The motions of a spar buoy in regular waves

Report 1499

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AERODYNAMICS
J. N. Newman
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STRUCTURAL
MECHANICS
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HYDROMECHANICS LABORATORY B)
RESEARCH AND DEVELOPMENT REPORT
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May 1963 Report. 1499

THE MOTIONS OF A SPAR BUOY IN REGULAR WAVES

J. N. Newman

Wom mn i AA

May 1963 Report 1499

TABLE OF CONTENTS

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Figure

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Figure

1

2

3

4

5

6

- The Coordinate Systems

List OF FIGURES

eo 8 © © © © © © © 8 8 we ee ee ee ew

-— Plot of the Surge Amplitude-Wave Amplitude

Ratio for the Circular Cylinder

Ce

- Plot of the Pitch Amplitude-Wave Slope Ratio

for the Circular Cylinder

e+ © © © © © © © © 8 8 ee ee te el

-— Plot of the Heave Amplitude-Wave Amplitude Ratio
forthe, Undamiped) Ginculaaz Gylind en. ss eae aisaeeeee set

- Plot of the Heave Amplitude-Wave Amplitude Ratio
for the Damped Circular Cylinder with R/H=0.1.....

- Plot of the Heave Phase Lag for the Damped
Gineuilarn Gyyilktaclor waldo IR//et SO, MGs eb. diced sce0c 0b.

NOTATION

Incident wave amplitude

Gravitational acceleration

Body draft

18

18

20

20

21

Body moment of inertia in pitch about the center of gravity

= Nil

Bessel function of the first kind of order zero

Wave number, w"/ 2g

Radius of gyration, «2 = I/m

y

Body mass

Unit normal vector into the body

0
-H

(z = ac S(z) dz

iii

Pp Pressure

Q,(k) = fe (z - 2G)" S(z) eK dz
R(z) Sectional radius of the body
S(z) Sectional area of the body

r Polar radius, re = x + y“
t Time

(35 Wp £4) Cartesian coordinate system

ZG Coordinate of the center of gravity
iC Heave displacement

¢* Free surface elevation

8 Polar coordinate

E Surge displacement

fe) Fluid density

o) Velocity potential

xX Vertical prismatic coefficient

co) Pitch angle

w Frequency of oscillations

iv

ABSTRACT

A linearized theory is developed for the motions of a
slender body of revolution, with vertical axis, which 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 except for the low frequency range

where very sharp maxima occur at resonance.

INTRODUCTION

The motions of a vertical body of revolution, which is floating in the
presence of waves, present a problem of interest in several connections.
The motions of a spar buoy, of a wave-height pole, and of floating rocket
vehicles are important examples of such a problem. The same methods
developed for 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 of motions 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 of applied
ship-motion theory. An alternative procedure is toformulate the (inviscid)
hydrodynamic problem as a boundary-value problem for the velocity po-
tential and to employ slender-body techniques to solve this problem. The
latter approach is followed in the present work, 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 of buoyancy and gravity coincide, is treated in

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

“References are listed on page 27.

pitch oscillations.

In deriving the hydrodynamic forces and moments acting on the body,
we shall assume that the incident waves and the oscillations of the body are
small, and thus we shall retain only terms of first order in these ampli-
tudes. We shall also assume that the body is slender. The analysis with
only first-order terms in the body's diameter leads to undamped 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 of a floating slender
body of revolution with a vertical axis in the presence of small incident
surface waves. Let (x,y,z) be a fixed Cartesian coordinate system with
the z-axis positive upwards and the plane z=0 situated at the undisturbed
level of the free surface. The x-axis is taken to be the direction of propa-
gation of the incident wave system, and the motion of the body is assumed
to be confined to the plane y =0. We shali also employ a coordinate sys-
tem (x',y',z') fixed in the body, with z' the axis of the body, so that with
the body at rest, (x,y,z) = (x', y',z'); and a circular cylindrical system
(r,9,z), where x =rcos 0 and y=r sin®@. If €, €, and w are the instan-
taneous amplitudes of surge, heave, and pitch, respectively, relative to

the body's center of gravity, it follows that
se (2 op oe) COS Wh ar (zB! — ZG) sin |

y=y' [1]

