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ABSTRACT OF DOCTORAL DISSERTATION HELSINKI UNIVERSITY OF TECHNOLOGY P.O. BOX 1000, FI-02015 TKK http://www.tkk.fi
Author Kristjan Tabri
Name of the dissertation Dynamics of Ship Collisions
Manuscript submitted 30.09.2009 Manuscript revised 10.12.2009
Date of the defence 09.02.2010
Monograph Article dissertation (summary + original articles)
Faculty Faculty of Engineering and Architecture Department Department of Applied Mechanics Field of research Naval Architecture Opponent(s) Professor Preben Terndrup Pedersen Supervisor Professor Petri Varsta Instructor Professor Jerzy Matusiak The thesis studies ship collisions computationally and experimentally on large and model scales. On the basis of the experimental observations a 3D simulation model is proposed that couples the motions of the ships to the contact force, and considers all the major hydromechanical forces that act on colliding ships. Additionally, the effects of sloshing and the dynamic bending of the hull girder are investigated and implemented into the simulation model. Large-scale experiments were analysed in order to get a deeper insight into the collision dynamics. On the basis of the large-scale experiments a model-scale test setup is designed using the Froude’s scaling law. There, the emphasis was laid on the external dynamics and the structural response, properly scaled from the large-scale test, was modelled using homogeneous foam in the side structure of the struck ship model. It is shown that the model-scale experiments illustrated the large-scale tests both qualitatively and quantitatively. A wide range of symmetric, both with and without sloshing, and non-symmetric collision scenarios are studied on a model scale. The experimental findings are exploited in the development of a coupled collision simulation model. The model is formulated in three-dimensional space, and the contact force between the colliding ships considers both the normal and frictional components. A discrete mechanical model for sloshing is implemented into this time-domain model. This linear sloshing model describes the fluid in partially filled tanks with a single rigid mass and with a number of oscillating mass elements that interact with the ship structure through springs and dampers. The dynamic bending of the ship hull girder is included by modelling it as an Euler-Bernoulli beam. Both the experiments and the simulations emphasised the importance of the coupling between the motions and the contact force. It was especially obvious in the case of non-symmetric collisions and in the experiments with sloshing. The penetration paths calculated with the developed time-domain simulation model agreed well with those from the experiments. The total deformation energy was predicted with a deviation of about 10%. The hydrodynamic radiation forces acting on colliding ships proved to have a strong influence on the energy distribution as at the end of the contact they accounted for up to 25% of the total available energy. However, if the interest is in the maximum deformation, the approach with the hydrodynamic damping ignored yields an error of about 5% in the deformation energy. The results of the large- and model-scale experiments with partially filled liquid tanks emphasised the importance of sloshing for collision dynamics. The structural deformation energy in the tests with sloshing was only about 70%-80% of that in similar collision tests without sloshing. The simulation method with the linear sloshing model overestimated the deformation energy by up to 10% for low filling levels of water, but in the case of medium filling levels the predictions agreed amazingly well. Keywords ship collisions, model-scale experiments, large-scale experiments, water sloshing
ISBN (printed) 978-952-248-272-3 ISSN (printed) 1795-2239
ISBN (pdf) 978-952-248-273-0 ISSN (pdf) 1795-4584
Language English Number of pages 51+60
Publisher Helsinki University of Technology, Department of Applied Mechanics
Print distribution Helsinki University of Technology, Department of Applied Mechanics
The dissertation can be read at http://lib.tkk.fi/Diss/ 2010/isbn9789522482730/
5
Preface This thesis is based on work done at the Schelde Naval Shipyard during 2003-2004
and at the Department of Applied Mechanics, Helsinki University of Technology during
2004-2009. During the thesis process I was financed within the framework of the Marie
Curie Intra-European Fellowship programme, by the Finnish National Graduate School
of Engineering Mechanics, the Finnish National project TÖRMÄKE, and the EU-
funded research project MARSTRUCT, and by the Finnish Maritime Foundation. This
financial support is greatly acknowledged.
I wish to express my sincere gratitude to my supervisor, Professor Petri Varsta, for
his continuous support and encouragement during the course of this work. His guidance
was crucial and helped me overcome many obstacles I encountered in my research work.
I would also like to thank my instructor, Professor Jerzy Matusiak, whose knowledge
and sense of exactness contributed significantly to the scientific quality of the thesis. I
am very grateful to Professor Jaan Metsaveer from Tallinn University of Technology
for his long support and for introducing me to Naval Architecture. Special thanks are
due to Leila Silonsaari, whose support and help with daily matters is highly appreciated.
I would like to take this opportunity to honour Mr. J.W.L. Ludolphy, who was the
initiator of the cooperation between the Schelde Naval Shipyard and Helsinki
University of Technology in the framework of the Marie Curie Intra-European
Fellowship programme. His involvement made it possible to obtain valuable
experimental data and to gain an insight into a more practical approach to ship collision
problems. He passed away unexpectedly on 22 September 2002. The guidance and long
discussions with my co-workers, Joep Broekhuijsen, Jan Jaap Nieuwenhuis, Marcel
Elenbaas, Rob de Gaaij, and Bob van de Graaf at the Schelde Naval Shipyard were
valuable and inspiring, especially during the earlier stages of the research work.
I would also like to thank my colleagues in the Ship Laboratory, Heikki Remes, Jani
Romanoff, Sören Ehlers, Alan Klanac, Jasmin Jelovica, Pentti Kujala and Pentti Tukia,
for their support and helpfulness, and for creating a pleasant working atmosphere. I
greatly appreciate the support and friendship of my Estonian colleagues D.Sc. Hendrik
6
Naar and Meelis Mäesalu. Special thanks go to my dear friends from Otaniemi for the
fantastic and unforgettable times we spent together.
Finally, I am deeply grateful to my parents for their irreplaceable support and
devotion. Last but not least, I thank my dear Kaia-Liisa for bringing immeasurable
happiness into my life and for being supportive throughout the whole thesis process.
Helsinki, January 2010 Kristjan Tabri
7
Contents Preface .......................................................................................................... 5Contents ........................................................................................................ 7List of Symbols ............................................................................................ 9List of publications and Author’s Contribution .................................... 14Original Features ....................................................................................... 161. Introduction ........................................................................................ 18
1.1. Background ...................................................................................................... 181.2. State of The Art ................................................................................................ 211.3. Scope of work .................................................................................................. 231.4. Limitations ....................................................................................................... 25
2. Experimental study ............................................................................ 272.1. Large-scale experiments .................................................................................. 272.2. Model-scale collision experiments .................................................................. 282.3. Model-scale collision experiments with sloshing effects ................................ 30
3. Collision dynamics .............................................................................. 323.1. Physical phenomena of ship collisions ............................................................ 323.2. Hydromechanical forces .................................................................................. 343.3. Contact between the ships ................................................................................ 363.4. Dynamic hull bending ...................................................................................... 373.5. Sloshing interaction ......................................................................................... 38
4. Time-domain simulation model ........................................................ 394.1. Equations of motion and time integration ........................................................ 394.2. Numerical solution procedure .......................................................................... 394.3. Comparison to a momentum conservation model ........................................... 42
5. Conclusions ......................................................................................... 456. References ........................................................................................... 47Errata ......................................................................................................... 51 Appendix
[P1] Tabri K., Broekhuijsen J., Matusiak J., Varsta P. (2009) Analytical
modelling of ship collision based on full-scale experiments. Journal of
Marine Structures, 22(1), pp. 42-61.
[P2] Tabri K., Määttänen J., Ranta J. (2008) Model-scale experiments of
symmetric ship collisions, Journal of Marine Science and Technology, 13,
pp. 71-84.
8
[P3] Tabri K., Varsta P., Matusiak J. (2009) Numerical and experimental
motion simulations of non-symmetric ship collisions, Journal of Marine
Science and Technology, doi: 10.1007/s00773-009-0073-2.
[P4] Tabri K., Matusiak J., Varsta P. (2009) Sloshing interaction in ship
collisions – An experimental and numerical study, Journal of Ocean
Engineering, 36, pp. 1366-1376.
9
List of Symbols a(ω) frequency-dependent added mass
a, b, c parameters defining the elliptical paraboloid
AL
A, B, C parameters defining the plane
lateral area
area
b(ω) frequency-dependent added damping
B ship’s breadth
cn
C
damping coefficient of the nth damped mass-spring element
y
[C] damping matrix
drag coefficient
D depth
E energy
EB
E
bending energy
C, ED
E
deformation energy
F
E
work against the friction force
K, EKIN
E
kinetic energy
SL
EI flexural stiffness
sloshing energy
F force and moment vector
FC
F
contact force
E
F
external excitation force in the sloshing model
F
F
fluid force in sloshing model
H
F
hydrodynamic radiation force
I
F
inertial force
K
F
velocity-dependent component of the radiation force
M
F
total force resulting from sloshing
n
F
Froude number
p
F
compressive force in the contact model
q frictional force in the contact model
10
FR
F
restoring force
µ
g gravity
inertial component of the radiation force; viscous shear force
h height; vertical distance; water depth;
hW
I number of partially filled tanks; moment of inertia
water height
J total number of degrees of freedom associated with oscillating masses
kn
k stiffness; radii of inertia
stiffness of the nth mass-spring element
K sum of roll moments acting on the ship; retardation function,
[K], [K] stiffness matrix
[Kb
l
] matrix of retardation functions
T
L length
dimension of the tank in the sloshing direction
im∗ generalised mass of ith mode
mx, my, mz
m
sum of ship’s and its added mass in the x-, y-, and z- directions
n
m
nth oscillating mass-spring element in sloshing model
R
ˆRm
a single rigid sloshing mass in a tank
sum of rigid sloshing masses
mT
m
total fluid mass in a tank
ST
M sum of pitch moments; ship mass including added mass
structural mass
[M] or [M] mass matrix
µΩ M matrix of non-linear acceleration and added mass, structural mass and
inertia terms
n normal
N number of oscillating masses per tank
NR
p,q,r angular velocities
number of rigid degrees of freedom
p pressure; normal coordinate (in vibrations); normal traction
11
plane
Q total number of degrees of freedom
q distributed loading; tangential traction
iq∗ generalised loading associated with ith mode shape
r position vector
R position vector
RN
surface
Reynolds number
t time
T ship’s draught
[T] matrix of transformation
tangent plane
u surge velocity at ship’s centre
u vector of translational velocity
v sway velocity at ship’s centre, velocity in general
V0
V
initial velocity of a sloshing tank
F
w heave velocity at ship’s centre
final velocity of a sloshing tank
W work
xn
x
displacement of the nth oscillating mass-spring element in the sloshing
model
R
x
displacement of the rigid mass in the sloshing model iyizi
x
local coordinate systems (i=A, B) 0y0z0
X, Y, Z sum of forces acting on the ship in the longitudinal, transverse, and
vertical directions
inertial coordinates
β collision angle
γ pitch angle
δ penetration; logarithmic decrement of damping
12
ε restoring coefficient
η horizontal vibration response of the hull girder
λ scaling factor AQλ direction vector in the contact model
µ non-dimensional added mass; viscosity
µd
ξ damping ratio; internal damping of hull girder
coefficient of friction
ρ density
σ stress
τ time parameter in retardation function, duration
ϕ roll angle
[ϕ] column matrix of Euler’s angles
ψ, θ , φ Euler’s angles
φ natural mode
ω frequency
Ω vector of rotational velocity
∇ volumetric displacement of the ship
Superscripts
0 inertial coordinate system
A striking ship
B struck ship
M model-scale
S ship-scale
Subrscripts
0 initial
E elastic
F fluid (force)
G gravitational (force)
13
H radiation (force)
i ith vibratory mode; index
Κ retardation (force or energy)
n nth sloshing mass; denotes sloshing matrices
P plastic
SL sloshing
µ added mass
Abbreviations
dof degree of freedom
VOF volume of fluid
CFD computational fluid dynamics
LED light-emitting diode
KG vertical height of the centre of gravity
COG centre of gravity
14
List of publicationsThis thesis consists of an introductory report and the following four papers:
and Author’s Contribution
[P1] Tabri K., Broekhuijsen J., Matusiak J., Varsta P. (2009) Analytical
modelling of ship collision based on full-scale experiments. Journal of Marine
Structures, 22(1), pp. 42-61.
The author gave a quantitative description of large-scale collision
experiments in which sloshing interaction on ships was included. The
manuscript was prepared by the author. Broekhuijsen provided large-scale
experimental data. Matusiak and Varsta made valuable recommendations
concerning the development of the theory and contributed to the manuscript.
[P2] Tabri K., Määttänen J., Ranta J. (2008) Model-scale experiments of
symmetric ship collisions, Journal of Marine Science and Technology, 13, pp.
71-84.
The author designed the test setup, validated its physical similarity to the
large-scale collision experiments, conducted the final analysis, and prepared
the manuscript. Määttänen took part in designing the test setup, conducted
the experiments, and provided the initial analysis of the results. Ranta
studied the modelling of structural resistance and provided the geometry of
the impact bulb to ensure dynamic similarity to the large-scale tests.
[P3] Tabri K., Varsta P., Matusiak J. (2009) Numerical and experimental
motion simulations of non-symmetric ship collisions, Journal of Marine Science
and Technology, doi: 10.1007/s00773-009-0073-2.
The author developed the theory and carried out the calculations and the
validation. The manuscript was prepared by the author. Varsta made
recommendations on the development of the contact model and contributed
to the manuscript. Matusiak contributed to the external dynamics model and
to the manuscript.
15
[P4] Tabri K., Matusiak J., Varsta P. (2009) Sloshing interaction in ship
collisions – An experimental and numerical study, Journal of Ocean Engineering,
36, pp. 1366-1376.
The author designed the test setup and conducted the experiments and the
development and validation of the simulation model. The manuscript was
prepared by the author. Matusiak and Varsta provided valuable comments
and contributed to the manuscript.
16
Original Features
Ship collisions are a complex phenomenon as they consist of transient ship motions and
structural deformations, commonly referred to as external dynamics and internal
mechanics. The majority of collision simulation models decouple the external dynamics
from the inner mechanics for the sake of simplicity. This thesis proposes a coupled
simulation model considering all six degrees of freedom for both colliding ships. The
ships are regarded as being rigid when their global motions are being considered. An
allowance for major local structural deformations is made when dealing with the contact
between the ships. The following features of this thesis are believed to be original.
1. The distribution of energy components during large-scale collision experiments
was calculated and presented in [P1]. These distributions present the quantitative
significance of different energy-absorbing mechanisms in ship collisions and
reveal the important phenomenon of sloshing.
2. A model-scale test setup for ship collisions was designed and scaled according
to the large-scale tests in [P2]. It was shown in [P2] that the model-scale tests
were physically similar to the large-scale ones. The test setup was exploited to
investigate collision dynamics in symmetric [P2] and non-symmetric collisions
[P3].
3. An experimental study on sloshing interaction in collision dynamics was
performed. The interaction was studied on a large scale in [P1] and on a model
scale for a wide range of collision scenarios in [P4].
4. A three-dimensional ship collision model, including the coupling between
external dynamics and inner mechanics, was developed with the help of the
kinematic condition [P3]. This condition is based on the mutual ship motions
and on the geometry of the colliding ships, giving the penetration and, thus, the
contact force. The elastic springback of structures deformed in a collision was
considered during the separation of the ships. The simulation model was
validated with the help of the experimental results of the symmetric collision
tests in [P1] and those of the non-symmetric collision test in [P3].
17
5. The effects of fluid sloshing in partially filled tanks are included in the
simulation model, using a discrete mechanical model for sloshing [P1 & P4].
The vibratory response of the hull girder of the struck ship is included for the
sway motion of the struck ship [P1].
18
1. Introduction
1.1. Background
Waterborne vehicles have been subject to a variety of accidents since their early
dawn. With the increasingly higher speeds and displacements of ships, the
consequences of accidents could be disastrous. Today’s society is more reluctant to
accept environmental damage and casualties. Therefore, significant marine accidents,
for example the collision of the Stockholm and Andrea Doria in 1956 or the grounding
of the Exxon Valdez in 1989, have often formed a basis for the development of
measures to increase the safety of shipping. Operational safety measures aim to reduce
the probability of accidents occurring, while structural safety measures in ships
concentrate on the reduction of the consequences. Regardless of the measures developed,
accidents involving ships can never be completely avoided – human errors, technical
malfunctions, or other unpredictable events continue to occur. Eliopoulou and
Papanikolaou
A reduction of the consequences of ship collisions implies the ability of colliding
structures to absorb energy without a rupture causing flooding or oil spillage. The
structure’s ability to withstand collisions is collectively called crashworthiness. A
crashworthiness analysis of ship structures combines two separate fields, external
dynamics and inner mechanics (Minorsky, 1959). The external dynamics evaluates the
ship motions, giving as a result the energy to be absorbed by structural deformations,
while the inner mechanics evaluates the deformations the structures undergo while
absorbing that energy. The first studies on the crashworthiness of ship structures date
back to the 1950s and deal with the collision safety of nuclear-powered ships. The
understanding of the physics involved was based on facts learned from actual collision
accidents (Minorsky, 1959) or on simplified experiments on the inner mechanics
(2007) studied the statistics of tanker accidents and showed that the total
number of accidents and the number of accidents causing pollution has decreased
significantly in recent decades. However, the accidents causing pollution have not
decreased to the same extent as the overall number of accidents. It has become obvious
that the safety measures to reduce the consequences in accidents, such as ship collisions
or groundings, should still be improved.
19
(Woisin 1979). Mainly because of the seriousness of the possible consequences, early
crashworthy structures implemented in nuclear-powered ships such as the NS Savannah
or NS Otto Hahn (see Soininen (1983) for the structural principles) were, however, far
too impractical to be implemented in typical commercial ships. A significant
improvement in crashworthiness came with the introduction of double-bottomed and,
later, double-hulled tankers. Spacious double walls provide an additional buffer zone
between the intruding object and the inner compartments of the ship. Knowledge about
the performance of these structures was based on large-scale structural tests in the
laboratory (Woisin, 1979; Amdahl and Kavlie, 1992). In recent decades many research
studies have been devoted to developing even more efficient and compact crashworthy
structures, for example a Y-core side structure (Ludolphy and Boon, 2000) or a buffer
bow (Kitamura, 2000).
Figure 1. Large-scale collision experiment in the Netherlands (photo taken by the
author).
The performance of the novel Y-core structure was studied, together with other
concepts, in a series of large-scale experiments (Carlebur, 1995; Wevers & Vredeveldt,
1999); see Figure 1. There, collisions between two river tankers with displacements of
an order of magnitude of 1000 tons were studied. These were the first collision
experiments that also included the external dynamics and its coupling to the inner
mechanics. The analysis of the experiment with the Y-core side structure revealed
20
shortcomings in the understanding of collision phenomena. Contrary to the predictions
made beforehand, the structure that was tested did not tear and was only slightly
damaged (Broekhuijsen, 2003). Not only the way in which the structure deformed, but
also the amount of energy to be absorbed was significantly lower compared to the
predictions. In addition to the contact and the hydromechanical forces, there were other
mechanisms absorbing a significant part of the available energy. Possible interactions
arising from partially filled cargo tanks and from the dynamic bending of the hull had
been excluded from the analysis. For the sake of brevity, we refer to these very relevant
phenomena as complementary effects. These depend on the time histories of the ships’
motions and on the contact force. The motions should be calculated in parallel to the
structural deformations to account for possible mutual interaction, i.e. the coupling
between the inner mechanics and the external dynamics has to be considered on a
reasonable level.
The decoupling is possible in symmetric ship collisions, where the striking ship
collides at a right angle with the amidships of the struck ship and where the
complementary effects are negligible. In such a collision the ship motions are limited to
a few components and the contact force as a function of the penetration can be
predefined. The actual extent of the penetration is obtained by comparing the area under
the force-penetration curve to the deformation energy from the external dynamics.
Statistical studies (Lützen, 2001; Tuovinen, 2005) have, however, indicated that the
majority of collisions are non-symmetric in one way or another. Often the collision
angle deviates from 90 deg or the contact point is not at the amidships. In non-
symmetric ship collisions the penetration path cannot be predefined with reasonable
precision, but it should be evaluated in parallel with the ship motions, implying once
again the need for a coupled approach. Model-scale experiments (Määttänen, 2005)
provide a first insight into the dynamics of non-symmetric collisions. These tests,
together with the large-scale experiments, make it possible to develop and validate a
coupled approach to ship collision simulations, including all the relevant energy-
absorbing mechanisms.
21
1.2. State of The Art
One of the first calculation models to describe the external dynamics of ship
collisions was proposed by Minorsky (1959). This single-degree-of-freedom (dof)
model was based on the conservation of linear momentum and there was no coupling
with the inner mechanics. The interaction between the ship and the surrounding water
was through a constant added mass. The model allowed fast estimation of the energy
available for structural deformations without providing exact ship motions. Woisin
(1988) extended the collision model to consider three dof – surge, sway and yaw. Later,
in 1998, Pedersen and Zhang included the effects of sliding and elastic rebounding.
These two-dimensional (2D) models in the plane of the water surface are capable of
analysing non-symmetric collisions and evaluating the loss of kinetic energy in a
collision absorbed by structural deformations. However, there only the inertial forces
are taken into account and thus, the effects of several hydromechanical force
components and the contact force are neglected.
Minorsky’s assumption of the constant added mass was investigated
experimentally by Motora et al. (1971). In their experiments a ship model was pulled
sideways with a constant force. They concluded that the constant value of the added
mass is a reasonable approximation only when the duration of the contact and the
transient motion is very short. A more precise method of presenting this so-called
hydrodynamic radiation force was proposed by Cummins (1962) and Ogilvie (1964).
Their method draws a clear distinction between two components of the force; one
component is proportional to the ship’s acceleration and the other is a function of her
velocity. The first of these components is what is described with the constant added
mass term. The velocity-dependent component, commonly referred to as the
hydrodynamic damping force, accounts for the memory effects of water. This approach
requires the evaluation of frequency-dependent added mass and damping coefficients,
which can be obtained by experiments (Vugts, 1968), by numerical methods (Journee,
1992), or by algebraic expressions based on a conformal mapping solution (Tasai, 1961).
In 1982, Petersen suggested a procedure for the time-domain simulations of ship
collisions in which the hydrodynamic forces were included, as suggested by Cummins
(1962) and Ogilvie (1964). Ship motions were again limited to the horizontal plane of
22
the water. This approach was validated with the model tests of Motora et al. (1971) and
good agreement was found.
As far as the author knows, the model of Petersen (1982) was the first time-domain
simulation model capable of treating non-symmetric collisions in 2D. Earlier time-
domain simulation models by Drittler (1966) and Smiechen (1974) required either
preliminary knowledge of the ship motions or were limited to symmetric collision
scenarios. In Petersen’s model the coupling between the motions and the structural
deformations was included quasi-statically using non-linear springs. Therefore, the
component of the contact force arising as a result of the relative velocity between the
ships, i.e. the frictional force, was disregarded. Brown (2002) described a similar
coupled approach and also included the frictional force between the ships. Still, the
hydrodynamic forces were considered only through the constant added mass. He
compared the calculated deformation energy to that evaluated with the decoupled
approach of Pedersen and Zhang (1998). It was concluded that, while the total
deformation energy was predicted well, the decoupled method results in a different
decomposition of the total deformation energy in the transverse and longitudinal
directions, compared to that given by the coupled method.
Le Sourne at al. (2001) formulated the external dynamics of collisions in a three-
dimensional (3D) space considering six dof for global ship motions and, similarly to
Petersen (1982), included the hydrodynamic forces on the basis of Cummins (1962) and
Ogilvie (1964). The coupling between the ship motions and the structural deformations
was carried out simultaneously with the help of structural analysis with the finite
element method. The method was used to simulate an eccentric ship-submarine
collision, where only small angular motions are excited during the contact. The analysis
concentrated mainly on the motions after the contact and revealed the importance of the
hydrodynamic damping force on the roll and yaw motion of the submarine.
As already discussed, the large-scale collision experiments (Carlebur, 1995; Wevers
& Vredeveldt, 1999) revealed that the existing collision models did not explain all the
possible energy-absorbing mechanisms. Tabri et al. (2004) included the sloshing model
of Graham and Rodriguez (1952), and were the first to present the importance of
sloshing interaction in collision dynamics. This sloshing model was also solved by
23
Konter et al. (2004), applying the finite element method. Sloshing in collisions was also
studied by Zhang and Suzuki (2007), applying the arbitrary Lagrangian-Eulerian
method. They concluded that for deep filling levels of water a simple mechanical model,
similar to that of Graham and Rodriguez (1952), tends to overestimate the energy
involved in sloshing motion.
The energy consumption resulting from the vibratory bending of the hull girder in
collisions has been studied by Tabri et al. (2004) and Pedersen and Li (2004). Both
studies revealed that the energy consumption in relation to the other absorbing
mechanisms was small.
1.3. Scope of work
The present investigation comprises both an experimental and theoretical study;
see Figure 2. Ship collisions are studied experimentally on a large and on a model scales.
On the basis of the experimental observations, a 3D simulation model is proposed that
couples the motions to the contact force, and considers all the major hydromechanical
forces that act on the colliding ships. This makes it possible to carry out the simulations
needed for the case of non-symmetric collisions. Additionally, the effects of sloshing
and the dynamic bending of the hull girder are investigated and implemented into the
simulation model.
Large-scale experiments were analysed in order to get a deeper insight into the
collision dynamics [P1]. This study was published in the year 2004 in the form of an un-
reviewed conference paper by Tabri et al. (2004). However, the data from the large-
scale experiments covered a limited range of symmetric collision scenarios and, in
particular, no information was obtained about non-symmetric collisions. Therefore, a
model-scale test setup was designed on the basis of these large-scale experiments, using
the Froude’s scaling law to preserve geometric, dynamic, and kinematic similarity as far
as possible. There, the emphasis is laid on the external dynamics. The structural
response, properly scaled from the large-scale experiments, is modelled using
homogeneous foam in the side structure of the struck ship model. This technique was
adapted from model-scale ship grounding experiments (Lax, 2001). The crushing
24
resistance of the foam, together with a geometrically properly dimensioned impact bulb,
gives a force-penetration curve similar to that in the large-scale tests. A wide range of
symmetric and non-symmetric collision scenarios are studied on a model scale,
allowing the influence of several parameters, such as the ships’ masses, the collision
velocity, the location of the contact point, the collision angle, and structural resistance
to be investigated [P3 & P4]. Moreover, the sloshing effects are studied by filling the
tanks on board the striking ship to different water levels [P4]. To improve the
understanding of the sloshing interaction, these tests are repeated with equivalent fixed
masses in tanks.
Figure 2. Outline of the investigation.
The experimental findings are exploited in the development of the coupled
collision simulation model. The model is formulated in 3D, and the contact force
between the colliding ships considers both the normal and frictional components [P3].
The formulation of the equations of motion follows the derivations by Clayton and
Bishop (1982) and, in addition to this, the hydrodynamic radiation forces are evaluated
using the theories of Cummins (1962) and Ogilvie (1964), together with the strip theory
25
(Journée, 1992). The kinematic condition between the colliding ships is served through
a common contact force. The contact force is evaluated as an integral over normal and
tangential tractions at the contact surface. The contact process is divided into three
phases according to whether the penetration increases, remains constant, or decreases.
During decreasing penetration the elastic springback of the deformed structures is
considered.
The complementary effects, including the sloshing forces and the dynamic
bending of the hull girder, are investigated both experimentally and computationally [P1
& P4]. Several symmetric collision scenarios are experimentally studied on a model
scale to reveal the dynamics of sloshing. A discrete mechanical model for sloshing
(Graham and Rodriguez, 1952) is implemented into the time-domain simulation model
[P4]. This sloshing model provides a simple means to include the sloshing forces in a
collision in the motion equations. It should be noticed that this model does not require
precise and time-consuming numerical calculations. The sloshing model describes the
fluid in partially filled tanks with a single rigid mass and with a number of oscillating
mass elements that interact with the ship structure through springs and dampers. The
dynamic bending of the ship hull girder is included by modelling it as an Euler-
Bernoulli beam with a certain set of physical properties such as mass, flexural stiffness,
and internal damping [P1].
1.4. Limitations
This thesis proposes a simulation model of ship collisions in which the approach to
consider different phenomena involved, is kept simple for the sake of time efficiency.
This sets certain limitations on the applicability of the model. These limitations are
discussed below.
All the experiments and calculations are limited to collision scenarios where the
struck ship is initially motionless. This is mainly due to the test setup of the model tests,
where the focus was on the physical phenomena. The dynamics and kinematics
involved with the forward speed of the striking ship are investigated. However, the
physical principles remain the same for the struck ship in spite of possible interaction
26
resulting from the wave pattern arising from the surge of the struck ship also inducing a
contact force component in that direction. The simulation model will not have this
limitation, even though it lacks proper experimental validation. A calm sea is assumed
throughout the experimental programme and the development of the simulation model.
