The motions of a spar buoy in regular waves

on,

ifn

tmC^J^rt^l&Hr

_{OF THE}

MAW

0AVIO

TAYUOR

MODEL

BASIN

HYDROMECHANICS

THE

MOTIONS

OF

A SPAR BUOY IN

REGULAR

WAVES

AERODYNAMICS by J. N.Newman STRUCTURAL MECHANICS

^

APPLIED

^d

HEMATICS HYDROMECHANICS

LABORATORY

RESEARCH AND

DEVELOPMENT REPORT

THE

MOTIONS

OF

A SPAR BUOY IN

REGULAR

WAVES

by

J. N.Newman

TABLE

OF

CONTENTS

Page

ABSTRACT

1

INTRODUCTION

1

THE

FIRST-ORDER

VELOCITY POTENTIAL

2

THE FIRST-ORDER FORCES

AND

EQUATIONS OF

MOTION

9

THE

DAMPING

FORCES

14

CALCULATIONS

FOR

THE CIRCULAR CYLINDER

17

DISCUSSION

AND

CONCLUSIONS

21

ACKNOWLEDGMENT

22

APPENDIX

2 3

LIST

OF

FIGURES

Page

Figure 1 - The Coordinate Systems 3

Figure 2 - Plot of the Surge Amplitude

-Wave

Amplitude

Ratio for the Circular Cylinder 18

Figure 3 - Plot of the Pitch

Amplitude-Wave

Slope Ratio

for the Circular Cylinder 18

Figure 4 - Plot ofthe

Heave

Amplitude

-Wave

Amplitude Ratio

for the

Undamped

Circular Cylinder 20

Figure 5 - Plot ofthe Heave

Amplitude-Wave

AmplitudeRatio

for the

Damped

Circular Cylinder vi^ith

R/H

=0.1

20

Figure 6 - Plot ofthe

Heave

Phase Lag for the

Damped

Circular Cylinder v/ith

R/H

=0.1

21

NOTATION

A

Incident

wave

amplitude

g Gravitational acceleration

H

Body

draft

I

Body

moment

of inertia in pitchabout the center of gravity

i = nTH"

Jq Bessel function of the firstkind oforder zero

K

Wave

number, oj /g

2

ky

Radius of gyration, k =

I/m

m

Body

mass

n Unit normal vector into the body

p Pressure

Qn(^) = _{/_H} <^ ~ ^g)"" S(z)eK2 ^^

R(z) Sectional radius of the

body-Si z) Sectional area ofthe body

2 2 2

r Polar radius, r = x + _y

t

Time

(x,y,z) Cartesian coordinate system Zp Coordinate ofthe center of gravity

t,

Heave

displacement

£,*

Free

surface elevation

9 Polar coordinate

^ Surge displacement

p Fluid density

$ Velocity potential

)( Vertical prismatic coefficient

<\> Pitch angle

ABSTRACT

A

linearized theory is developed for the motions ofa

slender body of revolution,with vertical axis, v/hich is

float-ing in the presence of regular waves. Equations of motion

are derived which are

undamped

to first order in the body

diameter, but second-order damping forces are derived to

provide solutions valid at all frequencies including resonance.

Calculations

made

for a particular circular cylinder show^

extremely stable motions exceptfor the low frequency range

where

very sharp

maxima

occur at resonance.

INTRODUCTION

The

motions of a vertical bodyof revolution, which is floating in the presence ofwaves, present a problem of interest in several connections.

The

naotions ofa spar buoy, ofa wave-height pole, and offloating rocket

vehicles are important examples of such a problem.

The

same

methods

developedfor these motions

may

be applied to find the forces acting on

offshore radar and oil-drilling structures.

A

theoretical discussion of this problem, which also treats the

sta-tistical problem ofmotions in irregular waves, has been presented by

Barakat. However, this analysis is restricted to the case of a circular

cylinder and is based upon several semi-empirical concepts ofapplied

ship-motion theory.

An

alternativeprocedure is toformulatethe (inviscid) hydrodynamic problem as a boundary-value problem for the velocity

po-tential andto employ slender-body techniques to solve this problem.

The

latterapproach is follow^ed inthe presentw^ork, leading to linearized

equa-tions of motion which

may

be solved for an arbitrary slender body with a vertical axis of rotational

symmetry.

The

particular case ofa circular

cylinder,

whose

centers ofbuoyancy and gravity coincide, is treated in

detailand curves are presented for the amplitudes of surge, heave, and

pitch oscillations.

