Proceedings of the 11th International Ship Stability Workshop

On the prediction of parametric roll

Michiel Gunsing,
MARIN The Netherlands

Reint Dallinga,
MARIN The Netherlands

ABSTRACT

The numerical assessment of parametric rolling by means of time domain simulations is

troublesome. This is due to a number of practical and conceptual problems. Therefore, simple
transparent methods that give immediate insight in the characteristics of a particular design are still
of interest. The present paper describes a method in which the results of linear calculations on the
ship motions are used to estimate hydrostatic stability variations. Following Dunwoody (1989a) the
stability variations are translated in a reduction of the roll damping and the safe operational limits of
the ship. Numerical results are compared with experimental data.
KEYWORDS

Seakeeping, Parametric Rolling, Stability Variations in Waves, Risk Assessment

INTRODUCTION resistance and prudent seamanship), the

modelling of active roll stabilisation, the
Over the past years the assessment of the risk steering and the modelling of the model
on parametric roll for new ships has received propulsion (including the reaction of the main
considerable attention. Despite these efforts the engines on the encountered propeller load
question how to perform a practical and variations). Secondly, and perhaps even more
reliable analysis has not been fully resolved. demanding, is the question what test duration
In general, tests with a scale model are will give a reliable assessment of the risk of
considered as the most reliable way to obtain encountering dangerous ship behaviour.
data since most of the underlying physical
phenomena are included. Think of relevant Numerical simulations offer a flexible
details like the natural speed variations in alternative for tests with a scale model. On one
irregular waves (France et al. (2001)), the hand in the spectrum of available tools one
effect of large roll angles on the added might consider CFD (Umeda et al(2008)),
resistance, the propulsive characteristics which incorporates potentially some (but not
(including propeller ventilation), loss of rudder all) details of the nonlinear ship behaviour in
and stabilizer efficiency and the natural non- waves if it covers details like appendages,
linearity‟s in the wave profile. moving rudders, fins and the steering. On the
However, the design of an experiment with a other end of the spectrum of tools are one
scale model is not entirely straightforward. degree of freedom (1-DOF) time domain
First of all there are is a host of practical issues models and methods based on statistical
regarding the representation of the wind and approaches (Archer et al. (2009)).
waves (spectral shape, directional spread), the
target mean speed (governed by the added
 Proceedings of the 11th International Ship Stability Workshop

The commonly used time domain, “non-linear” practical issues will be mentioned and an
potential flow calculations (see for example outlook to future developments will be given.
France et al. (2001)) are in the midst of this
spectrum of tools. They do account for 6-DOF STABILITY VARIATIONS AS THE CAUSE
OF PARAMETRIC ROLL
ship behaviour and the weak non-linear effects
related to rapid changes in hull geometry To illustrate the physics of parametric rolling,
around the waterline. However, they neglect a simple 1-DOF model with a time dependant
the non-linear diffraction and the non-wave
spring term is analysed.
making aspects of the roll damping. The latter
omission requires correction on basis of
empiricism. Furthermore, the added resistance
is not covered correctly because of the neglect (1)
of the non-linear diffraction, the sustained
speed and speed variations are not covered
adequately. Where represents the total of structural and
Because of the fact that the efforts of going hydrodynamic inertia, the damping and the
through time domain simulations do not restoring term. To understand why and how
necessarily bring the expected accuracy, there stability variations lead to large roll angles
is an interest in simpler transparent methods. we will assume a harmonic roll oscillation with
The approach adopted in the present paper is a the associated roll velocity.
combination of a new method to obtain the
variations in stability ( GM) and an existing
formulation Dunwoody (1989a) to translate (2)
these variations in a decrease in effective roll (3)
damping.
The first step in this method uses linear
frequency domain potential flow calculations to
We will also assume a harmonic stability
calculate the motion response and relative
variation of the transverse metacentric
wave elevation along the waterline (accounting
height GM with a different frequency and
for reflected and radiated wave components).
phase angle given by Equation (4)
These results are used to calculate the
hydrostatic stability variations, accounting for
the hull form variations above and below the
calm water line. (4)
In a second step this (non-linear) transfer
function of the stability variations is used in a
formulation developed by Dunwoody (1989a)
The roll moment exerted on the ship equals the
to estimate the apparent reduction in roll
product of the stability moment and the heel
damping in irregular seas.
angle:
In a last step this result, which depends on
significant wave height and mean wave period,
is compared with the roll damping of the hull
(estimated by means of an empirical method or (5)
existing model test data) to estimate the wave
conditions in which the effective roll damping
is negative. It is shown that the results of these The mean of the product of the roll moment
computations show a fair agreement the results and the roll velocity, the work, over a longer
from tests with a scale model. A number of period of time ( ) is given by:
 Proceedings of the 11th International Ship Stability Workshop

Fig. 3 and 4 show the same results, but now for

a phase shift of for . Fig. 4
(6) shows that on the average no energy is added
over one number of roll cycles.

