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Beitrag 10

Estimating Potential Danger of Roll Resonance for Ship Operation

Prof. Dr.-Ing. habil Knud Benedict, Dr. Michael Baldauf, Dipl.-Ing. Matthias Kirchhoff
Hochschule Wismar – University of Technology, Business and Design; Dept. of Maritime Studies

  1. Introduction into Dangerous phenomena for ships in heavy seas, some methods of calculation and aim of this paper
    1. Introduction and effects
    2. Brief overview on existing methods and aim of this paper
  2. Description of effects and methods / Theory Background
    1. Ships motion and Ship natural rolling periods
    2. Sea state and encounter period to waves
    3. Conditions for resonance – Synchronous Rolling Resonance
    4. Parametric Rolling Resonance
    5. Dangerous Stern Wave Encounter and high wave groups
    6. Broaching and "Surf-Riding"
  3. Summary of effects and formulas, general task description and example
    1. Summary of effects and formulas, general task description
    2. Example: Calculation for a Semi container ship / Specific Task description
    3. Solution Details for the specific Example Semi container ship
  4. References


Abstracts

Lately, there have been several instances of ships being badly damaged due to heavy rolling motion in sea state, which clearly shows the need for a method to estimate the potential danger in order to support the work of the ships' officers.

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 on-board 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 Surf-riding 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.


Kurzfassung:
Abschätzung potentieller Gefahr bezüglich Rollresonanz für Schiffsführung

In der letzten Zeit sind mehrfach große Schäden durch starkes Rollen von Schiffen im Seegang aufgetreten, so dass ein Bedarf für die Abschätzung von Gefahren besteht.

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. Surf-riding und Broaching entsprechend der Berechnungs-Vorschläge 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.


1. Introduction into Dangerous phenomena for ships in heavy seas, some methods of calculation and aim of this paper

1.1. Introduction and effects

Over the last few years several vessels have experienced the dangerous effects of rolling resonance. Two results of these harmful encounters are shown in Fig. 1. According to an article in SNAME [1] the damage to one ship on its left hand side was caused by a heavy storm on 26th of October 1998. The named C11 vessels are the second generation of Post-Panmax containerships and it is very important to find out why these situations occur and what the countermeasures are to avoid potential damage by proper changes of course, speed or stability of the vessel respectively.

Fig. 1: Representative Container Damages

The following phenomena can occur when a ship is affected by high sea state:

  • Synchronous rolling motion
  • Parametric rolling motion
  • Reduction of intact stability caused by riding on the wave crest amidships, especially in high wave groups
  • Surf-riding and broaching-to
  • Combination of phenomena listed above

The effects and occurrence of these phenomena are described in Tab. 1.

PhenomenaOccurrenceEffect
DirectionPeriods/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. Surf-riding and broaching-to 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


1.2. Brief overview on existing methods and aim of this paper

Over the last decade there have been many investigations into this and publications can be found on how to calculate these effects but either they do not cover all the effects which are mentioned or they were not designed to support the ship's crew in order to give effective guidance for the operation of ships. Some examples are given below:
Hilgert [2], [3] has developed a method to calculate the potential occurrence of synchronous resonance to manually prepare sector diagram as in Fig. 2, however this was done for single values of a ships' speed or course and does not give the full overview of the situation.

Fig. 2: Manual prepared sector diagram indicating the potential synchronous resonance sectors in wind waves from 23° for one specific ship speed value V = 8 kn (Hilgert 1990)

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

Fig. 3: Computerized Presentation of potential dangerous Areas of speed and heading in polar diagram as Stripes for synchronous resonance and interference effects for two wave directions (NCIS)

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.

Fig. 4: Presentation of roll angle amplitudes (maximum in red colour) for a ship in waves and comparison with stripe of synchronous resonance and sector for parametric resonance with the method developed in this paper (indicated by blue dotted lines)

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.

Fig. 5: Polar Diagram for estimation of potential dangerous situation in order to avoid synchronous and parametric Resonance for one single ships' natural rolling period (10 s) and sea state from North with several wave periods Tw

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:

  • IMO-Polar-Diagram Indicating Dangerous Zone Due to Surf-riding and Broaching and Marginal Zone (see Fig. 13) and
  • IMO-Polar-Diagram Indicating Dangerous Zone due to High wave group encounters (see Fig. 14).

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:

  • Simplified, user friendly approach for calculation of the effects, may be given manually for education and training and for use on board
  • An overall approach for presentation of all effects in one diagram using polar representation, including IMO recommendations for Surf-riding, Broaching-to und Wave group encounter for fast decision making for counter measures.