N
iT

Ce ater sin J + (z' — z7,) cos Wt 2,

where ZG is the vertical coordinate of the center of gravity in the body-
fixed system; see Figure 1. The displacements €, €, and J are assumed
to be small oscillatory functions of time; we shall consistently linearize
by neglecting terms of second order in these functions or their products

with the incident wave amplitude A. Thus Equation [1] may be replaced by

Figure 1 - The Coordinate Systems

x=

SB was) ae (A = ae)
y=y' [2]

m= (6 Sad cp aY

If an ideal incompressible fluid is assumed, there exists a velocity
potential, ®(x,y,z,t), satisfying Laplace's equation, such that its gradi-
ent is equal to the velocity of the fluid. This function must satisfy the

following boundary conditions:

(1) On the body, the normal velocity component of the body must equal
the normal derivative of ®. For a body of revolution defined by the equa-
tien r'=—=R(z'),, where r'= Nx'2 4 y'2, this boundary condition may be

expressed by the equation’

[rot S IRB") = e + VO. Vv) [r' — R(z')] = 0

[3]

Gin ? S IR( yg")

(2) On the free surface, the normal velocity component of the free
surface must equal the normal velocity component of the fluid particles
in this surface, and the pressure must equal atmospheric pressure. In
the linearized theory, these conditions reduc e” to

2
® co)
) i 0

— — = = O, 4
aD 8 Se 0 on z [4]

or in the case of a sinusoidal disturbance with frequency w,

Ran O83 @ on z= 0, [5]
Oz

where K = w/e.

(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 by a proper
distribution of these singularities. 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
lar /az'|<< 1. The depth H i Ae sumed 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 << l.

We wish to obtain the velocity potential of leading order in the small
parameters of slenderness and oscillation amplitudes in order to obtain a
consistent set of linearized equations of motion for the body. However,

it will turn out that the potentials of different phases of the motion are of

different orders of magnitude with respect to the slenderness parameter.
For example, the potential due to surge or pitch is of order R as R— 0,
whereas the potential due to heave is 0(R¢). Similar differences will oc-
cur in considering the components of each potential which are in phase
and out of phase with the respective velocities of the body. In order to
circumvent these difficulties without unnecessary higher order perturba-

tion analysis, we decompose the velocity potential in the following form:

Os y725 6) = o- (x,y,z; t) + by (x,y, 25 t) + by, (% yt)

[6]

+Al[g/w eKZ cos (Kx — wt) + d, (x, y, Zt)!

where be ; by , and $y, are linear in the displacements (€,€,) and their
time derivatives, respectively. The potential A g/w eKZ cos (Kx - wt)
represents the incident wave system and the potential Ad, (x,y,z; t) re-
presents the diffracted wave potential, corresponding to waves incident
on a restrained body. Each potential » in Equation [6] must satisfy the
free surface boundary condition and the radiation condition; the complete
potential © must satisfy the boundary condition on the body. This condi-
tion, Equation [3], is reduced as follows:

_ te) Ost | Ore Oy! a OTE OZ LEG RanOZE

)
££ 9Oe Pio Nia) se as
(se mt (a dx' dt dy' dt az' ot dz’ ot

OR) BS) de sae,
or' oz' dz?

or neglecting second-order terms in A, —€,{€, and y,

— -— —-[&+(z-2zG)b] cos 6 +(6- xb) dR/dz = 0
[7]

Gr, 72 & IBY (52),

where a dot denotes differentiation with respect to time. Substituting
Equation [6] into Equation [7] and separating terms according to their
dependence on different displacements, we obtain the following boundary

conditions on the body:

do : ob

2 = Ecos + 0(R =) [8]
or Zz
do i fea)
— = Uz - 2q)cos0 + 0(R =“) + 0(R2) [9]
dp toR do

2 = -= + 0(r—*) [10]
or Oz Oz
Oba

=- w eKzZ [cose sin wt - (KR cos’ 6+ <) cos at]

or dz

o(R a 0(R)
[11]

= wekz [ cos @ sin wt +(3 KR + = +3 + = KR cos 26) cos ut]
CLON 3
5. 0(R + 0(R2)
Oz

To satisfy the above boundary conditions, we employ slender-body
theory.° For example, the potential satisfying Equation [8] is an axial
line of horizontal dipoles, of moment density 3& [R(z)]@ per unit length.

