Report 1673.

AERODYNAMICS

Robert Taussig

STRUCTURAL
MECHANICS

HYDROMECHANICS LABORATORY

RESEARCH AND DEVELOPMENT REPORT

@ =
i

IYI AA
no 160

May 1963 Report 1673

MOTIONS OF A SPAR RAFT IN REGULAR WAVES

by

Robert Taussig

May 1963 Report 1673

TABLE OF CONTENTS

Page
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ii

NOTATION

Amplitude of incident surface wave

Radius of circle on which spar axes are located
Radius of ith spar as a function of Z.
Acceleration of gravity

Moment of inertia of iu spar about x-, y-, z-axes,
respectively

Wave number of incident waves = wife
Mass of one spar (assumed equal for each spar)
Mass per unit length of ula spar

Number of spars

Space-fixed cylindrical polar coordinate
Body -fixed cylindrical polar coordinate
Defined in Equations [13a], [13b]
Cross-sectional area of ith Spar = may (zs)
Components of force on nee spar
Space-fixed Cartesian coordinates
Body-fixed Cartesian coordinates

Linear displacements of structure
Components of moment on i spar

Angular displacements of structure about x-, y-, z-axes

Space-fixed cylindrical angular coordinate

iii

Body-fixed cylindrical angular coordinate

h :
spar axis

Equilibrium angular position of i
Kinematic viscosity of water
Density of water

Complete velocity potential
Velocity potential of incident wave
®- 2)

Circular frequency of incident waves

iv

ABSTRACT

A theoretical analysis is constructed for the hydro-
dynamic forces acting on a system of interconnected vertical,
slender, axisymmetric bodies which are floating in presence
of incident waves. The theory is based on linearized water
wave potential theory and the use of slender body techniques.
The resulting expressions for the hydrodynamic forces are
used to predict the motions of sucha system. The effects

of viscous damping are also estimated.

INTRODUCTION

A spar raft as defined here consists of several long thin bodies of
revolution rigidly interconnected so that they will float vertically in the
water and support a platform or submerged weight. When regular waves
are incident on sucha structure, it will generally oscillate in six degrees
of freedom. The purpose of this report is to provide an approximative
method for calculating such motions.

The assumptions are: (1) that the spars are identical, (2) that their
interconnections are made in such a way that the mass and the hydrody-
namic effects of the connecting members may be neglected, and (3) that
the individual spars are far enough apart for their hydrodynamic interac-
tions to be neglected. The motions of a single spar buoy have been treated
by Newman;! here his method is extended to the case of N spars arranged
ina circle. In addition to including the hydrodynamic and inertial forces
on several spars, it is necessary to extend Newman's analysis to allow
for all six degrees of freedom. (In his problem, only three degrees of
freedom involved nontrivial results.)

The basic assumptions of Newman's analysis are used here. In
particular, it is assumed that the wave amplitudes and body motions are

small enough that linearized free surface theory may be applied and that

"References are listed on page 30.

the spar radii are small enough compared to wavelength and spar sepa-
rations that slender body theory may be used. Equations of motion are
derived on these bases. These equations predict motions which are un-
damped; thus they are valid only for frequencies which are not near the
resonance frequencies.

Near the resonance frequencies, it is necessary to consider the
damping due to wave generation, and this report shows that forces of high-
er order in terms of spar radii must be included. The leading damping
forces are found, thus providing equations of motion valid near resonance.

In addition, this report indicates that viscous forces depend linearly

on the velocities for axial motions, and these forces are found explicitly.

GEOMETRY AND COORDINATES

It is convenient to define several coordinate systems. With the
structure floating at rest, we place the origin of a space-fixed reference
frame at the undisturbed free surface over the center of gravity of the
structure. Let the Cartesian coordinates of a point in this system be
(x,y,z), with the z-axis directed upwards. In this same system we de-
fine cylindrical coordinates (R, 9, z):

R% = x* + y4 sO) S tein

or

oS IN COS FF 47 S IN Gia

zis here the same as the Cartesian coordinate z.

Let the undisturbed axis of the jth

spar be located at R=ag, 9 = 0;.
We define another set of space-fixed coordinates, (xj, yj, Zi), with origin
at R=ag, 9=9;, z= 0. Let the cylindrical coordinates of a point in this
system be (124 2 ris Zi), with the latter having the same orientation as the
previous cylindrical system.

th

In the undisturbed condition, the surface of the i~ spar will be

specified by the equation:

Re = a; (Z;) S45 2 SS Jals
These quantities are all shown in Figure 1.

Finally, we introduce primed coordinate systems which correspond
to each of the systems just mentioned, but which are fixed in the body.
When the body is in its equilibrium position, the primed and unprimed
systems coincide.

The linear displacement of the raft will be described by three dis-
placement variables, Xo» Yor 293 the angular displacement IS Gy (Bp We

which are the positive rotations, respectively, about the x-, y-, z-axes.

