I Introduction

--A solution for the ship motion problem at seis requires the determination ofthe dynamic equilibrium of forces and moments. It is- generally accepted that for the fluid forces the influence of viscosity and surface tension is of 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. 1-t 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 any-case 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 three-dimen-sional 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, the former 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 cross-sections the hydrodynamic coefficients of two-dimensional 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 is-a 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 two-dimensional case are measured and compared to calculations as well. So a complete-set 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 confirmed-several characteristics- andshown-new lines for the investigations. However, it is ques-tionable whether the results are more correct and it does 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 mot-ions.

In the past 15 years most attention has been con centrated upon pitching and heaving. M-uch progress has been made by means of combining elementary two-dimensional solutions for the hydrodynamic forces in a modified strip theory. For many practical applica-tions the-matter may be regarded as solved in this way. Evidence that the hydrodynamic coefficients calculated by two-dimensional 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 at relatively 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 cross-sections; 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 three-dimensional 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

a priori that for these motions the neglect of viscous and non-linear 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 two-dimensional 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 well-rounded rectangle of BIT 2.52 roll damping would vanish. This theoret-ical result vas experimentally verified by McLeod and 1-Isiek [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 fairly well 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 two-dimensional 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 and are possibly influenced by wave reflection.

Further it is interesting to note that Dimpker found a departure from pure two-dimensional 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 section shape, 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 Lewis-forms, 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 experi-ment. Naturally this was first tried for the most simple case of heaving. The experimental difficulties are very

great, however, and it is not surprising that only a few experiments of the actual two-dimensional 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 Lewis-fòrms 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 extended to 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 measure-ment 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 r-axis is in the water surface, directed to the starboard side of the section. The z-axis 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 the cylinder 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 nota-tion 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)

a51 = hydrodynamic mass or mass moment of inertia in the. i-mode of motion,

a,j = mass coupling coefficient in the i-equation by motion in thej-mode,

b5 = damping coefficient against motion in the i-mode,

b = damping coupling coefficient in the i-equation by motion in the j-mode,

Cil hydrostatic restoring coefficient against a dis-placement in the i-direction,

Cli = hydrostatic coupling coefficient in the i-equation

by a displacement in the j-direction,

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, soe» = C:i = C#,= O.

The vertical motion is symmetric with respect to the z-axis 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. sway-roll 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 y-components 'in the z-equation may be expected to be extremely small with respect to the z-components. The experimental results will also show this clearly. In a potential flow the sway and roll problem is asym-metric with respect to the z-axis.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,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 set-up for 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 respect to the Ovz-axes, 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 Oyz-system; so K is 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.

### (,,,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 wave train 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 the cylinder' is equal to the same component of the cylinder velocity, both taken in the nican position of the cylinder.

The underlying assumptions are

the fluid is inviscid, incompressible and irrota tional;

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)'nethod [9]. The generalization of this method to the waying and rolling of arbitrarily shaped cylinders has been given by De Jong [18] in Delit. It was program-med in Algol-60 by the Computer Department of the Technological University at Deift, while the calculá-tions vere carried out on their TR-4 computer.

5 The experiments

5.1 The general set-up

The experiments were done in the main basin of the Delit Shipbuilding Laboratory. Its dimensions are 142 m length and 4.20 m width. As the draughts of the cylinders and the set-up for the various modes of mo-(ion were different the water depth varied between 1.80 m and 2.25 ni. About halfway its length a very stiff bridge was constructed across the tank. At this bridge the motion mechanism was built up. At both nds of the tank beaches damped out most of the

generated waves. A wave height meter of the resistance

Table 1. Principal data of cylindcrs

T -profUe BOx4Omm 0.30 re

'°! stiffening

E

o o

Fig. I. The cross-section of the

cylinders

type measured the outgoing waves at a distance of 10 m from the centre line of the model.

The cylinders occupied the whole width of the tank; the clearance between the end bulkheads and the tank walls being only some millimeters. The cylinders were constructed of wood with transverse and longitudinal stiffening. The principal data are summarized in table 1. The cross-sections are shown in figure 1. The trans-formation coefficients (rom the unit circle are given in the appendix.

Measurements were done for three amplitudes of

mo-tion over a frequency range ranging from w = 1 rad/sec to 12 rad/sec. For heave and sway the amplitudes were 0.01, 0.02 and 0.03 m; for roll 0.05, 0.10 and 0.20 radians (2.86, 5.73 and 11.46 deg.).

5.2 The heave experiments

All of the seven sections were tested in heaving. The sinusoidal motion was directly produced by a vertical oscillator.

By a method of harmonic analysis the in-phase and quadrature component of the first harmonic part of the force signal were obtained. This measuring tech-nique has been described in [2] and [17]. The outgoing waves were directly recorded on a UV-recorder and

read manually.

From equation (3.6) it is easily derived that the heave coefficients are obtained by the following relations

Z050 000S C - CZ0

a5

-w

### = +

Z0500sincrozo

5.3 The sway experiments

Except for the horizontal force now also the moment about the centre of gravity is required. Considering potential flow the problem is asymmetric with respect to the centre plane, so that no vertical forces are developed. To check this point vertical forces have been measured as well.

A harmonic analysis was applied separately to the signal from each dynamometer, just as in heaving,

nl

Cylinders

circle rectangle triangle 2 Lewis-forms

LcngthL 4.19m 4.19m 4.19m 4.19m Breadth B 0.30 m 0.40 m 0.3464 m 0.30 m Draught T 0.15 ni 0.20-0.10--0.05 m 0.30 m 0.15 m D/Tratio 2 2-4-8 1.155 2 Area coefficient 0.9992-0.9983-0.9966 0.50 :z/4 Displacement 148.1 kgf 335.2L167.6_83.8 kgf 217.7 kgf 148.1 kgf

and the ultimate results were combined to obtain thö required forces and the moment about the point of intersection of the centreline and the waterline; Just as in heaing the wave height signal was written on a U V-recorder. The two Lewis-forms were not inves-tigated in sway.

The equations (14), (3.5) and (3.6) show that the required coefficients can be derived from the measure-ments as follows

b..

### +

0SIfl C IW)'a

J

For the circle, the triangle and the rectangle at BIT = 2 is 0G = O and the coefficients are directly computed from the measured force and moment. For the rec-tangle at BIT = 4 and 8 the correction terms 0Ga,, and 0Gb,, play an important role. The two terms in each coefficient are Opposite in sign, thus making a, and b,the small diflèrence of two rather large quan-tities Therefore the experimental accuracy of the coupling coefficients in these cases will not be very great..

