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Date Author Address

March 2007

Tuitman, IT. and W. Trouwborst Deift University of Technology

Ship Hydromechanics Laboratory

Mekelweg 2, 26282 CD De!ft

TU'Delft

Deift University of Technology

Page /of 1/1

Derivation of slamming loads using the optimal state estimation method

by

iT. Tuitman and W. Trouwborst

Report No. 1521-P 2007 Proceedings of MARSTRUCT 2007, Thert International

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PROCEEDINGS OF MARSTRUCT 2007, THE 1ST INTERNATIONAL CONFERENCE ON MARINE STRUCTURES, GLASGOW, UNITED KINGDOM, 12-14 MARCH 2007

Advancements in Marine Structures

Editors

C. Guedes Soares

Instituto Superior Técnico, Technical University of Lisbon, Lisbon, Portugal

P.K. Das

Universities of Glasgow and Strathclyde, Glasgow, Unfted Kingdom

Taylor & Francis

Taylor & Francis Group

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Taylor & Francis is an imprint of the Thy/or & Francis Group, an infor,na business

© 2007 Taylor & Francis Group, London, UK

Typeset by Charon Tec Ltd (A Macmillan Company), Chennai, India

Printed and bound in Great Britain by Bath Press Ltd (a CPI-group company), Bath

All rights reserved. No part of this publication or the information contained herein may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, by photocopying, recording or otherwise, without written prior permission from the publishers.

Although all care is taken to ensure integrity and the quality of this publication and the information herein,

no responsibility is assumed by the publishers nor the author for any damage to the property or persons as a result of operation or use of this publication andlor the information contained herein.

Published by: Taylor & Francis/Balkema

P.O. Box 447, 2300 AK Leiden, The Netherlands

e-mail: Pub.NL@tandf.co.uk

www.ballcema.nl, wwwtaylorandfrancis.co.uk, www.crcpress.com

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Advancements in Marine Structures - Guedes Soares & Das (eds) 2007 Taylor & Francis Group, London, ISBN 978-0-415-43725-7

Table of Contents

Preface IX

Methods and tools for loads and load effects

Uricertainty of average wave steepness prediction from global wave databases 3

E.M Bitner-Gregersen & C. Guedes Soares

Wave induced global loads for a damaged vessel 11

L. Folsø, E. Rizzuto & E. Pino

Effect of ship length on the vertical bending moments induced by abnormal waves 23

N Fonseca, C. Guedes Soares & R. Pascoal

On extreme ship response in severe short-term sea state 33

T Fukasawa, H. Kawabe & T Moan

The 2D numerical modelling of slamming on a wedge and bow shape using the

finite volume method 41

A. loan & L. Domnisoru

Extreme value predictions and critical wave episodes for marine structures by FORM 51

.1 Juncher Jensen

Extreme response estimation for marine structures based on design contours and

response surface methods 61

B.J Leira

Ship weather routing based on seakeeping performance 71

MB. Pacheco & C. Guedes SOares

Time domain simulation of ship global loads due to progressive flooding 79

TA. Santos & C. Guedes Soares

Investigation of the variation of loads experienced by a damaged ship in waves 89

TWR Smith, K.R. Drake & S. Rusling

Derivation of slamming loads using the Optimal State Estimation Method 99

iT Tuijinan & W Trouwborst

Effective breakwaters for high speed container vessels 107 KS. Varyani & XE ?ham

Methods and tools for strength assessment

The racking analysis in the early stages of the structural design 117 VAmatulli, M Biot & L. Schffrer

Broadband excitation induced by propellers and ships comfort evaluation 127

F Besnier N Buannic, L. Jian, A. .Blanchet & S. Branchereau

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VI

Numerical analysis of eigen vibration modes for an orthotropic ship structural panel 135

I. Chirica & L. Domnisoru

Non-linear hydroelastic response analysis in head waves, for a large bulk carrier ship hull 147