Since the motions are assumed to be small enough that squares and prod-
ucts of these variables are negligible, the location of the raft is complete-
ly specified, and the three angular displacement variables can be treated

as the components of a vector.
Let r°~ be the position vector of a point fixed in the raft, where

B= (3G yo, 2). in terms) onthe space-—tixedycoordinates an —i(e<;ayn) 2),

we have, to first order in small quantities:
HS BW oy ese [1]
where
rq = (Xo. Yor 20)
a = (a, B,Y)
In terms of components, this equation is equivalent to

2

*
!

BS ore ap Xo + (Bg NING
Yir= eae Gy oN ES aA [ola

Z=2°t Zo + Isp 1/8) 3%

To first order also, it follows that

YO SX a aq = eee OL lea

The unprimed coordinates are related by

x = ag cos 0, + x;

K
|

= ag sin, + y, [2]

and the same equations hold if x, y, z and Xi, Yj, Z, are all primed.

It is assumed that there are incident waves which are described by

the velocity potential

Bo (Exmyiizint) = oy ge cos(kx — wt) [3]

where A is the amplitude of surface wave,
k is 27/wavelength= wave number, and
w is the circular frequency.

In the definition of the coordinates given above, the orientation of the x-
and y-axes was not specified, except for the orientation of the plane which
they defined. Now we specify that the x-axis points in the direction of
propagation of the surface waves and the y-axis completes the right-hand

system.

THE VELOCITY POTENTIAL

The surface of the ith spar can be specified by the equation:

(o)
iT]

BGs yy 2;)

=) Cita Vim aa (2.7)

: 2
[hci Xo 7 Bz; + Y(ag sin 8; + y;)]
@
+r Ly; = 9) = VWila@ GOS Oh ap ==) sr az. |

2
mg ay [z:

i — 29 — @(a¢ sin ©; + y;) + B(ag cos 0; + x;)]

th

The boundary condition on the i~ spar is then

OF.
i

ot

PW @) 2 Wa) 1s = 0 on FF. =0

where VY; indicates the gradientin the (X;,y;> z.) system, and = O(x,, Vy» 2> t)
is the velocity potential (viz., ®p plus a potential due to the presence and
motion of the structure). After some simplification, we find that the

boundary condition is

o® oO®
OR} TZ

= [Xo aF B(z, ar a, ay) - Yag sin 0; | cos As
+ [yo - &(2, + ajay) + vag cos 0; | sin), [4]

= [2o @ anl@ SAO, 6 cos Oar Ga R= e,

da;
where ay = an . Second and higher order terms in the motion variables
Ze
i
have been consistently dropped.
Let 6=@) + %,. We substitute this relation into the last equation

to obtain a boundary condition on 6). From Equations [2] and [3] we

note that
Gy = -— eke cos (kx - uot)
[5]
582 BAN eos [k(ag cos 9. + R, cos A.) - ot]
i i i i

It follows that

0® a®

SE eal = wy eR {l-2 + ka. a’ +2 a%) cos r:
OR; i Oz; F.<0 clea i i

+4 Ie a’ cos 3X,] sin (kag cos 6, — ut)

= [(a; + 4 ka,) + 2 ka, cos 2r;] cos (kag cos 0; — wt)

50k Sy) [6]

where the omitted terms are of third and higher order in terms of the

Spar radius a; and its derivative ay, or of second and higher order in

the motion variables. Neglecting now second-order terms in a; and a,,

we find the condition on 9):
0®) oD, : : ;
(+ - 33) = {x9 + 62, - Vag sin 0;
OR; 92; /R.=a.
a eal
IRVAR
+wAe =}! sin (kag cos 0; - ot)} cos 2.
+ {i - az, + yaqg cos ai} sin );
ic ay nega! k ) t) 22
z ka;wAe cos (kag cos 0; -w cos il
- {i gay + agar (é sin 8, - B cos Q;) (aca

- (ay + 3 ka, ) ire sai cos (kag cos 0; - wt) }

We note that the boundary condition is now applied at the surface of the
undisturbed spar, and the right-hand side is evaluated in the space-fixed

(x;, y;, Z,;) coordinate system.

If this ith spar were located alone in an infinite fluid with the above
boundary condition valid for -H< z; < 0, the solutions for 2) by slender

body theory would be

0)
of = Ib {3 a, | Za; + ag aj (a sinydSaPicos) 6.)

- oA (a; + 4k ka;) cos (kag cos 0; - wt) |

ee ee koa g
+3 af[ x) +60 - Vag sin®; +wAe * sin (kag cos 0; ~ wt)] ——
al
2 es é : 6)
oa ey Be oS ey enh) =
4 kt 92 2 2 =F
= 5 Be [wk Ae”? cos (kag cos re eles [RU + (a, - Sy] © ae
x.
1

da; (¢)
where a, = a;(C) and a, = Tar in the integrand. We adapt this type of

i
solution to the present case as follows: (1) [R? (Ba t)"] * is the poten-
tial for a source located at (0,0,€) in an infinite fluid.

