Ship motions in longitudinal waves

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SHIPMOTIONS IN LONGITUDINAL WAVES

by

IR. J. GERRITSMA

Publication no.-14 of the Shipbuilding Laboratory,

Technological University, Delft.

?EB3Qr

it

I

SUMMARY

In this study on shipmotions, three methods are used to determine

the frequency response function of a shipmodel in longitudinal

waves.

In the first place, heaving and pitching motions of three ship.

models of differeniful1ness have been measured in regular waves.

The second method is' based on the assumption that heave and pitch

can be de,Scribed with sufficient accuracy by a set of two coupled

linear d,ifferential equations of the second order.

The coefficients in these equations have been determined by means

of exprirzients, and the solution of the equations of motions gave

the fequenoy response function of the shipmodel. Here again, three

mode

were used to test, the assumptiàns.

The third method is based on the assuiption that the response of a

shipmodel in irregular waves may be found by a linear superpostion.

of motions in regular waves,. Consequently, the frequency response

function may be found by analysing the result of tests in irregular

waves. This has been done for one shipmodel only. The results of

the three methods are compared, For the sake' of completeness some

of the results of earlier work are also given in this publication.

II

INTRODUCTION.

The frequency response function of a ship in regular waves is a

basis for the further

analysis

of

shipmotions

in irregular seas.

St. Denis and. Pierson (1], showed that a detailed statistical

des-cription of the shipmotions in an irregular sea is possible when.

the principle of linear superpostion is valid both for waves and'

for shipmotions. Therefore, modeltests in regular waves are a

and

this point of view seems widely accepted at present.

Inhi&fani&us paper rloff (2] attacked the problem of ship response

in its full extent: an analytical approximation was given of the

motions of a ship with oblique heading to regular waves.

However, important bydrodynamic effects were neglected by Kriloff and no experimental investigation was made to test his theory. In any case, the facilities to test a model in oblique waves were not available at that time. Igonet t3) made a comparison between calculated and measured motions of a ship model for the simple case

of longitu&inal regular waves and. zero model speed. He found a good.

agreement between Kriloff's theory and the experiment, which is

remarkable because of the fact that

in

the evaluation of the exciting forces and moments a further simplification was made by neglecting th Smith effect.

A further contribution to the development of the theory of shipmotion was given by Weinbium and St. Denis (k,

₅₎

in which the important conceptions of hydrodynarnic mass and damping received more attention than in Kriloff's paper.

A mathematical evaluation of these hydrodynamic phenomena is diff i-cult owing to the free surface boundary conditions in the low

frequency range, which is valid for shipmotions, and only rough approximations were given in these papers.

An important experimentalmethod, for the determination of hydrôdy-xinic mass and damping of an oscillating shiplike body on the water surface, was published byHaskind and Riman (6], who forced a mathe-matical shipmodel to performan oscillating

motion with

one degree

of freedom. In this manner added mass and damping could be deter-mined experimentally. However, only the case of heave at zero speed was considered. A remarkable recent development of this technique is found in the planar motion mechanism of tbie Taylor Model Basin

(73.

This apparatus will be used to determine experimentally all

the' hydrodynaniic coefficients playing a part in the motion of a

body with six degrees of freedom. There is no 'doubt that such an apparatus will be extremely valuable in the axtalysis of hipmotions.

Golovato (8] published experimental data on hydrodynamic mass4 damping

and coupling

effects of a heaving mathematical shipmodel at various speeds of advance.

In general, however, it may be said that experimental data on

hydro--=---4yhamic massan dampingare=rather -so roe

theexis ting literature.

on the subject.

Further developments of the theory of shpmotions were published by Hasknd.

(9)

and by Korvin Kroukovsky. and oobs (10).

In the latter case a comparison was made between the calculated heaving and pitching shipmotions and the results of tanxperiments in regular waves. This was done for some widely different ship forms. In many cases a reasonable agreement was found between theory and experiment, but some significant differences were also shown.

Here again, a number of rather intuttive assumptions had to be _made in order to approximate some of the coefficients (mainly those

con-cerning added mass and damping) of the çuations of mtion.

In the absence of sufficient experimental data for these _coefficients the validity of the assumptions had to be judged by means of the

final result of the computation i.e. the shipmotions. However, a

total of fourteen coefficients makes it difficult to locate errors

when necessary.

In view of the need for experimental data in this field, a systematic

series Of tests was carried out in the Doift Shipbuilding Laboratory (ii, 12). For one particular model (parent _{form Series Sixty}

°B°60

heaving and pitching motions were measured in regular head _waves. In addition, a].l the coefficients of the differential equations were

determined experimentally and the resulting motion equations.were

solved.

A good agreement has been found between the calculated and measured motions.

The investigation mentioned above was continued by testing another two models of the Series Sixty with blockcoefficients CB = 0.70. and

0.80, and the results are given in the present publication. For these two models, the relation between wave-height and motIon amplitudes, and the influence of surge on heave and pitch wqs also

studied.

Finally, the frequency response function of the CB 0.70 - model in regular waves (including the phase relations) was determIned from tests in irregular long crested head seas by using the method of cross-spectral analysis.

It will be clear that the results of the model tests mentIoned above may give an insight in the applicability of a linear theory on the problem of heave and pitch. .

frequency for three different shipforms. These data may be used to check the results of future theoretical calculations.

---The tests in irregular waves show that the frequencyesponse functions obtained by cross-spectral analysis and the responses

which are measured directly in regular waves, are in oloseagreement. Similar work by Lewis [13) showed that 1iie method is valid for the amplitudes of the motions.

The tests under consideration here show that the phase relations of the motions wbth regard to the wave can also be determined,

provided that the linear superposition principle may be asSumed to be valid.

In the following chapters the results of the tests and _{their analysis} will be given in detail.

III Experimental determination of the motions in regular head seas.

1.. Main dimensions of the shipmodels.

The main dimensions of the three shipmodels under consideration _are given in table 1; the lines of the models are taken from the Series Sixty parent forms as published in (14).

TABLE I..