5.4 TIte roll experiments

It was to be expected that the measurements when rolling were. difficult. As the resulting hydrodynamic forces and momentsare small theroll amplitude should not be too small. lt was decided to üse 0.05; 0.10 and 20 radians. These angles also cover an important range in actual rolling. Since it is not unlikely that the hydrodynamicquantities will depend on roll amplitude, it is of importance that the motion isa purely sinusoidal rotation about a fixed axis to eliminate uncertainties resulting from a non-harmonic motiOn. lt was further considered that the rolling should be produced by a pure rolling moment without residuary forces, which would interfere with the measurement of the hydro-dynamic forces by rolling. The roll axis ¡s situated in the water surface, except for the rectangular sections with BIT = 4 and 8. The results can be modified,

however, to obtain the required quantities for rolling about the intersection of centreline and waterline. The equations are presented below. Just as in swaying it was not accepted a priori that vertical forces would be absent. The two Lewis-forms were not tested in rolling either.

The coefficients can be obtained with thé aid of equation (3.4), (3.5) and (16) as follows

Z0 Cose7 - C& 0G a.y W2& 5m

### 0G b:,

W4aFor the circle, the triangle and the rectangle at BIT = 2 is 0G = Oand the computations are straightforward. But when 0G O here even more than with swaying the experimental accuracy may be questionable. Many correction terms appear, which are subject to

experi-mental errors in themselves. Since the measurements of a,, and b,, are reliable, the a,, and b>. are computed by equations (5.3). For the other quantities it is pref-erable to calculate the theoretical coefficients about G instead of O and to compare those directly to the measurements. The values about G are given by (corn. pare equation (3.5))

### (a)G = (t

+### (t4 +3

(t### iö2

### a =

### = a+20Ga,+0G2.a.,

### (b)G = b +

b,» + ö?j. b,### b, =

### = b+20Gb,»+0G2b1.,

(aZ)G = (b.,j,)(;_{b:,j, + 0G «b.)}

For the measurement of the. wave heights produced by rolling, a different technique was tried. The wave height signal was now fed into the harmonic analyser just as the force signals. For the wave signal this

(5;4) (1%). -b, =

### a.

### -b, =

Yoc 3COS Cy ,fl (5.2) a, K'jq'j cos CK - (c### + 0Gng.)0

(5.3) 2### ¡

-W W'.0 'j sin C)### b=+

### + OGs,-0Ga-0G2

SiflC K0 COS C### +0G'b'0Gby#_0G2b;.y

+### K00 sinc,

= yosc acos ev### 0Ga,

(1)2 , W).0 Z00 cos e,### 0Gb,

sin s### = +

### 0G b,

2 W (i)4)method failed, howcvcr. At that moment it was not ssible to switch ovcr to a rccordcr and thereforc no ..ieasured wave heights are presented for (h is case.

5.5 The !;:easzrcInc',,i of tt'a ve forces

The wave forces and (lie wave moment were recorded on a UV-recorder. The recordings were analysed

ma nu a l'y.

The incoming waves were mçasurcd 30 ni in front of the models This distance could not be made shorter as the reflection against the model of the starting waves inflUenced the recording so soon that for the longer waves no time was left to obtain a stationary picture The wave moment lias to be corrected for the distance 0G as well. The right hand side of equation (3.5), being the wave moment about G, is measured as a whole. So that for the case of the BIT = 4 and 8 rectangle sections the measured K is to be compared to (lie theoretical valûe

(K = K%.+0GYW (5.5)

}Ç. and Z,,, are always measured directly.

lt has to be noted that (5.5) is not an algebraic but a'vector-equation. Due account has to be taken of the phase relations of the respective quantities.

56 Discussion of i/le experi!nenla! accuracy

In a complex test as this it is hardly possible to analyse the separate sources of possible errors and to estimate their magnitude. it is possible, however, to obtain an idea of the overall accuracy by the results of some special tests.

lii the first place the circular cylinder was rolled. The output of all dynamometers should be zero now, except for the inertia of the body itself. In the second

place some oscillation experiments were exactly cpeated in air. This was obtained by lowering the water level so that nothing was changed. It will be understood that in this way only sway and roll cxperi ments arc possible, for the buoyancy is indispensable to balance the weight for vertical force measurements. The only non-zero result now should be the mass or mass moment of inertia of the cylinder itself.

The rolling circle in water produced little output

indeed. The results are presented in figures 6.1 and 6.2.

The total error in dampingcoefflcient b and coupling coefficient b is very small; compare the results for the other sections in figures 6.3 through 6.10. In the low frequency range the mass coeflicients a and a, show distinctly that they are in error. A very small absolute error in the measured force or moment causes large deviations in the coefficients in this range. But

the error in a for the middle and higher frequencies is still fairly large. The supposed added moment of inertia is negative and this must probably be attributed to experimental errors or to a systematic mistake in the determination of the body inertia. The latter was

deter-mined dynamically by allowing the cylinder to roll freely in air when attached to a spring and measuring the period. Before doing so the spring-apparatus was calibrated by known weights. If this method should

turn out not to be accurate enough (in this case in-dicating too large inertia) the a for all sections should be too small. Indeed this is the general picture. It can-not be decided conclusively, however, although the rolling test in air (see below) produces another indica-tion in this respect.

The coupling coefficient a for the rolling circle deviates rather much from zero as well. But this was due to a weak point in the connection of the cylinder axis to the bridge across the tank. The construction hinged slightly, thus producing a small sway notion of the cylinder. The horizontal force measured is fully accounted for by the mass inertia force maw2, when a is a horizontal displacement of 0.5 x l0

### m for

4a = 0.05 and of 2x iO m for /'a = 0.20. As the circle was the first cylinder tested in rolling this point was detected soon and eliminated by considerable strengthening of the weak element. So for the other sections this error inay will be much smaller.

By these results it will also be clear that the coupling coefficients of the rectangle with BIT = 4 will be very inaccurate because they have the same order of mag-nitude as the measurements in the circle case.

The rolling test in air was done with the rectangle BIT = 4. Only the moment to sustain the oscillation was measured. The body inertia measured in this test was indeed some four per cent lower than that being determined With the spring apparatus. This is of the same order as a for the circle rolling in water; figure 6.1. lt supports the idea that the determination of I is not very accurate, as suggested above, and that thereby the measured a, will be too small for all sec-tions. Of course it is impracticable to determine the inertia of all sections by an oscillation test in air, so

that this discrepancy has to be accepted.

The measured b4, in air was of the same order as that of (lie circle; see figure 6.1.

In swaying the test in air was performed with the rectangle BIT = 8. The inertia force measured, com-pared to the ideal inertia force lflYaW2, was slightly smaller for low frequencies and slightly higher for high frequencies. The deviation at the low frequency side

Table il. Estimated crrors in sway coellicients.

turns out to be completely within the

### absolute;mea-suring accuracy, but at the high frequency side it is

meaningful (about + IO per cent).

### The in-phase moment about the centre of gravity

was less satisfactòry, especially for frequencies above

### co = 7 sec'. This suggests that the centre of gravity

of the ballasted model was not exactly in the required

### position and/or that the torsional

### rigidity of the

cylinder itself allowed for deformations and unpropor-tional loading of the gauges. This must make the a#,

### (at least for the rectangle with BIT = 8) rather

in-accurate.

