Fulltext01

Experimental investigation of wave
induced vibrations and their effect on

the fatigue loading of ships
A thesis submitted in partial fulfillment of the

requirements for the degree of
doktor ingeniør
by
Gaute Storhaug
Oslo, 11th of September, 2007
Department of Marine Technology

Faculty of Engineering Science and Technology
Norwegian University of Science and Technology
Abstract
This thesis represents an attempt to reveal and explain the mysterious excitation sources
which cause global wave induced vibrations of ships. The wave induced vibrations of
the hull girder are referred to as springing when they are associated with a resonance
phenomenon, and whipping when they are caused by a transient impact loading. Both
phenomena excite the governing vertical 2-node mode and possibly higher order modes,
and consequently increase the fatigue and extreme loading of the hull girder. These effects
are currently disregarded in conventional ship design. The thesis focuses on the additional
fatigue damage on large blunt ships.
The study was initiated by conducting an extensive literature study and by organizing an
international workshop. The literature indicated that wave induced vibrations should be
expected on any ship type, but full scale documentation (and model tests) was mainly
related to blunt ships. While the theoretical investigation of whipping mostly focused on
slender vessels with pronounced bow flare, full scale measurements indicated that whipping
could be just as important for blunt as for slender ships. Moreover, all estimates dealing
with the fatigue damage due to wave induced vibration based on full scale measurements
before the year of 2000 were nonconservative due to crude simplifications. The literature
on the actual importance of the additional fatigue contribution is therefore scarce.
The workshop was devoted to the wave induced vibrations measured onboard a 300m iron
ore carrier. Full scale measurements in ballast condition were compared with numerical
predictions from four state-of-the-art hydroelastic programs. The predicted response was
unreliable, and the programs in general underestimated the vibration level. The excitation
source was either inaccurately described or lacking. The prediction of sea state parameters
and high frequency tail behavior of the wave spectra based on wave radars without proper
setting and calibration was also questioned. The measurements showed that vibrations in
ballast condition were larger than in the cargo condition, the vibration was more correlated
with wind speed than wave height, head seas caused higher vibration levels than following
seas, the vibration level towards beam seas decayed only slightly, and the damping ratio
was apparently linear and about 0.5%. The additional vibration damage constituted 44%
i
ii
of the total measured fatigue loading in deck amidships in the North Atlantic iron ore
trade, with prevailing head seas encountered in ballast condition.
Four hypotheses, which may contribute to explain the high vibration levels, were formu-
lated. They include the effect of the steady wave field and the interaction with the unsteady
wave field, amplification of short incident waves due to bow reflection, bow impacts includ-
ing the exit phase and sum frequency excitation due to the bow reflection. The first three
features were included in a simplified program to get an idea of the relative importance.
The estimates indicated that the stem flare whipping was insignificant in ballast condition,
but contributed in cargo condition. The whipping was found to be sensitive to speed.
Simplified theory was employed to predict the speed reduction, which was about 5kn in
5m significant wave height. The estimated speed reduction was in fair agreement with full
scale measurements of the iron ore carrier.
Extensive model tests of a large 4-segmented model of an iron ore carrier were carried
out. Two loading conditions with three bow shapes were considered in regular and irreg-
ular waves at different speeds. By increasing the forward trim, the increased stem flare
whipping was again confirmed to be of less importance than the reduced bottom forces in
ballast condition. The bow reflection, causing sum frequency excitation, was confirmed to
be important both in ballast and cargo condition. It was less sensitive to speed than linear
springing. The second order transfer function amplitude displayed a bichromatic sum fre-
quency springing (at resonance), which was almost constant independent of the frequency
difference. The nondimensional monochromatic sum frequency springing response was even
higher. The sum frequency pressure was mainly confined to the bow area. Surprisingly, for
the sharp triangular bow with vertical stem designed to remove the sum frequency effect,
the effect was still pronounced, although smaller. The reflection of incident waves did still
occur.
In irregular head sea states in ballast condition whipping occurred often due to bottom
bilge (flare) impacts, starting with the first vibration cycle in hogging. This was also ob-
served in cargo condition, and evident in full scale. This confirmed that the exit phase,
which was often inaccurately represented or lacking in numerical codes, was rather impor-
tant. Flat bottom slamming was observed at realistic speeds, but the vibratory response
was not significantly increased. Stern slamming did not give any significant vibration at
realistic forward speeds.
The fatigue assessment showed that the relative importance of the vibration damage was
reduced for increasing peak period, and secondly that it increased for increasing wave
heights due to nonlinearities. All three bows displayed a similar behavior. For the sharp
bow, the additional fatigue damage was reduced significantly in steep and moderate to
small sea states, but the long term vibration damage was less affected. The effect of the
bulb appeared to be small. The contribution of the vibration damage was reduced signif-
icantly with speed. For a representative North Atlantic iron ore trade with head sea in
iii
ballast and following sea in cargo condition the vibration damage reduced from 51% at full
speed to 19% at realistic speeds. This was less than measured in full scale, but the damping
ratio of 1-3.5% in model tests was too high, and the wave damage in following seas in cargo
condition was represented by head sea states (to high wave damage due to too high en-
counter frequency). Furthermore, the contribution from vibration damage was observed to
increase in less harsh environment from 19% in the North Atlantic to 26% in similar World
Wide trade. This may also be representative for the effect of routing. The dominating
wave and vibration damage came from sea states with a significant wave height of 5m. This
was in agreement with full scale results. In ballast condition, the nonlinear sum frequency
springing appeared to be more important than the linear springing, and the total springing
seemed to be of equivalent importance as the whipping process, which was mainly caused
by bottom bilge (flare) impacts. All three effects should be incorporated in numerical tools.
In full scale, the vibration response reached an apparently constant level as a function of
wave height in both ballast and cargo condition in head seas. This behaviour could be ex-
plained by the speed reduction in higher sea states. The vibration level in cargo condition
was 60-70% of the level in ballast condition. Although common knowledge implies that
larger ships may experience higher springing levels due to a lower eigenfrequency, a slightly
smaller ore carrier displayed a higher contribution from the vibration damage (57%) in the
same trade, explained by about 1m smaller draft. Moreover, the strengthening of the
larger ship resulted in a 10% increase of the 2-node eigenfrequency. The subsequent mea-
surements confirmed that an increased hull girder stiffness was not an effective means to
reduce the relative importance of the vibration damage.
The relative importance of the excitation sources causing wave induced vibration may differ
considerably for a slender compared to a blunt vessel. Therefore, full scale measurements
on a 300m container vessel were briefly evaluated. The damping ratio was almost twice as
high as for several blunt ships, possibly due to significant contribution from the container
stacks. The reduced relative importance of the vibration damage with increasing wave
height for the iron ore carrier in full scale was opposite to the trend obtained for the con-
tainer vessel. Less speed reduction in higher sea states was confirmed, and the whipping
process was apparently relatively more important for the container vessel. Both for the
blunt and slender ship of roughly 300m length, the total fatigue damage due to vibration
was of similar importance as the conventional wave frequency damage. The contribution to
fatigue damage from wave induced vibrations should be accounted for, for ships operating
in harsh environment with limited effect of routing, especially when they are optimized
with respect to minium steel weight.
The four hypotheses were all relevant in relation to wave induced vibrations on blunt ships.
Further numerical investigation should focus on the sum frequency springing caused by bow
reflection and the whipping impacts at the bow quarter. The wave amplification, steady
wave elevation and the exit phase must be properly incorporated. When it comes to design
by testing, an optimized model size must be selected (wall interaction versus short wave
iv
quality). The speed must be selected in combination with sea state. The wave quality
must be monitored, and a realistic damping ratio should be confirmed prior to testing. For
the purpose of investigating sum frequency excitation, a large restrained bow model tested
in higher waves may be utilized to reduce uncertainties in the small measured pressures.
Acknowledgements
Prof. Torgeir Moan and Prof. Odd M. Faltinsen have been my supervisors. They took
different roles, and I am deeply grateful for their contribution. They were professional,
positive, supportive and understanding, and they serve as an inspiration for me and oth-
ers. I am also pleased to see that Prof. Moan and Prof. Faltinsen have continued the
research within this field at CeSOS.
The Department of Marine Technology granted me a four year scholarship including one
year as a research assistant, and DNV awarded me a scholarship to cover other expenses.
I am thankful for their funding, the opportunity I have been given and the flexibility they
have shown in order for me to finish this thesis. CeSOS, Marintek and Anders Jahre’s
fund sponsored the experiments. The experiments were expensive, resource demanding,
but rewarding. I would like to thank the staff of Marintek, in particular Ole David Økland,
for their comradeship and flexibility.
DNV gratefully initiated the workshop, which gave me the opportunity to meet many
good collegues: Prof. Jørgen Juncher-Jensen, Dr. Finn Rüdinger and Dr. Jelena Vidic-
Perunovic from DTU, Michiel van Tongeren from DUT, Jens Bloch Helmers, Gabriel Holts-
mark and Øyvind Lund-Johansen from DNV, Prof. XueKang Gu from CSSRC and Dr.
Ole A. Hermundstad and Dr. Mingkang Wu from CeSOS. Thanks for many fruitful dis-
cussions and positive attitude.
DNV provided free access to extensive full scale measurements, which were helpful to my
understanding of wave induced vibrations. This resulted in four papers together with
my good collegues Gabriel Holtsmark and Erlend Moe from DNV and PhD student Ingo
Drummen.
I enjoyed my stay in Trondheim, where I met many good colleagues, who made the time
memorable and the working atmosphere excellent. Special thanks to Torgrim Driveklepp,
Dr. Lars Rønning and Dr. Svein Ersdal for many evenings where world problems were
v
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solved.
I am honored to have Prof. Jørgen Juncher Jensen (DTU), Dr. Svein Skjørdal (Grenland
Group) and Prof. Sverre Steen (administrator, NTNU) as the doctoral committee. They
have a strong background within the same research topic, and their publications have also
served as an inspiration. I am grateful for their thorough comments and criticism of this
thesis, and I am looking forward to the discussions on the day of the defense.
Finally, I would like to thank my family for all support, patience and for listening to my
occasional frustration. I am also sorry for the days I was away from my kids, who were
born in Oslo in September 2004. Unfortunately, only one third of the model experiments
was finished by then, and there was still 9 months of the PhD period left.
This thesis is dedicated to my twins, Simen and Elias.

Contents
Abstract i
Acknowledgements v
Contents vii
Nomenclature xiii
1 Introduction 1
1.1 Background and motivation for the present work . . . . . . . . . . . . . . . 1
1.1.1 Recent experiences from a ship in operation . . . . . . . . . . . . . 2
1.1.2 Comparison between measured and predicted vibration response . . 3
1.1.3 Today’s rule requirements . . . . . . . . . . . . . . . . . . . . . . . 4
1.2 A dive into literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2.1 The beginning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2.2 Ship measurements considering wave induced vibrations . . . . . . . 6
1.2.3 Model experiments considering wave induced vibrations . . . . . . . 11
1.2.4 Springing and whipping theories . . . . . . . . . . . . . . . . . . . . 14
1.3 Purpose of the work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
1.4 Outline of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2 Theory 23
2.1 Hydrodynamic theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.1.1 Hypothesis 1: Steady wave elevation . . . . . . . . . . . . . . . . . 23
2.1.2 Hypothesis 2: Amplification of incident wave at the bow . . . . . . 26
2.1.3 Hypothesis 3: 3D impact forces . . . . . . . . . . . . . . . . . . . . 27
2.1.4 Hypothesis 4: Second order velocity potential and wave elevation . 28
2.1.5 Speed reduction due to added resistance in waves . . . . . . . . . . 29
2.2 Structural representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.2.1 Modal decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . 33
vii
viii CONTENTS
2.2.2 Analytical solution of the beam equation with shear deformation . . 35

2.2.3 Finite element approach . . . . . . . . . . . . . . . . . . . . . . . . 38
2.2.4 Segmented models . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
2.2.5 Single degree of freedom system . . . . . . . . . . . . . . . . . . . . 39
2.3 Damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
2.3.1 Linear damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
2.3.2 Nonlinear damping . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
2.3.3 Methods to estimate linear and nonlinear damping . . . . . . . . . 41
2.4 Transfer functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
2.4.1 Linear transfer functions . . . . . . . . . . . . . . . . . . . . . . . . 43
2.4.2 ’Nonlinear’ transfer functions . . . . . . . . . . . . . . . . . . . . . 44
2.4.3 Second order transfer functions . . . . . . . . . . . . . . . . . . . . 44
2.5 Time series analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
2.5.1 Estimate of amplitudes . . . . . . . . . . . . . . . . . . . . . . . . . 45
2.5.2 Spectral density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
2.5.3 Wavelet transformations . . . . . . . . . . . . . . . . . . . . . . . . 48
2.6 Waves and sea states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
2.6.1 Irregular waves and spectral density . . . . . . . . . . . . . . . . . . 49
2.6.2 Wave heights exciting linear and second order springing . . . . . . . 51
2.7 Fatigue damage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
2.7.1 Rainflow counting . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
2.8 Long term fatigue analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
2.8.1 Use of response surface to extrapolate results . . . . . . . . . . . . 54
3 Simplified analysis of vibration response and speed reduction 57

3.1 Simplified analysis of whipping from stem flare impacts and linear springing 57
3.1.1 Simulations in regular waves in ballast condition . . . . . . . . . . . 59
3.1.2 Simulations in irregular waves in ballast and cargo condition . . . . 62
3.2 Calculated speed reduction in different sea states . . . . . . . . . . . . . . 66
4 Experimental methodology and analysis procedures 69

4.1 Strategy of the test program . . . . . . . . . . . . . . . . . . . . . . . . . . 69
4.2 Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
4.2.1 Model description . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
4.2.2 Measurement devices and their locations . . . . . . . . . . . . . . . 75
4.2.3 Model scaling laws and nondimensional properties . . . . . . . . . . 75
4.3 Uncertainties in the experimental data . . . . . . . . . . . . . . . . . . . . 76
4.3.1 Visualization of wave quality using Wavelet . . . . . . . . . . . . . 79
4.4 Procedure for estimating damping and natural frequency . . . . . . . . . . 83
4.4.1 Damping by decay tests . . . . . . . . . . . . . . . . . . . . . . . . 83
4.4.2 Damping estimates in irregular sea . . . . . . . . . . . . . . . . . . 84
4.5 Estimation of mode shapes. . . . . . . . . . . . . . . . . . . . . . . . . . . 87
4.6 Procedure for estimating transfer functions . . . . . . . . . . . . . . . . . . 90
CONTENTS ix
4.6.1 Linear transfer functions . . . . . . . . . . . . . . . . . . . . . . . . 90

4.6.2 ’Nonlinear’ transfer functions . . . . . . . . . . . . . . . . . . . . . 96
4.6.3 Second order transfer functions . . . . . . . . . . . . . . . . . . . . 97
4.7 Procedure for calculation of fatigue damage . . . . . . . . . . . . . . . . . 99
4.7.1 Scatter diagrams and investigated sea states . . . . . . . . . . . . . 100
4.8 Method for identifying whipping . . . . . . . . . . . . . . . . . . . . . . . . 101
4.8.1 Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
4.9 The relative importance of linear and monochromatic second order excitation105
5 Model test results 111

5.1 Damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
5.1.1 Damping estimates for the 2-node mode from decay tests . . . . . . 112
5.1.2 Spectral method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
5.1.3 Damping in higher order modes . . . . . . . . . . . . . . . . . . . . 117
5.2 Wave amplification and added resistance in waves . . . . . . . . . . . . . . 118
5.2.1 Measured and predicted wave amplification in the bow . . . . . . . 118
5.2.2 Measured and predicted added resistance in head waves . . . . . . . 119
5.3 Transfer functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
5.3.1 Linear and ’nonlinear’ transfer functions . . . . . . . . . . . . . . . 122
5.3.2 Measured second order transfer functions . . . . . . . . . . . . . . . 130
5.4 Wave and high frequency fatigue damage . . . . . . . . . . . . . . . . . . . 132
5.4.1 Base case - vessel in selected sea states at full speed . . . . . . . . . 133
5.4.2 Effect of speed reduction and wave environment . . . . . . . . . . . 136
5.4.3 Effect of changed trim . . . . . . . . . . . . . . . . . . . . . . . . . 139
5.4.4 Wave conditions along the tank and the disturbance from the vessel 142
5.4.5 Extrapolation of results by the response surface method. . . . . . . 143
5.5 Relative importance of whipping and springing . . . . . . . . . . . . . . . . 149
5.5.1 Acceleration levels used to indicate whipping . . . . . . . . . . . . . 152
5.6 Linear versus monochromatic second order springing . . . . . . . . . . . . 153
5.7 Illustration of vibration events . . . . . . . . . . . . . . . . . . . . . . . . . 156
6 Full scale measurements 163

6.1 Damping and springing frequencies . . . . . . . . . . . . . . . . . . . . . . 163
6.2 Measured speeds in different sea states . . . . . . . . . . . . . . . . . . . . 167
6.3 Wave induced stress in different sea states . . . . . . . . . . . . . . . . . . 168
6.4 Prediction of whipping events from measurements . . . . . . . . . . . . . . 170
6.5 Wave and vibration frequency fatigue damage . . . . . . . . . . . . . . . . 176
7 Conclusions and recommendations 181

7.1 Brief summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
7.2 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
7.2.1 Literature studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
7.2.2 Workshop; Introductory assessment of full scale measurements . . . 183
x CONTENTS
7.2.3 Investigation by simplified theoretical methods . . . . . . . . . . . . 184

7.2.4 Model experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
7.2.5 Full scale measurements . . . . . . . . . . . . . . . . . . . . . . . . 191
7.2.6 Contribution/major findings . . . . . . . . . . . . . . . . . . . . . . 194
7.3 Suggestions of further work . . . . . . . . . . . . . . . . . . . . . . . . . . 196
References 201
Appendices 214
A Theory 215
A.1 Potential theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215
A.1.1 Calculation of high frequency added mass . . . . . . . . . . . . . . 217
A.1.2 Detailed numerical implementation of 2D BEM . . . . . . . . . . . 218
A.1.3 Example of added mass . . . . . . . . . . . . . . . . . . . . . . . . 223
A.2 Amplification of incident short waves . . . . . . . . . . . . . . . . . . . . . 223
A.3 Added mass for a cone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227
A.3.1 Added mass for a cone at any deadrise angle . . . . . . . . . . . . . 227
A.4 Resistance components . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229
A.4.1 The wave making resistance, Cw . . . . . . . . . . . . . . . . . . . . 229
A.4.2 The viscous resistance, CV . . . . . . . . . . . . . . . . . . . . . . . 230
A.4.3 The air resistance, CAA . . . . . . . . . . . . . . . . . . . . . . . . . 230
A.4.4 The resistance due to base drag, CBD . . . . . . . . . . . . . . . . . 231
A.5 Solving the beam equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 231
A.5.1 Assumptions made in the simplified procedure . . . . . . . . . . . . 232
A.5.2 Numerical scheme applied in the simplified procedure . . . . . . . . 234
A.6 Finite beam element formulation . . . . . . . . . . . . . . . . . . . . . . . 235
A.7 Analytical solution of beam equations . . . . . . . . . . . . . . . . . . . . . 237
A.8 Stiffness distribution for a segmented model . . . . . . . . . . . . . . . . . 238
A.9 Different features of the SDOF system . . . . . . . . . . . . . . . . . . . . 240
A.10 Estimate of the wave height in a sea state given the period . . . . . . . . . 244
A.11 The Beaufort scale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246
B Experimental set-up procedures and uncertainty analysis 247

B.1 Model setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247
B.1.1 Description of transverse cuts . . . . . . . . . . . . . . . . . . . . . 247
B.1.2 Mass properties of the segments . . . . . . . . . . . . . . . . . . . . 247
B.2 Scaling laws and nondimensional properties . . . . . . . . . . . . . . . . . 253
B.2.1 Scaling laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253
B.2.2 Nondimensional properties . . . . . . . . . . . . . . . . . . . . . . . 255
B.2.3 Measurement devices with locations . . . . . . . . . . . . . . . . . . 257
B.3 Uncertainty analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258
B.4 Uncertainties and calibration of different sensors . . . . . . . . . . . . . . . 262
CONTENTS xi
B.5 Calibration of waves prior to model testing . . . . . . . . . . . . . . . . . . 266

B.5.1 Sensitivity of springing response . . . . . . . . . . . . . . . . . . . . 266
B.5.2 Location of wave probes and tested wave conditions . . . . . . . . . 268
B.5.3 Results of regular wave tests . . . . . . . . . . . . . . . . . . . . . . 268
B.5.4 Irregular waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273
B.6 Sea states considered in the fatigue assessment . . . . . . . . . . . . . . . . 276
B.7 Modes shapes, sectional properties and mass distribution of the real ship . 277
C Miscellaneous results 281

C.1 Linear and ’nonlinear’ transfer functions . . . . . . . . . . . . . . . . . . . 281
C.2 Second order transfer functions . . . . . . . . . . . . . . . . . . . . . . . . 282
C.3 Wave conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288
C.4 Relative contribution of fatigue damage from sea states . . . . . . . . . . . 290
C.5 Whipping illustrated by measured time traces . . . . . . . . . . . . . . . . 293
C.5.1 Higher mode vibrations for bow 2 and 3 in ballast cond. . . . . . . 297
xii CONTENTS
Nomenclature
General Rules
The variables will be defined the first time they are used.
Subscripts
D Dipole
c Cradle
m Model scale
f Full scale, or flexible
T Total
st Static
s Springing
S Source
dyn Dynamic
max Maximum
sum Sum frequency
r Rigid or response
R Response
Superscripts
L Loading condition
S Sea state
env Envelope
T Torsethaugen
t Total
xiii
xiv NOMENCLATURE
Roman symbols
A33 3D added mass

AS Effective shear area
Av Vertical projected area of cross sectional structure
AT Transverse projected wind area
2D
a, ar , af , a33 Sectional added mass per unit length of rigid and flexible modes
ai Force amplitude
a Arm from centre of gravity to additional mass
a Coefficient in SN-curve
B, Bf Breadth, foil width
Bn Beaufort number
B Damping matrix
2D
b, b33 Damping and 2D damping in heave due to heave
bf , bhl Foil damping, hull lift damping
b1, b2 , btot Linear, nonlinear and total linearized damping
C Constant in Bernoulli’s equation
CT,w,V,AA,BD Total, wave, viscous, air and base drag resistance coefficient
CF Flat plate friction coefficient
∆CF Roughness allowance
Caw Coefficient of added resistance in waves
C Stiffness matrix
c Water particle speed in z-direction or foil coord length
c Water plane stiffness per unit length or effective wet width
m Element stiffness mass matrix
cV BM Whipping criterion
c∆ Addition to whipping criterion due to natural decay
D Depth or cummulative fatigue damage
D Diameter for propeller or dynamic amplification
D Influence matrix for unit dipoles
dn,i 3 hour damage in sea state i and in loading condition n
d Distance between centre of tank and side
d Force panel diameter
ds Surface element length
E Young’s modulus, 2.1 · 1011 P a
E(h|t) Mean nondimensional wave height given nondimensional period
Fs Impact force
Fd Damping force
Fh Harmonic oscillating linear
√ force
Fn Froude number = U/ Lg
F Xi Axial force in sensor number i
FY i Shear force in cut number i
NOMENCLATURE xv
FP Force on force panel

f Frequency in Hz [1/s], force, mode dependent factor or fraction
f (y) Gaussian probability density function; Gaussian Bell
G Shear modulus
g Acceleration of gravity ≈ 9.81[m/s2 ]
H(ω),|H(ω)| Transfer function and transfer function amplitude
H(ωi , ωj ), |H(ωi , ωj )| Second order transfer function and absolute value
HS Significant wave height [m] ≈ H0
H Regular wave height
Hr , Hµ Hull roughness height
h Submerged water height or nondimensional wave height
h Height from point of rotation to centre of gravity.
hkeel Height from point of rotation to keel
I, Iv , Ih Area moment of inertia abt. vertical and horis. neutral axis [m4]
It Total mass moment of inertia [kgm2]
Im Model mass moment of inertia about COG
I Diagional matrix
i, j, k Unit vector in x-, y- √
and z-direction
i Complex number = −1
ki Wave number = 2π/λ
k0 Form factor or wave number
kθ Rotational stiffness
kz Shear stiffness
k Constants in 3D added mass expressions
L Length, normally length between perpendiculars
Lpp Length between perpendiculars
LW L Length at water line
LOA Length overall
M Total mass or moment
M Mass matrix
Mi Mass for segment i
m Mass per unit length or SN-curve inverse slope
m Additional mass
m Consistent element mass matrix
m0 Standard deviation and zero moment of spectrum
m2 Second moment of spectrum
N Number of samples/elements/cycles to failure or normal force
ni Unit normal vector in direction i
ni Number of cycles in stress range i
n Normal vector on boundary (hull) pointing into the fluid
n Number of cycles or high frequency tail factor
n Propeller revolutions per second
P , PE Power in kilowatt and engine power
P Force per unit length
xvi NOMENCLATURE
Px Axial force
Pi Pressure sensor number i
P Force vector
p Point on boundary (hull)
p Probability density function
p(h, t) Joint probability density function of wave height and period
p(h|t) Conditional probability density function of wave height given period
pLn Probability of loading condition number n
pSi Probability of sea state number i
p Pressure
pav Average pressure from force panel
Q Shear force
q Field point
qi Principle coordinate number i
q Principal coordinate vector
RT Total wave resistance
Raw Added resistance in waves [N]
Rxx (τ ) Autocorrelation function
RELi Relative motion sensor i
R, ri Response amplitude (for component i)
r Radius of gyration
S Influence matrix for unit sources
S Wet surface area, or wave steepness
S(ω), Sx (ω) Wave spectrum and response spectrum
Sxx (ω) Spectrum of response x
Se (ω) Encounter spectrum
SB Base drag area
s Wavelet layer variable or springing build up length
s, n s- & n- local coordinates in S-N plane
T Draft or time
TAP , TF P Draft at aft and forward perpendicular
TP , TZ Peak and zero-upcrossing period
T0 Wave period or time at end of impulse
t t-factor from t-distribution, time or period
t nondimensional period
tr t-factor for rejection based on normal distribution
U Vessel speed, current speed or wind speed
Ue Tangential velocity along the hull at still water level
u Deflection in u-direction (vertical)
V Velocity vector for fluid particles
VR Relative velocity
V Vessel speed or wind speed
V Velocity vector of fluid particles
Wd Dissipated work
NOMENCLATURE xvii
WPi Wave probe number i

w Deflection
x, y, z x-, y- & z- global coordinates
x, y, z x-, y- & z- local coordinates
Xi Unspecified sample number i
X Mean value of unspecified parameter
X(ω) Complex spectrum of response x
y Vertical deflection
y Displacement vector for various degrees of freedom
zna Vertical distance from base line to neutral axis
zc Vertical distance from base line to apex of cone
Z Complex number; distance between field point and hull point
Greek symbols
α Philips constant in JONSWAP, dead rise angle of cone or stiffness ratio

βi Frequency/mode dependent term in the beam equation
βi Nondimensional frequency = ω/ωs
β1 , β2 ω/ωs corresponding to specific response level on each side of the resonance peak
β Heading angle relative to ship (00 is head sea)
Φ, Φs Velocity potential and total steady potential
φs Steady disturbance potential
φ1,2,T Linear, second order & total potential
γ JONSWAP peakness parameter
γ, β Newmark-β parameters
i Random phase number
η1,2,3,4,5,6 Motion in surge, sway, heave, roll, pitch & yaw
ηt Total effciency coefficient
η Maximum usage factor
ρ Density of water; 1.025/1.0 tonnes/m3 for sea water/fresh water
σ Stress, standard deviation or a constant
σ2 Variance
σX standard deviation of mean value
∆σ Stress range
δ, δr , δf,i Damping ratio in % of rigid and flexible mode number i
δi Phase angle number i
λ Wave length
Λ Scale; ratio of full scale dimension to model scale dimension
ζ, ζr Wave amplitude, radiated wave amplitude
ζs , ζa Steady & incident wave amplitude
xviii NOMENCLATURE
ζI , ζT Incident & Total wave amplitude

ζ1 , ζ2 Wave amplitude corresponding to linear and sum frequency springing
ω Vertical deflection or wetting factor (pile up)
ω0 Wave frequency [rad/s]
ωi Wave frequency number i or natural frequency for mode i
ωe Encounter frequency
ωc Critical wave frequency
ωd Damped natural frequency
ωp Peak frequency
ω2 , ωs 2-node natural springing frequency
ωEI Natural frequency due to pure bending stiffness
ω Vorticity vector or eigenvalue vector
Ω Submerged volume
Ψi Mode shape number i
ψ(t) Basic wavelet
Ψ Mode shape eigenvector
θ Angle, vertex angle or rotation in degrees
θi Phase angle (lag) for response component i
ν Poisson ratio = 0.3 for steel, or viscosity coefficient of water = 10−6
µ Friction coefficient
µD Mean value of damage
τ Time
Abbreviations
ACCZi Acceleration in z-direction at position i

ABS American Bureau of Shipping
AD Analog to digital
AP Aft Perpendicular
B Bichromatic
BEM Boundary element method
BL Base line
CL Centre line
COG Centre of gravity
CN Classification Note related to DNV rules
CoV Coefficient of Variation = standard deviation divided by mean value
CPU Central Processing Unit (time)
CSSRC China Ship Scientific Research Centre
DNV Det Norske Veritas
DOF Degree of freedom
NOMENCLATURE xix
DTU Technical University of Denmark

DUT Delft University of Technology
FE Finite element
FFT Fast Fourier Transform
FP Forward perpendicular
FS Full speed
GB Gerritsma & Beukelman strip teory
GPS Global Positioning System
HBM Horizontal bending moment
HBM Hoottinger Baldwin Messtechnik
HF High Frequency
HTS High Strength Steel (36 refers to yield stress of 355MP a)
IACS The International Association of Classification Societies
JONSWAP Joint North Sea Wave Project (wave spectrum)
jpdf Joint probability density function
kn Knots = 0.5144m/s
LAMP 3D sink source forward speed program developed at MIT
LCB Longitudinal centre of buoyancy from aft part
LCG Longitudinal centre of gravity from aft part
Lin Linear
LNG Liquified Natural Gas
LR Lloyds Register
LSQR Least square root
MIT Massachusetts Institute of Technology
M Monochromatic
MLER Most likely extreme response
NL Nonlinear
NTNU Norwegian University of Science and Technology
O() Order of Magnitude
Pa Pascal = N/m2
POSi Position sensor number i between rigid segments
PM Pierson-Moskowitz wave spectrum
RAO Response Amplitude Operator
RMS Root Mean Square
RO-RO Roll on roll off
RS Reduced speed
SCF Stress Concentration Factor
SDOF Single degree of freedom
SES Surface Effect Ship
SN SN-curve; Stress and Number of cycles to failure
STF Salvesen, Tuck & Faltinsen strip theory
TIMSER Time simulation program developed at DNV
TA Towing force sensor aft
TF Towing force sensor forward
xx NOMENCLATURE
TLP Tension Leg Platform

Torsethaugen Two peak wave spectrum
TRF Transfer function
UK United Kingdom
USSR Union of Soviet Socialistic Republics
VBM Vertical bending moment
VCG Vertical centre of gravity above keel
VSF Vertical shear force
VERES 2D strip theory program from Marintek
WAMIT 3D sink source zero speed program developed at MIT
WASIM 3D sink source forward speed program developed at MIT/DNV
WF Wave Frequency
WL Water line
2D 2-dimensional
3D 3-dimensional
Mathematical operators and other symbols
d Derivative
∂ ∂ ∂
∇ Mathematical differential operator = ∂x i + ∂y j + ∂z k
3
∇ Displacement [m ]
× Cross product of vectors
∂ partial derivative for a function with more dependents
D ∂
Dt
material (substantial) derivative following a particle in space, = ∂t +V·∇
0
Degrees
t Time derivative
II,III,IV
Second, third and fourth derivative in space
x,y,z Derivative in x-,y- & z-direction
` Length, integration along the water line; finite element length
˙ Time derivative
¨ Second time derivative
X Overline refers to mean value, vector symbol or nondimensional quantity
∗
Complex conjugate
Chapter 1
Introduction
Ships are as a minimum designed to withstand load effects due to still water loading and
wave loading. Other loads are considered depending on the ship type and intended opera-
tion. The ship is normally considered as a rigid body in the dynamic analysis carried out
to determine the deflection and stress in the structure. However, in reality a ship is flexible.
When the natural frequencies are sufficiently low structural dynamics may be excited by
different mechanisms. In some cases, the flexible vibration may play an important role in
the fatigue and ultimate strength as well as for comfort. The present thesis focuses on
the wave induced vibrations of the whole hull girder, in particular, for large conventional
monohull ships. The consequence in terms of accelerated fatigue damage is the main
concern.
1.1 Background and motivation for the present work

The motivation was threefold. Full scale measurements of the ship considered herein con-
firmed the presence of wave induced vibrations. The current ship rules disregard flexible
vibrations, and comparison between numerical predictions and measurements revealed poor
agreement.
The background and relevance of this work have been presented in more detail by Storhaug
et al. (2003), Moe et al. (2005) and Storhaug et al. (2006). DNV personnel and others were
strongly involved, thus, the references are not included in this thesis. Both the references
and this thesis consider the large blunt ore carrier in Figure 1.1, which is referred to as the
ship or the vessel. In the following the background is briefly summarized.
1
2 CHAPTER 1. INTRODUCTION
Figure 1.1: The large blunt iron ore carrier in cargo condition.
1.1.1 Recent experiences from a ship in operation

Inspection onboard the ship revealed cracks after one year of operation in the North At-
lantic. The ship, built to HTS36 standard, satisfied the IACS’s requirements. Obviously,
these requirements were insufficient to avoid the fatigue cracks. Figure 1.2 presents a typ-
ical crack observed after one year.
Extensive finite element (FE) and wave load analyses proved that the stress concentration
factor (SCF) was well above the commonly assumed SCF, and that the fatigue life was well
below 20 years. Still, there was a mismatch between the predicted and observed fatigue life.
As a result a DNV monitoring system was installed in 1999. The measurements confirmed
basically continuous wave induced vibrations, and more measurements were carried out to
determine its consequence.
The fatigue contribution from wave induced vibration became a concern (again), and DNV
developed a preliminary Guidance Note to account for the vibrations. A similar ship was
1.1. BACKGROUND AND MOTIVATION FOR THE PRESENT WORK 3
Figure 1.2: Picture of a longitudinal stiffener in deck in way of the transverse bulkhead,
and located in the ballast tank. The crack originated from the bracket toe termination.
built to increased strength standard, and the particular ship was repaired and strengthened
in 2000. Further history has been given by Moe et al. (2005).
1.1.2 Comparison between measured and predicted vibration re-

sponse
In January 2002 a DNV workshop on fatigue strength analysis of ships was arranged in
Finland. DNV presented preliminary results from full scale measurements, while Jensen
and Vidic-Perunovic (2002) presented suggested guidelines to avoid excessive springing vi-
brations. The ship was found satisfactory according to the guidelines, even so the ship was
vibrating heavily.
A DNV workshop on springing 2002 was initiated with participants from DNV, DTU,
DUT, NTNU/Marintek and CSSRC. Ten different sea states were selected from the full
scale measurements in ballast condition. The measured vibration and wave responses were
compared to predictions from four state-of-the-art nonlinear hydrodynamic codes, which

all include the effect of flexibility. Wave radar measurements were used as input. All
programs failed to predict reliable estimates of the hull vibrations, and in general, the
predicted vibration level was too low. It was concluded that the important excitation
mechanism was either missing or inaccurately described. The work was summarized by
Storhaug et al. (2003). Moe et al. (2005) indicated that the measured sea states, used
as input to the predictions, were overestimated. Since the calculations gave results on the
lower side, the differences are expected to increase.
1.1.3 Today’s rule requirements

The current IACS rules disregard wave induced vibrations as a contribution to extreme and
fatigue loading. Only some nonlinear effects are included in the extreme bending moment,
since bow impacts and whipping to a certain degree can be avoided by course changes and
speed reduction, (Mürer 1995). Classification societies such as DNV, ABS and LR have,
however, operated with their own additions related to bow impacts.
The IACS wave bending moments were also stated in the Joint Tanker Project Rules issued
by ABS, DNV and LR, (DNV et al. 2004), and in the Joint Bulk Project Rules issued
by IACS, (IACS 2004). In these new rules, the whipping addition from the individual
classification societies has actually been removed for tankers and bulk carriers. Without
adjusting the acceptance criteria accordingly, this may signify a reduction in the safety
level.
Based on extensive full scale measurements carried out by DNV, the preliminary Guidance
Note, proposed as a consequence of the preliminary measurements on the ship, was rejected.
A revised Guidance Note was proposed, (DNV 2005b).
1.2 A dive into literature

Originally, the intention was to investigate the effect of springing. During the research it
became evident that whipping must be included. In terms of the consequence for fatigue,
the high frequency response due to springing and whipping may be considered as one topic.
Before diving into the published documentation, and in the present context of this research,
a few main terms need to be defined.
Wave induced vibration: All local and global vibrations caused by waves, such as
springing, whipping, swinging (horizontal 2-node), double bottom vibration etc. In this
thesis, wave induced vibration refers to whipping and springing, and may simply be denoted
as vibration.
1.2. A DIVE INTO LITERATURE 5
Springing: Conventionally, springing is defined as stationary resonance vibration of the

vertical 2-node mode due to waves with encounter frequency coinciding with the natural
2-node frequency. In the present research this definition is refined. Springing is stationary
or apparently stationary resonant vibration due to oscillating wave loads and wave impacts
that excite the vertical 2-node mode. This includes nonlinear forces that may oscillate more
locally, and small wave impacts, which due to low damping will be difficult to distinguish
from stationary springing in irregular sea.
Whipping: Conventionally, whipping is defined as transient hull girder vibration caused

by an impact. It is not necessarily related to waves. In consistency with the definition of
springing, whipping is detectable transient vibration due to wave impacts. Whipping vibra-
tion may be identified by the presence of higher vibration modes, but the vertical 2-node
mode is normally dominant amidships. The whipping response depends on the pressure
time history of the impact, and the natural periods of the structure.
Hydroelasticity: The interaction between the hydrodynamic loading and the elastic de-
formation associated with the vibration is considered. The interaction may affect both the
loading and vibration level. The importance of the hydroelasticity depends on the nature
of the impact, the flexibility and the level of the flexible response.
There are also a few other terms often used that need clarification.
• Springing frequency (period) refers to the natural frequency (period) of the ver-
tical 2-node vibration.
• Wave frequency (WF) response refers to the range of wave frequencies related
to conventional rigid body response that is normally considered important. Wave
damage refers to the fatigue damage from the WF response, which is wave/low pass
filtered.
• Total damage refers to the fatigue damage from the total measured stress signal
(only filtered for noise).
• High frequency (HF) response refers to the dynamic flexible response in particular
at the springing frequency. High frequency in this context is related to the lowest
natural frequency of the structure, and may appear as confusing for ”noise and vi-
bration people”. The high frequency damage may also be referred to as vibration
damage, and constitutes more than the fatigue damage from the high pass filtered
stress. It is the difference between the total damage and the wave damage.
1.2.1 The beginning

The first digression of wave induced hull vibrations, irregular in time and quickly damped
out, came during 1880’ies, (Schlick 1884). Hydroelasticity as a term was introduced in
1959, (Heller and Abramson 1959), while the methodology had already been used within
the aeronautical field for several years, (McGoldrick 1960). Todd (1961) stated that tran-
sient vibrations due to slamming in case of large pitching and heaving motions were of
intermittent or minor importance, but his concern was engine/propeller induced vibra-
tions. Within this field elastic models were tested already in 1935, (McGoldrick 1935).
The terms springing and whipping were apparently introduced during the sixties, but the
phenomena were known earlier. Sauvalle (1960) measured 2-node vibrations in relatively
moderate sea conditions on the ore carrier Jean Schneider. This may be the first docu-
mentation of measured springing. Up to the mid seventies springing was sometimes used
to cover both terms.
The theoretical research of wave induced vibration, in particular springing, was a result
of observations made from full scale measurements carried out to investigate wave loading
on large ships. This was necessary due to the trend towards increased length beyond the
validity of the empirical ship rules. A ’shudder’ was observed from measurements on the
216m Great Lakes ore carrier Shenago II, (Yagle 1963), lengthened by 29m and renamed
Charles M. Beeghly in 1971. Cleary et al. (1971) revealed springing to be the most fre-
quent dynamic stress mechanism on the 217m Great Lakes carrier Edward L. Ryerson. The
amplitude was in a few occasions so high that it should be considered in design. These
vessels were narrow (length to width ratio of up to 10), blunt and sailing on small draft in
addition to being more flexible than ocean going vessels. The shallow waters on the Great
Lakes reduced the natural frequency further, and the steep shores caused reflection and
confusing seas, which contributed to springing.
The springing research began more or less independently and simultaneously in USSR,
considering similar ships sailing on inland waters. Full scale measurements were available
to Belgova (1962), who investigated springing by a theoretical approach after conducting
model tests. This may be the first attempt to predict springing.
In addition to increased ship size, which reduce the natural springing frequency, there may
be other factors that increased the awareness of springing during this period. The damping
reduced as a consequence of increased ship size, moreover, the damping reduced for welded
ships compared to riveted ships.
1.2.2 Ship measurements considering wave induced vibrations

During the mid thirties measurements were carried out onboard the vessel San Francisco,
(Hoffman 1972). Ten years later the vessel Ocean Vulcan was instrumented, but first dur-
ing the sixties the science of full scale measurements took off. Wave induced vibrations
were frequently observed, and occasionally it was considered as noise. Gunsteren (1978)
provided an overview of references with observed vibration including 4 publications on
Great Lakes bulk carriers, 10 on ocean going bulk carriers, 14 on tankers and 8 on con-
tainer ships. Additional references were provided by Storhaug et al. (2003) and Moe et al.
(2005). The amount of observations strongly suggest that wave induced vibration can be
anticipated on all ships, not proving that it is important, but that it may contribute to the
fatigue and extreme loading.
Only a few men explored wave induced vibrations in more detail. The Great Lake bulk
carrier S.S. Ontario Power, similar to Edward L. Ryerson, was investigated, (Mathews
and Kawerninski 1967). This vessel was strengthened for ocean going operations. Later
full scale measurements of three Great Lakes vessels, S.J. Cort (305m), W.C. Ford (192m)
and C.M. Beeghly (246m), were compared to linear theory, (Stiansen et al. 1978). Several
ocean going vessels were also subjects for attention illustrated by the following list.
? Bell and Taylor (1968) carried out measurements on a 47 000 dwt. tanker, partly riveted,
in ballast condition with and without bulbous bow.
? Nordenstrøm et al. (1970) reported extensive springing measurements from a 203 000
dwt. tanker, Berge Commander, in ballast and cargo condition.
? Goodman (1971) referred to full scale measurements of a 192 000 dwt. tanker, SS
Myrina.
? Little et al. (1971) carried out measurements on four tankers and one bulk carrier ranging
from 66 500 to 326 585 dwt. The purpose was to investigate ordinary wave loading, but
all revealed wave induced vibration (whipping on the bulk carrier).
? Whipping was measured on an 80000 dwt. tanker M.T. B-Maru, which had its bow
replaced with a bulbous bow, (Tasai et al. 1971).
? Skjørdal (1974) performed measurements on a 60 000 dwt. bulk ship in order to deter-
mine the cause of the vibration.
? Hure (1973) reported measurements on two 250 000 dwt. tankers, Emeraude and Jade,
in ballast condition.
? Gran (1974) carried out preliminary measurements on a 255 000 dwt. tanker, Esso Bonn,
on a ballast trip. Gran (1976) reported the consequence of vibration on the fatigue and
extreme loading from extensive measurements in ballast and cargo condition.
? Kagawa et al. (1977) carried out springing measurements on a 237 000 dwt. tanker.
? Lopes and Troymann (1989) instrumented an 18 000 dwt. tanker for the purpose of
evaluating the fatigue damage from springing based on a single ballast trip.
? Lacey and Chen (1995) reported the consequence of springing and whipping on the
fatigue and extreme loading from extensive measurements in ballast and cargo condition
based on a 190 000 dwt. ARCO tanker, California.
? Storhaug et al. (2003) reported the consequence of vibration on the fatigue damage from
one year of measurements in ballast and cargo condition of the ship.
? Moe et al. (2005) reported the consequence of vibration on the fatigue damage from
extensive measurements in ballast and cargo condition of the ship.
? Storhaug et al. (2006) compared the ship with a 175 000 dwt. Capesize bulk carrier
with respect to vibration damage.
What knowledge can be deduced from these publications? Relevant findings about the
wave induced vibrations may be summarized as follows.
• The springing vibration was higher in ballast than in cargo condition, e.g. (Bell and
Taylor 1968).
• The springing vibration was reduced 10-15% by increasing the ballast level in the
fore peak tank, (Lopes and Troymann 1989).
• The springing vibration was higher in head seas than in beam and following seas,
e.g. (Hure 1973). In general, the decrease towards beam seas was slight, and the
maximum may occur at off head seas, (Gran 1976).
• Nordenstrøm et al. (1970) indicated that the occurrence of vibration was almost
independent of the heading in ballast, but following the ’normal’ trend in cargo
condition.
• The RMS springing stress increased with increased significant wave height to a certain
value. The peak was roughly at 3 to 5m depending on ship size, and the decay was
less on the larger vessels, (Stiansen et al. 1978).
• The springing may come in ’bursts’ or ’beats’, (Little et al. 1971).
• The springing vibration was relatively low at high Beaufort strength on the tankers,
e.g. (Little et al. 1971).
• The vibration was sensitive to speed, e.g. (Skjørdal 1974). The vibration appeared
to increase with the square of the speed, (Bell and Taylor 1968), or more and may
be pronounced also at zero speed, (Gran 1974).
• The vibration increased with Beaufort strength, e.g. (Kagawa et al. 1977). The
occurrence of vibration was independent of Beaufort strength in ballast, opposite to
cargo condition, (Nordenstrøm et al. 1970).
• The vibration may occur without evidence of slamming, and may be unaffected by
slamming, e.g. (Bell and Taylor 1968).
• The vibration may exceed the wave bending, e.g. (Bell and Taylor 1968). The
springing stress amplitude was 51% in average of the wave bending stress, (Gran
1974). The wave bending moment exceeded the vibration stress on the smaller vessel
first for significant wave heights above 6m, (Stiansen et al. 1978).
• The vibration, sometimes unpleasant, was felt onboard, e.g. (Skjørdal 1974). It was
visible, e.g. (Castberg and Gran 1976).
• The bulbous bow did not seem to affect the vibration level, e.g. (Bell and Taylor
1968).
• The whipping vibration and slamming often occurred when the bow enters the water,
but the vibration may also originate on the up-pitching cycle resulting with a first
half cycle in hogging, e.g. (Bell and Taylor 1968). Saunders (1965) explained this by
rapid loss of buoyancy (one of his four excitation sources).
• Blunt ships may experience more vibration than slender ships, e.g. (Gran 1976).
This was supported by the number of references on blunt ships, but their large sizes
and lower damping may be contributing factors.
• Gran (1974) reported also about the apparently unstable front wave and the steady
wave field which increased the bending moment significantly and linearly with speed.
• Bi-directional sea was reported, e.g. (Vidic-Perunovic 2005), which may effect the
vibration level.
• The maximum of wave bending and vibration stress may not occur simultaneously,
(Bell and Taylor 1968). The maximum wave bending and vibration stress can and
do occur at the same time, (Stiansen et al. 1978). The combined amplitude process
deviates from the Rayleigh distribution, (Gran 1980), but the wave and springing
response were essentially narrow banded Gaussian processes when separated, (Gran
1977). The springing and wave stress can be considered as statistically independent
variables, (Gran 1976). The combined stress was 5% larger than the square root
relation, Kagawa et al. (1977).
• The heading distribution varies with trade. Even distribution was observed by Nor-
denstrøm et al. (1970), while dominating head seas in ballast and following seas in
cargo condition were observed by Lacey and Edwards (1993).
• 26 bottom slams with severe whipping response were reported by Lacey and Chen
(1995). It may also be rare events, (Storhaug et al. 2003).
• Wave conditions were in most cases based on visual observations, but in a few oc-
casions wave buoys were used, (Cleary et al. 1971). Wave radar measurements in
combination with fatigue assessment were first considered by Moe et al. (2005).
• The visual observations of the sea states overpredicted the height and underestimated
the length compared to meteorological calculations, e.g. (Hure 1973).
• The vibration was observed in most sea states, e.g. (Little et al. 1971).
• Whipping was detected without significant rigid body motions, e.g. (Tasai et al.
1971).
• The whipping contribution to the total stress may be important, e.g. 55% increase to
114MPa on the bulk carrier Fotini L, (Little et al. 1971). Due to the whipping, the
total response (127MPa static and 119NP a dynamic) may exceed the IACS design
bending moment already at a significant wave height of above 9m in cargo condition,
(Lacey and Chen 1995). Mineral Seraging and Jordaens showed an amplification
due to whipping of 50% from full scale measurements in significant wave height of
up to 8m, (Kawakami et al. 1977). Large whipping responses were also illustrated
on different warships, (Clarke 1986).
• Gran (1974) reported a 14% increase to 68MPa due to springing on a single ballast
trip. Stiansen et al. (1978) stated that the maximum combined nominal stress level
increased by 78% to 105MPa due to steady build up.
• A 7% increase of the 20 year extreme was estimated to come from springing, reaching
a total of 118MPa, (Castberg and Gran 1976). This was probably underestimated
with reference to the preliminary measurements on the single ballast trip, (Gran
1976).
• The voluntary and involuntary speed reduction may be significant in head seas, e.g.
(Tasai et al. 1971). The speed reduction differed for a bulk carrier and a container
vessel of similar size, (Moe et al. 2005).
• The wave spectrum energy in the high frequency tail represented an uncertainty, e.g.
(Hure 1973). A possible overshoot of energy compared to standard wave spectra may
correspond to a factor of two, (Goodman 1971).
• The uncertainty in damping was high, e.g. (Hure 1973). (Gran 1974) estimated the
damping ratio to 1.14% from whipping, while values from spectra were 25% higher.
The damping ratio in ballast condition was estimated to 0.5%, (Storhaug et al. 2006).
• The increased stiffness of the hull girder was said to be ineffective in order to reduce
the relative vibration level, e.g. (Skjørdal 1974).
• The maximum nominal springing stress amplitude was measured to 61MPa, (Stiansen
et al. 1978), and 40MPa, (Lacey and Chen 1995).
• Springing and whipping were difficult to separate, e.g. (Gran 1974), who also stated
that whipping contributed in moderate sea states. Both springing and whipping
contribute, (Little et al. 1971).
• The wave slope and natural period were defined as important parameters, (Gran
1977). The springing frequency for a given ship size had a standard deviation of
0.08Hz, (DNV 1985).
• The fatigue damage from wave induced vibration constituted roughly by 21%, (Gran
1976), 30% (Lacey and Chen 1995), 77% (Lopes and Troymann 1989) and 44% before
strengthening and 38% after, (Moe et al. 2005). The former three underestimated
the contribution.
• Whipping was said to provide the major contribution to fatigue damage on landing
crafts, (Clarke 1986).
• The fatigue damage rate increased when the springing share exceeded a certain limit,
(Gran 1980). This may be SN-curve dependent.
• The impact slams (breaking waves against stem flare) occurred often in cargo condi-
tion, but not with a significant whipping response, (Lacey and Chen 1995).
• The momentum slamming due to sudden and fast immersion of the bow was identi-
fied 427 times during 3 months causing a nominal whipping stress amplitude above
34MPa, (Lacey and Chen 1995).
1.2.3 Model experiments considering wave induced vibrations

Model tests were performed to investigate physics under more controlled environment, to
validate computer programs or as an alternative to less reliable computer programs. The
results have often be regarded as the ’truth’, without considering the uncertainties.
A method to measure the wave bending moment by a jointed model was developed by
Lewis (1954), but the first elastic model test was carried out by McGoldrick and Russo
(1956), who investigated ’whipping’ on SS Gopher Mariner. Belgova (1962) performed later
model experiments of an inland vessel with respect to linear springing in regular waves.
The findings from selected references are listed and commented upon in the following.
• The flexibility has been represented in three ways: Backbone segmented model,
(Achtarides 1979), continuous models, (Tasai 1974) and segmented model with lo-
cal springs, (Jullumstrø and Aarsnes 1993). The latter may provide a more reliable
estimate of the bending moment.
• Flexible models were not always Froude scaled, e.g. (Slocum and Troesch 1982).
Artificial low frequency will give higher hydrodynamic damping and possibly coupling
with rigid modes.
• Continuous stiffness models gave too high springing frequency, e.g.(Tasai 1974), or
too high damping, e.g. (Hashimoto et al. 1978).
• The wave quality in the model tank was poor, especially for short waves, e.g. (Hoff-
man 1972). It is necessary to investigate this prior to springing tests.
• Restrained models were investigated, e.g. (Slocum and Troesch 1982). These may
be used to indicate the excitation sources and the coupling effect.
• Strip theory was not applicable for short waves less than roughly half the ship length
and for high speeds, e.g. (Moeyes 1976). Strip theory may not be applicable for
predicting springing.
• Higher harmonics were investigated in regular waves, e.g. (Kumai 1974). The con-
tribution may come from whipping as well as springing, and the effect in irregular
sea is uncertain.
• The higher order effects reduced in magnitude compared to the second order effect,
e.g. (Hashimoto et al. 1978). The second order effect may be more important.
• Second order springing was investigated, e.g. (Kawakami and Kiso 1976). The im-
portance was 30% in one sea state, (Slocum and Troesch 1982). More sea states must
be considered.
• 2-segment models were used often, e.g. (Slocum and Troesch 1982). A small change
in the mode shape might be important for the springing level, e.g. (Hoffman and van
Hoof 1976), hence 2-segmented models may affect the springing response.
• The damping did not appear to be speed dependent for conventional designs, e.g.(Tasai
1974).
• The damping was speed and frequency dependent, e.g. (Hoffman and van Hoof 1976).
• The damping was often not properly addressed, e.g.(Goodman 1971).
• The elastic effect due to large whipping events was described as deflections traveling
from bow to stern in sagging and from stern to bow in hogging, (Kawakami et al.
1977). This represents the presence of several modes.
• The blockage was not often considered, e.g. (Moeyes 1976). This may make it more
difficult to capture the springing peak.
• Nonlinearities appeared to increase for shorter waves and higher speeds, e.g. (Moeyes
1976).
• Water tight sealing between segments was sometimes omitted, e.g. (Malenica et al.
2003). This may create additional damping and hydrodynamic forces.
• No aft part slamming was observed in low waves, but it was observed in high head
sea waves due to a flat aft part, (Takarada et al. 1993).
• The static loading due to forward speed might be significant, e.g. (Moeyes 1976).
• Model tests were performed in head sea with only a few exceptions, e.g. (Watanabe
and Sawada 1985).
• Bottom slamming might occur in ballast condition, but not in cargo condition. The
former gave more vibration, e.g. (Watanabe and Sawada 1985).
• The springing response increased with increased flexibility, (Kapsenberg and Brizzo-
lara 1999).
• The amplitudes including whipping deviated from the Rayleigh distribution, e.g.
(Watanabe and Sawada 1985).
• Springing increased with speed, but the linear springing was less sensitive to speed
than the second order springing, (Slocum and Troesch 1982).
• Hump-hollow behaviour of matched resonance or excitation forces was observed, e.g.
(Slocum and Troesch 1982). Hump-hollow behaviour was not observed by (Achtarides
1979).
• Coupling between rigid body modes and springing was insignificant, e.g. (Achtarides
1979).
• Coupling between rigid body modes and springing was significant, e.g. (Slocum and
Troesch 1982). This was derived based on unrealistic high natural period excited by
second order springing, while the coupling appeared insignificant for linear springing.
• The waves in the aft part in head seas were significantly modified, e.g. (Takarada
et al. 1993), possibly reducing the importance of the aft part.
• The excitation force in the aft part contributed significantly to the springing level,
e.g. (Achtarides 1979). This may be based on the lack of excitation rather than high
excitation.
• A transom stern gave twice the springing level compared to a cruiser stern, e.g.
(Hoffman and van Hoof 1976). This could be due to lower damping and better
matching of hump-hollow behaviour for the transom stern.
• The whipping doubled the total bending moment, e.g. (Takarada et al. 1993).
• Local hydroelastic effects in the bow part of a high speed vessel were insignificant,
e.g. (Kapsenberg and Brizzolara 1999), due to too long rise time.
• Fatigue damage contribution from vibration was estimated to 30-45% in a few head
and bow quartering sea states, but it was low in beam and following seas, e.g. (Dud-
son et al. 2001). This was the first model experiment where the vibration damage
was evaluated. The natural frequency was too low, and the ship was a prototype.
1.2.4 Springing and whipping theories

The hydrodynamic theories considering ship responses in waves evolved from the fifties,
and the superposition principle for irregular waves was first applied by St.Denis and Pier-
son (1953). A decade later flexible structure was introduced in the linear springing theory
by Belgova (1962), while Kaplan et al. (1969) was probably the first who considered whip-
ping. He solved the structural response both directly and with modal modes. A brief
overview of some of the theories is provided in the following.
Simplified theories were developed to understand the behaviour of the structural response,
and to indicate the importance of certain effects. Usually, analytical response of pris-
matic beams was considered without proper description of the excitation loads. Bishop
et al. (1973) explained the terms ship/wave matching and resonance encounter herein re-
ferred to as hump-hollow behaviour and linear springing. Faltinsen (1980) considered linear
springing from Froude-Kriloff excitation as well as simplified whipping. He illustrated the
small effect of the restoring forces on the springing period. Hamid (1980) considered higher
harmonics from second to fourth order and confirmed the importance of coupling with rigid
modes. Sele (2001) evaluated the springing response from second order Froude-Kriloff ex-
citation of bi-directional wave trains coming from arbitrarily directions. The effect may
be significant for opposite wave directions and similar wave frequencies. Senjanovic et al.
(2001) investigated whipping and pointed out that the maximum bending moment may
not occur at the midship section, and the bending moment converges faster than the shear
force with respect to number of modes included.
Linear springing strip theories constitute the dominating part of the springing references
starting with Belgova (1962). Salvesen et al. (1970) developed a strip theory, which was of-
ten used as a basis for hydroelastic theory. They used the Haskind relation to estimate the
diffraction forces. This theory is referred to as STF. Ertekin et al. (1995) confirmed that
the Haskind(-Hanaoka) relation also applied to hydroelasticity. Another strip theory that
has been widely used was developed by Gerritsma and Beukelman (1967), later modified
by Gerritsma et al. (1974). This theory is referred to as GB. The STF theory appears to
be more mathematical consistent accounting for the diffraction, while the GB theory was
based on a relative motion hypothesis (between the ship and incident waves). Diffraction
was approximated, often in combination with the obsolete Smith correction factor. Both
theories were extended to nonlinear theories by others to consider nonlinear springing or
whipping. The theories were not applicable for modified short waves relevant for springing,
and for this purpose more complex theories are necessary. The strip theories may be more
useful for whipping predictions, and to illustrate the importance of certain effects. They
are also far more efficient than CPU demanding 3D simulations.
Linear hydroelastic slender body theory considering the Helmholtz equation to account for
some modification of short waves along the ship was first considered by Skjørdal (1978) and
Skjørdal and Faltinsen (1980). The structural response was solved directly, and both the
hydrodynamic and structural damping were regarded as important. The predicted vibra-
tion deviated significantly from model experiments of a tanker, but still it was a significant
improvement compared to the GB theory. Slender body theory has also been applied by
others, (Price and Wu 1986; Wu et al. 1991).
The linear high speed strip theory, (Faltinsen and Zhao 1991b; Faltinsen and Zhao 1991a),
was refined to account for hydroelasticity by Wu et al. (1993). The high speed strip theory
accounts for interaction of the steady field stepped from the bow and backwards. The
steady field was known when the unsteady field was solved. The hydroelastic theory was
applied to academic ship shaped structures. This theory was also applied by Hermundstad
et al. (1994), and later refined by Hermundstad et al. (1999) to account for interaction
between hulls of a catamaran.
Several linear 3D theories have been considered, e.g. (Price and Wu 1985; Bishop et al.
1986; Keane et al. 1991; Du et al. 1998; Bingham et al. 2001). The theories consider both
mono and multihulls with different accuracy with respect to how the forward speed effects
were handled. The forward speed will in general enter all parts of the theory, e.g. free sur-
face condition, body boundary condition and source and force formulations. These parts
were handled in different ways. Irregular frequencies may cause difficulties in some for-
mulations as they may appear in the frequency range of the physical resonance frequencies.
Watanabe (1968) was probably the first who investigated nonlinear springing due to change
of buoyancy and momentum from relative motion on non vertical ship sides, in particular
at the bow. The ship was assumed to be restrained from moving in accordance to observa-
tions made on a 50 000 dwt. bulk carrier. The resulting force gave higher harmonics, which
was believed to be the important mechanism. Subsequently, several Japanese researchers
investigated higher harmonics effects, e.g. (Kumai 1972).
Jensen and Pedersen (1979) developed a quadratic strip theory in the frequency domain
including flexible effects. The second order effects were related to perturbation of added
mass, damping and non vertical sides at the still water line. The linear theory was option-
ally represented by the STF or GB theory. The flexibility was represented by a Timoshenko
beam and modal superposition. The theory was further described by Jensen and Dogliani
(1996), who predicted a 10% increase of the fatigue damage of a container vessel due to the
elastic effects. The second order effect dominated the elastic contribution. The computer
software has been widely applied in many references, e.g. (Jensen and Pedersen 1981;
Jensen and Petersen 1992; Jensen 1993; Hansen et al. 1995; Naess 1996; Jensen 1996;
Jensen and Wang 1998). The theory was recently modified to account for bi-directional
seas as an additional second order contribution, (Vidic-Perunovic 2005). Comparison with
full scale measurements confirmed that the second order effect from bi-directional sea im-
proved the predictions in wave systems of opposite directions. Nonlinear springing was
also considered by Søding (1975).
A second order perturbation theory was derived by Slocum (1983). Simplifications were
introduced by assuming slender body theory, wall sided ships and zero speed. Limited
calculation results were compared to model test results, (Slocum and Troesch 1982). He
et al. (2004) refined this theory. Different approaches were used to solve the Helmholtz
equation in head and oblique seas with empirical matching in between. The nonlinear
terms consisted of the velocity square term from Bernoulli’s equation, and solution of the
second order near field problem for diffraction. Correction factors based on model tests
were used for the latter.
The 3D program WAMIT developed at MIT include second order effects, but it was only
valid for zero speed. It was applied to the ship by Birkenes (2000), and significant sum
frequency springing was predicted. Problems with irregular frequencies and convergence
were encountered.
Nonlinear whipping theories predict the response in time domain. Most of these were strip
theories. Kaplan et al. (1969) developed a linear strip theory for oblique seas with bow
flare slamming. The linear motions were input to the nonlinear force predictions, which
accounted for the buoyancy and the material derivative of the added mass times the ver-
tical velocity. The damping was defined as speed dependent. The calculations indicated
that the 3-node mode was unimportant. Kaplan and Sargent (1972) extended the theory
to include bottom slamming and linear springing in head seas. Convolution integrals were
used to produce time domain results from frequency domain, but the three effects were
considered separately. Many of the strip theories employ the bottom slamming approach
by Ochi and Motter (1973), e.g. (Antonides 1975; Chen 1980; Gu et al. 1989; Domnisoru
and Domnisoru 1997; Gu and Moan 2002), while others apply the expressions by Stavovy
and Chuang (1976), e.g. (Aksu et al. 1991; Gu et al. 2003). Many other nonlinear strip
theories have also been presented, e.g. (Yamamoto et al. 1978; Schlacter 1989; Xia et al.
1995; Xia et al. 1998; Wu and Hermundstad 2002; Gu and Moan 2002; Gu et al. 2003),
but the presence of overlapping forces have in general not been properly dealt with.
Xia and Wang (1997) applied nonlinear hydroelastic slender body theory to a warship and
the S175 container vessel. Experimental data from the latter was often used. The nonlin-
earities consisted of nonlinear restoring, Froude-Kriloff and momentum forces, and it was
concluded that the linear 3D theory overestimated the rigid body motions and gave poor
predictions of the bending moments. The elastic effects were small in the investigated sea
states.
Wu et al. (1996) applied the high speed strip theory, and extended it to nonlinear hydroe-
lastic theory. The time series were derived by convolution integrals, and the predictions
were compared to experimental results of a catamaran. The nonlinear effects were signifi-
cant, but differences from model tests were observed. This theory was later compared to
linear hydroelastic theory in (Wu and Moan 1996).
There were only a few nonlinear 3D programs. Aksu et al. (1991) applied the slamming
theory by Stavovy and Chuang (1976) on predefined hull elements on 3 barges. Lin et al.
(1997) presented whipping predictions using the 3D nonlinear program LAMP(-2). This
program includes nonlinear Froude-Kriloff and restoring. The impact loads were applied
to a beam model, hence hydroelastic effects were disregarded. Comparison was made with
model experiments of a naval ship, showing that the calculations underpredicted the whip-
ping response, though the rigid body response was almost captured. A similar theory was
described by Tongeren (2002), who used the DNV program WASIM to calculate the hy-
droelastic response of the ship. The pressure was calculated up to the undisturbed wave,
and the nonlinear effects were basically the same as in LAMP(-2). The difference between
linear and nonlinear response was small.
Is there anything to learn from these theories and their application? Obviously, a great
deal of effort has been put into the development of these theories, validation against ex-
perimental data, application on different designs and comparison with other computer
programs. Some of these programs are commercialized, and may be used to predict design
loads. The main conclusion is, however, that no program can reliable predict the whipping
and springing response for a general design. They may still reveal that certain effects are
important, but the magnitude can not be trusted. Different projects have confirmed that
the difference between the programs are considerable, e.g. (Xia et al. 1995; Watanabe
and Soares 1999; Storhaug et al. 2003). Still some statements can be made from theories
considering springing and whipping.
Springing
A few statements based on the linear and nonlinear springing theories are provided.
• The response predictions was sensitive to the location of the springing frequency
versus the hump-hollow in the transfer function, e.g. (Bishop et al. 1973).
• Springing was only slightly affected by changing trim and mass distribution, e.g.
(Chen 1980).
• The springing was higher in low than high draft, (Chen 1980).
• The coupling between flexible and rigid body modes were disregarded, e.g. (Kawakami
and Kiso 1976).
• Hamid (1980) confirmed that coupling was important.
• The slender body theory appeared to be insufficient for predicting linear springing
of blunt ships, e.g. (Skjørdal and Faltinsen 1980).
• The hydroelastic effect increased with flexibility and speed, e.g. (Hermundstad et al.
1994). The former was not evident in (Hansen et al. 1995).
• The structural damping may be neglected for high speed ships due to large hull lift
damping, e.g. (Hermundstad et al. 1994).
• The bilge keel and transom stern damping in addition to wave making appeared to
be important, e.g. (Søding 1975).
• The interaction between the hulls of multihull ships may be important, e.g. (Bingham
et al. 2001).
• The second order effect from restoring due to non vertical walls appeared to be small,
e.g. (Søding 1975).
• The second order effects from strip theory appeared to give a small elastic contribu-
tion on tankers and a significant effect on container vessels, e.g. (Jensen and Pedersen
1981).
• The second order contribution to the elastic effects may be dominating on container
vessels, e.g. Jensen and Dogliani (1996).
• The second order nonlinear effects (without hydroelasticity) may increase the fatigue
damage by 50-100% on container vessels, e.g. (Jensen 1993).
• The springing was found unimportant for another large high speed container vessel,
(Jensen and Wang 1998). Significant whipping was estimated on this ship with
another theory.
• The high frequency tail of the wave spectra was not well defined, but it was important,
(Chen 1980).
• Springing may be important in developing seas, (Chen 1980).
• The importance of considering non uniform steady flow and translating-pulsating

Green’s functions were confirmed by Du et al. (1998).
Whipping
A few statements based on the time domain whipping theories are given in the following.
• The 3-node mode may be unimportant for whipping, e.g. (Kaplan et al. 1969). 3
modes may be sufficient for the dynamic response, (Antonides 1975). 5 modes were
enough to ensure convergence, (Senjanovic et al. 2001).
• A SES may experience significant whipping, e.g. (Brown et al. 1981).
• A small change on the pitch motion may significantly change the slamming force on
a SES, e.g. (Brown et al. 1981).
• The rigid body motions was insignificantly changed by slamming, e.g. (Antonides
1975; Yamamoto et al. 1978).
• The whipping effect was insignificant for a tanker, but significant for a container
vessel, e.g. (Yamamoto et al. 1978).
• The slamming process in time may be described by a Poisson process, e.g. (Ochi and
Motter 1973). This was not recognized by Hansen and Thayamballi (1995) due to
clustering effects.
• Whipping may be more sensitive to speed in high sea states, e.g. (Chen 1980).
• Whipping was larger in developing seas, (Chen 1980).
• Reducing speed was the most efficient way of reducing the whipping response, (Chen
1980).
• Whipping was more important in head seas, (Schlacter 1989).
• The bottom emergence force may improve the predictions, (Schlacter 1989).
• 50% difference was observed from four programs used to predict wave bending mo-
ment in regular head sea. The difference increased for increasing flare, (Xia et al.
1995). The difference between several programs increased with increased wave height,
and the agreement was poor, (Watanabe and Soares 1999).
• Inertia loads and memory effects must be included, (Xia et al. 1998).
• Nonlinear von Karman and nonlinear modification of radiation forces were important
as well as nonlinear restoring and Froude-Kriloff forces, (Wu and Hermundstad 2002).
• The nonlinearities increased the fatigue damage by 10-100%, while the elastic effects
increased the fatigue damage by 100-900%, but it was sensitive to damping and
natural frequency, (Gu and Moan 2002).
• Shorter vessels may experience more nonlinear effects, e.g. (Wu and Moan 1996).
• Slamming effects may contribute significantly to whipping from the aft part in strip
theories, e.g. (Storhaug et al. 2003). This effect may not be realistic.
• Kutta condition was important in the exit phase of wet deck slamming, (Ge 2002).
Comparing findings from full scale measurements, model tests and theories reveal several
contradictory results, and some were also obvious. Some findings were used as input to
the preparation of the model tests, and for comparison with subsequent results. Moreover,
they may increase the general knowledge of the phenomena, and may serve as references
for interested readers.
1.3 Purpose of the work

The original objective was to investigate fatigue damage related to springing, but no clear
approach was predefined. Several relevant topics were considered, e.g.
? Fatigue strength methodology including effects of mean stress, residual stress, SN-curves,
SCFs, crack propagation and counting procedures in relation to the small vibration.
? The structural representation and solution methods including a method to predict struc-
tural damping, and practical means to increase it.
? Statistical methods to estimate short and long term contribution to fatigue and extreme
loading, i.e. how the wave frequency and high frequency response should be combined.
? Excitation sources from wave loading by a numerical or an experimental approach.
? Description of the wave environment, i.e. wave energy spreading, energy distribution
and the energy in the high frequency tail, obtained from wave measurements by buoys,
radars, satellites and wind models.
The DNV workshop on springing and the literature investigation pointed out the spring-
ing/whipping excitation sources as the main uncertainty, though certainly several of these
points above deserve more attention. First after two and a half years (after one year with
subjects and one year as a teaching assistant), it was decided to carry out model exper-
iments with the primary objective to reveal the dominating excitation source of the wave
induced vibrations, which contribute to the fatigue damage. Secondary objectives were to
suggest how the excitation sources can be included in numerical predictions, how model
experiments ought to be carried out for the purpose of design verification, suggest improved
geometry design to reduce the vibration level and propose ship rule formulas to limit the
effect of the wave induced vibrations.
1.4 Outline of the thesis

The background, motivation and literature investigation were covered in this chapter, and
the remaining part of the thesis is divided into the following sections
? The hypotheses, which may explain the observed vibration, are presented in Chapter 2,
which also cover other theoretical issues employed in the present research.
? Simplified analyses of whipping from stem flare impacts are presented in Chapter 3 for
an idealized vessel in ballast and cargo condition in head sea. The speed reduction as a
function of sea states is also covered.
? The model design, setup and strategy are shown in Chapter 4. Moreover, practical pro-
cedures used to evaluate the time series with respect to damping and transfer functions,
and a couple of methods to distinguish nonlinear and linear springing, and springing
from whipping are described. It deals also with the calibration of sensors and waves.
1.4. OUTLINE OF THE THESIS 21
? Chapter 5 presents the results and discussion. The results consist of damping estimates,
wave amplification at the bow, added resistance in waves, transfer functions and fatigue
damage. Observations illustrated by time series of different responses are displayed.
? Results from full scale measurements, in particular damping, speed reduction and fatigue
damage as a function of wave height, are shown in Chapter 6. Comparisons with model
test data are also presented.
? Finally, Chapter 7 presents a brief summary followed by conclusions from the different
tasks, contribution and recommended future work.
A more detailed amount of theoretical aspects is presented in the appendices.