Thus in an infinite fluid,
') Ve ) 20
oe FE [ [R(z})] ips [ie4 + (z - 2i)° i, ZiT [12]

To satisfy the free ake and radiation conditions, we substitute

for the source potential [r2 ar (4 > Fay 217 2 , the potential of an oscillating

source under a free surface.” With this substitution we obtain, in place
of Equation [12]:
i

ne 7 2 9 Deine
Op S Be ie [R(z)] 2 {22+ -2) ee

(oo)
A f Isso Je) Js(0497 491) Jp (kr) ak} dz [13]
hy) pets

0
+ roke| Eee cee aye : ~ [Jy(Kr)] dz,

and, ina similar fashion,
: 2 y) 2 Daz
{ 2
dy=2¥ ii [R(z,)] (4 - eq) {Iz ig ((4 =F) ]
kt+tK ok +
+ f 28 —= (aay) y o (kr) ak } dz, [14]

0
+ roky [ [Rein (z= ze) ele tay) 2 J o(Kr)] dz,
aH

Be 1
$= 2t if R (2) [p24 (z-2)°] *

ee)
k+K k(z+z,)
i i ze 1 Jp (xr) ak dz, [15]

0
+ roKt 2) SES oS 9 Sul Wf (eae) ae
ns 0 1
-H Zi

0
da=-te [ eal le sad Beco!
-H dz
8 92 278
+ R@ sinwt — + 2 TRS coswt —— [r* + (2-2) ee

o [16]
k+K k(zt
A . k+K .k(z+ 2) ster) a} dz,

k-K

0)
= 7TwK | pee tall( KR +22 )R sin wt
Zi

2
- R* cosut Saag KR* a te Jo (Kr) dz,
Ox 9x2

where f denotes the Cauchy principal value. From the Appendix we see
that the potentials [13] to [16] satisfy the boundary conditions [8] to [11],
respectively, with a maximum fractional error of order R. Unfortunate-
ly, this error is not so small as in the classical slender-body theory for
an infinite fluid, where the error is of order R2 log R; for this reason

the present theory may not hold for as wide a range of slenderness as in

t

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 of the 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-
ing order, only the singular term moe (za 2,)2]72 contributes to the in-
tegrals over z), and the integrals may be evaluated directly since for

any continuous bounded function f(z,) and small values of r,

0 1
| MeV 4 (esas) 2 dz, = —2f(z)logr + 0(1)
Br

0) al
{ fla) = [e? 2 (2 oa © Gia = 2 1) cose + 0(1)
Ox
-H
0 2 Zul
| g [s2 4 (esate) ° dz = p 22) cos 26 + 0(1)
2 1 | 2
-H Ox ree

tO Clal<w< 0, wiK< Jal.

Thus on the body,

, = E R(z) cos + 0(R2) [17]
by = - | R(z)(z - zG) cos + 0(R*) [18]
6, = ~6 RF log R + 0(R?) [19]

A (hii 4 SS )R log R cos wt + Rcos® sin wt] + 0(R2)

[20]
e&Z R cos@ sin wt + 0(Ré log R)

THE FIRST-ORDER FORCES AND EQUATIONS OF MOTION

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

Ra oo Sa
on oes 8b
= - pez - pe pp pe - pal TE + ge? sin(KR cos 0 - at)

= pgz + pER(z)cos® + eWR(z)(z - Zc) cos 0 = pAw*eX2R cos® cos wt
te pgAeKz sin wt - ogAKe2R cos® cos wt + O(R2 log R)

= pgz + p€R(z)cos0 + oR (z)(z = #@) COS 8) wv pgAeX2 sin wt

- 2pw*Ae®4R cos® cos wt + 0(R% log R) [20]

The force and moment exerted on the body by the fluid are obtained
by integrating the pressure over the surface. In the absence of any other
external forces, the force or moment must equal the respective accelera-
tion times the mass or moment of inertia of the body. Thus, with % the

unit normal vector into the body, the equations of motion are

mé = ff pcos(n,x)dS [22]
m(t + g) = ff p cos(n, z)ds [23]
Iv = ff pl(z2- ZG)cos(n,x) - x cos(n,z)] dS [24]

where m is the body's mass, I its moment of inertia about the center of
gravity, and the surface integrals are over the submerged surface of the
body.