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.


2. Description of effects and methods / Theory Background

2.1. Ships motion and Ship natural rolling periods

The ships' motion can be generally subdivided into 6 degrees of freedom. For the problems handled within this paper we will mainly focus on rolling motion and the surge/sway and yaw motion for the surf riding and broaching.

Fig. 6: Ships motions related to the axis of the hull

To calculate the rolling period Tr of a ship one can apply the so called WEISS-Formula for small roll angles up to Φ ≈ 5 or even to Φ ≈ 10°:

With:

  • GM – Initial stability, metacentric height [m]
  • B – ship's beam [m]; Lpp – ship's length [m]
  • Cr – the inertia coefficient for rolling motion (or rolling time coefficient) is in the range of about 0,75 < Cr < 0,80. At ships with a high decks load Cr can have higher values, with RoRo-Ships Cr can be even ~1. It can be taken from the yard's ship documentation, from own observed data or can be calculated acc. to the IMO-Guidelines as to Cr = 2*c with
    • c = 0,373 + 0,023(B/d) − 0,043(Lpp/100)
    • or by other suitable determination methods.
    • It is advised to check the Cr data given in the ships database by the comparison of GM-results from the ships inclining test and roll time measurements. This enables precise Cr values which can be calculated by
    • Cr = √GM (inclining test) * Tr (roll time test) / B

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:

  • Nearly linear gradient up to the maximum, proportional to the tangent according to GM (i.e. Tr(10°) = Tr(40°)), indicated by the green broken line
  • Strong over-proportional increase of the up-righting lever up to the maximum, indicated by the blue full line, i.e. Tr(10°) > Tr(40°),
  • Strong under-proportional decreased curve, i.e. Tr(10°) < Tr(40°), indicated by the red dashed line.

Fig. 7: Different graphs of up righting lever GZ for the same GM versus roll angle Phi To calculate the rolling period Tr(40°) for rolling angles F up to 40° the following formula can be used according to [6]:

With:

v = 0,6 * GZ_40
w = GZ_20 + 4 * GZ_30 + 1,6 * GZ_40
x = w + 1,5 * GZ_10 − 3 * GZ_20 − GZ_30
y = w + 2,5 * GZ_10 + GZ_20
z = y + 1,5 * GZ_10

  • B – ship's beam [m]
  • Cr – the inertia coefficient for rolling motion (or rolling time coefficient)
  • GZ – Up righting lever [m] at indexed rolling angle _10 to _40 in [°]


2.2. Sea state and encounter period to waves

The sea state is approximated by a regular wave system with one characteristic direction, average wave height, described by average wave period Tw, wave length Lw and wave speed Cw,

Fig. 8: Regular Wave System and speed component due to ships and wave speed

The main characteristic types of wave systems are:

  • SWELL (full developed sea state, long crested; period about 12 s … 20 s, wave length: 200 m and more)
  • WIND SEA (new developing sea waves, short crested, period: from 6 s … 12 s, wave length up to about 200 m)
  • INTERMEDIATE SEA (intermediate type of sea state between both type above)

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:

Lw = k * Tw2 and Cw = k * Tw

where k denotes the coefficient for the wave system (wave number), which is:

  • k = 1,56 for full developed swell, long crested
  • k = 1,3 for heavy seas not fully developed in intermediate conditions
  • k = 1,04 for wind sea, short crested with new developing sea waves.
and
  • Lw: wavelength [m]
  • Tw: wave period [s]
  • Cw: wave speed / celerity [m/s]

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 type of the wave system and its wave period Tw or length Lw
  • the ships actual speed vector V
  • the encounter angle gamma (γ) between ships' course and direction of wave propagation

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:

1) The Ship makes no way and the speed is V = 0 (or ship is in beam sea with gamma = 90°):
Then the encounter period is equal to the wave period: TE = Tw
2) Encounter of Ship with Head waves:

TE = Lw / (Cw + V) that yields TE < Tw
3) Encounter of Ship with following waves:
There are three different conditions possible
Waves are overtaking the ship Cw>V:

TE = Lw / (Cw − V) that yields TE > Tw
b) Encounter of Ship with high speed when overtaking waves V > Cw:

TE = Lw / (−Cw − V) that yields negative values of TE
c) Encounter of Ship with the same speed as the waves V = Cw:
Then the encounter period TE becomes very large to infinite values.

For general encounter situation the encounter period TE can be calculated as to:

With:

  • k – Coefficient for the wave system (wave number)
  • V – ship's speed vector [kn] and component V * cos (gamma)
  • Tw – wave period [s]
  • gamma – encounter angle (γ = 0° for head sea; (γ = 180° for following sea)

(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).

