Nonlinear Hull Girder Loads in Ships

Nonlinear Hull Girder Loads in Ships

L. J. M. Adegeest

Nonlinear Hull Girder Loads in Ships

PROEFSCHR1FT

ter verkrijging van de graad van doctor aan de Technische Universiteit Deift,

op gezag van de Rector Magnificus Prof. ir. K.F. Wakker, in het openbaar te verdedligen ten overstaan van een commissie, door het College van Dekanen aangewezen,,

op dinsdag 9 januari 1995 te 14.00 nur

dOor

Leonardus Johannes Maria ADEGEEST

Scheepsbouwkundig Ingenieur

Dit proefschrift is goedgekeurd dóor de promotor: Prof. dr. ir. J. A. Pinkster

Acknowledgements

The information presented in this report reflects the result of four years of research performed at the Laboratory of Ship Hydromechanics of the Deift University of Technology. The discussions with, and encouragements by professor J. A. Pinkster and dr. J. A. Kenning were an indispensable motivating factor in the synthesis of the results.

It would have been impossible to produce the results without the help of the people working in the laboratory. Their assistance is very much appreciated.

I am very grateful also to the Royal Netherlands Navy who were respon-sible for the initiation and sponsorship of this project.

And last but not least, I'd like to thank Aima who, after suffering from my changing moods for such a long time, kept her patience and even read and checked the grammar used in this thesis. Any errors or omissions are, of course, my own.

Léon Adegeest Delft, October 1994

s amenvatting

De vertikale belastingen in een schip, varend in kopgolven, zijn bestudeerd. Een harmonische analyse van regelmatigesleeptank resultaten laten duidelijk tweede en derde harmonische componenten zien. _{Dit sterke niet-lineaire} gedrag treedt vooral op rond de resonantie piekvan de relatieve beweginen van de boeg. Uitwaaierende spanten in het voorachip vergroten de niet-lineariteiten dan ook aanzienlijk. In onregelmatige golven duiden de grote gemeten skew en kurtosis op dit niet-lineaire karakter van de belastingen.

Dit sterke niet-lineaire karakter van de belastingen heeft een

belangri-jke invloed op de bepaling van de statistische eigenschappen. Vanwege de vereiste rekentijd is een analyse op basis van een direkte oplossing in het tijddomein niet wenselijk.

In dit rapport worden de belastingen in onregelmatige gevolgen berekend met een derde-orde Volterra modellering. Deze efficiente methode vereist het bekend zijn van de eerste, tweede en derde orde overdrachtsfuucties in regelmatige golven. Het model is gevaiideerd aau de hand van experimentele resultaten van proeven in regelmatige en onregelmatige golven. De derde orde simulaties lieten een zeer goede overeenkomst zien. _{Dezelfde mate} van overeenkomst werd niet gevonden met de eerst noch de tweede orde

modellering.

Een volledige numerieke analyse van de statistische eigenschappen van de belastingen vereist 4e berekening van de overdrachtsfuncties in regelmatige golven voor een beperkt aantal frequenties. Door gebruik te maken van een niet-lineair tijddomein programma waarin vormveranderingen van het momentane ingedompeld volume worden meegenomen, kunnen al redeijke resultaten bereikt worden.

Een complicatie bij het toepassen van het derde-orde Volterra model is de bepaling van de vereiste niet-lineaire overdrachtsfuncties. De _amplitude afhankelijkheid in de genormaliseerde tweede harmonische _{respons duidt op} de aanwezigheid van hogere dan derde-orde effecten. Het blijkt echter dat de statische eigenschappen redelijk voorspelt worden met het derde-orde Volterra model wat inhoudt dat in de praktijk een redelijke schatting van deze functies al voldoet. HetVolterra model in combinatie met de bekende overdrachtsfuncties kan ook gebruikt worden orn de spektrale en statistische momenten van de respons in arbitraire golven direkt te berekenen zonder het uitvoeren van simulaties. Nadat deze momenten gevonden zijn biedt de mogelijkheid zich aan orn benaderende statistische modellen te gebruiken voor een direkte berekening van de langednur statistiek.

Abstract

The vertical hull girder loads in a ship in head waves are studied. Fourier analysis of new regular wave towing tank results indicated clear second and third harmonic responses. The severest nonlinear behaviour occurs around the resonance peak of the relative motions at the bow. An increase of the bow flare and of the forward speed both increase the nonlinear response components significantly. In irregular waves, large skew and kurtosis are. measured.

These nonlinear characteristics of the hull girder loads havean important impact on the strategy to be followed to determine statistical properties in arbitrary waves. A straightforward solution in the time domain is not acceptable due to the huge required computation time.

Applying a third order Volterra modelling, nonlinear responses in irreg-ular waves are calculated very efficiently using Fast Fourier Transformation techniques. The first order and the required approximations of the second and third order frequency response functions are derived from regular wave results only. A comparison of the power spectra and the probability density functions of. fifty towing tank recordings for two models with those of the reconstructed signals shows a good agreement. The same degree of corre-spondence is not found using a linear or second order modelling.

A complete numerical analysis of the statistical properties of the hull girder loads requires the calculation of the response functions in regular waves for alimited number of frequencies. Encouraging results are obtained using a nonlinear time dómain program in which variations of the wetted geometry are taken into account.

A complication in applying the third order Volterra modelling is the unique determination of the notilinear frequency response functions. Ampli-tude dependent trends in the normalised second harmonic responses show that higher than third order effects are' present. As the statistical prop-erties are predicted well by the third order Volterra modeffing, it follows that a reasonable estimate of the nonlinear frequency response functions already satisfies.. The Volterra modelling in combination with the known frequency response functions can also be used to calculate directly the spec-tral and statistical response moments in arbitrary waves without performing simulatiöns. Knowledge of these moments offers' the opportunity to apply approximate statistical, models for the calculation of lifetime statistics.

Over de auteur

De schrijver van dit proefschrift werd geboren op 30 augustus 1966 te Wa-teringen. Al snel verhuisden zijn ouders met hun twee zonen naar Dreumel. Daar bracht de auteur zijn verdere jeugd door. In 1984 deed hij eindexamen aan het Atheneum van het Pax Christ college te Druten.

Hij verhuisde weer terug naar het Westen van Nederland om Scheeps-bouwkunde te gaan studeren aan de Technische Hogeschool Deift. Begin 1990 behaalde bu het Ingenieursdiploma door de afronding van een afs-tudeeropdracht bij de vakgroep Scheepshydromechanica. Aansluitend heeft hij ruim vier jaar onderzoek kunnen verrichten bij diezelíde vakgroep op het gebied van de niet-lineaire belastingen in een scheepsdoorsnede. De resul-taten van dit onderzoek zijn weergegeven in dit proefschrift.