In computing the pressure integrals over the body surface, it is ex-
pedient to employ the (x', y',z') system, fixed in the body. The direction

cosines are

cos(n,x') = — cos 6 + 0(R2)

dR

2
i = = a) R
cos(n, z') ag (R“)

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

E

2 Fy cos b+ FPF, sin=F + F_, + 0(u%)

F, =F, cos $- Fy: sin= Fi: -h Fy: + 0(u7)

Thus the equations of motion may be written in the form

a 27 ce C+x'w dR
m€ = - cos 0+ J —]} pRdz'dé'
Id ( =)
0 -H
i" Big Ge ge ap
real (Gar ()) = J (SS + ¥ cos 0] pRdz'de'
0 4-H fl
: 2n pth Ct+x'
i = J | [(2'- 2G) cos(n,x') - x! cos(n,z')] pRdz' do!
0 -H

2m pCF-C+x'
is if | (z'- z¢)cos 8 pRdz'do' + 0(R>)
0 0

where (* is the free surface elevation at the body. Substituting Equation
[21] for the pressure and neglecting second-order terms in the oscilla-

tory displacements €,€,, and A, we obtain

: Zant 9 dR
mé -- vg [ J (- cos a+ y SS) (z'+ C - WR cos 6')Rdz' do!
0 bar dz

21 0) co -
=p || J cos 0'[€R cos 0' + JR(z-— ZG) cos 0!
0) -H

1
+ gAe®” sin wt — 2w“Ae’” R cos 0' cos wt] Rdz' do!

10

0
= -— Tog f (yR+2y2 oe) R ae
-H

0 oy oo Kz'

> (Our i] le + w(zt— ZG) DY OCI cos wt] R¢ dz'
-H

or, since

0
{ (yR + 2y2" oe) R az! = [i 1 (R22!) dz! = 0
-H H dz

it follows that
0 ee ee 2 K 3
i [é + f(z - 26) - 20 Ae” cos wt] S(z) dz + 0(R~ log R)
[25]

where

S(z) = 7 [R(z)]*

is the sectional area function.

In a similar manner we obtain

: 0 e ds
m(¢+g) =-pgOS(0)+ pg | ' S(z)dz + pgA sin wt f eKkz ©? az
-H -H dz
4
+ O(R~ log R) [26]
Ip =-pgy (z - 2) S(z)dz
IL, eee

0
- J [E +h (2 - 2¢)- 2u%Ae

KZ cos wt] (z- Zc) S(z)dz

+ 0(R° log R) [27]

11

From Archimedes! principle, or equivalently, satisfying Equation

[26] to zero order in ©,
0)
gaa | Oe i] S(z)dz
-H

and thus
- 0
mC =-pgtS(0)+ pgA sin wot |
-H
while, from Equations [28] and [25],

0

eKz = dz + 0(R* log R)
Z

Dre 2S p I [G(z - 26) - 2u2Ae® cos ut] $(z)dz + 0(R? log R)

Let us denote:

Tete mké
> Vertical Prismatic Coefficient
pHS(0)
p v n
12 = — J (2-2) S(z)dz (n = 1,2)
n m G
-H
v n
Q,(K)= £ [ cK@(z-2g)"S(z)dz (m= 0,1)
n ma J) ns

and note that €, ©, and \ must be sinusoidal with frequency w.

equations of motion then become
fats, op 129 Wh 2AQ) cos wt
(1 -XKH)¢ = A(l- XKHQ)) sin wt

200]
(P2+ky-=)4+ ES = = GINO} cos wt

12

The

[28]

[29]

[30]

[31]

[32]

[33]

Note also that surge and pitch are coupled, unless P, = 0 or unless
the centers of gravity and buoyancy coincide.