### By the latter experiment in air an estimate could be

made of the overall error in the various sway coeffi-cients. lt is presented in table 11.

Of course, strictly speaking, these numbers onlyapply

### to the rectangle BIT = 8. But it does not seem too

### crude an assumptiàn to suppose that the order of

magnitude is also valid for the other sections. By doing

so the results of the sway experiments, presented in

### figures 4.1 'through 4.8, are very satisfactory. The.

measured ay and for the circle for instance (which

### should, be zero just, as a, and b,) are practically all

within this error range. Since in general little scatter

was experienced and' the measurements could be repro-duced nicely as well the experimental accuracy has 'no

doubt been' considerable. In the opinion of the 'author

the results approach the limits of experimental

possibi-lities for complicated tests of this type. 6 Discussion of the results

6.1 For/zeaving

The non-dimensional added mass and damping coef-ficients in heaving are presented in figures 3.1 through

3.5.

The experimental points have directly been derived

### from the measurements without smoothing a force

### curve through the measured points. There is little

scatter and the consistency of the experiments is very

satisfactory. Only in the low

### frequency 'range'

### (coj(B/2g) < 0.50) deviations appear, especially in

the added mass. This is due to' experimental

inaccura-cies. No non-linear effects with the amplitude of

oscil-lation could be detected, not even for the triangle. The agreement with the theoretical computations is also good. The lines drawn represent calculations

ac-cording tothe best section fit, the dotted lines.acac-cording to the corresponding Lewis form. In general the

### experi-mental values arc very slightly higher at the high

frequency side. With this in mind 'all coefficients for all sections are systematically more closely predicted with the aid of the actual section fit than by a Lewis

form. This can best be demonstrated with figures 3.1,

### 3.2 and 3.3. The sections form a three parameter

family; the transformation coefficients are given in the 'appendix in table A-I. Within the two-parameter Lewis form they are identical. The 'dotted lines in figures 3.2

and 3.3 do correspond with the drawn ines in fig. 3.1. The differences, shown by the computations when all the coefficients are taken into' account, are fully con-firmed by the experiments.

The differences between the two theoretical

### calcula-tions are generally not of much importance for the

### sections tested. This proves that for heaving the

breadth-draught ratio and area coefficient are the most important parameters; at least for the cross-sections

considered.

### 62 For st'aying

The added mass and damping in sway, together with the two coupling coefficients into roll are presented in figures 4.1 through 4.8. The measurement of vertical forces to determine the coupling coefficients a, and b, of sway into heave showed a large scatter and the

results all fell within the experimental accuracy.

There-fore no significance can be attached to them and a:,

### and b, 'should be taken equal to zero.

H Max. exp. error Remarks

0.140 at w = 1.75,. increasing nearly linearly from zero at w

### =

0.40### b.

### Aa-

### / - <

### IB

0.025 IB . .### at w I = 0.85, decreasing towards both sides

A \! 2g 2g

< 0.043 ' .nearly constant ' ' '

AB

The rneasurng points showcd itL1c scatter and no ystematic discrepancy between the various amplitudes of oscillation, except for the triangle (figures 4.3 and 4.4). A smooth force curve has been drawn through the measured points, and this curve was further used for the determination of the cocrncients. Therefore the measured coefficients do determine a continuous curve and only one symbol is used in the graphs. This way of working was preferable because the roll coupling -coefficients are rather sensitive and it is logical to smooth the experimental data at the origin.

In swaying experimental errors in the low-frequency range are hardly present. Only the mass coefficients show some deviations below w/(B/2g) = 0.25. The coupling coefficients into roll for the circle (figure 4.2) should be zero, while for the rectangle BIT = 4 (figure 4.7) they are very small and will be very unreliable. These facts have already been discussed in the sections

.6 and 5.3.

In general experiments and computations correspond very satisfactorily. The agreement in added mass and

mass coupling coefficients is good. The b, and b,» do show the influence of viscosity, however. In this respect the results for the three rectangles and for the triangle arc convincing. In ligure 4.5 the theoretical and experi-mental bry coincide for BIT = 8 and get more oIT for BIT = 4 and 2 at the higher frequencies. It can be understood that the flow pattern about the deeper immersed sections suffers more from separation than about the shallow one. Difference with sway amplitude cannot yet be noted. lt culminates in a still stronger eddy formation at the sharp edge of the triangle, which also increases with sway amplitude (ligure 4.3). The b,» shows the same tendencies. From the above it may tentatively be concluded that apparently the generated eddy can roughly be considered as an additional phenomenon which hardLy disturbs the pressure dis-tribution over the section given by potential theory and thus predicts the added mass and mass coupling coefficients correctly.

Just as in heaving the experiments show a slight tendency to produce somewhat higher values for added mass and damping than the theory at the high frequen-cies. Probably this is a small systematic error at the high frequencies of motion, where high demands are imposed on the structural set-up.

Also in swaying the actual section fit improves the theoretical prediction. Generally the difference between the two computations is not great, however. The actual section contour obtains somewhat more im-portance for a correct prediction of the coupling

coefficients.

The two sections forming a three parameter family with the circle were not tested in sway.

63 For the generated waves

The amplitudes of the outgoing waves in swayingand

heaving are shown in figures 5.1 through 5.7. As said in section 5.4 no measurements of the waves in rolling are prcsent

The same,points as raised in the discussion ofswaying

and heaving apply here. The wave measurements are consistent with the force measurements. The only ex-ception is figure 5.2, where the measured waves seem to be closer to the Lewis prediction while the damping coefficient obtained from force measurement is closer to the actual prediction. An explanation cannot be

given. Probably it must be attributed to an error in the calibration of the wave height meter in this test. The scatter at the high frequencies in heaving with the rectangle BIT = 8 (figure 5.7) is caused by the fact that the cylinder starts slamming at the water surface here.

It is noteworthy that the scale for tjy0 in the upper half of the figures is twice that for C/; in the lower half. Swaying generates higher waves than heaving above a certain frequency. Below that frequency hardly any waves and damping appear inhorizontal motion.

6.4 For rolling

The coefficients, derived from the rolling tests are presented in figures 6.1 through 6.10. As in swaying the vertical force measurements did not lead to a value of a1 and different from zero.

Again as in swaying a smooth force curve has been drawn through the measuring points and further data have been taken from this curve. If necessary three curves for the various amplitudes of motion wereused. The experimental results for the circle have been dis-cussed in section 5.6.