Contents

Nomenclature ix

i

Introduction

i

1.1 Definition of Loads 2

1.2 Probability of Loads 6

1.3 Objective of this Study

2 Model Tests

ix

2.1 Historical Review of Some Experimental Work ix

2.2 Motivation for Present Model Tests x

2.3 Experimental Program xii

2.4 The Models xv

2.4.1 Geometries and Mass Distributions xv

2.4.2 Rigidity Properties xvii

2.5 Regular Wave Tests xviii

2.5.1 General Discussion of the Observed Phenomena xix

2.52 Effect of Forward Speed xxxvi

2.5.3 Effect of Bow Geometry xxxvi

2.6 Irregular Wave Tests xxxviii

2.7 Concluding Remarks xlvii

3 Volterra Modelling of Nonlinear Processes

xlix

3.1 The Third Order Volterra Model li

3.2 The Approximate Volterra Model liv

3.3 Validation of the Modelling lviii

3.3.1 Time Histories lviii

3.3.2 Response Spectra lxv

3.3.3 Sample Probability Densities lxix

3.3.4 Peak-Peak Probability Densities lxxv

Vi _Contents

3.4 Summary of the Results lxxvii

4 The Large Amplitude Relative Motion Problem

lxxix

4.1 Applied Solutions of the Bódy-Fluid Interaction Problem lxxxi

4.2 Setup of the Numerical Model lxxxiv

4.2.1 The Undisturbed Wave Pressure Integral lxxxvii

4.2.2 The Hydrodynainic Reaction Force xci

4.2.3 The Diffraction Force xcv

4.3 Results of the Simulations xcvi

4.3.1 Motions xcvi

4.3.2 Hull Girder Loads xcix

4.3.3 Results in Irregular Waves cvi

4.4 Summary of the Results cix

5

_{Discussion and Recommendations for Further Study}

₁₁₁

6 Conclusions 117

A Splash Zone Loads

127

A.1 Forces on a Panel in the Splash Zone . 127

A.2 The Probability Density Function 129

B Characteristics of the Approximate Volterra Modelling

133 B.1 Auto-Power Spectra and Spectral Moments 133

B.2 Statistical Moments 135

C Dynamics of a Ship in Large Amplitude Waves

139

C.1 Rigid-Body Motion Equations . . . 139

C.2 Structural Loads - 143

C.2.1 The Rigid Hull Girder Problem 143

C.2.2 The Flexible Hull Girder Problem 145

C.3 Towing Forces 148

D The Boundary Value Problem

149

D.1 The Exact Boundary Value Problem 149

D.2 The Linearised Boundary Value Problem 151

D.2.1 Radiation Forces . . . 153

Contents vii.

E Experiments with Divided Models

157

E.1 Forced Oscillation Experiments 157

E.1.1 First Order Reaction Forces 157

E.1.2 The Timman-Newman Relations 165

E.1.3 Higher Order Reaction Forces 167

E.2 Wave Force Measurements 169

E.2.1 First Order Wave Forces 169

E.2.2 Higher Order Wave Forces 172

Nomenclature

Nomenclature

Bold quantities dèfine vectors or matrices. Systems of axes:

(OXYZ) Righthanded earth-fixed coordinate system with positive Z-axe pointing upward (Gxyz)b Righthanded body-fixed coordinate system (Gxyz) Righthanded direction-fixed coordinate system,

traveling with the Vessel

Latin symbols:

g Acceleration of gravity constant

h Water depth

hi'(t1) First order time domain kernel

h2(t1, t2) Second order time domain kernel

h3(t1, t2, t3) Third order time domain kernel

k Wave number

m (3 X 3)-mass matrix

n Body-fixed normal vector 'pointing into the fluid = (ns, ,,

p Total pressure

PO Incident wave pressure

p(.) Probability density, function

t Time variable

t0 Initial time

y = (u, 'u,w)TVelocities along body-fixed axes

x

Coordinates of panel centres

y1,2,3(t) First, second, third order correlated responses

Ya,b,c(t) First, second, third order uncorre!ated responses

Mean response

A Wave amplitude

Added mass coefficients

B Bending moment-vector

x Nomenclature

B2(w) Second order frequency response function

B3(w) Third order high-frequency response function

B4 (w) Third order low-frequency response function

Bjk(w) Damping coefficients

D Wave directionality distribution function E Young's modnius of elasticity

E[.] Expected value of [.] F Body-fixed force-vector

F, Body-fixed force-vector component for j = i . ..6

Fn Froude number U/-/

G Ship geometry as a stochastic quantity G(w) Double-sided spectrum

H Wave height

H5 Significant wave height

Hi'(wi) First order frequency response function

H2(wi., w2) Second order frequency response function

H3(wi,w2,w3)Third order frequency response function I (3 X 3)-mass moments of inertia matrix

Kk(t)

Retardation functions

L Ship length

M Body-fixed moment-vector

M 5:hips mass

N Number of samples, cycles P(.) Probability distribution function

Q Shear force-vector

R

Transformation matrix

S Instantaneous wetted hnll surface

So Mean wetted hull surface

S(w) Single-sided spectrum

T

Transformation matrix

T Period; draught

Spectral peak period

U Steady forward speed

X Translations of COG in earth-fixed system X0 Initial position of COG in earth-fixed system

Y1,2,3(w) First, second, third order correlated responses

First, second, third order uncorrelated responses Z(w) Fourier transform of wave surface elevation

Nomenclature xi

3. Greek symbols:

a

Elilerian angles of rotation Damping of a joint

(3 Heading; wave direction

7

Stiffness of a joint

7jk Time-independent restoring coefficients Interval

6(.) Delta function

e Phase angle

((t)

Wave surface elevation Relative motion

k1 Average deviation

Ic3 Skew

K4 Kurtosis

A Wave length

Wave length of component in peak of wave spectrum Ak Time-independent damping coefficients

p, Mean

p,jk Time-independent added mass coefficients

Complex unity motion

p Density of water

a

Standard deviation

Total velocity potential

qs Steady forward speed perturbation potential

q5T Unsteady part of total velocity potential q5o Potential of undisturbed waves

Radiation potentials

q57 Potential of diffracted waves

'p Angular deflections of flexible joints Frequency of oscillation

Incident wave frequency Spectral peak frequency

= (p, q, r)T Angular velocities around body-fixed axes

4. Others:

BM1O Bending moment at station 10 (amidships)

BM15 Bending moment at station 15

(a quarter of the length aft of the bow) COG centre of gravity

FFT Fast Fourier Transformation pdf probability density function PDF probabifity distribution function

RAO Normalised amplitude of the jth harmonic component SF10 Shear force at station 10 (amidships)

SF15 Shear force at station 15

(a quarter of the length aft of the bow)

STF Salvesen-Tuck-Faltinsen strip method formulations

V gradient

Chapter 1

Intro dct ion

The probability of structural failùre of a ship should be as low as possible for a number of reasons. Considèr, for instance, the impact of environmental disasters and the factor of crew's safety. However, for reasons both practical and ecönomic, the probability of a ship failing does not have to be lòwer than acceptable. Acceptance criterIa have to be defined by governments., classification societies and other official institutions. The design procedure of a ship which is based on this second. approach requires the definition of an 'acceptable' probability of failure combined with a great sense of re-sponsibility, and tools for the accurate modelling of the loads in the ship's structure.

In practice, the design process of the ship construction still leans heavily on primarily semi-empirical rules. These rules were adapted oser the years based on practical experience and are easy to apply in the design process of conventional ships. In the 1992 rules, Lloyd's Register prescribes the de-sign vertical wave-induced midship bending moment by the following simple formula:

M = f1f2Mo

in which fi is a ship service factor, 12 is a wave bending moment factor which differs for hogging and sagging condition and M0 is the bending mo-ment as a function of the length, beam and block coefficient. In combination with the calculated still water bending moment and the sectional modulus of the construction, the maximum vertical bending stresses at deck and keel are calculated. The sectional modulus has to satisfy a minimum modulus, which is also prescribed as a function of the length, 'bam and block

2 _{Chapter 1. introduction}

cient and of the same ship service factor fi. Ships with large bow flareare treated in a special, but also empirical fashion.

For new types of ship designs, rules based on experience are not avail able and a need is felt to design, analyse and judge arbitrary ship designs with rational calculation techniques without or with a limited use of classic design-rules.

