2..09

I.5L

J.-O.(.O

o.iz

:O3

-/-/3z.. -!.SO -i.?J

®

ft:.2

q.l

2.'/

1..17 Ö.3

0.0?'!- Oj

0.71 ¡.7E/

3._ij

5 B"

, ft.

,3 .

.('f

.4I(

.C'

.4'j

.5 .3

H .tt.

.zZ . . ..

S ft.

. .0 .1L.f

.111!-./L&

.,/fL/

.t3

.177... .1 .O3s"

() Wt./ft..

Ib./ft.

3.jÇ

/o.L(

I

ZR

/1.0?..-)l.2..

I1.(O

/o.or

C.33

z.j

y :s/.?., ft.=® +

:

.2.-Y-U3

.q .?.q

.U1J L4

,fl. .5.Q .117

® BH ®X ()

,o1L 1g3'

IIs'

irs' jÇ

irs'

ì5'

u

C

S/B"H

®H .77

.?!

??5

91-:

15

u.c

.30 .733

t/7

B/H

O

-/.0

Z.LLL 2..2.(.c .Z.z6 2aJ,j.. ZU6 2.2..(.

j.9!s

I.oi?

fl

I

)Ll-j

iq7 I7 !7

UIW-I 9 7ç

ií

8"

02 ft2

13!

/,,,

4 ¡.j

SIto

9-LO' 'f Lo

'h.()

31(_

öo

e :: B2C2: (ì3 x ® ,ft2

130 .tII

.417

.6/7

.os'

'5I7

27 O

4) pSkp1r/ex(í),,Ibsec2/ft2

.Ö''

37

¿

'U9

»I(,?

1/57

. ?3

?.?_(. 0ÇZ_

() 8m =

.

Ib-sec/ft2

!It.

.

.35

,3-'37,3

(4

I1Ø

ÛTJ

I37(

.(:) C2: ®X

.tbsec2' .9g3 .

.y,5

'OO$ L

s-q

.5,Ç

: Y37

3.72. /8a6

I7

pqB*pgX®, lb/ft2

2l.f 4o.Io

yo

qV-o.qz.

be'íL 3só

¡Vil

33z. !72?

«j pgB"

x ©11b/ft

4-t.o

5'6 «3

2q2- q6 l533°S33S30

j-Uì3 -iz7

® pB2® X (îj, Ib

93.97 ?1.ì_ 1/7./t )Li5s'

j7

5'5 Z53

IIS7

1.z.6 5.7f 173.oc

(31)

TABLE A-2

c wave celerity, c (gX/2n)h/'2

2.26J5T

frequency of encounter, O) =

(V + c)2n/X

k4

factor to correct k2 for free surface effects (from Figure A-2)

=

ratio of amplitude of waves radiated by oscillating body to amplitude of body oscillation (from Figure A-3)

N()

damping force per unit of vertical velocity

d(pSk2k4)/d

slope measured on plot of pSk2k4 versus

(Figure A-4)

Row 5 for k2k

is not needed unless experimental values of section virtual masses are available, when

it may be of interest to compare the experimental results with the computed k2k4

k4C'(mB

/8s) or

preferable to use the experimental values instead of the theoretical.

When experimental values are

used, row 4 of Table A-2 and rows 11 through 14 of Table A-1 should not be computed. The coefficient B (references 1, 2, and 3) is actually

B

fN()2

d

- 2VD

d (5k2k4)

When the sectional areas go to zero at the ends of the underwater hull, integrating the last term by parts yields a term which is equal and opposite in sign to the second term.

Even when the sectional

areas are not zero at the ends, the algebraic sum of the last two terms is negligible compared with the first term.

Hence, the coefficient becomes

B

=J'N()2

[N] 2

(32)

TABLE

Â-2

COMPUFATI ON OF HYDRODYNAM Ic COEFFICI ENTS OF EQUATIONS OF MOTION FOR

HEAVE AND

PITCH Model

¡I'f.t

Model Speed

, V , ft /sec

Z.3'j.

Wove Length

, X

ft

+.O

Wove Celerity ,c

, ft/sec

h9'5

Frequency of Encounter , w, w2/29

i.15

p92/w5 7.,2$ STATION NUMBER I 3 5 7 9 II 13 15 7 19

()

e ft. from LCG. Z.Ò/

IS

h0

0M.-1/

37 I7O -.2

®

2

Zt/

).f7

Ó.3(,

0.Ói'4-0.7!

