hydrodynamic coeeficients

Ocean Engng. Vol. 8, pp. 25-63. 0029-8018/81/010025-39 $02.00/0 Pergamon Press Ltd. 1981. Printed in Great Britain

H Y D R O D Y N A M I C COEFFICIENTS FOR VERTICAL CIRCULAR CYLINDERS AT FINITE DEPTH

TARIK SABUNCU* and SANDER CALISALI" U.S. Naval Academy, Annapolis, MD, U.S.A.

Abstract--Hydrodynamic coefficients for vertical circular cylinders at finite water depth are obtained and presented for different depth to radius and draft to radius ratios. A summary of equations for computer application is presented. Limiting values for heave added mass for zero frequency are discussed. A matching technique is used to satisfy the continuity of pressure and normal velocities.

NOMENCLATURE a radius of the circular cylinder a~j added mass or moment coefficient bo damping coefficient d gap between the cylinder and the bottom of the sea D volume of the cylinder D ) a constant to satisfy equation (2.12) g gravitational acceleration h depth of water Hk J~ + i Yk Hankel function i x / - l I~ modified Bessel function, first kind of order k Jk Bessel function, first kind of order k K~ modified Bessel function, second kind of order k too, mq as defined by the equations (2.22) and (2.23) n normal vector r position vector t draft of the cylinder v velocity x, y, z cartesian coordinate system Yk Bessei function, second kind of order k

Zk function defined by equations (2.18) and (2.19) &j Kronecker delta 0 angular position in cylindrical coordinates ;, fluid density ~p, q~ velocity potentials 'J O~2a/g to radian fiequency

angular velocity of pitching motion

I. INTRODUCTION

VERaICAL circular cylinders are used in many oceanographic applicat ions such as buoys,

drilling rigs and ins t rumentat ion platforms for their simplicity in construction. The available data on their hydrodynamic coefficients is limited and the present numerical procedures are based on finite element solutions or the numerical solut ions of integral equations. The present method is relatively simple to formulate and the solut ion requires a very short computer time. The hydrodynamic coefficients such as added mass, damping coefficients for heave, sway and pitch motions are formulated, and the results for different depth to radius and draft to radius values are presented in graphical form.

*On leave from the Technical University of Istanbul, Turkey. tUniversity of British Columbia, Vancouver, B.C., Canada V6T IW5.

25

26 TARIK SABUr~CU and SANDER CALISAL

Havelock (1955) theoretically determined the added mass and damping coefficients for a sphere. Kim (1974) studied the hydrodynamic coefficients for ellipsoidal bodies oscillating at the free surface. Wang and Shen (1966) calculated the added mass and damping coefficients of sphere in infinite and finite depth of water. Garrison (1975) gave the general formulation of these coefficients for arbitrary forms in terms of distributed singularities and the numerical results for a vertical circular cylinder in infinite and finite depth of water. Bai and Yeung (1974) calculated added mass coefficients for horizontal and vertical cylinders. Bai (1976) gave the added mass and damping coefficients for axisymmetric ocean platforms. Kritis (1979) had applied the hybrid integral method of Yeung (1975)to axisymmetric bodies and gave numerical results for a circular cylinder.

The various methods developed for the solution of three dimensional axisymmetrical bodies can be summarized as follows. In the first group of methods Sources and Multipoles are distributed inside the body and their strength is calculated to satisfy impermeable boundary conditions of the body. The second set of solutions distributes the singularities at the surface of the body and an integral equation is used through the use of Green's theorem. The solution of the integral equation gives the strength of the singularities. Thirdly, the finite element formulation is used to find the velocity potential at specified node points. Possible combinations such as the hybrid method referred to above also exist, combining the above solutions and reducing the computational effort. The present formulation follows the general procedure outlined by Garrett (1971) who studied the scattering of waves at t he presence of circular docks.

Although it is of major concern to Naval Architects and ocean engineers very few data exist on the hydrodynamic coefficients of circular cylinders. Serving to this aim graphical results covering a large range of parameters and summary formulas for computer applications are presented in this paper.

2. F O R M U L A T I O N A N D S O L U T I O N O F T H E P R O B L E M

The coordinate system Oxyz is shown in Fig. 1. The origin is at the bottom and z is positive upwards. The region O<z<h is assumed to be filled by an incompressible fluid of

Ly

I

c E

I z=d

z. D B

o A FIG. 1.

Hydrodynamic coefficients for vertical circular cylinders at finite depth 27

density p. The undisturbed free surface is at z = h. The radius of the cylinder is a and draft T = h -- d, where d is the gap between the cylinder and the bottom. The standard small motion assumptions are made and the motion is periodic with frequency co. An irrotational flow is assumed to exist given by

• (r, 0, z; t) = Re{Up (r, 0, z) e -i®t} (2.1)

where r, 0, z are cylindrical coordinates and 0 = 0 corresponds to the positive x axis. 0 (r, 0, z) is a complex spatial velocity potential which satisfies:

a I 1 ~ 1 ~ 8s) ffrrS + - + - - - + • = 0 (2.2) r ~r r 2 ~30 ~

in the fluid region.

The boundary conditions for different motions of the cylinder are given below.

At the free surface

120 2 w

% - - - - t p = 0 a t z = h (2.3) g

% = 0 a t z = 0 (2.4)

% = v . n + t~. (r × n) (2.5)

on the body surface where n is the normal to the surface. Radiation condition is satisfied by keeping only the outgoing waves.

For heave motion

~rp ~z

and ~rp a r

i

- - V n o n z - - d (2.6)

-- 0 o n r = a f o r d < z < h (2.7)

where V n is the velocity of the cylinder due to heave motion alone.

For sway motion

~z - - 0 o n z = d (2.8)

B~P - - V~ cos 0 at r = a for d < z < h (2.9) c~r

where Vs is the velocity of the cylinder due to sway motion alone.

28 TARIK SABUNCU and SANDER CALISAL

For pitch mot ion

?UP -- ~ r cos 0 on z -- d (2.10) ?z

~Up = -Q (h -- z) cos 0 at r = a for d ~ z ~, h 8r

( 2 . t l )

where ~q is the angular velocity of pitch motion.

Following Garre t t ' s method the fluid is divided into two parts, namely the interior par t A B O D and the exterior par t A C E F as in Fig. 1. The appropr ia te solution of velocity potential in each region is found and the solutions are matched at the boundary so as to have the continuity in q~ and its first derivatives satisfied. The velocity potential in the interior domain is expressed as

Up = D k [q)kp (r, z) -47 Cpk,, (r, z)] cos kO for 0 < r < a 0 < z -< d (2.12)

where k = 0 refers to heave mot ion and k = 1 refers to sway and pitch motion, q~kp and q~k h are the particular and homogeneous solutions in the interior region.

