Some notes on the calculation of pitching and heaving in longitudinal waves

REPORT No. 22 S _{December 1955}

STUDIECENTRUM T.N.O. VOOR SCHEEPSBOUW EN NAVIGATIE

AFDELJNG SCHEEPSBOUW' - PROF. MEKELW'EG- DELFT

(NETHERLANDS RESEARCH CENTRE T.N.O. FOR SHIPBUILDING AND NAVIGATION) ÇSÑJPBU!LDJNG DEP1RTMENT - PROF. M EKEL WEG DELFT,I

*

SOM.E NOTES ON THE

CALCULATION OF PITCHING AND HEAVING'

IN LONGITUDINAL WAVES

by

Ir. ja GEIRRITSMA

'i

SOME NOTES ON TH:E CALCULATION

OF PITCHING AND HEAVING IN LONGITUDINAL WAVES

by Ir L GERRITSMA

Sniiimary

The exciting forces and' moments are calculated for a normal cargo-passengership wich zero velocity in

long-itudinal waves These calculations are based on the Froude Kriloff hypothesis and on the velocity potential for

waves of small amplitude according to Lamb. The influence of second ordcr quantities (such as the Smith effect

or the inclusion of relative velocity) on the exciting forces and moments is considered, as well as the influence of the coupling between heaving and pitching motions The same calculations are made for i mathematical

model, and a comparison is made with published experimental results.

i. 1,:Iroduction

Using the Froude-Kriloff hypothesis it is quite easy to calculate the so called exciting forces and moments which act on a ship in regular sinusoidal

waves.

The hypothesis states that the structure of the

wave is not influenced by the presence of the ship;

the hydrodynamic forces which arise due to the

presence of the ship are neglected.

'Based on these assumptions, Kriloff

[i, 2]

developed a method for calculating the forces

and moments created by waves.

Weinblum and St. Denis [3, 4] also begin with the Froude-Kriloff hypothesis after which an

estimation is made of the influence of damping and added mass on the exciting forces and moments. It is clear that the Froude-Kriloff hypothesis contains a rather far-reaching assumption.

For instance, the presence of the ship will

in-fluence the orbital velocities of the water particles in the wave, and this in turn will change the

pressure distribution.

'Waves are also generated as a result of the speed

of the ship and its motions in the sea, and these

are superimposed on the existing waves, resulting

in a varying amount of distortion.

It is clearly necessary to see to what extent the Froude-Kriloff' hypothesis leads to satisfactory results. 'For example, oñecan determine the heaving and pitching motions of a modél in waves and these can be compared with the theorçtical valües.

Igonet [5] carried out such an investigation; he

found fairly good agreement between theory and

experiment for 2/L> 0,7 where 2 = wavelength

and L = ship length.

This result is remarkable, considering that Igonet

did not include (among other things) the Smith

effect.

Weinbium mentioned that the Smith effect might be important [6] so that the agreement which Igonet found may not be conclusive.

In order to determine the corréctncss of methods' of calculation with' such a comparison it is

neces-sary to know the damping and added mass very accurately. Both these 'quantities influence the

magnitude of the ship motions. In general these

values are obtained from pitching and heaving tests at the natural frequency of the model. Duc to the' high value of damping the values .òbtaincd 'in this

way are quite inaccurate since the model is practically at rest after two, oscillations.,

Furthermore Haskind and Rieman [7] ' have shown that the damping and added mass are a, function of 'the frequency of the motion.

This' result was obtained by applying an exciting

force of varying frequency to a model in still

water.

The damping of the heaving 'and pitching motion in waves depends on the relative velocity of the water with respect to the 'ship and the added mass is dependant on the rclàtive acceleration, as shown by Weinblum and St. Denis [3, 4]. It now appears that the inclusion of relative motions also affects the' exciting, forces and moments.

One can determine these "corrected" forces and moments experimentally [7, 8].

For example the exciting moments can be found by the method shown in Fig. I.

'The shipmodel' is pivoted at t'he centre of gravity and is held by a 'rod at Q where the forces caused by the waves can be measured. From this the moments can be found., This is actually not a direct check on the moments calculaed" according to the Froude-Kriloff hypothesis since, as has, been

mentioned, the damping and added mass also in-fluence the exciting moments.

Korvin-Kroukovsky and Lewis [8, 9] show the 3

Fig. I. Mgusurçiiienl of exca/jo g ,no,ueisIs ii, n'aies

importance of the coupling between the heaving

and pitching motions: pure heaving causes pitching and vice-versa. This coupling is caused by the fact that the centre of gravity is not directly above the centre of flotation and also due to theasymmetrical distribution of the damping forces along the length of the ship.

The former has already been considered by Kriloff [i]. It is important to ascertain whether

the large differences between the coupled and

un-coupled motions given in [8] hold fOr a large

range of 2/L.

In this article a simple case is considered; namely the pitching and heaving of a ship with zero velo-city in longitudinal waves.

The relative magnitudes of the various second-order effects, mentioned in various publications,

are determined in order to make an appraisal of

their importance.

In particular the following effects are. treated: the Smith effect,

the iiifluence. of relative motion,

the coupling between pitching and heaving. The calculations arc made for a normal cargo-passenger ship in waves of various lengths

The exciting forces and moments are calculated alOng the lines suggested by Kriloff although the

velocity potential for sinusoidal waves of small

amplitude according to Lamb [IO] is actually used. The damping and added mass arc calculated in

a manner similar to that given in [4].

Here a two-dimensional strip theory is used The damping and added mass arc calculated for a given cross section. These values arc then integrated over the lèngth of the ship.

This method of calculation may be open to some criticism, particularly with regard to pitching [8].

However, the purpose of the calculation is to

determine, the order of magnitude of the various second-order effects. In this case the above

mentioned methods of calculation are accurate enough.

The experimental results are not given in this

article since the measuring technique was not suf-ficiently accurate.

In particular this was the case with the experi-ments to determine the damping. and added mass where the model was made to pitch and heave, in still water by means of rotating eccentric weights.

Reflections of the waves from the walls of the

tank caused by the ship motions, seemed to have a large effect.

The author knows of only one case where the

experimental values, of damping, added mass and motions for one model are published together [7]. These results are given for the heaving motion. of a mathematical shipform. The calculation of the exciting forces and of the heaving motions for this model has been carried out here in the same manner as for the cargo-passenger ship.

A comparison between experiment and theory is therefore possible in this case.

2. Calculation of heaving anti pitching

Consider a ship with zero velocity in longitudinal wavcs A system of orthogonal coordinates OXYZ is fixed in space; the origin O lies in the undisturbed

surface of the water, OZ is vertically upwards whereas OX is horizontal and perpendicular to the wave crests.

A second system of orthogonal coordinates ox'z is related to the ship; the origin o lies in the centre

of gravity G of the ship.

