Normal mode approach for ship strength experiments, a proposal, International Symposium on the dynamics of marine vehicles and structures in waves

LABORATORIUM VOOR

SCH EEPSCONSTRUCTI ES

TECHNISCHE HOGESCHOOL - DELFT

RAPPORT Nr

SSL 179

BETREFFENDE:

Normal mode approach for ship strength expeiments, a proposal.

SHIP STRUCTURES LABORATORY Deift University of Technology, Mekeiweg 2, Delft,

The Netherlands.

NORMAL MODE APPROACH FOR SHIP STRENGTH

EXPERIMENTS, A PROPOSAL

by

R. Wereldsma.

Paper to be presented at the International Symposium on the Dynamics of Marine Vehicles and Structures in Waves, London l97.

Content: Summary. List of symbols. Introduction. Elastic model. Stiff model.

L) Similarity of model tests on structural loading and

seakeeping tests.

Proposal for model tests based on Normal Mode Technique.

5.1. Realisation of the rigid model and the instruments. 5.2. Realisation of the elastic model.

Evaluation of the measurements.

Conclusive remarks.

List of references.

List of figures.

-2-List of symbols.

EI bending stiffness of beam.

F force on nth segment. n

F(x) distributed force.

transfer function.

length of beam.

m mass per unit of length.

RAO Response Amplitude Operator.

r participation factor (corresponds to the generalized force).

A length scale.

deflection pattern of z + i noded mode as a function of X.

w encounter frequency of the waves.

3

-Summary. _e

The rapid increase in size and/or speed of ships, the fundamental change in

structural behaviour of open containerships and multiple hull vessels arid the

application of structures other than ships call for a fundamental method for

strength analysis. The "normal mode method" has been adopted for this purpose

and an experimental technique has been outlined as a possibility to meet the

necessary completeness of the strength analysis.

Beside the regular "quasi-steady" strength calculations, normally accounted

for by the "standard wave technique" the method automatically includes effects

of a structural dynamic character such as whipping or springing, or effects

of the structural flexibility which tends to reduce the bending of the ships

girder. Not only vertical bending but also torsional-horizontal deformations

are taken into consideration. Two methods have been discussed, i.e.:

The rigid model experiment, followed by an analysis based on the

structural behaviour estimated from drawings.

The elastic model experiment, that results immediately in sectional

forces and moments.

For both experiments the similarity with ship motion experiments has been

-5-1) Introduction.

For many years experiments for the determination of the girder loading and

the dimensions of the scantlings have been carried out in seakeeping basins,

where the ship model was provided with a bending moment pick up at the

loca-tion of interest (midships) /1/. )

It was not necessary to pay attention to the elasticity 'of the ships hull.

The models were made simply of wood and the installed pick up was designed

for a proper sensitivity. The flexibility of the pick up and the natural

frequency of the model-pick up combination was not of any interest because

the excitation frequencies (frequency of encounter of the waves) were

sufficiently low. (See fig. 1).

With the increased size and/or speed of the ships and the increased f

lex-ibility due to more flexible constructions (open containerships) or

applica-tion of high tensile steel (reduced moment of inertia) new phenomena as wave

excited vibrations (2-noded vertical or one-noded torsional-horizontal) known

as springing or whipping have been observed /2/, /3/,

/t/.

This critical vibration affects the regular strength considerations and it is necessary to

obtain insight in the mechanism of these phenomena.

For a very large ship we also have the effect of its large deflections that

might affect the bending moments due to a sea loading and makes the results

of the conventional strength analysis questionable.

The problem to be solved is similar to that of irregular spatially loaded

3-dimensional elastic structures, able to vibrate in various natural modes.

This problem is also encountered in civil engineering where tall buildings

subject to wind forces or earth quakes, large bridges subject to irregular

traffic loads etc. need to be analysed. A powerful method to attack these

problems may be found in the normal mode method /5/, /6/. By this method,

the dynamics of the structure is broken down into a series of independent

systems, the sum of which represents the dynamics of the entire structureS

-6-again. The independent systems are represented by the elgenfunctions, each

having ari eigenfrequency, being the result of the solüt ion of the system of

equations describing the dynamics of the original structure.