The simulation model has both non-linear and linear features. Non-linearity arises
from the contact force and linearity is related to hydromechanical forces such as the
hydrostatic restoring force and the hydrodynamic radiation forces. Because of this, the
roll angle of the ships is limited to angles of approximately 10°. Moreover, the
hydrodynamic added mass and damping coefficients are calculated at the initial
equilibrium position of the ships and the effect of the change in actual position and
orientation is disregarded. These limitations are supported by the observations from the
large- and model-scale experiments. Effects arising from wave patterns around the
colliding ships and the hydrodynamic coupling between them are considered as being
secondary and thus they are disregarded. Flooding and the ensuing loss of stability are
not investigated as the time scale of a collision is small compared to these.
A study of the inner mechanics of ship side structures lies beyond the scope of the
thesis as the focus is on collision dynamics. Therefore, in the study the side structure of
the struck ship is replaced by homogeneous foam, with, however, the dynamic
similarity being maintained. The calculation model for the contact force covers the
kinematics between the ships and is based on the tractions at the contact surface
described by the normal and tangential components, which are easily obtained for
homogeneous material compared to that of real ship structures. The model is quasi-
static, but includes friction forces based on the relative velocity between the ships.
The sloshing interaction is due to sway and surge, and angular motions play a minor
role. The sloshing model applied in the simulations is based on linear flow theory and
the effect of possible roof impacts is ignored. However, the damping effect is included
with a simple viscous damping model.
The vibratory response of the hull girder is studied on a large scale, as in the model-
scale experiments rigid ship models were used. In the simulation model the bending is
considered as a vibratory response of the lowest eigenmode of an Euler-Bernoulli beam.
This response is included only for the sway motion of the struck ship.
27
2. Experimental study The collision dynamics were investigated through large- and model-scale
experiments. The large-scale collision experiments (Wevers & Vredeveldt, 1999)
provided an insight into symmetric collisions with sloshing interaction. The knowledge
from the large-scale experiments is favourable as it is free of scaling effects, but on the
other hand the tests are costly and thus a wide range of collision parameters cannot be
studied. Model-scale experiments offer an alternative as a wider parametric range can
be covered, but special attention has to be paid to scaling. A model-scale test setup was
designed and validated with the help of the results of the large-scale experiments. In the
model-scale tests a total of 46 collision experiments, including symmetric and non-
symmetric collisions, were carried out. Further, the collision experiments with water
sloshing were studied. For the sake of brevity, the experiments with water sloshing are
henceforth referred to as “wet” tests, while the other experiments without sloshing
phenomena are simply “dry” tests.
2.1. Large-scale experiments
Several full-scale collision experiments using two inland tankers were conducted in
the Netherlands by TNO in the framework of a Japanese, German, and Dutch
consortium (Wevers & Vredeveldt, 1999). These experiments had different purposes,
such as to validate numerical analysis tools, to investigate collision physics, and to test
new structural concepts. In this study the experiments with Y-core (Wevers &
Vredeveldt, 1999) and X-core (Wolf, 2003) side structures are of interest.
In the Y-core experiment both the striking and the struck ships had large amounts of
ballast water in partially filled tanks, providing the possibility of sloshing occurring
during the collision. On the other hand, in the X-core experiment the possible sloshing
effects were practically removed, as there was only a small amount of ballast water. On
the basis of these experiments, the time histories of different energy components
throughout the collision were evaluated in [P1] and the importance of sloshing
phenomena was revealed on the basis of energy balance analysis. The observations and
28
measured results of the large-scale experiments provided basic knowledge for the
design of the model-scale experiments and for the development of the simulation model.
2.2. Model-scale collision experiments
The model-scale experiments were performed to extend the physical understanding
of ship collisions. The large-scale experiments were scaled to model scale using a
scaling factor of 35 [P2]. The Froude scaling law was obeyed considering the
practicalities of the experiments, such as reasonable magnitudes of velocities and forces
involved in ship collision. This led to proper relation between inertia and gravity forces,
while viscous forces were overestimated as a result of a too small Reynolds number.
However, viscous forces have only a minor role in such a transient event as ship
collision. As shown later in this chapter, the prevailing forces in ship collision are
inertia forces together with the contact force.
The scaling resulted in ship models with the following main dimensions: length
LA =LB = 2.29 m, depth DA = DB = 0.12 m, and breadth BA = 0.234 m for the striking
ship and BB
The striking ship model was equipped with a rigid bulb in the bow and it collided
with the side structure of the struck ship model; see Figure 3. At the contact location a
block of homogeneous polyurethane foam was installed. The force-penetration curve
from the large-scale experiment was used to scale down the structural response of the
struck ship and, thus, maintain dynamic similarity. The scaling was based on the
crushing strength of the foam and on the geometry of the bulb [P2]. The selected foam
had a crushing strength of 0.121 MPa (Ranta and Tabri, 2007).
= 0.271 m for the struck ship. When compared on the same scale, the
flexural rigidity of the ship models was significantly higher than that of the large-scale
ships and thus, the hull girder bending was not studied on the model scale.
The dry model-scale experiments were divided into three different sets on the basis
of the collision scenario. The first set concentrated on symmetric collisions [P2], while
the second and the third sets contained non-symmetric collision experiments [P3]. In the
second set, the eccentricity LC of the contact point was varied between 0.13LB to 0.36LB
from the amidships towards the bow, but the collision angle β had the same value, 90°.
29
In the third set, the collision angles varied from 30° to 120°, but now the eccentricity
was kept constant at around ~0.18LB
.
Figure 3. Model-scale test setup.
During the collision all six motion components of both ships were recorded with
respect to an inertial coordinate system using a Rodym DMM non-contact measuring
system. Depending on the collision scenario, the contact force was recorded either in a
longitudinal or in the longitudinal and transverse directions with respect to the striking
ship model. These two separate measuring systems were not synchronised because of a
non-constant time lag resulting from the post-processing of data in the Rodym system.
Thus, an automatic correction of the time lag was not possible and the synchronisation
had to be carried out manually [P2].
The model-scale results of symmetric collisions, covering the motions, forces, and
energy distributions, proved that the test setup provided results qualitatively and
quantitatively similar to the large-scale tests [P2]. This fact made it possible also to
exploit the results of the non-symmetric model-scale tests for the validation of the
simulation model.
30
2.3. Model-scale collision experiments with sloshing effects
The sloshing phenomenon was experimentally studied on a model scale, with
several water filling levels in two onboard tanks being considered [P4]. On the basis of
the experience from the dry tests, the measuring systems were developed further to
improve the synchronisation. In addition, two water tanks were installed on board the
striking ship model; see Figure 4. The free surface elevation in the tanks was measured
with four resistive wave probes made of steel wire. Three probes were installed in the
fore tank and one in the aft tank.
Figure 4. Model-scale test setup for wet tests.
Four different even-keel loading conditions of the striking ship model were tested
under three velocities. The amount of water in the water tanks varied from 21% to 47%
of the total displacement of the model. To deepen the understanding of the effect of the
sloshing phenomena on collision dynamics, most of these wet tests were repeated with a
loading condition in which the water in the tanks was replaced with rigid masses of the
same weight.
31
The setup for the wet tests gave reliable and repeatable results [P4]. The results of
the model-scale experiments clearly emphasised the importance of sloshing in
connection with the collision dynamics. According to the results of the model
experiments, it can be stated that the sloshing made the striking ship behave like a
lighter ship, causing a reduction in the collision damage to the struck ship. This could be
seen from the deformation energy, which in the wet tests was only about 80% of that in
the dry tests with equivalent collision scenarios. This energy reduction value was clearly
influenced by the filling level, but not that strongly by the initial collision velocity of
the striking ship.
32
3. Collision dynamics
3.1. Physical phenomena of ship collisions
When two ships collide, the contact force arises as a result of the penetration of
structures, defined as a relative position between the striking and the struck ship. The
contact force causes the ships to become displaced from their current position. At any
time instant the force has to be in balance with the inertial and hydromechanical forces
associated with this movement. Ship motions are defined in the inertial reference frame
O0x0y0z0, while the forces are mainly defined with respect to local coordinate systems
Oixiyizi, which move with the ship; see Figure 5. Hereafter, the superscript characters A
and B denote the striking and the struck ship, respectively. If the superscript is omitted
or replaced by i, it means that the description is common to both ships.
Figure 5. Definition of collision dynamics and kinematics.
During a collision, the initial kinetic energy of the ships is transferred to the
work done by the ship motions and by the forces. However, to get a comprehensive
overview of collision dynamics a presentation through energies is more advantageous
compared to that through motions and forces. The time histories of the energy
components in the symmetric large-scale collision experiments with Y-core (Wevers
33
and Vredeveldt, 1999) and X-core (Wolf, 2003) structures were studied in [P1]. These
time histories are presented in Figure 6, including the energy EA associated with the
striking ship, EB associated with the struck ship, and their difference, i.e. the
deformation energy ED. In these tests, the contact between the ships lasted for about 0.6
s. For the sake of comparison, the total deformation energy *
DE , consisting of plastic
and elastic components, evaluated with the decoupled approach of Minorsky (1959) is
also presented in the figure. This energy is evaluated by exploiting the ships’ structural
and hydrodynamic added masses, as presented in Tables 1 and 2 in [P1]. With the help
of this energy, a large discrepancy between the Y- and X-core tests is revealed. In the
Y-core experiment, differently from the X-core one, both ships had partially filled liquid
tanks on board that caused the sloshing phenomenon during the collision. This sloshing
stores part of the energy and therefore there is less energy available for structural
deformations. Thus, the model of Minorsky fails to predict the deformation energy in
the case of the Y-core experiment, while in the case of the X-core test the prediction
agrees well with the experimental value.
a)
b)
Figure 6. The time history of energy components in relative form throughout a large-
scale collision with Y-core (a) and X-core (b) experiments.
To get a more detailed understanding of the physics of ship collisions, the time-
histories of EB and E
A in the Y-core collision are divided into their components in
Figure 7. There, the importance of the hydrodynamic damping B
KW and the kinetic
energy B
KINE of the struck ship compared to that of the striking ship, for which A
KW is
34
not even depicted, can be clearly seen. It should be noticed that the kinetic energy B
KINE
covers both the structural mass and the hydrodynamic added mass. Furthermore, Figure
7 illustrates the transformation of the kinetic energy into sloshing energy i
SLE during the
collision. The significance of the sloshing is clearly seen in the case of the striking ship
in Figure 7b, where at the time instant of the maximum contact force, around t = 0.5 s,
almost all of the energy is associated with the sloshing motions of the liquid in the tanks.
The energy B
FE to overcome the hydrodynamic drag resulting from sway motion, and
also the energy B
BE caused by the dynamic bending of the hull girder, are relatively
small compared to the other energy components.
a) b)
Figure 7. The variation in the motion energy components throughout the large-scale
collision in the case of the struck ship (a) and the striking ship (b).
3.2. Hydromechanical forces
The hydromechanical forces, especially the radiation force, together with the ship’s
inertia and the contact force, are the main components in collision dynamics. An
accelerating or decelerating ship encounters a hydrodynamic radiation force induced by
the relative acceleration between the hull and the water. The acceleration component of
this force is based on the constant added mass at an infinite frequency of motion
multiplied by the ship’s acceleration. In the presence of a free surface an additional
force component – hydrodynamic damping – arises. This is evaluated with the help of
35
the convolution of the velocity and retardation function, which accounts for the memory
effect of water (Cummins, 1962; Ogilvie, 1964). This approach requires the evaluation
of the frequency-dependent added mass and the damping coefficients, which are
calculated with the help of the strip theory (Journée, 1992).
The importance of the radiation force and the corresponding energy increases in
non-symmetric collisions as a result of the longer contact duration compared to that of
symmetric collisions. This is demonstrated by the time histories of the relative radiation
energies in Figure 8. There, these energies are presented over the duration of the contact
in model-scale tests with similar loading conditions and collision velocities. There, the
kinetic energy Eµ resulting from the added mass dominates during the first half of the
contact, while the work iKW resulting from the hydrodynamic damping becomes the
most important radiation energy component by the end of the contact, where it accounts
for about 17% of the total available energy in a non-symmetric collision and for about
10% in a symmetric collision.
a)
b)
Figure 8. The proportions of the radiation energy to the total available energy in a
symmetric model-scale test, No. 111 (a), and a non-symmetric model-scale test, No. 303
(b) (see Appendix B in [P3] for the test matrix).
The hydrostatic restoring force is assumed to be proportional to the angular
displacement of the ship from the equilibrium position, limiting the displacement to
angles of approximately 10° [P3]. The hydrodynamic drag is assumed to be proportional
36
to the square of the ship’s velocity and is considered for surge and sway. In the surge
direction only the frictional resistance is included with the ITTC-57 friction line
formula, as the magnitude of the other resistance components is assumed to be
negligible. The modelling of the hydrodynamic drag in the sway direction is based on
the formula presented by Gale et al. (1994).
3.3. Contact between the ships
The interaction between the ships during a collision is through a contact force
common to both ships. The interaction is modelled using the kinematic condition, which
describes the penetration path of the contact force from the ship motions in the time
domain. The penetration is solved piecewise in the time domain and then the
instantaneous contact force is calculated on the basis of the geometry of the colliding
bodies and using a simple model of contact mechanics relating penetration to the
contact force.
As assumed in Chapter 1, all the deformations resulting from a collision are
limited to the side structure of the struck ship, which deforms according to the shape of
the penetrating rigid bow of the striking ship. The contact process is divided into three
distinct phases [P3]. The contact starts with a loading phase, during which the
penetration depth increases. This loading is followed by a short stiction phase, during
which the direction of the relative velocity between the ships reverses and, thus, an
unloading phase starts, during which the ships separate as a result of the elastic
springback of the deformed structures.
The contact force at each time step is obtained by integrating the normal and
tangential tractions over the contact surface between the colliding bodies. The shape of
the contact surface is based on the geometry and the relative position of the ships. The
normal traction is equal to the known maximum normal stress on the surface of the
deformed side structure. The tangential traction is based on the Coulomb friction law
stating the proportionality between tangential and normal traction components.
For the integration of the contact force, the contact surface is divided into
integration elements. The normal traction occurs in compression and its direction is
37
determined by the normal of the element. However, the direction of the tangential
traction depends on the relative motion between the ships. During the loading and the
unloading phases this direction is determined as a projection of the relative velocity into
the plane of the integration element. In the stiction phase the direction of the tangential
traction is based on the relative acceleration to avoid singularity problems with the
reversing velocity (Canudas et al., 1995; Dupont et al., 2000).
When the penetration starts to decrease in the unloading phase, the contact is not
immediately lost because of the elastic springback of the deformed structure. This is
considered throughout the unloading phase in order to define the contact surface and the
direction of the relative velocity in the plane of the integration element. It was assumed
in [P3] that the struck ship’s structure in contact with the integration element recovers
along a path defined by a normal of the initial un-deformed surface. The direction of the
relative velocity on the contact surface is based on the rate of recovery along that path
and this direction determines that of the tangential traction.
3.4. Dynamic hull bending
In addition to rigid body motions in sway, impact loading caused by a collision
also induces the transverse dynamic bending of the hull girder of the struck ship [P1]. It
is assumed that the cross-sections of the hull girder remain plane, which allows the
modelling of the ship girder as an Euler-Bernoulli beam. The hull girder is modelled as
a body with free end boundary conditions. There, the major physical properties are its
length, flexural stiffness, mass per unit length, and internal damping.
The vibratory response of the ship hull girder is based on the superposition of its
eigenmodes. These, with the corresponding eigenfrequencies, are solved according to
Timoshenko et al. (1937). However, in a collision only the response resulting from the
lowest eigenmode is of interest. The equation of motion of vibratory response in time
domain is expressed with generalised coordinate (Clough and Penzien, 1993) exploiting
the corresponding generalised loading at each time step and the generalised mass, which
also includes the constant added mass in sway. The loading considers only the
component of the contact force transverse to the struck ship. The value of internal
38
structural damping in bending is based on the measured damping values of several ships
(ISSC, 1983).
3.5. Sloshing interaction
Sloshing covers a transient fluid motion inside an onboard tank caused by a rapid
movement of a ship hull during a collision. The effect of the sloshing on the collision
dynamics is based on time-varying loads on the tank bulkheads and, thus, causes a
change in the energy distribution during a collision.
Sloshing interaction in ship collisions was studied in [P1] and [P4] on the basis of
the large- and model-scale experiments, and on theoretical modelling, applying
computational fluid dynamics (CFD) and the mathematical analogy model developed by
Graham and Rodriguez (1952). This model is based on the linear potential flow theory,
which assumes linearised free surface conditions. There, the water in a partially filled
tank is divided into a rigid mass and into a finite number of oscillating masses. These
oscillating masses are connected to the bulkheads by springs and viscous dampers,
where the spring compression and the damper velocity give the sloshing force for the
equations of motion in sway and surge. Each of these oscillating masses with damping
describes one eigenmode of fluid sloshing.
To get full correspondence to the results of the potential flow theory, a mechanical
model of sloshing requires an infinite number of such oscillating elements. However, it
has been proven that the sloshing force induced by a spring-mass element decreases
rapidly with an increasing mode number (Abramson, 1966). On the basis of the CFD
calculations, it was concluded in [P1] that a sufficient number of oscillating masses is
three in collision applications. The damping coefficients for the viscous damping model
were evaluated by CFD calculations in [P1] and by model-scale experiments in [P4].
The damping coefficient increases as a function of the initial velocity of the striking
ship and decreases with the relative filling level in the tanks.
39
4. Time-domain simulation model
4.1. Equations of motion and time integration
A time domain simulation of collision combines all the forces discussed in Chapter
3 in a single calculation and gives the ships’ behaviour with the collision forces. The
relation between the forces and the ship motions is described through a system of
equations of motion for each ship. There, the contact force is derived with the help of a
kinematic condition based on the relative motion between the ships. The system of six
equations of rigid body motions was presented in [P3]. In [P4] this system was extended
to take the sloshing into account. There, it was assumed that the effect of sloshing on
the ship’s mass centre is negligible and thus it remains fixed to its initial position. The
transverse vibratory bending of the hull girder of the struck ship was included in [P1] as
an additional equation of motion in the sway direction. All the Newtonian equations of
motion are expressed in the local coordinate systems of the ships, allowing a fast
evaluation of the hydromechanical and the sloshing forces.
The time integration of the equations of motion is based on an explicit 5th
-order
Dormand-Prince integration scheme, which is a member of the Runge-Kutta family of
solvers (Dormand and Prince, 1980). Inside a time integration increment, seven sub-
increments are calculated. The hydrodynamic inertia force, the restoring force, the
sloshing forces, and the ship motions are updated in every sub-increment. On the other
hand, the contact force, velocity-dependent radiation force, and the hydrodynamic drag
are kept constant during the whole integration increment for the sake of time efficiency.
The results converged when the time increment was around 10 ms on a full scale.
4.2. Numerical solution procedure
The procedure of the time domain simulation is presented in Figure 9, where it is
divided into three steps. First, at time t, the position, velocity, and acceleration are
known for both ships. As a second step, the external forces are calculated for time t on
the basis of these values. The gravity force is constant throughout the collision and acts
along the global vertical axis z0. The hydromechanical forces are calculated in a local
40
coordinate system from the position and motions of the ships. For the contact force the
relative position and motions are presented in the local coordinate system of the striking
ship, where the contact force is calculated. Given the contact force, the vibratory
bending of the hull girder is evaluated. Sloshing forces ensue from the relative motion
between the sloshing masses and the ship. As a final step, the values of the initial
parameters are all substituted into the equations of motion, whence the values of the
ship motions are solved for time instant t+∆t.
The solution of the equations of motion for both colliding ships at time instant t+∆t
provides kinematically admissible motions given in the local coordinate system Oixiyizi.
In addition, the vibratory response of the hull girder of the struck ship is added to the
sway motion of the rigid body. The new position of the ship’s centre of gravity at t+∆t
with respect to the inertial frame is evaluated by transforming the translational
displacement increments to the inertial frame. After this, the orientation with respect to
the inertial frame is updated by the angular increments of Euler’s angles. The process is
repeated until the end of the collision.
41
1. Known solution at time t: position, orientation, and motions
, , , , , && &&t i t i t i t i t i t ix x x Ω Ωϕϕϕϕ
sloshing motions , ,& &&t i t i t i
n n nx x x
hull bending response ( ) ( )( , )η φ=t B
ix t x p t
2. Calculate external forces at time t
a. Gravity force [P3] GF
b. Hydrostatic buoyancy force [P3] BF
c. Hydrodynamic forces
(i) frictional resistance and hydrodynamic drag [P3] FF
(ii) hydrodynamic radiation force [P3] ( ) ( )µ + Kt tF F
d. Contact force = − = +A B A A
C C p qF F F F
(i) compressive force [P3] A
pF
(ii) friction force [P3] A
qF
e. Dynamic hull bending: generalised force [P1] iq∗
f. Sloshing interaction [P4]
[ ] [ ] ( ) [ ] [ ] ( )Tt T Tt Tt T t T
n n nnΩ +C x x K x x& & ϕϕϕϕ
3. Solve the equations of motion for t+∆∆∆∆t a. Equation of motions (EOM) (F- force, G- moment of forces) [P3]
µ µ µΩ
+ − = − −
&
& R R
u u FM M F F F
Ω Ω G
b. EOM in the case of complementary effects [P1] & [P4]
[ ][ ] [ ]
[ ][ ]
[ ] [ ][ ] [ ]
[ ]
( ) ( ) ( ) ( ) ( )2
1 1 1 1 1 1 1 1 1
0 0
0 0 0 0 ,
2
µ µµ
ξω ω
Ω
∗ ∗ ∗ ∗
+ + + + = − + + =
&
&
&
&& &
n n n
n n n
B B B B B Bm p t m p t m p t q t
u u xM M F
M Ω C Ω K FG
u u x
ϕϕϕϕ
Solution of equations of motions of ships’ response at time t+∆t
, , , , ,+∆ +∆ +∆ +∆ +∆ +∆ && &&
t t i t t i t t i t t i t t i t t ix x x Ω Ωϕϕϕϕ , , ,
+∆ +∆ +∆& &&
t t i t t i t t ix x x , ( )η+∆t t B x
Notations: [Mµ] mass and inertia matrix, also including added masses; µΩ M matrix of non-
linear acceleration and mass and inertia terms, also including added masses; [ ]nM matrix of
sloshing masses; [ ]nC matrix of damping coefficients for sloshing; [ ]nK matrix of stiffness
coefficients for sloshing; 1
∗Bm generalised mass; 1ωB lowest eigenfrequency of the hull girder;
ξ internal damping of the hull girder; vector; [ ] matrix
Figure 9. Solution process of time-domain simulations.
42
4.3. Comparison to a momentum conservation model
The deformation energies and penetration paths of the contact force are calculated
for four non-symmetric model-scale collision tests using both the developed time-
domain simulation model and a decoupled approach based on the momentum
conservation law. A calculation model based on the momentum conservation gives as
an output only the energy absorbed by structural deformations. Here, the decoupled
model of Zhang (1999) is exploited to evaluate this energy, and in Table 1 it is
compared to the energies obtained from the model-scale experiments and by the present
coupled approach. For the decoupled model only the total deformation energy is
presented, while for the other two methods the pure plastic deformation energy is also
presented. In the decoupled model the decomposition of the total energy into its plastic
and elastic components requires a knowledge of the ships’ velocities immediately after
the contact, which cannot be precisely defined on the basis of the decoupled approach
alone.
Table 1. Deformation energy obtained by different methods (total deformation energy/plastic deformation energy)
Test Experimental (total/plastic)
Decoupled model (Zhang, 1999)
Coupled model (total/plastic)
[J] [J] [J] 202 2.36/2.28 2.48/- 2.51/2.14
301 4.20/4.14 4.30/- 4.62/4.21
309 3.19/3.19 4.4(2.9*)/- 3.60/3.60
313 3.14/3.09 3.7/- 3.45/3.14
*- deformation energy assuming sliding contact in Zhang’s model
Computational models tend to overestimate the total deformation energy by
approximately 10% and both methods give a similar outcome, except in Test No. 309,
where there is significant sliding between the ships. The plastic deformation energy
evaluated with the coupled approach agrees well with the experimental measurements.
In the decoupled model the total deformation energy is the only outcome and the
penetration is assumed to follow the direction of the initial velocity of the striking ship
43
(Zhang, 1999). To solve the final value of the penetration corresponding to the
deformation energy obtained, the contact force model from [P3] is exploited.
a) 202 (β=90 deg, LC=0.83 m)
b) 301 (β=120 deg, LC=0.37 m)
c) 309 (β=145 deg, LC=0.46 m)
d) 313 (β=60 deg, LC=0.29 m)
Figure 10. Penetration paths of the bulb in the struck ship (see Appendix B in [P3] for
test matrix).
The penetration paths evaluated by two computational models are presented in
Figure 10. There, the penetration paths of the bulb into the side of the struck ship are
presented. The longitudinal extent of the damage is denoted by xB and, correspondingly,
the transverse extent by yB. For Test No. 202, presented in Figure 10a, the results of
44
both methods agree well with the measured one, even though the longitudinal
penetration is slightly underestimated. In other tests with an oblique angle the
differences between the results obtained with the different methods are larger. The
developed method estimates the penetration paths with good accuracy, but the
decoupled approach yields a deeper penetration, while the longitudinal extent of the
damage is smaller. This becomes especially clear from the results of Test No. 309 in
Figure 10c, where the striking ship slides along the side of the struck ship and thus the
penetration path deviates significantly from the direction of the initial velocity. A
circular marker in Figure 10c denotes the deformation energy when the sliding contact
is assumed to occur in the model of Zhang (1999).
45
5. Conclusions The dynamics of ship collisions have been studied experimentally on a large and on
a model scale. The experimental observation gave valuable information for the basic
assumptions in the 3D time-domain simulation model for collisions that was developed.
There, the ship motions and the contact force are treated simultaneously and all the
major external forces acting on the ships during a collision are considered at a
reasonable level of accuracy. The analyses of the large-scale collision experiments
revealed that the existing simulation tools did not include all the relevant effects.
Therefore, the simulation model that was developed also considers some
complementary effects, such as sloshing and the vibratory bending of the hull girder in
the transverse direction.
Both the experiments and the simulations emphasised the importance of the
coupling between the motions and the contact force, which resulted in a complex
motion kinematics that could not be handled on the basis of the initial input parameters
of the collision alone. It became especially obvious in the case of non-symmetric
collisions, where the penetration paths were heavily dependent on the actual ship
motions during the collision and on the structural properties of the ships. The
penetration paths calculated with the time-domain simulation model agreed well with
those from the large- and model-scale experiments, while the decoupled approach
predicted, as expected, penetration that was too deep and short. However, both the
coupled and decoupled approaches were able to predict the total deformation energy
with a deviation of about 10%. On the basis of the model tests it can also be concluded
that the hydrodynamic coupling between the colliding ships caused slightly higher
penetration in the vertical direction, compared to that predicted with the time-domain
simulations. The elastic springback of the deformed structures of the struck ship became
important when the unloading phase was modelled. While in symmetric collisions there
is a clear shift from the loading to the unloading at the time instant of the maximum
contact force and penetration, in non-symmetric collisions the unloading starts in some
regions of the contact surface before the maximum penetration occurs. Thus, in these
scenarios the elasticity also plays an important role in predicting the maximum
46
penetration. Furthermore, the hydrodynamic radiation forces acting on the colliding
ships proved to have a strong influence on the energy distribution as at the end of the
contact they accounted for up to 25% of the total available energy. However, if the
interest is in the maximum collision force and penetration depth, then the approach
based on the constant added mass and ignored hydrodynamic damping is still
reasonable, as the error in the deformation energy is about 5%. The energy absorbed to
overcome the hydrodynamic drag accounts for about 1%-2% of the total energy.
The results of the large- and model-scale experiments with partially filled liquid
tanks emphasised the importance of sloshing for collision dynamics. The structural
deformation energy in the wet tests was only about 70%-80% of that in similar dry
collision tests. This energy reduction is strongly affected by the amount of sloshing
water, while the effect of the collision speed is of secondary importance. The simulation
method that was developed with the Graham and Rodriguez linear sloshing model
overestimated the deformation energy by up to 10% in the case of the wet model-scale
tests, but in the case of the large-scale wet test the predictions agreed amazingly well.
The overestimation in the model-scale tests was due to the low relative filling levels of
water and thus, the linear sloshing model is close to the boundary of its validity.