It can be shown that if that 5

. Only if the work

performed by the stability variations affects the
energy contained in the roll motion. The phase 0
shift ( ) between the change in GM and the
roll motion determines whether energy is added
to or removed from the system. In Fig. 1 the
roll angle, roll velocity and moment are shown 5
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
for . In this case the moment
due to the stability variations and the roll time [s]
velocity always have the same sign. The related Roll Angle
positive work implies that energy is added to Roll Velocit y
Moment due t o Stabilisation variat ions
the roll motion. Fig. 2 shows the energy added
during in every roll cycle.
4 Fig. 3: The roll angle, roll velocity and righting moment during
one roll period of roll, when roll and wave show a 90 degree
2
phase shift.

0 5

2
Work

4
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
5
0 2.5 5 7.5 10 12.5 15 17.5 20 22.5 25
time [s]
Roll Angle T ime [s]
Roll Velocit y
Moment due t o Stabilisation variat ions
Fig. 4: Energy build up during 5 roll cycle (phase shift 90 deg)
Fig. 1: The roll angle, roll velocity and righting moment during
one roll period of roll, when roll and waves are in phase Assuming small roll angles Equation (5) can be
linearised in . In this case the average amount
10 of energy added in roll given by Equation (6)
can be rewritten as:
Work

(7)
0
0 2.5 5 7.5 10 12.5 15 17.5 20 22.5 25

T ime [s]
Where is the ship displacement, GMa the
amplitude of the stability variation and the
natural frequency of roll.
Fig. 2: Energy added during 5 roll cycles (phase shift 180 deg) Whether or not the above increase in
energy materialises in an increased roll angle
 Proceedings of the 11th International Ship Stability Workshop

depends on the average (per roll cycle) amount

of energy dissipated by the equivalent linear
roll damping bE, which is given by Equation
(8).
(11)

(8)
Noteworthy is the fact that the expected value
for the damping reduction increases with the
spectral density of the stability variations. This
Equating Equations (7) and (8) yields a implies it increases with the wave height
criterion in terms of roll damping. Parametric squared.
roll can start to develop when:
The practical implication of the above result in
terms of the risk of encountering parametric
roll in a given time frame is an issue that
(9) requires further work.
Fig. 5 shows the character of the non-
dimensional sum of the roll damping of hull
and appendages and the expected value for the
IRREGULAR WAVES reduction due to the stability variations for a
range of combinations of significant wave
As shown foregoing, the change in metacentric height and peak period. It is clear that a large
height due to the changing water height along negative damping is to be expected in the
the ship is driving parametric roll. It is also higher sea states in combination with a peak
shown that the increase and decrease in roll period of 16.5s, which is half the roll period of
amplitude can be written as a (negative) the subject vessel.
contribution to the roll damping.
In irregular waves the random phases of the
wave components yield slow variations in the ND Roll
contribution of the stability variations in the Damping
total damping. Dunwoody(1989a) gives an
expression for the reduction in irregular waves.
Assuming a broad band spectrum for - the
spectral density of stability fluctuations - and Tp
taking its value at the encounter frequency that
matches twice the roll frequency he arrives at
the following expression for the expected value Hs
for the reduction of the non-dimensional
damping.
Fig. 5: Typical non-dimensional roll damping ratio as function
of significant wave height Hs and wave peak period Tp.

(10) ROLL DAMPING

Obviously the roll damping of a vessel is an
important parameter in the assessment of
parametric roll. A relevant characteristic is the
Where is the encounter frequency. In a fact that it decreases with decreasing speed. In
dimensional format this result is given by: the lower speed range it is dominated by the
 Proceedings of the 11th International Ship Stability Workshop

non-linear eddy and bilge keel damping functions. The calculated relative wave
components. At moderate and higher speeds elevation accounts for the radiated and
the linear lift damping of the hull becomes a diffracted waves.
dominant factor (Ikeda et al.(1978), Dallinga et At low to moderate speeds, this method gives a
al. (1998)). In other words: the risk on good representation of the wave elevation
parametric roll decreases with increasing along the ship, offering a fair and efficient
speed. Other means that increase the ships roll estimate of the stability variations.
damping also decrease the risk on parametric The above calculation yields the transfer
rolling. For example an anti roll tank is a very function of the relative wave elevations (phase
effective way to increase the damping at low
and amplitude) at every waterline panel. From
speed. Fin stabilizers are effective as well, but the surface elevation the vertical position of the
only at some forward speed. buoyancy point above the baseline (KB) is
calculated by integrating over the underwater
MAGNITUDE OF THE STABILITY volume (Equation (13)) over the actual hull
VARIATIONS form. The actual waterline width and hull form
are also used to calculate the vertical distance
With the foregoing establishing the risk of
from the buoyancy point to the transverse
parametric roll has reduced to estimating .
metacentre (BM).
Assuming linearity this quantity is obtained
from a multiplying the square of the transfer
function GMa/ a with the wave spectrum S .
(13)