The rectangle BIT = 2 shows satisfactory

measure-ments a, and b, are in full agreement with the computations. In b the effect of viscosity can be noted, while a suffers from errors which has been discussed in section 5.6 as well. The data fit the trend, however. The triangle (figures 6.3 and 6.4) shows exactly the same picture, apart from the fact that the influence of viscosity is much more pronounced, as can be expected. Of course, the relative importance of the viscous effects in rolling will be much larger than in swaying, because the wave damping is of a smaller order. There is a large difference between the various amplitudes of motion in and The differences increase approximately linearly with 4, but the starting point is not necessarily the theoretical line. These facts suggest that the roll coefficients b and b, can be represented by

b,»,(w,40) = {b#$w)}heor + ib,»,(w) + c1(o) &

(6.1) by4,(W,lJa) = {byqs((0)}theor + E.b(w) + c2(w) .

The value of the /'s and oF c and c2 dillers from scc lion to section. For sccLons not cdgcd too sharp!y apparently only e1 difFers appreciably from zero. This is the case e.g. with the rectangle BIT = 2.

rS in Svaying the eddy formation does not seem to dtstrov ilic pressure distribution over the sectiOn, since

and arc satislictoriIy predicted by potcntitI theory.

: COI1t)IriSOfl between the actual section fit and thc Lewis form is not 'cII possible, since the experi-mental results are not accurate enotigh in this respect. The rciiii ni ng t'o sect ions : the recta nglcs with B,T = 4 and S present difficulties. Since the centre of rotation in these cases was not situated ir. the waterline the coefficients had to be corrected for the distance 0G as described in section 5.4. The results for a and are shown in figures 6.7 and 6.9 It is clear that the influence of the position of G on the coefficients is very large. In the damping coefficients a large contri-bution of viscosity is noted but they fit the trends fully. That the effect of viscosity is more pronounced than for the rectangle BIT = 2 is understandable because rolling takes place about a point well above the water-line. No explanation can be given for the fact that in figure 6.7 /a = 0.05 gives higher experimental values for b than = 0.10 or0.20, while infìgurc 6.9 the mutual

order is as might be expected. The measurements of a for both sect ions are very low and evidently greatly unreliable. A direct explanation is not available, but reasons may b found in the sections 5.4and 5.6.

The computed values of 2,j/B/Ja for thc rectanglö and thé triangle aie presented in figure 7; the circle does not generate any waves. The upper half of this figure shows a strong dependence on DIT-ratio. Ursell

[5], derived analytically for the very low frequencies that a well rounded rectangle of BIT = 2.52 would not generate waves in rolling and consequently have no

dam-ping. To check this some additional computations were made for the samesection at various draughts. They are given in figure 8. The following range of B/T-ratios was covered: 1-2-2.50-3.20-4-8 and 16, while five frequencies of motion were selected: w ..j(B/2g)

### =

0.50-0.75-1.00-1.25 and 1.50. Vanishing roll damping is found for all frequencies somewhere between BIT

_{=}

2.50 and 4.00. The actual point is dependent on
fre-quency. Below this point the roll damping of deep and
narrow sections increases sharply at low frequencies.
Above this point the shallow and broad sections offer
great advantage at the high frequencies of motion.
6.5 For rite wave forces and i/ic ii'arc Inomeilt

The theoretical computation of exciting forces on fixed bodies is generally based on the so-called Froude-Krylov hypothesis, which states that the presence of

the body in no way disturbs the incident wave system. Since the potential of this wave system is known the pressure distribution over the body surface is also known and' the forces required can be obtained by integration. lt is clear that this view can only provide a rather rough approximation of the actual situation. In fact the whole diffraction problem of the waves about the body has to be solved before a realistic pressure distribution can be obtained. This is a very difficult task and an approximate solution has been sought by applying correction terms to the forces obtained by t he FroudcK rylov hypothesis for the relative motion between the body and the water particles. lt has been shown several times that these corrections are quite large and that this procedure leads to reasonable results for the exciting force and moment in head waves.

Haskind found relations between the exciting forces and the far field velocity potential for forced oscillation in calm water. Thus an actual solution of the diffrac-tion problem can be avoided and the exciting forces are simply related to the damping coefficients. Newman [19] elaborated this further and found for

two-dimen-sional sections in beam waves the following expressions

### Ya=a[ui_b,y]}

### i

= (6.2)

=

### c[-2 b]i

Non-dimensionahized by the respective hydrostatic force and moment the results of these calculations are shown in figures 9.1 through 10.5. The computations have been performed with the damping coefficients fdr the actual section fit.

On the other hand Motora [20] used the method of applying correction terms to the FroudcKrylov force. Following his procedure the dotted lines in the graphs are obtained. Again the nmss and damping coefficients for the actual section fit have been used. No way is available for a computation of the moment according to these lines. How important the correction terms are is shown in figure 9.1 for the circle. The calculation by the FroudcK rylov hypothesis (including Smith-effect) is far off, especially in the horizontal force. When even the Smith-effect in the incoming wave is not taken into account the calculation has become purely static and a line at the point 1.00 over the whole frequency range will result. The mass correction term doubles the hori-zontal force in the very long waves for the circle and even triples it for the triangle (figure 9.5).

do fit the experimental results better. They arc much simpler and imicli moie straightforward as vclI.

The measurements of the moments arc presented in fiurcs 10.1 through 10.5, together with the Newman-curves. The computations for thc rectangle BIT = 2 agree well with the cxperimcnts, thosc for the triangle satisfactorily. That the circle does not exactlyexpericnce a zero moment must be attributed to viscous cllccts and to the same experimental errors as in rolling. For the rectangle BIT = 4 and 8 the moment has to be computed according to equation (5.5). This is a vector equation, but supposing that the phase dilicrence of K0. and Y. with the wave elevation at O is equal it reduces to a simple algebraic equation. Indications for this supposition arc found in the fact that both are proportional to the wave slope, so that in the long wave approximation the phases are indeed identical, and in the measurements of and. , shown in figures 11.2 through 11.5. In non-dimensional form the moment about G is then computed from

(K,) (K0)0 _{+0G 12A}

, (6 3)

'f1,ogB3k0

### '/12gB3k0

B B2 g,lk(0### where O= 0.10 m for B/T= 4and 0G = 0.15m

for BiT = 8 The results in figures 10.3 and 10.4 show a quite reasonable agreement with experiment.

The measured phase angles and c, are presented in figures I 1.1. through II .5. Calculations of C) and £; according to Motora'smcthodare presented as well. The measurements show much scatter and arc rather inaccurate because of the fact that the incoming waves had to be recorded at a distance of 30m in front of the model. The general trends arc in agreement, however.

lt is noteworthy that the recording of the transverse force and of the moment was not simply harmonic, contrary to that of the vertical force. A typical example is reproduced in figure 2. Apparently the higher order contributions do not significantly change themagnitude of these quantities, since the agreement with linear computation is good.

R.cngIe BIT 2 VB O

Fig. 2. Typical recording of wavecxciting forces

7 Conclusions

The added mass and damping coefficient in heaving can be computed accurately by potential theory. This statement is not only valid for the practically important middle frequency range, but also for very high frequencies of motion. At the low fre-quency side the experiments fail to check the theory. The influence of viscosity is negligible, perhaps with the exception of large bulbshaped sections where separation may occur at the

upper-side of the bulb. .