Chapter 2
Theory
This chapter describes briefly selected theoretical issues which are relevant and related to
springing and whipping, while Appendix A summarizes some of the basic hydrodynamic
theory. Section 2.1 deals with the hydrodynamic hypotheses, which may contribute to
explain the significant ship vibrations observed. The hypotheses constitute the contribution
in this chapter. Section 2.2 elaborates on the various formulations of structural behaviour
employed, while Section 2.3 describes the different damping mechanisms, which influence
the resonance response. The remaining sections outline theoretical aspects relating to time
series analysis, spectral analysis, sea state models, transfer functions and fatigue damage
analysis.
2.1 Hydrodynamic theory

The governing equations of potential theory are quoted in Appendix A.1. They were used
to calculate the high frequency added mass of general 2D sections. The added mass was
combined with structural formulations to calculate the mode shapes and natural frequen-
cies, both as input to hydroelastic analysis and to determine appropriate locations, stiffness
and design of the flexible joints in the model test.
In the subsequent subsections the main excitation hypotheses are outlined. A simple
approach to estimate the speed reduction, which reduces the vibration response, is also
included.
2.1.1 Hypothesis 1: Steady wave elevation

When a ship moves forward in calm water it creates steady waves, which are independent
of time from a reference system fixed to the ship. The steady wave field has four obvious
23
24 CHAPTER 2. THEORY
effects.
1. The unsteady waves are superimposed on the steady wave elevation. To incorporate
the steady water elevation in numerical predictions may affect the results.
2. The interaction between the steady and unsteady waves may be significant. The
interaction may vary along the hull, e.g. it may be more pronounced in the bow
region in head seas. This may increase the generalised springing excitation force.
3. The steady wave field contributes to trim and sinkage (squat), which also affects the
force distribution along the hull.
4. The steady wave may cause breaking bow waves and instabilities.
The relevance of these effects may differ for blunt ships compared to slender ships. The
dominating literature on springing considers blunt ships, and the steep bow waves may
cause the incident waves to break, which may result in stem slamming.
The total steady velocity potential can be written as the summation of the free stream and
the steady disturbance potential
Φs = −U · x + φs (2.1)
This expression assumes a coordinate system fixed to the ship with incoming current mov-
ing with the speed of U in the negative x-direction with the ship at rest. The Froude
number can be assumed low, and the flow may be solved based on linear theory repre-
sented by a double body approximation (horisontal flow at the still water line). Once the
disturbance potential is determined, the Bernoulli’s equation, Eq.(A.5), can be used to find
an estimate of the wave elevation around the ship relative to the still water line. In centre
line and water line at the bow, i.e. at the stagnation point, a simple expression for the
surface elevation is derived from Bernoulli’s equation.
U2
ζs = (2.2)
2g
The speed is important, and at 15kn the wave elevation is roughly 3m. Similarly, the
steady elevation along the hull can be expressed as
1
ζs = (U 2 − Ue2 ) (2.3)
2g
where Ue is the tangential velocity at the intersection between the still water line and the
hull.
For a cylinder or a sphere the tangential velocity can be written

Uecyl = 2U · sin(θ)
3
Uesphere = U · sin(θ) (2.4)
2
2.1. HYDRODYNAMIC THEORY 25
The 3D case has a maximum difference from the 2D case at 900 , giving a smaller wave
trough at the side, while the difference at stem and stern is zero. The trough may in this
case be larger in absolute value than at the crest. Turbulence and vortex shedding may
influence the results for blunt bodies.
The wave elevation around an elliptic bow, which is defined by a source and a free stream,
is more realistic. Both a 2D and 3D source are considered. The key point is to find the
stream function value at the stagnation point in order to find the shape of the geometry,
and finally the tangential velocity along the shape at water line. Linear theory and double
body approximation are implicitly assumed. The 3D stream function is only applicable
in the axisymmetric case. Figure 2.1 illustrates schematically the steady wave elevation
around the approximate bow geometry for a speed of 15.2kn. The singularity point for the
2D case is moved slightly away from the origin in order to get the same stagnation point.
The breadth at infinite x-position corresponds to the ship breadth of 53m. The rise up
of water is wider in the 3D case, but the bluntness is also larger. The trough at the bow
quarter is -1.0m in the 3D case and -1.8m in the 2D case. The elevation at the stem is
3.1m. The illustration is certainly in qualitatively agreement with observations on blunt
ships (but not necessarily on slender ships).
Steady wave elevation Steady wave elevation
2D 3D
2D WL 3D WL
4 4
3 3
30 30
Steady elevation [m]
Steady elevation [m]
2 2
1 20 1 20
0 0
10 10
−1 −1
−2 0 −2 0
30 20 30 20
10 Longitidinal x [m] 10 Longitidinal x [m]
0 −10 −10 0 −10 −10
−20 −30 −20 −30
Transverse y [m] Transverse y [m]
Figure 2.1: Schematic illustration of steady wave elevation around the bow in the 2D (left)
and 3D (right) case.
For a real ship, the whole wetted geometry influences the steady velocity field and the wave
elevation around the hull. Nonlinear theory is necessary to describe it accurately, and the
viscous boundary layer may affect the behaviour in the aft ship.
The behaviour of the stem wave depends on the loading condition. Baba (1976) showed
that wave breaking resistance contributes with about 5% to the wave resistance. In deep
draft condition, short waves with unstable crests in front of the bow were observed, while
in shallow drafts condition the short waves broke. The ’necklace vortex’ around the hull,
caused by the breaking waves in the bow, was said to have its origin from a phenomenon
similar to the hydraulic jump. A bulb may reduce the wave breaking resistance in ballast
condition, and if the wave breaking resistance is related to springing, it may reduce the
springing excitation. The bulb was originally believed to cause nonlinear springing excita-
tion due to its shape, (Ohkusu 1980).
The interaction between the nonlinear steady wave field and linear unsteady wave field
has been investigated by a few, (Bertram 1998; Nechita 2001). The interaction had a
significant effect on the motions and response, and the elevation was sensitive to the ship
shape. The interaction increases with speed.
2.1.2 Hypothesis 2: Amplification of incident wave at the bow

In head seas the bow reflects the incident waves causing an amplification of the wave am-
plitudes. A basic result can be derived for a 2D regular wave, which is reflected by a fixed
vertical wall of infinite extent. From linear theory, the solution corresponds to a standing
wave with double amplitude compared to the incident wave amplitude, ζa .
The amplification is affected by the finite ship size and forward speed. In short small waves,
the ship motions will in practise be zero, and the ship will appear to be of infinite size.
This problem has been solved earlier in relation to added resistance in waves, assuming
the hull to be vertical at the water line. The wave amplitude is derived in Appendix A.2
from the velocity potential expressed by Faltinsen et al. (1980).
A simple expression of the amplification at the stem in head sea, expressed as the ratio of
the total amplitude to the incident wave amplitude, is
ζ ωe 2
= (1 + ωe2
) (2.5)
ζa ω0 +1
ω02
where ωe refers to the encounter wave frequency, Eq.(A.28), and ω0 is the wave frequency.
This expression will later be compared to experimental data.
The amplification versus wave period at three different speeds is illustrated in Figure 2.2.
The amplification increases for lower periods and higher speeds. The wave periods causing
springing by first order (linear) and sum frequency (second order) excitation are defined
at the intersections with the two rising curves. A springing period of 2s is assumed, hence,
linear springing in head sea is excited by wave periods of 4s at a speed of 15kn, giving
an amplification of 3. For the same speed, waves of 7s cause sum frequency excitation
with an amplification of 2.7. This is higher than the amplification of 2 for a fixed wall.
The increased splash zone due to forward speed may also be predicted by linear 3D panel
theory, but the linear springing theory considers only the wet area below the mean still
water line. An inaccurate (smaller) splash zone is predicted by 2D strip theory. The splash
zone refers to the area which is partly dry and partly wet due to relative motion between
the ship and the surface motion. The effect of the splash zone, causing nonlinear forces,
are regarded as important.
6
15kn
10kn
5 5kn
Linear
4 Second order
ζT / ζI
0
0 2 4 6 8 10 12
T0 [s]
Figure 2.2: Amplification of head sea incident waves at the stem versus the wave period at
three speeds (decaying curves). Wave periods exciting linear and sum frequency springing
are defined at the intersections with the two rising curves. A springing period of 2s is used.
The amplification is reduced compared to Eq.(2.5) in longer waves, since the vessel starts
to move, and the ship can no longer be considered as large compared to the incident waves.
2.1.3 Hypothesis 3: 3D impact forces

The combined effect of the steady wave and amplification of the incident wave result in
a wetted hull surface area that differs from the area commonly considered in numerical
predictions. The velocity of the water surface increases due to the reflection, and impacts
may occur, since the bow geometry deviates from wall sided shape away from the still
water line. In particular flare effects may be significant, but also the bilge area at the bow
quarter (bottom flare) may induce impacts due to the steady wave trough and the negative
amplification.
The impact force in head seas may be written, (Faltinsen 1990)

dA33 dVR d2 ζ
Fs = max(VR2 , 0) − A33 + ρgΩ(t) + ρΩ(t) 2 (2.6)
dh dt dt
dη3 dη5 Xn
dζ
VR = −x + q̇i (t)Ψ(x = Lpp /2) + U η5 −
dt dt i=3
dt
where VR is the relative velocity, which is positive upwards, and ρ is the density of water.
Eq.(2.7) includes rigid body motion in heave, η3, and pitch, η5 , flexible modes such as
the 2-node springing mode (i = 3), angle of attack term due to pitch and forward speed
and movement of the water surface. The interaction between the steady and unsteady
flow is neglected. q is the time dependent amplitude of the flexible mode shape, and Ψ
is the time independent orthogonal mode shape at the position of the impact. The angle
of attack term for unrealistic flexible structures should also include the derivative of the
flexible modes at the bow.
The first term in the impact force is often referred to as the slamming force, assumed to
act only when the bow enters the water. The other terms are often neglected, but could
be important especially in the exit phase, e.g. (Ge et al. 2002). The second term is an
inertia term, while the third and fourth term are due to the hydrostatic and Froude-Kriloff
force. Ω is the volume of the body getting wet relative to the steady water level and A33
is the high frequency 3D added mass.
How is the added mass found? 2D expressions may be useful, e.g. Wagner’s analysis of a
wedge is well known, (Faltinsen 1990), and an improved analytical formulation for skewed
impact of a wedge with exact boundary condition was derived by Driveklepp (2000). 3D
expressions, which may be more relevant for blunt ships, are however rare. Faltinsen and
Zhao (1997) derived the added mass for a cone following Wagner’s approach, which is valid
for small angles, while Shiffman and Spencer (1951) derived the added mass for a cone of
any deadrise angle based on approximate formulations. The latter formulation is described
in Appendix A.3, and it may represent the stem flare geometry. Alternatively, Chezhian
(2003) used a generalised Wagner’s theory, and Zhao and Faltinsen (1998) used a fully
nonlinear theory to investigate slamming on academic and axisymmetric bodies.
2.1.4 Hypothesis 4: Second order velocity potential and wave

elevation
In perturbation theory the total velocity potential without forward speed can be expressed
as
φT = φ1 + φ2 + ... (2.7)
where the linear velocity potential, φ1 , is proportional to the wave amplitude, ζ, and the
second order velocity potential, φ2, is proportional to ζ 2. Inserting Eq.(2.7) into the free
surface boundary conditions A.8, and keeping terms up to second order, the free surface
condition for the second order potential can be written
∂ 2
φ2tt + gφ2z = − (φ + φ21y + φ21z ) +
∂t 1x
1 ∂
φ1t (φ1tt + gφ1z ) on z = 0 (2.8)
g ∂z
Faltinsen (1990) considered two waves moving in opposite directions in deep water
gζ1 k1 z gζ2 k2 z
φ1 = e cos(ω1 t − k1 x + δ1 ) + e cos(ω2 t + k2 x + δ2 ) (2.9)
ω1 ω2
and derived the solution for the sum frequency effect
2ζ1 ζ2 ω1 ω2 (ω1 + ω2 )
φ2 = e|k1 −k2 |z sin[(ω1 + ω2 )t − (k1 − k2 )x + δ1 + δ2] (2.10)
−(ω12 + ω22 )2 + g|k1 − k2 |
If the wave number, k, is the same, there is no decrease of the second order pressure with
depth, z, and no traveling wave.
A secondary effect of the second order perturbation theory is the additional wave elevation
increasing the crest and reducing the trough. Considering a single regular wave in deep
water, the second order potential is zero, but the wave elevation to second order can be
expressed as
1
ζ = ζa cos(kx − ωt) + kζa2 cos(2(kx − ωt)) (2.11)
2
Figure 2.3 illustrates the effect of the second term for a typical wave height and wave period
exciting sum frequency springing. This secondary second order term is apparently small,
and Storhaug et al. (2003) indicated that the excitation from the second order incident
waves did not explain the high measured vibration response on the blunt iron ore carrier
with vertical sides.
The expressions for the second order velocity potential were derived for arbitrary headings.
Helmers (2001) derived it for regular waves and applied it to a barge. Sele (2001) extended
it to irregular sea, still without diffraction and applied it to a barge, and Vidic-Perunovic
and Jensen (2004) implemented the undisturbed wave into a second order strip theory, and
compared numerical predictions with full scale measurements from the ship.
The bow will reflect incident waves as explained in Section 2.1.2. This effect is relevant for
all ships, not only in bi-directional sea. The effect is similar, but not the same as the second
order effect in Eq.(2.10). The real ship has finite extent, forward speed and 3D geometry.
The forward speed appears in the free surface condition, the body boundary condition
and possibly in the force formulations. At zero speed this has been investigated on a
TLP, (Winterstein et al. 1994), by the program WAMIT, (MIT 1998). The uncertainty
is related to how the pressure decays with draft and the extent of the bottom area with
effective pressure in the case of forward speed.
2.1.5 Speed reduction due to added resistance in waves

Full speed in high sea states is unrealistic, and the vibration level is known to be sensitive
to the speed. Accordingly, it is necessary to predict realistic speeds as input to the model
0.8
0.6
0.4
0.2
ζ [m]
−0.2
−0.4
−0.6
−0.8
−1
0 50 100 150 200
x [m]
Figure 2.3: The second order (cont. line) versus linear (dashed) regular wave train.
ζ = 1m, T = 7s.
experiments, also since one of the objectives is to compare model test results with full scale
measurements.
The total wave resistance, RT , represented by the total resistance coefficient, CT , is written
as
RT
CT = (2.12)
0.5ρSU 2
where S is the wet surface of the actual loading condition. The total resistance can be
divided into different components, and the following components are considered in calm
water
CT = Cw + CV + CAA + CBD (2.13)
Cw is the wave making component, CV is the viscous friction, CAA is the air resistance
and CBD is the base drag resistance. The form factor method is used on the viscous part.
Appendix A.4 provides a more detailed description.
The next resistance component is the added resistance due to waves, Caw . The added
resistance in waves depends on the sea state, and a speed reduction of 5kn is typical for
the sea states dominating the fatigue damage, (Moe et al. 2005). The added resistance
in waves has contributions from bow reflection in the short wave length regime and from
larger ship motions in the longer wave length regime. Assuming short waves, which are
totally reflected by the ship, an analytical expression was derived by Faltinsen et al. (1980)
Z
Raw,1 1 2ω0 U
= ρg(1 + ) sin2(θ)n1 d` (2.14)
ζa2 2 g `
θ is the local angle displayed in the coordinate systems of Figure A.8, where n1 is also the
unit normal of the hull geometry (corresponds to N). This expression is integrated around
the non-shadow part of the ship, `, which is the intersection between the still water line
and the hull as illustrated in Figure A.8. Assuming head sea, and the bow to be shaped as
a circle with a radius of B/2, the added resistance in short waves is written
Raw,1 1 2ω0 U
2
= ρg(1 + )B (2.15)
ζa 3 g
For a wedge section with half apex angle, θ, the expression becomes
Raw,1 1 2ω0 U
2
= ρg(1 + ) sin2 (θ)B (2.16)
ζa 2 g
Faltinsen (1990) showed some reference values for realistic ship shapes, and the value for
the ship is slightly smaller than Eq.(2.15). This reflection contribution may be valid up to
wave lengths of half of the ship length, but up to 70% of the ship length is used. Between
70 and 100%, a linear decrease to zero is assumed.
The second contribution comes from ship motions, when the relative motion in the bow
and stern region becomes pronounced. It is relevant for wave lengths from the ship length
and above in head seas, but it depends on the forward speed. Faltinsen et al. (1980) pro-
vided resistance in waves for a ship with block coefficient of 0.8 versus the nondimensional
encounter frequency. A similar shape is assumed for other speeds, and a Gaussian shape
is used to reproduce it
p
Raw,2 (ωe L/g − 3.25)2
Caw,2 = = 8.75 exp (2.17)
ρgζa2 B 2/L 2σ 2
where σ = 0.4. Caw,1 is made nondimensional in the same way as Caw,2, but based on
Raw,1 . The total added resistance in waves is the sum of these two components. The shape
of each contribution and the total wave resistance curve are shown in Figure 2.4.
The added resistance in irregular sea is the sum of the individual waves defining the sea
state, hence
X
n
B2
Raw,T = (Caw,1,i + Caw,2,i)ρgζi2 (2.18)
i=0
LW L
where the individual wave amplitude, ζi , is found from the wave spectrum, S(ω), as in
Eq.(2.90).
Short waves Long waves Total

10 10 10
9 9 9
8 8 8
/(ρgζ2B2/L) 7 7 7
6 6 6
a
5 5 5
aw
4 4 4
R
3 3 3
2 2 2
1 1 1
0 0 0
0 5 0 5 0 5
ωe√(LWL/g) ωe√(LWL/g) ωe√(LWL/g)
Figure 2.4: Nondimensional added resistance in waves versus nondimensional frequency.
The power due to the total resistance must be in balance with the total delivered (available)
effect of the propulsion system.
RT · U + Raw · U = P · ηt (2.19)
where P is the engine power and ηt is the total efficiency coefficient of the propulsion
system. At least two of the terms, and possibly also the still water part, are influenced by
the waves. The calculation of the total efficiency coefficient was rejected. The trust versus
speed was found unreliable. The calculations were sensitive to the propeller revolution. The
data were obtained from similar ships and were inaccurate. Therefore, a fixed efficiency
coefficient independent of speed and propeller revolution was applied, assuming that the
maximum power was utilized. Further description of the coefficients is given in Appendix
A.4.
2.2 Structural representation

The structural representation of the hull depends on the design and response of inter-
est. In many cases a beam model, such as the Timoshenko beam, is sufficient, while e.g.
cruise ships with many discontinuous decks require refined models. Responses as torsion
(warping), racking, pitch connecting moments may require FE models, but thin walled
beam elements may be an alternative, (Wu 2002). The approaches, which were used to
calculate mode shapes, natural frequencies and responses in this study, are presented in
the following. The approached includes modal decomposition of bending deformation in
Section 2.2.1, analytical formulations including shear deformation in Section 2.2.2, FE rep-
resentation in Section 2.2.3 and segmented beam formulation with rotational springs for
2.2. STRUCTURAL REPRESENTATION 33
comparison with model tests in Section 2.2.4. The analytical derivations were used to il-
lustrate the modes shapes and for comparison with numerical FE analysis. A program for
the latter was developed and used to provide input modes to the hydroelastic calculations
reported by Storhaug et al. (2003).
2.2.1 Modal decomposition

The modal decomposition is extensively used to analyse the elastic behaviour of ships.
The method is illustrated by describing the expressions used in the simplified analysis in
Section 3.1.
The equation of motion describing the vertical motion, y(x, t), of a flexible beam, when
bending deformation and elastic support are included, can be written
(m + a)ÿ + bẏ + EIy IV + cy = P (x, t) (2.20)
The equation is based on homogeneous sectional properties per unit length, e.g. mass m,
added mass a, force P and water plane stiffness c. EI is the sectional bending stiffness
and b is the damping. The dots represent the time derivative, while the superscript, IV ,
denotes the space derivative. The time independent boundary conditions
L L
y II (± ) = 0 and y III (± ) = 0 (2.21)
2 2
refer to zero shear force and moment at the ends.
The flexible eigenmodes are deduced from the homogeneous equation by setting P = b =
c = 0. The natural frequencies of the flexible responses are insignificantly influenced
by the water plane stiffness, (Faltinsen 1980), but the water plane stiffness is essential
in calculation of the rigid body response. The equation can be solved by separation of
variables
X
n
y(x, t) = qi (t)Ψi (x) (2.22)
i=1
where q is the principal coordinate or amplitude, and Ψ is the eigenvector or mode shape.
Five analytical mode shapes are displayed in Figure 2.5. The mode shapes must be able to
describe both the rigid body response (quasi-static) and the vibration.
By combining Eq.(2.22), Eq.(2.20) and performing some manipulations, the beam equation
is converted into a set of independent equations representing the flexible responses, i =
3, 4, 5...
Z L/2
EI 4 fi
q̈i (t) + 2δi ωi q̇i (t) + β qi(t) ≈ P (x, t)Ψi(x)dx (2.23)
m + ai i (m + ai )L −L/2
Mode shapes
0.8 Heave Pitch

Normalised mode shape, 1 at bow
0.6
0.4
0.2
−0.2
−0.4
−0.6
−0.8 2−node Springing

3−node 4−node
−1
−150 −100 −50 0 50 100 150

Distance from midship [m], positive forward
Figure 2.5: The five first mode shapes for a homogeneous Euler beam including heave and
pitch.
The ≈ sign indicates that the f i constants are approximated. The added mass, damping
ratio, δ, and the β-coefficient are mode (frequency) dependent, and the total response is
the sum of the individual responses. Further description is given in Appendix A.5.
The hydrodynamic theory, the representation of the structural flexibility and the model
set-up should be selected depending on the purpose of the model test. The right hand side
of Eq.(2.23) is defined as the generalised force. The integrand is a multiplication of the
force per meter times the mode shape. The force from the harmonic load exciting resonance
has typically 5 to 15 oscillations along the ship, while the springing mode has one oscil-
lation, hence the response is affected by the accuracy of the force and mode distribution.
The integration may be less sensitive to local loads in the bow or stern region. The ef-
fect on how to split the model in case of whipping was investigated by Økland et al. (2003).
The bending moment can be calculated as
∂ 2y X n
V BM = EI 2 = EI qi(t)ΨII (x) (2.24)
∂x i=3
The rigid body modes do not directly contribute to the moment, since the curvature is
zero, while the flexible modes contribute to both quasi-static response and vibration. The
alternative way is to obtain the moment from the difference of the inertia forces and the
excitation, radiation and restoring forces integrated forward of the transverse cut. This
is the conventional way of doing it and then the rigid modes contribute. The quasi-static
response actually requires many modes in order to provide an accurate distribution along
the ship, while the dynamic flexible response only requires a few. The stress is calculated
as
V BM Xn Xn
II
σ= (z − zna ) = E(z − zna ) qi(t)Ψ (x) = E(z − zna ) qi (t)ΨII (x) (2.25)
I i=1 i=3
where z is the distance from base line to the actual position considered, and zna is the
distance from base line to the neutral axis. Sagging gives positive stress in deck.
2.2.2 Analytical solution of the beam equation with shear defor-

mation
For long slender beams, with low height to length ratio, the bending stiffness dominates,
while for short high beams the shear deformation dominates. For a ship both effects
may be relevant. The ratio may be related to the distance between the nodes. The
beam equation, without damping but including shear deformation, was derived based on
procedures outlined by Bergan et al. (1981). The solution for a free-free beam was not
found in the literature, but was derived here to check the accuracy of the FE program, and
to illustration the importance of shear deformation on the natural frequency. The equation
is expressed as
∂ 2y ∂2 ∂ 2y EI ∂ 4 y
m 2 + 2 (EI 2 ) − m = P (x, t) (2.26)
∂t ∂x ∂x GAS ∂x2∂t2
where AS is the effective shear area and G = E/2(1 + ν) is the shear modulus. The water
plane stiffness is neglected.
The solution procedure is the same as for a beam with only bending stiffness, (Faltinsen
1980). The homogeneous equation is considered, and the method of separation of variables
is used, which gives one differential equation for time and one for space. The equation for
space is solved by inserting a solution on the form Ψ(x) = Cesx, which gives a characteristic
equation with four roots. The solution may be written
Ψ(x) = C1 sin a1 x + C2 cos a1x + C3 sinh a2 x + C4 cosh a2x (2.27)
where
s p
(ab)2 + 4a
ab +
a1 =
2
s p
−ab + (ab)2 + 4a
a2 =
2
m
a = ω2
EI
EI 2(1 + ν)I
b= = (2.28)
GAS AS
By using the four boundary conditions for the free-free beam, expressions for the eigenvalues
can be derived by combining these four equations to remove the unknown C-coefficients.
a1 tan(a1L/2) = −a2 tanh(a2L/2) for i = 1, 3, 5...
a2 tan(a1L/2) = a1 tanh(a2L/2) for i = 2, 4, 6... (2.29)
Eq.(2.29) must be solved numerically. Once a1 and a2 are estimated, a is determined as
a41
a= (2.30)
1 + ba21
and the natural frequency becomes
r
EI
ωi = a (2.31)
m
The four equations are combined again to remove the C-coefficients. The last C-coefficient
is removed by normalizing the mode shapes to give unit deflection at the bow, hence
( 1 cos a1 x
a
( a2 )2 +1 cos a1 L/2
cosh a2 x
+ ( a1 1)2 +1 cosh a2 L/2
i = 1, 3, 5...
a2
Ψ(x) = 1
1 sin a1 x (2.32)
a
( 2 )2 +1 sin a1 L/2
sinh a2 x
+ ( a1 )12 +1 sinh a2 L/2
i = 2, 4, 6...
a1 a2
These mode shapes approach the mode shapes of pure bending for small values of b or
large values of AS , when a1 ≈ a2.
To illustrate the importance of the shear deformation, a rough approximation is derived

assuming that a1 ≈ a2 . The eigenvalues are solved from
tan(a1 L/2) ≈ −1 for i = 1, 3, 5...
tan(a1L/2) ≈ 1 for i = 2, 4, 6... (2.33)
with the approximate solution
π
a1 ≈ (2i + 1) (2.34)
L
Finally, the natural frequency is related to the frequency from pure bending, ωEI
r s
EI 1
ωi = a = ωEI · (2.35)
m 1 + ba21
To provide a numerical example, the following properties are assumed

L = 300m, I = 700m4 , ν = 0.3, AS = 2m2
The springing frequency is 0.90 times the frequency from pure bending stiffness. For
the 3-node and 4-node mode, the reduction factors are 0.78 and 0.67. From accurate
calculations, the reduction factors are 0.94 for springing and 0.83 for the 3-node mode.
The corresponding mode shapes and bending moments are shown in Figure 2.6. The mode
shapes differ less than the moments, which is calculated from
∂ 2Ψ(x) 1
V BM = EI( 2
+ mω 2 Ψ(x)) (2.36)
∂x GAS
The moment from the normalized mode shape is reduced to 92% of the moment for pure
bending stiffness. The significant change in the frequency indicates that shear deformation
ought to be included, in particular for higher modes.
Mode shape with and without shear stiffness

1
EI and GAS
Deflection, [1m at bow]
EI
0.5
−0.5
−1
−150 −100 −50 0 50 100 150
4 Bending moment with and without shear stiffness

x 10
3
EI and GAS
2.5
EI
2
VBM [MNm]
1.5
0.5
−0.5
−150 −100 −50 0 50 100 150
Distance from midship [m], forward positive
Figure 2.6: The effect of shear deformation and moment on the springing (and the 3-node)
mode based on realistic dimensions.
From Figure 2.6, a few observations can be made. A bending moment of 3000MNm gives
a deflection of 10cm at the ship ends. Assuming a sectional modulus in deck amidships of
60m3 , which is a reasonable value for a 300m ship, gives a nominal vibration stress ampli-
tude of 50MP a. If the derivative of the mode shape is considered, and a man is standing
in one end of the ship, the 10cm at the other end would look like a vibration amplitude
of 42cm in his reference system. High vibration response will therefore be visible, and the
visible deflection (including the quasi-static effect) can be used to indicate the level of the
vertical bending moment amidships.
Minor effects of damping, rotational mass and end pressures are presented in Appendix A.7.
2.2.3 Finite element approach

A small program based on finite beam elements was developed to provide input of flexible
modes to the program VERES, (Buarø et al. 1999), which was used by Storhaug et al.
(2003). The equation of motion is written
Mÿ + Bẏ + Cy = P(t) (2.37)
where M, B and C are the mass, damping and stiffness matrices. y and P are the
displacement and force vectors. The eigenvectors and natural frequencies are derived from
the homogeneous equation
Mÿ + Cy = 0 (2.38)
by using the method of separation of variables
y = Ψq (2.39)
A harmonic solution of the time dependent principal coordinate vector, q, is assumed. The
natural frequencies and thereafter the eigenvectors, Ψ, are then derived from
(C − ω 2 M)Ψ = 0 (2.40)
The matrix element formulations of stiffness and mass are quoted in Appendix A.6. The
stiffness matrix includes bending and shear deformation. The mass matrix, which includes
rotational mass, assumes an even mass distribution within the element. Alternatively, the
masses are lumped to the element ends, which give a diagonal mass matrix with zero el-
ements outside the diagonal. The rotational masses may be skipped, and the equation
system may have to be reduced depending on the eigenvalue solver.
The FE program is verified against the analytical solution presented in Section 2.2.2 for a
homogeneous beam with lumped mass distribution and small rotational mass. Realistic
data corresponding to the ship are used. The convergence of the maximum bending moment
for the springing mode versus the number of elements is presented in Figure 2.7. Roughly
10 elements are necessary to represent the springing mode, and the convergence study
verifies the accuracy of both the FE program and the analytical solution.
2.2.4 Segmented models

The stiffness distribution in the model tests differs from the real ship. The model consists
of stiff segments produced by mounting the ship’s hull onto a rigid steel frame. In between
the stiff segments, flexible joints are inserted. The flexibility is therefore restrained to
specific locations, and the stiffness is discontinuous.
Ge (2002) used a segmented model consisting of rigid bodies connected by flexible beams.
In the present study the design of the flexible joints are refined to make the stiffness ad-
justable to achieve the requested natural frequency. It is convenient with a different stiffness
2.3. DAMPING 39
VBM convergence
1.35
1.3
1.25
1.2
1.15
VBM / VBManalytic
1.1
1.05
0.95
0.9
0.85
0 10 20 30 40 50 60 70 80 90 100
No. of beam elements
Figure 2.7: Convergence of the FE midship 2-node moment versus the number of elements.
formulation based on rotational and shear spring stiffness instead of beam elements. The
stiffness based on the moment M and the shear force Q is written
M
kθ = (2.41)
θ
Q
kz =
z
These properties were obtained from calibration measurements and from modelling the
joint by 3D BEAM, (DNV 2005d). The definition of the local stiffness and mass ma-
trices are presented in Appendix A.8. They are inserted into a matrix system similar to
Eq.(2.40). Appendix A.8 presents confirmation of the expressions versus SESAM (DNV
2005f), analytical solution and results by Økland (2002).
2.2.5 Single degree of freedom system

By invoking modal analysis the resulting set of dynamic equations of motion is uncoupled.
Characteristic features of the behaviour of SDOF systems with linear damping and stiffness
are presented in Appendix A.9.
2.3 Damping
The damping is a key parameter in resonance phenomena. It influences both the amplitude
and number of significant vibration cycles in case of whipping. The damping may be caused
by internal and external mechanisms, and it may be linear or nonlinear. In all cases
damping is dissipation of energy during a cycle. The lost energy (work) can be expressed
as the area inside a closed hysteresis in a force-displacement diagram, e.g. (Langen and
Sigbjørnsson 1979).
I
Wd = Fd dy (2.42)
where Fd is the total damping force and y is the displacement. A few relevant damping
mechanisms will be described.
2.3.1 Linear damping

Linear hydrodynamic damping is typically caused by wave generation due to the motion of
the vessel. It is frequency dependent, and typically approaches zero for high and small fre-
quencies. Faltinsen (1990) presented a linear 2D hydrodynamic damping, relating outgoing
waves to the motion of the vessel
ζr 2 g 2
b2D
33 = ρ( ) (2.43)
|η3 | ω 3
ρ is the density, η3 is the heave motion and ζr is the radiated wave amplitude. The damping
2D
coefficient is multiplied with the velocity in the equation of motion, i.e. b33 · ẏ.
Foils may introduce speed dependent damping, bf , due to lift and heave velocity. The
lift for a 2D flat plate is expressed by e.g. Newman (1977). Since the heave velocity will
change the effective angle of attack, the quasi-steady damping can be expressed as
bf = ρU cπBf (2.44)
where c is the cord length and Bf is the width of the foil.
Forward speed may give hull lift damping. Sele (2001) wrote it as
Z
2 2D ∂Ψ 2D
bhl = U Ψ(xtransom ) a33 (xtransom ) − U Ψ a dx (2.45)
L ∂x 33
This is further explained by Faltinsen (2005). Similar contributions may be identified from
coupling terms, as expressed by Faltinsen (1990) for rigid body motions in linear strip
theory.
A third speed dependent damping term is often used in theoretical investigations, e.g.
(Goodman 1971). It is similar to the prior term, and it is multiplied with the mode shape
squared and integrated along the hull. The contributing parts are located at the bow and
stern.
∂a2D
33
U (2.46)
∂x
2.3. DAMPING 41
The three latter expressions are all linearly dependent on speed.
Elastic materials do not have a hysteresis curve indicating dissipation of energy during a
cycle, because it follows the same path for both loading and unloading. However, steel
structures in practise have welds including cracks. These give hysteresis curves under
cyclic loading. Elliptic hysteresis curves for steady state harmonic loading indicate linear
damping. The hysteresis damping of materials is defined as independent of frequency, but
may be amplitude dependent. For instance, Bergan et al. (1981) expressed this as
ẏ
b·y (2.47)
|ẏ|
The structural damping may also be expressed as proportional to the displacement, but
in phase with the velocity. In this case it may be expressed as a complex stiffness, e.g.
(Langen and Sigbjørnsson 1979)
c(1 + ib)y (2.48)
2.3.2 Nonlinear damping

Viscous hydrodynamic damping is often nonlinear, and it may be written
b2ẏ|ẏ| (2.49)
where ẏ is here the relative vertical velocity. It has contributions typically from bilge keels,
skin friction, eddy-making and transom stern effects, and the forward speed influence the
results. The rudder may provide both viscous and foil damping, with vortex shedding as
a related phenomenon.
Coulombs damping is another nonlinear contribution due to friction. The damping force
may be expressed as
µNsign(ẏ) (2.50)
where N is the normal force and µ is the friction coefficient, e.g. (Langen and Sigbjørnsson
1979).
Finally, water impacts, when occurring in the flat stern geometry as a consequence to
violent vibration, may actually contribute as a nonlinear damping due to antisymmetric
loading.
2.3.3 Methods to estimate linear and nonlinear damping

The effect of the damping is illustrated for the linear SDOF system described in Ap-
pendix A.9. The linear damping can be estimated based on the decay curve after an
impulse loading. Hammering tests are practical for model tests, while dropping the anchor
or using a shaker are more convenient for a real ship. The damping ratio is estimated as
1 xi
δ= ln( ) (2.51)
2πn xi+n
where xi is the peak value of amplitude no. i. Alternatively, the half-power bandwidth
method can be used, (Clough and Penzien 1993). The damping ratio is estimated as
β2 − β1
δ= (2.52)
2
where β1 and β2 is the ratio of the frequency to the natural frequency placed on each side
of the resonance peak, and corresponding to the response level of the peak value divided
by the square root of two. The difference is therefore the width at this particular response
level of the transfer function.
In full scale, the methods above may not be feasible, and damping must be estimated based
on irregular response. The half-power bandwidth method, Eq.(2.52), can however also√ be
used in this case. The width should then be taken at half the peak value, and not at 1/ 2
times the peak value. Further, the damping in irregular sea states can be estimated by
a linear spectral approach outlined in (Rüdinger 2003; Storhaug et al. 2003). Basically,
it fits the analytical spectrum from the SDOF system to the spectrum based on FFT of
the time series. The analytical relation between the response spectrum, Sx(ω), and the
excitation spectrum, S(ω), is written
1
Sx (ω) = |H(ω)|2S(ω) = S(ω) (2.53)
(ωs2 − ω 2 )2+ 4δ 2 ωs2 ω 2
The excitation spectrum is assumed to be described by the high frequency tail of the wave
spectrum.
A
S(ω) = (2.54)
ωn
The slope parameter n, the coefficient A, the springing frequency ωs and the damping ratio
δ are determined by unconstrained nonlinear optimization using the MATLAB function
fminsearch. Guessed input values are necessary to control the procedure. A Hanning win-
dow is used with 4096 FFT points defining the resolution. The frequency interval is also
important, since the total response is not described by a SDOF system.
Storhaug et al. (2003) estimated the slope parameter n to be close to 4, which would be
representative for Torsethaugen wave spectrum rather than the JONSWAP spectrum. By
considering other loading conditions and ships, it was however confirmed that this slope
factor was not necessarily representative for the high frequency tail behaviour of the wave
spectra. This may be due to the hump-hollow behaviour of the excitation. To obtain a
2.4. TRANSFER FUNCTIONS 43
proper estimate of the high frequency slope, the expressions should be modified.
The following procedure may be used to determine the nonlinear damping of the viscous
type. It was used to check the nonlinearities of the damping in e.g. Figure 5.5. The
homogeneous equation of motion including viscous damping may be formulated as
mÿ + b1 ẏ + b2 ẏ|ẏ| + cy = 0 (2.55)
where b2 is the damping coefficient related to the viscous part, while b1 is related to con-
ventional linear damping. This equation can be linearized within each period by requiring
equivalent linear energy dissipation per cycle from the damping terms, e.g. (Faltinsen 1990;
Huse 1993). The equation is reduced to
mÿ + btot ẏ + cy = 0 (2.56)

where
8
btot = b1 + ωy0 b2 (2.57)
3π
y0 is the response level for the cycle considered. The total damping btot from the decay
8
curve can be plotted versus 3π ωy0 . The crossing with the y-axis defines the linear b1 damp-
ing, while the slope represent the viscous damping coefficient b2 .
2.4 Transfer functions

2.4.1 Linear transfer functions
The response, y(t), to a linear system exposed to a linear oscillating force, f (t)
f (t) = ai cos(ωi t) (2.58)
is described by
y(t) = ri cos(ωi t + θi ) (2.59)
where θi is the phase lag, ωi is the forced frequency and ai and ri are amplitudes. The
ratio of response amplitude to force amplitude for all frequencies are defined as the linear
transfer function amplitude, referred often to as the transfer function
ri
|H1 (ωi )| = (2.60)
ai
while H1 (ωi ) is the complex transfer function, which contains information about the phase
relations.
The transfer functions are made nondimensional as shown in Appendix B.2.2.

2.4.2 ’Nonlinear’ transfer functions

’Nonlinear’ quasi-transfer function is defined as the ratio of the maximum response ampli-
tude and the wave amplitude. The purpose is to illustrate the relative importance of the
higher harmonics at the encounter frequency. The force is assumed to create a response
consisting of the encounter response plus higher harmonic responses as
X
N
y(t) = ri cos(iωi t + θi ) (2.61)
i=1
The quasi-transfer function is now defined as
max(y) − min(y)
|HN (ωi )| = (2.62)
2ai
where the maximum and minimum values are identified within each encounter period. The
higher harmonics depend on the excitation level, hence, higher excitation amplitudes will
increase the contribution from the higher harmonics and the transfer function is only partly
nondimensional.
2.4.3 Second order transfer functions

If the excitation force consists of two frequency components
f (t) = ai cos(ωi t + θi ) + aj cos(ωj t + θj ) (2.63)
the response including second order effects can be written
y(t) = r1 cos(ωi t+θ1)+r2 cos(ωj t+θ2 )+r3 cos((ωi +ωj )t+θ3 )+r4 cos((ωi −ωj )t+θ4 )+r5
(2.64)
The second order transfer function is derived from a part of this response as
r3 r4
|H(ω1 , ω2 )|sum = |H(ω1 , ω2)|dif = (2.65)
ai aj aiaj
In this study only the sum frequency effects are considered, hence it represents only the
first quadrant of the 3D transfer function, which should be symmetric about the 450 line
where ω1 = ω2 .
Results are presented only for the vertical bending moment, which is made nondimensional
in the following way
V BM
HV BM (ω1,e , ω2,e ) = √ (2.66)
ζ1 ζ2 k1 k2 ρgBL2
2.5. TIME SERIES ANALYSIS 45
A monochromatic wave amplitude, ζ, is considered as equivalent to two waves of half am-

plitude. The wave numbers ki = 2π/λ, are necessary to make the moment nondimensional,
since the square of the wave amplitude is also employed.
A more detailed procedure on how the linear, ’nonlinear’ and second order transfer func-
tions are established is presented in Section 4.6.1 and 4.6.3. Nonlinear systems may also
be assessed in alternative ways, e.g. (Schetzen 1980).
2.5 Time series analysis

The methods used to evaluate the time series with respect to mean values, amplitudes,
phases and uncertainties are outlined in the following. The processes are assumed station-
ary, ergodic and Gaussian distributed. The Gaussian distribution of the vibration response
was illustrated by Storhaug et al. (2003).
2.5.1 Estimate of amplitudes

The linear transfer function is established by different methods. The first method removes
the mean value and defines the zero crossing process. Between the zero crossings, the
local maximum and the local minimum are identified, and these are coupled to define the
amplitude as half of the difference. By combining pairs of these, the mean amplitude, a,
is estimated.
1 X max(x) − min(x)
N
a= (2.67)
N i=1 2
This method is useful in order to estimate the uncertainty represented by the standard
deviations, Appendix B.3. Filtering may be necessary.
The second method assumes a sinusoidal signal, and finds the amplitude from the standard
deviation of the whole times series using all time samples,
√
a = 2 · σ where
1 X
n
2
σ = (xi − x)2 (2.68)
n − 1 i=1
which is useful when only one single frequency is present. Alternatively, filtering may be
applied.
The third method finds the amplitude from the standard deviation of the discrete wave
spectrum (when the mean value is zero).
√
a = 2 · σ where
Xn
σ2 = S(ωi )dωi (2.69)
i=1
This method is useful when several distinct peaks in the spectrum exist, and the ampli-
tudes corresponding to the different frequencies are to be determined.
The fourth method fits a series of sinusoidal components to the signal using a least square
fit.
XN
Ai sin(ωi tj ) + Bi cos(ωi tj ) = R(tj ) (2.70)
i=1
The subscript j refers to each sample in the selected time series of the response, R, while
subscript i depends on the chosen frequencies which are used in the fitting procedure. The
coefficients A and B are solved using the MATLAB function lsqr. The same resolution
as the FFT is used, and zero frequency is included to capture the mean level. For each
frequency the sin and cos pairs can be rewritten as
ri cos(ωi tj + θ) (2.71)
Hence, it is useful for finding the phases between components. Iteration on the frequency
is necessary for long time series.
The fifth method, referred to as unconstrained nonlinear optimization, finds the minimum
of a function containing several unknown parameters. The function is the mean of the
squared difference between the time series and the expression ri cos(ωi tj + θ). An initial
estimate of the unknown parameters is necessary, and the method becomes less robust
when the number of parameters increases. The MATLAB function fminsearch is applied.
In principle, all these methods should provide the same answer, which will confirm the
correctness of the methods used. In each case at least two methods are used.
2.5.2 Spectral density

The response spectrum illustrates distinct frequency effects, indicates noise and can be
used for filtering purposes. The spectrum is obtained from the time series. Two different
methods are considered.
The first method (Wiener-Khintchine Theorem) derives the spectrum by first establishing
the autocorrelation function. Assuming a continuous time series, this is written
Z T
1
Rxx (τ ) = lim x(t)x(t + τ )dt (2.72)
T →∞ 2T −T
Z
1 ∞
Sxx (ω) = Rxx (τ )e−iωτ dτ (2.73)
π −∞
It assumes that the stochastic process is ergodic, i.e. the ensemble average can be eval-
uated as a time average of a single record. The process should be at least weakly stationary.
2.5. TIME SERIES ANALYSIS 47
The second method applies Parseval theorem to find the autospectral density function,
herein referred to as the spectrum
1
Sxx (ω) = lim |X(ω)|2 (2.74)
T →∞ 2T
The second method is more efficient, and FFT is applied to the discrete finite time series.
The definition in MATLAB differs from Eq.(2.74) with reference to factors and integration
boundaries, which are accounted for. The inverse FFT reproduces the filtered time series.
For a finite time series, the end effects may become pronounced for short time series in
combination with a narrow band filter. A rough filter (cut off) is used to remove the part
of the spectrum outside the frequency interval of interest, and the applied filter did not
change the phasing between the harmonics. The resolution of the discrete FFT is
2π
∆ω = (2.75)
dtN
which may be rather large, if the time series is short.
The cross correlation function, Rxy , defined by replacing x(t) in Eq.(2.73) with y(t), is
used to define the phase between a response and a reference response, y. The period, T0,
of the reference response in regular waves is found by the autocorrelation at the first local
maximum, τ = T0 > 0. The response period is found in the same way, and the ratio defines
the phase lag.
Smoothing the raw FFT spectrum gives a better appearance, rather than the spiky ap-
pearence from the discrete spectrum. Moreover, using the smoothed spectrum will provide
more reliable estimates of certain characteristics such as the peak period. This method
has therefore been employed when necessary. A Gaussian distribution, referred to as a
Gaussian Bell, is defined as
1 (x − x)2
f (x) = √ exp (2.76)
2πσ 2σ 2
where σ is the standard deviation. The original spiky discrete spectrum was smoothed
by moving this Gaussian Bell through the frequency range and thereby weighting each
specific value based on the Gaussian distribution. A wide bell will remove all details, while
a narrow will not make a difference. A standard deviation of 0.125rad/s was used, and
95% of the interval considered was then confined within 0.5rad/s. The springing peak will
become lower and wider. The smoothing will provide end effects, reducing the values at
the lower and upper end to half. This does not matter, because it is above or below the
frequency range of interest.
2.5.3 Wavelet transformations
The wavelet transformations may be used to evaluate the quality of a time series, e.g. to
detect when the short waves break down in towing tanks, to identify relations between
waves and response, or between different responses, and to detect/count special events. In
this thesis it was used to evaluate the wave quality of short waves.
Wavelet analysis is a method to investigate properties of time series. While Fourier analysis
provides the spectrum, the wavelet analysis provides also a time or space variation of the
spectrum. E.g. for a stochastic process and for an impulse process, the Fourier analysis
may give the same spectrum. A single extreme peak in the time series is smeared out.
The wavelet analysis will give a spectrum which reveals the single extreme peak at other
frequencies than the main components indicating a special event. Fourier analysis may
alternatively be used over a time interval that is shifted gradually.
Newland (1993) considered orthogonal wavelets, which is useful, because efficient algo-
rithms similar to FFT can be utilized. Harmonic discrete wavelets were only briefly de-
scribed. His method involved a specific frequency resolution limited by the number of
time samples used. This is not convenient, since small changes in the frequency can be of
interest. Moreover, the square maps used to visualise the spectrum give a poor resolution
in time for the lower levels of low frequency.
Donoho et al. (1999) offers MATLAB routines, and cwt.m was modified to serve the
purpose of this research. Each octave band (according to Newland (1993)) was divided into
new layers to increase the resolution with corresponding new different wavelet components.
The same routine is also modified to visualise the square map for a continuous wavelet
moving through the whole time series rather than being distributed at specific points. The
basic wavelet used is expressed as
ω0 1 t−τ 2
ψ(t) = ei s
(t−τ )
e2( s
)
(2.77)
where 1 < s < 2 defines the width at each layer, and s is doubled for each octave. τ is the
mid point of the wavelet, which was moved through the time series by changing τ stepwise.
The wavelet method does not appear to be well understood from a physical point of view.
An alternative method, which may be more promising is the Hilbert-Huang transforma-
tions, (Trulsen and Krogstad 2001; Veltcheva 2001). Huang et al. (1998) developed the
method and provided a detailed description of it.
2.6. WAVES AND SEA STATES 49
2.6 Waves and sea states

2.6.1 Irregular waves and spectral density
A sea state consists of irregular waves. In linear theory the irregular waves is developed as
superposition of regular waves. The sea state is commonly represented by a wave energy
spectrum such as JONSWAP, (DNV 2000)
5 ω ω−ω
exp[−0.5( σω p )2 ]
S(ω) = αg 2 ω −5 exp[− ( )−4 ]γ p
4 ωp
5 HS2 ωp2
α= · (1 − 0.287 ln(γ))
16 g 2
σ = 0.07 when ω ≤ ωp and σ = 0.09 when ω > ωp (2.78)
where γ is the peakness factor, ωp is the spectral angular peak frequency and α is the
generalised Phillips’ constant. The significant wave height, HS , and peak period, TP = 2π
ωp
,
are input values defining the sea state. From a measured spectrum the significant response
and zero up-crossing can be estimated as
√
HS ≈ 4 m 0 (2.79)
r
m0
TZ = 2π (2.80)
m2
Z ω∗
m0 = S(ω)dω (2.81)
0
Z ω∗
m2 = S(ω)ω 2 dω (2.82)
0
This can also be estimated from the time series. A one side spectrum is assumed with
an upper limit, ω ∗ , which is limited by the length of the time series or noise. The zero
crossing period is more sensitive to the cut off at the ends than the significant wave height
due to ω 2. The relation between the peak period and the zero up-crossing period, TZ , for
JONSWAP is written
r
11 + γ
TP = TZ (2.83)
5+γ
JONSWAP is representative for developing seas, while Pierson-Moskowitz (PM) wave spec-
trum is a special case with γ = 1 representing fully developed sea and unlimited fetch. The
PM spectrum based on wind input is written
A − B4
S(ω) = e ω where
ω5
g
A = 0.0081g 2 and B = 0.74( )4 (2.84)
V
where V is the wind speed 19.5m above the water surface, (Myrhaug 1998).
Torsethaugen double peak wave spectra include both swell and wind sea, (Torsethaugen
1996). The high frequency tail behaviour differs from JONSWAP and PM. The exponent
n in ω −n from Torsethaugen may be closer to four rather than five in the other two spec-
tra. This affects the springing predictions as illustrated by Tongeren (2002). The wave
energy spreading, represented by cosp (θ) or bi-directional spectra, is relevant in full scale,
(Storhaug et al. 2003; Vidic-Perunovic 2005), but the effects are not considered during the
model tests.
The encounter response or encounter wave spectra are converted into response or wave
spectra by considering the encounter frequency relation in Eq.(A.28). Rewriting the fre-
quency relation, accounting for head to beam seas (not following sea; absolute signs), gives
s
g U
ω0 = (−1 + 1 + 4ωe cos(β)) − 900 < β < 900 (2.85)
2U cos(β) g
The relation between the wave and encounter frequency spectrum is given as
S(ω)dω = Se (ωe )dωe
U
S(ω) = Se (ωe )(1 + 2 ω cos(β)) (2.86)
g
and the wave and encounter frequency are related values through Eq.(2.85).
Wave breaking is avoided during the model tests to reduce the change of the spectra to a
minimum. The breaking (steepness) criterion for a regular wave is written
g 2π 2
λ ( )
2π ω0
H< = (2.87)
7 7
while for irregular sea, the following criterion is assumed applicable in the towing tank
2πHs
S= < 0.03 (2.88)
gTP2
to avoid a single wave breaking in the tank. The waves along the towing tank represent
an approximation to real sea states, since it is no wind in the towing tank. Especially, the
energy in the high frequency range is expected to be reduced along the tank due to lack
of wind. The breaking criterion
1
2πHS 10
for TZ ≤ 6s
S= 2
< 1 (2.89)
gTZ 15
for TZ ≥ 12s
stated in (DNV 2000) agree well with Eq.(2.88). Linear interpolation is used in between
the ranges.
To generate a time series of the wave elevation, the individual regular wave components
were derived by dividing the wave spectrum into many blocks. Each block had a specific
2.7. FATIGUE DAMAGE 51
frequency interval, dω. The individual wave amplitudes were estimated by a single block
as
p
ζi = 2S(ωi )dω (2.90)
where ωi is the mid frequency in the block. The sea state with waves moving in the negative
x-direction was assembled as
X n
ω2
ζ= ζi · cos( i x + i + ωi t) (2.91)
i=1
g
where i is a random number between 0 and 2π. The amplitude could also have been a
stochastic variable, (Wang 2001), but this was not important in this context.
2.6.2 Wave heights exciting linear and second order springing

It may be of interest to know the wave amplitudes that excite the linear and second order
springing response. The second order transfer function at resonance depends on the wave
amplitude squared as well as the damping level. In this thesis the sum frequency springing
response obtained from model experiments was corrected based on the corresponding ex-
citing wave height and period to illustrate the relative importance of the monochromatic
sum frequency springing and linear springing. This is further explained in Section 4.9.
An investigation by Myrhaug and Kvålsvold (1992) and Myrhaug and Kvålsvold (1995)
were utilized. The joint distribution of wave heights and periods was established by Weibull
fits to wave measurements. This means that it was possible to find an estimate of the
mean regular wave height given the wave period in a specific irregular sea state. This is
further described in Appendix A.10. Based on mid interval significant wave height in the
Beaufort scale in Appendix A.10, the mean regular wave heights exciting the linear and
monochromatic second order springing as a function of the Beaufort strength are shown
in Figure 2.8. The wave height exciting second order springing is roughly half a meter
higher and two seconds longer than the waves exciting linear springing in head seas. The
lower plot presents also the zero up-crossing period and the specific wave periods exciting
springing, and the middle plot displays the significant wave height corresponding to the
Beaufort strength. The wave heights increase with Beaufort strength, even though the
speed reduction from full scale measurements was accounted for.
2.7 Fatigue damage

Cracks causing leakage are unacceptable. Crack growth or fatigue may be divided into
three phases: Initiation, crack growth and sudden fracture (or crack arrest). In this con-
text, the crack growth phase due to high cycle fatigue is considered. Initial cracks will
always exist as small air voids in welded materials. According to current design practise,
4
3 Second order
H [m]
Linear
2
1
0
1 2 3 4 5 6 7 8 9 10 11
15
10
HS [m]
5 Tz
0 TSec
e
1 2 3 4 5 6 7 8 9 10 11
Lin
Te
15
10
T [s]
0
1 2 3 4 5 6 7 8 9 10 11
Beaufort number
Figure 2.8: Mean regular wave heights, H (upper plot), exciting linear and monochro-
matic sum frequency springing in head sea state at realistic speeds as a function of the
Beaufort strength. The speeds are interpoated between pars of speed in knots and signif-
icant wave height (15/0,15/4,10/5,7.5/6,5/7,2.5/8,0.1/9,0.1/20). The significant wave
height in middle plot and the zero crossing period, TZ , in the lower plot correspond to the
Beaufort number. In the lower plot the wave periods corresponding to linear, TeLin , and
monochromatic sum frequency, TeSec , springing are also shown.
the fatigue life is defined as the time it takes for an initial crack to grow through the
thickness of the plate. Such short cracks are difficult to detect by visual inspections.
The strength formulation is in accordance to DNV Classification Note on fatigue, (DNV

2005a). The cumulative fatigue damage is calculated by Palmgren-Miner’s rule
Xk
ni
D= ≤η (2.92)
i=1
Ni
where ni is the number of cycles for a specific stress range in block i, and Ni is the number
of cycles to failure at the same constant stress range. The summation should be less than
η = 1.0 or some lower value depending on consequence and access for inspection. Miner-
Palmgren’s rule disregards the effect of the loading sequence, and from Wirshing (1981)
the mean value and coefficient of variation (CoV) are estimated to µD ∼ = 0.8 and CoV ∼ = 0.3.
The number of cycles to failure as a function of the stress range is defined by the SN-curve
for a welded joint in air or with cathodic protection. The SN-curve is based on a welded,
2.8. LONG TERM FATIGUE ANALYSIS 53
but grinded joint without any SCF . The SN-curve corresponds to a 2.3% probability of
failure, hence the mean fatigue life is about twice. The SCF is multiplied with the nominal
stress range to define the total stress range, ∆σ. The two slope SN-curve is written
log(N ) = log(a) − m log(∆σ) (2.93)

log(a) = 12.65 m = 3 when N ≤ 107
log(a) = 16.42 m = 5 when N > 107
The mean stress level is disregarded, and the residual stress is assumed to be tensile at
yield, but the effects are otherwise disregarded. Thickness effects are also neglected as-
suming a thickness below 25mm.
The full scale measurements of the actual iron ore carrier were evaluated by (Storhaug
et al. 2006), and an equivalent slope of m = 4 for the wave frequency damage and m = 3.7
for the vibration damage were determined based on several years of measurements. This
is useful when scaling long term results due to different stress levels, and it is used in
Section 6.5.
2.7.1 Rainflow counting

Based on the stress time series established from the measurements, the stress cycles, which
result in fatigue damage, need to be determined. For a general wide band process, the
Rainflow counting method has been verified by experiments, (Strating 1973), to count
cycles that contribute to the crack growth. The rigid body response is a slightly wide band
process, while the vibration response is a narrow band process, but the combined effect is
certainly a wide band process. The counting procedure results in an exceedance diagram
of stress ranges. The standard fatigue calculations can subsequently be applied. Further
description of Rainflow counting may be found in for instance (Almar-Næss 1985; ASTM
1998; Glinka and Kam 1987), with an alternative in (Jiao and Moan 1990).
2.8 Long term fatigue analysis

The short term damage, during a typical sea state of tree hours, is established through
Rainflow counting of the measured time series. For shorter time series simple time extrap-
olation to three hours is carried out.
To estimate the fatigue damage during 20 years of service, the damage is estimated for
all the relevant sea states defined by a scatter diagram. The scatter diagram presents the
different sea states with their probability of occurrence. The sea states are characterised
by combinations of significant wave height and zero up-crossing period. The important
sea states in North Atlantic and World Wide trade (scatter diagram; i.e. combinations of
significant wave height and period) are illustrated in Section 4.7.1.
The current investigation only accounts for a vessel in head sea, but ballast condition, cargo
condition and port time were considered. The long term fatigue damage was calculated as
the weighted sum
NL NS
20 · 364 · 24 X X L S
D= pn pi dn,i (2.94)
3 n=1 i=1
where p refers to the probability of loading condition, L , and sea state, S . dn,i is the corre-
sponding fatigue damage for a specific three hour sea state and loading condition. For the
ship, the time in port, ballast and cargo condition were 33.3% each in the North Atlantic
trade, (Moe et al. 2005).
Since, there was not enough time to carry out experiments for all sea states in a scatter
diagram, extrapolation and interpolation was carried out based on a response surface
method, which is described in Section 2.8.1.
2.8.1 Use of response surface to extrapolate results

The model was tested in some representative sea states. These were selected to reveal the
trends of the fatigue damage due to vibration and wave excitation. A rough procedure
was derived to extrapolate the fatigue damages from the selected sea states to calculate
the total fatigue damage for the whole scatter diagram. The method is referred to as the
response surface method. Results from the equations below are illustrated in Section 5.4.5.
The hydrodynamic loading phenomena that cause the vibration damage in different sea
states are not fully known, e.g. both springing and whipping contribute. Hence, empirical
equations were tested to represent the distribution of the fatigue damage for the whole
scatter diagram. The empirical equations expressing the response surfaces were fitted to
the results of the fatigue damage in the different sea states. The WF damage and HF dam-
age were considered separately. Similarly, the ballast and cargo condition were handled
separately.
The total fatigue damage in the different sea states was described by the equation DT (h, t).
h refers to the significant wave height, while t refers to the zero up-crossing period. The
total fatigue damage was the sum of the WF damage, DW F (h, t), and the HF damage,
DHF (h, t). How the fatigue damage is calculated is explained in Section 4.7. Polynomial
expressions were chosen as
DW F (h, t) = a0 + a1 h + a2t + a3ht + a4h2 + a5 t2 + a6h2 t + a7ht2 + a8h2 t2

DHF (h, t) = b0 + b1h + b2 t + b3ht + b4h2 + b5 t2 + (2.95)
2πh
b6 h2 t + b7 ht2 + b8 h2t2 + b9 sin8 ( )) (2.96)
20
2.8. LONG TERM FATIGUE ANALYSIS 55
The equation for the vibration damage, similar to the equation for the wave frequency
damage, did not represent the vibration damage satisfactorily. More terms in the poly-
nomial series gave unreliable answers. Based on the behaviour of the vibration damage,
a sinusoidal term was added. The exponent of 8 was used due to a significant increase
in vibration damage observed between 3 and 4m significant wave height. Such a sudden
increase was not observed for the periods. The value of 20 gave a sinusoidal peak at 5m,
which appeared to give a natural peak for the reduced speed case. In full speed the sinu-
soidal term was less important due to a rapid growth of the vibration damage as a function
of sea state (and speed). The importance of this term may therefore vary for a blunt and
a slender ship with different speed behaviour and different excitation sources. It may also
vary with the dominating sea states, in case a less harsh envionment is considered. The
same procedure is however expected to be feasible also for other ship types.
The expressions for wave frequency and vibration damage are nonlinear in h and t, however,
the expressions are linear in the unknown coefficients ai and bi . A least square fit by the
MATLAB function lsqr(A,B) was performed to the measured data considering the matrix
relation, e.g. for the wave frequency damage
A·X =B (2.97)
Ai = [1 hi ti (hi ti )(h2i ) (t2i ) (h2i ti )
(hi t2i ) (h2i t2i )]
X = [a0 a1 a2 a3 a4 a5 a6 a7 a8]0
Bi = DW F (hi , ti)
X and B are column vectors, i denotes row number i, and 0 refers to the transposition
symbol. Matrix A is made up of all the rows of Ai , and h and t are chosen equal to
the values for the tested sea states. This matrix is therefore known, and row i refers to
damage from a specific sea state where h and t are fixed. 12 to 39 sea states were considered.
To control the behaviour of the parameters slightly, zero points were introduced along the
edge of the scatter diagram where the wave heights were zero. Such wave heights will cause
zero fatigue damage. In addition zero points were introduced along the edge where the
periods were zero. The latter is not really physical or have much meaning, since such sea
states do not exist. It however controls the prediction towards steep sea states, which are
unlikely to occur. When negative damage in a sea state was predicted, the damage was set
to zero. Basically, it is important to cover a good spread of the sea states in both period
and height for this procedure.
A few obvious uncertainties exist. The first uncertainty relates to the goodness of the least
square fit. This uncertainty is indicated by the difference in the predicted and measured
fatigue damage from only the selected sea states, and constitutes as much as 10% in
the original ballast condition. The second uncertainty relates to the equations ability to
extrapolate/interpolate the fatigue damage to the region outside/in between the selected
sea states. The method is applied to both few and many sea states in one loading condition.
The deviation in the results will indicate the magnitude of uncertainty related to the small
number of sea states selected. The third uncertainty relates to the stochastic behaviour
of the short time series in irregular sea. Despite these uncertainties, the response surface
method will illustrate the location of the important sea states, and the relative importance
between the WF and HF damage.
Chapter 3
Simplified analysis of vibration

response and speed reduction
The purpose of this chapter is to illustrate the contribution to vibration from impacts on
the upper bow/stem flare area based on the hypotheses in Section 2.1. The amplification
of the incident waves due to reflection from the blunt bow, the additional steady wave
elevation at the stem and the impact force on a half cone are considered. This effect is
referred to as stem flare whipping, which is simulated in regular and irregular head sea.
Nonlinear springing is not included, but Froude-Kriloff excited linear springing is included
in the simulations.
The speed is estimated in different sea states in Section 3.2. A set of speeds are selected
for the subsequent model tests.
3.1 Simplified analysis of whipping from stem flare

impacts and linear springing
Whipping may contribute to the high frequency response. Since the damping is low, it
is possible that whipping of small magnitudes superposed on the springing vibration still
appear as springing. The impacts may occur often, hence the whipping vibration may still
be present when a new impact occurs. The new impact will either increase or reduce the
vibration level depending on the phasing.
The simplified hydroelastic analysis employs the Euler beam formulation, modal super-
position and simplified strip theory. The linear excitation consists of the Froude-Kriloff
57
58CHAPTER 3. SIMPLIFIED ANALYSIS OF VIBRATION RESPONSE AND SPEED REDUCTION
pressure which may excite linear springing. The second loading comes from impacts on the
bow/stem represented by a half cone. Impacts exciting whipping occur when the combined
effect of steady water elevation, incident waves, reflected waves and ship motions make the
water surface hit the half cone. The steady water elevation, incident and reflected waves
are calculated based on an infinite vertical wall representing the bow and moving forward
with the vessel’s speed. Total reflection is assumed also for the longer waves, and it con-
stitutes a diffraction effect localized to the bow. The numerical analysis in time is solved
by a Newmark-β scheme with iterations to account for the hydroelastic effect. Iterations
are necessary to get feedback from the vibration response on the hydrodynamic excitation
forces. Further explanations are given in Section 2.1.1 - 2.1.3, 2.2, 2.6 and Appendix A.3
and A.5. A program was made for this purpose.
The dimensions and sectional properties are listed in Table 3.1. The added mass of the
flexible modes given in Table 3.1 is lower than the actual added mass for these frequen-
cies, but it is used to achieve a natural springing frequency close to 3.32rad/s in ballast
condition, as for the real ship.
Table 3.1: Properties of a rectangular ship used in simplified assessment.