L. Domnisoru & A. loan

Round Robin study on structural hot-spot and effective notch stress analysis 159

W Fri cke, A. BolIem, I. Ozirica, Y Garbato F Jancart, A. Kahi, H. Remes, CM. Rizzo,

H. von Selle, A. Urban & L. Wei

Fatigue strength assessment of floating production storage and unloading vessels 169

V Garbatov, S. Tomasevic & C. Guedes Soares

Effects of changes in curvature on the natural characteristics of curved structures 177

B. Hu, fT Xing, R.A. Shenoi & J Smith

Hydroelastic analysis of cantilever plate in tune domain 187

F Kara & D. Vassalos

Residual strength and survivability of ships under combined vertical and horizontal bending 197

l.A. Khan & RK. Das

Collapse strength of longitudinal plate assemblies with dimple imperfections 207

R.M Luls, C. Guedes Soares & RI. Nikolov

First - principles collision analysis for design 217 G. Mermiris, D. Vassalos & D. Konovessis

On the influence of boundary conditions on the vibrations of ship propulsion systems 225

L. Murawski

Development of a numerical simulation method for fatigue crack propagation in structures

under variable amplitude loading 233

T Okawa & V Sumi

A method for progressive structural crashworthiness analysis under collisions and grounding 241

JK. Paik & JK Seo

Methods for ultimate limit state assessment of marine structures: A benchmark study 249

f K. Paik, JK. Seo, Bi Kim, VS. Suh & B.S. fang

Bending response of laser-welded web-core sandwich plates 263

J Romanoff& P Varsta

Studies on the behaviour of bottom structures during grounding 273

M.S. Samuelides, J Amdahl & R. Dow

Global strength analysis in head waves, for a tanker with longitudinal uniform structure 283

L. Stoicescu & L. Domnisoru

A numerical approach for fatigue crack propagation in ship structures underwave

loading - A review of the development of CP-System 295

YSumi&TOkawa

The natural vibration characteristics of a water-shell tank interaction system 305

fTXing, YRXiong&M Tan

Natural dynamic characteristics of an integrated liquid - LNG tank - water interaction system 313

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VII

Analysis of sharp corners in structural details 323

L. Xu & N Baritrop

Formulation for ultimate shear strength of ship structure 331

S. Zhang & P Kumar

Experimental analysis of structures

Comparison of experimental and numerical impact loads on ship-like sections 339

A. Joan, S. Brizzolara, M. Vwiani, N Couly, R. Donner 0. Her,nundstad, T Kukkanen, S. Malenica & P Temarel

A plea for large-scale testing 351

B. Boon

Numerical study of the absorbed energy in clamped steel beams with different lengths

under transverse impact 357

D.M Dimas & C. Guedes Soares

Measuring damping properties of viscoelastic materials for marine applications 367

A. Ferrari & E. Rizzuto

Experimental evaluation of the behaviour of a mild steel box girder under bending moment 377

JM. Gordo & C. Guedes Soares

An experimental and numerical study on GFRP box girder under pure bending 385

C. Guedes Soares, NZ C'hen, FM Santos & C. Santos

Structural loads and response analysis for a Canadian frigate 391

D.C. Stredulins/cy, N G. Pegg & C.P Gardiner

A benchmark study on ductile failure criteria for shell elements in multiaxial stress state 401

K Tabri, H. Alsos, J Broekhujsen & S. Ehiers

Materials and fabrication of structures

Buckling strength of square composite plates with geometrical imperfections - Preliminary results 413

Berggreen, C. Jensen & B. Hayman

Vibration characteristics of smart sandwich beams embedded with magnetorheological

elastomer cores 421

WJ Choi, YR Xiong & R.A. Shenol

Improving the strength of adhesive butt joints for pultruded composites 429

S.A. Hashim, 0. Fozzard & EK. Das

A review of the causes of production defects in marine composite structures and their influence on performance

Hayman, C. Berggreen & N. G. Tsouvalis

Methods and tools for structural design and optimization

Benchmark on ship structural optinusation 453

N. Besnard, M. Codda, A. Ungaro, C. Toderan, A. Kianac & F Pécot

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A concept of 0mm-optimization for ship structural design 473