Main dimensions of shipmodels

= 0,60 = 0,70

B 0.80

Length between perpendiculars Breadth

Draught

Displacement Blockcoeffjcjent

Area of waterline plane Waterline coefficient

Mass moment of inertia for pitch (in air)

Radius of inertIa

Moment of inertia of waterline plane

Depth at midship section Sheer forward 2.438 in 0.325 in

0a130.

in

61,9 kg

0.600 2 0.561 m 0.706

2.3i.

kgmsec2 0.25 L 0.17Q in4 0.203 m 0.052 in 0.042 in A 0.0940 in A 0.0366 0.020 in 2.438 m 0.348 in 0.139 in 82.9 kg 0.700 2 0.669 in 0.785 3,14. kgznsec 0.25 L 0.229 in4 0.217 in

.052 in

0.042 in A 0.0409 in F 0.0122ni1 -.0.020 in I.

r

F 1? 2.4-38 in 0.376 in 0.150 in 109.9 kg 0.800 2 0.801 in 0.871 4.16kgme 0,25 L 0.321 0.232 0.052 in

Cotro

flotation

-0.020

0.042

-0009Cm

'0:0610 in in Of from .

Centre of houyancy from J Metcentrjc height

As indicated by Table 1, the three models

_{show a systematic}

I

variation in the blockcoefficient; therefore,in

_{the next pages}

/

the shipmdels will be indicated by their

_{CB value.}

_/

A systematic investigation into the

_{influence of ship form on tè}

motions was not aimed at, and the

_{variation of breadth and draught}

as given by the Sixty Sexies parent forms

_{was accepted.}

/

The models were made of wood, as paraffin wac Showed deforrnati/ozrs

in these tests of long

_duration.

_I

2. Experimental procedure and

_{test results.}

In these tests the model

_{was free to pitch, heave and surge and}

restrained for roll,

_{sway and yawing motions8 A gravity type}

dynamometer provided a constant towing force,

_{acting through the}

centre of gravity of the model.

This dynamometer has

a I

: 5

ratio between towing force and. towing

weight and, consequently, the acceleration forces due to surge,

acting on the towing weight,

_{are reduced in the same ratio,}

The mean speed of the model is obtained

_{by measuring the speed of}

the towing carriage and by adding to or subtracting from this speed

the relative speed of the model

_{with regard to the carriage.}

This is done by means of a phototransistor which is mounted on the

surge wheel of the dynamoineter; the transistor

_{moves along the}

circumfence of a circular disk with

_{one thousand slots. The}

periphe-rical speed of the disk is equal

_{to the carriage speed.}

The number of pulses obtained in this

_{manner is counted every 10 sec}

by an electronic counter and gives the

_{mean speed of the model}

through the water.

Heave and pitch were measured by

_{means of low friction potentiometers}

and for the recording, which included

_{a time base, a five ohaniel.}

"Rapidgraph" penrecorder

_{was used. (see also (12)).}

The first series of tests

_{was made in waves having a constant height}

of 1/4.8L and wave lengths

_{respectively:}

0.75 L, 1.00 L, 1.25 L,

1.50 L,

1.75 L

For all three the models, the

_{speed range between Pr = 0 and Fr =}

0.30wa iiestigáted, It will

_{the high}

speeds in this range

_{are considerably in excess of the service}

speeds of the fuller models. The service speeds of the three

ship-forms according to (ii

_{are given in table 2.}

7

Table 2

Service speeds, of Series Sixty Parent forma

From a practical point of view the high speeds for the full models may not be very interesting, but they are considered in order to

see how far a linearized theory will hold.

. Thb. test results' are given as dmensionlegs motion parameters:

ii,

Zj

ro/,and 'r where: j'= pitchamplitude 2= heave amplitude 111P

= maximum wave slope = r = wave amplitude

In addition the phase.lag 2/ between heaving and pitching is given (heave after pitch).

The experimental data for the three models are shown in

figure 1.

In the speed range between Pr 0 and FR ' 0,12 no experimental

points are shown in this figure, because wall influence caused some

scattering of the measured valuee. To avoid confusion, only the

_mean

trend ofthese points is represented by a dotted line.

As shown theoretically by Brard

_C15),

the waves generated by a moving pulsating source will proceed in front of the source if:

CA)eV

' (1)

where:

We=

circular

frequency of the motion'

V

=speed.

In our case the frequency of the wavea is given by (A) and. the

frequency of excounter follows from:

zvEy

0B ' 0.60 0.70 0.80

and this can be written as:

z'rr (F

)Z

+

For the limiting cases, where the expression is equal to , the following combinations of speed and wave length are found (see table

₃₎

Table ₃

Critical speed as a function of wave length

For a restricted tankwidth,wall effects will present themselves at speeds higher than those given in table 3. Below a certain speed the reflection of the wave system generated by the motions of the

shipmodel will ilifluence these _{motions, See also t16.}

The magnitude of this wall effect is a function of the Froude number,

the ratio of tà.n]yMidth and inode]/length _{and the ratio of wave14ength}

and modellength.

The Froude number and the wave length ratio;determjne the value of

(A)eV/5 _{and this in turn determines the}

angle of the sector in which the waves, generated by the model, are enclosed.

For

WeV/

_{+ tiiis angle is 90 and for WeV/g),Yjt will be}

smaller than _{(The angle is measured from the ship's centerline)}

When WeV/ 'j

_{, the influence of the reflected waves on the}

model motions depends on the tank width _ratio.

In our case the tank width ratio is

1.75,

_{and irregularities in the}

trend of the motion parameters were observed at speeds below

_Fr=

0.11 to 0.14, depending on the wave length ratio.

Results of a very interesting research on wall efföcts were presented by Numata _{C17) at the Twelfth meeting of the, American}

Towing Tank Conference n

_1959.

A ₅ feet model of the Series

_{Sixty with b1ockcoeffjcjentO.6O}

_was

'to*ediña

gti].ai' head. s éa In the _{9 feet wi e tank (tank width}

ratio 1.8) and in the 75 feet square basin (tan*idth ratio for this particular test 4.25) of the Davidson _Laboratory.

0.75

1.001,25

1.50

The wave dimensions were in both oases:

wave length 1,5 L, wave height 1/48 L.

Except in the region 0±'

= +,

tue measured heave and pitch amplitudes were in close areèment.

In the region of , corresponding to a Froude number 010, the results in the narrow tank showed rather large irregularThi

For speeds higher than Fr .0.14 the wall effect appeares to be very small. C16),

The analysis. o± our results will belirnited to those _caseawere

Pr. 0.15, and it is believed that wall effects may be considered

to be neg].igible in the speed range Fr _0.15.