The difference in the calculations for an actual section fit and foran approximate Lewis-form is small bit it is fully confirmed by the experiments. In swaying the experimental results agree well with the computations both for added mass and damping and for the coupling terms into roll. It may be concluded that the potential theory gives a reliable prediction ofall the hydrodynamic coef-ficients involvcd For relatively sharply edged sec-tions viscous effects are clearly perceptible, but they do not invalidate the practical usefulness of the theory at all. At the edges separation occurs, which means an energy loss due to eddy forma-tion. This increasès the values for the coefficients of thoc terms which represent the dissipation of energy, namely b,j' and b5'. The pressure dis-tribution over the contour is apparently hardly affected, so that and ay are still predicted cor-rectly. When the edges arc very sharp the strength of the generated eddy may increase with the sway amplitude, thus making b, and b, in fact non-linear quantities. However, the effect of non- lineariza-tion will not be serious. The wave damping is the. largest contribution in the total coefficients;. the viscous contribution increases them with about 10 to 40 per cent. Of course the higher the frequen-cy the larger the deviation and since the actual swaying is a low frequency motion it is not believed that the influence of viscosity is practi-cally of much importance.

For rolling the same comment as for swaying holds in principle. Since the wave damping part for the sectiòns considered is an order smaller than for swaying the viscous effects obtain much more importance, however. Here as well nearly the whole influence is restricted to the energy dis-sipating ternis, that is to the coefficients b and

### b.

The theoretical computations for the actual sec-tion fit always better agree with the experiments than for the corresponding Lewis-förm. Generally the differences are not enormous, but they may

obtain importance for the coupling cthcts of sway and roll. Since it is hard io judge about tite mag-flitLide of these coupling effects beforehand it is recommeñded that a better approximation of the section contour than a Lewis-form is used fòr theoretical predictions. When a computer pro-gram is present it requires little extra effort to

do so.

5. The wave exciting forces are well predicted by both theoretical methods. The Newman-method, which includes the diffraction of the incoming waves, is preferred because it isfundamentally more straight-forward and much simpler. lt also gives a correct theoretical prediction of the wave moment. A dis-advantage is that it does not present phase angles.

6,. The calculation of the horizontal force and _{}

mo-ment according to the FroudeKrylov hypothesis and the lóñg wave approximation largely under-estimates the actual wave force and nioment. The Smith-effect causes an important decrease in the exciting forces and moment, even for the fairly

long waves.

Since viscous effects will mainly be due to separa-tion and eddy formasepara-tion and not to skin fricsepara-tion it is believed that scale effects are. of minor im-portance for oscillation tests to determine the coefficients, or for ship motion tests.. At the size of the models, used in thc.experiments, also surface

tension will only have a negligible influence.

References

GERRITSMA, J. and SMITH, W. E., Full-scale destroyer mo-tion measurements. Journal of Ship Research, vol. Il, no. I,

March 1967. p. l-8. (Also Report 142 of the Shipbuilding

Laboratory, Technological University Delft, March 1966). GERRITSMA, J. and BEUKELMAN, W., The distribution of the hydrodynamic forces on a heaving, and pitching ship model in still water. Report 61 S of'the Ncthcrlands:Ship Research Centre TNO, September 1964.

SERAT. M. E., Effect of form on roll. Transactions of the

SNAME, vol. 41, 1933, p. 160-180.

BAUMANN, H., Schlingerversuch mit einem Kreiszylinder. Schiffbau, Schiffahrt und Hafcnbau, vol. 38, 1937, p.

371-376.

URSELL, F., On the rolling motion of cylinders in the free surface of a fluid. Quarterly Journal of Mech. and Applied

Math, 2, 1949, p. 335-353.

McLEon, W. C. and I'lsmii, T., Experimental investigation

of Ursclls theory on wavemaking by a rolling cylinder.

Schiffstechnik, Bd. IO. lIeft 50, 1963, p I7-22

DtMPKER, A., Ober schwingcnde Körper an der Oberfläche des Wassers. Verft-Rccdercj-1-lafcn IS, lIeft 2, lS Januar

1934, p. 15-19.

HOLSTEIN, H., Untersuchungen an einem Tauchschwingun-gen ausführenden Quader. Wcrft-Rcedcrcj-Hafen, 17, Heft 23, I Dezember 1936, p 385-389.

The inlluencc of B/T-ritio is large. Especially roll damping nearly vanishes in the importa ni p'-tical B/7range. The added moment of inertia is rather depending on BIT as well.

IO. The iiiathematical model, presented in 'section 3, will probably be very useful for actual computa-tions ofship motions in beam waves. All ofthe coefficients can be obtained by computation ac-cording to potential theory when the section

con-tour is known, 'but four of them will have to be corrected for viscous effects:

### h, b. h

and h,,.The correction to the first one is no doubt the most important, while that to the last one will be

negli-gible in several cases; the two coupling coefficients

can be taken equal.

### Il.

The coupling effects of sway into roll and vice versa are mixed up With the influence of the verti-cal position of the. centre of gravity.. Therefore they may be very different for various cases. 8 AcknowledgementFor those who are familiar with experiments in ship hydrodynamics it will be clear that this programme could not have been carried' out without the assistance

ofmany. There is hardly any member ofthe personnel

of the Shipbuilding Laboratory who has not attributed to it in some way or another.

The author wishes to express his sincere gratitude to all of them.

URSELL, F., On the heaving motion of a circular cylinder

on the surface of a fluid. Quarterly Journal of Mech. and'

Applied Math., vol. 2, 1949, p. 218-231.

GRIM, O., Berechnung der durch Schwingungen eines

Schiffskörpers erzeugten hydrodynamischen Kräfte. Jahr buch der Schiffbautcchnischcn Gesellschaft 1953. p. 277-299.

### Il.

GRI1bI, O., Dic hydrodynamischen Kräfte beim Roilversuch. Schilfstcchnik, Bd. 3, 1955/1956, p. 147.TASAI, F., On the damping force and added mass of ships

heaving and pitching. Rep. of Research Inst. for Applied

Mech., Kuyushu University, vol. VU, no. 26, 1959. TASAr, F., Hydrodynaniic force and moment produced by swaying and rolling oscillation of cylinders on the free

sur-fase. Rep. of Research Inst. for Applied Mech., Kyushu

University, vol. IX, no. 35, 1961.

PORTaR, W. R., Pressure distribution, added mass and

damping coefficients for cylinders 'oscillating in a free

sur-face. University of California, Inst. of Eng. Research.

Berkeley, July' 1960.

TASM, F., Measurement of the wave height produced by the forced heaving of the cylinders. Rep, of Research Inst. for Appl. Mccli., Kyushii University, vol. \'lll, no. 29. 1960.

