Wave and structural load experiments for elastic ships, Presented at the 11th Symposium on Naval Hydrodynamics, 1976

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LABORATORIUM VOOR

SCHEEPSCONSTRUCTIES

TECHNISCHE HOGESCHOOL

- DELFT

RAPPORT Nr.

SSL 204

BETREFFENDE:

Wave and structural load experiments

for elastic ships

by

Prof.Dr.Ir. R. Wereldsma & Ir. G. Moeyes

Paper to be presented at the

11th Symposium on Naval Hydrodynamics

University College London - Office of Naval Research,

28 March - 4 April 1976.

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R. Wereldama and G. :1oeyes. Delft University of Technology,

Department of Shipbuilding and Shipping.

WAVE AND STRUCTURAL LOAD EXPERIMENTS FOR ELASTIC SHIPS

SYNOPSIS

Measurements of wave exciting forces and total girder loadings on a large tanker model in head waves are described, with wave length varying from X/L = .085 to 2.0 and speed ranging from F _{.10 to .30.} Wave exciting forces may be predicted with strip theory for wave lengths greater than about half the ship length, but could not be calculated accurately for short waves.

The analysis of total girder loading is based on the "Normal Mode Method". Experimentally _determined par-ticipation factors do qualitatively not agree with predicted curves based on the assumption of a simple

sinusoidal longitudinal force distribution.

1. INTRODUCTION

The recent growth of maritime activities and its accompanying construction of big tankers, fast con-tainer ships, drilling platforms and other units according to new concepts, asks for a rational meth-od of strength analysis. In designing these units experience on equally sized prototypes is not avail-able and extrapolation of 8tendard load calculations is not free from risk. As an example springing of Great Lake carriers, big tankers and long container ships and the resulting higher frequency stress variations and fatigue problems, can be mentioned es a phenomenon which did not occur before. To in-clude frequency and resonance aspects independently in strength analysis the- "normal mode technique" might be used to handle any spatial structure, as has been demonstrated by van Gunsteren in ref. (i) for the analysis of springing of a supertanker. In ref. (2) it is shown how the experiments on models and the construction of models is affected by the application of this "normal mode technique". In this reference a description is given of the possibility to predict the stresses in the struc-ture as a result of the wave loading and of the mass-elastic behaviour of the structure. Also an experimental technique is proposed in ref. (2) in order to arrive at the generalized wave excitation forces.

In this paper the results of preliminary experim-ents with a captive and a free "rigid back bone" model in the Delft Ship Hydromechanics Laboratory are reported. Tests with the restrained model give the wave exciting forces, uhereas tho free model tests supply the generalized forces, e.g. the net loading of the ships' girder not affected by the

deflection of the giik nor by elastic resonance phenomena.

t

n'k and finance limitations only

head waves and the resulting vertìcallo:.

Where: ₌ [s] = =

M1=

tMJ =

=

I1=

been-considered. Measurements are carried

Out

on a large tanker model, which is extremely sensitive to such loads.

2. DISCUSSION OF THE OVERALL STRENGTH PROBLEM OF THE SHIPS GIRDER

When the ships girder is discretized in a sufficient number óf masses and elasticities and when linearity is assumed, the motions due to heave and pitch and vertical bending can be described by:

-[i] {x)

+ EI

ij +

[.i] ()

₌ = {F) - [M] {M} -

[J fi}

- []

_fxj

displacement vector

stiffness matrix of the mechanical system

damping matrix of the mechanical -system

mass matrix of the mechanical system wave excitation forces

matrix of hydrodynamic added mass, due to relative accelerations of the hull matrix of hydrodynamic damping, due to velocities of the hull

matrix of buoyancy forces, due to dis-placements of the hull.

The right hand side of equation (i) consists of all hydrodynamic forces, both exciting and reaction

forces, and is fully determined by the ships' form. The left hand part of equation (i) is closely

cou-nected to the ships' structure, which determines mass anl stiffness distribution and internal

(con-structional) damping. Possible vibratiet L.mpers must also be included in this part.

-During the design process of a ship ruin din'ns ions and body form are mostly established in

an

early atape, before the censtr ....n will be determined

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R. Wereldsma and G. Moeyes 2

and detailed. So the strength analysis method to be applied should preferably contain a one-time deter-mination of the right hand side cf equation (i), after which the total problem could be solved in a loop where the left hand constrct1onal side cf (i) is evaluated. (See ref. (3)). To facilitate this iteration and to pay full attention to frequencies of elastic motions and their related aspects of fatigue stress, a coordinate transformation of (i) to normal modes of the "dry ship" (i.e. the two rigid body motions heave and pitch and two and more node vibration modes) will be applied. This results

in a diagonalisation of the mechanical macsand

stiffness matrix (ref. (Ji)).

In general the damping matrix can not directly be diagonalized, depending upon the lengthwise damping distribution. However, diagonalization is possible when the damping distribution is supposed to be similar to the distribution of mass or stiffness,. whichis ger.eral practice in ship vibration anal-ysis.

The final result becones:

[s)

) +

[C] {I)} + [M] _f ₌

= {r)

_-

_{[M) (} -}

[c)

i] - [s)

{i (2)

where: = vector of normal mode displacements

f rj

= vector of generalized forces.