We replace this source potential by another source potential which,
in addition, satisfies the free surface condition. (2) We now impose the
condition that the radius a; is much smaller than the distance between
Spars; 1.e., a;/ag << 1. Then the potential obtained by satisfying the

boundary condition on the jth

spar will produce negligible fluid velocities
at the other spars, and the total potential can be expressed as a sum of
potentials, each satisfying the conditions on one spar. The resulting total
potential is

wA kz
—ee

Dixy Zt)! = cos (kx - wt)

NETO
+ y Lae a, [Z)a, + aga; (a sin 0, - B cos 0.)
n=] i

DUNE (az +4 ka.) cos (kao cos 8, — wt) ]

+4 a [xp + BL - ya sin 0; Ay ASS sin (kag cos 0; - ot)] a

i

Do 5 6 )
fm Oe [yo - a0 +yapq cos OD) i
2
eae lone cos (kag cos Oe eh] Oe
i 1 2
Ox
i
Re 4 2 ak eet le v(zitC) 5 R.)d d
[R; (4, = 6) ek o( vR; v C

NE AO
+ twk s [,, {boas aoav te sin 0; - B cos 8;)
i=] ¥

[8]

[8]
kt continued

th AS? (eg op 4 ka,) sin (Kap cos 0, - ot)]

2 e ; kt, ii 8
+a. [x9 + B84 - yag sin®; + Ae ” cos (kag cos 0; ete) ee

Bo

2 )
+a; [yg - 265 t+ vag cos il ee
a

2

+ 4 ap [k Aes sin (kag cos 0; - wt) |] So Leet) yar} dg
Ox’
i

In accordance with the assumptions of slender body theory, we eval-
uate these terms as R; — O and identify the values so obtained with the
potential on the body surface IU =a. Because a;/ag << 1, the value of ©

to lowest order on the jth

spar depends only on the first term above and
one term in the first sum. By the same approximation procedure, we find
that all terms in the second sum contribute amounts of higher order in
terms of a;. Later we shall reconsider the second sum when we calculate

damping forces.

FIRST-ORDER FORCES AND MOMENTS

The pressure is obtained from Bernoulli's equation in linearized

form
p=-p—-pgz Kon

Thus it will be necessary to evaluate @, on each spar and to integrate,
in an appropriate way, the result over all spars to find the forces and
moments.

Using slender body approximations again, we find that on the jth

spar

G= = ol cos (kag cos 0; — wt) = Da AorA a; cosX, sin (kag cos®, — wt)
~ [zgay ~ aga, (asin Oa = B cos 9.)
= OAC aes Lika.) eos (ka, cos 0. — wt)]a. log a
Re Se 0 i TS SA
= [esq 3 Bz. - ya, sin 0,] a, cos Ne = [yo - #2; + yag cos 6, ] a, sin d;

+ 0(a%) [10]

Also, we note that on the surface of the jth spar

ZZ Za at a(ag sin 0; + a, sin dg )) = Bag cos 0; +a; cosh; )

Thus the pressure on the jth spar is
kez.
p=-p¢gAe 1 sin (kag cos 0; — ot)

le
+2gkAe 44 a, cos A, cos (kay cos; — ut)

[11]

= [Zz a, + aja, (a@ sin 0, — 8 cos 0. )

k

- Zs ? 1 ° ie
gkAe 1(ai +3 ka,) sin (ka, cos 0. wt) ] a, log a,

0
= [x a Bz. Ya sin 0, Ja, cos we = [Fo = dizi ty Ya ROS 0, Ja, agra Nf

+ g lz, + Zo

+ a(a, sin 8, + a, sin Ax) = Bla cos 0, + a, cos Va)

Let the force be resolved along the space-fixed axes which corre-
spond to the coordinates (x,y,z). In particular, designate the components
of hydrodynamic force on the it? spar by XG, Yq. Zi. Wikewasie, Vet the com
ponents of hydrodynamic moment on the jth spar be denoted by A;, B;, T;
which correspond to the rotations a, B, Y. Note specifically that the

moments are taken with respect to the space-fixed axes at the center of

10

the whole structure. If we let n be a unit normal vector out of the fluid,

then

XxX; = | p cos(n, x) dS [122]
Si

Y; = iL p cos(n,y)dS [12b]
1

Zi = i p cos(n,z)dS [1 2e]
Si

A; = I ply cos(n,z) - z cos(n,y)] dS [12d]
i

B; = { p[z cos(n, x) - x cos(n,z)]dS [12e]
Si

ifs = i p[x cos(n,y) - y cos(n, x)] dS [iple2 Fei]
; S

The integrals are taken over the instantaneous surface of the fea spar.

Here cos(n, x) is the cosine of the angle between n and the x-axis, etc.

We find readily that, to first order in small quantities,

cos(n,x) = — cos r< ap Sy fialial d- + Ba,
cos(n,y) = —y cos AX; — sind, = aa;
cos(n,z) = 6B cos 2X, — a sind, + a,

For abbreviation, we also define two sets of integrals

0

sat , Did P ;
Sn = J S(z.)(z,°) dz; [13a]
-H
0 kz7 n
cig lis ep SizaNzaardc [13b]

11

th

, 2 i 3
where S(z;) = 7a; (257) = cross-sectional area of i~ spar at Zs.

