Stability Analysis of Parametric Roll Resonance B.J.H. van Laarhoven DCT 2009.062 Traineeship report Coach: prof.dr. T.I. Fossen Supervisor: prof.dr. H. Nijmeijer Eindhoven University of Technology Department Mechanical Engineering Dynamics and Control Group Eindhoven, June 2009
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Stability Analysis of Parametric Roll Resonance Analysis of Parametric Roll Resonance B.J.H. van Laarhoven DCT 2009.062 Traineeship report Coach: prof.dr. T.I. Fossen Supervisor: prof.dr.
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Stability Analysis of Parametric Roll
Resonance
B.J.H. van Laarhoven
DCT 2009.062
Traineeship report
Coach: prof.dr. T.I. Fossen
Supervisor: prof.dr. H. Nijmeijer
Eindhoven University of Technology
Department Mechanical Engineering
Dynamics and Control Group
Eindhoven, June 2009
1
Abstract
For investigation and developing fundamental knowledge about how and why ships and structures behave
in an ocean environment Norway started the Centre for Ships and Ocean Structures. One of the research
topics of this centre of excellence is parametric roll resonance. Parametric roll resonance is a phenomenon
that causes a roll motion to a ship due to longitudinal waves. This roll motion can become so large that
capsizing can occur. It is dangerous for ships to encounter this phenomenon. Even when capsizing does not
happen this roll motion could cause damage to the ship and can be a threat for the cargo and crew.
Although there are very specific conditions when parametric roll resonance can occur there are several
examples where it did happen and caused significant damage to the ship and cargo for millions of euro’s.
To describe this phenomenon the coupled 3 DOF model proposed by Neves and Rodríguez is used. The
pitch angle and vertical displacement are assumed to be harmonic and therefore and uncoupled 1 DOF
model is derived to describe the roll motion of the ship. For control it is interesting to have an indication
what the values of the control parameters could be to prevent parametric roll resonance to occur. To
determine these values of the control parameters first the stability regions of the uncoupled 1 DOF are
determined. These simulations can easily be used as an indication for the control parameters since it is easy
to see what the value of the control parameters should be to reduce the roll angle. Since in this study only
an uncoupled 1 DOF model is used, results can be more accurate when a higher degree of freedom model is
used to describe the roll motion.
Also attention has been paid to a simple velocity controller. One of the control parameters used is the
forwards velocity of the ship. Simulations show that an increase or decrease of the forward velocity would
lead to a reduction of the roll angle. Decreasing the velocity always leads to a reduction of the roll angle.
But increasing the velocity does not always lead to a reduction of the roll angle. The reduction of the roll
angle when increasing the velocity seems to be highly dependent on the forward acceleration of the ship.
No explanation for this has been found and further investigation is needed.
2
List of symbols
symbol description
t time φω natural roll frequency wρ density of water
g gravitational constant
∇ water displacement
bGM constant part metacentric height
xI ship inertia in roll direction
44A added mass in roll direction
eω encounter frequency 0ω wave frequency
U forward velocity
β heading angle
φ roll angle
z heave displacement
θ pitch angle
1d linear damping constant
2d nonlinear damping constant
aGM varying part metacentric height
3K nonlinear moment in roll direction
3
Contents
1 Introduction 4
2 Parametric resonance 6
2.1 Resonance in mechanical systems 6
2.2 Parametric resonance in mechanical systems 7
2.3 Parametric resonance in ships 8
3 Ship model 14
3.1 Ship model 14
3.2 Equilibrium point 15
3.3 Nonlinearities 16
4 Numerical analysis 18
4.1 Detection method 18
4.2 Simulation 18
4.3 Results 19
4.4 Simulation with simple speed controller 21
4.5 Results 22
5 Conclusions and recommendations 26
A Extended explanation used ship model 27
B Simulations near the equilibrium point 29
C M-files used for simulation 30
D Results of simulation done with varying ship parameters 34
E MATLAB simulink model 35
F Main characteristics of the container ship 36
References 37
4
Chapter 1
Introduction
The Centre for Ships and Ocean Structures is located in Trondheim Norway. It is a centre of excellence that
is founded to investigate and develop fundamental knowledge about how ships and structures behave in the
ocean environment, using analytical, numerical and experimental studies, [21].