The left hand part of equation (2) is now diagonal-ized and the ship motion problem can be separated from the ship deflection problem. Also the dynamic amplification of one cf the modes (e.g. the spring-ing phenomena for the 2-noded deflection) can be dealt with separately. For this purpose it is

neces-sary to diagonalize the entire system. Since this diagonalization requires knowledge of' the bydrodyn-.

8mb effects of the hull and the surrounding water it is proposed to make an approach by two steps. The expansion of equation (2) for two rigid body modes and one elastic mode reads as follows:

11

O M

O 2 + + O

O M3

O O C33 I3 s11 o

ol

+ 2 = o o s33

r1

M11 M12 M13 =

r2

-

M21 M22 M23 2

-r3

M31 M32 M33

If the ship is not restrained by bow hawsers, moor-ing lines and the like, and if no additional artif-icial heave or pitch damping devices are fixed, the damping coefficients C11 and C22 are zero. The first step for the solution of the set of equations re-presented by (3) is to determine the wave exciting forces

r} ,

either by calculation or by experiment. The strip-theory (ref. (5), (6)), which is based on

a two-dini'nsional potential calculation combined

with a three-dimensional coordinate transformation, is a proven tool- to calculate the forces due to waves which are not tco short compared to ship length (e.g. AIL > 0.50) (ref. (7)). The experiment-al determination of wave exciting forces, with a restrained segmented model, is described into more detail in the following paragraphs.

The second step in solving equations (3) is to deter-mine the hydrodynamic coefficients represented by the fM] , [C] and

[s]

matrices. These matrices have non-zero elements outside the diagonal, due to bydro-dynamic cross-coupling. The coupling coefficients have a significant influence on rigid body motions so analoguously their influence on elastic modes can not a priori be neglected. On the other hand rigid body displacements are small at the higher frequen-cies involved in elastic mode resonance, which does suspect less important influences of heave end pitch motion effects in the elastic modes. Rydrodynamic coupling between elastic modes depends upon the longitudinal distribution of damping and added mass at higher frequencies, about which no accurate re-cults are known for the three-dimensional case. Calculation of added mass and damping for two-dimen-sional sections is possible, either including free surface effects for a frequency range where signif-icant rigid body motions occur (see ref. (8)), or in the asymptotic case of very high frequencies where surface waves are neglected.

These two-dimensional.values provide a base for a sufficiently accurate prediction of coefficients in

[M] and [C] which influence rigid body motions, but their applicability in the case of' vibration modes becomes questionable because of three-dimensional effects.

The buoyancy matrix

[s]

can be calculated for all modes. A direct experimental determination of hydro-dynamic reaction forces could be done by means of forced oscillation in one of the modes, with a normal vertical oscillator for rigid body motions and a special sectional oscillator for the elastic motions. Hydrodynamic reaction forces may also be obtained indirectly with the free moving model tests

described in the following sections, when exciting

-forces are known from captive model tests, when the buoyancy matrix

[s]

is calculated and when heave and pitch are recorded.

However,these free moving model tests were primarily intended to measure directly the girder loading caused by the waves, including the effect of hull motions, i.e. inertia forces and hydrodynamic re-action forces.

3. DESCRIPTION OF THE SHIP AND THE MODEL AND ITS CONSTRUCTION

The particulars of the ship, a tanker, are given in

Figure 1 and in Table 1.

-Table 1. Particulars of ship. Length between P.P. Breadth moulded Depth moulded Block coefficient Prismatic coefficient L.C.B. in percentage of L Deadweight Horse power r.p.m. 310.0 m 16.q m

2.5 n

0.814 +3. 13 - 21',OOO.00 tOnG 28,000.00 HP 80 rpm + C11 C12 C13 2l 22 23 C31 C32 C33 i1, '2 13 -S11 S12 S13 2l 22 23 S31 S32 S33 'i '2 'p3

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The models have been built to a scale 1:200/3 and.

the test conditions are as follows:

ballast draft (even keel)

12.27

i

displacemeñt ballasted

1h6,000 tons.

Ballast condition has been investigated first

be-cause loads in short waves are more important in

ballast than fully loaded, due to the lower natural

frequency of the two-node vertical vibration. It is

realized that an even keel ballast condition is not

usual for tankers. Even keel, however, has been

chosen to ease the comparison with theory.

The cantive model consists of a stiff beam and 2h

sections connected to the beam. The beam is

support-ed rigidly to the towing carriage of the tank. Each

of the 2h connections of the segments to the beam

consists of a dynamozneter equipped with

semi-con-ductor strain gauges to measure the vertical force

excerted on the section. The total rigidity of' the

system is such that the lowest natural frequency is

high in comparison with the frequencies of interest,

so that elasto- and hydro-dy-namic effects are

ne-gligible. (See Figure 2).

The free model consists of a stiff beam, constructed

as light as possibile. To this beam are connected 12

sections, containlng as much of the required weight

as possible. The 'connections of sections to the beam

consist of a 3-component balance, in order to sense

the vertical loading, the horizontal and the

tor-sional loading (see Figure 3). For the experiments

reported in this paper only the vertical loading is

measured. The entire stiffness of the model is Such

that for the wave frequencies of interest no dynamic

effects will influence the measurement, except the

two possible motions of the model. More details

about the weight distribution and the effects of the

parasitic mass of the beam are dealt with in the

Appendix.

h.

MEASURENTS AND EVALUATION

a.

