An experimental analysis of ship motions in longitudinal regular waves

cRERT

-lO-P

REPORT No. 30S

December 195.8

STUDIECENTRUM T.N.O. VCOR SCHEEPSBOUW EN NAVIGATIE

AFDELING ScHEEPSBOUW PROF. MEKELW'EG DELFT

(rS4ETHERLLANDS' RESEARCH CENTRE TNO. FOR SHIPBUILDING AND NAVIGATION)

(SHIPBUILDING DEPARTMENT - PROF MEKLELJVEG - DELFT,

AN EXPERIMENTAL ANALYSIS OF

SUIT MOTIONS IN LONGITUDINAL

REGULAR WAVES.

by

Jr. j. GERRITSMA

AN EXPERIMENTAL ANALYSIS OF SHIPMOT'IONS IN

LONGITUDINAL REGULAR WAVES

by

Pr. J. -GERRITSMA

Publication no. 10 of the Shipmodel basin ofthe Shipbuilding Laboratory. Technological University Delft

Synopsis

In the current literature on seaworthiness it is generally assumed that the pitching md heaving motions of a

ship in longitudinal regular waves can be described with sufficient accuracy 'by a set of two coupled linear

'differential equations of the 'second order with- constant coefficients.

Using -a model, performing forced oscillations in 'still' water, the coefficients of these differential equations are determined experimentally for an 8 ft model of the Sixty series with a blockcoefficient of 60 The exciting forces and moments are measured' on the restrained model in waves.

Four modelspeeds and a wide frequency range are considered.

The solutions of the differential equations, determined in this manner, are compared with the measured motions of the same model in regular-waves.

A goo,d agreement is found between the computed and measured modëlmotions.

It is shown that some of the so called coupling terms, ,have an important influence on amplitude and phase

of the model motions.

I.

I'nlioduction-In a first approximation theoretical considerations on heaving and pitching motions of a ship in regular longitudinal waves lead -to' a system of two -coupled

linear differential equations of the second order

with constant coefficients.

The influence of the surging is assumed' not to

effect the' heaving and pitching motions.

The coefficients of t-hese differential equations are functions of the shipform, the speed' and the

frequency of 'the motions as are the exciting forces

and moments which in addition are, functions of

the wave dimensions.

This linear approximation is valid only for rela-t-ively small wave heights; strong non linearities,,

such as the shipping of excessive amounts of water

over the bow are excluded In their simplest form

these differential-' equations are:

az + bz ± cz ± di +

s.,; ,+

_{g'' =}

F0 cos -(wt + a)

-A-ip+Thp±Cip+Dz+Ez+Gz=

= M0 cosi(w/ + 1-)

(l)

where: I) pitch angle

z vertical' -displacement -of the centre-of gravity -centre-of the ship

U) = circular frequency

ag

-coefficients of the differential

equa-AG

tions-F0 = amplitude of exciting force

M0 amplitude of exciting moment

a,'fl = phase differences with respect to -the wave

The vertical displacement T of the water

sur-face -in -a cross section through the centre of gravity is given by:

= r cos- 0) 1

where r is the wave amplitude; the' circular, fre--quency w, is a function of the wave-length 1 and

the speed of the ship The coefficients of the

differ-ential equations (1) are known as- the mass and

the mass moment of-inertia -(a and B), the damping 'coefficients (b and B'), -the coefficients of the, re-storing functions (c and- C) and the cross coupling

-coefficients (d, e, g, D, E, G). The coefficients a

and A respectively, consist of the mass and the mass

moment of inertia of the model in air to which is added the hydrodyiiamic mass and' the hydro-dynamic mass moment of inertia u, respectively. Several methods of calculating t-he- coefficients of

the equations of motion, and of the exciting forces and moments have been published. These- methods make use of simplified -shipforms or other simply-fying assumptions [1,- 2, 3].

-Korvini Kroukovsky and Jacobs [2] applied- their

method to eight widely different shipmodels; the

resulting equations of motion- were solved and the

solutions (the computed motions) were compared

with the, motions -observ ed in towing .tank tests. In - general the correlation between experimental and

computed- results was -found- to be good but also

some rather large differences were

shown-.-In general such a comparison has a certain

dis-advantage: the solution of the differential'- equations is a function of a large number of parameters; each

of the coefficients contributes to the final result

and 'each -of these coefficients must be, computed using more or less simplifying assumptions.

fore, the reasons for! such a difference between

computed and measured' motions, which may arise in a certain, case, are hard to locate,

In this paper we will apply an experimental anl-ysis: the equations of motions are set up, using the experimental values which have been found for the

coefficients and the exciting, forces and motions. The solution of the. eq.uations. is then. compared

with measured values of the model motions. It is .the authot's opinion that in this way a

some-what more reliable base is obtained to judge the

applicability of the equations (1) in the' case of

heaving and pitching. Moreover, the experimental determination of the coefficients, forces and mo-ments provides the opportunity for a comparison with the values of each coefficient separately, as

calculated by applying the excisting theoretical methods or methods that will be developed in the. future.

It will 'be clear that the programm of this investi-gation can 'be split up in the following three parts:

the experimental determination of the motions

in, longitudinal regular waves;

the experimental determination of the coeffi-cients, exciting' forces and moments of the

differential equations;

the solution of the differential equations (1)

and .the comparison of this solution with the

corresponding measured values under a.

This programm is carried out for an 8 ft. model of the Sixty Series with a blockcoefficient of .60

(parent form) [4]. A part of the necessary data

was already available as a result of earlier investiga-tions [5., 8].

In' this study all the data which have been used

are given for completeness.

We shall now consider the various parts of the

progr.amm separately.

2. Experimental determination of the ',notion.

am-plitudes and the phase of pitching and hçavin,

The towing arrangement of the shipmodel is

shown in Fig. 1. The model is guided by means of

two pairs of light' tubular horizontal roller guides

of which one pair is fitted near the bow and the

other near the stern.

Two vertical rollers connected to the carriage

can slide between these guides and in this way f

tee-dom in pitch, heave and surge is ensured though the model is forced to stay on its cOurse without yawing (Fig. la).

Another arrangement has been used 'for some of

the tests (see Fig. lb.) ; here a so called

"nut-'cracker" prevents the yawing motions and the hori-zontal_and vertical guides can be omitted. The last

system is to .be preferred when_large_modelLrnotions_

are to 'be measured.

a,

GRAVITY

b

GVAVITY DYNAMOhIETEP

DY NAMOMET ER

Fig. I. Test arrangement for the ,,,easure,,,e,,t of pitch and heave in longitudinal waves

Heaving and pitching motions were measured by means of low friction precision potentiometers; the pitch angle was magnified' 15 times by a gear.