The particular solutions for different motions are given as

qg°p (r, z ) = - ! {z z - r ~ ) f o r heave* 2d2\

(2.13)

qgp (r, z) = 0 for sway (2.14)

1 ( z2 133,) for pitch. (2.15) ~o b ( , ' , z ) = ~ /~ r -

The part icular solutions satisfy the respective kinematic conditions at the bo t tom of the cylinder and the bo t tom of the fluid for r < a given by the equations (2.4) to (2.11). D k is a constant o f dimension [L~/T] and is chosen to fit the particular mot ion of the cylinder.

In the exterior region the velocity potential is given in terms of an eigen function ex-

pansion.

q~ = D ~ q~ (r, z) (2.1 6)

tp, = Bk0 Hk(mor ) Z0(z ) + ~ Bk¢ K~, (mcr) Z¢(z) q = l

(2.17)

where Ilk is the Hankel function of the first kind of order k -- 0. 1 and Kk is the modified Bessel function of the second kind of order k =: 0. 1. Bk0 and Bkq are complex unknowns. The or thonormal Zi functions in the interval 0 ~ z -~ h are defined as

*A particular solution for heave motion is suggested by Professor J. N. Newman of M.I.T.

with

Hydrodynamic coefficients for vertical circular cylinders at finite depth

Z o (z) = No 1/2 cosh (mo z)

Zq (z) = N o*'' cos (mqZ)

1 [ sin]~ c?_ m )l No = ~ 1 -I- 2moh 3

29

(2.18)

(2.19)

(2.20)

(,0 2 mo tanh (moh) -- (2.22)

g

0) 2 mq tan (m~h) = -- _ (2.23)

g

To satisfy the continuity of velocity potential at r = a for 0 < z < d the homogeneous solution defined above is used as follows

Cp (a, z) + Ch (a, z) = C , (a, z) (2.24)

~-r [~°kp (a, z) + cpk, (a, z)] = ~-cp*, (a, z) (2.25) o r

f o r 0 < z < d . The value of homogeneous potential at r = a can be expanded as

Cpkh (a, z) Ao k (ndz) = -~- + A*. cos . . . . (2.26)

For continuity of potential function coefficients Aki'S a r e obtained using (2.24) as:

d

(:) Ak" = 3 [¢¢', (a, z) -- Cpkp (a, z)l cos q z dz 0

(2.27)

The particular solution is a known function, therefore Ak. can be expressed as

(7) A*, = ~ ~o*, (a, z) cos dz -- ~*,, (2128)

where m0.and mq are the solution of the equations

Nq = 1 [ 1 + sin_(2mqh)l (2.21) 2 m,~h _1

30 TARIK SABUNCU and SANDER CAUSAL

where d 2f o~*kn = (t (~kp (a, Z)

0 cos (nnz/d) dz, for k --: 0.1. (2.29)

The interior homogeneous solution can be written now as

¢pk h (r, z) : - 2 - Ak° (~)k + Z~° Akn __Ik(nnr/h ) cos (n -h ) (2.30) .= l Ik(nrca/h)

i n r < a , O < z < d .

Multiplying the equation (2.25) by Zq (z) and considering that O~fi3r (a, O, z) = V cos 0 for d < z < h and integrating between limits in 0 and h one obtains

d

f ~q)kh (a, z) • Bko (h mo) Hk' (moa) = 0

Zo (z) dz ÷ ~k o (2.31a)

d

B% (m,h) . K'k (m~a) = f -~r ~kh (a, z). Z~ (z) dz + ~ 0

(2.31b)

for q = 1, 2 . . . .

where d h

[~ko ~---- J ~ Oq)kp(a'z)~r Z°(z) dz+ I f o d

V Zo (z) dz (2.32)

d h

~kq : f ~(pkp(a' Z) Zq(z) dz ~- [ f ,~r ~ v z , (z) dz (2.33) 0 d

Where V = 0 for heave, V = V~ for sway and V = ~ (h -- z) for pitch motion. Inserting the values of ~pk, from (2.17) into (2.27) and ~0kh from (2.28) into (2.31a,b)

we obtain

Ak. = { Bko • n~ (mo~)" Eo. = r. Bk~ • 0~1

f o r n = 0 , 1,2 . . . .

Kk (mqa) Eq, } -- ~k (2.347

Bko kdAko 4a

oo i • Eoo + Z Akn nn I k (n~a/d) . Eon}

n=l 2 Ik (nnald) + (mob) Hk' (moa) (redo ttk' (moot (2.35a)


kd__ Eqo B k = { 4a

• A*o + ~..4". nx I,' (n•a/d) Eq. } .= l 2 l k (nna/d) q_ f~*q

(rn~ K'k (rnqa) (rnqh) K k' (m,fl) (2.35b)

The equations (2.34) and (2.35 a, b) form a coupled system of equations where Akn and Bka are the unknowns. Substituting Bkq from (2.35 a, b) into (2.34), a set of linear equations for A, are obtained.

Thus:

"yd.i Akn ~-- - - hkn for n : O, 1, 2 , . . . J-O

(2.36)

where

f H k (i~loa) Eon " Eoj ~ K k (mjo) Eqm " Eqj] Yk"l = { LHk' (mo a) moh q- q . ! K k ' (mqa) mql~ d U~.- 8,,1} (2.37)

= (H, (moa) . §~" E~, ~ ~:E,,,,'~ (2.38) hkn \ H k' (moa) mffl d- q-t m~h / -- at*,,

where 8 , , j kronecker delta,

Ukl ----- -~ lk' (flta/d)/lk(jrca/d), fo r j = 1, 2, 3 . . . . (2.39)

kd Uk° = 4a ' (2.40)

d

2 f Zo(z ) cos (nnz/d) dz (2.41) Eo. = ;t 0

3. H Y D R O D Y N A M I C F O R C E S A N D M O M E N T S I N T E R M S O F T H E V E L O C I T Y P O T E N T I A L

The forces and moments are defined by the integrals taken over the body surface as follows:

F = - - p ~ ~ • n a ~ (3.t)

M ---- - - p ~ ~ • (r × n) ,is (3.2)

32 TARIK SAnUNCU and SANDER CALISAL

The added mass and damping coefficients are calculated in the following manner.

an + i b11-- Dlff pD COpD pD ~p~(r, O, z) nl as (3.3)

aP--~-i b l e - - olff UP~(r, 0, z ) ' ( r × n) ds p Da COp Da p D

s

(3.4)

a~2 @ i b22 - - Doff p-D cop D p D ~pO( r, O, z) n~_ as $

(3.5)

+ , = f f ~pl(r, O, z)(r × n)ds p Da 2 cop Da 2 p D s

(3.6)

q6a + i b s x _ D ff Upl(r,O,z)n~ds p Da COp Da 9 D

$

(3.7)

where D is the volume of the cylinder, axx and bn are sway added mass and damping coefficients, ale and ble are sway induced pitch added moment and damping coefficient, a22 and b22 are heave added mass and damping coefficients, a6x and b~x are pitch induced added mass and damping coefficients. Here ~1 in (3.3) and (3.4) is the potential for sway motions, and ~1 in (3.6) and (3.7) is the potential for pitch motion.