In still water the stern of the ship is directed in' the positive direction of OX and the two coordinate

systems coincide. From this we assume that the

centre of gravity G is situated at a distance h above the keel, where h is the draught of the vessel.

The error due to this assumption is very small as can be seen later.

The velocity potential of plane irrotational waves of small surface amplitude r and of length 2, moving in the negative direction of OX is given by Lamb [IO].

iii =-cre sin (kX + w!) _(I)

where: L 2 A

2'r

w = -

_t. t. C wave period wave velocity

acceleration due to gravity. According to Bernoulli's Law the pressure p at

any point is given by:'

/=;,;I +i

I,.

Fig. 2.

DIRECTIOP4 OF WAVES

in which V0 =voiumeof the ship below thewater-line 'in still water,

V1 = a certain part of 'the volume which is variable due to the ship motions..

The 'volüme Vi will be small compared. with Vo as we,. consider waves with small amplitude.

Furthermore a = is small (for instance'. when the waveheigh't-wave length ratio iS I : 30,

(1

Therefore 'i nglcct the integrals à /whkh arc

vI

small compared with the similar integrals/'.

ro

This gives:

¡p cos (nZ) dO. =

0,

=

!1'±

0gfdV

osaf eI2coskX+ wI)dV

As we arc considering small' wave heights, 'p and are small' and cos ip' I, thus x X

Therefore: +12 Z, _+t2 Z0 / dz = _,, _C+FX . -i +12 ' +12' +12

= - 2 C/y0 dx - .2

tpfxy0 dx + 2/Z0j'0dx

li

,Ii

-ti Also 'from (4): Z0 r cos. kx + w!) =

= 'r cos (kx) cos w! - rsin (kx)sin w! Therefore:

-

- ,pS0 ± ra0 cos w! - rb0 sin w!

I;,

in which: O area of water.piane,

S0 = statical moment of waterplanc,

a0 = -

2Jo cos (kx) dx

-f '2

b0= 2 /jsin (kx)dx

-

ti

The integration element dV over the volume V1

is chosen as:

dV=j'0dxdz

It is assumed that the breadth ordinate 'y,, of the'

ship's surface' is a function of x only, over the volume V,, in this region the ship is wall sided

The limits of integration are shown in Fig. 2

2. In' the same manner we find:

f

e cos

kX + w!) dv

vo +1, 0 2 11, tos (kx + w!) efr/ei3 dz =

-'I

= A0coswf' B0sinw!

'5

where: = density of the fluid. From (1) and (2) it follows that:

p =

ogre cos (kX + w!), - 0gZ (3) Thé wave profile is:

Z0 r cos (kX ± w!)

...

(4:)

and the isóbár at a mean distance "a" below the

undisturbed surface is given by:

Za=rè_'&3cos(kX+wi ...(5)

The pósitio oftheshipas'regards the heaing and. pitching motións is defined by the ordinate

of the centre of gravity and the pitching angle 'j'

which is positive as drawn in Fig. .2.

The following forces are acting on the ship: a.. displacement forces,

b. damping forces,

c inertia forces.

a. Displacement forces.

A force PdO, is acting on a surface elçment dO3 of' the.ship's hull, in. the direction n of the normal

on dO, which is assumed to be positive going

inwards.

In 'Z direction this force will be:

Pcos(nZ)dOe

and in, X direction:,

p cos (t;X) do,

Integration over the 'ship's hull gives the totál

force in Z directión: 'cos(nZ) dO.

=

_f!

dv

where dV is a volume element of the ship, or with (3):

fp cos' nZ) dO. =

ô'

= os ¡dv -

osa

_f

e cos (kX + w) dV.

2r

with a = riz = ---

maximum wave siope..

Now we split up the integrals over the volume V into two parts

where:

+'2 0

A0 = a/y cos (kc) dx eks dz

B0 = 2 f j'.sin (kx) dx f e dz

The breadth ordinate y of the ship's hull is now

a function oLz and t as given by hc shipform

Also:

3.

jdV=V0

ro

As the weight of the ship gives a force gVo we find as the total force in the positive direction of Z: gV, ± cos (nZ) dO1 o,

,=eg(

CO0tpS0+r(a0kA)coswt-- r(b0 CO0tpS0+r(a0kA)coswt-- kB0) sin wt

...,

(6) in which:

Z., = the isobar at the mean draught a of the cross

section.

The total damping force in the positive direction

of Z will be:

+.12

The total pitching moment can be found in a

_{f N(x)}

?-,xop _{+ C} Za) dx

similar way, starting from the expression: dt

.! (X) cas (nZ) .+ Zp cos (ÑX» dO, 0e

J,;'.

iv

_f4dv

+ I

One can find:

f (Xp cos (Z) + Zp cas QiX)) dO

ès

gfxii V+ üg/ldV_ gafxeko cos.(kx+ wI)dV+

Yo Vi Vo

+ ga

(zeks sin, (kx+ wi) dV

=

= o.g CS0 ip!0 + r(â1

kA1 4 kD.1) cas wi

-. r(b1 - kB1 - kC) sin w!)

where:.

statical moment of waterplane: Io = inertia of. waterplane,

+ '2 a.= 2/j' x cos (kx) dx

I

- 12 b1 = 2 fj'° x sin (kx) dx -t-li O

2 fj xcas

(kx) dx dz

1î

h

+12 0 = 2 fj',x sin (kx.dx /ek dz -"I O

C= 2fycos(kx)dx/ze"dz

h

(7) D1 +2 <kx) dx (ze dz b. Dampiiig forces

The damping, farce acting on a prismatic floating body of length dx due to a unit heaving velocity will be called:

N(x.)dx

- The -nicai-i--relative velocity of the- water with

respect' to a. cross section at a distance x from G is equal 'to:

Now we have from (5)':

wre_ sin (kx + wi)

Thus:

N(x)

_{(x + C}

Z5) dx

+12

- wr

sin wi /N.(x) e-ka cos (kx) dx

--

'I

+12 +1i

-

/xN (x) dx fN'(x)dx

...

(8)

'i

.

,I

+12

The term nr cos wtfN (x) e' sin (/zx) dx

can be' neglected since sin (Izx) is an odd function which is small for normal shiplines As the whole first term of expression (8) can be considered as a second ordèr quantity this 'is justified.

In a. similar way we find the moment produced by the damping fôrces:

-

wr cos.wlfxN(x) e-kasin (kx) dx

-_./x2N'xdx_/xNx)dx

_.(9)

An expression for N (x') is given by St' Denis in reference (4),. namely.:

N '(x) 4 o sin2 (kj') e2k1

where: In a, similar way the moment of the inertia

forces equals:

+12

-

(K

_{+ cl!jcJo2-xdx.) ±}

+ w2r,C'2 sin w! / Gj02x e_ka sin (kx)L'

-II

- (11)

where K = inertia of the shIp.

