A Simplified Method for the Estimation of the Natural Roll Frequency ...

A Simplified Method for the Estimation of the Natural Roll
Frequency of Ships in Heavy Weather
S.Krüger, F. Kluwe, TU Hamburg- Harburg
1. Background and Motivation
Several incidents where either cargo was lost or the ship capsized have drawn the
attention of the Maritime community to the fact that under certain conditions, sudden
large roll angles can occur. These sudden large roll angles typically occur in head or
following seas, and they are related to significant alterations of the stability of the ship
in waves. In extreme cases, the stability may become negative on the wave crest and
the ship experiences large roll angles simply due to the related stability loss.
However, it is also possible that even if the ship has sufficient (or excessive) stability,
large rolling angles can occur which then result in large transversal accelerations and
cargo damage. Typically, these situations occur if the encounter frequency of the
waves is in resonance to the natural roll frequency of the ship. As the rolling of ships
in heavy weather is equivalent to a forced oscillation, it is obvious that a resonance
situation will always occur if the excitation frequency is equal to the system’s
eigenfrequency or to higher orders of that eigenfrequency. Most dangerous is the so
called 2:1 resonance, where two pitch cycles coincide with one roll cycle. If the
stability of the ship is low, the related roll period takes large values and this situation
can occur in following sea scenarios at slow or medium ship speeds, where also the
encounter period takes large values. If on the other hand the roll period is small due
to high stability, this situation can be found at slow or medium speeds in head seas.
In the so called 1:1 resonance, the eigenfrequency of the ship coincides with the
encounter frequency of the waves, which results in a situation where one pitch cycle
coincides with one roll cycle. For typical ships, this situation is possible in followings
seas at higher speeds when the stability is more on the low side. However, not all
possible resonances or close to resonance situations must necessary result in the
occurrence of large roll angles. If the exciting forces are sufficiently small (this means
that stability alterations are below a certain threshold value) or if the roll damping is
large enough, the roll motion may remain moderate. On the other hand it is also
possible that large roll angles may occur in situations that are far beyond a
resonance situation. Therefore, the only accurate way to actually judge upon the roll
angles in a specific head or following sea scenario is actually a full numerical
simulation.
On the other hand, there are some practical applications where it would be useful if a
decision could be made to check if a critical resonance situation is actually met. This
would require a comparison of the encounter frequency (or period) and the natural
roll frequency (or period) of the ship. The latter is mostly computed based on the still
water situation for small roll angles. Based of the comparison of that natural roll
period and the encounter period it is then often decided whether a situation may be
potentially dangerous or not. In some cases, this procedure may be successful, but
there have been also many cases been identified where this rough assumption does
not lead to correct results. This is due to the fact that even in still water, the roll period

may actually depend on the roll amplitude, and further, that the ship may significantly
alter its roll period when operated in waves. As a matter of fact, for these reasons it is
practically not possible to judge upon a potentially dangerous situation if the
estimation of the ship’s natural roll period is based on the still water approach for
small roll angles only. But it may be useful to improve this simple estimation of the
natural roll period to get closer to potentially dangerous situations in possible
resonances, although the method which is suggested below will not be accurate
enough to fully replace a dynamic simulation.
2. The Principle of Natural Roll Period Determination
Fig. 1: Determination of natural roll period by roll decay test (left) and relevant righting
levers for Stillwater, crest and trough
The principle determination of the natural roll period is shown in Fig.1. The left side
shows a typical roll decay test which may also (theoretically) be performed with a full
scale ship. The ship is inclined to an initial heel φo (which is small for practical
reasons) and then released. The result is an oscillation around the static heel of
equilibrium, and the ship oscillates in its own natural roll period. As the oscillation is
damped, the amplitudes decrease. From such a test, the natural roll period and the
damping increment can be determined. As the ship may be idealized as a one
degree of freedom oscillation system, its natural period can also theoretically be
determined based on a characteristic restoring moment and a characteristic mass
moment of inertia. This leads to the well known formula of Weiss:
GM denotes the metacentric height of the ship and i the roll radius of gyration, which
is obtained from the mass moment of inertia around the x- axis of the ship and the
actual deplacement. For most ships, the roll radius of gyration (including section
added mass) can be approximated by 0.4 times ship’s beam, but for some ships
having a high superstructure or large amount of deck cargo, i can amount up to
0.45B. It is very important to note that the formula expresses the restoring lever by
the metacentric height, which is only possible if the righting lever follows roughly the