The combination of these formulas and definitions with the previous
pressure results yields for the force and moment components
X; = 2pgkAcos(Kay cos®, — wt)Ty) — p(%q — Yap sin9,)S) - pBS, [14a]

Y; =- pl(¥g + Y ag cos 9;)Sq + paS, [14b]

Z, = pgkA sin(kap cos 0; - wt)Ty) + pg Sp

[14c]
-pg[A sin(kao cos0; — wt) + zg + @apQ sin®, - Bag cos 0. | S(0)
A, = pgkag sin 0; A sin(kag cos 0; - wt)To [14d]
+ pglyg t ag Sin 8; + yag cos 6;)SQ) + pl¥g + Y ag cos 8; - ga)S) - paS,
- pgay sin®; [z) + wagsin®; - Bajcos0, + Asin(ka, cos ®@, - wt)] S(0)
Be Ss pgkay cos®, Asin(kajy cos 0, — wt) T 9 1 pgkAcos(kag cos 0; — wt) T)
- pg(Xq + ag cos; - yag sin®;) So - p(X - Y ag sin®, + gB)S)-e BS,
+ pPgagcosO; [Zo + @ag sinQ; - Bag cos 0; + A sin(kag cos 0; - wt)] S(0)
[14e]
PD, =-2pgkag sin 0; A cos(kag cos 0; — ot) Tp
+ p[ag sin 0; xg - ag cos 9; yg - apy] So
[14¢ ]

+ p[ag sin 0, B+ ag cos 8,a] S)

S(0) is the cross-sectional area at z; = 0 when the whole system is at

rest and in equilibrium.

12

SIMPLIFIED INTERPRETATION OF FIRST-ORDER
FORCES AND MOMENTS

These expressions for the forces and moments can be viewed from
a simple point of view. Consider, for example, the x-component of force,

Equation [14a], which when written out becomes

0

H

S(z.7) {2 eke cos (kag cos 8; - wt)
- Xp + Ya sin 0; - 2/6 paz [14a7]
From Equation [5] we see that

a* &

kz
= gkA 1 cos (ka, cos @. — wt
ot ox g e ( 0 pease Ties )

on the equilibrium position of the aoa spar. Thus the first term in the

bracket in Equation [14a“] is just twice the local acceleration that the
water would have at the mean position of the spar axis if the spar were
not present. The terms, - Xq + ag sin ae re (8 give the negative ac-
celeration (in the x-direction) of the point on the spar axis. The quantity
pS(z;) is the added mass per unit length of a cylinder accelerating nor-
mally to its axis. Thus the x-component of force is the integral over the

mean spar length of

(added mass per unit length) times (2 times local
water acceleration on spar axis due to incident wave

alone minus acceleration of point on axis of spar).

It may appear strange that the water particle acceleration is doubled
in this formula. However, the cause is seen on examination of Equations
[8] and [10]. Inthe latter equation, the terms containing the factor
(cos A,) give rise to x-components of force. Here the term due to the
incident waves (the second term) is already doubled. Half of this con-
tribution comes directly from the first term of Equation [8] (i.e., direct-

ly from the incident wave potential) and half comes from the term

13

iP 3 ag ole <> cin (kag cos 0; - ut) a [Ro P(e = eyey 2 dt

-H el

The latter is effectively a diffraction potential; it is part of the singularity
potential which offsets the normal velocity component of the incident wave
on the spar. These results can also be regarded as a special case ofa
general body, accelerating in a time-varying (but spatially constant) in-
finite field of fluid. It follows from consideration of the forces, both in
the fixed and moving coordinate systems,’ that the hydrodynamic force
on the body is the added mass times the relative acceleration plus the
displaced mass of fluid times the spatial acceleration of the (undisturbed)
fluid. For a circular cylindrical section, the added mass and the dis-
placed mass are equal, and the above relation for the x-component of the
force on the spar follows immediately.

The z-component of force, Equation [14c], consists of three parts:

(a) [pg So];
(b) [-pg(zq + wap sin 8; - Bag cos 0;) S(0)];

(c) pgA sin (kag cos 0; - wt) [kT 5 = S(O) |)

Part (a) is just the hydrostatic force. Part (b) is the decrease in buoyancy
which occurs when the spar is raised an amount (Zo + aag sin 8;-Bagcos Sia ))
Part (c) is the integral over the undisturbed spar surface of the vertical
pressure force due to the incident wave alone. This is easily seen by

noting that

0
kT) - $(0) = - okzi dS :
-H dz,
and, thus, that Part (c) is equal to
o eae ds
| [=soreeSien = sin (kag cos 0; — wt)] > dz = J Pg cos (n,z) dS
-H dz; S;

14

to first order, where pg is the first term of Equation (ete:

The moments are obtained by calculating the force per unit length
along each spar, multiplying by the appropriate lever arm, and integrating
along the lengths of the spars. It should be noted again that the moments
are calculated with respect to the space-fixed axes. Thus a point located

th

at z; on the i'’-axis has space-fixed coordinates (see Equation [17]}:

x = ag cos 0; + x) + Bz, - y ag sin 0, ;

o .

