I
Introduction
A solution for the ship motion problem at seis requiresthe determination ofthe dynamic equilibrium of forces
and moments. It is generally accepted that for the fluid
forces the influence of viscosity and surface tension isof minor importance compared to pressure and wave
effects. At the present state of knowledge this proposi
tion has not been disproved by model or full scale
experiments, at least as far as ship motions are con
cerned. Possibly the manoeuvring problem is suffering
from viscous effects. 1t has further been supposed that
the whole motion problem can be regarded as linear.
Again up to the present state of development this has
been ¿onflrmed surprisingly, in anycase for engineering
purposes and apart from very special objects.

By these circumstances the determination of the
hydrodynamic forces acting on the ship's hull forms a
linear boundary value problem in potential theory The
superposition principle holds and the actual phenom
enon can be split up into the sum olharmonic oscilla
tions of the ship in still water and waves coming in on
the restrained ship. The two fields can be investigated
entirely separately. Considering only the first field the
problem can be stated as the oscillation ola rigid body,
moving with a certain specd in the surface of a heavy,
ideal fluid. The solution supplies the six transfer func
tions of the ship, which are composed of both rigid
body characteristics und hydrodynamic quantities. Un
fortunately the solution of this general threedimensional problem, including fòrward spccd, ¡s not yet
possible.

Two theories have been developed to find an ap
proximate solution: the slender body theory and the
strip theory. Both have limits as to their validity, theformer giving better results at low frequencies of mo
)Rcport 112 S Nèthorlands Ship ilcsoarchContrc fl40.
i Shipbuilding L,boratory Deift Techni,Iogicii Univcrsity.
THE HYDRODYNAMIC COEFFICIENTS FOR SWAYING,
HEAVING AND ROLLING CYLINDERS IN A FREE SURFACE*)

by
Ir. J. H. VUGTS **)
Summary
For various crosssections the hydrodynamic coefficients of twodimensional cylinders are determined by forced oscillation tests
and by theoretical computations. The.purposeof thisstudy is to check the theoretical basis of the computations for all three possible
modes of motion and to establish the influence of section shape in this respect.
Theresults show good agreement for heaving and forswaying, while there isa fair correspondence for rolling. Apart from deviatitins
due to experimental inaccuracies appreciable differences between theory and experiment only exist for the coefficients of those terms
hich dissipate energy in sway and roll There viscous effects are distinctly present especially for sharply edged sections in roll
The wave exciting terms in the twodimensional case are measured and compared to calculations as well. So a completeset of hydro
dynamic quantities for the coupled motions of cylinders in beam waves is presented.
tian, the latter at relatively high frequencies of motion.
Accepting the basic assumptions the development of
the slender body theory is mathematically much more
rigid. It has confirmedseveral characteristics andshown
new lines for the investigations. However, it is ques
tionable whether the results are more correct and itdoes not look very promising for practical purposes
either. The strip theory has been developed by physical
and intuitive reasoning and is much simpler in use.
Moreover, it gives better results in the frequency range,
which is important for the longitudinal motions.In the past 15 years most attention has been con
centrated upon pitching and heaving. Much progress
has been made by means of combining elementary
twodimensional solutions for the hydrodynamic forces
in a modified strip theory. For many practical applica
tions thematter may be regarded as solved in this way.Evidence that the hydrodynamic coefficients calculated
by twodimensional potential theory are correct
is
scarce, however, although the ultimate resùlts for the
ship and for parts of the ship are in good agreement
with experiments [1, 2]. It is acceptable that the theo
retical solution will be confirmed in a wide range by
suitable experiments. As a matter of fact some experi
mental evidence is available. But the performance atrelatively low and high frequencies of motion and the
influence of an accurate representation of section shape
are unsettled details. As a contribution to this matter
a series of experiments was carried out in heaving with
cylinders of 7 different crosssections; see figure 1.