A gravity type dynamometer with a reduction of

1 : S was fitted' to provide a constant towing force

to. the model by means of a very 'light subcarriage. This subcarriage is guided by a polished' horizontal rod using ball bearings to minimize friction.

The main dimensions of the shipmodel are as.

follows;

length between perpendiculars L - 2.43 8 m

breadth 0.325 m

draft 0.130 rn

displacement .61.9 kg

'block coefficient 0.60

area of waterline plane 0.561 m2

mass moment of inertia (in air) (corresponding with a radius of

inertia of '/ L) 2.25 kgmsec2 moment of inertia of waterline

plane 0.170 m4

A forecastle of 15 % L was fitted

The wave dimensions in which the model was

tested are:

wave height in all cases: 1/8 L

The wave height was measured in front of the modelat a distance of 4 meters by means of a

re-sistance type wave height recorder.

Fig. 2 shows the test results in the form of

di-mensionless motion amplitudes p/a,0, zo/r and the phase angle 0 between heaving and pitching mo-tions (heave after pitch).

where:

= pitch amplitude

zo = heave amplitude

- maximum wave siope

2r

1

In the low, speed range (FR <'0.10 a 0A2) no test results are shown in Fig. 2; the measurements

in this range are not reliable as a result of walleffect. The dotted lines represent a mean tendency of some

widely scattered points. Further particulars about this wall effect are given in [5].

These tests formed part of an investigation

con-cerning the influence of modelscale and tank width on the motions of a shipmodel in waves.

Therefore, a series of three geometrical similar

shipmodels (with lengths respectively 6.15 ft., 8 ft.

Fig. 2. Motion amplitudes ann' phases

and 10 ft.) was tested in regular waves of uniform height.

From the results it appeared that the

dimension-less motion-amplitudes and the phase lags. of the

motions were the same for the three models, except for small deviations which were of the same order

of magnitude as the measuring errors. Similar

re-sults of other investigators [6] showed also that

scale need not be expected to have any significant

influence on the motions.

This very small influence of the scale is easily explainable when it is. assumed that the viscous

forces acting on the model may be neglected in

comparison with other forces, such as inertia and

restoring forces.

Basing .on the results of the above mentioned

tests on scale effect it is generally assumed that. this is true for heaving and pitching motions. Therefore the test results can be used for the prototype with-out .ccrrection.

Moreover, an elementary approximative

estima-tion of theinfluence of the viscous forces in

com-parison with other forces which play a role in

ship-motiOns leads to a similar conclusion.

3. The experimental determination of the

coeffi-cients and of the exciting forces and moments

of the differential equations

The equations (1) are used to determine the

coef-ficients a - g, A - G and the exciting forces and

moments. It is assumed that some terms, introduced

by the relative thotion of the ship with respect to

the wave, are absorbed in the right band side of the

equations.

For instance the damping force of the heaving

motion in waves is: b (z_{- c).,} where is the

verti-cal component of the watervelocity. The term b

of this expression is included. in the' right hand side:

F1) cos (w t -4- a).

A similar reasoning can be. given for the other

terms of the differential equations.

It is generally assumed that the coefficients a

-and .A - G are the same for the motion in still

water and in waves.

We shall use this. approximation. The model is forced to oscillate in calm water and the resulting motion amplitudes and the phases with respect to

the-excitator provide the necessary data to compute

the coefficients a, b, d, 'a, A, -B, D and F; Three

cases are to be considered:.

1. The pure heaving motion

All motions, with the exception of vertical

.dis--placements are prevented.

An excitator, driven by the DC. motor-is acting - on the model via a spring of known stiffness.

The motion of the excitator is sinusoidal as

re-sUt-of the use of a Scotch yoke (see Fig. 3 a.)..

1-5

:

05, L54 PITCH y=1i4

-.

' 126

,"

N

. VL=O.78 '1.54 HEAVE

---',(

0__

A/L=Loo 13 - PHASE I

LAG . HEAVE AFTERPITQ-1

PURt HEAVING MOTION

PURE PITCHING MOTION

c.COMbINED PITCHING AND HEAVING MOTION

Fig. 3. Test arra,,ge,neni for the measnreme,,l of ,,,ass, mass mome,,f of i,,ertia, damping a,zil the coefficients of the con 1,11,15 Ier,,,s

The pure pitching motion

The model is free to pitch but is restrained for

heave. The pivot is located in the models' centre

of.gravity and the excitator is connected to the model near the bow at a known distance from

the pivot (see Fig. Yb).

The combined heaving and pitching motion Now the model is free to pitch and heave. As for the pure pitching case these motions are

introduced by an excitation near the bow of

the model (see Fig. 3c)..

In all three the cases the amplitude(s) of the

motion(s) and the phase of the motion with respect

to the excitation are measured at certain values of

the frequency of motion. The magnitude of the

exciting force and moment is determined by the stroke of the Scotch yoke, the model motion and

the stiffness of the spring.

The measured values are used to determine some of the coefficients of the differential equations (1). Adetailed-description of this method, which was

introduced by Haskind and Rieman [7] is given in

[8 j where some results are given for the same ship-

-model as used for our present investigations.

The first test gives us the coefficients a and b, mass and coefficient of damping of the heaving

motion, respectively. For "a" the following relation

is valid:

a

6.31 + ,u,

the constant term being the mass of- the model and the 'added hydrodynamic mass.

The second test gives us A and B: the mass mo-ment of inertia and the coefficient of damping of

the pitching motion.

The mass moment of inertia of the model in air

is 2.25 kgmsec° and A = 2.25 + u,, where ,u is

the added mass moment of inertia.

Finally, the third test provides the necessary data to compute d, D, c and E.

The determined values of u, b, and B are given in Fig. 4 as- a function of the speed of the

model and the circular frequency of the motion.

In the computation of these values the

coeffi-cients c and C are involved. Both áf' them may be

assumed to be constant over the speed range which is investigated here.

Their magnitude follows from the well known

statistical calculations:

c = y' S0

561 kgm C " K 170 kgm

where:

specific gravity of water

so area of waterplane

K= moment of inertia of waterplane

Actually there is some dependency on speed, but

in our case the variation of c and C with speed is

very small and may be neglected as will be shown.

To find the order of magnitude of this

depend-ency on speed the following experiments were

carried out.

In the case of pure heaving, the vertical

displace-ments of the model resulting from accurately known vertical forces were determined. Similarly

in the case of pure pitching, the trim angles caused

by applying moments of known magnitude were measured.