T14s view is also supported by the results of ear1erwork; the motion parameters of three geometrically similar models appeared

to be the same for Fr.. 0.15, wIthin the experimental accuracy of

the tests, whereas the tankwidth ratio 'varied from 1.40. to 2.28(16).

In table LI., the results concerning the motion amplitudesas presented

in figure 1, are given in dimensional form for four speeds and a

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-10.-Although this investigation was not intended to deal with the influence of shipform bn shipmotlone,. attentton may be drawn to the remarkable differences in motion amplitudes between the three shipmodels.

In most eases the

0B 0.80 model has the smallest pitch amplitudes.

Heave amplitudes are also smaller, except in the case of eoessive model speed. Especially for wave lengths equal to or less than 1.25 L the reduction in pitch and heave is quite substantial.

A further remarkable point is the small variatton of the phase lag (heave after pitch) as a function of shipform.

An analysis of those facts, however, is hardly possible because Of the simultaneous variation of blockcoffioint breadth and draught of the models.

Secondly, the influence of the wave height on the motion parameters has been investigated.

In first instance, only the C 0.70 model was tested in regular waves with three heights :- L, L and - L and. lengths 1,00 L,

1,25 L, i,50L and 1,75 L..

In a later stage it seemed desirable to judge also the linearity of the CB = 0.80 model motions.

The test results are given in figure 2 in a similar form as in figure 1.

In general, the differences in and

Zq

for the wave

heihts

L and 4. L are very small: they approach the measuring

error. The L wave causes some moderate non linearities.

Especially in the case of the CB 0.80 mOdel, excessive amounts of water were shipped at high speeds (Pr> 0.20) and ±n long waves. Even in these rather improbable conditions a linear approximation

seems reasonable for the motions of a ship in waves of hetght3 = less than 1/40 L, and. in many cases even the motions in 1/30 L waves may be treated in the same way.

As shown by fig, 2 the phase lag appears to be independnt Of the wave height, the discrepancies being well within the range of expert-mental errors.

In figures 1 and 2, arrows indicate the conditions of resonance, _'\

whereas the resonanøe factors

A

_,

determined by the natural frequency of the motion, the wave lengbh and the model speed, are

Given in table 4.

The

natural

frequencies of the motions

Will

be Given in chapter

IV.

The cases, where the model shipped water,

are

Indicated by

underlIning

the corresponding resonance factors.

As

can

be seen from table 4, this occurs only in or near to the resonance condition, as was to be expected.

A third series of tests was carried out in order to

determine

the

influence of

surge

on heave and. pitch.

These tests were carried out with the = 0.70

and

the 0B 0.80 model. For this purpose, the models were restrained for

surge

and

only pitching

and

heaving motions were possible.

Comparison with the results of the models which were free to surge

may reveal

the coupling between surge

and the other two motions, as

will

he

shown in chapter IV where these

tests

will be analysed.

IV The equations of motion for heae and pitch.

1.

Determination

of

the coefficients of the differential equations. The second method to determine the frequency response function of a shipmodel is the solution of the differential equations

which

describe the motions. As a first approximation the equations for heave and pItch canbe given in the following form;

(9,10)

++C++DiEz+GzM0e

(2)

where heave

=

pitch angle

circular frequency

of the motion

cL_g1

coefficients, depending on hull geometry,

A.-GJ

speed

and frequency

exciting force with phase

M=

exciting moment with phaseJ3

and

the elevat±on of the water surface in a cross section through

the centre of'gravity Is given by

Ii.

where

r =

wave amplitude

..:. 12

The linearity of the equations implies that the motion amplitudes depend linearly on the amplitudes of the forcing functions. When

these in turn vary linearly with wave height, the heave and pitch amplitudes will be proportional to the wave height.

The results of the tests in regular waves show that this linear desoriptio of the motions may be considered to be approximately

valid and may lead to reasonable results

for

the case of

moderate

wave height a.

The equations for the motions in

calm

water are obtained equating the right hand sides of (2) to zero. In a first approximation it is

assuxne.d that the coefficients a - g and A - G are the same. for the

motion in waves and in calm water.

This approximation enables us to determine the coefficients by

exciting the model in øalni water.

The exciting forces and moments due to the waves

can

be measured with the aid of a restrained model.

The experimental techniques of those tests are described ±n (11,12), whereas a short account of the principles is given below.

The values of c, C, g and G are determined from two common inclining

tests.

For the case: CA) 0, pure pitching and pure heaving we find:

cz0 = F0

CqM0

As the force or the moment is known and the values of z0 and

are measured, the coefficients c and C can be determined

as

functions

of speed. This has been done for the three ahipmodels and the results are presented in a dimensionless way in figure 3 where c/oat and

C/C are shown as a function of the Froude number.

The values c C , respectively, are the area -and the' longi-tudinal moment of inertia of the waterline plane multiplied by the specific gravity of wter.

In the speed range which is normal for the shipform under oonsid.era-tion (see table 2) the variaoonsid.era-tion of o and C is small as is shown by

fig. 3.

For higher speeds this variation, and its influence on the other

--

13

In a similar way g and G can be determined whè1 the model is free to pitch and heave simultaneously and = 0.

Than:

C,0-,..Gz0=M0

g and G are the coefficients of the statiOal coupling terms; foz zero speed they can be calculated from:

G=j-'S0

V

where: -e.. horizontal distance between centre of

V bouancy and centre. of flottation

V

V area of water plane V

V

V

-.'=

specific gravity of the water. V

The distance between the centre of bouancy and the centre of

V

flotation is about 2% of the length (see table 1) for each of our

models and consequently the statical coupling can be considered

to be a second order effect. From this it follows that the variation

of g and. G with speed is a third order

effect,

and. indeed, the

experiments showed that thts variation could not be meásüred with sufficient aocurcy. Therefore, the calculated values aocording to the formula will be used; they are given in table

_5.

able ₅

Calculated values o g

g ama. are given in g V V

In order tobe able to determine theotherooefficients,a, A, b

and B, another two series of tests are required.

V

Again, for the first series of tests the model is free tó. pitch

and restrained .1' or heave or free to heave and restrajned or pitch;

Now the model is excited via a spring

by a sinusoidal

_varyng

force

or moment of known frequency and the resulting motion is mesured.