Inderiving the hydrodynamic forces and

moments

acting on thebody,

we

shall

assume

that the incident

waves

and the oscillations of thebody are

small, and thus

we

shall retain only terms of first order inthese

ampli-tudes.

We

shall also

assume

that the body is slender.

The

analysis w^ith

onlyfirst-order terms in the body's diameter leads to undamiped resonance

oscillations of infinite amplitude.

To

analyze the motions near resonance,

it is necessary to introduce

damping

forces which are of second order with

respect to the diameter-length ratio.

THE FIRST-ORDER VELOCITY POTENTIAL

We

shall consider the hydrodynamic problem ofa floating slender

body of revolution with a vertical axis in the presence of snnall incident

surface waves. Let (x,y,z) be a fixed Cartesian coordinate system with

the z-axis positive upwards and the plane z = situated at the undisturbed

level ofthe free surface.

The

x-axis is taken to be the direction of

propa-gation ofthe incident

wave

system, and the motion of the body is

assumed

to be confined to the plane y = 0.

We

shall also employ a coordinate

sys-tem

(x',y',z') fixed in the body, with z' the axis ofthe body, so that with

the body at rest, (x,_y,z) = (x',y',z'); anda circular cylindrical system

(r,9, z),

where

x = r cos 6 and

y=rsin6.

If£,t,, and 4^ a-re the

instan-taneous amplitudes of surge, heave, and pitch, respectively, relative to

the body's center of gravity, it follows that

X = ^ + x' cos ijj + (z' - Z/-) sin \\i

y=y'

[1]

z = t, - x' sin \\) + (z' - Zp) cos ^ +

zL

where

zL is the vertical coordinate ofthe center of gravity in the

body-fixed system; see Figure 1.

The

displacements ^, t,, and ijj are

assumed

to be small oscillatory functions of time;

we

shall consistently linearize

by neglecting terms of second order inthese functions or their products

Figure 1 -

The

Coordinate Systems

X= ^ + x' + (z' - Zq) l]j

y = y'

z = t, — x' l|j + z'

[2]

Ifan ideal incompressible fluid is assumed, there exists a velocity potential, $(x,y,z,t), satisfying Laplace's equation, suchthat its

gradi-ent is equal to the velocity ofthe fluid. This function

must

satisfy the

following boundary conditions:

(1)

On

the body, the

normal

velocity component ofthe body

must

equal

the normal derivative of $.

For

a body of revolution defined by the

equa-tion r' =R(z'),

where

r' = \/x"^ + _y , this boundary condition

may

be

^

[r-

-R{z')]H(A

+

v$V)[r'

-R(z')]=

on r' = R(z')

(Z)

On

the free surface, the

normal

velocity connponent ofthe free

surface

must

equal the

normal

velocity component ofthe fluid particles

in this surface, and the pressure

must

equal atnnospheric pressure. In

the linearized theory, these conditions reduce to

^+g^

= on

z.O.

[4]

at2 ^ 9z

or in the case ofa sinusoidal disturbance with frequency oo,

K$-|-

=0

on z = 0, [5]

oz

where

K

= oj /g.

(3) At infinite distance

from

the body, the

waves

generated by the body

are outgoing (the radiation condition).

The

free surface condition, Equation [5], and the radiation condition

are satisfied by the potential of oscillating singularities beneath the free

surface; the boundary condition on the body

may

be satisfied bya proper

distribution of these singiilarities. This distribution

may

be found

from

slender-body theory but

some

care is required in linearizing the present

problem. If r' = R(z') is the equation of the body surface over its

sub-merged

length

(-H

< z' < 0),

we

shall

assume

that

R

and its first

deriva-tive are continuous, that

R(-H)

= 0, and that the magnitude of the slope

|dR/dz'|« 1.

The

depth

H

is

assumed

finite, and it follows that

R

is

small of the

same

order, as dR/dz'. In the analysis to follow

we

shall

also require that

R

be small

compared

to the wavelength of the incident

wave

system, or that

KR

«

1.

We

wish to obtain the velocity potential ofleading order in the small

parameters of slenderness and oscillation amplitudes in order to obtain a

consistent set of linearized equations ofmotion for the body. However,

different orders ofmagnitude with respect to the slenderness parameter. For example, the potential due to surge or pitch is oforder

R

as

R

—

0,

whereas the potential due to heave is 0(R ). Similar differences will

oc-cur inconsidering the components of eachpotential which are inphase

and out of phase with the respective velocities ofthe body. In order to circumvent these difficulties withoutunnecessary higher order

perturba-tion analysis,

we decompose

the velocity potential in the following form:

$(x,y,z;t) = <t)t(x,y,z;t) + cj)^(x,y,z;t) + cj), (x,y,z;t)

[6]

+

A

[g/w e^^ cos (Kx- cot) + <^j^(x,y, z; t)]

where

<\>t , (^v > ^^'^ ^

i

^^^ linear in the displacements {^,1,,\\)) and their

time derivatives, respectively.