The total structural damage over the lifetime of a ship can be expressed in terms of fatigue damage induced by many small- and moderate-amplitude stress cycles in combination with extremely large-amplitude stress cycles caused by slamming impacts, which possibly inflict local damage due to overstressing, as well as additional fatigue damage caused by induced vibra-' tions.

Both commercial ship owners and navies, for different reasons, require a level of operability of their vessels which is as high as possible. The Oper-ability of the vessel is maitily determined by its motion behaviour in various wave conditions. The reason for a study performed by Bales et al (1982) was thè observation that Soviet warships did not appear to suffer the same degree of degradation as those of the U.S. Navy did in extreme sea states. The final objective was the design of ships with an improved seakeeping performance. This demand of 'a high operabifity of a vessel requires that. the ship's structure has a high damage-resistant capacity to prevent it from becoming damaged in dynamic wave load conditions. Scale enlargement of commercial vessels, required by the ship owners, was another reason for more detailéd load analyses.

From the ship builders point of view,, an accurate modéffing of the hull girder may result in a different distribution of material in the construction. 'Fatigue loads as well as extreme loads have to be considered in the design process. Especially when using less traditional materials such as there are high tensile steel or aluminium alloys, care has to be taken for the fatigue loads.

1.1

Definition of Lóads

FOr a more detailed analysis of the experienced load's in a ship's construction, it is helpful to make a distinction between the different types of hull girder loads In this study 'hull girder 'loads' are defined as the forces and moments in a hull' cross-seçtion,, i.e. internal loads, whkh make equilibrium with the excitation forces and moments, i.e. external loads, and the 'iñertial reactions

1.1. Definition of Loads 3

of the ship. Six load components are distinguished of which the vertical bending moment and shear force are the most important two in case of slender vessels sailing in head and bow waves. The other components are the lateral bending moment and shear force, the torsional moment and the longitudinal tension or compression force. Torsional moments are important in case of open container ships sailing in oblique waves for instance.

The total hull girder loads can be considered to be the sum of the fol-lowing contributions:

Still water loads at zero fOrward speed. These loads are the result of the difference between the weight and buoyancy distributionin stifi water. In vessels which are symmetric in the longitudinal vertical plane this load only has a significant meaning fOr the vertical bending moment and shear force. For heeled vessels, the load components in lateral and longitudinal direction can be of relevance as well. Caused by changes in cargo distribution for instance, this load can vary drastically within a short period.

Almost time-invariant loads due to the speed induced ship wave pat-tern. The consequence of a moderate forward speed of a vessel, i.e. sinkage and trim, mainly effects the vertical bending moment and shear

force.

Dynamic loads caused by the environmental conditions. These wave-induced hull girder loads are the net results of the excitation forces and the ship responses to these excitations as a rigid body, requiring the solution of the seakeepiug problem.

Slamming-induced loads or whipping loads. These loads are the inter-nal vibratory moments and shear forces which arise from an impulsive excitation of the hull structure causing dynamic elastic deflections of the ship. Global loads as well as local loads are affected by slamming. Springing loads. The presence of these loads is restricted to flexible hulls in which hull-deflection-induced shear forces and bending mo-ments occur as a distortion on the the rigid hull girder loads. In

fact every ship construction will experience springing loads since these loads can theoretically not be separated from the dynamic behaviour

of a ship. If the natural frequency of the hull girder is low enough, résonance in dynamic wave conditions is possible.

4 Chapter 1. Introduction

In the present study, the attention is focused on the third contribution in this list, ie. the hull girder loads in a rigid ship. Usually these loads are referred to as. wave-induced loads although this definition is a10 suitable

to describe springing loads, which are the wave-induced loads in an elastic vessel. Calculation of the waveinduced hull girder loads can be based on the widely available ship motion programs, varying from two dimensional linear strip theory programs to nonlinear three dimensional computer codes. l_n the early sixties already, Daizell (11964a) published hi observatións on measured niidshiip bending moments in severe wave conditions. Fròm those results it was obvióus that nonlinear phenomena played an important role. Figure 1.1 shows for two forward speeds as a function of the wave steepness the amplitudes of the bending moment in hogging and sagging condition as well as the measured dóuble heave and pitch amplitude

In most of the developed ship motion computer programs, the ship is modeled as a rigid body. An extensive description of the physical and me-chanical behaviour of a flexible body interacting with the surrounding fluid,

i.e. the concept of hydroelasticity, was given by Bishop and Price (1979). This type of modeffings allows the study of springing and slamming phe-nomena.

To obtain insight into the local stress distribution, a more détailed stress analysis has to be performed employing results from the hydrodynamic inter-aclion of the ship with the surrounding fluid in combination with an accurate structural model of the ship. Realizing this, Lin et al (11992) (American Bu-reau of Shipping) suggested the use of finite element methods to analyse a limited number of selected extreme stress conditions in a tanker. Identifica-tion of an extreme stress condiIdentifica-tion was based on estimated maximum hull girder bending moments and accelerations in irregular waves, in combina-tion with expected environmental condicombina-tions. At the instant of occurrence of this estimated extreme condition, the situation was frozen and an equiv-alent regular wave was determined. The. pressures on the hull as well as the resulting dynamics of the vessel were used as input for the stress analysis program. lin this semi-static analysis of the. extreme loads, it is implicitly assumed that the ship wifi not fail due to fatigue damage as long as the hull girder can withstand the most severe conditions. This omittance of a fatigue analysis is a shortcoming in a rational structural design process.

The general problem with the use of finite element methods for the anal-ysis of wave-induced stresses in a ship is the calculation of the distribution of the fluid pressure over the hull surface and the unknown inertia forces on the individual elements of the ship. To overcome this problem, an

in-tegral solution of the hydrodynamic problem describing the fluid-structure interaction and of the' elasticity problem is in fact the only theoretically jus-tifiable direct method. Amongst others, this subject was studied by Vorus and Hylari'des (1981), Nestegrd and 'Mejlaender-Larsen (1994,) and Price et al (1994). Practical' complications arise when the excitation due to the surrounding fl:uid becomes strongly nonlinear..

-a -.3 N N o I.. bi 4 u II- .00a

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Figure 1.1: Hogging and sagging midship bending moment coefficients _11H and jhs and double motion amplitudes as measured by DaIzell in a Mariner model (1964a) 04 00 .0 3,. o .0* .01 JO-* -I -S 040 .0000 0000 N N 0000 4 o 4.0030 1.1. Definition of Loads 5 .00 .00 .00 .30 n,

Il-:

00M o t-si 4 u JO - J.-0030 040

6 Chapter 1. Introdùction

1.2

Probability of Loads

The fatigue problem is closely related with the lifetime distribution of loads and induced stresses. Analysis of the most extreme stress condition alone does not satisfy in that context. Calculation of the probability of a ship failing requires the calculation of the lifetime distribution of the hull girder loads, local loads and the strength of the construction. A compact way to describe these loads is in terms of long-term probability density or distri-bution functions. The severity of the experienced loads in a seaway and consequentially the shape of their probability density functions depends on a number o parameters:

The environmental conditions. In an irregtilar seaway, the main exci-tations a vessel is exposed to are determined by the wave conditions. Parameters such as current and wind cannot be neglected in the 'anal-ysis of offshore structures but they are of minor importance for a ship underway. The wave cortthtions are characterised by the wave spec-trum parameters such as a significant wave height H3, a characteristic period T, a spectral shape S«(w) and a directionality function D. These wave conditions are of a probabilistic nature but are usually modeled as an irregular process stationary over a limited' period. The ship's geometry G, rigidity R and mass distribution M. These pa-rameters may slowly change over the lifetime and even during a journey of the ship caused by fuel consumption for instance. Different load-ing conditions as well as damage cause a change in mass distribution or geometry. Because of corrosion of the steel plating, slow changes in the rigidity occur. These parameters can usually be considered as constant during a relatively long period.