The equations of -,izo/ion

With the- aid of the expressions (6) to (li) the

following equations of motion arc found:

a + b + c + d + eip =

= rF1, cas w! ± rF2 sin (0/ = rF cos (w! - e)

= rM1 cos w! ± pM2 sin mt = rM cos(òi1 - à) (.12) wherc: a = UV0 + /L +12 b

₌

f Nx) dx

-II

e = 5

+12

d- =/x N(x) dx

e=S0

F1 = g (a0 - kA0) - G0

F2 = g (b0 kB0)

-G0 =

wre/

Cj'2 eka cos kx) dx +'2

=

w/ Nx) e_ka cos (kx) dx

=

C Cj,2 dx A = K + -p, -I; '2

B =/x2N(x)dx

C 0510 +12,

D = /xN(x)dx

- li

E=

M1

= g

- kAi +

kV1)

_{- Ji}

M2 = - US'Q'1 - kB1 - kG1) + I-Ii

Yo = the offset of the load waterline as ,a function

ofx,

a = the mean depth of the cróss section

con-sidered

= circular frequency of the motion. c. hier/ja forces

According to Lewis the added mass of a pris--matic -floating--body-of'1engthdx due to a- unit of

heaving acceleration is equal to

where C, coefficient depeñdent on the sectional area coefficient fi (x,) and the breadth-depth ratio (see Fig 12 of reference [3]).

Since Lewis' work considers high circular

fre-quencies (vibratión problems) his expression is

clearly not applicable to ship motions, and must be consLdered as a fairly rough approximation in this

field.

-The mean relative acceleration of the water with respect to a cross section at a distance x from G: amounts to

d2

+ - Za)

According to [81 the cross coupling inertia terms can be neglected; therefore the inertia force in the positive direction of Z produced by the added mass can be written as follows:

4h

- C1

fcxYo2(

Za)dX

where C1, is a coefficient which takes into account the three dimensional character of the flow around

the ship (see [4])-.

From () we find:

d2Za

= - w reu cos (kx ± w!)

The inertia force of the ship itself in the positive:

direction öf Z will 'be: VöC and thus the total

inertia force is:

+'Z ¿2 - C1

J

Cy02( - Za)'dX

4h

= -

(v0+ c1f/ CJ2dx)

-_hiì +12

- (02f ci cos (01fCyò2 e': cos(kx) dx -Ii

-(10) A-gain the asymmetricál term is neglected.

8

ji

wfx N(x)

e sin (kx) dx

4- '2 H1 =

¶!4

c2fx

Cy02 e- kx)dLc - II. + '2 xy02.dx

Sili,tio,z of ihr iiffcreniial .cqt:alioiis

There are several methods for solving the above mentioned simultaneous differential equations.

The method of. successive approximations as used

in [1] is very useful for our purpose:

Simultaneously it gives the solution for the

un-coupled and thç un-coupled motion. A comparison

between the two solutions is then possible.

So that end we first solvethe followingequations:

a + b +c = rFcos (ut

e)

A + B + C' = rM cos (wi -

ô)

The resulting values for and p (and their derivatives) are then substituted in the coupling

terms f the simultaneous differential equations. Then, each of these equatiOns can be solved easily, after which a new iteration is possible.

Two iterations are quite sufficient as the coup-ling terms can be considered as second order effects. As a check the method given in [81 is also used. The solutions can be presented in the following form:

= 0/r cos (ut - y)

and

v'la = v'0/a cos wt - fi)

in which the heaving motion is related to the wave amplitude r, and the pitching angle is divided by

the maximum wave slope u;

fi and ;' are the phase differences with the wave

motion.

3. Numerical results

a. Exciting forces and moments

For a normal cargo-passengership the exciting forces and moments and the resulting motiòns are calculated along the above mentioned lines..

The main particulars of this ship arc given in

table 1. L,ri = 145.57 m Le,, = 144.75 m

B = 21.03 m

T =

8.50m TABLE I V0 = 16,763 m 2,318 m2

So = 4,230 m3

= 2,663,091 m4

centre of flotation 1.827 m from 1/2 L,,

centreofbuoyancy+0.081 m from V2

The wave length ratio L/A varied f rom 0.5-2.0. The exciting forces are considered first.

Formula (12) gives the maximum exciting force

r F rVF,2 +F22

where:

F, = C&ao - gkAo - G0

F2

gbo + gkBo

-To show the importance of the varioUs factors which influence the exciting forces, the following

quantities are calculated: F1' = egao = g(ao - kA0) = Qg(au - kA,1) - G1

= gb,,

= g(b,, - kB11)

= g(bu - kB11) - E,,

With r F = r i/p13 + F22; the maximùm force is known in each ofthe three cases.

The Smith effect and the influence of relative

. . . . E.! /Cl1

IZ

motion us not included in r i

= r

y j i The former is introduced by the terms kIt0 and kB11; this will be cleat when we consider formula

(3) and the derivation of formula (6).

The iñclusion of the Smith effect only, gives the following maximum heaving force:

r F'1 = r /F1t2 ±

F2!'2

The influence of relative motion is given by the terms E0 and G,, (see.fórmulae (8), (10) and (12)) and the inclusion of these terms gives:

r F" =

rv/.Fh12 + F2

In order to compare the influence of E,, with

regard' to G,, we also calculated:

r F" = r ',/F

¡2

+ F2"

(E,, is introduced 'by the damping and G,, by the added mass when relative motions arc considered.) Comparison of r F" and r F" gives an idea of the importance ofE,.

In fig. 3 the above mentioned maximum heaving forces arc given in a dimensionless form:

namely on a base of L/A (thusF isdivided by the area of the waterplane).

The same procedure is followed with the calcul-ation of 'the pitching moments.

We have:

M,' =

M," = Qgai ,g(kA1 - kD,,.) M,'11 = vga, -

g(kA, - kD,) - J,

MJ' jgb M2"

gb + g(kBi + kC,')

M.?" = gb, + e,g(AB1 ± AC1) + H1

1.0 I I .4 .5 .6 Fig. F'/00 IA In this case: r M' = r v'.M1" + M21W

as J' is introduced by the damping and H, by the added mass, when relative motions are considered. To make a dimensionless plot the moments are re-lated to the maximum wave slonc and the inertia of the waterplane Io

3. Hraiing force cuc/fieleul for saryillg L

Fig. 4. Pikbing nsorncsIl cocí! lele,,! for I'aryislg L

---.' - I I - I I I I -I r I I P.O 1.1 12 13 1.4 _!.5- 1.6 1.7 1.8 1.9 2.0

-rM

rM

MA

2r1

2yc10 A ° MA

Fig. 4 gives2 on a base of L/ì. for the various cases mentioned above.