Generai

A framework of the measured speed and wave length

values is presented in Table 2. The design speed of

the considered tanker is F = 0.15. Higher speeds

are not realistic for the full ship form, but have

been included to have a wider range for studying

speed effects.

For the free model tests F

= .15 and .25 have been

omitted. The shortest wave length was dictated by

the available wave maker and electronic equipment,

the longest wave length by model and tank

dimen-sions. For some free model tests additional

wave

lengths have been investigated, besides those

in-dicated in Table 2, in order to determine the

wave

length dependency of the measured quantities more

accurately.

Some of the low speed, long wave runs suffered from

wall influence. Boundaries calculated according to

ref. (9) are indicated in Table 2.

The wave height has been chosen so that a

reason-able signal could be recorded in connection with

the sigrial-noisu ratio, and that linearity was

as-sured (A/n

about 60).

b.

jvo

MoOd

Tests

For the measurements in genera] it

_{can be stated}

that the output of the pick-ups coincides with the

stiffness matrix multiplied with the displacement

vector. Referrf'-- to formula (3) it can be stated

that due to the stiff structure and the high

sen-sitivity of the pick-ups:

This is allowed because the mass-elastic and

hydro-dynamic effects are negligible and substantial

dis-placements-will not take place.

Returning to the original coordinates {xl ve find:

(5)

so that for the captive model test each section

pick-up has a signal u equal to the hydrodynamic

force exerted on the section, and we obtain 2h

con-centrated forces representing the wave load.

c.

Free Model Tests

-Referring to formula (3) it can now be stated that

S11 and S22 are equal to zero (free model) so that:

Oi

000

02

000

03 00S3

_'p3

r1

M1 O O

r2

-

O

M22 O

+

r3

,O

0M33

M12 M12 M13

C11 C12 C23

-

M21 M22 M23 2

-

C21 C22 c23

M31 M32 M33

c31 a32 C33

'i,1

s11 s12 s23

p1 -

-

s21 s22 s23

j,2

(6)

S31S32 S33

_p3

From this it follows for the case the model is

suf-ficiently stiff so that deflection modes such as

and higher are negligible:

1!11

IM11

i

J1l

j.'

2I

-

IO

M2

-12J

[r3J

[o

o 01

02

03

= S11 q)1

S22

S33

'p3

r1

r2

r3

(h)

01 02

03 = 40

I

(o

'2

-L S th = I

[M11

M121

ii1

IC11

c121

IM MI

C

cj

+ 21 M22

j'2

-

[c21 c22

_12I

₊

L 31

.52j 31 32

R. Wereldsina and G. Moeyes

₃

s12

'p

-

21

22

q2

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R. Wereidsina and G. Moeyes 14

order to get a Fresentation which can physically better be understooth

The distribution of wave exciting forces has also been calculated with the strip-theory as developed by errisma et al. (ref. (5)), however, due to lack of time, without adapting it to the numerical de-mands put forward by the short waves and without ta]ng into account the finite section length as mentioned above. As an example the results of these preliminary calculations are shown for

_{A/L = 0.75}

and .1.0, where the nentioned effects do not yet sig-nificantly count, in Figures 13 and 114.

The agreement between measurement and theory is good except for the higher speeds, as could be expected with the excessive wave formation along the full tanker body. Another deviation of theory and meas-urements is apparent at the prallel mid-body, where strip theory predicts a constant lengthwise distrib-ution, but where experiments show a decay in amplit-ude due to three-dimensional effects (ref. (11), (12)).

For.shorter wave lengths the disagreement between preliminary calculated and experimental results was so large that presentation was not considered use-ful. The measured forces do not show the consider-able decay in amplitude from fore to aft which is sometimes expected in literature.The force peak at the afterbody might cause a significant increase of springing stresses.

An apparent feature of the exciting force distribu-tion at shorter wave lengths and higher speeds is the rapid increase in phases from fore to aft, com-bined with a fluctuating amplitude. This typical behaviour might result from diffraction terms in the wave force expression, containing wave circular frequency, speed and sectional damping and added mass.

b. Free Model Test Results

Although the experiments have originally been initi-ated to investigate the mechanics of springing, i.e. a dynamic phenomenon caused by the time vari-able short wave loadings, interesting results have been obtained also for larger waves, this because the springing problem can be seen as a special case of the general strength problem of ships. The re-quired method of analysis for springing covers also the regular low frequency analysis in a generalized version. Therefore the experiments are also extend-ed to. longer waves and static phenomena.

Still Water Test Results

Figure 17 shows for 3 speeds of the model the stat-ic load distribution caused by the wave system gen-erated by the model. The model has adjusted itself for heave and pitch so that only higher mode load distributions could be developed.

It can be concluded that for a ship model speed of

1.35

m/sec a static sagging loading is generated which is comparable to the standard wave sagging/ hogging loading. From the measurements appears that this static load distribution is not essentially affected by waves, so that the superposition prin-ciple may be applied.

Wave Test Results in Ship Coordinates In Figures 18, 19 and

20

the dynamic part of the girtr lnìding due to incoming headwaves Is shown 01 = o = r -

M111

-- '12'2 --

-+

c122 -

-

(8)

02 = O = r2 - I4222 - M211 -

-

-+ C22l2 -

- S22,2

.( 9)

03 =

r3 +

-

M311 - M322 -

c?J1

-+ C32,i2 -

S32J1

-

32'2

(io)

The outputs 01 and 02 are zero, because the general-ized wave forces are fully consumed to generate the ship motions. Then, (8) and (9) are the equations of motion describing the rigid body motions heave and pitch.