I.7'/

t9

® ?w2/2g

.5

fz./

3O 3D

?3b ,'13Ò .'1

Z7

Y,i

® k

1q T 5

5iì

311 5? 'S?

.5U..

.,7b

k2 k4 (see notes)

® pSk2k4, Ib-sec2/ft2

.O(.'.. 7..'

UI .2.I

.zi

.7..S( .2.7.5 .7.-3

13'

i.io'/

(Ï)

's

o3

Rt1

.o34- -Ic'! .231 31û -,23 -,OO

0I

.û7

psk2k4e

Dx®,Ib;

® pSk2k4CC®x®,Ib_sec2 ,g7

,73

Çi/f

_-7...fl

32..7

_-f3f

.1"!

c'0I

o

.031, o

.I/

Ö/

Ho

.I3

.L7

i0

;.-

.I3

lItf

® d(pSk2k4)/dC, Ib-sec2/ft3 vd(psk2k)/de, lb-sec/ft2 .

7f .5',3 -.o7'1

o

O

.10)

3'h

U

¡

%'lO

.37

L/ ¿j

3t0

7(p)

L(

¡2

j .afl'

.1(4.

I(.

.l(.

I(

V7

5i.. .(

@)

.V?? SI 375 .37S

.31S' 3S Yòi

(DIO h3I5' I.t.t?

3171

() N(e)Xe2

¿.077 15TS

.q,7

.133' .00'3

oY? ,a/ !.o

L1L 1Lt

71i3'

([i)

N()-V d(pSk2k4)/dC 1.177

j.i11 ? .37.5' 7s

.37$

C( .L1'

I7 hii

® [r(e)-v d(pSk2k4)/dC]C

7J/öj

Y .z..i4'

.j33

-.1Ç/ ,s' (.Lj1! -i3' .oî'

Coefficients of Equations of Motion a m + (pSk2k4) = 2.,3'1I B [N(e)] 2 gI.S c = e (I,9B*e2) -VE

272. y

A J + (psk2k4e2) =3,010 E e

[Ne) -v d(pSk2k4)/dC]C. ô.r

G (pgBC)

-/a.S7

d D (pSk2k4C) = e

E -2V[e

(pSk2k4)] =

_I/.7j.

g G -Vb =

0. ir

b =

N()

17/ c =

!(pgB) =

172..3

(33)

Exciting force

F

1/F12

+

F22

cos (ot - arc tan

F2/F1)

Exciting moment M =,/M12 +

M22

cös (cet - arc tan M2/%1)

TABLE A-3

h wave amplitude, ft 2.ncJX -wave frequency

-(ßF12/ )E12

excfting -(-heav-i-ng)fórce-components (dF1 2/dx)

M12 =

(34)

TABLE

A-3

CALCULATION OF EXCITING FORCES AND MOMENTS Model

/II7.?-Model

Speed , V

ft /sec

2_.3?-/

Wave Length , X

ft

1O Wave

Amplitude

h

ft

O. O.'

Wove

Celerity. c,,

. ft/sec

'ii.

S-2r/X (.O1

Wove Frequency w0

2rc/X

.

¿/ r

:

lb Ib

ft-lb

F cos[wt-orctan F2/FI] =

.37&l cos(Jt-J7/°)

M

=,/2+M22 cos[wt_

arcton M2/M1} STATION NUMBER I 3 5 7 9 lI 13 15 17 19

(I)

¿

,ft from L.C.G.

i.e4

¡.g,

¡.01 0.(,O

0.JL -O.3

-ö.V/ -/3? -1S

-2.i.f

®

2yr /X , rod. Z.1,-1b

o'ft 1.'.It'j

.'1S

IS7 -1ii -!.Ioc -!.72

-2..3Sh 2.(y

®

sin(2i-/X)

I./çl/

j

7o7 .'L

-Jsv

-271 -.?T-.7o7-.,s

() cos(2,r/X)

-4'/

.1/yy

.iY.

7o7 ?ff

7f,'

s'sv

-.7o7 -.y.fT

2ry/X

.273 ,FJ'I'Y 372.

372

.372. '372. '3L?

'397

'27f "

®

e277Y/X 7V

Í

'$7

.t'?