Operational parameters such as the ship's speed U and relative heading /3 with respect to the wave direction.

The short-term probability density function is defined as the probabifity density function in a stationary sea condition for a given geometry, relative heading and speed. Defining an arbitrary structural load as a stochastic quantity L, the long-term probability distribution function of the loads can be written as the summation of the conditional probability density func-tions multiplied by the joint probability density function of the environmen-tal, geometrical and operational parameters listed above. Symbolically, this summation can be written as

1.3. Objective of this Study 7

N

PL(L) = >2PL(LIC = c1) pc(C = c) (1.2), j=1

in which the condition C covers any combination of relevant parameters. In the range of linear responses, the conditional short-term probability den-sity functions of the loads can be calculated by application of the theory on stätistics of Gaussian processes in combination with linear ship motion programs.

Depending on the combination of the geometry, the operational condi-tions and the natural environment, the wave-induced hull girder loads can show a considerable nonlinear behaviour and therefOre, other techniques have to be applied to calculate the short-term probability density functions. For the long-term prediction of the load probability distribution function, information on the service profile of the vessel as well as on theenvironmental conditions in the ship's service area has to be available. The servicè profile is determined by parameters such as the service area or routes, seasons, loading condition, speed and course. These data have to be obtained from the statistics provided by ship operators. Data on the environmental conditións in the specific areas along the specific routes are provided by wave climate databases, which are an important source of uncertainties. In this context, the influence of the use of different wave climate databases, spectral shape, heavy weather maneuvering and voluntary speed reduction on the long-term probability: distribution function is a matter of interest too.

1.3

Objective of this Study

The present study is primarily concerned with the wave-induced loads in a hull cross-section in those conditions which introduce a nonlinear behaviour. This nonlinear behaviour affects the statistical characteristics of the loads in terms of probability density functions, important in a fatigue analysis, and the extreme wave load prediction.

A new extensive series of towing tank experiments was set up to evaluate the hull girder loads in regular and irregular waves. The bow geometry, the forward speed and the wave conditions were varied. The. experimental procedures and the observed trends in the results are described in chapter 2. Using the results obtained from the tests in regular waves, a method was derived to predict the shortterm probability density functions of the hull girder loads in irregular nonlinear conditions. The results of this method

8 Chapter 1. Introduction

were compared with results of irregular wave tests.. The derivation and validation of the method is described in chapter 3.

In chapter 4, a nonlinear ship motion program is described. This tool was used to approximate the nonlinear frequency response functions which were required to perform the statistical analysis described in chapter 3.

In chapter 5, a procedure is discussed that is the synthesis of the earlier developed tools. Following this procedure, it becomes possible to calcu-late the probability density functions òf the nonlinear hull girder loads in arbitrary wave spectra without performing simulations.

Chapter 2

Model Tests

2.1

Historical Review of Some Experimental

Work

Since many years the validation of ship motion computer codes is based on experimental results., obtained from towing tank experiments. The first reported model tests concerning the experimental determination of bending moments in waves were carried out by Sato (1938). This test program was started after the breaking of two Japanese destroyers in 1935. An unseg-mented brass model was used for the measurement of stresses in the hull bottom. From these stresses the bending moments were derived. This way the influence of the material and construction properties cannot be elirni-nated in the calculation of the hull girder loads.

Lewis (1954) recognised this problem as he proposed a method to mea-sure the pure bending moments by using a segmented model of a T2-SE-A1 tanker. An extensive series of experimental results on wave load measure-ments in a container vessel was presented by Tan (1972). The experimeasure-ments covered a range of speeds, frequencies and

headings.-The first results of systematical experiments, focussed on the nonlin-ear vertical hull girder load responses, were presented by Daizell (1964a, 1964b). These results were obtained by subjecting models of three variants of a Mariner ship, a tanker and a destroyer to a range of regular waves over a range of wave lengths and heights. The vertical bending moments were presented in hogging and sagging condition separately, not providing infor-mation about the harmonic components in the response signais. However, it was proved without a doubt that the sag/hog-ratio was not equal to unity,

IO _{Chapter 2. Model Tests}

which should be the case for linear signals. Furthermore the experiments showed that the sag/hog-ratios tended to be larger for the destroyer model and the Mariner variants than for the full tanker model.

Similar conclusions followed from a very extensive model test series,

re-ported by Murdey (1972). Thirty-three models representing a_{range of ship} forms with block coefficients ranging from 0.50 to 0.88 were towed. From these tests the shift in zero level and the double amplitude of the bending moments were published. Nethercote (1981) presented the model test re-sults for a frigate and again only the double amplitude and zero shift values for the vertical bending moments were presented. Irregular wave tests were performed too. Using the regular wave results, double amplitude values were predicted in irregular waves in conjunction with the measured irregular wave spectra.

Watanabe et al (1987, 1989) studied experimentally the noilinear effects of bow forms with varying flare on a model of the S-175 container ship. They concluded that an increased bow flare reduced the relative bow motion and. deck wetness. The vertical bending moments however increased with an increase of the bow flare.

Hay et al (1994) studied combined wave- and slam-induced loads from model tests and full scale trials of a CG-47 frigate. Vertical and lateral bending moment frequency response functions were derived from the com-bined measurement of the responses and waves in different sea states. It was concluded that the results for the response. amplitude operators did not depend on the sea state. However, when considering the vertical bending moment response functions in head wave conditions, differences with a max-imum of fifteen percent could be observed in the results from different sea states around the peak frequency. Results from regular wave tests were not presented.

2.2

Motivation for Present Model Tests

Actually the few data sets presented in literature and discussed above are not sufficient in the study of nonlinear hull girder loads. Most of the exper-iments were performed in regular waves only. From 'those. test results, too much information was lost due to the presentation of the results in terms of hog/sag-ratiós or double amplitudes. Until now no systematical results were presented showing the harmonic components of a response experienced in regular wave conditions in order to investigate the actual order of the

2.2. Motivation for Present Model Tests 11

process. It is clear from the reported results that we are not dealing with a linear process. An important question is whether it is possible to

approxi-mate the hull girder loads by a second order theory or whether even higher order contributions have to be taken into account.

It was frequently suggested that the heave and pitch motions of a ship hardly show any nonlinear response. O'Dea, Powers and Zselecsky (1992) reported measured higher harmonic heave and pitch responses for a S-175 model of only a few percent of the first harmonic response. Compared with

the earlier experienced nonlinearities in the hull girder loads, this seems to be a negligible effect. It has to be realised, however, that the accelerations are more strongly nonlinear than the displacements when we compare them with the magnitude of their linear components. This can be ifiustrated on the assumption of a third order, zero mean periodic displacement, which is written in terms of the first three harmonic components as

y(t) = y (iwt i) ₊ y2e(2iwt+2) _{+ y3e(31t3) (2.1)}

Hence, the displacement, velocity and acceleration are given in matrix no-tation by

It can easily be seen that relative to the. first harmonic component, the second harmonic acceleration is four times as large as the second harmonic displacement while the third harmonic component is even nine times larger. This much more pronounced nonlinear inertia effect directly influences the hull girder loads behaviour.

To generate data that can be studied and compared with numerical solutions in much more detail, new experiments had to be performed in regular waves. An extensive description of the tests in regular waves was given by the author (1993, 1994a) in which the results from more than 150 regular wave towing tank runs were collected.