One of the research topics nowadays is parametric roll resonance in ship. Parametric roll resonance in ships
is a resonance phenomenon that gives a roll motion due to longitudinal waves acting on the ship. This roll
motion, which can achieve roll angles so high that the ship could capsize, can be dangerous for the ship.
Even when parametric resonance does not lead to capsizing this can be dangerous for the crew, cargo and
ship itself. This phenomenon has been known by the maritime society since the fifties. In those days only
small ships, like fishing vessels, encountered this phenomenon. Nowadays there are examples of accidents
with container vessels that cause significant damage to cargo and ship for millions of euro’s, [11]. For
example in October 1998 the APL China sailed from Taiwan to Seattle and during this trip it experienced
parametric roll resonance. When the vessel arrived in Seattle more than sixty percent of its cargo was lost at
sea or damaged, see figure 1.1. A more recent example of ships encountering parametric roll resonance is
the Maersk Carolina. In January 2003 the ships encounters a storm at the Atlantic sea. It alternates course
to minimize roll, in other words the ship was heading into the sea. In a few cycles the roll angle increased
up to 47 degrees and lasted a few minutes. The ships lost 133 containers and another fifty experienced
severe water damage. The cargo claims reached up to almost 4 million dollars, see figure 1.2.
Figure 1.1: The APL China arriving at Seattle [2]
5
Figure 1.2: The Maersk Carolina encountering parametric roll resonance [2]
To see what parametric roll resonance is and what could happen to a cruise ship the reader may visit [22].
Because not only container vessels but also destroyers, cruise ships and roll on roll off paxes encounter
parametric roll, this phenomenon recently attracted scientists, to understand the physics, detect parametric
roll resonance during sailing and eventually prevent large motions of the ship by active control. For
controlling this phenomenon it is interesting to know what the influence is of certain ship parameters on the
stability of the ship. The main goal of this study is to define the stability regions of the ship model proposed
by Neves and Rodríguez [8]. In other words define for which values of the ship parameters the ship
encounters parametric roll resonance and for which not. The results can be used as an indication for the size
of the control parameters.
In the second chapter it is explained what parametric roll resonance is and under which conditions a ship
can encounter this phenomenon. Also previous studies according to parametric roll resonance are
discussed. Then the used ship model is presented and analytical analysis to this model is done. In the fourth
chapter numerical analysis are presented and an analysis according to the control of parametric resonance
using a velocity controller is done.
6
Chapter 2
Parametric resonance
In this chapter attention is paid to parametric resonance in mechanical systems. It is explained what it is,
why and under which conditions this phenomenon happens. Specific attention will be paid to parametric
roll resonance in ships sailing in head seas. A list of empirical criteria is given to indicate when ships
encounter parametric roll resonance. Also different models that describe this phenomenon are showed
together with detection and stabilizing methods.
2.1 Resonance in mechanical systems
In mechanical systems often oscillations and resonances occur due to for example actuators in the system.
To explain this first is attention is paid a simple mechanical system an undamped, undriven pendulum, see
figure 2.1. A mass is connected to the end of a rod. The other end of the rod is pinned at to point A, which
acts as the pivot point of the system. Such a mechanical system is called an oscillator.