Output 03 is the generalized version cf the girder loading, composed of the wave excitation

r3

and corrected for the hydrodynamic reaction forces, added mass, damping and buoyancy, caused by the ship motions *1 and

Outputs 0 are similar generalized loads,

how-'5,

ever now referring to higher node displacement func-tions (for the sake of Simplicity not shown in the formulae), including hydrodynamic effects originat-ing from the rigid body motions ,J,. and 'p2 too, but iot from other elastic modes.

d. Instrumentation Figures 1 _and

5 show the instrumentation used for the measurements. The method of signal filtering, based on the frequency of encounter, is for each

case (captive versus free model) adapted to the par-ticular requirements. For the captive model tests multiplication is carried out through rotating

re-solvers, driven by a servo-mechanism following the wave zero-crossings (ref. (io)). For the free model tests the demodulated signals are weighted and sum-med before being multiplied with the references,

i.e. before being Fourier-snal,yzed on a wave signal base. Multiplication takes place with electronic multipliers. In both cases average velues are ob-tained by integration and time division over a chosen number of wave periods.

5. MEASURE4ENT RESULTS

a. Captive Model Test Results

Figures 7 through 16 show amplitude and phase of the sectional wave exciting forces, each for a partic-ular wave length and the whole range of speeds. Pm-plitudes have been divided by wave amplitude and section length to make them better comparable to each other and to calculations. In this respect it must be realized that most calculations provide a continuous lengthwise distribution of the liisit force values on infinitely short sections. The for-ces on sections with a finite length must be ob-tained by vectorial integration of the former ones. When the wave length approaches more and more the section length the deviation between the continuous basic force distribution and stepped sectional for-ces as shown in Figures 7 - 16 grows. Phases have not been determined with respect to the wave at the ships' center of gravity, as is usual, but at the

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R. Werei.oua azi U. Moeyes

₅

by means of an instantaneous distribution and an

es-timated envelope of the amplitudes of the variations.

For those experiments where the ship moticns are not

negligible i.e. in the range of about 0.5

A/L < 2.0

a small correction is necessary the to the

accelera-tion forces of the beam. (See Appendix). This

correc-tion has not been made, but the general picture will

not be affected by this correction.

An example of the total loading in ship coordinates

i.e. the statio and the dynamic wave loading, is

shown in Figure 21. Although it is interesting to

have this complete picture of the girder loading for

head

waves, not very much can be done with it,

unless a rational method of analysis is applied.

This method is found in the "normal mode method" as

described in ref. (2) and a further evaluation of

the measurements is made according to this method.

Wave Test Results in Generalized Coordinates

As has been outlined in ref. (2) a special modal

in-tegíation of the 12 section forces results in the

participation of the wave excitation in the various

elastic modes, or in the excitation forces for the

two- and multinode deflection andes, coinciding with

the vibration modes.

Together with the mass-spring response function of a

particular mode an "Elastic Response Amplitude Oper-.

ator" can be formulated, similarly as the regular

"Response Amplitude Operator" used for ship motions.

The modal integration consists of the summation of

the mass- and deflection-weighted section forces,

measured during the experiments (see Figure 5).

Results of the experiments are shown in Figures 22,

23 and 214 for the two-, three- and four-noded

de-flections for the various ship speeds. These figures

show that the Bystematic pattern, expected when a

sinusoidal loading moves along the ship girder, is

not realistic. The hydrodynazaic loading caused by

waves and ship motions obviously disturbs the

deter-ministic picture to be expected.

6.

FINAL REMARKS

Further theoretical and experimental study of

wave exciting forces in the short wave range is

necessary because of the disagreement between

results 0±' existing calculation procedures and

the experiments described in this paper. This

may be greatly influenced by diffraction terms

in the wave exciting forces.

Added mass and damping matrices should be

in-vestigated more thoroughly, especially in the

three-dimensional higher frequency case.

e)

The static girder loading caused by the ship

generated wave system amounts to the order of

the standard wave sagging-hogging loading.

d)

The originally expected picture (see ref. (2))

of the modal participation of the wave loading

as a function of wave length is in strong

dis-agreement with the measured functions. As far

as wall effects were not encountered during the

experiments, this nisy be due to the difference

between the assumption of simple Froude-ICrilof f

exciting forces for the prediction (2) and the

real forces (see also remark a). It may be

con--

clu'id

Nöre study is necessary in the range

of shorter waves to understand the total

physic-al mechanism.

For a complete picture of the loading on ships

it is necessary to investigate oblique waves

for the introduction of torsion and horizontal

bending, which is essential for the analysis of

container and open ships.

A further study is necessary, either

theoretic-al or experimenttheoretic-al, to introduce statistics in

the girder-load-pro

aused by irregular

waves.

ACKNOWLEDGEENT

Complicated experiments as have been described in

this paper can not be carried out without the

sup-port and team-work of many people.

The authors are indebted to the technical and

sci-entific staff of the Delft Ship Hydromechanics

Laboratory and Delft Ship Structures Laboratory for

their enthousiastic and creative cooperation.

REFERENCES

van Gunsteren, F.F.; Springing; wave induced

ship vibrations; Intern. Shipb. Progress; 1910;

Vol. 19, No. 195, Nov.