We = AG Sia Os a5 7 a Wag COS OF = Cra 5

N
iH]

Zi + Zq + @av, sin 6; — Bag cos 8, .

FIRST-ORDER EQUATIONS OF MOTION

Let M be the mass of a spar. (The N spars are assumed to be iden-

h spar about the

tical) wet I) J, i) be the moments of anentiaof ithe it
X-, y-, Z-axes, respectively. Moreover, let Mo; Ip; Jo» and Ko denote
the mass and moments of inertia for any additional superstructure, and
assume (0, 0,z )) to be the center of gravity thereof. Then the equations

of motion are

(My + NM) xo

N
ee. xe [15a]
i=l

(My + NM)¥9 = [15b]

5

(My) + NM)Zo = > Z, - g(NM + Mo) [Lee]

N N N
- MggzZo+ @ y lig = S Ai - 2 y I. m (z.")y dz,” [15a]
i=0 i=l i=l

15

N N N
-Mopgzo+ B y J; = > Bet s s IE m (z,°) x dz [15e]
i=0 ae i=l
N N
Vv » Les Y DG Les |
i=0 i=l

m (z;") is the mass per unit length of the spar itself. The integral is taken
over the length L of the spar. This length generally extends from 2; =-H
to some value of Zs greater than zero. x and y are the distances from the
fixed reference frame to a point on the axis of the sen spar.

The moment of inertia I; about the x-axis is
I; = Ie m(2/)(y* + 2°) dzz

2

where y = ag sin 0; + yo t+ y aq COS Pe <= CRA SP oo On and
Z=Z + Zq + @aq sin 0; - Bag cos 0, + gdh ta
the omitted terms being of higher order in the small motion variables.

Since I; is multiplied by @, we need keep only the zero-order terms in

Iso

;- Clearly then,

2a? aes

I; = I m (2) [a4 sin i
L

to the required order in small quantities; that is, 1; has the same value as

in the equilibrium position. Similarly,

qy
iH

i (s m (2) (x2 oF 2*) dz.

= { m (z;) laa eos” 0; + rae) dz,
L

16

= | = 2)(x° + y? ) dz.

ie

L

The integral terms in Fquations [15d] and [15e] can also be written

explicitly by expanding x and y in the integrands. Thus

{ m (z.7) [ay sin 8; + yg + yag cos 8, — az. | dz

m(z-)y dz.”
iP i i 4, i

(ay sin 0; + yo + y aq Cos 8,)M -a [) m(z.) 2° dz-
J m(z.)x dz. = | m (2.7) [ag cos 0, + xX) + B2°- yap sin 9. | dz
L L
= (ag cos 0; + Xo — Y ag sin 8,)M+B i‘ m(z/)z° dz

We note that the final integral terms here would have vanished if z had

been measured from the spar center of gravity.

Now let us write the equations in full. Equation [15a] becomes

N
(My + NM)x_ = Z2pgkAT9 y cos (kag cos 0, - wt) - pN Xp Sg
i=l
15a*
. [15a']
- pNBS) + pagVSo y sin 0;
i=l

Clearly, Mg + NM = pSQoN, since at rest the buoyancy of the total raft

equals its weight. Also, we now impose the condition that the spars have

a regular angular spacing. Thus

if ps mt)
CA UE ee Nee 5 it — Wl eye tay cl ree AN

17

If N=1, then ao = 0, and the last term in Equation [15a”] vanishes.

ibe IN| S> IW
: N-1l 27 N 27
N N-1 oni sin (0, + al =) sin(® : =|
y sin 0; = y in (04 | se oh eG
N 1
71 | i=0 sin —
N

and again the last term in [15a“] vanishes.

Therefore, the equation of motion for xg is

N
2(Mo + NM) Xo + pNS,6 = 2pgkATo y cos (kag cos 8, - wt) [16a]
i=]
By similar arguments, the equation of motion for yg, Equation
[15b], becomes
2(My) + NM)¥q - pNS) @ = 0 [16b]
The equation for Zs written out, is
N
(My +r NM) (Zo ap fef)) = pgkATo y sin (kag cos Q; - wt) + NpgSo
i=]
N
- pg A5S(0) > sin (kao cos 8; - wt) - pg Nz S(0)
i=l
N N
- pgaay,S(0) > sin 0; + pgBa,)S(0) y cos 0;
i=l ie

Again we note that Mp + NM = pS)JN, which enables us to eliminate the

gravity term on the left. Also,

18

and ag = 0 for N=1. The equation becomes

N
(Mj)+NM) Zz, =-NpgZp S(0) - pgA[-kT, + S(0)] » sin (kag cos 8; — wot)

i=l

[ 16c]

The first term on the right-hand side is just the change in buoyancy which
accompanies a vertical displacement of the raft. The remaining terms
correspond to the vertical force obtained by integrating the dynamic pres-
sure due to the incident wave over the surface of the spar. This is the
Froude-Krylov hypothesis: To a first approximation, the presence of the

body does not distort the incident wave or the pressure associated with it.