.I2'

?/ 7o7

'7ô

.r-r

® pgBh

/.jiJf

Z.OS Z.O2I 2.,O?j 2,074

i.J

2.024 Z.OL

/.7L

13s

®

pSk2k4(-hw02) -.130

-7z -.?o-.s'qô-590 -.io.577 -.930'1

® ®+®

.(7 1.533

/.L/3J ¡.93/

1,1.13) /.L/3/

I./9

l.Sa( /.4,7

[N (C) -Vp dSk2k4 /dC] (h

w)

, 37

/4',Ç .1 z.t I z.ì. .

I u.

.o?7 . . 3T(...

® ®X®

L/g(/

I.'3(4'

I.1fr/

/.Oi2-.2.2-3 -,5tj

-J.2j'7-IS'Ö -1.03? -.139

®

m33 -.1i4

Ol.3 û7L

.111

loI

.o'j-.oi9 -.I'7

-.311

@ ®@

.I15

¡.q3

,.o

.,y -.,-qj _pj -_12.j-.giÇ

®

,Ò-.(=?(ô

.223

!.OI?..

I.LI

I.V75'

.S-,-/3I.o37-.SS2.

® @®

,73

.3fg .I3

.Ö

'0I' -.Ö(.S-.0T

.ô1(.

-.I7-.o0

® 3-®

-l.o

-:.ô'fl

»7z

I.315 ¡.330

-. sz -.gç2 -7q

@ @® dF1/dx

.O7

??o

75

27

-.373

.5'TI.O'77

32.

'770

'370

®

dF2Idx _.-lc?3 -.13'

.osr

3r

.j

.71t

.513 -.!ó7 -.L97 -.g'ô

.ii?

oS'7

®

dM1/dx (75' (.'317 1.O4

vsy .oz.

.139

.7zt

H/z..

I.IO7T I.öo .Óo7 3.(9. ( dM2/dx

-i. wr 1.IS,

Ö.7 313

.giS -.330 -.93/

.iqi

1./LS l.ssb

-.1 l'i -.as'4

(35)

where

F,/12

+ F22, M ç,/M12 + M22, a = arc tan F2/F1, 'r

arc tan M2,%1, (from Table Â-3).

Only the real part of the solution will be taken, that is, heave

- ict i(t - 5) / z Re Ze Re z e Z COS ,(&)t -5 o o and pitch

- it

i(t

-e - R-e 8-e

- Re 8e

e cos (cet -o o

is taken as Z1 - iZ2 and

as 81 - i82, rather than Z1 + i22 and 81 + i82, so that Sand c will

be lags.

TABLE A-4

z

maximum single amplitude of heaving motion

80

maximum single amplitude of pitching motion

ô

s

phase lag of heaving motion after wave node at c.g.

-phase lag of pitching motion after wave node at c.g.

[- ict

i(ct-a)

Since the excitina force and moment are given as complex quantities

Fe (-F e ) and

- it

i(t-'r)l

- it

°

- it

Me (-M e )J

in the equations of motion, by substituting z

Ze

and e = 8e

and their time

o

derivatives these equations are reduced to two simple algebraic equations that can be easily solved.

--io

F=Fe

-F (cos a-i sina)

o

Mei'r

M(cos 'r - i sin

(36)

TABLE A-4

SOLUTION OF

EQUATIONS

OF MOTION FOR HEAVE

AND PITCH

Model

I'1I'le2-Model

Speed ,V

, ft /sec

2,39/

Wave

Length

, X ,

ft

Wave

Amplitude

. h ,

ft

OÖ,S-Phase log with respect to wove node at C.G.

Frequency

of

Encounter ,w

u2

'f2..5

F0

/F + F

= .37V

PS

-= arc tan F2/F1

/7/0

from Table A-3

OiR

.-.

o.3L

M0 :/M+M

3, PS-OR

r

arc fan M/M1 1°

PS-OR

= F(coscr- ¡sino-) =

(PS-oR)(ps-QR)

3O z.57 3S'/

(I)

M M0(ÇOST -Islnr) FS

7. 31.o3.L

Mo

77.L/_ l77.3V

p =

aw2 + c + ¡bu -17, FS - Mo

.-S Aw2 ê-C f-IBw from

(?j

(_Mo)(5 -OR)

-?l,Oo

1-25IZO-.

Q

-du2 + g +lew

-Table''A-2

R

-Du2 +G +iEu

3.(j -fr O.4

R

'f.?(

-i--®

_R)(Ps _QR)

-- z4,000

¿

-=®/

z1 - ¡z2

.o3o7 +.öoT#

Z0/Z+Z

ft

Z

-=

*8

= orcton (z2/z,)

1i .S

i

= 8, l82

.00i5/-./32.4

o=

rad

orcton (2/i)

lz. z.