The differential equations (1) for the case

(2) 0 and no coupling will be:

cz0F0

C Ipo Mo

(2) where:

F0

=

the vertical -force M0 the moment.

The measurements were- carried out' -in -the speed

range FR = 0 to FR 0.30 and the mean heave ,and,,,pitchamphtucIesajpeared tohe '1 cm. and

2 degrees respectively.

4

Fig. 4. Added mass, mass ,,zo,,,enl of inertia and coefficients of

dam i;zg for heave and pitch

Both the coefficients c and C were found to vary slightly with' 'the model speed.

The variations with respect to the statistical

values are given in table 1.

TABLE 1 0 Fit 0.15 0.20 0.25 0.30 0

1%

1%

C 0

1%

1%

2%

4%

Although the percentages of table 1. are based on a, sufficient number of measurements, the

differen-ces they represent can hardly be considered to be

'significant.

After all, a variation of c by two percent results in a 'variation -of a heave amplitude of -i 'cm, by

0.02 cm, and a variation of c by four per cent.

effects a two degrees pitch amplitude by only 0;08

degrees. '

For speeds higher than considered in this

in-vestigation, there certainly is a marked decrease 'of C, 'but these higher speeds are not important as the

design speed' of the model is approximately

FR = 0.25. -'

Moreover, non linearities will be introduced in the case of larger -amplitudes and these will corn-pensate more or less for the slight variations 'of c

and C with speed. -'

The reasolls mentioned above lead to the assump-tion that the statical values of c and C represent a sufficiently accurate approximation in the speed'

range under consideration.

It may be emphasized however that this

approx-imation may not be valid for other shipforms.

The coefficients of the coipling terms are deter

mined by means of the 'third test.

Ih principle, the coefficients g and G can be

computed from the statical displacement and

trim when a known force and a known moment

are applied to the shiprnodel.

This can be done by using the following equa-tions:

c z0 ± g Po =

C ± G z0 = M0

(3) For normal shipforms' the horizontal distance of

the centre of gravity to the centre of' 'flotation of the waterplane is máll and coflsequently the

coupling terms g sp and Gz are small compared

with cz and C sp.

As a resul't, a reliable experimental determinatiOn of the variation of g and G with speed could not be

made. Therefore it has been assumed that their

statical values- are valid over, the speed range under

consideration.

Thus:

g=G=y'i S0±=-3:2kg'

where I is the horizontal distance between the centre of gravity-of the model and' the centre of flotation.

In principle the coefficients of the dynamic

coupling terms, e, E, d and D, can also be

deter-mined from the third test. For one speed only

(FR = 0.20)' the results have been given already in '[.8]. '

'It was shown' there that the terms dz 'and D 'y

are small compared with the. other terms of the

equations of motion. The relative importance of these terms can 'be judged with the aid of tabl 2

where the amplitudes of all' the terms of equations (1) are given for the following case:

FR = 0.20; w 7 (resonance), z0 = 1 cm; -vio=2° o E PITCH '

Fr.15

-Fr=20----20

E,

so 10 5

-.'

,.

I.

\\

'

".

Fr.25---.

'Fr=.30--- "-...-0 5 _,--sec 10 '

0

40 HEAVE .

'8'

p1 £ I "

-'

--.____fI 20' J 4 0 5c.d.-sec 10

TABLE 2

Fig. L Coefficients e and E

However, by estimating the experimental errors in d and D it could be shown that small errors in the measured values of phase angles and possibly existing small variations of g and G with speed, result in very large variations of d and D.

So it appears that the experimental determination of d and D is not very reliable.

However, the absolute magnitudes of both d and

D remain very small and, therefore, it is assumed. here that d D 0.

The values for e and E are given in Fig. 5 as a

function of the circular frequency w and the.dimen-sionless Froude number of the model.

Again the result is sensitive for errors in the

phase measurement but less than in the case of d

and D.

Therefore, the phase measurement has been car-ried out with great care and moreover the measured phase angles have been faired before they were used

for the calculation of e and E.

The results of these tests show some differences

from the values determined in earlier tests for the case FR. = 0.20 {8]. However, at and near

reso-nance (w ' 7), where the coefficients e and E may be important, the agreement is quite good.

Generally, the coupling terms are small as

cOm-pared with the other terms as can be seen from

table 2.

Their influence on the motions of the model

cannot be judged immediately from table 2 as the

Fig. 6. Testarrangcmezt for the ,neasnre,ne,ziofthe exciting forces and moments

phase difference between the heaving and pitching

motion is very important in this respect as will be

shown later.

The exciting forces and moments, which form the

right hand sides of he equations of motion (1) are

measured on a-restrained model in waves. The test arrangement is given in Fig 6.The vertical motion of the model is restrained by a polished rod, which

is guided by means of linear roller bearings. This

rod is connected to the carriage by means of a strain

gage dynamometer which measures the heaving forces.

The rod- is connected to the model by means of a pivot. The pitching motions are restrained by a

bracket construction, connected to the rod. This

bracket is connected to the model by means of a

second dynamometer, and the forces measured here,

multiplied by the distance to the pivot give the

pitching moment.

The natural frequency of the system model-dy-namometer is in the order of magnitude of w0 =

50, both for heave and pitch.

In our case the circular frequency of encounter varies between w = 5 and w = 11; consequently

only a negligible dynamic magnification of the

magnitudes of the exciting forces and moment will

exist.

The damping of the system is less than 0.1 and,

therefore, the error in phase is also negligible. The measured values of the heaving force

ampli-tude F0, the ampliampli-tude of the pitching moment M0

arid the phase differences a and fi with respect to

thewae are given in Fig; 7 as

funct-ions-of-the-wave length ratio 2/L and the dimensionless Froude number FR.

4 The solution of the differential equations and

the coin parison of the solutions with measured heaving and pitching motions

YFhe solution of the differential equations, (J is 'giié'ñ in an elegant form by Korvin-Kroukovsky

+10

Fr.I5-

Fr.,.20

Fr.25--F___,I_,_

\

-,--______ 0 Ui 10 \_ / /

.,

\

0 5 i..i.sec' 10 Heave Pitch a (',2

=

5.7 kg

4 w2 -

5.7 kgm b w 2.2

Bw =

1.8 c

=

5.6 C

=

6.0 4 (s) e w g

= 0.2

0.6-O.3 ,, ,, D ü2 = 0.1

Ew =

0.4 G

=-1.1

,, ,,

Fig 7 Heaving forces and pitching moments

[2]. For this purpose the equations care written in

the following form:

az + bz ± cz + dsp + 'ev' ± gap. F9kot

where F = Po and, M = M0

It will be understood that only the real patt' of

'the solution is to be taken.