Property Value Unit
Density of water, ρ 1025 [kg/m3 ]
Acceleration of gravity, g 9.81 [m/s2]
Young’s modulus, E 2.1 · 1011 [N/m2 ]
Length, L = Lpp 294 [m]
Breadth, B 53 [m]
Draft, T , Ballast/Cargo 11.33/18 [m]
Deck height above BL in bow, z 27 [m]
Neutral axis above BL, zna 10.8 [m]
Distance from BL to apex of cone, zc 18.8 [m]
Area moment of inertia, I 680 [m4]
Speed, U 7.8 [m/s]
Deadrise angle of cone, α 50 [◦ ]
Wave heading, β (0=head sea) 0 [◦ ]
Sectional mass, m= ρBT 615502/977850 [kg/m]
Sectional water plane stiffness, c = ρgB 532928 [kg/(ms2)]
Damping ratio for rigid modes, δr 0.4 [−]
Damping ratio for flexible modes, δf,i 0.005(2i − 5) [−]
Added mass for rigid modes, ar 2m [kg/m]
Added mass for flexible modes, af 0.5m [kg/m]
Natural springing frequency, ωs 3.22/2.56 [rad/s]
Natural rigid body frequency, ωr 0.54/0.43 [rad/s]
The cone geometry is an idealization of the real geometry. Figure 3.1 illustrates the differ-
3.1. SIMPLIFIED ANALYSIS OF WHIPPING FROM STEM FLARE IMPACTS AND LINEAR
SPRINGING 59
ence. The cone is shown with an angle corresponding to 60 and 40◦ . The former is close
to the stem angle, while the latter is close to the flare angle. Due to the 3D geometry an
angle of 50◦ was chosen. The effective angle differs due to the slope of the water surface
and the flow along the bulb. The added mass formulation by Shiffman and Spencer (1951)
was applied. The cone is assumed with infinite extent upwards, hence water on deck is
disregarded. 3 flexible modes in addition to pitch and heave are considered.
Stem geometry Flare geometry

30 30
60 deg 40 deg
40 deg
25 60 deg 25
20 20
z from BL [m]
z from BL [m]
Steady WL
15 15
10 10
Still WL
5 5
0 0
285 290 295 300 0 5 10 15
x from AP [m] y from CL [m]
Figure 3.1: Stem and flare geometry at FP of the real ship compared with two cone angles.
The steady wave elevation at 15kn is illustrated.
3.1.1 Simulations in regular waves in ballast condition

In regular waves the impact event is well defined in case the water hits the half cone. Linear
springing occurs only if the encounter frequency is equal to the springing frequency. The
impact force refers to all the load components in Section 2.1.3 that constitute an impact
of short duration. For illustration, a typical impact force distribution is provided in Fig-
ure 3.2. The wave height corresponds to the response at 15kn in Figure 3.5. It is seen that
the conventional slamming is only a part of the total force, and the exit phase may be as
important as the entry phase. The water level on the cone is indicated by the hydrostatic
force. The maximum force corresponds to 200kN, and it is observed that the hydroelastic
effect on the loading is insignificant (the thick continuous line is on top of the dashed line).
The small hydroelastic effect means that the global vibration of the hull girder does not
significantly affect the magnitude of the impact load. The duration of the impact is about
0.8s, hence it acts almost like an impulse, since the natural period is 2s. The important
feature is the difference between the negative and positive area below the total force curve,
resulting in an effective negative force. For other impacts the relative magnitudes varies,
and the force may neither be impulse like nor effectively negative. Note, it appears from
the Figure 3.2 that the inertia and ’Froude-Kriloff’ term is similar. If the vessel is not
moving much the relative velocity and the surface velocity would be similar according to
Eq.(2.7), and if also the added mass are similar to the displaced mass, then the inertia
term would be similar to the ’Froude-Kriloff’ term. This is actually the case here.
4 Hydrodynamic impact forces

x 10
0
Forces in N
−5
Total
−10
Total without hydroel.
Inertia
Hyd
−15 Slam
FK
35.4 35.6 35.8 36 36.2 36.4 36.6

Time in seconds
Figure 3.2: Illustration of the total impact force distribution on the half cone (α = 50◦ )
in regular waves. Slamming=Slam, ’hydrostatic’=Hyd, inertia=inertia and ’Froude-
Kriloff’=F K; H = 5m, U = 15kn, T = 5.9s.
The steepness criterion for regular waves in Eq.(2.87), given in Section 2.6, is used to in-
vestigate the sensitivity of vibration in full speed of 15kn as a function of the regular wave
height, H. It provides an upper limit of the vibration response in head waves. The results
are presented as nominal stress amplitude of both the high frequency (HF) vibration and
the wave frequency (WF) response in Figure 3.3. The HF vibration may include both linear
springing and stem flare whipping, but both excitation sources may not contribute simul-
taneously in the simulations. The hump-hollow behaviour is observed for both the WF
and HF response. Moreover, the linear resonance and the second harmonic are observed
at an encounter period of 2 and 4s.
It is not evident when the water touches the cone in Figure 3.3. Due to the short waves,
the vessel does not move significantly in the simulations. If an incident wave amplification
SPRINGING 61
40
WF [MPa]
20
0
3 4 5 6 7 8 9 10 11 12
60
HF [MPa]
40
20
0
3 4 5 6 7 8 9 10 11 12
6
4
Te [s]
0
3 4 5 6 7 8 9 10 11 12
Regular wave height, H [m]
Figure 3.3: The upper limit of HF due to linear springing and whipping and WF nominal
stress amplitudes in regular head waves based on the steepness criterion (1/7); U = 15kn,
α = 50◦ .
of 3 is assumed, a wave height of 2.9m is enough to touch the cone’s apex. The ’linear’
springing peak will then include a small effect from the impact force, hence nonlinear
forces may contribute also when the encounter frequency coincides with the springing fre-
quency. The submergence of the cone becomes up to 1.6m in the case of ’linear’ springing.
The second harmonic peak has an amplification of 2.7, and the motion of the vessel is
still small. The water reaches the deck level at 9.3m wave height, and the speed will be
reduced considerably. Hence, the predicted HF amplitude is most conservative. Accord-
ing to the steepness criterion, it is theoretically possible to induce large stem flare whipping.
To investigate the sensitivity to wave periods, a regular wave height of 5m is chosen in

combination with full speed of 15kn. Figure 3.4 presents the WF and HF nominal stress
amplitude versus the encounter period. The lowest wave period does not satisfy the break-
ing criterion, while the upper wave period of 15s captures the rigid body response peak.
The springing peak and hump-hollow behaviour are evident. A sudden drop of the vi-
bration level is observed at roughly 3.5s, since the amplification of the incident waves are
reduced, and the waves do no longer cause impacts on the cone. The second harmonic at 4s
encounter period (lower plot) is therefore not present. This indicates that only short high
waves may contribute to stem flare whipping in ballast condition, and no significant contri-
bution is expected from long waves. The simulated wave periods include those which can
cause linear springing, hence linear springing is contributing in addition to stem whipping
at 2s encounter period. The apparent vibration at higher wave periods above 6s is partly
due to transient effects in the time series, and due to end effects in the FFT procedure
used to estimate the regular filtered response. The vibration level above 6s is therefore
somewhat conservative.
100
WF [MPa] 80
60
40
20
0
0 2 4 6 8 10 12
15
10
HF [MPa]
0
0 2 4 6 8 10 12
Encounter period, Te [s]
Figure 3.4: WF and HF vibration amplitude versus the wave encounter period; U = 15kn,
H = 5m. The HF response above 6s is affected by transients in short simulations.
The effect of speed on the HF response is investigated for a wave height of 5m and wave
period of 5.9s. This is a steep wave, which reduces the speed, and for this wave the second
order effect should appear at 8.5kn. Figure 3.5 presents the HF nominal stress amplitude.
The magnitude of the impact force at 8.5kn indicated that the water surface barely touched
the cone, hence, in this case the HF response is insignificant. The vibration shows a strong
speed dependence at higher speeds, and below 10kn the HF response is insignificant in
ballast condition.
3.1.2 Simulations in irregular waves in ballast and cargo condi-

tion
Real waves are irregular, even though there may be clustering of waves with similar am-
plitude and frequency, e.g. (Wist 2004; Hansen and Thayamballi 1995). JONSWAP wave
spectra with a γ factor of 2 is used. Results for irregular sea states with a peak period
of 10s and varying significant wave height are shown in Figure 3.6 for ballast and cargo
condition in full speed. The stress is presented as nominal standard deviation stress. For
ballast condition, the HF vibration is linear up to a significant wave height of 4m, where
the slope seems to change due to whipping. Up to this level the response is due to linear
springing from Froude-Kriloff pressure along the ship. The vibration levels are in general
low and the whipping appears to be insignificant. In cargo condition, the HF response is
nonlinear due to whipping, and the vibration level becomes significant in higher sea states,
SPRINGING 63
5
HF [MPa]
0
5 10 15
Vessel speed, U [kn]
Figure 3.5: HF vibration amplitude versus speed; H = 5m, T = 5.9s.
e.g. above 4m significant wave height. The WF response shows a linear behaviour with
increasing wave height, without influence from whipping (meaning that the whipping has
no effect on the global motions as the coupling is nonlinear). The curves in Figure 3.6
should not be expected to be smooth due to the stochastic nature of the simulated sea
states and the length of the simulations.
25 20
WF std.dev. [MPa]
WF std.dev. [MPa]
20
15
15
10
10
5
5
0 0
1 2 3 4 5 6 1 2 3 4 5 6
2.5 15
HF std.dev. [MPa]
HF std.dev. [MPa]
2
10
1.5
1
5
0.5
0 0
1 2 3 4 5 6 1 2 3 4 5 6
Significant wave height, HS [m] Significant wave height, HS [m]
Figure 3.6: Standard deviation of nominal stress for WF and HF (due to springing and
stem flare whipping) versus significant wave height. U = 15kn, TP = 10s, Ballast=left,
Cargo=right.
The sensitivity of the peak period is investigated at a significant wave height of 5m and
speed of 15kn. The lower peak period corresponds to the breaking criterion for irregular
sea. Figure 3.7 shows a nonlinear decreasing trend of the HF stress for increasing peak pe-
riod in both ballast and cargo condition. The vibration level is in general low for ballast,
but it is significant for cargo condition in particular at low peak periods below e.g. 12s.
40 40
WF std.dev. [MPa]
WF std.dev. [MPa]
30 30
20 20
10 10
0 0
7 8 9 10 11 12 13 14 15 7 8 9 10 11 12 13 14 15
2.5 20
HF std.dev. [MPa]
HF std.dev. [MPa]
2
15
1.5
10
1
5
0.5
0 0
7 8 9 10 11 12 13 14 15 7 8 9 10 11 12 13 14 15
Peak period, TP [s] Peak period, TP [s]
Figure 3.7: Standard deviation of nominal stress for WF and HF (due to springing
and stem flare whipping) versus the peak period. U = 15kn, HS = 5m, Ballast=left,
Cargo=right.
The vessel speed in 5m significant wave height is less than 15kn. The HF vibration versus
the speed is shown for a peak period of 10s in Figure 3.8 for ballast and cargo condition.
The HF vibration increases with speed in a slightly nonlinear manner, while the WF stress
remains almost unchanged. A speed reduction of a few knots reduces the vibration to half.
Again the HF response is low in ballast, but may be significant in high speeds above 10kn
in cargo condition.
Impacts in real sea states may be complex and sensitive to various parameters. An im-
pact event in irregular waves is shown in Figure 3.9. After the first slam, a large negative
peak is observed followed by an oscillatory behaviour without any significant conventional
slamming. The impact force may or may not cause significant vibration depending on the
magnitude and integrated effect. This simply illustrates the complexity of the excitation,
and it indicates that this oscillating behaviour may provide a whipping response that is
difficult to distinguish from conventional springing.
Questions have been raised whether slamming on anchor bolsters could contribute to the
vibration response on the iron ore carrier considered as the main case in this study. The
two bolsters have a diameter of about 4m, and they are 1m deep as illustrated in Figure 1.1.
The lower part is located about 22m above base line. They are expected to constitute a
minor effect in ballast condition, since they are located well above the apex of the cone.
SPRINGING 65
18 15.4
WF std.dev. [MPa]
WF std.dev. [MPa]
17.8 15.2
17.6 15
17.4 14.8
17.2 14.6
17 14.4
5 10 15 5 10 15
2 8
HF std.dev. [MPa]
HF std.dev. [MPa]
1.5 6
1 4
0.5 2
0 0
5 10 15 5 10 15
Vessel speed, U [kn] Vessel speed, U [kn]
Figure 3.8: Standard deviation of WF and HF vibration versus vessel speed, HS = 5m,
TP = 10s. Ballast=left, Cargo=right.
5 Hydrodynamic impact forces

x 10
0
Impact forces [N]
−5
−10
Total
Inertia
−15
Hyd
Slam
−20 FK
89.5 90 90.5 91 91.5 92 92.5

Time [s]
Figure 3.9: Illustration of the different force components of an impact in irregular sea.
HS = 5m, TP = 15s, U = 15kn in ballast condition.
The angle between the water surface and the anchor bolster may however cause a slamming
like impulse and contribute in cargo condition. The slamming force on the anchor bolsters
may be included as a refinement of the procedure.
3.2 Calculated speed reduction in different sea states

Since the fatigue damage due to wave induced vibrations is sensitive to the speed, it is of
interest to estimate the involuntary speed reduction in various sea states. The theory in
Section 2.1.5 is employed to estimate the involuntary speed reduction in irregular waves,
which are represented by a JONSWAP spectrum with peakness factor, γ, of two. Table 3.2
presents the input values.
Table 3.2: Overview of input parameters.

Parameter Notation Value Unit
Density of salt water ρ 1025 kg/m3
Acceleration of gravity g 9.81 m/s2
Viscosity coefficient ν 10−6 m2/s
Roughness height Hr 150 µm
Diameter propeller D 8.4 m
Beam at water line B 53 m
Draft at AP, Cargo / Ballast TAP 19.04 / 12.06 m
Draft at FP, Cargo / Ballast TF P 18.01 / 11.33 m
Length at water line, Cargo / Ballast LW L 288.7 / 287.907 m
Centre of buoy. forw. of midship, Cargo/Ballast LCB 10.93 / 12.498 m
Displacement, Cargo / Ballast ∇ 237553 / 145963 m3
Wet calm water surface, Cargo / Ballast S 23160/18441 m2
Base area, Cargo / Ballast SB 12.32 / 0 m2
Maximum engine power PE 19966 kW
Figure 3.10 shows an example of the speed estimate for a total propulsion efficiency factor
of 0.7 in cargo condition and for a sea state with 9m significant wave height and 11s zero
up-crossing period. The calm water resistance increases with speed to the power of 3, while
the added resistance in waves increases in a more linear fashion. The maximum speed is
estimated at the intersection between the total required effect and the horizontal line de-
noting the maximum available power, and it is about 6kn in this case. The maximum
speed in calm water was estimated to 15.7kn. The speed reduction is defined relative to
the maximum speed in calm water.
Table 3.3 shows the speed reduction in different sea states. The maximum power was as-
sumed independent of speed and sea state. The maximum speed in calm water in ballast
condition is estimated to 16.6kn based on a realistic propulsion efficiency coefficient of 0.7.
3.2. CALCULATED SPEED REDUCTION IN DIFFERENT SEA STATES 67
4 Estimating maximum speed in Hs=9 and Tz=11

x 10
9
(RT+Raw)V
8 RTV
R V
aw
7
PEmax * ηt
Effect in cargo condition [kW]
0
0 5 10 15 20
knots
Figure 3.10: Calm water resistance and added resistance in waves in cargo condition
versus the maximum available efficient power, P Emax · ηt ; HS = 9m, TZ = 11s.
A constant total efficiency coefficient is questionable, hence, two values were considered.
The result indicates that the speed reduction is insensitive to the total efficiency coeffi-
cient. Some sensitivity to the zero up-crossing periods was observed in higher sea states.
It is expected that the speed reduction is more sensitive to the zero up-crossing period
than reflected in the predictions, since the nondimensional added resistance in waves was
assumed independent of speed.
The speed reduction was considered in the model tests, except in the original ballast
condition. Based on the uncertainties in the calculations and possible voluntary speed
reduction, the following speeds for the different sea states were chosen as shown in Table 3.4.
Table 3.3: Estimated speed loss due to added resistance in waves.

HS TZ Cargo ηt = 0.7 Cargo ηt = 0.9 Ballast ηt = 0.7 Ballast ηt = 0.9
Umax 15.7 16.8 16.6 17.6
3 6 1.2 1.1 1.3 1.1
3 7 1.2 1.0 1.2 1.1
3 8 1.2 1.0 1.3 1.0
3 9 1.3 1.1 1.4 1.1
3 10 1.5 1.2 1.6 1.3
4 7 2.1 1.8 2.3 1.9
5 8 3.5 2.9 3.8 3.2
5 9 4.2 3.4 4.6 3.6
5 10 4.5 3.8 4.9 4.1
5 11 4.0 3.5 4.4 3.8
7 9 8.3 7.3 9.2 8.2
7 10 8.0 7.4 8.7 8.1
7 11 7.0 6.6 7.6 7.1
9 11 9.4 9.2 10.1 9.8
Table 3.4: Selected speeds in different sea states and loading conditions.
HS Speed in cargo condition, U [kn] Speed in ballast condition, U [kn]
3 13.2 15
4 13.2 15
5 13.2 and 10 15 and 10
7 13.2 and 5 15 and 5
9 13.2 and 0 15 and 0
Chapter 4
Experimental methodology and

analysis procedures
The strategy of the model tests is described in Section 4.1 followed by a brief description
of the experimental model setup with sensors and the applied scaling laws in Section 4.2.
The main uncertainties are listed in Section 4.3 before the procedures to evaluate the data
are outlined, i.e.: Damping in Section 4.4, mode shapes in Section 4.5, linear, ’nonlinear’
and second order transfer functions in Section 4.6 and fatigue damage in Section 4.7. A
new method, developed to identify whipping, is presented in Section 4.8, while a method
to illustrate the relative importance of linear and second order springing is described in
Section 4.9. In general, results from these procedures are presented in Chapter 5.
4.1 Strategy of the test program

The literature study, simulations based on simplified theory, the use of existing hydroelas-
tic computer codes and the hypotheses provided the background to establish a strategy,
which is outlined below. However, it was not possible to answer all of the questions raised.
The main goal was to identify the causes of the wave induced vibrations on large blunt
ships. A set of conditions, listed in the following, were defined to approach this problem.
? The hull geometry in (Storhaug et al. 2003) was selected so as to make it possible to
compare model and full scale results, which is unique. The literature indicated that large
size and blunt bows were important parameters, and the selected ship represents many
Capesize bulk carriers and tankers.
69
70 CHAPTER 4. EXPERIMENTAL METHODOLOGY AND ANALYSIS PROCEDURES
? The model should have realistic structural dynamic properties including higher order
modes. This implied Froude scaling and distributed stiffness.
? The hydrodynamic damping should be separated from the structural damping, since the
literature was indistinct on this point. The damping may be frequency dependent also
in air, and a design that made it possible to achieve the same natural frequency in air
and in water was desirable.
? The moment should be measured along the hull. The moment should be converted into
deck stress for comparison with full scale measurements.
? The neutral axis should be realistic, to avoid false contributions from horizontal impact
forces, e.g. breaking waves against the vertical stem.
? The model should not be restrained from rigid heave and pitch motions, since restrained
models may provide less response than unrestrained models due to coupling effects.
? Towing forces should be measured to be able to relate relative motion in the bow to
added resistance in waves.
? Short waves, exciting springing, should be considered. Short waves have poor quality
making it desirable to use a large model. The wave quality should be monitored.
? Head waves and high speed produce large vibrations. This is also the first step for
comparison with numerical predictions (and other experiments). Hence, the tests could
be performed in a towing tank. The sensitivity to speed should be investigated.
? Realistic speed should be used to achieve realistic relative importance of excitation loads.
? Ballast condition produces higher vibrations than cargo condition, but the stem flare
may contribute significantly in cargo condition. Hence, both loading conditions should
be considered. A change in trim was of interest and may increase the loading on the
stem (bow) flare, while reducing the springing loads.
? Different wave heights, which may ”attack” different geometry with different nonlinear
effects, should be considered (for each wave length).
? Both regular and irregular waves should be considered. They may reveal different phys-
ical effects, and indicate the combined effect. Representative sea states for long term
predictions and trends should be selected.
? The bow and stern were areas of main concern. Pressure cells, slamming panels, relative
motion sensors, shear force transducers and video cameras should be used to reveal the
mechanisms. Special attention should be given to the bow based on the hypotheses.
4.2. EXPERIMENTAL SETUP 71
? The bulb geometry (and flare) is one cause of second order excitation. The blunt bow
reflects incident waves, and this constitutes another second order excitation. The effect
of bow geometry should be investigated, first by testing the original geometry, then by
testing a bow without bulb and flare, and finally by testing a sharp bow with reduced
bow reflection.
4.2 Experimental setup

4.2.1 Model description
A large model of a bulk carrier was constructed in model scale of 1:35 (Λ = 35), Figure 4.1.
The hull was made of divinycell foam with glassfibre coating. Plywood was used internally
to increase the local strength, and a large heavy triangular steel frame was mounted inside
the hull. The purpose of the steel frame was to make each segment rigid. Flexible joints
were inserted in between the rigid segments to obtain the requested natural frequencies,
and the flexibility was confined to the flexible joints. To make the model rigid, the flexible
joints could be locked. The 1500kg steel frame reduced the need for and served as supports
for the ballast weights. The displacement in cargo condition was roughly 5.6tonnes.
The hull geometry is presented in Figure 4.2. The ship has vertical sides at the ballast
water line (11-12m above BL) in the fore ship, while in the aft ship the geometry is rather
flared. The stem includes both a bulb and a stem flare, which starts just above the draft in
cargo condition (18.5m above BL). The geometry is conventional, and the main dimensions
of the model are presented in Table 4.1.
Table 4.1: Main ship particulars of the model with different bows
Length overall, LOA , bow 1 and bow 2 in ballast cond. 8.70m
Length overall, LOA , bow 2 in cargo condition 8.55m
Length overall, LOA , bow 3 8.84m
Length between perpendiculars, Lpp , bow 1 and bow 2 8.40m
Length between perpendiculars, Lpp , bow 3 8.69m
Breadth moulded, B 1.51m
Depth moulded, D 0.77m
Design draft, T 0.54m
Draft ballast condition, T 0.334m
Trim ballast condition, TAP − TF P,bow1 0.021m
Draft cargo condition, T 0.529m
Trim cargo condition, TAP − TF P,bow1 0.029m
Neutral axis above BL, zn.a. 0.22m
The model was divided into six segments assembled to allow measurements of forces in
five cuts. Moreover, three of the middle cuts were made flexible, reducing the number
Figure 4.1: Picture of the 8.7m model with the steel frame and transverse triangular end
plates, which defined the location of the flexible joints and global load sensors.
25
20
18.5
15
11.5
10
Figure 4.2: Ship geometry with bulb, stem flare and aft ship flare. The drafts are typically
11-12m and 18.5m in ballast and cargo condition.
of stiff segments to four. The segments and cuts were numbered from the aft. The first
rigid segment consisted of segment 1 and 2. The second rigid segment was segment 3, the
third was segment 4 and the fourth consisted of segment 5 and 6. The positions of the five
cuts are presented in Figure 4.3, which shows the steel frame design inside the segmented
model. Cut 1 and 5 were rigid cuts included to measure shear forces with possible nonlinear
stern and bow forces. Shear forces were also measured at cut 2 and 4, while moments were
measured at cut 2, 3 and 4 at the quarter lengths where the joints were flexible. The stiffness
of the flexible joints could easily be adjusted by a sophisticated arrangement. Further
description of the measuring positions and flexible joints are described in Appendix B.1.1.
Figure 4.3: Work drawing of steel frame, flexible joints and measuring positions. The
cuts are indicated relative to AP. The neutral axis is indicated in the top figure.
Three different bow shapes were tested as shown in Figure 4.4. The difference between
bow 1 and bow 2 was related to removal of the bulb in ballast condition, and removal of
bulb and stem flare in cargo condition. Bow 3 differed considerably. The bow was sharp
triangular with a vertical stem. Less than 10% of the total length was modified when
the bows were replaced. Bow 3 was placed on the cradle, which was used to estimate the
mass moment of inertia in pitch and COG, as seen on the lower right picture in Figure 4.4.
Figure 4.4: The three bows. Bow 1 = Upper, Bow 2 (cargo) = Lower left, Bow 3 = Lower
right. Bow 3 (segment 5 and 6) is placed on the cradle. All bows have significant bilge
radius (bottom flare).
The mass properties of the segments and the pendulum test procedure are presented in
Appendix B.1.2. The uncertainty in the mass moment of inertia of each rigid segment
was less than 6%. The difference between the three ballast conditions or the three cargo
conditions exceeded the individual uncertainty, due to slightly different mass distribution
and bow design. The maximum difference of 19% was related to the rigid bow segment for
ballast 1 and 3 condition. Appendix B.1.2 presents also the discrepancy between the mass
properties of the model and the down scaled values from realistic loading conditions. The
agreement was satisfactory for the cargo condition. The mass moment of inertia in ballast
condition was more than twice the down scaled values for segment 3 and 4. The reason
was the number and weight of the end plates. The discrepancy affected the higher order
modes, changing the relative ratio of the three natural frequencies and mode shapes. The
springing mode was only slightly affected, since the requested frequency was still achieved.
4.2.2 Measurement devices and their locations

Different measuring devices and sensors were used to measure wave heights, axial forces,
shear forces, accelerations, global motions, local deflection, towing forces, speed, relative
motions, pressures and slamming forces. Moreover, two video cameras were employed.
Totally about 60 channels were used.
The axial force sensors were the most important sensors, and the axial forces were con-
verted into hull girder bending moments. From these sensors, vibration stress and fatigue
damage were estimated. Heave, pitch, speed and waves were measured as well as the shear
forces, pressures, slamming forces and relative motions, which may reveal something about
the physics. Local deflections and accelerations (and moments) were used to estimate the
mode shapes, while cameras were used to indicate wall interaction, steady waves, breaking
waves and green seas on deck. The local deflections between the rigid segments were also
used to estimate the rotational stiffness.
The moments were measured at 2.24, 4.34 and 6.44m from AP corresponding to 0.27, 0.52
and 0.77Lpp . The axial force transducers were placed in the top of the steel frame adjacent
to the springs as illustrated in Figure 4.3. The locations of all sensors as well as their type
and manufacturer are described in Appendix B.2.3.
4.2.3 Model scaling laws and nondimensional properties

Scaling laws are an essential part of model experiments. For investigation of rigid body
motions and global loads, Froude scaling is the basic scaling law (ratio of inertia to gravity).
The Froude number, Fn , is defined by
U
Fn = √ (4.1)
Lg
The Froude number is required to be the same in model, m, and full scale, f . This defines
the model speed
s
Lm Uf
Um = Uf =√ (4.2)
Lf Λ
The frequencies of gravity waves on infinite water depth follow the dispersion relation
2π
ω 2 = kg = g (4.3)
λ
where k is the wave number and λ is the wave length. The dimensions such as the wave
length is scaled by Λ. Making the frequency nondimensional with respect to the dispersion
relation and requiring that the ratio should be the same in model and full scale, gives the
scaling of the frequency and time (used to slow down videos)
p q
ωm λm /g = ωf λf /g
√
ωm = ωf Λ (4.4)
The second basic scaling law relates to the flexibility of the hull girder. The natural
frequency of a homogeneous beam Eq.(A.55) and the frequency scaling law Eq.(4.4) are
combined to derive the scaling law for the bending stiffness.
ρm
EIm = EIf (4.5)
ρf Λ5
The shear stiffness was disregarded, but may be included, (Achtarides 1983). By using
the sophisticated design of the flexible joints, it was not necessary to care much about
the exact stiffness. This was achieved by adjusting the stiffness to get the requested fre-
quency according to Eq.(4.4). Further scaling and nondimensional quantities are listed in
Appendix B.2.
Other physical effects such as viscous effects may be present, but scaling was disregarded.
A thin wire was glued around a section in the bow area to initiate turbulence in the
boundary layer, and the bilge keels were omitted to minimize the damping for the 2-node
mode, (Sele 2001).
4.3 Uncertainties in the experimental data

In general uncertainties are inherent in experimental results, but they are also present
in numerical predictions due to inaccurate idealization and solution procedures. The un-
certainties are determined to indicate the goodness of the results. It is necessary when
previous experience is lacking, and it may constitute a significant part of the total time
necessary to carry out the experiments. Without assessment of the uncertainties, the qual-
ity of the results are reduced and the results may become worthless.
Some of the equipment and sensors, which were used in this experiment, have already been
used in earlier projects, and convincing experience has been gained. The uncertainty anal-
yses were therefore concentrated on those sensors which were new, and those which were
of significant importance for the main results. For the remaining sensors and equipment,
the author relied on the good workmanship of the staff at MARINTEK.
It is customary to divide the total uncertainty into bias errors and precision errors. The
former is a systematic, fixed or constant error, while the latter is a random component
4.3. UNCERTAINTIES IN THE EXPERIMENTAL DATA 77
related to repetitions.
The bias error is often related to the equipment for measurements and undesirable physical
effects which may not be fully realized. One may not be aware of its presence. A large
part of the bias error was removed by calibration. Some bias error was still present, and
it was indicated by using different sensors to measure the same quantity. All sensors were
calibrated before they were mounted, while a few critical sensors were calibrated also after
mounting. In any case, zero setting, by use of the measurement system CATMAN (HBM
2004), was carried out before each run to reduce effects of drifting and changed initial
conditions. In general uncertainty analysis based on bias errors has not been carried out.
The precision error was related to the measurement procedure and the stochastic nature
of properties. Each repetition was based on slightly different initial conditions. The preci-
sion error was handled by repeating the tests to establish the distribution and confidence
intervals including rejections of wild readings. The precision error was established by a
procedure outlined in Appendix B.3. It resulted in a mean value, a 95% confidence interval
of sample values and a 95% confidence interval of the mean value. It was realized that
the assessment used to determine the numerical results also contains uncertainties, and
alternative procedures were utilized to determine the same quantity.
A number of uncertainties are listed in the following. These are put into groups related
to wave and tested conditions (circle), to instrumentation (dot) and to model design (di-
amond). Some uncertainties are related to random errors, while others are related to
gross systematic errors. The importance is referred to as insignificant (I), minor (M) or
significant (S), although all uncertainties have not been quantified.
◦ The output wave elevation and frequency differed from the requested values (S).
◦ The effective distance with good wave quality was not always sufficient to achieve max-
imum steady state vibration response in short waves (S).
◦ The wave elevation and wave frequency changed along the tank in particular for short
steep waves, but changes were also observed in long waves probably due to water depth
difference (S).
◦ Scattered waves from the model were reflected by the side wall and affected the loading
in the stern part for long waves and low speeds (S).
◦ Undesirable low frequency surge motion changed the encounter frequency initially (M).
◦ Wave disturbances originated from many sources: Previous runs, wave maker, dampers,
rough surface along the tank, reflection from the beach and changes in water depth (S).
◦ Limited number of repetitions (M).

◦ Inaccurate combination of speed and wave frequencies requested to excite the resonance
peak (S).
• Wrong calibration factors inserted into the measurement system (M).
• Round off and limited number of significant digits written to file in relation to poor range
setting of sensors, in particular pressure sensors (M).
• Sampling frequency (100 Hz) which was too low to capture slamming and acceleration
peaks (M).
• Switching of cables between the 60 sensors (M).
• Wrong use of AD cards (M).
• Loss of sensor signals during a run (M).
• Calibration of relative motion devices on idealized straight geometry (M).
• Numerical analysis procedures used to estimate properties, e.g. damping estimates from
spectral methods (S).
The damping was mostly nonlinear and the behaviour differed (S).
The stiffness was weakly nonlinear (S).
Guidance springs to keep the model on a straight course affected the towing resistance
occasionally (M).
The predicted rigid body motion was affected by the flexible response and location of
the ”Nypos-tree” (M).
Mounting of some sensors and loading of the model caused changes, which were disre-
garded in the calibration (M).
Leakage and absorbed water changed draft and trim and introduced damping (M).
Rigid motions affected the towing forces (I).
Overloading of sensors or material (M).
Some sensors may be affected by temperature changes (I).
The design of the wave probes was not optimal for its use as moving wave probes (circular
instead of foil like) (M).
Wall effects due to blockage causing increased effective speed in a channel, and different
boundary conditions than in numerical simulations (M).
The elevation of the track and towing carriage close to the wave maker introduced changes
in estimated motions and wave heights (I).
Improper quality of initial produced bearings (S) and improved bearings (M).
The number of uncertainties is large, but most of them were insignificant, minor, reduced,
removed or quantified. E.g. switching of cables was discovered based on physical insight,
and the results was corrected. If this had not been detected, it would have introduced a
severe bias error into parts of the experiments. A more detailed description of the uncer-
tainties and calibration procedures are provided in Appendix B.4 focusing on the global
force transducers. In addition differences in physical properties like mode shapes, mass
distribution and high frequency tail of sea states may deviate from observed on a real ship,
but this is an uncertainty related to the idealization error.
Several authors pointed out the difficulties with model testing in short waves due to poor
wave quality, e.g. (Hoffman and van Hoof 1976; Goodman 1971). Some also failed to
produce waves exciting linear springing, (Dong and Lin 1992; Domnisoru and Domnisoru
1997). Before the present tests started up, MARINTEK was requested to assess the length
of the tank which exhibited homogeneous wave conditions for the purpose of springing
tests, (Stansberg 2004). A separate study was conducted to calibrate the waves prior to
the model tests, and results are presented in Appendix B.5 and Section 4.3.1. The sensi-
tivity of springing response to changes in speed and wave frequencies is also discussed.
Uncertainty analysis of experiments within marine technology was carried out more thor-
oughly by Ersdal (2004), and a general description was provided by Coleman and Steele
(1989).
4.3.1 Visualization of wave quality using Wavelet

The assessment of regular and irregular wave quality in Appendix B.5 confirmed that it
was necessary to measure the waves during the experiments. The location of the dras-
tic change in wave quality was not identified in Appendix B.5, but Figure B.10 gave an
indication in one specific case. Wavelet analysis was carried out to determine the length
with satisfactory quality, referred to as the effective length measured from the wave maker.
Wave probe WP3 on the towing carriage was used. A clear criterion was not defined, but
the effective length was determined based on visual inspection of the Wavelet spectra.
Figure 4.5, 4.6 and 4.7 present the Wavelet spectra as compared to the wave energy spectra
for the shortest waves supposed to excite linear springing. Even though the quality based
on the low amplitude waves in Figure 4.5 appears poor, the frequency is quite regular. The
quality decreased with increasing wave height. The Wavelet spectra provided a visualiza-
tion of the effective length, while the wave spectra only indicated that the higher waves had
more disturbances. The end of the time series represented a distance 15m from the wave
maker. The speed was 1.3m/s, and the effective length was estimated by visual inspection
to be 27m, 47m and 66m for the highest, middle and lowest wave. The wave quality was
better than expected from Stansberg (2004), and it was decided to capture also the linear
springing response in the model tests.
WP3 [m]
0.01
−0.01
165 170 175 180 185 190 195 200 205

1
Wavelet Ψ
−1
165 170 175 180 185 190 195 200 205
25
ω [rad/s]
20
15
e
10
165 170 175 180 185 190 195 200 205
Time [s]
−4
x 10
1.4
1.2
1
S(ωe) [m2s]
0.8
0.6
0.4
0.2
0
10 15 20 25
ωe [rad/s]
Figure 4.5: Wavelet (upper three: Time series, extended nondimensional basic wavelet
and Wavelet spectrum) and FFT wave spectrum (lower) for short waves intended to excite
linear springing for H = 0.7m, T = 4.14s in full scale.
0.02
WP3 [m]
−0.02
190 195 200 205 210 215 220 225

1
Wavelet Ψ
−1
190 195 200 205 210 215 220 225
25
ω [rad/s]
20
15
e
10
190 195 200 205 210 215 220 225
Time [s]
−4
x 10
3.5
2.5
S(ωe) [m2s]
1.5
0.5
0
10 15 20 25
ωe [rad/s]
linear springing for H = 1.4m, T = 4.14s in full scale.
The time series of the lowest wave displays low amplitudes at 53m, possibly caused by
observed disturbance from side-wall dampers located at a distance 40-45m from the wave
maker. The dampers were removed in the subsequent model tests. The highest waves
0.04
WP3 [m]
0.02
0
−0.02
205 210 215 220 225 230 235 240

1
Wavelet Ψ
−1
205 210 215 220 225 230 235 240
25
ω [rad/s]
20
15
e
10
205 210 215 220 225 230 235 240
Time [s]
−4
x 10
2
S(ωe) [m2s]
0
10 15 20 25
ωe [rad/s]
linear springing for H = 2.1m, T = 4.14s in full scale
display also slightly lower frequency than requested. The reason is somewhat unclear, but
could be both wave maker and wave propagation related. Based on the estimated distance
of 30m to build up maximum linear springing response, the two highest waves were ex-
pected to provide poor results. The fixed wave probe, WP1, was moved from 10 to 5m
4.4. PROCEDURE FOR ESTIMATING DAMPING AND NATURAL FREQUENCY 83
from the wave maker to increase the effective running distance. Fortunately, whipping from
running the model into the larger front waves of the wave trains caused often a more rapid
build up of the response, but still it was observed occasionally that the springing response
did not reach its maximum. For linear effects it was however not critical to capture the
highest waves.
The highest waves intended to excite second order springing did not confirm instabilities for
the investigated distance of 85m, except when waves were breaking continuously (requested
wave height above steepness criterion was also tested).
4.4 Procedure for estimating damping and natural

frequency
4.4.1 Damping by decay tests
Decay tests were carried out in water and in air to estimate the damping according to
Section 2.3.3. The axial force at midship section was considered. In air the model was
placed on two supports located below the two node points, or by hanging the model in
ropes. The damping in calm water was evaluated in zero and forward speed. The vibration
was excited by a sledge hammer or by hand.
A number of decay curves were produced in each run, and the whole time series was fil-
tered to remove the mean level and other vibration modes. Three methods were applied
to estimate the damping.
The first method utilized cycle-by-cycle decay from Eq.(2.51). A n = 10 cycle interval
was found appropriate to reduce the uncertainty. The damping values are presented versus
the amplitude in Figure 4.8 as expressed in Eq.(2.57), and the mean value and standard
deviations were calculated from the scatter.
The second method considered the envelope process by fitting a decay curve as
F X3(t) = R(t0) exp−δωmax (t−t0 ) (4.6)
to the positive peak values. t0 represents the start of the interval with the start amplitude
R(t0 ). The frequency ωmax corresponds to the vibration peak of the spectrum. The start
amplitude R(t0 ) and the damping ratio δ were determined by a nonlinear fit using the
MATLAB function fminsearch. Fits were performed at both high and low response levels
in each decay interval, and the damping ratios at high and low levels were compared to
indicate the presence of nonlinearities. Figure 4.9 shows an example of fits made nondi-
mensional with respect to the start value.
0.03
0.025
0.02
δtot
0.015
0.01
0.005
0
0 2000 4000 6000 8000 10000 12000 14000
8ω x0/3π
Figure 4.8: Nonlinear damping ratio estimated from cycle-by-cycle method for ballast 2
condition in calm water at zero speed. The slope indicates the nonlinear contribution, which
is close to zero in this case. x0 is the vibration amplitude in N from the midship axial force
transducer, and ω is about 20rad/s.
The third method was based on fitting harmonic decay curves to the filtered decay curves
according to
F X3(t) = R(t0) exp−δω(t−t0 ) cos(ω(t − t0) + θ) (4.7)
The natural frequency ω, the damping ratio, the start value and the phase, θ, were esti-
mated again using fminsearch. An example is shown in Figure 4.10. Results are presented
in Section 5.1.1.
4.4.2 Damping estimates in irregular sea

The damping in irregular sea states was estimated using the spectral method described in
Section 2.3.3. The analytical expression of the response spectrum was fitted to the mea-
sured spectrum derived by the MATLAB function psd. A mowing Hanning window was
used with a initial width of 2048 time samples, and the interval was chosen from 0.5rad/s
below to 1.0rad/s above the natural frequency (in full scale values).
Different effects may introduce uncertainties
• The width of the frequency interval.

4.4. PROCEDURE FOR ESTIMATING DAMPING AND NATURAL FREQUENCY 85
1
Peaks
0.9 Fit
0.8
0.7
0.6
FX3 [N/N]
0.5
0.4
0.3
0.2
0.1
0
0 1 2 3 4 5 6 7 8
Time [s]
Figure 4.9: Double envelope fit to peak values from filtered decay curves for ballast 2
condition in calm water at zero speed. 8s corresponds to 26 cycles.
250
Filtered
200 Fit
150
100
50
FX3 [N]
−50
−100
−150
−200
−250
0 1 2 3 4 5 6 7 8
Time [s]
Figure 4.10: Harmonic fit to a filtered decay curve for ballast 2 condition in calm water
at zero speed.
• The width of the Hanning window (number of sample points).
• The sampling frequency and length of the time traces.
• The hump-hollow behaviour of the excitation, which was not included by the analyt-
ical SDOF system.
• Low springing response compared to the surrounding wave frequency response.
• White noise in the measured signal.
• Rigid body motions changing the wet surface and added mass, thereby changing the
natural frequency and making the damping apparently larger.
The length of time series corresponded to more than six minutes with a sampling frequency
of 100Hz, and the model was tested in head seas with significant vibration response. The
main uncertainties were therefore related to the change of the natural frequency in the
waves, and the effect of the number of sample points and width of the considered fre-
quency interval, which will be illustrated in the results.
An example of a fit from the model tests is shown in Figure 4.11. The springing peak is
clear and a second peak from the 3-node vibration is observed. Results are presented in
Section 5.1.2.
2
10
Measured spectrum
SDOF Fit
1
10
10
0 ωx=3.33 δ=0.011
Sx(ωe) [MPa2s]
−1
10
−2
10
−3
10
−4
10
0 1 2 3 4 5 6 7 8 9
ωe [rad/s]
Figure 4.11: Fit of analytical spectrum to measured spectrum in ballast 2 condition for
the sensor amidships, HS = 2m, TZ = 5s, U = 15kn.
4.5. ESTIMATION OF MODE SHAPES. 87
4.5 Estimation of mode shapes.

The modes shapes in the experiments represent an idealization of the real mode shapes. A
few approaches were used to study the effect of various idealizations, including the following
mode shapes, which was investigated for ballast 1 condition
1. The real mode shape measured onboard and provided by DNV based on accelerom-
eters.
2. The measured segmented mode shape from experiments.
3. The calculated segmented mode shape based on model mass properties and stiffness
with and without 2D added mass distribution.
4. The calculated continuous mode shape from FE model based on full scale mass and
stiffness distribution and 2D added mass distribution.
The mass distribution, mode shapes and cross sectional properties are listed in Appendix B.7.
The mode shapes in the model tests were derived from the accelerations or moments. The
mode shape from acceleration, when the excitation was harmonic with frequency ω, were
estimated from
ẅ
w=− (4.8)
ω2
while the deflection from the moment was estimated based on the rotational stiffness. A
narrow band filter was applied to remove disturbance from swell in the regular waves,
which were supposed to excite springing. This was necessary when the mode shapes were
derived from the moment.
Figure 4.12 compares the calculated and measured mode shape of the segmented model,
which was run in ballast 1 condition in a regular wave exciting springing. The stiffness in
the calculations was based on the mean rotational stiffness in Table B.6. The mass was
based on the measured mass of ballast 1 condition in Table B.1, while the 2D added mass
was predicted at infinite frequency based on the theory in Appendix A.1.1. The 95% confi-
dence interval of the mean value from the measured mode is shown. It was estimated based
on the peak values from the individual cycles during the last 10s of the run. The agree-
ment of the mode shapes is satisfactory, but the calculated natural frequency exceeds the
measured by 5% (3.49rad/s versus 3.30rad/s as full scale values). The large discrepancy
is explained by the large uncertainty in the measured rotational stiffness, which appears
to be too high also since the added mass was too high (without 3D effects). The dry mode
shape differs from the wet mode, which ought to be considered in numerical predictions.
The calculated three wet mode shapes of the segmented model in ballast 1 condition are
shown in Figure 4.13. The 3-node mode may produce a contribution to the high frequency
1.1
0.9
Wet
0.7 Dry
Normalised mode shape
Measured
0.5
0.3
0.1
−0.1
−0.3
−0.5
−150 −100 −50 0 50 100 150
Distance from midship section [m], positive forward
Figure 4.12: Wet and dry calculated 2-node mode compared to measured ”resonance”
mode from ballast 1 condition, H = 1.4, T = 4.14 , ωe = 3.30rad/s in full scale. The
modes are normalized, and the 95% mean confidence interval is shown for the measured
mode.
response at the midship section in case of whipping. The shear deformation did not influ-
ence the results for the segmented model, since the shear stiffness was chosen deliberately
high.
0.8 2−node
3−node
0.6 4−node
Normalised mode shape
0.4
0.2
−0.2
−0.4
−0.6
−0.8
−1
−150 −100 −50 0 50 100 150
Figure 4.13: The calculated wet segmented mode shapes in ballast 1 condition with natural
frequencies of 3.5, 9.6 and 15.9rad/s.
4.5. ESTIMATION OF MODE SHAPES. 89
The segmented model was an idealization of the real flexibility. The wet continuous mode
shapes were calculated based on realistic distributions of mass, 2D added mass, bending
and shear stiffness using 50 finite beam elements. The first three mode shapes are shown
in Figure 4.14. In contradiction to the segmented model, the 3-node mode displays only a
small moment at midship section. Moreover, the ratio of the natural frequency from the
segmented and continuous model increases for the higher modes. Changes in the number
of flexible cuts, locations and stiffness distribution of the segmented model were considered
initially, without finding a significantly improved design compared to the chosen one.
0.8 2−node
3−node
0.6 4−node
Norm. mode shape
0.4
0.2
−0.2
−0.4
−0.6
−0.8
−150 −100 −50 0 50 100 150
Figure 4.14: The calculated wet continuous mode shapes in ballast 1 condition have natural
frequencies of 3.0, 6.6 and 10.8rad/s.
The first two mode shapes for the segmented and continuous FE model are compared to the
real measured full scale mode shapes in Figure 4.15. The segmented modes were based on
the model mass in ballast 1 condition, while the realistic mass was used in the calculations
of the continuous FE model. The natural frequencies from full scale measurements were
3.15 and 6.78rad/s. The natural frequency of 3.15 differs from results presented elsewhere
of 3.32rad/s. The measurements were carried out in the English Channel, which has an
average water depth of about 54m. If a 2D cross section of a rectangle with draft of 12m
and a beam of 53m is assumed, figures in (Faltinsen 1990) suggest that the added mass
is roughly twice of the displaced mass, and that the added mass in heave is increased by
about 10% on 54m water depth. This yields a reduction from 3.32 to 3.21rad/s. It is
therefore reasonable to conclude that the reduced frequency was a result of shallow waters
rather than an unconventional ballast condition. The difference in the continuous mode
shapes may come from inaccurate shear stiffness distribution and lack of 3D added mass
effects in the calculations, but the ratios of the frequencies for the 2 and 3-node modes
shapes are good. Despite the difference in mode shapes and frequencies, the chosen con-
figuration with three flexible cuts with identical stiffness distributed to the quarter lengths
was considered acceptable.
1
Norm. mode shape
Calc. cont.
Meas. full scale
0.5
Calc. segm.
−150 −100 −50 0 50 100 150

1
Norm. mode shape
0.5
−0.5
−150 −100 −50 0 50 100 150

Figure 4.15: Comparison of the 2 and 3-node mode shapes obtained from the segmented
model (based on model mass), the continuous FE model (based on real mass and estimated
stiffness) and full scale ballast condition. The natural frequencies are 3.5, 3.0 and 3.15rad/s
for the 2-node, and 9.6, 6.6 and 6.78rad/s for the 3-node mode.
Note that the 3-node mode from the segmented mode shape has a higher frequency relative
to the continuous FE model and the real mode shape, despite that the mass moment of
inertia for the rigid segments were too high. This is a general problem for segmented mod-
els, and the relative increase of the frequency for the higher modes is due to ”restrained”
deformations.
4.6 Procedure for estimating transfer functions

4.6.1 Linear transfer functions
According to Section 2.4.1, the linear transfer functions were estimated from regular wave
responses, and they were made nondimensional according to Appendix B.2.2, e.g. the
4.6. PROCEDURE FOR ESTIMATING TRANSFER FUNCTIONS 91
moment was made nondimensional as

V BM
V BM = (4.9)
ρgζa BL2
The transfer functions were determined at full speed of 15 and 13.2kn in ballast and cargo
condition, respectively, and the frequencies covered the range from linear springing to rigid
body response. In the following, the procedures to establish the transfer function will be
explained using some results for illustration.
Each regular wave run defined a single point in the transfer function. How this single point
was determined is explained in the following. A time interval with a duration of 10 to
80s was selected, and the FFT was applied to identify the encounter frequency from the
spectrum. A filter, with a width of about 20% of the encounter frequency, was applied to
remove disturbances. The inverse FFT reproduced the harmonic time series. Thereafter,
different procedures were utilized to determine the amplitudes, phases, mean values and
frequencies (the expressions in parentheses refer to the notation used in the figures for the
different methods).
1. The zero crossing and peak process were identified. The amplitude from each couple
of maximum and minimum was determined (Peak), and the mean and standard
deviation (uncertainty) were calculated.
2. Amplitudes were estimated based on the standard deviation of the filtered time series
(Time).
3. Amplitudes were estimated based on the standard deviation of the filtered spectra
(Spec).
4. The amplitude, phase and frequency were derived using a nonlinear fit (Fit) to the
filtered time series for each cycle along the selected time series. The mean and
standard deviation of the three parameters were calculated.
5. The least square fit (LSQR) was used to estimate the amplitude, frequency and phase
from the selected raw time series. The resolution from the FFT was not sufficient to
ensure that the best fit was achieved, and iteration was used within the resolution of
the FFT with a small step to ensure a good fit.
6. The mean value was obtained from this least square fit and the sample mean of the
selected raw time series (Raw). The former method was more accurate.
7. The frequencies were determined from the peak in the spectrum (Max spec), the
nonlinear fit and the least square fit.
8. The phase lags were found by a spot check (Spot) in the middle of the selected
interval, from the nonlinear fit, the least square fit and from the auto correlation
function (AutoC) derived by the MATLAB function xcov. The phases were defined
as a lag to the relative motion at COG. A narrow band filter resulted in larger end
effects for amplitudes, but better estimates of the phase relations. The end effects
were accounted for in the estimate of the amplitudes.
The transfer functions were plotted versus
q the encounter frequency along the x-axis, and
Lm
not the nondimensional frequency ωe g
. Further description is given in Section 2.5.1.
Some results will be shown to illustrate the alternative methods described above. The
heave transfer function (amplitudes) are presented in Figure 4.16 for ballast 1 condition.
The transfer function approaches one for long waves and zero for short waves. Notice that
the heave motion in practice is zero above 6rad/s. The 95% confidence interval of the
mean amplitude is typically less than 10% when the heave motion is significant. Satisfac-
tory agreement is observed between the different methods.
1
Peak
0.8 Time
Spec
0.6
η3/ζa
Fit
LSQR
0.4
0.2
0
1 3 5 7 9 11 13 15 17 19 21
95% mean conf. int. [%]
40
Peak
30 Fit
20
10
0
1 3 5 7 9 11 13 15 17 19 21
ωe [rad/s]
Figure 4.16: Linear transfer function of heave motion for ballast 1 condition. 95% con-
fidence interval of the mean amplitude is illustrated. Definition of methods is described in
the text.
The phases corresponding to the heave transfer function are presented in Figure 4.17. The
95% confidence interval of the mean phase is typically below 100 . The uncertainties are
less at lower frequencies. The phases change significantly with frequency, but also to some
extent with amplitude due to nonlinear responses in higher waves. The auto correlation
method is considered as the more robust method.
6
Spot
5 AutoC
4 Fit
θ [rad]
LSQR
3
2
1
0
1 3 5 7 9 11 13 15 17 19 21
95% mean conf. int. [rad]
0.8
0.6
0.4
0.2
0
1 3 5 7 9 11 13 15 17 19 21
ωe [rad/s]
Figure 4.17: Phases for linear heave motion of ballast 1 condition. 95% confidence inter-
val of the mean phase is illustrated. Definition of methods is described in the text.
The ratio of the frequency is illustrated in Figure 4.18. The ratio indicates when distur-
bances, e.g. swell from previous runs, affect the results. The 95% interval of the mean
frequency confirms that the uncertainties in the frequencies in general are low, but that
higher waves tend to increase the uncertainty.
Finally, the mean heave motion is shown in Figure 4.19. The mean motion was caused by
forward speed with trim and sinkage in the order of 2-3cm. The track was also elevating
towards the wave maker, increasing the apparent sinkage for shorter waves. The elevation
of the track was maximum 3cm, but the average during the measured runs was about 1cm.
The mean values may also indicate improper zero setting or nonlinear additions in higher
waves.
Figure 4.20 presents the transfer function of pitch including phases. The pitch approaches
the wave slope for long waves and zero for short waves. The pitch motion is in practice
zero above 5rad/s. The model did not move in pitch and heave for encounter frequen-
cies above 6rad/s, and the first, second and third order springing in Figure 4.22were not
expected to be significantly affected by coupling between global ship motions and hydroe-
lastic resonance response. This does however not mean that there is no coupling between
rigid ship motions and whipping, when even higher order harmonics may be involved and
1.2
1
0.8
ωe/ωe 0.6 Max spec/LSQR
Fit/LSQR
0.4
0.2
0
1 3 5 7 9 11 13 15 17 19 21
10
0
1 3 5 7 9 11 13 15 17 19 21
ωe,m [rad/s]
Figure 4.18: Ratios of frequencies corresponding to heave motion of ballast 1 condition

and the 95% confidence interval of the mean frequency from the nonlinear fit. Definition
of methods is described in the text.
0.05
Raw
LSQR
0.04
0.03
η [m]
3
0.02
0.01
0
1 3 5 7 9 11 13 15 17 19 21
ωe [rad/s]
Figure 4.19: Mean sinkage (positive) in forward speed corresponding to heave motion of
ballast 1 condition (some of the lower points at 9 and 20rad/s corresponds to lower speeds).
the encounter frequency is lower.
Figure 4.21 presents the transfer function for the bending moment at midship section in-
cluding phases. The bending moment shows a rigid body response peak at 4rad/s and a
Spot
1 6
Peak AutoC
0.8 Time 5 Fit
Spec LSQR
4
0 a
0.6
θ [rad]
η /k ζ
Fit
LSQR 3
5
0.4
2
0.2 1
0 0
1 3 5 7 9 11 13 15 17 19 21 1 3 5 7 9 11 13 15 17 19 21

50 0.8
Peak
40 Fit 0.6
30
0.4
20
10 0.2
0 0
1 3 5 7 9 11 13 15 17 19 21 1 3 5 7 9 11 13 15 17 19 21
ωe [rad/s] ωe [rad/s]
Figure 4.20: Linear transfer function of pitch including phases of ballast 1 condition and
the 95% confidence intervals of the mean amplitude and phase.
linear springing peak at 19.5rad/s. The first and second cancellation period in the bending
moment may be visible. The 95% confidence interval of the mean moment amplitude is
typically in the order of 4%, and the nondimensional value of the rigid body moment peak
is about 0.02. The wall interaction appears to affect the transfer functions at encounter
frequencies below 3.5rad/s.
0.05 Peak 6
Spot
VBM /(ρgζ BL2)
Time 5 AutoC
0.04
Spec 4 Fit
a
θ [rad]
0.03 Fit LSQR

LSQR 3
3
0.02
2
0.01
1
0 0
1 3 5 7 9 11 13 15 17 19 21 1 3 5 7 9 11 13 15 17 19 21
10 0.8
Peak
8 Fit 0.6
6
0.4
4
2 0.2
0 0
1 3 5 7 9 11 13 15 17 19 21 1 3 5 7 9 11 13 15 17 19 21
ωe [rad/s] ω [rad/s]
e
Figure 4.21: Linear transfer function of vertical bending moment at midship section
including phases for ballast 1 condition, and the 95% confidence intervals of the mean
amplitude and phase.
A first estimate of the critical frequency of wall interaction is formulated as

Lg
ωc = (4.10)
U 4d
where d is the distance between the center of the ship and the side wall. It assumes that
the radiated wave from the bow is reflected by the wall and hits the stern, and that the
generated wave front moves with the group velocity in deep water. This gives a wave
frequency of 3.0rad/s corresponding to an encounter frequency of 4.2rad/s (4.8rad/s in
cargo condition). A frequency below this is necessary to provide a significant effect on
the bending moment, and the observed 3.5rad/s agrees well with this estimate in ballast
condition. Converted to full scale, the wall interaction starts at a period above 14s. The
wall interaction must be anticipated in longer waves and at lower speeds. Appendix C.1
shows that the apparent false predicted transfer function values at low periods actually
are well reproduced in the different loading conditions, confirming that they are a result
of physical wall interaction due to this unconventional large model.
4.6.2 ’Nonlinear’ transfer functions
’Nonlinear’ is quoted, because the transfer function is defined to illustrate the total response
(NL) at the encounter frequency, as explained in Section 2.4.2. A narrow band filter is also
used to determine the transfer function value of the higher harmonic (HF), which may
coincide with the resonance frequency. The ratio of the HF frequency to the encounter
frequency displays the relative value of the nth order multiple. The second order multiple
is related to the hypothesis 4 and the higher order multiples are related to hypothesis 3.
This ’nonlinear’ transfer function is compared to the linear transfer function. In all cases,
the peak analysis described in Section 4.6.1 is applied. In this way the relative importance
of the total nonlinear and wave induced vibration effect is displayed.
Figure 4.22 illustrates the ’nonlinear’ transfer function and the frequency ratios. All results
are made nondimensional based on the linear wave amplitude. The higher harmonics, in
particular the second and third order multiple, are pronounced. The frequency plot shows
that the spectral peak at twice the encounter frequency is largest for the second order
effect (hypothesis 4), while the spectral peak at three times the encounter frequency is
largest for the third order effect. The third order effects become important especially
for the highest waves (hypothesis 3), while the second and first order effect appear to be
strong also in small waves, especially since the second order effect is also present outside
resonance. For fourth and fifth order effects, the nonlinear effect come from nonlinear
geometry rather than the elastic effects, since the linear and HF response do not add up to
the total nonlinear response. The uncertainty in the HF components is higher than for the
encounter and nonlinear response, which may be partly explained by low vibration values
in longer waves. The maximum peak in the response spectrum at a frequency of 17rad/s
occurred at a slightly different frequency than the encounter frequency, and this indicated
that the surge motion was too large (and was controlled in subsequent runs).
4
0.05 Lin
Max spec/LSQR
NL
VBM3/(ρ g ζaBL2)
0.04 3 Fit/LSQR
HF
ωe/ωe
0.03 2
0.02
1
0.01
0 0
1 3 5 7 9 11 13 15 17 19 21 1 3 5 7 9 11 13 15 17 19 21

30 1.5
25 Lin
20 NL 1
HF
15
10 0.5
5
0 0
1 3 5 7 9 11 13 15 17 19 21 1 3 5 7 9 11 13 15 17 19 21
ωe [rad/s] ωe [rad/s]
Figure 4.22: ’Nonlinear’ transfer function of vertical bending moment (left plots) taken
as maximum range divided by wave height for midship section in ballast 1 condition. The
frequency ratios are shown to the right. The 95% mean confidence intervals for the linear,
high frequency and nonlinear response and frequency are displayed.
4.6.3 Second order transfer functions

Two component waves were run in order to investigate the sum encounter frequency effect
at ω1,e + ω2,e . This sum frequency was supposed to coincide with the 2-node springing fre-
quency, ωs . Monocromatic waves with sum encounter frequency 2ω1 equal to the springing
frequency were also considered. The intention was to observe how the sum frequency effect
related to hypothesis 4 changed with increasing ratio ω2,e /ω1,e . The ratio of one corre-
sponds to monochromatic waves. Further explanation is given in Section 2.4.3.
Nonlinear damping affected the vibration level, except in ballast 2 condition. The non-
linear damping deteriorated the appearance of the second order transfer function when
different wave heights were considered. Therefore, only waves of the same height were
analyzed. Figure 4.23 and 4.24 show the spectra and the filtered time series components
in one run. The two wave components (W P 3) are located with encounter frequencies at
8 and 12rad/s. The bending moment (F X3) at midship section is small at the encounter
frequencies, while it is large at the sum frequency of 20rad/s. Small sum frequency effects
are observed on the pressure at the stem (P 3), but not on the wave or relative motion
(REL1) in the bow due to small steepness. This sum frequency pressure comes from the
radiation due to the vibration of the hull and from the sum frequency excitation due to
the incident waves. It was difficult to separate these two small pressure components to
illustrate the importance of the second order excitation from the reflected waves, so this is
not shown.
The determination of the amplitude at the encounter frequencies and sum frequency were
based on three methods.
4
x 10
10
(ω )
e
5
FX3
S 0
5 −4 10 15 20
x 10
4
SWP3(ω)
2
0
5 −3 10 15 20
x 10
4
SP3(ωe)SREL1(ωe)
2
0
5 4 10 15 20
x 10
2
1
0
5 10 15 20
ωe [rad/s]
Figure 4.23: Spectra for a bichromatic wave exciting resonance in ballast 1 condition (run
1071). FX3 = Axial force (bending moment), WP3= Wave elevation, REL1= Relative
motion in the bow and P3= Pressure at stem. Unit is basic unit squared times time, e.g.
N 2 s for F X3.
1. The first estimate was taken from the standard deviation of the filtered time series,
Eq.(2.68).
2. The second estimate was taken from the peak analysis of the raw signal based on
zero crossings for the longest encounter period, Eq.(2.67). This represents the total
nonlinear combination.
3. The third estimate came from the least square fit to the raw signal, Eq.(2.70). Iter-
ation for each of the three frequencies was necessary to find good peaks. The least
square fitting procedure provided also phase information, which was disregarded.
The least square procedure is illustrated in Figure 4.25. The raw signal is compared to
the fitted signal using all components and using only the three dominating components.
Satisfactory agreement is observed.
The second order transfer function was presented with the frequencies in the xy-plane and
the responses in z-direction. The encounter responses were presented, for comparison, on
the 450 diagonal in the xy-plane where ω1,e = ω2,e . The unidirectional transfer function for
sum frequency is symmetric about the 450 diagonal, since |H(ω1,e , ω2,e )| = |H(ω2,e , ω1,e )|,
and two points were plotted for each sum frequency combination. The results are presented
in Section 5.3.2.
4.7. PROCEDURE FOR CALCULATION OF FATIGUE DAMAGE 99
ω1e
FX3 [N]
200
0 ω2e
−200 ω1e+ω2e
140 141 142 143 144 145
0.01
WP3 [m]
0
−0.01
140 141 142 143 144 145
REL1 [m]
0.04
0.02
0
−0.02
−0.04
140 141 142 143 144 145
P3 [N/m2]
50
0
−50
140 141 142 143 144 145
Time [s]
Figure 4.24: Filtered time series for a bichromatic wave exciting resonance in ballast
1 condition (run 1071). FX3 = Axial force (bending moment), WP3= Wave elevation,
REL1= Relative motion in the bow and P3= Pressure at stem.
4.7 Procedure for calculation of fatigue damage

The procedure to determine the short term fatigue damage is outlined in the following.
1. √
The different
√ runs in each sea states were merged, and the time was scaled with
Λ = 35. Each sea state amounted to about 38 minutes in full scale.
2. The time series of the axial force were converted into full scale hot spot stress in
deck. A moment arm of 0.88m and scaling according Eq.(B.10) to full scale bending
moment were applied. A section modulus derived by NAUTICUS, (DNV 2005c),
of 53.92m3 (43.92/46.52) was used at midship section (aft quarter/fore quarter) to
convert moment to stress. The nominal stress was multiplied with a SCF of 2.0,
representative for e.g. the hot spot on the longitudinal stiffener flange in way of the
transverse web frame.
3. The total time series was derived by filtering the measured time series to remove
everything above 2.0Hz. The wave stress time series was derived by filtering above
0.35Hz. Smooth filters were applied.
200
FX3 [N]
0 Raw
−200 LSQR
−400 3 comp.
140 141 142 Time [s] 143 144 145

300
Coef. [N]
200 cos
100
0 sin
−100
8 10 12 14 16 18 20
Phase [rad] Ampl. [N]
300
200
100
0
8 10 12 14 16 18 20
2
0
−2
8 10 12 14 16 18 20
ωe [rad/s]
Figure 4.25: Least square fit to time series of bending moment (FX3) due to a bichromatic
wave giving resonance in ballast 1 condition (run 1071). The coefficients in front of the
sinusoidal terms, amplitudes and phases are illustrated.
4. The stress time history relevant for fatigue analysis was obtained by Rainflow count-
ing of the total and wave frequency (WF) stress time series.
5. A two slope SN-curve for welded joints in air or with cathodic protection, (SN-I in
(DNV 2005a)), and Miner Palmgren’s rule were applied.
6. The total and WF damage were calculated from the total and wave stress.
7. The vibration damage was calculated as the difference between the total and WF
damage.
The rainflow counting was performed utilizing the DNV program TIMSER, (Mathisen
1988; Mathisen 2001). The procedure has been validated against other procedures with
good agreement, and the current procedure is referred to as the hot spot stress approach.
Further description is given in Section 2.7.
4.7.1 Scatter diagrams and investigated sea states

The model was tested in different sea states listed in Appendix B.6. An overview of these
sea states compared to the North Atlantic and World Wide scatter diagrams is presented
4.8. METHOD FOR IDENTIFYING WHIPPING 101
in Figure 4.26. The scatter diagrams illustrate the probability of occurrence, which were
taken from DNV CN 30.5, (DNV 2000). The selected sea states were chosen from areas
with high and low probabilities. A ship operating in World Wide trade will in general
encounter lower sea states with shorter periods than in the North Atlantic trade. One key
issue relates to how these scatter diagrams influence the vibrations and wave damage.
North Atlantic World Wide Selected

18 18 18
16 16 16
14 14 14
12 12 12
TZ [s]
TZ [s]
TZ [s]
10 10 10
8 8 8
6 6 6
4 4 4
0 3 6 9 12 15 0 3 6 9 12 15 0 3 6 9 12 15
H [m] HS [m] HS [m]
S
Figure 4.26: Scatter diagrams (HS versus TZ ) for North Atlantic and World Wide trade
compared to the investigated sea states. The light cells were tested in all combinations of
bow design and loading condition at full speed, while the darker cells were also tested for
ballast 2 and cargo 2 condition.
4.8 Method for identifying whipping