The above equations of motion are not unexpected. The restoring
forces on the left-hand side consist of hydrostatic and inertial forces plus
entrained mass terms 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 of Equations [31] to [33] could have been
obtained by neglecting the presence of the free surface. Moreover, the
exciting forces on the right-hand side of these equations are those which
follow from the "Froude-Krylov" hypothesis that the pressure in the wave
system is not affected by the presence of the body. These results are, of
course, a consequence of the fact that the body is slender.

The solutions of Equations [31] to [33] are

1 - XQ) KH
= A sinwt | ————_ 34
: ( 1 -XKH ee
2
P,Q, -Q,(P5+k = 12%) 5S)
b= 2Acos otf E22 yt | [35]
2(P, + LS - P,/K) - PY
P{Qy9- 2Q)
wb = 2A cos wt as a ween ee [36]
2(P, + IS - P,/K) - PY
We note that when
Ke [37]
XH
there is resonance in heave, and when
Pi
Renee eer TT Pe

Is)

there is resonance in pitch and surge. To determine the oscillation am-
plitudes in the vicinity of these resonance frequencies, it is necessary to
consider the damping mechanism 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 in the radius of the body.

THE DAMPING FORCES

The damping forces will follow by considering the last terms in
Equations [13] to [16] and will consequently be of higher order in R than
those terms which we retained in the previous analysis. This procedure
is nevertheless consistent, since at resonance the lower order restoring
forces vanish. In other words, we are retaining the lowest order force
or moment of each phase separately. Fora further discussion of this
point, see Reference 4.

We proceed, therefore, to study the damping forces, or the forces
in phase with each velocity. The only contribution from Equations [13]

to [16] is the potential
K v dR 2, @ Kz
ec! Zz
o* = twKe [ye (eG t [E + (z) - 2G)]R me 1 Jo (Kr) dz,
[39]

Since Jo (Kr) =l- i (Kr)@ + ..., it follows that on the surface r = R(z),

the leading terms are

II

Jo (Kr)
and
0 1

al Jp (Kr) = 24 Kea = -1K2R cos 0
Ox

Thus, to second order in R the damping potential on the body is

14

i 3 lé+y (z) = Zc)] K* R(z)R(z)) cos o} eh41 dzy

0
=1yKeK2zz J oKZ1 ds_ dz [40]
-H

dz, :
3 K : K
- + wK’R(z) cos 0e"™” J [€ + b(z, - zc)Je al S(z}) dz)
-H
The damping pressure on the body is

* : 0
pe Sop 20% = -4 wp Kel? ¢ | eee ee dz,
ot _H

dz,
[41]

0
+ + op K?R(z) cos 0 eX [6 + U(z, - Zc)] eal S(z)) dz)

Then the heave damping force is
0 27 0 2
Pein = J J Drak Rvdeldz = — S wp Kt eKz 9S a, [42]
Z -H (0) dz BE dz

Similarly, the surge damping force and the pitch damping moment are

0 27
Fe oul J p* cos @RdO dz
-H Jo [43]
0 OR i: Kz,
2 - top> f s(z) eK az) | [é + p(Z)- zc)le “1 8(2,)42]
-H -H
and

I)

0 27
M* =! p*(z - 2G) cos 6 R dé dz
-H “0

(z- 2q) S(2) eK? az ) [ 44]

or in terms of the integrals Pj; Po, Qo: and Q):

Ee = -i wp Kt[K = Q(K) - 5(0)]*
45
_ i eK 2 "al

o 1 wm? 3 : ‘
pe Bos K~ Qo(K)[€ Qo(K) + Qy(K)] [ao])

eet Moca ;
My = - 3 —— K°Q)(K)[6 Q9(K) + $Q)(K)] [47]

In place of Equations [31], [32], and [33], we obtain the damped equa-

tions of motion

(a {2 9p Pi¥ =- 24 Q)(K) cos wt
ne | } [48]
+5 op K Q)(K)[€ Qo(K) a pQ,(K)]
(1-xKH)¢ = A sin wot[1 -XKH Q,(K)]
m KC 2 [
U(P - Py/K+ Ke) +P) & = - 2A Q,(K) cos ot
[50]

pspiadh K? Q,(K) [€ Qo(K) + JQ,(K)]

The damping terms of these equations of motion are given by the terms
linear in the velocities é, t , and wb. 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 of motion, [31] to [33], and the resulting
displacements, [34] to [36].