The regular wave results can only supply a limited amount of information about the nonlinear behaviour in irregular waves. To examine the statistical properties of nonlinear hull girder loads in a realistic sea state, large data sets obtained from irregular waves tests were required as well.

y(t) (t) = i i

_2

i 2iw

_2

i 3iw

9w2

yi(t) y2(t) y3(t) (2.2)

12 Chapter 2. Model Tests

2.3

Experimental Program

Experiments were carried out with a Wigley hull form as basis geometry. The models were connected to the towing carriage by a tow cylinder, thus allowing heave motions and providing a constant forward speed. A defini-tion of the experimental setup and of the sign coiivendefini-tions used is given in figure 2.1.

st. O

u

10 15 18 20

Figure 2.1: Experimental setup and sign conventions

The parameters

The objective of the new experiments was to investigate the vertical hull girder loads in the nonlinear regime as a function of the most important parameters in regular and irregular waves. The variables investigated in regular waves were

the wave conditions, defined by wave amplitude and frequency, the forward speed, and

2.3. Experimental Program 13

3. the bow geometry of the model above the mean draught. Fixed test conditions were

the geometry of the mean under water hull form, the mass distribution along the length of the hull, the rigidity of the hull girder,, and

the heading, i.e. only tests in head waves were performed.

In irregular waves, only the effect of a variation of bow geometry was exam-ined. This was done in one wave spectrum and at just one forward speed.

The measured quantities

The quantities measured included the heave motion of the overall centre of gravity, the pitch motion around the overall centre of gravity, the vertical bending moment and shear force amidships (station 10) and at a quarter of the length aft of the bow (station 15), the towing force and the relative mo-tion at the bow (stamo-tion 18). Heave and pitch momo-tions were measured using potentiometers. The hull girder löads and the towing force were measured using strain-gauge type dynamometers. In the present study, the towing forces are not discussed. The wave surface elevation was measured using two wire-type wave probes mounted on the towing carriage, one alongside the model in the longitudinal centre of gravity and a second one diagonally in front ofthe model. The linear functioning of all transducers was checked.

Choice of the hull forms

To investigate the hull girder load responses as a function of the variables listed above, it was decided to perform tests on two Wigley hull variants. There were several reasons for the choosing of a Wigley hull form as the test object. First of all the Wigley hull form has a water plane area that does not change much around the mean draught. This means that the restoring forces are in principle linear as long as the motions are small. In addition, this type of hull form has a high resonance peak for pitch and for the relative motions at the bow in head wave conditions. This implies that in moderate wave conditions in which nonlinear wave effects do not play a significant role, large relative motions are experienced and the nonlinear behaviour of interest is induced. To investigate the effect of additional bow fiare, a second Wigley hull variant with a similar under-water hull form and

14 Chapter 2. Model Tests

main particulars was used but this time with extrapolated extremely flaring cross-sections in the bow region. It was reasoned that the choice of these two variants should result in pronounced differences between the two models in experienced higher harmonic responses due to the combination of large amplitude relative motions at the bow and the alternatively wall-sided and extremely flaring bow cross-sections. A more detailed description of the models is given in section 2.4.

The test conditions

A short overview of the test conditions is given, in table 2.1. The effect of the forward speed was only investigated for the original Wigley hull form. The influence of the bow geometry variation on the results was studied by comparing the results obtained for the two Wigley variants at Froude number 0.3. A more detailed description of the specific test conditions is given in the sections discussing the regular wave results,, section 2.5, and the irregular wave results, section 2.6.

The facilities

The experiments were performed in towing tank I of the Ship Hydromechan-ics Laboratory of the Deift University of Technology. The dimensions of this tank are: L x B x T = 142.0 x 4.3 x 2.5 meter. The towing tank has been equipped with a piston-type wave maker and a towing carriage can travel

along the length of the tank with a maximum speed of 7 rn/s.

Table 2.1: Test conditions

Model Speed Regular waves Irregular waves

Original Wigley Fn = 0.2 (0.99 m/s) w0 = 2.5 . . .7.0 rad/s

H/A=0...max

Original Wigley Fn = 0.3 (1.49 m/s) wo = 2.5 ...7.0 rad/s

H/)t=0...max

= 3.6 rad/s H3 = 8 cm Wigley with bow flare Fn = 0.3 (1.49 m/s) wo = 2.5.. .7.0 rad/s

H/\=0...max

H3 = 8 cm= 3.6 rad/s

Z4. The Models 15

2.4

The Models

2.4.1

Geometries and Mass Distributions

Based on the Wigley hull form definition, two towing tank models were designed. The normalised beam y of the under water ship is described by a polynomial in the x and zcoordinate according to

y = (1 z2)(1 - x2)(1 +0.2x2) + z2(1 - z8)(1 - x2)4 x E [-1,11, z E [-1,0] (2.3) Both models satisfied this under water hull form formulation, scaled to the main dimensions of L = 2.5 meter, B = 2.5/7 meter and T = 2.5/18 meter. One model had a constant waterline from the mean draught up to the deck, resulting in the traditional Wigley hull form definition with wall-sided cross-sections. From now on the model with the wall-sided cross-sections in the bow will be called the 'original' Wigley hull form. These characteristics cause the model to have a typical linear restoring coefficient in heave as well as in pitch mode.

The second hull form was a variant of the original' Wigley hull form. The cross-sections in the bow were extrapolated with bow flare. The bow contour from the mean draught up to the deck made an angle of 45 degrees with the still water plane. Table 2.2 shows the main characteristics of the Wigley geometries. The bow form variation is clearly illustrated m'figure 2.2.

Length (L) 2.500 m Beam (B) 0.357 m Draught (T) 0.139 m Displacement volume 69.57 m3 LIB 7.000 LIT 18.000 Block coefficient (CB) 0.561 Midship coefficient (CM) 0.909

Water plane coefficient 0.693

k/L

0.25

Figure 2.2: Plans of both Wigley variants

The mass distribution was arranged in such a way that the total centre of gravity in longitudinal and vertical direction corresponded with the centre of the water plane in still water. The individual centres of gravity, themasses and the mass-moments of inertia were calibrated for each module separately. These data are collected in table 2.3. The tow cylinder, mentioned in this table, did not contribute to the mass moment of inertia but did contribute to the model's mass.

Table 2.3: Characteristics of the individual modules

Section Mass [kgj XG [m] zG [m] Io [kgm2] I (aft) 31.51 -0.550 0.139 3.70 II 19.72 0.311 0.139 0.67 Ill (bow) 11.75 0.954 0.139 0.66 tow cylinder 6.60 - -1+11+111+ tow cylinder 69.60 0.000 0.139 27.17

2.4. The Models 17

2.4.2

Rigidity Properties

Since we were interested in the measurement of nonlinear wave-induced hì1l girder loadsrather than in the measurementofrealistic whipping and spring-ing loads, as high as possible natural frequencies of the construction were desired. In order to measure the first three harmonic components accurately, a lowest natural frequency was required which is a factor of at least six times the highest first order excitation frequency. The combination of the maxi-mum wave frequency, the condition of head waves and the highest forward speed resulted in the highest excitation frequency, which was approximately 2.3 Hz. This corresponded with a frequency of the third harmonic equal to 6.9 Hz and involved a required minimum natural frequency in vertical vibration mode of about two times 6.9 Hz or 13.8 Hz.

The models were built-up out of three rigid modules, connected to each other in longitudinal direction by combined shear force/bending moment strain-gauge dynamometers. The models were constructed in such a way that the stiffness of the hull girder was an order lower at the location of the dynamometers recording the hull girder loads than at an arbitrary cross-section in the individual cross-sections. This. way the hull girder can be considered as a series of rigid bodies, connected by hinged joints with a well-defined stiffness and damping. A general view of the model with bow flare is shown on photograph 2,1.