Also for ships it is possible to apply this method and to analyse the final

girder loading caused by the waves and acceleration forces, including the.

static and dynamic response /6/.

In this paper an outline is given of a testing technique for ships models,

based on the normal mode method, taking into account the quasi-steady

sagging-hogging loading, the dynamics due to springing', the hull flexibility

and the irregularity of the waves.

Two approaches will be discussed. The first realisation is the elastic model,

where the experimental results can be directly extrapolated to full size

values. A second technique results in a stiff model, not able to respond

dynamically, but now combined with an additional calculation of the final

dynamic response of the hull. This last possibility is considered to be more

practical when these studies take place in the design stage.

2) Elastic Model.

For the design of the model, where beside the geometrical and hydrodynatnic

requirements, also the structural dynamics are reflected, it is necessary

to build a model with a prescribed flexibility so that the ratio of the natural

frequencies and the frequency of wave encounter are the same for the full size

structure and the model. Since the hydrodynamics require Froude-law scaling,

which means that the frequency of wave encounter increases with IX (X length

scale) for the model, the natural frequencies of the girder have to increase

-

-7-The 2-noded natural frequency of a vertical hull vibration is proportional to

tIRI

the factor

V where m equals mass per unit of length.

Now holds:

EI

T7modei

(len9th)f il size

EI

_-

_8t12)model

ifuli

size

(EI)

dei (length)f11

si-se

. .

(m.l.13)

model

(EI)fuil

(length) .

(m.l.l)

si-se

model

full

Bi-Ze

According to the geometrical requirements, m.l is proportional to i

(displace-ment of the water). We find:

(EI)

_model

_i

_model

(EI)

_{full- size}

- l

full size

-If we should build a geometrical similar construction out of the same material,

we should have:

(EI)

dei I model i

full size I full size

-which means that the bending stiffness of our model is A times to high.

Consequently we have to choose another material having a reduced E-modulus

(reduced by a factor A), or build a distorded model of any material so that

the product

EI

obtains the proper value for the model (i.e. p- times the

full size value).

For the vertical elasticity of the model a "back bone" can be designed having

the scaled

EI

value. Attached to this "back bone" are model segments in order to obtain the geometrical similarity. (See fig. 2 and ref. /3/).

For the torsional-horizontal direction a similar approach can be made, although

the design of the "back bone" is then more complicated.

För the case we waTit tò test a model in oblique seas, it is necessary to have

proper elasticities in all 3 directions and in that càse the constrùction of

the "back bone" is, due to all elastic boundaries, difficúlt to realize.

Another possibility is to have the continuous elasticity lumped to a reduced

number of elastic connections between the segments of the model so that the

overall flexibility and dynamic performance is acceptably approximated (see

fig. 3). In that case thére exists aso the advantage to design the elastic

hinges between the segments such that the vertical, horizontal and torsional

flexibility can independently be adjusted to the model requirements. When the

elastic hinges are provided with strain gauges and a calibration in advance

of the measurements is carried out for the various components of interest,

e.g. vertical and horizontal bending moment, torsional moment and side forces,

(as also can be done for the continuous "back bone"), a measurement in oblique

seas gives the necessary information about the sectional forces.

The measurement on the elastic model can be seen as simUltaneous

measure-ment of the excitation forces generated by the waves and the dynamic response

of the structure. The transfer function of waves to sagging-hogging deflection

can be seen as a Response Amplitude 2perator. A similar technique as has been

developed for ship motions can be applied for sagging-hogging arid girder

vibration phenomena.. (See fig.5).

It has to be realized however that in these cases of models having scaled

flexibility, the tests being made hold only for that particular flexibility,

being a reflection of that particular ship. If, as a result of the tests, it

is decided to change the main structure of the ship, resulting in other elastic

characteristics, another test with a new elastic element (or "back bone") is

necessary to s.tudy.the effect of the alteration.

This lack of "flexibility" is the main drawback of these. types of tests,

particularly. when the ship to be. studied is in the design stage: and elastic

.3.) Rigid Model.