The solution is of the form:

z - z e°' and ' , e'°

where z z0 e and y = 5Po eu.

The vaiues of z and p and, consequently, the

values of z0, Po, y and 5 are determined from the

following expressions:

-

M.QF.S

- F.RM.P

Z

= Q.RP.S

and

1'Q.RP.S

where:

-P = - aw +' ibw + c

Q = -

dco2 ± jew ± &

R = - Dw-f-'iEw+G

S = - Aw2

+

iBw + C

By substituting the experimental values of the

coefficients into the differential equations, they can be solved for the following20 casesf

Speed FR = 0;15,0.20,0.25.,0.30 Wavelength ratio A/L = 0.75, 1.00, 1.25., 1.150,

1.75

Wave height

2 r =

The solutions giving the 'amplitudes of, motion and phases, are summarized in table 3 together

with the' corresponding experimental values.

In order to. judge the influence of the coupling terms, the solution of the' uncoupled equations is

also given. This implies that: d e = E

= = G

0.

A similar comparison is given in 'Fig. 8, where t'he motion amplitudes are represented in the.

di-mensionless form: spo/av 'and z0/ r (a is the maxi-mum wave slope).

In the Figs. 9 and 10 the computed phase-angles

' and are compared with the corresponding

meas-ured values.

The' experimental values of ' and a have not been measured in' the Deift Tank, 'but were kindly

sup-plied by the' Netherlands Ship Model ,Basin.'

The following conclusions may be drawn from

table 3 and Figs. 8, 9 and 10.

1 The differences between computed and meas-ured motion amplitudes and phase angles are

small in the case of the coupled differential

equations. 1 1 ' I 7'

FrJ5

' ' 0 35 -_4

- - -

900 4:

F

Lip' ' ',-, A' 1.

'/

' 7.-' 45

'I.

.0t: 7 :,

Fr20

. ' . .F0 ' F t

9.

'S

'''

. ' ft ' I, ui

P. '.-.----

900 4

F

LIP ' t ¶2, I , ,50 0' Fr 25

\\,___

__.:

I

-

---.M0 Is" I

'.4

,50

E

0 o , 0 I I

F

¶5 Fr=3O ' , ' I I35 U, 2I 9 .4 450 -: ' -'

0.

05 1.0 "YL 15 20

10

Fig. 8. Comparison of corn pu/ed and observed ,no/ion anipliludes and phases

t EXPERIMENT MOTION MOTION COUPLED .._...UNCOUPLED 135°-' 9d'- - - --45°-. 0 1 I I I I

,-:--'1.0-

,,//

0.5-0 Fr=0L20 I 0.75 LI 100 l;25I 1.50 I.75-'VL' - EXPERIMENT MOTION MOTION COUPLED

_.__._

135°-I I I 1 I 15- 1.0-1.0

/ °

Q5-.

/

-Fr=oi5

I I I I I 0.75 I.00 1.25 1.50 175 -'VL 135°-EXPERIMENT MOTION _...o....COUPLEMOTiON ₁₇ 135°-- EXPERIMENT MOTION MOTION ___ ..5.._UNCOUPLED _0_-..0....COUPLED ___..._UNCOUPLED II ___ -

-Fr0.25

_,d#!'

/

Fr0.30

II

TABLE 3. Comparison, of the computed and measured motion anL'litudcs and phase angles

For pitch the difference is in the order of magnitude of 0.2° - 0.3° and for heave: 0.2

a 0.4 cm.

Generally the cothputed values are somewhat

smaller than the experimental ones.

The difference in the phase angles is smaller

than 10 - 20 degrees as well for s9 = - as

for y and seperately.

2. The comparison shows a different.picture when

uncoupled equations are considered The char-4acter of the curves (see Fig. 8) is now

corn-'.pletefy different from the experimental curves

in particular for the heaving motion The

dif-ferences for, the pitching motions, are small but

they also show a different character of the

curves in some cases. -.

The error. in phase may amount to, about 45

degrees.

The sOlution of the coupled equationsshows some smal differences. between experiment and calcula-tión. Thesedifferences are only slightly larger than

the experimental errors '[5].

It is shown, however, that in most cases the

motion is underestimated' a little, in particular at

or near resonance. It is possible that,the measure-ment of the exciting forces and momeasure-ments using a

restrained model in waves, gives too small values at

or near resonance. At resonance the phase of the

motions with respect to the wave is causing a bow

and stern immersion' which is larger than in the case of a restrained model. Consequently larger

exciting forces and moments can be expected "be-cause of the shipform above the waterline.

For 'medium wave heights we iiiay 'conclude that

the linear differential equations according to (1)

give a sufficiently accurate description of the

heaving and pitching motions of a ship in longi-tudinal' waves.

Some of the coupling terms are necessary to

obtain satisfactory results.

A first' approximation, in which these coupling terms are neglected, does not give good results in

our case, in' particular for heaving.

It. is emphasized that our conclusions are based

on the test results of one particular model.

In the near'future a similar investigationon two other shipforms (w'ith blockcoefficients .70 and

.8 0) will be 'completed. 11 AlL s10 in degrees - Z in cm 0 = - 7 in degrees Experi-ment ' Coupled motion Uncoupled motion Experi, ment Coupled . motion ' ' Uncoupled motion Experi-ment Coupled motion Uncoupled motion 0.75 1.1 .1.1 1.2 04 0S ' 0.2 - -d 1.00 3.3 2.6 3.0 2.3' 1.9 0.8 43 37 44 1.25 3.6 32 3.6 2.1 1.9 1.4 62 40 95 1.50 2.9 28 2.6 2.0 1.8 1.6 74 59 92 1.75 2.2 2.2 2.0 '2.0 2.0 1.9 79 71 94 0.75 0.7 0.9 0.9 0.1 0.3 0.1 '

-

-d 1.00 2.'8 2.5 3.2 2.6 ' ' 2.1 0.8 63 64 ' 90 II 1.25 3.8 3.6 4.0 2.9 -' 2.8 1.7 51 43 75 1.50 1.75 3.2 2,3 3.4 2.7 3.0 2.4 2.6 ' 2.2 2.5 ' 2.1 2.1 ' 2.2 68 71 4,8 62 98 101 0.7'S 0.4 0.8 0.7 0.2 0.5 0.2

-'