It is difficult to separate whipping and springing, e.g. (Gran 1974). This is not an issue
when evaluating the implication of specific measurements, however it is necessary to know
which effects to include in theoretical predictions when the results are to be generalized.
For this reason a method was derived to distinguish the phenomena by identifying whip-
ping events from the model test time series. The intention is to illustrate the importance of
hypothesis 3 related to whipping impacts relative to the linear and second order springing
(hypothesis 4).
The method is based on the envelope process of the filtered 2-node vibration response.
If the slope of the envelope curve exceeds the maximum possible slope from springing, a
whipping event is identified. The key point is to establish a reliable or sensible criterion
for the maximum springing slope.
Assuming a lightly damped SDOF-system, originally at rest, produces the following enve-
lope process
uenv (t) 1
= (1 − e−δωs t) (4.11)
ust 2δ
where δ is the damping ratio, u is the response in time, and ωs is the springing frequency
of the forced loading. The subscript st refers to the static deflection. The maximum
slope of the envelope process is given by the derivative at t = 0. It is difficult to define
the start of the impact, and the derivative after one springing period is chosen as a good
approximation.
uenv (t = Ts )0 ωs
= e−δ2π (4.12)
ust 2
The response and envelope process, representing the 2-node springing of the model, are
shown in Figure 4.27.
50
u(t)/ust
40
uenv(t)/ust
30
20
10
st
u(t)/u
−10
−20
−30
−40
−50
0 1 2 3 4 5 6 7 8 9 10
Time [s]
Figure 4.27: Dynamic response and envelope curve from a constant harmonic loading of
a SDOF-system, which initially is at rest, δ = 0.01 and ω = 20rad/s.
The static response can be related to the maximum springing response as
ust = 2δudyn,max (4.13)
and to define a criterion, the maximum springing response must be determined. The
springing vibration is assumed to come from linear and sum frequency excitation, and the
4.8. METHOD FOR IDENTIFYING WHIPPING 103
two processes is assumed to be independent, which is reasonable. The maximum of the

total response is approximated by
q
udyn,max = u2dyn,max,lin + u2dyn,max,sum (4.14)
The maximum values refer to the linear springing peak from the linear transfer function
and the nonlinear springing peak from the second order transfer function. The second
order transfer function is assumed to contain only springing excitation, and higher order
effects are assumed to be caused only by whipping. The static bending moment is written
r
ζ2
V BMstatic = 2δ (V BMlin,max ζ1 ρgBL2 )2 + (V BMsum,max ( )2 k2 ρgBL2 )2 (4.15)
2
The subscript 1 refers to the wave amplitude at the encounter frequency of linear springing,
while subscript 2 refers to the wave amplitude (wave number) at the encounter frequency
of the sum frequency monochromatic springing. The criterion for the slope of the envelope
process becomes
r
ζ2
cV BM = ωs δe−δ2π (V BMlin,max ζ1 ρgBL2 )2 + (V BMsum,max ( )2 k2 ρgBL2 )2 (4.16)
2
This criterion is used to identify a sudden increase of the response, i.e. whipping.
For a whipping event causing a sudden decrease in the vibration level due to opposite
phasing with the existing vibration, the natural decay due to the damping is assumed to
contribute to the decrease (as a small additional effect). The additional decrease is strictly
dependent on the initial vibration level. In order to define a general initial level, the decay
is related to the reduced response during one vibration cycle at an initial vibration level
of two times the significant vibration amplitude for the sea state. The correction based on
this chosen reference level is written
q
∆
c = 4 · E(V BMHP 2
)(1 − e−δ2π ) (4.17)
The subscript HP refers to the high pass filtered bending moment capturing the dynamic
2-node response, while E(V BM 2 ) is the variance of the bending moment time series with
zero mean. When applied to a sudden decay of the positive envelope process, the decrease
exceeding −cV BM − c∆ indicates a whipping event, which reduces the vibration level.
The wave amplitudes are estimated as two times the expected mean wave height for the
linear and sum frequency wave period in accordance with Appendix A.10. The combination
of the linear and sum frequency response at full speed with high wave amplitudes makes
the criterion strict. The probability of springing to exceed the criterion is low, while it is
likely with whipping response below the criterion. The criterion is therefore intended to
identify detectable and significant whipping events.
4.8.1 Procedure
The procedure is illustrated for ballast 2 condition in a sea state of 4m significant wave
height and 7s zero up-crossing period at full speed of 15kn. The time series is filtered, keep-
ing the interval between 12.5 and 25rad/s, and the envelope process is identified between
local maxima above zero. The slope is calculated by the Euler method. The nondimen-
sional peak values estimated for the bending moment amidships are V BM max,lin = 0.065
and V BM max,sum = 1.6 taken from Figure 5.15 and 5.20, respectively. Figure 4.28 presents
the detected whipping events marked with circles at the second end of the slope. The mo-
ment is represented by the axial force transducer at midship section, F X3 , and the slopes
are displayed in the lower plot. An axial force of 1000N corresponds to 25MP a nominal
stress level in deck.
2000
1000
FX3 [N]
−1000
−2000
0 50 100 150 200 250 300 350 400
4000
Slope [N/s]
2000
−2000
0 50 100 150 200 250 300 350 400
Time [s]
Figure 4.28: Whipping events (both decrease and increase) identified for bending moment
(FX3) at midship section, HS = 4m, TZ = 7s, U = 15kn. 1000N corresponds to 25MPa
nominal stress amplitude in deck amidships. Note time in model scale.
The cumulative probability distribution of the envelope process is derived from 1400 slopes.
The criteria, which are not symmetric about zero, are shown in Figure 4.29. 12 slopes ex-
ceed the criteria with some margin, and intuitively these represent the actual whipping
events. Note that the exact number of whipping events could not be determined in these
model tests, so that this methods only indicates the number of significant whipping events.
The total number of real whipping events including the small whipping events are un-
known. The goodness of the method can therefore be debated, but Figure 4.29 indicates
that the criterion appears to be reasonable. The criterion fits well into the distribution of
slopes where the largest slopes are well separated from the rest.
4.9. THE RELATIVE IMPORTANCE OF LINEAR AND MONOCHROMATIC SECOND ORDER
EXCITATION 105
0.9
0.8
0.7
0.6
CPD [−]
0.5
0.4
0.3
0.2
0.1
0
−2000 −1000 0 1000 2000 3000
FX3 [N/s]
Figure 4.29: Cumulative probability distribution of the envelope slopes of F X3 versus the
criteria, HS = 4m, TZ = 7s, U = 15kn. 3000 N/s corresponds to about 25MPa nominal
stress amplitude in deck amidships.
To estimate the relative importance of springing and whipping, the identified whipping
events are used to define an artificial whipping process. This is created based on the am-
plitudes at the aft end of the slopes exceeding the criteria. This implies a sudden increase
or decrease followed by a logaritmic decay. To separate the whipping and springing con-
tribution, it is assumed that the processesq are independent. The standard deviation of
springing stress is calculated as σspring = σT2 ot − σW
2
hip , where T ot refers to the measured
vibration and W hip refers to the artificial whipping process. The probability of a slope
indicating whipping is also calculated. This procedure will be applied to all sea states in
Section 5.7.
The method may also be applied to the negative envelope process to check whether there
is a difference. As illustrated in Figure A.15 the whipping maxima differ in sagging and
hogging, and whipping may be caused by both an upward and a downward force.
4.9 The relative importance of linear and monochro-

matic second order excitation
A simple approach is applied to investigate the importance of the monochromatic second
order effect (2ωe = ωs ), as compared to the linear springing as a function of the wind
speed. The second order effect in this simplified procedure is handled as a linear effect, but
the second order resonance peak must then be corrected for each sea state. The analysis
is not consistent with the correct formulation of general second order analysis, but the
intention is to illustrate the importance of the hypothesis 4 related to nonlinear springing
relative to linear springing, and for this purpose the methodology is sufficient. The relative
importance is shown in Section 5.6.
An artificial (simplified) ’nonlinear’ transfer function is established based on Figure 5.15,

which shows the nondimensional ’nonlinear’ midship bending moment from ballast 2 condi-
tion as a function of the encounter wave frequency. The rigid body response is represented
by a transfer function of the following Gaussian shape
(ωe −ωp )2
V BM r = 0.02e− 2σ 2 (4.18)
where σ = 1.25 denotes the width. The magnitude of 0.02 is taken from Figure 5.15.
The linear resonance peak is defined by the SDOF system

0.065 · 2 · 0.01
V BM s = p (4.19)
(1 − β 2)2 + (2δβ)2
where β = ωωes and ωs is the springing frequency. The value of 2 · 0.01 reflects the resonance
level for the damping ratio in ballast 2 condition, and 0.065 corresponds to the nondimen-
sional magnitude of the linear resonance peak in Figure 5.15.
The second order contribution is established by replacing the frequency ratio in Eq.(4.19)
with β2 = 2ωωs
e
, and replacing the 0.065 factor with 0.048, corresponding to the peak of the
second order resonance in Figure 5.15. The frequency ratio defined as this makes the ’non-
linear’ peak at the encounter frequency of half thickness compared to the linear resonance
peak, and this is necessary since the actual vibration due to the second order effect occurs
at the frequency corresponding to the linear resonance. The second order effect is also
proportional to the wave amplitude squared, and therefore the second order vibration has
to be corrected for the wave amplitude at the encounter frequency. The correction used
is ζ/1.05, where 0.048 is the nondimensional second order peak value in Figure 5.15 corre-
sponding to the wave height of 2.1m. The wave amplitude, ζa , in an irregular sea state is
estimated as the expected mean wave amplitude according with Appendix A.10. The mean
wave height versus Beaufort numbers are presented in Figure 2.8. The artificial monochro-
matic second order contribution must accordingly be corrected for every sea state, but the
response spectrum can be established assuming a linear transfer function.
The second order resonance curve in the transfer function is displayed at the encounter
frequency, consistent with the ’nonlinear’ transfer function. The peak is of half width com-
pared to the linear resonance curve, and this shape is combined with the wave spectrum
to produce the response spectrum. The vibration will still occur at the same frequencies
as the linear resonance. An example of the ’nonlinear’ transfer function, with the second
order effect corrected for a Beaufort number of 6, is shown in Figure 4.30. Torsethaugen
EXCITATION 107
wave spectrum is assumed, and it requires the significant wave height as input, which is
taken from the Beaufort scale in Table A.1. Torsethaugen requires also the peak period.
The relation between significant wave height and zero up-crossing period in Appendix A.10
is utilized, and the zero up-crossing period is converted to the peak period using Eq.(2.83)
assuming a γ of two. The wave spectrum is converted to encounter spectrum to consider
the correct encounter wave frequency. The measured speed reduction in full scale taken
from Moe et al. (2005) is used in the estimate of the wave amplitude, which is necessary
for correction of the second order response.
0.07
Linear springing
0.06
Monochromatic second order springing
0.05
VBM/(ρgζ BL2)
0.04
a
Rigid body response
0.03
0.02
0.01
0
0 0.5 1 1.5 2 2.5 3 3.5 4
ωe [rad/s]
Figure 4.30: Illustration of the artificial nondimensional ’nonlinear’ transfer function of

the midship bending moment (VBM) in a typical sea state for Bn = 6; U = 10.9kn, δ =
0.01. The monochromatic second order springing peak is corrected based on the estimated
mean wave amplitude for each sea state (Beaufort number). The dotted line is the total
nondimensional response. The continuous lines refer to the rigid body, the monochromatic
sum frequency and linear springing peak.
The wind based PM wave spectrum in Eq.(2.84) assumes fully developed sea. It predicts
higher significant wave heights than the Beaufort scale at high Beaufort strengths (19m
reference height instead of 10m). For comparison, the Torsethaugen and PM wave and
encounter spectra at Beaufort 6 are shown in Figure 4.31. The Torsethaugen wave spec-
trum is more peaked with a higher peak period than the PM spectrum. The encounter
spectra are moved towards higher frequencies due to the forward speed, and at 1.66rad/s
corresponding to the second order springing, the wave energy from PM is higher.
The wave spectra are combined with the artificial transfer function to calculate the re-
sponse spectrum by SR (ωe ) = |H(ωe )|2 S(ωe ), where H(ωe ) is the artificial encounter trans-
PM
3 S
PM
Se
2.5 HS=3.5m, TP=11.9s ST
STe
S(ω) , Se(ωe) [m s]
2
H =3.2m, T =9.0s
S P
1.5
0.5
0
0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
ω , ωe [rad/s]
Figure 4.31: Illustration of Torsethaugen (T) and PM wave and encounter spectra for
Bn = 6, U = 10.9kn.
fer function. This can be done since the ’nonlinear’ monochromatic springing is consid-
ered as a linear effect herein. The response spectra are shown in Figure 4.32 based on the
Torsethaugen wave spectrum. The second order effect is displayed at the frequencies where
the vibration actually occurs instead of at the encounter frequency corresponding to the
monochromatic peak in Figure 4.30. The energy from the wave spectrum is however taken
from the encounter frequency corresponding to the monochromatic ’half width’ peak. The
monochromatic second order effect is about twice as high as the linear contribution for this
sea state.
The nondimensional bending moment is converted into full scale stress in MP a (for com-
parison with full scale measurements). The nondimensional bending moment is multiplied
by ρgBL2 /106 /53.92, where 53.92 is the section modulus in deck amidships and 106 is used
to get the stress in MP a. The standard deviation of the filtered high pass nominal stress
is calculated for the linear and second order response. The second order response refers
to the sum of the second order and linear response. The phasing of the wave frequency
components are generated by a random phase lag, hence the response from the linear and
sum frequency effect will also be independent, and they can be combined accordingly. Note
that this is not the case for the second order effect of the incident waves, i.e. the second
order effect related to the incident waves satisfying the second order free surface condition,
as illustrated in Figure 2.3. The results are displayed in Section 5.6.
EXCITATION 109
−4
x 10
Linear
8 Second order
6
S(ωe) [s]
0
0 0.5 1 1.5 2 2.5 3 3.5 4
ωe [rad/s]
Figure 4.32: Illustration of spectra for linear response, linear springing and monochro-
matic second order springing based on Torsethaugen spectrum, Bn = 6 (HS = 3.5m and
TP = 11.9s) U = 10.9kn. The monochromatic second order effect is displayed at the actual
vibration frequency (opposite to at the encounter frequency in Figure 4.30).
Chapter 5
Model test results
The main results from the model tests are provided in this chapter, beginning with the
damping in the first section. Section 5.2 deals with the wave amplification in the bow and
the added resistance in waves compared to analytical formulations, followed by transfer
functions in Section 5.3 and fatigue damage in Section 5.4. Section 5.5 illustrates the relative
importance of whipping and springing, while Section 5.6 illustrates the difference between
linear and monochromatic sum frequency springing. Finally, a few whipping and springing
events are displayed in Section 5.7. Mainly results from ballast 2 condition are covered.
Results from other loading conditions and bow shapes are shown in Appendix C, and full
scale measurements are described in Chapter 6.
5.1 Damping
The damping ratio was estimated in air and in calm water to investigate the following
effects
? Forward speed.
? Rubber sealing and water leakage (in air).
? Effect of mode shape (distinct natural frequencies)
? Springing frequency in air.
? Wall effects due to radiated and reflected waves.
? Mounting gear; Safety ropes, cables, towing and guidance lines.
111
112 CHAPTER 5. MODEL TEST RESULTS
? Changed number of bearings and design.
The damping ratio was estimated by procedures explained in Section 4.4. The damping
contributions from wall reflection, mounting gear and effect of frequency in air were small
and regarded as insignificant.
5.1.1 Damping estimates for the 2-node mode from decay tests
The damping was in most cases nonlinear due to insufficient quality (tolerance) and tight-
ening of bearings, loose weights and supports (in air). During the first tests in ballast 1
condition, deficient aft lower bearing was identified in high waves. A second lower bearing
was mounted as shown in Figure B.1, and the previous runs were repeated. Figure 5.1
illustrates the effect of using two lower bearings instead of one for the three joints in bal-
last 1 condition. The damping level and its nonlinearity was reduced (resulting in larger
vibration levels). The effect of speed in the model with only one bearing is also shown
in Figure 5.1. Apparently, the effect of speed is small. This agrees with Tasai (1974) and
Hashimoto et al. (1978), except that they indicated an unexplained jump at zero speed.
This jump was not evident from the present measurements.
0.02
0kn
0.018 y = 9e−006*x + 0.0039 5kn
10kn
0.016
15kn
New bearings 0kn
0.014
linear
0.012
0.01
δ
0.008
0.006
0.004
0.002
0
0 100 200 300 400 500 600 700 800
FX3 [N]
Figure 5.1: Damping ratio of the 2-node mode from harmonic fits at different speeds in
ballast 1 condition. A linear regression (the equation is presented) is applied to the data
from the new bearings. The x-axis refers to the axial force, F X3 , which can be converted
into a moment at the midship section.
Figure 5.2 shows a nonlinear damping in air from ballast 1 condition. The estimates are
associated with significant uncertainty, and the difference between the damping in air and
water appears to be low. The supports contributed to the damping, and the hydrodynamic
contribution was therefore difficult to estimate.
5.1. DAMPING 113
0.02
0.018 y = 1.6e−006*x + 0.0064
0.016
0.014
0.012
0.01
δ
0.008
0.006
0.004
0.002
0
0 200 400 600 800 1000 1200 1400
FX3 [N]
Figure 5.2: Damping ratio for the 2-node mode in air (on supports) from harmonic fits
for ballast 1 condition. Linear regression is performed.
The rubber sealing and water leakage (small amount of water accumulated between the
flexible cuts below the pump) contributed to the damping in water (and in air after the
model had been in water). The contribution, illustrated in Figure 5.3, was estimated from
tests in air for cargo 1 condition. The rubber sealing and water leakage were of similar
importance, and their total contribution was about 0.15%. The tested force range was
small, but nevertheless the contribution was significant.
−3
x 10
7
y = 2.2e−005*x + 0.0022
6
4
δ
2
With rubber/water
linear
1 With rubber
Without
0
0 50 100 150 200
FX3 [N]
Figure 5.3: Damping ratio for the 2-node mode from harmonic fits in air for cargo 1
condition. Linear regression is applied to the data with rubber sealing and water leakage.
Prior to cargo 1 condition, all bearings were modified to include a split and set screws. The
purpose was to tighten the bearings to the steel springs (rods). In water the model was
pretensioned due to still water sagging or hogging moment, and it was difficult to tighten
the bearings sufficiently well. This tightening was in the subsequent ballast and cargo 2
condition done in air without any pretension, resulting in linear damping. Unfortunately,
for cargo 2 condition the bearings were by a mistake adjusted in water, resulting again in
a nonlinear damping behavior. Also for ballast and cargo 3 condition improper tightening
resulted in nonlinear damping. The nonlinear damping reduced the quality of the results
and made the assessment difficult. Low and linear damping must be ensured prior to tests
hereafter, and modification of the flexible joint design should be considered. Set screws
must be used with care, since they produced a nonlinear damping behaviour different than
observed in ballast 1 condition.
The damping ratio for ballast 2 condition is presented in Figure 5.4. It is in principle linear
with a mean of 1.0% from the left plot. The uncertainty to the mean value is low. The right
plot confirms that the damping is speed independent. This contradicts several theories,
e.g. (Kaplan et al. 1969; Goodman 1971), and model experiments, e.g.(Hoffman and van
Hoof 1976; Troesch 1984), but agrees with model experiments by Achtarides (1979).
0.014 0.014 0kn

15kn
0.012 0.012
0.01 0.01
0.008 0.008
δ
0.006 0.006
0.004 0.004
0.002 0.002
0 0
0 1 2 3 4 5 6 7 8 0 200 400 600 800 1000 1200
Test no. FX [N]
3
Figure 5.4: Damping ratio for the 2-node mode obtained by cycle-by-cycle decay with 95%
confidence interval of samples and mean (left) and from harmonic fits (right) for ballast 2
condition. Test no. 7 refers to 15kn, while the rest refers to 0kn.
The damping tests in air after ballast 2 condition were not carried out by the author. The
model was hanging in air from two strops, intended to be located at the two node points
of the 2-node mode. The results indicated a damping ratio of 0.5%, but the response level
was only about 10N (too small impulse loading). This damping ratio differs by 0.5% from
the damping in water in Figure 5.4. Accounting for the rubber sealing and water leakage,
the hydrodynamic damping constituted about 0.35%. This estimate is only an indication
of the hydrodynamic contribution, since the response level was low.
Table 5.1 lists the approximate damping values obtained in the different tests. The rather
unexpected damping behaviour in water for ballast 3 condition and cargo 2 condition is
5.1. DAMPING 115
illustrated in Figure 5.5. The strong nonlinear contribution was dominated by the insuffi-
cient tightening of the bearings.
Table 5.1: Damping ratio for the 2-node mode in water obtained by different decay tests.
Load. cond. δ-range [%] Load range [N ] Comment
Ballast 1 0.7-1.3 300-800 Weakly nonlinear increasing
Ballast 2 1.0 200-1250 Linear
Ballast 3 3.0-1.2 100-700 Nonlinear, decaying
Cargo 1 1.0-3.0-2.5 200-700-2500 Nonlinear, increasing and then weaker decreasing
Cargo 2 1.0-3.5-2.0 50-200-800 Nonlinear, increasing and then weaker decreasing
Cargo 3 3.5-1.7 100-800 Nonlinear, decaying
Total damping ratio versus amplitude Total damping ratio versus amplitude
0.06 0.045
0.04
0.05
0.035
0.04 0.03
0.025
δtot
δtot
0.03
0.02
0.02 0.015
0.01
0.01
0.005
0 0
0 2000 4000 6000 8000 10000 12000 0 2000 4000 6000 8000 10000 12000
8ω x0/3π 8ω x0/3π
Figure 5.5: Nonlinear 2-node damping ratio determined by a cycle-by-cycle decay for
ballast 3 condition in 15kn (left, run9885) and cargo 2 condition in 13kn (right, run9862).
According to Eq.(2.57), the x-axis refers to the vibratory response level, and the slope refers
to the equivalent damping coefficient due to the nonlinear damping.
5.1.2 Spectral method

The damping in irregular sea states was estimated utilizing the spectral method explained
in Section 2.3.3. At least twelve sea states times three flexible joints were considered in
each combination, and the 95% confidence interval of the sample and mean values were
estimated. The natural frequency and JONSWAP high frequency tail factor were included.
The results are shown in Table 5.2. The damping ratios for ballast 1 condition were most
uncertain, and they were well above the estimates from the decay curves. The uncertainty
in the natural frequencies was low, but the confidence interval of the mean value from
ballast 1 condition exceeded the other estimates by a factor of two. This indicates a
nonlinear stiffness, which explains the relatively high damping estimated for ballast 1
condition. The factor n does not represent a reliable estimate of the high frequency tail
factor, which is typically taken as 4 to 5 in standardized wave spectra, e.g. (DNV 2000).
This is illustrated in Table 5.2 by a value of 0.9 estimated in the first row, while it becomes
more reasonable by changing the frequency interval. Moreover, if there should be a hump
hollow behaviour of the excitation, such behaviour is not captured by the spectral method.
The estimated values are therefore not reliable.
Table 5.2: Damping ratio, natural frequency and high frequency tail factor for the 2-node
mode with 95% confidence interval of sample estimates (s.) and mean value (m.) obtained
by the spectral method. Fitted frequency interval below/above the natural frequency and
number of FFT sample points are indicated. Bold is default values.
Loading cond. δ [%] /95% s./95% m. ωs[rad/s] /95% s./95% m. n /95% s./95% m.
Ballast 1, 2048 -0.5/1 2.76 /2.44/0.41 3.20 /0.12/0.02 0.9 /4.3/0.7
Ballast 1, 4096 -0.5/1 2.54 /2.42/0.40 3.20 /0.12/0.02 0.9 /4.3/0.7
Ballast 1, 2048 -1/1.5 2.80 /2.71/0.45 3.21 /0.10/0.02 2.2 /3.9/0.7
Ballast 1, 2048 -0.3/0.5 2.08 /1.22/0.21 3.21 /0.12/0.02 2.9 /7.2/1.2
Ballast 1, 4096 -0.3/0.5 1.94 /1.23/0.21 3.21 /0.12/0.02 3.3 /6.7/1.1
Ballast 2 1.39 /0.56/0.06 3.39 /0.08/0.01 4.2 /4.8/0.5
Ballast 2, 4096 -0.3/0.5 1.30 /0.32/0.03 3.39 /0.09/0.01 4.8 /6.1/0.6
Ballast 3 2.14 /0.76/0.09 3.33 /0.04/0.01 4.2 /5.5/0.7
Cargo 1 2.97 /0.82/0.10 2.77 /0.10/0.01 5.0 /5.9/0.7
Cargo 2 3.02 /0.81/0.10 2.95 /0.09/0.01 5.1 /4.7/0.6
Cargo 3 2.68 /0.79/0.10 2.88 /0.04/0.01 5.0 /4.1/0.5
In Table 5.2 the sensitivity of the input parameters to the procedure is illustrated for bal-
last 1 condition. The frequency range of the fit was by default taken from 0.5rad/s below
to 1.0rad/s above the natural frequency. The default number of sample points in the FFT
procedure was 2048. Increasing the number of sample points and using a narrower band
produced lower and better estimates. The damping for ballast 2 condition was roughly
30% higher than predicted by the decay test. This was in accordance to Gran (1974), who
indicated a 25% increase. Gunsteren (1978) concluded that the method can not be used
due to nonstationary response! Based on these findings, the spectral method in case of
linear damping and stiffness may only be useful as long as a correction factor of about 0.8
is included in the damping estimates. For nonlinear damping and stiffness it is less useful.
Figure 5.6 shows that the damping increased in higher and longer waves, while the nat-
ural frequency decreased. The latter indicates that the stiffness was reduced. Moreover,
the variation in the added mass/wet area in larger waves may contribute to explain the
decrease and variation in the springing frequency with increasing response level. This re-
duced natural frequency in higher waves was also observed in the ballast 2 condition, and
consequently, the damping ratio from the spectral method was overestimated due to the
resulting wider spectral resonance peak.
Due to the nonlinear damping behaviour in ballast and cargo 3 condition shown in Fig-
ure 5.5, the low vibration amplitudes were damped out faster than the high vibration
amplitudes. If for instance the low vibration amplitudes came from springing and the
5.1. DAMPING 117
Damping ratio=0.019391 95%=0.012326 95% mean=0.0020544 Springing frequency=3.2104 95%=0.11661 95% mean=0.019435
0.04 3.5
0.03 3.45
0.02 3.4
δ
0.01 3.35
0 3.3
ωs [rad/s]
0 5 10 15 20 25 30 35
Sea state number * 3 3.25
15 3.2
Hs [m]
Tz [s]
V [kn] 3.15
10
3.1
5
3.05
0 3
1 2 3 4 5 6 7 8 9 10 11 12 0 5 10 15 20 25 30 35
Sea state number * 3
Figure 5.6: The 2-node damping ratio versus sea states (left) and springing frequency
(right) for ballast 1 condition for all three flexible joints.
larger amplitudes came from whipping, the springing vibration was partly filtered by this
damping mechanism. The opposite may be the case for ballast 1 condition.
5.1.3 Damping in higher order modes

The decay tests in ballast 2 condition were assessed to determine the damping of the 3-
and 4-node modes. The damping of the 3-node mode (48.8rad/s) was evaluated at the for-
ward quarter length. The cycle-by-cycle method confirmed a speed independent damping.
The three methods gave a mean damping ratio of 1.43% with a 95% confidence interval of
the mean of 0.12% in average. The fit to the harmonic decay curve displayed a nonlinear
decreasing damping for increasing response from about 1.7% at 300N to 1.2% at 1100N .
This force was measured by the axial force transducer, and the level was comparable to
values in Figure 5.4.
The 4-node mode (71.0rad/s) was evaluated at midship. The estimates from the cycle-
by-cycle method became unreliable due to low sampling frequency. The time series were
noisy with sharp maxima. A large scatter, also with many low values, were observed. The
damping ratio from the two other methods was estimated to have an average mean of
3.23% with a 95% confidence interval of the mean of 0.15%. The fit to the harmonic decay
curve showed again a nonlinear decaying trend of damping with increasing response from
3.5% at 300N to 3.0% at 700N .
The amplitudes of the 2-, 3- and 4-node modes were in the same order of magnitude,
which are not expected to occur during wave impacts. The damping ratio was frequency
dependent, i.e. a function of the mode shape. Strong nonlinear viscous effects, indicated
by Hoffman and van Hoof (1976) at higher frequencies, were not evident from these tests.
5.2 Wave amplification and added resistance in waves

5.2.1 Measured and predicted wave amplification in the bow
The analytical estimate of the wave amplification in Section 2.1.2, Eq.(2.5), is compared
with the measured relative motion at the bow. The results of ballast 2 condition are shown
in the left plot of Figure 5.7. The analytical solution starts where the wave length was equal
to the ship length, which implies large ship motions. Above 5rad/s the pitch and heave
motion were insignificant. In general, the measurements yield somewhat larger values than
the analytical linear estimate, but a few lower values are also observed. There are several
reasons for the difference.
1. In the regular wave tests, a few of the highest waves struck the flare area causing
an artificial increase in the relative motion, since the change to non-vertical side was
not accounted for in the calibration of the relative motion sensor. It was relevant for
wave heights of 3.5m and above, with frequencies supposed to excite springing by
sum frequency effects.
2. Some water jets or run ups were observed.
3. In short waves it appeared to be a strong interaction between the steady and unsteady
wave field causing breaking front waves and slamming (hypothesis 1), especially for
the steeper incident waves. The water spray in particular at the bow quarter exceeded
occasionally the deck level, which was 16m above the still water level, even though
the wave height was less than 2.1m.
4. Lower/higher speeds than 15 knots were also considered for the wave frequencies
supposed to excite the linear and second order resonance. This explains a few of the
lower/higher values.
A similar difference between analytical estimates and measurements is displayed in the

right plot of Figure 5.7, which presents the results for cargo 2 condition. The bow had no
flare and the speed was reduced to 13.2kn. The maximum exceedance of the measured
compared to the analytical wave amplification is in the order of 20% for both loading
conditions. The uncertainties in the wave amplitudes increase somewhat for the shorter
waves as illustrated in Appendix B.5.3, still the linear analytic estimate of the amplification
can be considered as a good first approximation for the surface motion which may cause
impacts (related to hypothesis 3).
The nonlinear contribution appeared to be moderate, but some sum frequency effects were
observed at e.g. the wave lengths supposed to excite springing by sum frequency effects.
Figure 5.8 illustrates the sum frequency effect in the spectrum and time series for a wave
supposed to excite linear springing. The rise up of water in the bow is more rapid than
the decay (third order effects were also present). The sum frequency effect in the bow is
larger than observed for the wave elevation amidships.
5.2. WAVE AMPLIFICATION AND ADDED RESISTANCE IN WAVES 119
Non−dimensional "nonlinear" R.A.O. Non−dimensional "nonlinear" R.A.O.

5 6
Lin Lin
4.5 NL NL
HF HF
5
4 Analytic Analytic
3.5
4
3
1 a
REL1/ζa
REL /ζ
2.5 3
2
2
1.5
1
1
0.5
0 0
0 5 10 15 20 0 5 10 15 20
ωe,m [rad/s] ωe,m [rad/s]
Figure 5.7: Nondimensional measured relative motion and analytical wave amplification,
Eq.(2.5), in the bow of ballast 2 condition in 15kn (left) and cargo 2 condition in 13.2kn
(right).
−3 Spectra
x 10
0.2
1.5
0.15
REL1 [m]
SREL1(ωe)
1
0.1
0.5
0.05
0
84 85 86 87 88 89 5 10 15 20 25 30 35 40
−4
x 10
0.06 8
0.04
SREL3(ωe)
REL3 [m]
6
0.02
0 4
−0.02 2
−0.04
0
84 85 86 87 88 89 5 10 15 20 25 30 35 40
Time ωe [rad/s]
Figure 5.8: Unfiltered time series (left) and spectra (right) of the relative motion in the
bow (upper) and midship (lower) in ballast 2 condition, H = 2.1m, T = 4.2s and U = 15kn
(run1508).
5.2.2 Measured and predicted added resistance in head waves

Added resistance in waves is partly related to the relative motion in the bow and stern.
In Section 5.2.1 the measured amplification (relative motion) in short waves was found to
exceed the analytical estimate. If the measured exceedance compared to the analytical for-
mulation of the amplification was due to bulk water rather than water spray, the measured
added resistance would also be expected to exceed the analytical estimate.
The added resistance in waves was predicted by deducting the calm water resistance from
the total resistance. The measured calm water resistance is listed in Table 5.3. No wire
was applied to initiate turbulence for bow 2 and 3, and may explain the increased calm
water towing resistance compared to bow 1. The calm water towing resistance for bow
3 was higher partly due to larger wet area. It was expected that the calm water towing
resistance in cargo condition was higher than in ballast condition due to larger wet area,
but the opposite is observed for bow 1. The measured calm water towing resistance in
ballast 2 condition was larger than measured in small regular waves, and it was corrected
approximately based on the difference obtained from ballast 1 condition (0.2N ). Ballast
and cargo 3 condition did not have transverse lines. The effect of the transverse supporting
lines is indicated by comparing the ratio of ballast and cargo 3 condition with the other
two bow designs. The effect is significant.
Table 5.3: Measured calm water towing forces (CWTF). The numbers in parenthesis
indicate adjusted values to other speeds assuming a speed square relation. Mean towing
forces obtained in small amplitude regular waves are also displayed for comparison.
Load. cond. CWTF [N ] U [m/s] Comments
Ballast 1 55.4 / 55.6 1.30 run1220 calm water / run1010 in small waves
Cargo 1 20.8 (46.6) / 35.9 0.87 run1433 (adjusted to 1.30) / 1.15 run1301-3; small waves
Ballast 2 61.3 (58) / 58.4 1.30 run9813 (corrected) / run1500-2; small waves
Cargo 2 49.3 (63.0) / 51.4 1.15 run9864 (adjusted to 1.30) / run1700-2; small waves
Ballast 3 66.5 / 67.2 1.30 run9884 / run2323-5; small waves
Cargo 3 62.7 (80.1) / 65.6 1.15 run2222 (adjusted to 1.30) / run2100-1-3; small waves
Figure 5.9 shows the results of ballast and cargo 2 condition. The analytical estimates
show some resemblance to the mean value of the measurements, but the uncertainties are
considerable. Several factors contributed to the uncertainties.
1. Faltinsen (1990, pp. 146) illustrated that the added resistance in waves was sensitive
to the exact bow shape. The analytical formulation was based on a circular cylindrical
bow, while the actual bow shape was more elliptic, which will reduce the resistance.
2. Baba (1976) illustrated that the effect of breaking steady waves differed in ballast and
cargo condition, indicating that the interaction with the steady potential differed.
3. The added resistance in small short waves was small (e.g. 1N ) compared to the calm
water towing resistance (e.g. 50N ), and improper zero setting of the towing force
introduced a large uncertainty in the nondimensional added resistance in waves. E.g.,
1mm surge corresponded to 4N assuming a towing spring stiffness of 2000N/m at
bow and stern.
4. The same uncertainty as in point 3 was introduced in case improper zero setting
was carried out prior to calm water testing, used to obtain the calm water towing
resistance.
5. The calm water resistance was obtained at a different speed than considered in regular
wave tests for cargo 1 condition. The calm water resistance was adjusted by a U 2
5.2. WAVE AMPLIFICATION AND ADDED RESISTANCE IN WAVES 121
relation (speed square) to obtain a value to be used as input to the estimate of added
resistance in waves.
6. The transverse supporting lines contributed to the longitudinal forces when the model
moved, or when they were not perfectly transverse. They were skipped in ballast and
cargo 3 condition without loosing directional stability.
Added resistance in waves Added resistance in waves

11 10
Raw Raw
10 9
LSQR LSQR
9 Analytic Analytic
8
8
7
7
Raw/(ρgζaB /L)
Raw/(ρgζaB /L) 6
2 2
2 2
6
5
5
4
4
3
3
2 2
1 1
0 0
0 5 10 15 20 0 5 10 15 20
Figure 5.9: Nondimensional added resistance in waves for ballast (left) and cargo 2 con-
dition (right) in 15kn and 13.2kn.
The peak value of the nondimensional added resistance in waves due to the rigid body
response was about 8.5 for ballast 2 and 6 for cargo 2 condition as seen in Figure 5.9. The
assumed nondimensional value in the calculation of the speed reduction in Section 2.1.5
was 8.75. Considering the uncertainties, the discrepancies are considered acceptable.
Figure 5.10 shows the nondimensional added resistance in waves for bow 3 in ballast and
cargo condition. The agreement between the analytical solution and measurements for this
wedge bow with half apex angle of 360 was improved as seen in Figure 5.21. The rigid body
response was similar to the blunt bow with a nondimensional maximum of 10 in ballast
and 9 in cargo condition. The maximum may have been affected by wall interaction. Bow
3 had less added resistance in waves, accordingly, the involuntary speed reduction in short
waves will be less than compared to the blunt bows.
The measured relative motion exceeded the analytical estimate, but a similar difference
was not observed for the added resistance in waves. This fact suggests that the measured
exceedance of relative motion was not related to bulk water.
Added resistance in waves Added resistance in waves

12 12
Raw Raw
LSQR LSQR
Analytic Analytic
10 10
8 8
Raw/(ρgζaB /L)
Raw/(ρgζaB /L)
2 2
2 2
6 6
4 4
2 2
0 0
0 5 10 15 20 0 5 10 15 20
Figure 5.10: Nondimensional added resistance in waves for ballast (left) and cargo 3
condition (right) in 15 and 13.2kn.
5.3 Transfer functions

5.3.1 Linear and ’nonlinear’ transfer functions
The ballast 2 condition is considered in this section, while the other test results are pre-
sented in Appendix C.1. The ’nonlinear’ transfer function is compared to the linear and
the high frequency contribution. The linear transfer function refers to a response that
is proporsional to the wave amplitude, while the ’nonlinear’ transfer function may have
perturbations proporsional to the wave amplitude squared etc., but it is still made nondi-
mensional with respect to the linear wave amplitude (at the encounter frequency). The
high frequency contribution is mainly the result of the filtered vibration response. The
linear (Lin), ’nonlinear’ (NL) and high frequency (HF) part are illustrated for the heave
and pitch motion in 15kn in Figure 5.11. The high frequency contribution to the nonlinear
response is small, but some flexible response is observed in heave at the first and second
harmonics, due to the location of the motion sensors. In high and long waves, the ship mo-
tions were influenced only slightly by nonlinear forces. In heave, the apparently nonlinear
effects in short waves were due to poor resolution (1mm) of the measured heave motion
and possible disturbances.
The shear force transfer functions are presented in Figure 5.12, the bending moments in
Figure 5.15, the accelerations in Figure 5.16 and the pressures in Figure 5.17. In general,
the higher harmonics are visible, and the first, second and third order resonance are pro-
nounced. The higher harmonics have significant contribution from the high frequency
response, which occur between 19 and 22rad/s. The higher harmonics are also observed
outside resonance, e.g. for the shear force at cut 5 between 6 and 10rad/s. The second
harmonic relates to the hypothesis of second order contribution from bow reflection (hy-
pothesis 4). The higher order harmonics with high frequency vibration response relates

1
Lin Lin
0.9 NL 1 NL
HF HF
0.8
0.8
0.7
0.6
η5/(kζa)
0.6
3 a
η /ζ
0.5
0.4
0.4
0.3
0.2 0.2
0.1
0 0
0 5 10 15 20 0 5 10 15 20
Figure 5.11: ’Nonlinear’ transfer function of heave (left) and pitch (right) in ballast 2
condition. ’Nonlinear’ refers to the peak process of the total signal made nondimensional
as in linear theory.
mostly to the bow impact hypothesis. It should be noted that for e.g. the monochromtic
sum frequency effect (second order) the displayed ’nonlinear’ response refers to different
incident wave heights. Thereby, for the same frequency the ’nonlinear’ response amplitude
will vary.
The relative contribution of the springing in the shear forces in Figure 5.12 is higher in the
aft than in the fore ship, due to higher flexible response and lower rigid body response. At
the forward cross section between the first and second harmonic, the difference between
the linear and ’nonlinear’ response exceeds the high frequency contribution, indicating the
importance of the second harmonic from hypothesis 4. Figure 5.13 presents the time series
and spectra of the shear forces at the aft and forward cut in short regular waves. The
second and third harmonics do not coincide with the resonance frequencies, and they are
pronounced despite the small incident waves. The nonlinearities in the forward cut exist
even though the geometry at water line is close to vertical. The observation agrees with
the excitation forces measured in short waves by Moeyes (1976). He indicated a small
nonlinear behaviour in the bow of a tanker, but also at some wall-sided midbody sections.
The second order effect in the aft ship is also pronounced in Figure 5.13, but it is unclear
if this is caused by the inertia as an overall effect, or by the second order effects due to
nonlinear geometry in the stern part. From Figure 5.14, the pressure in the bow area has a
stronger second order effect than measured in the pressure at the stern. This indicates that
the second order contribution to the shear force in the stern part is at least not significantly
contributed by sum frequency effects from excitation pressure locally.
The relative importance of the linear and second order effect at resonance compared to the
rigid body response peak are displayed for the three quarter lengths in Figure 5.15. The
rigid body response at the forward quarter length is about 2/3 of the midship moment,

0.1
0.25
Lin Lin
0.09
NL NL
0.08 HF HF
0.2
0.07
VSF1/(ρgζaBL)
VSF2/(ρgζaBL)
0.06
0.15
0.05
0.04 0.1
0.03
0.02 0.05
0.01
0 0
0 5 10 15 20 0 5 10 15 20

0.1
0.25 Lin
Lin
0.09 NL
NL
HF
HF 0.08
0.2
0.07
VSF4/(ρgζaBL)
VSF5/(ρgζaBL)
0.06
0.15
0.05
0.1 0.04
0.03
0.05 0.02
0.01
0 0
0 5 10 15 20 0 5 10 15 20
Figure 5.12: ’Nonlinear’ transfer function of shear forces from aft (upper left) to fore
(lower right) in ballast 2 condition.
while the linear springing response is about 1/3 compared to the midship section. For the
aft quarter the ratio compared to amidships is apparently about 1/3 for both the rigid
body response peak and the springing peak. The relative importance of the 2-node spring-
ing is then indicated to be about equal at the aft and midship section, while it is lower at
the forward quarter length. This has to do with the envelope of the rigid body response
along the ship compared to the 2-node bending moment distribution. The former is due
to traveling wave loads passing through the ship, while the latter comes from a stationary
vibration mode.
The pressure from the rigid body response increases towards the bow, while the vibration
pressure decreases in Figure 5.17. The excitation pressure at the resonance is not signif-
icantly larger in the resonance area than outside, hence the measured pressures at linear
and sum frequency resonance were apparently a consequence of the vibration, rather than a
cause of it, and the sum of the radiation and restoring pressures appeared to be large. The
Spectra
0
400
−20
300
SFY5(ωe)
FY5 [N]
−40
200
−60
−80 100
−100 0
73 74 75 76 77 78 79 5 10 15 20 25 30 35 40 45 50
20 50
10 40
SFY1(ωe)
FY [N]
0 30
1
−10 20
−20 10
−30 0
73 74 75 76 77 78 79 5 10 15 20 25 30 35 40 45 50
Time [s] ωe [rad/s]
Figure 5.13: Time series and spectra of shear forces at forward (upper) and aft (lower)
cut in ballast 2 condition (run1550, 15kn, H = 2.1m, T = 5.09s).
10000
100
SFX3(ωe)
FX3 [N]
−100
0
60 61 62 63 64 65 10 4 15 20 25 30
x 10
400 6
P3 [N/m ]
2
SP3(ωe)
200 4
0 2
−200
0
60 61 62 63 64 65 10 15 20 25 30
100 4000
P13 [N/m ]
2
SP13(ωe)
−100 0
60 61 62 63 64 65 10 15 20 25 30
Time [s] ωe [rad/s]
Figure 5.14: Time series and spectra of moment at midship (row 1) pressure at stem
(row 2) and pressure below stern at centre line (row 3) in regular waves (run1550, 15kn,
H = 2.1m, T = 5.09s).
restoring pressure can be roughly estimated by ρg times the vertical acceleration divided
by the frequency squared in regular waves. This constitute only a part of the difference
between the total pressure in resonance and outside resonance, hence the dominating part
of the difference is due to radiation pressure, which in general is small, but was not small
in the presences of wave induced vibrations. The ’nonlinear’ and linear pressure differ in
Lin Lin
0.06 0.06
NL NL
HF HF
0.05 0.05
VBM2/(ρgζaBL )
VBM4/(ρgζaBL2)
0.04 0.04
2
0.03 0.03
0.02 0.02
0.01 0.01
0 0
0 5 10 15 20 0 5 10 15 20
Non−dimensional "nonlinear" R.A.O.
0.06 Lin
NL
HF
0.05
VBM3/(ρgζaBL )
0.04
2
0.03
0.02
0.01
0
0 5 10 15 20
ωe,m [rad/s]
Figure 5.15: ’Nonlinear’ transfer function of bending moments in ballast 2 condition.

Upper left= Aft, upper right= Forward and lower= Midship.
general at high frequencies and low pressures (outside resonance). In between the linear
and second order resonance, the factor is close to two. The second order excitation pressure
from reflection contributed (hypothesis 4), but the low pressures were also affected some-
what by poor resolution. Figure 5.18 presents the time series and spectra of the pressure
in the stem and below the bow. The pressures are not sinusoidal as the incident waves.
Higher harmonics are present and the fourth harmonic coincides closely with the 3-node
resonance in this case. The second, but also the third harmonics, are significant. The
non-sinusoidal behaviour of the stem pressure seems to be apparent also in the low bottom
pressures, even though the quality of the bottom pressure is poor.
A method employed to separate the excitation pressure from the remaining pressure failed.
It was based on evaluation of pressure from decay tests in calm water, and ’deducting’ this
from the total pressure in the regular wave tests. The remaining part was expected to be
the excitation pressure, but the method was not found reliable.

35 35
Lin Lin
NL
NL
HF
30 30 HF
25 25
ACCZ4L/(gζa)
ACCZ8L/(gζa)
20 20
15 15
10 10
5 5
0 0
0 5 10 15 20 0 5 10 15 20
Figure 5.16: ’Nonlinear’ transfer function of vertical acceleration at midship (left) and
bow (right) in ballast 2 condition.
The vertical acceleration in Figure 5.16 displays the relative importance of the acceleration
due to resonance and due to the rigid body motions. In this case, actually the relative
importance is more important amidships, where there is the contribution from pitch is
close to zero.
The behaviour of the linear and second order resonance versus speed were investigated. The
midship bending moment is displayed in Figure 5.19. The second order resonance remains
high for a considerable change in the speed, while the linear part is more sensitive to speed.
The second order resonance is still significant at zero speed. The quality of the vibration
response was, however, observed to be poor, i.e. the vibration was not very regular at low
speeds due to disturbances and reduced wave quality. The reduction in the linear part was
partly due to deficient wave quality, and only a moderate speed reduction was possible. It
was difficult to measure a quantity that proved the importance of the hypothesis 1 related
to the interaction of the steady wave field. The results, however, suggest that the linear
springing is more sensitive to speed than the second order springing excitation. This may
be caused by two effects: A) The second order excitation may be more localised to the bow
region and not affected by oscillations along the hull, and B) the linear springing excitation
may be more affected by the steady wave interaction. The hump-hollow behaviour versus
speed was indicated by Hoffman and van Hoof (1976) in high speeds. This was not evident
here, in particular not for the sum frequency effect, suggesting again that it is a local effect.
Few repetitions, wide frequency spreading of the regular waves, inaccurate estimates of the
resonance peaks at varying speed, and low speed introduce uncertainties. The sensitivity
to speed in these model tests contradicts the findings by Slocum and Troesch (1982), who
indicated that the linear springing was less sensitive to speed. Still, due to the pronounced
difference in the current model tests, the sum frequency effect is still considered to be less
sensitive to speed compared to linear springing in this case.

2.5 2.5
Lin Lin
NL NL
HF HF
2 2
1.5 1.5
P7/(ρgζ )
P2/(ρgζa)
a
1 1
0.5 0.5
0 0
0 5 10 15 20 0 5 10 15 20

2.5 2.5
Lin Lin
NL NL
HF HF
2 2
1.5 1.5
P6/(ρgζ )
P1/(ρgζa)
a
1 1
0.5 0.5
0 0
0 5 10 15 20 0 5 10 15 20
Figure 5.17: ’Nonlinear’ transfer function of bottom CL pressures in ballast 2 condition.

Upper left=0.92Lpp , upper right=0.95Lpp , lower left=0.96Lpp , lower right=0.97Lpp.
4 Spectra
x 10
10
1600
8
1400
P3 [N/m ]
2
SP3(ωe)
6
1200
4
1000
2
800
0
95 96 97 98 99 100 5 10 15 20 25 30 35 40 45 50
100 300
80
P1 [N/m2]
SP1(ωe)
60 200
40
100
20
0
0
95 96 97 98 99 100 5 10 15 20 25 30 35 40 45 50
Time ωe [rad/s]
Figure 5.18: Time series and spectra of stem (upper) and bottom (lower) CL pressure
of ballast 2 condition (run1551, 15kn, H = 2.1m, T = 5.75s, outside resonance). Stem
pressure is taken at FP, 10m above BL, while bottom pressure is taken at 0.97Lpp outside
resonance.
Linear resonance, non−dimensional

0.06
Lin
VBM3/(ρgζaBL2)
NL
0.04 HF
0.02
0
0 0.2 0.4 0.6 0.8 1 1.2 1.4
2nd order resonance, non−dimensional

0.06
Lin
VBM3/(ρgζaBL2)
NL
0.04 HF
0.02
0
0 0.2 0.4 0.6 0.8 1 1.2 1.4
Speed [m/s]
Figure 5.19: Nondimensional bending moment amidships in ballast 2 condition at linear

(upper) and second order (lower) resonance versus speed. H = 2.1m is used for the second
order excitation.
5.3.2 Measured second order transfer functions

Many wave frequency combinations can cause sum frequency excitation. The sum fre-
quency resonance response as a function of the frequency difference of the bichromatic
waves was investigated. Results for ballast and cargo 2 condition are presented in this
section, while result plots for other loading conditions are presented in Appendix C.2.
Figure 5.20 presents the nondimensional second order transfer function of the bending mo-
ment at the midship section, |H(ω1,e , ω2,e )|. The sum frequency responses from the least
square fit and from the standard deviation of the filtered time series are similar. The
peak process of the raw time series, which includes the encounter response, gives higher
estimates. The maximum encounter and sum frequency response appear to add up to the
maximum nonlinear response. The sum frequency response of the filtered time series is
considered in the following.
ω1 TS ω1 TS
ω2 TS ω2 TS
ω1+ω2 TS ω1+ω2 TS
ω2+ω1 TS ω2+ω1 TS
ω1+ω2 LSQR ω1+ω2 LSQR
ω2+ω1 LSQR ω2+ω1 LSQR
VBM3/(ρgBL ζ1ζ2sqrt(k 1k2))
VBM3/(ρgBL2ζ1ζ2sqrt(k 1k2))
1.5 0.8
ω1+ω2 NL ω1+ω2 NL
ω2+ω1 NL 0.6 ω2+ω1 NL
1
2
0.4
14
0.5 12
12 0.2
10
10
0 0 8
14 8
12 12 6
10 10
8 6 ω1,e [rad/s] 8 ω1,e [rad/s]
6 6
ω2,e [rad/s] ω2,e [rad/s]
Figure 5.20: Nondimensional second order transfer function for ballast 2 condition (left)
and cargo 2 condition (right) in 15 and 13.2kn. The methods to obtain the various re-
sponses, TS (time series), LSQR (least square fit) and NL (nonlinear), are outlined in
Section 4.6.3.
The nondimensional response was larger in ballast 2 than in cargo 2 condition, but the
damping differed. The damping ratio in ballast 2 condition was 1%. By correcting the
response level of the bichromatic waves in cargo 2 condition, corresponding to a reduction
in the damping ratio from about 1.8% to 1%, the nondimensional level increased from 0.1
to 0.2, which was still lower than in ballast 2 condition. Similarly, the monochromatic
peak in cargo 2 condition corresponded to a damping ratio of 2.5%, and adjusted down to
1%, the nondimensional value became 2.0, which was higher than in ballast 2 condition.
The bichromatic response is apparently constant in Figure 5.20. The trend of |H(ω1,e , ω2,e )|
away from the diagonal, |H(ω1,e , ω1,e )|, differs slightly in ballast and cargo 2 condition. For
a small frequency difference of the bichromatic waves, the springing response became beat
shaped when the model ran into the periodic wave clusters. The nondimensional response
was based on the time series covering both the beats and the ’dead zones’. This behaviour
in combination with the change of damping at different vibration level may explain the
difference in ballast and cargo 2 condition close to the monochromatic peak.
The nondimensional second order bending moment for the monochromatic and bichromatic
waves are compared in Table 5.4. Since the damping varied, a correction has been made to
the measured response, which was corrected to a damping ratio of 1% for comparison with
ballast 2 condition. The damping, taken from Table 5.1, was estimated at different response
levels in Figure 5.20, Figure C.5 and Figure C.6. Typical response levels are indicated
together with the corresponding damping and the corrected nondimensional second order
bending moment in Table 5.4. The nondimensional second order transfer function should be
basically independent of the wave heights squared, but for reference, the wave heights of the
bichromatic waves are included in parenthesis in Table 5.4. For the monochromatic peak
the wave heights in most cases was twice the individual wave heights of the bichromatic
waves. The following observations are made:
? The monochromatic level is higher in ballast than in cargo condition except for bow 2.
? The difference of the monochromatic response for bow 2 indicates that the draft does not
play an important role, but the draft appears to be more important for the bichromatic
waves.
? The difference in the monochromatic response for bow 1 and 2 may be caused by the
submerged bulb, which may reduce the sum frequency bottom pressure.
? The whipping contribution from the stem flare in cargo 1 condition appears to be small
(when compared to cargo 2 condition without stem flare).
? The monochromatic response of bow 3 is smaller due to reduced reflection, but the effect
does not vanish. This may partly be explained by the fact that the steady wave pattern
around the pointed bow was more elliptic, and because of interaction with the steady
pattern still some waves were reflected back as illustrated in Figure 5.21.
? In general, the bichromatic response is higher in ballast than in cargo condition. The
difference is small except for bow 2.
? The bichromatic response for bow 1, 2 and 3 in ballast or cargo condition displays only
a moderate difference, except in ballast 2 condition.
Uncertainties were introduced due to the nonlinear damping, nonlinear stiffness in ballast
1 condition and inaccurate resonance conditions away from the diagonal. Second order
incident waves may also contribute, but this effect should be small as the incident waves
are not very steep.
Table 5.4: Nondimensional second order bending moment for different bow designs for
bichromatic (B) and monochromatic (M) waves. The numbers in parenthesis for the bichro-
matic waves refer to the wave heights in full scale.
Load. cond. B Resp. [N] B δ [%] B V BM M Resp. [N] M δ [%] M V BM
Bal. 1 250 (2x0.7m) 0.7 0.4 400-550 1.0 1.2
Cargo 1 150 (2x1.0m) 1.0 0.2 300-400 1.6 1.0
Bal. 2 600-1000 (2x1.05m) 1.0 1.0 1500 1.0 1.6
Cargo 2 100 (2x1.05m) 1.8 0.2 400-700 2.5 2.0
Bal. 3 700-800 (2x1.75m) 1.2 0.4 800-1600 1.0 0.9
Cargo 3 175 (2x1.75m) 3.3 0.3 500-700 1.9 0.6
Figure 5.21: Pictures of bow reflection (including radiation) for bow 3. The half apex
angle is roughly 360 .
The constant level of the sum frequency response away from the diagonal was earlier
identified by Slocum (1983). The current monochromatic peak deviates from his finding.
The ratio of the monochromatic to bichromatic response is (only) 1.6 in ballast 2 condition,
but this ratio may depend on the ship design and sea conditions.
5.4 Wave and high frequency fatigue damage

Cumulative fatigue damage for longitudinal details in deck was calculated by Rainflow
counting on the total and wave frequency pass filtered signal, which gave the total and
wave frequency (WF) damage. The latter is also referred to as wave damage. The dif-
ference constituted the high frequency damage (HF) also referred to as vibration damage.
A SCF of 2.0 was assumed with a two slope SN-curve for welded details protected from
corrosive environment. Reduction in the fatigue damage due to the mean stress effects
was disregarded. The vibration level was not corrected for the varying damping ratios
measured in the six combinations of bow shapes and loading conditions.
5.4. WAVE AND HIGH FREQUENCY FATIGUE DAMAGE 133
5.4.1 Base case - vessel in selected sea states at full speed

The vessel was run in head sea at 15 and 13.2kn in ballast and cargo condition. The wave
and vibration damage were determined from twelve sea states, shown in Figure 5.22 and
Table B.10.
15
14 HS
13 TP
12
11
10
HS [m], TP [s]
9
8
7
6
5
4
3
2
1
0
1 2 3 4 5 6 7 8 9 10 11 12
Figure 5.22: Numbering of the twelve sea states (HS =left and TP =right) tested at full
speed and used in the wave calibration.
Figure 5.23 displays the relative importance of the midship wave and vibration damage,
which were normalized by the total damage. Their sum is normalized as one for each sea
state. The vibration damage contributes more in ballast than in cargo condition. The
results differ only moderately with respect to bow geometry. Bow 3 has less contribution
from the vibration in sea states with short and low waves, while in more severe sea states
the difference is small. The reflection causing sum frequency excitation is reduced for the
sharp pointed bow, while in more severe sea states whipping may contribute. The damp-
ing behaviour may also reduce the springing response for bow 3. Despite the geometry
of bow 3 differs considerably from the blunt bows, no drastic change is observed in the
relative magnitude of the overall vibration damage. In general, for a given wave height
the vibration contribution reduces for increasing periods, while at a given peak period
the vibration contribution increases for increasing wave heights. The latter indicates the
increasing importance of nonlinear forces.
The fatigue damage ratios at the two quarter lengths are compared to the midship section
in Figure 5.24 for ballast and cargo 2 condition. The aft and forward quarter length dis-
play the same trend as the midship section, but the relative importance is slightly changed,
but not necessarily reduced at the aft quarter. This is consistent with observations and
comments made to Figure 5.15 in Section 5.3.1.
1 1
0.9 0.9
0.8 0.8
0.7 0.7
0.6 0.6
Ratio [−]
Ratio [−]
0.5 0.5
0.4 0.4
0.3 0.3
0.2 0.2
0.1 0.1
HF HF
WF WF
0 0
1 2 3 4 5 6 7 8 9 10 11 12 1 2 3 4 5 6 7 8 9 10 11 12
1 1
0.9 0.9
0.8 0.8
0.7 0.7
0.6 0.6
Ratio [−]
Ratio [−]
0.5 0.5
0.4 0.4
0.3 0.3
0.2 0.2
0.1 0.1
HF HF
WF WF
0 0
1 2 3 4 5 6 7 8 9 10 11 12 1 2 3 4 5 6 7 8 9 10 11 12
1 1
0.9 0.9
0.8 0.8
0.7 0.7
0.6 0.6
Ratio [−]
Ratio [−]
0.5 0.5
0.4 0.4
0.3 0.3
0.2 0.2
0.1 0.1
HF HF
WF WF
0 0
1 2 3 4 5 6 7 8 9 10 11 12 1 2 3 4 5 6 7 8 9 10 11 12
Figure 5.23: Ratio of WF (top) and HF (bottom) damage to total fatigue damage for
midship section at full speed. Left= Ballast, right= Cargo. Row 1= Bow 1, row 2= Bow
2, row 3= Bow 3.
1 1
RatioL/4 [−]
RatioL/4 [−]
0.5 0.5
HF HF
WF WF
0 0
1 2 3 4 5 6 7 8 9 10 11 12 1 2 3 4 5 6 7 8 9 10 11 12
1 1
RatioL/2 [−]
RatioL/2 [−]
0.5 0.5
HF HF
WF WF
0 0
1 2 3 4 5 6 7 8 9 10 11 12 1 2 3 4 5 6 7 8 9 10 11 12
1 1
Ratio3L/4 [−]
Ratio3L/4 [−]
0.5 0.5
HF HF
WF WF
0 0
1 2 3 4 5 6 7 8 9 10 11 12 1 2 3 4 5 6 7 8 9 10 11 12
Figure 5.24: Ratio of WF (top) and HF (bottom) damage to total fatigue damage at aft
(row 1), midship (row 2) and forward (row 3) quarter length at full speed. Left= Ballast
2, right= Cargo 2 condition.
The absolute value of the half hour fatigue damages at the three cross sections of ballast
and cargo 2 condition are presented in Figure 5.25. The wave damage increases moderately
from ballast to cargo condition, except at the aft quarter length which displays a higher
increase. Even though the vibration damage is smaller in cargo than in ballast condition, it
is still pronounced in the more severe sea states in cargo condition. Moreover, the vibration
and wave damage increase rapidly with increasing wave height.
−4 Fatigue damage, Di, for each 38 minute sea state −4 Fatigue damage, Di, for each 39 minute sea state
x 10 x 10
HF HF
WF WF
2 2
DL/4
DL/4
1 1
0 0
1 2 3 4 5 6 7 8 9 10 11 12 1 2 3 4 5 6 7 8 9 10 11 12
−3 −3
x 10 x 10
1 1
HF HF
WF WF
DL/2
DL/2
0.5 0.5
0 0
1 2 3 4 5 6 7 8 9 10 11 12 1 2 3 4 5 6 7 8 9 10 11 12
−4 −4
x 10 x 10
3 3
HF HF
2 WF 2 WF
D3L/4
D3L/4
1 1
0 0
1 2 3 4 5 6 7 8 9 10 11 12 1 2 3 4 5 6 7 8 9 10 11 12
Figure 5.25: HF (left) and WF (right) fatigue damage at full speed. Row 1= aft, row 2=
midship, row 3= forward quarter length; Left = Ballast 2, right= Cargo 2 condition.
The fatigue damage of the individual sea states was combined with the probability of oc-
currence in different trades. The fatigue damage in each sea state, assuming 10 years head
sea North Atlantic trade in ballast or cargo condition, is presented in Figure 5.26 for the
three bows. The selected sea states have a total probability of occurrence of 26.8%. The 5
and 7m significant wave heights contribute more than the 3 and 9m waves.
The total fatigue damage from the twelve sea states is summarized in Table 5.5. The
vibration damage in head sea at full speed constitutes roughly 1/4 of the total damage in
cargo condition, and 2/3 in ballast condition. Ballast 2 condition has higher fatigue damage
from the vibration, which may be explained by smaller damping and larger springing
excitation. The wave damages in ballast and cargo condition are insensitive to the bow
design accounting for the different ship length. The increase in wave damage from ballast
to cargo condition is about 70%.
Table 5.5: Cumulative fatigue damage for the vessel during 10 years head sea North
Atlantic trade covering 27% of the sea states. The fatigue damages are based on full speed
and SCF of 2.0 for coated details in deck at midship section.
Fatigue damage Cargo 1 Ballast 1 Cargo 2 Ballast 2 Cargo 3 Ballast 3
HF 0.5849 2.0101 0.6657 2.9617 0.5754 1.7744
WF 1.7039 1.0107 1.7029 1.1081 1.9457 1.0640
Total 2.2888 3.0208 2.3686 4.0698 2.5211 2.8384
HF contribution [%] 26 66 28 73 23 63
A first estimate of the fatigue life in North Atlantic trade considering bow 2 gave 1.4 years
accounting for the probability of the sea states and 33% harbour time. Head sea was
assumed in ballast condition for 33.5% of the time, and following sea was assumed in cargo
condition the remaining time. In cargo condition the vibration damage was disregarded,
and the wave damage in head sea was assumed to represent the wave damage in following
sea. The contribution from the vibration damage constituted 51%. The North Atlantic
tanker trade with prevailing following seas in cargo condition plus accounting for the mean
stress effect in deck reduces the total damage and the vibration contribution compared to
the iron ore trade.
5.4.2 Effect of speed reduction and wave environment

At full speed in the North Atlantic iron ore trade, the vibration damage was dominating.
The realistic speed depends on the sea state, and the speed was estimated in Section 3.2.
The involuntary speed reduction was considerable, and the speeds in Table 3.4 has been
assumed in the following.
The fatigue damage at midship section in 19 North Atlantic head sea states with and
without speed reduction is presented in Figure 5.27 based on ballast 2 condition (left plot).
The vibration damage is reduced considerably in the more severe sea states due to the
speed reduction. Only a small change is evident in the wave damage, which is influenced
0.5 0.5
HF HF
WF WF
0.45 0.45
0.4 0.4
0.35 0.35
0.3 0.3
DL/2
DL/2
0.25 0.25
0.2 0.2
0.15 0.15
0.1 0.1
0.05 0.05
0 0
1 2 3 4 5 6 7 8 9 10 11 12 1 2 3 4 5 6 7 8 9 10 11 12
0.5 0.5
HF HF
WF WF
0.45 0.45
0.4 0.4
0.35 0.35
0.3 0.3
DL/2
DL/2
0.25 0.25
0.2 0.2
0.15 0.15
0.1 0.1
0.05 0.05
0 0
1 2 3 4 5 6 7 8 9 10 11 12 1 2 3 4 5 6 7 8 9 10 11 12
0.5 0.5
HF HF
WF WF
0.45 0.45
0.4 0.4
0.35 0.35
0.3 0.3
DL/2
DL/2
0.25 0.25
0.2 0.2
0.15 0.15
0.1 0.1
0.05 0.05
0 0
1 2 3 4 5 6 7 8 9 10 11 12 1 2 3 4 5 6 7 8 9 10 11 12
Figure 5.26: HF (left) and WF (right) fatigue damage at midship section assuming 10
years head sea North Atlantic trade. Left column= Ballast, right column= Cargo condition,
row 1= Bow 1, row 2= Bow 2, row 3= Bow 3.
by wall effects at small speeds and for long wave periods. The small change represents,
however, a reduction as expected, since the number of encountered cycles per time unit is
reduced. Therefore, the overall effect of the wall effects does not appear to be significant.
The significant wave height that dominates the contribution to fatigue damage is reduced
to 5m, and the vibration damage is still pronounced.
0.5 0.5
HF HF
L/2
Full U, DL/2
0.4 0.4
WF WF
Full U, D
0.3 0.3
0.2 0.2
0.1 0.1
0 0
0 5 10 15 20 1 2 3 4 5 6 7 8 9 10 11 12 13 14
0.5 0.5
L/2
Reduced U, DL/2
Reduced U, D
0.4 0.4
0.3 0.3
0.2 0.2
0.1 0.1
0 0
0 5 10 15 20 1 2 3 4 5 6 7 8 9 10 11 12 13 14
HS HS
15 15
HS [m], TP [s]
HS [m], TP [s]
TP TP
10 10
5 5
0 0
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 1 2 3 4 5 6 7 8 9 10 11 12 13 14
Figure 5.27: Pairs of HF (left) and WF (right) fatigue damage in head sea North Atlantic
trade for midship section of ballast 2 (left) and cargo 2 (right) condition. Row 1= Full speed,
row 2= Reduced speed, row 3= Sea states. U = 10, 5, 0kn in 5, 7, 9m HS .
The 9m sea state in ballast 2 condition was also tested at 15, 10, 5 and 0kn, resulting in
a standard deviation HF stress of 38.0, 20.1, 9.1 and 3.6MPa, respectively. The follow-
ing curve, 3.6 + 0.26 · U 1.8 is a good fit, indicating that the vibration increases with the
speed to the power of slightly less than 2. Considering the uncertainties, the slope is in
some agreement with the factor of 2, which has been indicated in the literature, (Bell and
Taylor 1968), but the vibration is still present at zero speed in agreement with (Gran 1974).
The effect of the speed reduction are presented in the right plot of Figure 5.27 for cargo 2
condition in 14 sea states. The same trends are observed as for ballast 2 condition, but the
relative reduction in the vibration damage for the dominating sea states of 5m significant
wave height appear as smaller than in ballast condition. The change in speed is however
smaller, from 13.2 to 10kn, and the vibration is small to begin with.
Table 5.6 presents a summary of the wave, vibration and total fatigue damage at full and
reduced speed in the six combinations of loading conditions and bow shapes. Both the
North Atlantic and World Wide trade are considered. The number of sea states and their
probability of occurrence vary. The calculations were based on the same assumptions as
in the previous Section 5.4.1. Several observations are made
? The speed reduction reduced the vibration contribution in the order of 25 to 50%.
? The contribution from vibration damage was about 15% in cargo and 40% in ballast
condition in North Atlantic head sea trade.
? The fatigue life for bow 2 with realistic speed, estimated in the same way as in Sec-
tion 5.4.1 for North Atlantic trade, increased from 1.4 to 3.6 years. The vibration con-
tribution reduced from 51 to 19%.
? The tested sea states in ballast 3 condition had twice the probability of occurrence
compared to ballast 1 condition, still the vibration contribution was not significantly
altered.
? The total fatigue damage was reduced significantly in World Wide trade compared to
North Atlantic trade. By considering bow 2 and the same head sea conditions as in
the North Atlantic, the fatigue life was increased from 3.6 to 6.6 years in the World
Wide trade (a different heading profile in the World Wide trade may further increase
the fatigue life).
? The contribution from vibration damage increased in World Wide trade compared to
North Atlantic trade. Considering bow 2, the vibration contribution increased from 19
to 25%.
The fatigue damage, fatigue life and vibration contribution in the North Atlantic trade
were derived in the following way (for bow 2 with no HF damage in following sea in cargo
condition):
1.8953 · 2 · 0.335 1.451 · 2 · 0.335
D= + = 5.58
0.526 0.307
20
= 3.6 years
D
0.8185
DHF · 2 · 0.335
100 = 0.526 100 = 19% (5.1)
D D
The cut off frequency for noise in full scale was set at 2.0Hz, which means that the 2-, 3-
and 4-node modes were included in the estimate of the HF damage. To estimate how much
the 3- and 4-node mode affected the HF damage, the cut of frequency was taken at 1.0Hz
in ballast 2 condition. This reduced the HF damage with less than 1% amidships and less
than 6% at the quarter lengths at full speed. The effect at reduced speed was slightly
smaller. It is seemingly not necessary to consider the higher order modes in the fatigue
assessment of blunt ships when it comes to wave induced vibrations. If confirmed also for
extreme loads, the sampling frequency and necessary storage capacity can be reduced.
5.4.3 Effect of changed trim

Lopes and Troymann (1989) reported a reduction in the springing stress of 10-15% by
increasing the amount of ballast water in the fore peak tank. The sea states were possibly
not stationary during this operation.
The effect of changed trim was explored in ballast 2 condition at full speed. 100kg was
placed in the bow, resulting in an increased draft of 1.6m at FP, including increased mean
Table 5.6: HF, WF and total fatigue damage in North Atlantic and World Wide trade for
all combinations at full and reduced speed. The probability of occurrence of the tested sea
states, and the contribution from the vibration damage are presented. 10 years of head sea
trade was assumed in each combination. Extrapolation of results are based on direct time
extrapolation of only the tested sea states.
North Atlantic World Wide
Full Reduced Probability Full Reduced Probability
Combination speed speed [%] speed speed [%]
Ballast 1 HF 2.0101 26.8 0.9180 18.3
Ballast 1 WF 1.0107 26.8 0.4147 18.3
Ballast 1 Total 3.0208 26.8 1.3327 18.3
HF contribution[%] 67 69
Cargo 1 HF 0.6003 0.2873 30.7 0.2626 0.1584 24.5
Cargo 1 WF 1.7100 1.4329 30.7 0.6851 0.6103 24.5
Cargo 1 Total 2.3103 1.7202 30.7 0.9477 0.7687 24.5
HF contribution[%] 26 17 28 21
Ballast 2 HF 3.1095 0.8185 52.6 1.6042 0.6654 59.3
Ballast 2 WF 1.1277 1.0769 52.6 0.4977 0.5059 59.3
Ballast 2 Total 4.2371 1.8953 52.6 2.1018 1.1713 59.3
Cargo 2 HF 0.6811 0.2795 30.7 0.2830 0.1500 24.5
Cargo 2 WF 1.7076 1.4510 30.7 0.6726 0.6175 24.5
Cargo 2 Total 2.3887 1.7305 30.7 0.9556 0.7675 24.5
Ballast 3 HF 2.1585 0.9952 69.1/69.7 0.9412 0.5324 59.2/58.2
Ballast 3 WF 1.4599 2.1577 69.1/69.7 0.6407 0.9224 59.2/58.2
Ballast 3 Total 3.6184 3.1530 69.1/69.7 1.5819 1.4548 59.2/58.2
Cargo 3 HF 0.5791 0.2582 30.7 0.2205 0.1253 24.5
Cargo 3 WF 1.9493 1.8211 30.7 0.7765 0.7846 24.5
Cargo 3 Total 2.5283 2.0793 30.7 0.9970 0.9099 24.5
draft of about 0.35m. The change was expected to increase the flare forces and reduce the
bottom excitation, and the change in the vibration would indicate the dominating source.
Four sea states were considered, corresponding to the steepest sea states tested at 2, 3, 4
and 5m significant wave height.
Figure 5.28 shows the ratio of the fatigue damage from the original draft to the increased
draft at FP. The WF and HF damage ratios are displayed in pairs. In general, the ratios
are above one with one exception, which refers to the smallest sea state. The smallest sea
state, HS = 2m and TZ = 5s, with increased bow draft was run soon after a larger sea
state, HS = 5m and TZ = 8s, and the WF damage became affected by swell from the
previous sea state.
4
3
HF
DL/4
2
1 WF
0
1 2 3 4
4
3
DL/2
2
1
0
1 2 3 4
4
3
D3L/4
2
1
0
1 2 3 4
H [m], T [s]
12
10
P
8 Hs
6
4 Tp
2
S
0
1 2 3 4
Figure 5.28: WF (right) fatigue damage ratio of original to increased forward draft and
HF (left) fatigue damage ratio, D, of original to increased forward draft in ballast 2 con-
dition at aft (L/4), midship (L/2) and forward (3L/4) cross section. 4 sea states were
considered as indicated on the abscissa axis: 1) HS = 2m, TZ = 5s, 2) HS = 3m, TZ = 6s,
3) HS = 4m, TZ = 7s and 4) HS = 5m, TZ = 8s.
The vibration damage ratio indicated that the bottom excitation reduced more than the
increase of the flare force for increased forward draft. The WF damage was also reduced
for increased draft/forward trim. To illustrate their relative magnitudes, the increase of
the flare forces in the highest sea state of HS = 5m and TP = 10.9s was estimated ac-
cording to the simplified analysis in section 3.1. By increasing the draft at FP with 1.6m,
the standard deviation nominal stress in deck of the vibration response increased from
0.8M P a to 1.4MP a, while the measured stress reduced from 10.8MP a to 7.3MP a. The
reduction in the springing stress was therefore significant, and the whipping contribution
was relatively small in this sea state (and the simplified calculations certainly lack the
important excitation source). The measured vibration stresses from the original and in-
creased forward draft are presented in Figure 5.29. The whipping events also appeared to
be reduced, indicating that the excitation sources were located at the bottom (bilge) flare
rather than at the stem/bow flare.
Even though only a few sea states were considered, the finding suggests that changing
the operational condition slightly may be effective in order to reduce unacceptable fatigue
loading of such ships designs. Fuel consumption is another matter.
40 40
30 30
20 20
10 10
σ [MPa]
σ [MPa]
0 0
−10 −10
−20 −20
−30 −30
−40 −40
100 200 300 400 500 600 700 800 900 100 200 300 400 500 600 700 800 900
Time [s] Time [s]
Figure 5.29: Nominal vibration stress in deck amidships for original (left, run1600) and
increased draft at FP (right, run1660). HS = 5m, TP = 10.9s and U = 15kn.
5.4.4 Wave conditions along the tank and the disturbance from
the vessel
The waves in irregular sea states were calibrated as described in Appendix B.5.4. The pres-
ence of the large vessel caused scattering, and the continuous runs performed in irregular
seas implied disturbances and residual waves in the subsequent runs. The effect of the
disturbance was estimated by comparing the measured waves during the model tests with
the measured waves during the wave calibration. The difference was expressed in terms
of the wave spectrum parameters in Eq.(2.78), such as the significant wave height, peak
period, peakness-, shape- and tail factor.
Table 5.7 presents the results from ballast 2 condition, which was tested in 33 sea states
shown in Table B.12. The spectrum parameters were determined for all sea states and
three wave probes. The mean and 95% confidence interval of the mean were presented
as the ratio of the measured to the requested value for both the raw and the smoothed
spectra. Values were derived both from a nonlinear fit to the spectra and from spectral
moments. Comparison was made with wave calibration results in Table B.11, which was
derived from the twelve sea states in Table B.10.
The wave height of the wave probe in front of the vessel, WP3, in Table 5.7 is increased by
10%. It is about 6-7% above the requested value, while the peak period is still as requested.
The peakness factor is as requested, and the uncertainty is still significant. The tail factor,
n, is increased slightly compared to the wave calibration tests, while the shape factor, m,
is somewhat reduced below the requested value.
It was observed that the vibration response had a tendency to increase towards the wave
maker, and moreover to be higher for the first run than for subsequent runs in each sea
state, indicating that the 3D wave disturbance in subsequent runs actually reduced the
Table 5.7: JONSWAP parameters from tested sea states in ballast 2 condition with 95%
confidence interval of the mean value. Values were obtained from nonlinear fit to raw and
smoothed (sm.) wave spectra and from spectral moments. Results are presented as the ratio
of the measured to requested value.
Parameter Mean (raw) 95% mean (raw) Mean (sm.) 95% mean (sm.)
HS fit WP3 1.063 0.022 1.067 0.024
HS moment WP3 1.072 0.022
HS fit WP1 1.125 0.022 1.132 0.024
TP fit WP3 0.994 0.007 0.997 0.007
TP fit WP1 0.979 0.008 0.977 0.007
γ fit WP3 1.017 0.094 0.733 0.055
γ fit WP1 0.937 0.086 0.657 0.042
n fit WP3 1.032 0.008
m fit WP3 0.973 0.006
n fit WP1 0.995 0.007
m fit WP1 0.988 0.004
vibration level. This is illustrated in the merged HF response for 3 runs in the first plot
of Figure C.16. The observation indicated a reduced energy of short waves along the tank,
and reduced quality in continuous runs. The fatigue damage at midship section for ballast
2 condition was determined based on the runs in the first and second half of the tank, the
second half being closest to the wave maker. The WF and HF damage from the second
half divided by the first half of the tank are displayed in Figure 5.30 for the sea states at
full speed. The first two sea states have a HF ratio of more than 10, and the HF damage
close to the wave maker is higher in 18 of 19 sea states. The HF damage in ballast 2
condition at full speed in North Atlantic head sea trade increased by 12% from 3.11 in
Table 5.6 to 3.49 based on the second half of the tank, confirming that the HF damage was
underestimated based on runs along the whole tank length. The linear springing response
was the one being most affected and reduced due to lack of stationarity along the tank for
the high frequencies.
In Appendix C.3 the JONSWAP parameters obtained from the other loading conditions
and bow shapes are presented without any significant new findings. In general, the sea
states were acceptable, except for the linear springing excitation.
5.4.5 Extrapolation of results by the response surface method.