Therefore, it can be concluded that this linear sloshing model gives results which are
completely satisfactory within a certain range of water depths. The vibratory bending of
the hull girder contributed to the sway velocity and acceleration at the amidships of the
struck ship; however, its contribution to the energy balance is small, around 1%-2% of
the total energy.
The model developed here can be used to estimate the deformations in non-
symmetric collisions or when the ships are prone to sloshing in collision. However, the
contact model should be extended to consider ship-like side structures, which have more
complex deformation mechanisms in comparison to the one used in the simulations of
the model-scale experiments. This, together with motion simulations, would improve
the accuracy of collision analyses and, thus, allow the crashworthiness of different
structural arrangements to be increased. More advanced sloshing models could enlarge
the validity range of the sloshing and even include the different geometries and
structural arrangements inside the tanks.
47
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51
Errata Publication Page Place Instead of Read
[P1] 59 above Table 2 68% 72%
[P1] 52 equation 22 ( ) ( )( )22ˆB BRm x xµ+ ( ) ( )( )22ˆ 1B B
Rm x xµ+
[P2] 77 Table 2, column 9 (µroll
838%
(%))
878%
439%
495%
708%
278%
12%
11%
23%
20%
12%
36%
[P3] below Table in Appendix 2 u0 u, contact velocity 0
[P4]
, contact velocity
1375 below Tables C1 and C2 is the mB m=30.5 kg B=30.5 kg
[P1] Tabri K., Broekhuijsen J., Matusiak J., Varsta P. (2009) Analytical modelling
of ship collision based on full-scale experiments. Journal of Marine Structures,
22(1), pp. 42-61.
Analytical modelling of ship collisionbased on full-scale experiments
Kristjan Tabri a,*,1, Joep Broekhuijsen a, Jerzy Matusiak b, Petri Varsta b
a Schelde Naval Shipbuilding, P.O. Box 555, 4380 AN Vlissingen, The NetherlandsbHelsinki University of Technology, Ship Laboratory, P.O. Box 5300, 02015 TKK, Finland
Keywords:
Ship collisions
Full-scale experiments
External dynamics
Water sloshing
Elastic bending of hull girder
a b s t r a c t
This paper presents a theoretical model allowing us to predict the
consequences of ship–ship collision where large forces arise due to
the sloshing in ship ballast tanks. The model considers the inertia
forces of the moving bodies, the effects of the surrounding water,
the elastic bending of the hull girder of the struck ship, the elas-
ticity of the deformed ship structures and the sloshing effects in
partially filled ballast tanks. The study focuses on external
dynamics. Internal mechanics, presenting the collision force as
a function of penetration, was obtained from experiments. The
model was validated with two full-scale collision experiments, one
with a significant sloshing effect and the other without it. The
comparison of the calculations and the measurements revealed
that the model predictions were in good agreement, as the errors
at the maximum value of penetration were less than 10%.
Ó 2008 Elsevier Ltd. All rights reserved.
1. Introduction
Regardless of continuous work to prevent collisions of ships, accidents still happen. Due to serious
consequences of collision accidents, it is important to reduce the probability of accidents and to
minimize potential damage to ships and to the environment. Better understanding of the collision
phenomena will contribute to the minimization of the consequences. This paper describes a mathe-
matical model for ship–ship collision simulations and uses the results of full-scale collision experi-
ments for validation.
* Corresponding author.
E-mail addresses: [email protected] (K. Tabri), [email protected] (Joep Broekhuijsen), [email protected]
(J. Matusiak), [email protected] (P. Varsta).1 Currently research scientist in Ship Laboratory of Helsinki University of Technology, Finland.
Contents lists available at ScienceDirect
Marine Structures
journal homepage: www.elsevier .com/locate/
marstruc
0951-8339/$ – see front matter Ó 2008 Elsevier Ltd. All rights reserved.
doi:10.1016/j.marstruc.2008.06.002
Marine Structures 22 (2009) 42–61
Many authors approach the collision problem by separating it to external dynamics and internal
mechanics. External dynamics determines ship motions while internal mechanics concentrates on the
structural response. One of the earliest reported attempts to predict a ship’s response in collisions was
made by Minorsky [1]. In his study, the energy absorbed in collision, i.e. the loss of kinetic energy, was
based on the momentum conservation. Interaction between the ships and the surrounding water
was modelled by the additional inertia force proportional to the increase in the ship’s mass due to the
surrounding water, i.e. the added mass. Motora et al. [2] investigated the validity of Minorsky’s
assumption of the constant added mass in a series of model tests. They concluded that this assumption
is a reasonable approximation only with a very short-term impact – less than 0.5–1 s. For collisions
with a longer duration, the value of the added mass increases and can reach a value equal or even
higher than a ship’s own mass. This problem was solved by making a clear distinction between two
components of the radiation force, one component proportional to the acceleration and the other one
related to the velocity. Cummins [3] and Ogilvie [4] investigated the hydrodynamic effects and
described the force arising from arbitrary ship motions using unit response functions. Like in the work
of Motora et al. [2], these approaches require that the frequency dependent added mass and damping
coefficients of a ship are evaluated.
Smiechen [5] proposed a procedure to simulate sway motions in the incremental time domain for
central, right-angled collisions. Hydrodynamic forces were considered by impulse response functions.
In 1982, Petersen [6] continued using impulse response functions and extended the analysis techniques
to consider all the ship motions in the waterplane. Petersen also simulated multiple collisions between
two similar ships to investigate the effects of different force penetration curves and collision condi-
tions. These simulations revealed that Minorsky’s [1] classical method underestimates the loss of
kinetic energy.
Woisin [7] derived simplified analytical formulations for fast estimation of the loss of kinetic energy
on inelastic ship collisions by using the constant addedmass value. In such collisions, it is assumed that
at the end of the collision both ships are moving at the same velocity. A few years later, based on
experimental data, Pawlowski [8] described the time dependency of the added mass and presented
similar analytical formulations. Pedersen and Zhang [9] also examined the effects of sliding and
rebounding in the plane ofwater surface. Also, Brown et al. [10] described a fast time domain simulation
model for the external dynamics and compared the results obtained by different calculation models.
Though there are many tools to predict the outcomes of the collision, they tend to lack a relevant
validation. A series of full-scale collision experiments conducted in the Netherlands allow for a deeper
understanding of the collision phenomena. As the existing tools have failed to predict the outcomes of
the experiments at a sufficient accuracy, a new study on collision interaction was initiated. The goal of
the study is to analyze these full-scale experiments and as a result, propose a mathematical description
of the phenomena. The analysis of the full-scale measurement data indicated that in order to describe
the experiments, the effects of free surface waves, i.e. water sloshing, and the elastic bending of the
struck ship hull girder have to be included in the model as well.
This paper concentrates on the external dynamics of the collision and the internal mechanics of the
colliding ships, giving the collision force, is obtained from the experimental test data. The behaviour of
both ships is described separately and combined by the common collision force based on the kinematic
condition. The aim is to simulate ship motions during and immediately after the contact. The analysis is
limited to a case in which an unpowered ship collides at the right angle with another ship.
2. Analysis of collision interaction
2.1. Formulation of the collision problem
Colliding ships experience a contact load resulting from the impact between the striking ship and
the struck ship. This force induces ship motions, which in turn cause hydromechanic forces exerted by
the surrounding water. While the striking ship is handled as a rigid body, the struck ship’s motions
consist of rigid body motions and the vibratory response of the hull girder. Furthermore, a ship’s
motions are affected by the sloshing forces arising from the wave action at the free surface in partially
filled ballast tanks. The collision situation under the investigation is idealized assuming that
K. Tabri et al. / Marine Structures 22 (2009) 42–61 43
- the striking ship is approaching perpendicular to the struck ship,
- the contact point due to the collision is at the midship of the struck ship,
- the propeller thrust of the ships is zero during the collision,
- the bow of the striking ship is rigid and does not deform,
- the collision force as a function of penetration is known a priori and it is independent of the
penetration velocity,
- the collision force excites the dynamic bending of the hull girder of the struck ship,
- ballast tanks in both ships are partially filled.
The first two idealizations state that only symmetric collisions are investigated. Consequently, all
the motions and forces are on the x0z0-plane. Fig. 1 presents collision dynamics with motions and
penetration. Here and in the subsequent sections superscript characters A and B denote the striking
and the struck ship, respectively. Coordinate systems xAyAzA and xByBzB have their origins fixed to the
ship’s centre of gravity. These coordinate systems are used to describe the motions of the colliding
ships relative to an inertial Earth fixed coordinate system x0y0z0. At the beginning of the collision, the
Earth fixed coordinate system coincides with the xAyAzA system. Forces acting on the striking ship are
denoted as XA, ZA andMA for surge, heave and pitch. For the struck ship, they are YB, ZB and KB for sway,
heave and roll, respectively. All of these forces are acting on a ship coordinate system.
During the contact, the collision force, i.e. the response of the ship structures, is equal to the force
required to displace the ship. The collision force thus depends on the ships’ motions, and the collision
problem is basically formulated with the displacement components. Relative displacement between
the striking ship and the struck ship, i.e. the penetration depth,
dðtÞ ¼
Z t
0
nhuAðtÞ þ _gAðtÞhA
icos gAðtÞ ÿ
hvBðtÞ ÿ _4BðtÞhB þ _hB
icos 4BðtÞ
odt (1)
forms the time t dependent kinematic condition for the collision process. Here uA and _gA are the surge
velocity and the pitch rate of the striking ship, vB and _4B are the rigid body sway velocity and the roll rate
of the struck ship, respectively. Velocity _hB describes the horizontal vibration response of the hull girder
of the struck ship. The vertical distance between the ship’s centre of gravity and the collision point is
denoted by hA and hB. It should be noted that Eq. (1) assumes small rotationalmotions. All of thesemotion
components depend on the forces acting on the ships. The following sections present the formulations,
where the outcome will be the time history of the penetration validated with the measured one.
2.2. Hydromechanic forces and moments
Hydromechanic forces and moments acting on a floating object consist of water resistance,
hydrostatic restoring forces and radiation forces expressed in terms of hydrodynamic damping and
Fig. 1. Coordinates used in the analysis.
K. Tabri et al. / Marine Structures 22 (2009) 42–6144
added mass. A ship moving in water encounters frictional and residual resistances. Residual resistance
is not included in the study because it is considered small compared to other phenomena. Frictional
water resistance is approximated with the ITTC-57 friction line formula.
Hydrostatic restoring forces exerted on the ship are proportional to its displacement from the
equilibrium position. Linear dependency between the displacement and the resultant force is given by
a constant spring coefficient. This simplification holds when the displacements from the equilibrium
position are small.
It is a common practice to model the radiation forces by the added mass and damping coefficients.
These coefficients are frequency dependent. In the frequency domain, the force due to acceleration _v
and velocity v is evaluated as
FHðuÞ ¼ ÿaðuÞ _vðuÞ ÿ bðuÞvðuÞ; (2)
where a(u) and b(u) are the added mass and damping coefficients. For the sake of brevity and clarity
a single translational degree of freedom motion is considered here. However, this representation
applies for six degrees of freedomwhen discussing the radiation forces. Eq. (2) is only valid in the case
of pure harmonicmotion. Therefore it does not suit well for the time domain simulations with arbitrary
motions. To represent the radiation forces in the time domain, it is useful to split them into a part Fmproportional to the acceleration and a velocity dependent damping part FK:
FHðtÞ ¼ FmðtÞ þ FK ðtÞ: (3)
The force proportional to the acceleration is calculated as
FmðtÞ ¼ ÿmrV _vðtÞ; (4)
where
m ¼ limu/N
aðuÞ
rV: (5)
Force Fm given by Eq. (4) would almost be the full representative of the radiation forces if the duration
of the motion is short. If the duration exceeds 0.5–1 s, damping starts to play a role [2]. This is
considered by force FK. In the time domain, this force is represented by the so-called convolution
integral [3].
FKðtÞ ¼ ÿ
Z t
0KðsÞvðt ÿ sÞds; (6)
where K(s) is a retardation function, taking into account the memory effect of the force:
KðsÞ ¼2
p
ZN
0bðuÞcosðusÞdu: (7)
Retardation functions were evaluated by the Fast Fourier Transformation algorithm, as described by
Matusiak [11].
For the rotational motions the moments of added mass and damping coefficients are used instead of
their linear motion counterparts. Also the corresponding rotational accelerations and velocities are used.
As the full-scale experiments were carried out in relatively shallow water, the effect of the depth of
water on the frequency dependent coefficients was investigated. For comparison, the coefficients were
evaluated by the Frank close-fit theory [12] and by the finite element (FE) method based on the two-
dimensional linear potential theory [13]. In Frank’s theory, the velocity potential has to fulfill the
Laplace equation in thewhole fluid domain and the boundary conditions at the free surface, at the body
surface and infinitely far away from the body. The FE method fulfils additional boundary conditions at
the bottom of the sea, thus the effects of shallow water are included. For comparison, the coefficients
are calculated for a two-dimensional rectangular cylinder with the breadth to draught ratio B/T¼ 4.4
and the water depth to draft ratio h/T¼ 2.8.
K. Tabri et al. / Marine Structures 22 (2009) 42–61 45
Effects on the waterplane motion components like surge, sway and roll were small. The most
significant increase in the added mass and damping values can be seen in the coefficients of heave
motion, see Figs. 2 and 3. As in the experiments analyzed later, the prevailingmotion components were
surge and sway, Frank’s theory was considered sufficient.
2.3. Water sloshing in partially filled tanks
Sloshing is a violent flow inside a fluid tank with a free surface. Sloshing is induced if the tank’s
motions are in the vicinity of some of the natural periods of the fluid motion inside the tank. Several
numerical methods have been developed to calculate such fluid-structure interaction, but their
disadvantage is a long computational time. For convenience, it may be desirable to replace the fluid by
a simple mechanical system. This section describes a mechanical system that produces the same forces
as the sloshing fluid.
In a simplified mechanical model, sloshing water is replaced with a number of oscillating masses.
According to the potential theory, a complete mechanical analog for transverse sloshing must include
an infinite number of suchmasses. It has been shown by the analysis that the effect of each spring-mass
element decreases rapidly with the increasing mode number [14]. The sufficient number of mass-
spring elements is evaluated by comparing the results with those obtained by computational fluid
dynamics (CFD). Fig. 4 presents the idea behind the equivalent mechanical model.
The effect of sloshing is considered only in the case of horizontal motions. Sloshing effects on the
rotational motions could be incorporated by evaluating the equivalent height h between the fluid
centre of gravity and the mass-spring element, including their effects at the equilibrium of the
moment.
Every eigenmode of fluid motions inside the tank is represented by one mass-spring element,
a damper and one rigid mass. The equation of the translational motion for a single mass mn connected
to the tank walls by a spring of stiffness kn and a damper with a damping coefficient cn becomes
mn€xn þ cnÿ_xn ÿ _xR
þ knðxn ÿ xRÞ ¼ 0; (8)
in which xR and xn present the motions of a rigid mass and those of an oscillating mass in respect to an
inertial coordinate system. Here the sloshing damping is described as a viscous damping and the
damping force is always proportional to the relative velocity between the oscillatingmass and the rigid
mass. In reality, sloshing is damped out due to the viscosity of water and due to water impacts on the
tank structure. For a precise description of the sloshing, more complicated damping models should be
used. As a detailed investigation of the sloshing behaviour exceeds the limits of this study, viscous
damping is considered to be a sufficient representative of the phenomenon.
Fig. 2. Effect of shallow water on the heave added mass.
K. Tabri et al. / Marine Structures 22 (2009) 42–6146
From Eq. (8), the reaction force of a single oscillating mass to the tank structure is simply
ÿcnð _xn ÿ _xRÞ ÿ knðxn ÿ xRÞ. The total force FM acting on the tank structure can be expressed as a sum of
the forces due to the rigid mass and the force due to the N oscillating masses
FM ¼ mR€xR þXN
n¼1
mn€xn: (9)
Properties mn, cn and kn for the spring-mass elements were derived so that the mechanical model
would give a force identical to the fluid force FF. The fluid force in a moving tank was obtained by
integrating the pressure over the tank boundaries
FF ¼ #S
pðx; y; z; tÞdS: (10)
Fig. 3. Effect of shallow water on the heave retardation function.
Fig. 4. Simplified mechanical model for sloshing.
K. Tabri et al. / Marine Structures 22 (2009) 42–61 47
Pressure distribution p(x,y,z,t) for an irrotational flow of inviscid and incompressible fluid is obtained
from Bernoulli’s equation using the concept of velocity potential. The velocity potential is evaluated by
satisfying the Laplace equation, the kinematic body boundary condition, the linearized kinematic and
dynamic free surface boundary conditions.
Given the formulations for FM and FF, the propertiesmn, cn and knwere evaluated by the equilibrium
of the forces. Refs. [15] and [14] present lengthy derivations and give formulations for mn and kn. Total
fluid massmT in a tank with breadth B and fluid height hW is divided into N oscillating masses and into
a single rigid mass
mR ¼ mT ÿXN
n¼1
mn: (11)
Sloshing damping was evaluated using the logarithmic decrement of damping d, defined as a ratio
between two successive velocity peaks
dhlnvi
viþ1¼
2pxffiffiffiffiffiffiffiffiffiffiffiffiffiffi1ÿ x2
q ; (12)
where x is a damping ratio. The logarithmic decrement was evaluated by the CFD calculations. Due to
simplicity, in the collision simulations presented later, it is assumed that d is the same for each mass-
spring element. The damping coefficient cn of the nth mass is
cn ¼un
pd: (13)
When a shipwith I fluid tanks is under consideration, the total rigidmass bmR of the ship is the sum of
the ship structural mass mST and the rigid part of the fluid mass in each tank. Using Eq. (12), the total
rigid mass bmR is
bmR ¼ mST þXI
i¼1
mR;i ¼ mST þXI
i¼1
mT;i ÿ
XN
n¼1
mn;i
!: (14)
It follows that the total number of oscillating masses is J¼ IN. The motions of the whole ship can be
presented by a system similar to that depicted in Fig. 4. The complete system consists of J oscillating
masses and the rigid mass bmR . If such a system is subjected to an external excitation force FE acting on
the rigid mass, the equation of motion combining Eqs. (8) and (9) is expressed as
2
664
bmR 0 / 00 m1 / 0« « / «
0 0 / mJ
3
775
8>><
>>:
€xR€x1«
€xJ
9>>=
>>;þ
2
666664
PJj¼1 cn ÿc1 / ÿcJÿc1 c1 / 0« « / «
ÿcJ 0 / cJ
3
777775
8>><
>>:
_xR_x1«
_xJ
9>>=
>>;
þ
2
666664
PJj¼1 kn ÿk1 / ÿkJÿk1 k1 / 0« « / «
ÿkJ 0 / kJ
3
777775
8>><
>>:
xRx1«
xJ
9>>=
>>;
¼
8>><
>>:
FE0«
0
9>>=
>>;: (15)
Matrices in Eq. (15)were composed in awayconsistentwith themotion definitions of Eq. (8). According
to that definition, all the motions are defined with respect to an inertial frame, and the interaction
between the rigid body and the oscillating masses is through the damping and stiffness matrices.
K. Tabri et al. / Marine Structures 22 (2009) 42–6148
The necessary number of oscillatingmasses per tank and the damping properties were evaluated by
comparing the results of the mechanical model with those of the numerical CFD calculations. Two-
dimensional (2D) calculations were made applying the CFD program Ansys Flotranwith the volume of
fluid (VOF) method [16]. The verification was done for a two-dimensional tank with breadth lT¼ 10 m
and water height hW¼ 0.95 m. To simulate the sloshing comparable to that in the case of the collision
experiments analyzed later, two different calculations were carried out as follows:
- For a decelerated tank, with an initial velocity V0¼ 3.5 m/s decelerated to zero in 0.5 s. Simulations
were carried out with and without transversal stiffeners with a height of 0.3 m. Sloshing direction
was transversal to stiffeners.
- For an accelerated tank, with the velocity increasing from zero to the final velocity VF¼ 2.5 m/s. No
stiffeners were modelled.
Fig. 5 presents the sloshing force obtained by the CFD calculations in the case of the decelerated
motion. Results show that sloshing is damped out significantly due to the first impact at the tank wall.
As a result the amplitude of the sloshing force decreases to almost a quarter of its maximum value.
After the first impact, damping decreases and changes in force amplitudes during one period are small.
The effect of the stiffeners in those 2D calculations is not significant and it causes only slight changes
in the peak values of the sloshing force. In reality, the effect of the stiffeners is higher, as in 2D CFD
calculations the stiffeners at tank sides are not taken into consideration. Furthermore, the effect of the
stiffeners increases as the water depth in the tank decreases.
As the contact in real collision is only of a short duration, damping values and the sufficient number
of masses are evaluated considering approximately for the first 5 s of the CFD calculations. Fig. 6
presents part of Fig. 5 together with the results of the mechanical model. The thick solid line shows the
results of the VOF calculations and thin lines show the results of the mechanical model in the case of
different damping coefficients. Our analysis revealed that the damping coefficient x ¼ 0:2.0:3 is
suitable for the decelerated motion.
The analysis of the accelerated tank motions shows that x ¼ 0:05.0:1 is a suitable damping
coefficient. Damping is higher in the case of the decelerated motion, indicating that higher velocity is
damped out faster. Also, stiffeners increase damping in the case of the decelerated motion. Still, it
should be remembered that those values depend on the velocity and should be reconsidered if the
velocities differ from those described above. Furthermore, our analysis revealed that a sufficient
number of oscillating masses in a single tank are three.
Fig. 5. Sloshing damping analysis with the VOF method for a decelerated tank at an initial velocity of V0¼ 3.5 m/s. Sloshing direction
transversal to stiffeners.
K. Tabri et al. / Marine Structures 22 (2009) 42–61 49
2.4. Elastic bending of a ship hull girder
Impact loading on a ship induces not only the rigid body motions, but also the dynamic bending of
the ship hull girder. Dynamic bending covers the hull girder vibration where the cross-sections of the
beam remain plane. This allows the modelling of the ship hull as an Euler–Bernoulli beamwith the free
ends. Major physical properties of the beam are its length L, flexural stiffness EI(x), internal damping x,
andmass per unit lengthm(x). The transverse loading q(x,t) is assumed to vary arbitrarily with position
and time, and the transverse displacement response h(x,t) is also a function of these variables.
The total dynamic response of the ship hull girder is regarded to be a superposition of the responses
of the different eigenmodes. The essential operation of the mode-superposition analysis is the trans-
formation from the geometric displacement coordinates to the normal coordinates. This is done by
defining the bending response h as
hðx; tÞ ¼XN
i¼1
fiðxÞpðtÞ; (16)
which indicates that the vibration motion is of a natural mode fiðxÞ, having a time dependent normal
coordinate p(t). Mode shapes and corresponding eigenfrequencies ui were evaluated as presented in
[17]. The equation of motion for the ith vibratory mode is expressed as in [18]
mi€piðtÞ þ 2xuim
i_piðtÞ þ u2
i mi piðtÞ ¼ qi ðtÞ; (17)
where the generalized mass of the ith mode is
mi ¼
Z L
0fiðxÞ
2mðxÞdx; (18)
and the generalized loading associated with the mode shape fiðxÞ is
qi ðtÞ ¼
Z L
0fiðxÞqðx; tÞdx: (19)
The value of internal bending damping x for ships is usually obtained experimentally. If no empirical
values exist for a particular ship, the measured internal damping values from many ships are reported
in [19]. These values indicate that the internal damping is practically independent of the frequency and
a value x ¼ 0:05 may be used.
Fig. 6. Evaluation of damping coefficient x by comparing the results of the VOF method to those of the mechanical model. Decel-
erated tank at V0¼ 3.5 m/s and N¼ 3. Sloshing direction transversal to stiffeners.
K. Tabri et al. / Marine Structures 22 (2009) 42–6150
2.5. Contact force
During a collision, both ships experience a common contact force FC arising from the bending,
tearing and crushing of the material in the ship structures. The best way to obtain contact force for
highly non-linear process such as a collision is by a sophisticated finite element (FE) analysis or by
experimental testing. FE calculations for the contact force exceed the limitations of the paper and are
not analyzed here. The emphasis of the study is on the external dynamics and the contact force as
a function of penetration is assumed to be known a priori from the experimental data or from the other
sources. Due to symmetric collisions, only one force-penetration curve is necessary. An example of an
experimentally measured force-penetration curve is depicted in Fig. 7 with a solid line. The dotted line
shows the fitted curve.
When the collision force reaches its maximum, ships start to separate and the penetration
decreases. Due to the elasticity of the deformed structures, contact is not lost immediately and the
collision force still has some value. This elasticity, i.e. the elastic spring-back, is modelled by a single
variable a, which gives the inclination for the spring-back line. Fig. 7 shows two examples of the
spring-back lines, both having the same inclination, but a different starting point. This starting point
is equal to the maximum penetration in the collision and its location is determined by the external
dynamics. If the penetration starts to increase again after decreasing, the collision force follows the
original path.
2.6. Formulations of motion equations
For the sake of brevity, only the equations of motions for the struck ship are presented here in detail.
For the striking ship, the equations are simpler as they do not include the component of the vibratory
motion. The effect of the bending motion is assumed to be small compared to the rigid body motions
and therefore only the first eigenmode is included. Furthermore, only the bending in the sway direction
is considered.
It is assumed that the sloshing and the hydromechanic forces are distributed uniformly along the
ship length. Concerning the first eigenmode of a uniform beam with free ends, it holds that
Z L
0f1ðxÞdx ¼ 0 (20)
Fig. 7. Measured and approximated force-penetration curve.
K. Tabri et al. / Marine Structures 22 (2009) 42–61 51
and it follows that the uniformly distributed sloshing and hydromechanic forces need not to be
included in the bending analysis. Thereby the generalized force for the first eigenmode was evaluated
by
qB1 ðtÞ ¼
Z L
0f1ðxÞFCðx; tÞdx: (21)
In reality, those forces are never perfectly uniform. With detailed knowledge available about the
distribution of these forces, their effect can easily be included.
A generalizedmassmB1 was evaluated using the sway addedmass mB22 and the total rigidmass bmB
R of
the ship as
mB1 ¼
Z L
0fiðxÞ
2bmB
RðxÞ þ mB22ðxÞ
dx: (22)
For convenience, the equations of motions were first evaluated by neglecting the effects of sloshing.
Furthermore, assuming that there is no coupling between the motion components and using the
notations presented in Fig. 1, the equations were written as
8>>>><
>>>>:
rVB_vB ÿ _4BwB
¼ YB þ rgVBsin 4B
mB1€p1ðtÞ þ 2xuB
1mB1
_p1ðtÞ þÿuB1
2mB
1 p1ðtÞ ¼ qB1 ðtÞ
rVB_wB þ _4BvB
¼ ZB þ rgVBcos 4B
IBx €4B ¼ KB
;
(23)
whereVB is the volumetric displacement of the ship and IBx is themoment of inertia with respect to the
x-axis. Sway force YB, heave force ZB and rolling moment KB were described in a ship’s coordinate
system and were evaluated as the summation of forces and moments described in the previous
sections:
YB ¼ FC þ YBF þ YB
H
ZB ¼ ZBR þ ZBHKB ¼ FCh
B þ KBR þ KB
H;
(24)
with subscripts C corresponding to the collision force, F to the frictional resistance, H to the radiation
force and R to the hydrostatical restoring force. It should be noted that all the forces due to the
surrounding water are included in YB, ZB and KB. Thus, the added mass does not appear explicitly in Eq.
(23), but is included through YBH, Z
BH and KB
H.
In the presence of sloshing, the total mass rVB of the ship was divided into a single rigid mass bmBR by
Eq. (14) and into JB oscillating masses. All the forces and moments presented by Eq. (24) are acting on
the rigid mass. Sloshing is induced by the coupling terms in the stiffness and damping matrices in Eq.