It has to be considered now whether a solution in
the field of the lateral motions and rolling, which cover
a frequency range from very low to rather high,
depending on the size and type of ship, can be obtained
in a way similar to that in heave and pitch. An analytic
threedimensional solution of the hydrodynamic prob
lem for an arbitrary case does not appear to be possible
in the near future. Moreover, it was not established
25J
.252
a priori that for these motions the neglect of viscous
and nonlinear effects is just as permissible as for the
symmetric heaving. Possibly eddy formation plays a
more important role. For both reasons a theoretical
and experimental investigation of the basic infinitely
long cylinder is ofgreat valuc.Cylinders with 5 different
crosssections have been oscillated in sway and roll and
the measured hydrodynamic coefficients are compared
with those, computed by potential theory. Coupling
terms of sway into roll and vice versa are included.
To complete the picture also the wave exciting forces
and moment on the restrained cylinders have been
obtained by measurement. They are compared with
theoretical results as well. So a complete set of hydro
dynamic quantities is presented, by which the two
dimensional case, that is the motion pattern of infinitely
long cylinders in beam waves, can be analysed.
2
Historical development
The subject rolling has an important place in the
literature since 1860. Attention concentrated especially
upon roll damping by determining extinction curves
for free floating models and even for actual ships. The
value of severaFof these tests may be questioned when
they were performed in small basins or at full scale in
docks or harbours. Besides the influence of the induced
swaying and of the position of the centre of gravity i
a certain condition of loading upon the results of the
tests does not seem to have been recognized fully. In
1933 Serat [3] used small cylindrical models for extinc
tion experiments to study the effect of different forms
and of the position of the centre of gravity on roll
damping more fundamentally. Although he used cylin
ders in principle he did not imitate twodimensional
conditions. But in 1937 Baumann [4] did for a large.
circular cylinder. He recognized that added mass,
moment of inertia and damping in roll were zero for
this section, a fact which allowed him to determine the
added mass and damping in sway for his freely floating
model. He only investigated one frequency of motion,
but his results are of a remarkably correct order. Ursell
(1949) [5] calculáted the outgoing waves for a forced
rolling motion at very low frequencies by potential
theory. He found that for a wellrounded rectangle of
BIT
2.52 roll damping would vanish. This theoret
ical result vas experimentally verified by McLeod and
1Isiek [6]. For the first time now the roll axis was fixed
in space and situated in the water surfice. Their experi
ments were not fully convincing, but they agreed fairlywell with the predicted results, despite the fact that the
tests were carried out at the natural frequency of the
cylinders, which was not so low that the condition
w e O of Ursell could simply be considered satisfied.
MeLeod
iitl Ilsich on the other lind found tlit the
wave damping only accounted for 20 to 50 per cent. of
the total damping.
In heaving some experiments vcre performed in the
thirties by Dirnpker [7] and Holstein [8]. They imi
tated twodimensional conditions for a circle, a wedge
a nd a recta agIe
i n a sinn Il ta n k . The fornìer author
only investigated free oscillations. He determined the
damping decrement, while the increase in
natural
period with respect to the period calculated allowed
him to give an indication of the added mass Holstein.
on the other hand, also perforiiied forced oscillations
and measured the progressing waves. His results may
still be useful, although they arc not very accurate andare possibly influenced by wave reflection.
Further it is interesting to note that Dimpker founda departure from pure twodimensional conditions in
forced heaving. For a combination of heave amplitude
and frequency, exceeding a certain limit, a standing
wave system along the length of the cylinder developed.
According to his investigations this is a pure hydro
dynamic phenomenon, not depending upon sectionshape, surface tension or accidental circumstances
during the test.In 1949 Urscll [9] had also indicated a general way
to come to a theoretical solution of (lie boundary value
problem in two dimensions. Grim [IO, Il] and Tasai[12, 13] extended this principle from the circular to
elliptic cylinders and Lewisforms, while Porter [14]
ultimately formulated the solution for heaving of an
arbitrarily shaped cylinder. Now in principle the way
was free to investigate the influence of form, of fre
quency of motion, and of the coupling effects between
sway and roll in detail. But first the validity of the
theoretical approach had to be established by experiment. Naturally this was first tried for the most simple
case of heaving. The experimental difficulties are verygreat, however, and it is not surprising that only a few
experiments of the actual twodimensional case are
known. Tasai [15] measured the wave heights produced
by forced heaving cylinders. Porter [14] measured the
total vertical force on a heaving circular cylinder and
the pressure in a number of points along the contour.