The fatigue damage from the individual sea states was extrapolated to cover the whole
scatter diagram according to the procedures outlined in Section 2.8.1. No correction for
varying damping was made.
3
HF
Ratio of DL/2 [−]

WF
2
0
0 5 10 15 20
15 HS
TP
H [m], T [s]
10
P
5
S
0
0 5 10 15 20
Sea state number
Figure 5.30: The midship HF (left) and WF (right) fatigue damage ratio between the
second half and the first half of the tank for ballast 2 condition in sea states at full speed.
Polynomial expressions were fitted to the hourly WF and HF damage from the tested sea
states. The fitted fatigue damage of the individual sea states were combined with the
probability of occurrence from the scatter diagram. The results are shown in Figure 5.31
for North Atlantic and Figure 5.32 for World Wide head sea wave environment for ballast
3 condition at realistic speeds. The dominating sea states (peak) of the HF damage cover
a smaller number of sea states than the peak of the WF damage. The peak of the HF
damage in North Atlantic corresponds to significant wave heights of 4-5m with zero up-
crossing periods of 9-10s. The peak of the WF damage is located at 5-6m and 10-11s. In
World Wide head sea trade, the peaks of the HF and WF damage were moved towards
lower wave heights and periods.
In full speed the peak of the HF damage of ballast 3 condition in head sea North Atlantic
trade was moved towards higher sea states with significant wave heights of 6-7m and zero
up-crossing periods of 10-11s, while the location of the WF peak was insensitive to speed.
This is shown in Figure 5.33. The speed affected also the total damage and vibration dam-
age. The wall effects were present in particular at low speeds. Considering the dominating
sea states, the wall effect was more important for the WF damage than the HF damage,
since the speed reduction and peak periods were lower for the dominating sea states con-
tributing to the HF damage.
In Figure 5.31 39 sea states with realistic speed were utilized. Results based on 14 sea
states, similar to those run in other combinations, are presented in Figure 5.34 for ballast 3
condition in the North Atlantic head sea wave environment. The total fatigue damage was
North Atlantic HF fatigue damage North Atlantic HF fatigue damage

20
HF damage:0.74593
15
0.06
Tz [s]
10 0.04
d
0.02
20
5 0
0
10
10
0
0 5 10 15 20 20 0 Tz [s]
Hs [m]
Hs [m]
North Atlantic WF fatigue damage North Atlantic WF fatigue damage

20
WF damage:2.0445
15
0.1
Tz [s]
10
0.05
d
20
5 0
0
10
10
0
0 5 10 15 20 20 0 Tz [s]
Hs [m]
Hs [m]
Figure 5.31: Midship HF (upper) and WF (lower) fatigue damage for ballast 3 condi-
tion in North Atlantic head sea wave environment at realistic speeds. 39 sea states were
considered.
increased by 8%, and the contribution from HF damage was reduced by 2% by considering
more sea states.
The difference in location of the dominating sea states in ballast and cargo condition was
small. The dominating sea states of the HF damage were moved towards higher wave pe-
riods in cargo condition, which may indicate that whipping, due to more heave and pitch
in longer waves, was more dominant. The WF damage was located at the same position.
Results for realistic speeds in other combinations are shown in Appendix 5.4.5.
Table 5.8 presents the results for the HF and WF damage in full and realistic speed in
North Atlantic and World Wide head sea wave environment. For ballast 3 condition re-
sults based on 39 (All) and 14 sea states can be compared. The damages represent 7 year
continuous trade without port time in the respective combinations. Assuming 30% port
time, the damage in ballast and cargo condition can be summarized to produce the cumu-
lative damage for a 20 years service. As noted Table 5.8 includes the fatigue damage from
all sea states, while Table 5.6 only includes the extrapolated fatigue damage based on the
tested sea states only. The former table will then indicate higher fatigue damages, and for
fatigue life estimates from the latter table the percentage of the tested sea states has to be
World Wide HF fatigue damage World Wide HF fatigue damage

20
HF damage:0.43432
15
0.04
Tz [s]
10
0.02
d
20
5 0
0
10
10
0
0 5 10 15 20 20 0 Tz [s]
Hs [m]
Hs [m]
World Wide WF fatigue damage World Wide WF fatigue damage

20
WF damage:0.8876
15
0.04
Tz [s]
10
0.02
d
20
5 0
0
10
10
0
0 5 10 15 20 20 0 Tz [s]
Hs [m]
Hs [m]
Figure 5.32: Midship HF (upper) and WF (lower) fatigue damage for ballast 3 condition
in World Wide head sea wave environment at realistic speeds.
accounted for (direct time extrapolation).
From Table 5.8 different observations were made
? Full speed gave too conservative HF damage, but also higher WF damage.
? World Wide environment gave higher relative HF damage than North Atlantic.
? Ballast condition gave higher relative vibration damage than cargo condition (con-
tributed by lower damping).
? The increased ship length for bow 3 increased the WF damage in cargo condition, but
not in ballast condition.
? The HF damage of bow 3 was reduced compared to bow 2, but it was not significantly
reduced compared to bow 1.
? The difference in the HF damage between bow 1 and 2 in ballast was caused by different
damping, but possibly also by slightly different excitation due to the bulb.
? The fatigue life for bow 2 in North Atlantic bulk trade was estimated to 2.9 years, which
is comparable to 3.6 years obtained by the simple time extrapolation in Section 5.4.2.

20
HF damage:4.1327
15
0.4
Tz [s]
10
0.2
d
20
5 0
0
10
10
0
0 5 10 15 20 20 0 Tz [s]
Hs [m]
Hs [m]

20
WF damage:2.5239
15
0.2
Tz [s]
10
0.1
d
20
5 0
0
10
10
0
0 5 10 15 20 20 0 Tz [s]
Hs [m]
Hs [m]
Figure 5.33: Midship HF (upper) and WF (lower) fatigue damage for ballast 3 condition
in North Atlantic head sea wave environment at full speed.
The contribution from the HF part was reduced by 1% to 18% compared to the simple
time extrapolation.
? The fatigue life in the World Wide trade was 5.9 years, twice the fatigue life in the North
Atlantic. The HF contribution was 29%.
For bow 3 the estimated fatigue life was 2.8 years in the North Atlantic. Accounting for
the reduced length from 304 to 294m, assuming the stress was proportional to the length
squared and that the fatigue life was inverse proportional to the stress in the power of 3,
the corrected fatigue life became 3.4 years. The HF damage contribution was 10%. The
pointed bow was not an effective design to reduce the vibration damage relative to bow 2,
considering the difference in damping.
Some features of the testing tend to underestimate the relative importance of the vibration.
1. The damping in the model tests was too high (e.g. reducing the damping to ap-
proximately half for a container vessel increased the ratio of the vibration damage to
wave damage by about 40% for a typical sea state according predictions performed
by Drummen et al. (2006)).
2. The springing response reduced with increased distance from the wave maker.

20
HF damage:0.76622
15
0.1
Tz [s]
10
0.05
d
20
5 0
0
10
10
0
0 5 10 15 20 20 0 Tz [s]
Hs [m]
Hs [m]

20
WF damage:2.255
15
0.2
Tz [s]
10
0.1
d
20
5 0
0
10
10
0
0 5 10 15 20 20 0 Tz [s]
Hs [m]
Hs [m]
Figure 5.34: Midship HF (upper) and WF (lower) fatigue damage for ballast 3 condi-
tion in North Atlantic head sea wave environment at realistic speeds. 14 sea states were
considered.
3. Subsequent runs in a single sea state resulted in lower springing response.
4. The low number of selected sea states resulted in slightly higher HF damage contri-
bution.
Other factors not considered in the test program were
1. Storhaug et al. (2003) showed that the contribution from the HF damage was reduced
for corrosive environment considering a single slope SN-curve.
2. Storhaug et al. (2003) showed that the decrease of the vibration stress from head to
beam sea was less than the decrease of the stress from the conventional wave loading.
3. Routing tending to reduce the occurrence of high wave heights, will increase the
relative contribution from vibration. Good seamanship may reduced the vibration
damage when this is a concern when in adverse weather, while the contribution may
be increased when other issues are of more importance, e.g. to void excessive roll.
4. Head sea cargo condition, representing following seas, reduced the vibration contri-
bution due to too high wave damage mainly caused by lower encounter periods.
5.5. RELATIVE IMPORTANCE OF WHIPPING AND SPRINGING 149
Table 5.8: Overview of 7 year midship HF and WF fatigue damage for the different bow
and loading combinations in full (FS) and realistic speed (RS) for World Wide (WW) and
North Atlantic (NA) head sea wave environment. Extrapolation of results are based on the
response surface method covering all sea states.
Condition HF WF HF WF Tot. Tot. % HF contr. % HF contr.
NA NA WW WW NA WW NA WW
Ballast 1, FS 4.10 2.41 1.94 0.86 6.51 2.80 63 69
Ballast 2, FS 6.51 2.67 3.03 0.93 9.18 3.96 71 77
Ballast 3, FS 3.92 2.46 1.58 0.90 6.38 2.48 61 64
Ballast 3, FS, All 4.13 2.52 1.68 0.92 6.65 2.60 62 65
Cargo 1, FS 1.69 4.61 0.67 1.46 6.30 2.13 27 31
Cargo 2, FS 2.29 4.90 0.97 1.57 7.19 2.54 32 38
Cargo 3, FS 1.91 5.30 0.74 1.85 7.21 2.59 26 29
Ballast 2, RS 1.24 2.22 0.98 0.98 3.46 1.96 36 50
Ballast 3, RS 0.77 2.26 0.44 0.99 3.03 1.43 25 31
Ballast 3, RS, All 0.75 2.05 0.43 0.89 2.80 1.32 27 33
Cargo 1, RS 0.58 3.46 0.32 1.42 4.04 1.74 14 18
Cargo 2, RS 0.62 3.41 0.32 1.45 4.03 1.77 15 18
Cargo 3, RS 0.57 4.38 0.30 1.82 4.95 2.12 12 14
5. Storhaug et al. (2003) indicated also some small vibration levels in following seas.
The 18% contribution from the HF damage can be considered as a lower limit for the iron
ore trade in the North Atlantic for this vessel design.
5.5 Relative importance of whipping and springing

A procedure to identify whipping was outlined in Section 4.8. It is applied to the ballast 2
condition in the following.
The total, whipping and springing standard deviation stress in deck are presented as a
function of the sea state in Figure 5.35. The estimates in sea states with reduced speeds
were biased, since the springing criterion used to define the occurrence of whipping was
based on full speed. Disregarding the cases with reduced speed, whipping occured in all
sea states, but it became comparable to springing first around 4m significant wave height.
Whipping was then comparable to springing in the dominating sea states for the HF dam-
age displayed in Section 5.4.5.
The slope between two local peaks of the filtered 2-node response was used to indicate whip-
ping events. The probability of a slope to indicate whipping is presented in Figure 5.36
based on the moment. The probability of whipping appears to increase with increasing
Whip
Std. dev. stress [MPa]
15 Spring
Tot
10
0
0 5 10 15 20 25 30
15
HS [m], TP [s], U [kn]
10
0
0 5 10 15 20 25 30
Sea state number
Figure 5.35: St.dev. of total (right), whipping (left) and springing (middle) nominal
stress in deck amidships of ballast 2 condition shown versus HS (left), TP (middle) and U
(right).
wave heights and possibly also with increasing peak periods. For the sake of interpretation,
the probability of whipping for 5m significant wave height was about 0.03 (3 out of 100 vi-
bration cycles indicate whipping; 1 vibration cycle has a duration of 2 seconds in full scale,
i.e. 3 events during 200 seconds). This means that every minute there was a whipping
event at full speed in full scale. The peak periods of the tested sea states were between
11 and 15s. In full speed the encounter peak period was about 10s, and a whipping event
occured then for every sixth wave in average. This is about 5-6 times the rule of thumb
criterion for voluntary speed reduction, which is related to the encounter waves rather than
the whipping response, e.g.(Faltinsen 1990, pp. 282).
A few factors may influence the capability of the method to detect whipping.
• The whipping events may occasionally have contribution from springing.
• Low whipping response may occur without being identified.

5.5. RELATIVE IMPORTANCE OF WHIPPING AND SPRINGING 151
0.03
Probability [%]
0.02
0.01
0
0 5 10 15 20 25 30
15
10
0
0 5 10 15 20 25 30
Sea state number
Figure 5.36: Probability of whipping from individual slope estimates in ballast 2 condition
at midship section shown versus HS (left), TP (middle) and U (right).
• The frequency range in the filtering process was chosen rationally, and changing it
affects the results.
• Other responses may be more suitable to identify whipping, e.g. acceleration.
• The method required transfer function values, which were input to the criterion.
• The linear and second order springing values should be obtained in realistic speeds.
• The criterion was based on long crested sea and may be too strict for short crested
sea.
• The bichromatic sum frequency response was disregarded.
Despite these factors, the results suggest that both whipping and springing ought to be
included in theoretical investigations. It indicates that the impacts related to hypothesis
3 is of equivalent importance as springing.
5.5.1 Acceleration levels used to indicate whipping

In hull monitoring systems, vertical accelerations in the bow may be used to identify bow
impacts. When the acceleration exceeds a predefined criterion, a warning should tell the
captain to take action. The rules for hull monitoring, (DNV 2005e; ABS 1995), do not
provide well defined criteria. ABS figures the following criterion: When three times the
standard deviation of the acceleration exceeds 0.4g, bow impacts occur. This criterion is
compared to the model test results in the following.
The translatory accelerations in the model tests do not need to be scaled to full scale. The
ABS’s criterion was compared to the accelerations from ballast 2 condition. Figure 5.37
presents the total, three times the total and the HF standard deviation of the vertical bow
acceleration. The high pass filtered acceleration included higher modes in this case. In
addition, single maximum amplitudes versus three times the total standard deviation are
shown. The speed, significant wave height and peak period are included to characterize
the operational condition.
3.92
3*Tot
Std.dev. [m/s2]
3
Tot
2
HF
1
0
0 5 10 15 20 25 30
3.92
3 3*Tot
AZ [m/s2]
max Tot
2
max HF
1
0
0 5 10 15 20 25 30
15
10
0
0 5 10 15 20 25 30
Sea state number
Figure 5.37: 3 times the total (left), total (middle) and HF (right) standard deviation of
measured bow acceleration in ballast 2 condition for the different sea states characterized
by HS (left), TP (middle) and U (right). In row 2 the maximum total (middle) and HF
(right) amplitude are compared to 3 times the total standard deviation (left).
5.6. LINEAR VERSUS MONOCHROMATIC SECOND ORDER SPRINGING 153
The criterion was not exceeded in any of the sea states, but it was no doubt that bow
impacts and whipping did occur. In the highest sea state at full speed, a value close to
3.92m/s2 was reached with a maximum single amplitude exceeding the acceleration of
gravity. It may be possible for the measured acceleration to exceed the acceleration used
in design without providing an alarm. For the highest sea state the ratio between the
maximum total amplitude and three times the standard deviation was relatively high, but
it was reduced drastically with speed. Figure 5.38 confirms bottom exit followed by bot-
tom slamming for a relatively large bottom area. The nominal stress in deck was about
150M P a, which is high, and the contribution from whipping was significant. This was a
conventional flat bottom impact, which gave a significant increase of the vibration cycle in
sagging. The slamming peaks are ahead (in time) of the acceleration, which again is ahead
of the bending stress amidships. Note also the clustering of the large responses.
A better choice may be to relate the criterion to amplitudes rather than the standard
deviation, and base the criterion on some fraction of the design value with an appropriate
safety margin to account for the stochastic nature of such extreme events. Moreover, a
high ratio between the maximum acceleration amplitude and three times the standard
deviation may actually be an indication of severe bottom impacts. The local loading due
to the impacts is another matter.
5.6 Linear versus monochromatic second order spring-

ing
The measured transfer function was used as input to a basic method, which predicted
the difference between monochromatic second order springing and linear springing. The
method was outlined in Section 4.9. The results from this method, in terms of standard
deviation of the HF nominal stress in deck at midship section, is considered in the following
based on ballast 2 condition.
Results from Torsethaugen and PM (wind based) wave spectra are compared in Figure 5.39,
and the predictions were based on realistic speeds provided in Chapter 6. The estimated
significant wave height and peak period from the two spectra differ as displayed in the
lower plots. The trend of the linear and monochromatic second order HF stress differs as
well.
The maximum linear HF stress, located at Beaufort 5 to 7, is roughly 3MP a in terms of

nominal standard deviation stress. The nonlinear contribution is increasingly important
above Beaufort 5. The nonlinear HF stress reaches a plateau at 5MP a for PM, while
Torsethaugen displays an increasing trend exceeding the PM predictions.
Stress [MPa]
100
0
−100
26 27 28 29 30 31 32 33 34
10
AZ,8 [m/22]
−10
26 27 28 29 30 31 32 33 34
5000
P3 [N/m2]
0
26 27 28 29 30 31 32 33 34
10000
P1 [N/m2]
5000
26 4 27 28 29 30 31 32 33 34
x 10
P2 [N/m2]
2
1
0
26 27 28 29 30 31 32 33 34
Time [s]
Figure 5.38: Full scale nominal stress in deck (sagging upwards), vertical bow accelera-
tion (positive downwards) and bottom pressures at stem, 0.972 and 0.945LP P in ballast 2
condition at full speed; HS = 9m, TP = 15s (run1630). Model time scale.
The model tests in ballast 2 condition was performed based on different sea states and
speeds as shown in Figure 5.40. Some conditions are comparable to those in Figure 5.39.
Torsethaugen with Beaufort 7 is similar to sea state 13 and 14. The former predicts a
standard deviation stress of 5.5MP a, while the latter gives 4 to 5MP a. Torsethaugen
with Beaufort 9 predicts 6.5MP a, while sea state 19 gives 2MP a. The former is based on
5kn higher speed. Torsethaugen with Beaufort 5 predicts 3MP a, while sea state 3 gives
2.5M P a.
Perfect agreement between predictions and model tests were not expected, since
5.6. LINEAR VERSUS MONOCHROMATIC SECOND ORDER SPRINGING 155
10 10

Lin Lin
8 NL 8 NL
6 6
4 4
2 2
0 0
1 2 3 4 5 6 7 8 9 10 11 1 2 3 4 5 6 7 8 9 10 11
25 25
H H
S S
20 20
T T
HS [m], TP [s]
HS [m], TP [s]
P P
15 15
10 10
5 5
0 0
1 2 3 4 5 6 7 8 9 10 11 1 2 3 4 5 6 7 8 9 10 11
Beaufort Beaufort
Figure 5.39: Pairs of linear (left) and second order monochromatic (right; including
linear) HF standard deviation nominal stress versus Beaufort strength. PM (left plot) and
Torsethaugen (right plot).
8
HF [MPa]
6
4
2
0
0 5 10 15 20
WF [MPa]
20
10
0
0 5 10 15 20
H [m], T [s], U [kn]
15
10
P
0
S
0 5 10 15 20
Figure 5.40: Measured nominal standard deviation HF and WF stress in ballast 2 con-
dition at reduced speed in different sea states characterised by HS (left), TP (middle), U
(right). JONSWAP wave spectrum with a γ = 2 is used.
1. The measurements include whipping, which contributes according to Section 5.5.
2. The predictions lack second order response from bichromatic waves displayed in Sec-
tion 5.3.2.
3. The input transfer function values in the predictions were based on full speed.
4. The spectra differ.
The predictions illustrate that the linear springing dominates in low wind speeds, while
nonlinear effects dominate in higher winds speeds. For the sea states dominating the
vibration damage, the sum frequency springing is considered of similar to higher importance
than the linear springing, thereby comfirming the importance of hypothesis 4. Moreover,
the high frequency tail behavior of the wave spectra differs and affects the level of the HF
response and the trends for increasing wind strength. The importance of sum frequency
effects contradicts (Slocum 1983), who only indicated a moderate (30%) importance in a
single sea state.
5.7 Illustration of vibration events

Section 5.6 illustrated the relative importance of linear and second order springing, while
Section 5.5 confirmed the contribution from whipping. The excitation sources causing
whipping were not well explained in Section 5.5. A few measured time traces are displayed
in the following to indicate the excitation sources for bow 2 in ballast and cargo condition.
The selected time traces consider ’extreme’ whipping events. Bow 3 differs considerably
from bow 2, and plots of whipping events for ballast and cargo 3 condition are shown in
Appendix C.5.
Figure 5.41 presents whipping events from ballast 2 condition at full speed in small sea
states (3m). The left plot of the figure refers to a sea state with a peak period of 8.2s,
while the right plot refers to a peak period of 13.6s. Section 5.5 indicated that similar
whipping events occured once every hundred springing cycle. The different rows display
the total and wave pass filtered midship stress (sagging=positive), the relative motion in
the bow (zero at still water level in zero speed), the pitch motion (bow up=positive), the
pressure below the bow and the stern (zero at calm water in zero speed). Note that ac-
celeration is not shown in these figures as in Figure 5.38, since the acceleration does not
necessarily involve strong bottom slamming events in these cases.
For clarification, some details are provided to ease the interpretation of the displayed time
traces for ballast and cargo 2 condition.
? Bow 2 in ballast condition was without bulb, but with stem flare. The bottom (bilge)
flare ended 15cm above BL in the region 20 to 60cm aft of FP.
? The stem flare was removed in cargo 2 condition.
? The draft at FP in ballast and cargo condition was 32cm and 51cm.
? The stem flare and deck level started 21cm and 45cm above the still water line in ballast
condition.
5.7. ILLUSTRATION OF VIBRATION EVENTS
REL [m] Stress [MPa] 157
REL1 [m] Stress [MPa]

20 20
0 0
−20
−20 −40
82 83 84 85 86 87 88 89 90 106 107 108 109 110 111 112 113 114
0.3
0.2 0.2
0.1
1
0
0
−0.2
82 83 84 85 86 87 88 89 90 106 107 108 109 110 111 112 113 114
0.1
Pitch [0]
Pitch [0]
0 0.5
−0.1 0
−0.2 −0.5
82 83 84 85 86 87 88 89 90 106 107 108 109 110 111 112 113 114
P3 [N/m2]
P [N/m2]
2000 2000
1000 1000
0
3
0
82 83 84 85 86 87 88 89 90 106 107 108 109 110 111 112 113 114
P13 [N/m2]
P13 [N/m ]
2
300 400
200 200
100 0
82 83 84 85 86 87 88 89 90 106 107 108 109 110 111 112 113 114
Time [s] Time [s]
Figure 5.41: Whipping events in ballast 2 condition in HS = 3m, U = 15kn. Left:

TP = 8.2s (run 1582), right: TP = 13.6s (run 1594). Row 1= Total (continuous) and
wave pass filtered (dashed) midship stress, row 2= Relative motion in bow, row 3= Pitch,
row 4= Stem pressure, row 5= Stern pressure.
? The measured relative motion in cargo 2 condition was limited to a height of 18.5cm
above the still water line corresponding to 7.5cm below deck level.
? A pitch angle of 10 corresponds to a motion of 7.3cm in the bow.
? The CL bow pressure, P3 , was measured 29cm above BL at FP.
? The CL bottom pressure, P1 , was measured 23cm aft of FP.
? The CL bottom pressure, P5 , was measured 11cm aft of FP.
? The stern pressure, P13, was measured in the stern flare 5.6cm above the still water line.
? The value of 100N/m2 corresponds to a water head of 1cm.
? The vertical line represents the possible initiation of whipping.
Returning to the left plot of Figure 5.41. The relative motion in the bow increases, and the
water hits the stem flare. The initiation of whipping does not agree in time with the pos-
sible stem flare force. The bow pressure decays earlier than the relative motion, indicating
that the measured relative motion may include water spray/run up. The pitch motion is
small, and the relative motion is solely due to the motion of the water. The stern pressure,
P13 , is moving in and out of the water, and no slamming is observed in the stern area.
The positive stern pressure does not correlate well with the time of the impact. The wave
elevation in the stern is small compared to the relative motion in the bow, but the flare
in the stern is more pronounced. The first strong vibration amplitude occurs in hogging,
and correlates to some extent with both the decay in the bow pressure and the decay of
the stern pressure. It is likely that the contributing force was a downward pull at the bow,
since the troughs of the stern pressure are not correlated with changes in the vibration
response. The example illustrates that it is not easy to reveal the excitation source from
measurements by visual inspection of the responses.
The right plot of Figure 5.41 presents a similar time trace in longer waves. In this case, the
bow has maximum draft (negative pitch) at the start of the impact. The pressure at the
bow agrees well with the pitch motion, and the pressure decay starts at the same instant
as the first half vibration cycle in hogging. Again, the maximum pressure is reached about
0.1s before the maximum relative motion. The pressure at stern is increasing at the time
of the impact in this case, and the hogging cycle indicates that the impact was due to a
downward pull in the foreship only. The vibration amplitude is less than in the short waves.
Figure 5.42 presents whipping events from ballast 2 condition in high sea states (9m, 15s).
The left plot assumes full speed, while realistic speed of 5kn is considered on the right side.
The left plot of Figure 5.42 presents a large whipping event. The pitch is positive and the
bow is still going up at the initiation of the impact. The bottom pressure, P 1, has a rapid
increasing suction pressure at the start of the whipping event, resulting in a hogging cycle.
Subsequently, flat bottom impact with slamming occurs, and an increase of the following
vibration cycle in sagging is observed. The stern pressure and the observed slamming at
68.5s does not affect the vibration response. The source of the current whipping event is
regarded as a conventional flat bottom impact, but it has contributions from both the exit
and entry phase as illustrated in Figure A.15.
Stress [MPa]
50 50
0 0
−50 −50
−100
62 63 64 65 66 67 68 69 70 277 278 279 280 281 282 283 284 285
0.4
REL1 [m]
0.2 0.2
0 0
−0.2 −0.2
62 63 64 65 66 67 68 69 70 277 278 279 280 281 282 283 284 285
Pitch [0]
2 2
Pitch [ ]
0
0 0
−2 −2
62 63 64 65 66 67 68 69 70 277 278 279 280 281 282 283 284 285
P1 [N/m ]
2
P3 [N/m2]
2000
2000
0
1000
−2000 0
62 63 64 65 66 67 68 69 70 277 278 279 280 281 282 283 284 285
P13 [N/m2]
P13 [N/m2]
1500
1200
1000 1000
800 500
62 63 64 65 66 67 68 69 70 277 278 279 280 281 282 283 284 285
Time [s] Time [s]
Figure 5.42: Whipping events in ballast 2 condition in HS = 9m, TP = 15s. Left:

U = 15kn (run 1629), right: U = 5kn (run 1636). Row 1= Total (continuous) and wave
pass filtered (dashed) midship stress, row 2= Relative motion in bow, row 3= Pitch, row
4= Bottom/stem pressure at bow, row 5= Stern pressure.
5.7. ILLUSTRATION OF VIBRATION EVENTS 159
The right plot of Figure 5.42 presents a whipping event in realistic speed when the bow
is going down. The pressure probes are going in and out of the water and the pitch mo-
tion is significant, indicating large relative motions both at the stern and bow. Small
slamming events occur in the aft ship at this speed, but do not appear to produce signifi-
cant whipping. It is no clear explanation for the initiation of the whipping towards hogging.
A couple of vibration events from cargo 2 condition in realistic speeds are presented in
Figure 5.43. The left plot refers to a sea state with a significant wave height of 3m and
peak period of 8.2s, while the sea state to the right refers to 5m and 15s. In the left plot of
Figure 5.43, the pitch motion is small, while the relative motion and stem pressure display
rapid fluctuations. The upper limit of the relative motion is exceeded at 91.5s, but the
water surface did not reach the deck level. The relative motion is far from the non-vertical
bilge area, and the stern pressure is small. The period of the relative motion oscillation is
about 0.7s both at 92 and 96s, and the stem pressure at 92s displays a rapid oscillation,
which is not reflected in the relative motion. This represents apparently springing excited
by sum frequency effects.
Pitch [0] REL1 [m] Stress [MPa]
10 50
0 0
−10 −50
91 92 93 94 95 96 97 98 99 100 73 74 75 76 77 78 79 80 81
0.2 0.2
0.1 0
0 −0.2
−0.1 −0.4
91 92 93 94 95 96 97 98 99 100 73 74 75 76 77 78 79 80 81
Pitch [0]
0 1
−0.1 0
−0.2 −1
91 92 93 94 95 96 97 98 99 100 73 74 75 76 77 78 79 80 81
2000
P3 [N/m2]
P13 [N/m2] P5 [N/m ]
1500
2
1000 0
500
0 −2000
91 92 93 94 95 96 97 98 99 100 73 74 75 76 77 78 79 80 81
P13 [N/m ]
400 800
2
300 600
200 400
200
100
91 92 93 94 95 96 97 98 99 100 73 74 75 76 77 78 79 80 81
Time [s] Time [s]
Figure 5.43: Vibration events in cargo 2 condition. Left: HS = 3m, TP = 8.2s, U = 15kn
(run 1746), right: HS = 5m, TP = 15s, U = 10kn (run 1798). Row 1= Total (continuous)
and wave pass filtered (dashed) midship stress, row 2= Relative motion in bow, row 3=
Pitch, row 4= Bottom/stem pressure at bow, row 5= Stern pressure.
The right plot of Figure 5.43 presents a whipping event. The whipping starts when the
bow is in the up-pitching cycle and and moves upwards, while the relative motion moves
downwards. The bottom pressure, P 5, displays a change in the suction pressure at the
initiation of the impact. It occurs prior to the radiation pressure from vibration, observed
in the stern pressure. The minimum surface level is 11cm above the BL at the stem, but
even lower at the bow quarter due to the steady wave elevation. The estimated difference
was 5cm in 10kn, and the water surface at the bottom flare may move down to about 6cm
above BL. The whipping appears to come from water moving down at the bottom flare
area at bow quarter and then moving up again with duration comparable to the vibration
period. This may cause a dynamic amplification up to a factor of three according to Fig-
ure A.15 for a sinusoidal full period impact.
Figure 5.44 presents another whipping event in cargo 2 condition. The significant wave
height is 9m with a peak period of 15s. The left plot represents full speed, while the right
plot represents zero speed, which includes voluntary speed reduction. The left plot shows
actually a flat bottom impact also in this loading condition. The whipping starts first with
a half cycle in hogging. The second cycle in sagging is not significantly increased indicating
that the suction force in this case was more important than the subsequent slamming.
REL [m] Stress [MPa]

100 100
0 0
−100 −100
151 152 153 154 155 156 157 158 159 456 457 458 459 460 461 462 463 464
0.2 0.2
0 0
1
−0.2
−0.4 −0.2
−0.4
151 152 153 154 155 156 157 158 159 456 457 458 459 460 461 462 463 464
5
Pitch [ ]
0
Pitch [0]
2
0 0
−2
−5
151 152 153 154 155 156 157 158 159 456 457 458 459 460 461 462 463 464
P13 [N/m2] P3 [N/m ]
[N/m2] P [N/m2]
5000
2000
0 0
5
−2000
−5000
151 152 153 154 155 156 157 158 159 456 457 458 459 460 461 462 463 464
2000 1000
1000
0 0
13
−1000 −1000
P
151 152 153 154 155 156 157 158 159 456 457 458 459 460 461 462 463 464
Time [s] Time [s]
Figure 5.44: Whipping events in cargo 2 condition in HS = 9m, TP = 15s. Left: U =

15kn (run 1786), right: U = 0kn (run 1787). Row 1= Total (continuous) and wave pass
filtered (dashed) midship stress, row 2= Relative motion in bow, row 3= Pitch, row 4=
Bottom/stem pressure at bow, row 5= Stern pressure.
In the right plot of Figure 5.44, representing zero speed, no flat bottom impact is observed.
The pitch angle is negative and a high wave caused water on deck. There is a rapid change
in the stem pressure, which may explain the vibration, but more likely it was caused by im-
pacts against the ’superstructure’. The vibration was about 8.5MP a in amplitude, which
constitutes a small contribution to the WF stress. From observations, the bow design
without a stem flare gave much more water on deck.
In this section significant vibration events have been illustrated, supporting hypothesis
3 related to impacts and hypothesis 4 related to sum frequency excitation due to bow
reflection. It should be noted that the impact related to hypothesis 3 is related to bottom
flare impacts in ballast condition rather than bow flare impacts. The first vibration cycle
often occurs in hogging due to a downwards pull. This is in agreement with Saunders
5.7. ILLUSTRATION OF VIBRATION EVENTS 161
(1965), who explained it by rapid loss of buoyancy, and it is also observed in time series
presented by Bell and Taylor (1968). Later publications do in general not reflect these type
of excitation sources.
Chapter 6
Full scale measurements
Full scale measurements of the iron ore carrier was carried out by DNV, and a workshop was
initiated where measurements were made available. The evaluation of the measurements
have so far resulted in three publications, (Storhaug et al. 2003; Moe et al. 2005; Storhaug
et al. 2006). The ship’s geometry is identical to the model with bow 1, which again is similar
to bow 2. Full scale measurements of the response for the vessel in head sea ±300 , are
compared to ballast 2 condition. Damping, involuntary speed reduction in waves, stress
versus sea states, whipping response and fatigue damage are presented in the following
sections. Focus is given to the wave induced response after strengthening made in 2000,
because the wave radar was adjusted after strengthening to produce more reliable wave
height estimates, (Moe et al. 2005). Vidic-Perunovic (2005) did also assess measurements
from the large iron carrier with respect to bi-directional sea states and vibration response.
Unfortunately, the period before the wave radar was adjusted was chosen.
6.1 Damping and springing frequencies

The damping ratio was estimated for a large number of records measured onboard the ship.
Measurements from other ships were assessed as well. The spectral method was used to
obtain the damping ratio as described in Section 2.3.3 and Section 4.4.2. In most cases it
was applied to half hour records.
Figure 6.1 presents estimates of the damping ratio in ballast and cargo condition after
strengthening the vessel. The damping appears to be linear in ballast condition, and linear
to weakly nonlinear in cargo condition. The possible nonlinear contribution may come
from the iron ore cargo. The mean damping ratios are 0.80 and 0.86% in ballast and cargo
condition, confirming a small difference. The 95% confidence intervals of the sample values
163
164 CHAPTER 6. FULL SCALE MEASUREMENTS
0.04 0.04
0.035 0.035
0.03 0.03
0.025 0.025
0.02 0.02
δ
δ
0.015 0.015
0.01 0.01
0.005 0.005
0.008583 0.003956 0.0001949
0.0080198 0.0036438 0.00015196
0 0
0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 0 0.5 1 1.5 2 2.5
HF standard deviation stress [MPa] HF standard deviation stress [MPa]
Figure 6.1: Damping ratios as a function of HF standard deviation nominal stress in

ballast (left) and cargo condition (right) from the measurement period of June 04 to May
05.
are significant, while the 95% confidence intervals of the mean are small.
The 2-node natural frequency was estimated simultaneously with the damping ratio. The
natural frequency estimates in ballast and cargo condition are presented in Figure 6.2. The
mean springing frequencies are 3.63rad/s in ballast and 3.23rad/s in cargo condition. The
uncertainty is twice as large in cargo condition due to slightly different cargo conditions,
but in general the uncertainty is insignificant.
3.7
ω2 [rad/s], ballast
3.65
3.6
3.6349 0.018769 0.00078272
3.55
8 10 12 14 16 18 20 22 24
3.3
ω2 [rad/s], cargo
3.25
3.2
3.2341 0.038925 0.001913
10 11 12 13 14 15 16 17 18
WF standard deviation stress [MPa]
Figure 6.2: The natural 2-node frequency as a function of the WF standard deviation
stress in ballast (upper) and cargo condition (lower) for the measurement period of June
04 to May 05.
6.1. DAMPING AND SPRINGING FREQUENCIES 165
The natural frequency in Figure 6.2 appears to be insensitive to the WF response level.
The expected change of the natural frequency, due to the changed wet area and added
mass in high and long waves, is not evident. The constant frequency as a function of the
WF stress level differs from the decay in higher sea states observed in model tests and
displayed in Figure 5.6. The decay in model test was explained by a combination of higher
WF responses in long crested sea and somewhat nonlinear stiffness.
The same procedure was employed on 30 records before strengthening in ballast and cargo
condition. In addition a few other ships were considered. An overview of the ships is given
in Table 6.1.
Table 6.1: Overview of ships considered in damping assessment.

Ship name Lpp [m] T [m] B [m] Ship type
Ship 1, ’The ship’ 294 18 53 Ore carrier
Ship 2 300 18.3 50 Bulk carrier
Ship 5 281 17.4 48 Ore carrier
Ship 6 285 12.2 32.2 Container vessel
The results of the damping estimates are given in Table 6.2. The average damping in cargo
condition is about 10% higher than in ballast condition. The damping for ship 1 is also
higher after strengthening, which increased the natural frequency by 10%. The natural
frequencies of the ship before strengthening were used as basis for the model tests. The
difference between the damping before and after strengthening may be affected by the
low number of records considered before strengthening. However, the increased damping
after strengthening may also be caused by an increase in the structural damping, while the
increased natural frequency will tend to reduce the linear hydrodynamic damping. The
damping of the container ship (ship no. 6) is noticeably higher than the damping estimated
for the bulk carriers.
It was not easy to obtain good damping estimates from the spectral method. Half hour
records rather than 5 minutes records were preferred as a basis, and the response spectrum
fit was based on a high resolution with 4096 FFT samples and a narrow interval of 0.3 to
0.5rad/s below and above the natural frequency. With reference to Section 5.1.2, a reduc-
tion factor should be included for the spectral method due to effects, which will widen the
spectral vibration peak (e.g. varying natural frequency due to change of added mass in
waves).
In Figure 6.3 the damping ratio is estimated based on the decay curve from a whipping
event in ballast condition after strengthening. The damping is 0.51%, and the average of
several such events was 0.50%, while the spectral method predicted 0.80%. The ratio of
Table 6.2: Spectral estimates of damping ratio for different ships and loading conditions
Ship no., remark δ ω2 [rad/s] Loading cond.
1, before strengthening 0.70 3.32 Ballast
1, before strengthening 0.70 2.91 Cargo
1, after strengthening 0.80 3.63 Ballast
1, after strengthening 0.86 3.23 Cargo
2, June-Oct.-03 0.69 3.23 Ballast
2, June-Oct.-03 0.87 2.78 Cargo
3, 5 min. time traces, April-03-July-04 0.61 3.23 Ballast
3, 5 min. time traces, April-03-July-04 0.67 2.75 Cargo
4, Jan-Dec.-03 0.96 3.62 Ballast
4, Jan-Dec.-03 1.12 3.16 Cargo
5, Nov-04-May-05 0.81 3.94 Ballast
5, Nov-04-May-05 0.70 3.51 Cargo
6, April-June-02, Feb-April-05 1.52 3.92 Cargo and ’ballast’
0.625 suggests strongly that a correction factor is necessary based on the damping esti-
mates from the spectral method. The whipping events were normally disturbed by wave
action after the initial whipping events, and it was not easy to find many good quality
whipping responses as input to the damping evaluation. A practical way is to drop the
anchor, but then any speed effect on the damping will not be detected. The latter may be
especially important for container vessels.
40
HF
30 Fit
δ=0.0051 ωs=3.63
20
HF stress [MPa]
10
−10
−20
−30
−40
1500 1510 1520 1530 1540 1550 1560
Time [s]
Figure 6.3: Damping estimated from the decay curve of the high pass filtered stress after
a whipping event, 04.12.15.23:30 on starboard side.
The damping estimates in Table 6.2 may be multiplied with a factor of about 0.8 suggested
by Gran (1974). An average value of the damping ratio for the bulk carriers, disregarding
6.2. MEASURED SPEEDS IN DIFFERENT SEA STATES 167
the effect of loading conditions, is then 0.63%. The ratio of 0.625 from the ship in ballast
condition suggests an even lower average value of 0.49%. Such large blunt ships are there-
fore expected to have a damping ratio in the range of 0.5 to 0.6%.
Limited data are available for the container vessel. The higher damping for the con-
tainer vessel may be explained by contribution from the container stacks with its twisting
locks and lashing system, and possible from higher hydrodynamic damping due to for-
ward speed effects. The difference in damping between container vessels and other types
of vessels should be investigated further in order to obtain realistic damping values for
future numerical predictions. In case the estimated damping level for the container vessel
is representative for such ships, this high damping may be beneficial from a fatigue point
of view compared to the bulk carriers.
6.2 Measured speeds in different sea states

The wave induced vibration level is sensitive to speed. In order to check if the predicted
speeds in Section 3.2 and the speeds applied in the model tests were reasonable, a compar-
ison is made with full scale measurements.
The speed was measured in full scale by a GPS, which measured the speed over ground
and disregarded the current or drift. The speed as a function of the significant wave height
after strengthening is presented in Figure 6.4. The speed reduction in head sea is similar
in ballast and cargo condition, and the speed reduction in following sea is also similar
in ballast and cargo condition. The behaviour in head and following sea differs, but a
noticeably speed reduction is observed also in following sea probably due to the relative
motion effect. The uncertainty in predicted significant wave height is about 10% according
to Miros (2003) assuming proper X-band radar setting. Moe et al. (2005) confirmed good
agreement between measured wave heights and alternative data after strengthening, so the
10% uncertainty is reasonable in this case.
The measured speed may be compared to the speeds applied during the model tests,
given in Table 3.4, and the predicted speeds by the simplified theory, given in Table 3.3.
The applied speeds were 15, 10, 5 and 0kn in 3, 5, 7 and 9m significant wave height.
The corresponding full scale measured speeds were 13.8, 9.6, 5.3 and 4.5kn in ballast
condition, and the predicted speeds were 14.2, 11.2, 7.4 and 6.3kn. The predictions were
somewhat high compared to the measurements, which may be explained by the assumed
propulsion efficiency factor in the predictions, and possible drift/current in the full scale
measurements. The simplified method was however useful to make a first estimate of the
speed reduction for the purpose of model tests. The applied speeds during the model
testing were reasonable, except for the highest sea state where the applied speed was too
low. The full scale measurements suggest that the voluntary speed reduction is not an issue,
20
18
16
14
Speed [kn]
12
10
8 Ballast head sea
Ballast following sea
6
Cargo head sea
4 Cargo following sea
2
0
0 1 2 3 4 5 6 7 8 9 10
HS [m]
Figure 6.4: Measured full scale speed in head and following sea in ballast and cargo
condition versus measured significant wave height. Extension of Figure 30 in Moe et al
(2005).
and that all available power is basically used at all times. The vessel speed in head seas
for even higher extreme sea states is however still an issue for further investigation (rarely
measured, but when measured it is available through the VDR(voyage data recorder) for
a certain period of time on ”all” ships).
6.3 Wave induced stress in different sea states

The standard deviations of the WF and HF nominal stress as a function of the measured
significant wave height are presented in Figure 6.5 for ballast condition in head sea ±150 .
The measured HF and WF stresses were obtained from the period after strengthening,
but they were multiplied with 1.39 to represent the stress levels during the period before
strengthening. The 1.39 factor denotes the difference in the section modulus at the mea-
surement position at the deck level before and after strengthening. The nominal stress
level in Figure 6.5 can then be compared to the stress obtained from the model tests. The
WF stress increases as a function of the significant wave height. The HF stress increases
up to 4m, where it appears to reach a constant level. This is partly a consequence of the
speed reduction in higher waves. A change in the dominating excitation sources may be an
alternative reason, however, since this trend was also predicted in Figure 5.39 just based on
the speed reduction, the speed reduction appears to be the main cause of the behaviour.
It is noted that Stiansen et al. (1978) presented similar trends as shown in Figure 6.5 from
measurements onboard three Great Lakes bulk carriers. There was however some variation
depending on vessel size, and the current trend resembles the trend for the largest of the
three vessels with almost the same length (1000ft), but more shallow draft.
In Table 6.3, the results in Figure 5.40 are compared with those in Figure 6.5. The most
6.3. WAVE INDUCED STRESS IN DIFFERENT SEA STATES 169
25 9
20
7
6
WF std.dev. stress [MPa]
HF std.dev. stress [MPa]

15
5
4
10
2
5
0 0
0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7
Significant wave height [m] Significant wave height [m]
Figure 6.5: WF and HF std. dev. nominal stress versus HS in head sea ±150 for
ballast condition ’before’ strengthening. The stress is derived based on the period after
strengthening, but it is multiplied with a factor of 1.39 to represent the nominal stress
before strengthening. The standard deviations within half meter intervals are displayed.
probable zero up-crossing periods from the North Atlantic scatter diagram are indicated in
bold numbers, (DNV 2000). The HF stress is obtained from the strengthened vessel with
the natural frequency that is 10% higher than prior to strengthening. According to Moe
et al. (2005) the corrected standard deviation HF stress (multiplied with 1.39) was maxi-
mum 0.8MP a lower than prior to strengthening. The WF stress from short term statistics
was also calculated by WASIM simulations, (DNV 2004), which were based on full speed,
ballast condition, long crested head sea and the same sea conditions as considered in the
model tests.
Table 6.3: Comparison of head sea WF and HF std. dev. nominal stress in MP a from
ballast 2 condition (MS), full scale measurements (FS) and WF stress from WASIM (WA).
The most probable TZ periods are indicated in bold. Sea states refers to numbering in
Figure 5.40.
HS WF FS HF FS Sea state WF MS HF MS TZ [s] WF WA
2 2.3 2.7 1-5 1.2/1.9/3.3/4.4/5.9 4.1/2.7/2.3/1.8/1.4 5/6/7/8/9 0.6/1.3/2.4/4.0/5.3
3 4.1 4.2 6-10 3.1/4.7/7.8/8.7/10.3 6.4/5.0/4.4/3.5/2.7 6/7/8/9/10 1.9/3.6/5.9/8.0/9.1
4 7.8 5.4 11 6.3 9.0 7 (9) 4.8 (10.6)
5 12.9 5.4 12-15 12.2/17.8/16.2/17.1 5.2/5.0/3.7/3.5 8/9/10/11 9.9/13.2/15.2/15.8
7 23.0 4.9 16-18 20.8/20.7/21.0 3.6/3.5/2.6 9/10/11 18.6/21.3/22.1
The HF stresses obtained in the model tests for the most probable wave periods in Table 6.3
are significantly below the full scale average values (when accounting for 0.5-0.8MP a ad-
dition in the full scale values). The model was run in 15kn in 4m significant wave height
with a zero up-crossing period of 7s, while the most probable values were 12kn and 9s.
The HF stress from the model tests is not representative for this sea state. The full scale
results were also affected by wave energy spreading.
The WF response obtained from the model tests in Table 6.3 are in some agreement with
the WASIM simulations. The difference is explained by wall effects and disturbance from
residual waves in the smallest sea states with short peak periods. In the highest sea state
it is fair agreement in the WF stress from the full scale, model scale and WASIM results,
while in lower sea states the full scale WF stress is lower. This may again be due to more
wave energy spreading in lower sea states.
The ratio between the WF and HF stress in full scale was less than in the model tests,
both because the HF stress in the full scale was higher and the WF stress in full scale was
lower. This will increase the overall vibration damage contribution in full scale compared
to the model tests.
Figure 6.6 presents standard deviation of the WF and HF stress versus significant wave
height for cargo condition. The WF stress is similar in magnitude as the WF stress from
ballast condition. The HF stress increases with wave height, but it reaches a plateau
already at 3m significant wave height. The vibration stress at the plateau is about 60%
of the similar plateau in ballast condition. The vibration level in head sea for cargo
condition is still significant. The number of half hour records indicated by dots in Figure 6.6
confirms that head seas within ±150 occured occasionally in cargo condition, even though
the prevailing condition was following seas, (Storhaug et al. 2003).
6.4 Prediction of whipping events from measurements

A method to identify whipping from the envelope process of the vibration response was
outlined in Section 4.8. In Section 5.5 the method was shown to detect whipping in most
of the sea states tested. The same method is applied for a number of full scale records of
the vessel (ship 1) in the following.
The criteria for identifying whipping were corrected compared to the model tests to account
for
• A damping ratio of 0.64% in full scale, rather than 1% from model tests.
• Wave periods exciting linear and second order springing were based on the relative
direction of the primary sea, rather than pure head sea.
6.4. PREDICTION OF WHIPPING EVENTS FROM MEASUREMENTS 171
30 7
6
25
5
20
WF std.dev. stress [MPa]
HF std.dev. stress [MPa]

4
15
10
2
5
1
0 0
0 1 2 3 4 5 6 7 8 0 1 2 3 4 5 6 7 8
Significant wave height [m] Significant wave height [m]
Figure 6.6: WF and HF std. dev. nominal stress versus HS in head sea ±150 for cargo
condition obtained after strengthening, but multiplied with 1.39 to represent the period
before strengthening. The standard deviations within half meter intervals are displayed.
• A section modulus of 73.8m3 at the measurement position, rather than 53.9m3 .

Characteristic data from a few records are listed in Table 6.4. The heading, β, indicates
that the vessel was running slightly off head seas with a significant wave height of about
4.4-5m and a speed of 9.7-11.9kn. All data were obtained during the same day, and they
represent sea states and response levels, which give the main contribution to the fatigue
damage, (Moe et al. 2005). The peak period of the response, TP,r (corrected based on
speed and heading), is compared to the peak period from the wave measurements, TP .
The peak period of the response, considering the shape of the transfer function of the
bending moment, was expected to give an alternative estimate of the peak period of the
waves, although tending to produce higher response periods than wave periods, when the
wave period is lower than the peak period of the transfer function. The response peak
period is consistently larger than the measured peak period from the waves. Due to the
significant discrepancy by 24% in average between the two estimates, and that the stan-
dard deviation of response periods is 43% higher than from the wave radar, the measured
wave periods may be somewhat deficient. This was also indicated by Moe et al. (2005),
who illustrated a limited variation of the peak periods within a small range of wave heights
compared to the North Atlantic scatter diagram. It should be mentioned that the North
Atlantic scatter diagram is also related to significant uncertainties.
Table 6.4: Overview of some records from ballast condition after strengthening (not cor-
rected).
YY.MM.DD.Hour HS [m] TP /TP,r [s] Bn U [m/s] β [0 ] HF/WF Std.dev. stress [MPa]
05.01.10.09:00 4.5 10.9/13.1 5 6.1 353 3.6/10.8
05.01.10.10:00 4.5 10.9/14.1 5 5.9 344 2.8/10.6
05.01.10.11:00 4.5 10.9/14.6 5 5.4 339 2.8/11.4
05.01.10.12:00 4.7 11.0/14.5 5 5.3 336 2.6/11.4
05.01.10.12:30 4.8 11.4/15.7 5 5.1 338 3.0/11.8
05.01.10.14:00 5.0 12.0/15.2 5 5.0 348 3.0/12.1
05.01.10.18:00 4.4 11.9/13.3 5 5.1 349 3.0/10.2
05.01.10.19:00 4.5 12.2/14.9 5 5.5 347 3.1/11.6
05.01.10.20:30 4.7 12.4/13.4 5 5.3 348 3.3/10.5
05.01.10.21:30 4.9 12.0/13.9 5 5.6 339 3.1/10.9
The method predicts whipping in two of the ten sea states (at 14:00 and 18:00). Figure 6.7
presents the high pass filtered 2-node nominal vibration stress and the corresponding slopes.
Several whipping events seem to be present, but only one whipping event was detected.
The vibration stress level was about 30% of the wave frequency stress for these sea states.
15
Whipping
10
HF stress [MPa]
5
0
−5
−10
−15
0 200 400 600 800 1000 1200 1400 1600 1800
5
4
Slope [MPa/s]
3
2
1
0
−1
−2
−3
−4
−5
0 200 400 600 800 1000 1200 1400 1600 1800
Time [s]
Figure 6.7: The HF stress time series with indication of whipping events and slopes in
ballast condition after strengthening, 05.01.10.18:00 on starboard side.
Figure 6.8 presents the corresponding cumulative probability distribution of the slope pro-
cess. The criterion on the positive slopes is apparently a bit strict.
A few records with higher HF responses are listed in Table 6.5. In general, these records
were taken from higher sea states with lower speeds. There is still a discrepancy between
0.9
0.8
0.7
0.6
0.5
P
0.4
0.3
0.2
0.1
0
−5 −4 −3 −2 −1 0 1 2 3 4 5
Slope value [MPa/s]
Figure 6.8: The cumulative probability distribution, P , of the slope process versus the
criteria for stress in ballast condition after strengthening, 05.01.10.18:00 on starboard side.
the measured peak period and the peak period from the response. In this case the variation
of the periods are similar, but the average ratio of the response peak period (corrected for
heading and speed) and the wave radar peak period of 1.19 is still high.
Table 6.5: Overview of some records from ballast condition after strengthening (not cor-
rected).
YY.MM.DD.Time HS [m] TP /TP,r [s] Bn U [m/s] β [0 ] HF/WF std.dev. stress [MPa]
04.12.15.22:30 5.7 10.8/14.5 8 3.0 4 4.7/14.8
04.12.15.23:30 6.1 11.5/13.2 8 2.8 11 5.2/10.2
04.12.16.03:00 7.2 13.1/14.1 8 3.3 34 4.6/11.1
04.12.16.06:00 7.2 13.3/16.2 8 2.7 18 4.6/17.1
04.12.16.07:00 6.9 13.7/11.3 8 3.2 20 4.3/10.6
04.12.16.08:30 6.8 13.7/13.5 8 2.9 9 4.0/14.6
04.12.16.09:30 6.4 12.1/14.7 8 3.0 22 4.3/20.1
04.12.16.10:00 6.4 13.6/14.0 8 3.3 22 4.1/17.4
04.12.16.14:30 4.9 12.1/14.8 8 4.9 354 4.2/10.5
04.12.16.16:30 4.6 9.1/16.6 8 4.3 339 4.3/16.5
04.12.16.18:00 5.2 9.1/12.0 7 3.7 331 5.1/12.3
05.01.14.05:00 5.8 13.7/14.9 6 3.5 16 4.3/14.7
05.01.14.05:30 5.4 13.4/17.5 6 4.3 21 4.5/13.2
05.01.14.09:00 5.4 11.3/14.7 6 4.4 20 4.2/12.5
05.01.14.09:30 5.5 12.1/15.3 6 4.2 24 5.1/14.1
05.01.14.13:00 5.9 13.7/13.6 6 3.8 31 4.1/15.0
05.01.14.18:00 6.6 14.3/14.8 6 3.3 35 4.1/14.2
Whipping was detected in three of these 17 sea states (23:30, 14:30, 05.01.14.09:30), while
some apparent whipping events were not identified as in Figure 6.9 (the slope was high but
just below the criterion). The whipping criteria appear to be too strict, explained by the
30
20
HF stress [MPa]
10
0
−10
−20
−30
0 200 400 600 800 1000 1200 1400 1600 1800
7
5
Slope [MPa/s]
3
1
−1
−3
−5
0 200 400 600 800 1000 1200 1400 1600 1800
Time [s]
Figure 6.9: The HF stress time series with no indication of whipping events and slopes
in ballast condition after strengthening, 04.12.15.22:30 on starboard side.
following uncertainties
1. The measured peak period may be too low, resulting in too high calculated wave
heights for linear and second order resonance.
2. The criteria were based on transfer function values from model tests at full speed,
while significant speed reduction was observed in the different sea states.
3. The criteria were based on transfer function values from model tests in head sea,
while in many cases the direction was slightly off head sea.
4. The criteria were based on transfer function values from model tests in long crested
seas, while significant spreading was present in the different sea states, e.g. (Storhaug
et al. 2003).
The criteria need to be improved in order for the method to be useful for prediction of
whipping events in full scale measurements.
To reduce the uncertainty related to point one, the response peak period was given as in-
put instead of the wave peak period. Whipping was then detected in eight of 17 sea states
(instead of three). From the detected whipping events an artificial whipping process was
established and compared to the total vibration process, as illustrated in Figure 6.10 for
a single sea state. The standard deviation of the artificial whipping process was 2.0MP a,
while the total measured HF process gave 4.5MP a. The corresponding cumulative proba-
bility distribution compared to the criteria is presented in Figure 6.11. It still appears that
the criteria are a bit too strict, hence other effects such as speed and spreading needs to
be accounted for as well.
20
Whipping proc. [MPa] 10
−10
−20
0 200 400 600 800 1000 1200 1400 1600
20
HF stress [MPa]
10
−10
−20
0 200 400 600 800 1000 1200 1400 1600
Time [s]
Figure 6.10: The HF stress time series compared to an artificial whipping process based
on the detected whipping events in ballast condition after strengthening, 05.01.14.05:30 on
starboard side.
0.9
0.8
0.7
0.6
0.5
P
0.4
0.3
0.2
0.1
0
−5 −4 −3 −2 −1 0 1 2 3 4 5
Slope value [MPa/s]
Figure 6.11: The cumulative probability distribution, P , of the slope process versus the
criteria for stress in ballast condition after strengthening, 05.01.14.05:30 on starboard side.
A strict criterion was applied to detect whipping, and still whipping was detected. This
indicated that the whipping process was of similar importance as the springing process
also in full scale.
A whipping event is illustrated in Figure 6.12. It is observed that the first large vibration
half cycle occurs in hogging, which agrees with the model tests. Moreover, the vibration
contribution to the total stress level is evident, and the total stress amplitude in hogging
reaches almost 60MPa in this sea state of 5 meter significant wave height. The nominal
stress level in deck is 6% higher, and the ship after strengthening refers to a section modulus
far beyond rule minimum. Therefore, a nominal stress level of about 90MPa in deck may
be expected in this sea state on a similar ship with more standard optimized scantlings.
This is considerable, and the effect of the wave induced vibrations on the extreme loading
should be evaluated further. The importance of whipping occurring during the hogging
cycle due to a downwards force should also be further assessed, and numerical predictions
should be able to reproduce this effect before they can be considered as reliable.
50 50
Total
40 WF 40
30 30
20 20
HF Stress [MPa]
10 10
Stress [MPa]
0 0
−10 −10
−20 −20
−30 −30
−40 −40
−50 −50
−60 −60
1470 1475 1480 1485 1490 1470 1475 1480 1485 1490
Time [s] Time [s]
Figure 6.12: A whipping event after strengthening, 04.12.15.23:30 on starboard side.