(15). Furthermore, it is assumed that all the sloshing masses have their centre of gravity at the ship’s
centre of gravity and therefore they do not contribute to the rotational motions. Denoting motions of
the sloshing masses by xB, a new equation of motions is formulated by combining Eqs. (15) and (23),
yielding
hMBi
8>>>>>>>>>>><
>>>>>>>>>>>:
_vB ÿ _4BwB
€xB1«
€xBJB
€pB1ðtÞ
_wB þ _4BvB
€4B
9>>>>>>>>>>>=
>>>>>>>>>>>;
þhCBi
8>>>>>>>>><
>>>>>>>>>:
vB
_xB1«
€xBJB_p1ðtÞwB
_4B
9>>>>>>>>>=
>>>>>>>>>;
þhKBi
8>>>>>>>>><
>>>>>>>>>:
yB
xB1«
xBJB
p1ðtÞzB
4B
9>>>>>>>>>=
>>>>>>>>>;
¼
8>>>>>>>>><
>>>>>>>>>:
YB þ rgVBsin 4B
0«
0qB1 ðtÞ
ZB þ rgVBcos 4B
KB
9>>>>>>>>>=
>>>>>>>>>;
; (25)
K. Tabri et al. / Marine Structures 22 (2009) 42–6152
where
hMBi¼
2
666666666664
bmBR 0 / 0 0 0 00 mB
1 / 0 0 0 0« « 1 « « « «
0 0 / mBJB
0 0 0
0 0 / 0 mB1 0 0
0 0 / 0 0 bmBR 0
0 0 / 0 0 0 IBx
3
777777777775
(26)
and
hCBi¼
2
6666666666664
PJB
j¼1 cBj ÿcB1 / ÿcB
JB0 0 0
ÿcB1 cB1 / 0 0 0 0« « 1 « « « «
ÿcBJB
0 0 cBJB
0 0 0
0 0 / 0 2xuB1m
B1 0 0
0 0 / 0 0 0 00 0 / 0 0 0 0
3
7777777777775
(27)
and
hKBi¼
2
66666666666664
PJB
j¼1kBj ÿkB1 / ÿkB
JB0 0 0
ÿkB1 kB1 / 0 0 0 0« « 1 « « « «
ÿkBJB
0 0 kBJB
0 0 0
0 0 / 0ÿuB1
2mB
1 0 00 0 / 0 0 0 00 0 / 0 0 0 0
3
77777777777775
: (28)
For the striking ship, the equations of motion without the effects of sloshing are
8>><
>>:
rVA_uA þ _gAwA
¼ XA ÿ rgVAsin gA
rVA_wA ÿ _gAuA
¼ ZA þ rgVAcos gA
IAy €gA ¼ MA
;
(29)
where VA denotes the volumetric displacement of the ship and IAy denotes the moment of inertia in
respect to y-axis. Forces for surge, heave and pitch are
XA ¼ FC þ XAF þ XA
H
ZA ¼ ZAR þ ZAHMA ¼ FCh
A þMAR þMA
H:
(30)
The second order differential equations of motion, Eqs. (25) and (29), are non-linear due to the
coupling in acceleration terms. Equations can be linearized within a time increment Dt assuming that
the changes in the velocities and the time derivatives of the velocities are small within the time
increment [6]. Furthermore, all the forces were assumed constant during Dt. Under these assumptions,
the solution at the time (t0þDt)can be found if the solution at the time t0 is given. Equations were
solved using the fourth order Runge–Kutta method.
K. Tabri et al. / Marine Structures 22 (2009) 42–61 53
3. Validation of the theory by full-scale collision experiments
To validate the collision model, the results obtained from a series of full-scale collision experiments
were used. Several full-scale collision experiments using two inland vessels have been conducted in
the Netherlands by TNO (Dutch Institute for Applied Physical Research) in the framework of a Japanese,
German, Dutch consortium of shipyards and a classification society. The experiments conducted had
different purposes, such as to validate numerical analysis tools, to investigate various aspects in
collision and to prove new structural concepts. In this study, the following two experiments were used
to verify the analytical model:
- collision experiments with the Y-core side structure.
- collision experiments with the X-core side structure.
Those two experiments differ, as in the experiment with the Y-core ship side structure, both ships
contained large amounts of ballast water and therefore the effects of sloshing were significant. In the
experiment with the X-core side structure, sloshing effects were practically removed, as only a small
amount of ballast water had a free surface.
3.1. Experiment with the Y-core test-section
The experiment with the patented Y-core test section designed by Schelde Naval Shipbuilding was
conducted in the Netherlands on 9th of July 1998. A detailed description of the collision experiment
with a preliminary analysis is presented in [20].
Two moderate size inland waterway tankers were used in the collision test. The striking ship was
named Nedlloyd 34 and the struck ship was called Amatha. The main dimensions of both ships are
given in Table 1. In the table and in the following figures, subscript 11 corresponds to the surge motion
and 22 to the sway motion.
The striking ship, which was equipped with a rigid bulbous bow, impacted the struck ship at
amidships on the course perpendicular to the struck ship. Due to that, very small yaw motions were
expected. At the moment of the first contact, the velocity of the striking ship was 3.51 m/s.
A comparison between the measured and the calculated results is presented in Figs. 8–11. Figures
also include the calculations, where the effects of sloshing are neglected. The time history of the
collision force presented in Fig. 8 shows a good agreement at the beginning of the collision. The first
force peak is predicted at good accuracy both in terms of the absolute value and the duration. After the
first force impulse, the ships separated and the force decreased to zero. Due to a higher resistance of the
struck ship and due to the sloshing effects, a second contact occurred. The calculation model predicts
the absolute value of the second force peak, but delays it for 0.5 s. In the experiment, also a third
contact occurred, which was not predicted by the calculations. The first peak becomes higher and the
subsequent contacts do not occur when the sloshing effects are neglected.
Velocities of the ships are presented in Figs. 9 and 10. The general behaviour of the striking ship
velocity was the same in the experiment and in the calculation. In the beginning, the velocity decreased
Table 1
Main dimensions and loading conditions of the ships
Striking ship Struck ship
Length, L 80 m 80 m
Beam, B 8.2 m 9.5 m
Depth, D 2.62 m 2.8 m
Drafta, T 1.45 m 2.15 m
Displacement, D 774 tons 1365 tons
Added mass of prevailing motion component m11¼0.05 m22¼ 0.24
Number of tanks 2 5 2 6
Ballast water with free surface 303.5 tons 545.0 tons
a In the report [20], the exact draft of the ships is not given, it only contains their total displacements. The draft presented here
is evaluated by lines drawings and the reported displacements.
K. Tabri et al. / Marine Structures 22 (2009) 42–6154
significantly due to the contact force. When the collision force decreased to zero, the ship started to
accelerate. This acceleration is mainly due to sloshing, as the sloshing force is preceded by the collision
force, see Figs. 6 and 8. In the calculations, the velocity decreases to 0.2 m/s instead of 0.65 m/s, which
was measured in the experiment. The calculations with different sloshing properties revealed that the
duration of the deceleration is strongly dependent on the sloshing damping coefficient x. As the same
damping coefficient was used for every tank, regardless of the water height, the source of the inac-
curacy is obvious. The second decrease in the velocity, indicating the beginning of the second collision,
is also delayed, which in turn results in a delayed second force peak in Fig. 8. When the sloshing is
neglected, the ship deceleration is similar to the experimentally measured, but the acceleration is
significantly lower. Here the acceleration is only due to the surrounding water, which effect is low
compared to that of the sloshing.
Fig. 8. Collision force FC as a function of time.
Fig. 9. Velocity of the striking ship.
K. Tabri et al. / Marine Structures 22 (2009) 42–61 55
The time history of the struck ship velocity is presented in Fig. 10. The agreement between the
measurement and the calculation is better than in the case of the striking ship. This is mainly due
to the fact that in the case of the struck ship, the effects of sloshing were smaller and the changes
in the sloshing damping values did not appear so significant. Differences between the measured
and the calculated value increased after the second collision, but the general behaviour still
remained the same. Again, the sloshing effects are obvious. Without sloshing, the first velocity peak
is higher and as there are no subsequent contacts, the velocity remains oscillating around constant
level. By the end of the observed time period, the energy involved in the sloshing is almost fully
transformed to the kinetic energy of the ship, see Fig. 13, and the calculated velocities approach to
each other.
Due to the bending of the ship hull girder, the velocity signal has an oscillatory behaviour. Those
oscillations, especially the frequency, are predicted well with the Euler–Bernoulli beam theory. It
Fig. 10. Velocity of the struck ship.
Fig. 11. Penetration as a function of time.
K. Tabri et al. / Marine Structures 22 (2009) 42–6156
also indicates that the added mass value for the sway motion can be predicted well with Frank’s
method.
The integration of the relative velocity between the ships results in a penetration time history
presented in Fig. 11. Similar effects, which were also seen in the force and velocity time histories, are
seen in the penetration value as well. The maximum penetration value determining the damage in the
struck ship is predicted with a very good accuracy. After the first peak, the penetration, decreasing too
much, also delays. Reasons for that lie in the errors in the velocity of the striking ship. Without sloshing
the maximum penetration is higher and after the first peak it decreases and remains zero.
The total energy involved in the collision is divided into three components. These are energy EA
involved in the motions of the striking ship, energy EB involved in the motions of the struck ship and
energy EC absorbed due to the deformation. Component EA again consists of several energy compo-
nents, as presented in Fig. 12. The work against the friction and the damping force is not presented as
they were insignificant compared to the other energy components.
Fig. 12 reveals the importance of sloshing in the case of the striking ship. The energy involved in
sloshing is at its maximum at the time instant when the collision force and the penetration have
reached the peak values. After the maximum value, part of the sloshing energy is returned to the
kinetic energy of the ship and part of it is absorbed by damping. Kinetic energy EKINA describes the
energy involved in the rigid body motions only. This energy is calculated using the total mass of
the ship and the addedmass of the correspondingmotion component. This means that 5% of EKINA is due
to the added mass. The energy involved in sloshing, ESL, is evaluated using the relative motions
between the oscillating masses and the rigid body. When the effects of the sloshing are neglected, the
kinetic energy of the striking ship decreases almost to zero and at the end of the contact the ship
possesses significantly less energy compared to the case where the sloshing is included. As seen from
the Fig. 13–15, this energy difference is absorbed by the deformation of ship structures and by the
motions of the struck ship.
In the case of the struck ship, the total energy EB is divided between more components, as the work
against the damping and friction forces is more important, see Fig.13. Also, the transformation from the
sloshing energy to the kinetic energy of the ship happens faster.
According to Table 1, 24% of EKINB is due to the added mass. That value compared to workWK done to
overcome force FK shows the importance of the damping. In a later phase of the collision, damping
energyWK is larger than the energy involved in the motions of the addedmass. Still, it should be noted
that the damping energy starts to play an important role after the maximum penetration value is
reached, i.e. its importance on predicting the maximum value is not very significant here. Neglecting
Fig. 12. Variations of relative energy components throughout the collision in the case of the striking ship (EA total energy; EKINA
kinetic energy involved in rigid body motions; and ESL energy involved in sloshing).
K. Tabri et al. / Marine Structures 22 (2009) 42–61 57
the sloshing increases the total energy of the struck ship with the largest gain in the kinetic energy.
Also the other velocity dependent energy components increase slightly as the velocity becomes higher.
For simplicity, a detailed distribution of energy components for the calculations without sloshing is not
presented in the figure.
Variations between EA, EB and EC are presented in Fig. 14. Figure shows that only 43% of the initial
energy is absorbed by the structural deformation. Energy distribution for the case where sloshing is
neglected is presented in Fig. 15, which reveals that the deformation energy is 58% of the initial energy.
A simple closed form method based on the momentum conservation [9] gives 65% for the relative
deformation energy. This energy was calculated considering the elasticity of the ship structures,
described by the relative velocity between the ships immediately after the contact is lost. The
Fig. 13. Variations of relative energy components throughout the collision in the case of the struck ship (EB total energy; EKINB kinetic
energy involved in rigid body motions; ESL energy involved in sloshing; WK work against the damping force FK; EB bending energy;
and EF work against the friction force).
Fig. 14. Variations of relative energy components throughout the collision with sloshing effects included (EA energy involved in the
striking ship; EB energy involved in struck ship; and EC deformation energy).
K. Tabri et al. / Marine Structures 22 (2009) 42–6158
differences between these results clearly indicate the importance of sloshing in the prediction of the
deformation energy.
3.2. Experiment with the X-core test-section
The second example demonstrating the application of the model is the calculation of collision
interaction in the experiment conducted in April 2003. The striking ship used in the test was the same
as in the earlier Y-core test, but the struck shipwas a slightly larger inlandwaterway barge. The effect of
sloshing was removed, as only one tank in the striking ship had water ballast with a free surface. The
striking ship was equipped with the same bulbous bow as in the earlier tests. The tested section was
a laser welded X-type sandwich structure designed in cooperation with the EU Sandwich and EU
Crashcoaster projects. Main dimensions for both ships are given in Table 2. At the moment of the first
contact, the velocity of the striking ship was 3.33 m/s.
A comparison between the calculated and the measured penetration is shown in Fig. 16, which
shows a good agreement, as the maximum penetration is predicted well. Again, at the later stage of the
calculation, the error increased. The measured penetration started to rise again, while the calculated
penetration kept decreasing slightly.
The variation of relative energy components presented in Fig. 17 differs from the Y-core collision.
Without the sloshing water, more energy is absorbed by the deformation of the ship structures. This
indicates that the sloshing water ‘‘stores’’ the kinetic energy of the striking ship and therefore the
energy available for the deformation is decreased.
According to the momentum conservation method [9], the relative amount of the deformation
energy becomes 68%. The value calculated from the experimental measurements is 75%, see Fig. 17.
Fig. 15. Variations of relative energy components throughout the collision with sloshing effects neglected (EA energy involved in the
striking ship; EB energy involved in struck ship; and EC deformation energy).
Table 2
Main dimensions and loading conditions of the ships
Striking ship Struck ship
Length, L 80 m 76.4 m
Beam, B 8.2 m 11.4 m
Depth, D 2.62 m 4.67 m
Draft*, T 1.3 m 3.32 m
Displacement, D 721 tons 2465 tons
Added mass of prevailing motion component m11¼0.05 m22¼ 0.29
Number of tanks 2 5 2 7
Ballast water with free surface 44.6 tons 0 tons
K. Tabri et al. / Marine Structures 22 (2009) 42–61 59
Larger value reveals the effect of the surrounding water, especially the part FK of the radiation force,
which is not included in the momentum conservation method. Force FK is given with Eq. (6) and the
work done to overcome this is presented as WK in Fig. 13. This force is an additional resistance to
the ship motions and can thereby be considered as an additional mass. A larger ship mass increases the
inertia of the ship, and it cannot be displaced so easily.
4. Conclusion
This paper presents a model allowing for predictions of the consequences caused by the collision
where large forces arise due to sloshing in ship ballast tanks. Motions of both the ships as well as the
penetration depth during and after the collision were predicted in good agreement both in terms of
time and absolute values. Furthermore, the vibrations of the hull girder of the struck ship corresponded
Fig. 16. Penetration as a function of time.
Fig. 17. Variations of relative energy components throughout the collision (EA energy involved with the striking ship; EB energy
involved with the struck ship; and EC deformation energy).
K. Tabri et al. / Marine Structures 22 (2009) 42–6160
well with the measurements. The comparison of the energy balances revealed the significance of
sloshing, as in the experiment with the sloshing effects, sloshing ‘‘stored’’ the kinetic energy. Therefore,
only 43% of the total energy was absorbed by the structure. In the experiment without the sloshing
effects, the amount of the absorbed energy was 75%. The importance of damping due to the
surrounding water was large, as in the later phase of the collision, damping engagedmore energy than
the added mass. Damping started to play an important role after the maximum penetration value was
reached.
The simplified closed form method overestimated the deformation energy with sloshing water and
underestimated it without sloshingwater. The closed formmodel needs some assumptions to bemade,
especially when elasticity needs to be considered. The simulation model is almost free of assumptions
and only needs initial collision conditions and a collision force as a function of penetration. Therefore,
the presented model is suitable for more precise collision simulations where the forces arising by the
surrounding water and sloshing are to be included.
Acknowledgement
This work was carried out under the Dutch National Veilig Schip project funded by SENTER and in
the framework of the Marie Curie Intra-European Fellowship program. This financial support is
acknowledged.
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September. University of Trieste; 2001. p. 12.[12] Journee JMJ. Strip theory algorithms. Delft University of Technology; 1992. Report MEMT 24.[13] Kukkanen T. Two-dimensional added mass and damping coefficients by the finite element method. Report M-223. Ota-
niemi, Finland: Helsinki University of Technology; 1997. p. 61.[14] Abramson, editor. Analytical representation of lateral sloshing by equivalent mechanical models, the dynamic behaviour of
liquids in moving containers. NASA SP-106. Washington; 1966. p.199–223.[15] Graham EW, Rodriguez AM. The characteristics of fuel motion which affect airplane dynamics, Trans. Of ASME, Series E. J
Appl Mech Sept 1952;19(no.3):381–8.[16] Hirt CW, Nichols BD. Volume of fluid (VOF) method for the dynamics of free boundaries. J Comput Phys 1981;39:201–25.[17] Timoshenko SP. Vibration problems in engineering. 2nd ed. New York: D. Van Nostrand Company Inc.; 1937. p. 470.[18] Glough RW, Penzien J. Dynamics of structures. 2nd ed. New York: McGraw-Hill Book Company; 1993. p. 740.[19] ISSC. Steady state loading and response, report of committee II.4. Proceedings of the eighth international ship structures
congress; 1983. p. 64.[20] Wevers LJ, Vredeveldt AW. Full scale ship collision experiments 1998, TNO. report 98-CMC-R1725. The Netherlands: Delft;
1999. p. 260.
K. Tabri et al. / Marine Structures 22 (2009) 42–61 61
[P2] Tabri K., Määttänen J., Ranta J. (2008) Model-scale experiments of symmetric
ship collisions, Journal of Marine Science and Technology, 13, pp. 71-84.
1 3
ORIGINAL ARTICLE
J Mar Sci Technol (2008) 13:71–84
DOI 10.1007/s00773-007-0251-z
K. Tabri (*) · J. Määttänen · J. RantaHelsinki University of Technology, Ship Laboratory, P.O. Box 5300, 02015 TKK, Finlande-mail: [email protected]
Model-scale experiments of symmetric ship collisions
Kristjan Tabri · Jukka Määttänen · Janne Ranta
growing steadily. Collision risk can be reduced either by minimizing the probability of collisions or by reduc-ing the consequences; however, reduction of the conse-quences requires a good understanding of the physical phenomena. Recent large-scale experiments of symmet-ric ship collisions have addressed defi ciencies in under-standing and discovered that the existing calculation tools fail to predict the outcomes with suffi cient accu-racy.1 Inaccurate predictions indicate that the calcula-tion models either do not include all the required phenomena or include them poorly.
Although there have been several studies of analytical models designed to simulate ship collisions, such as those of Minorsky,2 Petersen,3 and Pedersen and Zhang,4 experimental data with which to study the phenomena and to validate the calculation models have nevertheless been scarce. Some of the few model-scale tests to have been carried out are reported in Motora et al.5 In these tests, the ship model was pulled sideways with a constant force. This resulted in rather low and slowly changing acceleration with a maximum value of below 0.15 m/s2. In real collisions, the acceleration changes rapidly over a short time. To acquire data from real collisions, several large-scale collision experiments were performed in the Netherlands. All the large-scale experiments were symmetric, which means that only a limited number of motion components were excited. Furthermore, due to ballast loading conditions in some of the large-scale experiments, severe water sloshing occurred and affected the collision dynamics.
The lack of experimental data initiated a new study with the main aim of obtaining validation data. Data from large-scale experiments are favourable as they are free from scaling effects; however, due to their expensive and complicated nature they are not feasible for
Abstract This study was initiated due to the lack of experimental data on ship collisions. The feasibility of model-scale ship collision experiments was examined and a series of model-scale ship collision experiments is presented. The theoretical background for the analysis of experiments is given together with the principles of scaling. Proper scaling should assure physical similarity to the large-scale experiments conducted in the Nether-lands. The Froude scaling law was followed, resulting in the improper scaling of some forces: the effects of this are discussed. The study concentrates on the dynamics of collisions. The structural response, properly scaled from the large-scale experiments, was modelled using polyurethane foam as the ship’s side structure. The collision process was analysed and the results of model-scale tests, large-scale experiments, and a simple analytical model were compared, showing that there was both quantitative and qualitative agreement in the results of the experiments conducted at different scales. The analytical model yielded good quantitative assessment of the deformation energy.
Key words Model- and large-scale collision experiments · Froude scaling law · Physical similarity
1 Introduction
Due to the continuous increase in the amount of water transportation, the risk of ship collisions has been
Received: January 19, 2007 / Accepted: June 6, 2007© JASNAOE 2008
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72 J Mar Sci Technol (2008) 13:71–84
studying a wider range of collisions. Cheaper and simpler model-scale experiments offer an alternative, but atten-tion has to be paid to scaling in order to include properly all the relevant effects. A series of model-scale collision experiments was performed in the test basin of Helsinki University of Technology. The purpose of the tests was to provide experimental validation data for various collision scenarios. The tests aimed at obtaining precise measurements not only during contact but also for a period of time after contact.
This article concentrates on symmetric collisions and studies the feasibility of model-scale collision experi-ments. One aim was to provide a set of experimental validation data and to prove physical similarity between the experiments conducted at different scales. The theo-retical background for analysing the experiments and the principles of scaling are presented. Scaling and designing the experimental setup were carried out using the information from large-scale experiments. Proper scaling is important to maintain physical similarity to the large-scale experiments. The Froude scaling law was followed and its advantages and disadvantages are discussed. The ship models were geometrically similar to large-scale ships with a scaling factor of 35. The force–penetration curves from the large-scale experi-ments were used as a reference to model the structural response. In the model-scale experiments, part of the side structure of the struck ship was replaced with poly-urethane foam to produce the required structural resis-tance. During the tests, the motions of the models and the contact force were recorded. In this article, a single experiment is analysed thoroughly to explain the phe-nomena of a ship collision. The results of the model-scale tests were compared to those of large-scale experiments and to a simple analytical calculation model. Comparison with large-scale experiments was performed both in a non-dimensional form, to study the overall similarity, and also in a dimensional form, to study the quality of the experiments. Here, only the
most important results will be presented and discussed; an elaborate description of the tests and the analysis is presented by Määttänen.6
2 Formulation of the collision problem
A collision between two ships is a dynamic process involving the motions of two bodies. The main laws covering the dynamics and the kinematics are the con-servation of momentum and the equilibrium of force and energy. This article presents suffi cient defi nitions for the analysis of symmetric collision experiments and describes the background of the process. A ship collision is called symmetric if the striking ship hits the struck ship amid-ships at an angle normal to it.
2.1 Collision dynamics
In symmetric ship collisions, all the motions and the forces are assumed to be limited to a single plane. Figure 1 presents the kinematics with the motions and the pen-etration. Here, and in subsequent sections, superscript characters A and B denote the striking and the struck ship, respectively. If a variable is described with super-script i where i = A, B it means that the description is common to both ships. Coordinate systems xAyAzA and xByBzB have their origins fi xed to the centre of gravity of the ships. These coordinate systems are used to describe the motions of the colliding ships relative to an inertial Earth-fi xed coordinate system x0y0z0. At any instant, the ship’s position with respect to the inertial frame is deter-mined by vector Ri
C.Angular changes gA and jB from the equilibrium posi-
tion are given with respect to the inertial coordinate system. When subscript 0 is added to a motion compo-nent, its projection to the inertial coordinate system is considered, e.g., u0
A = x. A · i0 = uA · i0. Displacement com-
Fig. 1. Defi nition of collision kinematics
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J Mar Sci Technol (2008) 13:71–84 73
ponents are projected in a similar way. The translational velocity in the inertial coordinate system is thus:
ɺR u wC = +0 0 (1)
As the motions of the ships are limited to the x0–z0 plane, there is only one rotational motion component for both ships and the vectors of angular velocity are simply:
W
W
A A
B B
=
= −
γ
ϕ
j
j
0
0 (2)
Contact force FC describes the response of the ship structures. When in contact, there is always a balance between the force and the ship motions. In Fig. 1, the letter P denotes the point where the ships fi rst make contact. This point is described in the ship coordinate system using position vector ri
P. The point is fi xed to the ship and does not follow the deformation of ship struc-tures. The relative displacement between the ships, i.e., the penetration depth
d = + − +( )R r R rCA
PA
CB
PB (3)
forms the kinematic condition for the collision process. Given the contact force as a function of penetration, the kinematic condition forms an important link, combining inner mechanics to external dynamics.
2.2 Energy distribution in the collision process
The analysis of ship collisions is often based on energies. Also, the proposed classifi cation procedure by Ger-manischer Lloyd7 for novel crashworthy side structures compares the energy absorption capacity of a new struc-ture to that of a conventional structure.
There are several energy absorbing mechanisms in a collision. The major part of the energy is divided between the kinetic energy EK of the system and the energy ED absorbed by structural deformation. The latter is a com-bination of energies absorbed in different deformation processes such as tearing, stretching, crushing, and fric-tion between the structures. However, due to the empha-sis of this article, these processes are not treated separately and thus are simply referred as deformation energy. The kinetic energy of the system is the sum of the kinetic energies of both ships, which are evaluated using the velocities given in Eqs. 1 and 2. Hereinafter, when dis-cussing the kinetic energy of a ship, the energy involved in the motions of the ship and its added mass is consid-ered. This kinetic energy does not include the increase of the added mass over time, which is referred to as the damping part of the radiation force. Figure 2 presents a
time history of the main energy components throughout contact with a duration of tC.
Let us assume that the struck ship is standing still and the striking ship is approaching with velocity R
.AC ≅ uA
0, later referred to as the contact velocity. When the two ships collide, contact is established and the contact force obtains some value. During contact, the striking ship decelerates and loses its kinetic energy, while the struck ship gains energy through acceleration. In this energy transfer, a part of the energy is stored in deformed ship structures. This deformation energy ED is calculated as the integral over the product of the contact force FC and the penetration d. When the velocities of both ships have equalized, the deformation energy, consisting of both elastic ED,E and plastic energy ED,P, is at its maximum, while the kinetic energy of the system is at its minimum, see Fig. 2. The elastic energy stored in the structures is transferred back to kinetic energy as the elastic force starts separating the ships. The separation continues until the contact force decreases to zero. Disregarding the slow elastic recovery of the side structure, the defor-mation energy at that instant is equal to the plastic deformation energy.
Energy is also absorbed to overcome hydrodynamic forces such as water resistance and the damping part of the radiation force. These velocity-dependent forces do not play a major part during contact of short duration. Their importance increases when analysing the phe-nomena after contact. In addition to hydrodynamic forces, there are hydrostatic restoring forces. The restor-ing forces are signifi cant if the displacement from the equilibrium position is large.
Fig. 2. Distribution of main energy components throughout the contact. EK, kinetic energy of the system; ED, total deformation energy; ED,E, elastic deformation energy; ED,P, plastic deformation energy; tC, duration of the contact
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74 J Mar Sci Technol (2008) 13:71–84
3 Scaling of collision experiments
3.1 Conditions for similarity
The forces involved in a collision are hydrostatic and dynamic, gravitational, inertial, and the contact force. In order to recreate a large-scale collision at model scale, the ratio of any two forces acting on the ship model must be equal to the corresponding ratio of forces in the original, i.e., dynamic similarity must be maintained. This dynamic similarity presupposes geometric similar-ity, as the force and pressure distribution should be geo-metrically similar at both scales. The importance of geometric similarity is evident as it determines the water pressure on the ship models and thus affects their position and motions. Geometric similarity is achieved by scaling the large-scale dimensions with a scaling factor l:
xx
MS[ ] =
[ ]λ
(4)
where [x] represents a number of coordinates suffi cient to defi ne the shape of the hull. Subscripts M and S correspond to model and ship scale, respectively. The dimensioning of all the other parameters is thus based on l. Dynamic similarity also presupposes kinematic similarity owing to the presence of the inertia forces. Kinematic similarity is similarity of motion, which implies both geometric similarity and similarity of time intervals. The forces present in the system must be char-acterized according to their origin and their relationship to some set of parameters. These reference values are the ship length l, the velocity v, the density r and the viscos-ity m. The forces acting on the ship must be described with these parameters. Inertia forces FI arise due to the acceleration of the ship and the surrounding water, therefore:
F l vI ∼ ρ 2 2 (5)
Gravitational forces FG and the hydrostatic forces are due to an increase in the potential energy of the ship and can be described as:
F glG ∼ ρ 3 (6)
and the viscous shear forces Fm can be described as:
F vlµ µ∼ (7)
The square root of the ratio between the inertia force and the gravity force gives the Froude number:
FF
F
l v
gl
v
gln
I
G
= = =ρ
ρ
2 2
3 (8)
whereas the ratio between the inertia force and the viscous force is the Reynolds number:
RF
F
l v
vl
lvN
I= = =µ
ρ
µ
ρ
µ
2 2
(9)
Both of these are non-dimensional values and thus do not depend on the actual scale. Keeping the Fn value the same for both the ship model and the ship leads to the scaling law for the velocity:
vv
MS=λ
.
