f10
From earlier publications some simple but qualitative methods are known for estimating the potential of resonance and therefore high rolling amplitudes, based on the comparison of the ships natural rolling period and the period of wave encounter. The results of these methods were presented in several types of diagrams, though lacking a clear presentation and therefore understanding of direct countermeasures. In a 1990 Nautical Consulting and Information system a polar diagram representation of results was used, which was close to Radar like presentation mode already. However, the calculation of potential dangerous situations for synchronous resonance was based on an evaluation of each single encounter situation and so called parametric resonance and other wave effects were not yet considered. In the paper a simple method is shown for the onboard calculation of the information necessary to prepare a polar diagram for synchronous and parametric resonance and other wave effects from basic data of the ship and the sea state, even by manual calculation. It is also possible to include the potential danger of high wave group encounter or Surfriding and broaching respectively.
As a result the tendencies of several effects like the ships' natural period and seastate parameters can be shown. The effect of counter measures, for instance changes of speed, course and ships' stability/roll time period can be discussed.
Aus der Literatur sind einfache, qualitative Verfahren bekannt, mit denen man durch einfachen kinematischen Vergleich der Eigenschwingungsperiode des Schiffes mit den Erregungsperioden durch Seegang auf potentielle Resonanznähe und damit große Amplituden schließen kann. Nachteilig ist bisher u. a. die mangelnde Übersichtlichkeit der Darstellung in verschiedenen Diagrammformen zur leichten Ableitung von Maßnahmen für die Schiffsführung. Für ein Nautisches Beratungssystem wurde 1990 erstmals eine Darstellung als übersichtliches Polardiagramm ähnlich einer Radarspinne gewählt, allerdings erfolgte die Berechnung der Darstellung durch aufwendige Einzelauswertung für eine Vielzahl von Einzelsituationen und es mangelte an der fehlenden Einbeziehung der so genannten parametrischen Resonanz und anderer Effekte.
Es wird im Vortrag eine einfache Methode zur Berechnung der notwendigen Informationen für die Darstellung von potentiell gefährlichen Bedingungen für synchrone und parametrische Resonanz in einem Polardiagramm aus den Grunddaten des Schiffes und des Seeganges gezeigt, die sogar auch manuell an Bord durchgeführt werden kann. Informationen zur Beachtung der Gefahr der Wirkung von hohen Wellengruppen bzw. Surfriding und Broaching entsprechend der BerechnungsVorschläge der IMO können einbezogen werden.
Als Ergebnis lassen sich Tendenzen für die Wirkung von bestimmten Einflüssen wie Seegangsperiode und Eigenperiode des Schiffes und von Maßnahmen zur Gefahrenabwendung wie Änderungen von Kurs, Geschwindigkeit und Anfangsstabilität/Rollperiode erkennen.
Fig. 1: Representative Container Damages
The following phenomena can occur when a ship is affected by high sea state:
The effects and occurrence of these phenomena are described in Tab. 1.
Phenomena  Occurrence  Effect  

Direction  Periods/Encounter  
1. Synchronous rolling motion  All directions possible  Natural rolling period of a ship coincides with the encounter wave period.  Heavy oscillations with high amplitude 
2. Parametric rolling motion  Specifically for head and stern wave conditions  Wave encounter period is approximately equal to half of the natural roll period of the ship  Heavy oscillations with high amplitude 
3. Reduction of stability riding on the wave crests of high wave groups  Following and quartering seas  Wave length larger than 0,8 x ship length and significant wave height is larger than 0,04 x ship length  Large roll angle and capsizing 
4. Surfriding and broachingto  Following and quartering seas  The critical wave speed is considered to be about 1,4√L ~ 1,8√L with respect to ships' length  Course deviation and capsizing 
Tab. 1: Table of dangerous phenomena
In 1990 a Nautical Information and Consulting System (NCIS) was developed at the Maritime Academy Warnemuende und manufactured by STN Atlas Elektronik (SAM). As one of the most important parts of this system the resonance module could give advice on potential danger of synchronous resonance, even for two wave systems in parallel and it could derive countermeasures for it. However, this was not possible by manual calculation anymore
Linnert presented in 1997 a method to predict the roll angle amplitudes for a ship in waves in a quantitative way by means of transfer functions applied to a specific wave spectrum [5]. However this method could not predict potential effects for direct head or stern wave encounter. The comparison with the method to be presented here in this paper in Chapters 2.3 and 2.4 shows clearly that the stripes for synchronous resonance cover the result in a sufficient way: the stripe is located at the maximum of the roll amplitudes calculated by the method of Linnert, whereas the parametric roll resonance effects are not achieved by his method.