The

equations

of motion in this case are:

V V

2+kz+cz

V .4A1

VBVCV

MoeLt

V 0.60 _0.70 0.80

-32

.-36

42.

The solution is:

z=z

0

a'

=

4fr

-

14-The values of

Z1 _,

d

( ,

M0

and CO

are measured

during the tests and, as c and 0 have been determined already,

the equations can be solved for a, b, A and B by equating the real

and imanginary parts on both sides.

The coefficients b and. B are the damping coefficients of heave and

\ pitch respectively, whereas a and A denote mass. and mass moment of

inertia.(in both cases the hydrodynamic mass Is included)

These coefficients depend on the frequency of the motion, the

Froude number and the hull geometry. They are given in figure 4 as

the following dimensionless coefficients:

.

The total mass "a" isdivided by the mass of the model:

hi.

ht

where in

mass of the model =

= hydrodynamic mass

4

weight of mode].

The total mass moment of inertia "A" is divided by the mass moment

of inertia of the model in air "I":

I

_I

T..

_L12A

where

I

mass moment of inertia in air

L).

,LL,= hydrodynainic mass moment of inertia.

The .

damping coefficient for heave is given by:

b14L

where

weight of model.

L

length of model.

S =

gravity constant

äiiznilarly the damping coefficIent

4

for pitch is written as:

.a16L

/1

15

are expressed in this way only to obtain a dimensionless presentation The factors and do not have any further physical

meaning.

Figure 4 shows that the damping ooefficents, the total mass and maes moment of inertia depend on speed and frequency and on the

hull geometry. The variation with speed is fairly small in the speed range under consideration1

\ Due to wall effect the measurements below a frequency (A) = 4 are

not very reliable. .

.

.

'Therefore, the dependency of the coefficients on the parameter opuld not be investigated in the region where is. about equal

to 1/4.. Have].ock C18) showed in a oa1cu1ationof the damping moment I of a long narrowplank, making forced pitching oscillations in water,

that a steep rise and fall in the damping moment occurs at

HwvyZ

\In our case this effect was masked by wall effect.

Finally, the shipmode]. is left free to heave and pitch simuitaneoualy and is forced to oscillate in still water.

Then the equations of motion are: a

-a.z+bz+cz+Lp+e1 i-qi

F0,e

The solution can be expressed

Z

Z

'ct+JJ

Substitution of the solution in the equations of motion results in four identities from which the w*iown coefficients d, e, D and E can be determined, as the other quantities are already known from previous experiments (a, b,

C,

g, A, B, C, G) or will be measured during the teats

(201.p0,

_,

4 F,

M0)

The results for e-:

and E are

given in fIgure ₅ in the dimensionless

form; and as a function of the frequency and

the blooeZficie

16

-opposite sing.

Figure ₅ indicates that our experiments confirm his theory

qualitatively, at least for the higher frequencies. Te three shipmodels are not symmetrical and as may be expected, this causes a shift in the zero line between e and E.

The values for d and D appeared to be.very small. They are largely affected by experimental errors.

An estimation of the relative importance of' the terms dqi and DZ.

with regard to the other terms in the equations. of motion showed that they can be neglected without eausing appreciable errors in the calculation of the' motion amplitudes and phases.

L/ti-cLJ

L,tt(J

Finally the forcing

functions

F0e'

and

M0e

of the

models in waves were measured by making use of a restrained model. The experimental technique is described in (12]. The tests were done in regular waves 11th a height of g

L

or 5.08 cm.

The dimensionless amplitudes of the exciting forces and moments

are:

NI0

CM=.IKw

where = amplitude of heaving frce in a wave with

amplitude r.

lvi amplitude of pitching moment in a wave with

amplitude

I.

specific gravity of water. area of waterplane

1< longitudinal moment of inertia of waterplane area.

06,

= maximum slope of a wave with, amplitude r.

The phase angles & and with 'regard to the wave are shown

in figure 6.

From this figure it appears that the influence of speed on the forcing functions is very small.Unfortunately the range of the

dynamometers which were used to measure F0 and was not sufficient

C'

_.F and

-

17

-consequently, it was. impossible to judge the linearity of the forcing functions with respect to wave height.

2. The solution of the differential eQuations.

Th results so far enable

us,

to solve the equations of motion

(equation (2))as all of the coefficients are

_known.

Their solution

is given by Korvin Kroukovaky

and Jacobs

in

(10]

in the following

form;

-zz1,e

q'=0e

The values of .Z

and

91 and, consequently, the values of _ZO,fIO.T

and.

J

can

be determined from the following expressIons:

p=

j;;p

Qk'PS

The. equations of motion were solved for the following cases: Wave length:

0.75

L,

I

00 L, 1.25 L, 1.50 L and

_1.75

L Wave height: 1/Ll.8 L.

Speed : Fr .=

0.15,

0.20 0.25, 0.30

In order to show the influence of the coupling between pitching

_and

heaving motions, the differential equations were solved also for the case where:

The results of these calculations are presented in the figures 7a,

7b

and 7c

together with the experimental values obtained by cross plotting in figure

1.

It is shown that in general a good agreement exists between the calculated

and

experimental values. For the 0.60

_and

the

MQ-F5

Qi?_PS

.1 where:

P _--LLW+C

c

2+(.e.c.4)

=

_.Dw2Ew

-G

-

5 -Aw2LBi-,C

18

-the 0B = 0.70 model the differences are of the same order of magnitude as the experimental errors, both for amplitudes and phases.

The CB 0.80 model shows some significant discrepancies for wave lengths larger than 1.25 L.

At speeds above Fr 0.20 the model shipped large quantities of water over the bow ator near resonance conditions. It is quite possible that In this region, the coefficients of the 'differentIal

equations obtained by forcing the model to oscillate in still water diff'er considerably from those which should be taken into account

for .the motions in waves.

For instance, the stability coefficients, or spring constants, c and C will be smaller when deck immersion or forefoot emersion occurs,, and this could be one of the reasons why the motion ampli-tudes of the CB 0.80 model are somewhat underestimated in the conditions at or near resonance.

Wetness' for the CB 0.80 model is more critical than for the other

two models because at high speeds an excissive bow wave builds up, due to the large angle of entrance of the waterline (total ang]q 86 degrees ) The bow wave reduce the freeboard in' this cash

more than in the case of the other two models.