Wereldsma, R.; Normal mode approach for ship

strength experiments, a proposal; Symp.'The

dynamics of marine vehicles and structures in

waves'; London, April 19714.

Wereldsma, R.; Modell-Versuchsprobleme bei

elast jachen, seegehenden Konstruktionen;

Sehiffstechnik; 1975; Bd. 22; pp. 29-146.

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_{Bishop, R.E.D., Eatock Taylor, R. and Jackson,}

K.L.; On the structural dynamics of ship hulls

in waves; Trans. R.I.N.A.; 1973.

Gerritsma, J. and Beukelman, W.; Analysis of

the modified strip theory for the calculation

of ship motions and wave bending moments; Neth.

Ship Rea. Center TNO; 1967; Report No. 96 S,,

June.

-Salvesen, N., Tuck, E.0. and Faltinsen, 0.;

Ship notions and sea loads; Trans. S.N.A.M.E.;

1970.

(T)

Waab, R. and Vink, J.H.; Wave induced motions

and loads on ships in oblique waves; Neth. Ship

Res. Center TNO; 19714; Report No. 193 S, May.

Vugts, J.H.; The hydrodynanic forces and ship

motions in waves; Doctor's thesis, Delft Univ.

of Technology; 1970.

lianaoka, T.; On the velocity potential in

Michell's system and the configuration of wave

ridges due to a moving ship; Japan Soc. of Nay.

Architects; 1953.

(lo) Buitenhek, M.; Phase-locked loop servo-system;

Delft Ship Hydromech, Labor.; to be published.

(ii) Grim, 0.; Die Deformation eines regelmässigen,

in Lingsrichtung laufenden Seegang durch ein

fahrendes Schiff; Schiffstechnik; 1962; Vol. 9,

No. 146.

(12) Mamo, Il.; On the wave pessure acting on the

sui: -.

L'iìlcr -'p fi -ji h.»ad seas;

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R. Wereldsma and G. Moeyes

APPENDIX

Construction of Model

Measurement of Structural Loadin

Corrections for Parasitic Mass of the Beam. Ideal-isation

The Bhip model for the free model tests is composed of a light stiff beam, representing the bending stiffness of the model. Hooked to the beam are the model sections. These sections represent the shape of the ship, and do contain all the mass of the model (see Figure A-1). The connection of the sec-tions to the beam is equiped with strain gauges so that three force components (i.e. vertical force, horizontal force and torsional moment) can be meas-ured. The stiffness of the entire structure (inclu-ding beam, pick-ups and sections) is such that the lowest natural frequency is about 23 c.p.s., which is high in comparison with the highest frequency of interest, the frequency of wave encounter, about 5 c.p.a. The Sections of the model are as a matter of fact large pressure pick-ups, measuring the inte-grated pressure over the surface of the section. The measurements can be analyzed according t a technique of generalized coordinates. Because two rigid body motions are performed during the measure-ment of this free model it is necessary to analyze the dynamics of the system through normal modes which are in this case assumed. These modes are given in Figure A-2, together with their mass-Spring representation.

Originally the break-down in normal modes consists of a regular decomposition of a beam with mixed mass and elasticity into its normal modes. Because the sections are elastically connected to the beam, and this elasticity is applied for the measurement, it is necessary to describe the elastic deflections of the sections relative to the beam in order to anal-yze the output of the measuring system. Therefore

each original mode has been split into two new modes. Besides the rigid body heave- and pitch-motion we have now introduced an elastic motion governing the elastic deformation of the measuring springs in a modal fashion. Each of the elastic modes1 i.e.

sag-ging-hogging and the multiple node deflections, are now split into two new modes, composed of "in phase motion distribution" and "opposite motion distribu-tions". Since the model consists of 12 sections we need for a complete description 2i elastic modes and

1 modes related to the rigid body motions, in total

28 degrees of freedom.

Fora description of the behaviour of the various modes, in connection with the signal output of the pick-ups, two types of systems as shown in Figure A-3 have to be analyzed. The output of the instrum-ent equals x1 - X2.

In the case of Figure A-3a referring to pitch and heave modes, the transfer of r1 to C1 (x1 - x2) equals

C1 (x1 - x2)

r1 M

12 22

M M2

-a -

C1

(Al)

For the case M2 - O (mass of beam of the model tends to zero) the transfer equals:

C1 (z, - z2) r1

-This means that all the excitation r1 is used for

the acceleration of the mass of the sins.

For the case M2 O but C1 is sufficiently high (which means a stiff spring, related to a suffici-ently high natural frequency, which is indeed the case for our model) the following transfer function is obtained:

C1 (z1 - x2) M2 r1

M1+M2

independent of the frequency,

M 2 = M . modal acce].-or; C1 (x1 - z2) = r1 M1 + M2 2 eration (An) This part of r1 is an inertia force, that has not been subtracted from the hydrodynamic forces (as should have been the case for M2 o). This distrib-uted parasitic force is affecting the output of elastic modé excitation, and, if a substantial error is introduced, corrections need to be applied. The second system of importance is that of Figure A-3b. This system is involved in all the elastic mode measurements as well as the rigid body modes of the captive model. When the stiffness of the springs involved is sufficiently high (i.e. natural frequencies high in comparison with the maximum frequency of interest) a quasi-static approach can be made and it follows:

C1 (x1 - x2) = r1 (A5)

so that in all cases a measuring output equal to the generalized force is obtained. In general the fol-lowing conclusions can be made:

For the Captive Model

The output of the instruments equals the generalized forces for all the modes, and a reconstruction to the distributed loading results in a proper measure-ment of the load distribution.