(10)
However, dynamic similarity also demands that the Reynolds numbers for the ship model and the ship are the same, in which case the model-scale velocity must be:
v vM S= λ. (11)
Evidently, Eqs. 10 and 11 cannot simultaneously be satisfi ed, and thus model tests with the proper relation-ship between inertia, gravity, and viscous forces are impossible. Fortunately, viscous forces do not play a very important role in the dynamics of ship motions during such a transient event as a collision. In the colli-sion process, inertial forces are large due to high accel-erations. Velocities, especially in the case of a sideways moving struck ship, are rather low, and thus the viscous forces are small compared to the inertial forces. Consid-ering this, the Froude scaling law (Eq. 10) was used to scale the experiments. What results is a Reynolds number that is too small and thus frictional forces are induced that are too high. As the gravity and density are con-stants, rough dynamic similarity is assured if the param-eters are scaled accordingly:
length
area
volume
force
time
velocity
12
3
3
1 2
1 2
∼
∼
∼
∼
∼
∼
λ
λ
λ
λ
λ
λ
A
V
F
t
v
eenergy E ∼ λ 4 (12)
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J Mar Sci Technol (2008) 13:71–84 75
3.2 Modelling of structural resistance
The only force not considered so far is the contact force. This force depends on the ship structures, and when scaling, the cross-sectional properties of the impact bulb and the side structure of the struck ship must be con-sidered. Model-scale experiments concentrating on the external dynamics and the precise deformation mechan-ics of the structures was outside the scope of this study. However, in order to maintain dynamic similarity, the resistance has to be similar to that of large-scale ship structures. Several options to model structural resis-tance in model-scale ship grounding experiments were studied by Lax.8 Considering the feasible scale for the experiments and the resistance level of different materi-als, polyurethane foam was chosen as a suitable material.
Several quasi-static penetration tests were conducted to fi nd the relationship between the shape of the impact head and the contact force.9 These tests revealed that the compression ratio for the foam was approximately 75%. For larger compression values, densifi cation begins and the force increases rapidly. Such an increase did not occur in the large-scale experiments and thus it had to be prevented also in the model-scale tests. The dimensions of the foam block were chosen so that the predicted maximum penetration would not cause den-sifi cation. The friction coeffi cient between the foam and the painted impact head was determined to be 0.2, which is close to the friction between two objects made of steel.
Based on the material tests, an analytical formulation was developed to estimate force–penetration curves for an impact head of arbitrary shape. Estimated curves were compared to those from the large-scale experiments with Y-core1 and X-core10 structures. These structures are depicted in Fig. 3 and their force–penetration curves scaled down to model scale are presented in Fig. 5. Con-sidering the resistance level of the foam and the dimen-sions of the test basin, the feasible scaling factor l was determined to be 35. Two bulbs of axisymmetric shape were manufactured with the dimensions presented in Fig. 4. The force–penetration curves obtained with these
two bulbs are given in Fig. 5, in which the results of collision experiments no. 101 and 113 are presented (see Table 4). The curve of bulb 1 corresponds well to the X-core experiment, while bulb 2 produces higher resis-tance compared to the large-scale structures.
Obeying the Froude scaling law means that the process is recreated almost precisely over the contact duration, as then the inertia and the contact force are the domi-nant forces. When contact is lost, the main external forces will be the radiation force and the sway steady motion resisting force. The fi rst behaves as an increase in the ship mass and has properties similar to inertia forces. The latter is a viscous force and, due to improper scaling, is too high. Therefore, the model-scale experi-ments will slightly deviate from the original behaviour
Fig. 3. Experimentally tested large-scale structures used as refer-ences for model-scale tests: Y-core structure (a), X-core structure (b)
Fig. 4. Geometry of the impact bulbs used in model-scale experiments
Fig. 5. Force–penetration curves in model- and large-scale experi-ments. Large-scale curves (X-core and Y-core) are scaled to the model scale using l = 35
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76 J Mar Sci Technol (2008) 13:71–84
as time increases. This deviation is mostly absent for the contact force and the deformation energy, but increases with respect to the motions after the contact. Due to higher resistance, the ship models decelerate more and the velocities at the later stages of the collision are slightly lower compared to the original.
4 Model-scale experiments of ship collisions
4.1 Ship models and the test matrix
Ship models were scaled according to the line drawings of the ships participating in the Y-core experiment. The main particulars of the ships and the corresponding models are given in Table 1. The depth of the models was increased in order to use a larger ship mass for the different collision scenarios.
The parallel middle body of the models was made of plywood and had several transverse bulkheads to obtain suffi cient stiffness. Bow and aft parts had a more complicated shape and were made of wood. Part of the port side of the struck ship model was made of polyure-thane foam, as shown in Fig. 6 and Fig. 8. The foam block extended from the bow to 15 cm past amidships, amidships being the aimed contact point. The foam blocks were glued to a plywood plate that was screwed to the model with brackets. The bracket connections were of high stiffness to minimize motions between the foam block and the model.
The striking ship model was equipped with the impact bulb connected to the bow through a force sensor and an aluminium frame, see Fig. 7. The bulb and its connection to the striking ship model was essen-tially rigid and subject to insignifi cant deformations,
which were thus disregarded. The vertical position of the bulb was adjustable to control the height of the contact point.
During collision tests, three different loading condi-tions were used for both ships. Table 2 presents drafts, masses, the vertical height of the centre of gravity (KG), and the radii of inertia kii for different loading condi-tions. The longitudinal centre of gravity of the struck ship was always located amidships. The table also pres-ents the values of non-dimensional added mass coeffi -cients m, calculated as:
µω
ρωi
a=
( )∇→∞
lim
for translational motions such as sway and heave and:
µω
ρωii
ii
a
k=
( )∇→∞
lim2
Table 1. Main linear particulars of the ship models and reference ships (l = 35)
Ship/model Length (m) Breadth (m) Depth (m)
Model A 2.29 0.234 0.12Model B 2.29 0.271 0.12Large-scale ship A 80 8.2 2.62Large-scale ship B 80 9.5 2.80
l, scaling factor; A, striking ship; B, struck ship
Fig. 6. Model of the struck ship
Fig. 7. The force sensor and the bulb
Fig. 8. General arrangement of the model tests. dof, degree of freedom; LED, light-emitting diode; COG, centre of gravity
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J Mar Sci Technol (2008) 13:71–84 77
for rotational motions such as roll and pitch. Frequency-dependent added mass coeffi cients [a(w)] were evaluated using strip theory.11 The surge added mass for both ship models was taken as 5% of the total mass.
The test matrix is presented in Table 3, which gives the main parameters defi ning the collision scenario: the masses of the ship models; the contact velocity uA
0; the bulb type; and the coordinates XC, ZC of the actual contact point, as depicted in Fig. 8. The table also gives the vertical distance between the centre of gravity and the contact point ZC − KG, which is an important param-eter deciding the initial rolling direction of the struck model.
4.2 General arrangement of the test setup and the measuring devices
The general arrangement of the test setup is presented in Fig. 8. The striking ship model was launched towards the struck model, which was kept motionless with line reels. These reels were released just before contact. The
launching of the striking ship was performed using impulse loading from a pneumatic cylinder. Loading was transferred to the striking ship model in a location close to its centre of gravity to avoid any initial pitch motion. The contact velocity was varied by adjusting the pressure in the cylinder.
Two separate measuring systems were used, one to measure ship motions and the other to record the con-tact force. Motions were recorded with the Rodym DMM non-contact measurement system (Krypton N.V., Leuven). This consists of a camera and infrared light emitting diodes (LEDs). The diodes blink with a certain frequency and the camera records their position. According to the position and the centre of gravity of the model, the exact location and the orientation in iner-tial frame x0y0z0 is calculated. Three diodes were installed on the struck ship model and one diode was installed on the striking ship model. Three diodes allow evaluation of all six motion components, whereas one diode gives the translational components without any correction for angular motions. The sampling rate of the system was 125 Hz. In the large-scale experiments, the duration of
Table 2. Physical parameters of the ship models
Model Draft (cm) Mass (kg) KG (cm) kXX (cm) kYY (cm) msway (%) mheave (%) mroll (%) mpitch (%)
Striking 4 20.5 7.4 19 77 17 300 838 182Striking 6 28.5 6.4 15 72 23 210 878 147Striking 8 40.5 5.1 9 70 28 170 439 125Struck 4 20.5 7.4 19 93 16 376 495 155Struck 6 30.5 7.3 17 83 21 238 708 128Struck 8 44.5 5.1 9 70 27 190 278 144
KG, vertical height of the centre of gravity; kXX, radii of inertia with respect to longitudinal axis; kYY, radii of inertia with respect to transversal axis; msway, added mass coeffi cient for sway motion; mheave, added mass coeffi cient for heave motion; mroll, added mass coeffi cient for roll motion; mpitch, added mass coeffi cient for pitch motion
Table 3. Test matrix
Test no. mA (kg) mB (kg) uA0 (m/s) Bulb ZC (cm) XC (cm) KGB (cm) ZC − KGB (cm)
101 28.5 30.5 0.39 2 7.1 11.0 7.3 −0.2102 28.5 30.5 0.86 2 6.8 −1.5 7.3 −0.5103 28.5 30.5 0.91 1 6.8 0.3 7.3 −0.5104 28.5 30.5 0.45 1 7.2 6.4 7.3 −0.1105 28.5 30.5 0.66 1 7.0 2.3 7.3 −0.3106 20.5 30.5 0.90 1 6.1 3.3 7.3 −1.2107 40.5 30.5 0.83 1 6.5 2.0 7.3 −0.8108 40.5 20.5 0.83 1 4.8 1.0 7.4 −2.6109 40.5 20.5 0.45 1 4.7 6.0 7.4 −2.7110 28.5 20.5 0.92 1 4.8 5.0 7.4 −2.6111 28.5 44.5 0.93 1 9.5 2.0 5.1 4.4112 20.5 44.5 1.01 1 8.5 2.5 5.1 3.4113 20.5 44.5 0.58 1 8.7 4.8 5.1 3.6
mA, mass of the striking ship; mB, mass of the struck ship; u0A, contact velocity; ZC, vertical coordinate of the actual contact point; XC,
longitudinal coordinate of the actual contact point; KGB, vertical height of the centre of gravity of the struck ship; ZC − KGB, vertical distance between the centre of gravity and the contact point
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78 J Mar Sci Technol (2008) 13:71–84
the contact was about 500 ms, which corresponds to 85 ms in the model scale. Thus, about 10 measurement points were obtained in a model-scale collision during contact.
The contact force was measured in the longitudinal direction; vertical and transverse components are assumed to be small in a symmetric collision. To mini-mize these force components, the tests were carried out so that the aimed contact point was not only amidships, but also vertically close to the centre of gravity of the struck ship. The force sensor, displayed in Fig. 7, con-sisted of an aluminium frame and a displacement sensor. The force was evaluated based on the deformation of the frame. These deformations were still negligibly small in the presumed force range. The sampling rate was 1250 Hz. After the experiment, the permanent deforma-tion of the foam was measured with a sliding gauge to accommodate the slow elastic recovery of the foam.
4.3 Post-processing of the measurement data
The two systems were not synchronized because of a time lag resulting from the post-processing of the data in the Rodym system. According to the supplier, the duration of the time lag differs from test to test. Thus, automatic correction for the time lag was not possible and the synchronization was carried out manually. Two signals were synchronised looking at the changes in the measured signals due to contact.
Before contact, the model either is stationary or moves with a constant velocity. This is described by the straight line in Fig. 9a, which shows the sway motion of the struck ship model. The inclination of this line is the initial velocity of the model. In the fi gure, the line is almost horizontal, indicating that the initial velocity of the struck model was zero. When contact occurs, the
behaviour changes and the straight line was replaced with a polynomial line. The point at which the polyno-mial met the straight line, or was closest to it, was considered to be the approximate beginning of contact. Because the frequency of the position measuring was 125 Hz, the actual beginning of contact could lie between two measurement points. The combination of linear and polynomial fi t allowed the starting point to be placed between two actual measurement points. Fitted polyno-mials described the displacements of the models with respect to their position at the beginning of the contact. Velocities and accelerations were obtained by taking the analytical derivative of the polynomials.
When contact was initiated, the force signal rose rapidly from its initial level. As the measuring frequency for the contact force was 1250 Hz, the starting point could be detected with good accuracy, see Fig. 9b. The fi gure shows raw measurement data with a thin dotted line and fi ltered data with a bold line. The force signal had oscillatory behaviour in the vicinity of its maximum at tM. A possible source of the noise was vibration in the force sensor. The noise was fi ltered out using nonrecur-sive fi ltering12 with a fi ltering window corresponding to the frequency of the noise. The time axes for the motions and the force were shifted so that time t = 0 s corre-sponded to the beginning of contact.
The quality of the synchronization was ensured by comparing the normalized force and the accelerations, see Fig. 10. The force and the accelerations were normal-ized and made dimensionless with respect to their maximum values. As the ships were accelerated by the contact force, all three signals should be similar and have their peak values in the same place. The water surround-ing the ship models acts as a time-dependent force and thus, after reaching peak values, the accelerations do not correspond to the contact force.
Fig. 9. Curves fi tted through the measured points for the sway position of the struck ship model (test 103) (a) and the contact force (test 103) (b)
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J Mar Sci Technol (2008) 13:71–84 79
5 Analysis of the results
5.1 Analysis of a collision experiment
The phenomena of a symmetric ship–ship collision is now analysed in detail by looking at a typical collision experiment. Test 103 was chosen for thorough analysis. In many subsequent fi gures, a vertical line is drawn at time tM = 0.065 s, when the contact force attained its maximum value. At this time, the deformation energy, and thus also the deformations in ship structures, are supposed to be at a maximum. Figure 11 presents the main translational velocity component surge for the striking ship model and sway for the struck ship model. The force history of the test is presented in Fig. 9b. When
the contact started at t = 0 s, the velocities of both ships started to change rapidly. After contact was lost, roughly at t = 0.1 s, the rapid change ceased and the velocities started to decrease slowly. The next contact occurred during the interval t = 0.2–0.4 s, causing some change to the velocities. Until time tM, the change in velocities was almost linear.
The velocity of the struck model was also infl uenced to some extent by rolling, even though the Rodym system considers the effects of angular motions on translational velocity. The Rodym system assumes that the rolling takes place with respect to a predefi ned centre of gravity. Differences arise as the centre of rolling is not exactly at the centre of gravity and, even more, the centre changes depending on whether contact is occurring or not. These two distinct phases are seen in the roll motion of the struck model presented in Fig. 12. During contact, the rolling is strongly affected by the contact force. The direction of the rolling motion depends on the distance between the centre of rolling and the contact point, given as ZC − KGB in Table 3. The centre of rolling in a tran-sient contact process is different from that of a free fl oating body. The location of the centre of roll motion is governed by the combined effect of inertia and contact and hydrodynamic forces. However, based on the exper-iments, a trivial conclusion can be drawn: if the contact point is clearly above the centre of gravity of the struck ship, the initial rolling angle is positive and vice versa. If the contact point is close to the centre of gravity, the rolling direction varies and the amplitude of the motion is smaller. The rolling amplitude increased when contact was lost and the contact force did not prevent free rolling. Before the maximum contact force, the angular motions were small—less than half a degree—but they increased signifi cantly after contact.
Fig. 10. Normalized accelerations in comparison to normalized contact force (test 103)
Fig. 11. Velocities of the colliding ship models (test 103). tM, instant of maximum contact force
Fig. 12. Angular motions of the struck ship model (test 103)
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80 J Mar Sci Technol (2008) 13:71–84
decrease the penetration value. Considering the distance between the centre of gravity and the tip of the force sensor, an error of the given magnitude would be caused by an angular motion of less than 3°. Another source could be the connection of the foam blocks to the struck ship model. Under the impact load, the whole foam block might displace slightly, and thus the actual penetration will be smaller than calculated from the displacements.
5.2 Test results and physical similarity to the large-scale experiments
The model-scale tests were intended to be physically similar to the large-scale experiments. Two different large-scale collision experiments were used to validate the similarity: collisions with X-core and Y-core side
Fig. 13. Penetration depth as a function of time (a) and the force–penetration curve (b) (test 103)
Fig. 14. Velocities of the colliding ship models, normalized contact force, and normalized penetration (test 103). FC, contact force; d, penetration depth
The struck model usually turned slowly due to the different hydrodynamic properties fore and aft. In side-ways motions, the resistance of the aft body was larger. Furthermore, as the actual location of the contact point was often located slightly away from amidships, some yaw motions were excited, as can be seen in Fig. 12.
Given the ship motions, the penetration time history was calculated using Eq. 3 and is depicted in Fig. 13a. Combining the contact force and the penetration history results in a force–penetration curve, see Fig. 13b. The penetration reached its maximum of 45.8 mm at t = 80 ms. After this, the penetration started to decrease. Contact was still maintained for some time due to imme-diate elastic recovery of the foam. When the contact force became zero, the penetration had a value of 42.7 mm. The fi nal penetration value measured with the sliding gauge was 34 mm. The difference between the measured and the calculated result shows that the elastic recovery of the foam lasted longer than the duration of contact. Similar to the ship motions, the penetration increased linearly until time tM and then changed to more complex behaviour.
The velocities, the normalised contact force, and the normalized penetration were combined to form an over-view in Fig. 14. Obviously, the penetration is at its maximum when the relative velocity between the models is zero. The fi gure reveals that the maximum of the pen-etration does not exactly coincide with that of the contact force. The same can be seen from the force–penetration curve in Fig. 13b. Looking at the fi ltered force signal, the difference in penetration was 1.6 mm and in time it was 13 ms.
With monotonously increasing force–penetration curves, these points should have coincided. The differ-ence could have originated from two sources. The angular motions were not considered for the striking ship model, but any pitch or yaw increment would reduce the actual displacement of the force sensor and thus
1 3
J Mar Sci Technol (2008) 13:71–84 81
Table 4. Large-scale collision experiments
Test mA (tn) mB (tn)
M
M
A
B (−) uA
0 (t = 0) (m/s) E0 (MJ)
max(d ) (m)
d/BB (−)
max(FC) (MN) ED (MJ)
E
E
D
0 (−)
tc (ms)
X-core 721 2465 0.24 3.33 4.20 0.84 0.088 6.4 3.19 0.76 696Y-core 774 1365 0.48 3.51 5.01 0.58 0.061 6.7 2.15 0.43 545Y-core (calc.) 774 1365 0.48 3.51 5.01 0.70 0.073 7.7 3.25 0.65 490
M
M
A
B
, mass ratio (including added mass); E0, initial kinetic energy; max(d ), maximum penetration depth; d/BB, relative penetration depth;
max(FC), maximum contact force; ED, total deformation energy; E
E
D
0
, relative deformation energy
Table 5. Model-scale collision experiments
Test
M
M
A
B
(−)
uA0 (t = 0) E0 max(d) d/BB
(%)
max(FC) EDE
E
D
0
(−)
tc
MS(m/s)
LS(m/s)
MS(J)
LS(MJ)
MS(mm)
LS(m)
MS(N)
LS(MN)
MS(J)
LS(MJ)
MS(ms)
LS(ms)
101 0.81 0.40 2.34 2.3 3.5 9.7 0.34 4 231 9.9 1.5 2.2 0.62 60 355102 0.81 0.88 5.18 11.5 17.2 22.7 0.80 8 540 23.1 6.2 9.3 0.54 58 341103 0.81 0.91 5.38 12.4 18.6 45.9 1.61 17 298 12.8 7.1 10.7 0.57 101 595104 0.81 0.46 2.75 3.2 4.8 21.3 0.74 8 147 6.3 1.7 2.5 0.52 108 637105 0.81 0.65 3.82 6.2 9.4 33.2 1.16 12 221 9.5 3.6 5.5 0.58 98 582106 0.58 1.00 5.91 10.7 16.1 46.1 1.61 17 296 12.7 7.7 11.6 0.72 100 592107 1.16 0.83 4.90 14.6 21.9 40.0 1.40 15 309 13.2 7.3 10.9 0.50 105 621108 1.78 0.82 4.82 14.1 21.2 34.8 1.22 13 260 11.2 7.2 10.8 0.51 86 506109 1.78 0.45 2.65 4.3 6.4 24.9 0.87 9 134 5.7 1.8 2.8 0.43 102 606110 1.26 0.92 5.45 12.7 19.1 42.2 1.48 16 278 11.9 7.0 10.6 0.55 90 532111 0.53 0.94 5.58 13.3 20.0 61.2 2.14 23 245 10.5 9.2 13.7 0.69 157 928112 0.38 1.01 5.95 10.9 16.4 51.6 1.80 19 301 12.9 7.8 11.7 0.72 100 592113 0.38 0.60 3.53 3.8 5.7 25.3 0.88 9 179 7.7 2.6 3.9 0.68 100 592
MS, model scale; LS, large scale
Fig. 15. Relative deformation energy (EDEF/E0) as a function of the mass ratio (MA/MB)
structures.1,10 Information about these two experiments is presented in Table 4.
The mass ratio
M
M
A
B is calculated including the surge
added mass for the striking ship model and the sway added mass for the struck ship model. In the Y-core experiment, these were 0.05 and 0.24, respectively, while in the X-core experiment, the corresponding values were 0.05 and 0.29. Table 4 also presents the contact velocity uA
0, the initial kinetic energy E0 of the striking ship, the penetration depth d, the maximum contact force, the combined plastic and elastic deformation energy ED, and the duration of contact tc. In Table 5, the same parame-ters, both in model scale (MS) and equivalent large scale (LS), are presented for the model tests. In the following discussion, the model-scale tests are analysed using large-scale representation of the results.
The main parameters describing the collision dynam-ics are the contact velocity uA
0 and the masses of the par-ticipating ships, and these parameters have the strongest infl uence on the relative deformation energy. This energy is shown as a function of the mass ratio in Fig. 15, where
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82 J Mar Sci Technol (2008) 13:71–84
the results of the model-scale tests are presented together with the results calculated using the simple analytical formula of Minorsky2:
E
E
M
M M
D B
A B0
=+
(13)
The fi gure reveals that the relative deformation energy decreases with increasing mass ratio. The X-core experi-ment follows the overall trend, whereas in the Y-core experiment, the relative deformation energy was lower. In the Y-core experiment, there was a large amount of water with free surface in the tanks of both ships. Water sloshing absorbs kinetic energy and thus affects the collision dynamics. Because energy was absorbed in the motions of the water, there was less energy available for structural deformation and the relative deformation energy became smaller. In Tabri et al.,10 the Y-core experiment was recalculated without the sloshing effects and the relative deformation energy became 65%, which corresponds well to the model-scale experiments. This recalculation is referred as Y-core (calc.) in Table 4 and in Fig. 15. The simple analytical approach agrees well with the general behaviour, but as it determines the outcome based on the masses only, it does not include the effects of the velocity and structural resis-tance. In the model-scale experiments, these effects become apparent as experiments with the same mass ratio yield slightly different outcomes.
This non-dimensional comparison is good for quan-titative assessment because the differences in input parameters vanish and the comparison is comprehen-sive. Even though the relative deformation energies follow the same trend, the non-dimensional representa-tion does not yet assure the similarity. The deformation energy was calculated from the contact force and the penetration, but the penetration does not defi ne the magnitude of the displacements. Furthermore, scaling errors might be present in both values so that they cancel each other and the error might not be obvious. Thus, it is necessary to look at dimensional values to assure the similarity. The X-core experiment was free of sloshing, but had a rather low mass ratio. Also, data from the X-core experiment is scarce, as only the force–penetration curve is presented in the literature. In the Y-core experi-ment, the mass ratio is similar to that of the model-scale tests and detailed data about the motions is available. Considering this, the Y-core experiment is used in the comparison.
None of the model-scale tests was an exact repetition of the Y-core experiment. Model-scale experiment number 113 was selected because of its suitable mass ratio and contact velocity. Experiments were compared,
looking at the displacement and force time histories. However, the input parameters for the tests were still quite different and the histories do not match perfectly. The structural resistance in the Y-core experiment was higher than that in test 113, see Fig. 5. In addition, the model masses scaled to large-scale were higher at 879 and 1908 tons compared to 774 and 1365 tons in the large-scale experiment. The contact velocity was practi-cally the same in both tests.
These differences are clearly refl ected in the results, but the general behaviour still remains the same. The contact force depends on the inertia of the ships and on the structural resistance. The inertia of heavier ship models was larger and the changes from the initial veloc-ity were slower. This, combined with the softer structural response, yielded smaller displacements for the struck ship model and higher displacements for the striking ship model, as seen in Fig. 16. The maximum penetra-tion in the model-scale experiment was equivalent to 0.88 m, which was clearly higher than the 0.58 m in the large-scale experiment, as depicted in Fig. 17. The softer responses and the heavier ship models also extended the duration of contact. This can be seen from the penetra-tion history, in which the maximum penetration occurred later, and also from the force time history in Fig. 18. The force peak was wider and the maximum force was higher. A second force peak in the large-scale experiment occurred due to sloshing. Obviously, this was not present in the model-scale experiments and the force remained at zero.
Even though a dimensional comparison was made for experiments with quite different input parameters, they still clearly demonstrate the same physical behaviour.
Fig. 16. Comparison of displacements in the model- and large-scale experiments
1 3
J Mar Sci Technol (2008) 13:71–84 83
The comparison confi rmed that the model-scale experi-ments agreed with the large-scale experiments both quantitatively and qualitatively.
6 Conclusions
A series of symmetric model-scale ship collision experi-ments is presented. The feasibility of such experiments and their agreement with large-scale tests were exam-ined. The Froude scaling law was applied with a scaling factor of 35. The ship models were geometrically similar to the large-scale ships. The structural response was
modelled using polyurethane foam and an impact bulb dimensioned to result in a properly scaled force. Two measuring systems were used to record ship motions and the contact force.
Post-processing of the data resulted in some compli-cations using the Rodym position measuring system in a transient collision process. The Rodym system modi-fi es data before the output, causing a time lag, the length of which is hard to estimate precisely. Thus, the auto-matic synchronization of the two systems was impossible and so it was performed manually. Synchronization problems could be relieved by adding an acceleration sensor with a high sampling rate to the measuring system and comparing the motions obtained from it to those of the Rodym system.
A single experiment was analysed in detail to show the behaviour of ships in a collision. The analysis showed that ship motions were almost linear up to the instant when the contact force reached its maximum. After contact is lost, the behaviour becomes more compli-cated. The presence of linear behaviour until the maximum deformation depth allows signifi cant simplifi -cations in the analysis of ship collisions. Rolling and other angular motions were small during contact, because the contact force prevented large amplitude rolling of the struck ship model. After contact was lost, the rolling amplitude increased signifi cantly. The effect of the rolling was also visible in the signals for translational motions. The Rodym system corrected the translational motions to a predefi ned centre of rotation, but because this posi-tion changed during the collision process, the correction did not work perfectly.
Comparison with large-scale experiments showed the physical similarity between experiments using different scales. In the non-dimensional comparison, in which the relative deformation energy was studied as a function of the mass ratio, the large-scale experiments followed the same trend as the model-scale tests. Analysis showed that the mass ratio is the most important parameter in determining the portion of total energy absorbed by ship structures in symmetric collisions. Other parameters, such as the collision velocity and structural response, have a secondary effect on the relative deformation energy. A qualitative comparison was carried out by looking at the dimensional time histories of model-scale test number 113 and of the Y-core experiment. Input parameters for those two tests were not identical, and thus the results also deviated in the expected manner. The force time history was nevertheless very similar at both scales. The comparison indicated that the model-scale experiments not only followed the same general behaviour as the large-scale experiments, but also the magnitudes of the motions and the forces agreed. Also,
Fig. 17. Comparison of penetration in the model- and large-scale experiments
Fig. 18. Comparison of the contact force in the model- and large-scale experiments
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84 J Mar Sci Technol (2008) 13:71–84
the simple analytical model agreed well with the general behaviour of the experiments.
With the test setup validated using large-scale experi-ments and an analytical formula, the setup can be exploited to cover an even wider range of collision sce-narios. Possible effects rising from arbitrary collision angles and collision locations can be studied and imple-mented as simulation tools.
Acknowledgments. This experimental study was carried out under the Finnish national research project TÖRMÄKE, funded by TEKES through R&D program MERIKE. This contribution and fi nancial support is gratefully acknowledged.