Paulling and Richardson [16] carried out the most
extensive experiments so far. For fourdifferent sections
the vertical force and the pressure in 4 to 6 locations
was recorded, both its magnitude and phase. Wave
heights were measured as well.
The results of these experiments vere such that the
theoretical prediction was contirmedsubstantially. Thus
for heaving only some details remain to be investigated,
as discussed in the introduction.
Now time has certainly come to direct the attention
again to rolling and.swaying. The situation is quite dif
fcrcnt with respect to these motions. Theoretical predic
tions for some Lewisfòrms have been presented by
Tasai [:13], but as far as the author knows not a single
experimental check in these fields is available. And it
has already been stated in the introduction that the
validity of the theory cannot automatically be extendedto these cases. lt is especially questioned whether the
flow condition when rolling can be described ade
quately by potential theory.
As experimental procedure two methods are possible.When the cylinder has aforced oscillation in one mode
of motion, either the pressure along the cylinder con
tour or the force required to sustain the motion can
be measured. Pressure measurement has the advantage
that it allows the most direct and most detailed com
parison with theory. On the other hand itrequires high.
accuracy, complicated equipment and extensive anal
ysis
The force measurement involves the prcssûre integra
tion over the body surface, so only the overall result
can be compared with theoretical predictions.
In DeIft experience has been gained with the latter
way of testing and an advanced measuring technique
has been developed [li]. Therefore the force measurement as to amplitude and phase has been accepted to
obtain the experimental results.
3
The mathematical model for motions in two
dimensions
Let Or: be a coordinate system whichisfixedinspace.
The raxis is in the water surface, directed to the
starboard side of the section. The zaxis is vertical,positive downwards The origin O is the intersection
of the centreline of the sectionand the Waterline.
Suppose now that the centre of gravity G of thecylinder is situated in O. The most general way of
describing the motion of a linear system is a set of
three coupled. equations of motion. In. a formal notation this set can be put down as
(ni + a1)i +by;.j
(J _f.
+ 'sz + Cy:Z +
+ a)
+ bq+ cçb = Y.
(ni + a.)z + b.:±
+ C:
+
Ci; +
b2q+ C:q,(l +
= z.
(i
+ a) + b4/ + c/ +
+
+ c,.,y +
+
(1.:'+ 1)4
+=
K.
Where
in = mass of the cylinder sectioti,
¡
= mass moment of inertia about G,
(3.1)
253
a51 = hydrodynamic mass or mass moment of inertia
in the. imode
of
motion,
a,j = mass coupling coefficient in the iequation by
motion in thejmode,
b5 = damping coefficient against motion in the
imode,
b = damping coupling coefficient in the iequation
by motion in the jmode,
Cil
hydrostatic restoring coefficient against a dis
placement in the idirection,
Cli
= hydrostatic coupling coefficient in the iequation
by a displacement in the jdirection,
Y = horizontal wave force (Y) when freely floating
in waves or external force (Y0,0) when forcedly
oscillated in still water,
z = ditto in the vertical direction,
K
= ditto, moment about O.
The coefficients c1, and c53 can by definition be deter
mined by pure hydrostatics.
By simple reasoning the equations (3.1) can be
simplified greatly. The horizontal displacement is not
opposed by any restoring force, so
e» = C:i =
C#,
=
O.
The vertical motion is symmetric with respect to the
zaxis and. canñot produce any lateral forces or mo
ments; therefore
= b, = c, = a,s = b#:
C
=
O.
A static heel j does not generate a horizontal force,
or c, = O. But due tO differences in the immersed and
emerged wedge when heeling about a fixed axis in
space, in general C:#
O. Then the mathematical
model is reduced to
(m + a,)j+ b»j' + ay
+b,,4'
= Y
(I ±
+ bçb + c+ aj + b,j' = K
(in + a2)ï +b2 +cz+
a24,i/.
+
+ c,çb +
= Z
Heave does not influence the coupled. swayroll motion,
as is seen by (3.2), but the reverse need not be true:
see (3.3). It will be clear, however, that the 4' and
ycomponents 'in the zequation may be expected to
be extremely small with respect to the zcomponents.
The experimental results will also show this clearly.