GMφ - line. For the given roll decay test with a small initial roll angle of 5 Degree, the
roll oscillation takes place between the points denoted by 1 and 1’ in Fig. 1, right. For
this oscillation with a small amplitude, the linearization of the righting h lever by GMφ
is sufficiently accurate, as both points 1 and 1’ lie more or less exactly on the GMφline.
But it is well known that the righting lever h follows that linearization only for
small roll angles, and consequently, the roll period determined by such kind of
linearization becomes then inaccurate for a large amplitude roll motion. E.g. for a roll
oscillation between the points 2 and 2’ the actual restoring moment would be
significantly smaller than the linearization, as the actual righting lever is completely
below the GMφ-line. Consequently, the related natural roll period must be larger.
This is shown in Fig.2.
Fig. 2: Roll decay test for a large initial Roll angle (left) and principle of the
determination of an effective GM
There, a roll decay test for the same ship as Fig.1, left was simulated, but for a large
initial roll angle. In comparison to Fig.1, the roll damping was reduced to keep the
large roll amplitudes for a longer time. The initial roll oscillation is equivalent to the
points denotes by 2 and 2’ in Fig. 2, right. It can clearly bee seen that initially, the
natural roll period amounts to roughly 3 roll cycles per 50 s (16.7s), whereas after the
decay of the roll amplitude the natural roll period is decreased to about 4 roll cycles in
50 s (12.5s). This is in line with the non linear characteristics of the related righting
lever curve in Fig.1, right: At larger heeling angles, the righting lever is significantly
below the GMφ-line. Consequently, if the initial GM-value is used to determine the
natural roll period, this will result in a roll period being too small for a large angle roll
oscillation. From the resulting roll periods it can be seen that a correct roll period
determination for the large roll oscillation should be based on a GM value of about
56% of the initial GM valid for small roll angles. Taking into account the
characteristics of the righting levers, this becomes obvious.
If on the other hand the ship has a significant positive amount of form stability (the
righting levers will be above the GMφ- line), then the linearization of the problem with
respect to the initial GM will always result in roll periods being too large.
As the area below the righting lever curve is a measure of the energy which is stored
in the oscillating system, a so called effective GM can be determined by the principle
of area balance as shown in Fig 2 right: The effective GM for the oscillation 2 – 2’ is

determined by obtaining the same area under the righting lever curve A2 compared
to A1.
3. Natural Roll Periods in Waves
If the ship operates in either following or head seas, the still water righting lever curve
is not relevant any more. The effect of the waves may be approximated by the crest
curve (the wave crest is at mid ship) or the wave trough curve (the trough is at mid
ship). As it can be seen from Figs 1. or 2., right, the righting levers may differ
significantly from the still water curve. In extreme cases, the initial stability may even
become negative on the wave crest. If the initial GM is negative, a roll period
determination by the Weiss’ formula is impossible. In this condition, the ship will not
oscillate around the unstable upright condition, but around the static equilibrium
which is denoted by EQ in Figs.1 or 2. But with respect to the roll motion in waves,
the oscillation around that static equilibrium is not of interest, as the ship in waves
oscillates roughly around the upright position at about zero heel. With respect to that
condition, the momentary value of the roll acceleration is not obtained from an
oscillation, but simply from the formula:
The roll acceleration is obtained from an effective heeling moment (based on the
negative righting lever and the damping) and the vessel simply heels to the static
equilibrium EQ and then starts to oscillate around this condition. For this oscillation
around that equilibrium, the Weiss’ formula could again be used. But as mentioned
above, this kind of oscillation is not of interest in this context. Because for all
situations where the ship is close to a critical resonance, the crest is at mid ship when
the vessel is nearly in an upright position, and the vessel simply starts to heel close
to the static equilibrium. The ship is then rightend up again when the crest passed
towards her end. In this condition, the trough curve is valid. As in this position the
ship has sufficient positive initial stability, the roll oscillation is again well defined.
So the vessel permanently is in one extreme situation where the roll oscillation
around the upright position is not existent or the other one where the roll oscillation is
governed by the trough righting levers. Therefore, it is suggested to use an averaged
crest- trough righting lever curve instead of the still water righting lever to better take
into account the effects of the sea state. This concept follows a well known approach
by Wendel that was used to define the stability standard of the German Navy BV
1033. Therefore, according to the present proposal, the natural roll period of a ship
in a natural seaway should be determined as follows:
• The natural roll period shall be based on an artificial stability curve which is the
average of the crest and trough righting lever curve in a reference wave.
• Not the initial GM shall be used as input for the Weiss formula, but an effective
GM which is based on the area equality below this artificial righting lever curve
and the linearization for a certain angle of interest.