-d ' 1.00, 2.4 ' ' 2.3 2.6 ' 2.2 , 2.0 0.7 83 70 73 II 1.25 3.5 ' 3.2 4.4 3.7 3.5 2.0 61 50 89 1.50 1.75 3.5 2.5 3.1 2.4 3.5 2.7 3.4 2.5 30 2.5 ' 2.5 2.5 ' 59 72 60 73 102 105 © 075 0.2' 0.4 0.4 0.1 9.2 0.,1

-

-

-d 1.00 ' 2.0 2.1 1.9 1.7 1.6 0.5 96 ' -10 70 1.25 3.1 2.7 ' 4.2 38 3.3 '1.7 76 , ' 61 79 '1.50 3.6 2.9 4.1 - 4.0 ' 3.2 2,3 58 47 95 1.75 2.7 26 3.1 2.8 2.5 2.4 73 ' 58 ' 101

12

Fig. 9. Comparison of computed and observed phase angles

-(uncoupled -motion)

The author is greatly indebted to Mr; E. Baas and

Mr. W. Beukelman, who carried out the greater

part of the tests and the calculations.

List of symbols

a h c e coefficients of the differential

equa-ABCDEGJ tions of motion

- F heaving force

-Ft .= dimensionless Froude number

K = moment of inertia of waterplane L = length between perpendiculars

M pitching moment - r= wave amplitude

S3 -arc a of waterplanc

.1 time

z - vertical displacement of the models'

ctrc of gravity

fi I) -phase angles

(Lw maximum wave slope

specific gravity A wavelength -

-= pitch angle

- t =veticaldisplacernent of water surface

Fig. 10. Comparison of corn putcil and observed phase angles

(coupled -motion)

= added mass

addèd mass moment of inertia

w circular frequency

= natural circular frcquency of a

system

References.

I. St. Denis, M. and Picrso,,, W. 1.: "On the nsotons of thips in confused -seas". SN.A.M.E. 1953. I Korvin-Kroukovsky, B. V. and Jacobs, -W. K.: "Pitching and

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

1957.

-Haskind, M. D.: The hydrodynamic theory of the oscillation of a ship in waves". Prikladaya Matensatikai Mekhaniki, Vol. 10, No. I, 1946.

4 Todd F H Some further experiments on single screw mer

chant ship forms - Series 60". S.N.A.M.E., 1953. Gerri/sma, I.: "Seaworthiness tests with three geometrical similar

shipmodels". N.S.M B Symposium, 1957, Wageningen. Szébehely, V. G., Bled soe M. D. and Ste/un, G. P.:- "Scale effects

in Seaworthiness'.: 11th A.T.iLC., DT.M:B., Sept. 1956.

Haski,sd, M. D. and Rie,nan, I. S.: "A method of determining pitching and heiving charactetitics of a ship" Bulletin de l'Academie des- Sciences do. U.R.S.S.,. -Classe des Sciences

- techniques 1946, No. 10. (translating Russian-Dutch liv

Ir. G. Vossers).

Gerrllsma, J.: "Experimental determination -of damping, added

mast and addi nass. moment of inertia of a shipmodel".

International Sh!pbuitding Progreu1957, No. 58.

+9d- Fr.15

+9o0j_,L...

Fr=

+45°-j-

Fr.25 7 +900 +45°-.

-.

COUPLED MOTION N$.M.B. Fr=.30

-.-:

O75 i!oo l25 L15o

I75-#..t

-

_r.

9O0_ o +90- I

I-

I I' -J Fr=.20 ;0. I

IllIFr

2- ,90q_- +45°-1 I I UNCOUPLED MGTION N.S.M.B Fr-30 - - 0.75 -LaO 1.25 1.50

'EPOR1 -o

AN EXPERIMENTAL ANALYSIS OF SHIPMOTIONS IN LONGITUDINAL

REGULAR WAV.

by

IR. J. GERRITSMA

Ptiblication no. 10 of the Shipmodel basin of the Bhipbuil-ding Laboratory Technological University Deift.

8ynoDsiB:

In the current literature on seaworthiness, it is generally assumed that the pitching and heaving motions

of a ship in longitudinal regular waves can be described with sufficient accuracy by a set of two coupled linear differential equations of the second order with const4nt coefficients.

Based on this assumption, the coefficients of these differential equations and the exciting forces and moments are determined experimentally for an 8 ft.model of the

Sixty Series with a blockcoefficient of .60.

Four modelepeeds and a wide frequency range are

con-sidered.

The solutions of the differential equations, deter-mined in this manner1 are compared with the measured motions of the same model in regular waves.

A good agreement is found between the computed and measured modelmotions.

It is shown that some of the so called coupling terms, have an important influence on amplitude and phase of the model motions.

I. Introduction.

In a first approximation theoretical considerations on heaving and pitching motions of a ship in regular longi-tudinal waves lead to a system of two coupled linear diff e-rential equations of the second order with constant

coeffi-eienta. - - --

-The influence of the surging is assumed not to effect the heaving and pitching motions.

-2-The coefficients of these differential equations are funct-ions of the sbipform, the speed and the frequency of the motion as are the exciting forces and moments which in addition are functions of the wave dimnnwlons.

This linear approximation is valid only for relatively small wave heights; strong non linearities, such as the shipping of excessive amounts of water over the bow are

excluded In their simplest form thee, differential equations

are i

+ b + oz + di'+ ei'+ gti F0cos (Lot + o( )

A)'+ B'+ C+ D+ EZ.+ Gz

M0cos (wt

(1)

where: 'p pitch angle

vertical displacement of the centre of gravity of the ship.

W

circular frequency.

a

-A - G) coefficients of the differential equations

F0 amplitude of exciting force

M0 amplitude of exciting moment

0 ,9 phase differences with respect to the wave.

The vertical displacement of the water surface in

a cross section through the centre of gravity is given by:

r cog

W t

where r is the wave amplitude; the circular frequency (' is

a function of the wave-length A and the speed of the ship.

The coefficients of the differential equations (1) are

known as the mass and the masnoinent of inertia (a and B), the damping coeflicientR (b and B), the coefficients of the restoring functions (c and C) and the cross coupling

coeffi-cients (d, e, g, D, E, G). The coefficoeffi-cients a and A respec-tively, consist of the mass and the mass moment of inertia of the model in air to which is added the hyd.rodynamic mass

fr

and the hydrodynamic mass moment of inertia

--

-3-Several methods of calculating the coefficients of the equat-ions of motion, and of the exciting forces and moments have been published. These methods make use of simplified ship-forms or other simplifying assumptions (1, 2, 3)

Korvin Kroukovsky arid Jacobs (2) applied their method to eight widely different ahipmodels; the resulting equations of motion were solved and the solutions (the computed motions were compared with the motions observed in towing tank tests. In general the correlation between experimental arid computed results was found to be good but also some rather large dif-ferences were shown.

