A Simplified Method for Estimating Roll Frequency

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A Simplified Method for the Estimation of the Natural RollFrequency 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 theattention of the Maritime community to the fact that under certain conditions, suddenlarge roll angles can occur. These sudden large roll angles typically occur in head orfollowing 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 andthe 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 andcargo damage. Typically, these situations occur if the encounter frequency of thewaves is in resonance to the natural roll frequency of the ship. As the rolling of shipsin heavy weather is equivalent to a forced oscillation, it is obvious that a resonancesituation will always occur if the excitation frequency is equal to the system’seigenfrequency or to higher orders of that eigenfrequency. Most dangerous is the socalled 2:1 resonance, where two pitch cycles coincide with one roll cycle. If thestability 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 theencounter period takes large values. If on the other hand the roll period is small dueto 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 theencounter frequency of the waves, which results in a situation where one pitch cyclecoincides with one roll cycle. For typical ships, this situation is possible in followingsseas at higher speeds when the stability is more on the low side. However, not allpossible resonances or close to resonance situations must necessary result in theoccurrence of large roll angles. If the exciting forces are sufficiently small (this meansthat stability alterations are below a certain threshold value) or if the roll damping islarge enough, the roll motion may remain moderate. On the other hand it is alsopossible that large roll angles may occur in situations that are far beyond aresonance situation. Therefore, the only accurate way to actually judge upon the rollangles in a specific head or following sea scenario is actually a full numericalsimulation.

On the other hand, there are some practical applications where it would be useful if adecision could be made to check if a critical resonance situation is actually met. Thiswould require a comparison of the encounter frequency (or period) and the naturalroll frequency (or period) of the ship. The latter is mostly computed based on the stillwater 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 bepotentially dangerous or not. In some cases, this procedure may be successful, butthere have been also many cases been identified where this rough assumption doesnot 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 significantlyalter its roll period when operated in waves. As a matter of fact, for these reasons it ispractically not possible to judge upon a potentially dangerous situation if theestimation of the ship’s natural roll period is based on the still water approach forsmall 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 possibleresonances, although the method which is suggested below will not be accurateenough 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 rightinglevers for Stillwater, crest and trough

The principle determination of the natural roll period is shown in Fig.1. The left sideshows a typical roll decay test which may also (theoretically) be performed with a fullscale ship. The ship is inclined to an initial heel φo (which is small for practicalreasons) and then released. The result is an oscillation around the static heel ofequilibrium, and the ship oscillates in its own natural roll period. As the oscillation isdamped, the amplitudes decrease. From such a test, the natural roll period and thedamping increment can be determined. As the ship may be idealized as a onedegree of freedom oscillation system, its natural period can also theoretically be

determined based on a characteristic restoring moment and a characteristic massmoment 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, whichis obtained from the mass moment of inertia around the x- axis of the ship and theactual deplacement. For most ships, the roll radius of gyration (including sectionadded 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 to0.45B. It is very important to note that the formula expresses the restoring lever bythe 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, theroll oscillation takes place between the points denoted by 1 and 1’ in Fig. 1, right. Forthis 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 GMφ-line. 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 oflinearization becomes then inaccurate for a large amplitude roll motion. E.g. for a rolloscillation between the points 2 and 2’ the actual restoring moment would besignificantly smaller than the linearization, as the actual righting lever is completelybelow 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 largeinitial roll angle. In comparison to Fig.1, the roll damping was reduced to keep thelarge roll amplitudes for a longer time. The initial roll oscillation is equivalent to thepoints denotes by 2 and 2’ in Fig. 2, right. It can clearly bee seen that initially, thenatural roll period amounts to roughly 3 roll cycles per 50 s (16.7s), whereas after thedecay of the roll amplitude the natural roll period is decreased to about 4 roll cycles in50 s (12.5s). This is in line with the non linear characteristics of the related rightinglever curve in Fig.1, right: At larger heeling angles, the righting lever is significantlybelow 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 rolloscillation. From the resulting roll periods it can be seen that a correct roll perioddetermination for the large roll oscillation should be based on a GM value of about56% of the initial GM valid for small roll angles. Taking into account thecharacteristics of the righting levers, this becomes obvious.

If on the other hand the ship has a significant positive amount of form stability (therighting levers will be above the GMφ- line), then the linearization of the problem withrespect 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 principleof 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 comparedto A1.