For the Free

For the case the beam is indeed massless the output of the force pick-ups represents the inertia cor-rected ships' girder loading, composed of the hydro-dynamic wave loading, the hydrodynamio loading due to the ship motions and the inertia forces. For the practical case that the beam also has a mass, corrections have to be introduced because part of the forces is necessary to accelerate the mass of the beam, and is therefore not included in

the correction for the inertia forces.

For the reconstruction of the distributed loading it is in principal necessary to apply an inertia correction, caused by the mass of '

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R. Wereldsma and G. Moeyes 7

Force Distributions to be Apied for the Correction

in the Case of the Free Model

The weight distribution of beam, sections and addit-ionaj. equipment, necessary for the guidance of the model under the carriage, is shown in Figure A-14. The correction to be applied is evaluated in Figure A-5. The distributed forces shown in this figure

-have to be decomposed in generalized forces opera-ting in the applied elastic modes.

The distributed forces are as a matter of fact the result of beam acceleration for the rigid body modes (heave and pitch) and need therefore be applied as corrections. The magnitude of this correction de--pends on two -factors, i.e. the magnitude of the

ac-celeration during the tests and the participation of these force distributions in the modes of interest. Since the area of interest is in the short waves, the ship motions and the accelerations will be small. Also the participation in the 2-noded elastic mode which is the most important for ship strength anal-ysis will be small because of the almost constant distribution. For the evaluation of the measurements it is assumed that, for the time being) the correc-tions are negligible.

(9)

R. Wereldsma and G. Moeyea

Table 2.

Tested wave lengths and steeds.

C = carried Out with captive model F = carried out with free model

not tested because of equipment restrictions - boundary of region with wall influence,

according to ref. (9). AIL F n .10 .15 .20 _.25 .30 V in n/sec. 0.68 1.01 1.35 1.69 2.03

(10)

R. Wereldsma and G. )4oeyeB 9 LiST OF FIGUIIES Fig. 1: Fig. 2: Fig. 3: Fig. L: Fig. 5: Fig.

6:

Fig. 7: Fig. 8: Fig. 9: Fig. 10: Fig. 11: Fig. 12: Fig. 13: Fig. 1: Fig. 15: Fig. 16: Fig. 17: Fig. 18: Fig. 19: Fig. 20: Fig. 21: Fig. 22: Fig. 23: Fig. 21e: Fig. Al: Fig. A2: Fig. A3: Fig. AIS; Fig. A5:

Body plan of investigated ship. Captive model test arrangement. Free model test arrangement.

Instrumentation for the captive model test. Instrumentation for the free model test. Sectional wave exciting forces, ballast: A/L = .085.

Sectional wave exciting forces, ballast: A/L = .120.

Sectional wave exciting forces, ballast: AIL = .215.

Sectional wave exciting forces, ballast: AIL = 1.00.

Sectional wave exciting forces, ballast: A/L 1.25.

Sectional wave exciting forces, ballast: A/L = 1.50.

Static girder loading due to ship gener-ated wave system.

Sectionalized girder loading per unit wave height at a forward speed of 0.68 rn/sec.

(force/wave height).

Combined loading due to the standard wave loading and the ship generated wave load-ing.

Generalized forces per unit wave height for various wave lengths at a forward model speed of 0.68 rn/sec.

Generalized forces per unit wave height for various wave lengths at a forward model speed of 1.35 rn/sec.

Generali zed forces per unit wave height for various wave lengths at a forward model speed of 2.00 rn/sec.

Photograph of free model with hoisting rig.

Modal representation of the sectionalized free model, takiñg into account the elas-ticity of the force pick-ups.

Generalized representation of the measure-ment of:

-the rigid body modes, and the elastic modes.

Mass distribution of the ship, the model, the beam with appendages.

Distributed forces generated by the para-sitic mass of the beam, and the rigid body motions of the free model tests. These distributions are the basis for the cor-rections to apPly on the measured general-ized girder loading.

(11)

FiG.1 BODY

(12)

FREE MODEL TEST

Fig. 3.

Free model test arrangement.

LN]lD3] I MII NJ

CAPTIVE MODEL TEST

(13)

MULTIPUERS

STRAIN GAGE

INSTRUMENTS

FORCE PICK UP's

WITH SEMICONDUCTOR

STRAI NGAGES

AV E RA GE R S

RIGID BEAM

2 MODEL SECTIONS

Fig.

.

Instrumentation for the captive model test.

Ç2L AVERAGED IN PHASE

iFORCE SIGNALS

f

2 AVERAGED QUADRATURE

i. FORCE SIGNALS

IN PHASE

QUADRATURE

WAVE SYNCHRONIZED

.fEFERENCE GENERATOR,

I

¡INCOMING

-

1WAVES

WAVE

SIGNAL

(14)

12 STRAINGAGE

SIGNALS

()5

MODAL

WEIGH TI NG

OF 12

SIGNALS

43 12 STRAINGAGE

INSTRUMENTS s'

12 FORCE.

I

PICK UP'S

12 WEIGHTED

STRAINGAGE

SIGNALS

RIGID LIGHT BEAM

NNNN

12 MODEL SECTIONS

SUMMATION OF

12 WEIGHTED

SIGNALS

Fi& 5.