A. Kianac & J Jelovica

Least cost optimization of a large passenger vessel 483

T Richir N Losseau, E. Pircalabu, C. Toderan & P Rigo

Structural optimization in fatigue-life assessment concerning a DDG of Ammiragli Class of

the Italian Navy 489

G. Vacca, Mi Galliussi & S. Simone

Decision support problem formulation for structural concept design of ship structures 499

V Zanic, P Prebegn & S. Kitarovic

Structural reliability, safety and environmental protection

Holistic analysis of ship's sustainability 513

0. Cabezas-Basurko, E. Mesbahi & S.R. Moloney

Inland water going ship risk assessment 523

G.VEgorov

Impact of the new common structural rules on the reliability of a bulk carrier 529

A. W Hussein, A.R Teixeira & C. Guedes Soares

Practical reliability assessment method of ship's hull girder in longitudinal bending 539

KIjima,YFujii&TYao

Structural system reliability using FEM and reliability processors 547

MShahid&RK.Das

Structural reliability levels in Ice Class rules 557

X Wang,HSun&R.Basu

Factors affecting the non-destructive inspection of marine structures 565 A. Zayed, l Garbatov & C. Guedes Soares

Author index 577

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Advancements in Marine Structures - Guedes Soares & Das (eds)

2007 Taylor& Francis Group, London, ISBN 978-0-415-43725-7

Conference Chairmen

Prof. Carlos Guedes Soares, 1ST, Technical University of Lisbon, Portugal

Prof. Purnendu K. Das, Universities of Glasgow & Strathclyde, UK

Technical Programme Committee

Prof. N. Baritrop, University of Glasgow & Strathclyde, UK Dr. N. Besnard, Principia Marine, France

Dr. M. Codda, CETENA, Italy

Prof. R.S. Dow, University of Newcastle upon Tyne, UK

Prof. W Fricke, TUHH, Germany Dr. S. Gielen, TNO, Netherlands

Prof. J.M. Gordo, 1ST, Technical University of Lisbon, Portugal

Prof P.F. Hansen, DTU, Denmark Dr. B. Hayman, DN\ç Norway

Prof A. Incecik, University of Newcastle upon Tyne, UK Prof T. Jasirzebski, TUS, Poland

Prof J.J. Jensen, DTU, Denmark

Dr. J. McGregor, Bureau Veritas, France

Prof T Moan, NTNU, Norway Prof V. Papazoglou, NTUA, Greece

Prof R.A Shenoi, University of Southampton, UK Prof P. Temarel, University of Southampton, UK

Prof A. Ulfvarson, Chalmers University of Tech., Sweden

Prof P. Varsta, Helsinki University of Technology, Finland

Advisory Committee

Dr. R.I. Basu, ABS, USA Prof. WC. Cui, CSSRC, China

Dr. M. Dogliani, RENA, Italy

Prof S. Estefen, COPPE, Brazil Dr. P. Hess, NSWC, USA

Dr. M.L. Kaminski, MARlIN, The Netherlands Prof. J.K. Paik, Pusan National University,. Korea

Dr. N.G. Pegg, DND, Canada Dr. R. Porcari, CETENA, Italy

Dr. B.C. Simonsen, DN Norway

Prof Y Sumi, Yokohama National University, Japan

Dr. P.M. Videiro, Petrobras, Brazil Dr. S.G. Waestberg, DNV, Norway Prof T. Yao, Osaka University, Japan

Conference Secretariat

Alma Moise, 1ST, Technical University of Lisbon

Maria de Fatima Pina, 1ST, Technical University Of Lisbon

Nicola Pollock, Universities of Glasgow & Strathclyde

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Advancements in Marine Structures - Guedes Soares & Das (eds) © 2007 Taylor & Francis Group, London, ISBN 978-0-415-43725-7