Fig. 9: All conditions with the same speed component (blue), i.e. encounter period are on that one line (red) rectangular 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.


2.3. Conditions for resonance – Synchronous Rolling Resonance

Resonance – the phenomenon of building up extreme rolling amplitudes caused by waves – develops when the ship's natural rolling period coincides with the excitation period of the waves (the encounter period).
The rolling amplitudes of the ship may be stimulated depending on the ratio between the ships' natural period Tr and encounter period TE. There are two significant types of resonance: Synchronous and parametric resonance.

Synchronous resonance occurs when the ships' natural period Tr and the encounter period TE have nearly the same value:

  • Direct Resonance where the maximum amplitudes are to be expected:
    Tr = TE or Tr / TE = 1,0
  • Range with still up to about 50 % higher amplitudes
    0,8 ≤ Tr / TE ≤ 1,1

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.


2.4. Parametric Rolling Resonance

Parametric Rolling occurs specifically in head or stern seas when the ships natural period Tr and the encounter period TE have nearly double or half values:
  • Direct Resonance Parametric Rolling Resonance:
    Tr = 2 * TE or Tr / TE = 2,0
  • Range with still up to 50 % higher amplitudes
    1,8 ≤ Tr / TE ≤ 2,1
  • Parametric Rolling occurs specifically in head or stern seas

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:

  • There is more bow flare and transom stern shapes with higher up-righting moments
  • The change of up-righting moments between positions of the wave crest at the bow or stern (high moment) and midship (low moment) is much larger than before, that means stronger wave effect!

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.


2.5. Dangerous Stern Wave Encounter and high wave groups

When a ship is riding on the wave crest, the intact stability will be decreased substantially according to the ship form. The amount of stability reduction is nearly proportional to the wave height and the ship may lose the stability when the wave length is one to two times of ship length and wave height is large. This situation is especially dangerous in following and quartering seas, because the duration of riding on wave crest, i.e. the time of inferior stability, becomes longer – and specifically when there is danger of parametric resonance as described in 2.4.

Fig. 11: Wave encounter in Head/Stern Sea and two different positions of wave crest at midships (red) and bow and stern (green)

For the two different position of wave crest at the ship length in Fig. 11 the effect on up-righting forces and moments is shown in Fig. 12. The up-righting moment is:

  • higher (more stable) for positions of the wave crest at the bow or stern (ship in wave trough) and
  • lower (less stable) when the wave is at midships position

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: IMO-Diagram Indicating Dangerous Zone due to High wave group encounters [4]

Definition of Symbols Used:

  • V – actual ship speed [kn]
  • Tw – mean wave period (second)
  • Te – encounter wave period (second)
  • χ – encounter angle of the ship to wave (degree, χ = 0° seen from stern)

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.


2.6. Broaching and "Surf-Riding"

When the ship speed is so high that its component in the wave direction approaches to the phase velocity of wave, the ship will be accelerated to reach surf-riding and broaching condition. That means the ships will be lifted by a following wave at the stern and accelerated; if then the ship is affected by small course change a yawing/swaying motion can occur followed by large heel angels up to capsizing.
The critical speed for the occurrence of surf-riding considered to be 1,8√L (kt), where L is ship length. It should be noted that there is a marginal zone (1,4√L ~ 1,8√L) below the critical speed, where a large surging motion may occur, which is almost equivalent to surf-riding in danger. In these situations, a significant reduction of intact stability may also be induced with longer duration.

Fig. 14: IMO-Diagram Indicating Dangerous Zone Due to Surf-riding 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 surf-riding and broaching-to 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 surf-riding in danger.


3. Summary of effects and formulas, general task description and example

3.1. Summary of effects and formulas, general task description

The method presented here allows for calculating and plotting a polar diagram for ship operation in a very simplified way as indicated in Fig. 15: Using a Radar Plotting sheet (with speed values at the axis instead of distances) 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) on courses with direct head or following sea.

Fig. 15: Resulting Polar diagram with dangerous course and speed vectors based on the example ship and calculated with the respective formulas from Tab. 2, indicated by coloured circles.
(Example-Ship: Semi container Lpp = 113 m, B = 17,6 m; rolling coefficient Cr = 0,74; i.e. Tr = Tr(10°) = 10 s; Sea from 23° with Tw = 8 s in Wind sea (k = 1,04)

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:

PhenomenaDirection/Section/AreaEquations to Calculate the speed values as basis for the Diagram Elements
1. Synchronous rolling motion Stripe segments over diagram; all directions possible

For: 1.2.