Instrumentation for the free model test..

MULTIPLI ERS

(n

w

IN PHASE AND

QUADRATURE

COMPONENTS

OF GENERALIZED

SHIPSGJR DER

LOADING

IN PHASE REFERENCE

QUADRATURE REFERENCE

WAVE SYNCHRONIZED

REFERENCE GENERATOR

INCOMING WAVE,.

WAVE PROBE

(15)

200 o

=-F =.10

F =15

F =20

F =.25

=

F =30

200 -360

O

F'

L

FIG. 6

: SECTIONAL WAVE

FORCES: BALLAST

AIL = 0.85

200 o

-360

(16)

200 o

200

20 o

o

200

O

'J

JHIH

cL

-._l-_

-

200'-F'

L

n

F =.15

F =20

F =.25

F =.30

- 360

-- 360--

360-- 360

360

(17)

200 o

200

200 o

200

O O

J-J-1

-F'

L

F =10

F =.15

F =.20

F =.25

F =.30

-- -

360--

360-

_- 360_-

360-FIG. 8

: SECTIONAL WAVE

FORCES: BALLAST

X/L =160

(18)

200

20 o

o

200

O

-

J-F =10

F =15

F =20

F =.25

F =.30

Ii'

ç-o

-360

O

-360

O

-360

o

-360

FIG. 9

: SECTIONAL WAVE. FORCES: BALLAST

AIL = .215

-360

(19)

F'

L

F =.10

O

F =15

F =.20

F =.25

F =30

360

O

-360

O

-360

O

-360

o

-360

FIG. 10 :SECTIONAI WAVE

FORCES: BALLAST

X/L=.28

400 o

-J

400

O

400 o

(20)

400 F'

L

F =10

F =15

F =20

F =.25

F =.30

-360'-

-

-360--360

FIG.11

:SECTIONAL WAVE

FORCES: BALLAST

X/I =375

W.'.

O---

---WALL

INFL

ÂCC.

TO(9

-

r-

f-=

rTJ

J---_J_

-¿O ¿O

40 o

600

(21)

400 ¿00

0

400 o

400 - WALL INFL.

ACC. TO (9)

F'

F =20

F =.25

O

360

O O

360 FIG. 12

: SECTIONAL

WAVE

FORCES: BALLAST

X/L=.55

F =.10

W.'.

O

360 F =.15

O

360

360 F =30

o

400 o

(22)

40 40(

40

400 F'

aL

FIG. 13 : SECTIONAL

WAVE

FORCES: BALLAST

AIL .75

F =.10

F =15

F =20

F =25

F =30

O

-360

O

-360

O

I

'J

J i I i

i

wL

- STRIP THEORY

WALL INFL.

ACC. TO (9)

i

'.:

L

WI.

0

400

O

-360

o

-360

O

tH

-360

(23)

40

400 kW

F'

L

FIG. 14

: SECTIONAL WAVE

FORCES: BALLAST

X/L = 1.00

F =.10

F =.25

k

O O O

-360

O

360 THEORY

ACC.TO( 9)

- STRIP

-

WALL INFL.

L

WI-.

WL.

---36

F =15

-360

F =20

-360

F =.30

-0

400

O

400

O

400

O

(24)

aL

F =.10

F =.15

F =20

F =.25

F =30

360

360 FIG. 15

: SECTIONAL WAVE

FORCES: BALLAST

X/L = 1.25

NO ACCURATE EXP

(0eV

W.'.

-NO ACCURATE EXP

1v_.27

-WALL

INFLUENCE

ACC TO(9)

k-

--

-

-=

-- ----

f----1

_j

ç

i

40 o

40

40 o

40

O

400

360

(25)

40

400

40 4ti0

F'

L

FIG. 16

: SECTIONAL WAVE

FORCES: BALLAST

X/L

1.50 F =.1O

O O

o

-360

P-.

_{NO ACCURATE EXP}

wv_.25

--WALL INFLUENCE

ACC.TO (9)

-HW.I

-jJ

F-

--

a

--

-NO ACCURATE

WeV

EXP

-g

-.25

-

_WI.

--

-w.'.

-360

F =.15

-360

F =.20

-360

F =.25

-360

F =.30

o

400 o

(26)

TON

1200-300

O-

50-1500-

50-M

FWD

1<

FULL SiZE

Fig. 17.

Static girder loading due to ship generated wave system..

i

MODEL

rn/sec.

L

V=2,OOm/sec.

V = 1,35

V066

!

Ï

O'.

-e--FULL SIZE

V=5,55

V=11.02

rn/sec.

V=1633m/sec

_\

"t-/

I

i.

DEL

\

I

(27)

I

1.

I

H

I

i

I

1

J

I

-- fd._+_4.

i.

r

I

_I

.1

I. I...

i

.1

I

-I

I

i ..l

I

--

+---I---±----+----i

I

I. I

i

I

P

AMPL.OF 501/rn

(FULL SIZE)

-t

4--0415

¿g

-

I4

1.__--

VP"

J

X/L

tOO

075 0,28

Fig, 18.

Sectionalized girder loading per unit wave height

at a forward speed of 0.68 rn/sec. (force/wave height).