The a-equation is

N N
oo 2 . 2 va BNC 2
a {onal y sin OL aiat > il m (Z,°)(z,7) anc}
: : L
i=l i=l

N

=- pga jA[S(0)- kTo] y sin 8; sin (kag cos 0; — wt)
i=]

+

N
a“ S(0) La sin? D4 S B sin 0. cos 9. |
P8409 i i i
i=]

+

N
ve] > f m (27) 27 def + My zo
Sle

If N>2,
N N
5 Pepe 2: = S 2 £4
sin 0; = cos Q; = N
i=l test

19

N

’ sin 9; cos 0; = 0
13]
IIS, sieAe IN) S45

a {anmag + NpSz 1p + N m (27) (22)° any bs G3 {wess; [16d]
L

2 7 ce a
+ >NpgapS(0)- Mpgz)- Ng ib m (z,7) z. az; }
N
— IN OSG) = = pga jA[S(0)- kTo] > sin 0; sin (Kap cos 0; - ut)

i=1

Similarly, for N > 2, the B-equation becomes
8 41NMa*~+NpS,+J,+N salar a’ das los ein aas
2 0 a2 nO c HL US i PS

2, cA 7 Ce
+ 4NpgapS(0) - Ng { m (2,7) 2, az }
L

N
+ N pS) Xg = pgagA[S(0)- kTo] y cos 8; sin (kag cos 0; — wt)
i=]
N
+ 2pgkAT, y cos (kag cos 0, — ut) [1l6e]
i=]

Under the same assumptions, we obtain for the last equation

N

. 2 \ ete 5 ; ws
i {Ky + 2NMaj} = 2pgka,ATy sin0, cos(kap cos 0. wt) [16£]
i=l

In the case of N = 1, the above equations reduce to Newman's equa-

tion for a single spar. If N = 2, these equations do not hold. However,

20

the special conditions that follow from N=2 can easily be applied here to
obtain simple equations. The case is not considered sufficiently interest-
ing to warrant writing out the equations here.

The heaving motion can be obtained immediately, if desired, since
the z )- equation contains no coupling terms. In addition, the equation for
rotation about the z-axis is not coupled to the other equations. However,
the yg and @ motions are coupled; also the xj and 6 motions. Similarly,
the couplings are simple enough for these equations to be solved directly.

We should note that there are two resonance frequencies. In heave,

there is resonance when

2 PESO) [17a]
M+Mo/N

In either of the coupled motions there is resonance when

2 a ia 7
2Me[eS + 2 Pag S(0) - Mo Z0/N ib, m (25°) 2; az; |

| | a eam es To CO) yh
I,/N

2 0 2 Bne 2 Df
am [tas + es, + {ic in f+ [, menes dzi]- S}

Since the equations contain no damping forces, infinite response ampli-
tudes are predicted when resonance occurs. Of course, this is meaning-
less in the linearized model and so the above equations ([17a] and [17b])
can be valid only in frequency ranges, not including neighborhoods of the
two exceptional frequencies. When such neighborhoods are excluded from
consideration, the predictions should be fairly accurate if the small ampli-
tude and slenderness restrictions are observed, since damping forces are
of higher order than the forces considered. Near the resonance frequen-
cies, however, the damping forces are important, evenif small. This

problem is considered in the next section.

21

EQUATIONS OF DAMPED MOTION

In the previous section, we considered only buoyancy and accelera-
tion forces on the spars because, generally, these were the forces of
lowest order in the small parameter a,(z). At resonance, these forces
cancel each other, thus they are no longer the lowest order forces. We
must re-examine the previous analysis and include terms of a higher
order in a;(z) to obtain equations of motion which have meaning at and
near resonance.

The boundary condition, Equation [7], was valid only to first order
in a,;(z). If we now include second-order quantities (in a;(z)) in the bound-
ary condition, Equation [7], and add the necessary corresponding terms in
the potential function, Equation [8], we simply obtain more terms in the
acceleration and buoyancy forces. Since these terms are much smaller
than those already considered, they can alter the response only slightly,
principally by changing the resonance frequencies somewhat. They still
contribute no damping forces.

Nevertheless, the desired damping forces can be obtained from the
second summation in the potential function, Equation [8]. These terms
were discarded earlier because they contributed forces of higher order
than those being considered. It is easily seen, however, that these terms
do lead to damping forces, which will be the lowest order forces at reso-
nance. We also see that the terms in this second sum which involve the in-
cident wave amplitude A do not contribute damping effects. They simply
affect the driving force, again by an amount of higher order.