16 PAULINO, J. R. and RICHARDSON, R. K., Measurement of pressures,forcesand,radiatjng waves for cylinders oscillating

in a free surface. University of California, Inst. of Eng.

Research, ilerkeley, June 1962.

17. ZuNniawotu', l-1. J. and BIaThNHEK, M.. Oscillator-tech. niques at (lie Shipbuilding Laboratory. Report 11 1 of the

Shipbuilding Laboratory. Technological University Dclft,

November 1963.

l8. J0NG, B. DE, Bcrekcning van dc llydrodynanlischc coëflì-cienten van oscillcrcndc cylinders. Report 174 of the Ship-building Laboratory, Technological University Delft, March

1967..

The transformation coefficients

The tanslormation coefficients for all of the seven sections arc presented in the tables below.

TableA-I. Three-parameter family

Table A-Il. Other sections as Lewis forms

### APPENDIX

NIWMAN, J. N., The exciting forces on fixed bodiesin waves. Journal of Ship Research, vo!.. 6, no. 3, December 1962, p. 10-17.

MOTORA, S., Stripwisc calculation of hydrodynamic force.s dueto beam waves Journal of Ship Research, vo!. 8, no. 1, June 1964, p. 1-9.

Table A-III. Other sections as actual section fit

Circle Lewis form

(hg. 12)
Lewis form
(fig. 3.3)
BIT 2 2 2
/1 T/4 r/4 n/4
a1 0 -0.036500 -1-0.036500
a5 O -4-004028 _{+0.004028} ¡
a, O +0.036500 -0.036500
Rectangle
Triangle
B/T=2 B/T=4 B/T=8
0 +0.306944 +0.562879 -0.333428
a3 -0.160702 -0.142628 -0.107010 +0.148725
a, 0 -0.0227ll -032234 +0.009924
a7 +0.014776 +0.007352 -0.003475 +0.016740
a,

### -

### -

-l0.003070: a,,### -

### -

### -

+0.005562 a,,### -

### -

### -

+0.001575 a,,### -

### -

### -

+0.002743 a,,### -

### -

### -

+0.001163, a,,### -

### -

### -

+0.001733 Rectangle Triangle BIT = 2 ¡J/.T = 4 BIT = 8 BIT 2 4 8 1.155 fi 09992 0.9983 9966 0.50 a1 O +0.292531 +0.547939 -0.314152 a, -0.139447 -0.122406 -0.086768 +0.172370o» PA ' Y2g 150 S -wYi- -200

Fig. 3.1. Added mass and darnpingcocfficient in heaving

### 0:11

### ii

_{.0}Q o i. 0.01 m 002m o 003m

### P PII

### :.iìî.J

Q 02 050### _

075 100 ¶ 25 bfW pAY 29### i:

0.o 125 100 pAT 2g o 075 1,00 125 1.50 1.75 200

### Trw.

92g O z,.0.OIm O O.02m O 003m 050 075 100 125 1,50 175 2.00 rwFig. 3.2. Added mass-and damping,cocfficicnt in heaving

pAT 2g

### t::

t o 2 .5 t 0. 0 25 050 075 100 125 1.50 1.75 200 00Fig. 3.3. Added massand dampingcocflicicnt in heaving Fig. 3.5. Added mass and damping coefficient in heaving,

011.2 1. _{8}
p,

### 4

### 1h

O oO. o### u-\V

### D

--o"### ..

-I .0 i..O.O1 in o O02m o 003m### f

o### 1 if -II

### TE

0 25 050 075 100 125 150 175 200 0.25 050 075 1.00 1.25 150 1.75 200 0.25 OSO 0.75 100 1,25 150 t75 2.03 -wYi 025 050 075 100 125 150 __175 2MO IB wvi.Fig. 3.4. Added mass-and' damping'coefficient.in heaving

2. 2. pA

### i'::

050 0 Q" pA 125 100 ;AVZ5 075 OSO 0.2 0 125 100 b..lrr pA Y29 075 050 o opAS

-0

-0

wvi

Fig. 4.1. Added mass and damping coefficient_{in swaying}

### rr

-Fig. 4.2. Coupling coefficients of sway into roll

Su pA 300 200 pA 72g 01 1 pAD

### r:

00 o 060 500 2. 2.0### i',:

o o 1.00---0.25 050 075 1.00 I 25 1 50 1.75Fig. 4.3. Added mass and damping_{coefficient in swaying}

i lb o 1.75

wni-.50 1.75

### irr

### --wvi

Fig. 4.4. Coupling coefficients of sway into roll

### rr

200 200 200 0 .0 2 11### 151ll,l...

0### r

### 0.0o

7 s 05 2 5### .61012

S l ..### tOy..l

.2.O00n### o0

### ---00 000

0 1 05 2### 05000510

### .515

### .5'

### _'.uuu000000

5.0 s 2 02### .00510

### .515

### .52

010 o### ?tII

pAD 12g -070 000 0 O03m 00### 000000 o

00000 030_{D0oa:_ ....}

DOD
01.0 _{flc}nçn 175 200 pA

2. 050 o o 0 25 0.50 0.75 1 00 1.25 1.50 1.75 2.00 ifW

### -uv

125 150 175### rr

- wyi 2.00 p pAB -±z pAD### rr

w w### rr

- w -wYk o H h -6 o### ---.,.-

_{,}

00### -

### '

ThD### .í

-o 00 O.lOo I0o5_-__ -O### 00000°

0.25 OSO 07S 1.00 1.25 1.50 1.75 2.( aià### T

0.05 0.100 000 o 0.25 0.50 OIS 1.00 t2S ISO 1.75 aI O 00_{o}0. BIT 2 O

### .

B 0' . . o O o### -L

### 0

### Í

### L.1

### :

005 0.1OQ o O 00 0 -0.25 OSO OIS 1.00 1.25 1.50 175 2.1### L

o 0.05### _I

0.25 0.50 075 1.00 1.25 150 175 21 oo### _o

### _p-.

### 00000

### _u_

Fig. 4.5k Added mass.anddamping coefficient in swaying Fig. 4.7. Coupling coefficients of sway into roll

Fig. 4.6. Coupling coefflcients of sway;into roll Fig. 4.8. Coupling coefficientsof sway into roll.

o 025 ,050, 0.75 IDO 1.25 1.50 1.75 200 -. wv. 050 075 100 0 25 050 075 loo 175

### irr

Yi 200 125 150 1.25 1,00 bMfr 0AV29 07S 050 025 o o 001 pAB 010 -OEs - 020 0.25 020 .tL pAB t### *

### t

L

0

-wyli.