Moreover, the exciting forces and moments as experienced by the - model in a very wet condition, may differ from the values obtained

from a restrained. model in waves.

The natural frequencies for pitch (t.t),,) and heave (W), which are used for the calculation of the resonance factors

and

J\,-

(see table LI.), are determined by using the

Wnz

experimental values of the mass (a) or the mass moment of inertia (A), and the spring constants c and C.

Neglecting the damping, the natural frequencies are approximated

by:

and

The natural frequencies obtained in this manner are summarized in

19

-Table 6

Natural circular frequencies for heave and pitch

With these natural frequencies the conditions for resonance could be calculated and they are indicated by arrows in figures 7a, 7b

and7c.

The measured phase differences 27 (heave after pitch) also show a good agreement with the calculated values; the differences _are of the same order as the experimental errors which are 10 - 20

degrees approximately.

The phases of the motions with regard to the wave ( and

I

were not measured during the first series of teats in regular waves. However, phase values for the 0.60 model kindly supplied by

the Netherlands Shipmodel Basin, where . 10 feet model had been tested under the same conditions (see table 7b),

In a later stage of our investigations it was decided to test the CB = 0.70 and the 0.80 model, while restrained for surging motions. Then a comparison with the results of the earlier tests might show the influence 0±' surge on heave and pitch. Moreover,

the phase relations could be measured and compared with the calcu-lations. In order to avoid confusion in figures

_7,

the results of these latter tests are presented in table

7a

(amplitudes) and 7b

(phases). The ainplii4es are given in dimensional form so as to

make comparison with the values in table I more easy.

0B=°60

_CB=Oa?O

Fr o.15 7.2

7.o

6.5

6.9

6.2

_6,7

0.?0.

7.0

7.1

6.5

6.7

6.2 _6.5 0.25

7.0

7.2

_{6.7 6.9 6.2 6.6} 0.30

7.1

7.2

6.8

6.8

6.2 6.9

Table 7a

Motions in regular

_{waves; watre height 1/48 L}

_5.08cm

Surging motion restrained

in degrees; Z0 in cm.

in degrees.

CB

0.70

'

03 = 0.80

27

z0.iY

p0.75

0.7

0.2

-

0.4

0.2

O ioo

2..7

1.8

50

_2.3

_1.3

57

°

'1,25

1.50

3.1

2.6

2.3

2.1

48

71

3.1

2.9

2.4,

2.1

53 71

1.75

2.2

2.0

81

2.4

2.1

₇₇

0.75

0.5

0.1

-

0.2

0.2

di.00

2.3

1.4

65

1.7

0.9

66

"

1.25

3.3

2.9'

'52

3.0

2.7

62

1.50

3.2

2.6

60

3.1

2..5

63

1.75

2.5

.2.2

75

.2.6

2.5

0

u

0.75

0..3

0.1

-

0.2

0.2

1.100

1.8

1.0

70

1.4

0.6

68

,

1.25

1.50

3.1

3.2

3.1

3.1

62

56

2.8

3.4

2.3

3.3

73

64

1.75

2.7

2.6

68

_2.9

3.0

64

0.75

0.3

0..2

.. -

_0,3

0..2

O

_1.00

1.4

0.7

₆₅

1.0

0.4

₆₁ u

_1.25

2.9

2.9

77

2.6

2.1

77

1.50

3..1

3.8.

61

3.1

_3.5

78

1,75

2.9.

3.2

61

2.8

_3.7

62

Phase differences of motion in

regular

waves

with

.respect to the wave. Surging motion restrained

except for CB = 0.60 model.. Table .7b

and Cc are given in degrees.

From tables 7a and 7b and. 4. it an be concluded that for the 0.70

model the influence of surge is negligible both for amplitudes and,. phases. The small differences which are present even have a tendonc'y to improve the agreement between calculation and experiment.

The correlation for heave of the 0.80 mode], improves and in particular at wave lengths larger than 1.25 L, but for pitch the

differences become slightly larger in this region. OB =.

0.60

ca]i1rn

= 0.70 CB' 0.80

ke'int

T

exp&t

V

S

r

1.00-29 -32 +5

-39

+11

041

+11-50 +7

..59+11

'1.25 -11 -8 +32 -.

+42-5

+39-5

+4.8 -16 .e41 I 1.50 +11 .

-12

+4.7 +5

+76-7.

+53 +5

76 -.17

+54 1.75 - -18

+53 -1

+80 -1.

58+2

+79.

-.19 +58 1.00 -.611..

-.3

-71

-7

-76

-II

-.75

-.3

-80

..ILI. -.99 -25 1.25 -32 +21 -.25 +19 -31 +21 -26 +20 -40 +22

55 +20

1450 +3 +60

-7

+4.1 +62

-7

+4.3 0 +63 -23 +4.0 ti _{1.75 - - -10} +52

-3

+72 -8 +50 -s-I +71 -14 +51 1.00 -96 -19

-95

.-25 -102 -32 -101 .23 -.102 -311- -14.7 _11.3 1.25

-55

+6

-4.6

+4-

-54.

+8 -43 +7

-79

-'6

-86

-1

1.50 -10

+4.5 -26 +34 -15

.i41 -.3

+27 -.19

.e45 -.4.2 +31

1.75

-

-

-25

+4.8 -7

+61 +10 +4.8

-6

+58-51

+45

2

1.00 -120 .-33

-117

-.37 -117

-52

-.1,19 ..42 -12 -65 -.131 -59

d

1.25 -75

+15 -61

'0

. ...77

0

-65 +2

-.77

0

-101 -20

iL

1.50 -27

+1

-27

+20 -LIG +15 ...211.

+12 -43

+35 -56

+18

1.75

-

-

-18

+L1.Ô -16

+4.5 +8'

+30-19'

-.43 -.26

+33

- 22

In all cases, the calculated phases ,

f

and show a

satis-factory agreement with the experiments.

As shown in figures 7a, 7b and 70, it appears thay the 2/ values calculated under the assumption of no coupling, disagree with the experimental values; the coupling seems to be important in this respect.

For shortness the calculated and.