The shell of the individual modules was made of glass fibre reinforced pölyester. Extra ribbons of high density foam were glued into the model to stiffen the shell especially in vertical direction. For the construction of the bulkheads an aluminium, honeycomb structure with a Kevlar top layer was used, resulting in a very stiff basis plate. on which the dynamometers were mounted. The gaps between the different sections were made watertight with a very thin rubber seal.

The final construction resulted in two models with a lowest structural natural frequency in vertical mode in water of around 13 Hz and a nondi-mensional damping coefficient, derived from the exponential decay curve in water, equal to 0.013. The structural stiffness coefficient against bending was determined in air by measuring the static deflection as a function of an imposed moment, and was found to 'be equal to 21,200 Nm/rad. More details on the construction of the models. are given by Adegeest (1994a).

18 _{Chapter 2. Model Tests}

Photograph 2.1: The assembled model with bow flare, consisting of three rigid modules connected by strain gauge type dynamometers

2.5

Regular Wave Tests

The regular wave tests were performed in at least four different wave heights for each combination of model, speed and wave frequency. The wave heights the models were subjected to during these tests were more or less uniformly distributed over the range between the practically minimum possible steep-ness and the maximum steepsteep-ness. The maximum wave steepsteep-ness was de-terinined by the maximum allowable vertical motions with respect to the free space in the experimental setup (in the low wave frequency range), the degree of deck wetness or bottom slamming (around the resonance peak for pitch) or the wave breaking limit (in the high wave frequency range). Green water on deck and bottom slamming were only allowed to a limited extent. Bow flare slamming however occurred frequently and almost continuously in the steepest waves at the resonance peak for pitch. The measured responses were analysed up to the third harmonic component. All responses were ref-erenced to the wave surface elevation in front of the model. A correction of the phase shift was applied to account for the distance between the wave probe in front of the model and the model's centre of gravity.

2.5. Regular Wave Tests 19 .004 .003 002 .001 o Wave, 2nd harm. A1[mJ

Figure 23: Amplitudes of the measured. second harmonic regular wave com-ponents compared with the second order Stokes wave

2.5.1

General Discussion of the Observed Phenomena

At first the nonlinearities in the regular waves were investigated. It was ex-pected that the wave surface elevation satisfied the Stokes wave formulation, which is until the second order given by

((t) = A cos(wt - kx) + kA2 cos(2wt - 2kx) (2.4)

In figure 2.3, the amplitudes of the second harmonic components are collected for all measured waves during the whole set of experiments. In order to compare the results for arbitrary wave frequencies, the amplitudes were divided by the wave number k. Comparison of the measurement points with the plotted polynomial y = 1/2A2 shows that the nonilnearities in the waves were not more severe than those defined by the Stokes wave formula-tion. This means that in the steepest waves, a maximum second order wave amplitude was measured of only a few percent of the first harmonic.

20 _{Chapter 2. Model Tests}

a

Relathie MotionsL18, istharm.

I V V V V

V;

6 e SV . . E 4 a t

e.

_e V 2 I 2 3 4 e w0 (radial y Original (FnO.3) Original (Fn-O.2)

Figure 2.4: Relative motions at the bow measured on the original Wigley

Before concentrating on the hifluences onthe results caused by variations in the forward speed and bow fiare, a general impression on the experienced responses is obtained by discussing some examples of recorded signals, pre-sented below. The recordings show the responses of both Wigley hull vari-ants at Fn = 0.3 in regular waves with a similar amplitude. Examples are given for three wave frequencies varying from a wave length to ship length ratio A/L of approximately three to one. These recordings were selected based on the measured relative motion frequency response function given in figure 2.4.

The lowest frequencyjs a typical long wave condition. The ship smoothly follows the wave surface and the relative motions at the how are small com-pared with the wave amplitude. The intermediate frequency corresponds with the resonance peak frequency for the relative motion at the bow, re-sulting in a severe nonlinear behaviour of geometrical origin which is even in small amplitude waves. The highest frequency is somewhat higher than the resonance peak frequency. Both the heave and pitch response afready decreased considerably at this frequency.

o t

2.5. Regular Wave Tests 21

in all hull girder load time recordings., a high. frequency distórtion signal can be observed. This low amplitude distortion is caused by structural vibrations, representing the springing loads.

As a result of the chosen sign conventions, defined in figure 2.1, the bending moments are positive in sagging condition whereas the shearforces

are positive iñ hogging condition.

Figure 2.5 and 2.6 show the time traces as recorded in the longest waves. A nonlinéar pitch response is clearly observed for the Wigley with bow flare. A corresponding nonlinear behaviour of the bending moment responses is

also noticed. The original Wigley hull showed somewhat flattened peaks in the midship bending moment and shear force in sagging cnditiàn. A similar behaviour, but less pronounced, is observed in the cross-section at the bow. Generally the experienced loads are small since the models are following the wave, resulting in instantaneous dynamic conditions which are comparable with the still water condition.

In the intermediate frequency no significant nonlinear behaviour is ob-served in the motion responses for either one of the Wigley models. See the figures 2.7 and 2.8. The hull girder loads in the Wigley with bow flare, how-ever, clearly show a periodic nonlinear behaviour.. The bending moments amidships as well as in the forward are sharply peaked in sagging condition. Values in sagging are respectively twice the hogging bending moment amid-ships and almost three times the hogging moment in the forward. A similar behaviour is found for the shear force in the forward. In contrast with the other loads, the shear force in the midship cross-section has a large peak in hogging condition.

While the hull girder loads for the Wigley with bow flare increased dra-matically in sagging condition, the bending moment responses in theoriginal Wigley show somewhat flattened peaks in sagging condition. Differences be-tween the maximum experienced loads in sagging and hogging condition are of minor importance although it is obvious that the signals do contain higher order components as indicated by the flattened peaks and different upward and downward slopes of the recordings.

For the recordings showing results obtained in a wave with a wave length equal to A/L 1, figure 2.9 and 2.10, the responses resulting from the Wigley with bow flare are extremely peaked. It is worth noting that the bending moments in the bow almost reach the same magnitudes as the midship bending. moments. This is not the case in the original Wigley model in which the measured midship bending moment is almost twice the bending moment at the bow in sagging as well as in hogging condition.

22 _{Chapter 2. Model Tests} .05 E o .IOO -.05 Io Wave Heave Pitch

Bending Moment ordlO

Shear ForceordlO

Bending'Moment ord.15

Io

Time IJ

Figure 2.5: Responses of Wigley model with bow flare, Fn = 0.3, AlL = 2.74, H/A 1/55

2.5. Regular Wave Tests 23 .05 e Wave Heave Pitch

Bending Moment ord.1O

ShearForce ord1O

Bending Moment oud.15

to

ShearForceord.15

Time (e)

Figure 2.6: Responses of original Wigley model, Fn = 0.3, A/L = 2.74, H[À 1/55

o b0

24 Ghapt er 2. Model Tests .030 .015 -.015 -.030 0 30 Io 5 -5 I0 Wave

AAÂAAAA

Heave I0 Pitch

Bending Moment ord.1O

kÀÁÀÁÁAA.A.A.A

Shear Force ord.1O

Bending Moment ord.15

Shear Force ord.1 5

IO

Time (51

Figure 2.7: Responses of Wigley model with bow flare, Fu = 0.3, A/L = 1.36, HI) 1/75