In general such a comparison has a certain disadvantages the solution of the differential equations is a functiori of a large number of parameters; each of the coefficients con-tributes to the final result and each of these coeffic.ents

must

be

computed using more or less simplifying assumptions.

Therefore, the reasons for such a difference between computed and measured motions which may atise in a certain ease are hard to locate.

In this paper we will apply an

experimental analysis: the

equations of motions °are set up, using the experimental values which have been found for the coefficients and the

exciting forces

and. motions. The solution of

the equations

is then compared with measured values of the model motions.

I1 is

the author's opinion that in this way a somewhat more

reliable base is obtained to judge the applicability of the equations (1) in the case of heaving and pitching. Moreover, the experimental determination 01 the coefficients, forces

and moments provides the opportunity for a comparison with the values of each coefficient separately, as calculated by applying the excisting theoretical methods or methods that will be developed in the future.

It will be clear that the programm of this investigation can be split up in the following three parts:

--the experimental determination of --the motions in longitudinal regular waves.

the experimental determination of the coefficients, exciting forces and momenta of the differential

equations.

the solution of the differential equations (1) and the comparison of this solution with the correspon-ding measured values under a).

This programm is carried. out for an 8 ft. model of the Sixty

Series with a blockcoefflcient of .60 (parent form). (k).

A part of the necessary data was already available as a result of earlier investigations (5, 8)

In this study all the data which have been used are given for completeness.

We shall now consider the various parts of the programm seperately.

II Experimental determination of the motion amplitudes and the

phase of pitching and. heavinp.

The towing arrangement of the shipmodel is shown in figure 1.

The model is guided by means of two pair's of light tubular horizontal roller guides of which one pair is fitted near

the bow and the other near the stern.

Two vertical rollers connected to the carriage can slide between these guides and in this way freedom in pitch, heave and. surge is ensured though the model is forced to stay on its course without yawing (figure Ia).

Another arrangement has been used. for some of the tests (see figure Ib); here a so called "nutcracker" prevents the yawing motions and the horizontql and vertical guides can be omitted. The last system is to be preferred when large model motions are to be measured

Heaving and pitching motions were measured by means of low friction precision potentiometers; the pitch angle ws nag-nified 15 times by a gear1

fitted to provide a constant towing force to the modeL by

means of a very light suboarriage. This subcarriage is guided

br a polished horizontal rod using ball bearings to minimize friction1

The main dimensions of the shipmodel are as follows:

length between perpendiculars L

breadth draft

displacement

block coefficient

area of waterlin plane

mass moment of inertia (in air)

(corresponding with a radius of inertia

of +L)

moment of inertia of waterline plane A forecastle of 15% L was fitted.

-5-238 m.

0,325 rn 0,130 in. 61.9 KG. 0.60 0.561 m2. 2.25 kgmsec2 0.170 in4

The wave dimensions in which the model was tested are:

wavebeight in all cases z 1/48 L

wave lengths respectively : 0,75, 1.00, 1.25, 1.50,1175L

The wave height was measured in front of the model at a

die-tance of 14. meters by means of a resisdie-tance type wave height

recorder.

figure 2 shows the test results in the form of dimensionless

motion amplitudes "o/oc , 24/. and the phase angle ir

between heaving and pitching motions. (heave after pitch)

where

pitch amplitude

zo heave amplitude

maximum wue slope

_-

_A

In the low speed range (FR < 0.10 1 0.12) no test results

are shown in figure 2; tie measurements in this, range are

not reliable as a result0f walleffect. The dotted lines represent a mean tendency of some widely scattered points.

-7-in the right hand side of the equations.

For instance the damping force of the heaving motion in

waves is: b ( - ), where is the vertical

component

of the watevelocity. The term b of this expression is

included in the right hand side: F0 cos (w t

+ o. ).

A aimilar reasoning can be given for the other terms of the differential equations.

It is generally assumed

that the

coefficients a - g and

A - 0 are the same for the

motion

in still water and in

waves.

We shall use this approximation. The model is forced to

oscillate is calm water

and

the resulting motion amplitudes

and. the phases

with respect to the excitator provide the necessary data to compute the coefficients a, b, d, e, A, B, O and E. Three cases are to be considered:

The pure heaving mat ion.

All motions, with the exception of vertical displacements are prevented.

An excitator, driven by de D.C. motor is acting on the

model via

a

spring of known stiffness.

The motion of the excitator is senusoidal as result of the use of a Scotch yoke (see figure 3a).

The pure pitching motion.

The model is free to pitch but is restrained for heave. The pivot is located in the modelé' centre of gravity

and, the excitator is connected to the model near the

bow at a known distance from the pivot. (see figure 3b) The combined heaving and pitchin motion.

Now the model is free to pitch and heave.

As for the pure pitching case these motions are intro-duced by an excitation near the bow of the model.

(e..

figure 3c)

In all three the cases the amplitude(s) of the motion(s) and the phase of the motion with respect to the excitation are measured at certain values of the frequency of motion. The magnitude of the exciting force arid moment is determined

by the stroke of the Scotch yoke, the model motion and tile stiffness of the spring.

The measured values are used to determine some of the

coef-ficients of the differential equations

(1)

A detail.d description of this method, which was introduced

by Haskind and. Rieman [i'] is given in [8) where some results

are given for the same shipmodel as used for our present investigations.

The first test gives Es the coefficients a and b, mass and coefficient of damping of the heaving motion, respectively. For "a" the following relation is valid:

a 6.31 + , the constant term being the mass of

the model and the added hydrodynamic mass.

The second test gives us A and B: the mass moment of inertia and the coefficient of damping of the pitching motion.

The mass moment of inertia of the model in air is 2.25 kgm55c1

and. A - 2.25 + , where is the added mass moment

of inertia.

Finally, third test provides the necessary data to compute

d, D, eandE.

The determined values of/' , b ,,, and B are given in

figure k as a function of the speed of the model and the circular frequency of the motion.

In the computation of these values the coefficients c and C are involved. Both of them may be assumed to be constant over the speed range which is investigated here.

Their magnitude follows from the well known statical

calcu-lations:

c

-I'S0 .56lkgin1

C -ix -l7Okgm

where:

specific gravity of water area of waterplane

K moment of inertia of waterplane

Actually there is some dependency on speed, but in our case the variation of c and C with speed is very small and. may

be neglected as will be shown.