Derivation of slamming loads using the Optimal State

Estimation Method

IT. Tuitmafl

Delfi UniversitY of Technology, Ship Hydromechanics and Structures, The Netherlands

W. TrouwbOrSt

TNO, Centre for Mechanical and Maritime Structures, The Netherlands

ABSTRACT: This paper shows the use of the Qptimal State Estimation Method (OSE) for estimation of

slamming forces for model experiments. The OSE estimates the external loads using the response of the model

and calculated natural modes. The response is measured with several accelerometers to allow the OSE to estimate

the response of the flexible modes which is needed for an accurate estimation of the external forces. When the calculated natural modes are close to the actual natural modes and with well conditioned measurement signals, the OSE estimates the external force accurately.

1 INTRODUCTION

To measure slamming loads for model experiments -usually pressure gauges, load cells or segments with force transducers are used. Another way to derive the

slainmiEiig load is the Optimal State Estimation Method

(OSE). The OSE method uses the measured response

of the model to predict the slamming loading by

combining numerical and experimental results. For the OSE method the model needs to be equipped with accelerometers only, which normally give no restric-tions to the model except that one should be able to calculate the eigen modes accurately.

The OSE method is developed during the joint

US-NL research program DYCOSS (Dynamic Behavior of

Composite Ship Structures) (Trouwborst & Costanzo

1999). The OSE method has been used for the analyses

of full scale shock trial and slamming measurements. This is the first time the method is used to derive the slammmg impulse for model experiments.

This paper will first present the theoretical back-ground of the OSE method. Some results of the OSE

method for full scale measurementswill be shown.

Finally the use and results of the OSE method for the model slamming experiments will be discussed.

2 OSE METHOD

The theory of the OSE method will

be discussed

briefly based on the discussion that can be found

99

in (Thouwborst & Costanzo 1999). An extensive

discussion can be found in (Molengrafi 1990).

The OSE method uses the modal equations and measurements to make an estimate of the response

and external loading.

2.1 Modal equations

The common dynamic equations are used as starting point:

M+C±+Kx=f

(1)

The mass matrix includes the added mass of the fluid. The infinite, frequency independent, added mass can

be used because of the high frequency of natural modes

of ship structures. Using a FE-package the eigen value

problem- is solved:

(wiVI + K)

<p = 0 (2)

The results are the eigen frequencies o and eigen

modes .

The eigen modes are normalized with

respect to the mass matrix:

JVIço=1, çoK,çc=w2, fori=1,n

(3)

The dynamic equation is transformed to modal

coor-dinates using the transformation matrix T which

contains a subset of eigen modes.

(11)

With:

= TMXT.

C,, = TCT

(5)

Kp=TtKT,

.fpTtf

The transformation to modal coordinates allows for a

significant reduction of the number .of degrees of

free-dom. Only the eigen modes below a cut-off frequency and with sufficient energy will be used.

2.2 Measurement equations

A vector with measurements signals Zm Cfl be

introduced:

zm Z8p + Zji + Zaji + ZuU (6)

with p,j and )E3 the modal displacements, velocities

and accelerations according to equation (4). The

math-ces Z, 1, and Z are the participation matrimath-ces for

the mesured displacements, velocities and

acceler-ations as present in Zm. In fact, based on the kept eigen

modes, these matrices transform the modal- quantities p,Ab and j5 to physical displacements, velocities and

accelerations.

Vector u are the measured external forces, if any. The boolean matrix Z selects which measured forces

are used.

2.3 The optimal state estimation method

There will always be differences between the

measure-ments and the mathematical model. To account for

these differences residuals are introduced for all equa-tions. The OSE estimates the response and external loads of the model by minimizing these residuals.

First a residual is introduced for the time-üerivative

equations:

=i,+ii,

v=a+2,

(7)

Where , and a are the estimated displacements,

velocities and accelerations respectively, based on the set with modal degrees of freedom.

By replacing the p, and j5 in the model equation

(4) by the estimation variables , and a the residual

is introduced.