20

Io

o

2.5. Regular Wave Tests 25

E

z

Wave

Pitch

Bending Moment ord.1O

Shear Force ord.1O

to

Shear Force ord.1 5

Timo (81

Figure 2.8: Responses of original Wigley model, Fn = 0.3, A/L 1.36,

H/A 1/75

BendlngMoment ord.15

26 _{Chapter 2. Model Tests} 5.0 2.5 C, o U -2.5 5.0 0 30 20 z E z 40 o -IO Wave Heave Pitch

Bending Moment ord.1O

Shear Force ordlO

Bending Moment ord.15

Shear Force ord.1 5

I0

Io

Time [s]

Figure 2.9: Responses of Wigley model with bow flare, Fn = 0.3, A/L = O99, H/.\ 1/30

2.5. Regular Wave Tests 27 40 30 20 o -lo E z Wave Heave Pitch

Bending Moment ord.1O

Bending Moment ord.15

Shear Force ord.1O

'o

Shear Force ord.1 5

Time I°J

Figure 2.10: Responses of original Wigley model, Fn = 0.3, AlL = 0.99, H/A 1/30

28 _{Chapter 2. Model Tests}

Heave Pitch

o

w (radis] w (radis]

Figure 2.11: Fast Fourier Transforms of some responses of the Wigley with bow flare, Fn = 0.3, À/L = 1.36, H/A 1/75

o. 5000 4000 3000 2000 1000 o 503 o 3000 2000 1000 15 15 5 w (radis] BM1 O 10 o 5 u) (rad/s] BM1 5 lo IS o Q t Q

I

IO 5 IO IS 15 to

2.5. Regular Wave Tests 29

To get a better understanding of the characteristics of the ship's re-sponses in extreme conditions, the signals were transformed to thefrequency domain by means of a Fast Fourier TransformatiOn (FFT) in which no data window was applied. It was verified whether the results were sensitive to the application of different types of windows. This was not the case As an example the responses shown in figure 2.7 and 2.8 are transformed to the frequency domain. These recordings were obtained in a nearly green-water. on-deck-condition, or a maximum immersion of the bow. Some transformed results are shown in the figures 2.11 and 2.12 for the Wigley with and with-out bow flare respectively.

The graphs showing the FFT's of the heave and pitch motion show mi-nor higher harmonic peaks for the Wigley with bow flare. In case of the original Wigley hull form, only the second harmonic pitch motion can be distinguished graphically,

Observing the FFT's of the bending moment responses however, it is obvious that all possible harmonic components are present in the recordings. For the Wigley without bow flare, the peaks of the higher harmonic responses gradually become smaller. Responses up to the third harmonic are clearly distinguished and although higher harmonic components are very small, they can stifi be located. In case of the Wigley with bow flare, not only significant second and third harmonic components are experienced, but also a peak at the frequency of the fourth harmonic that is even higher than the peak of the third harmonic component. Spikes at five and six times the frequency of the first harmonic are significantly higher than for the original Wigley responses,

The natural frequency for structure vibrations, which was measured be-fore and was about 13 Hz for both models, can also be located. From the graphs showing the FFT's, it is clear that the structural natural frequency is far enough beyond the frequency of the third harmonic component at these particular wave frequencies to avoid excessive structural resonance vibra-tions.

The results obtained for the second speed, Fn 0.2, are not discussed. The trends observed for that speed were similar with the behaviour of the corresponding model at Fn = 0.3, see Adégeest (1994a).

30 Chapter2. Model Tests

1500

o

w (rad/s] (û Erad/sl

Figure 2.12: Fast Fourier Transforms of some responses of the original Wigley, Fn = 0.3, AlL = 1.36, H/A 1/75

2000 2000 1500 1000 50' 5 w (radial BM1 O IO 15 o 1000 800 000 400 200 5 w (radial BM1 5 lo '5 0 5 Q - 2 a IO - Q Q Q 2 i L1 IS 6 IO '5 Heave Pitch 12 0 Q 8

£

2.5. Regular Wave Tests 31

It has to be realised that the Fourier components contain mixed

con-tributions from different orders of resonses. The first harmonic response for instance also contains a contribution of a cubic response, i.e. the low frequency third order response. The presence of a fourth order component implies, besides a fourth harmonic, a contribution to the mean value as well as to the second harmonic whereas the first and third harmonic components are affected by a possible fifth order contribution and so on.

Although harmonic components higher than the third one are clearly identified, from now on the recordings obtained from regular wave tests are analysed up to the third harmonic component. The classical Fourier analysis was applied, based on a minimum of twenty fundamental periods exactly. Because of the mixed character of a harmonic component, which contains contributions from different orders of responses, the responses are carefully called first, second and third harmonic responses instead of first, second and third order responses. In the graphs containing the processed data, the first, second and third harmonic components were divided respectively by the first order wave amplitude, the square of the amplitude and the cube of the amplitude. The mean value measured in waves was first corrected for the measured speed-dependent zero offset in still water. The corrected mean value is presented as a frequency response function value,, divided by the square of the first order wave amplitude. These normalised harmonic components are referred to as the Response Amplitude Operators of the jth harmonic component, in symbolic notation written as RAO.

32 Chapter 2. Model Tests

For some responses, the RAO's as defined above are shown in the fol-lowing figures. The first harmonic heave and pitch amplitudes are shown in figure 2.13. Beíiding moment responses are shown in figure 2.14 and 2.15 for the measured mean value and the amplitudes of the first three harmonic components. For more information on measured phases and other responses, a reference is made to Adegeest (1994a).

As can be seen in the figures presenting the RAO's of the. different har-monic components of the bending moments, considerable 'scatter' seems to occur in the normalised experimental results. However, if the processed' results are plotted as a function of the wave amplitude, it becomes clear that this deviation of measurement points in the frequency domain shows a consistent trend as a function of the wave amplitude. This is illustrated by an examplé, figure 2.16, showing some results obtained for

o = 4.5

radis, which is close to the resonance peak frequency for relative motions at the bow. The origin of this phenomenon lies in the aforementioned mixing behaviour of different orders of responses.

s a E 200 ISO ISO o

° Bow tiare (FnO.3) V Originai(FnO.3)

OrIginaI(FnO.2)

Figure 2.13: First harmonic heave ¿ud pitch response

3 4 5 7 2 3 5 e

% a, [radial

Heave, istharm. Pitch, ist harm.

2.0,

V.5:

E: I.0

E E z o 400 200 s D0 H V O

os

D I . H

t

D H0 D D D o H o H 2 3 4 5 a (radJ e 7 2 3 I 5 (radial

BM1O, 2nd harm. BM1O, 3rd harm.

4000 av L5x105 3 2000 B Do O D a- OD V a- I.0xI00 E 2000

B;

_u, V V D D V

Dv

05x103 V

'u

1000, H I V.

Vv

a '

$ s. V

D9

Pio o o

i .i

.. E z 4000 300° D D D O D D o2000 V D 1000 D

BM1O, istharm. BM1O, Mean

2 3 4 5 e 7 2 3 4 5 7

a (radis]

D Bow flare (En-03) a (radis]

O Original (Fn0.3)

Original (Fn02)

Figure 2.14: Measured harmonic components of midship bending moment

2.5. Regular Wave Tests 33

600 5000

34 Chapter ₂ Model Tests 300 loo oo o 2 D D BD s a. sa D D! .u..

'..