'In the case of a stiff model., the structural flexibility :f the original, ship

is avoided in the model. Iñ this case we will measure the trué girder loading

instead of the. result of this loading.,, i.e. a bending .momènt, whether 'or' not dynamically amplified. The model is not able to deform. in any direction. This

means that the "back bone" with the model segments hooked on to it, needs to

have natural frequencies far higher, say 10 X higher, than the frequency of

interest. This means that the natural frequency of the model segment and its

connection to the "back bone" also needs to have a value high in comparison

with the frequency of interest. ' . .. . . .

The measurement of the wave loading of this system does, in contradistinction

with the elastic model technique, not take place on the "back bone" itself

but on the connections of model segments to the. "back bone"

-So the vertical, horizontal and torsional forces executed by the segments

to the back bone are recorded. Because the mass and mass-distribution is also.

properly scaled and is concentrated, in the modél segments, the forces mn

tioned, are automatically corrected for inertia forces, as introduced by the..

ship motions. See fig. 4. . . , ,

The segments. and their connection operate .as a matter of fact as a. large.

pressure pick up or as a system that automatically produces the surface

in-tegrated pressures,, acting on the ships hull at the location of the segment.

In this way we .obtain the. ntt girder loading from which by integratfon, the

sectional. forces and moments can be obtained, as directly measured in the

earlier bending moment seakeeping tests.

In order to obtain from the nett girder loading that'' portion, that is

respons-ible for the sagging-hogging deformation or the natural 2-noded vertical

'de-flection (both are assumed' to be the same), it is necessary to carry out

another integration accbrding to the normal mode technique, i.e.

lo

-participation factor for the 2-noded deflection

2-noded vertical deflection

F(x) _{vertical distributed force acting on the ships girder as a}

result of water pressure and inertia forces.

This integration results in the participation factor being the generalized

force, and the input for the calculation of the dynamic response of the elastic

ship.

For the construction of the model a number of difficulties can be encountered.

The "back bone" construction is a compromise between the two requirements of

large stiffness and small weight. A small weight is necessary to have as much

as possible of the inertia forces included in the measurement, i.e. in the

segments of the model in order to have a proper inertia correction due to

ship motions.

-A similar problem exists for the design of the connection between the model

segments and the "back bone" that need to be sufficiently stiff for natural

frequency requirements but sufficiently flexible for the purpose of the force

measurement. For the force pick ups it is therefore necessary to make an

effective use if the flexibility (strain gauges as sensors) and to avoid

elas-ticity not being used for measuring purposes. (Generally used bending springs

for the pick ups are not recommended).

It- is also necessary to have the rigidity of the model segments in accordance

with the overall stiffness requirements. The generally used glassfiber

rein-forced plastic structure needs to be stiffened by other means, such as steel

inserts, until the required stiffness is obtained. The weight may not be larger

Li) Similarity of iodel tests on structural loading and seakeeping tests.

As the construction of the RAO for ship motions can be broken down into two

steps i.e. the determination of the excitation forces (to be measured on a

captive model) and the determination of the dynamic properties of the model

by means of oscillation tests (planar motion mechanism), the RAO for elastic

girder deformations can be split in a similar way into two portions i.e. the

determination of the excitation forces for that particular shape of

deforma-tion and the dynamic properties for that deformadeforma-tion. An illustradeforma-tion is given

in fig. 5.

In this case the captive model has to be replaced by a model stiff against

f lexural deformations (i.e. the natural frequencies of the various modes need

to be high in comparison with the frequency of encounter of the waves but the

model needs to be free to carry out the ship motions).

The planar motion mechanism tests have to be replaced by a vibration test of

the elastic modèl or a calculation for the case only drawings are available.

The results of the captive model tests (vertical force and pitching moment)

represent the participation of the wave force to these excitations. Similarly

the results of the stiff model experiments represent the participation of the

waves solely causing the deflection under consideration.