The

potential

A

g/w e cos (Kx - cot)

represents the incident

wave

system and the potential Acj)^(x,y,z;t)

re-presents the diffracted w^ave potential, corresponding to

waves

incident

ona restrained body.

Each

potential ^ in Equation [6]

must

satisfy the

free surface boundary conditionand the radiation condition; the complete

potential $

must

satisfy the boundary condition on the body. This

condi-tion. Equation [3], is reducedas follows:

, 9 , v^;^ ^ r . o/ -\i ar' ax', ar' ay' ^ ar' az'

dR

az'

{-T— +

V$

•

V

[r'- R(z')l = + i- +

^ at '^ ' '- _{9x' at ay' at az' at dz' at}

^ a$ a$

dR

. . D/ .\

+

=0

on

r=R(z),

ar' az' dz'

or neglecting second-order terms in A, ^ , L, , and ljj,

||--|f

g-

[|+

(z -Zg)*^] C°S

e+(t-x4;)dR/dz=

[7]

on r =

R

(z).

where

a dot denotes differentiation with respect to time. Substituting

Equation [6] into Equation [7] and separating terms according to their

dependence on different displacemients,

we

obtain the following boundary

= - £ c\_Lnlr> ^

9r

= _i cose + _(^^-57) t»i

—

-i- = 4.(2 -

ZG)cose

+

o(r—

-i: _{+ o(R2y}

[9]

or \ dz/

or dz \ dz /

^^A

_K7

_{r / ?}

dR\

1

= - Gje^^ cose sinwt-(

KR

cos'^9 + Icosoot

ar _[ \

dz/

J

(94>A

\ -,

R^—

j+

0(r2)

[11]

= CO

e^^

[- cose sinojt +

[ i

KR

+ + i

KR

cosZeJcoswt

j

/ 94)Av

+

0[R

1+ OIR"^)

To

satisfy the above boundary conditions,

we

ennploy slender-body

theory.^

For

example, the potential satisfying Equation [8] is an axial

line ofhorizontal dipoles, of

moment

density j | [R(z)] per unit length.

Thus inan infinite fluid,

4>£ =

ieJ°

[R(z')]^|- [r2 +

(z-z'/]''

dz; [12]

To

satisfy the free surface and radiation conditions,

we

substitute for the source potential [r'^ + (z - z^)'^] ^, the potential of an oscillating

source under a free surface.^ With this substitution

we

obtain, in place

of Equation [12]:

/"

-+

j:"^^e^(-+-l)j^(kr)dk)dzi

[13]

+ TTcKI

r°

[R(zi)]2

e^<^+^l)^[j

(Kr)] dz^

and, ina similar fashion,

-H

+

X"M^e^(^+^l)jo(kr)dk\dzi

[14]

+ TTu^K^ f [R(zi)]^(zi-zc) e^<^"^^l^f-|-J (Kr)ldz^

J_H '- -'

+

^°'k^e^(-+-l)j^(kr)dk}dzi

[15]

+ ttcK;

r°

R{zi)^

eK{z+Zi) j^(Kr)dZj^

J_H

^^1

^^

^ _ i<^ r e^'^l

<U

KR

+

^)

R

coso.t

[16] +

r2

sinojt

—

+ ^KR"*

cosGot-^

W[r2

+ (z - z,)^] 9x _9x2j V ^ ^

r"k+K

k(z+zi) ^ ,,

,j,l,

^

fo^^

Jo(kr)dk|dzi

-.cK

f'

e^(^-^2zi)|(iKR+

|^)Rsin.t

-

r2

coswt

•^+

^ KR"*

sina3t-^l

jQ(Kr)dzj

where

4- denotes the Cauchy principalvalue.

From

the Appendix

we

see

thatthe potentials [13] to [16] satisfy the boundary conditions [8] to

[H],

respectively, with a

maximum

fractional error of order R.

Unfortunate-ly, this error is not so small as inthe classical slender-body theory for

aninfinite fluid,

where

the error is of order

R

log R; for this reason

the present theory

may

nothold for as wide a range of slenderness as in

the aerodynamic case. However, for the slender floating bodies which

are envisaged at present (viz., a rocket vehicle or one support of a stable

platform), this is not expected to cause practical problems.