References
1. Wevers LJ, Vredeveldt AW (1999) Full-scale ship collision experiments 1998. TNO- report 98-CMC-R1725, Delft, the Netherlands, p 260
2. Minorsky VU (1959) An analysis of ship collision with refer-ence to protection of nuclear power plants. J Ship Res 3:1–4
3. Petersen MJ (1982) Dynamics of ship collisions. Ocean Eng 9:295–329
4. Pedersen PT, Zhang S (1998) On impact mechanics in ship collisions. Mar Struct 11:429–449
5. Motora S, Fujino M, Suguira M, et al (1971) Equivalent added mass of ships in collision. J Soc Nav Archit Jpn 7:138–148
6. Määttänen J (2005) Experiments on ship collisions at model scale. Master’s Thesis, Helsinki University of Technology, p 145
7. Zhang L, Egge ED, Bruhns H (2004) Approval procedure concept for alternative arrangements. In: Proceedings of the 3rd international conference on collision and grounding of ships, ICCGS2004. Izu, 25–27 October, pp 87–96
8. Lax R (2001) Simulation of ship motions in grounding. Hel-sinki University of Technology, Report M-260, p 129
9. Ranta J, Tabri K (2007) Study on the properties of polyure-thane foam for model-scale ship collision experiments. Helsinki University of Technology, Report M-297
10. Tabri K, Broekhuijsen J, Matusiak J, Varsta P (2008) Analyti-cal modelling of ship collision based on full-scale experiments. J Mar Struct (in press)
11. Journée JMJ (1992) Strip theory algorithms. Delft University of Technology, Report MEMT 24
12. Press WH, Flannery BP, Teukolsky SA, Vetterling WT (1986) Numerical recipes: the art of scientifi c computing. Cambridge University Press, Cambridge, p 818
[P3] Tabri K., Varsta P., Matusiak J. (2009) Numerical and experimental motion
simulations of non-symmetric ship collisions, Journal of Marine Science and
Technology, doi: 10.1007/s00773-009-0073-2.
ORIGINAL ARTICLE
Numerical and experimental motion simulations of nonsymmetric
ship collisions
Kristjan Tabri • Petri Varsta • Jerzy Matusiak
Received: 13 January 2009 / Accepted: 12 October 2009
Ó JASNAOE 2009
Abstract A calculation model to simulate nonsymmetric
ship collisions, implying an arbitrary impact location and
collision angle, is described in the paper. The model that is
introduced is based on the time integration of twelve
equations of motion, six for each ship. The motions of the
ships are linked together by a mutual contact force. The
contact force is evaluated as an integral over the surface
tractions at the contact interface. The calculation model
provides full time histories of the ship motions and the
acting forces. Physical understanding of the underlying
phenomena was obtained by a series of model-scale
experiments in which a striking ship collided with an ini-
tially motionless struck ship. In this paper, numerical
simulations of four nonsymmetric collisions are presented
and the calculations are validated with the results of the
experiments.
Keywords Collision dynamics Nonsymmetric ship
collisions Model-scale collision experiments Time domain simulation model
1 Introduction
Ship collisions continue to occur despite the development
of numerous preventive measures. Human error, technical
failures, and other unpredictable events can never be
completely avoided. Eliopoulou and Papanikolaou [1]
studied the statistics of tanker accidents and revealed that
the total number of accidents and the number of accidents
causing pollution has decreased significantly in recent
decades. However, the accidents causing pollution have not
decreased to the same extent as the overall number of
accidents. It has become obvious that the measures applied
to reduce the consequences of collisions should be
improved, together with the preventive ones. To reduce the
consequences, one has to understand the nature and the
character of the underlying phenomena. A statistical study
by Tuovinen [2] analysed more than 500 collision acci-
dents, the data on which were gathered from published
investigation reports, the IMO, database and damage cards.
The statistical data presented the current trends, as the
majority of the accidents studied (388) were registered
during or after the year 1997, and only 115 accidents were
registered before that. The study revealed that, with respect
to the collision angle, only about every fourth collision can
be considered symmetric, i.e. the striking ship collides with
the amidships of the struck ship at right angles. In the
studied cases, about 25% of the collisions occurred at
a right angle or in its vicinity (±10°). All of the other
accidents were in one way or another considered to be
nonsymmetric. This fact emphasises the importance of
understanding the nature of nonsymmetric collisions. Thus,
this paper is focused on the physical phenomena of non-
symmetric ship collisions, and a calculation model is pro-
posed to predict the ship motions and the structural damage
in such a collision.
Several calculation models for simulating the dynamics
of ship collisions exist. The calculation models can be
classified as closed form expressions or as time domain
simulations. Closed form models are based on the conser-
vation of momentum and allow a fast estimation of struc-
tural deformation energy without providing the time
histories of ship motions, which leaves the exact penetration
K. Tabri (&) P. Varsta J. Matusiak
Department of Applied Mechanics, Marine Technology,
Helsinki University of Technology,
P.O. Box 5300, 02015 TKK, Espoo, Finland
e-mail: [email protected]
123
J Mar Sci Technol
DOI 10.1007/s00773-009-0073-2
path of the colliding bodies unknown. In closed form
models, there is no coupling between the external dynamics
and the inner mechanics, and therefore no coupling effects
are included. One of the first closed form models for ship
collisions for a single degree of freedom (dof) was proposed
by Minorsky [3]. This model has been modified and
extended by many authors to consider up to three dof in the
horizontal plane [4, 5].
In time domain simulation models, the system of equa-
tions of ship motion is solved using a numerical integration
procedure. A precise description of the whole collision
process, together with the full time histories of the motions
and forces involved, is achieved. Motion-dependent forces,
such as the radiation forces arising from the interaction with
the surrounding water, can be included in the analysis. As a
result of the complexity involved in solving the equations,
these simulation models are often reduced to include the
motions in the horizontal plane only [6, 7].
The proposed calculation model is based on a time
domain simulation that takes into account six dof for each
ship and is thus capable of handling arbitrary collision
angles and locations. The inner mechanics and the external
dynamics are coupled, preserving the interaction between
the colliding ships, which makes it possible to estimate the
time history of the common contact force. This force is
evaluated by considering the geometry of the colliding
ships and calculating the surface tractions at the contact
interface. The striking ship is assumed to be rigid, and all
of the structural deformations are limited to the struck ship.
The calculation model that is developed aims to predict the
ship motions until the contact between the ships is lost.
This requires precise modelling of the effects of the sur-
rounding water, and the popular approach of using constant
added mass as a representative of the radiation force does
not suffice; thus, in addition, the retardation functions [8, 9]
are used to include the time dependency of the radiation
force. The model adopts a linear approach to the restoring
force, limiting the angular displacements from the equi-
librium position to small angles, i.e. below 10°. More
accurate methods, such as the precise integration of
hydrostatic pressure over the ship’s hull, allow larger dis-
placements and angles, but the integration process is very
time-consuming and therefore is not considered here.
Frictional water resistance and hydrodynamic drag are
assumed to be proportional to the square of the ship’s
velocity. Effects arising from the wave pattern around the
colliding ships and from the immediate consequences of
the collision, such as flooding and loss of stability, are not
included in the calculation model.
The physical phenomena are studied and the calcula-
tions are validated with non-symmetric model-scale colli-
sion experiments. The test setup for the experiments was
designed and validated with large-scale collision tests [10].
The emphasis in the model tests was placed on the external
dynamics, and thus the side structure of the struck ship was
modelled using polyurethane foam. The scaling of the
contact force was based on the results of the large-scale
experiments. During the collision, the motions of both
ships in all six dof were measured, as was the contact force
in the longitudinal and the transverse directions with
respect to the striking ship. Several collision scenarios,
with different collision angles and locations, were tested in
order to obtain not only large translational motions but also
relatively large angular motions. The tests were limited to
those cases where the striking ship approaches an initially
motionless struck ship. This was mainly due to the test
setup of the model tests, where the focus was on the
physical phenomena. In all of the tests, the contact point on
the struck ship was above the waterline and was located in
the parallel middle body.
2 Calculation model
The calculation model assumes a situation in which the rigid
striking ship approaches, at a certain angle, a specific loca-
tion on the struck ship. As soon as the ships are brought into
contact, the contact force starts to play a major role in the
collision dynamics. When the contact is lost, the force
decreases to zero and the hydromechanical forces will gov-
ern the collision process. The equilibrium between the forces
acting on the ship and the resultant ship motions is described
through a system of six equations of motion. These Newto-
nian equations are expressed in a local coordinate system of
the ship, allowing convenient description of the hydrome-
chanical forces. The numerical time integration of these
equations yields the ship motions in the local coordinate
system. Though the motions are conveniently presented in
the local frames, the position and orientation of the ships can
only be specified by reference to an inertial coordinate sys-
tem that is fixed with respect to the Earth.
2.1 Motion kinematics
The convenient description of the mutual motions and the
kinematic connection of the colliding bodies requires five
different coordinate systems, which are presented in Fig. 1.
Hereafter, the superscript characters A and B denote the
striking and the struck ship, respectively. If the superscript
is omitted or replaced by i, it means that the description is
common to both ships. Superscript 0 indicates the inertial
frame. The origins Oi of two sets of local axes are fixed to
the mass centre of gravity of the ship. Positioning the
coordinate systems at the centre of gravity simplifies the
analysis, as the acceleration components resulting from
mass eccentricity disappear. For ship position, reference is
J Mar Sci Technol
123
made to the inertial coordinate system O0x0y0z0 with a
position vector Ri, but when describing the ship motions it
is common to refer in the local coordinate system to surge,
sway, heave, roll, pitch, and yaw. The first three are
translations along the axes and the last are rotational
motions. The instantaneous translational velocity ui of the
ship’s centre of gravity is given as a time derivative of the
position vector Ri
ui ¼ _Ri ¼ _x0i0 þ _y0j0 þ _z0k0 ¼ uiii þ viji þ wiki ð1Þ
where _x0; _y0 and _z0 are the velocities of the ship’s centre of
gravity in the inertial frame and u, v and w are their pro-
jections onto the local coordinate system Oixiyizi.
The second set of local axes, referred to as the horizontal
body axes Oinigifi, are used for a more convenient pre-
sentation of the orientation of the ships. At the onset of the
contact process, the directions of the OAnAgAfA system
coincide with those of the inertial frame O0x0y0z0, while
OBnBgBfB is rotated with respect to O0x0y0z0 to obtain the
collision angle b. The collision angle b is defined as the
angle between i0 and the direction of nB. The positions of
OAnAgAfA and OBnBgBfB at the onset of the contact are
considered to be the reference positions of the ships. Both
sets of local axes, Oinigifi and Oixiyizi, are coincident at the
beginning of the contact. During the calculation, these
horizontal body axes are only subjected to translational
motions, and the body fixed coordinate system Oixiyizi is
reoriented with respect to Oinigifi.
The orientation of the ship is defined using the method
of modified Euler angles [11]. In this method, the rotations
must be handled in a certain order: first a ‘‘swing’’ w to the
actual azimuth, then a ‘‘tilt’’ h to the actual elevation, and
finally a ‘‘heel’’ / to the actual orientation. The relation
between the velocities in the inertial coordinate system and
their projections onto the local axis is given [11] by an
orthogonal matrix of transformation [T]:
_x0
_y0
_z0
8>><
>>:
9>>=
>>;¼ T½
ui
vi
wi
8>><
>>:
9>>=
>>;
¼
cosw cos hcosw sin h sin/
ÿ sinw cos/
cosw sin h cos/
þ sinw sin/
sinw cos hsinw sin h sin/
þ cosw cos/
sinw sin h cos/
ÿ cosw sin/
ÿ sin h cos h sin/ cos h cos/
2
66666664
3
77777775
ui
vi
wi
8>><
>>:
9>>=
>>;ð2Þ
Matrices are denoted by [] and vectors by in
equations where both types of objects are present. In
equations consisting only of vectors, or when the actual
type is obvious, the above notation in omitted for the sake
of brevity. Position and force vectors can be transformed in
a similar manner as the velocities in Eq. 2. The angular
velocity Xi is defined in the local coordinate system as
Xi ¼ piii þ qiji þ riki ð3Þ
where pi, qi, and ri are the angular rates of roll, pitch, and
yaw in the local coordinate system. The derivatives of the
Euler angles depend on the angular rates as [11]
_/
_h
_w
8>><
>>:
9>>=
>>;¼
1 sin/ tan h cos/ tan h
0 cos/ ÿ sin/
0 sin/= cos h cos/= cos h
2
4
3
5pi
qi
ri
8><
>:
9>=
>;ð4Þ
It should be noted that even though the angular rates _/,_h, and _w are vector quantities, the Euler angles cannot be
presented as a vector. For the sake of simplicity, they are
still collectively referred to via a column matrix
½u ¼ ½/ h w T.The translational velocity uiP of a point P positioned by
the vector rPi in the local coordinate system is evaluated as
uiP ¼ ui þXi riP ð5Þ
The relative position between the ships is described by a
penetration vector d that is defined in the inertial frame as
d0 ¼ RA þ rAP ÿ RB þ rBP
ÿ ð6Þ
It is also useful to express this penetration in the body
fixed coordinate systems to account for the orientation of
the ships:
dA ¼ d
0 iA þ d0 jA þ d
0 kA ð7Þ
dB ¼ d
0 iB þ d0 jB þ d
0 kB ð8Þ
Fig. 1 Definition of coordinate systems and position vectors
J Mar Sci Technol
123
2.2 Fluid forces and gravity
Hydromechanical forces and moments acting on a surface
ship consist of water resistance, hydrostatic restoring for-
ces, and radiation forces expressed in terms of hydrody-
namic damping and added mass. A ship moving in water
encounters frictional and residual resistance. Residual
resistance is not included in the study because it is con-
sidered to be small compared to other phenomena. The
frictional water resistance is considered only for surge and
sway. The friction force FF,x for surge is approximated with
the ITTC-57 friction line formula. For sway, the hydro-
dynamic drag force FF,y is calculated as [12]
FF;y ¼1
2qv2CyAL ð9Þ
where q is the water density, v is the sway velocity, Cy is a
drag coefficient that depends on the shape of the hull and
on the angle between the ship’s longitudinal axis and its
velocity vector, and AL is the lateral underwater area of the
ship. In the model tests reported by Gale [12], it is shown
that the drag coefficient Cy varies between 0.5 and 1.2, and
Cy = 1 is used in the subsequent calculations.
Buoyancy loading FB is split into the buoyancy qgrk0 at
theequilibriumposition and the hydrostatic restoring forceFR:
FB ¼ ÿqgrk0 þ FR ¼ ÿqgrk0 þ ½K fxg½u ð10Þ
where g is the gravitational acceleration,r is the volumetric
displacement of the ship, [K] is a matrix of linear restoring
terms, x presents a vector of translational displacements
from the equilibrium position, and [u] are the Euler angles
already described in the previous section. The terms of the
matrix [K] are presented in Appendix 1. As these restoring
terms are based on small angular displacements, the same
linear restoring terms can be applied both in the local and the
inertial coordinate systems. Gravity loading FG opposes the
buoyancy in the equilibrium position:
FG ¼ qgrk0 ð11Þ
It is common practice to model the radiation forces by
frequency-dependent added mass a(x) and damping b(x)
coefficients. To represent the radiation forces in the time
domain, it is useful to split them into a part Fl proportional
to the acceleration and into a velocity-dependent damping
part FK: [8]
FlðtÞþFKðtÞ¼ÿ½a1 _uðtÞ_XðtÞ
ÿZ t
0
½KbðsÞuðtÿ sÞXðtÿ sÞ
ds
ð12Þ
where t denotes time, s is a dummy variable, [a?] is the
matrix of added masses a(x = ?) at infinite frequency,
and [Kb(s)] is a matrix of retardation functions, which
account for the memory effect:
½KbðsÞ ¼2
p
Z1
0
½bðxÞ cosðxsÞ dx ð13Þ
where [b(x)] is a matrix comprising of added damping
terms. The retardation functions Kb(s) are evaluated by fast
Fourier transformation [13].
2.3 Contact process between ships
During a collision, the ships interact through the contact
force arising from the deformations of their structures. The
interaction model presented in this chapter considers a
homogeneous side structure whose stiffness is significantly
lower than that of the striking ship. Thus, it is reasonable to
assume that all of the deformations are limited to the struck
ship and that the striking ship can be treated as rigid. The
deformed shape of the side structure of the struck ship is
restricted to following the shape of the penetrating bow.
Throughout the derivation of the contact model it is
assumed that the stress state in the deformed structures can
be obtained easily, and so the derivation of the contact
force concentrates on the contact kinematics.
When two nonconforming bodies are brought into con-
tact, they initially touch each other at a single point or line.
As the contact proceeds and the bow penetrates further into
the struck ship, the contact interface expands. To predict
the shape of the contact interface––see Fig. 2––the geom-
etries of the colliding bodies are defined in the local
coordinate system of the striking ship. Therefore, the sur-
face S of the axisymmetric bulbous bow of the striking
ship is defined as
Fig. 2 Contact geometry with kinematics and surface tractions
J Mar Sci Technol
123
xA ¼ f A a; b; c; yA; zAÿ
ð14Þ
where a, b, and c are the shape parameters. It is assumed
that Eq. 14 has first-order partial derivatives inside the
contact interface. The geometry, position, and orientation
of the struck ship B are transferred to OAxAyAzA and are
approximated as a plane P by
xA ¼ f B A;B;C; yA; zAÿ
ð15Þ
where A, B, and C are the parameters defining the plane.
Subtracting Eq. 15 from Eq. 14 yields the curve C of
intersection that bounds the contact interface S in the yAzA
plane; see Fig. 2. The area bounded by curve C is denoted
as area A. Contact between the ships exists when curve Cexists and thus area A has a real positive value.
The magnitude and the direction of the contact force FC
depend on the structural geometry and on the relative
motions of the colliding ships. As the bow penetrates into
the side structure, surface tractions are formed on the
contact interface. The normal traction—referred to as
pressure—is denoted by p and the tangential traction
caused by friction by q in Fig. 2.
The resultant contact force FC is resolved into a com-
pressive part Fp and a frictional part Fq. It is assumed that
this resultant force acts at point P, which is the centre of
area A. At the exact centre of the contact, the resultant
moment of surface tractions is zero. Even though this
condition is not always satisfied at point P, it is still used as
the centre. The moment lever that is caused is considered
small compared to the dimensions of the ship, and thus the
additional moment is neglected.
2.3.1 Compressive and friction force
Inside the contact interface S, every infinitesimal area dS*
described by its central point Q in the right-hand picture in
Fig. 2 is subjected to normal compressive traction p and to
tangential traction q. The force resulting from the normal
traction is evaluated by integrating over the interface S
FAp ¼
ZZ
S
pnAQ dS ð16Þ
where nQA is an unit normal at Q pointing outside the bulb,
and is defined as
nAQ ¼iÿ of A
oyAjÿ of A
ozAk
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ of A
oyA
2þ of A
ozA
2r ð17Þ
The surface integral over S can be expressed as a more
convenient integral over area A:
ZZ
S
dS ¼ZZ
A
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1þ of A
oyA
2
þ of A
ozA
2s
dyA dzA ð18Þ
The bulb geometry and the distribution of p in the
contact interface define the direction of FpA. The frictional
force component Fq acts on a tangent plane T at point Q;
see Fig. 2. The direction of this force is denoted by the
vector kQA and it depends on the contact kinematics. The
discussion about the exact value of kQA is presented in the
next section. Assuming that the tangential traction is
proportional to the pressure, the frictional force can be
evaluated as
FAq ¼
ZZ
S
qkAQ dS ¼ZZ
S
lqpkAQ dS ð19Þ
where lq is the coefficient of friction. The resultant contact
force acting on point P is
FAC ¼ ÿFB
C ¼ FAp þ FA
q ð20Þ
Within the loaded interface at any point Q, there is
equilibrium between the normal traction and the stresses in
the deformed material at the contact interface:
p ¼ ÿrn ð21Þ
where rn indicates the normal stresses in the direction of
the surface normal.
2.3.2 Contact phases
The contact process is divided into three distinct phases.
The contact starts with the loading phase, during which the
penetration increases. The loading is followed by a short
stiction phase, during which the penetration remains
roughly the same. The contact process ends with the
unloading phase, during which the penetration decreases as
a result of the separation of the ships. All three phases
differ with respect to contact kinematics, which is defined
by the relative velocity between the ships at the integration
point Q
uABQ ¼ uAQ ÿ uBQ ð22Þ
and additionally by the relative acceleration _uABQ . An inte-
gration element dS, presented in Fig. 2, is assumed to
undergo the loading phase when the following three con-
ditions are fulfilled:
1. The angle between uQAB and nQ
A is less than 90°
2. The magnitude juABQ j is larger than the threshold
velocity uAB03. The penetration with respect to the transverse direction
of the struck ship is increasing, i.e. uABQ jB[ 0.
J Mar Sci Technol
123
In the loading phase, the direction of kQA opposes the
projection of the relative velocity uQAB to the tangent plane
T of the indenter, and is defined as
kAQ ¼ ÿprojT
uABQ
uABQ
ð23Þ
where projT u indicates the projection of a vector u onto the
plane T , and can be written as
projT u ¼ n u nð Þ ð24Þ
where n is the unit normal vector of the plane.
During the transition phase from the loading to the
unloading, the angle between uQAB and nQ
A increases and
becomes larger than 90°. If during this transition the magni-
tude juABQ j falls below the threshold velocity u0AB, the integra-
tion element dS undergoes the stiction phase. Figure 3a
presents a situation in which a surface element close to the
boundary of the contact interface is in the stiction phase.
The time instant when the first surface element enters either
the stiction or the unloading phase is denoted as t = te1.
In the stiction phase, uQAB approaches zero and possibly
changes its sign. Therefore, evaluating the direction of kQA
by the relative velocity uQAB may yield the singularity
problem in Eq. 23. Despite the small relative velocity, the
friction or (more properly) the stiction force does not dis-
appear, as the bodies—even though they are not sliding—
still undergo small reversible elastic deformations that
result in an interaction force [14]. In order to avoid using
complicated theoretical models for stiction, the proportional
friction is still used, but the direction is based on the relative
acceleration _uABQ at Q, as suggested by several authors [14,
15], and thus the direction of kQA is defined as
kAQ ¼ ÿprojT
_uABQ
_uABQ
ð25Þ
In the unloading phase, either (1) or (3) or both
conditions are not satisfied. When the penetration starts
to decrease, the contact is not immediately lost as a result
of the elasticity of the structures. The deformation of the
structure recovers to some extent in order to restore its
initial undeformed shape. This elastically recovered region
is simply referred to as the recovered region. It is assumed
that the structure recovers along the shortest possible path,
i.e. along -jB. The final shape of the recovered region
follows the surface geometry of the penetrating bow. The
amount of restoration is controlled by the relative thickness
of the recovered region e and by the transverse penetration
d0 jB in the struck ship. The shape of the elastic region is
given in a general form as
xA ¼ f eða; b; c; e; d0 jB; yA; zAÞ ð26Þ
In Sect. 3.2, the functions f A; f B; f e given by Eqs. 14,
15, and 26, the parameters a, b, c, A, B, and C, and the
material properties rn, lq, and e are presented for the
contact configurations used in the model-scale experiments.
In the unloading phase, the direction of kQA has to take into
account the velocity of elastic recovery. As the recovering
structure maintains the contact with the indenter, and as the
Fig. 3 Reversal of the velocity
vector during the transition from
the loading to the unloading
phase and the storing of the
elastic regions. a Stiction phase
starts at t = te1. b Maximum
penetration is reached at
t = te2[ te1. c Two recovered
regions at t = tj[ te2[ te1
J Mar Sci Technol
123
recovery is along -jB, the velocity of the elastic recovery is
uABQ jB
jB, and thus the relative velocity between the de-
penetrating bow and the recovering structure is
uAQ ÿ uBQ þ uABQ jB
jB
¼ uABQ ÿ uABQ jB
jB ð27Þ
and the direction kQA of the friction force is evaluated as
kAQ ¼ ÿprojT
uABQ ÿ uABQ jB
jB
uABQ ÿ uABQ jB
jB
2
64
3
75 ð28Þ
The recovered region is evaluated at two time instants,
as indicated in Fig. 3: first at t = te1, when the stiction or
the unloading phase occurs for the first time in some
integration element, and for the second time at t = te2 when
the transverse penetration d0 jB in the struck ship reaches
its maximum value. The first region is always updated
when the bow penetrates further. During later stages, when
t = tj[ te2, at each integration point Q the recovered
region is determined by the larger penetration value.
2.4 Equation of motion
A ship with its mass described in a mass matrix [M] and
moments of inertia in an inertia matrix [I] is subjected to
the external force
F ¼ FFþFlþFKþFC þ FB þ FG ð29Þ
and to the moment G of the external force about the centre
of gravity of the ship. All of the forces are described or
transferred by the matrix [T] to the local coordinate system
Oixiyizi, where the general equation of translational motions
is written according to Newton’s law as [11]
M½ dudt
þ M½ X u ¼ F ð30Þ
and correspondingly for rotational motions as
I½ dXdt
þX I½ X ¼ G ð31Þ
The time integration of these equations is based on an
explicit fifth-order Dormand–Prince integration scheme
[16], which is a member of the Runge–Kutta family of
solvers, where the calculation advances from tj to
tj?1 = tj ? dt with seven subincrements. For time-efficient
integration, the forces on the right-hand side of the equations
are kept constant during time step dt, while the motions on
the left-hand side are updated in every subincrement. Thus,
the preciseness of the integration can be increased bymoving
some force components from the right-hand side to the left.
This is donewith the forces that do not require history values
from former time steps and are linear with respect to
motions. Therefore, the hydrostatic restoring force FR
proportional to the displacements and the radiation force
component Fl are moved to the left. The latter results in a
full added mass matrix containing several terms, which
couple the translational and rotational motions. This fully
coupled added mass matrix and the coupling term X u in
Eq. 30 require the simultaneous solution of translational
and rotational motions. In their general form, the two
equations of motion are combined to give
M½ 0
0 I½
_u_X
þ M½ X u
X I½ X
ÿ FR ÿ Fl
¼ F
G
ÿ FR ÿ Fl ð32Þ
and the component form of this equation that is suitable for
numerical integration is presented in Appendix 1. The
solution of Eq. 32 provides kinematically admissible
motions at the end of the integration increment at t = tj?1,
and the external forces on the right-hand side are updated
accordingly. There, the new position of the ship with
respect to the inertial frame is evaluated by integrating with
respect to time over the velocities in the local coordinate
system and transforming the translational displacement
increments that are obtained to the inertial frame by the
matrix [T]. These increments are added to the position
vector Ri. The orientation is updated by simply adding the
angular increments to the Euler angles. Given the positions
and the orientations, the penetrations are calculated from
Eqs. 6 to 8. The equations of motion, Eq. 32, are estab-
lished for each ship. These equations are treated separately
in the integration during the time increment dt, and after
each step the mutual contact force FAC ¼ ÿFB
C is updated for
both ships in order to maintain the kinematic connection.
3 Model-scale experiments of ship collisions
The full-scale experiments [17] provided validation data
for symmetric collisions in which the ship motions are
limited to only a few components. In order to gain a deeper
insight into the dynamics of nonsymmetric collisions, a
series of model-scale experiments was performed at the
Helsinki University of Technology. The model tests were
designed to be physically similar to the large-scale exper-
iments. The design, scaling, and validation of the test setup
for the model-scale experiments are explained in detail by
Tabri et al. [10] and Maattanen [18].
3.1 Test setup and measuring systems
The general arrangement of the test setup is presented in
Fig. 4. The models were geometrically similar to the ships
participating in the full-scale experiments. Considering the
level of structural resistance of the ship models and the
J Mar Sci Technol
123
dimensions of the test basin, the feasible scaling factor k
for the Froude scaling law was determined to be k = 35.
This resulted in models of length L = 2.29 m, depth
D = 0.12 m, and breadth B = 0.234 m for the striking
ship and B = 0.271 m for the struck ship.
Table 1 presents draughts, masses, the vertical height of
the centre of gravity KG, and the radii of inertia kii for
different loading conditions in the ship models. The lon-
gitudinal centre of gravity of the ships is always located at
the amidships. The table also presents the values for the
added masses. The nondimensional added mass coeffi-
cients l are calculated as
lj ¼ limx!1
ajðxÞqr ; with j ¼ x; y; z ð33Þ
for translational motions such as sway and heave and
ljj ¼ limx!1
ajjðxÞqrk2jj
; with j ¼ x; y; z ð34Þ
for rotational motions such as roll, pitch, and yaw. The
frequency-dependent added mass a(x) was evaluated with
strip theory [19]. The coefficients were first evaluated in a
coordinate system with its origin located in the water plane.
As the equations of motion are established in the Oixiyizi
system, the origin of which is fixed to the mass centre of
gravity, the added mass and the damping coefficients were
transferred to Oixiyizi considering the distance between the
water plane and the centre of gravity. The surge added
mass for both models is taken to be 5% of the total mass of
the model.
The striking model, equipped with a rigid bulb, hit the
struck model at a location where a block of soft foam was
installed. The structural response in these model-scale
experiments was scaled so as to be similar to that in the large-
scale ones. The scaling is described in the next section.
Two separate measuring systems were used in the exper-
iments, one to record the ship motions and the other to
measure the contact forces in the longitudinal and the trans-
verse directions with respect to the striking ship. The motions
were measured with the Rodym DMM noncontact mea-
surement systemwith a sampling rate of 125 Hz. This system
provided the position and the orientation of the models with
respect to the inertial frame O0x0y0z0. Taking the time
derivatives of the position and the orientation signals yielded
velocities in the inertial coordinate system, and the values in
the local frames were obtained using Euler’s angles, as given
by Eqs. 2 and 4. It was estimated that the positions of the
models were measured with a precision of ±0.05 mm and
their orientations with a precision of ±0.5°.