(8)

Vector i are unknown external forces which the USE

should estimate. Matrix H is the participation matrix for the external forces.

The last residual 2 is introduced for the measure-ment equation (6):

Zm

Z8.+ ZIJ+Z0â+ Zü+(

(9)

100

Equations (7) to (9) form the so-called

identifica-tion model, which is the basis for the optimal state

estimation method. Following (Molengraft 1990), the

identification model can be formulated more

com-pactly using the augmented state estimator and the

augmented input estimator :

x=(s,v)

- ,

p=(u,a)

,'t -t t

--St -St t

(10)

Using these estimators, equations (7) to (9) can be

rewritten as: (11)

Ezm =F+F+(

with: A

(

0 0)'

'

B

(

o I)'2

F

C'\

-

,

Z)

F

(H M'\

- Z Z0)'(=)

For an optimal state estimation, and b have to be

determined such that the residuals and are

min-imized. For the minimization of the residuals the

following penalty function is introduced:

J() =

1

((t)w(t)

+Ct(t)VC(t))dt

+ ((to).qo)tR0((to) - qo) (14)

This function is minimized using the conditions of

equations 11 and 12. Vector q are the initial conditions

at the starting time t0 as defined by the user. The term

((t0) qo) is a residual on the initial conditions at

start-ing time t0. The user supplies the weightstart-ing matrices

W, V and R0 which express the confidence in the time

derivative equations, model equations, measurements and initial conditions.

The approach to minimize function (14) can be

found in reference (Molengraft 1990). The estimated

response and external forces are known after the

optirnisation.

3 PREVIOUS USE OF THE USE METHOD

The OSE method has been used for estimation of the

response and external loading for full scale shock trials

and slamming events.

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oy

4 EXPERIMENTS

In,g,Ced md o4000 en e,nJod 4odty

Figure 1. OSE results for rn-frigate.

Figure 1 shows the velocity at a location of a Dutch

M-frigate during a shock trial. This shock trial was analysed using the OSE method. The measurement at the location is not used for the OSE estimation of the response. The velocity signal is reconstructed using

the- estimated response.

The shock load excitated many high order modes

which are not used for the OSE estimate. Also

--there--will. always be differences between the struc-tural model and the actual ship. Therefore the

esti-mated velocity response will differ from the measured velocity

Figure

1 shows that the estimated response is

reasonable close to the measured response at the

location.

Estimation of slamming impulse for a full scale

slamming experiment showed also reasonable results.

In the summer of 2006 experiments were held in the towing tank of the Technical University of Deift. The goal for the experiments was to determine the

slam-ming loads and added mass for flexural modes fora

simplified hull shape. The OSE methodwas chosen to

derive the slamming loads after the experiments.The

experiments where setup such that it would beeasy to

apply the OSE method.

Table 1. Hydrostatics

Length over all 3.8m Length waterline 3.32 m

Length parallel section 3.0 m

Beam 0.75 m Draught 0.08 m Depth 0.20 m Volume of displacement 0.130m3 Displacement mass 130 kg Area of waterplane 2.38 m2

Figure 3. Model constrains.

4.1 Design

The hydrostatics of the modelare presented in table 1.

The cross sections can be found in figure 2. Figure 3

shows the applied constraints. Thesway, roll and yaw

motions are suppressed and springs control the surge

motion. Figure 4 isa photograph of the actual model.

The natural modes have to be calculated using a

FE package. In order to obtain accurate results the

material propei.ties should be well known. Therefore Figure 4. Used model.

101

I

0

-0 4 -0.3

-0.2 -0.1 0 0.1 0.2 0.3 0 4

Figure 2. Cross sections.

0.4

0.3 I

0.2 -t

0.1 4

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aluminium is used instead the normal material for

building models.

The OSEmethod predicts the response of the model

by estimation of the participation of the natural modes. The better the OSE can make a distinction between the

different natural modes the more accurate the estima-tion will be. Local modes will give only response at a few, or in the worst case none, accelerometers which

make the estimation of the participation of these modes

difficult. For an accurate estimation of the response

and the external forces most of the energy of the

response should be in the response of the global natural

modes.