3 4 5 m (radJ o 7 n e UD D0 V V: (rad/s] -2500 .5000 2 15x100

D Bow haze (FnO.3) V Original (FnO.3) Original (FnO.2) a D D D : I 5V

e

D a D D D D Da V V D VVV V V0

:I

aU

Î

.

w 3 4 5 (rad/sJ

Figure 2.15: Measured harmonic components of bending moment at a quar-ter of the length from the bow

o 7 V

.,

E'

JSSlSÇ

_{I !}

o o o 2 3 4 5 o 7 2 s 200 E E z o D OD D D o U D

'

L 2500 a-E E z O

i

BMI5,2ndharm BMi5, 3rd harm

o

4 5

0 (radis]

7

300 O 1500 6000 ¶4500 13000 1500 o BM15, 2ndharm 4000 a-E 2000 00 o t.04I0

a

E E 05x100 o

A

/

i

..____ -9- -., BM15, 3rd hann.

Figure 2.16: Measured normalised harmonic components of the bending moment at the bow as a function of the wave amplitude, wo = 4.5 rad/s

2.5; Regular Wave Tests 35 /

o .01 .02 .03 .04 o .01 .02 .03 .04

Wave amplitude [ml Wave amplitude [ml

0 .01 .02 .03 .04 0 .01 .02 .03 .04

Wave amplitude 1ml

Original (Fn=O.2) Wave amplitude (mJ

'--c Original (FnoO.3)

D---a Bow fiare (Fn=O.3)

e-.

----I.-.

BM15, ist harm. BM15, Mean

200

E

a:

36 _{Chapter 2. Model Tests}

2.5.2

Effect of Forward Speed

Studying the results showing the first harmonic heave and pitch RAO's for Fn = 0.2 and Fn = 0.3, common trends are observed such as an increase in heave response for higher speeds. A similar behaviour was found for the pitch motion, which also became more severe at higher speeds. As a consequence, the same- result was. obtained fòr the relative motion frequency response functions, shown already in figure 2.4.

During the experiments, it Was observed that the peak values of the first harmonic ' hull girder load frequency response functions are slightly higher for the higher speed. In the low and high frequency range, the curves for both Froude numbers tend to the same values.

The peak frequency of the higher harmonic responses consistently coin-cides with the resonance peak frequency of the relative motion at the bow. The heights' of the peaks also appear to be higher for the highest speed. Both observations emphasise the close relationship between the relative motion of the bow and the degree of nonlinearity in the hull girder load responses.

2.5.3

Effect of Bow Geometry

A comparison between the heave frequency response functions for the Wigley hull form with and without bow flare shows that some differences occur between the first harmonic heave- motions of the two models in the reso-nance peak. Differences between the first harmonic pitch motions of the two models are negligible. Despite the presence of bow flare, an almost per-fect correspondence is found. Apparently, the mass moment of inertia of the model is so big that the additional bow flare is not enough to change the pitch motion noticeably. O'Dea and Troesch (1986) already discussed the- sensitivity of the heave response to bow flare. They explained similar deviations between the measured heave responses in waves of different am-plitude as'non]inearities, induced by the bow geometry. It was also found that the pitch motions were less sensitive to those nonlinear effects. Both observations are in correspondence with the present experimental results.

In the low and high frequency range the first harmonic hull girder load responses are similar for the Wigley with and without bow flare. For the Wigley with bow flare, slightly higher values are found for the bending mo-ment in the forward as well as-for the midship bending momo-ment at frequencies in the relative motion resonance peak. An explanation for these differences is the low-frequency contribution from third or even fifth order responses.

2.5. Regular Wave Tests 37

Later in this report, this will be confirmed in chapter 4.

Noteworthy results were found for the mean value of the bending mo-ments. While only very small mean values were found for the Wigley without bow flare, clear sharp peaks were measured, especially for the bending mo-ment responses, as is shown iñ figure 2.14 and 2.15. The clifferrtces in higher harmonic responses are particuhrly clear in the bending moment response in the cross-section at station 15. The peak values of the frequency response functions for the Wigley with bow flare are more than three times as high as those for the Wigley with wall-sided cross-sections. The same phenomenon, but less pronounced, can be observed in the midship region.

38 Chapter 2. Model Tests

2.6

Irregular Wave Tests

Experiments in irregular waves were performed with both Wigley models at one forward speed corresponding to Fu = 0.3. Per model 50 runs were conducted with a length of 50 seconds each at a sample frequency of 40 Hz. This corresponded with 2500 seconds of data per model, or 100,000 data samples per measured response. For a ship with a length of 150 meter this period is equivalent to over five hours of recordings.

All tests were conducted in different realisations of one single wave spec-trum. The realisations were stored on tape. The wave spectra the models were subjected to are shown in figure 2.17(left).

.00016 o 20 IS w C a Q Io S o

Figure 2.17: Realised wave spectra during experiments with both Wigley hull variants (left) and the probability density of the samples compared with the equivalent Gaussian probability density function (right)

The graphs were constructed by taking the average of the fifty realised wave spectra after finishing all the runs. It is shown that the realised spectra resembled each other reasonably well. The peak frequency of the incident wave spectrum, around 3.6 rad/s, was just below the peak frequency of the

2 4 6 B IO .jß -.05 0 .06 .10

o (rad/s] Wave Surf a Elevation (mJ

Realized Wave Spectra Wave Surface Elevation

Bow Rare, Fn=O.3 - Gauss (theory)

2.6. Irregular Wave Tests 39

relative motion response function in regular waves at the same speed. The variance of the wave signal corresponded with a significant wave height of about eight centimeters, which resulted in a smooth wave spectrum with a significant wave steepness H3/À 1/60. Using this wave spectrum, large amplitude relative motions at the bow were induced already in mildly sloped waves. To verify the Gaussiannity of the wave surface elevation, both the experimentally found probability density function and the theoretical Gaussian equivalent were plotted in figure 2.17 (right). This result did not give any indication that the incoming wave itself contained significant nonlinear components.

Examples of some recorded irregular responses are shown in the fig-ures 2.18 and 2.19. The figfig-ures show the responses of the original Wigley and the Wigley with bow flare respectively in a comparative realisation of the wave spectrum. Both models experienced a severe slam after about 7.5 seconds. This is clear from the sharp transient-load peaks in the bending moment recordings and the induced decaying whipping vibrations. Wave-induced springing vibrations can also be observed. By filtering the signals at a cutoff frequency of 4 Hz, the high frequency structural responses were eliminated from the recordings causing an important decrease of the maxi-mum peak values as well. Filtering wasperformed by taking the Fast Fourier Transform of the signal as ne block of data, cutting off the high frequency tail of the resulting spectrum and inverse Fourier transformation of the spec-trum to a new, filtered time history.

It can be seen that in the Wigley with bow fiare, the maximum bending moments in sagging condition are considerably larger than in hogging condi-tion. Especially in the forward, sagging bending moments up to three times the hogging value were found. Furthermore, it appeared that the sagging moments in the forward reached similar magnitudes as the sagging moments amidship. The same behaviour was not observed in the original Wigley hull form.

40 _{Chapter 2. Model Tests} -5 15 -5 Wave o

i

Bending Moment ordlO (filtered)

Bending Moment ord.l 5 (unfiltered)

25

Timefsj

Figure 2.18: Some responses of the original Wigley in irregular waves, Fn =

0.3

BendingMoment ord.15'(flltered) 25

Héave 25

25

2.6. irregular Wave Tests 41 .10 .05 z z 30 30 20 lOo Bending Momentord.1O(unflltered)

Bending Moment ord1O (filtered) Wave

Bending Momentord.15 (filtered) Bending Moment ord.1 5 (unfiltered)

w»

25

20 25

Turno. (s]

Figure 2.19: Some responses of the Wigley with bow flare ¡n irregular waves,

Fn=O.3

Heave 25 o Pitch 25 5 lo Io 20 Io