Cl

values for uncoupled motions

are not given here. However, it appeared that coupling has a major effect on but not on Cl

In conclusion it may be stated. that the equations of motion, caB

expressed. by (2), give a statisfactory description of heave and pitch in long crested regular head seas except for the case of the very full model when the wave length exceeds 1,25 L and the speed of the model

is much le.rger than the design speed.

The influence of surge on heave and. pitch is very small.

V. Tests with the CB 0.70 model in jrrelar waves.

1. Measuring method.

A third method to determine the frequency response function o a

shipmodel is based upon the spectral analysis of measured model motions in irregular waves.

The input (= the irregular waves) and the output (= the irregular motions of the shiprnodel) are related by eans of the time invariant

frequency response function of the model. By applying spectral

analysis, the amplitude and. phase response functions can be determined from experiments in irregular waves.

The necessary irregularity of the waves is obtained, by changing the period of the valve mechanisme of the pneumatic wavemaker at constant

time intervals.

This produces waves of different length in succession and by the dispersion of these regular waves an irregular wave is obtained at some distance from the wave maker.

In our case the period of the valve mechanisme was changed one hundred times, the changes taking place at 1.5 second intervals,

-23-which resulted in a total time of 150 seconds for one wave program. The periods were chosen at random from the range 0.8 -

1.75

seconds, corresponding to a wave length range of 1.00 - LI..78.m, which is

im-portant for the motions of an 8 ft1 shipmodel.

The tests in the irregular waves were carried out with the CB

_0.70

model. Only two speeds were considered: 'r 0.15 and Fr 0.20.

Three runs of 50 seconds were made at each of these Speeds, and the total length of the recordings was considered. tobe sufficient for the determination of the frequency response function of the model.. The wave height meter was located abreast of the centre of gravity of the model and at a distance of 1.5 m. The model itself was res-trained for surge so as to facilitate the determination of the phase relations. As has been shown already the effect o surging motIons on heave and pitch is negligible, at least for the = 0.70 model,

and. therefore, the conclusions thatinay drawn from the present tests re not restricted by the absence of surge.

2. Analysis of the records.

A spectral analyèis has been made of the simultaneous recordings of heave, pitch and the wave by using the digital methods as indicated

by Tukey [19) an by Press and Tukey

_[203.

The fIrst step in this process is the digitizing of the analog

recordings and this was done by hand at 0.2 seconds intervals, resul-ting in

750

readings for each recording. .

In a later stage a three channel digital recording system became

ava1abie.

For the theory of the spectral analysis the reader is referred to the literature covering this field (for instance [21]).

However, a short summary is given below. .

The irregular wqves and the resulting irregular shipmotions may be described approximately as stationary Gaussian processes.

If x(t) is the variable of one of these processes, the following approximation is used;

-24-It is assumed that the mean value of x is equal to zero.

The coefficients C,, _{are constants where as the phase angles}

have a random distribution in the range O_ Z1T

Now the autoovarianoe function is defined by:

k'

('v)

T

4J

Y() x

-rJ Lt

(4J

The.following relatiois are valid forR

'J2(tJ=

_c'&t).

2,, (a)

O= the inec*n vcJue of x2

.,i (o) _

RO. (t)

From (3) and(4) it follows that:,

- d

()

Thepower spectrumG@')

Of )C Is now defined by:

c= C4'j

e-/

1 or if

T. oo

C11 _{xx (6)} GO COnsequently

12(tJ

=

G,,,

(Wri) CS(4LT)AC4) or in limit form:

=

CoE,

(ca./t)cL.w

(7)

0

-The Fourier transform of

₍₇₎

will be:

(eJ

coLAJt

8)

The expressions (4) and (8) can be approximated by digital methods, when the recordings are available ind.lgita]. form, as will be shown

later.

Now equation (3) can also be written in the following form which is used by St. Denis and Pierson in [1):

x (t).=

cs (cu +EV2G

(o1)4w

= 25

Assumenow that.a linear time invariant systemT(W) is subjected

to the input (3).

The linear system is characterized by the frequency response function:

T(w=\T(w)e"3

where:

amplitude response function and phase response function

en the linear system is subjected to a sine input of unit amplitude, the output will be a sine of the same frequency and the amplitu4e is

Tu)

; the phase with regard to the input wil]. be: '1(w)

By superposition the output y(t) due to the input (3) will be:

y(t)

=

c

Co

(cA1+E-l-,7)

t)

$ince

E

has a random distribution the phase angle (Epitii) ha

a random distribution as we].]. and y(t) is also a stationary random

process.

The power spectrum of the output,

G,

, is defiied by:

GYY(C4)ctL&

'

(10)'

and with

(6) we find:

G, (w)

G()

This expression may be used to determine the amplitude response function of the shipmodel when the power spectra of wave 'and motion &re available.

The phase response. function cannot be evaluated by means of

this procedure.

In order to find 'x(UJ) , we have to start from the covariance of

xandy:

' (12)

With (3) (9and(12):

GO

-26.-(13)

If we

denote: (.11i.) and

_'

G,(w)J

Su4 (C4.)=

Q (w)

12 (/c(w)

coSw w+)s c

d

('k)

is called the eQ spectrum and _{is called the quadrature} spectrum and combined they form a complex functions

G(w tGxy(w)e"

of which the real part is equal to the oospectruni and the imaginary part is formed by the quadrature spectrum.

The Pouries transform of (15) results in:

CoS )t

and

c'

ckC

.Agaii these equations will enable U8

to compute Cand %

from the

recordings.

Prom (10) and (13):

)T(w)L=

Y('!

Expression (17) is a second method to determine _{the amplituderesponse}

function. In combination with (11) and (Ill.) it _{can be shown that fOr}

a time invariant linear system the following relation is iralid:

(18)

or in continuous form:

Rxy(t)J1.Gy(cu)I:Cc6(ct1+)ctw

where:

Prom (1i.) we find the expression which determines

the

_{phase response} function:

1Cn

27

-Q(c).

cxy ()

Finally the following relations may be mentioned:

The telationa between input and output were used to amplitude and phase response functions of pitch and irregular sea.

The input in our case

is the

wave r(t) and the outputs are

the

pitching and

heaving motions (respectively

tfi(t)

and. z(t) )

The amplitude response functions _'

/rG)

_{and the phase} response functions cJ.)) and are found from

the

equations:

±

r

_Grr()

VGrr(W)

and

tan ciw)

,Qrj' '°?