The model is designed such that the global natural

frequencies are as low as possible and the local natural frequencies are as high as possible to obtain a response

that is dominated by the global modes. This resulted in-a large model for the towing tank with little

free-board. The complete model was build from plates with

a thickness of 5 mm to obtain high frequencies for the local modes.

4.2 Measurements

Twenty accelerometers with a range of 20 g are

mounted on the model. Four are oriented horizontally, fourteen are oriented vertical and two are placed on the bottom plate u the bow which has a small angle.

Frequencies up to 300 Hz will be of interest for

application of the USE method because most

impor-tant natural modes have a frequency up to this

fre-quency. The signals are sampled with a sample rate of

10 kHz to record responses .up to 300 Hz accurately.

For determination of the natural frequencies a

modal hammer is used which measures the applied

force.

The wave elevation in front of the model and the rigid body motions are also measured.

4.3 Experiments

in the first part of the experiments the natural

fre-quencies of the model were measured for both dry and wet, with and without forward velocity. These mea-surements are used to check and correct the FE results and the added mass calculation.

The second part of the experiments bonsisted of

slamming in regular waves. Slamming in a peak wave of a wave train was the third part of the experiments.

The last part of the experiments was excitated

using a large hammer (10 kg) to simulate a slamming event. The force applied with the hammer was

mea-sured. in this way the results of the USE method

can be checked. The impulse of the hammer was

tuned by rubber between the hammer and model to

obtain a response that was comparable to the measured

slamming response.

102

Figure 5. FE model.

5 PREPARATION FOR USE

Before the USE method can be applied the natural

modes should be calculated. The offset and drift of the

acceleration signals should also be removed.

5.1 Natural modes

The natural modes are calculated using the FE package TRIDENT. The used FE mesh is shown in figure 5. The

small mesh size is chosen to make the meshing of the small eometrica1 details possible.

The small blocks are the supports on which the accelerometers where mounted. These blocks are

added to the mesh to obtain the eigen vectors as accu-rate as possible for the locations of the accelerometers.

Boundary elements are placed on all wetted ele-ments for calculation of the added mass in still water.

Hydrostatic springs are added to obtain heave and pitch

modes, but there will be a small error in these modes

because only the infinite added mass is used in the FE

calculation.

The density of the aluini.nium is scaled abit to obtain

the exact weight as the real model. This accounts for the weight of cables, accelerometers, paint, etc.

The natural modes are measured during the first part of the experiments. The frequency response

function (FRF) between the hammer and the 14

vertical accelerometers are calculated. The results

of the measurements in still water are shown iii

figure 6.

The FRF shows the actual frequency of the natural modes very clearly. The measured frequency of the

first dry longitudinal bending mode is used to scale the

elastic property of the aluminium. This compensates for the differences between the FE model and the real

(14)

1.4 1.2 1 0.8 0.6 0.4 0.2 0 Frequency [Hz] wet 20 40 60 80 100 120 140 160 180 200

Figure 6. FRF between hammer and 14 accelerometers.

Table 2. Comparison between measured and calculated

frequencies.

Table 2 shows a comparison between the measured and calculated frequencies. Both the wet as well as the dry calculated natural frequencies are close to the

measured values. This shows that the FE model is con-sistent and that the calculated added mass is close to the real added mass. The calculated mode shapes are used

for the OSE calculation but the measured frequencies are used to obtain better results.

5.2 Measurement signals

The accelerations, velocities and displacement are

esti-mated by the OSE method. The error in the velocities

and displacements can not be estimated if only

acceler-ations are used. During the experiment the rigid body displacement is measured using an optical tracking

system. The signal f this system can not be used

directly because the sample frequency is only 60 Hz and the signal does not include the displacement by

the flexible modes.