(37)

TABLE A-5 DEFINiTIONS AND NOTES

z

cos (cet - 6)

=

(cos

t cos 6+ sin cjt Sin ô)

E df 1/dx

-E-df 1/dx

sin (cet - 6) =

+-z

.(cos

t si.n

- Slfl

(*)t COS ô

= ZC0S (cat &) =

-(cos

t cos--6- +-s-jn

t js-in

-ô-)-e =

e

cos (t

e)

= e

(cos cüt cos e + sin wt sin e)

o o .

-ô = -e

sin (cat - e) = +

ùe

(cos ct sin e

- sin

t cos e)

o o e

e0cos (cit

=

-e0

(cos

t cos e + sin

t sin e)

1..

Shear-amplitude

;1/f1Z

+ f2

and phase lag. after wave node at the c.g, is arc tan f2/f1

Edf2/dx -df 1/dx

=Eadf/dx = _Eadcï/dx

m2 = E adf2/dx =

-adf/dX

Bending moment amplitude

m12

+

m22

and phase lag after wave node. at thec.g, Is arc tan m2/m1

The sums ofhe loads

df/dx

over the entire length of the hull--that isE(df/dx}-_should equal zero9

-as should

E(adf/dx).

Small non-zero answers are ascribable to round-off

nd fairing errors.

For this

reason, summations ovêr -forebody and. afterbody are averaged to obtain shear force and bending moment at

midships.

-If the weight distribution 6m is not given, let row 5 .= row 4.

The shear force and bending moment due to

mass loading can then be obtained from the total mass (rn), LCG position, radius of gyration of the whole

(k) and radii of f orebody and- afterbody.

(The radi.us of gyration of each half can be approximated by

k/2.

(38)

-TABLE

A-5

CALCULATION. OF SHEAR AND BENDING MOMENT AT MIDSECTION Model ¡?1?.2. Model Speed , y

ft /sec

Wave Length / Model

Length.. XI.L 1.0 Wove Amplitude , h

ft

Frequency of Encounter

w - w2

qa..5-z0.ft 03ø7 sin 8 -.62.L2

cos a-.q9?7

8 rad O'/O/ sin ¿

'?4'93 COSE Shear, lb

,/fi

cos(wt - arcton f2/f1)

.67v casct-qz.°)

B.M ,ft-Ib

/m22 cos(wt

-arctan m2/m1) t1 , lb m,,ft-lb f2, lb

rn2ft-lb

STATION NUMBER I 3 5 7 9 II 13 15 17 19 F:1 9191

-)je

(Ô]

zl

j. I.oS

o,Lô

O.!?.. -ö.3L -o4'/ -/.3L

:.

[O]

I.6f

1.2.0 0.72.

0.2.4/ -ô.2

-o.7z -/..ò

.lf(.

.33/

.3

.37/.

.373 .3

4Ö .33

[O1p5"2'4

O)

OZ.

.2. 2.T -Zu .Z7. .

[O][OflO

,

7 riri

S L

.t3Ç

.33

[0.1 p98

22.sr

I. lJ ô.k3

'ir:zì

32r

YO? .¼.

_37r

ö.9a.. Yo.9. .2 ¿f

3JaÇ

f t

/1.7/

J I

[O][ØO

.

rr

j

.o

t -z!i

-.7j .5

.ri

¿s 9 ¿r L'j1

-/jsr

-oi

y a

a.c .0(-. ir . ¿,ç

[J

. .

-o

-j.

-i.r

-i.-°

1.531., -.

[]

: .

O-.ÖS2.

-.

ç

-.o

-.05' -.0'!

j .ioI

ÖTL

.03'

[E] (2wV805in.c) x .s_z.

.i

ID] (-z,cos8)

x OaO]

j 1.131 reiroroj__'io'p a a. a a

EiIm

JdF,/dx

..

'

4 a

L]LO ' adf /dx

ij

--i./z.!

.ag'9

3f .030

.01'

?5

O2-j.'122.

f. o! +.!9.

-.oq

-.0'!' -.6

'

.0 f -.0'

-.o'!i -.O2S-.00?

()](-2wV8.cose)x_0.oi'

(w28OsinE)

x OO]II

0]

.00

3.2.L!Z,5i!

DI.

02. .02.0 .ö .OU

.ojo

.01

.oio

.O,3

.oif

.63?..