(19)

(20)

determine the heave in an

The calculation of the necessary spectra was carried out by means

of digital methods [19, 20J.

In order to make this possible, the analog reoordins had to be digitized, as mentioned before.

The autocovariance function is approximated by:

J

x(t\.x (t.

₎

( p 0, 1, 2

w1i

N

750 total number of readi!ngs

3P number of time sifts

- 28

The Fourier transformation can be approximated by making use of

the. trapezoidal rule.

L

h

2i

{

k

where;

_f

1,

4 op.rn

tf poorr

This approximation is usually called the "raw" spectrum.

A smoothing process. is now applied to the raw.spectrumby.using the relations:

L0.+L,

=422

(21)

Gmc

L,

_Lu-i

By

_means

of this procedure an evaluation of the spectral density

function is obtained.

A similar procedure is followed to find the cross s:pectra.

The covarlance function Ixyt) is approximated by:

Thesame

b(2i) jsapp=ijed

thee

two spectra to obtain the ultimate approximation for and

Q(')

I;;,=;

)c (±).y (t4)

...,+m)

The raw co.- and quadrature spectra are computed. from:

c=

.Kr(DI 2)

CoS

where:

-

*

{

P9

29

-The results Of the calculations, which were carried out on the "Zebra" digital computer 'of the Mathimatical Department of the Deift Technological University are shown in figures 8 and

_9.

The frequency response functions are given in gø figures 10

1;ifi.

For easy comparison with the previous experImental results, the amplitude response functions for pitch are given in the form:

The results of tests in regular waves are.als9 shown in figures

ijj

4/,2/@.

It appears that

a

close agreemont is found between

the amplitudes and phathes as obtained from the tests in regular waves and those derived from the tests in irregular waves,

Particularly, the agreement for the phase angles is stdlcing. However, the two methods for the determination of the amplitudes (a. spectral analysis, b. cross spectral analysis) show very slight

differences as might be expected. As is obviotis, only in the. ideal

case of an irflnite length of recording and

an

exact evaluation

of

G,

and'Gxy

' the derived amplitudes wIll be the same according

to the formula (18). .

The derived frequency response functions are, not much affected by errors in the estimation of the spectra and. cross spectra, because the same procedure is followed in the calculation of input and output spectra.

An estimation of the effect of the sampling errors (due to the finite length of the recordings) on the. evaluation of the spectral density functions is given by the coefficient of variatioxi according to 120). In our case the coefficient of variation, which is the ratio of the root mean square deviation of the spectral densIty to the average

value '(or by approximation: tO the measured. value), is expressed by:

I.

(m\z (o

_.0.

In this case it may be assumed that the spectral density as

estimated, has a normal distribution. This means that at a proba-. bility level of 0.68 the root 'mean square deviation equals 20% of

the measured value.

The' agreement between the results in regular waves and those n

irregular waves is such that a further analysis of the small diff

3Qa.

the cross 8pectra the interested reader may be referred to _22]

Acknowledgement1'

The Author is indebted to . E. Baas and Mi, W. Beukelman, who

List of symbols:

,/3

1éEil

phase angles.

maximum wave slope specific gravity

I) _{wave length}

pitch angle

amplitude of ptoh

vertical d.isplacemént of water surface

,4(. added mass

added mass moment of inertia

t

_{time shift.}

weight of model

resonance factor, for heave

resonance factor for pitch circular frequency

circular frequency of encounter

natural circular frequency for heave natural .circular frequency for pitch

ct1c&ec3

coefficients of the differential equations

ABCDEG I

for heave and ptoh

blockcoefficient

dimensionless amplitude of heaving' force

. dimensionless amplitude of pitching moment

C(w)

Co spectrum

F0 . , amplitude of heaving force

Fr

. Froude number

spectrum

I

, longitudinal mass moment of 'inertia in air

K

longitudinal moment of inertia Of

water-plane

L

'Vie

T(w)

-

31 -.

length between perpendiculars amplitude of pitching moment quadrature spectrum

frequency response function

acceleration. of gravity. .

eentre of. bouancy and. centre of flottation

mass of shipmode]. .

List of symbols:

r

wave amplitude

±

time

Z

_heave..

Z0

_{heave amplitude}

. 32

33

-References:

St. Denis, M. and Pierson W.J.:

"On the motions of ships in confused seas't S.N.A.M.E.

_1953.

Kriloff, A.:

"A general theory of the oscillation of a ship on waves" I.N.A. 1898.

Igonet, 0.:

"Experiences de tangae an point fixe".

A.T.M.A.

_1939.

I. Weinb].um, G. and St. Denis, M.:

"On the motions of ships at sea".

S.N.A.M..

_1950.

St. Denis, M.:

"On sustained sea speed". S.N.A.M.E.

_1951.

6. Haskind, M.D1 and Rirnan, 1.5.:

"A method of determining pitching and heaving characteristics of a ship".

Bulletin de l'Academie des Sciences de U.R.S.S. Classe des Sciencestechniques,

19kG, no. 10

(translation Rusàian-Dutch by Ir. G.Vossera).

Gertler, M.:

"The DTMB planar-motion-mechanism".

Symposium on the towing tank facilities, instrumentation and, measuring technique, Zaeb

_1959.

Go].ovato, P.:

"The forces and moments on a heaving surface Ship" Jourhal of Ship Research

_1957.

Haskind, M.D.:

"The bydrodynamic theory of the oscilla1ion of a ship in waves".

Prikladaya Matematikal Mekhanika. Vol. 10, no.1, 1946

Korv1.i Kroukovsky, B.V. and Jacobs, W.R.

"Pitching and heaving motions of a ship in regular waves" S.N.A.M.E.

_1957.

Gerritema, J.

"Experimental determination of damping, added mass and added mass. moment of inertia of a shipmodel".

International Shipbuilding Progress

_1957.

Gerritsma, J.

"An experimental analysis of shipmotlons in longitudinal regular waves'.

International Shipbuilding Progress

_1958.

Lewis, E,V.;

"Ship Speeds in irregular Seas".

S.N.A.M.E.

_1955.

14.. Todd, F.H.:

"Some further experiments on single screw merchant ship forms - Series Sixty".

S.N.A.M.E.