Fig. 5.1. Wave amplitude ratio in swaying and heaving

wv Fig. 5.2. Wave amplitude ratio In heaving

w

Fig. 5.3. Wave amplitude ratIo in heaving

o O y.O,Olm O O.O2m O 003m

### Dy

0 0 025 flcO 075 flfl### ,n

,, 1.00 075 O i..0.OIm O 002m D 003m _____ . .### PIO

0 025 OcA 075 inn

### ic

### in

### i,.

### :'

H 50 e 0 0,25 050 0.75 1.00 1.25' 1.50 1.75 2.0 O z..O.OIm O 002m O 003m 0 0.25 050 0.75 1.00 125 t co i 75 72. 050 o Io o o wY y. 1.25

### k

Z. 025 o 12 1.00### i::

0.25 o### irr

yi. o### Irr

yiFig. 5.5. Wavc amplitude ratio ¡n swaying and heaving _{Fig. 5.7. Wave amplitude ratio in swaying and heaving}

200 ESO I00 250 o O y..0.Olm o 002 m D 003m

### J,?

0 025 050 070 1.00 1.25 1.50 1.75 2l O y.. 001m o 002m O 003m 0 0 25 0.50 075 1.00 1.25 1.50 1.75 2.0 O Z..00lm### :

### :

O 0O D z.. 001 m 002m 003m O y.001m O ________. O O. 2.00 ISO l.00 050 O O y.001m O 002m D 003m I I b o 0.50 075 1.00 1.25 1.50 I iS 20 O z..0.OIm O 002m D 003m### ,

### ,.

### /

o o o o o### /

### /0

o O 0 D o. .50 I. 2.00Fig. 5.4. Wave amplitude ratio in swaying and heaving

1.50 1.75 2.00

### irr

wY-i. Fig. 5.6. Wave amplitude ratio in swaying and heaving

1.50 1.7$ 2.00

### tir

### -

_{yi.}1.50 1.75 2.00

### irr

Ith### rr

075 1.00 125 1. 1.00 I ;. 025 ob,.

pAß

### -wv

Fig. 6.2. Coupling coefficients of roll into sway

### i i:

005 o 75 1.011 1.25 ISO 1.75 200### __wIr

_{-}

'2g
115 1.50 1.75
### rr

- " vi 205 00 1.25 150 175 200### -C)

o * $,.0.05 o 0.10 a 0.20 o * o o o pou V### o e

o a D J### V

### 1i

.-o u -O O### 20

### 70o000000000

- - -D o .-.-o c 000° o 00 O### O ,

0,05 o 0.10 D 0.20 O### -D

### 0*0

0.050 2g 0.025 O 0.025e### 0(OflO00C00O

### oc

O o 025 050 0.75 1.00 1.25 1.50 t75 2 o### -e'-,

_o ,. 0 D O.OS 0.10 0;20 O O o o o O COO C. 0000 ODD000000IIUE V O RO o### 0O

D.### M

### .111

### Ii

005 0,10 0.15 220e o Q...______.### o0OO'0000000d000c o

c 025 0.50 075 1.00 1.25 1.50 1.75 2.01 a### 300

o O o o D_{0o.Qo0C2.&o}o o o,. o.oÇ

_{0}0.10 o o 0.20

Fig. 6.1. Added mass momcnt of inertia and damping _{Fig. 6.3. Added mass moment of inertia and damping}

*.uciiiici,, iii gull _{coefficient in roll}

urn, u Q - 1.25 1.50 1.75 200

### -w

Fig. 6.4. Coupling coefficients of roll intosway

1,50 1.75 2.00 1.2 1.50 1.75 2.00 02 02 0.1 o,. 02 o

### t:::

005 o o 005 -010 o 0.1### ut

_{pAß}

### i:

o 005 -0.10 o 0.1### ut

pAß### i:

0.00 -050 -0.60 o 0.1 pAß ij### i::

OhO0125 0,100 0025 o -02 0--0 005 pAS 2g -02 025 025 050 050 075 075 1.00 1.00 Ir-w 1.25 1.50 175 2.00

Fig. 6.5: Added mass moment of inertia and damping coefflcient in roll 525 1.50 1.75 200 - WY3I 0.525 0.100 a..

### i::

0.025 0.525 0100 b...lrr 0075 0050 0025 ois 0.10 pAS 5j### t :

005 0 0 025 0.50 0.75 1.00 1.25 5.50 175 2 00 0.25 050 075 1.00### - wi

1.25 1.50 1.75### rr

### -wvi

2.Fig. 6.7. Added mass moment of inertia and damping

coefficient in roll 0.50 0.75 1.00 1.25 150 1.75 .. 200

### °i

D### ;-t_

### or

o o ° n o o o. m Io s.. 0.05 0.0; 020 0 o### :

o o s.. 0.05 o 0.10 o ,020 C### L0JjJr88888 8

### -'

### :11 i

### 'fl

### --

### ---

.-..' O o -010 D 0.20 o, . o o O û D o o### °?;

!i-o-D 00### -O,,

oo0 o o .. 0:05 0.10 ; 0.20 -### ._.it::

- __o__ . 00_00000Fig. 6.6. Coupling coefficients of roll into sway Fig. 6.8. Coupling coefficients of roll into sway

025 050 0.75 1.00 1.25 1.50 175 2.00

### _wvi

0.50 0.75 1.00 5Th ISO 1.15 200 lr-o 050 05 1,00 1.25 1.50 1.75 2.00 0125 o too### t:::

0025 o o -005 .!zi A9 -010 0.55 0.io pAS### i :

0. -0100.150 0 125

### 0i

### i::

0050 0.025 0.100 0.15 0.10 -Sit pAD 005 o 005 -0.10 OIS 010 pAD 12g 005 -005 010 o o 025 0.25 0.50 075 1.00 1.25 150 1.15 2.O0 050 0.50 0.75 .75 1.00 1.00 1.25 25 1. 0 1.75 .50 ifä -wyi. 2.00 2C, BI, 0.50 0.'0### t::

0.10 o 0.3 0.2 0.l o 0.150 0.050 0.025 o 0.25 050 0.75 loO 72 0.25 0.50 075 1.00 7.25 1.50 1.75 lr-- wviilr-- wvii-10 12 n B### -'r

200 16 12 iL B### --r

16### r-

'-H--..-### '---

-.### - -

-.1### \

o o -o_{O O O}

*
D
o o
D
D o,.o.os
0.10
,bo,tO'1
0.20
## i

F### fo

o Lo ..0.0S 0.10 0.20 t o-.---### _2

### -.

D D D### ,/_

_, O . o 0### 800000

0 O### --

0-q 88### &00

o o D o o u o O 050### .---

TI### w'Q:W

-1.00### 7

### -/AP»

I 1.50 050Fig. 6.9. Added mass moment of inertia and damping Fig. 7. Wave amplitude ratio in rolling

coefficient in roll

Fig. 6.10. Coupling coefficients of roll into sway Fig. 8. Added mass moment of inertia and damping

coefficient in roll vs. B/T-ratio 0.25 0.50 075 1.00 1.25 1.50 1.75 2.00 D. 0.'0 2C BI, 0. 0.20 0.10 o