A displacement signalcan also be obtained by

dou-ble integration ofan acceleration signal. The calculated displacement is shown in figure 7. Line d(t) is the

dis-placement measured by the optical tracking system and line ff a(t) is the double integration of the accel-eration signal. There is a clear difference between the two signals. The difference is mainly due to the fact

that the initial velocity and displacement are not known

for the doubleintegration, but the changing angle with

103 C 1.5 0.5 -0.5 40 20 30 1 0 -10 0 0 0.5 1 1.5 2 2.5 3 3.5 4 Time [s]

Figure 7. Displacements at accelerometer I.

11j

a(t)

a(t)

-0.5 1 1.5 2 2.5 3 3.5

Time [s]

Figure 8. Acceleration at accelerometer 1.

respect to the gravity field and inaccuracies of the

accelerometer will also cause drift.

Before the OSE is used the error in both the accel-eration and displacement signals are minimized using

the displacement signals from the optical tracking

system.

The first step of the correction is the subtraction of the additional measured accelerations due to gravity when the pitch angle is not zero. In the second step a fifth order polynomial fit on the differences between the double integrated acceleration and the measure-ment is done. The first coefficient will be the initial displacement of the point, which can not be known from the acceleration only. The initial velocity is the second coefficient. Third coefficient of the polyno-mial fit is the mean offset of the accelerometer. Only the fourth and fifth coefficient are used to remove real

drift.

The original acceleration signal a(t) and the

cor-rected a(t) are shown in figure 8. Only a small change

ofthe.mean value is visible but the integration as shown

in figure 7 is better.

The double integral of the corrected accelerations at the bow and stern are also supplied as additional

signals to enable the OSE to estimate the error in

displacement and velocity.

lth tors. 45.3 42.6 29.9 28.1

Ith long. 47.0 46.9 24.8 25.3

2ndturs. 111 112 72.8 70.8 2nd long. 126 128 63.1 61.2

dry

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6 RESULTS

First a validation case will be shown, thereafter a

com-parison between OSE results and calculations for a

slam event will be shown. For these calculations only the acceleration signals of the 14 vertical

accelerom-eters are used. Two additional displacement signals at the bow and stern are also used. The first ten natural modes which have significant displacements in

verti-cal direction are the subset for the estimation of the

response.

6.1 Validation

For validation purpose the model is hit with a large

hammer to obtain a comparable response as measured at slamming events. The hammer force is measured using an accelerometer,f = m a. This will result in a non-zero force before and after the impact because the hammer still accelerates.

The measured force and the estimated force

obtained with the USE method are shown in figure 9.

An unknown force at the location of the hammer

impact is used for the USE calculation. Theestimated

force by the USE during the actual hit is very close

to the measurement. The OSE estimation shows some oscillations after the hit because the actual damping

of the modes is not known. The USE .uses the external

force to compensate for the differences between the real damping and the user defined damping.

6.2 Slamming event

An experiment in regular waves where large accelera-tions due to slamming were measured is selected for analyses with the OSE method. The forward velocity for this experiment was 1 rn/s with a wave height of 0.18 rn and wave frequency of 0.64 Hz.

Two unknown forces, one at the bow and one at the stern, are added to the USE calculation. The unknown force at the bow is located at the estimated center of the slamming force. In this case at least two forces are

needed because also the hydrodynamic heave force and

0.4 0.45 0.5 0.55 0 6

Time [sJ

104

pitch moment should be estimated by the OSE method

to obtain the correct motions. Using more tmknown force locations would give more insight in the actual

force location, but this was not done to reduce the

number of unknowns and problem size.

In total 3.5 seconds were analysed. With a sample rate of 10 kHz the USE has to minimize function (14) for 35.000 time steps using 16 measurement signals, 10 unknown modal responses and 2 unknown forces.

The initial velocities and displacements are user

defined variables. At the start of the measurement they are not zero, but for simplicity the initial conditions in

the USE calculation are set to zero. At the first stages of the calculation the USE method will use the exter-nal forces to obtain a velocity and displacement that is close to the measured velocity and displacements. Therefore the force estimation of the first second will be incorrect.