So now we consider the potential

Ie 20

% , , F

®” = tmwk s ie {a [29/4 aga; (a sin 8, - B cos 9;)]
iil

2 : )
+ av [x9 + BC - Yag sin Oe
i

4 an L¥g- at + Ya, cos 9; ] Zh etait syteR, 1 dt

Yan

22

(See the second sum in Equation [8].) Here a; and as are functions of ¢.
*

To calculate the associated force on the ie spar, we must evaluate
Om Ie = a;(z;).
For small values of (kR,;), the Bessel function in é* can be approx-

imated by the beginning of its Taylor series, that is,

2
TOMESR a) = Lie a (ORE)

Similarly,
ony (GRE) y= = BIR, Gos ty 4
etl es Zi eee
alt
SAE) ee OR cine
Oy: i i i
al

Keeping a one-term approximation in each case, we find
ok = kz:
® = zwk[z, + agla sin 0, — B cos 8.)][S(0)-kTp]e 1
3 kz; :
- twk a.(z;)e i {Lixp - vag sin 9.)T) + BT, ] cos r.

+ [(yg + Yag cos 8;) Tp) - aT, J] sin as}

The pressure due to this potential is, when evaluated on R; =a,

: tee : lez
= - zpwk[zp + angle sin 0, - B cos 9.)] [S(0)- kTole a

+ A pants? a, (z,)e74 {['% - Y aq sin 0,)T, + ea | cos i,
+ [(vg + ¥a_q cos 9;) Ty - aT, }sin yt

23

In the expressions for e* and Dae several terms which are of higher
order in small variables have been omitted.

The forces and moments due to this pressure distribution are calcu-
lated from Equations [12a] through [12f], with asterisks inserted where

appropriate. The results are as follows:

xe a5 book? {lig - Vag sin @),)) a8 + oan,
we = Se Sabie {6 + yap cos 0.) TS - 2g |
* i ; » AME : 2
Z, = - Epak {294 aq (4 sin 8. —- B cos 0,) }[8(0) - kT]
A nth 2
Ay = -Zpwkag sin 0; {yt agla sin Ons 8B cos 9,)} [5(0) - kT]
ea pepe ine. a @.) ore S awe
A POS Og v Veg COS EE Lg Ey = els
2

By = Zz pwkag cos 6, {ig + ag (a sin 0. - B cos 03) } [S(0) - kTo]

3 , ss i ANT Ue)
es + pwk {(sig - a9 sin 0,)T 5T, + Boe

There is no need for a damping moment De since the y-motion has no

resonance in any case.
N
The modified equations of motion are obtained by adding DX
i=1
etc., to the right-hand sides of the previous equations, [15a] through

s

H
.

[15f], or alternatively, to [16a] through [16f]. After simplification,

the equations become, for N >2,

i Bulow a 3 pein
N {[2(M + Mo/N) p + 3 pok Too] + [pS,B +4 pwk ee ay}

N [18a]

= 2pgkAT) s cos (ka, cos 0; — ot)

i=l

24

0° BD 0 = 3 :
N {[2(M + Mo/NI¥q + pak Do Yo! ales + towk T)T ay} = 0 [18b]

n {im + M,/N)Z, + 3 pwk[S(0) - reales ray p£5(0)25 }

N

= - pg A[S(0)- kT ] y sin (kay cos 0, - ut) [18c]
i=1

N fesver, + pS, + igi] mz dz] a +(3 pwokag LO) = kT]

+4 pak? Te +[pgsy +1 9gaés(0) - EMozo/N-al_ mzdz|la

# 3 :
= [pS] ¥9- [Fp TT] ¥o }
N
= - pga jA[S(0)- kT] » sin 0, sin (kag cos 0, — wt) [18d]

i=l

N fesver, + pS,+K,/N+] mz“dz]6 +(3 pwk a4 [S(0)-kT)]°
ib

+2 pok? T2)p + [peS, +4 pgass(0) - EMy%o/N-af mz dz] 6

. 8 :
+ [p S)] x + [4 pwk iy Mull |

N
= pga jA[S(0)- kTo] » cos 0; sin (kag cos 0; — ut)
i=l
N
+ 2pgkAT, S cos (Kay cos 0; — wt) [18e]
i=l

25

N
(Ko + NMas) ¥ = - pgkagATg S sin 0, cos (Ka, cos 0, ~ ut) [18¢]
i=l

Of course these equations can be solved very easily by substitution
of XQ = Cy sin(wt + 51); Yo = Co sin(wt + 55), etc. The results will not

be written because no additional perspicuity seems to follow.

VISCOUS DAMPING

All damping forces introduced so far correspond to the energy lost
through radiation of surface waves. In addition, energy will be lost through
the mechanism of viscosity. The viscous damping forces, in general, will
be of second order in the motion variables. As an example, suppose that a
right circular cylinder translates in a direction perpendicular to its axis.
The viscous drag is proportional to the velocity squared, and so is negli-
gible by the standards already assumed.

If the cylinder has an axial motion, however, the viscous force will
be linear in the velocity. For example, if a right circular cylinder of
radius a has an axial velocity Re yee} =Wo cos(wt + « ), then we can

see easily from elementary fluid mechanics that the velocity anywhere in

the fluid is

w (r,t)

"
w
@
=
0)

[ ker Je » + ikei J

Vv

ei (otte),
[ ker JZ a ate iei V2 a]

Wo Re

where Ko (x) is the modified Bessel function of argument x and order
zero, and the second expression gives Kj)(x) in the Kelvin representation

of its real and imaginary parts. The only velocity component is that which

26

is parallel to the cylinder axis, and this component depends in space only
on the distance r from the axis.