.032-03

.03 .0 .03

03 .ôf ai(

[] (-8sin)

x

-1.1

1I -.fo

-.

l t. .0 .C)

-.0 1 .a

-.2.5

-.3 -.ôì.t -.00

.00

0.

.00 .02.7..

0 O

-.71,3 -.731 .ozÇ

--.

-.

0 df2/dx

-.zfo

I.of

-.

--. tSf i--.?I IW

-.

°

.7o

2.3e . q

i - j.

lt -. i

r

(39)

IO

0.9 c0.8

4C

co

U,

SI

Q

0.6

0.5 o

2

3 *

B/H

Fig. A I - Sectional

inertìa

coefficients G' as

functions

of

the

B/H

ratio

and

section

coefficient

C. From

Prohaska.

4 4O

1.30 LIO

1.05 C=I,OO_

-0.95

(40)

2.0

1.5

0.5

& I I t

i

I I I

till

I I. I i

Ti

I I -t I I

I

i

I I I I I

i

0.5 LO

L5

2.0.

2.5

3.0 *2

_Bw

2g

Fig. A- 2

-Ursel11s k4 for

circular

cylinder.

(Free

surface

effect

on

virtual

mass)

k4

1.0

(41)

2.0

1.5 ¡

1.0 _0.5

o

_2.0

I .5

¡

1.0

_0.5

o

Fig. A-3a

GRIM'S

¡

vs SECTION COEFFICIENT B2*gW2O25 1

(

BH

I.33 Bw2

0.35 2g

B/H

3.0

2.0

_I.33L -I

Bw2050

2g

BH.

BH

3.0 Bw2

0.75 2g

ji

0.4

0.5

0.6

0.7

0.8 '0.9

1.0

0.4

0.5

0.6

07

0.8

0.9 t.0

Cs

S/BH

C =S/BH

(42)

4.0

3.0.

¡

2.0

I,. O

o

4.0

3.0 ¡

2.0 l.0

o

B'H

3.0

2.0 _{I .33}

_0.4

&I 2

2g

0.4

05

0.6

0.7

08 O9

C,s

S/B'H H

Fig. A-3b

GRIM'S

¡

vs SECTION. GOEFFIÔ lENT , C5

l.0

0.4 a-0.5

0.6

0.7

C5 u S/B' 1.1.

0.8

0.9

I .0 BW2g 1.0

2g

3.0

NOTE CHANGE VERTICAL IN SCALE

20

jA

0.4

H

Bw2

B'H=

2.5

.

2g

3.0

2.0

1.33

04

(43)

STA. N O.

19

17

15

13 II

9 0.3-

o.,'-.

o

_.,

FT.

Fig. A 4

-Virtual mass pSk2k4 vs. ¿

(44)

Length L.B.P., ft.

TABLE A-1

MODEL CHARACTERISTICS

_Model

Maximum Beam ,ft

Draft ,ft

L. C. G;

Position

With

Respect

to

Station IO

-Model

Displacement, L Ib

Gyradlus. k0 . ft

m:h/g

Jg k STATION NUMBER I 3 5 -7 9

II

3 15-17 19

1ft. from LCG

® ® B,ft.;

V

® H,ft.

V

®1'2

-Wt./ft., Ib./ft.

(13

y

SI B', ft g () + ()

® B"H

-Cs

SIB"H g

+ ()

B"/H

® +®

® C'!

2 2 --.

® B'tC © x

-,

ft2

V

-«

pSli2.pY/.X6i ,lb-sec2/ft2

-, Ib-sec2/ft2 V

6J (Sm) ¿ ®x ® ,lb-sec2

-1D pBpgX®, lb/ft2

pQBC ®X ®, lb/ft

V pa

(13 X (íJ. lb

(45)

TABLE A-2

COMPUTATION OF HYDRODYNAM IC COEFFICIENTS OF EQUATIONS OF MOTION FOR

HEAVE AND

PITCH Model Model

Speed ,.V

ft /sec

Wove Length

,

, ft

Wove

Celerity ,c, ft/sec______

Frequency

of

Encounter, , oit/2g pg26i3 STATION NUMBER -I 3 5 T 9 II 13 -15 17 19

1()

(îj

¿ , ft.- from LCG

t,

®

Bw1/2 g

®k4

-k.