To find the order of magnitude of this dependency on speed the following experiments were carried out.

In the case of pure

heaving,

the vertical displacements of

the model resulting from accurately known vertical forces were determined.. Similarly in the ease of pure pitching, the

trim

angles

caused by applying moments of known magnitude were measured.

The differential equations

(1) for the case LQ

0 and no

coupling will be:

C *

where:

-

the vertical force.

- the moment.

The measurements were carried out in the speed range FR * 0

to FR - 0.30 and the mean

heave and pitch aiiiplitudee

appeared

to be 'b-' cm. and '- 2 degrees respectively.

Both the coefficients c and C were found to vary slightly with the model speed.

The variations with

respect to the statical values are given

in table 1.

TABLE I

-9-(2)

Although the p.z'centages of table I are based on a sufficient number of measurements, the differences they represent can

hardly be

considered

to be significant.

AtteraI1,

variatiofl rby two

eentrèrnilts Tha

FR C C 0 0 0.15

+2%

-1%

0.20

+1%

-1%

025

-1%

-2%

0.50

-1%

-4%

In principle the coefficients of the dynamic coupling terms,

e, E, d and D, can also be determined from _{the third test.}

For one speed only (PR - 0.20) the results have been given already in (8].

It was shown there that the term. di and D are small

compared with the other terms of the equations of motion. The relative importance of these terms can be judged with the aid of table 2 where the amplitudes of all the terms of equations (1) are given for the following case:

FR - 0.20 ;

u

- 7 (resonance), z0 - 1 cm ; - 20 TABLE 2 -

5,7

kg

Aw- 5,7

kgm. - 2,2 " _I c

_-5,6

_{C -6,0}

- -0,2 ..Vw 0,1 ' - -0,6 " _{- 0,} g

_--0,3

' G --1,1

However, by estimating the _{experimental errors in d and D}

it could be shown that small errors in the measured values of phase angles and possibly existing small variations of g and G with speed, result in very large variations of d

and 1).

So it appears that the experimental determination of d and D is not very reliable.

However, the absolute magnitudes _{of both d and D remain very}

small and, there2ore, it is assumed here that d. - D - 0.

The values for e and E are given in figure _{5 as a function}

of the circular frequency a and the dimensionless Froude number of the model.

Again the result is sensitive for errors in the phase

measurement but less than in the case of d and D.

Therefore, the phase measurement has carried out

with great care and moreover the measured phase angles

12

-ofe andE.

The results of these tests show some differences from the values determined in earlier tests for the case PR - O.2008

However, at and near

resonanoe (w.'7),

where the

coeffi-cients e and. E may be iniportañtthe agreement is quite

good..

Generally, the coupling terms are small as compared with the other terms as can be seen from table 2.

Their influence on the motions of the model cannot be judged immediately from table 2 as the phase difference between the heaving and pitching motion is very important in this respect as will be shown later.

The exciting forces and moments, which form the right hand sides of the equations of motion (1) are measured on a restrained model in waves. The test arrangement is given

in figure 6. The vertical motion of the model is restrained by a polished rod, which is guid.d by means of linear

rol-ler bearings. This rod is connucted to the carriage by means of a strain gage d.ynamometer which measures the hea-ving forces.

The rod is connected to the model by means of a pivot.

The pitching motions are restrained by a bracket construct-ion, connected to the rod. This bracket is connected to the model by means of a second dynamometer, and the forces

measured here, multiplied by the distance to the pivot give the pitching moment.

The natural frequency of t system model-dynamoxneter is in

the order of magnitude ot - 50, both for heave and

pitch.

In our case the circar frequency of encounter varies

between (U 5 and

'U -

11; consequently only a

negli-gible dynamic magnification of the magnitudes of the exci-ting forces and moment will exist.

The dampin

of the systemis less thanO.1 antherfore,

13

-The measured values of the heaving force amplitude F0, the

amplitude Qf the pitching moment and. the phase

differen-ces 0C and with respect to the wave are given in

figure

₇

as functions of the wave length ratio and

the dimensionless Froude number FR4

IV. The solution of the differential equations and the

compari-son of polutioxxs with measured heaving and pitching

mQtiOfls.

The solution of the differential equations (1) is given in

an elegant form by Korvin-Kroukovaky

(2)4

For tbis. purpose

the equations are written in the following form:

+ b

+ cz + d+ e+

e

Ayi+ By+ C'+ D+

+ Ga = Me

iwt

where

Fe'

and. M0e1

It will be understood that only the real part o± the solu-tion is to be taken.

The solution is of the form:

z=et

and

where =

er

and.

= °

The values of and C' _{and, consequently, the values of}

z0, y,

and

are determined from

the following

expressions:

-

-= and - - P.S where:

P

- aw+ ibw + o

Q

- dw+ iew + g

R - - Dc0L+

IEW + G

S - AWL+

iBv + Q

By substituting the experimental values of the coefficients into the differential equations, they can be solved for the fallowing 20 cases:

Speed PR - 0.15, 0.20, 0.25, 0.30

Wavelength ratio = 0,75, 1.00, 1.25, 1.150, 1.75

Waveheight

2r=1/48L.

The solutions giving the amplitudes of motion and. phases,

are si,nmrized in table ₃ together

with

the corresponding

experimental values.

In order to judge the influence of the coupling terms, the

solution of the

uncoupled equations Is also given. This

impliestbatdxDeE+gG0.

TABLE 3

Comparison of the computed and measured motion ani1ituc1es and phase angles.