Figure 2.1: Undamped and undriven pendulum
The oscillation of the pendulum can be described as
02
2
2
=+ θωθ
dt
d (2.1)
Starting at an initial value 0θ this equation describes the movement of the pendulum. For the damped
undriven pendulum a rotational damper is added around the pivot point and the oscillation can be described
as
02
2
2
=++ θωθ
γθ
dt
d
dt
d (2.2)
Due to this damping term all solutions decay to zero independent of the initial value 0θ . The damped and
undamped pendulum are examples of unforced oscillations. A forced oscillation can occurs when a
mechanical system is driven by for example an actuator. In relation to the pendulum this means that an
actuator generates a periodic force, which results in a moment around pivot point A, to give an excitation to
the pendulum. In the model this results in a force term added to the right side of the equation. This is called
a forced oscillation. The equation that describes this forced oscillation becomes
)cos(2
2
2
tKdt
d
dt
dΩ=++ θω
θγ
θ (2.3)
7
Resonance occurs when K is large or when the frequency of the driven force Ω is near one of the
resonance frequencies of the system. This means that the system oscillates with high amplitude. The
resonance frequencies of a system are the natural frequency eω of the system and multiples of that
frequency. When the frequency of the driven force is near the natural frequency of the system, eω≈Ω ,
the oscillation is called primary or main resonance. In mechanical systems it is preferred to avoid resonance
due to the fact that a small driving force can cause large amplitude vibrations and therefore could cause
damage to the system.
2.2 Parametric resonance in mechanical systems
Resonance as described in the previous section, leads to differential equations with constant or slowly
varying parameters. Parametric resonance differs from resonance, because it is an instability phenomenon.
This instability is caused because parametric resonance gives differential equations with rapidly changing
system parameters. This leads to large variations in restoring forces and therefore instability. Parametric
resonance occurs when a system is parametrically excited by a periodic force and the frequency of this
force is near one of the resonance frequencies of the system. Faraday (1831) was the first who recognized
parametric resonance. During experiments with surface waves in a fluid-filled cylinder he noted that when
the cylinder was vertically excited the surface waves had half the frequency of the vertical excitation. In
1859 Melde did the first experiments relating to parametric resonance. He tied a string between a rigid
support and the extremity of the prong of a massive tuning fork of low pitch. For several combinations of
mass and tension of the string and frequency and loudness of the fork, he noted that the string could
oscillate laterally while the exciting forces work in longitudinal direction at twice the time period of the
fork, [1].
In the example of the pendulum parametric resonance occurs when instead of actuating the pendulum by a
moment around turning point A, now point A is periodically translated in vertical direction see figure 2.2.
Figure 2.2: Parametrically excited pendulum
Figure 2.3: Definition of ship motions [2]
The oscillation can be described by die following equation
8
0)cos(2
2
=++ θθ
tqpdt
d (2.4)
Where p is a function of the ratio of the forcing and natural frequency and q represents the amplitude of the
parametric excitation. This equation is generally know as the Mathieu equation and was discovered by
Mathieu in 1868 while studying vibrations of an elliptic membrane. The Mathieu equation can be used for
describing the response of many mechanical systems under the influence of parametric excitation, [1].
2.3 Parametric resonance in ships
Ships in calm water can be externally exited by for example wind. This can lead to certain motions of the
ship. For definition of the different motions of a ship see figure 2.3. If the ship encounters a roll motion due
to wind than, due to the roll damping of the ship, the roll motion decays to zero after a few time periods,
see figure 2.4. However when the sea is not calm and a ships encounters parametric roll resonance,
instability is caused due to large variation is a model parameter. The ship starts to roll until it capsizes or
stabilizes up to a certain roll angle, see figure 2.5. Because of this phenomenon sailing in head seas can be
dangerous. Also when parametric resonance does not lead to capsizing this can be dangerous for the crew,
cargo and ship itself.
Figure 2.4: Roll in calm sea [2]
Figure 2.5: Example of parametric roll resonance [2]
Although this is a dangerous phenomenon it does not happen to every ship at any time. The environmental
and physical conditions that simultaneously need to happen to cause parametric roll resonance area
• The encounter frequency of the ship and waves must be approximately two times the natural roll
frequency of the ship.
The natural roll frequency of the ship is defined as
44AI
GMg
x
bw
+
∇=
ρωφ (2.5)
• The length of the waves should be equal to the length of the ship.
9
• Due to the previous criterion the ship needs to sail in heads or stern seas. Especially for large
container vessels only waves in head or stern seas can reach such a length.
• The wave height hw needs to be larger than a ship dependent threshold value ht.
• Ships should have the correct hull shape.
Figure 2.6: Ship hull on a wave (yellow line) crest (left) and on a wave trough (right) [2]
The first criterion contains the encounter frequency. This frequency eω is defined as the frequency at
which the ship and the waves meet. The following function describes the encounter frequency [3].