The planar motion mechanism tests result in added mass, added moment of

in-ertia, damping components and buoyancy (spring forces). Similarly the

vibra-tion test results in added mass terms, damping and for special cases of long

flexible ships in an additional spring stiffness due to displacement forces.

Based on this separation it is possible to measure for one hull shape the

modal excitations and to calculate the resulting elastic response whether

or not amplified by resonance. If the structure is responding unfavourably

and needs to be modified no new excitation test (wave test) is needed but

only a new elastic response calculation, which is, in contradistinction with

the elastic model, more "flexible".

Pro.osal for model tests based on Normal Mode Techni.ue.

12

-(the model built e.g. on the lumped parameter approximation), which is theit

comparable with the regular seakeeping test with a seifpropelled model being

free to carry out its motions.

In order to gain experience in this technique where beside the regular strength

investigation also critical dynamic phenomena such as springing are included

in the method, a program of experiments is proposed. This program includes

experiments with a stIff model as well as with an elastic model.

The measurements will take place in regular and irregular waves, for both

models in the same waves.

From the. stiff model the participation factor of the wave forces r,1(w) for

the 2-noded deflection will be determined. The shape of the 2-noded deflection

will be experimentally determined from the elastic model by means of a separate

experiment as well as the transfer function for the dynamic amplification and

the damping. The shape of the deflection is necessary to calculate the

par-ticipation factor of the wave forces on the rigid model. Further by using the

transfer function H(w), the magnitude of the deflection to be expected on the.

elastic model can be analysed. From this. magnitudç and the shape of the

de-flection the resulting bending moments can be found. The direct measurement

of the bending moments to be achieved by a measurement with the elastic model

in the same waves opens the possibility to compare the results obtained with

the rigid and the elastic model. Fig. 6 illustrates the proposed tests and

13

-5.1. Realisation of the z'ißid model and the instruments.

An introductory analysis of the stiff model shows that in connection with the

various conflicting requirements about the construction, as outlined above,

a compromise canbe achieved within a reasonable accuracy.

For the ratio of the weight of the ship against the weight of the stiff "back

bone" a figure of 5% has been obtained for a steel thin-walled structure of

i min. thick plate.

The natural frequency of the two-noded vibration of this beam with the

con-nected segments is about 30 c.p.s. which is within the requirements as stated

above.

The design of the force pick ups, connecting the segments to the back-bone,

is based on ari earlier statement, i.e. to have all flexibility used for the

measurement. In fig. 7 and 8 the force pick up is shown. The complete balance,

enabling us to measure the various components (vertical force, horizontal

force and torsional moment) is built up from 5 of these pick ups and extra

connections. See fig. 9 and 10. The stiffness of the total system must be

such that for the mass-spring (model segment - balance) system a natural

frequency of 30 c.p.s. -is obtained.

The integration of the forces of the various segments takes place by a

multiplier and an adder as shown in fig. ii and is an electronic realisation

of formula (i).

The arrangement of the model and the towing carriage for these measurements

is different for symmetric tests and non-symmetric tests. For symmetric tests

(vertical forces, head waves) no automatic steering device is required.

For oblique waves an automatic course keeping device is necessary to perform

5.2. Realisation f the elastic model.

-The elastic model is a lumped parameter estimation of the elastic ship.

As a first approximation the ship can be seen as a slender beam where the

first three normal modes in vertical and horizontal-torsional direction are

assumed to be significant. The hull is cut into segments, hooked together

with elastic hinges. A possible realisation of the adjustable intersegmental

elastic hinge is schematically shown in fig. 12 where the torsional and

bend-ing elasticities as well as the location of the shear centre can be

individu-ally adapted to different requirements.

It is proposed to have the elasticity lumped in three equidistant locations.

The magnitude of the elasticity (say for the vertical direction) is such that

the 3 natural frequencies belonging to the four-segmented model are equal to

the first three natural frequencies of the continuous system. The mass and

its distribution of the original ship and of the model segments are kept

similar, so that the net girder loading (i.e. wave forces and mechanic and

hydraulic acceleration forces) is properly reflected in the model.

The elastic hinges are provided with e.g. strain gauges in order to sense

the sectional moments of interest.