Y. - p9Ak 2 Y. pgAk o 0 .0 025 050 0.75 tOO 125 1.50 1.75 2.0 - Newman -- Motora 2. 3. I pl .gd.d n... dn.p..g .1n w 0 025 050 075 100 125 150 175 wvnr I I

-I### ---

Newman Motora 0 025 I I 050 I 0.75 .1.00 -I I I 1.25 I 1.50 I 175 I### i

I 200 . O 0 025 0.50' 075 1.00- 1.25 1.50 I.7S 2.0 200 ° . L 150_{--9-.3---.}O 0.25 0.50 0.75 1.00 1.25 1.50 1.75 20 - --- Newman -- Motora 025 O - 1 0.25

### j

050 j_ 0.75 1.00 I. I_{j_}1.25

### -wv

100 1.75 I I I### ii

o 0.25 OSO 0.75. 1.00 I _I_### I-L

I I 1.25 1.50 1,75 I I I -I O### 0

0.2 03 0.4 0.5 06 0.7 0,9. 0.9 1.0### lt

Fig. 9.!. Horizontal and vertical wave exciting forces

050 075 1.00 1.25 1.50 1.75 2.00

-0 0.1 02 03 os 06 07 08 .1.0

Fig. 9.3. Horizontal.and vertical wave exciting forces

.1 U. U.J .4 0.5 06 0.7 0,9 09 1.0 1.1

### ff

Fig. 9.4. Horizontal and vertical wave exciting forces

.1 02 0.3' .5 0.6 1.0 tI,

### .if

Fig. 9.2. l-lorizoñtal and vcrtical waveexcitihg forces

250 200 Y. eOAk

### t.;:

050 O Y. pgAI t. o 075 o so 025 o 1.00 0.75 z.. PgBC. 050 025 .5.00 0.75 z. 0.50 0.252

Fig. 10.1. Wave exciting moment

Fig. 12. Wave excitingmoment

Fig. 9.5. Horizontal and vertical wave exciting.forces

,yi

u tj uo U.B 0.7 OB 0.9 9.0 1.1

n n. n, --

### ..

---Fig. 10.3. Wave exciting moment

K,3.

kC

.12 '

Fig. 10.4. Wave exciting moment

w

.1 OB _{0: 1.0} 1.1

Fig. 10.5. Wave exciting moment o 00 0 -A- -02S 0.50 0.75 1.00 1.25 1.50 1.75 2.0: I t J - Newni% -O meuSur.d ebmtG.

### IIu

0 0.25 0.50 .75 1.00 1.25 9.50 9.75### trr

-I I I. I I I' I I -I i Newman -- Motora 0.25 0.50 075 1.00 1.25 1.50 .75 I J I I I :I_ I- I.- I### 'w

### _-_wy.

2 1### ':

c. 0 0.25 0,50 075 1.00 9.25 ISO I 75 0### U

o 'i 00### 00

9 0' nl n 0.25 0.50 0.75 9.00 1.25 .1.50 1.75 I J I I I I I J I _wyii### V.

Newman O### :P0:0

0 025 0.50 0.75 1.00 1. 5 9.50 I7S i I I I I I 1 i i I n ni n,### n,n...___

-.1 1 I### Z\N±iN.wm00

0 025 050 0 75 1.00 9.25 1.50 I 'S a Ye pØAk I. o z. PoeC. 05 02 o 2.0 1.5### t:

o### t:

0 2.51 2 1. 1.0 05 o.7 08. 09 1.

L

e

Fig liI.4., Phase anglesof the excitingforcesandmoment

lTg-wyi.

Fig. 11.2. Phase angles of the exciting forces and moment

o Q '.. -- Hotoro O O C w o

### -- ---&

o 0 -o O 0.25 0.50 ois loo .1.25 _1 _l_ I I 1.50 wIn I -. I 1U rj1 90### Is

### J.

### 80.

### _...---e--

_{00}---4-go.

### .---

. o-___,L.o 80o0 O o 0.25 0.50 0.75 100 1.25 1.50 1.75 2.0 I I - .0.O_1 o o 8 .### --

Motora O O### .

e.### .

### _f:b_;_._O_

0 0.25 0.50 o 0.75 1.00 .1.25 I I I I 1.50 1.75 w-'w I -I I au leo. 00_{0t0}

.2 _{0.25}0.50 0.75 1.00 1.25 1.50 1.75 2.0 __S_o,-

### u:.

### -e

### - -

.--- Motora 0 t, jO o### -1__

__!oi o### Ù0

° -r 0 0.25 0.'0 I I 0.75 1.00 I I I I 125 1.50 1.75 i### .wIn

I 0. 0.8 0.7 .8 1. 1.Fig. 11.3. Phase anglesof the exciting forces and moment

a'. 0.2 0.3 01. 0.5 0.5 0.7 0.8 1.o 1.1

Fig. 11.5. Phase angles of the exciting forces andmomeu 90

t

### L.

Fig. 11.1. Phase angles of the exciting forces

180

go.

L

90

A Area of cylinder section

B Breadth oía section at the waterline.

G Cèntre of gravity of a section

### i

Polar mass moment of inertia of a cylinder per unit length about G### K

External moment about OK0 Moment about G as a result of oscillation Wave excitingmoment about O

L Length of the cylinder

### M

Transverse metacentre of a sectionO Origin of coordinate system; intersection of centre line of section with waterline

### T

Draught ofa section at the centre line Y External horizontal forceY0,. Horizontal force as a result of oscillatiOn HOrizontal wave exciting force

### Z

External vertical forceZ0 Vertical force as a result of oscillation Vertical Wave exciting force

a1, a3, ...Transfòrmation coefficients

### a,

Hydrodynamicmass or massmoment of inertia in the ¡-mode of motiona11 Mass couplingcoeflicient in the i-force (moment) equation by motion

in the j-mode

b,, Damping coefficient against motion in the i- mode

b,1 Damping coupling coefficient in the ¡-force (moment) equation by motion in the j-mode

e,, Hydrostatic restoring coefficient against a displacement in the i-direction

C1j _{Hydrostatic coupling coefficient jn the i-force (moment) equation by}

a displacement in the j-direction

g Acceleration of gravity

w2

### k = - Wave number

g

nI:QA Mass of the cylinder section

yz. Coordinatesystem fixed inspace; the origin is point O, Ozis vertically downwards, Oy to starboard

Sway amplitude z Heave amplitude

Phase angle of K0, With respect to 4, Phase angle of Y0, with respéct to y Phase angle of Z0 with respect to z

Phase angle of K with respect to wave elevation at O Phase angle of Y,. with respect to wave elevation at O Phase angle of Z,, with respect to wave elevation at O Wave amplitude

Specific mass of water

4, RoIl angle .

Roll amplitude

w Frequency of motion or wave frequency