(12)

(14)
A correction for forward speed yields the
spectral density at the right encounter
frequency. Regarding the evaluation of the The overall values are obtained by integrating
stability variations experienced by the hull over the length of the ship.
Dunwoody(1989b) uses a method that omits The above evaluation is repeated for the full
the diffracted and radiated waves. In Umeda et range of phase angles phase between 0 to 2π,
al(2008) a CFD approach was followed to each resulting in different waterlines. For the
obtain the roll moment, but they also showed amplitude GMa the half of the dynamic range
that this was heavily overestimated by their between the maximum value and the
method. This is probably due to the grid size. minimum value was taken.
Very fine grids are needed capture all relevant
details but this leads to unacceptable time
consuming calculations. (15)
In the present paper, the transfer function of
the stability variation GMa/ a is obtained by
means of hydrostatic considerations from the
relative wave elevations along the ship. The
latter calculated by means of a linear three- Sample results
dimensional frequency domain potential flow A typical result for different wave amplitudes
code PRECAL. This code calculates the wave is given in Fig. 6. Since these data are given in
induced excitation and the motion induced a non-dimensional form, the data can be used
reaction forces using zero speed Greens
 Proceedings of the 11th International Ship Stability Workshop

for a range of ships with a comparable hull

shape.

Fig. 7: Amplitude the sectional data on BM and KB

This is in line with theory. If one assumes a

half submerged rectangle (draught=T m), the
Fig. 6: Typical transfer function for the GM as function of
buoyancy point is located half way the baseline
frequency(top) and as function of wave length over ship length and the waterline, at . If the relative wave
λ/LPP(bottom) for wave amplitudes between 0.01 and 5m. elevation increases the local draft by , the
height of the centre of buoyancy changes by
Fig. 6. shows that the transfer function GMa/ a . Because the waterline width is constant
reaches its maximum value at a wave length of in this example BM changes only due to the
about 80% of the ship length. change in displacement. The total change in
GM for this example is given by Equation (16)
A second observation is that the peak value is
relatively insensitive to wave height variations.
The remark that the linear contribution to the
stability variations is small compared to (16)
contribution of the hull shape is therefore not
justified. If it would hold, this non-linearity
would have been more pronounced in Fig. 6. It is clear from this equation that if T is small, a
given has a large impact on the GM. This
Sectional contribution is in line with observations that ships with a
In the calculation of the stability variations the flat pram type stern have a higher risk on
sectional contribution of KB and BM became parametric rolling (Levadou et al. (2006) )
available. Fig. 7 shows typical distribution of
sectional amplitude of KB and BM over the Note on the method to calculate the relative wave
ship‟s length. The graph clearly shows that elevation
those parts of the ship that have a small
Because of the use of zero-speed Greens
draught, have large BM variations and
functions in PRECAL, the prediction of the
contribute significantly more to the change in
relative wave elevations becomes less accurate
GM than those sections that have a large draft.
at higher speeds. Because the mis-prediction
are particularly large in the diverging flow at
the bow and the converging flow at the stern,
they have a relatively large effect on the
derived stability variations. At increasing speed
a more accurate description of the dispersion of
the waves should be considered (e.g. Bunnik
(1999)).
 Proceedings of the 11th International Ship Stability Workshop

ESTIMATED OPERATIONAL LIMITS establish what are the essential physical aspects
of the problem. The second problem concerns
Fig. 8 shows a typical result from a the statistical aspects.
computation. The lines show the calculated In the near future we will address the
results. The markers indicate experimental statistical aspects first. The first step will
data. It can be seen that the numerical method investigate the relation between Dunwoody‟s
predicts a threshold wave height of 2.5 to 3 m estimate of the maximum excitation level with
significant wave height at a wave peak period results of simple 1-DOF time domain
of around 16.5 seconds. calculations on the variations in the roll
The markers in Fig. 8 show test results for a damping. A second step will investigate the
290m container ship. Green triangles indicate related onset of parametric roll (limiting
tests that showed (within the adopted test ourselves to the growth up to moderate roll
duration) no parametric rolling; blue squares angles), including the effects of non-linear roll
mark the position where large roll angles were damping. Efforts will be made to relate the
found. frequency of critical events to the decrease in
roll damping and spectral estimates of this
quantity.
Depending on the outcome of the foregoing it
may be of interest to improve aspects of full
blown numerical simulations. The adequate
modelling of the natural speed variations in
high waves as well as the effects on non-linear
diffraction in the parametric excitation come to
mind.
Fig. 8: Comparison of test results to calculated using the References
broadband approach.
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the stability variations, offers a practical Sixty-Seventh European Study Group Mathematics
estimate of the operational limits. with Industry, Dynamic Models of Extreme Rolling
of vessel in head Waves, 2009

Bunnik, T, PhD-Thesis TU-Delft, Seakeeping

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 Proceedings of the 11th International Ship Stability Workshop

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