The

values ofthe potentials [13] to [ 16] on the body

may

be found

by setting r = R(z) and retaining the leading

terms

for small R.

To

lead^-2 ?

--ing order, only the singular

term

[r + (z - z,) ] 2 contributes to the

in-tegrals over Zi, and the integrals

may

be evaluated directly since for

any continuous bounded function f(zi) and small values of r.

f(zj) [r'^ + (z - z^)''] ^ _dz^ = -2f(z)logr + 0(1)

H

f(z^)

i_[r2

+ (z

-z/]~'

dzj -

-Z^cose

+ 0(1)

H

9x f(z,)

-^

[r2 + (z - z,)^]"^ dz, = 2

^

cos 20 + 0(1) -H ax^ r2 for

-H

< z < 0, r

«

H.

Thus on the body,

<^t = i R(z) cosO + 0(r2) [17]

4>^ = - >]; R(z)(z - Z(3)cos0 + 0(r2) [18]

4>. =

_-i

R^log

R+

0(r2) [19] '=' dz

=

we^^

[(^

KR

+

4^)R

log

R

cos wt+

R

cos sin wt] + 0(r2)

dz

e^^

R

cos sin wt + 0(R^ log R)

THE

FIRST

-ORDER

FORCES

AND

EQUATIONS

OF

MOTION

From

Bernoulli's equation, the linearized pressure on the body is

9$

P = - P _3^ - Pgz

94>t d<^A. Sff'r _{r ^<^A}

- _Pg- -

P^-

_p-9^-

P^

-

PA[^^+

ge^^

sin(KR cos 9

-.t)J

- pgz + p|r(z)cos9 + pi|iR(z)(z-Z(--)cos9 - pAco^e

R

cos9 cos cot

+

pgAe^^

sin wt -

pgAKe^^R

cos9 cos cot + 9(R^ log R)

= - pgz + p|R (z)cos9 + pijjR(z)(z -

Zq)cos9

+ pgAe sin ut

-

Zpcj^Ae^^R

cos9 cos wt + 0(r2 log R) [21]

The

force and

moment

exerted on the body by the fluid are obtained

by integrating the pressure over the surface. In the absence ofany other

external forces, the force or

moment

must

equal the respective

accelera-tiontimes the

mass

or mioment of inertia ofthe body. Thus, with n the

unit

normal

vector into the body, the equations ofmotion are

m^

= //

pcos(n,x)dS

[22]

^i'i + g) = // P cos(n,z)dS [23]

14* =

//p[(z-ZQ)cos(n,x)-xcos(n,

z)]dS [24]

where

m

is the body's

mass,

I its

moment

of inertia about the center of

gravity, andthe surface integrals are over the

submerged

surface ofthe

body.

Incomputing the pressure integrals over the body surface, it is

ex-pedient to employ the (x',y',z') system, fixed in the body.

The

direction

cos(n,x')= - cos + O(R^)

cos(n,z')=

-^

+ 0(r2)

dz

and the forces along the (x, z) axis are related to the forces along the

(x',z') axis by

^x

" ^x' ^°^ "^ ^

^z' ^'" ^ = F^, +4) F^, + 0(4;2)

F^. = F^, cos 4j - F^, sin ^=

F

^, -i\j F^, + 0(^^)

Thus

the equations of motion

may

bewritten inthe

form

rZn

,^*-t.+

x>

.

mi

= _{(- cos} 9 + 4j

^j

pRdz'

de'

.Ztt ^^*-^,+

x>

_.^ .

m(t,+ g) = I

j^+

^j; cos e]

pRdz'

de'

.Ztt _-;*-C+x'4;

14/ = [{z'- Zp) cos(n,x')- x' cos(n,z')]

pRdz'

de'

-'0

J-H

.Ztt rtj^-l + x'^

= - \ (z'- Zq)cos e

pRdz'

d0' + O(R^)

where

^ is the free surface elevationat the body. Substituting Equation

[21] for the pressure and neglecting second-order

terms

in the

oscilla-tory displacements |, ^, 4'. and A,

we

obtain

m^

= - _Pg

I I

[- cos 9' + 4^

^]

(z' + ; - 4jR cos e')Rdz'de'

j cos 0' [|

R

cos

9' _{+ L|JR(z}- Zq) cos 9'

•'-H

+

gAe^^'

sin wt -

Zw^Ae^^ R

cos 9' cos wt]Rdz'd9'

- TTpg r (4iR + 2^z'

-^J

R

dz' - _{pTT r} il + '^(z'- Zq) - 2co^Ae^^' cos wt]

R^

dz' J-H or, since

C{*^''*^'^h'^"*

c

—

dz' (R^z') dz' = it follows that .0

i = - P f ii + 4'(z- ^r) - 2oj^Ae^^ cos wt] S(z) dz + 0(R-^ log R)

J-H

[25]

\where

i2

S(z) = TT [R(z)]'

is the sectional area function.