In a potential flow the sway and roll problem is asym
metric with respect to the zaxis.and these contributions
theoretically even vanish. Thus heaving becomes an
uncoupled, one degree of freedom motion. For the
time being the equation (3.3) is retained, however.
The cylinders are harmonically oscillated in one of
the three modes of motion
y = Y0$)t,
z = 4' = O
z =
z0sinwl;
çA = y = O
4'
=
4'05mn0)1
z = y = O,
while Y, Z and K are measu red simultaneously
i ft
amplitude and phase.In principle the cylinders were oscillated in roll about
thc point G, coinciding, with 0. But for the rectangle
it was very impractical to change the mechanical setupfor the three draughts investigated. Thereforethe centre
of gravity and the point of rotation remained at the
highest position and G was no longer situated in O for
BIT = 4 and 8.
It, is preferable to define the hydro
dynamic quantities a,
and C.j always with respectto the Ovzaxes, because they have nothing to do with
rigid body characteristics. On the other 'hand body
motions are most logically introduced as translations
of and rotations about the centre of gravity. Therefore
the equations (3.2) and (3.3) are rewritten for aGyz
system with 0G
O, but with a1»
and, Cj defined
for motion of the point 0. This is obtained by trans.
forming.
YG = Yo
ö. (J)o
G
0
ZG = Z0
KG = K0+0G
(hor. forces.).
Working this out and dropping the index G for motions.añd exciting forces results in.
(m + a»)i +.b)j' + {.a
+
j. a} +
= Yw
{I +a+0Ga,.+0G2 a,,+0Ga#}+
+.{b+ 0G b.,,+OG2
+ {c
+ 0G
g)4 + {a#+OG
a}$ +
=
.0"
+a:)±+ b::+c:zz+(az,+0Gaz,}+
+ { b +ö. b.,}. + c4,4 +
aj +
(3.6)
+b:yi = Zw
.
J
Regarding these equations the following remarks are
made:
 0G is positive when G is below 0, negative when G
is above the water surface.
 The right hand sides are expressed as wave forces,
which (being hydrodynamic quantities) arc also
determined with respect to the Oyzsystem; so Kis the wave moment about 0. When oscillating the moment to sustain the motion
is directly measured about G, so the right hand side
of (3.5) is to be replaced by K0, only.
 The restoring
coefficient in (3.5) is equal tomg . GM =
,,zg(OG  0M) = .i;ig .OG  nig OAt; the latter term,
(,,,q.OM) is nothing else than c, so this con
tribution is negative when M is below the watet
surface and positive when M is above O.
4
.A theoretical solution of the hydrodynamic problem
The hydrodynamic problem arising from the motion of
an infinitely long cylinder in the free surface of'an ideal
fluid is uniquely deteriruned and solvable. In an ideal
fluid, being initially at rest, a velocity potential must
exist satisfying Laplace's equation in two dimensions
and the respective initial and boundary, conditions.
When linearization is considered permissible the har
monic motion of the cylindér becomes of primary im
portance. In that case only the boundary conditions
for the velocity potential remain when the transient
phenomena have died out. Summarizing, the velocity
.potential must satisfy the following requirements in
two dimensions
Laplace's equation;the linearized free. surface condition;
the radiation condition, which states that a wavetrain of constant amplitude progresses from. the
cylinder to infinity;
every disturbance in the fluid must vanish at in
finite depth;
the normal component of the fluid velocity at thecylinder' is equal to the same component of the
cylinder velocity, both taken in the nican positionof the cylinder.
The underlying assumptions are
the fluid
is
inviscid, incompressible and irrotational;
the surface tension may be neglected;
the fluid domain is infinitely large;
the motion amplitude is small with respect to the.
dimensions of the section and. the generated waves
have amplitudes which are small with respect to
their length.
It is well known that these conditions are approximat
ely fulfilled in many problems associated with ship
motions, so that (lie theoretical solution is of significant
value both qualitatively and quantitatively.. Perhaps
some reserve is justified regarding the rolling motion
as far as viscous efThcts and small motion amplitudes
are concerned. The results of the experiments will have
to show to what extent this invalidates the theoretical
corn putat ions.
The solution has been obtained according to Ursell's
}
(3.4)
(3.5)