O .1 .2 .3 .5 .6 .7 10 1.1 1.2 1.3 -1.4 15 - 16 1.7 .18 19 20

lo

.3

For L/A - O (A -- 00), the heaing force

coef-ficients and the pitching moment coefcoef-ficients ap-proach i which can be verifiéd quite easily.

Figs 3 and 4 show the importance of the Smith

f

F'

F"

AM' AM"

effect (comparc

with - and

with

00 00 27r10

2rî0

Also the influence of the relative motion is no-I

F"

F1Í1 AM"

1M"

table (compare - with - and,

with

0

27rí0 22v10

However, this is mainly duc tó the inclusion of

added mass when we consider: relative motions

I

F"

F"

AM'!'

(compare --- with -- 'and' --- with

0,, 0,, 2.2vI 2.2vi

Heaving and pitching motions

The solution of the simultaneous differential

equations ('12) given in Fig. 5 where ipo/a and are plotted on' a base of LIA..

For .a comparison thé solutión for the case

d = e = D = E = O (uncoupled motion) is also

shown.

Table2 gives the phase difference of the motion

and the wave, both for the coupled and the

un-coupled motions.

Comparison 'between calculated and experimental results

In reference [7] experimental results of a mathe-matical shipform are published. The' fóllowing dâta are given: s. -L/2 1.94, 1.46 1.16 0.97

.73

0.58 0.49' 0.00

Fig. S. Pikbh,g anSI bcavi,,g n,oljo,s for varying L

UNCOUPLED MOTION _____ COUPLED MOTION 42 103 67 2

1

i

3

o fi

TABLE 2. Angles given in degrees

coupléd motion i 58 68 68 76 80 83, 85 90 Wave Z0 = .r cos(wt ± kx)

Heaving' motion Pitching motion'

,,/r cos'(wI - r)' V'la = w,,/ucos (o1 fi)

uncoupled motiOn 60 1110 88

r

fi

5

₉

8

₆

o

4

64 68 78 84 88 90 90

I. damping and added 'mass of 'the heaving motion

in 'still water,;

2. exciting forces and heavingamplitudes in waves; .3. phase differences of the exciting forces' and' the

'heaving motions' with respect to the wave

motiOn.

The equation of the model surface is:

i 13 B j

i2x

i I Z 'i

= ±

_{t1 - (yJ J il}

+

_{b - ¿ibf} for - h ( Z ( ¿J h

.for-- Ih(Z(o

.8 1.1' ' 12 I3 14 15 1.6 1.7 1:8 19 2:0

CALCULATED EX P ER IME NT (HASKIND R'IEMAN)

,r.

DAMPING COEFF. b i 1 1 i i- i i I I i I 0 5 I0 IS w SCC

Fig. 6. Added tnass and damping coc/fcienl for varying circular frequency

where:

L200cm

h=l3Jcm

B

2cm

¿Jb=

1 cm

The exciting forces,, the damping and the added mass are calculated for this model' with the same method as used for the cargo-passenger ships.

In this case the différential equation will be:

a ± bC + c = rFcos

(w! - a)

and the solution

C/r = 0/r cos w! - y)

The coefficients and b are known from the experiments. The value of F is determined as

fol-lows: As we have seen above, the damping and

added mass for each cross section must be known to calculate the exciting force,

The experimental results give these values for,the whole model only.

0,---.--.

_Io -f.. .3O -.40 0 CALCULAI 2 .3 LA... .5 .6 .7 .8 .9 1.0

Fig. 7. Phase iliffcre,iee bei,&'eg,, bcai'h,g force mid wate

Therefore it is assumed that the distribution of

these quantities over the length of the modèl is

'equal to that ofthc calculated values.

This applies to. the correction for the relative

motion which ¡s a second-order effect: an error in

the assumptions made, vilI not affect the result

greatly.

Figs 6-10 give the results of the calculations; the experimental results are taken from [7].

Fig. 6 shows that the calculated added mass ap-proaches the experimental values at high circular

frequencies only; this can be cxpected as stated

aboye.

The damping coefficients dònot agree accurately; but there may be errors in the experimental values. too: it is extremely difficult to determine damping coefficients experimentally.

Also the experimental and the calculated phase-differences show a certain disagreement; however, the absolute errors are about 10 dègrces, which is not very much (see Figs 7 and '8).

Fig. 9 gives the exciting force coefficients. It ¡s shown here that the inclusion of the Smith effect

and the effect of relative motion is necessary to get

agreement between experiment and calculation.

Still there is some discrepancy between the two

curves, which is also the case in Fig. lO where' the

heaving motion amplitudes are plotted on a base

of L/Â. e 40-' 30-t- -Io tXPERIMENT

-

CALCULA1ED

--4--

_.

r4cm

e r=3cm 0 .1 .2 ' .3 L/A.-.S .6 .7 8 .9 1.0 Fig. 8. Phase di/fm-reisce be/wee,, beating auj1 watet

11

Ï2

0

O"

4

O

Fig. 9. flea ri;sg force' coc/fick,,! (malbemalicei issosici)

-for varylsig L 4. Conclusions

The calculations show that the second order

effects, such as the Smith effect and the inclusion of the relative motion when determining the inertia forces, arc fairly important.

However, the influence of the damping on the exciting forces and: moments 'is- small and can be neglected. Therefore it is perhaps possible to use experimental values, of damping coefficiénts for the. whole model in still water (coefficients. b and B sec formula 12).

Ekccpt for the heaving motion when LIA ) . 9

the influence of the' coupling between heaving and

pitching motions is very small, but perhaps this

statement is valid only for a ship with zero velocity. Future experimental research will be necessary to verify the calculated exciting forces and moments. The determination of the added mass should be based on experiments with prismatic bodies in the low circular frequency range.

Also the influence of three-dimensional flow

(coefficients Ci and G2) must be determined. The comparison between calculation and experi-ment for the mathematical model is not quite satis-factory. For the greater part this will be due to the rough assumptions being made.

Extensive modeltests in waves are necessary to check the calculation of ship motions and experi-mental accuracy is essential for this purpose.

Ackno w! ed gement

The author would like -to thank Messrs J., J. van

den Bosch, Tii. Resink and L. van der Pias who

carried out the greater part of the calculations. References

I. Kriloff, A.: "A new theory of the pitching motion of ships on

waves and of the stresses produced by tisis motion."

INA

1896.-Krilof J, A.: "A general theory of the oscillation of -a ship on

waves."1NA I 898.

'X'chiblssm, G. and SI. Dessi,, M.: On the- motions òf ships at sea." SNAME- 1950.

SI. Dessi:; M.:"On sistaincd sea- speed." SNAME 1951.