Krüger and others [9], [10] published methods and results for calculating critical conditions in waves in polar diagram presentation mainly for the probability of capsizing. These methods are based on full simulations taking into account hydrodynamic effects to provide the wave heights the ship can stand before capsizing: They are dedicated more to the ship design process than to ship operation and need high computer power and are time consuming.
Ammersdorffer published examples on how to calculate ships natural rolling periods for large roll amplitudes and a diagram for estimating the speed limits for avoiding parametric resonance in stern sea [6]. However, this diagram is not a user friendly approach to derive countermeasures in a convenient way for ship operation for fast decision making.
This year (2003) a draft of a new German guideline for stability on board ships was prepared [8]. Some of the results of Ammerstorffer were implemented into the draft. This guideline also provides polar diagrams for the estimation of potential dangerous situations in order to avoid synchronous and parametric resonance; however they have to be prepared for each ship separately and for sets of the ships' natural rolling period (see for Tr = 10 s). Furthermore the sea state is defined from North with several wave periods, therefore one would have to calculate and transfer course and wave direction to actual ship conditions before using it for decision making.
Additionally the IMO has published guidelines [4] to the master for avoiding dangerous situations in following and quartering seas to be aware of several effects due to sea state which results in:
For both of these diagrams which are provided in dimensionless form it is not suitable to have to calculate the ratio V/T and to transfer Course and wave direction beforehand.
Summarizing all the methods reviewed above, we then come to the conclusion that the current methods are lacking:
This report describes how to effectively find out the potentially dangerous situations. Simplified calculation methods are given which were developed at the Department of Maritime Studies Warnemuende and presented e.g. at the Meeting of Seamanship Lecturers of Germany in 1997. These methods aim to manually calculate a polar diagram presentation e.g. on RADAR Plotting Sheets. The methods are designed for the specific ship situation to allow for an effective analysis of the potential danger of the ship in sea state. In contrary to the methods presented in the new guideline for stability [8] the use of many ship specific predefined forms can be avoided, the overview is related to the ships heading and speed in a direct way. Furthermore information on dangerous situations with regard to surfriding and broaching as well as high wave group encounter will be included to get a full overview of the ships' condition.
To calculate the rolling period Tr of a ship one can apply the so called WEISSFormula for small roll angles up to Φ ≈ 5 or even to Φ ≈ 10°:
With:
For large roll angle amplitudes up to Φ ≈ 40° or more the roll period can change, compared to the period Tr(10°) for small angles. The magnitude of the difference is according to the type of the stability curve. There are three types of curves:
With:
The main characteristic types of wave systems are:
The wave period Tw is the period a fixed observer would time between the passing of two consecutive wave crests or two consecutive wave troughs. The wave period directly corresponds to the wavelength Lw. The following relation holds between the wave length and wave speed for harmonic waves:
where k denotes the coefficient for the wave system (wave number), which is:
The encounter situation between ship and waves is very important for the wave impact: The ship will be forced into oscillation exited by the encounter period TE between ship and sea. This period depends on:
The following typical conditions for Encounter periods TE of a ship with speed V in waves with wave speed Cw and wave period Tw can be distinguished:
For general encounter situation the encounter period TE can be calculated as to:
With:
(For conditions where the ship is overtaking the waves the wave speed has to be considered as negative in the denumerator)
Speed vectors of the ship should be represented in polar diagram like a Radar screen format, but instead of the distances the speed is used as coordinate axis. The magnitude of the encounter period is dependent on the component V * cos γ of the ship speed V in direction onto the waves. Therefore all of the speed vectors V on the different courses (thin blue arrows) have the same component length (thick blue line).
All conditions with the same encounter period are on that one line (red) orthogonal to the direction of the waves (green).
This allows for calculating and plotting a polar diagram for assessing wave effects for ship operation in a very simplified way as in Fig. 15: The only task is to draw a line in the direction of the wave propagation and to calculate the encounter speed values (indicated by small circles) for several wave effects on the specific courses with direct head or following sea only. Then the shapes of areas with potential danger can be drawn. In the following chapters these effects will be explained and formulas will be given to easily calculate these basic speed values.