2. Parametic rolling motion Segment for direct head and stern wave conditions ±30°

For: 3.4.

3. Reduction of stability riding on the wave crest of wave groups Segment for direct Following and quatering seas ± 45° 5.


6.

4. Surf-riding and broaching-to 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:

  • The synchronous resonance area will be drawn as a stripe over the whole angle area of the polar diagram, rectangular to the sea direction.
  • The parametric excitation will be drawn in the same way but only for a sector segment of ±30° around the direction of stern sea or against the sea respectively.
  • Additionally the areas for surf-riding and encounter of wave groups in zones of +45° to -45° around stern sea directions will be drawn.

By means of the polar diagram the following general tasks can be identified:

  • Assessment of situation: The ships' speed and course will be indicated by drawing the speed vector in the respective course direction. Assessment of conditions and areas for synchronous and parametric rolling as well as conditions for Surf-riding and Broaching is now very easy: if the speed vector is within one or more of the areas then potential danger of the respective effect exists.
  • Estimation of countermeasures for improving sea keeping if resonance exists:, i.e. by the following actions:
    • Change of course or speed: to be taken visually from the polar diagram.
    • Measures to change stability, that means for instance: calculation of alternative GM-values (and thereby a change in the ships roll time period T) to avoid resonance, if course and speed V and therefore the same encounter period TE0 shall be maintained.


3.2. Example: Calculation for a Semi container ship/Specific Task description

A semi container ship (Lpp = 113 m, B = 17,6 m; inertia coefficient for rolling motion Cr = 0,74) is cruising with natural rolling period Tr = Tr(10°) = 10 s in wind sea coming from 23° with a wave period Tw = 8 s.
The following tasks are to be solved for situation assessment and decision making:
  1. Prepare a RADAR plotting sheet as a Polar Diagram for assessment of wave effect for the given wave direction and ship speed range.
    1. Calculate the basic speed values for synchronous rolling resonance on courses with head sea or following sea for near-resonance conditions V0.8 and V1.1 as well as for direct resonance V1.0! Complete the polar diagram with the speed values and the stripes for synchronous resonance condition.
    2. Determine the basic speed values for parametric rolling resonance and draw the segments into the polar diagram accordingly!
    3. Determine the basic speed values for the potential danger of surf-riding and encounter with high wave groups and draw the respective segments into the polar diagram!
  2. Assess the following situations and make suggestions for potential countermeasures:
    1. The ship cruises at 8 kn: For which courses is direct synchronous resonance to be expected and for which courses is a 50 % reduction of the roll amplitude to be considered at the border lines of the resonance stripes? Is there potential danger of parametric resonance?
    2. The ship shall run on course = 140°. Which speed changes are necessary to avoid synchronous resonance?
    3. The ship shall run on course = 140° and with speed V = 8 kn. If this course and speed are to be kept constant which change of initial stability (ΔGM) and therefore roll period Tr(10°) would be necessary to achieve a better sea keeping behaviour of the ship?
  3. What will be the ship's natural roll period Tr(40°) for large roll angle amplitudes of 40° given a value GM = 1,7 m and the values of uprighting levers GZ for roll angles 10° to 40°: GZ_10 = 0,32 m; GZ_20 = 0,8 m; GZ_30 = 1,6 m; GZ_40 = 1,9 m. Which effect has this change of roll period for the resonance condition for this sea state in the polar diagram?


3.3. Solution Details for the specific Example Semi container ship

  1. Prepare Polar Diagram (see Fig. 15): A Radar Plotting sheet is to prepare by adjusting the scale of the axis according to the speed range (in this case 25 kn) with appropriate speed values instead of distances. Then draw the straight wave direction line for the wind sea (k = 1,04) coming from 23 ° over the whole range of the diagram (this is the blue line).

    1. Synchronous rolling resonance zone: The basic speed values for synchronous rolling excitation on courses with head sea or following sea for near-resonance conditions V0.8 and V1.1 will be calculated by using the formula for the speed from Tab. 2 for the number 1. and 2. with values:

      and


      The basic speed V1.0 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:

      • Positive with γ = 0°. That means the speed is to be drawn for head waves or
      • Negative, then the velocity with γ = 180°, that means it is to be drawn on courses with following waves measured from the centre of the diagram.

      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.

    2. Parametric rolling resonance zone: The procedure for calculating the basic speeds speed from Tab. 2 is the same as for a) but this time for the number 3. and 4. for other ratios

      and <



      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.