For reasons of symmetry

ale : ae l

ble = b61

a44 = a66

b44 = ha6

The heave added mass and damping coefficients in particular are given by

9-D CO pD pD ~p° o (r, d) + Cp°h (r, d) dr 0

(3.8)

Using q~op and ~OOh from equations (2.13) and (2.30) into (3.8) and integrating

p~ COpD h - - d 2 - - 8 + 2 A°° + - . (3.9) rt ~=1 n Io(nna/d) j


Similar calculations are made for other added mass and damping coefficients and these formulas are given below.

~ Kl(moa)[sin(mJO -- sin(mqd)] "~ a~A + i bn _ B'o H~(m°a) [sh(m°h) -- sh(m°d)] + ~ Blq pD co pD [ Nolt2mo(h -- d) q..l Nql/~mo(h -- d) j

(3.10)

] pOa 2 O~pDa~ (h - d)a 2 a[t"'N-~"znl(m°a)" Bl° + q-IZ t o No-"2gl(moa) BZ o +

--[~ (d)' (1- 6kd]l-[a~'~, l ( d ) , ~o (-1)" 12(n~a/d)]} + 8 Alo + - Z A.

lZ o " t" rl l I (nna/d)

.al, + i b , , _ d ~rtoN.-'"H,(moa) B'o + ~: ,oN¢-*K,(m~a) B'0I + p~Da ~pDa h ---d LL ,,-, ~ J

_ A~ 0+_1 y (-- 1) n AZ .l.,(nna/d) n .-1 n Ii(nrca/d) '

(3.11)

where

t 0 = { ( 1 - ~)sh(moa)mod + ch(moh) -- ch(mod )

(rood) 2 3

tq = { ( l -- h) sin (mod) moa

cos (moh) -- cos (m¢d) ], (3.12)

(mod) ~ I

ael -F i b61 -- [d~2 f Blo Hl (moa) No-l~z sh(m°h) -- sh(m°d) pDa ¢opDa \ a / [ mo(h -- d)

+ {. slo &(m,,r) Uq-l,3. q m l

sin mqh -- sin mqd

mq(h -- d) (3.13)

4. L I M I T I N G V A L U E S F O R H E A V E A D D E D M A S S A N D D A M P I N G C O E F F I C I E N T S F O R e ) = 0

As the frequency of the motion co approaches to zero the values of m0 given by (2.22) go to zero while meh values tend to qn. The coefficients of the linaer equation (2.36) become real and the formulation corresponds to the case where rigid boundary conditions exist at the free surface. The only imaginary value is the imaginary part of Ao which is equal to

7ta 2

lm ,4o = Aio . . . . (4.1) 2dh

34 TARIK SABLINCU and SANDER CAL|SAL

From which, using (3.9) the damping coefficient is immediately obtained as:

lim b-3~ = na°" (4.2) (o-->0 o)pD 4h(h - d)

The above result can also be obtained by using the Haskinds relations given for infinite depth by Newman (1976). The equations (2.36), (2.37) and (2.38) for heave motion are

A°o = - - 7. 7o~A°/+h% (4.3)

7O

7. 7°m A°/ = h°,, (4.4)

{ [Ho (moa) Eo~'Eoj + ~ Ko (mja). Ea~EQj] } yo/ = L-Ho'-~oa) " moh q=, Ko'(mja) m a h J U° - - shy

(4.5)

h°° : 110 ' (moa) -,-no-h + -- -- (4.6) q=l m~h 2 \ d l l

hO = i f Ho (moa) [5°oEo. . . . . . . ~ {~% Eq~ } 2(--1) ~ Ho'(moa) mob + q-17. ----rn~h - ~ (4.7)

The above equations for very small mo values can be written as follows

70i = ~(0, j ) + i m0~t~ (0, j ) (4.8)

7.j = a(n,j) + i m o' ~(n,j) (4.9)

where

4jd2h ~ l~(jna/d) ( _ iy_ . ~ Ko(qga/h) :t(n, j) = ~"/ + r~ 2 Io(jxa/d ) q-i K,(qga/h)

q (sin (qnd/h)) ~ [(qn)~ -- (nh)*] [(qd) 2 -- (./h) ~]

ho = {2n s ah'd s a ~'=, Ke(qna/h)'lKl(q~a/h) qS (sin q~d/h)'--(~-- 2\d]! (a'~'~ } - , ] {!~ ~aS}~- (4.10)

f 2ah~ ~ Ko(qlta/h) 1 (sin qnd/h) 2 h . : ~ ~ ( - - 1 ) " q=, Kl(qna/h) q[(qd) 2 - ( n n ) 8]

2(--1) ~ } { (-- l)~al~o2 } (nn) ~ + i 2nhn I •

( 4 . l l )


After these preparat ions taking the limit o f (4.4) for very small m 0 values and using (4.8), (4.9), (4.1 I) we obtain

~- (Ahl + i Aij) [~ (n,./) + i m4o [~ (n,j)] = h,. + i m~o hi, (4.12) I R I

Here subscript r and i refer to the real and imaginary part o f the values. Separat ing into real and imaginary parts

[A n rn4o [~ (n, j ) + Aii ~ (n, j ) ] - = hi,, m2o 1 - 1

oo

X [A n ~(n,j) - Ai~ m% [3(n,j)] = h,,, i = 1

Taking the limit o f the right hand side of the first equations as m 0 -* 0 we obtain

and

cza

Y. Arj [~t (n, j) + rn4o ~3 (n, j)] = h,n

A ij ~ (n, j ) = - - m ' o [3 (n, j ) A ~j

One can see therefore that all Aijs, j = 1, 2 . . . have real values.

As a special case take a floating disc. This can be expressed as a limiting case where d --~ h. One can show that for this case the equat ion (4.4) has a diagonal matr ix and the unknown A . ' s are given by

AO 2(-- 1 )" = - - n = l , 2 . . . (n~)'

! ' ) (4.13)

since all sine terms in equation (4.12) are equal to zero. The added mass o f the circular disc in heave mot ion for to -~ 0 is obtained as

( ! ) I ( ! ) (~) ~ 1 ll(n~a/h) t lira an2 _ pn 1 1 4 F (4.14) o~--~0 a 8 3 + 8 -- ~ , - t n s Io (nxa/h)

In ano ther limiting case where d goes to zero, again all the sine terms in equat ion (4.12) are equal to zero and the coefficient matrix becomes diagonal. In this case one can show that as to and d go to zero the heave added mass in fact is infinite. According to this present theory one can conjecture that f o r d < h and at finite h the heave added mass remains finite at low frequencies but goes to infinity for infinite depth.