(28)

J..

k

a

-iN cm

MODEL)

AMPI. OF 50 T/m

(FULL SIZE)

I

I I ¡

i:

-A

--

-(I

A/L

= 2,0

1,5

1.0

0.75 Fig. 19.

Sectionalized girder loading per unit wave height

at a forward speed of 1.35 rn/sec. (force/wave height).

"k--.---,

(29)

I

s

AFT.

MODEL SPEED 2,00 rn/sec.

FWD.

T

Fig. 20.

Sectionalized girder loading per unit 7ave height

at a forward speed of 2.00 in/sec. (force/wave height).

0.5

0.6

0.28

(30)

VARIABLE LOADINQ DUE TO

INCOMING WAVE À/L1,

WAVE HEIGHT 14cm.STANDARD WAVE

STATÌC LOADING DUE TO

SHIPGENERATED WAVE

AT FORWARD MODEL

SPEED OF 1,00 rn/sec.

Fç1 21

-

/

.

CALIBRATION

I

=1200 T/SECTION.(FtJLL SIZE)

Fig. 21.

Combined loading due to the standard wave loading

and the ship generated wave loading.

-.

(31)

-20

T

15 w

_N

U)

I

J

u-I I

t,

'V.

3-NODED

PARTICI PATION

5 cJ

û

o

z

-10

I'

Fq

t

¿

r

4

1.

4-NODED

PARTICIPATION

o

MODEL SPEED 0,68 rn/sec.

2

3

4

5

6

7 $'ig. 22.

Generalized forces per unit wave height for various

wave lengths at a forward model speed of

_o.68

rn/sec.

4U

N/cm

30

10 2 NODED

PARTICIPATION

t

2 o-

₄₀

i

5-

₃₀

i

0-

₂₀

5-

_lo

15 _{- 30}

10 ₂₀

4--.--,-e t

I. L'

- - 0

_-e-

(32)

e-s-f

20

15 MODEL SPEED 1,35 m,5

J

/ /

aNODED

PARTICIPATION

'f

V.--G'

L

I ,

2NODED

PARTICIPATION

¿

I

q

I

I ' i,

I. j

tf

ft

-LO

30

i-I-J

10

i'

q

3NODED

PARTICIPATION

¿

II

I,'

I I ¡ 'I '

h

I ,'

r

/

r'

I I

\i

I

.-___

f

S. I?

1 5 30

1 0

₂₀

A

p?

a

II

? it

-i' t

4

¿ 1 I

-

10

' / I

2

5

6

L,

Fig. 23.

Generalized forces per unit wave height for various

wave lengths at a forward model speed of' 1.35 rn/sec.

(33)

t

20 lo

5 w

r..4

u)

-J

Li.

25-

20-i

10

5 N/cm

40

30

20 lo

-J

w,

û

o

-

10

' I

'4

4-NODED

q,

_{PARTICIPATION}

t t

I

-G.

f

»0,,

-o

MODEL SPEED ZOO m/se

0.

4

5 _L,'

-s

Fig. 2I.

Generalized forces per unit wave height for various

wave lengths at a forward model speed

01 2.00 rn/see;

I

_e

2-NODED

iPARTICIPATION

I

t

I

X

»50

30 -20

-

10 I.'

f'

_'1

'i

3-NODED

PARTICIPATION

i

.1

¿

I 1 t J

7

t I t

_/

s

/

1 i

5-

₃₀

I I I I I

i

(34)

'V,

MODE

CAPTIVE

MODEL

FREE

MODEL

m,EI

BEAM

WITHOUT

BEAM

WITH

BEAM

WITHOUT

BEAM

WITH

M'.

.

M3

Ei

..M

NUMBER

F1IF21

IF3

MASS

I

_¶

Il

I]

HEAVE.

i

MOTION

F

i

a________

_Ir1

W

_

I

PITCH

L]

___

MOTION

b

I I

tr2

î

i

tv.

fr2

_fr2

.

SAGGING-[*1

HOGGN6

DEFLECTION

i

-

L J

C

___J._-

tr

MULTINODE

DEFLECTION

L i

I I

MULTINODE

Rq

MULTINODE

i

:

DEFLECTION

ri

i'

Ii

DEFLECTION

_tVR

_1m

_ÌTR

(35)

b)

p..

Fig. A3.

Generalized representation of the measurement of:

the rigid body. modes, and

the elastic modes.

MASS OF BEAM

MODEI(HEAVE)

FORCE DISTRIBUTION DUE

TO ACC. OF MODE I

MODE IL (PITCH)

FORCE DISTRIBUTION DUE

10 ACC. OF MODE 11

un u u u

. nuiiu,uu.d Illiliftuu u'

IIIII1II11!rnhutulID'.ulp,

!!.!.1.11oiiiIIIñOi

Fig. A5. Distributed forces generated by the parasitic mass of the

beam,

and the rigid body motions of the free model tests. These

dis--tributions are the basis for.the/crections to apply on the

measured generalized girder

6ading.

(36)

KG

40

30

20

10 Fig. A4.

Mass distribution of the ship, the model, the beam with appendages.

PI9,Á 4

r

MASS

DISTRIBUTION OF MODEL

I MASS DISTRIBUTION

OF FULL SIZE SHIP

MASS DISTRIBUTION OF

! 1BM AND

APPENDAGES

ts

s e e elsli

e