Excessive rolling of cruise ships in head and following waves

EXCESSIVE ROLLING OF CRUISE SHIPS IN HEAD AND FOLLOWING WAVES Authors: R.P. DalUnga, J.J. Blok and H.R. Luth

Maritime Research Institute Netherlands (MARIN)

SUMMARY

Model tests and calculations were performed on atypical generic present day cruise ship. It was brought to light that roll angles above 40 degrees can fairly easHy be obtained in seastates that can hardly be qualified as "strong gales', provided that the waves are taken head-on or stem-on and that the speed is reduced. Because of the non-linear character these large roll amplitudes occur virtually without warning.

To investigate the nature of the behaviour and to develop ways to assist in reducing the associated nsks, dedicated model tests were performed with a model of a typical cruise ship By means of tests in irregular waves, the operational limits from a point of view of rolling were established for a range of. speeds; The merits of bilge keels and fin stabilizers to suppress the rolling were investigated.

The understanding of the observed phenomena was tested on basis of a simple one degree of freedom model and time domain; simulations with a numerical model with six degrees of freedom.

1. A QUALITATIVE APPROACH.

Rolling is perhaps the most important ship motion because it has a tendency to take large amplitudes The reason for this is twofold In the first place rolling is a motion that has low damping, even if bilge keels are fitted, in the second place the natural penod, at which dynamic amplification occurs, is usually nght in the range of prevailing seaway wave penods

Natural period

A typical large cruise ship will have a transverse metacentnc height of about 2 m In combination with the relatively large transverse radius of gyration of these ships this results in natural periods of roll around 25 S;

This period is way outside the range of prevailing wave periods present in wind-driven seas, even up to gale strength. Reasoning within the concepts of linear seakeeping theory this means that, apart from the steady heel introduced by the relatively large wind force and the dynamic response to wind gusting, the roll motions will generally be quite limited

In swell waves, in particular at unfavourable combinations of heading and speed, "tuning" between the dominant wave, encounter frequency and the natural frequency of roll may not always be avoided. In these cases dynamic amplification effects may introduce relatively large roll amplitudes.

Non-linear rollinq

Linear seakeeping theory associates a high roll response with. unfavourable tuning between the encounter frequency and the natural frequency of roll. However, there are two other non-linear mechanisms which may yield large roll angles. Both are associated with the fact that the effective metacentric height of a ship moving among waves must be regarded as a dynamic quantity,

ratherthanasastaticvalue. .

---.---TECIIMSCIIE UNIVERSE1T

Scheepshydraznehica

For ships in following seas the GM stability variations are long period, i.e.almost quasi-static in character As the waves slowly overtake the ship the flat transom stem of present day hull forms, which introduces relatively large stability variations, may cause temporarily loss of stability. As was observed during several model test programs for fully loaded container ships, the associated

roll amplitudes can be very large.

A second non-linear mechanism which introduces rolling is observed in conditions in which the dominant encounter period is close to half the natural period of roll. If this unfavourable .tuning is combined with relatively large stability variations large roll angles may be obtained. The relative wave elevation at the vessels bow and stern (associated with pitching) in relation to the large variations in the wetted hull geometry are important parameters in the stability variations.. The phenomenon is referred to as "half.cycle" roll or "parametric"

roll, and may show up

conspicibusly in waves from ahead.

Tuninq

Observations from various test programs have shown parametric roll is a phenomenon which starts quite. unexpectedly and very quickly attains very large amplitudes.

The event appears to be tnggered by a particular sequence of wave components of penod and height This means that it may be quite difficult to define a "safe" threshold wave height, although less frequent a lower wave of sufficient duration may triggerthe phenomenon just as well.

2. A QUANTITATIVE APPROACH.

To asses the extent of the problem and to obtain insight in the separate contributions from rudders, bilge keels and fins to counter the rolling motions a series at model tests were performed with a 1:50 scale model of a 240 m cruise ship.

The body plan of the vessel is shown in FigUre 1, the loading condition is given in Table 1 The hull form and weight distribution are typical for coritemporary large cruise ships.

Dunng the tests the model was self propelled and steered by means of an autopilot It was equipped with bilge keels and one pair of 13 81 m2 active fin stabilizers of the retractable type with a 20% chord flap. An 80 grain size 10% chord sand strip at one-quarter chord from the nose was used to stimulate turbulence on the fin. Bilge keel heights of 30 and 80 cm were used during the tests.

Measurements comprised the motion response and the forces acting on the individual rudders, fins and bilge keels.

Test program

All tests were performed in realistic irregular head and following waves. At zero speed half an hour (prototype) test duration was adopted. At non-zero speed the test duration was limited to one run over the length of the Seakeeping Basin In these cases the test duration vaned from around from about 1300 s at 4 knots to 450 s at 12 knots. A JONSWAP type wave spectrum characterizes the irregular wave conditions during the tests.

The main variables in the test program were the .significant wave height and the peak period of the waves, the forward speed and the heading, bilge keel height and fin control gain.