Two contact force components were measured in the
striking ship: the longitudinal force FC,xA and the transverse
force FC,yA . The vertical contact force was not measured
because of the limitations of the measuring instrumenta-
tion. However, with the contact point being close to the
waterline, only small vertical forces were expected. The
sampling rate was 1250 Hz and the precision of the mea-
sured forces was estimated to be ±0.01 N. The synchro-
nisation of the force and the motion measurements in the
time domain is described in Tabri et al. [10].
3.2 Force response
The model-scale experiments concentrated on the external
dynamics, and the precise deformation mechanics of the
side structures were beyond their scope. However, in order
Table 1 Physical parameters of the models
Model Draft (cm) Mass (kg) KG (cm) kXX (cm) kYY (cm) kZZa (cm) lsway (%) lheave (%) lroll (%) lpitch (%) lyaw (%)
Striking 4 20.5 7.4 19 70 70 17 300 12 220 14
Striking 6 28.5 6.4 15 67 67 23 210 11 170 20
Striking 8 40.5 5.1 9 65 65 28 170 23 146 27
Struck 4 20.5 7.4 19 77 77 16 376 20 231 10
Struck 6 30.5 7.3 17 69 69 21 238 14 184 17
Struck 8 44.5 5.1 9 65 65 27 190 36 164 25
a It is assumed that kZZ = kYY
Fig. 4 General arrangements of the model tests
J Mar Sci Technol
123
to maintain dynamic similarity, the structural resistance has
to be similar to that of the full-scale ship structures. As the
properties of the polyurethane foam used as the side
structure of the struck model are constant, the level of the
contact force can only be modified by changing the shape
of the impact bulb. In a preliminary material study, a
number of penetration tests were carried out in order to
study the crushing mechanisms of the foam [20]. The
crushing strength of the foam is assumed to be equal to the
stress rn at the contact interface and it was determined to
be rn = 0.121 MPa. The friction coefficient between the
foam and the painted surface of the bulb was lq = 0.15-
0.2 [20]. Three different axisymmetric bulb shapes pre-
sented in Fig. 5 were evaluated [20]. The shapes of the
bulbs were defined as an elliptical paraboloid
xA ¼ f Aða; b; c; yA; zAÞ ¼ ÿ ðyAÞ2a2
þ ðzAÞ2b2
þ c
!
ð35Þ
with the semi-axes a and b having the following values:
a = b = 0.129ffiffiffiffim
p½ for bulb 1;
a = b = 0.258ffiffiffiffim
p½ for bulb 2;
a = b = 0.169ffiffiffiffim
p½ for bulb 3.
Parameter c describes the coordinate value in OAxAyAzA,
where the surface of the bulb intersects with the xA axis and
c = -LA/2 as the centre of gravity of the ship was at the
amidships.
The force–penetration curve of bulb 1 corresponds well
to the X-core large-scale experiment, as seen in Fig. 6. The
thick line presents the large-scale measurement [10, 21],
which is scaled down with a scaling factor of k = 35. The
thin line is measured from the model-scale test with bulb 1,
and the dashed line is obtained using the approach pre-
sented in Sect. 2.3. The other two bulbs provided signifi-
cantly higher resistance compared to that of the large-scale
experiment. It should however be noted that such a
presentation of structural resistance is very general, and only
presents realistic structural behaviour to a certain extent.With
extensive deformations, the actual force–penetration curve
could be different from the monotonously increasing curve of
the current setup. However, the approach is still valuable due
to its simplicity and is sufficient to maintain the physical
similarity with respect to the external dynamics.
In the time simulations of the collision experiments, the
recovered shape of the deformed foam was evaluated on
the basis of the shape of the impacting bulb, the relative
thickness of the recovered region e, and the maximum
translational penetration d0 jB in the struck ship. With the
point of contact located in the parallel middle body, the
struck ship can be presented as a plane in OAxAyAzA :
xA ¼ ÿ AyA þ BzA þ Cÿ
ð36Þ
The parameters A, B, and C depend on the current position
and on the orientation of the struck ship. Given definitions for
the geometries of the ships (Eqs. 35 and 36), the penetration
depthd0 canbeevaluated, and thus, basedonEq. 26, the shape
of the recovered region in OAxAyAzA is written as
xA ¼ f e a; b; c; e; d0 jB; yA; zAÿ
¼ ÿ yAð Þ2
1ÿ eð Þa½ 2þ zAð Þ2
1ÿ eð Þb½ 2þ cÿ e d
0 jBÿ
!
ð37Þ
Upon comparing the experimentally measured force–
penetration curves to those evaluated with the approach
presented in Sect. 2.3, the relative thickness of the
recovered region e was determined to be around e = 0.03.
3.3 Test matrix
The model-scale experiments were divided into three dif-
ferent sets on the basis of the type of collision scenario.
Fig. 5 Geometries of the axisymmetric impact bulbs used in the
model-scale experiments
Fig. 6 Force–penetration curves obtained in the large- and the
model-scale experiments and by calculations. Large-scale measure-
ments are scaled to the model scale with k = 35
J Mar Sci Technol
123
The first set concentrated on symmetric collisions and is
not discussed here. The second and the third sets consisted
of nonsymmetric collision experiments. In the second set,
the location of the contact point was changed, ranging from
an eccentricity of 0.13LB (30 cm) to 0.36LB (83 cm) from
the amidships towards the bow, but the collision angle b
was still 90°. The motions of the striking ship were mainly
translational, while the struck ship was subject to yaw
motions in addition to translations. In the third set, the
collision angles varied from 30° to 120° and the eccen-
tricity was around *0.18LB (*40 cm). The yaw motions
of both ships were significant, yielding diverse motion
dynamics. The complete test matrix containing the exper-
iments in the second and third sets is presented in
Appendix 2.
4 Validation
The model-scale experiments provided a vast amount of
data for the validation. Here, this amount is limited to a set
sufficient to provide qualitative and quantitative validation.
Four experiments presenting distinctively different colli-
sion scenarios were chosen for thorough analysis: test 202
from the second set, and tests 301, 309, and 313 from the
third set. Test 202 was a collision at right angles with an
eccentricity of 0.36LB (83 cm). In tests 301, 309, and 313,
the eccentricity was about 0.16LB - 0.19LB (37–44 cm)
and the collision angles were 120°, 145°, and 60°,
respectively. More detailed information about the selected
experiments is presented in Appendix 2.
The time histories of the calculated and the measured
longitudinal contact force FC,xA and the transverse contact
force FC,yA as experienced by the striking ship are presented
in Fig. 7. In addition to the measured forces, the calculated
time history of the vertical contact force FC,zA is also given
in the figure. In test 202, the longitudinal force is clearly
dominant and only a minor transverse force arises as a
result of the yawing of the struck ship. The transverse force
increases significantly when the collision angle is other
than 90°. In tests 301 and 313, the transverse force changes
its direction as the bow gets stuck in the foam. With the
large collision angle in test 309, the bow slides along the
struck ship and the transverse force remains positive
throughout the contact. The calculations predict the loading
phase well; the deviation from the measurements increases
in the unloading phase. This consists of more complicated
mechanisms, which cannot be precisely predicted with
robust calculation models. In the calculations, the stiction
phase appears as a somewhat unrealistic drop in the force
values. The stiction phase can clearly be seen in test 202 at
t = *55 ms. When the stiction is not considered in the
Fig. 7 Calculated and
measured components of the
contact force FCA as experienced
by the striking ship (superscript
A and subscript C are omitted
for brevity)
J Mar Sci Technol
123
calculations, the drop in the force value is more significant
and has a longer duration, as shown in Fig. 8, which pre-
sents two calculations of FC,xA in test 202, one with stiction
included and the other without.
The calculated vertical contact force is clearly lower
than the other force components. Even though minor roll-
ing of the struck ship occurs in the calculations, the
distribution of the contact pressure in the vertical direction
is rather symmetric, and thus the resultant force becomes
low.
The motions of the ships are presented in Fig. 9 as the
position of the origin Oi in the inertial frame. Their posi-
tions are scaled to start from zero. A circular marker is
drawn every 20 ms to include the time scale. Displacement
along the x0 axis is an order of magnitude larger compared
to the y0 displacement. This is to be expected, as the initial
velocity of the striking ship was along the x0 axis. Dis-
placement along the x0 axis is predicted well in terms of
both distance and time. In the transverse direction the
relative differences are larger, but are acceptable, consid-
ering their small magnitude.
The yaw angle of the struck ship is presented in Fig. 10.
The magnitude of the angle at the end of the contact is
about 1°, but it is still very important for evaluating the
penetration history. The yaw is the largest in test 202, as a
result of the high eccentricity. The calculation model tends
to somewhat underestimate the angular motions.
Considering the position and the orientation of the
models, the bulb’s penetration paths can be evaluated.
Figure 11 presents the path of the tip of the penetrating
bulb inside the side structure of the struck ship. In test 202
the penetration has almost only the transverse component.
During the short transient contact phase, the angular
motions of the ships are still too small to extend the
Fig. 9 Position of Oi in the
inertial frame (marker spacing
20 ms). Note the different scales
of the y0 axis
Fig. 8 Calculated contact force FC,xA with and without stiction phase
included (test 202)
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123
Fig. 10 Yaw angle of the
struck ship
Fig. 11 Penetration paths of the
bulb in the struck ship
J Mar Sci Technol
123
penetration in other directions. Test 309 presents an
opposite scenario, in which the striking ship slides along
the side of the struck ship, causing shallow but long pen-
etration. The penetrations in tests 301 and 313 extend
almost equally in both directions. While in the other tests
the calculation model predicted the penetration paths well,
in test 313 the penetration in the longitudinal direction is
underestimated.
The above analysis focused on the motions in the plane
of the water surface. When looking at the other motion
components, such as the rolling of the struck ship and the
pitching of the striking ship, the calculation model misses
the effects arising from the waves generated by the moving
ships. This situation is analysed by looking at the results of
test 309, where these effects were clearly present. The
calculation model considers only the roll and the pitch
caused by the vertical and the horizontal eccentricity
between the contact point and the centre of gravity.
However, the measured angular motions clearly exceed
these values, as shown in Fig. 12. As a consequence, the
penetration in a vertical direction is also underestimated in
the calculations; see Fig. 13. Some conclusions can be
drawn on the basis of the visual analysis of the videos of
the model-scale experiments. In these, the pitching of the
striking ship is due to the wave trough left behind by the
accelerating struck ship, and (through the contact between
the ships) this also increases the rolling of the struck ship.
5 Conclusions
The developed calculation model predicts the motions and
the forces in the plane of the water surface rather accu-
rately. The character of the contact forces is highly
dependent on the collision scenario, i.e. on the exact angle
and location of the collision. In an eccentric collision at
right angles, the contact force, and thus also the penetra-
tion, is transverse to the struck ship and has only a minor
longitudinal component. In collisions at small or large
angles, the extent of the damage to the side of the struck
ship is long and shallow. In all of the calculated scenarios
the forces are predicted well in the loading stage, while
during the unloading some deviations from the measure-
ments occur. This, as well as a somewhat unrealistic and
rapid drop in the force values in the stiction phase, indi-
cates that the interaction model cannot fully describe these
complex phenomena. However, these effects are small
when looking at the penetrations.
The longitudinal and the transverse penetration inside
the struck ship are properly predicted with the calculation
model, while they are slightly underestimated in the ver-
tical direction. This is due to the pitching of the striking
ship, as induced by complex wave patterns during the
collision. The effect of this is not considered in the cal-
culation model. This hydrodynamic interaction as well as
the effect of the forward speed of the struck ship is left for
future studies.
The model for evaluating the contact force could be
extended to consider ship-like side structures whose
deformation mechanisms are extremely complex in com-
parison to that of the side structure used in the model-scale
experiments. This, together with motion simulations, would
Fig. 12 Pitch of the striking
ship (a) and roll of the struck
ship (b)
Fig. 13 Penetration in the vertical direction in test 309. Note the
scaled zB axis
J Mar Sci Technol
123
improve the accuracy of collision analyses and thus allow
the crashworthinesses of different structural arrangements
to be enhanced.
Appendix 1: Scalar form of equations of motion
For numerical integration, the equation of motion
M½ 0
0 I½
_u_X
þ M½ X u
X I½ X
ÿ FR ÿ Fl
¼ F
G
ÿ FR ÿ Fl ð38Þ
is rearranged to obtain a more convenient form:
Ml
_u
_X
þ MX
l
h i u
X
ÿ T½ T K½
Rf gu½
þ K½ dxf gdu½
¼F
G
ÿ FR ÿ Fl ð39Þ
where the matrices have the following component form:
½Ml ¼
mþ ax 0 0 0 0 0
0 mþ ay 0 ayx 0 ayz0 0 mþ az 0 azy 0
0 axy 0 Ix þ axx 0 ÿIxz0 0 ayz 0 Iy þ ayy 0
0 azy 0 ÿIxz 0 Ix þ azz
2
6666664
3
7777775;
ð40Þ
MX
l
h i¼
0 ÿðmþ axÞr ðmþ axÞq 0 0 0
ðmþ ayÞr 0 ÿðmþ ayÞp 0 0 0
ÿðmþ azÞq ðmþ azÞp 0 0 0 0
0 0 0 0 I45 ÿI460 0 0 ÿI45 0 I560 0 0 I46 ÿI56 0
2
6666664
3
7777775
ð41Þ
with
I45 ¼ ðIzz þ azzÞr ÿ ðIzx þ azxÞpÿ Izyq;
I46 ¼ ðIyy þ ayyÞqÿ Iyzr ÿ Iyxp;
I56 ¼ Ixx þ axxpÿ Ixyqÿ Ixz þ axzr
and
½K ¼
0 0 0 0 0 0
0 0 0 0 0 0
0 0 ÿqgAW 0 qgAWxF 0
0 0 0 ÿGMTgm 0 0
0 0 qgAWxF 0 ÿGMLgm 0
0 0 0 0 0 0
2
6666664
3
7777775
where m is the structural mass, ai and aii are the transla-
tional and rotational added masses, AW is the waterplane
area, xF is the longitudinal centre of flotation, q is the water
density, g is the gravitational acceleration, GMT is the
transverse metacentric height, and GML is the longitudinal
metacentric height. The subscript characters in the mass
and inertia terms follow the common notation; a single
character refers to a value involved with translational
motions, and two characters refer to rotational motion or to
a coupling between two motion components.
The restoring force is divided into a constant part FRjtjevaluated at the beginning of the time increment at t = tj, and
into the change dFRjtjþ1in the force during the increment:
FRjtjþdFRjtjþ1¼ ½TT ½K fRg½u þ ½K fdxg½du ð42Þ
This split is necessary, as the restoring force depends on
the ship’s position with respect to the inertial coordinate
system given by the position vector R and Euler’s angles.
The increase in the force during the time increment is still
evaluated via the displacements in the local coordinate
system, but because of small angular displacements the
error will be negligible.
During the time increment, the matrices and vectors on the
left-hand side of Eq. 39 are updated several times within the
increment, while the right-hand side is kept constant.
Appendix 2: Test matrix for second and third sets
Test b
(deg)
Bulb LC(m)
mA
(kg)
mB
(kg)
u0(m/s)
FC,xA
(N)
FC,yA
(N)
ED,P
(J)
201 90 1 0.82 28.5 30.5 0.87 226 25 3.91
202 90 1 0.83 28.5 30.5 0.71 179 17 2.36
203 90 1 0.83 28.5 30.5 0.38 91 13 0.75
204 90 1 0.45 28.5 30.5 0.91 300 36 6.30
205 90 1 0.48 28.5 30.5 0.38 115 6 0.95
206 90 1 0.38 28.5 30.5 0.71 221 13 3.43
207 90 1 0.80 28.5 20.5 0.90 200 27 4.2
208 90 1 0.41 28.5 20.5 0.89 229 33 4.92
301 120 1 0.37 28.5 20.5 0.87 172 47 4.20
302 120 1 0.32 28.5 20.5 0.30 59 14 0.51
303 120 1 0.3 28.5 44.5 0.84 204 52 6.50
304 120 1 0.38 28.5 44.5 0.37 89 21 1.01
305 (sliding) 145 1 0.32 28.5 20.5 0.34 44 29 0.54
306 (sliding) 145 1 0.44 28.5 20.5 0.87 115 65 3.91
307 (sliding) 145 1 0.38 28.5 44.5 0.84 118 67 5.47
308 145 1 0.34 28.5 44.5 0.28 45 27 0.52
309 (sliding) 145 3 0.46 28.5 20.5 0.87 120 86 3.19
310 (sliding) 145 2 0.44 28.5 20.5 0.88 142 94 3.19
311 120 3 0.42 28.5 20.5 0.88 217 69 4.25
312 120 2 0.41 28.5 20.5 0.86 313 104 4.64
313 60 1 0.29 28.5 20.5 0.76 177 41 3.14
314 60 1 0.32 28.5 20.5 0.36 80 16 0.81
315 60 1 0.38 28.5 44.5 0.75 202 52 4.35
316 60 1 0.4 28.5 44.5 0.43 104 24 1.17
Absolute maximum values are presented for FC,xA and FC,y
A
m, ship mass; u0, contact velocity; ED,P, plastic deformation energy
J Mar Sci Technol
123
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[P4] Tabri K., Matusiak J., Varsta P. (2009) Sloshing interaction in ship collisions –
An experimental and numerical study, Journal of Ocean Engineering, 36, pp.
1366-1376.
Sloshing interaction in ship collisions—An experimental and numerical study
Kristjan Tabri , Jerzy Matusiak, Petri Varsta
Helsinki University of Technology, Department of Applied Mechanics, Marine Technology, P.O. Box 5300, 02015 TKK, Finland
a r t i c l e i n f o
Article history:
Received 25 March 2009
Accepted 30 August 2009Available online 10 September 2009
Keywords:
Collision dynamics
Sloshing interaction
Model-scale experiments
a b s t r a c t
Sloshing interaction in ship collisions is studied both experimentally and numerically. The rapid change
in ship motions resulting from contact loading in collisions initiates violent sloshing inside partially
filled liquid tanks on board. Sloshing affects the collision dynamics and reduces the amount of energy
available for structural deformations. An understanding of this interaction phenomenon was obtained
by a series of model-scale experiments, in which a striking ship, with two partially filled tanks, collided
with an initially motionless struck ship without any liquid on board. The influence on the structural
deformation of the fluid mass in the tanks and the velocity of the collision was studied. Numerical
simulations of the cases were performed, combining a linear sloshing model with a theoretical collision
model. The simulation model was validated with experimental results and good agreement was
achieved in the case of medium filling levels in the tanks, while correspondingly the deformation
energy was overestimated by up to 10% for shallow filling levels.
& 2009 Elsevier Ltd. All rights reserved.
1. Introduction
Large-scale ship collision experiments (Wevers and Vrede-
veldt, 1999) revealed that fluid sloshing in partially filled tanks
had a significant effect on the dynamics of ship collisions (Tabri
et al., 2009a, b). These large-scale experiments between two river
tankers aimed to study the performance of different structural
concepts and to gain insight into the collision dynamics. To obtain
the desired ship drafts, some tanks of the ships were only partially
filled, making it possible for water sloshing to occur. Here, we
exploit the outcomes of two large-scale experiments—Y-core test
with a large number of partially filled tanks on board of both
ships (Wevers and Vredeveldt, 1999) and X-core tests with only
insignificant amount of water in partially filled tanks (Wolf, 2003).
It is known that under external dynamic excitation partially filled
tanks with fluids are prone to violent sloshing, in which the
eigenperiods of the fluid oscillations depend on the depth of the
fluid and on the horizontal dimensions of the tank. The effect of
sloshing on collision dynamics is through time varying loads
exerted on the containing ship structures and by storing part of
the kinetic energy. In a collision, the sloshing is initiated by a rapid
change in the motion of the ship. If the amount of sloshing fluid is
significant compared to the total mass of the ship, the distribution
of energy components in the collision is changed. Conventional
calculation methods used to simulate ship collisions do not
normally include the sloshing interaction.
Numerical simulations of large-scale collision experiments
(Tabri et al., 2009a, b) included the sloshing interaction for
the first time, exploiting a mathematical model developed by
Graham and Rodriguez (1952). This method is based on a linear
potential flow theory. The correspondence between the results of
the numerical simulations and those of the experiments was
found to be good, proving the suitability of the sloshing model for
collision simulations. In that large-scale experiment the ratio
of the depth of the water to the dimensions of the tank in
the sloshing direction was above 0.2, which is considered a
lower bound for the applicability of the linear sloshing theory
(Chen et al., 2008). The calculations revealed that the sloshing
‘‘stored’’ part of the kinetic energy and thus reduced the amount
of energy available for structural deformations (Tabri et al.,
2009a, b). The sloshing was more significant in the striking ship
compared to the struck ship due to larger kinetic energy involved.
Zhang and Suzuki (2007) studied the sloshing interaction during a
collision numerically using the Lagrangian and arbitrary Lagran-
gian–Eulerian methods. On that basis, they concluded that the
linear sloshing model underestimates the energy available for
structural deformations by about 17% compared to more advanced
numerical models, which include non-linear effects. However, in
their simulations the tank was filled to up to 95% of its height and,
therefore, the influence of roof impacts might be significant,
presenting a rather unfavorable setup for the linear model.
In the large-scale experiments the sloshing took place in a
number of partially filled tanks on board both ships and, thus, the
concurrent interaction of several sloshing processes made it
cumbersome to evaluate the pure sloshing phenomenon. To gain a
deeper insight into this important phenomenon of sloshing, it was
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journal homepage: www.elsevier.com/locate/oceaneng
Ocean Engineering
0029-8018/$ - see front matter & 2009 Elsevier Ltd. All rights reserved.
doi:10.1016/j.oceaneng.2009.08.017
Corresponding author. Tel.: +35894513392; fax: +35894514173.
E-mail addresses: [email protected], [email protected] (K. Tabri).
Ocean Engineering 36 (2009) 1366–1376
ARTICLE IN PRESS
decided to carry out unique model-scale collision experiments, in
which the direct influence of sloshing could be evaluated by
comparing the results of tests with and without water in tanks.
Hereafter, these experiments are referred to as wet or dry tests.
The focus was placed on symmetric collision tests, where the
striking ship model collided at a right angle to the amidships
of the struck ship model. This simplified the influence of ship
motions on the sloshing. Two tanks were installed on board the
striking ship model, while the struck model did not carry any
liquids on board.
In the present study, the linear sloshing model of Graham
and Rodriguez (1952) is implemented in the collision model
presented by Tabri et al. (2009), where the contact force model
was improved compared to that of Tabri et al. (2009a, b). The aim
of the study is to validate the calculation model with different
water-filling levels and collision velocities, thus providing
a validated numerical calculation model applicable for the
structural design of ships in the conceptual stage.
2. Modeling of sloshing during collision
In a collision, the momentum of the striking ship is transmitted
through contact loading to the initially motionless struck ship.
When these two ships collide, their masses remain constant and
the change in momentum is caused by the change in velocity. In
collision dynamics, the total energy is composed as a sum of the
kinetic energy EK, deformation energy ED, and of the work done to
overcome the hydrodynamic forces. For symmetric collisions
the deformation energy in respect to the initial kinetic energy
of the striking ship, EKA|t=0, can traditionally be estimated by
Minorsky’s (1959) formula:
ED
EAK jt ¼ 0
¼aB2þmB
ðaA1þmAÞþðaB2þmBÞ; ð1Þ
where the superscript A refers to the striking ship and B to the
struck ship. Term a1 refers to the hydrodynamic added mass
associated with the surge motion, a2 is the sway added mass, and
m is the mass of the ship. This model considers the inertia forces
resulting from the total ship mass m and the hydrodynamic added
mass a. Other components of the hydromechanical forces, such as
frictional resistance, the restoring force, and the hydrodynamic
damping, are excluded.
If one of the ships, or both, has partially filled liquid tanks with
a free surface, the fluid inside the tank starts to slosh during the
collision – see Fig. 1 – and interacts with the containing structure
over a longer time span than would a rigidly fixed mass. Thus, the
participation of the sloshing mass in the momentum transmission
is delayed and Eq. (1) is not valid any more.
Time domain simulation models such as those of Petersen
(1982), Brown (2002), Tabri et al. (2009a, b), etc. also consider
only fixed masses and no fluids on board, but as they evaluate the
whole time history of ship motions, the sloshing interaction can
be included in these.
2.1. Equations of motion of ‘‘dry’’ ship
In Tabri et al. (2009b) the equations of motion of a dry ship
subjected to an external force F and moment G were presented in
a form convenient for numerical simulations:
½Mm_u_X
þ½MOm
u
X
¼F
G
ÿ Fm: ð2Þ
The scalar form of Eq. (2) considering all six degrees of freedom
is presented in Appendix A. Besides the mass and inertia terms,
the matrix [Mm] also contains the added mass coefficients
describing the acceleration component of the radiation force
Fm, which makes it possible to subtract it from the total force
F (Tabri et al., 2009b). Non-linear cross-coupling velocity terms,
together with appropriate mass, inertia, and added mass
coefficients, are given in [MmO]. The translational velocity
ui=uiii+viji+wiki and angular rate X=piii+qiji+riki are given
in a local ship-related coordinate system Oixiyizi; see Fig. 2. The
superscript i=A, B indicates that a definition or description is
common for both ships. The ship’s position with respect to an
inertial coordinate system O0x0y0z0 is defined by a position vector
Ri, and the orientation by Euler’s angles ½u ¼ ¼ f y ch iT
between Oixiyizi and the horizontal body axes OixiZizi. The
horizontal body axes are subjected to translations only and thus
their orientation with respect to the inertial coordinate system
remains constant throughout the collision.
Here it is assumed that the effect of sloshing on the ship’s mass
center is negligible and that thus the origins Oi of these two local
coordinate systems are both fixed to the initial mass center of
gravity of the ship. This simplification is believed to have only a
minor influence on the ship’s response.
2.2. Extended equations of motion to consider sloshing
The linear sloshing model replaces the fluid mass mT in a tank
with a single rigidly fixed mass mR and with a number of
discrete mass-spring elements mn; see Fig. 3. Each oscillating
mass corresponds to one sloshing mode. The effect of a spring-
mass element on the sloshing force decreases rapidly with an
increase in the mode number. It was shown by Tabri et al. (2009a)
that in collision applications it suffices if the total sloshing
response is derived as a superposition of responses of three lowest
Fig. 1. Sloshing of liquid in partially filled tank. Fig. 2. Definition of coordinate systems and position vectors.
K. Tabri et al. / Ocean Engineering 36 (2009) 1366–1376 1367
ARTICLE IN PRESS
sloshing modes i.e. the number of oscillating masses in a single
tank is three. The equation of translational motion for a single
massmn connected to the tank walls by a spring of stiffness kn and
a damper with a damping coefficient cn is
mn _unþcnðun ÿ uÞþknðxn ÿ xÞ ¼ 0; ð3Þ
where in a local coordinate system u, u, and x present the
acceleration, velocity, and displacement of the tank, and un, un,
and xn, correspondingly, those of the oscillating mass on the
direction of a certain motion component. Here it is assumed that
the angular motions do not influence the sloshing dynamics and
vice versa. Thus, the sloshing interaction is considered only for
sway and surge motions presented in the local coordinate system
of the ship. On the basis of Graham and Rodriguez (1952) and
Fig. 3, the formulations for the mass mn and stiffness kn of nth
oscillating mass element are
mn ¼mT8lT
p2ð2nÿ 1Þ3hW
!
tanh ð2nÿ 1ÞphW
lT
; ð4Þ
kn ¼o2nmn ¼mT
8g
p2ð2nÿ 1Þ2hW
tanh ð2nÿ 1ÞphWlT
2
; ð5Þ
where hW is the water height in the tank and lT is the dimension of
the tank in the sloshing direction. Correspondingly, the damping
coefficient cn is
cn ¼on
pd; ð6Þ
whereon is the eigenfrequency of the nth oscillating mass and d is
the logarithmic decrement.
The total number of degrees of freedom, Q, associated with a
ship is the sum of the rigid degrees of freedom, NR, and the total
number of oscillating masses J
Q ¼NRþ J ð7Þ
and J=3I where I is the number of partially filled fluid tanks.