βω
ωω cos2
00 U
ge += (2.6)
Where 0ω is the frequency of the waves, g is the gravitational acceleration, U is the forward speeds of
the ship and the heading angle of the ship. The heading angle is defined as the angle between heading of the
ship and the direction of the wave, see figure 1.7.
The last criterion, the vessel should have the correct hull shape, needs some more attention. The geometry
of the hull is critical for parametric roll resonance to happen. Figure 2.6 shows the ships hull of specific
vessels. These hull designs are based on years of investigation of what is the optimal design looking at
economical aspects for example maximum cargo and minimal water resistance. The result is a hull design
that is like a box in the middle and towards the head and back of the ship the geometry has large gradients.
This results in large difference in water plane area dependent on the waves. When the ship is on a wave
crest the water plane area is given by the yellow line in the left part of figure 2.6 and when the ship is on a
wave trough the water plane area is given in the right figure. Hull designs like this are common in fishing
and container vessels due to this hull shape these ships susceptible to encounter parametric roll resonance.
Figure 2.7: Heading angle of a ship [2]
Parametric roll resonance is a result of waves acting on the ship. It is obvious that beam waves, waves that
come towards the side of the ship, cause roll movement of the ship. Parametric roll resonance however is
caused by head or longitudinal waves, waves that come to the head of the ship. The stability of the ship
10
depends on the waterline of the ship. Due to the specific hull shape shown in figure 2.6 the waterline
changes when sailing in longitudinal waves. This can be explained by mean of figure 2.8. In situation 1 due
to a wide waterline there is more stability than in a still water situation (this means that it a bigger force is
needed to push the ship away from it’s equilibrium point 0=ϕ ) so the push back force, the force that is
generated to get the ship back in its equilibrium point 0=ϕ is large. A half time period later the ship is in
situation 2 where the stability is decreased, the push back force is smaller so the roll speed increases. At last
the ship ends up in situation 3 where the stability is again large which lead to a large push back force but
because the roll speed was increased in situation 2 the ship rolls more over which leads to a larger roll
angle. This repeats itself until the ship capsizes or stabilizes up to a certain roll angle.
Figure 2.8: The effect of waves on a ship [2]
2.3.1 Modeling
For better understanding of the physical behavior, the detection of parametric roll resonance and to reduce
this phenomenon by active or passive control this calls for the development of mathematical models that
describe the ship behavior in head seas. In the past few years several mathematical model are suggested by
scientists. Most of the proposed models are based on the Mathieu equation, see (2.4). For ships it is
common to, under certain assumptions, decouple body motions in longitudinal modes (surge, heave and
pitch) and lateral motes (sway, roll and yaw). This leads to uncoupled one degree of freedom models to
analyze parametric roll resonance. France et al. [4] and Shin et al. [5] point out that a Mathieu type one
degree of freedom model can easily be used to show when ships encounter parametric resonance. In 2006
Bulian [6] came up with a 1.5 DOF model where the assumption of quasi-static heave and pitch leaded to
an analytical description of the GZ curve. This model is valid for moderate ships speed in head seas and
gives reasonable results for the prediction of parametric roll resonance.
To get better understanding of the ship behavior Neves [7] derived a 3-DOF nonlinear model where heave,
roll and pitch were coupled. By using Taylor expansion up to second order the restoring forces and
moments in heave, roll and pitch were described. This model however predicted a roll angle that was too
large compared to experimental results. These results are obtained by experiments done with a 1:45 ship
model in a towing tank. Therefore in 2005 Neves and Rodríguez [8] expanded the model found in 2002 by
using Taylor expansion up to third order. In this model the nonlinear coupling coefficients are derived as
functions of the characteristics of the hull shape. This model was designed to predict roll motions of a
fishing vessel. Also papers of Neves and Rodríguez [9, 10] show that this model matches the experimental
11
results better than the earlier proposed second order model. Holden et al. [11] used the third order model
from Neves and Rodríguez for the prediction of the roll angle of container vessels. The validation discussed
in the paper showed good agreement between the third order model and the experimental results for the
situation where parametric roll resonance occurred and as well where it did not occur.