36 TARIK SABUNCU and SANDER CAL|SAL

,.J

Q I

"o "o ~ 0 5

H y d r o d y n o m i ¢ c o e f ' t t c i e n l - s f o r heove 2 . 5

d-- ~ GOlf f ISSON

2 0 ~-- • K r t l i s

- n Tr , is 1 , h e o r y

-\ • - - A d d e d moss coe f "F

[3

t ~ -- \ a

• " ~ - a D o m p i n g coedS

- ~'A[:~l k

I .... I L [ ~ , , ~ _ i . , - , I ,,I ,-, I 0 12 2 4 3 6 4 8

Omeqo . Omeqo *, A / G

F I G . 2 .

5. NUMERIC SOLUTION AND RESULTS

The specific formulas used for the calculation of added mass and damping coefficients are given in the Appendix. The required routines for the calculation of Bessel functions and solution of linear equations are obtained from the IMSL computer library. The computations were done at the U.S. Naval Academy. The results for heave were first compared to the results published by Garrison (1975) and Kritis (1979). The results are given in Fig. 2. Added mass values obtained by this theory compared well with those of Kritis while Garrison's numbers are observed to be higher. The damping coefficients for heave are observed to be less than the values reported by Kritis and they were observed to agree better with those of Garrison.

The heave added mass and damping coefficients are also compared with the experimental data reported by McCormick et al. (1980).

Figure 3 shows experimental data and theoretical values computed for co =- 3 rad/s, In this diagram added mass value is non-dimensionalized by the total mass of the cylinder of height equal to the depth of water. This is expressed in the diagram as ANU/MH. The experimental data are observed to remain above the theoretical curve; the best correlation is observed at about depth/radius values equal to 7.

Experimental and theoretical damping coefficients are compared in Fig. 4. Experi- mental values remained above the theoretical calculation while showing similar trends, This discrepancy can be possibly explained by the viscous damping neglected by the theory. The numerical results are also tested with those reported by Bai (1976) and the results for added mass and damping coefficients are observed to agree within a derivation of 3 ~. In all calculations the infinite series are represented by 20 terms. The reader should note that the capital letters used in these figures correspond to small letters in the nomenclature and all variables are nondimensionable.

0

I

c

54

3G

1 8

0 45


Added moss rat io for d r a f t s T = 3 F T , T - - 3 . 5 FT o

~ g

rj Exper imenta l o (McCormick)

E3

I e

Th,s theory ~ " J ~ l ~ . ~ . • ~ •

Omega =3 .0 rod/sec

I i I l ~ l I 5 2 59 6,6 7.5

Oepl-h / radius

FIG. 3.

@0

37

o _(2 #

i-

o ~ c}

-g

E C3

65 Damping c0ef ficient

5 2

3 9

2 6

L3

0 4 5

E x p e r i m e n t ( M c C o r m i c k )

- - Theory

Omega = 3.0

I I I I I I 1 I l 52 5 9 6 6 73

H / K * D ~ ( H - D ) * 1 0 0

F I G . 4.

8.0

3g TARIK SABUNCU a n d SANDER CALISAL

60 Ver t i ca l cy l inder heave H / A = 0 5

64

4~

32

L

"~.A

~ o ~ o ~ o , o ~ o ~ o ..... o

I i ! ! ! I i 0 1.2 2.4 3 6

O m e g a - ~ O m e g o * A /G

FIG, 5.

A T / ~ = O I

0 Tt/A toO,3

o-" [

,I 8

40 Ve r t i ca l cyhnder heave H / A = I

32

24

o

<

L O ~2

• T/ /~. - 0 . I

0 T / A - 0 . 3

[] T/ .& " 0 . 5

I':" i: I C l I - 2 4 36 48

Omega w- Omegc * - A / G

Ftc~. 6.

6

Hydrodynamic coefficients for vertical cirCular cylinders at finite depth

Vert ica l cylinder heave H / A = 2 25

EO

e; .u

o = u

o E

5

• T / A = O I o T / A ==O.3 E] T / A - 0 .5 • T / A = I + T /A -- 1.5

.

1.2 2 4 36

Omega ~Omega ~ A/G

F I G . 7 .

=,, •

. I

48 6

3 9

5.6

~" 42

==

2 8

7.0

].4

Verl" icol cyl inder heave H / A = 3

• T/A-O.3

o T / A - 0 . 5

[] TIA-I

& ~ A . . . . . A • A . - - - - - - ' ~ - " - A

3~j,,,,~ 0 '~ .'1 O 0

I I I ~ I l [ 2 2.4 3.6

Omega ~ Omega -M A/G

TIA-2

- - &

(3 ,

1 t a 8

Fla. 8.

6

40 TARIK SABUNCU a n d SANDER CAL1SAL

!40

112

o ~ 8 4

{

E

2 8

V e r t i c a l c y l i n d e r h e a v e H / A = 5

\ -- N, A

• [ / A = O I

; T / A = 0 3

E] T / A = O . 5

" T / A ~ I

+ T / A - 2

0

-t I 3 6 4 . 8 I 2 2 4

O m e g a ~ O m e g o ~-A/G

FIG. 9.

,,,o

I ~ ! 6

9e r-

4-

_o

t)

~ 4 8

2 4

V e r t i c a l cy l i nde r heave H / A ~ tO

.I:. •

\ \ •

• T / A = 0 I

q T / A : 013 [1 T / A : O . 5 • T / A = I

T / A : 2 T / A z 3

• - - &

- -"~-~'L~-'C-. ~ C- ~ ~J "C --~ ............ ~; :-----~

12 2 4 36 4 8

Omega * O m e g a * &/G

FIG. 10.

Hydrodynamic coefficients for vertical circular cylinders at finite dep th

Ver'l'ical cylinder heave deep wa?er 12

9.6

7.2

¢:8

2.4

• T IA =OA

,- T / A = 013

T I A = 0 . 5

• T I A = 3 \ \

• ,,I,

° " - - ' ¢ " "--" - 1 " - ' - - - I - ° - ' - - t " I 1 I 1 2 2.4 3,~,

Omega * Omega* AIG

FIG. 11.

• ,, •

D

48

41

3

E

12 8

96

Vertical cylinder heave HIA-05

I _

k\ \

• T / A = O I

o T/A =0.3

6.4

32 k ~'~'~., "~,

C 1.2 2,4 3 6

Omega. Omega,~, A/G

4 8

FIG. 12.

42

8

6.4 i

0 4 8

i -

~ " 3.2

16

TARIK SABUI~CU and SANDER CALTSAL

Vertical cylinder heave H / A = I

A • T/AmO, I

o T I A I O . 3

\ -- \ [] T /A=0 .5

- \

0 I 2 2.4 3 6 48 G

Omega -~ Omega ~ AIG

FIG. 13.