14 (see flotes)

-® pSk2k4, Ib-sec2/ft2

OE Ibsec2

pSk2k4Ca(ix(îJ

® pSk2k4C2îx,Ib_sec2

-d(pSk2kj/dC, Ib-sec2/ft5

6j Vd(pSk2k4)/dC,lbSec/ft2

®A2

6i

NCC) lb-sec/ft2

® N(dxe2

1 N(e)-V. d(pSk2k4)/d( -N()-V d(pSII2k4)/dC] e -Coefficients of EquatIons a am + (psk2k1) a A a

J +

AC (pShI k4 ¿2) a d a D v A( (,Sk2k4C) a

bsACN(C)a

of Motion -B a AC [N(e)] ¿ c a e pgB*e2) -VE E a V d(,Sk2k4)/dC]C'. G a A ¿ '(pa BC) a 5 E -2V[AC T(J,$ksk4)] a

g a

G -Vb a

ca_AeJ(pQB)'

(46)

-TABLE

A-3

CALCULATION OF EXCITING FORCES AND MOMENTS Model Model

Speed . V , ft /sec

Wave Length , X V Wave

Amplitude , h

ft

Wove

Celerity, c,,

, ft/sec

2r/ X

Wave Frequency w0

2vc/X

w02 F

=/Ç2

cos[wt-arctan F/F,]

M,,/22co8[wt_

arcton M214b11} F1, Ib F21Ib

M,, ft-lb

M2, ft-lb. STATION NUMBER I 3 5 7 9 II 13 IS I? 19

(D ¿

ft

from L.C.G. V V V V

(D 2,r(/X ,

rad.

® sIn(2v/X)

® cos(2,4/X)

2iry/X

V

j e">'

(, póB"h V pSIi2k4(-hw02) V [N () -Vp dSk2k4/dC] (h we)

.® ®x®

V V V V V V V V

®

V V V cí

®x® dF2/d,i

x (D

dM, V V

X (D

dM2 /thi V V

(47)

TABLE A-4

SOLUTION OF.

EQUATIONS

OF MOTION FOR

HEAVE AND PITCH

Modet

Model

Speed

V , ft /sèc

Wove Length

k

ft

Wove

Amplitude

, h

ft

Frequency

of

Encounter w

w2

*

Phase lag with respect to wove node at C.G.

F0./F+F

.

PS

r

arc tan F2/F,.

from Table A-3

. OR .

M0 /M+ M

.. PS-OR

r

arc tan M2/M1. PS-QR F0(CO8cr-ISlno-)

(Ps-QR)(Ps-QR)

Mr,(cosr-lslnr)

MO

.P_aw2+C+Ibwc

. S Aw2 + C + ¡9w from

(s

-o)(s -OR)

Q

-dw2 + g + leo,

Table A-2

R . - Do + G + fEw ?R

MP-FR

(P-R)(Ps-oR)

-i

.

zo=v/z+4

f'

-a

arctan (z2,';)

(/®= 8II82

rad ¿

orcton (2/,)

(48)

TABLE A-5

CALCULATION OF SHEAR AND BENDING MOMENT AT MIDSECTION Shear Ib s

cos (ut - arcton

B.M., ft-Ib: cos(wt.orctan m2/m,) t111b ft-lb lb

ft-lb

STATION NUMBER I 3 5 7 9 II 13 IS 17 19

[.)][

]

O ¿

ft

from LCG

2 o ft from Sta. IO

O 8m

Table A-1 ,

ts)

4 pSk2k

Table A-2, 6)

6 pB (Table A-1, (I))

-.

7 fN)-VPdSkak4/4Toble A-2,@)

_{Q ®xQ (Table A-1,}

IO (7)xi

(Table A-2, Il

(w2z0cos8)x ®

s

x()

2

w280 cos4x 8

8 13 (2wVO0sIne)x 4 4 (-z0cos8) x L 15 (-90cos)

X 9

s 9 -16 (-wz.sIn8+v8.cose)xO X 7 17

4w80sIn) X

IO .xtO dF /dx (Table A-5, 19 Z613 through (íJ dfgldZ

x (j)

o df /dx (w2z,sln8)

X O

s

xQ

(w2O0sine) X O s

O

3 (.2wV80cose)x O

O

(-z0sln3)

X O

O

(-01sln)

x O

O

(wz.cos8+vO.sinc)xQ)s

O

EJ (wO0 cose) x . dF2/d (Table A-5, Q3) Z 21 !hrouqh

)

df2/dx

XO

0df2/ds Model Model Speed , V

ft /sec

Wove Length I Model

Length , X/L Wove Amplitude . h . ft Frequency of Encounter , w u2 z0 .ft sin 8 cos 8

9. rad

sin cos