Io o'. 0' .1 .2 EX PER IME NT

3L/5

CALCULATED o r3crn -. r,,4Cm .6 .7 .8 .9 1.0

a =

Fig. IO. Caleulalcil -asid cxperisszen!al heasing ans ¡siiludes

S. ¡gosse!, C.: "Experiences dc Tangage au point fixe." ATMA 1939.

WeIs,blsuss, G.: "Recent progress in theoretical studies on the

behaviour of ships in a seaway 7e International conference

on ship hydrodynamics 1954.

Hash iss1!, M. D -and Ricn,an. I. S.: "A method of determining

pitching and heaving characteristics ol a ship." Bulletin de

l'Académic des Sciences. de URSS, Classe des Sciences

tech-niques. 1946, no. IO (translation Rus,ian-Dütch by ir. G.

Vossers);' - -

-Kor.in-Kro,,kovsky, & V. and Lest-is, E; V.: "Ship motions in

regular and irregular seas." Experimental Towing Tankr

Technical Memorandum no. .106, 1954.

Koriiss-Krosskos.'shy and Lcwss: "Suggested Research in the sea-keeping qualities of ships." International Shipbuilding

Progress. I9-5.

10 Lamb, H.: "Hydrodynamics", 6th edition.

List of main symbols 2Trr

2 = maximum wave slope' = phase difference fi, , e,-â = pitch- angle sp = ordinate of heaving = density of fluid Q wave period T = circular frequency wave length = velocity potential = wave velocity = pressure = draught of ship = wave amplitude = coordinates breadth of ship = arca -of waterplane

-statical moment of waterplane inertia of waterplane

= volumetric displacement = -coordinates

= length of ship

= maximum heaving force. = maximum pitching moment

O, 4 .5 L/A .6 .7 6 .9 LO o) a C p h r xyz B

0.

so Io' Vo x-Yz L rF rM

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8ME NO

ON THE CALCVLAT ION OF PITOli ING AND

_{HIAviwe IN}

WNGi2uiIN&J,1 WAVE.

by

Ix, J. GRRITSMA

=

Rapport No

_{van hat Laboratorium voox' Scheepsbouwkunde}

te Delft

SUVAT

Th. exciting torces and moments

_{are calculated for a normal}

cargo-passengershjp with zero velocity in longitudinal

waves.

These calculations are based on the Froude-Kriloff hypothesis

and. on

the velocity potential

for waves of small amplitude

according tot Lamb.

The influence of second order

_{çiantities (such as the}

Smith effect or

the inclusion of relative velocity)

_{on the}

exciting fox,ces and moments ta considered,

_{as well as the}

influence of the coupling between heaving

_{and. pitching motions.}

The same calculations are made for a mathematical model,

and a comparison is made with

published

_{experimental results.}

I. _IlTROD1TION

tYsing the Froude-Kriloff hypothesis

_{it is quite simple to}

calculate the so called exciting torces

_{and moments which}

act on a ship in regular

_{sinusoidal waves.}

The hypothesis states that the stiicture of the wave ta not

influenced by the presence of the ship;

the hydrodynamic forces which arise due to the presence of

the ship are neglected.

Based on these assumptions,

_{Kriloff (1,2] developed a method}

to calculate the forces and.

_{momenta created by waves.}

Weinblum and St. Denia (3,

4] also begin with the

_Froude-.

Kriloff hypothesis after which an estimation is

made of the

influence of damping and added

_{masa on the exciting forces}

It is clear tbat the Froude.Kx'jloff

_{bypothesis contains}

a rather far-.reaching aesumption.

For instance, the

_{presence of the ship will influence}

the orbital velocities of the

_{water particles in the}

wave, and thia in tum will change the

_{pressure distri-.}

butio.

Waves are also generated

_{as a result of the speed of the}

Ship and ita mottons in the sea, and these are aupei'.

imposed on the excisting

_{waves, reilting in a vazying}

aaount of distortion.

It is clearly necessary to

_{see to ithat extent the}

Proude-Krileff hypothesis leads to

oôci resulta, Por

example,

on4 can determine the heaving and

_{pitching mottons of}

a model in waves and. theas can be compared

_{with the}

theeretical values. Igonet t5) carried out such an

invas-tigation; he found fairly good.

_{agreement between theory}

an&exper1ment for À/L ) 0,7

_{where A = wavelength and.}

L,

= ship length,

This result is remarkable,

_{considering that Igoet did}

not include (among other things)

_{the Smith effect.}

Weinbium ment1oned that the Smith

_{effect might be important}

so that the agreement which Igonet

_{found may not be}

con-clusive,

In order to determine the correctness of calculating

methods With such a comparison it is necessary to loiow

the damping and. added

_{masa very accuratly, Both these}

quantities influence the magnitude of

_{the ship motions,}

In general these values

_{are obtained from pitching and}

heaving teats at the natural

_{frequency of the model.}

De to the high value

_{of damping the values obtained.}

in this

_{w&y are quite lnaocurate since the model la}

practically still after two

_osculations,

Furthermore Raskid and. Rteman

_{7) have shown that the}

damping and added

_{mass are a function of the feueney}

of the motion,

This result

_{was obtained by applying an exciting force}

of varying frequetcy to

_{model in still water.}

-3-The damping of the heaving and. pitching _{motion in waves}

depends on the relative velocity of the water with respect

to the ship and the added _{mass is}

_{dependant on the}

rela-tive acceleration, as shown by Weinbium

_{and St. Denia t3,4]}

It now appears that the inclusion

of relative motions also

affects the exciting forces and

moments0

One can determina these "corrected"

_{forces and moments}

ex-perimentally

7, 8].

For example the exciting momenta can be found with

_the

method shown in fig, i

The shtpmodai is pivoted at the

centre

_{of gravity and. is}

held. by a rod at _{where the forces caused by the waves}

can be measured. Prom this

_{the momenta can be found.}

Thia is actually not

a direct check

on the momenta

calcu-lated. acoording to the Froude-Irtioff

_{hypothesis since,}

as has been ment iònsd,

the damping

and added mass also

influence the exciting momenta.

Korvin-Kroukoyalçy and Lewis £8,

₉

_{show the importance of}

the coupling between the heaving

and

pitching motions:

pure heaving causee pitching and vice-versa0

_{This coupling}

is caused by the fact that the centre of gravity io not

directly above the centre

of flotation and also due to

the asymmetrical distribution of the

damping forces

along

the

length of the ship.

The former has already been considered by Kriloff (1).

It is important to ascertain whether the large

_differences

between the coupled and. uncoupled

_{motions given in (8)}

hold for a large range of A/L

In tute

article a simple case is

_{considered; namely the}

pitching and heaving of a ship with zero velocity in longi-tudinal waves.

The relative

*gnitudee of the

various second-order effects,

mentioned in literaturé,

_{are determined tú order to make}

an appraisal, of their importance.

In particular the following effects are treated: the Smith effect,

the influence of the relative motion,

-k-The calculations are made for a normal

cargo-passenger ship

in waves of various

lengths.