Vertical cyhnder heave H i& =2

:~ 2 4 . -

? g, ~- 1.6 E g~

a~ A • T t / ' . =01 52

\A,~A o T/,a =03

X A u T/A=O 5 \

& • TIA = I

*- T / A = I 5 A

o ~ 2 2 . 4 3 e 4 ~ G

Omega~Omego~- A/G

FIG. 14.

Hydrodynamic coemcients for vert ical c i rcu lar cyl inders at f in i te depth 43

Verl ical cylinder heave H /A = 3 I

08 ~ , • T / A = 0 . 3

o T/A = 0.5

~ O 6 o T / A = I ,4-

°~ ,, T/A=2

& O.4 E

0.2

0 1.2 2.4 36 4.8 6

Omega ~-Ome go,x- A/G

FIG. 15,

4.-

u N-

H 8

Vert'icot cylinder heave H / A , , 5

c TIA=0.3 2.4

T /A -0 .5

• T / A = I

+ T I A = 2 : \ - \

"E t.2

E ID 0

0,6

1.2 2 4 3.6 4.8

Omega ~ Omega ~ A/G

F l o . 16.

I 6


.o_ I, 8

"& 1,2 E o c3

Vertical cylinder heave H / A = I O 3

;/-%\ • T,~=o 2.4 / &

• \ o TIA=O 3 -/ \ • u T / A = O.5

A ~ + T / A = 2

0.6 ~ A

,, i n i I ~ 0 I 2 2 4 3.6 a 8 6

Omega * Omega -w- A/G

FIG. 17.

2.4

4-

8 u ~Z 18

u

o

0.6

Vert ical cylinder heave deep water

:i,\ • T / A ~ O I

J ' ~ o T/A=0 3

N L~ T/A=0 5

• • TIA-3

N

I. 2 24 3.6 4

Omega .'4- Omega* AIG

FIG. 18.

Hydrodynamic coefftcients for vertical circular cylinders at finite depth 45

.u

1=. o

'-t-

Verl"ical cylinder sway H / A = 6 0 75

06 f +

/ • b1~ / x +. - * \ t

x , , o.., t I I O. 12

- - I S S H I K I

+ L.F.E.M(BAI)

x 'L.F.E .M (BAI)

• This theory

This Theory

D o o } 1 t I I

2.4 3.6 48 Omega ~ Omeqa, A/G

FIG, 19.

0.75 [ • Ver t ica l cylinder sway H/A=I

o T / A = 0 . 5

0.45 ~ ~ D T/A = 0.'5

~ '~q , T /A- - 0 . '

03 ,#..,.+-+--+ t~ + ,

• ~

0 1.2 2 .4 3.6 4 .8 6

Omega ~Omeqo -~ A / G

FIe. 20.

46 TAalK SABUNCU and SANDER CALtSAL

o8[ , / \

0.48

0 32

Ol6

v e u r i c e l cy l inder sway H / A = 2

n T / A = 1.0

T / A - O , 5

,. T /A=O.3 i

+ T / A = O 2

c T / A = O t I

5 ~ A / * / A / !

[ 2 24 3 6 4 8 6 Omego ~ Omega~ A/G

F]G. 21.

Ver'ticol cylinder swoy H/A,-3

0 .8 /'"\ • T / A = 2 ~. T / A = I

T / A = O 5 if. T /A =-013

o E

~ 0.4

02

~,~_ x : T / A = 0 2

- ~ , , ~ . l t ~ . \ ~ \ 4 . _ . - - - , - ~ - ,

×

L, E I t i L 4 ~ - - ' ~ . - ' r ' ~ , ! - " ~

r2 2'4 36 4 8" Omeg n ~-Omege-~ A/G

Flo. 22.

C, 8

.~_ ~_

0 6

0 4

3 2

H y d r o d y n a m i c co¢tticients fo r vert ical c i rcular cyl.inders at f ini te depth

Vert ical cylinder sway H / A = 5 I

• T / A = 3

~: T / A = 2 / / ~ :-'- T / A = I

+7 \ \ ,+ -,-,,A=o~ +,-+ % ~ T , A = O S

• T / A = 0.2

, \ V . . _ , _ . . , _ - - , - - -

o ~

I l i I I I 1 2 2 4 3 6 4 8

Omega -~ Omeqo* A/G

Fla. 23.

47

Ver~'ical cyl inder sway H / A = I O

I L~:;~ll - J, T / A - 5

× T /A == 3

o T /A ,= 2 0 . 8

+ /" 0.6 • - . - - - - - • ~ J ' ~ ~ *

~ 0.4 ~ " x ~ x ~ :'-

0 I 2 3 4

Omega-~ Omeoo ~ A/G

o T / A - 0 . 5

• T / A - - 0 . 3

+ T / A ==0.2

st T /A == 0.1

T ,

Fla. 24.


Vert ica l cyl inder sway H/A " -20

0 8

i~' o e

o E

0 4

0 2

0 .5

• T / A I I O

x T / A = 5

o T / A - 3

* T / A 1 2

o T / A - I

• T / A - 0 . 5

+ T / A I O . 3 "~ T / A i O.2

I I l 2 3 4

Omega ~ Omega ~-A/G

FIG. 25.

Ver l" ical cyl inder H / A = I sway

0.4

.~ 0 3

o u

& 0.2 E g

QI

• • T I A = O 5

, A o T / A = O 2

* ' I , 4 I E I I 1

I, 2 2,4 3 6 4.8 6

Omega K- Omega K- A / G

FIG. 26.

0.6

Hydrodynamic coefl~ients for vertical circular cylinders at f inite depth

Verticol cylinder swoy H/A - 2

49

048

.~ o 36

u

024

o

0.12

0-

/ ~ , • T , A - ,

x ~ × TIA 0.5

/ ×I A~,~ . T IA"' O. 2

1.2 4 . 8

0.36 "0

u

"~- 0.24 &

0.12

2.4 3.6

Omega ~ Ome(la ~ A/G FxG, 27.

Vertical cylinder sway H / A - 3

0'6 L • TIA-2 x TIA,- I

/ ~ , ~ T,, o.~

lJ oT, .o i l . T /A, -O. ,

0 1 . 2 2 . 4 3 . 6 4 . 8 " " 6

Omeqa ~ Omego. A/G

Fze. 28.

50

Q~

TARIK SABUNCU and SANDER CALISAL

Vert ical cy l inder sway H / A =5

0.4~

+ - c -

.~_

• -~ 0.3E 4-

0 u

01

" 024 E O

c ~

4--

v

&

O.12

065

052

0 . 5 9

O Z 6

0.13

.2 2.4 3.6 4 .8 6

Omega ,Omega. A/G

FIG. 29.

Vertical cylinder sway H /A= IO

• TIA-5

.,#~ x T,A-3

i'I~\ \. o ,,~.~

/ / i ' ~ X " ~ X o , , , -o ,~

Ii/ "\_Z~"~. . ~ yi .-f'--~:_~~-----~ ~

1

i ; i I r ~ I J

0.8 1 6 2 . 4 3 2 4.