Positive stress is sagging (compression in deck). Cut off frequency of WF response at 0.08
and 0.3Hz.
6.5 Wave and vibration frequency fatigue damage

Rainflow counting was applied to calculate the fatigue damage from the stress records ob-
tained from full scale measurements. The calculation procedure was basically the same as
for the model tests, and did not introduce new uncertainties. A few selected results of in-
terest are presented in the following. Further details may be found in (Moe and Holtsmark
2005; Moe et al. 2005; Storhaug et al. 2006).
The measured total fatigue damage, wave and vibration damage for the vessel before
strengthening are presented in Table 6.6 in addition to the number of half hour recordings.
The vibration damage constituted 61% of the damage in ballast condition and 13% damage
in cargo condition. The ship encountered head seas in ballast and following seas in cargo
6.5. WAVE AND VIBRATION FREQUENCY FATIGUE DAMAGE 177
condition most of the time, which explained the lower contribution in cargo condition.
Table 6.6: Measured fatigue damage from half hour recordings in the period July 1999 to
June 2000 before strengthening in North Atlantic trade. Taken from (Moe and Holtsmark
2005).
Load. cond. No. of rec. Tot. damage Wave damage Vibration damage
Ballast 3069 0.0515 0.0199 0.0316
Cargo 2827 0.0280 0.0243 0.0037
The time in ballast, cargo condition and port was roughly 33% each in this North Atlantic
iron ore trade. A time extrapolation based on the recordings suggested a fatigue life of 6.3
years, which was roughly twice the fatigue life predicted based on the model tests in head
seas. The difference was explained by the effect of wave headings, wave energy spreading,
routing and the annual variations encountered in the full scale measurements. The routing
was indicated to reduce the fatigue damage to half, (Storhaug et al. 2006). The model
experiments and full scale measurements were then in satisfactory agreement considering
the sensitivity of fatigue life estimates to the various sources of uncertainties.
The fatigue damages based on the period after strengthening are presented in Table 6.7 for a
period of more than 4 years. The annual damage before strengthening is 4.1 higher than the
annual damage after strengthening based on the number of recordings. Moe et al. (2005)
indicated that the annual variations was moderate (maximum about 1.5 times minimum
for both ballast and cargo condition) from the measurement period after strengthening,
hence the annual variations for a trading ship may be somewhat less than expected for a
stationary FPSO. The sectional modulus increased from 52.7m3 before strengthening to
73.2m2 after strengthening at the measuring position. The ratio of 1.39, representing the
difference in stress level for the same bending moment, to the power of 4 is 3.7. The power
of 4 was estimated by (Storhaug et al. 2006) for relevant stress levels and the two-slope
SN-curve. The expected ratio of 3.7 is in fairly good agreement with the measured ratio of
4.1, considering the natural annual variations of the wave climate. The vibration damage
constitutes 52 and 18% of the total damage in ballast and cargo condition, respectively.
The total contribution of vibration damage considering all records drops from 44% to 40%,
and the change considering the 10% increase of the natural frequency is small. I.e. the
springing level was possibly reduced after strengthening, but 10% more vibration cycles
counteracted the reduction of the vibration damage. Increasing the strength was not
effective to reduce the relative contribution from vibration, but it reduced the nominal
stress level and thereby the overall fatigue damage.
Fatigue damage for ship 5 in Table 6.1 was calculated at a position with section modulus
of 70.6m3 . The ship operated in different trades such as North Atlantic, Europe-Brazil
and Europe-South Africa. The resulting fatigue damage from one and a half years of mea-
surements are presented in Table 6.8. The vibration damage constitutes 65 and 33% of the
Table 6.7: Measured fatigue damage in North Atlantic for measurement period October
2000 to May 2005 after strengthening. Taken from (Moe and Holtsmark 2005).
Ballast 12679 0.0531 0.0255 0.0275
Cargo 12079 0.0290 0.0235 0.0053
total damage in ballast and cargo condition, and 55% overall. The contribution exceeds
that from ship 1. One reason is that ship 5 operated in less severe trade, which is also
reflected in less total fatigue damage. Moreover, (Storhaug et al. 2006) showed that during
the same period in North Atlantic trade, ship 5 vibrated relatively more than ship 1, when
the stress level was corrected with respect to their dimensions. This may be caused by
slightly different bow shape, less operation draft for ship 5 and different encountered sea
states. In any case, the full scale measurements reveal a considerably higher contribution
to the fatigue damage from wave induced vibrations than indicated from the model tests
in head seas.
Table 6.8: Measured fatigue damage for measurement period November 2003 to May 2005
for ship 5 in all trades. Taken from (Moe and Holtsmark 2005).
Ballast 5508 0.0062 0.0021 0.0040
Cargo 5195 0.0027 0.0018 0.0009
The vibration damage constitutes 18% of the total damage in cargo condition in Table 6.7,
which also indicates that the ratio of the wave damage between the cargo and ballast
condition is close to 0.9. The model tests indicated that the ratio of the wave damage
between the cargo 2 and ballast 2 condition in head sea was 1.4 (average of ratio at full and
reduced speed). The large observed difference between 0.9 and 1.4 is mainly a consequence
of the number of cycles encountered in following seas compared to head seas. Accordingly,
the prediction of the vibration contribution in the North Atlantic iron ore trade based on
the model tests was underestimated compared to the full scale measurements for several
reasons.
1. HF stress in ballast condition was higher in full scale than in model scale, mainly
due to lower damping in full scale and low wave energy in the high frequency tail in
the model tests.
2. The WF stress in ballast condition was lower in full scale than in model scale, due to
wave energy spreading and possibly due to slightly off head seas to reduce pounding.
3. The cargo condition contributed to the vibration damage in full scale, but the vibra-
tion damage contribution was neglected in the predictions based on the model tests
under the assumption of following seas.
6.5. WAVE AND VIBRATION FREQUENCY FATIGUE DAMAGE 179
4. The wave damage from cargo condition in head seas in the model experiments exceeds
the wave damage in following seas by roughly 50% due to the number of cycles
encountered.
5. See also reasons listed in Section 5.4.5.
The wave (WF) and vibration (HF) half hour damage as a function of significant wave
height for the ballast condition of the ship (ship 1) after strengthening is presented in Fig-
ure 6.13. The relative importance of the vibration is reduced for increasing wave heights
mainly due to the speed reduction. Moreover, the wave damage as a function of the wave
height is apparently linear at higher wave heights. Assuming that the wave frequency stress
is proportional to the significant wave height, a more rapid growth would be expected. The
number of encountered cycles are however reduced in higher head waves, the peak periods
of the encountered sea states influence the wave frequency stress level and Figure 6.13 cov-
ers all encountered headings. With reference to the latter, it has been said that the captain
tries to avoid running straight into the waves in higher sea states to reduce the hull girder
loading (reduce pitching and pounding and accept some small roll instead), which are
displayed onboard by the monitoring system. This has so far not been documented by as-
sessing the higher sea states more thoroughly, but it is indicated in Table 6.4 and Table 6.5.
−5
x 10
8
Wave Damage
7 Vibration Damage
Average half hour fatigue damage
0
1 2 3 4 5 6 7 8 9
HS [m]
Figure 6.13: Avarage half hour WF and HF (top) damage as function of the significant
wave height in ballast condition after strengthening covering all headings. Taken from (Moe
et al. 2005).
The total contribution to fatigue damage is presented in Figure 6.14 for ballast condition
after strengthening. A significant wave height of 5m is the most important wave height
for the total fatigue damage in this trade. This agrees well with the model experiments in
Section 5.4.5.
0.012
Wave Damage
Vibration Damage
0.01
Total fatigue damage
0.008
0.006
0.004
0.002
0
1 2 3 4 5 6 7 8 9
HS [m]
Figure 6.14: Total WF and HF (top) damage as function of the significant wave height
in ballast condition after strengthening covering all headings. Based on 13695 half hour
records during a 4 year period. Taken from (Moe et al. 2005).
Chapter 7
Conclusions and recommendations
7.1 Brief summary

This study was carried out in the following phases
1. Literature study.
2. Participation in a workshop.
3. Definition of hypotheses (based on activities 1 and 2 and other efforts).
4. Testing some of the hypotheses through simplified analysis.
5. Testing of the hypotheses through model experiments.
6. Comparison of model and full scale measurements, and
7. Conclusions and suggestions for further work.
The literature study was summarized in a number of findings and observations related to
full scale measurements, model tests and springing and whipping theories.
The purpose of the workshop was to compare the predicted vibration response from four
state-of-the-art computer codes with measured vibration response in order to identify the
dominating excitation source. The computer codes were representative for the various ex-
isting software, and the comparison was made based on ballast condition simulated in 10
different sea states.
181
182 CHAPTER 7. CONCLUSIONS AND RECOMMENDATIONS
For blunt ships in particular, four hypotheses were defined. The force distribution along
the hull implied by these hypotheses are antisymmetric, e.g. due to wave reflection in the
bow but not in the stern. They may increase the generalized force and thereby increase
the vibration. The first hypothesis was related to the steady potential and wave elevation
around the bow (hull) in forward speed. This includes instabilities and interaction between
the steady and unsteady potential. The second hypothesis was related to water surface
motion in the bow area due to reflected waves and forward speed. The third hypothesis
included impact forces due to the former two, and the fourth hypothesis was associated
with second order sum frequency excitation due to bow reflection. The decay of the sum
frequency pressure with draught and the extent below the bottom for a vessel of finite
extent and forward speed were main issues.
The first three hypotheses were incorporated into a simplified computer code to assess the
importance of whipping due to excitation on a half conical stem flare in ballast and cargo
condition. Linear springing from undisturbed incident waves was included. The sensitiv-
ity to speed, wave period and wave height in regular and irregular waves was investigated.
Moreover, a simplified procedure was employed to estimate the speed reduction in different
sea states.
Prior to the model tests, the wave quality in the towing tank was evaluated to ensure that
it was possible to produce linear and second order springing in short waves.
A large four-segmented flexible model of the iron ore carrier was made. The towed model
had flexible joints at the three quarter lengths, and Froude scaled springing frequency was
achieved by adjustable bearings. Various responses were measured. The model tests were
carried out in ballast and cargo condition in regular and irregular waves. The effects of
speed, wave height and wave period were investigated on three different bow shapes, the
original, the original without bulb and flare, and a sharp wedge shaped bow with vertical
stem. The effect of trim was briefly considered. The results were assessed to determine the
damping, linear and second order transfer functions and fatigue damage from conventional
wave loading and vibration. Moreover, different procedures were employed to distinguish
the linear and second order contribution, and to separate whipping from springing.
Finally, results from full scale measurements were compared with model test results to
indicate the validity of the model tests.
7.2 Conclusions
Separate conclusions were derived from the different phases of the study as outlined in the
following. The most important ones are finally summarized in Section 7.2.6.
7.2. CONCLUSIONS 183
7.2.1 Literature studies

The following conclusions were derived from the literature investigation.
? Wave induced vibrations have been observed (frequently) in ships, and whenever an
attempt has been made to measure them, the measurements have confirmed the presence
of wave induced vibrations. The question is not related to whether vessels vibrate or
not, but to the level and consequence of the vibration.
? Important parameters affecting the magnitude of vibration are dimensions, natural pe-
riods, nominal stress level (strength standard and steel type), draft, bow shape (stern
shape and midship shape), damping, speed, heading, sea condition (spreading, peak
period, wave height and tail behaviour) and trade.
? The springing investigations were mostly related to blunt ships, while the whipping
investigations were related to slender ships such as container ships with significant flare.
Most of the full scale measurements were related to blunt ships, and some of the extreme
whipping events were actually measured on these ships. Public literature of measured
full scale vibration damage of high speed vessels, slender ships or offshore ships was not
identified.
? The full scale vibration damage has been underestimated in previously published litera-
ture. The vibration damage was estimated based on the filtered high frequency response,
rather than the difference between the total fatigue damage and the filtered wave dam-
age.
? The elasticity of the hull girder tends to reduce the wave bending moment in the wave
frequency regime. This will be beneficial from a fatigue point of view.
7.2.2 Workshop; Introductory assessment of full scale measure-

ments
The following main conclusions were made based on the assessment of the first year of
measurements from the iron ore carrier.
? Bottom exit was not detected during the measurements.
? Wave induced vibrations were basically continuous.
? The wave induced vibration increased for increasing Beaufort strength, and wind was a
better measure for the vibration level than the measured waves.
? The difference of the vibration response for head and beam seas was small, while the
vibration response in following sea was much smaller.
? The vibration level in ballast condition was larger than in cargo condition for similar
head sea conditions.
? The vibration damage constituted 44% of the total fatigue damage in deck amidships.
? For this iron ore trade, the ship encountered prevailing head seas in the ballast condition
on the westbound leg and following seas in the eastbound cargo leg.
? The accuracy of the wave radar measurement system was questioned. The high frequency
tail was most deficient.
? The nonlinear damping method gave lower predicted damping ratio (0.44%) than the
spectral method.
? It was the first time the fatigue damage from wave induced vibrations from full scale
measurements was handled in a proper manner, i.e. taking the difference between the
total damage and the wave frequency damage by Rainflow counting.
Comparison with numerical predictions was the main part of the workshop. A number of
people were involved. The main conclusions were
? The four computer codes failed to predict the high measured vibrations, and the observed
trend in the full scale measurements was not captured.
? The dominating excitation source was either missing or not accurately described.
? The computer codes were sensitive to the location of the natural frequency compared to
the humps and hollows in the excitation (from cancellation effects), in particular from
the strip theory programs.
? It was the first time the fatigue assessment from full scale measurements was documented
utilizing a wave radar for the purpose of determining the sea conditions.
The sea states applied in the numerical predictions was later shown to be overestimated
by a deficient wave measurement system during the period before strengthening. Conse-
quently, this fact is expected to increase the difference between the measurements and the
predictions by the four programs.
7.2.3 Investigation by simplified theoretical methods

The theoretical investigation consisted of defining the hypotheses, testing three of the
hypotheses in a simplified computer code, estimating the speed reduction, and assessing
different structural representations of the hull’s flexibility. In the following the results ob-
tained for the large iron ore carrier are described.
The surface elevation of the steady wave generated by the ship was 3m at 15kn at the
stagnation point and about minus 1m at the bow quarter. The amplification factor of
the incident head waves at the stem exciting linear and second order springing was about
three at full speed (and two at zero speed). These effects may consequently imply more
whipping excitation.
The steady wave elevation, the wave amplification at the bow and the impacts on the
stem flare were included in the simplified predictions of whipping. Linear springing by
undisturbed wave pressure was also included. A homogeneous ship was considered, and
the following conclusions were derived based on ballast and cargo condition in head seas.
? The stem flare impacts contributed insignificantly to vibration in ballast condition.
? The stem flare impacts contributed to vibration in cargo condition, and appeared to
be much more important than the linear springing for sea states, which contributed
significantly to the fatigue damage. The stem flare impacts did only partly explain the
measured vibration level.
? The bow exit force exceeded occasionally the entry force, and the conventional slamming
force might be only a minor part of the maximum force.
? The vibration was confirmed to be sensitive to speed, and realistic estimates of the speed
in different sea states were necessary.
The simplified method used to estimate the speed reduction was robust. The results were
not sensitive to the total efficiency coefficient of the propulsion side. It predicted a signifi-
cant speed reduction in relevant head sea conditions, e.g. 4-5kn reduction in 5m significant
wave height.
An analytical expression for the natural frequency and deformation shape of a homogeneous
beam including shear deformation was derived. The effect of realistic shear deformation
on the natural frequency was in the order of 6% for the 2-node vibration mode, which
was in perfect agreement with FE representation of the beam. Other effects such as axial
end pressures due to waves as well as rotational mass and damping on the mode shape
and frequency can be disregarded. A stiffness formulation of the segmented model with
rotational springs was also proven to be in agreement with results from the model tests.
An estimate of the wave heights that will excite springing from linear excitation and sum
frequency effects was made as a function of the Beaufort strength. The mean wave height
is in the range of 1-2m for Beaufort 6-8, which contributed significantly to the total fatigue
damage. The waves causing the sum frequency effect was in the order of half a meter
higher than the waves causing linear springing.
A simple analytical solution of the dynamic response from a sinusoidal (a full period) im-
pact was derived, and it illustrated that the dynamic response amplification can be up to
3 rather than 2 from conventional slamming (transient impulse like) impact load.
Predictions of linear springing require an accurate description of the oscillating loads along
the hull and the deflection shape of the 2-node mode. Reliable linear springing predictions
require a 3D hydrodynamic model, which includes the interaction between the steady and
unsteady velocity potential. The springing predictions from sum frequency effects localized
to the bow due to reflection of waves, or whipping predictions due to the excitation in
the bow or stern will be less sensitive to the vibration shape. For linear springing using
3D theory, the antisymmetric mode shape and antisymmetric distribution of oscillating
loads along the hull will make the numerical predictions less sensitive to the hump-hollow
behaviour of the excitation versus the natural frequency, since the difference between humps
and hollows will be reduced compared to 2D strip theory. 2D strip theory is thereby not
considered as applicable for springing predictions of blunt vessels.
7.2.4 Model experiments

The wave calibration measurements confirmed the following features:
? The wave frequency was as requested, but the amplitude deviated from the requested
value. All waves must be measured.
? The short waves had poor quality, and a large model was desirable to capture linear
springing. The length with good quality was less than 80m, and decreasing for increasing
wave amplitude. The effective tank length for stationary springing response was less.
? The measured irregular sea state parameters were as requested (HS was slightly low),
but the high frequency tail of the wave spectrum was uncertain.
The model was unconventionally large and cumbersome to handle. This resulted in a
number of consequences.
? In ballast condition the mass moment of inertia of each segment was too high, affecting
the 3- and 4-node mode shapes. The stiffness of the three flexible joints were adjusted
to achieve the requested Froude scaled 2-node springing frequency.
? The same stiffness in all three joints was found suitable based on numerical predictions.
The segmented model gave in any case somewhat high frequencies for the 3- and 4-node
mode shapes.
? The calibration revealed a significant uncertainty in the rotational stiffness.
? The flexible joints, in particular the bearings, were not produced with satisfactory quality,
and they gave both slightly nonlinear stiffness and nonlinear damping, which varied for
the different combinations of bow shape and loading condition. The damping ratios were
between 1 and 3.5%.
? In one of the six combinations, i.e. ballast 2 condition, a linear damping ratio of 1.0%
was achieved.
? Disturbance of the wave pattern due to reflection from the walls were detected in the
response at full speed in cargo condition, while in ballast condition the rigid body re-
sponse peak of the bending moment transfer function was captured. Wall effects affected
the rigid body response in lower speeds and irregular waves with long periods, but the
effect on the overall fatigue damage was small.
The damping was an important factor influencing the springing level and the decay of the
whipping cycles. The findings from ballast 2 condition confirmed that:
? The damping was insensitive to speed.
? Nonlinear hydrodynamic viscous effects were small and not detectable.
? The rubber sealing and water leakage contributed with a damping ratio of about 0.15%.
? The structural damping ratio was about 0.5%.
? The hydrodynamic linear damping was about 0.35%, and cannot be disregarded in nei-
ther model or full scale.
? The damping in the 2-node mode in air was insensitive to the frequency.
? The damping increased with the number of nodes; 1.4% and 3.3% for the 3- and 4-node
modes, respectively.
? The spectral method predicted about 30% higher damping than the decay curves in the
model tests and 60% higher in full scale.
? The natural frequency decayed in higher sea states due to nonlinear stiffness and possibly
due to changes in the average wet length and added mass.
? The exponent n in the spectral method (used to predict the damping) was not represen-
tative of the high frequency tail behaviour of the wave spectrum neither for the model
nor the full scale measurements.
The following results were obtained in the regular wave tests
? The linear, second (sum frequency) and third order (3 · ωe = ωs ) resonance were easily
excited in ballast condition, but the third order was less easily exited in cargo condition.
? The linear springing peak in the bending moment transfer function exceeded the rigid
body response peak by a factor of three in ballast condition in a speed of 15kn. The
sum frequency springing response level was also higher than the rigid response peak.
? The linear springing peak was sensitive to speed.

? The sum frequency springing peak reduced slowly with speed, and it was still present at
zero speed. Hump-hollow behaviour is expected in linear springing from the integration
along the ship according to the generalised force in Eq.(2.23), since resonance is achieved
at different wave lengths at different speeds. No hump-hollow behaviour was observed
for the sum frequency effect, and this indicates that the dominating sum frequency effect
was localized to the bow.
? The hump-hollow behaviour in the bending moment transfer function was observed, but
the hollows were far from zero.
? The analytical surface motion in the bow at forward speed was slightly smaller than
estimated from measurements in short waves, as explained by small run up not accounted
for in the analytical estimate.
? The added resistance in waves at forward speed agreed fairly well with the analytical
short wave approximation. The measurements were sensitive to proper zero setting
and measured calm water resistance. The nondimensional peak due to maximum rigid
body motions in heave and pitch was in fair agreement with assumptions made in the
involuntary speed reduction assessment.
The sum frequency transfer function was investigated in more detail. It was necessary to
correct the response for the different damping levels. Moreover, only a few bichromatic
waves were tested. This made it difficult to conclude, but the following was observed:
? The second order transfer function gave an apparently constant springing response for
increasing frequency difference of the bichromatic waves, but the monochromatic sum
frequency waves gave a local peak above the bichromatic level.
? The monochromatic response was higher in ballast than in cargo condition for 2 out of
3 bows, but the difference was moderate.
? The bichromatic response was higher in ballast than in cargo condition, and the relative
difference was larger than for the monochromatic waves.
? Apparently, the bulb reduced the monochromatic response slightly.
? The monochromatic response was reduced only moderately by using the wedge bow 3 in
stead of the blunt bow.
Tests in irregular sea states were conducted at full and reduced speed. The following
conclusions were derived:
? At full speed in ballast condition the vibration damage was dominating (2/3 of the total),
while it was reduced to 1/4 in cargo condition. The wave frequency damage was higher
in cargo than in ballast condition.
? There was no significant difference in the vibration damage between the three bow shapes
considering the difference in the damping.
? The vibration damage was smaller in the first part of the towing tank than in the second,
indicating that the energy of the waves exciting linear springing was reduced along the
towing tank. The vibration damage was also less in subsequent runs compared to the
first run for each sea state due to 3D disturbance of the wave condition.
? The vibration damage was confirmed to be sensitive to (the realistic) speed reduction.
Considering the North Atlantic ore trade with head seas in ballast condition and following
seas in cargo condition the vibration damage constituted 19% of the total at close to
realistic speeds. The figure at full speed was 51%. In similar World Wide trade at
realistic speed the vibration damage increased from 19 to 26%, but the total fatigue
damage decreased to about half.
? The effect of trim was investigated in four steep sea states in ballast condition. An
increased forward draft of 1.6m reduced the vibration and wave damage by about 1/3,
hence the reduced bottom forces were more important than the increased flare forces.
This was in agreement with the simplified simulations indicating that the stem flare was
unimportant in ballast condition.
? Fair agreement was found between predictions of standard deviation WF stress by

WASIM and the model tests in the different sea states.
An extrapolation procedure was employed to estimate the total fatigue damage from all
the cells in a scatter diagram. Empirical relations were fitted to the measurements. The
wave and vibration part was considered separately, and the following findings were found:
? The dominating seas states for the wave frequency damage for ballast condition in head
seas at realistic speeds in the North Atlantic trade were located at a 4-6m significant wave
height and 9-11s zero up-crossing period, while for the vibration damage the dominating
sea states were located between 3-5m and 8-10s.
? For cargo condition the dominating contribution to wave frequency damage were located
at the same place as in ballast condition, while the dominating sea states for the vibration
damage was located between 4-5m and 9-10.5s.
? At full speed, the location for the vibration damage was moved towards higher sea states
and periods.
? The dominating sea states were located at lower wave heights and periods for less harsh
environments (i.e. for World-wide operation).
? The locations of the dominating sea states differed only slightly for the different bow
shapes.
? For bow 2, the fatigue life in the North Atlantic was estimated to be 2.9 years by the
response surface method (compared to 3.6 years from the simplest time extrapolation
based on only the sea states tested). 5.9 years was estimated in World-wide trade. The
contribution from the vibration damage was reduced from 18% for bow 2 to 10% for bow
3. Considering the higher damping for bow 3, the bow 3 was not an effective design to
avoid the vibrations.
A method was developed to separate high whipping response from springing response by
considering the envelope process of the 2-node vibration. Such a method is relevant in con-
nection to further development of numerical tools. Based on input values of the springing
peak from the linear and second order transfer function a criterion for the maximum slope
of the envelope process was defined. A whipping event was detected when the maximum
springing slope was exceeded. The method was applied to ballast 2 condition in head seas,
and it was evaluated based on the amidship bending moment.
? Whipping was identified in 31 of 33 sea states. The two sea states without whipping
corresponded to tests at reduced speeds, and the criterion was not strictly valid. This
means that whipping was identified in all valid sea states. The whipping events were
also interpreted as whipping based on visual inspection, while some apparently smaller
whipping events were not identified.
? The magnitude of the whipping stress was of similar magnitude, in terms of standard
deviation stress, as the springing stress already at a 3m significant wave height.
? The relative importance of the whipping process increased with peak period and wave
height.
? The criterion was based on springing values obtained at full speed, hence it was somewhat
strict when used at reduced speed. Still, the method detected whipping also at reduced
speeds except in two sea states.
? The probability of whipping per vibration cycle was up to 3%, meaning that a whipping
event would occur in average once every minute in full scale.
The measured acceleration may also be used to indicate bow impacts. A criterion was
tested in ballast 2 condition. It was based on the standard deviations of the acceleration.
The criterion was not exceeded in any sea state, even though the acceleration amplitude
was found to exceed the acceleration of gravity in one case. It is proposed that such a
criterion should be based on amplitudes, since single important events may be smeared
out by considering standard deviations. The ratio of the amplitude to three times the
standard deviation was above one in case of flat bottom impacts.
A method was applied to compare the relative importance of the linear springing and
the monochromatic sum frequency effect. It was presented as a function of the Beaufort
strength, and two wave spectra were considered.
? Both the linear and sum frequency excitation contributed.

? The sum frequency effect started to dominate from Beaufort strength 6 and above.
? The behaviour and magnitude depended on the wave energy spectrum. The Torsethau-
gen spectrum predicted higher vibration response than the wind based Pierson-Moskowitz
in high wind speeds.
Different time series were displayed to reveal whipping and springing events in different
situations, e.g.
? Often, the first significant vibration half cycle in whipping corresponded to hogging
indicating a downward force rather than an upward force.
? Whipping occurred both when the water rised high up in the bow, and when the water
surface reached the bottom bilge/flare area. The latter gave higher whipping response
in both ballast and cargo condition, and it occurred also for bow 3.
? Whipping occurred also for the conventional bottom impact, which was detected in
realistic speeds in ballast, but not in cargo condition. Still, the first vibration half cycle
occurred in hogging, and the next half cycle in sagging could increase in magnitude
due to the subsequent slamming. The increase due to the upward force was not always
observed, even though slamming pressures in the bow bottom was measured.
? Slamming pressures in the aft ship were observed at realistic speeds, but came possibly
from wall interaction. These slams did not give significant vibration responses.
? Green seas were observed to give vibration, but the magnitude was small. The effect
was somewhat unrealistic, since violent slamming occurred when the water struck the
vertical superstructure, which was used to prevent the water from entering the model.
This occurred in particular in cargo 2 condition when the stem flare was removed.
? Violent horizontal slamming impacts were observed for the blunt bows in short small
waves and in high forward speed. For regular wave amplitudes below 1m, the water
spray went occasionally above the deck level at the bow quarter in ballast condition.
This was apparently caused by the combined effect of steady and unsteady waves and
possible instabilities causing breaking waves and subsequent small relative angles be-
tween the stem and water surface (high speed camera may be utilized in future tests).
The slamming itself was not considered important for the 2-node vibration.
7.2.5 Full scale measurements

Extensive full scale measurements of the large iron ore carrier were carried out by DNV.
The measurement periods covered one year before and four year after strengthening in
the North Atlantic iron ore trade. The evaluation was summarized in three publications.
Recently, further investigation has also been carried out on a container vessel. First a few
general comments are made.
? Evaluation of wave data after a certain measurement period is not common practice.
Proper calibration/setting/adjustment of the wave radar measurements was necessary
to obtain reliable wave heights. Uncertainties were also related to the measured peak
period, which was suggested to be compared by the peak period from the measured
stress response spectrum for each half hour record.
? The wind should be measured simultaneously with waves. The measured wind speed
and angle should be corrected for the forward speed of the vessel, and the wind speed
should also be corrected down to 10m above the still water level, which is the reference
level of the Beaufort scale.
? The time in ballast, cargo condition and in port was 33% each.
? The routing reduced the fatigue damage to half considering the difference between the
directional scatter diagram for North Atlantic trade (from the supplier Argoss) and
the directional wave scatter diagram measured onboard. Accounting for the effect of
routing, the fatigue lives from full scale and model tests were in fair agreement, although
the relative importance of vibration damage differed.
? The contributing vibration damage came from the vibration cycles superposed on the
wave frequency stress cycles, rather from the high pass filtered damage isolated.
? Based on the assessment of measurements it was found that the total damage based on
the two slope SN-curve should be scaled by the ratio of stress level to the power of 4
rather than 3 for the relevant measurements (in case a higher section modulus or SCF
were considered).
? The full scale measurements carried out by DNV confirmed that the wave induced vibra-
tions on a container vessel was of similar importance as documented for the ore carrier.
The following conclusions were derived based on the comparison between full scale and
model tests.
? The damping ratio in full scale was in the order of 0.5-0.6% for several ships, while the
damping in the model tests was 1.0% in ballast 2 condition. The damping was basically
linear.
? The damping appeared to be only slightly higher in cargo than in ballast condition and
only slightly higher after strengthening than prior to strengthening. The differences in
damping ratio appeared to be in the order of 0.1%.
? The damping predictions from the spectral method was well above the damping from
whipping decay curves (0.8 versus 0.5% in ballast condition in full scale), and the spectral
method should be used with care, i.e. the damping estimates should be corrected.
? The measured speed reduction in head sea waves was in fair agreement with the predicted
speed reduction from the simplified method. The measured speed reduction did not
indicate voluntary speed reduction for this ship, and too low speed (0kn) was applied
for the highest sea state of 9m significant wave height in the model tests.
? The standard deviation of the wave frequency stress in head seas in full scale was some-
what below the WASIM predictions and model test results in ballast condition, probably
due to wave energy spreading. Cargo condition displayed a similar magnitude of the wave
frequency stress as ballast condition as a function of the wave height.
? The standard deviation of the vibration stress obtained in the full scale measurements
was higher than that of the model tests. The difference was due to lower damping in full
scale and less wave energy in the high frequency tail in the model tests.
? Whipping was predicted also in full scale. The whipping criterion was valid for pure
head sea at full speed, which partly explained the smaller numbers of detected whipping
events in full scale. The criterion needs to be refined to detect whipping in full scale
accounting for realistic speed and wave spreading.
? Prior to the strengthening of the ship, the vibration damage measured in full scale was
44% of the total damage for typical details in deck, compared to 18% according to the
model tests. The difference was explained by e.g. smaller wave frequency damage and
higher vibration damage in ballast condition in full scale, less wave frequency damage
in cargo condition in full scale (encounter period effect), and significant contribution
of vibration damage in cargo condition in full scale. The relative importance of the
vibration may also increase for a range of off head sea directions in full scale.
? The estimated fatigue life based on full scale measurements was 6.3 years, while it was
2.9 years from model tests (with bow 2). In addition to the effects mentioned above, the
effect of routing was significant.
? After strengthening, the natural period increased by 10% and the total damage decreased
by 78% (based on SCF of 2.0). The vibration damage now constituted 40% of the
total damage. Considering the annual variations, which were moderate, it can not be
concluded that the difference due to the increased natural frequency was significant.
? The most important significant wave height for both the wave and vibration damage for
the vessel in ballast condition was 5m. This was in fair agreement with the model tests,
which predicted the dominating sea states to be from 4 to 6m.
? The fatigue damage in ballast condition increased rapidly with increased wave height
in the full scale measurements, but the relative importance of the vibration damage
decreased. This was also the case for the model tests.
? Whipping from the model tests was observed to increase the hogging moment of up to
60% increasing the nominal stress level up to almost 100MP a at full speed in a sea
state with a significant wave height of 5m. The importance of this elastic amplification
was confirmed in a similar sea state in full scale, indicating 90MP a amplitude in a
sea state with a significant wave height of 6m. This suggests that further assessment
of extreme events should be carried out, when such high levels are possible already at
relative moderate sea states.
7.2.6 Contribution/major findings

A unique set of data related to the effect of wave induced vibrations on the fatigue damage
was obtained based on extensive model tests in head seas considering two loading con-
ditions and three bow shapes. New hypotheses were presented intended to increase the
understanding and explaining a significant part of the vibration. The results were discussed
in relation to published information, and the model test results were actually compared to
measured stress results supplemented by wave radar recordings of the same vessel operat-
ing in a North Atlantic iron ore trade. Many uncertainties have been emphasized. The
major findings are listed in the following.
? Conventional ships experience wave induced vibrations, which contribute to the fatigue
damage and extreme loading. Based on this investigation, the fatigue damage contri-
bution from the wave induced vibrations is of similar importance as the fatigue damage
caused by the wave frequency response. This is confirmed for blunt ships, and it has
also been indicated for more slender vessels.
? Existing prediction tools are not capable of predicting reliable vibration responses for
fatigue assessment (and extreme loading) in particular for blunt ships.
? Earlier full scale measurements/publications underestimated the contribution of fatigue

damage due to the wave induced vibrations.
? The vibration cycles superimposed on the conventional wave frequency cycles constituted
the important contribution in harsh environment.
? Linear theory is found to explain a significant part of the wave amplification measured
at the stem line in the bow.
? The added resistance in waves was sensitive to accurate measurements in the total re-
sistance in waves and in calm water. It was confirmed that the linear theory for short
waves was in reasonable agreement with the measurements.
? The observations confirmed a significant modification of the waves along the hull giving
small relative motion in the aft part, and the important vibration excitation was related
to the bow area. The 3D effects must be accounted for to capture both the linear and
second order excitation due to short waves.
? Detectable whipping was identified in both the model tests and full scale measurements.
The model tests indicated that the whipping process in ballast condition was of equivalent
importance as the springing process in the fatigue dominating sea states.
? Both springing and whipping excitation must be included in reliable numerical predic-
tions of fatigue damage.
? The total damping ratio of these large bulk carriers was in the order of 0.5-0.6% in
ballast condition, and up to 0.1% higher in cargo condition. The damping from container
vessels appeared as distinctly higher possibly due to the structural damping induced by
the container stacks. If this is a general finding, this is beneficial for container vessels
from a fatigue point of view.
? The damping was basically linear and speed independent. The hydrodynamic damping
appeared to be the dominating source in full scale based on the blunt vessels.
? Increasing the natural frequency of the ship did not effectively reduce the relative im-
portance of the vibration damage.
? The involuntary speed reduction of importance for fatigue damage was considerable in
the head seas in the North Atlantic trade. Realistic speeds must be estimated and used
in reliable predictions of the vibration damage (and the wave frequency damage due to
the number of encountered cycles). The involuntary speed reduction behavior in cargo
and ballast condition as a function of the significant wave height was similar.
? Model tests underestimated the contribution of vibration damage and overestimated the
total damage for various reasons. Such tests need refinements to be used directly in
design predictions.
? The sum frequency effects were of equivalent to more important than the linear springing
in the fatigue dominating sea states in the North Atlantic environment.
? The most important sum frequency effect for blunt ships appeared to be caused by
bow reflection, and the monochromatic sum frequency effect gave higher vibration than
the bichromatic sum frequency effect, which had apparently a constant nondimensional
resonance level as a function of the frequency difference.
? Increasing the forward draft reduced the vibration level significantly, confirming that the
bottom forces were more important than the stem flare forces.
? The bulb may reduce the monochromatic sum frequency effect slightly, and the draft
appeared to affect the bichromatic sum frequency effects more than the monochromatic
sum frequency effect.
? The stem flare impacts did not contribute significantly to the vibration in ballast con-
dition, but the stem flare impacts contributed in cargo condition. The high whipping
responses originated in particular from rapid surface motion in the bilge area at the bow
quarter in particular in ballast condition, but also in cargo condition. The whipping
cycles started with its first half cycle in hogging due to a downwards force. This was
also shown to be the dominating behavior in bottom impact, which occurred occasion-
ally in ballast condition at realistic speeds in the highest sea state. The combined effect
of steady wave elevation (negative) and amplification of the surface motion due to bow
reflection increased the probability of bottom bilge flare impacts.
? The whipping was often excited by first a downward pull and then an upward push,
which may increase the dynamic amplification with a factor of 3 (rather than 2).
? Whipping was often initiated when the bow was going upwards, but did also occur when
the vessel had only small heave and pitch motions in steep and moderate sea states.
? The sharp bow did not result in a considerable reduction in the vibration damage. Strong,
but reduced, sum frequency vibration was confirmed as well as whipping excitation in
the bilge area at bow quarter. Reducing the springing effect by a drastic change in the
bow geometry, was not an effective means to reduce the vibration damage, since the
whipping was still present.
? Based on the results of the sharp bow and vibration damage measured on an old de-
sign container vessel, vibration damage will contribute also on slender ships with high
powering.
? Routing reduced the total fatigue damage, but will increase the relative importance of
the vibration damage.
? Routing should not be taken advantage of if the vibration damage is disregarded in the
fatigue assessment.
7.3 Suggestions of further work

The objective of the present research was to reveal the important mechanisms causing wave
induced vibrations and thereby additional fatigue damage. The findings and experience
gained serve as basis for suggestions of further work. A few new questions were raised, and
recommendations to increase the likelihood of success in future model tests are provided
in the following.
The linear springing, sum frequency effects and whipping were identified as significant
sources to the wave induced vibrations. The next step is to capture these effects in numer-
ical simulations. The following efforts are suggested.
? The importance of the interaction between the nonlinear or linear steady and linear
unsteady potential should be investigated in forward speed for the purpose of more
7.3. SUGGESTIONS OF FURTHER WORK 197
accurate linear springing predictions. A 3D code is necessary to capture the modification

of the linear excitation along the hull.
? The second order free surface condition (and body boundary conditions) should be in-
cluded in a 3D code with forward speed. To limit the computational effort yet maintain
the accuracy, a sub domain in the bow area, where the reflection effect is important,
may be utilized, while the remaining mesh can be more coarse. Reflection effects due to
different mesh size must be handled properly.
? The whipping effect may be based on a 3D ’slamming’ approach with a high frequency
surface condition applied from zero draft to maximum draft. The time derivative of the
added mass times the relative velocity may be determined from the wet panels below
the disturbed water surface rather than the undisturbed incident waves often considered.
The steady wave elevation ought to be included. The nonlinear effect considering Froude-
Kriloff and restoring must also be applied below the disturbed water surface.
The decay of the sum frequency pressure versus the depth and the effective extent below the
bottom for a vessel with forward speed were not successfully obtained from the pressure
cells. Three approaches, which may be compared, are suggested to be used in further
investigation.
? A second order perturbation theory for total reflection against an infinite vertical wall
moving forward with constant speed may be solved analytically, similar to the linear
theory.
? A 2D numerical model in the longitudinal direction may be analyzed with second order
free surface condition and body boundary condition. A restrained model, which is not
pitching or heaving, is a useful approximation for large ships.
? Model testing may be carried out to measure the second order pressure of a restrained
model (to avoid the high pressures from radiation). An academic bow shape with a small
stem/bilge radius is recommended for comparison with the two theoretical approaches
given above.
It may be possible to investigate the correlation between waves, bow bottom pressure and
bending/vibration response to confirm which effects that contribute most. This may be
carried out by Hilbert-Huang transformations or alternatively by Wavelet transformation.
It was confirmed that the real vessel vibrated significantly in head sea, but the vibra-
tion did not decay significantly with heading towards beam seas. The effect of wave energy
spreading was also disregarded. These effects may be investigated in the ocean basin based
on the same model. The pointed bow intended to reduce the reflection effect in head seas
may also be less effective in quartering seas. The effect of various spectra is also of interest.
To differ between the linear and second order effects (and other effects), wave spectra may
be modified to include or remove different part of the spectra to distinguish between the
important effects in irregular sea states, e.g. the high frequency tail of the spectra may be
removed to exclude linear springing, or part of the spectrum may be removed to reduce
the sum frequency excitation.
The horizontal forces due to fluctuations in the propeller trust or due to stem slamming
were not considered as a contribution to the vertical vibration. It should be confirmed
that these effects are small in amore sophisticated manner. The former was not relevant
in the present model tests. The latter effect can not be excluded as a contribution to
springing, even though the horizontal slamming force attacks the vertical stem close to
the neutral axis in ballast condition. This has been indicated as a minor contribution in
previous research, but it can be confirmed in model experiments by shifting the neutral
axis by a refined flexible joint. For low speed (rpm) engines the half frequency effect due
to pitch moment from the engine is also a possible excitation source to be investigated in
heavy seas. This is e.g. relevant for large container vessels, but it is normally disregarded.
Stern impacts are also a matter of concern, though not believed to be important for the
sea states with high forward speed dominating the contribution to the fatigue damage. It
may however be an issue in extreme events with lower forward speeds. It is relevant on
ships with flat overhanging sterns.
Whipping due to flat bottom impacts was observed at realistic speeds in moderate sea
states in ballast condition. Significant increase of the total bending moment due to the
flexibility of the hull was observed. The effect of whipping on the extreme loading should
be investigated in particular in ballast condition in head seas at realistic speeds. To reduce
the influence of wall interaction in a towing tank, MLER waves may be utilized. Alterna-
tively, an ocean basin may be used for this purpose.
The effect of higher modes on the fatigue damage and the maximum whipping response
may be determined in order to simplify the model design dependent on ship type and ex-
citation source. E.g. one flexible joint may be used if linear springing is insignificant and
the excitation is localized in the bow or stern area.
The sharp pointed bow did not reduce the wave induced vibrations to an insignificant level.
Considerable contribution from the vibration damage has also been confirmed for an old
design container vessel. Hence, the effect of wave induced vibration on the fatigue damage
should also be investigated by measurements (model and/or full scale) on slender ships
such as new design container vessels (wider, longer and with more flare), Ro-Ro vessels,
car ferries, cruise vessels and LNG vessels, which are all operating with low drafts. Real-
istic speeds should be considered in different typical seas states, preferably representative
for harsh environments. Some of these ships may have limited possibilities for routing, and
the effect of routing should be determined. All these ship may operate in hogging condi-
tion with tension in deck, hence in particular deck may be vulnerable to fatigue damage.
Realistic damping ratios should be estimated by full scale measurements of different types
of ships.
7.3. SUGGESTIONS OF FURTHER WORK 199
Based on the extensive model experiments carried out some practical recommendations
are given with respect to the experimental procedures
? For the purpose of investigating linear springing, the wave quality along a towing tank
must be evaluated. The appropriate length of the tank with good quality waves should
also be determined for the irregular sea states.
? Based on the quality of the short waves, an optimal model size must be selected to have
as much linear springing excitation as possible, but still such that wall effects at the
realistic speeds encountered for the most important sea states do not contribute to the
total fatigue damage.
? Three flexible joints are proposed in order to include a realistic shape for the purpose of
linear springing and higher order modes for whipping, but if the 2-node mode is dominant
for whipping and nonlinear springing one flexible joint may be sufficient.
? The neutral axis should be located as in the full scale ship.
? An improved design of the flexible joints should be considered.
? The mass distribution of each segment should be realistic, including the mass moment
of inertia.
? Towing lines at the bow and stern are sufficient, and transverse lines can be skipped
as long as the pretension is enough for directional stability. The stiffness should give
a surge period above the periods of interest. The towing forces should not affect the
bending moment. Alternatively a self propelled model may be useful in order to obtain
realistic speed and surge acceleration levels in various sea states, including variations in
the same sea state.
? Linear (or weak nonlinear) damping and linear bending stiffness should be confirmed
prior and after the model tests to confirm the quality of the flexible joints. The damping
ratio as low as 0.5 to 0.8% should be achieved. A higher damping may be acceptable, if
this is properly documented based on full scale measurements.
? Realistic speeds and vibration frequency must be considered.
? Moment should be evaluated at 3 quarter lengths.
? Using a heavy steel frame inside the model, the stability should be checked for slender
vessels.
? Calibration of the main sensors must be carried out incluing uncertainty analysis (as for
the other main parameters).
? The candidate must participate in all parts of the experiments. The candidate may be
the only one who understands the intention of the experiments, and how it should be
carried out. Common practise may not be sufficient.
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Appendix A
Theory
A.1 Potential theory

Hydrodynamic theory for the purpose of calculating global loads on ships in an engineering
context assumes, in general, potential theory, which is based on an incompressible and
inviscid liquid with irrotational flow. The velocity vector, V(x, y, z, t), can be expressed
by the velocity potential, Φ, as
V = ∇Φ (A.1)
The assumption of irrotational liquid states that the curl of the velocity field is zero
ω = ∇×V = 0 (A.2)
The assumption of incompressible liquid with constant density gives a simple relation for
conservation of mass stating that the divergence of the velocity field is zero
∇·V = 0 (A.3)
The additional assumption of invicid liquid is related to lack of shear forces on a liquid
element.
Combining the definition of the velocity potential and the zero divergence yields Laplace
equation. It expresses the conservation of the mass in the liquid domain based on the
velocity potential. This is the governing equation in potential theory, and it must be
satisfied in the whole liquid domain.
∇2 Φ = 0 (A.4)
215
216 APPENDIX A. THEORY
Laplace equation does not have much meaning in practice unless boundary conditions are
applied, and these may be highly nonlinear. Before considering the boundary conditions,
it is necessary to introduce Bernoulli’s equation, which can be expressed as
∂Φ 1
p + ρgz + ρ + ρV · V = C (A.5)
∂t 2
where z is the distance from the still water line (positive upwards). The equation relates
the hydrodynamic and ”hydrostatic” (−ρgz) pressure to the velocity field and velocity
potential. C is a constant.
The boundary condition at the free surface states that a particle on the surface will remain
on the surface. This is referred to as the kinematic free surface condition and can be
written
∂ζ ∂Φ ∂ζ ∂Φ ∂ζ ∂Φ
+ + − = 0 on z = ζ(x, y, t) (A.6)
∂t ∂x ∂x ∂y ∂y ∂z
It is nonlinear, since both the wave elevation, ζ, and the velocity potential are unknown.
Bernoulli’s equation is used to express the second free surface condition referred to as
the dynamic free surface condition. The insignificant surface tension is neglected. The
dynamic free surface condition states that the pressure on the surface is identical to the
atmospheric pressure, pa , which is normally assumed to be constant (zero) and consequently
disregarded.
∂Φ 1
gζ + + (∇Φ · ∇Φ) = 0 on z = ζ(x, y, t) (A.7)
∂t 2
The second free surface condition is also nonlinear. It may be convenient to merge the two
surface conditions into one written as
Dp
= 0 on z = ζ(x, y, t) (A.8)
Dt
D
where Dt is the material (substantial) derivative giving the change of pressure when fol-
lowing a liquid particle on the surface.
The body boundary condition on the hull surface is generally defined as

∂Φ
=U·n (A.9)
∂n
where n is the normal vector on the hull, pointing into the liquid volume. The boundary
condition states that there is zero flux through the hull. The U defines the velocity vector
of the vessel in its rigid and flexible modes. The coordinate (reference) system is normally
body fixed, which means that the velocity potential includes the free stream velocity repre-
senting the vessel’s speed. The bottom (or wall) boundary condition is handled in a similar
A.1. POTENTIAL THEORY 217
manner.
The last conditions are the far field conditions and radiation conditions (in the time do-
main). These imply that waves created by a moving ship or by reflection from the ship
radiate away from the ship, and the disturbance from the ship approaches zero when the
distance from the ship goes to infinity.
The Laplace equation with boundary conditions can be solved by various methods. The
boundary element method (BEM) is the basis of the programs WASIM, VERES, SINO
and SOST, which were used by Storhaug et al. (2003). The governing equation is the
Green’s theorem, which is written
Z Z
1 ∂φ ∂ψ
φ(x, y, z, ) = − (ψ − φ )dS (A.10)
4π S ∂n ∂n
where ψ is a source and ∂ψ

∂n
is a dipole. The term φ becomes the strength of the dipole and
∂φ
the term ∂n becomes the strength of the source.
To solve Green’s theorem, the surfaces are divided into segments with a certain distribution
of the singularities. The strength is solved by letting the field point approach each of these
segments and by using the boundary condition. Further description of basic potential
theory and liquid mechanics can be found in many text books, e.g. (Faltinsen 1976;
Newman 1977; Mei 1989; Faltinsen 1990; White 1999).
A.1.1 Calculation of high frequency added mass

The added mass, represented by the high frequency asymptote, was used to estimate the
natural frequencies of the flexible modes. A 2D BEM theory was derived to calculate the
sectional added mass. This is briefly explained in the following.
The asymptotic high frequency added mass in heave was solved by a double body approxi-
mation in heave in infinite liquid. The wet surface up to the still water was mirrored about
the still water line. The surface was divided up into straight elements. On each element
a linear distribution of dipoles and sources were assumed. The 2D the Green’s theorem is
written
Z
1 ∂ ln(Z(p, q)) ∂φ(p)
φ(q) = − (φ(p) − ln(Z(p, q)) )dS(p) (A.11)
2π S ∂n(p) ∂n(p)
where Z is the complex notation of the distance between the field point q and the body
coordinate p. The use of complex sources and dipoles was found convenient in 2D.
The source and dipole expressions were integrated for each element in its local coordinate
system. Thereafter, the local coordinate system was transformed into the global coordinate
system. The boundary condition defines the strength of the source as

∂φ(p)
= U · n = η̇3 n3 (A.12)
∂n(p)
where the velocity potential is the unknown. Letting q approach p on each end of each
element, a set of linear equations were established and could be solved for the unknown
strength of the dipole. Continuity of the strength from element to element had to be
ensured. The equation to be solved is written
∂φ
(D + 2πI)φ = S (A.13)
∂n
where D is the matrix expression for the unit dipoles, S is the matrix for the unit sources
and I is the unit diagonal matrix. φ and ∂φ∂n
are vectors with N + 1 unknowns for the N
elements. The method is further described in Appendix A.1.2, since the formulation differs
slightly from what is often encountered in the literature.
A.1.2 Detailed numerical implementation of 2D BEM

A linear strength distribution for dipoles and sources was chosen, as illustrated in Fig-
ure A.1. The singularity strength at element i is expressed as
x ds − x
σ(x) = φi + φi+1 = B + Ax for 0 ≤ x ≤ ds (A.14)
ds ds
fi+1 fi
x
ds
y
=
1.fi+1
x
ds
y +
1.fi
x
ds
Figure A.1: Strength distribution of singularities for an element in the local coordinate
system. φi+1 and φi are singularity strengths for the dipole.
The integral for the source is written

Z ds
FS (Z) = (B + Ax) ln(Z − x)dx =
0
1 2 1
B[−(Z − x) ln(Z − x) − x]ds 2 2 ds
0 + A[ (x − Z ) ln(Z − x) − (Z − x) ]0 (A.15)
2 4
Similarly, the integral for the dipole is written
Z ds
i
FD (Z) = − (Cx + D) dx =
0 Z −x
Z − ds Z − ds
iD ln( ) + iC[ds + Z ln( )] (A.16)
Z Z
where C is similar to A, and D is similar to B, but in general with different values. The
complex function F (Z) = φ + iψ describes both the velocity potential and the stream
function in 2D. The velocity field was found from dFdz(Z) = u − iv. For the source this
becomes
dFS (Z) Z − ds Z − ds
= −B ln( ) − A[Z ln( ) − ds] (A.17)
dz Z Z
while for the dipole it becomes
dFD (Z) 1 1 Z − ds
= i[ − ](D + CZ) + iC ln( ) (A.18)
dz Z − ds Z Z
For the same strength distribution A = C and B = D, the potential of the dipole can be
related to the velocity of the source as
dFS (Z)
FD (Z) = −i (A.19)
dz
The velocity fields for a unit source and dipole strength are shown in Figure A.2 and A.3.
The dipole distribution looks like two end vortices, which can be shown.
When arranging segments to define the geometry, it was necessary to convert the global
coordinate system into a local coordinate system. These coordinate systems are shown in
Figure A.4. The Z = X + iY in the global system is converted to the z = x + iy in the
following way
(Z − Zi+1 ) · (Zi − Zi+1 )∗
z= (A.20)
ds
∗
The denotes the complex conjugate.
The numbering convention in Figure A.5 was chosen for a closed geometry. The geometry
stretched out along the X axis is displayed in Figure A.6. If the geometry of the hull and
Velocity field from a source element

2.5
2
1
1.5 1
1
1 1.5
1.5 2
0.5
2
3 3
1.5
3
0
y
3
2
−0.5
1.5 2
−1 1 1.5
−1.5 1
1
−2
−2.5
−2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5
x
Figure A.2: Velocity field for a segment stretching from (-1,0) to (1,0) with unit source
distribution.
Velocity field from a dipole element
2
0.5
1.5 0.5
1 1
5
0.
1
1.5 1.5
0.5
3 3
1
2
1
3
2
2
0
2
3
1.5
y
1.5
2
−0.5
1.5
1 1
0.
−1
5
−1.5 0.5
0.5
−2
−2.5
−2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5
x
Figure A.3: Velocity field for a segment stretching from (-1,0) to (1,0) with unit dipole
distribution.
X
Z
zi+1 z
zi
x n y
Figure A.4: Local and global coordinate system with numbering of element ends in the
global system.
Y
n3
Z3, f3
3 2
Z2, f2
Z4, f4 n2
n4 X
4 1
y
n
Z5, f5
Z1, f1
n1,n5
x
Figure A.5: Local and global coordinate system with numbering of elements and specifi-
cation of unknowns
surface is divided into discrete elements, the system will follow the numbering in Figure A.6,
while for a closed geometry continuity is required, hence Z5 = Z1 , n5 = n1 and φ5 = φ1.
The discrete equation, which must be solved, is written

X
4 X
4
∂φ(p)
2πφ(q) + φ(p)FD,i = FS,i (A.21)
i=1 i=1
∂n(p)
where the velocity potential is replaced by
φ = φη˙3 (A.22)
S
F D,1 FSD,4
x Z1,f1 Z2,f2 Z3,f3 Z4,f4 Z5,f5
y +
FED,1 F
E
D,4
Z1,f1 Z2,f2 Z3,f3 Z4,f4 Z5,f5

Figure A.6: Geometry stretched out showing unit singularity distribution, which should
be multiplied with the unknown strengths, φi . Note the local coordinate system. The lower
row denotes distribution with unknown strengths at the End, while the upper row denotes
distribution with unknown strengths at the Start.
Hence, φ is the unit velocity potential per m/s heave motion. Similarly, the body boundary
condition is expressed as
∂φ
= V · n = η˙3 n3
∂n
∂φ
= n3 (A.23)
∂n
For simplicity, φ hereafter means φ. By letting q approach Z1 at Z10 = Z1 + n1 , where
is a small number, the following equation can be established
E S E S
2πφ(Z10 ) + φ1 FD,1 (Z10 ) + φ2FD,1 (Z10 ) + φ2FD,2 (Z10 ) + φ3FD,2 (Z10 )
E S E S
+φ3 FD,3 (Z10 ) + φ4 FD,3 (Z10 ) + φ4 FD,4 (Z10 ) + φ1 FD,4 (Z10 ) =
E S E S
FS,1 (Z10 )n3,1 + FS,1 (Z10 )n3,2 + FS,2 (Z10 )n3,2 + FS,2 (Z10 )n3,3
E S E S
+FS,3 (Z10 )n3,3 + FS,3 (Z10 )n3,4 + FS,4 (Z10 )n3,4 + FS,4 (Z10 )n3,1 (A.24)
1
Superscript E denotes End where A = C = ds and B = D = 0, while superscript S
1
denotes Start where A = C = ds and B = D = 1. Subscript D denotes the dipoles, while
E
subscript S denotes the sources. E.g., FD,1 (Z10 ) means the dipole distribution at element
1 with unknown strength at End of the element evaluated at field point Z10 .
It should be understood that it is the real part of the source and dipole distributions,
which were used in the expression above. The normal component in the heave direction
was found as the imaginary part of the normal, which depended on two adjacent elements.
The normal is found as
−i(Zi −Zi−1)
dsi−1
+ −i(Zi+1ds−Zi−1)
i
ni = −i(Zi−Zi−1) −i(Zi+1 −Zi−1)
(A.25)
abs( dsi−1 + dsi
)
If this is done for all four points, there are four linear equations with four unknowns, and
this is written as a matrix system
(D + 2πI)φ = Sn3 (A.26)
A.2. AMPLIFICATION OF INCIDENT SHORT WAVES 223
It was solved by using the arithmetic matrix operator ”\” in MATLAB, and the whole
program to calculate added mass of a geometry defined from bottom centre line to the side
at the water line was less than 40 lines long.
The theory was extended to time domain analysis, including discrete elements on the water
surface. Numerical beaches were used on the surface far away from the body. The CPU
time was found unacceptable in the linear case with 100 sections of roughly 400 elements
each solved with a time step of 0.1 seconds for a period of 3 hours. By using Fortran
or coupling MATLAB and FORTRAN, the CPU time may be reduced. Moreover, for
comparison with towing tank experiments, the CPU time in MATLAB can be reduced
by considering head sea and symmetry conditions, and only half of the body and surface
elements need to be included. This can be investigated further.
A.1.3 Example of added mass

The method was compared to the analytical formulation of the high frequency added mass
for a circular cylinder
a2D
33 = ρπr
2
(A.27)
where r is the radius. The error in percent of the exact added mass was expressed versus
the number of elements, as illustrated in Figure A.7. A log-log scale is used, and a straight
line represents the error. This indicates that there were no numerical problems, but the ge-
ometry represented by the straight segments is an approximation. 8 segments on a quarter
of the cylinder gave an error of 1%, hence, this can be considered as a minimum number
of elements for a simple shape.
Towards the bow and stern the 3D effects are pronounced. To illustrate the order of
magnitude, Figure 4.8 in (Newman 1977) is considered. The overall added mass of an
ellipsoid is normalised by the displaced mass. The length is 2a and the maximum width
is 2b, and the mass becomes M = 43 πρab2. The nondimensional values are shown versus
the ratio ab . A ratio of zero corresponds to a 2D case, and the nondimensional added mass
in heave becomes 1.0. For real ship dimensions, the ratio is approximately ab = 16 . The
nondimensional value becomes 0.93, indicating an overall 3D effect of 7%. The sections
at the ends experience more 3D effects than amidships, hence a 7% reduction is a lower
limit towards the ends. The distribution of added mass influences the natural springing
frequency, but as a first approximation a reduction in the added mass of 7% can be assumed.
A.2 Amplification of incident short waves

The theory by Faltinsen et al. (1980) is briefly outlined. Firstly, the encounter frequency,
ωe , needs an explanation. Either the ship is moving with a velocity U , or the current
is moving with the same speed. In either case, the ship will oscillate with the meeting
Convergence of added mass versus number of elements on a quarter part

2
10
1
10
0
10
Relative error in percent
−1
10
−2
10
−3
10
−4
10
0 1 2 3
10 10 10 10
Number of elements on a quarter of the circle
Figure A.7: Convergence of high frequency added mass using 2D BEM method for a
circular cylinder versus number of elements on a quarter part.
frequency (frequency of encounter), ωe , and not the wave frequency, ω. The relation
between these two is
ω2
ωe = |ω + kU cos(α)| = |ω + U cos(α)| (A.28)
g
where α is the heading angle. In head sea α = 0, and the encounter frequency is larger than
the wave frequency, and visa versa in following sea. The encounter frequency is assumed
positive.
The speed U is assumed small, and only the incident flow and the stationary potential, φs ,
are considered. Using the boundary condition on the ship, it can be shown that φs = O(U ),
where φs is of the same order of magnitude as U . From Bernoulli’s equation, the steady
wave elevation ζs = O(U 2 ). The linear kinematic free surface condition by Taylor expansion
becomes
∂φs
|z=0 = 0 (A.29)
∂z
where the rest of the terms is of order U 3 . It represents a rigid wall, i.e. the velocity field at
the surface is horizontal in the xy-plane. If the waves are assumed small compared to the
A.2. AMPLIFICATION OF INCIDENT SHORT WAVES 225
ship, and the ship side is vertical and infinitely long, the ship side will give total reflection
of the incident waves. The total potential can be written in the local coordinate system as
Φ = V (x0, y0 ) · s + φI + φD (A.30)
where φD is the velocity potential of the diffracted (or reflected) waves, and V is the local
tangential velocity in the local coordinate system illustrated in Figure A.8. The resulting
velocity potential is based on an incident wave and incident velocity potential of
ζI = ζa sin(k0 x cos(α) + k0 y sin(α) + ωe t)

gζa k0 z
φI = e cos(k0 x cos(α) + k0 y sin(α) + ωe t) (A.31)
ω0
in the global coordinate system. The local and global coordinate systems are shown in
Figure A.8. Note that the global coordinate system differs from (Faltinsen et al. 1980).
x0, y0 is the origin of the local coordinate system. Transforming the incident waves into
U N X
(x
0 ,y
0 )
θ
S Y
α
Figure A.8: Global and local coordinate system for short waves. The capital N and S
correspond to small n and s in the equations.
the local coordinate system gives
ζI = ζa sin(k0 n sin(θ + α) + k0 s cos(θ + α) + k0 x0 cos(α) + k0 y0 sin(α) + ωe t)

gζa k0 z
φI = e cos(k0 n sin(θ + α) +
ω0
k0 s cos(θ + α) + k0 x0 cos(α) + k0 y0 sin(α) + ωe t) (A.32)
The solution is expressed as ((Faltinsen et al. 1980) but with some different signs).
Φ=φ+Vs
gζa k0 z
=− e sin(k0 s cos(θ + α) − k0 n sin(θ + α) + α0 + ωe t)
ω0
gζa k1z
− Be sin(k0 s cos(θ + α) + k2 n + α0 + ωe t)
ω0
+V s (A.33)
where
α0 = k0 x0 cos(α) + k0 y0 sin(α)
2k1 k0
B= sin(θ + α)
k1 + k0 k2
(ωe − V k0 cos(θ + α))2
k1 =
g
q
k2 = k12 − k02 cos2 (θ + α)
From the total potential the surface elevation may be calculated through Bernoulli’s equa-
tion again
∂Φ ρ ~ ~ 1
p + ρgz + ρ + V · V = pa + ρU 2 (A.34)
∂t 2 2
The constant on the right hand side comes from the far field condition at the surface.
Considering the surface, ζ, where p = pa , Taylor expand to z = 0 and keeping only the
linear terms, the total elevation becomes
U 2 − V 2 1 ∂φI ∂φD 1 ∂φI ∂φD
ζ= − ( + )− V( + ) (A.35)
2g g ∂t ∂t g ∂s ∂s
The linear wave elevation is written
U 2 − V 2 ωe
ζ= + ζa (sin(a) + B sin(b))
2g ω0
k0 V
− cos(θ + α)ζa (sin(a) + B sin(b)) (A.36)
ω0
where
a = k0 s cos(θ + α) − k0 n sin(θ + α) + α0 + ωe t
b = k0 s cos(θ + α) + k2 n + α0 + ωe t
If the behaviour at the stem is considered for head sea waves, then y0 = 0, α = 0 and
θ = π2 . This simplifies the expression to
U2 ωe 2
ζ= + ζa (cos(−k0 n + k0 x0 + ωe t) + k1
cos(k1 n + k0 x0 + ωe t)) (A.37)
2g ω0 k0
+1
A.3. ADDED MASS FOR A CONE 227
Transforming back to the global coordinate system using n = x0 − x (remembering that

we consider head sea) gives
U2 ωe 2
ζ= + ζa (cos(k0 x + ωe t) + ωe2
cos(−k1 x + (k0 + k1 )x0 + ωe t)) (A.38)
2g ω0 +1
ω02
where
ωe2 ω2
k1 = and k0 = 0
g g
At zero speed the expression represents a standing wave.
At the stem x = x0 and the amplification of the dynamic wave motion relative to the
incident wave finally becomes
ζ ωe 2
= (1 + ωe2
) (A.39)
ζa ω0 +1
ω02
This expression has been used for comparison with experimental data.
A.3 Added mass for a cone

Added mass for a cone was expressed by Faltinsen and Zhao (1997) for small deadrise
angles, while Shiffman and Spencer (1951) expressed it for any deadrise angle. The latter
was used in the whipping assessment in Section 3.1, and the approximate added mass
formulation will be outlined in Section A.3.1, since this reference can be difficult to obtain.
A.3.1 Added mass for a cone at any deadrise angle

The stem angle or bow flare angle may be in the range of 40 to 600 . An analytically
formula, derived by using expressions for an ellipsoid, is presented. The dimension of the
ellipsoid was varied depending on the actual submergence of the cone as illustrated in
Figure A.9. The velocity potential around the ellipsoid was used at each instance of the
submerged cone.
The added mass with respect to the vertex angle, θ, is written (considering only a half
cone a factor of 0.5 is used)
1 θ
A33 = kρ tan3 ( )h3 (A.40)
2 2
2π µ h
k = kw − ks , kw = kE · w3 , kE = γ ,γ=
3 2−µ c
initial level C
X
q/2
h
Figure A.9: Approximation of the cone by an ellipsoid at each time instance.
w is a wetting factor due to pile up of water, while ks is a free surface correction where
also the square of the velocity on the free surface is no longer neglected. µ, w and ks are
dependent on the ratio γ in the following way
 √

 2γ
[
1−γ 2
− acos(γ)] 0 < γ < 1
 (1−γ 2 )1.5 γ
2
µ= 3 √ γ=1 (A.41)


 2γ
[acosh(γ) −
2
γ −1
] γ>1
(γ 2 −1)1.5 γ
 2γ 2(1−γ)

 (2−µ)(1−γ 2 )1.5
[√ − acos(γ)] 0<γ<1
 1−γ 2
1
w−1= γ
γ=1 (A.42)

 2(γ−1)
 2γ
(2−µ)(γ 2 −1)1.5
[acosh(γ) − √ ] γ>1
γ 2 −1
 2

 log( γ1 )− 1−γ


π 2
w γ{ 2
− 12 } 0 < γ < 1

 3 [√
acos(γ) 2
]
1−γ 2 −γ
π 9 2 1
ks = ( )
3 8 16
γ=1 (A.43)

 2

 π 2 log( γ1 )− 1−γ

 3
w γ{ 2
acosh(γ) 2 − 12 } γ > 1
[√ ]
γ 2 −1−γ
The k-factors and wetting factor for different vertex angles are shown in Figure A.10.
The change of the added mass is calculated as (with θ being constant for the cone)
dA33 dA33 dh 3 θ
= · = VR · kρ tan3( )h2 (A.44)
dt dh dt 2 2
The draft including water pile up is written
H = wh (A.45)
A.4. RESISTANCE COMPONENTS 229
2.5
k
2.25 kW
2 k
E
1.75 k
s
w, k , k , k , k
w
W
1.5
E
1.25
S
0.75
0.5
0.25
0
0 20 40 60 80 100 120 140 160 180
Vertex angle θ [0]
Figure A.10: k-factors and wetting factor versus vertex angle.
while the effective width is written

θ
c1 = wh tan( ) (A.46)
2
The volume becomes
1 πc21H π θ
Ω= · = w3 tan( )h3 (A.47)
2 3 6 2
The ”hydrostatic” pressure has been considered up to the undisturbed free surface without
pile up.
A.4 Resistance components

Various resistance components were used in prediction of speed reduction in irregular sea
states. A part of the resistance components, which are related to resistance in calm water,
are explained in the following.
A.4.1 The wave making resistance, Cw

This component considers the wave making resistance, which is in the order of 5 to 10% of
the total calm water resistance for a blunt ship in calm water at F n ≈ 0.15, with reference
to Figure 10 in (Baba 1976). It includes wave pattern resistance created by the stern and
bow as well as wave breaking resistance created by the bow. A bulb may reduce the latter
and adjusting the length may reduce the former.
According to Walderhaug (1991) pp. 2.55, this component can be written simplified for a
large variation of ships as
B 3/2 1 + 2000F n10

1000Cw = 0.0032(LCB − 10CB + 30)( ) 2.3
(10F n)6CB (A.48)
T MCB
LW L
M = 1/3
5
as long as the flow separation at the stern is small. The block coefficient CB , the Froude
number F n, the length L, draft T and distance of centre of buoyancy forward of midship
LCB are all based on the actual water line.
A.4.2 The viscous resistance, CV

This is the main resistance part in calm water for a large ship, (Fathi and Dahl 2000), and
the total viscous resistance is written
CV = (1 + k0 )(CF + ∆CF ) (A.49)
where the friction coefficient for a flat plate is
0.075
CF = (A.50)
(log10(Rn) − 2)2
and the roughness allowance is
∆CF = (110.31(Hµ · V )0.21 − 403.33)CF2 (A.51)
with a hull surface roughness Hµ = 150. The form factor, k0 , is estimated by
k0 = 0.6ϕ + 145ϕ3.5 (A.52)

CB (LW L ) p
ϕ= (TAP + TF P )B
LW L
The method is referred to as the form factor method.
A.4.3 The air resistance, CAA

The air resistance is small and calculated as
AT
CAA = 0.001 (A.53)
S
where AT is the transverse projected area, (Fathi and Dahl 2000).
A.5. SOLVING THE BEAM EQUATION 231
A.4.4 The resistance due to base drag, CBD

This is due to the transom stern effect, and Walderhaug (1991) pp. 2.30 expressed this as
r
1 SB 3
CBD = 0.029 ( ) (A.54)
CF S − SB
where SB is the transverse area of the transom stern below still water line. This contri-
bution is small for the actual ship in cargo condition and zero in ballast condition. The
contribution may be included in the form factor method.
A.5 Solving the beam equation

The mode shapes are found from solving the eigenvalue problem of the structure (Euler
beam equation in Eq.(2.20)). Once the eigenvalues are known the eigenvectors can be
derived. For a homogeneous beam the mode shapes in head sea may be expressed as
Ψ1(x) = 1 , heave
x
Ψ2 (x) = 2 , pitch
L
cos(βi x) cosh(βi x)
Ψi (x) = + i=3,5,7...
2cos(βi 2 ) 2cosh(βi L2 )
L
sin(βix) sinh(βi x)
Ψi (x) = + i=4,6,8...
2sin(βi 2 ) 2sinh(βi L2 )
L
π
βi ≈ (2(i − 2) + 1)
2L r
c
ω1,2 =
m + ar
s
EI
ωi = βi4 for i > 2 (A.55)
m + af
The n mode shapes represent heave, pitch and n − 2 flexible modes (i=3 is the 2-node
mode). The added mass differs for the flexible modes and the rigid body motions. The
former can be approximated by the high frequency asymptotic value, while the latter is
frequency dependent.
The beam equation can be solved by introducing separation of variables. The equation is
multiplied with a mode shape and integrated over the length.

Z L/2 Z L/2
(m + ai) Ψi (x)Ψj (x)dxq̈i (t) + b Ψi (x)Ψj (x)dxq̇i(t)
−L/2 −L/2
Z L/2 Z L/2
+EI ΨIV
i (x)Ψj (x)dxqi (t) + c Ψi (x)Ψj (x)dxqi(t)
−L/2 −L/2
Z L/2
= P (x, t)Ψj (x)dx (A.56)
−L/2
The property of orthogonal modes is utilized


Z L/2 
 0 if i 6= j

L if i=1
Ψi (x)Ψj (x)dx = L (A.57)
−L/2 
 if i=2
 3 L
≈ 4 if i>2
Similarly,

Z L/2  0 if i 6= j
ΨIV
i (x)Ψ j (x)dx = 0 if i = 1, 2 (A.58)

−L/2 ≈ βi4 L4 if i > 2
The load per meter, P (x, t), consists of the local bow impact load, but also the harmonic
oscillating pressure along the ship. The impact load is assumed to act only locally, and
the final equations for the principal coordinates may be expressed as
Z L/2
B 1
q̈1 (t) + 2δr ω1 q̇1 (t) + q1(t) = [Fs(t) + Ph (x, t)dx]
m + ar (m + ar )L −L/2
Z L/2
B 3 2
q̈2(t) + 2δr ω2 q̇2(t) + q2(t) = [Fs(t) + Ph (x, t) xdx]
m + ar (m + ar )L −L/2 L
Z L/2
EI 4
q̈i (t) + 2δi ωi q̇i (t) + βi4qi (t) ≈ [Fs(t) + Ph (x, t)Ψi(x)dx]
m + af (m + af )L −L/2
(A.59)
where L = Lpp and the width of the ship at water line is B.
A.5.1 Assumptions made in the simplified procedure

The generalized harmonic force, Fh (t), was based on the dynamic pressure under the
incident wave
Z L/2 X
m
Fh,i (t) = ρgB ζa,j e−kj T cos(kj x + ωe t + j )Ψi (x)dx (A.60)
−L/2 j=1
A.5. SOLVING THE BEAM EQUATION 233
T = TF P is the draft of the ship floating without trim. The irregular sea was represented by
linear superposition. The phase angle =< 0, 2π > was randomly chosen. The diffraction
force was neglected, and the radiation forces were represented by approximate values of
hydrodynamic added mass and damping forces tabulated in (Faltinsen 1990).
The impact force outlined in Section 2.1.3 was based on the added mass from a half cone
as described in Appendix A.3.1. The water level above the apex of the cone was estimated
based on the relative distance between the water surface and the vertical vessel motion at
the bow. The wet draft h of the cone is estimated as

0 P
h = max (A.61)
ζ + ζs + TF P − zc − ni=1 qi (t)Ψ(x = Lpp /2)
0 refers to the water surface being below the apex of the cone. zc is the distance from base
line to the cone’s apex. The summation term refers to the vertical motion of the ship at FP.
The relative velocity is calculated as
X
n
∂ζ Xn
VR = q̇i (t)Ψ(x = Lpp/2) − −U · qi (t)ΨI (x)|x=Lpp /2 (A.62)
i=1
∂t i=1
while the acceleration is written
∂VR X X
n n
∂ 2ζ
= q̈i(t)Ψ(x = Lpp /2) − 2 − U · q̇i (t)I (x)|x=Lpp /2 (A.63)
∂t i=1
∂t i=1
The last terms in the expressions of relative velocity and acceleration represent the angle
of attack term.
The velocity and acceleration of the surface profile due to the incident and reflected wave
considering forward speed in head sea were given by
∂ζ ω2 2
= −ζa e (sin(k0 x + ωe t) + ωe2
sin(−k1 x + (k0 + k1 )x0 + ωe t))
∂t ω0 +1
ω02
∂ 2ζ ωe3 2
= −ζa (cos(k0 x + ωe t) + ωe2
cos(−k1 x + (k0 + k1 )x0 + ωe t)) (A.64)
∂t2 ω0 +1
ω02
ωe2 ω2
k1 = and k0 = 0
g g
and in these cases x = x0 = Lpp/2 considering the location of the bow.