The exciting forces and moments are

calculated along the

lines of Krilof

f although the velocity potential of

sinusoidal waves of

small amplitude according to Lamb

lO)

is actually used.

The damping and added mass are calculated in a

manner

similar to

that given in t4].

Here a two-dimensional strip theory is used. The damping

and added mass are calculated for a given cross section.

These values

are then integrated:

over the length of the

ship.

This method of calculation may be

opento some criticism,

particularly as regards

to pitching

t83.

However, the purpose of the

calculation is to determine

the

order of

magnitude of

the various

second-order effects.

In this case the above

mentioned

calculation

methods are

accurate enough.

The experimental

results are

not given in this article

since the

measuring techiiique was

not sufficiently accurate.

In particular this was the case with the

experiments to

determine the damping

and

added mase where

the model was

made to pitch

and

heave in still water by means

of

rotating excentric weights.

Reflections of the waves

from

the walls of the tank caused by the ship

motions, seemed to

have a large effect.

The author knows

of only one case where

the experimental.

values of damping, added mass and

motions

for one model

are published together t?].

These results are given for

the heaving motion of a niatheinatical shipform. The

cal-culation of the

exciting forces

and of the heaving motions

for this model has been

carried out

here in the same

manner as for the cargo-passenger ship.

A comparison between experiment and

theory is therefore

-5-.

II. CALCULATION OP HEAVING AND PITCHING.

Consider a ship with zero velocity in longitudinal

waves. A system of orthogona]. coordinates OXYZ is fixed

in space; the origin O lies in the undisturbed surface of the water, OZ is vertically upwards whereas OX is

horizontal and perpendicular to _{the wave cresta.}

A second system of 0rtbogonal coordinates oxyz is relateL to the ship; the origin o lies in the centre of gravity

G of the ship.

In still water the stern of the ship le directed in

positive direction of OX and the two _{coordinate systems}

coincide. From this we assume that the centre of gravity G is situated at a distance h above the keel, where h Is

the draught of the vessel.

The error due to this assumption Is very small as can be

seen later.

The velocity potential of plane irrotational waves of

small surface amplitude r and. of length A _{,moving in the}

negative direction of OX is given by Lamb [10].

cre

sI1(AX+wt)

(1)

where

k

r 1IÍ

wave period

C _{wave velocity}

According to Bernoulli's Law the pressure p at any point

is given by:

(2)

where p

density of the fluid. Prom (i) and (2) it follows that

iO7

e

cos(k X c4L,07Z

(3)

6

The wave profile

IB:

Z0

'cos (kX.wt)

and the

isobar at

_{a mean distance "a" below the undisturbed}

surface is given by:

-7

L4 = e

_{coS(kX#û t)}

The position of

the ship as regards to the heaving

_and

pitching motions is defined by the ordinate of the

centre of gravity and the pitching _{angle Ç11 which is}

positive as drawn in 1lg.2.

The following forces _are

acting

_{on the ship:}

displacement forces, damping forces,

inertia forces.

a) Displacement forces.

A force

,ôdC5

is acting on a surface

_element

_dO

of the

ship's hull, in the direction

n of

the

normal on

which is assumed to be positive _{going inwards.}

In Z direction this force

will be: _{and in}

X direction:

bCøS()c(

Integration

over the ship's hull

gives the total

_{force in}

Z direction:

/,cosZ)dq

V

yaz

where tV

is a volume element of the ship, _{or with}

(3)

¡frca5 (nZ)dOpyfd

_{V_?p(Je}

kz:cs

_(X

_9L

ô;

t,

_L'

with t*

¿'h:. =

_{a maximum wave slope.}

Now we split up the integrals

_{over the volume V Into two}

parts:

I,

in which V0

volume

of

the ship below the waterline

_In

still water,

(4)

(5)

V1 - a certain part of

the volume which is

variable due to the ship

motions.

The volume V1 will be small compared with V0

a we

consider waves with small

amplitude.

Furthermore

is small (for intance

when the

wave height

wave length

ratio is i

:

30,

).

Therefore we neglect the integrals

o.J

which are small

V,

compared with the similar integrals

f

This gives:

J,ocosnZ)dq _

ecos(kXi't4 ¿V

As we are considering small wave heights, Ç' and

are small and

cøSD/, thus

X X

Therefore: 1) z0

z0

J0V 2/,zf&c

-4

i49X

1j0cc

a5IjXyd.3*

JZ,y0/

i .4

aleo

with(1l.)

Z

,' cos ,' ccs14x

CaS wt

1 1I$IfriW

Therefore:

JdV:::

_ - 1-

,a0coswt_r4,Sìnwt

in which:

surface of waterline,

S0

statical moment of waterline1

2f

cas(kx)cA

4

=

The integration element

dV over

the volume V1 is

chosen

-8-It is assumed that the breadth ordinate ,of

the ship's

surface

is a function

of

z only, over the volume V1;

in

this region the ship

is

wall/sided.

The limits of interetion are shown in fig.2.

2) In the same manner

we find: ,14

fe

Cos

(kX,w4)dV 2ffco5(h7ci4)e

=

A

cos w*

¿B

/a

wt

where # _o 2

Jycos(kzJc*je

'd:z

-B6

=

2/y Sin(dx/ec(z

o

,

The

breadth

ordinate

y of

the

ship's hull

is

now a

function

of

z and x

as

given by

the

shipíorm.

Also:

3)

fdV

As the weight of

the ship

gives a force

_pV0

we

find

as the total force in the positive Z direction:

-(i

fpcos(#,Z»O

p

{rswL.t17I_kS0) sin wtJ...()

The

total pitching moment

can

be

found in a similar way,

starting from the expression:

,

pfxdVpjdV_

V#pp/re'sis

(kx1wÍ)dt4

(Ç

where: S0

statical moment of

waterplane,

R _{inertia of}

waterplane,

I'

J

j0GO5

.8, . =

X

s/n 'h x,J ¿t

o

A 2/x ôS&&)cIzJe1z

-o

_/

=

2J; z s' (x)chcf e

o CI

/

ycoìÁac)dXfZc

'at

-'

2/7 s/ia

(/c.ic)dx/Zecz

b) Damping forces.

The damping force acting on a prismatic floating body of

length 4edue to a unit heaving velocity

will

_{be called:}

IY(x)Q

The

mean relative velocity of the water with respect to

a cross section at a distance a from _{G is equal to:}

(Xu*.Z4)

_{(x51#_Z)}

in which

Z.

the isobar at the

mean draught a

_{of the cross}

o

-lo-.

The total damping force in the positive Z direction

will

be:

:f''

_¡

(XpZ)dz

Now we have from

(5)

dt

wr'e

sin(kxwif)

Thus;

4

..JiV(c)

#-Z)ax

4. /

-

¿r si,, w

f

tl(r) e -

c,s

(kr) dx_ J'xìy(,t)dfM(x)co

e

k¿

The

terni

weosciif

N(%)c

(kxJa(r can

be neelected

-sincé siii('kx)ia an odd function which is small for normal

ehiplines. As the whole first term of expression(8)

can

be considered as a second order quantity

this

neglection

is justified.