O m e g a * O m e g a ~ A / G

Fi6. 30.

0,6

Hydrodynamic co, dficients for vertical circular cylinders at finite depth

Ver t ico l c y l i n d e r sway H / A = = 2 0

51

4"- C

qJ

4- (p o u

¢:

Q.

E 0 r~

0,48

0.36

0.24

O.12

• T / A ' I O

x T / A == 5

o T / A = = 3

w T / A , = 2.

0 T / A " I

• T / A " 0 .5

+ T / A - 0 . 3 ~" T / A =" 0 . 2

0

0.8

"~ 0.6

o" u

0.4

0.2

0.8 1.6 2.4

OmeQa ~ O m e g o ~ A / G

FIG. 31.

Vertico{ cylinder pitch H/A-, I

3,2

H--M'.,- + ~ ÷ ~ ÷ ~ ~ - f ,, - . ÷ +

-- , IL

7

• T / A - O . I

o T / A - 0 . 2

o T /A-0 .3

' A

÷ T /A-0 .5

I I I 1.5

I I 3 4 .5

Omeqa. Omeqa. A / G

FIo. 32.

7.~

52 TARIK SABUNCU an d SANDER CALISAL

Verticol cy l inder pil-ch H / A ~ 2

0.8

+-

0.6

~ 0.4

0.2

0 8

E

~ 0.6

E

~ 0 4

0.2

"AL

,~OIO~o ~

o - , - ~ o ~ • - o •

,.+-,+ - ..,- - + - - ~ a h , q r ~ . - - - + ~ + - - - - - + I -~, " ~ " " " ' . , l ~ , l ~ " l l . - "I~ ~ ' '

i , , , i i L I I I

1,5 3 4.5

Omega.Omega ~ A / G

FIG, 33.

Ver t i ca l cy l inder p i t ch H / A = 3

• T / A ~ , O I

i TIA-0,2

D T/A'O.3

+ TIA = 05

TIA=I

,Ic

r_ ~ , ~ f ' l w j

~ m . - 6 - , - ~ - ~ . . , ~ , ~ , ~ * ..... •

L i i i I 15 3 4.5

Ome,ga ~ Omega ~- A /G

FIG. 34.

• T / A ~ O . I

T / A =O 2

z T / A =C, "~

• T / A =;" 5

+ T / A = I

o ~ T / A = 2

I 6

7.5

5

32

0~ "G 2.4

E ~s

0,8

7'.5


Verlicol cylinder pitch H/A=5

4 . 5

,J

\

~ e e ~ e ~ t ~ 4

1 .5

[ - ]

$ 4.5

Omega *Omega * A/G

Fro. 35.

Ver'tical cylinder pitch H/A =IO

I. 5 ~ I - - " " " ~ * - " ~- . . . . +

/

r-- o ~ e A ~ A x ',c ~ I - - e , • , J

0 I 2 3

Omega. Omega ~ A/G

FIG. 36.

• T I A - O . I

:( T I A I O i 2

o TIA =O.3

. TIA-0,5

o TIA~ I

• T I A - Z

÷ T / A - 3

A - 4

Y o.

6

A T / A - O . I

× T /A " 0 . 2

o T / A - O . 3

* T / A ~ 0 . 4 AAZ5

"~ T / A s I

• TIA-2

+ T/A-3

TIA-5

, d ",

4 5

53

54 TARIK SABUNCU a n d SANDER CALISAL

4-

u

E

"o

7.5

4 , 5

1.5

Vert ical cylinder pitch H I A = 2 0

A T /A - 0 . 2

T / A - O . 5

o TIA=O 5 " -I-

~ ' T / A - I

n T / A - 2

• T / A - 3

+ T/A," ,5

0.3

0.4 0.8 ( .2

O m e g a , O m e g a , A /G

F i o . 37.

Vertical cylinder pitch H/A,11

1.6

0 . 2 4

P_ .~ O. 18 ,.p. -g 6

~" 0.12

0 .06

0

/ /

/ \,\

t

1.5 3 4 5

Omega .Omega ~ A/G

FiG. 38.

z TIA-O.I

o TIA-0.2

c T / A - 0 . 3

+ T / A =O.S

7.5

0.25


Verticol cylinder pit"oh H/A = 2

55

0.2

~ 0.15

'~ O. I

0.05

, f ' \ • × " T / A - O. I

~ × - o T / A - 0 . 2

1.5 3 4.5 6 7.5

Omega ~Omego,~ A/G

FIG. 39.

0.3 Vertical cylinder pitch H / A - 3

• T / A I O . I

o T / A - 0.2

o T / A - 0 . 3

0.24 , ~ " A ~ A

\

'~- 0. t2

006

o 1.5 3 4.5 6 7.5 Omega ~ Omega ~ A/G

FZG. 40.

5fi

0.6

TARIK SABUNCU and SA~ER CALISAL

Vertical cylinder pitch H/A= 5

E

,p_ if-

0 .48

0 .36

0 .24

0.12

1.5 3 4 .5 6

O m e g a * O m e g a , A / G

Fx6, 41. Vert ical cy l inder patch H I A - I 0

A T / A = 0 I

× T I A - O . 2

o T / A - O . 3

* T / A - 0 . 5

u T / A - I

• T/A- 2 ~- TIA-3

T/A~4

0.75 I~

A T/A=O I

:' T / A = O 2

7 . 5

0.6

o T / A ~ 0 . 3

"~ .~ T / A - O . 5

io

/ ~ \ • 1/A-2 ~" 0 .3

~ TIAI3 / / •

O. :15 r o ~ t ~ x ~.

0 0,8 I 6 2 4 3 Z 4

Omega * O m e g o * A / G

FtG. 42.

{p

.&

cz~

0.7,~

0.6

0.45

0.3

0,15

O

0 8


Ver t icc l cyl inder pitch H / A - 2 0

• T / A " 0 . 1

x T I A - 0 . 2

! ~ ~ o T / A ' O '

u T I A - 2

• * - " ~ " • T I A - 3

;3;0

~ ' - ' ~ , " - - f " ~ "A . .~ ~ [ - .~ l I ~ ' ' ' ~ , 0.4 0 8 I. 2 I .6

Omega, Omega ~ A/G

FKi, 43.

Vertical cylinder pitch induced sway H/A -3

57

G

u

0.6

04

0.2

0

-0.2

\

""- F

O 1.5 3 4 $

Omega, Omega , A/G

FIG. 44.

A TIA =O. I

c TIA=O.2

"~ ~-: TIA=O.3

• TIA- 0.5

-~ T/A- I

T/A- I

I t 6 Z5

58 ̧

1.5

,,.a

"u 0,5

- 0 , 5 O

0.4

0 2 8

5 0.16

0.04

£3

- 0 . 0 8

- 0 , 2

TARIK SABUNCU and SANDER CAUSAL

Ver t i ca l cyl inder p i tch induced sway H / A = 5

\

k

A T / A = 0 . I

~. T / A - O . 2

c T t A - 0 .5

-,'- T / A = O . 5

o T / . & - I

+ • T I A ~ 2

+ T I A = 3

T / A m 4

i I I I t I 1.5 3 4.5

Omega ~ Omega ~ A/G

F I G . 4 5 .