CALCULATIONS FOR THE CIRCULAR CYLINDER

As a special case, we shall consider the circular cylinder R(z)=R

= constant. Then

¢ 2 1.0
1 0
Bos || (z-2zG)dz = -FH- Ze
-H
1 y 2 2 2
Bp ig lL, (oe dz = 3H + Hzq4+ 26
1 v K 1 KH
ee Z Ss eu eae
oreo iia a a edz sax Vi e )
1 Kz 1 -KH 1 -KH
Op (SS) Se i o(4 2 aq) G4 B Se Sera iS Yl + Kzc)
Ho Jy K K°H

We shall assume, moreover, that the centers of buoyancy and gravity co-
incide, or ZG ais H/2, so that the equations of motion are uncoupled and

there is no resonance in pitch or surge.

Then

2
Sa, lie sellin 2 &
12 0; P5 Game QS) ae ;

17

cn

Figure 2 - Plot of the Surge Amplitude-Wave Amplitude
Ratio for the Circular Cylinder

agg
eS

ie ea eas|

Ea
ne
Bs
LI
Be
Hl
Ei
ics
ie

3 4 5
La]

Figure 3 - Plot of the Pitch Amplitude-Wave Slope
Ratio for the Circular Cylinder

and it follows that

E = - 532 (1 - e KH) cos ut [22]
th a 528 cos wt oe EES [52]
(az) +]
ae vote 5 ; [a SER) Shr bs
(1 - KH)@ +[4 KH le) ae
H

[53]

Plots of the 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
amplitude to wave amplitude. For zero frequency this ratio is one and
for increasing frequencies it decreases monotonically to zero. Figure 3
shows the ratio of pitch angle to the maximum wave slope KA, multiplied
by the coefficient C = 4 + 6 (ky /H)*. This coefficient is equal to one if the
mass in the cylinder is uniformly distributed throughout its submerged
length. The ratio starts at one for zero frequency and decreases mono-
tonically to zero. Thus the pitch amplitude is always less than the wave
slope. Figure 4 shows the ratio of heave amplitude to wave height for
frequencies away from the vicinity of resonance. Near resonance, the
amplitude is shown in Figure 5 and the phase angle in Figure 6 for the
particular case R/H=.0.1. The ratio of heave amplitude to wave ampli-

tude is unity for zero frequency, rises to a maximum of

2 2
£(2) ~ 0.865 (7)
™\R R

at the resonance frequency KH = 1, and then decreases monotonically to

zero. The phase angle is similar to conventional one-degree-of-freedom

LY)

2.8

PEERS

24

O |

: | me
; | me
| Fa
a | - Rese |
Ds ee ~
2 3 4 ls) 6 v
KH

Figure 4 - Plot of the Heave Amplitude-Wave Amplitude
Ratio for the Undamped Circular Cylinder

0.5 0.6 0.7 0.8 0.9 10 11 I.
KH

Figure 5 - Plot of the Heave Amplitude-Wave Amplitude Ratio
for the Damped Circular Cylinder with R/H= 0.1

20

200

ae

é IN DEGREES

” i
80 -
40 wll

es

0.6 07 0.8 x : : L if ie)

Figure 6 - Plot of the Heave Phase Lag for the Damped
Circular Cylinder with R/H = 0.1

harmonic oscillators with linear damping; for low frequencies the heave
displacement and wave height are in phase, at resonance they are in quad-

rature, and at high frequencies they are 180 deg out of phase.