In a similar

manner

we

obtain

,0

m(C

+ g) =

-pg;S(0)+

pg r S(z)dz +

pgA

sincot f e^^

^

d

J-H

•'-H ^^

+ 0(R'* log R) [26]

Iifi

=-pg^i

I (z - Z(^) S(z)dz

J-H

-P

f [^ +i[/ (z - Z-) - 2oj^Ae^^ coscot] (z- z^)S(z)dz

+ 0(R^ log R) [27]

From

Archimedes' principle, or equivalently, satisfying Equation [26] to zero order in t,,

gm

= pg S(z)dz [28] and thus

JO

gKz dS _^^ ^ Q^j^4 ^Qg p^j j29]

H

^^

while,

from

Equations [28] and [25],

2m|

= - p r [i|/(z - Zq) - 2u)^Ae^^ cos cot] S(z)dz + 0(R^ log R) [30]

Let us denote: I =

mk^

X =

pHS(O)

.0

Q

Pri =

—

( (z - z^)"" S(z)dz

(n=l,2)

.0 (K) = -H-

e^z

_(z _

_z^r

S(z)dz

{n=0,l)

and note that ^, t, , and i^i

must

be sinusoidal with frequency co.

The

equations ofmotionthen

become

2^ + Pj4j = -

2AQ

cos wt [31]

(1 -

XKH)^

= A(l -

XKHQq)

sin Got [32]

{^2 +

ky

-] 4j + Pi ^ = -

2AQj

cos wt [33]

Note also that surge and pitchare coupled, unless P^ =.0 or unless

the centers of gravity and buoyancy coincide.

The

above equations ofmotion are not unexpected.

The

restoring

forces on the left-hand side consist ofhydrostatic and inertial forces plus

entrained

mass

ternms which double the inertial force at each section.

This might have been deduced as a consequence of slender-body theory and the fact that the entrained

mass

of a circular cylinder in an infinite fluid is just equal to the displaced mass. In other words, the

hydrodynam-ic forces on the left-hand side ofEquations [31] to [33] could have been

obtained byneglecting the presence ofthe free surface. Moreover, the

exciting forces onthe right-hand side ofthese equations are those which follow

from

the "Froude-Krylov" hypothesis that the pressure in the

wave

system is not affected by the presence ofthe body. These results are, of

course, a consequence of the fact that the body is slender.

The

solutions ofEquations [31] to [33] are

1 -

Xq^kh

/1 - AUqI^JtL \ L =

A

sin wt I

—

I \ 1

-XKH

/ 34" I = 2

A

cOS cot

^iQi

-Qo(P2

+

4~^i^^^

2(P2 +

^y-

Pi/K)-

Pf

I 35] r

PiQq-^Qi

1 ijj = 2

A

cos wt

—

L

2(P2

+

k^

-

Pj/K)

- _Pf J 36]

We

note that

when

K

=

XH

37]

there is resonance in heave, and

when

K

=

Pz +

_kJ-ipf

[38]

there is resonance in pitch and surge.

To

determine the oscillation

am-plitudes in the vicinity ofthese resonance frequencies, it is necessary to

consider the

damping

mechanisnn due to energy dissipation in outgoing

waves. Thus, for these frequencies,

we

must

consider the free-surface

effects on the restoring forces.

For

this purpose

we

must

retain

some

terms

which are of second order inthe radius ofthe body.

THE DAMPING FORCES

The damping

forces will follo^wby considering the last

terms

in

Equations [13] to [l6] and will consequently be ofhigher order in

R

than

those

terms

which

we

retained in the previous analysis. This procedure

is nevertheless consistent, since at resonance the lo^wer order restoring

forces vanish. In other words,

we

are retaining the lo^west order force

or

moment

of each phase separately.

For

a further discussion ofthis

point, see Reference 4.

We

proceed, therefore, to study the danriping forces, or the forces

in phase with each velocity.