2.3.2 Detection
To prevent ships from encountering parametric roll resonance there are several options, discussed in the
next section. However how perfect these options will be, maybe the most important issue is the detection of
parametric roll resonance. It is nice if there are systems which can reduce parametric resonance when it
already visually happens but usually the ship and cargo are already damaged. In the example shown in
figure 2.5, parametric resonance is build up in a few time periods. While long before it is visually
noticeable, parametric resonance already starts to evolve. The problem with this is when is certain behavior
classified as parametric resonance and when as normal roll disturbance. To reduce or avoid damage to ship,
cargo and crew it is obvious that fast detection is needed so the observation of the skipper is not good
enough. An automatic detection system is needed to detect parametric roll resonance sufficiently early
enough to take suitable precautions. Nowadays ships use on-line detection schemes that use numerical
calculations to predict parametric roll resonance [12]. For the detection of parametric roll resonance two
studies have been done one by Christian Holden about frequency-motivated observer design and another
one by Roberto Galeazzi about the prediction of the encounter frequency.
Frequency-motivated observer design for the prediction of parametric roll resonance
This detection scheme is based on the power spectral densities of data series that describe the ship roll
behavior. In the situation when parametric roll resonance does occur, the PSD shows one major frequency
component and in the situation that parametric roll resonance does not occur the PSD shows two
frequencies. Based on this observation the output of the system is modeled as a linear second-order
oscillatory time-varying system, with time dependent system parameters, driven by white noise. This
system is discretized to a discrete-time model for further analysis. Since the system is linear-time-varying
this method shows, at best, when parametric roll resonance is probable. For estimation of the time
dependent systems parameters three different methods are used, a discrete Kalman Filter, the method of
Recursive Least Squares and a Particle Filter. Then the eigenvalues of the system matrix are determined. A
conclusion according to the probability of parametric roll resonance can be drawn based on changes in the
eigenvalues. These methods have been validated using a 1:45 scale model of a 294 m tanker. The ship
model was exposed to regular en irregular waves in a towing tank. In regular waves all method where able
to predict the occurrence of parametric roll resonance with an accuracy of 100% while in irregular waves
the accuracy of the best method was 87.5% The Recursive Least Squares method is the fastest method
while the Particle Filter appeared to be the most accurate one, [12].
Prediction of the encounter frequency
Since it takes a long time to build up parametric roll resonance it is unlikely to provide an early warning
just by looking at the roll angle. Therefore also the heave and pitch motions (see figure 2.3 for definition of
these motions) of the ship are taken into account to get more information about parametric roll resonance.
Based on these time series of the heave and pitch motion of the ship an estimation of the encounter
frequency is made. Parametric roll resonance detection is based upon sinusoidal detection in white
Gaussian noise. This method is validated with the same experiments used to validate the methods of
Holden. In regular waves also an accuracy of 100% is reached but in irregular waves it is hard to predict
parametric roll resonance, [13].
2.3.3 Avoidance/Reducing parametric roll resonance
Avoidance or reducing of parametric roll resonance can be achieved in several ways. The most convenient
way is to reduce the likelihood of parametric roll resonance. This can be done by making sure that not all of
the conditions, explained in section 2.3, occur at the same time. This can be done for example by
modifications on the hull form in terms of reducing the large gradients at the stern and bow of the ship and
12
adding roll damping to the ship by placing keels at the ship hull. Because modifying ship hulls is quite
expensive most research has been done on stabilization of a ship by active systems. Roberto Galeazzi looks
at stabilization of a ship by using fins and a combination of fins and modifying ship speed, [14, 16].
Christian Holden does research at stabilization by using a U-tank, [15].
Stabilization using fins
This method exploits that when adding significantly enough damping to the ship, the roll motion will
detune. There are two ways to increase the roll damping of a ship, adding fins to the ship hull and increase
the speed of the ship. Since increasing the ship speed in not always applicable only attention has been paid
to stabilization with fins. To avoid an extra drag force while travelling normally the fin can pulled into the
ship and only come out when the ship encounters parametric roll resonance, see figure 2.9.