I 1 6

Ver t i ca l c y l i nde r p i tch induced sway H / / ~ 5

I I "It • TI~-O.I u T / A - 0.3

• T / A - 0 . 5

+ T / ~ - f

T / A - 2

: i i

1.5 3 4.5 .6

Omega ~ Omega ~ AVG

F l a . 46.

~ s

7.5


0.5

0.35

._~ 0.2

._~

u

• & 0.05 E o

-O, I

Vert ical cyl inder p i tch induced sway H / A - 5

• T / A =O.I

× T / A =O.2

o T /A =O.3

* T / A - 0 . 5

D T / A = I

• T / A - ?

+ T / A - 3

T / A - 5

-0.25 I T I ] : T I 0 1.5 3 4,5 6 7.5

Omega ¢,Omega ~ A/G

FIG. 47.

The nondimensionable heave added mass values for different water depths and draft values are given in Figs 5-11. The general behavior of the curves is that as the frequency increases the values remain constant. At zero frequency (numerically at least) the added mass values are observed to increase. The behavior of the curves at small frequency is discussed in section 4. Damping coefficients for heave are given in Figs 12-17. At high frequencies these values are observed to tend to zero while at zero frequency the values are finite at shallow depth and tend to approach zero as the depth increases. Higher values are observed to correspond to small draft to radius values. Deep water cases correspond to h/a = 20.

The sway calculations are compared with the results published by Bai and Yeung (1974) and are presented in Fig. 19. This computation tends to follow the values computed by Bai (1976) while the values reported by Isshiki and Hwang (1973)remained low especially at peak values. Figures 20-25 show the sway added mass values. Added mass values are observed to increase as draft to radius is increased at all depths. These values also remained finite at small frequencies. All curves are observed to have a local maximum at about v ---- 1. Damping coefficients for sway are equal to zero at zero frequencies and are observed to increase as the draft increases. The curves for damping coefficients are given in Figs 26-31. At high frequencies the curves show a decreasing slope and the maximum values are again observed at about v = 1. Pitch computations are first checked with those reported by Bai (1976) and Garrison (1975) and a good agreement is observed.

Pitch moment of inertias are presented in Figs 32-37. Except at very small drafts the curves have a very small slope. Inertia coefficients are observed to decrease as draft increases at shallow waters while in deep water higher coefficients correspond to higher drafts.

60 'I-ARIK SABUNCU and SANDER CALISAL

Figures 38-43 give the pitch damping coefficients. At shallow water (h/a -- 1 h /a : : 3) high damping coefficients correspond to small drafts while at deep water (h /a ~- 20) high damping coefficients are observed to correspond to high drafts. A peculiar curve is seen in Fig. 42 for T/ A .... 0.1 which suggests that the damping coefficient for very shallow discs increases as draft decreases even at moderately deep waters. Pitch-induced sway added mass (a~l) and pitch-induced sway damping coefficients (bG0 are given in Figs 44-46. It is interesting to note that some of the values are in fact negative for low drafts at finite depths such as h/a = 3 but as the draft increases the values become positive.

CONCLUSION

The solution presented in this paper offers a quick calculation of the hydrodynamic coefficients for a simple vertical circular cylinder. The results are compared to some available experimental numerical results. The agreement with numerical results are observed to be satisfactory. The comparisons with the experimental data showed that even though the trend is well represented, the amplitudes are not. This is partially acceptable, at least for the case of damping coefficients where the viscous resistance must be effective. The input variables for the calculations are water depth, radius of cylinder and its draft. Well docu- mented routines can be used for the calculation of special functions and the solution of the linear equations. The necessary formulas for computer application are also presented.

The hydrodynamic coefficients necessary to study the motion of the cylinder are presented in graphical format. It is hoped that this will increase the efficiency of future designs and that the designer will be able to estimate these coefficients precisely for his computations.

The limit o f the heave added mass value for zero frequency is also discusssed. It is shown that this quanti ty remains finite for finite depth and goes to infinity for infinite water depth. Special formulations are seen to be required for this limiting case.

The present formulation is currently extended to cylinders with variable cross sections.

Acknowledgements--Professor T. Sabuncu would like to thank the Council for International Exchange of Scholars for the support he received through his stay at M.I.T. as a visiting professor. Thanks are also due to Professors J. N. Newman, R. Yeung and F. Noblesse of M.I.T. for their interest and discussions, and to Professor P. F. Wiggins of the U. S. Naval Academy whose interest made the computer application possible.

Special thanks are also extended to Ms. Virginia Christensen and Miss Sharon Vaughn for typing this report.

REFERENCES AaRAMOWlXZ, M. and STEGUN, I. A. (1964) Handbook o f mathematical functions. National Bureau of Stan-

dards, Washington, D.C. BAI, J. (1976) The added mass and damping coefficients o f and the excitation Jorces on /our axisymmetric

ocean platforms. Naval Ship Research and Development Center, Report No. SPD-670-01, April. BAI, K. J. and YEUNG, R. W. (1974) Numerical solutions to free-surface flow problems. Tenth Symposium

Naval Hydrodynamics, pp. 609-647, Cambridge, Mass. GARRISON, C. J. (1975) Hydrodynamics of large objects in the sea, Part 11. Motion of free-floating bodies.

J. Hydronautics 9 (2), 58-63. GARRET'r, C. J. (1971) Wave forces on a circular dock. J. Fluid Mech. 46, 129-139. ISSHIV, I, H. and HWANG, J. H. (1973) An axi-symmetric dock in waves. Seoul National University, Korea,

College of Engineering, Dept. of Naval Architecture, Report No. 73-1, 38 pp. Jan. KLM, W. J. (1974) Ort the harmonic oscillations of a rigid body on a free surface. J. Fhdd Mech. 21,427-451. Karrls, IR. B. (1979) Heaving motions ofaxisymmetric bodies. The Naval Architect, p. 26, Jan. MACCAM¥, R. C. (1961) On the heaving motion of cylinders of shallow draft. J. Ship. Res. 5 (3).


McCORMICK, M. E., COFFEY, J. P. and RICHARDSON, J. B. (1980) An experimental study of wave power conversion by a heaving vertical circular cylinder in restricted waters. U.S. Naval Academy, Engineering Report EW 10-80, March.