The

only contribution

from

Equations [13]

to [l6] is the potential

4.* = TTuKe^^

j

/t,R|^

+ _{[e +4^{zi} -

_Zc)]R^

g^je^^l

jQ(Kr)dzi

[39]

Since J^(Kr) = 1

-;f (Kr)^ + .... itfollows that on the surface r = R{z),

the leading

terms

are

Jo(Kr)

S

1

and

—

J„ (Kr)

S

_ i

_K^x

= _{- 1}

K^R

cos

9x ^

Thus, to second order in

R

the

damping

potential on the body is

J-H '^ _d^l

- _{i [^ +}v|j (z^ - z^)]

k2r(z)R^(z^)

cose

I

e^^l dz^

1 x^

Kz.

r° Kz, dS , [40]

J-H dz^ 1

iK^R(z) cos e e^^ r [| + 4j(zj - zq)] e^^l S(zj) dz^

1 ..T^3

The

damping pressure on the body is

,0 -H P* = 9* - 1

K«Kz:

f Kz, dS ,^ -

-P^-

-aCopKe

^ J_^ e 1

—

dz^ [41]

+ _I a)pK^R(z) cos e e^^ [I + 4j(z, - Zq)] e 1 S(z,) dz^

Then

the heave damping force is

Similarly, the surge damping force and the pitch damping

moment

are

,0 .Ztt

J-H

Jo

p*cos

eRde

dz

[43]

=

_-icopK3^f

S(z)e^^dzVj

_[e + 4;(z^- z^)]e^^lS(z^)dZj'\

and

-Ztt

M*

= -I I p*(z - Zq) cos e

R

dedz --in - u .0

-LI

=

-^<^pK^lj

(z -

z^)S(z)e^^dzj

[44]

(j

[e +4^(Zi- z^)]

e^^lS(z^)dzjj

or in

terms

ofthe integrals Pi , P^' Qq' ^^"^ ^1"

fJ

=

_-ia;pKi[K

^

_Qo(K)-

S(0)]2 ,

wm

K

. , ,2 =

_-i

—

_rr

_{^ t^}

_-Qo(K)xKH]^

[45] 2

F*

=

_-i

^

_{K3Qo(K)[eQo(K)}

+ _4iQl(K)] [46]

My

= ~ 2

^

K^Qi(K)[|Qo{K)

+ 4;Q^(K)] [47]

In place of Equations [31], [32], and [33],

we

obtainthe

damped

equa-tions ofmotion

2| + P, 4; = -

2AQq{K)

cos ut

, . [48]

+ ^

^

K^Qo(K)

[eQo{K)

+ _4^Qi(K)]

(1-XKH)^

=

A

sinojt[1

-XKH

Qq(K)] ,2 [49] + i

—

-^

[1

-Qo(K)XKH]'

a)p _;j.2j^2

^(P2

-

Pj/K

+ k^) + Pj^ = -

2A

Qj(K) cos wt [50] +

i^

_{K^Ql(K)[eQo(K)+}

_4;Qi(K)] 16

The

damping terms of these equations ofmotion are given by the terms linear in the velocities 4, t , and ijj. It should be noted that for a slender

body

m

-* 0, and thus the damping coefficients will be small, as

was

to be

expected.

To

solve these equations for the three

unknown

displacements

and their phases is a straightforward but tedious matter. For applications

in ranges not including a resonance frequency, it is

much

simpler to

em-ploy the

undamped

equations ofmotion, [31] to [33], and the resulting

displacennents, [34] to [36].

CALCULATIONS

FOR

THE CIRCULAR CYLINDER

As

a special case,

we

shall consider the circular cylinder R(z) =

R

= constant.

Then

X = 1.0

P

=

1

r (z - zc)dz = -

_Ih

- Z( 1 r° 7 ?

^2

=

_H

J

^^"

Zg^^^

=

iH^

+

Hzc+

Z( 1 r'

Qo(K)

-H

_{J_}

Ql(K)

=

_^

C

eK-(z-Zc)dz

=

le-K^--4-(l-e-^^)(l

+

Kzc)

,0 -H ,0

H

.0 e

dz-—

(1-e

)

We

shall assume, moreover, that the centers ofbuoyancy and gravity co-incide, or

zq

= - H/2, so that the equations ofmotionare uncoupled and

there is no resonance in pitch or surge.