Synchronous resonance occurs when the ships' natural period Tr and the encounter period TE have nearly the same value:
These conditions are represented in red colour in the diagram Fig. 10:
Fig. 10: Amplitudes for synchronous roll resonance (red) and parametric resonance (brown, dotted): Ratio a of rolling and exiting wave amplitudes versus ratios between ships rolling period Tr and wave encounter TE
In the resulting polar diagram synchronous resonance conditions are to be seen as red stripes whereas specifically the Direct Resonance condition is represented as a red line nearly in the middle of the stripes and the conditions for 50 % lower amplitudes are at the outer border lines.
These conditions are represented by the graph in brown colour in the diagram Fig. 10. In the resulting polar diagram Fig. 15 they are to be seen as red sector segments in head or stern seas where the Direct Resonance conditions are represented as a black line nearly in the middle of the segment and the conditions for 50 % lower amplitudes are at the outer border lines. These conditions are represented in the polar diagram as red sector segments ± 30° off the wave direction.
This type of rolling can occur in head and bow seas where the wave encounter period is exiting the ship preferably by the effects due to the stability change when on wave crest or in wave trough as indicated in Fig. 11 and Fig. 12. Therefore the excitation is high specifically for those types of vessels with large differences of the stability at the respective wave positions as for instance modern container vessels. Today's ship hull forms are different from earlier designs:
For new container vessels with a "pontoon" stern shape and tremendous bow flare this exiting effect is larger than for the conventional ship hull form in earlier times.
For the two different position of wave crest at the ship length in Fig. 11 the effect on uprighting forces and moments is shown in Fig. 12. The uprighting moment is:
in comparison to still water conditions.
Fig. 12: Comparison of transverse stability curves for different wave positions at midships
Besides the danger of reduction of stability when the ship is riding on the wave crest for a long time there is also an exciting effect of waves in Head/Stern Sea when the waves are travelling along the ships hull periodically – this will yield potential for parametric rolling described in chapter 2.4. This leads to extreme dangerous situation when several high waves will trigger the ship coming as a group.
The IMO has given in the guidelines [4] a diagram highlighting the potential occurrence of high wave group encounters; however, the information is given in a dimensionless format only by a ratio of ships speed V and wave period Tw, χ is the encounter angle seen by 0° from stern.
Fig. 13: IMODiagram Indicating Dangerous Zone due to High wave group encounters [4]
Definition of Symbols Used:
Here the new polar presentation can have its benefit by relating the data to the current values of ships speed and wave period/direction with the potential of High wave group encounter as for example is given in Fig. 15: The segment for direct following and quartering seas ± 45° is shown as blue dot and dash area.
Fig. 14: IMODiagram Indicating Dangerous Zone Due to Surfriding and Marginal Zone in dimensionless form [4]
Here a new polar diagram can have its benefit by relating the data to the current values of ships speed and length as well as wave direction with the potential of surf riding/broaching to as for example is given. The segment for direct following and quartering seas ± 45° is shown in green colour in Fig. 15.
The dangerous surfriding and broachingto conditions are indicated by a green sector filled with full line, the sector with broken lines is representing the marginal zone below the critical speed, where a large surging motion may occur, which is almost equivalent to surfriding in danger.
The following Tab. 2 summarizes the effects and formulas for calculating the circles with the respective colours to the numbers of the formula in the table:
Phenomena  Direction/Section/Area  Equations to Calculate the speed values as basis for the Diagram Elements  

1. Synchronous rolling motion  Stripe segments over diagram; all directions possible 
 
2. Parametic rolling motion  Segment for direct head and stern wave conditions ±30° 
 
3. Reduction of stability riding on the wave crest of wave groups  Segment for direct Following and quatering seas ± 45°  5.
6.  
4. Surfriding and broachingto  Segement for direct Following and quartering seas ± 45°  7.
8. 9. 
Tab. 2: Summary of effects and formulas for calculation of basic polar diagram values
The results will be used to draw specific shapes of areas with potential danger in a Polar Diagram (see example) taking the speed values (in the circles) as a basis for the diagram elements:
By means of the polar diagram the following general tasks can be identified:
The basic speed V_{1.0} for direct resonance we get from:
Enter basic speed values: Now we have to complete the polar diagram with the basic speed values at angles of encounter at γ = 0° or γ = 180°. The term cos γ is not itemized in the above formulas because cos 0° = 1 or cos 180° = 1 respectively. As the case may be the result of V is:
In this task we get negative results, therefore all speed values above will be drawn in the polar diagram in direction with following sea measured from the centre (magenta circles)
Draw stripes for synchronous resonance zone: The synchronous excitation zone will be drawn as stripe over the whole angle area of the polar diagram, through the speed values in the magenta circles, rectangular to the sea direction.