In

a similar

way wefind the moment produced by the

damping

forces:

eL

#4

w,. coz

w'.tìY(/

sù(h4 z

An expression

for

M(x)ia

given by St Dents in reterence(Le),

namely

EZ/

N(x)

4/4) 31M

d

where:

- the offset of at the load waterline as a

function of

x,

s the

mean depth of the croas section considered,

-.11-c) Inertia force.

According to Lewis the added mass of a prismatic floating

body of length dx due to

a

unit of heaving acceleration is

equal to C

where

Ç

s coefficient dependent on the sectional

area coefficientp(') and the

breadth-depth ratio (see £ig.12 of reference(3). Since Lewis' work considere high circular frequencies

(vibration problema) his expression is cleary not

applicable to ship motions, and must beconsidered ae

a

fairly rough approximation in this field.

The mean relative acceleration of the water with respect to a cross section at a distance * from G amounts to

(.Z)

According to 181 the cross coupling inertia terms can be neglected; therefore the inertia force in the positive Z direction produced by the added mass can be written as

follows:

-

CfÇ(Z4jcc

_e1

where C, . coefficient which takes into account the three dimensional caracter of the flow around the ship (see

c41.)

With (5) we find:

The inertia force of the ship itself in the positive

Z direction will be; _flV and thus the total inertia

force is:

_c

12

44.

_(pLC,

fç die)

_-

w't'C.

co

wtfC,

e5(h2)dx

-

_-

...(lO)

Again the asymmetrical term is neglected.

In a similar way the moment of the inertia

_{forces equals}

to:

44

_{_k}

ç

rf

_x

cx) #co

C ! si'nwi/Çze

_sIf4/d

_11 W/le-e

k

_{¡nerlicL of t/e}

The

equations of motion.

With the aid of the expressions

₍₆₎

_{to (11) the following}

equations of motion are found:

'+bccc.e

ço

cass/rcos(J_f)

Aç 8y',-C

L2'

#E

where:

'

I

=J

/Y(x)

Cp;0,

c

el

L

I

G

?1rí' -

co('hc) x

O #eI

_ka.

=

wf

_{tV(r) e}

_{cos (k}

_{x) dx}

-Za

A

1'

a

f

x2N(r)d

C

D

-4

- 13

ltl, =r_kkz).Z

ML

_,(b,...AßkC»/1

J'

H,

'L

w/XN_4GL

(ç)e

S"7(b7c)ct2c

--e'

Mi'1rf3 Ç/rç Ze

_(dr

Solution of the differential _eouations.

There are several

methods to solve the _above

simultaneous differential _equations.

The method _{of the euccesive approximations}

1:1] i very useful for our

purpose:

Simultaneously it gives the solution for the

and the coupled motion. _A

_{comparison between}

solutions is then possible.

Po that end we first solve the following _equations:

+c

_{=P'FcoS(t_t)}

A4h 8*C,

The resulting values for end _W (end

their

_direvatives)

are

_{then substituted in the coupling terms of the}

simultaneous differential _equations.

Then, each of these equations can be solved _{easely, after}

which a new iteration _{is possible.}

Two iterations are quite sufficient as the coupling

terms can be considered as Second order effects. As _a

check the method given in t8] Is also used.

mentioned

as used in

uncoupled

The solutions can be preserted in the following form:

/r COS(L.Jt

-an1

=

cas

-ja)

in which the heaving motion is related to the wave amplitude r, and the pitching angle is divided by the

maximum wave slope DC ;

and are the phase differences with the wave motion.

III .NT.ThAERICAL RESULTS.

a) Exciting forces and moments.

For a normal cargo-passengership the exciting forces and moments and the resulting motions are calculated along the above mentioned lines.

The main particulars of this ship are given in table 1.

Table 1. LWL

1's5.57 m

iqz1.75 m

B T

B.50m

V0 =

161G3n

The wave length ratio

_LIA

varied from 0,5 - 2,0.

The exciting forces are considered first. Formula (12) gives the maximum exciting force

where:

F1

-kA0

-

G0

F2

-b0+ a

- E0

To show the importance of the various factors, which influence thö exciting forces, the following qualities

are calculated:

14

-00 =

a318

S0 =I423O

P)

i

centre of riotationf.8a7m from 1/L_{'z pp}

centre of buoyancy

P11

pa0

iL

_-(3a0

-

1cA)

1li].

=j3(a0

-

-With r F =

rVpi2

+ F22, the

_{maximum force is known}

in each of this

three cases.

The Smith effect and the

influenqe1atjv5

_motion

is not included

in r F

_r\/11

₊ p2l

The former is introduced by

_the

_{terms IcA0}

_{and kB0;}

this will be clear when we consider _{formula (3) and}

the derivation of _{formula (6).}

The lnclj

_of

_{the Smith eIfect}

_{only, gives the}

following maximum _{heavi force:}

r F11

-

F22

The influence of relative motion i _{given by the terms}

and G0 (see formulae (8), (10) and (l2) and the

1nc1sj

_{of these terms pives:}

111

rF

-r

15

-F21 _-

-(3b0

F2LL

'f(b

_-

_1cß0)

111

F

--j(b0

-

_Ic80)

_{- E0}

1)_r-

lii

i

In order to compare the influence of _{E0 as regards to}

G0

we also calculated:

r

_r

+

(E0

_{is introduced by the damping and G0 by the added}

massa when relative motions are considered).

Comparison of r F11' and r pIV gives an idea about _the

importance of E0.

In fig.3

the

_{above mentioned maximum}

heaving forces

are

given in a dimensionless _form:

namely _{on a base of}

_{L/A (thus P}

is divided by

the area of

_tRe

_waterp1ane). k

The sane _{procedure is .teìlowed}

_with

the calculation _of

the pitching _moments.

o

-16-i

P1

2

M1a'fa1

-j9ç(kA1-kD1) M21

fb1

_+p(kB1+kC1)

M11ja1

_{-((kA1-kD1)....31 M21.rpb1} +rp(kB1+kc1)+H

from which r M1, _{r M11, r M111 and r M}

are known.

In this case:

r a

rVMi2

M2l2

as _{is introduced by the damping}

and R1 by the added

mass, when relative

_{motions are}

_considered.

To make a dimensionless plot the moments _{are related to the}

maximum _{wave sloped and the inertia of the}

waterplane I:

rU

_rU

Ot To

o

Fig.4 gives

_{on a base of L,,}

_{for the Various}

cases

o A

mentioned above.

For _(A...00)

the heaving force _{coefuicjents and the}

pitching moment

_{coefficients approach}

1 which can be

verified quite _easily.