Then

h2

1 +

e-K^

1 -

e"^^

*p

Figure 2 - Plot of the Surge

Amplitude-Wave

Amplitude

Ratio for the Circular Cylinder

\

08

\

\

0.6

\

V

\

t. 0.4

\

V

_N

0.2

\

^

^-, ""^*" 9 KH

Figure 3 - Plot of the PitchAmplitude

-Wave

Slope

Ratio for the Circular Cylinder

and it follows that e =

"2^(1

-

e-K^)coswt

[51]

2A

cos oit I 1 +

e"^^

_ 1 -

e"^^

"| j-^^l ^ " "

H

_,^

.2

^1

2KH

" (KH)2 J ^ jt r 1 + r le ^ T L ^

?Ap"^^

r t, = =^-^ (1 -

KH)

sinojt (1

-KH)2 J'^

--/RX2

..KKl2

L [53] 2

Vh/

J

Plots ofthe above amplitudes and the heave phase angle are shown

in Figures 2 to 6 as functions of

KH.

Figure 2 shows the ratio of surge

amplitudeto

wave

annplitude.

For

zerofrequency this ratio is one and

for increasingfrequencies it decreases monotonically to zero. Figure 3

shows the ratio ofpitch angle to the

maximum

wave

slope

KA,

multiplied

bythe coefficient

C

= |^ + 6(ky/H) . This coefficientis equalto one ifthe

mass

in the cylinder is uniformly distributed throughout its

submerged

length.

The

ratio starts at one for zero frequencyand decreases

mono-tonically to zero. Thus the pitchamplitude is always less than the

wave

slope. Figure 4 shows the ratio ofheave amplitude to

wave

height for

frequencies

away from

the vicinity of resonance.

Near

resonance, the

amplitude is shown in Figure 5 and the phaseangle in Figure 6for the

particular case

R/H

= 0.1.

The

ratio ofheave amplitude to

wave

ampli-tude is unity for zero frequency, rises to a

maximum

of

fdf—

(0

atthe resonance frequency

KH

= 1, and then decreases monotonically to

zero.

The

phase angle is similar to conventional one-degree-of-freedom

Figure 4 - Plot ofthe

Heave

Amplitude

-Wave

Amplitude

Ratio for the

Undamped

Circular Cylinder

05 0.6 0.7 0.8

Figure 5 - Plot ofthe

Heave

Amplitude

-Wave

Amplitude Ratio

for the

Damped

Circular Cylinder with

R/H

=0.1

20

1=1

Figure 6 - Plot ofthe

Heave

Phase Lag for the

Damped

Circular Cylinder with

R/H

=0.1

harmonic oscillators with linear damping; for low frequencies the heave displacementand

wave

height are inphase, at resonance they are in

quad-rature, and at highfrequencies they are 180 deg out of phase.

DISCUSSION

AND

CONCLUSIONS

The

damped

equations ofmotion as givenby Equations [48] to [50]

may

be solved for anarbitrary bodyof revolution to obtain the oscillation

amplitudes and phases. Except inthe vicinity of the resonance frequencies

defined by Equations [37] and [38], it should be sufficient to use the

sim-pler

undamped

equations; the resulting oscillations are given by Equations

[34] to [36]. Plots ofthese oscillations are shown inFigures 2 to 6 for

a circular cylinder, with the important restriction that the centers of

buoy-ancy and gravity coincide. If this restrictionis relaxed, a resonance will

be introduced into the equations for pitch and surge, but the frequency of

this resonance

may

be kept small byballasting.

The

annplitudes at

reso-nance are extreme, but the resonance frequencyfor heave is quite snnall

and can be kept out of the practical range of ocean

waves

by

making

the

draft sufficiently large. Itwould

seem

wise to do this in practice and to

provide appropriate ballast so that the pitch resonance occurs at or below

the heave resonance frequency.

From

Equations [37] and [38] this

re-quires that

< JL

P^

+

k^-iP^

XH

The

advantage of spar-buoy-type bodies lies in their very small

motions in the higher frequency range.

By

proper design this advantage

may

be utilized; thus very

calm

motions can be expected inw^aves.

ACKNOWLEDGMENT

The

author is grateful to

Mrs.

Helen

W. Henderson

for computing

the results

shown

in Figures 2 to 6 and to Dr.

W.

E.

Cummins

for his critical review ofthe manuscript.

APPENDIX

Here

the potentials cf't. 4't' . 't'j, a-nd cj)., definedby Equations [l3]

to [l6], are shown to satisfy the boundary conditions [8] to [11],

respec-tively, to leading order in R.

For

this purpose, let us consider the

po-tential

*f„°^

="""'''

^0l^')<=^}^-l [54]

+ TTcoK f f(zpt) e^^^"*"^1^ jQ(Kr)dZj

where

f{z,,t) has sinusoidaltime dependence with circular frequency to.