For direct resonance the value is in between these results:
Enter basic speed values: In this task we get positive results for the basic speed values, therefore all speed values will be drawn in the polar diagram onto the wave direction line with head sea measured from the centre (red circles)
Draw sector stripes for parametric resonance zone: The parametric excitation will be drawn only for a sector segment around the sea direction of ±30° for directions of stern sea or against sea respectively, through the speed values in the red circles, rectangular to the sea direction.
 
 

Since they only occur in stern seas they will be drawn as negative speed on the wave direction line for the course with following seas. This applies for the next areas and speed as well:
Enter basic speed values: In this task we get negative results for the basic speed values, therefore all speed values will be drawn in the polar diagram onto the wave direction line with following sea measured from the centre (blue or green circles respectively).
Draw sector stripe segments for surfriding and encounter of wave groups zone: The zones for surfriding and encounter of wave groups will be drawn only for a sector stripe segment around the sea direction of ±45° for directions of stern sea, through the speed values in the blue or green circles respectively, rectangular to the sea direction.
By the way: These sectors are the same as to be seen in Fig. 2, where for the same example the calculation had to be done separately for these sectors only with much more time consumption and less overview on the total situation.
Fig. 16: Polar diagram for semi container and course sectors with synchronous resonance conditions at speed V = 8 kn (left) and speed range for avoiding resonance (right) at course = 140° (GM = 1,70 m: Tr(10°) = 10 s, Tw = 8 s) 
To assess the suitability of the result the stability has to be discussed for these roll periods calculation the respective GM values:
Another criterion is the required change ΔGM which is necessary to achieve the new GM_{0.8} in comparison with the initial GM_{init}
This relative small change might be better achievable than the larger change required in case of using the GM_{0.8} instead.
Fig. 17: Shifted resonance areas after changes of rolling period due to change of GM:  
a) Results for Tr(10°) = 11,3 s GM = 1,32 m  b) Results for Tr(10°) = 8,24 s at GM = 2,50 m. 
For the calculation of the ship's natural roll period for large roll angle amplitudes we will use the following formula:
The given values of GM = 1,7 m and the uprighting levers GZ for roll angles10° to 40° (GZ_10 = 0,32 m; GZ_20 = 0,8 m; GZ_30 = 1,6 m; GZ_40 = 1,9 m) will be used for the calculation of the following formulas:
This result is the new ship's natural roll period for large roll angle amplitudes:
It is smaller than the period for smaller roll angles Tr(10°) = 10 s, because the Uprighting lever is higher then the tangent with respect to GMvalue.
For the smaller period Tr(40°) = 7,86 s for large roll amplitudes the areas for synchronous and parametric resonances are indicated by brown colour in Fig. 18; they have been shifted towards the wave direction in comparison to the red coloured areas for the smaller amplitudes. This is important to know: If in the red areas the roll angles starts to increase due to resonance effect, it is for instance not to recommend to increase the speed against the waves because one would enter the resonance for higher roll amplitudes now!
[1]  France, William a.o.: An Investigation of HeadSea Parametric Rolling and its Influence on Container Lashing Systems. SNAME Annual Meeting 2001. 
[2]  Hilgert, H. a.o: Nautische Stabilitätsbilanzen (Heft 2). Lehrmaterial, Hochschule f. Seefahrt WarnemündeWustrow, 1991. 
[3]  Scharnow, U.: Schiff und Manöver – Seemannschaft 3, Transpress, 1987 
[4]  IMO Guidance to the master for avoiding dangerous situations in following and quartering seas, MSC circular 707, adopted on 19. October 1995. 
[5]  Linnert, M.: Schiffsführung abgestimmt auf das Seegangsverhalten?, HANSA 134. Jahrgang, Heft 7 1997, S. 13 ff. 
[6]  Ammersdorffer, R.: Parametrisch erregte Rollbewegungen in längslaufendem Seegang. in Schiff&Hafen Hefte 1012, 1998. 
[7]  IMO Code on Intact stability for all types of ships, Resolution. A.749(18) Nov 1993 
[8]  BMVBW / SeeBerufsgenossenschaft: Richtlinien für die Überwachung der Schiffsstabilität. Draft version 2003. 
[9]  CRAMER, H., KRUEGER, S. (2001) Numerical capsizing simulations and consequences for ship design JSTG 2001, Springer. 
[10]  HASS, C. Darstellung des Stabilitätsverhaltens von Schiffen verschiedener Typen und Groesse mittels statischer Berechnung und Simulation Diploma Thesis, TU HamburgHarburg (2001). 