The equations of motion (Eq. (2)) are now extended to consider
the sloshing interaction. Three new QQ matrices [Mn], [Cn], and
[Kn], containing the mn, cn, and kn terms and the vectors un, un,
and xn containing the oscillating degrees of freedom in Eq. (3),
are formed and combined with Eq. (2). Therefore, six J J zero-
matrices, denoted by [0], are formed and added to [Mm] and [MmO]
in order to account for the oscillating degrees of freedom. As a
result this gives new equations of motion with Q degrees of
freedom:
½Mm ½0
½0 ½0
" #
þ½Mn
! _u_X
_un
8
>
<
>
:
9
>
=
>
;
þ½MO
m ½0
½0 ½0
" #
þ½Cn
! u
X
un
8
>
<
>
:
9
>
=
>
;
þ½Kn
fxg
½u
fxng
¼F
G
ÿ Fm: ð8Þ
Eq. (8), containing in addition to rigid body motions also six
oscillating degrees of freedom, is presented in the scalar form in
Appendix B. Displacements are evaluated as the projections of the
position vectors R and Rn to the local coordinate system Oixiyizi
giving x=[T]TR and xn=[T]TRn, where the orthogonal
matrix of transformation (Clayton and Bishop, 1982) is, in
component form
x0
y0
z0
8
>
<
>
:
9
>
=
>
;
¼ ½T
xi
yi
zi
8
>
<
>
:
9
>
=
>
;
¼
cosccosy coscsinysinfÿ sinccosf coscsinycosfþsincsinf
sinccosy sincsinysinfþcosccosf sincsinycosfÿ coscsinf
ÿsiny cosysinf cosycosf
2
6
4
3
7
5
xi
yi
zi
8
>
<
>
:
9
>
=
>
;
: ð9Þ
The time integration of the equations of motion is based on an
explicit 5th-order Dormand–Prince integration scheme (Dormand
and Prince, 1980), which is a member of the Runge-Kutta family of
solvers. For time-efficient integration, the forces on the right-hand
side of the Eq. (8) are kept constant during time step dt, while the
motions on the left-hand side are updated in every
sub-increment. Also the matrix [MmO] is updated in every sub-
increment as it contains the velocity terms. Definitions for the
external forces F and G resulting from the surrounding water and
the contact between the ships are provided in Tabri et al. (2009b).
The solution of Eq. (8) provides motions at the end of the
integration increment. There, the new position of the ship with
respect to the inertial frame is evaluated by time integrating over
the velocities in the local coordinate system and transforming the
translational displacement increments that are obtained to the
inertial frame by the matrix [T]. These increments are added to
the position vector Ri. The orientation is updated by adding the
angular increments to the Euler’s angles. Given the positions and
the orientations, the penetration between the ships and thus, the
contact force is calculated. Also the other external forces on the
right-hand side are updated.
The equations of motion (Eq. (8)) are established for each ship.
These equations are treated separately during the time integration
increment and the mutual contact force is updated after each
integration step in order to maintain the kinematic connection.
3. Model-scale collision experiments with
sloshing interaction
An understanding of the sloshing physics was gained through a
series of model-scale experiments carried out in the test basin
of Helsinki University of Technology; see Fig. 4. An elaborate
description of the ship models and their scaling is given in
Tabri et al. (2007), where symmetric collision experiments
without sloshing interaction are presented. Here, the additional
modifications to the test setup and the measuring system
necessary for the wet tests are described.
Fig. 3. Simplified discrete mechanical model for sloshing.
K. Tabri et al. / Ocean Engineering 36 (2009) 1366–13761368
ARTICLE IN PRESS
3.1. Test setup
The model tests were designed to be physically similar to
the large-scale experiments (Wevers and Vredeveldt, 1999). The
feasible scaling factor based on the Froude scaling law was l=35
(Tabri et al., 2007).
To maintain the geometric similarity to the ships used in the
large-scale experiments with a Y-core structure (Wevers and
Vredeveldt, 1999), the models had the following main dimen-
sions: length LA=LB=2.29m, depth DA=DB=0.12m, and breadth
BA=0.234m for the striking ship and BB=0.271m for the struck
ship. Two fluid tanks were installed on board the striking ship
model, as shown in Figs. 4 and 5. There, the side height of the
tanks was greater than that of the ship used in the large-scale test,
so as to reduce the effect of roof impacts. In the model tests the
length of each tank was about 1/4 of the total length of the model.
The length of the tanks was exaggerated compared to the large-
scale tests to get clearer picture of the phenomena. With
these tank dimensions the striking ship resembled a large crude
oil carrier or a novel river tanker where increased crashworthiness
of the side structures allows larger and longer cargo tanks
(van de Graaf et al., 2004).
The striking ship model was equipped with an axi-symmetric
rigid bulb, the dimensions of which are given in Fig. 6. In the
struck ship model a block of polyurethane foam was installed at
the location of the collision. The crushing strength of the foamwas
s=0.121MPa (Ranta and Tabri, 2007). The model-scale
force–penetration curve was built up on the basis of this value
and the shape of the contact surface. This force–penetration curve,
presented in Fig. 7, corresponds well to that measured during the
large-scale experiment with Y-core side structure (Tabri et al.,
2009a, b) and scaled down with l=35. Thus it can be stated that
the dynamic similarity between the model-scale and large-scale
experiments was roughly maintained.
The striking model was connected to the carriage of the test
basin and it was accelerated smoothly to the desired collision
velocity u0A to prevent sloshing before the collision. There was a
connection between the carriage and the model situated close to
the model’s center of gravity in order to avoid initial pitch motion.
Fig. 6. Dimensions of the bulb used in the model-scale experiments.
Fig. 5. Test setup with two sloshing tanks.
Fig. 4. Ship models with sloshing tanks on board the striking ship.
K. Tabri et al. / Ocean Engineering 36 (2009) 1366–1376 1369
ARTICLE IN PRESS
The struck model was fixed to the basin with line reels. Both
models were released just before the contact.
All six motion components of the striking ship were recorded
with respect to an inertial coordinate system with a Rodym DMM
non-contact measuring system. This required as an input value
the location of the center of gravity of the ship model. This value
was considered constant during a collision. As discussed in Tabri
et al. (2007), the post-processing in the Rodym system caused a
time lag in the measured signals. In order to have precise
synchronization with other measured signals, the longitudinal
acceleration of the striking ship was also measured with a
Schaevitz LSBP-1 accelerometer. All six motion components of
the struck ship model were recorded by a Crossbow DMU-FOG
motion pack installed on board the ship. This measuring system
also provided time histories for the stabilized pitch and roll
angles.
The free surface elevation in the sloshing tanks was measured
with four resistive wave probes made of steel wire. Three probes
were installed in the fore tank and one in the aft tank; see Fig. 5.
The rigid bulb was connected to the striking ship via a force
transducer (Tabri et al., 2007). The force was measured only in the
longitudinal direction with respect to the striking ship model
as the other components were expected to be negligible in
symmetric collisions. All the signals were recorded with a
sampling rate of 1.25 kHz, except those with the Rodym system,
where the rate was 125Hz.
Given the ship motions, the penetration time history was
calculated based on the relative position between the ships, see
Eq. (3) in Tabri et al. (2007). Combining the measured contact
force and the penetration history results in a force–penetration
curve, and the area under that curve gives the deformation energy
ED at the end of the collision process.
3.2. Test matrix
Four different loading conditions of the striking ship model
were tested. These were based on the amount of water in the
sloshing tanks, which varied from 21% to 47% of the total mass of
the model. The tanks were filled in such a way that the model
maintained an even keel condition, which resulted in slightly
different amounts of water in the fore and aft tanks. Each loading
condition was tested with three different collision velocities:
0.4, 0.7, and 1.0m/s. Table C1 in Appendix C presents the test
matrix for the different collision scenarios of the wet tests.
The lightship mass of the striking ship without water was
mLSA=22.1 kg, except in the last three tests, where it was increased
to 26.3 kg. The relative filling level, defined as a ratio between
the water depth and the tank length hW/‘T, varied from 0.08 to
0.17. The mass of the struck ship was 30.5 kg throughout
the tests.
To gain a deeper understanding of the influence of the sloshing
interaction on the collision, most of the tests were repeated
with a rigid mass replacing the water in tanks; see Table C2 in
Appendix C for the full test matrix. These masses were positioned
in such a way that the position of the ship’s center of gravity
remained unchanged.
The physical parameters of the ship models during the tests are
presented in Table 1. The table presents the drafts, total masses,
vertical height of the mass center of gravity KG measured from the
base line of the model, longitudinal of gravity LCOG measured from
the amidships, the radii of inertia kxx and kyy in relation to the
x and y axes, and the calculated values for the ships’ added
masses. The radii of inertia with respect to the z-axis, kzz, are
assumed to be kzz=kyy. The radii are evaluated on the assumption
of a still water level in the tanks. The non-dimensional added
mass coefficients m are based on
mj ¼ limo-1
ajðoÞ
r=;with j¼ 2;3; ð10Þ
for translational motions such as sway and heave, and
correspondingly
mj ¼ limo-1
ajðoÞ
r= k2j
;with j¼ 4;5;6; ð11Þ
for rotational motions such as roll, pitch, and yaw. The water
density is denoted by r and the volumetric displacement of the
ship by r. The frequency-dependent added masses a(o) are
calculated with strip theory (Journee, 1992). The coefficients were
first evaluated in a coordinate systemwith its origin located at theFig. 7. Measured force–penetration curves in the large-scale and model-scale
experiments. The large-scale measurement is scaled to model scale with l=35.
Table 1
Physical parameters of the ship models during the test.
Model Draft (cm) Total mass (kg) KG (cm) LCOG (cm) kxx (cm) kyy (cm) m2 (%) m3 (%) m4 (%) m5 (%) m6 (%)
Striking (wet) 5.5 28.1 8.1 ÿ9.4 18 90 23 237 8 99 10
Striking (wet) 6.5 33.1 8.9 ÿ9.0 18 94 27 203 7 77 11
Striking (dry) 6.5 33.5 8.7 ÿ8.8 18 94 27 203 7 77 11
Striking (wet) 6.75 35.0 7.5 ÿ6.6 18 82 28 195 6 99 15
Striking (dry) 6.75 35.1 6.8 ÿ6.7 18 82 28 195 6 99 15
Striking (wet) 7.5 40.7 7.0 ÿ7.0 18 78 31 179 6 99 18
Striking (dry) 7.5 41.1 6.8 ÿ6.9 18 78 31 179 6 99 18
Struck 6 30.5 7.3 0 17 69 21 238 14 184 17
K. Tabri et al. / Ocean Engineering 36 (2009) 1366–13761370
ARTICLE IN PRESS
water plane and transferred to Oixiyizi considering the distance
between the water plane and the center of gravity. The surge
added mass m1 for both ship models is taken as being 5% of the
total mass of the model.
4. Sloshing interaction in collision model tests
First, emphasis is laid on the effect of the sloshing interaction
on the collision dynamics. The experimental results from a
wet and a dry test are compared. Typical force time histories of
a wet experiment (S1-V7) and of an equivalent dry test (N1-V7)
are shown in Fig. 8. In the wet experiment the water in the tanks
accounted for 37% of the total ship mass and the filling level hW/‘Tin the fore tank was 0.08 and that in the aft tank 0.13. The collision
speed in both experiments was 0.7m/s.
A comparison of the contact force histories of these two tests
indicates that the sloshing interaction lowered the magnitude of
the first force peak. Thus, as a result of the sloshing, the striking
ship seems to behave like a lighter ship. In the wet test in Fig. 8a,
after the first peak the contact was lost for about 0.1 s and the
ships came into contact again as the striking ship regained speed
as a result of the sloshing wave hitting the front bulkhead of each
tank; see Fig. 9. There, the time histories of free surface elevation
are presented for the probes S1 and S4, which are close to the
front bulkheads of both tanks.
Surge motions of the striking ship are presented in Fig. 10 and
these are in obvious correlation with the fluid motions inside the
tanks. The effect of sloshing was not that obvious on the other
motion components.
Fig. 9 shows that the main period of sloshing is about 1.7 s, a
value which is an order of magnitude longer than the duration of
the first force peak, being about 0.1 s. This fact reveals that the
interaction of the water is delayed with respect to other major
phenomena, such as the contact between the ships. Fig. 9
indicates that the sloshing motion is damped out strongly during
the first period, after which the sloshing amplitude continues to
decrease slowly. The damping during the first contact force
peak clearly affects the collision dynamics. The damping is
physically determined by the logarithmic decrement, which
can be calculated by comparing the amplitudes of the first two
consecutive sloshing waves:
d¼ lnX1
X2
; ð12Þ
Fig. 8. Force history of a wet and an equivalent dry experiment (tests S1-V7 and
N1-V7; see Tables C1 and C2).
Fig. 9. Free surface elevation close to the front bulkhead of fore and aft tanks in
test S1-V7.
Fig. 10. Time histories of the acceleration (a), velocity and displacement (b) of the
striking ship model in test S1-V7.
K. Tabri et al. / Ocean Engineering 36 (2009) 1366–1376 1371
ARTICLE IN PRESS
where X1 indicates the amplitude of the first wave and
correspondingly X2 the second wave.
The decrement depends on the relative filling level hW/‘T and
on the initial velocity of the striking ship model, as shown in
Fig. 11. There, the calculated damping decrement d is presented for
each wet test and 2nd order polynomial regression curves are
drawn for different initial velocities. The 10% error bounds
presented with dashed lines and the R2 values reveal relatively
large deviation of the evaluated damping values. The sloshing is
damped out faster in the case of a high initial velocity and a low
relative filling level. It should be noted that this damping has only
a minor effect on the maximum deformation energy in such a
transient contact process. Its importance increases when dealing
with the ship motions after the contact between the ships has
been lost.
From the ship motions the relative displacement at contact,
referred to as penetration, is calculated and its time history is
presented in Fig. 12a. In the dry test the maximum penetration is
deeper than that of the wet test. However, the second penetration
peak is higher in the case of the wet test as a result of the sloshing
interaction. The lower first penetration peak in the wet test givesFig. 11. Logarithmic damping decrement d versus relative filling level and initial
velocity as a parameter.
Fig. 12. Penetration (a) and relative deformation energy (b) in tests S1-V7
and N1-V7.
Fig. 13. Deformation energy (a) and relative deformation energy (b) as a function
of mass ratio.
K. Tabri et al. / Ocean Engineering 36 (2009) 1366–13761372
ARTICLE IN PRESS
a smaller force value and thus the relative deformation
energy ED/EKA|t=0 is only about 80% of that of the dry test; see
Fig. 12b.
5. Influence of ship and water mass relation
The results of the numerical simulations are compared to those
of the experiments. The numerical simulations are performed
using the same parameter values as in the experiments and
the damping coefficients cn – see Eq. (6) – in the wet tests are
evaluated using the decrement values taken from Fig. 11. The
sensitivity of energy to the ship and water mass is presented in
Figs. 13 and 14. There, the values are taken from the tests with a
constant initial collision speed u0A=0.7m/s.
The sensitivity study is based on two different approaches.
First, the lightship mass is kept constant as mLSA=22.1 kg and the
amount of water in the tanks varies; see Fig. 13. There, the
structural deformation energy and the relative deformation
energy versus the mass ratio of the ships are presented both for
dry and wet tests. Fig. 13 demonstrates that the deformation
energy in the wet tests is only about 80% of that of the dry tests.
The correspondence between the calculated and the measured
values is, on average, good. However, some overestimation in
energy exists in the case of the wet tests, possibly as a result of the
limitations of the sloshing model based on the linear potential
flow theory. This discrepancy decreases asmA and the water depth
in the tanks increases.
This shortcoming of the sloshing model does not appear when
we apply it to calculate the Y-core large-scale experiment
(Tabri et al., 2009a, b); see Fig. 13b. In that large-scale collision
experiment the relative filling levels in the tanks were higher,
ranging from 0.1 to 0.4, and the calculations agree well with the
measurements. The relative deformation energy of the Y-core
experiment does not follow the trend of the wet model-scale tests
as the sloshing took place in several tanks in both ships, yielding a
greater energy reduction. In the other large-scale test with the
X-core side structure (Tabri et al., 2009a, b), the situation was
different as there was only a negligible amount of water in the
tanks and thus the relative energy corresponds well to the trend of
the dry model-scale tests. However, the numerical simulations
slightly overestimate the deformation energy in the X-core test.
This previous presentation combines the effect of the ships’
total masses and that of sloshing on the deformation energy.
The pure effect of sloshing becomes more obvious when the tests
are compared in which the ratio mA/mB and the initial velocity are
constant, and the water or rigid mass in the tanks varies. This
comparison is given in Fig. 14, where the results of the tests S1-V7,
S4-V7, N1-V7, and N4-V7 are presented. As expected, in the
dry tests the relative deformation energy remains unchanged as
the mass ratio is constant. In the wet tests the relative
deformation energy is reduced as the amount of the sloshing
water increases.
6. Influence of initial collision velocity
The effect of the collision velocity on the deformation energy is
studied by comparing the test results where all the parameters
except the initial velocity are kept constant. The comparison is
presented in Fig. 15.
Obviously, the velocity has a strong influence on the deforma-
tion energy in Fig. 15a. However, the relative deformation energy
remains almost unchanged, as revealed by Fig. 15b. Thus, it can be
concluded that the relative amount of kinetic energy ‘‘stored’’ in
Fig. 14. Relative deformation energy as a function of sloshing mass (mA/mB=
const.).
Fig. 15. Deformation energy (a) and relative deformation energy (b) versus initial
velocity of the striking ship.
K. Tabri et al. / Ocean Engineering 36 (2009) 1366–1376 1373
ARTICLE IN PRESS
the sloshing does not depend on the initial velocity of the striking
ship. In addition, the deformation energy in the wet tests is still
about 80% of that in the dry tests and the calculations continue to
overestimate the deformation energy in the wet tests. This
overestimation seems to be independent of the initial collision
velocity.
7. Conclusions
The test setup developed for the wet tests proved to give
reliable and repeatable results. The measuring equipment
provided the opportunity to obtain valuable data for the under-
standing of the sloshing phenomenon and for the validation of the
calculation models.
The results of the model-scale experiments emphasized the
importance of sloshing interaction for collision dynamics. The
influences of sloshing on the collision were due to the delayed
transmission of momentum. Thus, the ship carrying liquid
appears lighter and the resulting collision damage becomes less.
The deformation energy in the wet test was only about 80% of that
in the dry tests. Therefore, for the precise estimation of
deformation energy in ship collision the sloshing effects have to
be included. The relation between the masses of the participating
ships had a significant effect on the deformation energy both in
the dry and in the wet tests. The initial collision speed of the
striking ship also had a strong influence on the deformation
energy. However, this was not the case when the relative
deformation energy was considered, as it remained almost
unchanged as the velocity increased.
The calculation method that was developed overestimated the
deformation energy by up to 10% in the case of the wet model-
scale tests, but in the case of a large-scale wet test the predictions
agreed well with the measurements. This overestimation is
attributed to low relative filling levels, at which the linear
sloshing model is at the boundary of its validity. In the large-
scale experiment the filling levels were higher and the predictions
agreed well. Thus it can be concluded that this discrete
mechanical model of the sloshing gives satisfactory results for a
certain limited range of water depths. However, more advanced
CFD methods could enlarge the range of validity and also include
the different geometries and structural arrangements inside the
tanks.
More advanced calculations should provide the distribution of
the sloshing pressure on the containing structure and thus, allow
the strength of the tank walls to be analyzed. As the large amount
of energy contained in sloshing is transmitted through the walls,
the sloshing loads due to the collision could exceed the design
sloshing loads of normal seagoing conditions.
Appendix A. Scalar form of equations of motion of a dry ship
The scalar form of equation of motion is based on the Newton’s
law and is given in the local coordinate system as (Clayton &
Bishop, 1982)
mð _uþqwÿ rvÞ ¼ Fx
mð _vþruÿ pwÞ ¼ Fy
mð _wþpvÿ quÞ ¼ Fz
Ix _p ÿ Ixy _q ÿ Ixz _rþðIzr ÿ Izxpÿ IzyqÞqÿ ðIyqÿ Iyzr ÿ IyxpÞr¼Mx
ÿIyx _pþ Iy _q ÿ Iyz _rþðIxpÿ Ixyqÿ IxzrÞr ÿ ðIzr ÿ Izxpÿ IzyqÞp¼My
ÿIzx _p ÿ Izy _qþ Iz _rþðIyqÿ Iyzr ÿ IyxpÞpÿ ðIxpÿ Ixyqÿ IxzrÞq¼Mz
:
8
>
>
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
>
>
:
ðA:1Þ
is presented in matrix form as
½M 0
0 ½I
" #
_u_X
þ½MX u
X ½IX
( )
¼F
G
: ðA:2Þ
For numerical integration, Eq. (A.2) is rearranged to obtain a
more convenient form by introducing matrices [Mm] and [MmO],
which contain the added mass and the non-linear acceleration
terms:
½Mm_u_X
þ½MOm
u
X
¼F
G
ÿ Fm; ðA:3Þ
As the acceleration component of the radiation force Fm is
already included through the matrices in the left-hand side, it
should be subtracted from the right-hand side. The matrices have
the following component form:
½Mm ¼
mþa1 0 0 0 a15 0
0 mþa2 0 a24 0 a26
0 0 mþa3 0 a35 0
0 a42 0 Ixþa44 0 ÿIxz
a51 0 a53 0 Iyþa55 0
0 a62 0 ÿIxz 0 Ixþa66
2
6
6
6
6
6
6
6
6
6
4
3
7
7
7
7
7
7
7
7
7
5
;
ðA:4Þ
½MOm ¼
0 ÿðmþa1Þr ðmþa1Þq 0 0 0
ðmþa2Þr 0 ÿðmþa2Þp 0 0 0
ÿðmþa3Þq ðmþa3Þp 0 0 0 0
0 0 0 0 I45 ÿI46
0 0 0 ÿI45 0 I56
0 0 0 I46 ÿI56 0
2
6
6
6
6
6
6
6
6
6
4
3
7
7
7
7
7
7
7
7
7
5
;
ðA:5Þ
with
I45 ¼ ðIzþa66Þr ÿ ðIzxþa64Þpÿ Izyq;
I46 ¼ ðIyþa55Þqÿ Iyzr ÿ Iyxp;
I56 ¼ ðIxþa44Þpÿ Ixyqÿ ðIxzþa46Þr:
and the following concept is used to abbreviate the definitions for
moments of inertia:
Ix ¼ rðy2þz2ÞdV ;
Ixy ¼ rxydV ;
where V is the volume of the body. For the sake of clarity, the
subscripts 1, 2, 3, 4, 5, 6 are used to denote the added masses for
surge, sway, heave, roll, pitch, and yaw, respectively.
Appendix B. Scalar form of extended equations of motion of a
wet ship
The equations of motion of a wet ship are defined as
½Mm ½0
½0 ½0
" #
þ½Mn
! _u_X
_un
8
>
<
>
:
9
>
=
>
;
þ½MO
m ½0
½0 ½0
" #
þ½Cn
! u
X
un
8
>
<
>
:
9
>
=
>
;
þ½Kn
fxg
½u
fxng
¼F
G
ÿ Fm; ð13Þ
The mass matrix [Mm] includes the total ship mass and inertia
without distinguishing between the lightship mass mLS and the
oscillating masses mn. The sloshing mass matrix [Mn] removes the
K. Tabri et al. / Ocean Engineering 36 (2009) 1366–13761374
ARTICLE IN PRESS
oscillating masses from the total mass matrix and assigns them to
appropriate sloshing degrees of freedom. The sloshing mass
matrix is defined as
½Mn ¼
ÿP3
n ¼ 1 mn;x 0 0 0 0 0 0 0
0 ÿP3
n ¼ 1 mn;y 0 0 0 0 0 0
^ ^ & ^ ^ ^ ^ ^ ^
0 0 m1;x 0 0 0 0 0
0 0 0 m2;x 0 0 0 0
0 0 0 0 m3;x 0 0 0
0 0 0 0 0 m1;y 0 0
0 0 0 0 0 0 m2;y 0
0 0 0 0 0 0 0 m3;y
2
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
4
3
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
5
ðB:1Þ
The subscript indices x and y denote surge and sway,
respectively. The degrees of freedom associated with heave and
angular motions are truncated for the sake of brevity as it is
assumed that they are not affected by the sloshing.
According to Eq. (3) each mass element appears twice for a
corresponding degree of freedom in [Cn] and [Kn], accounting first
for the rigid body motion and second for the oscillatory motion.
Considering this, the damping matrix is defined as
½Cn ¼
P3n ¼ 1 cn;x 0 ÿc1;x ÿc2;x ÿc3;x 0 0 0
0P3
n ¼ 1 cn;y 0 0 0 ÿc1;y ÿc2;y ÿc3;y
^ ^ & ^ ^ ^ ^ ^ ^
ÿc1;x 0 c1;x 0 0 0 0 0
ÿc2;x 0 0 c2;x 0 0 0 0
ÿc3;x 0 0 0 c3;x 0 0 0
0 ÿc1;y 0 0 0 c1;y 0 0
0 ÿc2;y 0 0 0 0 c2;y 0
0 ÿc3;y 0 0 0 0 0 c3;y
2
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
4
3
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
5
;
ðB:2Þ
and the stiffness matrix as
½Kn ¼
P3n ¼ 1 kn;x 0 ÿk1;x ÿk2;x ÿk3;x 0 0 0
0P3
n ¼ 1 kn;y 0 0 0 ÿk1;y ÿk2;y ÿk3;y
^ ^ & ^ ^ ^ ^ ^ ^
ÿk1;x 0 k1;x 0 0 0 0 0
ÿk2;x 0 0 k2;x 0 0 0 0
ÿk3;x 0 0 0 k3;x 0 0 0
0 ÿk1;y 0 0 0 k1;y 0 0
0 ÿk2;y 0 0 0 0 k2;y 0
0 ÿk3;y 0 0 0 0 0 k3;y
2
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
4
3
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
5
:
ðB:3Þ
Table C1
Collision experiments with sloshing.
Name u0A (m/s) mLS
A (kg) Water mass hW/‘T Amount of water (%) mA/mB (–) ED
fore (kg) aft (kg) fore (–) aft (–) exp. (J) calc. (J)
S1-V4 0.4 22.1 5.0 8.0 0.08 0.13 37 1.15 1.1 1.2
S1-V7 0.7 22.1 5.0 8.0 0.08 0.13 37 1.15 3.3 3.6
S1-V10 1.0 22.1 5.0 8.0 0.08 0.13 37 1.15 6.7 7.3
S2-V4 0.4 22.1 8.4 10.5 0.14 0.17 47 1.34 1.3 1.2
S2-V7 0.7 22.1 8.4 10.5 0.14 0.17 47 1.34 3.5 3.7
S2-V10 1.0 22.1 8.4 10.5 0.14 0.17 47 1.34 6.5 7.1
S3-V4 0.4 22.1 0.0 6.0 0.00 0.10 21 0.92 1.1 1.1
S3-V7 0.7 22.1 0.0 6.0 0.00 0.10 21 0.92 3.2 3.5
S3-V10 1.0 22.1 0.0 6.0 0.00 0.10 21 0.92 6.4 6.9
S4-V4 0.4 26.3 6.8 0.0 0.11 0.00 21 1.08 1.2 1.2
S4-V7 0.7 26.3 6.8 0.0 0.11 0.00 21 1.08 3.6 3.8
S4-V10 1.0 26.3 6.8 0.0 0.11 0.00 21 1.08 7.3 7.7
u0A is the collision velocity.
mLSA is the lightship mass of the striking ship.
is the mB=30.5 kg throughout the experiments.
Table C2
Collision experiments with fixed masses.
Name u0A (m/s) mLS
A (kg) Water mass hW/‘T Amount of water (%) mA/mB (–) ED
fore (kg) aft (kg) fore (–) aft (–) exp. (J) calc. (J)
N1-V4 0.4 22.1 5.0 8.0 – – – 1.15 1.3 1.3
N1-V7 0.7 22.1 5.0 8.0 – – – 1.15 4.1 4.1
N1-V10 1 22.1 5.0 8.0 – – – 1.15 8.6 8.8
N2-V4 0.4 22.1 8.5 10.5 – – – 1.35 1.5 1.6
N2-V7 0.7 22.1 8.5 10.5 – – – 1.35 4.5 4.4
N2-V10 1 22.1 8.5 10.5 – – – 1.35 8.6 8.9
N4-V4 0.4 26.5 7.0 0.0 – – – 1.10 1.3 1.4
N4-V7 0.7 26.5 7.0 0.0 – – – 1.10 4.1 4.2
N4-V10 1 26.5 7.0 0.0 – – – 1.10 8.2 8.7
u0A is the collision velocity.
mLSA is the lightship mass of the striking ship.
is the mB=30.5 kg throughout the experiments.
K. Tabri et al. / Ocean Engineering 36 (2009) 1366–1376 1375
ARTICLE IN PRESS
Appendix C. Model-scale experiments
See Appendix Tables C1 and C2.
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