This concept is tested on two extreme cases in the following sections.
4. Test applications
4.1. Negative form stability
Fig. 3: Polar diagrams of numerical capsizing simulation of a capsizing accident (left)
and related righting levers (right).
The concept is applied to a full scale capsizing accident of a RoRo ferry. The ship
capsized in following seas at about 16 kn speed and an encounter angle of abt.15
degree in waves having a significant period of abt. 7.5-8.5s (equivalent significant
wave length 88- 113m). The polar diagrams, left, show the limiting significant wave
heights as function of course and speed which lead to a capsize. The numerical
simulations of that accident (Fig.2, left) show two areas in following seas where the
ship is extremely endangered: One area at about 6-8 knots and one area at about
15- 17 knots. These are related to the typical resonances, namely 1:1 at about 16 kn
and 2:1 at about 6 knots (the rough centre of these areas). From Fig. 2, right, the fact
can be derived that the initial GM of the still water righting lever curve amounts to
1.68 m and is valid for small angles only. Based on the still water righting lever curve
and the related initial GM, a still water natural roll period of 13.3. s can be computed
for small angles. If the 1:1 or 2:1 resonance in a wave having a significant period of
8.5 s shall be met, this would result in the following critical ship speeds:
The 1:1 following sea resonance should be found at a speed of 9.63 kn, and the 2:1
should not be possible in following seas (it is computed at –7.5 kn which means head
seas). Compared to the directly computed resonance areas, these results are
completely wrong. The reason for this becomes clear when the righting levers are
regarded, as the still water curve does not at all represent either the trough or crest
condition, and that further, the initial GM does not represent even the still water
stability at all.
Based on the approach suggested in this paper, the problem should be linearized
based on the crest trough average (the magenta curve in Fig. 3, right) and the area
equivalent. This would in fact lead to an effective GM of about 0.68m (if an angle of
40 Degree is used as reference) and a natural roll period of 21s. Based on these 21s

natural roll period computed for 113 m wave length and 5m wave height, the 1:1
following sea resonance is computed at 15.9 knots and the 2:1 resonance at 5.1
knots. This fits much better to the real situation and shows that a simple approach
can in principle give useful results if the problem is linearized more accurately.
4.2. Positive form stability
Fig. 4: Polar diagrams of numerical roll motion simulation of a cargo loss accident
(left) and related righting levers (right). The right polar diagram is valid for a roll angle
of 15 Degree, the left for a roll angle of 40 Degree.
As alternative test case, a cargo loss event is analyzed. Large roll angles on a large
container vessel occurred in following seas at a speed of about 21 kn, encounter
angle 10 Degree, equivalent significant wave length about 170m, significant wave
height abt 5-6 m. The polar plot indicates the 1:1 following sea resonance clearly at
about 18 kn and the 2:1 following sea resonance roughly at about 4-5 knots. Based
on the still water righting lever curve and an initial GM of 0.65m, a natural roll
Stillwater roll period of 32.1 s is calculated (for small angles). Based on this value, the
1:1 resonance should be found at 21.7 kn and the 2:1 at 11.30 knots. These values
are actually not as bad as for the previous case, but still remarkably inaccurate It
must be noted that the difference between the still water righting lever and the
trough crest average is not very large for this example, which makes the
approximation better than before. However, if the natural roll period is determined
from the crest – trough average and 40 degree area equivalent, this results in an
effective GM of 1.1 m and a roll period in waves of about 24.4 s. Based on these 24.4
s, the 1:1 resonance should be at 18.7 kn and the 2:1 resonance should be at about
4.7 knots. Both values are much better in line with the numerical simulations for
both large and smaller roll angles compared to the estimation by using still water
initial metacentric height.
4. Conclusions
A simple straightforward method was suggested to better estimate the natural roll
period of a ship in heavy weather. The method is still based on the linearization of the
roll motion in waves, but uses an effective GM which is based on the crest trough

average of the righting lever curves in a specific reference wave. The reference wave
length may be derived from the significant period of a given wave spectrum. As the
method still linearizes the roll problem, the results are not correct enough to actually
replace fully nonlinear simulations. But two practical applications have shown that the
estimation of critical resonance situations will in most cases be more reliable than a
simple guess based on the still water roll period for small roll angles.