A.5.2 Numerical scheme applied in the simplified procedure

The equation of motion was solved in time using a Newmark-β scheme. The procedure is
illustrated for one mode, hence the subscript i is omitted. The initial condition is given by
f
q̇0 = 0 , q0 = 0 and q̈0 = (Fs,0 + Fh,0) at t = 0 (A.65)
L(m + a)
where f = 1 for i = 1, f = 3 for i = 2 and f = 4 for i > 2. At subsequent time steps the
acceleration is calculated from the equation of motion
f
q̈j = (Fs,j + Fh,j ) − 2ωδ q̇j − ω 2 qj (A.66)
L(m + a)
The Newmark-β scheme can be expressed in general form (where subscript j refers to time
step no. j)
q̇j+1 = q̇j + (1 − γ)dtq̈j + γdtq̈j+1

1
qj+1 = qj + dtq̇j + ( − β)dt2q̈j + βdt2q̈j+1 (A.67)
2
To achieve a stable system γ = 12 and β = 14 were used. This corresponds to the case of
linear acceleration and does not include an artificial damping of the response. The third
formula comes from the equation of motion.
f
q̈j+1 = (Fs,j+1 + Fh,j+1 ) − 2ωδ q̇j+1 − ω 2 qj+1 (A.68)
L(m + a)
The three last equations may be written in an explicit form
f 1
qj+1 = [Fs,j+1 + Fh,j+1 ] + [aj + 2ωδbj ]
cj (m + a)L cj
γ
q̇j+1 = qj+1 − bj
βdt
1
q̈j+1 = qj+1 − aj (A.69)
βdt2
1 γ
where cj = 2
+ 2ωδi + ω2
βdt βdt
γ γ
bj = qj + ( − 1)q̇j
βdt β
1 1
aj = (qj + dtq̇j + ( − β)dt2q̈j )
βdt2 2
The procedure was as follows:
1. Calculate the impact force and harmonic force at time step j, Fs,j , Fh,j
2. Calculate the acceleration for time step j, q̈j

A.6. FINITE BEAM ELEMENT FORMULATION 235
3. The initial conditions for deflection and velocity at time step j is known, qj , q̇j
4. Calculate the harmonic force at time step j+1, Fh,j+1 . This is straight forward since
the motion does not influence this term in linear theory.
5. Assume that the impact force at time step j+1 is equal to the impact force at time
step j, Fs,j+1 = Fs,j
6. Calculate the deflection, velocity and acceleration at time step j+1, qj+1 , q̇j+1 , q̈j+1
7. Based on these quantities at time step j+1 the relative deflection, relative velocity
and relative acceleration can be calculated, hence the updated impact force at time
instant j+1 is calculated, Fs,j+1
8. Finally the deflection, velocity and acceleration at time step j+1 are calculated once
again, qj+1 , q̇j+1, q̈j+1
9. The loop is ended and the procedure is applied to the next time step.
The procedure above includes one iteration on the impact force, and this was considered
as sufficient.
A.6 Finite beam element formulation

The basic beam element is illustrated in Figure A.11. At each end it has two degrees of
freedom. The stiffness matrix, (Langen and Sigbjørnsson 1979), for the element is written
 12EI 
`3 (1+α)
− `2 6EI
(1+α)
− `312EI
(1+α)
− `2 6EI
(1+α)
 − `2 6EI (4+α)EI 6EI (2−α)EI 
 (1+α) `(1+α) `2 (1+α) `(1+α) 
c=  (A.70)
 − `312EI
(1+α)
6EI
`2 (1+α)
12EI
`3 (1+α)
6EI
`2 (1+α) 
(2−α)EI (4+α)EI
− `2 6EI
(1+α) `(1+α)
6EI
`2 (1+α) `(1+α)
where
12EI E
α= and G = (A.71)
GAS 2(1 + ν)
The consistent mass matrix for the element is written
 
156 −22` 54 13`
m`  −22` 4`2 −13` −3`2  
m= (A.72)
420  54 −13` 156 22` 
13` −3`2 22` 4`2
Both the mass matrix and stiffness matrix are symmetric.
For a long slender ship a lumped mass system is convenient. For each element half of the
mass is placed at each element end. If the rotational mass is neglected, manipulation of the
equation system may be necessary to solve the eigenvalue problem. Firstly, the vector of
the degrees of freedom is separated into vertical degrees of freedom and rotational degrees
of freedom. The stiffness and mass distribution are altered accordingly. The system is
written

C11 C12 2 M11 M12 vt 0
( −ω )· = (A.73)
C21 C22 M21 M22 vr 0
Utilizing that all mass matrices except M11 is zero, the system is reduced to
[(C11 − C12 C−1 2

22 C21 ) − ω M11 ]vr = 0 (A.74)
The subscript t and r refers to vertical and rotational degrees of freedom. Solving this
eigenvalue problem for n elements give n + 1 degrees of freedom and n + 1 eigenfrequencies.
The eigenvalues are found by requiring that the determinant is zero. The eigenvectors are
found by introducing the eigenvalues into Eq.(A.74). The eigenvector was made nondi-
mensional by requiring that the aft end of the FE beam model had unit displacement. The
rotations are found by
vr = −C−1
22 C21 vt (A.75)
The elements in the stiffness matrix can be considered as forces per unit displacement.
When the displacements are determined, the sectional forces at element i based on the
global degrees of freedom are derived as
 i
  
V SFEnd1 v(i−1)2+1
 V BM i   v(i−1)2+2 
 End1 
= c ·   (A.76)
 V SFEnd2 
i i  v(i−1)2+3 
i
V BMEnd2 v(i−1)2+4
v1 v3
v2 EI, m, GAS v4
l
End1 End2
Figure A.11: Basic beam element with length `, mass per meter m, bending stiffness EI
and shear stiffness GAS .
A.7. ANALYTICAL SOLUTION OF BEAM EQUATIONS 237
A.7 Analytical solution of beam equations

The approach to derive analytical expression for the eigenfrequencies including shear de-
formation was illustrated in Section 2.2.2. Similar rough approximations were derived for
the beam equation including other effects. Axial force, Px , due to e.g. wave crests in both
ends of the ship, modify the beam equation to
∂ 2y ∂2 ∂ 2y ∂ 2y
m + (EI ) + P x = P (x, t) (A.77)
∂t2 ∂x2 ∂x2 ∂t2
with the rough approximation of the natural frequency
Px
k=
s EI
k
ωi = ωEI 1 − 2 (A.78)
a1
This indicates that changes in the frequency due to axial forces are rather small for the
springing frequency. This may however increase the damping estimates slightly when the
width of the springing response peak is considered.
The beam equation with rotational mass becomes

∂ 2y ∂2 ∂ 2y m1 r 2 ∂ 4 y
m + (EI ) − = P (x, t) (A.79)
∂t2 ∂x2 ∂x2 EI ∂x2∂t2
m1 = κm
where r is the radius of gyration of a vertical slice (cross section) about the y-axis. The κ
factor is included to differ between m, which may include added mass and m1, which does
not include added mass. The rough approximation for the natural frequency is
s
1
ωi = ωEI · (A.80)
1 + a21 κ2r2
This is also insignificant for the springing frequency, but the importance increases for higher
modes.
If external damping is considered, the beam equation is written

∂ 2y ∂2 ∂ 2y ∂y
m 2
+ 2
(EI 2
)+b = P (x, t) (A.81)
∂t ∂x ∂x ∂t
where b is the linear damping coefficient from external damping such as wave generation
damping. Solving the equation confirms that the linear damping has no effect on the mode
shape, but the equation for time after using separation of variables is affected.
The beam equation with internal structural damping gives complex and traveling mode
shapes. Lekkerkerker (1979) investigated analytically springing based on internal damping,
and solved the equations including hydroelastic effects. The force expression and the
geometry were simplified. It may be shown that the effect for a lightly damped structure
is small, since damped modes are essentially the same as undamped modes.
A.8 Stiffness distribution for a segmented model
A formulation was derived for the stiffness and mass matrix based on a direct approach.
The equilibrium of forces was applied directly for each segment. There were two springs in
between each element, one vertical spring, kz , representing shear stiffness and one rotational
spring, kθ , representing the bending stiffness. A segmented model with shear and rotational
springs are illustrated in Figure A.12. Continuity was required at the cuts between the
segments.
v2i-3 kz,i v2i-1 kz,i+1 v2i+1

lcgi-1 kq,i lcgi kq,i+1 lcgi+1
v2i-2 v2i v2i+2

Qi Qi+1
li-1 Mi li Mi+1 li+1
Figure A.12: Notations for a segmented beam model.
The local stiffness matrix for element number i, which is connected to element i − 1 and
i + 1 by springs i and i + 1 respectively, is written
   
0 0 0 0 0 0 v2i−3
 0 0 0 0 0 0   v2i−2 
   
 k31 k32 k33 + C11,i k34 + C12,i k35 k36   v2i−1 
~k · ~v =  ·  (A.82)
 k41 k42 k43 + C21,i k44 + C22,i k45 k46   v2i 
   
 0 0 0 0 0 0   v2i+1 
0 0 0 0 0 0 v2i+2
A.8. STIFFNESS DISTRIBUTION FOR A SEGMENTED MODEL 239
where the local stiffness factors, k, are
k31 = −kz,i
k32 = kz,i (ì−1 − lcgi−1 )
k33 = kz,i + kz,i+1
k34 = kz,i lcgi − kz,i+1 (ì − lcgi )
k35 = −kz,i+1
k36 = −kz,i+1 lcgi
k41 = −kz,i lcgi
k42 = −kθ,i + kz,i lcgi (ì−1 − lcgi−1 )
k43 = kz,i lcgi − kz,i+1 (ì − lcgi )
k44 = kθ,i + kθ,i+1 + kz,i lcgi2 + kz,i+1
2
(ì − lcgi )2
k45 = kz,i+1 (ì − lcgi )
k46 = −kθ,i+1 + kz,i+1 lcgi+1 (ì − lcgi ) (A.83)
ì is the rigid body length of element i and lcgi is the length from aft end to COG of element
no. i. The element numbering is going from the stern to bow. When the system stiffness
matrix is assembled, element number 1 does not have a flexible connection aft of the stiff
segment, hence kz,1 and kθ,1 are set equal to 0. Similarly, for the last element n, which has
the flexible connection number n + 1 to the right of the stiff segment, kz,n+1 = kθ,n+1 = 0.
The C coefficients represent the hydrodynamic restoring coefficients due to heave and pitch
of segment i, hence C21 is the restoring coefficient in pitch due to motion in heave. The
local mass matrix is written
   
0 0 0 0 0 0 v̈2i−3
 0 0 0 0 0 0   
   v̈2i−2 
 0 0 Mi + A11,i A12,i  
0 0   v̈2i−1 
M~ · ~v¨ =  
 0 0 A I + A 0 0  ·  v̈2i 
 21,i i 22,i   
 0 0 0 0 0 0   v̈2i+1 
0 0 0 0 0 0 v̈2i+2
where A is the added mass.
The segmented beam model was validated against results from Økland (2002), pp. 201-
202. The natural frequencies for 2-node and 3-node vibration were calculated based on
data from his model tests. The calculated periods were 0.17 and 0.068s versus 0.17 and
0.070s in his thesis.
A segmented model using a vertical and rotational spring between each segment was also
tested. The dimensions of the model were
Lpp = 294m, B = 53m, D = 2T = 23.3m, Iship = 694m4

An equivalent rotational stiffness was estimated based on considering the static deflection
at the middle of the beam, which was simply supported at the ends and exposed to a
lateral force at the middle. By requiring the same deflection at the middle, the rotational
stiffness becomes
Iship
kθ = 3E (A.84)
Lpp
This gave a 2-node natural frequency of 0.474Hz, and the mode shape is illustrated in
Figure A.13. 0.474Hz was also the exact natural frequency from the analytical solution,
which is written
s
kθ (I1 + I2)
ω= (A.85)
I1 · I2
1 Lpp 3
I1 = I2 = (2ρBT )( )
12 2
The mass moment of inertia includes a factor 2 in front of ρ, just because the added mass
was assumed equal to the mass.
−4
x 10
1
0.8
0.6
0.4 Vector with ω=2.9799rad/sec

Deflection [m]
0.2
−0.2
−0.4
−0.6
−0.8
−1
−150 −100 −50 0 50 100 150
Distance from midship [m]
Figure A.13: Flexible mode of a 2-segmented beam.
A.9 Different features of the SDOF system

The SDOF system for a harmonic loading with forcing frequency ωf is written
mẍ + bẋ + cx = F0 sin(ωp t) (A.86)
This system may represent the 2-node springing mode. However, while the force amplitude,
F , is assumed constant in the SDOF system, the real excitation on a ship is frequency and
A.9. DIFFERENT FEATURES OF THE SDOF SYSTEM 241
speed dependent, F (ωp , U ), displaying a hump and hollow behaviour as a function of the
frequency.
The solution to the SDOF system consists of summation of the homogeneous and particular
solution
x(t) = e−δωt [A sin(ωd t) + B cos(ωd t)] +

F0 1 2
1 [(1 − β ) sin(ωf t) − 2δβ cos(ωf t)] (A.87)
c [(1 − β ) + (2δβ) ] 2
2 2 2
ωf
β=
ω
The damping ratio is defined as
b
δ= (A.88)
2mc
and the damped frequency and natural frequency are expressed as
√
ωd = ω 1 − δ 2
r
c
ω= (A.89)
m
The coefficients A and B are determined from the initial conditions ẋ(t = 0) = ẋ0 and
x(t = 0) = x0. After a while the transient part dies out due to the damping, but the forced
vibration remains. The dynamic amplification of the forced excitation relative to the static
deflection is written
xmax 1
D=| |= 1 (A.90)
xstatic [(1 − β 2)2 + (2δβ)2] 2
The damping ratio for the flexible modes is in the order of 1%, hence the system is lightly
damped. This will give ωd ≈ ω and
1
Dmax ≈ (A.91)
2δ
The latter gives a resonance amplification of 100 times the static deflection for a damping
ratio of 0.5%.
The response to a system initially at rest is written

F0 −δωt ω
x(t) = [e (δ sin(ωd t) + cos(ωd t)) − cos(ωf t)] (A.92)
c ωd
Considering a lightly damped system, the response at resonance excitation is
x(t) 1
= (e−δωt − 1) cos(ωt) (A.93)
xstatic 2δ
The envelope curve of this process, or rather the slope of the non-oscillating process, is
expressed as
1
(1 − e−δωt ) (A.94)
2δ
This may define the increase of springing when entering a harmonic wave train in the tow-
ing tank.
By considering the maximum amplitudes of the homogeneous part decaying with time, an
estimate for the damping ratio can be made assuming a lightly damped system
1 xi
δ= ln( ) (A.95)
2πn xi+n
Another important issue is the behaviour of the response to an impulse loading. An impulse
loading is often considered to be an impact with duration less than 25% of the natural
period. Whipping is not necessarily an impulse loading. The Duhamel (convolution)
integral can be used to calculate the response at time t. This is written
Z t
1
x(t) = e−δω(t−τ ) P (τ ) sin(ωd (t − τ ))dτ (A.96)
mωd 0
for a damped system where P (τ ) is the general loading. The solution of an impulse of the
form sin(ωp t) with a duration of Tp/2 = π/ωp was derived analytically. At time t = Tp /2
the resulting displacement and velocity was estimated and used to define the free vibration
phase after the end of the impact. The results are presented as the maximum response
ratio to the static response 1/c versus the nondimensional time in Figure A.14. The nondi-
mensional time is the impact duration, which is Tp/2, divided by the natural period, Ts .
Similar shapes were presented by Bergan et al. (1981), with a maximum amplification
of 2. An example of the time series where the impulse duration is 0.9 times the natural
period is displayed in the right plot of Figure A.14. The plot does also show the minimum.
The minimum in absolute value is less than the maximum independent of the length of the
impact in this case. The time series also illustrates that the largest amplitude is achieved
after slightly more than half a cycle, and for short impacts the amplification follows a
seemingly straight line. The maximum amplification is about 1.7.
The impact may however have a different shape than the conventional slamming impact
similar to the one considered in Figure A.14. Due to the inertia part of the impact load
(and possibly other contributions) both a positive and a negative load, or the opposite,
may occur as illustrated in Figure 3.2. This was represented by a whole sinusoidal impact
sin(ωp t) with a duration of Tp = 2π/ωp . The convolution integral was again solved and
the initial conditions for the displacement and velocity was applied at t = Tp for the
free harmonic vibration. Figure A.15 presents the results of the amplification both as a
maximum and a minimum versus the nondimensional impact time. The amplification
may exceed a level of 2 and reach a level of 3 in this case. Moreover, the absolute value
A.9. DIFFERENT FEATURES OF THE SDOF SYSTEM 243
1.8 0.1
Max
1.6 − Min 0.08
1.4 0.06
1.2 0.04
static
1 0.02
(t)/x
x(t)
max
0.8 0
x
0.6 −0.02
0.4 −0.04
0.2 −0.06
0 −0.08
0 0.5 1 1.5 2 2.5 3 3.5 4 0 1 2 3 4 5 6 7 8
Tp/(2Ts) t/Ts
Figure A.14: Maximum amplification from a half sinusoidal impact with time series
illustrated by an impact duration of 0.9Ts; ωS = 19.7rad/s, m = 0.05 and δ = 0.02.
of the minimum may exceed the maximum. The absolute amplitude will in any case
exceed the static deflection, also in the negative cycle. This is actually the beginning of
a resonance case similar to Figure 4.27, and the absolute maximum appears at the second
local maximum or minimum instead of the first. A time series is included in the right plot
to illustrate this. From the measured time series either two of these impact cases may
occur, and the behaviour of the vibratory response may then indicate which loading that
occured.
3.5 0.15
Max
− Min
3 0.1
2.5 0.05
static
2 0
(t)/x
x(t)
max
1.5 −0.05
x
1 −0.1
0.5 −0.15
0 −0.2
0 1 2 3 4 5 6 7 8 0 1 2 3 4 5 6 7 8
Tp/Ts t/Ts
Figure A.15: Maximum amplification from a whole sinusoidal impact with time series
illustrated by an impact duration of 0.9Ts; ωS = 19.7rad/s, m = 0.05 and δ = 0.02.
A.10 Estimate of the wave height in a sea state given

the period
For the simplified nonlinear estimate of springing, and for the method used to separate
whipping and springing, the second order contribution was corrected with respect to the
wave amplitude. The refined method was used to estimate the wave height. It was based
on the joint probability density function (jpdf) of the wave height and wave period pre-
sented in (Myrhaug and Kvålsvold 1992; Myrhaug and Kvålsvold 1995). Their expressions
are presented in the following.
The jpdf for the wave height and wave period, p(h, t), is written
p(h, t) = p(t|h)p(h) (A.97)
where the marginal pdf for the wave height is
2.39h1.39 h 2.39
p(h) = 2.39
exp[−( ) ] (A.98)
1.05 1.05
(note that it differs from Rayleigh, which has an exponent of 2 inside the brackets) and
the conditional pdf for the period given the wave height is
√ √
β(t − 0.12 h)β−1 t − 0.12 h β
p(t|h) = exp[−( ) ]
ρβ ρ

0.78h + 0.26 for h ≤ 0.9
ρ(h) = (A.99)
0.962 for h > 0.9
β(h) = 2atan(2(h − 1.2)) + 5
h and t are nondimensional zero down-crossing wave height, H, and period, T . They are
made nondimensional as
H
h= √ where γH = 2.8582
γH m0
T
t= q where γT = 1.2416 and (A.100)
γT 2π m 0
m2
Z ∞
mi = ω i S(ω)dω
−∞
Once the wave spectrum is specified, the mean amplitude is calculated by the following
procedure. Firstly, the marginal pdf for the period is calculated as
Z ∞
p(t) = p(h, t)dh (A.101)
−∞
A.10. ESTIMATE OF THE WAVE HEIGHT IN A SEA STATE GIVEN THE PERIOD 245
Thereafter, the conditional pdf for the wave height given the wave period is estimated from
p(h, t)
p(h|t) = (A.102)
p(t)
and finally, the conditional mean value for the wave height given the wave period is deter-
mined by
Z ∞
E(h|t) = hp(h|t)dh (A.103)
−∞
A plot is shown in Figure A.17. This is the same curve as shown by Myrhaug and Kvålsvold
(1992), which verifies the expressions used. The jpdf is shown in Figure A.16.
Figure A.16: Joint probability density function of nondimensional wave height and period.
This has been used to estimate the wave height for a specific period that excites either
resonance by linear or second order excitation. It was printed versus the Beaufort wind
scale, however the PM wave spectrum based on wind speed may give unrealistic high
waves compared to the Beaufort scale, e.g. HS = 19.9m in Beaufort 11 with wind speed
of 30.6m/s. A JONSWAP wave spectrum was used instead, but the peak period must
then be defined. The relation from DNV (2000) giving TZ = 6HS0.3 is used with a peakness
factor γ of 2. The results were shown in Section 2.6.
1.2
E[h|t] [−] 0.8
0.6
0.4
0.2
0
0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8
t [−]
Figure A.17: Nondimensional conditional mean wave height for a given period in a sea
state, ref. Eq.(A.101).
A.11 The Beaufort scale

The wind speed and wave height for the Beaufort scale are listed in Table A.1. The wind
speeds, U = 0.836Bn1.5 [m/s], were taken from www.met.no and relates to wind 10m above
the sea surface. The numbers in parenthesis were taken from www.globalwavestatisticsonline.com
as well as the significant wave heights. The first number refers to probable wave height,
while the second refers to the maximum. The corresponding wave height depends on the
duration of the wind. Assuming unlimited fetch and 25kn wind speed, the wave height
(=1.8) increases by a factor of 1.33 by increasing the duration from 3 to 6 hours. For 9
and 12 hours duration the factors are 1.9 and 2.5, (www.bom.gov.au).
Table A.1: Beaufort scale of wind speed and significant wave height.
Beaufort no. Bn Wind speed, U [m/s] (kn) HS [m]
1 0.3-1.5 (1-3) 0.1-0.1
2 1.6-3.3 (4-6) 0.2-0.3
3 3.4-5.4 (7-10) 0.6-1.0
4 5.5-7.9 (11-16) 1.0-1.5
5 8.0-10.7 (17-21) 2.0-2.5
6 10.8-13.8 (22-27) 3-4
7 13.9-17.1 (28-33) 4-5.5
8 17.2-20.7 (34-40) 5.5-7.5
9 20.8-24.4 (41-47) 7-10
10 24.5-28.4 (48-55) 9-12.5
11 28.5-32.6 (56-63) 11.5-16
12 Above 32.6 (64) Above 14
Appendix B
Experimental set-up procedures and

uncertainty analysis
B.1 Model setup

B.1.1 Description of transverse cuts
The five transverse cuts were divided into one stiff cut at each end and three flexible cuts
at the quarter lengths. Examples are shown in Figure B.1. The circular steel rod/spring
had a diameter of 4.0cm, and the distance between the locks/bearings were about 31cm.
By adjusting the length between the bearings the springing frequency could be changed.
The numbering of the cuts starts from the aft. The design of cut 5 was identical to cut 1,
and cut 4 was identical to cut 2. The shear force was not measured at the same location
as the bending moment in cut 2 and 4, but about 8cm forward of the moment.
B.1.2 Mass properties of the segments

To compare model experiments and numerical calculations, mass, mass moment of inertia
and centre of gravity were determined for each of the four rigid segments. Moreover, the
mass was measured for segment 1 and 6 for the purpose of shear calculations. The mass
was estimated using a weight, while the mass moment of inertia, longitudinal and vertical
COG were obtained through a pendulum test using a cradle.
Pendulum tests
Huse (1993) described the pendulum test in principle. The full set of equations were de-
rived in this research to account for the properties and design of the cradle. The procedure
247
248 APPENDIX B. EXPERIMENTAL SET-UP PROCEDURES AND UNCERTAINTY ANALYSIS
Figure B.1: The joints at 3 cuts. Left picture = Cut 3 at midship section, middle pic. =
Cut 2 at aft quarter (similar to cut 4) and right pict. = Cut 1 at stern (similar to cut 5).
is briefly described in the following.
The LCG was determined by placing the model on the cradle and balancing it on even
keel. The cradle with model was oscillated to determine the period of oscillation. However,
both the VCG and the moment of inertia, I, were unknowns. Therefore, two additional
weights with known equal masses, m, were placed horizontally a distance a on each side of
the knife edge. The knife edge defined the centre point of rotation. The oscillation period
was determined again, and the I and VCG could be calculated.
It was necessary to separate the Im and VCG for the model from the measured total I t
and VCG that included the cradle. The cradle was heavy with a low centre of gravity to
ensure short periods and positive distance, h, from the knife edge to the centre of gravity
measured downwards. This was a requirement, but high h was also an advantage from an
accuracy point of view. 600 to 800kg was placed in the bottom of the cradle due to the
heavy mass of the model segments.
Firstly, the VCG and mass moment of inertia of the cradle, Ic , are determined.
8π 2ma2
hc =
Mc g(T12 − T02)
2ma2 T02
Ict = 2
T1 − T02
Ic = Ict − Mc h2c (B.1)
Mc is the total mass of the cradle, hc is the distance from the knife edge to the COG being
positive downwards, and T1 and T0 are the oscillation period with and without the addi-
tional weights. The oscillation period was measured by a crystal clock with three significant
decimals (milliseconds). Additional weights were hanged a fixed distance, a = 3.000m,
B.1. MODEL SETUP 249
from the knife edge. m = 10kg (measured to 10.000 and 9.990kg by an accurate weight) or
m = 15kg (14.986 and 14.967kg) on each side were used. g = 9.81m/s2 is the acceleration
of gravity.
Secondly, the model was placed on the cradle with the keel at a distance hkeel from the
knife edge. The pendulum test was repeated following the same procedures and using the
same equations. T1 and T0 differ from above.
8π 2 ma2
h=
Mg(T12 − T02)
2ma2T02
It = 2
T1 − T02
I = I t − Mh2 (B.2)
where
M = Mm + Mc
Mc
hm = h + (h − hc )
Mm
Im = I t − Mh2 − Mm (h − hm )2 − Ic − Mc (h − hc )2
V CG = hkeel − hm (B.3)
The results of interest are Im and VCG for each stiff segment. Im is about its COG.
Mass properties
The results are collected in Table B.1. For clarification, LCG was defined from the aft
part of the segment, hence from the dry transom stern 15.1cm aft of AP for segment 1
as displayed in Figure 4.3. In cargo condition the number in parenthesis indicates when
the calculation was based on cradle estimates with 10 or 15kg additional mass. This gives
an indication of the uncertainty. Rigid segment 1 was based on a slightly different cradle
mass, since the segment was supported from tilting.
Pendulum tests for bow 2 in ballast and bow 2 and 3 in cargo conditions were not performed.
The intention was to keep the mass distribution identical in the three ballast conditions
and in the three cargo conditions. Pendulum test for ballast 3 condition was performed
to provide an estimate of the uncertainties, since also the bow geometry was changed.
Results for ballast 3 condition are presented in Table B.2. The results are in satisfactory
agreement with the ballast 1 condition, considering that the mass distribution and bow
geometry differ. A picture of the cradle with segment 4 is shown in Figure B.2.
Uncertainties in the mass properties

No detailed uncertainty analysis was carried out. There were, however, some uncertainties
related to the pendulum tests, mass weights, the model mass in water versus in air and
Table B.1: Mass properties of rigid segments for original ballast and cargo 1 condition.
Rigid Segment Segment Mm [kg] LCG [m] VCG [m] Im [kgm2] m [kg]
Cradle case 1 B. all 782.1 n.a. hc =0.5422 209.4 15
Cradle case 2 B. 1 and 2 784.0 n.a. hc =0.5427 210.0 15
1 Ballast 1 and 2 556.2 1.67 0.523 152.4 15
2 Ballast 3 990.5 0.98 0.505 513.1 15
3 Ballast 4 936.9 1.04 0.498 426.5 15
4 Ballast 5 and 6 859.4 0.86 0.496 339.4 15
4 Ballast 5 and 6 859.4 0.86 0.498 331.2 10
1 Cargo 1 133.6
4 Cargo 6 115.3
Cradle case 1 C. all 980.9 n.a. hc =0.547 241.1 10
Cradle case 1 C. all 980.9 n.a. hc =0.548 241.0 15
Cradle case 2 C. 1 and 2 982.9 n.a. hc =0.546 241.8 15
1 Cargo 1 and 2 849.0 1.73 0.494 170.2 10
1 Cargo 1 and 2 849.0 1.73 0.496 167.3 15
2 Cargo 3 1689.0 1.01 0.491 555.6 10 (10)
2 Cargo 3 1689.0 1.01 0.491 554.8 10 (15)
2 Cargo 3 1689.0 1.01 0.492 546.3 15 (10)
2 Cargo 3 1689.0 1.01 0.492 545.6 15 (15)
3 Cargo 4 1761.0 1.04 0.479 615.4 10 (10)
3 Cargo 4 1761.0 1.04 0.480 614.7 10 (15)
3 Cargo 4 1761.0 1.04 0.480 609.9 15 (10)
3 Cargo 4 1761.0 1.04 0.480 609.2 15 (15)
4 Cargo 5 and 6 1224.5 0.87 0.500 353.0 10 (10)
4 Cargo 5 and 6 1224.5 0.87 0.500 352.3 10 (15)
4 Cargo 5 and 6 1224.5 0.87 0.501 348.4 15 (10)
4 Cargo 5 and 6 1224.5 0.87 0.501 347.6 15 (15)
1 Cargo 1 143.6
4 Cargo 6 119.0
Table B.2: Mass and inertia properties of rigid segments for ballast 3 condition.
Rigid Segment Segment Mass[kg] LCG [m] VCG [m] Im [kgm2] m [kg]
Cradle case 1 B all 981.3 n.a. hc =0.545 240.6 15
1 Ballast 1 and 2 535.0 1.72 0.526 156.6 15
2 Ballast 3 982.6 1.12 0.524 494.1 15
3 Ballast 4 981.0 1.06 0.492 454.0 15
4 Ballast 5 and 6 873.3 0.83 0.517 285.0 15
division of masses between segments.
For each segment about twenty cycles were counted. This was done 5-6 times for each
B.1. MODEL SETUP 251
Figure B.2: Cradle with segment 4 and with the crystal clock shown in front.
additional weight (0, 10 and 15kg). The average of the repeated tests was taken to find
the oscillation period. The standard deviation was within one millisecond, and the uncer-
tainty was in the order of 0.02%. Moreover, the height from the keel to the knife edge of
the pendulum introduced some uncertainty. This was related to the divinycell thickness
between the model and the aluminum frame of the cradle. This distance was measured
by a folding rule with an inherent uncertainty of 0.5mm. The thickness of the dyvinicell
was about 0.5 ± 0.2cm, while the height from the frame to the knife edge was 30.0cm.
The uncertainty was less than 1%. The additional weights were determined by a weight
with accuracy within 1 gram. 6 repeated tests with 1 of the additional masses gave for
instance all 9990 grams. Their arm, a, was estimated to 3.006m with a measuring tape
with an inherent uncertainty of 0.5mm. Basically, the pendulum tests was related to an
uncertainty within about 2.5%.
The mass of each segment was weighed with a resolution of ±0.2kg with an inherent un-
certainty of 0.1kg. Repeated tests gave the same, but there was no knowledge about the
bias. The ”100kg” ballast weights were weighing between 99.8 to 100.8kg. The average of
10 weights were 100.225kg, and the expected bias was about 0.2%.
The mass of each segment differed slightly from when it was placed in water with all the
equipment. Examples of lacking or added weights were pressure sensors, accelerometers,
top frame locks, ”Nypos tree”, cables (9.5kg), screws, bow flare aluminum foils (3kg), and
small weights from the towing arrangement. The total mass was probably maximum 10kg
less than in water. The uncertainty may be up to 1%.
The mass at the ends of each segments may be off by about 4kg (half the spring), which
gave an addition of up to 4kgm2 . Moreover, the ballast weights were taken out of the
model in between each test and also between the model tests and the pendulum tests.
The weight positions were marked and pictures were taken to reduce this uncertainty. All
together, this introduced an uncertainty of 1 to 2% for the different segments.
A rough estimate of the 95% sample confidence interval of the mass moment of inertia
from the Table B.1 is in the order of 2.0%. Totally, it was roughly estimated that the 95%
sample confidence interval was less than 6% for the smallest segment and less than 5% for
the largest.
For the different loading condition there were differences in the mass distribution. For the
bow 2 in cargo condition and bow 3 in both cargo and ballast, plywood was fitted to the
deck at the ship side to increase the deck height and to avoid water ingress in larger waves.
Water on deck was only allowed on the deck at the bow. The weight of the additional
plywood was in the order of 20kg. There were also some differences to the weights placed
inside in the different segments, and the weight of the bow segments. Table B.3 shows the
amount of free weights located inside each of the stiff segments. The model tests with the
different bows ought to be considered as separate loading conditions. Ballast 1 and ballast
3 referred to in Table B.1 and B.2 displays the largest difference.
Table B.3: Ballast weights inserted into the different segments in the different cases.
Segment Ballast 1 Ballast 2 Ballast 3 Cargo 1 Cargo 2 Cargo 3
1 and 2 170 150 130 450 410 400
3 320 300 300 1000 1000 1000
4 300 300 300 1100 1100 1100
5 and 6 250 230 300 610 620 630
Approximate full scale data scaled down

The output from the loading computer onboard the vessel was reviewed, and two repre-
sentative loading conditions from the loading manual were selected as basis for the model
tests. Detailed mass distribution was derived from NAPA (Napa 1989) as input to WASIM
(DNV 2004), which was used by Storhaug et al. (2003). The mass distributions for ballast
B.2. SCALING LAWS AND NONDIMENSIONAL PROPERTIES 253
and cargo condition were scaled down as a reference in the planning process. The full scale
data underestimated the mass moment of inertia slightly, since the contribution from the
vertical mass distribution was disregarded. The additional ballast weights of more than
one ton in ballast condition and three tonnes in cargo condition were placed inside the
model to represent the full scale mass moment of inertia, the realistic longitudinal mass
distribution and trim. The comparison with the model test conditions are shown in Ta-
ble B.4.
Table B.4: Mass properties of rigid segments for real versus model mass (m ) distribution.
Rigid Segm. Segm. M [kg] LCG [m] I [kgm2] Mm [kg] LCGm [m] Im [kgm2]
1 Ballast 1,2 602 1.53 164 556 1.67 152
2 Ballast 3 912 1.25 233 991 0.98 513
3 Ballast 4 963 1.09 195 937 1.04 427
4 Ballast 5,6 859 0.90 178 859 0.86 335
1 Cargo 1,2 872 1.64 179 849 1.73 169
2 Cargo 3 1677 1.04 596 1689 1.01 551
3 Cargo 4 1712 1.03 636 1761 1.04 612
4 Cargo 5,6 1285 0.81 305 1225 0.87 350
The comparison displays a large difference in the moment of inertia in ballast condition.
Because of the many steel end plates used, the radii of gyration became too large for the
segments. For future tests, end plates made of thicker aluminum plates were proposed.
The data in cargo condition was in satisfactory agreement. The inaccurate mass moment
of inertia in ballast condition may have affected the springing response slightly, but it was
expected to be important for the higher order modes.
B.2 Scaling laws and nondimensional properties

B.2.1 Scaling laws
The basic model scaling laws were shown in Section 4.2.3. These were the Froude scaling
implying the dispersion relation and the scaling of the sectional stiffness. Scaling of other
properties were derived from these based on the length scale, Λ. The subscript f denotes
values in full scale, while m denotes values in model scale.
Translatory motions ηi as surge (i = 1), sway (i = 2) and heave (i = 3) are scaled as
η1,f = η1,m Λ
η3,f = η3,m Λ
wf = wm Λ
ζa,f = ζa,m Λ (B.4)
w is the deflection or some translatory motion, and ζa is the incident wave amplitude. The
angular motion as roll (i = 4), pitch (i = 5) and yaw (i = 6) are not scaled. Translatory
velocities are scaled as
√
η̇1,f = η̇1,m Λ
√
η̇3,f = η̇3,m Λ
√
ẇf = ẇm Λ
√
ζ˙a,f = ζ˙a,m Λ (B.5)
while, rotational velocities are scaled as

η̇5,m
η̇5,f = √ (B.6)
Λ
The translatory accelerations are not scaled. Standing in the bow of the model will give
the same acceleration as in full scale, but the period differs.
η̈1,f = η̈1,m
η̈3,f = η̈3,m
ẅf = ẅm
ζä,f = ζä,m (B.7)
The rotational accelerations are scaled according to

η̈5,m
η̈5,f = (B.8)
Λ
Pressures, p, are scaled based on the hydrodynamic (static) pressure under a wave crest
according to linear theory
ρf
pf = pm Λ (B.9)
ρm
where ρ is the density of the water.
The vertical bending moment, V BM, and vertical shear force, V SF , are scaled as
ρf 4
V BMf = V BMm Λ
ρm
ρf
V SFf = V SFm Λ3 (B.10)
ρm
The panel force, F P , follows the same scaling as the pressure when the average pressure
is used. The force can be scaled as
ρf
F Pf = F Pm Λ3 (B.11)
ρm
The total towing force is not easily scaled, since the viscous parts depend on the Reynolds
number Rn = U x/ν and the rest depends on the Froude number. The viscous part can be
manipulated by initiation of turbulence by a small core placed around a transverse bow
section. Scaling the added resistance, RAW , in waves can be done by
ρf
RAW,f = RAW,f Λ3 (B.12)
ρw
The stiffness of the segmented model is represented by rotational stiffness, kθ , and shear
stiffness, kz . These are scaled in the following way
ρf
kθ,f = kθ,m Λ4
ρm
ρf
kz,f = kz,m Λ2 (B.13)
ρm
The damping ratio is nondimensional, and does not have to be scaled.
B.2.2 Nondimensional properties

The results should be made nondimensional wrt. the important physical variables for easy
comparison with other results, calculations and in order to check if the results are inde-
pendent of changes in the important physical variables.
The nondimensional transfer functions may be defined without first scaling the results.
E.g. considering the nondimensional transfer function of the pitch angular velocity gives
|η̇5,f | |η̇5,m |/Λ |η̇5,m |
η̇5,f = = = = η̇5,m (B.14)
kf ζa,f g/Lf 2π/(λm Λ)ζa,m Λg/(Lm Λ) km ζa,m g/Lm
where k = 2π/λ is the wave number, λ is the wave length and g is the acceleration of grav-
ity. The other responses yield the same agreement between transfer functions in model
and full scale.
Basically, the motions of interest in relation to the model tests were the two translatory
motions heave, surge and the rotational motion pitch. These are made nondimensional in
the following way
η1
η1 =
ζa
η3
η3 = (B.15)
ζa
η5
η5 =
kζa
The velocities of a harmonic response is equal to the frequency, ω, times the displacement
(with a phase shift). The wave length is made nondimensional by the length between per-
pendiculars, Lpp . The frequency and time, t, are made nondimensional using the dispersion
relation Eq.(4.3).
λ
λ=
L
ω
ω=p (B.16)
g/L
t
t= p
L/g
The translatory and rotational velocity are then made nondimensional as

η̇1
η̇1 = p
ζa g/L
η̇3
η̇3 = p (B.17)
ζa g/L
η̇5
η̇5 = p
kζa g/L
while accelerations are made nondimensional based on ω 2 times the displacement, hence
η̈1
η̈1 =
ζa g/L
η̈3
η̈3 = (B.18)
ζa g/L
η̈5
η̈5 =
kζa g/L
Pressure may be difficult to make nondimensional, especially when different physics are
involved. The pressure under the wave crest is made nondimensional by use of the hydro-
static pressure height.
p
p= (B.19)
ρgζa
The physics of the slamming pressure is assumed to be independent of the gravity due to
high accelerations, hence it is made nondimensional as
p
p= (B.20)
0.5ρVR2
where VR is the relative velocity between the structure and the wave. In this research the
ship had forward speed, different steepness of the waves and 3D geometry with different
”deadrise” angles. This made it difficult to find the relative velocity from the measure-
ments. The former method was therefore used.
The vertical bending moment, V BM, is made nondimensional by

V BM
V BM = (B.21)
ρgζa BL2
where B is the breadth. Similar the vertical shear force, V SF , is made nondimensional as
V SF
V SF = (B.22)
ρgζa BL
The force panels were positioned to measure the slamming pressures, but they measured
also ordinary wave forces. The force is made nondimensional as
4 · FP
FP = (B.23)
ρgζa πd2
where the disk diameter, d, in the model tests was 10.0cm. The forces on the panel was
divided by the area and presented as average pressure, pav , for comparison with the pressure
from pressure cells.
4 · FP
pav = (B.24)
πd2
The average pressure is made nondimensional in two ways
pav
pav =
ρgζa
pav
pav = (B.25)
0.5ρVR2
The towing force measured the total force from viscous skin friction, wave resistance in
calm water, wave breaking resistance and added resistance in waves. The latter is made
nondimensional by
RAW ·
RAW = (B.26)
ρgζa2 B 2 /L
B.2.3 Measurement devices with locations

Approximate positions of different sensors are listed in Table B.5. The coordinates are
given relative to AP, BL and CL with x pointing forward, y pointing to port side and z
pointing upwards. The position may vary for the three bows when indicated.
The NYPOS system measured the motions relative to the COG. The program, devel-
oped at MARINTEK, calculated the motions of the vessel in 6 DOF. This was based on
a ProReflex Motion Capture System with two cameras that captured infrared light from 3
active markers on the ”Nypos tree”. A separate PC was used for this purpose. COG did
not exactly coincide with the COG of either the real ship or the model. However, it was
sufficient for evaluating the motions. The ”Nypos tree” was placed on the segment aft of
the COG, and deflections introduced small ”errors” in pitch and heave. In the extreme
case the errors were estimated to 0.3◦ in pitch and 0.8mm in heave based on the relative
deflection between the segments.
Digital BM Hottinger Baldwin Messtechnik (HBM) MGC Plus amplifiers were used in com-
bination with the program CATMAN from HBM. CATMAN was modified by MARINTEK
to serve their needs. The fixed wave probes were measured by an analog DHI Water and
Environment amplifier and converted into a digital signal by an AD-card inserted in one of
the digital amplifiers. All channels were sampled at 100Hz using a Butterworth low pass
filter at 40Hz. This was insufficient to capture the slamming peaks, but the intention was
only to confirm its presence.
The pressures cells measured the absolute pressure. Two types were used with rated
pressure of 1.7 and 3.5 bar, respectively. Both were produced by Kulite Semiconductor
Products, Inc. The location of the pressure cells on bow 2 is illustrated in Figure B.3. The
bow had vertical sides and a natural bottom bilge curvature.
The axial sensors were produced by Entran, and the type ELKM-D2M was used with a
limit of 10 and 25kN, respectively. These were ordered based on the maximum expected
loads from still water and dynamic bending calculated by VERES for the highest sea state.
For bow 1 and 2, accelerometers produced by Lucas Schaevitz (UK) were used. The type
was A433-0001. One of these broke due to hammer decay test, and new accelerometers
produced by MARINTEK were used for bow 3.
The inductive transducer measuring the relative position between the segments were also
produced by HBM, while the rest of the sensors were manufactured by MARINTEK.
B.3 Uncertainty analysis

In experimental analysis it is necessary to repeat experiments to find the uncertainties,
referred to as precision error, in the measured quantity. If a specific quantity X is measured
N times, the mean value, the standard deviation of the samples and the standard deviation
of the mean value are found by
1 X
N
X= Xi
N i=1
v
u
u 1 X
N
σ=t (Xi − X)2 (B.27)
N − 1 i=1
σ
σX = √
N
B.3. UNCERTAINTY ANALYSIS 259
Figure B.3: Bow 2 run in cargo condition with locations of the pressure probes and relative
motion device. The first black horizontal line is located 6m above BL and marked for every
second meter upwards.
The latter assumes a Gaussian parent distribution. For a Gaussian distribution with infi-
nite number of samples, N , the 95% sample confidence interval is found within X ± 1.96σ.
However, for a finite number of samples the 95% sample confidence interval is found within
X ± tσ, where the factor t is found from the t-distribution with N − 1 DOF. The 95%
mean confidence interval is given in the same manner by X ± tσX . For N larger than 30,
t = 2.0 may be used as an approximation.
Wild readings may occur for various reasons. The criterion used herein is referred to as
Chauvenent’s criterion (Coleman and Steele 1989), which defines an acceptable scatter. It
1
says that points within a probability band of 1 − 2N is kept. This criterion applies the
Gaussian distribution instead of the t-distribution also when N is small. After rejecting
measurements outside X ± tr σ the mean and standard deviations are finally calculated
again, and no more rejections are considered.
The probability band versus number of samples is shown in the right plot of Figure B.4.
In the left plot the t-factors for rejection and for the 95% confidence interval are shown.
For N = 1, 6 and 30 samples the probability band is 0.5, 0.92 and 0.98. The t-factors for
rejection become 0.67, 1.73 and 2.39, while for the 95% confidence interval they become
12.7, 2.57 and 2.042. 12.7 is taken from N = 2, since the t-distribution for N = 1 is not
given. Six samples may be a suitable number of tests.
Factor for rejection and confidence interval Probability band for rejection
14 1
t−reject.
t−conf.
0.95
12
0.9
10
0.85
0.8
8
Probability
t−factor
0.75
6
0.7
0.65
4
0.6
2
0.55
0 0.5
0 10 20 30 40 0 10 20 30 40
N, number of samples N, number of samples
Figure B.4: The probability band for rejection (to the right), the corresponding t-factors
for rejection (to the left) and the t-factors for the 95% confidence interval (to the left).
B.3. UNCERTAINTY ANALYSIS 261
Table B.5: Position of devices.

Name Unit x [m] y [m] z [m] Comment
FX2 N 2.24 0 1.16 Measuring axial force in x-direction, sag=positive
FX3 N 4.34 0 1.16 Measuring axial force in x-direction
FX4 N 6.44 0 1.16 Measuring axial force in x-direction
FY11 N 1.26 0 1.16 Measuring vertical shear force in z-direction
FY12 N 1.26 -0.28 0.26 Measuring vertical shear force in z-direction
FY13 N 1.26 0.28 0.26 Measuring vertical shear force in z-direction
ACCZ1 m/s2 -0.14 0 0.80 Measuring vertical acceleration, downwards=positive
ACCZ2 m/s2 1.17 0.1 0.80 Measuring vertical acceleration
ACCZ8 m/s2 8.54 0 0.80 Measuring vertical acceleration
P1 N/m2 8.40/8.17/8.65 0 0.54/0/0 Bow 1/2/3, pressure cell
P2 N/m2 8.42/7.94/8.50 0 0.40/0/0 Bow 1/2/3, pressure cell
P3 N/m2 8.48/8.40/8.49 0/0/0.16 0.29 Bow 1/2/3, pressure cell
P4 N/m2 8.28/8.17/8.43 0/0.08/0 0 Bow 1/2/3, pressure cell
P5 N/m2 8.55/8.29 0 0.14/0 Bow 1/2, pressure cell
P6 N/m2 7.96/8.05 0 0 Bow 1/2, pressure cell
P7 N/m2 7.74 0 0 Pressure cell
P8 N/m2 8.44/8.05 0/0.08 0.63 Bow 1/2, pressure cell
P9 N/m2 4.17 0 0 Pressure cell
P10 N/m2 4.17/8.05 0/0.17 0.29/0 Bow 1/2, pressure cell
P11 N/m2 1.11/7.94 0/0.09 0 Bow 1/2, pressure cell
P13 N/m2 0.14 0.02 0.40 Pressure cell
P14 N/m2 0.00/7.94 0.02/0.17 0.43/0 Bow 1/2, pressure cell
P15 N/m2 0.13/7.94 0.02/0.26 0.47/0 Bow 1/2, pressure cell
COG m 4.48/4.51/4.35 0 0.50/0.43/0.49 ballast 1/cargo 1/ bow 2 and 3, used in Nypos
Surge m COG Positive forward
Sway m COG Positive towards starboard
Heave m COG Positive downwards
Roll ◦ Positive starboard down
Pitch ◦ Positive bow up
Yaw ◦ Positive turning to starboard
REL3 m 4.41 Relative motion at ”COG”
REL1 m 8.38+ -0.1± Bow 1, relative motion in bow, skewed
REL2 m 8.29± 0.35± Bow 1, relative motion at bow side, skewed
REL1 m 8.40 Bow 2, relative motion in bow stem
REL1 m 8.62 0.06 Bow 3, relative motion in bow
WP1 m 5-10 0 Wave probe from wave maker and CL of tank
WP2 m 164 3.3 Wave probe from wave maker and CL of tank
WP3 m 4.0 0 Wave probe from bow of model and CL of tank
WP5 m -3.6 3.4 Wave probe from bow of model and CL of tank
FP0 N 8.38 0.17 0.57 Lower slamming panel, 10cm diameter
FP1 N 8.42 0.16 0.67 Upper slamming panel, 10cm diameter
FP13 N 0.05 0.06 0.43 Aft slamming panel, 10cm diameter
U m/s Speed of towing carriage
TA/TF N Towing force ring AP/FP, positive in tension
POS2 mm 2100 76 825 Relative deflection from centre of I-bar, cut 2
B.4 Uncertainties and calibration of different sensors

Calibration was divided into two tasks. The first task was to calibrate the sensor for a
range of different loadings to determine the slope, or response per unit loading. This was
done by starting at zero load and then add loads stepwise up to the maximum load, and
step down to zero again. A straight line through the points was estimated by a least square
fit. It did not indicate the uncertainty in the slope or in the load at high or low load levels.
The second task was to perform zero setting of the sensors between each run. In practice,
after each run there was a waiting time until the water surface was calm. The responses
were then measured for a couple of minutes, and the time average was taken representing
the new zero point. Subsequently, the new test could be run. When only considering
the dynamic behaviour, the mean response was of less importance. However, the mean
value was used for different purposes, e.g. to determine added resistance in waves and to
determine zero level for nonlinear bending moments.
Uncertainty measures can be obtained in various ways. Calibration certificates may pro-
vide such information. Another way is to remove some points along the line and perform
a least square fit on the remaining points to estimate the slope. This can be done several
times, and the standard deviation and confidence intervals can be established according to
Appendix B.3. In the present research, tests were repeated several times. The confidence
interval was then based on independent estimates of the slope.
Calibration is normally performed under ideal conditions. For instance, calibration of the
axial force sensor was carried out under perfect centric and axial loading. These data were
also obtained from the calibration certificate provided. Calibration of the force transducers
was performed using small weights (5, 10, 15kg), which confirmed very good linear rela-
tionship at low load levels. Since both the shear and axial force transducers were mounted
in the model and eccentricities could exist, all shear and axial sensors were calibrated when
mounted using small and large loads. This was also done to estimate the rotational stiffness.
The calibration of the axial force versus the applied moment is illustrated in Figure B.5 for
the midship section. The data were derived from a single loading and unloading sequence
displaying an almost perfectly straight line. The weight measured by an axial sensor was
applied by a crane. The governing uncertainty was related to the moment arm.
Figure B.6 shows the calibration of the rotational stiffness, kθ = V BM/θ. The rotational
stiffness was difficult to determine accurately. The linear regression line did not necessarily
go through zero. A quadratic fit in the right plot represents seemingly a better fit through
the measured points. The damping tests indicated that the natural frequency varied more
with the response level than the full scale measurements. This fact in combination with
the linear regression line failing to go through zero support that the stiffness was truly
weakly nonlinear. This is possible explained by bearings below required tolerance, mixture
of steel and aluminum and redistribution of forces in the flexible connections.
B.4. UNCERTAINTIES AND CALIBRATION OF DIFFERENT SENSORS 263
Calibration curve
8000
data 45
linear
7000
y = 0.9181*x − 7.016
6000
5000
4000
mga [Nm]
3000
2000
1000
−1000
0 1000 2000 3000 4000 5000 6000 7000 8000
FX3 [N]
Figure B.5: Calibration curves for the moment amidships.
Linear curves as shown in Figure B.5 were established by repeated tests both for high and
low load levels with positive and negative loading to establish the uncertainties. Figure B.7
shows the estimated slopes with rejected values for all three axial sensors. The results were
merged, since they represent the same design. The expected value was the ratio between
the moment, V BM, and the axial forces measured, N . This was equal to the arm from the
neutral axis, which was slightly above the I-beam, to the centre of the screw in Figure B.1.
The arm was 0.90m amidships, which was close to the estimated mean values. Table B.6
presents the uncertainties for the different shear forces, moments and rotational stiffness.
F Y 5 denotes the shear force in cut 5. The rotational stiffness is rather uncertain. The data
were derived after cargo 1 condition. The factor of 0.88 was used to define the moment
and stress in the subsequent fatigue calculations and for transfer functions. The factors
for the shear forces were generally far from the expected value of 1.0, and confirmed the
importance of the calibration in assembled condition. The shear force sensors referred to
as guide towers were of slightly different design at the different cuts, hence these results
were not merged.
It should be noted that although the axial force sensor is calibrated as explained above,
there is a physical bias introduced when the model is placed in water. This is caused by
axial forces, that contribute to the measured axial force, which is converted into the verti-
cal bending moment. The vertical bending moment has contribution from axial forces, but
the axial force is also a separate global force. The measured ”vertical bending moment”
therefore contains a contribution from the global axial force. This effect is considered as
Calibration curve
7000
data 46
linear
y = − 1.324e+006*x + 201.2
6000
5000
4000
mga2 [Nm]
3000
2000
1000
0
−5 −4.5 −4 −3.5 −3 −2.5 −2 −1.5 −1 −0.5 0
θ3 [rad] x 10
−3
Calibration curve
7000
data 46
quadratic
2
y = − 6.235e+007*x − 1.607e+006*x + 55.1
6000
5000
4000
mga2 [Nm]
3000
2000
1000
0
−5 −4.5 −4 −3.5 −3 −2.5 −2 −1.5 −1 −0.5 0
θ3 [rad] x 10
−3
Figure B.6: Calibration curves for rotational stiffness amidships.
insignificant for the vibration response, but it will contribute to the nondimensional ver-
tical bending moment. The effect of the axial stress can not be separated with the single
axial force sensor in ”deck”, but two sets of sensors must be used to separate these global
loads. This is illustrated in (Storhaug and Moe 2007), which also shows that the axial
stress component is small for a container vessel. It should be kept in mind that most (if
not all) references on model tests and full scale measurements do not explain or handle
B.4. UNCERTAINTIES AND CALIBRATION OF DIFFERENT SENSORS 265
Calibration factors. Uncertainty sample / mean (95%)= 9.4643 % / 2.7321 %

1
0.9
0.8
0.7
0.6
FX
0.5
0.4
0.3
0.2 µ =0.87918 σ =0.037804 σµ =0.010913
0.1 µ =0.85428 σ =0.093561 σµ =0.025949
0
0 2 4 6 8 10 12 14
Run number
Figure B.7: Calibration factors for the moment for different sensors and load sequences.
Table B.6: Calibration factors; mean value and 95% mean confidence interval in %.
Response Mean value (slope) 95%
FY1 1.13 3.0
FY2 1.19 2.0
FY4 0.93 5.3
FY5 0.99 0.2
FX2 0.86 7.5
FX3 0.91 7.1
FX4 0.86 1.7
FX 0.88 2.7
kθ,2 0.88E6 53
kθ,3 1.58E6 145
kθ,4 1.23E6 100
kθ 1.22E6 22
this properly. Therefore there exist a lot of model experiments which is somewhat biased,
hence the reported vertical bending moment is not the true vertical bending moment, but
slightly different as in these model tests. From a fatigue consequent point of view, this is
not important. This additional effect is also recently illustrated by Clauss et al. (2007).
The expected maximum pressures were estimated in order to define a range to be used by
the amplifier. E.g. the maximum pressure was estimated to 70kP a for slamming includ-
ing a safety margin. This was defined as the range. The amplifier operates in the range
of ±10V . To achieve maximum resolution in the measurements using the amplifier, 10V
should give 70kP a. The resolution becomes important at low amplitudes. E.g. with a
16 bits amplifier, used in these experiments, the range from −10V to +10V was divided
into 216−1 = 32768 levels, and the resolution became 4.3P a. The use of filters reduced the
stepwise behaviour somewhat. Another important uncertainty was related to the numbers
written to file. In this case, three significant digits were used in the ASCII files. The
pressures were written in P a, but if it had been written in kP a, the uncertainty would be
0.5P a. At the bottom of the ship the pressures were low and in the order of 10-100P a
depending on waves and motions. For these pressures, the resolution became poor. To
determine the second order excitation pressure became difficult.
The relative motion sensors as seen in Figure 4.4 was calibrated by fitting the two copper
fibers on a vertical plate that was lowered into the water at known depths. A specific
distance between the copper fibers was used. Similar pairs were fitted on the model. The
hull may not be plane and vertical, and it was difficult to achieve exact the specified dis-
tance between the copper fibers. Some uncertainties were then introduced, but a few rough
calibrations were performed. E.g. by the water exceeding the upper limit of the copper
fibers, the measured relative motion was compared to the distance from still water level to
the upper part of the copper fibers. The agreement was within a couple of millimeters or
about 1%. The wave probes were calibrated in the same way as the relative motion sensors.
The accelerometers, the local deflection sensors, the slamming panels and the motions by
the NYPOS system were calibrated before conducting these experiments, while the towing
force sensors were calibrated by hanging known weights from the sensors.
B.5 Calibration of waves prior to model testing

A few questions must be answered before proceeding with springing tests. How sensitive is
the springing response to inaccuracies in wave frequency and speed? What is the necessary
distance to build up the springing response and to produce enough cycles at a ’maximum’
level for subsequent assessment? What is the accuracy of the requested wave frequency,
and how far down the tank are the waves of acceptable quality? The answers to these
questions will be elaborate in the following.
B.5.1 Sensitivity of springing response

The sensitivity of resonance response to changes in wave frequency and/or speed was illus-
trated utilizing basic theory. The relation between the wave frequency and the encounter
frequency in head sea waves with forward speed was found in Eq.(A.28), and this expres-
sion solved for ω0 was shown in Eq.(2.85). The change of the encounter frequency due to
B.5. CALIBRATION OF WAVES PRIOR TO MODEL TESTING 267
a change in speed and wave frequency can be expressed as

2ω0 ω2
δωe = (1 + U )δω0 + 0 δU (B.28)
g g
The SDOF system was used to find the √ necessary change in speed and/or wave frequency
causing a drop from resonance to 1/ 2 times the maximum level. This was given by the
half-power bandwidth method, Eq.(2.52), which was one way to determine the damping
ratio, λ, from the transfer function. It is rewritten
2λωs = ω2 − ω1 (B.29)
where ωs is the natural frequency,

√ and ω2 and ω1 is the frequency on each side of the peak
that corresponds to the 1/ 2 level.
√
The change in encounter frequency that gives 1/ 2 times the resonance peak based on a
damping ratio can now be written
2ω0 ω2
δωe = λωs = (1 + U )δω0 + 0 δU (B.30)
g g
Considering each component separately, and making them nondimensional gives
δω0 λωs
= (B.31)
ω0 (ω0 + 2g ω02 U )
δU gλωs
= 2 (B.32)
U ω0 U
The damping ratio is set to 1%, the speed to 7.7m/s (15kn) and the natural frequency to
3.33rad/s (0.53Hz). The wave frequency which gave resonance in head sea was 1.518rad/s
(4.14s). The relative change necessary to reduce the response to 70% corresponds to 0.6%
in wave frequency or 1.8% in speed, confirming the significant sensitivity to both, but that
the wave frequency might be critical.
The towing spring were chosen to provide a surge period, T , of about 6s (35s in full scale),
which was well above the periods of interest. A spring stiffness of 2000N/m at each end
was used in ballast condition, while 3000N/m was used in cargo condition. Considering
the surge motion under harmonic resonance condition, the surge velocity can be estimated
by
2π
η̇1 = ωη1 = η1 (B.33)
T
A surge motion of 2cm gave a maximum additional speed of ±1.6% at full speed. Hence,
the surge motion had to be controlled to a minimum, roughly less than 2cm, to avoid a
beat shape of the springing response. The controlling of the surge motion was confirmed
necessary from the tests, and if not done properly, the response became beat shaped with
a difference in the two spectrum peaks corresponding to the natural frequency in surge.
A third point was to estimate the necessary distance to build up the response to e.g.
x = 99% of the maximum response. This was estimated based on Eq.(A.94) as
x
ln(1 − 100 )
s = −U (B.34)
ωs δ
Roughly 30m was necessary in the model experiments assuming a damping ratio of 1% and
service speed. A sufficient number of cycles with a constant and ’maximum’ level should
also be available from the time series. For linear springing 10 cycles may be regarded as
sufficient, and this required a distance of only 4m, hence the build up phase was critical.
B.5.2 Location of wave probes and tested wave conditions

Five wave probes (WP) were used in the wave calibration. Two were fixed, one close to
the wave maker and one half way down the tank. Three others were placed on the towing
carriage and were moved down the tank with the model speed to cover a spatial distribu-
tion of the towing tank. A few measurements were also carried out at fixed positions.
The positions of the wave probes are shown in Table B.7. The wave probes consisted of
Table B.7: Channels and positions for wave calibration tests.

Name Channel Position
Time 1
WP1 6 Fixed 10m from the wave maker
WP2 7 Fixed 164m from the wave maker and 3m from the side wall
WP3 2 Towing carriage, 3m in front of vessel
WP4 3 Towing carriage, COG and at centre of the tank
WP5 4 Towing carriage, COG and 2.25m from the side wall
SPEED 5 Speed of the towing carriage
two thin cores and one circular column going down into the water. The column generated
a wake, which was somewhat disturbing in a visual sense. A foil shaped coumn is preferred.
The regular waves, as listed in Table B.8 in full scale values, were proposed for the spring-
ing model tests in ballast condition. The regular waves were chosen to excite linear and
higher order effects with three different wave amplitudes.
B.5.3 Results of regular wave tests

The wave amplitude was estimated by three methods. The first method was based on the
zero crossing process and local maxima. The second was based on the standard deviation
Table B.8: Regular wave test proposed for ballast condition.