    3. Zones for conditions of surf-riding and encounter of high wave groups.

      • Dangerous encounter with high wave groups: They occur in stern sea in zones of +45° to -45° around sea direction. The basic values for speeds in the range of 0,8 < V/Tw < 2 will be calculated from Tab. 2 for the number 5. and 6. as follows:


      • Dangerous surf-riding conditions occur in zones of +45° to -45° of the direction of the sea in stern seas conditions. The basic values for the speed can be calculated as follows for the number 7, 8. and 9.:

        • Start marginal zone

        • Start critical zone:

        • End critical zone:

        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 surf-riding and encounter of wave groups zone: The zones for surf-riding 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.

  2. Assessment of situations and suggestions for potential countermeasures.

    1. Determination of resonance sector for the speed V = 8 kn:
      The ships' speed will be indicated by drawing a speed circle (blue colour) with the constant speed of 8 kn around the centre. Assessment of conditions and areas for synchronous and parametric rolling resonance is now very easy (see Fig. 16): if the speed vector is within one or more of the areas then potential danger of the respective effect exists.

      • Direct synchronous resonance exists at intersection of resonance line with the speed-circle V = 8 kn:
        i.e. at course = 269° und course = 137° (magenta dotted lines)
      • At the intersection of outer boundaries of resonance stripe with the speed circle V = 8 kn we have the 50 % decrease (magenta broken lines)
        i.e. at 127° < course < 160° and 246° < course < 279°
      • The potential of parametric rolling resonance exists where the speed circle V = 8 kn is crossing the sector segment for head wave encounter,
        i.e. between 53° < course < 353°

      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)
    2. Determination of Speed for avoiding resonance at course = 140°:

      • This can easily taken from diagram (see Fig. 16): Draw a straight speed line for course = 140° (direction like blue speed-arrow)
      • The part of the line (dotted) that is inside the resonance area is the speed range that is to be avoided: The speed at the boundaries of the stripe is about
        4,3 kn < V > 12,8 kn.
      • The ship has to run slower than 4,3 kn or faster than 12,8 kn, to avoid resonance.

    3. Measures for change of initial stability for avoiding resonance at course = 140°:

      • If the ship shall run on course = 140° and with speed V = 8 kn then this would be a situation with synchronous resonance. Therefore a change of initial stability (ΔGM) would be possible to achieve a better sea keeping behaviour of the ship: By changing the natural rolling period of the ship the resonance stripes will be shifted thus far, that the ship can run without resonance phenomena.
      • For this condition the encounter period TE0 needs to be calculated for the given values of V0 = 8 kn and course = 140° as well as the encounter angle γ0:

      • Calculation of the ships' rolling period T, which would be at the boundaries of the stripes for this TE0 yields:

        ⇒ T0.8 = TE0 * 0,8 = 8,24 s;

        ⇒ T1.1 = TE0 * 1,1 = 11,35 s.

        To assess the suitability of the result the stability has to be discussed for these roll periods calculation the respective GM values:

           

      • The lower rolling period T0.8 seems not advisable, because such a short rolling period is coupled with large accelerations; since the ships is very stiff.
      • The second result with the higher value T1.1 is better suited as rolling period; this can be seen by the GM value (and also because as a rule of a thumb, the roll period Tr = 11,35 s is closer to the ships beam B = 17,6 m).

      Another criterion is the required change ΔGM which is necessary to achieve the new GM0.8 in comparison with the initial GMinit

      This relative small change might be better achievable than the larger change required in case of using the GM0.8 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 mb) Results for Tr(10°) = 8,24 s at GM = 2,50 m.

  3. Calculation and effect of Ship's natural roll period for large rolling amplitudes of 40°:

    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 up-righting 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:

    v = 0,6 * GZ_40 = 1,14
    w = GZ_20 + 4*GZ_30 + 1,6*GZ_40 = 10,24
    x = w + 1,5*GZ_10 − 3*GZ_20 − GZ_30 = 6,72
    y = w + 2,5*GZ_10 + GZ_20 = 11,84
    z = y + 1,5*GZ_10 = 12,32

    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 GM-value.

    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!

    Fig. 18: Up righting lever curve for semi container to calculate the 2 rolling periods and Polar diagram for 2 rolling periods depending on roll amplitude (GM = 1,70 m: Tr(10°) = 10 s and Tr(40°) = 8 s)


4. References

[1]France, William a.o.: An Investigation of Head-Sea 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 Warnemünde-Wustrow, 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 10-12, 1998.
[7]IMO Code on Intact stability for all types of ships, Resolution. A.749(18) Nov 1993
[8]BMVBW / See-Berufsgenossenschaft: 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 Hamburg-Harburg (2001).