Figure 2.9: Stabilizing fins can be fold out when necessary [2]
Stabilization was established by influencing the location of bifurcation point in the used ship model by
increasing roll damping with fins. The feasibility of this method was demonstrated using a four degree of
freedom model of a container vessel. The demonstration shows that stabilizing the ship using fins is
possible, [14].
Stabilization by using a U-tank
The principle of a U-tank stabilizer is that the water in de tank counteracts the movement of the ship. There
are two kinds of U-tanks, active and passive ones. Passive U-tanks can be useful to reduce roll of the ship if
the frequency of the roll motion is right. If the frequency is to high or low than the passive U-tank even
contributes to the roll motion and thus makes it worse. An active U-tank however always stabilizes the roll
motion under any circumstances, but due to the pump that is needed, consumes a lot of energy. To compare
both U-tank types Holden simulated with Lloyds U-tank model. These simulations showed that an active
U-tank always stabilizes and a passive one only reduces roll under certain conditions. For simulation
movies the reader could visit [18] for the passive U-tank and [19] for the active U-tank. Since Lloyds
model has its limitations Holden achieved a new Lagrangian U-tank model for better calculations and
larger validity range, [15].
Stabilization of Parametric Roll Resonance by Combined Speed and Fin Stabilizer Control
One of the criteria when parametric roll resonance occurs is that the encounter frequency should be twice
the natural roll frequency of the ship. The encounter frequency depends on the speed, see (2.6). Changing
the speed gives a change in the encounter frequency and this way does not satisfy the first criterion and so
prevent parametric roll resonance from happening. To increase the roll damping fin stabilizers are added to
the hull. The used velocity controller is based on Lyapunov’s stability theory. The fins are controlled by
using backstepping as control method for stability, [16].
2.3.4 Stability analysis of parametric roll resonance
Neves and Rodrìguez analytically derived equations for the stability regions for an uncoupled version their
3-DOF ship model, [8], which basically can be described by Hill’s equation. Based on the uncoupled ship
model a numerical simulation is made and compared to the analytical solution. To investigate the
influences of nonlinearities and coupling terms the 3-DOF ship model as presented in [8] is used to
13
numerically define the stability regions for parametric resonance. In both cases, uncoupled and coupled
simulations, there is looked at the influences of the encounter frequency and wave height on the stability
regions. The nonlinear terms have little effect on the shape of the stability limits. The nonlinear coupling
terms however are relevant in determination of the roll amplitude. Initial conditions play an important role
and there is evidence of typical nonlinear behavior: jump effect and bifurcation, [20].
14
Chapter 3
Ship model
In the first chapter an introduction to parametric roll resonance was made and clarified previous and current
research topic of the scientific community. In this chapter the ship model used for the numerical analysis is
explained and an analysis according to the stability of this ship model is done.
3.1 Ship model
To analyze parametric roll resonance in ships a mathematical model is needed. Parametric resonance occurs
in systems that can be describes as an autoparametric system. An autoparametric system consists of two
subsystems, a primary and secondary system. The primary system can be externally forced, self-excited,
parametrically excited or a combination of those. The secondary system on the other hand is coupled to the
primary system in a nonlinear way and not under the influence of any external force.
To describe parametric roll resonance as an autoparametric system, the model proposed by Neves and
Rodríguez [8] is used. It is a three degree of freedom model that combines the primary (heave and pitch)
and secondary (roll) system into one model. The primary system is externally exited by wave motion. The
secondary system is parametrically excited by the primary system. Holden et al. [11] showed that this
model, which was originally designed for describing the motions of a fishing vessel, also can describes the
motions of a container vessel in head seas. The model is shortly presented. For a more detailed description
the reader should read references [8, 11].
The generalized coordinate vector is defined as
Ttttzt ])()()([)( θφ=s (3.1)
Where )(tz , )(tφ and )(tθ represent respectively the heave, roll and pitch motions of the ship which are
defined according to figure 2.3. The equations of motion for heave, roll and pitch are defined as