NEWMXN, J. N. (1975) Marine Hydrodynamics. The M.I.T. Press, Cambridge, Massachusetts and London. NtWM^Iq, J. N. (1976) The interaction of stationary vessels with regular waves. Eleventh Symposium Naval

Hydrodynamics, London, pp. 491-501. W^NG and StlI~N (1966) The hydrodynamic forces and pressure distributions for an oscillating sphere in a fluid

of finite depth. M.I.T. Department of Naval Architecture and Marine Engineering, Doctoral Thesis. YEUNO, R. W. (1975) A hybrid integral-equation method for time harmonic free surface flow. 1st. Int. Conf.

Numer. Ship Hydrodynamics, Gaithershurg, Maryland, pp. 581-608.

APPENDIX

FORMULAS FOR COMPUTER APPLICATIONS

(a) Heave

The linear set of equations for complex coefficients

-/,,jAj = h,

is solved first. The expressions for y.j and h. are given below

7.j = (8.j + 16 U~ (L Po (n,]) + <30

Z Ko(rca) pq(n.])) + i 3--2-UjTp-°(n-'J-) q=l Kl(mfa) n mo.

where

4a.Lpo(i,O) + ~Ko(mfa) p.(i,O ) 2 ( - 1)" [n~od ] h ~ - d ( q=l K~(m¢a) -(n-n-)T + i TPo(n,O)

,.,, _ *t / ' ( i T ) - 2 l o ( j d ) j = 1 , 2 , 3 . . .

Po (n,j) - ( - l)"+J(med sh (me(t))~ [2moh) + sh(2moh)] [(moat) 2 + (nnp] [(rood) 2 + (yn) ~]

Pq (n,j) = ( _ 1 ) . + j ( m q d s i n ( m r , / ) ) 2 [(2n~0 + sin(2n'~h)] [ ( n ~ p - (nnP [(mpd) ~ - (j~)2]

L

T

Jo(moa) Jt (moa) + Y, (moa) Y~(moa) [J~(m0a)p + [Y~(moa)P

1

[J~(moa)]' + [Y~(moa)P

The added mass and damping coefficients are then computed by the following equation :

where azt and b22 are the added mass and damping coefficients for heave and D is the volume of the cylinder in water.

62 TARIK SABUNCU a n d SANDER CALISAL

(b) Sway

The linear set of equat ions for sway can be written in the form:

7~j Aj = -h , ,

where

~(.j = { - 6 . j + 1 6 U j [ L - Po(n,]) + ~ De" Y~(n , / ) ] } - i 3 2 q = 1 ~ m ~

Uj. T. Po(n.j)

h,, = 8 d- { L . P o ( n , 0 )Ko + ~ D,.Pt(n,O) Kk} - i 16 T. Potn, O) Ko a k ~ 1 gmoa

and where

{ jro 1, d d d Uj = 2- ~ 2a} f o r j = 1, 2 . . . . and Uo - 4a

1 | T = ([Jo(m'w) + Yo(m'oa)] + ( ~ ) [Ja(m'oa)

2 + Y~(m'oa)] - - -

gnoa

[Jo(moa) Jx(moa) + Yo(moa) Yt(moa)]) -1

L = {[Jo(moa) Jt (moa) + Yo(moa) Yl(moa)] - 1 [j2(moa ) + y l(moa)]}" T moa

Ko = { sh(moh) - sh(mod)}, Kq = { sin(inch) - sin(reed)} sh(mod) sin(mvd)

Dk mta = - , q = 1 , 2 , 3 . . . . 1 + inca Ko(mca)

K~(m~a)

The definition for Po and Pc remain the same as in heave.

The sway added mass and damping coefficients are obta ined as

aj_~ + i bu {BoHdmoa) [sh(moh) -sh(mod)] + #D co~D No~mo(h - d) q - 1

Bt Ka(m~a) [sin(inch) - sin(mcd)] N,½m,(h - d)

where

1 Ao dEqo + ~ An U,, Ee,} + ~qx B~ hm~F'q(a) { 4a n-I hrr~ F~(a)

n = 0 , 2 , 2 . . . . q = 0 , 1 , 2 . . . .

Eo, ~ ( - l )~N°-t2m°d sh(mQd) Fo(a) = H~ (mor) [(mod)' + (n~)']

Eq, = ( - l)"N~-½2m¢/sin(rood) F~(a) = K~ (m,r) [(re, d ) ' - (nn)']

1 No-~ [shmoh-shmod] ~gl 1 Ng-t [sin(mfh) - s i n ( % d ) ] I~°1 - a mo ' = a m ,


and 1 ( I + sinh(2meh)~

No - ~ 2meh ]

1 (I + sin(2m~)~ . N . - ~ ~ ]

(c) Pitch

The linear set of complexed valued equations is T.~ A s - - h . n - 0 , 1 , 2 , j ~ 0 , 1 , 2 . . .

where 0O

T. - { - 8.s + 16 U s [L" PO(n,j) + ~" D,. ee(n,j)]} - i 32 U s. T" Po(n,j) q == ! ~moa

h . - S { L ' P o ( n , 0).0o + ~ D , ' P , ( n , O ) . O J - a . - i 16 "T. eo(n, 0) 0o ¢ - I ~moa

~ lo(ra/d) d } d 0 n # j Us •{ /~(~):?_a forj = 1,2 . . . . U. =~4a ,8. j " { I n=q

h Oo = { - ~' - 8~d/ + (~-.3~ ] 1 ( - m - ~ s ~ - 3 )

O , - { [ ~ h 3/,a~' 1 cos(m~h)- 2ch(m~) - c l - 8 ~ d ! (mcd)' ] (n,~d) sin (m~d) }

.2(-1)" ' l (d) '} for n = 0 a. ~ ffi ~ - ~ - - ~-)} for n ffi 1,2,3 . . . . aoX ={:3 ( d ) - 4

The expressions for Po, P~, L, Tand Df remain the same as defined for sway. The roll (pitch) added moment of inertia is calculated as

d ~ t , K , . . a,, + i b,,/m d' to H, ('~°') Bo + ~ -~, ~,tm~a) B,] + pDa' a~(h - d) { [ ~'oll s # - I

1 1 1 ~ ( - I ) " I ' ( T ) - [~ ( ~ ) ' " - ~ ( ~ ) ' ) +S (~) Ao ÷ ~ ~ - - - - - ~ . " .- , . , ,

The expressions for Bo, B¢ remain the same except the B9 ' values for roll are given by

h 3{o ~' l ~o'=No-' {[~ ~ S'~:+C~V]~(m'd) , , ' - N,-½ { [ ~ h 3[~a~ * 1 sin(rood)

- a - s ~ : ( m ~ - ~ ] (m@')

artd the expressions for to and t~ are

, o - { ( , - -m-~-+ t , - ( ( 1 - -hd)sin(m~)

m~

ch(moh) - 2ch(med) + (rood)' J

[

cos(re, h) - 2 cos(m,/) ~m,d~ }

c~m~) _- ~ (m~ E (m~'P J

c o ~ m , h ) - ~(m,a)'~ . (re.d)' J

hydrodynamic coeeficients

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