Run T [s] H [m] Comment
8002 4.14 0.7 Linear resonance
8003 4.14 1.4 Linear
8004 4.14 2.1 Linear
8005 6.60 1.4 2nd order
8006 6.60 2.1 2nd order
8007 6.60 3.5 2nd order
8008 8.83 2.1 3rd order
8009 8.83 3.5 3rd order
8010 8.83 4.9 3rd order
8011 10.953 2.1 4th order
8012 10.953 4.1 4th order
8013 10.953 9.8 4th order
8014 13.019 2.1 5th order
8015 13.019 4.9 5th order
8016 13.019 11.8 5th order
of the time series, and the third was based on the standard deviation obtained from the
spectra. A filter was used in all cases to remove noise and low frequency components. The
latter was also partly due to elevation of the track along the tank (3cm difference). The
three methods gave in practice identical answers. The uncertainty represented by the 95%
sample confidence interval, was based on two times the standard deviation derived from
the zero crossing process.
The ratios between the requested input amplitudes in Table B.8 and the measured values
are shown in Figure B.8 versus the measured wave frequency for WP1. Waves that will
excite first to fifth order resonance were included. Both the mean value (dot) and the
95% sample confidence interval (bar) were presented. The uncertainty increased for higher
frequencies, even though the shorter waves were evaluated closer to the wave maker. The
uncertainty did not seem to increase for higher waves (the highest waves at second order
wave frequencies for WP2 were biased and shouold be disregarded). The uncertainty was
higher for the moving wave probes than WP1 close to the wave maker. The WP5 at the
side displayed more uncertainty than those at the centre. This may come from wall inter-
action, especially for the shorter waves. The shortest waves did not reach WP2, because of
the time necessary for the short waves to travel 164m. It was also noted that the highest
waves for linear springing showed a shift in the frequency. The mean amplitude differed
from the ratio of one, and no clear trend was displayed. A wave amplitude different than
requested, e.g. by 17% for the longest and highest wave, was found acceptable as long as
the wave height was measured, and the model test results were made nondimensional with
respect to the measured wave.
WP5 [−] WP4 [−] WP3 [−] WP2 [−] WP1 [−]
1.5
Low
1
Middle
0.5 High
2 3 4 5 6 7 8 9 10
1.5
1
0.5
2 3 4 5 6 7 8 9 10
1.5
1
0.5
2 3 4 5 6 7 8 9 10
1.5
1
0.5
2 3 4 5 6 7 8 9 10
1.5
1
0.5
2 3 4 5 6 7 8 9 10
ωWP1 [rad/s]
Figure B.8: Mean wave height value with 95% sample confidence interval (bar) versus
input wave height and wave frequency (WP1) for three wave heights.
Accurate wave frequency is essential. Figure B.9 shows the ratio between the dominating
frequency in the wave spectra and the input frequency for the different wave input. For
the moving wave probes, the encounter input frequency was used. This was based on the
measured speed, which was accurately measured. The speed was set to 1.300m/s, but the
measured velocity of the towing carriage was 1.3093±0.0004m/s as the 95% mean confi-
dence interval. The ratio was within 1% except the shortest waves for the moving wave
probes. WP4 and WP5 results were not shown because they gave identical results as WP3.
The discrepancy for the shortest waves may partly come from the resolution of the wave
spectra. Only 10.24s with acceptable quality were considered for the shortest and high-
est waves resulting in N = 1024 samples. The resolution was 0.628rad/s as a maximum
according to Eq.(2.75), while down to 0.06rad/s for longer waves. The maximum discrep-
ancy from a ratio of one was only about 4%. The resolution indicates an uncertainty of
maximum 3.5% for the fixed probes and maximum 1.6% for the moving probes. Taking
the resolution into consideration, the wave frequency was in general good.
The mean value of each wave run was considered as an independent estimate, and the
uncertainty of a random wave amplitude or wave frequency was estimated based on the
15 individual wave runs. The results are listed in Table B.9. Basically, this shows that the
wave amplitudes were as requested at the wave maker, but were too low at other positions.
Moreover, the frequency was as requested.
1.04
WP1 [−] 1.02
1
0.98
0.96
2 3 4 5 6 7 8 9 10
1.04
Low
WP2 [−]
1.02
1 Middle
0.98 High
0.96
2 3 4 5 6 7 8 9 10
1.04
WP3 [−]
1.02
1
0.98
0.96
2 3 4 5 6 7 8 9 10
ωWP1 [rad/s]
Figure B.9: Ratio of measured frequency from spectra to input frequency shown for dif-
ferent wave probes and wave heights.
Figure B.10 illustrates the difficulties with the short waves at fixed positions along the
tank. It was observed that between 40 and 60m from the wave maker, the waves became
unacceptable. At 60m the regular wave turned into a spectrum, which evolved continuously
towards longer wave components possibly by some interaction. It was observed that a
damper intended to damp out transverse waves, and located on each side about 40m from
the wave maker, initiated disturbances and possibly instabilities. It was decided to remove
this damper in the subsequent model tests.
Table B.9: Ratio of measured parameter to input parameter. Mean of amplitude or fre-
quency with their respective 95% mean confidence interval for the different wave probes.
Response Mean 95%
WP1, amplitude 1.012 0.035
WP1, frequency 0.999 0.002
Wave Prope 3 Wave Prope 3
0.02 20m 0.02 40m

ζa [m]
0 0
−0.02 −0.02
90 95 100 105 110 190 195 200 205 210
0.02 60m 0.02 80m
ζa [m]
0 0
−0.02 −0.02
300 305 310 315 320 430 435 440 445 450
0.02 100m 0.02 120m
ζa [m]
0 0
−0.02 −0.02
550 555 560 565 570 700 705 710 715 720
0.02 140m 0.02 160m
ζa [m]
0 0
−0.02 −0.02
820 825 830 835 840 950 955 960 965 970
0.02 180m 0.02 200m
ζa [m]
0 0
−0.02 −0.02
1050 1055 1060 1065 1070 1220 1225 1230 1235 1240
Time [sec] Time [sec]
Wave Prope 3 Wave Prope 3

3 3
S(ω) [m2s]
20m 40m
2 2
1 1
0 0
6 8 10 12 6 8 10 12
S(ω) [m2s]
60m 80m
0.2 0.2
0.1 0.1
0 0
6 8 10 12 6 8 10 12
S(ω) [m2s]
100m 120m
0.2 0.2
0.1 0.1
0 0
6 8 10 12 6 8 10 12
S(ω) [m2s]
140m 160m
0.2 0.2
0.1 0.1
0 0
6 8 10 12 6 8 10 12
S(ω) [m2s]
180m 200m
0.2 0.2
0.1 0.1
0 0
6 8 10 12 6 8 10 12
ω [rad/s] ω [rad/s]
Figure B.10: Wave amplitudes and spectra at 20m steps from the wave maker for the
shortest waves; H = 1.4m (in full scale). Only WP3 was considered.
B.5.4 Irregular waves

The irregular waves represented by JONSWAP, Eq.(2.78), shown in Table B.10 were defined
for the purpose of the springing tests. A peakness factor of γ = 2 was chosen. This may be
a realistic value for waves encountered in open ocean. Higher γ may give breaking waves,
which may change the spectrum. A steepness criterion defined by Eq.(2.88) was used as
a rule of thumb value to avoid breaking waves. This was satisfied for all sea states, and
breaking waves were only observed a couple of times in run 8100 and 8108.
Table B.10: Irregular sea states considered for wave calibration (at full speed).
Sea state no. Run TP [s] HS [m] HS /λP
1 8100 8.177 3.0 0.029
2 8101 10.902 3.0 0.016
3 8102 12.265 3.0 0.013
4 8103 13.628 3.0 0.010
5 8104 10.902 5.0 0.027
6 8105 12.265 5.0 0.021
7 8106 13.628 5.0 0.017
8 8107 14.990 5.0 0.014
9 8108 12.265 7.0 0.030
10 8109 13.628 7.0 0.024
11 8110 14.990 7.0 0.020
12 8111 14.990 9.0 0.026
The irregular waves were measured using two sweeps along the tank at full speed (1.3m/s).
This gave a time trace equivalent to 26 minutes in full scale. The duration was a bit short
to establish a reliable spectrum, nevertheless, the measured spectra were compared to the
requested spectra for the 12 different sea states.
FFT was used directly on the measured time trace to obtain the spectra. The spectra were
spiky, and they were smoothed by the Gaussian Bell in Eq.(2.76). The encounter spectra
from the moving wave probes were converted into wave spectra for comparison with the
fixed wave probes and input spectra. This gave a finer resolution for the converted spectra.
HS and TZ were calculated directly from the spectra, while TP was found at the maximum
peak. The smoothed spectra gave more reliable estimates of TP than the raw spectra. HS ,
TP , γ, n and m were also estimated by a nonlinear fit. This method was less sensitive
to cut off frequencies. n = 5 and m = 4 were tail and shape factors in the JONSWAP
spectrum. These were estimated based on previously estimates of HS , TP and γ. Fits to
both the raw and smoothed spectrum were considered, except for n and m.
The results for the 5 wave probes and 12 sea states were presented as ratios between the
estimated and requested input, as illustrated in Figure B.11, B.12, B.13 and B.14 for HS ,
TP , γ and n and m, respectively.
WP1
1.1 WP2
WP3
Hsfit/Hsinp
WP4
1 WP5
0.9
0 2 4 6 8 10 12
1.1
Hsfit/Hsinp
0.9
0 2 4 6 8 10 12
Sea state number
Figure B.11: Ratio of measured and input HS for raw (upper) and smoothed fit (lower).
1.1
1.05
Tpfit/Tpinp
1
WP1
0.95 WP2
WP3
0.9 WP4
0 2 4 6 WP5 8 10 12
1.1
1.05
Tpfit/Tpinp
0.95
0.9
0 2 4 6 8 10 12
Sea state number
Figure B.12: Ratio of measured and input TP for raw (upper) and smoothed fit (lower).
WP3 was relevant for describing the wave quality located in front of the model. The uncer-
tainties of the different sea state characteristics based on WP3 and also on all wave probes
4
WP1
3 WP2
WP3
γ/2 [−]
2 WP4
WP5
1
0
0 2 4 6 8 10 12
2.5
2
γ/2 [−]
1.5
1
0.5
0
0 2 4 6 8 10 12
Sea state number
Figure B.13: Ratio of measured and input γ for raw (upper) and smoothed fit (lower).
1.3 WP1
WP2
1.2 WP3
WP4
nfit/5
1.1 WP5
0.9
0 2 4 6 8 10 12
1.2
1.1
mfit/4
0.9
0.8
0 2 4 6 8 10 12
Sea state number
Figure B.14: Ratio of measured and input n and m for smoothed fit.
merged are listed in Table B.11. It was evident that the peakness factor should not be
estimated based on the smoothed spectrum. This factor had the largest variation, but the
mean value was as requested. The JONSWAP tail, n, and shape, m, parameters were as
requested, but the fit method may be somewhat deficient for the n factor when the whole
frequency range was considered. The wave heights were in general too low and the peak
periods were in general acceptable.
Table B.11: Uncertainty in JONSWAP parameters from fit to raw and smoothed spectrum,
and data derived directly from the moments of the smoothed (sm.) spectrum. Results are
presented as ratio of measured value to requested input.
HS fit WP3 0.959 0.020 0.963 0.020
HS fit WP1-5 0.961 0.014 0.967 0.014
HS moment WP1-5 0.956 0.012
TP fit WP3 1.011 0.016 1.010 0.015
TP fit WP1-5 1.011 0.007 1.009 0.007
TP moment WP3 1.008 0.018
TP moment WP1-5 0.998 0.011
γ fit WP3 1.030 0.148 0.890 0.100
γ fit WP1-5 1.001 0.069 0.896 0.060
n fit WP3 1.020 0.018
m fit WP3 0.999 0.026
n fit WP1-5 1.011 0.012
m fit WP1-5 1.001 0.012
The results were consistent with the regular wave tests indicating too low wave heights, but
with acceptable periods. There were still uncertainties related to how the shorter waves
were developing down the tank in absence of wind, but Figure B.10 indicated that the
energy was ”lost” after roughly 100m. It was expected that the contribution from linear
springing would be reduced along the tank in the subsequent model tests. Other effects
such as wall interaction, scattering from the model, wave-wave interaction, reflection from
the beach and depth changes may also influence subsequent results. In any case, the wave
heights must be measured during the experiments.
B.6 Sea states considered in the fatigue assessment

Table B.12 presents all sea states investigated for different combinations of bow design,
loading condition and speed. The sea states were defined by the significant wave height,
HS , and the zero-upcrossing period, TZ . The latter was converted to peak period, TP , for
input to the wave maker based on the Eq.(2.83).
B.7. MODES SHAPES, SECTIONAL PROPERTIES AND MASS DISTRIBUTION OF THE REAL
SHIP 277
Table B.12: Overview of sea states run for combination of loading conditions and bow
designs. The notation T refers to runs with additional trim by placing 100kg on the bow,
giving an increased draft of 1.6m. In case two speeds are indicated, the higher was used in
the ballast and the lower in cargo condition.
HS /TZ U [m/s] Ballast Cargo Ballast 2 Cargo 2 Ballast 3 Cargo 3
2/5 1.3 1639,1640,1641
2/5 T 1.3 1663,1664,1665
2/6 1.3 1642,1643,1644
2/7 1.3 1645,1646,1647
2/8 1.3 1648,1649,1650
2/9 1.3 1651,1652,1653
3/6 1.3/1.15 1100,1101,1102 1357,1358,1359 1582,1583,1584 1745,1746,1747 2350,2351,2352 2147,2148,2149
3/6 T 1.3 1654,1655,1656
3/7 1.3/1.15 1405,1406,1407 1585,1586,1587 1748,1749,1750 2353,2354,2355 2150,2151,2152
3/8 1.3/1.15 1110,1111,1112 1360,1361,1362 1588,1589,1590 1751,1752,1753 2356,2357,2358 2153,2154,2155
3/9 1.3/1.15 1120,1121,1122 1363,1364,1365 1591,1592,1593 1754,1755,1756 2359,2360,2361 2156,2157,2158
3/10 1.3/1.15 1130,1131,1132 1366,1367,1368 1594,1595,1596 1757,1758,1759 2362,2363,2364 2159,2160,2161
4/7 1.3/1.15 1408,1409,1410 1597,1598,1599 1760,1761,1762 2365,2366,2367 2162,2163,2164
4/7 T 1.3 1657,1658,1659
5/8 1.3/1.15 1140,1141,1142 1369,1370,1371 1600,1601,1602 1763,1764,1765 2368,2369,2370 2165,2166,2167
5/8 T 1.3 1660,1661,1662
5/8 0.87 1372,1373 1612,1613 1791,1792 2395,2396 2193,2194
5/9 1.3/1.15 1150,1151,1152 1374,1375,1376 1603,1604,1605 1766,1767,1768 2371,2372,2373 2168,2169,2170
5/9 0.87 1377,1378 1614,1615 1793,1794 2397,2398 2195,2196
5/10 1.3/1.15 1160,1161,1162 1379,1380,1381 1606,1607,1608 1769,1770,1771 2374,2375,2376 2171,2172,2173
5/10 0.87 1382,1383 1616,1617 1795,1796 2399,2400 2197,2198
5/11 1.3/1.15 1170,1171,1172 1384,1385,1386 1609,1610,1611 1772,1773,1774 2377,2378,2379 2174,2175,2176
5/11 0.87 1387,1388 1618,1619 1797,1798 2401,2402 2199,2200
7/9 1.3/1.15 1180,1181,1182 1389,1390,1391 1620,1621,1622 1775,1776,1777 2380,2381,2382 2177,2178,2179
7/9 0.43 1392 1633 1788 2392 2190
7/10 1.3/1.15 1190,1191,1192 1393,1394,1395 1623,1624,1625 1778,1779,1780 2383,2384,2385 2180,2181,2182
7/10 0.43 1396 1634 1789 2393 2191
7/11 1.3/1.15 1250,1251,1252 1397,1398,1399 1626,1627,1628 1781,1782,1783 2386,2387,2388 2183,2184,2185
7/11 0.43 1400 1635 1790 2394 2192
9/11 1.3/1.15 1260,1261,1262 1402,1403,1404 1629,1630,1631 1784,1785,1786 2389,2390,2391 2186,2187,2188
9/11 0.87 1637,1638
9/11 0.43 1636
9/11 0 1401 1632 1787 2300 2189
4/8.5 1.3 2403,2404,2405
4/9.5 1.3 2406,2407,2408
4/10.5 1.3 2409,2410,2411
2/7.5 1.3 2412,2413,2414
2/8.5 1.3 2415,2416,2417
2/6.5 1.3 2418,2419,2420
6/9.5 0.65 2421,2422
6/8.5 0.65 2440,2441
6/10.5 0.65 2442,2443
6/7.5 0.65 2444,2445
5/6.5 0.87 2446,2447
7/11.5 0.43 2448
3/11 1.30 2449,2450,2451
4/11.5 1.30 2452,2453,2454
5/11.5 0.87 2455,2456
6/11.5 0.65 2457,2458
8/9.5 0.22 2459
8/10.5 0.22 2460
7/7.5 0.43 2461
6/6.5 0.65 2462,2463
4/5.5 1.30 2464,2465,2466
2/9.5 1.30 2467,2468,2469
9/7.5 0 2470
9/9.5 0 2471
12/10.5 0 2472
B.7 Modes shapes, sectional properties and mass dis-

tribution of the real ship
The 2 and 3-node vertical mode shapes in one ballast condition were provided by DNV,
and numerical values are presented in Table B.13.
Distributions of bending stiffness and shear stiffness were derived for 10 cross sections along
Table B.13: Measured mode shape in full scale in ballast condition.

Position forward of midship [m] Nondimensional 2-node Nondimensional 3-node
-150.0 1.0 -1.0
-133.0 0.83 -0.75
-119.4 0.71 -0.60
-104.1 0.54 -0.40
-98.5 0.47 -0.28
-87.2 0.35 -0.13
-75.9 0.25 0.03
-65.1 0.14 0.14
-53.9 0.02 0.23
-42.6 -0.09 0.28
-31.8 -0.15 0.28
-20.5 -0.22 0.24
-9.2 -0.27 0.16
1.6 -0.29 0.06
12.9 -0.29 -0.06
24.2 -0.28 -0.16
35.0 -0.25 -0.23
46.3 -0.17 -0.28
57.5 -0.10 -0.24
68.4 -0.02 -0.23
79.6 0.08 -0.12
90.9 0.18 -0.02
101.7 0.28 0.10
113.0 0.40 0.27
124.3 0.53 0.38
136.3 0.62 0.53
150.7 0.71 0.65
the ship. This was input to the continuous FE beam model. A few simplifications were
made
1. Linear interpolation was used between the cross sections.
2. The structure was assumed to be 100% efficient and continuous without small open-
ings (except manholes).
3. The effective flanges were disregarded.
4. Gross scantlings (as built) were assumed.
5. The effective shear area was based on the projected vertical side and bulkhead struc-
ture.
B.7. MODES SHAPES, SECTIONAL PROPERTIES AND MASS DISTRIBUTION OF THE REAL
SHIP 279
The sectional properties are provided in Table B.14, and they were obtained from the DNV
Program NAUTICUS Section Scantlings. For clarification, Ih is the sectional area moment
of inertia about the horizontal neutral axis, Iv is about the vertical neutral axis at CL,
A is the total cross sectional area, Av is the vertical projected area without stiffeners and
zna is the position of the neutral axis above BL. Av was assumed to represent the effective
shear area As .
Table B.14: Sectional properties derived for the real ship.

Distance from AP [m] Ih [m4] Iv [m4] A [m2] Av [m2] zna [m]
17.0 185 191 3.39 1.30 15.91
44.5 437 1108 5.41 1.97 12.59
68.9 533 1640 5.89 1.88 11.85
81.1 603 1818 6.31 1.90 11.90
146.9 689 2021 6.78 1.92 11.83
204.3 665 1989 6.71 2.00 11.73
227.8 593 2008 6.37 1.84 11.12
253.2 498 1845 5.97 2.00 10.71
263.5 445 1485 5.56 1.95 11.05
285.7 270 399 3.92 1.46 14.02
The mass distribution as point masses based on the ballast condition is provided in Ta-
ble B.15 with FP and BL as references. More detailed mass distribution was used in the
numerical predictions.
Table B.15: Point mass distribution for ballast condition. Total mass is 146637tonnes.
Mass [tonnes] Distance from FP [m] Distance above BL [m]
229.2 2.74 13.85
372.9 9.10 14.70
472.8 16.52 14.76
4053.8 23.70 13.45
4631.1 30.48 13.42
9563.4 41.32 13.42
10843.2 55.81 13.01
7245.8 67.62 12.77
3693.7 88.02 13.05
11412.0 98.74 12.65
11443.0 113.13 12.65
11642.6 127.39 12.67
4461.9 138.33 12.88
6667.5 158.97 12.82
11396.2 170.61 12.67
11090.4 184.80 12.34
7842.2 199.33 9.43
1358.6 212.83 13.25
7545.1 229.07 14.21
8365.4 242.38 14.23
6088.0 255.62 14.50
2099.3 267.72 14.80
1515.3 274.18 14.38
1272.4 281.99 15.61
631.3 -0.29 15.45
29.1 298.69 13.85
670.7 0.62 13.85
Appendix C
Miscellaneous results
This appendix contains additional material about results presented in Chapter 5.
C.1 Linear and ’nonlinear’ transfer functions

The heave motion transfer function amplitudes for the different combinations are presented
in Figure C.2. The trend for longer waves differs in ballast and cargo condition, while the
different bow shapes do not affect the transfer functions significantly. The trend for the
longer waves is surprising. This is affected by wall interaction and the placement of the
’Nypos tree’. A few wild readings for the nonlinear estimates are also observed for the bow
2 in cargo conditions.
The pitch transfer function amplitudes are presented in Figure C.3. The trend differs for
ballast and cargo condition, and again they display a peculiar behaviour for longer waves.
The wall interaction were obviously important.
The transfer function amplitudes of the midship bending moments are shown in Figure C.4.
The rigid response peak of the cargo condition shows an odd behaviour due to the wall
interaction. The rigid body response peaks of the ballast condition are as expected, but
for bow 3 the nonlinear and linear estimates differ considerably at low frequencies. The
nonlinear estimates left of the peak must be disregarded, since they were obtained from
four-component waves rather than a regular wave train. This is why only the linear plots
were shown for heave and pitch of ballast 3 condition in Figure C.2 and Figure C.3. Except
for bow 1 in ballast condition the nonlinear effect is small. Bow 1 in ballast condition was
run in higher regular waves than the other runs, explaining the significant nonlinearities,
281
282 APPENDIX C. MISCELLANEOUS RESULTS
also contributed by wave induced vibrations.
The wall interaction caused difficulties for lower speeds in particular for cargo condition,
which displayed unexpectedly high nondimensional values for the rigid response peak. This
possibly affected the relative importance of the wave damage somewhat. Moreover, the
longest waves were possibly disturbed by the change in draft 80m from the wave maker
(deeper close to the wave maker). The draft changed from 10 to 5m. Figure C.1 presents
the measured wave and the corresponding full scale stress along the tank. The wave was
measured by wave probe 3 in front of the model, moving with a speed of 1.15m/s. The
requested wave length was 17.8m, and waves with length exceeding 10m may be affected.
This corresponded to encounter frequencies below 3.2/3.3rad/s in cargo/ballast condition
at full speed. A change in the measured amplitude is visible around 60s, which corresponds
to the location of the changed draft. The amplitude close to the wave maker is almost
40% higher than far from the wave maker. A similar change is not observed in the stress
amplitude. This will subsequently influence the nondimensional results depending on the
length of the time selected time trace.
C.2 Second order transfer functions

Figure C.5 presents the nondimensional second order transfer function for the bending
moment at the midship section, |H(ω1 , ω2 )|, for ballast 1 and cargo 1 condition, while
Figure C.6 presents the same for ballast 3 and cargo 3 condition.
0.04 20
15
0.02
10
0 5
Stress [MPa]
0
WP3 [m]
−0.02
−5
−0.04 −10
−15
−0.06
−20
−0.08 −25
−30
−0.1
20 30 40 50 60 70 80 90 100 110 20 30 40 50 60 70 80 90 100 110
Time [s] Time [s]
Figure C.1: Variation of measured raw wave amplitude in front of the model, (T = 20s,
H = 3.5m, V = 1.15m/s), and the corresponding midship deck stress. 60s corresponds to
the change in draft 80m from the wave maker.
C.2. SECOND ORDER TRANSFER FUNCTIONS 283
1
Lin Lin
0.9 1.6 NL
NL
HF
HF
0.8 1.4
0.7 1.2
0.6
1
3 a
η3/ζa
η /ζ
0.5
0.8
0.4
0.6
0.3
0.4
0.2
0.1 0.2
0 0
0 0.5 1 1.5 2 2.5 0 0.5 1 1.5 2 2.5 3
λ/L λ/L
1
Lin Lin
0.9 NL NL
HF 1 HF
0.8
0.7
0.8
0.6
3 a
η3/ζa
η /ζ
0.5 0.6
0.4
0.4
0.3
0.2
0.2
0.1
0 0
0 0.5 1 1.5 2 2.5 0 0.5 1 1.5 2 2.5
λ/L λ/L
1
Lin
0.9 NL
1.2
HF
0.8
0.7 1
0.6
0.8
3 a
η3/ζa
η /ζ
0.5
0.6
0.4
0.3 0.4
0.2
0.2
0.1
0 0
0 0.5 1 1.5 2 2.5 0 0.5 1 1.5 2 2.5
λ / L [−] λ/L
Figure C.2: Nondim. heave transfer function for ballast (left column) and cargo condition
(right column) in 15 and 13.2kn. Row 1= Bow 1, row 2= Bow 2, row 3= Bow 3. L = LP P
1
Lin Lin
0.9 NL NL
1
HF HF
0.8
0.7 0.8
0.6
η /(kζ )
η5/(kζa)
a
0.5 0.6
5
0.4
0.4
0.3
0.2
0.2
0.1
0 0
0 0.5 1 1.5 2 2.5 0 0.5 1 1.5 2 2.5
λ/L λ / L [−]
Lin Lin
1
NL NL
HF 1 HF
0.8
0.8
η /(kζ )
η5/(kζa)
0.6
a
0.6
5
0.4
0.4
0.2 0.2
0 0
0 0.5 1 1.5 2 2.5 0 0.5 1 1.5 2 2.5
λ/L λ/L
1
Lin
1 0.9 NL
HF
0.8
0.8
0.7
0.6
η /(kζ )
η5/(kζa)
0.6
a
0.5
5
0.4
0.4
0.3
0.2 0.2
0.1
0 0
0 0.5 1 1.5 2 2.5 0 0.5 1 1.5 2 2.5
λ / L [−] λ/L
Figure C.3: Nondim. pitch transfer function for ballast (left column) and cargo condition
(right column) in 15 and 13.2kn. Row 1= Bow 1, row 2= Bow 2, row 3= Bow 3.
0.025
Lin
0.05 Lin
NL
NL
HF
0.045 HF 0.02
0.04
VBM3/(ρ g ζaBL2)
VBM3/(ρ g ζaBL2)
0.035
0.015
0.03
0.025
0.01
0.02
0.015
0.01 0.005
0.005
0 0
0 5 10 15 20 0 2 4 6 8 10 12 14 16 18
0.035
Lin
0.06 Lin
NL
0.03 NL
HF
HF
0.05
0.025
VBM3/(ρ g ζaBL )
VBM3/(ρ g ζaBL2)
2
0.04
0.02
0.03
0.015
0.02
0.01
0.01 0.005
0 0
0 5 10 15 20 0 2 4 6 8 10 12 14 16 18
0.04 0.03
Lin Lin
NL NL
0.035
HF HF
0.025
0.03
0.02
VBM3/(ρ g ζaBL2)
VBM3/(ρ g ζaBL2)
0.025
0.02 0.015
0.015
0.01
0.01
0.005
0.005
0 0
0 5 10 15 20 0 2 4 6 8 10 12 14 16 18
Figure C.4: Nondim. midship moment transfer function amplitude for ballast (left col-
umn) and cargo condition (right column) in 15 and 13.2kn. Row 1= Bow 1, row 2= Bow
2, row 3= Bow 3.
ω1 TS
ω2 TS
ω1+ω2 LSQR
ω2+ω1 LSQR
1.2
0.8
0.6
15
0.4
0.2
10
0
15
10 ω1,e [rad/s]
5
5
ω2,e [rad/s]
ω1 TS
ω2 TS
ω1+ω2 LSQR
ω2+ω1 LSQR
0.7
0.6
0.5
0.4
0.3 12
0.2
10
0.1
8
0
12 6
10
8
6 ω1,e [rad/s]
ω2,e [rad/s]
Figure C.5: Nondim. second order transfer function for ballast 1 condition (upper) and
cargo 1 condition (lower) in 15 and 13.2kn.
ω1 TS
ω2 TS
ω1+ω2 LSQR
ω2+ω1 LSQR
0.8
0.6
0.4
14
0.2 12
10
0
14 8
12
10 ω1,e [rad/s]
8 6
6
ω2,e [rad/s]
ω1 TS
ω2 TS
ω1+ω2 LSQR
ω2+ω1 LSQR
0.35
VBM3/(ρgBL ζ1ζ2sqrt(k 1k2))
0.3
0.25
0.2
2
0.15 12
0.1
10
0.05
0 8
12
10
8 6 ω1,e [rad/s]
6
ω2,e [rad/s]
Figure C.6: Nondim. second order transfer function for ballast 3 condition (upper) and
cargo 3 condition (lower) in 15 and 13.2kn.
C.3 Wave conditions

The model was large and may affect the irregular sea states in subsequent runs, and the
wave conditions were evaluated. Table C.1 presents the sea state characteristics from wave
probe 3 (WP3) for other loading conditions and bow shapes, while results for ballast 2
condition were presented in Section 5.4.4. For low speeds the exceedance of the requested
wave height was observed to be somewhat higher than for high speeds, possibly due to
reflection from the bow. The wave height increased slightly for bow 3, possibly caused by
slightly smaller waiting time between runs, larger model and different bow shape reflection.
Based on the mean and mean confindence interval it can be concluded that the waves were
slightly higher than requested for bow 3. It should be noted that this is mainly due to 3D
disturbance, hence it may not cause a similar increase in the vibratory response. Actually,
it was observed that it caused a slight decrease for subsequent runs.
The same wave probes were used during the different tests. It was noticed that the amplifier
setting differed slightly, maybe because the same wave probes were not used at the same
positions. This uncertainty was regarded as small, but the bias was not estimated.
C.3. WAVE CONDITIONS 289
Table C.1: JONSWAP parameters with 95% mean confidence interval from tested sea
states. Values were obtained from nonlinear fit to raw and smoothed (sm.) wave spectra
and from moments. Results are presented as the ratio of the measured to requested input
for WP3.
Ballast 1
HS fit 1.014 0.028 1.019 0.025
HS moment 1.032 0.023
TP fit 0.991 0.008 0.993 0.008
γ fit 0.986 0.191
n fit 1.022 0.013
m fit 0.970 0.018
Cargo 1
HS fit 1.028 0.039 1.032 0.039
HS moment 1.034 0.038
TP fit 0.999 0.011 1.001 0.012
γ fit 0.947 0.099
n fit 1.023 0.009
m fit 0.975 0.008
Cargo 2
HS fit 0.975 0.045 0.960 0.032
HS moment 0.960 0.032
TP fit 0.993 0.012 0.999 0.012
γ fit 0.921 0.098
n fit 1.027 0.012
m fit 0.970 0.011
Ballast 3
HS fit 1.057 0.024 1.072 0.028
HS moment 1.071 0.027
TP fit 0.989 0.008 0.993 0.007
γ fit 1.060 0.097
n fit 1.030 0.009
m fit 0.973 0.007
Cargo 3
HS fit 1.044 0.029 1.051 0.032
HS moment 1.055 0.032
TP fit 0.998 0.011 1.002 0.010
γ fit 0.958 0.112
n fit 1.019 0.008
m fit 0.981 0.007
C.4 Relative contribution of fatigue damage from sea

states
The location of the fatigue dominating sea states in North Atlantic head sea trade in
realistic speeds is shown in Figure C.7 for ballast 2 condition and Figure C.8 for cargo 2
condition. The response surface method described in Section 2.8.1 were used.

20
HF damage:1.244
15
0.2
Tz [s]
10 d 0.1
20
5 0
0
10
10
0
0 5 10 15 20 20 0 Tz [s]
Hs [m]
Hs [m]

20
WF damage:2.2226
15
0.2
Tz [s]
10
0.1
d
20
5 0
0
10
10
0
0 5 10 15 20 20 0 Tz [s]
Hs [m]
Hs [m]
Figure C.7: Midship HF (upper) and WF (lower) fatigue damage for ballast 2 condition in
North Atlantic head sea wave environment at realistic speeds. 19 sea states were considered.
Ballast 1 condition was only tested at full speed, Figure C.9, while cargo 1 condition was
tested at realistic speed, Figure C.10.
Results for cargo 3 condition are displayed in Figure C.11, while results for ballast 3 con-
dition are displayed in Section 5.4.5.
C.4. RELATIVE CONTRIBUTION OF FATIGUE DAMAGE FROM SEA STATES 291

20
HF damage:0.61609
15
0.06
Tz [s]
10 0.04
d
0.02
20
5 0
0
10
10
0
0 5 10 15 20 20 0 Tz [s]
Hs [m]
Hs [m]

20
WF damage:3.4112
15
0.2
Tz [s]
10
0.1
d
20
5 0
0
10
10
0
0 5 10 15 20 20 0 Tz [s]
Hs [m]
Hs [m]
Figure C.8: Midship HF (upper) and WF (lower) fatigue damage for cargo 2 condition in
20
HF damage:4.0973
15
0.4
Tz [s]
10
0.2
d
20
5 0
0
10
10
0
0 5 10 15 20 20 0 Tz [s]
Hs [m]
Hs [m]

20
WF damage:2.4105
15
0.2
Tz [s]
10
0.1
d
20
5 0
0
10
10
0
0 5 10 15 20 20 0 Tz [s]
Hs [m]
Hs [m]
Figure C.9: Midship HF (upper) and WF (lower) fatigue damage for ballast 1 condition
in North Atlantic head sea wave environment at full speeds. 12 sea states were considered.

20
HF damage:0.58423
15
0.06
Tz [s]
10 0.04
d
0.02
20
5 0
0
10
10
0
0 5 10 15 20 20 0 Tz [s]
Hs [m]
Hs [m]

20
WF damage:3.4559
15
0.2
Tz [s]
10
0.1
d
20
5 0
0
10
10
0
0 5 10 15 20 20 0 Tz [s]
Hs [m]
Hs [m]
20
HF damage:0.57084
15
0.06
Tz [s]
10 0.04
d
0.02
20
5 0
0
10
10
0
0 5 10 15 20 20 0 Tz [s]
Hs [m]
Hs [m]

20
WF damage:4.3773
15
0.4
Tz [s]
10
0.2
d
20
5 0
0
10
10
0
0 5 10 15 20 20 0 Tz [s]
Hs [m]
Hs [m]
C.5. WHIPPING ILLUSTRATED BY MEASURED TIME TRACES 293
C.5 Whipping illustrated by measured time traces

Time traces from bow 2 in ballast and cargo condition were presented in Section 5.7 dis-
playing possible excitation sources of the vibratory response. A description of how to read
the different responses was included. This section considers the pointed bow 3 in ballast
and cargo condition. The bow had vertical sides and a vertical sharp stem with half apex
angle of about 360 . Some bottom flare was necessary to provide a natural transition from
the original cross sectional geometry at cut 5. The bottom flare started at BL at FP and
increased almost linearly to 37cm above the BL and 70cm aft of FP. The bottom flare
close to FP was less than on bow 2, and 70cm aft of FP it was significant first below 17cm
above BL with reference to Figure 4.4.
Figure C.12 presents vibration events at full speed of ballast 3 condition in low waves with
significant wave height of 3m. The left plot refers to a sea state with peak period of 8.2s,
while the right plot correspond to 13.6s. The ’stem’ pressure, P 3, was measured 16cm to
the side, 21cm aft of FP and 29cm above BL. The relative motion probe was located 9cm
aft of FP. The draft in ballast was 32.4cm, while the draft in cargo condition was 51.4cm.
10 20
0 0
−20
−10 −40
117 118 119 120 121 122 123 124 125 51 52 53 54 55 56 57 58 59
0.2 0.2
0.1 0.1
0
0 −0.1
117 118 119 120 121 122 123 124 125 51 52 53 54 55 56 57 58 59
Pitch [0]
Pitch [ ]
0.3
0
1
0.2 0
0.1 −1
117 118 119 120 121 122 123 124 125 51 52 53 54 55 56 57 58 59
3000
P3 [N/m ]
P3 [N/m2]
2
1000
500 2000
0 1000
0
117 118 119 120 121 122 123 124 125 51 52 53 54 55 56 57 58 59
P13 [N/m ]
P13 [N/m2]
2
400 1200
350 1000
300
250 800
117 118 119 120 121 122 123 124 125 51 52 53 54 55 56 57 58 59
Time [s] Time [s]
Figure C.12: Whipping events in ballast 3 condition at HS = 3m and U = 15kn. Left:

row 4= ’Stem’ pressure, row 5= Stern pressure.
The left plot presents a vibration event from forces acting on the bow. The pitch motion
is small, but it is influenced by a large wave striking the bow. The stem-side pressure
shows a rapid variation at the initiation of the vibration and at 122s, when the vibration
dies out due to a similar event. The difference in time corresponds to approximately 5.5
vibration cycles. The time between the peaks of the relative motion is about twice the
natural period, and it indicates sum frequency exitation. The zero level of the relative
motion is 8.6cm above the still water line, which agrees with the steady wave elevation in
the stationary point. The minimum measured relative motion is -3.7cm, hence the trough
is 12.3cm below the steady surface. At the bow quartering, the steady surface is expected
to be about -8.6/3 below the still water line. It may be expected that at the bow quartering
the trough reaches down to 17cm above the BL. The upper part of the bottom flare at a
height of 20cm was located 49cm behind FP. If the relative motion is ’frozen’ and moved
40cm due to forward speed and traveling trough, the trough will attack the bottom flare
0.2s later. Actually, 0.2s later the hogging cycle of the vibration appear possibly due to a
downward force followed by an upright force. The source therefore appears to come from
bottom flare and from sum frequency effects.
The right plot of Figure C.12 shows a similar event in longer waves with significant pitch
motion. Again, the bottom flare forces appear to be the source of the whipping. The vi-
bration response is much smaller, and it is apparently in phase with the hogging moment.
The phasing may be more well defined in sea states with long periods.
Figure C.13 shows whipping events from ballast 3 condition in realistic speed in 7 and 12m
significant wave height. On the left plot bottom flare forces appear to be the source of
the vibration, and the bottom pressure, P 4, located 27cm aft of FP, is negative at the
time of the impact. Slamming does occur in the aft ship, possibly contributed by wall
interaction, but it does not influence the vibration level. On the right plot bottom impacts
are observed even at zero speed, and give whipping from the exit and entry phase, but
slamming pressures appear to be small. The whipping response is about 20MP a, but the
maximum does not occur simultaneously as the maximum from the WF response, still the
maximum hogging is significantly increased.
P3 [N/m2] Stress [MPa]
50 100
0 0
−50 −100
285 286 287 288 289 290 291 292 293 201 202 203 204 205 206 207 208 209
0.4 3000
0.2 2000
0 1000
−0.2 0
285 286 287 288 289 290 291 292 293 201 202 203 204 205 206 207 208 209
4
P4 [N/m ] Pitch [0]
Pitch [0]
1 2
0 0
−1 −2
−4
285 286 287 288 289 290 291 292 293 201 202 203 204 205 206 207 208 209
P1 [N/m2]
2
1000 2000
0 0
−1000 −2000
−4000
285 286 287 288 289 290 291 292 293 201 202 203 204 205 206 207 208 209
P13 [N/m ]
P13 [N/m2]
2
1000 2000
1000
500
0
285 286 287 288 289 290 291 292 293 201 202 203 204 205 206 207 208 209
Time [s] Time [s]
Figure C.13: Whipping events in ballast 3 condition. Left: HS = 7m, U = 5kn, TP =

12.2s (run 2392), right: HS = 12m, U = 0kn, TP = 14.3s (run 2472). Row 1= Total
(continuous) and wave pass filtered (dashed) midship stress, row 2= Relative motion in
bow/’stem’ pressure, row 3= Pitch, row 4= Bottom bow pressure, row 5= Stern pressure.
Figure C.14 shows vibration events from cargo 3 condition in realistic speeds with a signif-
icant wave height of 3m. The left plot has a peak period of 8.2s, while the right plot has a
peak period of 13.6s. In the left plot there is no sign of impact, but the encounter period
of the waves are roughly twice the natural period, hence this may be springing from sum
frequency effects. The right plot displays a whipping event in longer waves. The steady
water elevation is 7cm above still water line, and the measured relative motion is 21cm
below the still water line. Based on a draft of 51cm, the water surface is expected to move
down to 20cm above the BL. It gives the same level of vibration with an amplitude of
10M P a as in ballast 3 condition in the same sea state. Again, the vibration cycle starts
in hogging and increases in sagging before decaying.

Pitch [ ] REL1 [m] Stress [MPa]
5 20
0 0
−5 −20
−40
44 45 46 47 48 49 50 51 52 96 97 98 99 100 101 102 103 104
0.2
0.2
0.1 0
0 −0.2
44 45 46 47 48 49 50 51 52 96 97 98 99 100 101 102 103 104
0.1 2
Pitch [0]
0
0 0
−0.1 −2
44 45 46 47 48 49 50 51 52 96 97 98 99 100 101 102 103 104
P3 [N/m ]
P3 [N/m2]
2
1000 2000
500 0
0 −2000
44 45 46 47 48 49 50 51 52 96 97 98 99 100 101 102 103 104
[N/m2]
P13 [N/m2]
400 1000
300 500
200
13
0
P
100
44 45 46 47 48 49 50 51 52 96 97 98 99 100 101 102 103 104
Time [s] Time [s]
Figure C.14: Vibration events in cargo 3 condition with HS = 3m and U = 13.2kn. Left:
row 4= ’Stem’ pressure, row 5= Stern pressure.
Figure C.15 presents two whipping events from cargo 3 condition in full and zero speed in
9m significant wave height. Both whipping events are excited due to bottom flare impacts
in the bow, and the magnitude of the whipping response is sensitive to speed. The radiation
pressure due to the vibration is pronounced in the stern pressure.
296 REL1 [m] Stress [MPa] APPENDIX C. MISCELLANEOUS RESULTS

100 50
0 0
−100 −50
−100
141 142 143 144 145 146 147 148 149 422 423 424 425 426 427 428 429 430
0.5 0.4
0.2
0 0
−0.2
−0.5
141 142 143 144 145 146 147 148 149 422 423 424 425 426 427 428 429 430
4
Pitch [0]
Pitch [ ]
2
0
2
0 0
−2 −2
−4
141 142 143 144 145 146 147 148 149 422 423 424 425 426 427 428 429 430
P1 [N/m2]
[N/m2] P [N/m2]
4000 2000
2000
0 0
−2000 −2000
1
−4000
141 142 143 144 145 146 147 148 149 422 423 424 425 426 427 428 429 430
P13 [N/m2]
1000 1000
0 0
13
−1000
P
−1000
141 142 143 144 145 146 147 148 149 422 423 424 425 426 427 428 429 430
Time [s] Time [s]
Figure C.15: Whipping events in cargo 3 condition with HS = 9m, TP = 15s. Left:
U = 13.2kn (run 2188), right: U = 0kn (run 2189). Row 1= Total (continuous) and wave
pass filtered (dashed) midship stress, row 2= Relative motion in bow, row 3= Pitch, row
4= Bottom bow pressure, row 5= Stern pressure.
C.5.1 Higher mode vibrations for bow 2 and 3 in ballast cond.

The 2-node and higher mode vibration are presented in Figure C.16 and C.17. The vibra-
tion from higher modes may display evidence of whipping. The left plot refers to ballast
2 condition, while the right plot refers to ballast 3 condition. The same sea states are
considered at full speed. The axial force at midship section is used to represent the 2-node
bending moment, while the forward quarter length is considered for the higher modes.
500N corresponds to 12.2 MP a in deck in full scale, while 100 N corresponds to 2.4 MP a
(range of y-axes).
Ballast 2 condition experienced more vibration than ballast 3 condition. Three obvious
reasons were
• The high damping in ballast 3 condition at low vibration levels may work as a filter
reducing small springing and whipping decay responses.
• The sharp pointed bow may have reduced the sum frequency effect from reflection
of incident waves
• The bottom flare causing whipping excitation was modified and somewhat reduced
for bow 3.
Spikes, visible in the higher modes at 133 and 266s, came from discontinuities in the merged
time series.
The whipping process is apparently more evident in the ballast 3 condition, but from
the higher modes both bow geometries experience whipping now and then. The standard
deviation of the midship nominal deck stress of the sea states for the two conditions are
presented in Table C.2. The HF and WF values are shown with their respective zero up-
crossing periods in parenthesis. The stress is significantly higher for the HF part in ballast
2 condition, while the WF part is similar.
Table C.2: Nominal standard deviation stress in deck at midship section for ballast 2 and
ballast 3 condition. HF and WF zero up-crossing periods are displayed in the parenthesis.
Sea state Ballast 2 Ballast 2 Ballast 3 Ballast 3
WF std. [MPa] HF std. [MPa] WF std. [MPa] HF std. [MPa]
HS = 3m TZ = 6s 3.09 (6.10) 6.37 (1.84) 2.73 (5.88) 2.69 (1.89)
HS = 3m TZ = 7s 4.73 (6.95) 4.95 (1.85) 4.27 (7.22) 2.42 (1.90)
HS = 3m TZ = 8s 7.75 (7.93) 4.35 (1.84) 6.55 (7.97) 2.07 (1.89)
HS = 3m TZ = 9s 8.75 (8.60) 3.45 (1.84) 8.30 (8.65) 1.58 (1.90)
HS = 3m TZ = 10s 10.34 (9.20) 2.73 (1.86) 10.67 (9.28) 2.16 (1.90)
HS = 4m TZ = 7s 6.33 (7.63) 8.96 (1.85) 5.64 (7.05) 4.87 (1.90)
HS = 5m TZ = 8s 12.19 (7.79) 11.37 (1.86) 10.84 (7.79) 6.13 (1.90)
2−node and higher mode vibration 2−node and higher mode vibration
500 500
HF [N]
HF [N]
0 0
−500 −500
0 50 100 150 200 250 300 350 0 50 100 150 200 250 300 350
100 100
50 50
Higher mode [N]
Higher mode [N]

0 0
−50 −50
−100 −100
0 50 100 150 200 250 300 350 0 50 100 150 200 250 300 350
Time [s] Time [s]
500 500
HF [N]
HF [N]
0 0
−500 −500
0 50 100 150 200 250 300 350 400 0 50 100 150 200 250 300 350
100 100
50 50
Higher mode [N]
Higher mode [N]
0 0
−50 −50
−100 −100
0 50 100 150 200 250 300 350 400 0 50 100 150 200 250 300 350
Time [s] Time [s]
500 500
HF [N]
HF [N]
0 0
−500 −500
0 50 100 150 200 250 300 350 400 0 50 100 150 200 250 300 350
100 100
50 50
Higher mode [N]
Higher mode [N]
0 0
−50 −50
−100 −100
0 50 100 150 200 250 300 350 400 0 50 100 150 200 250 300 350
Time [s] Time [s]
Figure C.16: 2-node vibration at midship (range ±500N ) and higher mode vibration at
forward quarter length (range ±100N ) for ballast 2 condition (left) and ballast 3 condition
(right) for HS = 3m and U = 15kn. Row 1 and 2: TP = 8.2, row 3 and 4: TP = 9.5, row
5 and 6: TP = 10.9s.
500 500
HF [N]
HF [N]
0 0
−500 −500
0 50 100 150 200 250 300 350 400 0 50 100 150 200 250 300 350
100 100
50 50
Higher mode [N]
Higher mode [N]

0 0
−50 −50
−100 −100
0 50 100 150 200 250 300 350 400 0 50 100 150 200 250 300 350
Time [s] Time [s]
500 500
HF [N]
HF [N]
0 0
−500 −500
0 50 100 150 200 250 300 350 400 0 50 100 150 200 250 300 350
100 100
50 50
Higher mode [N]
Higher mode [N]
0 0
−50 −50
−100 −100
0 50 100 150 200 250 300 350 400 0 50 100 150 200 250 300 350
Time [s] Time [s]
500 500
HF [N]
HF [N]
0 0
−500 −500
0 50 100 150 200 250 300 350 400 0 50 100 150 200 250 300 350
100 100
50 50
Higher mode [N]
Higher mode [N]
0 0
−50 −50
−100 −100
0 50 100 150 200 250 300 350 400 0 50 100 150 200 250 300 350
Time [s] Time [s]
Figure C.17: 2-node vibration at midship (range ±500N ) and higher mode vibration at
forward quarter length (range ±100N ) for ballast 2 condition (left) and ballast 3 condition
(right) for U = 15kn. Row 1 and 2: HS = 3m, TP = 12.2, row 3 and 4: HS = 3m,
TP = 13.6, row 5 and 6: HS = 4m, TP = 9.5s.
RAPP0RTER
UTGITT VED
INSTITUTT FOR MARIN TEKNIKK
(tidligere: FAKULTET FOR MARIN TEKNIKK)
NORGES TEKNISK-NATURVITENSKAPELIGE UNIVERSITET
PhD dissertations (not all) from the Department of Marine Technology, NTNU
UR-79-01 Brigt Hatlestad, MK: The finite element method used in a fatigue evaluation of fixed
offshore platforms. (Dr.Ing.Thesis)
UR-79-02 Erik Pettersen, MK: Analysis and design of celluar structures. (Dr.Ing.Thesis)
UR-79-03 Sverre Valsgård, MK: Finite difference and finite element methods applied to nonlinear
analysis of plated structures. (Dr.Ing.Thesis)
UR-79-04 Nils T. Nordsve, MK: Finite element collapse analysis of structural members considering
imperfections and stresses due to fabrication. (Dr.Ing.Thesis)
UR-79-05 Ivar J.Fylling, MK: Analysis of towline forces in ocean towing systems. (Dr.Ing.Thesis)
UR-80-06 Nils Sandsmark, MM: Analysis of Stationary and Transient Heat Conduction by the Use of
the Finite Element Method. (Dr.Ing.Thesis)
UR-80-09 Sverre Haver, MK: Analysis of uncertainties related to the stochastic modeling of
Three-Dimensional Flow Past Lifting Surfaces and Blunt Bodies.
(Dr.Ing.Thesis)
UR-85-46 Alf G. Engseth, MK: Finite element collapse analysis of turbular steel offshore structures.
(Dr.Ing.Thesis)
UR-86-47 Dengody Sheshappa, MP: A Computer Design Model for Optimizing Fishing Vessel Designs
Based on Techno-Economic Analysis. (Dr.Ing.Thesis)
UR-86-48 Vidar Aanesland, MH: A Theoretical and Numerical Study of Ship Wave Resistance.
(Dr.Ing.Thesis)
UR-86-49 Heinz-Joachim Wessel, MK: Fracture Mechanics Analysis of Crack Growth in Plate Girders.
(Dr.Ing.Thesis)
UR-86-50 Jon Taby, MK: Ultimate and Post-ultimate Strength of Dented Tubular Members.
(Dr.Ing.Thesis)
UR-86-51 Walter Lian, MH: A Numerical Study of Two-Dimensional Separated Flow Past Bluff
Bodies at Moderate KC-Numbers. (Dr.Ing.Thesis)
UR-86-52 Bjørn Sortland, MH: Force Measurements in Oscillating Flow on Ship Sections and
Circular Cylinders in a U-Tube Water Tank. (Dr.Ing.Thesis)
UR-86-53 Kurt Strand, MM: A System Dynamic Approach to One-dimensional Fluid Flow.
(Dr.Ing.Thesis)
UR-86-54 Arne Edvin Løken, MH: Three Dimensional Second Order Hydrodynamic Effects on Ocean
Structures in Waves. (Dr.Ing.Thesis)
UR-86-55 Sigurd Falch, MH: A Numerical Study of Slamming of Two-Dimensional Bodies.

(Dr.Ing.Thesis)
UR-87-56 Arne Braathen, MH: Application of a Vortex Tracking Method to the Prediction of Roll
Aug. 2007 IMT

Damping of a Two-Dimension Floating Body. (Dr.Ing.Thesis)
UR-87-57 Bernt Leira, MR: Caussian Vector Processes for Reliability Analysis involving
Wave-Induced Load Effects. (Dr.Ing.Thesis)
UR-87-58 Magnus Småvik, MM: Thermal Load and Process Characteristics in a Two-Stroke Diesel
Engine with Thermal Barriers (in Norwegian) (Dr.Ing.Thesis)
MTA-88-59 Bernt Arild Bremdal, MP: An Investigation of Marine Installation Processes - A Knowledge-
Based Planning Approach. (Dr.Ing.Thesis)
MTA-88-60 Xu Jun, MK: Non-linear Dynamic Analysis of Space-framed Offshore Structures.

(Dr.Ing.Thesis)
MTA-89-61 Gang Miao, MH: Hydrodynamic Forces and Dynamic Responses of Circular Cylinders
in Wave Zones. (Dr.Ing.Thesis)
MTA-89-62 Martin Greenhow, MH: Linear and Non-Linear Studies of Waves and Floating Bodies. Part I
and Part 11. (Dr.Techn.Thesis)
MTA-89-63 Chang Li, MH: Force Coefficients of Spheres and Cubes in Oscillatory Flow with and
without Current.(Dr.Ing.Thesis)
MTA-89-64 Hu Ying, MP: A Study of Marketing and Design in Development of Marine

Transport Systems. (Dr.Ing.Thesis)
MTA-89-65 Arild Jæger, MH: Seakeeping, Dynamic Stability and Performance of a Wedge Shaped
Planing Hull. (Dr.Ing.Thesis)
MTA-89-66 Chan Siu Hung, MM: The dynamic characteristics of tilting-pad bearings.
MTA-89-67 Kim Wikstrøm, MP: Analysis av projekteringen for ett offshore projekt.
(Licenciat-avhandl.)
MTA-89-68 Jiao Guoyang, MR: Reliability Analysis of Crack Growth under Random Loading,
considering Model Updating. (Dr.Ing.Thesis)
MTA-89-69 Arnt Olufsen, MK: Uncertainty and Reliability Analysis of Fixed Offshore Structures.
(Dr.Ing.Thesis)
MTA-89-70 Wu Yu-Lin, MR: System Reliability Analyses of Offshore Structures using improved
Truss and Beam Models. (Dr.Ing.Thesis)
MTA-90-71 Jan Roger Hoff, MH: Three-dimensional Green function of a vessel with forward speed in
waves. (Dr.Ing.Thesis)
MTA-90-72 Rong Zhao, MH: Slow-Drift Motions of a Moored Two-Dimensional Body in Irregular
Waves. (Dr.Ing.Thesis)
MTA-90-73 Atle Minsaas, MP: Economical Risk Analysis. (Dr.Ing. Thesis)
MTA-90-74 Knut-Aril Farnes, MK: Long-term Statistics of Response in Non-linear Marine Structures.
(Dr.Ing. Thesis)
MTA-90-75 Torbjørn Sotberg, MK: Application of Reliability Methods for Safety Assessment of
Submarine Pipelines. (Dr.Ing. Thesis)
MTA-90-76 Zeuthen, Steffen, MP: SEAMAID. A computional model of the design process in a
constraint-based logic programming environment. An example from
Aug. 2007 IMT

the offshore domain. (Dr.Ing. Thesis)
MTA-91-77 Haagensen, Sven, MM: Fuel Dependant Cyclic Variability in a Spark Ignition Engine - An
Optical Approach. (Dr.Ing. Thesis)
MTA-91-78 Løland, Geir, MH: Current forces on and flow through fish farms. (Dr.Ing. Thesis)
MTA-91-79 Hoen, Christopher, MK: System Identification of Structures Excited by Stochastic Load
Processes. (Dr.Ing. Thesis)
MTA-91-80 Haugen, Stein, MK: Probabilistic Evaluation of Frequency of Collision between Ships and
Offshore Platforms. (Dr.Ing. Thesis)
MTA-91-81 Sødahl, Nils, MK: Methods for Design and Analysis of Flexible Risers. (Dr.Ing. Thesis)
MTA-91-82 Ormberg, Harald, MK: Non-linear Response Analysis of Floating Fish Farm Systems. (Dr.Ing.
Thesis)
MTA-91-83 Marley, Mark J., MK: Time Variant Reliability Under Fatigue
MTA-91-79 Hoen, Christopher, MK: System Identification of Structures Excited by Stochastic Load
Processes. (Dr.Ing. Thesis)
MTA-91-80 Haugen, Stein, MR: Probabilistic Evaluation of Frequency of Collision between Ships and
Offshore Platforms. (Dr.Ing. Thesis)
MTA-91-81 Sødahl, Nils, MK: Methods for Design and Analysis of Flexible Risers. (Dr.Ing. Thesis)
MTA-91-82 Ormberg, Harald, MK: Non-linear Response Analysis of Floating Fish Farm Systems. (Dr.Ing.
Thesis)
MTA-91-83 Marley, Mark J., MK: Time Variant Reliability Under Fatigue Degradation. (Dr.Ing. Thesis)
MTA-91-84 Krokstad, Jørgen R., MH: Second-order Loads in Multidirectional Seas. (Dr.Ing. Thesis)
MTA-91-85 Molteberg, Gunnar A., MM: The application of system identification techniques to Performance
Monitoring of four stroke turbocharged Diesel Engines. (Dr.Ing.
Thesis)
MTA-92-86 Mørch, Hans Jørgen Bjelke, MH: Aspects of Hydrofoil Design; with Emphasis on Hydrofoil
Interaction in Calm Water. (Dr.Ing. Thesis)
MTA-92-87 Chan Siu Hung, MM: Nonlinear Analysis of Rotordynamic Instabilities in High-speed
Turbomachinery. (Dr.Ing. Thesis)
MTA-92-88 Bessason, Bjarni, MK: Assessment of Earthquake Loading and Response of Seismically
Isolated Bridges. (Dr.Ing. Thesis)
MTA-92-89 Langli, Geir, MP: Improving Operational Safety through exploitation of Design
Knowledge - an investigation of offshore platform safety. (Dr.Ing.
Thesis)
MTA-92-90 Sævik, Svein, MK: On Stresses and Fatigue in Flexible Pipes. (Dr.Ing. Thesis)
MTA-92-91 Ask, Tor Ø., MM: Ignition and Flame Growth in Lean Gas-Air Mixtures. An
Experimental Study with a Schlieren System. (Dr.Ing. Thesis)
MTA-86-92 Hessen, Gunnar, MK: Fracture Mechanics Analysis of Stiffened Tubular Members. (Dr.Ing.
Thesis)
Aug. 2007 IMT

MTA-93-93 Steinebach, Christian, MM: Knowledge Based Systems for Diagnosis of Rotating Machinery.
(Dr.Ing. Thesis)
MTA-93-94 Dalane, Jan Inge, MK: System Reliability in Design and Maintenance of Fixed Offshore
Structures. (Dr.Ing. Thesis)
MTA-93-95 Steen, Sverre, MH: Cobblestone Effect on SES. (Dr.Ing. Thesis)
MTA-93-96 Karunakaran, Daniel, MK: Nonlinear Dynamic Response and Reliability Analysis of Drag-
dominated Offshore Platforms. (Dr.Ing. Thesis)
MTA-93-97 Hagen, Arnulf, MP: The Framework of a Design Process Language. (Dr.Ing. Thesis)
MTA-93-98 Nordrik, Rune, MM: Investigation of Spark Ignition and Autoignition in Methane and Air
Using Computational Fluid Dynamics and Chemical Reaction
Kinetics. A Numerical Study of Ignition Processes in Internal
Combustion Engines. (Dr.Ing.Thesis)
MTA-94-99 Passano, Elizabeth, MK: Efficient Analysis of Nonlinear Slender Marine Structures.
(Dr.Ing.Thesis)
MTA-94-100 Kvålsvold, Jan, MH: Hydroelastic Modelling of Wetdeck Slamming on Multihull Vessels.
(Dr.Ing.Thesis)
MTA-94-101 (Dr.Ing.Thesis) Ikke godkjent.
MTA-94-102 Bech, Sidsel M., MK: Experimental and Numerical Determination of Stiffness and Strength
of GRP/PVC Sandwich Structures. (Dr.Ing.Thesis)
MTA-95-103 Paulsen, Hallvard, MM: A Study of Transient Jet and Spray using a Schlieren Method and
Digital Image Processing. (Dr.Ing.Thesis)
MTA-95-104 Hovde, Geir Olav, MK: Fatigue and Overload Reliability of Offshore Structural Systems,
Considering the Effect of Inspection and Repair. (Dr.Ing.Thesis)
MTA-95-105 Wang, Xiaozhi, MK: Reliability Analysis of Production Ships with Emphasis on Load
Combination and Ultimate Strength. (Dr.Ing.Thesis)
MTA-95-106 Ulstein, Tore, MH: Nonlinear Effects of a Flexible Stern Seal Bag on Cobblestone
Oscillations of an SES. (Dr.Ing.Thesis)
MTA-95-107 Solaas, Frøydis, MH: Analytical and Numerical Studies of Sloshing in Tanks.
(Dr.Ing.Thesis)
MTA-95-108 Hellan, øyvind, MK: Nonlinear Pushover and Cyclic Analyses in Ultimate Limit State
Design and Reassessment of Tubular Steel Offshore Structures.
(Dr.Ing.Thesis)
MTA-95-109 Hermundstad, Ole A., MK: Theoretical and Experimental Hydroelastic Analysis of High Speed
Vessels. (Dr.Ing.Thesis)
MTA-96-110 Bratland, Anne K., MH: Wave-Current Interaction Effects on Large-Volume Bodies in Water
of Finite Depth. (Dr.Ing.Thesis)
MTA-96-111 Herfjord, Kjell, MH: A Study of Two-dimensional Separated Flow by a Combination of the
Finite Element Method and Navier-Stokes Equations. (Dr.Ing.Thesis)
MTA-96-112 Æsøy, Vilmar, MM: Hot Surface Assisted Compression Ignition in a Direct Injection
Aug. 2007 IMT

Natural Gas Engine. (Dr.Ing.Thesis)
MTA-96-113 Eknes, Monika L., MK: Escalation Scenarios Initiated by Gas Explosions on Offshore
Installations. (Dr.Ing.Thesis)
MTA-96-114 Erikstad, Stein O.,MP: A Decision Support Model for Preliminary Ship Design.
(Dr.Ing.Thesis)
MTA-96-115 Pedersen, Egil, MH: A Nautical Study of Towed Marine Seismic Streamer Cable
Configurations. ( Dr.Ing.Thesis)
MTA-97-116 Moksnes, Paul O., MM: Modeling Two-Phase Thermo-Fluid Systems Using Bond Graphs.
(Dr.Ing. Thesis)
MTA-97-117 Halse, Karl H., MK: On Vortex Shedding and Prediction of Vortex-Induced Vibrations of
Circular Cylinders. (Dr.Ing. Thesis)
MTA-97-118 Igland, Ragnar T., MK: A Thesis Submitted in Partial Fulfilment of the Requirements for the
Degree of "Doktor Ingeniør". (Dr.Ing. Thesis)
MTA-97-119 Pedersen, Hans-P., MP: Levendefiskteknologi for fiskefartøy. ( Dr.Ing. Thesis)
MTA-98-120 Vikestad, Kyrre, MK: Multi-Frequency Response of a Cylinder Subjected to Vortex

Shedding and Support Motions. (Dr.Ing. Thesis)
MTA-98-121 Azadi, Mohammad R.E., MK: Analysis of Static and Dynamic Pile-Soil-Jacket Behaviour.
(Dr.Ing. Thesis)
MTA-98-122 Ulltang, Terje, MP: A Communication Model for Product Information. (Dr.Ing. Thesis)
MTA-98-123 Torbergsen, Erik, MM: Impeller/Diffuser Interaction Forces in Centrifugal Pumps. (Dr.Ing.
Thesis)
MTA-98-124 Hansen, Edmond, MH: A Descrete Element Model to Study Marginal Ice Zone Dynamics and
the Behaviour of Vessels Moored in Broken Ice. (Dr.Ing. Thesis)
MTA-98-125 Videiro, Paulo M., MK: Reliability Based Design of Marine Structures. (Dr.Ing. Thesis)
MTA-99-126 Mainçon, Philippe, MK: Fatigue Reliability of Long Welds Application to Titanium Risers.
(Dr.Ing. Thesis)
MTA-99-127 Haugen, Elin M., MH: Hydroelastic Analysis of Slamming on Stiffened Plates with
Application to Catamaran Wetdecks. (Dr.Ing. Thesis)
MTA-99-128 Langhelle, Nina K., MK: Experimental Validation and Calibration of Nonlinear Finite Element
Models for Use in Design of Aluminium Structures Exposed to Fire.
(Dr.Ing. Thesis)
MTA-99-129 Berstad, Are J., MK: Calculation of Fatigue Damage in Ship Structures. (Dr.Ing. Thesis)
MTA-99-130 Andersen, Trond M., MM: Short Term Maintenance Planning. (Dr.Ing.Thesis)
MTA-99-131 Tveiten, Bård Wathne, MK: Fatigue Assessment of Welded Aluminum Ship Details.
(Dr.Ing.Thesis)
MTA-99-132 Søreide, Fredrik, MP: Applications of underwater technology in deep water archaeology.
Principles and practice. (Dr.Ing.Thesis)
MTA-99-133 Tønnessen, Rune, MH: A Finite Element Method Applied to Unsteady Viscous Flow Around
Aug. 2007 IMT

2D Blunt Bodies With Sharp Corners. (Dr.Ing.Thesis)
MTA-99-134 Elvekrok, Dag R., MP: Engineering Integration in Field Development Projects in the
Norwegian Oil and Gas Industry. The Supplier Management of
Norne. (Dr.Ing.Thesis)
MTA-99-135 Fagerholt, Kjetil, MP: Optimeringsbaserte Metoder for Ruteplanlegging innen skipsfart.
(Dr.Ing.Thesis)
MTA-99-136 Bysveen, Marie, MM: Visualization in Two Directions on a Dynamic Combustion Rig for
Studies of Fuel Quality. (Dr.Ing.Thesis)
MTA-2000-137 Storteig, Eskild, MM: Dynamic characteristics and leakage performance of liquid annular
seals in centrifugal pumps. (Dr.Ing.Thesis)
MTA-2000-138 Sagli, Gro, MK: Model uncertainty and simplified estimates of long term extremes of
hull girder loads in ships. (Dr.Ing.Thesis)
MTA-2000-139 Tronstad, Harald, MK: Nonlinear analysis and design of cable net structures like fishing gear
based on the finite element method. (Dr.Ing.Thesis)
MTA-2000-140 Kroneberg, André, MP: Innovation in shipping by using scenarios. (Dr.Ing.Thesis)
MTA-2000-141 Haslum, Herbjørn Alf, MH: Simplified methods applied to nonlinear motion of spar platforms.
(Dr.Ing.Thesis)
MTA-2001-142 Samdal, Ole Johan, MM: Modelling of Degradation Mechanisms and Stressor Interaction on
Static Mechanical Equipment Residual Lifetime. (Dr.Ing.Thesis)
MTA-2001-143 Baarholm, Rolf Jarle, MH: Theoretical and experimental studies of wave impact underneath
decks of offshore platforms.
(Dr.Ing. Thesis)
MTA-2001-144 Wang, Lihua, MK: Probabilistic Analysis of Nonlinear Wave-induced Loads on Ships.
(Dr.Ing. Thesis)
MTA-2001-145 Kristensen, Odd H. Holt, MK: Ultimate Capacity of Aluminium Plates under Multiple
Loads, Considering HAZ Properties. (Dr.Ing. Thesis)
MTA-2001-146 Greco, Marilena, MH: A Two-Dimensional Study of Green-Water Loading. (Dr.Ing. Thesis)
MTA-2001-147 Heggelund, Svein E., MK: Calculation of Global Design Loads and Load Effects in Large High
Speed Catamarans. (Dr.Ing. Thesis)
MTA-2001-148 Babalola, Olusegun T., MK: Fatigue Strength of Titanium Risers - Defect Sensitivity.
(Dr.Ing. Thesis)
MTA-2001-149 Mohammed, Abuu K., MK: Nonlinear Shell Finite Elements for Ultimate Strength and Collapse
Analysis of Ship Structures. (Dr.Ing. Thesis)
MTA-2002-150 Holmedal, Lars E., MH: Wave-current interactions in the vicinity of the sea bed. (Dr.Ing.
Thesis)
MTA20-02-151 Rognebakke, Olav F., MH: Sloshing in rectangular tanks and interaction with ship motions
(Dr.ing.thesis)
MTA-2002-152 Lader, Pål Furset, MH: Geometry and Kinematics of Breaking Waves. (Dr.Ing. Thesis)
MTA-2002-153 Yang, Qinzheng, MH: Wash and wave resistance of ships in finite water depth. (Dr.Ing.
Aug. 2007 IMT

Thesis)
MTA-2002-154 Melhus, Øyvin, MM: Utilization of VOC in Diesel Engines. Ignition and combustion of
VOC released by crude oil tankers. (Dr.Ing. Thesis)
MTA-2002-155 Ronæss, Marit, MH: Wave Induced Motions of Two Ships Advancing on Parallel Course.
(Dr.Ing. Thesis)
MTA-2002-156 Økland, Ole D., MK: Numerical and experimental investigation of whipping in twin hull
vessels exposed to severe wet deck slamming. (Dr.Ing. Thesis)
MTA-2002-157 Ge, Chunhua, MK: Global Hydroelastic Response of Catamarans due to Wet Deck
Slamming. (Dr.Ing. Thesis)
MTA-2002-158 Byklum, Eirik, MK: Nonlinear Shell Finite Elements for Ultimate Strength and Collapse
Analysis of Ship Structures. (Dr.Ing. Thesis).
IMT-2003-1 Chen, Haibo, MK: Probabilistic Evaluation of FPSO-Tanker Collision in Tandem

Offloading Operation. (Dr.Ing.Thesis)
IMT-2003-2 Skaugset, Kjetil Bjørn, MK On the Suppression of Vortex Induced Vibrations of Circular
Cylinders by Radial Water Jets. (Dr.ing.Thesis)
IMT-2003-3 Chezhian, Muthu Three-Dimensional Analysis of Slamming (Dr.ing. Thesis)
IMT-2003-4 Buhaug, Øyvind Deposit Formation on cylinder Liner Surfaces in Medium Speed
Engines (Dr.ing.thesis)
IMT-2003-5 Tregde, Vidar Aspects of Ship Design; Optimization of Aft Hull with Inverse
Geometry Design (Dr.ing.thesis)
IMT-2003-6 Wist, Hanne Therese Statistical Properties of Successive Ocean Wave Parameters
(Dr.ing.thesis)
IMT-2004-7, Ransau, Samuel Numerical Methods for Flows with Evolving Interfaces (Dr.ing.thesis)
IMT-2004-8, Soma, Torkel Blue-Chip or Sub-Standard. A data interrogation approach of identity

safety characteristics of shipping organization (Dr.ing.thesis)
IMT-2004-9 Ersdal, Svein An experimental study of hydrodynamic forces on cylinders and cables
in near axial flow (Dr.ing.thesis)
IMT-2005-10 Brodtkorb, Per Andreas The Probability of Occurrence of Dangerous Wave Situations at Sea
(Dr.ing.thesis)
IMT-2005-11 Yttervik, Rune Ocean current variability in relation to offshore engineering

(Dr.ing.thesis)
IMT-2005-12 Fredheim, Arne Current Forces on Net-Structures (Dr.ing.thesis)
IMT-2005-13 Heggernes, Kjetil Flow around marine structures (Dr.ing. thesis)
IMT-2005-14 Fouques, Sebastien Lagrangian Modelling of Ocean Surface Waves and Synthetic
Aperture Radar Wave Measurements (Dr.ing. thesis)
IMT-2006-15 Holm, Håvard Numerical calculation of viscous free surface flow around Marine
structures
Aug. 2007 IMT

IMT-2006-16 Bjørheim, Lars G. Failure Assessment of Long Through Thickness Fatigue Cracks in
Ship Hulls (Dr.ing.thesis)
IMT-2006-17 Hansson, Lisbeth Safety Management for Prevention of Occupational Accidents

(Dr.ing.thesis)
IMT-2006-18 Zhu, Xinying Application of the CIP Method to Strongly Nonlinear Wave-Body
Interaction Problems (Dr.ing.thesis)
IMT-2006-19 Reite, Karl Johan Modeling and Control of Trawl Systems
IMT-2007-20 Storhaug, Gaute Experimental Investigation of Wave Induced

Vibrations and Their Effect on the Fatigue
Loading of Ships (Dr. ing. thesis).
IMT-2007-21 Sun, Hui A Boundary Element Method Applied to

Strongly Nonlinear Wave-Body Interaction
Problems. (CeSOS).
Aug. 2007 IMT

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Experimental investigation of wave

induced vibrations and their effect on

A thesis submitted in partial fulfillment of the

Oslo, 11th of September, 2007

Department of Marine Technology

This thesis is dedicated to my twins, Simen and Elias.

2.2.2 Analytical solution of the beam equation with shear deformation . . 35

3 Simplified analysis of vibration response and speed reduction 57

4 Experimental methodology and analysis procedures 69

4.6.1 Linear transfer functions . . . . . . . . . . . . . . . . . . . . . . . . 90

5 Model test results 111

6 Full scale measurements 163

7 Conclusions and recommendations 181

7.2.3 Investigation by simplified theoretical methods . . . . . . . . . . . . 184

B Experimental set-up procedures and uncertainty analysis 247

B.5 Calibration of waves prior to model testing . . . . . . . . . . . . . . . . . . 266

C Miscellaneous results 281

A33 3D added mass

FP Force on force panel

WPi Wave probe number i

α Philips constant in JONSWAP, dead rise angle of cone or stiffness ratio

ζI , ζT Incident & Total wave amplitude

ACCZi Acceleration in z-direction at position i

DTU Technical University of Denmark

TLP Tension Leg Platform

Mathematical operators and other symbols

1.1 Background and motivation for the present work

1.1.1 Recent experiences from a ship in operation

1.1.2 Comparison between measured and predicted vibration re-

compared to predictions from four state-of-the-art nonlinear hydrodynamic codes, which

1.1.3 Today’s rule requirements

1.2 A dive into literature

Springing: Conventionally, springing is defined as stationary resonance vibration of the

Whipping: Conventionally, whipping is defined as transient hull girder vibration caused

1.2.1 The beginning

1.2.2 Ship measurements considering wave induced vibrations

• The springing may come in ’bursts’ or ’beats’, (Little et al. 1971).

1.2.3 Model experiments considering wave induced vibrations

• The damping was often not properly addressed, e.g.(Goodman 1971).

1.2.4 Springing and whipping theories

• Hamid (1980) confirmed that coupling was important.

• Springing may be important in developing seas, (Chen 1980).

• The importance of considering non uniform steady flow and translating-pulsating

• A SES may experience significant whipping, e.g. (Brown et al. 1981).

• Whipping was larger in developing seas, (Chen 1980).

• Whipping was more important in head seas, (Schlacter 1989).

1.3 Purpose of the work

1.4 Outline of the thesis

A more detailed amount of theoretical aspects is presented in the appendices.

2.1 Hydrodynamic theory

2.1.1 Hypothesis 1: Steady wave elevation

For a cylinder or a sphere the tangential velocity can be written

Steady elevation [m]

2.1.2 Hypothesis 2: Amplification of incident wave at the bow

2.1.3 Hypothesis 3: 3D impact forces

The impact force in head seas may be written, (Faltinsen 1990)

2.1.4 Hypothesis 4: Second order velocity potential and wave

2.1.5 Speed reduction due to added resistance in waves

CT = Cw + CV + CAA + CBD (2.13)