₁₉₅₃

Brard, R.:

"Introduction & l'tude theorique dii tangage en marche".

ATMA, 19'18. Gerritenia, J.:

"Seaworthiness tests with three geometrical similar

shipmodels".

Proceedings of the Symposium on the behaviour of ships in a seaway.

Wageningen,

_1957.

Numata, E.:

"Influence of tank width on model tests in waves".

Note no. ₅₅₁ with addend.um, presented at the Twelfth meeting

of American Towing Tank Conference,

1959.

Havelock, T.H.:

"The effect of speed of advance upon damping of heave and pitch".

I.N.A.

_1958.

Tukey, J,W.:

"The sampling theory of power spectrum estimates". Symposium on applications of auto correlation analysis

tophysicalprobiems.

* 35

Press, H., Tukey, J.W.:

"Power Spectral methods of analysis and their application

/ to problems in Airplane dynamic".

Agard Flight Test Manual. Vol. IV, part. IV C.

Bendat, J.S.:

/

"Principles and. applications of random ]aoise theory"

/ Iew York, John Wiley & Sons.

Goodman, N.R.

"On the joint etiination of the spectra, co spectrum and quadrature spectrum of a two-dimensional stationary 'Oaussian process".

Scientific paper nO.10

Engineering Statistics Laboratory, :New York.

-36-.

Onderschriften van de figuren:

.figuur I : Motion amplitudésánd phases

in

regular waves.

figuu.r InfluenOe of wave height on motion parameters.

figuur 3 : Restoring force and moment coefficients as a function

of speed.

uiguur I. : Damping, mass and mass moment of inertia.

figuur ₅ : C:oss coupling coefficients

vtE

figuur 6 : Forcing functions for heave and pitch..

figuur 7a : Comparison qf calculated and measured motions CB= 0.60

7b

: ft . . .11

°B 0.70

7c

: H It It It H _CB 0 80

figuur 8 : Spectra and cross spectra for Fr .= 0.15

f-igutr

₉

: Spectra, and cores spectra for Fr = 0.20

figuur 10 : Comparison of.frequenc

response

functions determined

from tests in regular waves and. in irregluar waves.

- I :PIC,H I A/I17 F

I

t5 AI9T ___ç_ t74 -. 4',' -X4.i07fr

PHASE LAG EAJE At PIT

0 03 PITCH J;Q70

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SH'IPMOTIONS IN- LONGITUDINAL WAVES

by

Ir. J. GERRITSMA

Publication no. 14 of the Shipbuilding Laboratory, Technological University, DeIft

I. Su-i,i'inary

In this study on shipmotions, three methods are used, to determine the frequency response function of a shipmodel in longitudinal waves.

In the first place, heaving and pitching motions of three shipmodels of different fullness have been measured in regular waves. The second method is based on the. assumption that heave and pitch can

be described with sufficient accuracy by a set of

two coupled. linear differential equations of the

second order.

The coefficients in these equations have been

determined by means of experiments, and the solu-tion of t.he equasolu-tions of mosolu-tion gave the frequency response function of the shipmodel. Here again, three models were used to test the assumptions.

The third method is based on the assumption

that the response of a shipmodel in irregular waves

may be found by. a linear superposition of motions

in _{regular waves. Consequently, the frequency}

response function may be found by analysing the result of tests in irregular waves. This has been done

for one shipmodel only. The results of the three

methods are compared. For the sake of completeness some of the results of earlier work are also given in

this publication.

H Introduction.

The frequency response function of a ship in

regular waves is a basis for the further analysis of shipmotions in irregular seas. St. Denis and Pierson

[1] showed that a detailed statistical description of the shipmotions in an irregularsea is possible when

t.,he principle of linear superposition is valid both fort waves and for shipmotions. Therefore,

model-tests in regular waves are a valuable tool for the

study of the motions of a ship in waves, and this point of view seems widely. accepted at present.

In his famous paper Kriloff [2] attacked the problem of ship response in its full extent: an

analytical approximation was given of the motions of a ship with oblique heading to regular waves.

However, important hydrodynamic effects were

neglected by Kriloff and no experimental investiga-tion was made to test his theory. In any case, the facilities to test a model in oblique waves were not available at that time. Igonet [3] made a

compari-soi between calculated and measured motions of a ship model for the simple case of longitudinal reg-ular waves and zero model speed. He found-a good

agreementbetweeii Kiiloff1s rheory'imd the

experi-ment, which is remarkable because of the fact that

in -the evaluation of the exciting forces and moments

a further simplification was made by neglecting the Smith effect.

A further contribution to the development of

the theory of shipmotions was given by Weinblum and St. Denis' [4, 5] in which the important con-ceptions of hydrodynamic mass and damping re-ceived more attention than in Kriloff's paper.

A mathematical evaluation of these hydro-dynamic phenomena is difficult owing to the free surface boundary conditions in the low frequency range, which is valid for shipmotions, and only rough approximations were given in these papers.

An important experimental method, for the

determination of hydrodynamic mass and -damping of an oscillating shiplike body on the water surface, was published by Hask.ind and Riman [6], who forced a mathematical shipmodel to perform an

oscillating motion with one degree of freedom. In this manner added mass and damping could -be de-termined experimentally. However, only the ease of heave at zero speed was considered. A remarkable recent development of this technique is found in the., planar motion mechanism of the Taylor Model Basin [7]. This apparatus will be used' to determine experimentally all the hydrodynamic coefficients

playing a part in the motion of a body with six degrees of freedom. There is no doubt that such an

apparatus. will be extremely valuable in the

anal-ysis ' of shipmotions. Golovato [8] published

experimental data on hydrodynamic mass, damping a,nd coupling effects of a heaving mathematical shipmodel at various speeds of advance.

In general, however, it may. be said that' experi-mental data on hydrodynamic mass and damping are rather scarce in the existing literature on the subject.

Further developments of the theory of shipmo-tions were published by Haskind [9] and byKorvin Kroukovsky and Jacobs [10].

In the latter case a comparison was made between the calculated heaving and pitching shipmotions

and the results of tank experiments itt regular waves This was done for some widely different

ship forms. In many cases a reasonable agreement

was found between theory and experiment, but

some significant differences were also shown. Here again, a number of rather intuitive assumptions had

to be made in order to approximate some of the

coefficients (mainly those concerning added mass