Figures 3 _{nd 4 shop the importance of}

the Smith effect

P']. _M'

AU11

(compare

- with r-

_{and with}

)'

Also the influence _{of the}

_relative

motion is notable

11

A

_A111

(compare

- with

and

itI0 with _2it10

Jiowever, this is _{mainly due to th}

inc1usion1ç added mass

when we consider _{relative motions (compare}

p with

IV

A111

IV an

2iI

b. Heavixg _{nd pitchj}

_motjns.

The soltjO

_{of the}

_{simullaneous differential}

equations

(12) is given in fig.5 where _and

/,are plotted on

_a

base of _L/A

For a comparison the solution

_{for the case U}

_{e = D a}

₌

c. ConiDarjeon between _{calculated and experimental} results.

In reference

_r73

_{experimental results of a}

mathematical

shipforin are published. _{The following data}

are given:

danipin and

added mass of

_{the heaving motion}

in still water,

exciting forces and heaving _{amplitudes in waves,}

phase differences of the _{exciting forces arid the}

heaving motions with respect to the wave

_motion.

The equation of the models surface is:

y .±

fi

()'{i

+

for -h<Z<- A h

where: L 200 cm,

B 25 cm,

17

-Table 2 gives the phase difference of the motion and

the wave, both for the coupled and the uncoupled _motions.

Pable 2.

Angles given in degrees.

wave

_z0

_{r coswt +} lcz)

h

_{13,5 cm,}

¿b

1

cm.

for - A bZO

L/A coupled motion uncoupled motion

s3 b' ₁₃ 1,94 'IZ ₁₅₈

o

1,46

103 68 110 1,16 _$ 6$

0,97

a _{76 -S} 78 0,73 _i

_8o

_-9

0,58 1 83 88 0,49 ₃₅ - go

0,00

2 0 - 90

o

90

heaving motion _{pitching motion}

Y

18

-The exciting forces,

the damping and. the added mass are

calculated for this model with the same method _{as used}

for the _{cargopassenger ships.}

In this _{case the differential equation}

will be:

I,

and the solution

COB (Wt

-The coefficients _{a and b are known from the}

experiments, The value of F Ic determined as follows:

As we have seen above, _{the damping and added}

mass for

each Cross section must be known to calculate the

exciting force,

The experimental results give these values _{for the}

whole model only.

Therefore _{t Is assumed that the distribution of}

these

quantities over the length of the model is equal _{to that}

of the calculated _values.

This applies to the correction for the relative motion

which is a second-order effect: an error in the

assumptions made, will not _{affect the result}

greatly.

The fIgures 6 - 10 give the resulte of the

calculations;

the experimental results are taken from

_7j.

Figure 6 shows that only at high circular _frequency

the calculated added

_{mass approaches}

the experimental

values; this

can be expected as stated _above.

The damping ccÍicients do not agree _{accurately; but}

there may be _{errors in the experimental}

values too:

It is extremely difficult to determine damping coefficients experimentally.

Also the experimental _{and the}

_calculated

phase-differences show a certain disagreement; however, _the

absolute

_{errors are about 10 degrees,hjch Is not very}

much (see figures 7 and 8).

Figure 9 _{gives the exciting force coefficients.}

It is

shown here that the inclusion of Smith _{effect and the}

19

-between experiment and calculation. Still _{there is some}

discrepancy between the two curves, which _{is also the}

case in fig.1O where the heaving motion _{amplitudes are}

plotted on _{a base of}

L/A,

IV. CONCLUSIONS.

The calculations _{show that the second order effects,}

such as the Smith effect and the _{1fle1ujon of the}

relative

motion when determining _{the inertia fcrces,}

are fairly important.

However, the influence of the damping on the exciting

forces and moments is small and can be _neglected.

Therefore it is perhaps possible to use _{experimentei.}

values of damping coefficients of the whole _{model in}

still water (coefficients b and B see _{formula 12).}

Except for the heaving motion when

_{L/A> .9}

_{the influence}

of the coupling between heaving and _{pitching motions is}

very small, but perhaps

this statement is valid only for

a ship with

_zero

_velocity.

Future experimental research will be

_{necessary to verify}

the Calculated exciting forces and _moments.

The determination of the added mass should be based

on experiments with prismatic bodiee in the low circular

frequency range.

Also the influence of three-dimensional

_{flow(coejcjents}

C1 end C2) must be _determined.

The comparison between calculation and experiment for the mathematical model is not

quite satisfactory. For _the

greater part this will be due to the _{rough assumptions}

being made.

Extensive modelteets in waves are necessary to _{check the}

calculation of ship motions and _{experimental accuracy is}

essential for this _purpose.

The author would like to thank

_Messrs

J.J.van den Boech, Th.Resink and L.van der Pias who

11]

Kriloff, A. (2] Kriloff, A. Weinbium, G. St.Denjs, L St.Denis, M. Igonet, C. Weinbium, (8] Korvjn-Kroukovsky, Lewis, E.V.

(9)

Korvj.n-. ICroukovsky Lewis Clo)Lamb, 1:1 20

-A new theory of the pitching notion of

ships on waves and of _{the stresses}

produced

by this motion. INA

_1896.

A general theory of the oscillation _{of a}

ship on waves. INA _1898.

On the motions of _{ships at sea.}

SNAIvIE

1950.

On Sustained _{sea speed.}

SNAME 1951.

(7)

Haskind, LD. A _{method of determining pitching and}

Rieman, 1.8. _{heaving caracterjstjcs of}

a ship.

Bulletin de l'Académie des Sciences de

URSS, Classe des _{Sciences tehniques}

- 1946

no.10 (translation Russian-Dutch by

Ir G.Vossers).

B.V. _{Ship motions in regular and irregular}

seas. Experimental Towing _{Tank, Technical}

Memorandum no.106

_195k.

Suggested Research in the seakeeping

qualities of ships. _{Internatione]. Shipbuild...}

ing Progress

_1955.

HydrodynamIcs, 6th _edition.

Experiences de Tangage _{au point fixe.}

ATMA

_1939.

G. _{Recent progress in theoretical studies}

on

the 'behaviour of ships _{in a seaway.}

7e

_{International conference on ship}

-21-List of main sjmbols.

=!

_{maximum wave slope}

- phase difference pitch angle ordinate of heaving

f

density of fluid wave period W

= circular frequency

A

_{- wave length}

0

velocity potential

e

= wave velocity

p

pressure

= draught of ship r wave amplitude

xyz

_coordinates

B = breadth of ship 00 = area of waterplane

-

statical moment of

waterplane

- inertia of

waterplane

V0 _{- volumetric disp1cement}

XYZ coordinates

L

- length

of ship

rl' _{- maximum heaving force}

FIG 1.

_MEASUREMENT

_OF