DISCUSSION AND CONCLUSIONS

The damped equations of motion as given by Equations [48] to [50]
may be solved for an arbitrary body of revolution to obtain the oscillation
amplitudes and phases. Except in the 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 of these oscillations are shown in Figures 2 to 6 for
a circular cylinder, with the important restriction that the centers of buoy-
ancy and gravity coincide. If this restriction is relaxed, a resonance will
be introduced into the equations for pitch and surge, but the frequency of
this resonance may be kept small by ballasting. The amplitudes at reso-
nance are extreme, but the resonance frequency for heave is quite small

and can be kept out of the practical range of ocean waves by making the

21

draft sufficiently large. It would 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

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 in waves.

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 of the manuscript.

22

APPENDIX

Here the potentials E> y ' Pi. and da» defined by Equations [13]
to [16], 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
0) of(z,,t) -
pat J a fr te - 271
-H

oe)

k+K

+ + olay ai) Jp (kr) dk } dz, [54]
o k-K

tle

K(z+t 2)

0
+ mwK I f(z,,t)e Jo (Kr) dz,
-H

where f(z),t) has sinusoidal time dependence with circular frequency w.
By appropriate choice of the function f, the potentials Pe» by ; yp and
oa can all be obtained from and dy/dx. Thus it is sufficient to estab-

lish that the following conditions are satisfied on the body surface r = R:

©
R=
Ile

f(z, t) [55]

a — f(z,t 56
arIoe my ia Sena o!

Employing an alternative form of the source potential,” we write

in the form

1 ok (44 2))

[57]

23

[57]
Jo(Kr)dz, continued

= Shy hea
where
b) = $ ip ~- ir ap ((v4p, S £4) 2 +[r~+ (z, + zy} *} az,
-H

0
TONS J ea pO) Co ae Jy (Kr) dz,
a

The potential vy corresponds to an axial distribution of simple sources
together with an image distribution above the free surface z= 0. To

emphasize this fact we write yy) in the form

pl 2 “2
= £(-|z,|.t) [> + (z= 2,)7] > okay [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, differentiating with aspect to r and
neglecting terms which are of order R* or Ros @ inthe neighborhood

of the body r = R, we have

a
or

+ (z= 2,)°] dz) [59]

= es peek Ret SROs ans, Lense RS Sr continued
Sh Ciera, aye
r go bs (FA 2) -H
mS il he
a at
and similarly

2
Oh mn COED OF [60]
Or Ox a Be

Thus on the body the potential satisfies the conditions [55] and [56] to
leading order in R. To establish that the same is true of J, we now show
that the contributions from > and db>/ Ox are of higher order in R.

Since

fs)
Be Jo (kr) = -kJ) (kr)
it follows that
dY> 0 ©
Buisie ot Ub eh eel ay (teriidled 2
or Er ot ig) Mc aiK 1

3 [61]
+ Tw Ke { f (z,,t) or 2 eel TK) dzn

We wish to show that

Ny)
= O(f
a (f)
and
2
ay
t= off
Or Ox R

25

as R-— 0, and thus that

2 2
OOM Be Oh
<<
or or Or Ox Or Ox

for R/H<< 1. From the series expansion of the Bessel function,
J, (kr) = kr + 0(k?r?)

and thus, where this expansion is permissible 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, in the neighborhood of r=R,

ow af(0,t Qos
GS ge CENCE) J f is e871 3, (kr) dkdz,
Or at jan d@ ie IX l
+ 0(fR)
A WE OOo) f u J, (kr) dk
im at ) Rom OE
af(0,t) fe J, (kr)
Sif K eae SE we ak
at 0 k
oOo ae 0(£)
at
Similarly,
2
a”
t+ of)
Or Ox R

Thus, onthe body,

26

and

a bo / a(n ai)

or Ox Or Ox

Therefore, the potential \ satisfies the conditions [55] and [56] witha

fractional error of order R.

REFERENCES

1. Barakat, Richard, ''A Summary of the Theoretical Analysis of a
Vertical Cylinder in a 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 and Aeronautics,'' Journal of the
Royal Aeronautical Society, Vol. 64, No. 595 (Jul 1960), pp. 375-394.

4. Newman, J.N., "A Linearized Theory for the Motions of a Thin
Ship in Regular Waves,'' Journal of Ship Research, Vol. 5, No. 1 (1961).

27

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