A tangential (axial) stress on the cylinder surface is given by

) fae Wore 3
pais =— pNwv cos(ur+e +p-y +37)
or c 4

Ga!

where b,c, 6, Y are real numbers defined by

on
(0)
=p
=
i

Ker) Roe + ikei) «| Die
v v
ent = ker Jos + ikei V2a

Thus there is an axial force component per unit length on the cylinder:

27paNwv Wob

= =< sont s NSE |
c 4
2mpaNwv Wob 37
= = [costut + «) cos (p-y + 7)
Cc

- sin(wt + «) sin (6 -vyt >t)

The first term in brackets in the last equation gives the drag force.
For the slender bodies considered in this report, it is consistent to
calculate the viscous force in a stripwise manner. Thatis, at each cross

section of the jth

spar, where the radius of the section is a; (2,7), we con-
sider that particular section to be part of an infinitely long right circular
cylinder translating axially, calculate the axial viscous force per unit
length, and integrate such results over the length of the spar. To first

h

order in the motion variables, the axial velocity of the it spar is

zo + ag(a sin 8, - 6 cos 6.)

27

Thus we add to Equations [14c], [14d], and [14e], respectively, the addi-

tional force and moments

. a,(z;)b,(z;)
Ze* = -27pNov ale wet. cos |B(z;) - y(z,) + =|

(Zo + ag @ sin 0; - amie cos 0;)dz

1

0 a.-(z-)b.(z-)
aS mena a : 31
+ (ie eC) sin| B(2)) - v(2) +3]

(Zo + ag @ sin 0, - ab cos 0;)a2;}

>
MT

: SKK
ag sin 0; Z;

ah ek
= ag cos 0; Z;

If these additional quantities are inserted into the equations of mo-
tion, we can still obtain solutions by the same method used previously.

Although the viscous forces thus obtained are linear in the veloci-
ties, they do not fit properly into the perturbation scheme in terms of
small radius. The modified Bessel functions encountered have singulari-

ties when the argument approaches zero. In fact,

b 1

= HB BIS Ee SO

a aloga

Thus, as the slenderness of the spars is accentuated, the viscous forces

increase. This is in contrast to the potential flow forces, which become

smaller and smaller. No general conclusion can be drawn concerning the
relative importance of the viscous and nonviscous damping forces as far

as dependence on radius is concerned. Calculations should be made in

individual cases.

28

DISCUSSION

Equations of motion have now been obtained, based on the assumption
that wave amplitude, body motions, and spar radii are small quantities. If
damping is neglected, there are two resonance frequencies (as given by
Equations [17a] and [17b]) at which infinite amplitudes of motion will oc-
cur. Of course, there does exist damping which prevents the responses
from actually being infinite. That part of the damping due to generation of
outgoing waves is small (of second order in terms of spar radii) and so
is of importance only near resonance. Generally there is also a viscous
damping which may be of importance throughout the interesting range of
frequencies, or possibly only near resonance, or perhaps not at all. This
damping is associated only with axial velocities of the spars, but its ef-
fects appear in each of the modes of motion for which there is a resonance.
The relative importance of viscous damping can be determined in individ-
ual cases only by actually carrying out solutions of the equations of motion.

If it is desired to minimize the motions of the structure over a range
of practical wave lengths, the most effective procedure is to attempt to re-
move the natural frequencies from the desired range. If this is not possi-
ble, the natural frequencies should be chosen as frequencies for which the
incident wave amplitudes are smallest. In practice, this will generally be
equivalent to reducing the natural frequencies as low as possible.

From Equation [17a] we see that the heave natural frequency is
proportional to the radius of the spars at the undisturbed waterline. (S(0)
is the cross-sectional area at the equilibrium waterline.) Assuming that
the total mass, My)+NM, is approximately fixed, we have no other param-
eter to adjust; thus we would minimize S(0) as far as possible.

In the case of the other resonance frequency, we see from Equation
[17b] that, apparently, there are several parameters available for adjust-
ment. We note that if the center of buoyancy and center of gravity coin-

cide, then

pS] - MjpZ9 - I m(z)zdz = 0
L

29

By raising the center of gravity and/or lowering the center of buoyancy,
we can make this difference negative and thereby possibly decrease this
natural frequency. However, in general, the exact amount by which such
adjustments will affect the natural frequency is not certain, since the de-
nominator in Equation [17b] will also be affected. In any case, reductions
in S(0) (for lowering the heave frequency) will also lower the pitch-surge

frequency.

REFERENCES

ia

1. Newman, J.N., ''The Motions of a Spar Buoy in Regular Waves,
David Taylor Model Basin Report 1499 (in preparation).

2. Cummins, W.E., ''The Force and Moment on a Body in a Time-
Varying Potential Flow, '' Journal of Ship Research, Vol. 1, No. 1
(Apr 1957), pp. 7-18.

30

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32

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