A further refinement can be made by means of a similar summation of the

bending of the three locations as used for the rigid model experiments.

In this way the amplitude of the modal deflections can be recorded directly

from the deflections of the three elastic hinges. When these amplitudes are

correlated with the encounter frequency and amplitude of the waves, a direct

Conclusive remarks.

The proposed strength experiments can also be applied on other seaborn

struc-tures such as multiple hull ships, drilling platforms, mooring systems etc.

15

-Evaluation of the Tneasurements.

The measurements of the elastic model and those of the rigid model followed

by calculations, result in amplitudes of modal displacements.

From these displacements the longitudinal distributions of the sectional

forces such as bending moments, and torsional moments can be determined each

as a function of the longitudinal coordinate.

From these moment- and force-distributions the longitudinal stresses and shear

stresses along the girder can be analysed when the fundamentals of the

con-struction are known.

The total stress distribution can be found by addition of all modal

contribu-tions as well those of the vertical bending as those of the

horizontal-tor-sional deflections. For oblique waves all three types of deflections are

generated simultaneously by the wave loading and a summation can be made

by taking into account the proper phase relation of the various modes in

various directions. In ref. /7/ is indicated that, under certain

simplifica-tions, a systematic relation exists between the lower and higher modes in

vertical and horizontal-torsional direction. It is then possible to determine

a transfer function (dependent on the heading angle of the waves) between

the wave force amplitude and the stress amplitude at a certain location,

which opens the possibility to calculate by means of spectral analysis the

irregular stress and its statistical parameters as a result of a long crested

16

-The strength desig-i of these types of structures will then be based on

real-istic loadings as encountered in reality. The method iill not only include

the familiar static phenomena but also the dynamic arid vibratory behaviour

of the structure /8/.

The final result can be presented as a stress spectrum giving information

about the mean value, the frequency content and the deviation. A problem to

be solved is the resistance of the material against this type of loading and

the prediction of lifetime /9/. It might be necessary to include higher order

moments of deviation in the presentation of the statistical properties of the

stress pattern and to design more sophisticated material testing techniques

in order to answer the questions of failure probability.

List of references.

/1/ Report of Committee 2: Hydrodynamic Wave Loads, 5th I.S.S.C. Hamburg 1973. /2/ Goodman, R.A. Wave excited main hull vibrations in large tankers and

bulk carriers, Trans. R.I.N.A., April 1970.

/3/ Hoffman, D. and van Hooff, R. Feasibility study of springing model

tests of a great lakes bulk carrier, July 1972 (Webb Inst.

of Nay. Arch.).

/i1/ van Gunsteren, F.F. Springing, wave-induced ship vibrations, mt.

Ship-building Progress 17, 333_3L7

/5/ Hurty, W.C. and Rubinstein, M.F. Dynamics of structures 196L (Prentice-Hall, Inc.).

/6/ 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.

17

-stresses in a simplified ship's girder, due to a long crested

irregular oblique sealoading, ProceedThgs of the Symposium

"Development in merchant shipbuilding", Deift 1972, Dept. of

Nay. Arch. Delft University of Technology.

/8/ Muga, B.J. and Wilson, J.F. Dynamic analysis of ocean structures ,

1970 (Plenum Press, New York).

/9/ Nibbering, J.J.W. Fatigue of ship structures, 1963 (Report no. 55 S

NSS-TNO Deift); mt. Shibui1ding Progress, 10, no. 109, Sept.

List of figures. Fig. 1. Fig. 2. Fig. 3. Fig. tf. Fig. 5. Fig. 6. Fig. 7. Fig. 8. Fig. 9.

Regular seakeeping test arrangement for the determination

of wave generated Midship Bending Moment.

Hull flexibility and elasticity of the pick up are not

considered.

Segmented model with scaled elastic "back bone".

Hull flexibility included in the model /3/.

Lumped parameter approximation of elastic model.

Hull flexibility included and represented by the

inter-segmental elastic hinges.

Arrangement of stiff "back bone", force balance and model

segments for the "stiff model test".

Similarity of ship motion and hull deflection tesls.