3. Natural Roll Periods in Waves

If the ship operates in either following or head seas, the still water righting lever curveis not relevant any more. The effect of the waves may be approximated by the crestcurve (the wave crest is at mid ship) or the wave trough curve (the trough is at midship). As it can be seen from Figs 1. or 2., right, the righting levers may differsignificantly from the still water curve. In extreme cases, the initial stability may evenbecome negative on the wave crest. If the initial GM is negative, a roll perioddetermination by the Weiss’ formula is impossible. In this condition, the ship will notoscillate around the unstable upright condition, but around the static equilibriumwhich 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 wavesoscillates roughly around the upright position at about zero heel. With respect to thatcondition, the momentary value of the roll acceleration is not obtained from anoscillation, but simply from the formula:

The roll acceleration is obtained from an effective heeling moment (based on thenegative 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 oscillationaround that equilibrium, the Weiss’ formula could again be used. But as mentionedabove, this kind of oscillation is not of interest in this context. Because for allsituations where the ship is close to a critical resonance, the crest is at mid ship whenthe vessel is nearly in an upright position, and the vessel simply starts to heel closeto the static equilibrium. The ship is then rightend up again when the crest passedtowards her end. In this condition, the trough curve is valid. As in this position theship has sufficient positive initial stability, the roll oscillation is again well defined.

So the vessel permanently is in one extreme situation where the roll oscillationaround 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 averagedcrest- trough righting lever curve instead of the still water righting lever to better takeinto account the effects of the sea state. This concept follows a well known approachby Wendel that was used to define the stability standard of the German Navy BV1033. Therefore, according to the present proposal, the natural roll period of a shipin a natural seaway should be determined as follows:

• The natural roll period shall be based on an artificial stability curve which is theaverage 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 effectiveGM which is based on the area equality below this artificial righting lever curveand 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 shipcapsized in following seas at about 16 kn speed and an encounter angle of abt.15degree in waves having a significant period of abt. 7.5-8.5s (equivalent significantwave length 88- 113m). The polar diagrams, left, show the limiting significant waveheights as function of course and speed which lead to a capsize. The numericalsimulations of that accident (Fig.2, left) show two areas in following seas where theship is extremely endangered: One area at about 6-8 knots and one area at about15- 17 knots. These are related to the typical resonances, namely 1:1 at about 16 knand 2:1 at about 6 knots (the rough centre of these areas). From Fig. 2, right, the factcan be derived that the initial GM of the still water righting lever curve amounts to1.68 m and is valid for small angles only. Based on the still water righting lever curveand the related initial GM, a still water natural roll period of 13.3. s can be computedfor 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 2:1should not be possible in following seas (it is computed at –7.5 kn which means headseas). Compared to the directly computed resonance areas, these results arecompletely wrong. The reason for this becomes clear when the righting levers areregarded, as the still water curve does not at all represent either the trough or crestcondition, and that further, the initial GM does not represent even the still waterstability 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 areaequivalent. This would in fact lead to an effective GM of about 0.68m (if an angle of40 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 1:1following sea resonance is computed at 15.9 knots and the 2:1 resonance at 5.1knots. This fits much better to the real situation and shows that a simple approachcan 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 angleof 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 largecontainer vessel occurred in following seas at a speed of about 21 kn, encounterangle 10 Degree, equivalent significant wave length about 170m, significant waveheight abt 5-6 m. The polar plot indicates the 1:1 following sea resonance clearly atabout 18 kn and the 2:1 following sea resonance roughly at about 4-5 knots. Basedon the still water righting lever curve and an initial GM of 0.65m, a natural rollStillwater roll period of 32.1 s is calculated (for small angles). Based on this value, the1:1 resonance should be found at 21.7 kn and the 2:1 at 11.30 knots. These valuesare actually not as bad as for the previous case, but still remarkably inaccurate Itmust 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 theapproximation better than before. However, if the natural roll period is determinedfrom the crest – trough average and 40 degree area equivalent, this results in aneffective GM of 1.1 m and a roll period in waves of about 24.4 s. Based on these 24.4s, the 1:1 resonance should be at 18.7 kn and the 2:1 resonance should be at about4.7 knots. Both values are much better in line with the numerical simulations forboth large and smaller roll angles compared to the estimation by using still waterinitial metacentric height.

4. Conclusions

A simple straightforward method was suggested to better estimate the natural rollperiod of a ship in heavy weather. The method is still based on the linearization of theroll 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 wavelength may be derived from the significant period of a given wave spectrum. As themethod still linearizes the roll problem, the results are not correct enough to actuallyreplace fully nonlinear simulations. But two practical applications have shown that theestimation 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.

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A Simplified Method for Estimating Roll Frequency