By

appropriate choice ofthe function f, the potentials <j)t, cj)^ , 4",, and

<))a canall be obtained

from

i|i and 94j/9x. Thus it is sufficient to

estab-lish that the follow^ing conditions are satisfied on the body surface r = R:

a^4j

^

cos e a 9r8x j^2 at

^f{z,t)

[56]

Employing an alternative

form

ofthe source potential,^

we

write _^

inthe

form

4^=i

J

^([r2

+

(^-^l)^]"^+

[r2 +

(z+zi)2]"^

+

2K

j^"

_-^-^

_{e^^^'^^l) Jo(kr)dk\dzi} 57] 23

.0

K(z+z

)

[^"^1

+ 7rcoK I f(z,,t)e 1 jQ(Kr)dZ|^ continued

4^1 + _^2

where

+ TTcoK r f(zi,t) e^^^"*"^!^ J(.(Kr)dz,

The

potential 4^, corresponds to anaxial distribution of simple sources

together withan

image

distributionabove the free surface z = 0.

To

emphasize this fact•we write ij^i i^ the

form

H

1

4^1 = i _j ^f(-|z;L|,t)[r2 +

(z-

zi)2]"^dzi [58]

From

the conventional slender-body theory of aerodynamics,

we

may

ex-pect this potential to satisfy the boundary conditions [55] and [56] on the

body to leading order in R.. In fact, differentiatingwith aspect to r and

neglecting

terms

which are of order

R

or

R

cos in the neighborhood

ofthe body r = R,

we

have

d^, r

H

- 3 '1 "87 = - i f |-f(-Ui|,t) r[r2 +

(z-

zi)^]'^ dz^ J _{_ll} dt

II

H

-- --i^f(--|z|.t)

_f r[r2 +

(z-

zj)^]"

dz [59] 24

1 9£ ^ _9t

/T^

(z

-z^r

H

-H [59] continued

1

f^

r at and similarly 9r 8x cos 9 9f r2 at [60]

Thus on thebody the potential ijj satisfies the conditions [55] and [56] to

leading order in R.

To

establish that the

same

is true of ijj,

we

now

show^

thatthe contributions

from

i|j2 ^^^ d\\)y/dx are ofhigher order in R.

Since

^

Jo(k^) = -kJ^(kr) it follows that d^2

r!l=_K

r

^l

^L-e^(^+^l)j,(kr)dkdz,

9r _{J_H} at

_Jo

k-K

1 1 + ttojK^ r f(z, ,t) e^^^+^1^ _Jj(Kr)dz, [61]

We

wish to show that

~a7

= 0(f)

and

drax =

»(i)

as

R

-* _0, and thus that

«

and

«

9r 9r 9r 9x 9r9x

for

R/H

«

1. Fromi the series expansion ofthe Bessel function,

Jj(kr) = |-kr + O(k^r^)

and thus, -where this expansion is peirnissible in Equation [61], the

re-sulting

terms

are clearly of order fK. However, in the neighborhood of

z = 0, the

power

series expansion is not permissible in the integral over

k. It follows that, inthe neighborhood of r = R,

94^7'2 Sffo.t^9f 0,t rO r"' = -

K

—

^—

9r 9t r

I

-±—

e^^l J (kr)dkdzi

H

•'O + O(fR) 9f(0,t) _C^ 1 9t 9f(0,t)

_f^

Jl(^^)

S

-

_K

—

^—

-—

dk at Jn k Sinriilarly, =

_KJ£i£iil

= o(f) 9t 8^4;

—

=

oil]

r9x

VR/

Thus, on\the body.

9i|/2 / 9i|ji

"97

/ ^^1 \

and

9 \\)y . d \\i,

=

8rax \ 9r ax/

Therefore, the potential \\i satisfies the conditions [55] and _[56] with a

fractional error of order R.

REFERENCES

1. Barakat, Richard, "A Sumnnary of the Theoretical Analysis ofa

Vertical Cylinder ina Regular and an Irregular Seaway," _{Reference No.}

57-41,

Woods

Hole Oceanographic Institution (Jul 1957), Unpublished

Manuscript.

2. Wehausen, J.V., "Surface

Waves,"

Handbuch der Physik, Springer

Verlag, Section 13 (1961).

3. Lighthill, M.J. , "Mathematics andAeronautics," Journal ofthe

Royal Aeronautical Society, Vol. 64, No. 595 (Jul I960), pp. 375-394.

4.

Newman,

J.N., "A Linearized Theoryfor the Motions of a Thin

Ship in Regular

Waves,"

Journal of Ship Research, Vol. 5, No. 1 (1961).

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