Relation of elastic and stiff model tests and comparison

of results.

Fundamentals of force pick-up.

Photograph of force pick-up.

Arrangement of force pick-ups to 3-component balance.

Fig. 10. Photograph of assembly of "stiff back bone"; force pick-ups

and model segments.

Fig. 11. Electronic circuitry to calculate the modal integration. 18

1g

-Fig. 12. _{Schematic presentation of intersegmental elastic hinge with}

Bending Moment Pick-up

FIG.1

elastic "back bone"

straingauges for the measurement

of the midship Bending Moment

FIG.2

elastic hinges provided with straingauges

for Bending Moment measurements.

beam with negligible weight and high stiffness.

one segment of the 10-segmented

model.

-weights for proper mass-distribution

of shipmodet.

stiff force balance for the measurement of the resulting

water pressure and inertia forces,transferred from the

segment to the beam.

_forces

measured

V

measured

I force

Heave and pitch excitation _{wave height}

Notion response to wave excitation (R.A.O.)

motion 'force motion

-wave height wave height force

wave excited captive model. planar motion mech. FIGS

-2-.noded excitation [i-noded participatuo]_{wave height}

J

wave wave

excited

free stiff free elasticexcited

model.

modeL.

2-noded deflection response to wave excitation (R.A.O.) 2-noded deflection 2-noded deflection participation

x

wave height participation .wave height

wave excited

stiff model.

Vibrating

modeL.

Response function; added mass,added inertia,damping

[motioni natural frequencies for

LforceJ Heave and Pitch.

[2-noded deflectio added mass, added inertia,damping

natural frequencies for 2-noded vibration.

- ELASTIC MODEL.

+ i, Amplitude of Spectrum of

bending moment bending moment at 3 Locations, at 3 Locations.

RAfl-Bending moment

wave height

g

-E:--<-E J

Dynamic transfer function

of zil noded deflection.

COMPARISON

Normalized deflectior.s Wz at Locations i through m

COMPARISON

r- participation factor - i-1,(t)- (t)'

Fourier analyser

H',(w)2 iT7,.0(w)

AmpLitude °i

zil noded dei Lectuon FHWZ(w).rW0(w)

(R.A.Oj Bending stiffness ampLitude of bend moment at 3 locations I 4i 1+1 noded defLection I Vectona.

adder-'çt;iÇ

+ STIFF MODEL. see figli F1(t) F2(t) Fat) F(t) in waves Measurement Ioeterministk I

II

Istochasticl FFnt FFn(t) spectrtaa ofi participation factor -spectrum v'Z(14 S7z ail roiled lIH

del Lection

spectrum of amplitudel

of bending moments ati 3 Locations due to I z+l noded deflection Spectral adder-m b,0(t1 (t)VZn s Spectrum analyser Bending stiffness

T

Drawings of ship

,.'- 0 andIts construction.

t

Measurements in waves.

Measurements In

stilL water with

mechanical excitation. Cornputeri2ed analysis of normal modes and frequencies.

*

'-4.. IDetarrninistic I IStochastici deflection 1,2,3,1,

Distribution of zil noded

FLEXBL

UNG-'

ME5URING

£LMENT

FIG.?

i

L

FOROE PICK-UP RIGI1 BACK BONE

SEGMENT OF SI-HP MOFEL

Reference signal

(only for

-experiments

with

deterministic waves).

Signals from segmental force pick-ups.

F1(t) F2(t)

F(t)

_{Fm (t)}

i

Multipliers

electronic adder

Potentiometer settings

_'2

_m according to modaL deftections _{2 n}

m

Participation

_{7(t) ='Pz,n. F(t) is a constant voltage when,}

for regular waves,a reference signal is applied.

FI6.11

AdjustabLe spring for stiffness of hinge

and provided with straingauges.

)Mod

e t

segment

vertical

I

,..'

centre

of-rotation

F(G.12

Section:B-B

centre of rotation

bending spring

bending sprìng

I ¿ p' p' I t

I.

shear centreL

Section: A-A

torsional

horizontal

B

Model

segment