Figure 10 shows the estimated force at the bow of

the model. The question is whether this estimation

is correct. The estimation will not be as accurate as

the estimation of the hammer force because the added

mass will change in the waves and the USE method uses only modes calculated with the added mass of still water. During the slam event the added mass will change and consequently the modal shape and fre-quency. Because the USE method will still use the natural modes for the still water case there will be an

error in the estimation, but for the total estimated slam

impulse this error is expected to be small. First of all

the shape of the natural modes is not influenced much

by the added mass. Besides, the USE analyzes a time

history, in whichthe added mass will change during the

slam event and the average frequency of the structure during the slam event will be close to the still water

condition.

A calculation of the slamming force is done as ver-ificatidn whether the estimated slamming force is in

the expected order. For this estimation the time domain sea keeping program PRETTI (Veer & van Daalen 2006) is used. PRETTI is coupled with CRS-SLAM which is

a boundary element method based on the theory of

(Zhao & 0. Faltinsen 1997).

(16)

The calculation of the rigid body motions using pprrI showed significant differences with the

mea-surements. This could be expected. PRETTI uses the linear hydrodynarnic coefficients for the calculation

of the radiated and diffracted forces, which imply very small wave height and steepnesses with hydrodynamic

loading U to the still water line. The model was at

some time instances more than fifty percent out of

the water. Therefore the radiated and diffracted forces

can never be calculated correctly using the still water

hydrodynamic coefficients.

Because the slamming force is very sensitive to the relative velocity the calculated motions are not

used. The measured motions were imposed on the sea

keeping calculation. In this way the calculated relative

velocity for the slamming calculation will be correct. The OSE estimation for the external force does not

include the linear hydrostatic term because hydrostatic

springs are used for the FE model. The linear

hydro-static force is re-calculated and subtracted from the sea

keeping force for the comparison.

The time and amplitude of the calculated and

esti-mated slamming force compare very well in figure 10.

It seems very likely that both the estimate of the OSE

method and the slamming calculation are close to

the actual slamming force. The comparison between the sea keeping force, force between the slam peaks, is. worse. The estimated external force by the OSE method will not be correct because the infinite added mass approach is not valid for sea keeping motions.

An other reason for the differences is that the cal-culated sea keeping forces are not correct because otherwise the calculated motions would have been

more comparable to the measurements.

7 CONCLUSIONS

The OSE method is used to estimate the slamming loads for model experiments. The model usedwas

105

designed to allow for an optimal use of the OSE

method. Both the FE calculations and preparation of the measurement signals were done with care. With this effort the OSE method was able to make a very

good estimate of a hit with a large hammer. The

estimate for a slamming event was comparable to

the, results of calculations using a boundsry element

method. Based on these results it can be concluded

that the OSE method is able to estimate the slamming force based on the response of the model.

At the moment of writing this paper only one ham-mer hit and one slam event was evaluated using the OSE method. Therefore the performance of the OSE method for the other conditions is not yet known.

ACKNOWLEDGEMENT

'The OSE method software was developed by

Dr.ir. R.H.B. Fey, currently working at the

Eind-hoven University of Technology, Department of

Mechanical Engineering, using the previous work of

Dr.ir. M. Molengraft. The authors like to thank Mr. Fey

for his support in utilizing the OSE method.

REFERENCES

Molengraft, M. 1990. Identification of non-linear

mechan-ical systems. Ph. D. thesis, Eindhoven University of

Technology.

Trouwborst, W. & Costn7o, F. A. 1999. The optimal state estimation method, a tool to integrate full scale shock trial measurement data and numerical models. In 70th

Shock and Vibration Symposium, 15-19 November 1999,

Albuquerque NM USA.

Veer, R. van 't & van Daalen, E. 2006. Pretti vl.3 theory

manual. Technical report, CRS.

Thao, R. & Faltinsen, J. A. 0. 1997. Water entry of arbi-trary two-dimensional sections with and without flow

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