Y In degrees Z0 in Cm

Z:6y

in de:ree

Ex-- othu-. ixicou- Ic.- cou- uncou- Ex- cou-

uncou-pen-

pled pled

pen- pled

pled pen- pled pled

ment aotioncnotion ment motion motion ment

_{motioninotLon}

0,75

1.1

1.1

1.2 0.4 _0.5 0.2 - - -1.00 _3.3 2.6 3.0 _{2.3 1.9} 0.8 _{43 37} 44 1,25 3.6 3.2 3.6 2.1 1.9 1.4 62 40 ₉₅ 1.50 2,9 2.8 2.6 2.0 1.8

16

74 ₅₉ 92 1.75 2.2 2.2 2.0 2.0. 2,0 1.9 ₇₉ 71 94 0.75 _{0.7 0.9 0.9} 0.1 _0.3 0.1 - -1.00 2.8 _2.5 3.2 2.6 2.1 0.8 ₆₃ 64 75

°

_1.25

3.8

_3.6

4.0 _2.9

28

_1.7 51 43 90

1.50

3.2

3.4

3.0

26

_2.5

2.1

68

48

98

1.75

2.3

2.7

2.4

2.2

2.1 2.2 71 62 101 0.75 0.4 0.8

0.7

0.2 0.5 0.2 - - -i 1.00 2,4 2.3 2.6 2,2 2.0 0,7 83 70 73

o

1,25 3.5 3.2 4.4 _3.7 3.5 2.0 61 50 ₈₉ 1.50

3.5

3.1

3.5

3.4

3.0

2.5

₅₉

60

102

1,75

2.5

2.4

_2.7

2.5

2.5

25

72

₇₃

₁₀₅

o

0.75 0.2 0.4 0,14. _{0.1 0.2 0.1 - -}

-d

1,00 2.0 2.1 1.9

17

1.6 _0.5 96 80

70

1,25

3.1

1,7

76 61

₇₉

p 1.50 3,6 2.9 4,1 £1.,0 3,2 2,3 58 11.7 95

1.75

27

2.6

31

2.8 2.5 2.4

₇₃

58

01

. 15

-A similar comparison is given in 1 inure 8, where the motion

amplitudes are represented in the dimensionless formrS'%<w

and.

(o p.

is the maximum wave slope)

In the figures 9 and 10 the computed phase-angles

-

and é

are compared. with the correepondin

measured values1

The experimental values of Y and have not been

measured in the fleift Tank, but were kindly supplied by the Netherlands Ship Model Basin.

The following conclusions may be drawn from table 5 and

f&gures

8

9

and 10.

The differences between computed and measured motion

amplitudes and phase

angles are small in the case of the

coupled differential equations.

For pitch the difference is in the order of magnitude of

0.20

-

0.30

_{and for heave: 0.2}

₀ CIII.

Generally the computed values are

somewhat smaller than

the experimental ouea.

The difference in the phase angles is smaller than

10 - 20 degrees as

well for 2) S

as for

'

and

1

seperately.

The comparison shows a different picture when uncoupled

equatio

are considered.

The character of the curves

(see figure 8) is now completely different from the

experimental curves in particular for the heaving motion4

The differences for the pitching motions are small but

they also show a different character of the curves in

some cases.

The error in phase may amount to about 5 degrees.

The solution of the coupled equations shows some small dill

e-rences between experiment and calculation. These differences

are only slightly larger than the experimental errors

It is shown, however, that in most cases the motion s

under-estimated a little, in particular at or near resonano.

-16

and moments using a restrained model in wares, gives too

small values at or near resonance. At resonance the phase of the motions with respect to the wave is causing a bow and

stern immersion which is larger than in the case of a

res-trained model. Consequently larger exciting forces and

moments can be expected because of the ehipform above the

waterline.

For medium wave heights we may conclude that the linear

differential equations according to (1) give a sufficiently accurate description of the heaving and pitching motions of a ship in longitudinal waves.

Some of the coupling terms are necessary to obtain

satis-factory results.

A first

approximation, in

which these coupling terms are

neglected, does not give good results in our case, in parti-cular for heaving.

It is emphasized that our conclusions are based on the test results of one particular model.

In the near future a similar Investigation on two other

shipforma (with blockcoefficlents .70 and .80) will be

completed.

The author is greatly indebted to

Mr. E. J3aas and

Mr. W. Beukelman, who carried out the greater part

of the

17

-References.

[1) St. Denis, M and Pierson W.J.

On the motions of ships in confused seas.

SNAME

_1953.

[2 orvin Kroukovaky By, and Jacobs, W.R.

Pitching and heaving mot$.ons of a ship in regular

waves.

8NAME 1957.

(3) Haskind, M.D.

The hydrodynamic theory of the oscillation f a ship

in waves.

Priklad.aya Mateivatikai Mekhanika, Vol. 10 no.1.1946

[)

Todd, F.H. 0

Some further experiments on single screw merchant ship forms - Series 60.

SNAME 1953. Gerritsma, J.

"Seaworthiness tests with three geometrical similar

shipmodels".

N.SIM.B. Symposium 1957. Wageningen.

Szebehely V.G., Bledsoe, M, D en Stefun,G.P. Scale effects in Seaworthiness.

11th. ATTC, DTMB, September 1956.

(7)

Haskind M.D. en Rieman, 1.8.

A method of determining pitching and heaving cMraote-ristics of a ship.

Bulletin de l'Acaclemie des Sciences de URSS. Classe des Sciences techniques 1911-6. no. 10

(translating Russian - Dutch by Ir. G. Vossers)

[8) Gerritsina, J.

Experimental determination of damping,

added mass and

added mass moment of inertia or

a shipmodel.

List of symbols.

abcdeg

coefficients of the differential

ABC DEG

equations of motion.

P heaving force

PR dimensionless Froud.e number

K moment of inertia of waterplaxe

L length between perpendiculars

M pitching moment

V wave amplitude

so area of waterplane

t time

z vertical displacement of the models'

centre of gravity

o( ,9 phase angles

0< maximum wave slope

specific gravity

A

wavelength

pitch angle

3

vertical displacement of water surface

added mass

added mass moment of inertia

W _{circular frequency}

We natural circular frequency of a system.

-18-TOfl rt 0 r

'1ft.] IJ c)

',t,,,,/

,,,,,,,,,,t/

T:OWI.NG

CARRIAGE

HEAVE POt METER

GUIDE

DOUBLE GUIDES. ON BALL BE*IINGS

a

PITCH poi METE

,,,,///,

,,,,/,,/,,

,OyING,>ARRIAGE,/

,j/ ,jj

HEAVE POEMETER GUIDE

b

GRAVITY DYNAMOMETER GRAVITY DYNAMOMETER

(/

o4)

ITCH POT.METER

////////

15-

Lo-PITCH

%/L=I54 + L i

15-

,

'

HEAVE

)/L126

Xii74

1'

as

0,

'

=o78

PHASE 'LAG'

HEAVE AFTE PITCH

1'

45..

0

' H C -a A . '

ol

.-Fr

02

0:3

PURE HEAVING MOTION

PURE PITCH ING MOTION

c7

Lf

PITCH

Fr.I5

Fr.20---Fr.25

Fr.3O-"--20

E

I,

-I

U

E

F

5-!

':

/

NR

S

-5 (-a.--5ec'. 10

HEAVE

jI'

I .

-8

1

-4

H

II

'-_

.

20-H

0

J

0

5c.isec

II

10

+10

(I)

F

0

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