Characterof the roll res'øonSe

During the tests it became quite clear that the rolling in the tested head-on and following seas cases must be regarded as a "threshold" phenomenon. Below a particular wave height the roll 'is

roll motion. The character is illustrated in Figures 2 and 14. The threshold wave height was found to be dependent on the s.hip heading, peak period and ship speed.

Extent of the roll response

Zero speed proved to be the most severe test condition. A wave peak periodT of 12.2 s yielded the lowest threshold wave height. In following waves (denoted by 0 deg. heading) a significant wave height as low as 2 m was sufficient to introduce rolling in head seas (180 deg.) the threshold was around 2.75 m. See Figure 3.

The effect of wave height proved quite dramatic; a 7.5 m irregular wave with a peak period of 14.1 s yielded maximum roll amplitudes around 42 deg; the difference between head and following seas was negligible in this wave condition. The effect of wave period was less strong. Within the range of applicable periods at the same wave height the 12.2 s peak period yielded a maximum roll amplitude of around 34 deg. See Tables 2 and 3.

Forward speed affected both the threshold wave height as well as the maximum roll amplitudes. At 5 knots the threshold wave height in following seas rose from 2 to around 2.75 m, in head seas from 2.75 to around 5 m, see Figure 4. In the 7.5 m, 12.2 s following seas the maximum roll amplitude increased from 34 deg at zero speed to 39 dég at 5 knots; in head seas it dropped from 33 deg to 12 deg.

At 10 knots the threshold exceeded 10 m significant wave height.

Bilqe Keels

Repeat tesis with higher bilge keels (80 cm instead of 30 cm) indicated that their effect on the threshold wave height .was only limited; at zero speed the increase from 30 to 80 cm increased the threshold significant wave height by less than 1 m. See Figu.re 5. At 5 knots the effect was less obvious and smaller. Above the threshold wave height the effect was considerably larger in the 7.5 m 14.1 s wave the maximum roll amplitude reduced from 42 to32 deg.

Relevance of the test conditions

Situations with low speeds in higher wave conditions, although generally avoided, may occur in particular circumstances when there are propulsive problems or dunng standby and recovery operations Figure 6 indicates the limiting wave height (the threshold wave height above which

rolling occurs) at 0 and 5 knots speed in a typical annual Lower Caribbean wave climate. The threshold wave height is exceeded in about 6% of time when the ship sailsat lOw speed (5 knots)

in following seas.

3. MODELLING IN ONE-DEGREE OF FREEDOM.

3.1 GENERAL

One way to explain the observed behaviour of the vessel is by recognizing the fact that the normal stability variations experienced by the ship in waves can lead to excessive roll under special circumstances [Undeman, 8] The nature of this behaviour ("parametric roll!) is described in terms of a so called Mathieu equatiOn according to:

m.+b.+(c+csin(ot)).4=0

in which m represents the sum of the structural and added mass in roll, b the damping, c the restoring term (the product of transverse stability GM and the displacement i) and c the variations of the restoring term.

* _{the oscillation frequency of the spring term variations is about twice the natural frequency}

and:

*

the variations in the restoring term are sufficiently large in relation tO the available damping. If the variations of the restoring term are below a certain threshold value no rolling is observed.

At first glance the conditions fOr parametric roll are satisfied by large cruise ships. The large structural inertia in roll implies that the non-dimensional roll damping is, despite the presence of large stabilizers, relatively low. In addition, a wave length slightly longer than the ship (which yields relatively high stability variations) corresponds with a wave period Which is very close to half the natural period of roll of the present class ofvessels.

3.2 MAGNITUDE OF THE GM VARIATIONS

Hydrostatic calculations were performed to obtain an impression of the stability variations in waves. To this purpose the righting arm was calculated for 4 wave lengths (0.75, 1.00, 1.25 and 1 5 times the ship length) and a range of wave heights (0 to 5 m) The GM vanation was estimated by considering the GZ value (at a given heel) with a wave crest and a wave trough amidships dividing the difference in GZ by the heel angle yielded the vanation in GM Dividing the GM range by the wave height H yielded a kind of "transfer1' function" of the GM variation.

GZ, - GZ,

/

GMa 4,

H

The GM transfer function is, to some extent, non-linear in character; it depends both on the adopted heel angle for the GZ calculation and the wave height

Yet, in practice the non-linearity in the righting area GZ with heel angle seems relatively low for angles up to 20 deg. At large heel angles the GM variations in short waves increase in

magnitude. . .

-Varying the wave height shows that relatively low waves yield the highest GM vanations the vanations are largest in relatively long waves See Figure 7 At a wave length over ship length around one the GM vanations are around 0 8 m per m

Considering Froude's Law of similitude the above levels are independert of ship size at the same wave length-Ship length ratio.

3.3 ROLL DAMPING

In-house data

Experience from model tests shows that, the roll damping of the .present class of vessels can be described as the sum of a "linear" contribution which is independent of the roll amplitude and a "non-linear" contnbution which increases with the roll amplitude The linear contnbution is often associated with the lift of (active or passive) stabilizers and rudders, the skeg and the hull itself as well as wave making effects. The non-linear contribUtion is often associated with appendage drag and vortex shedding along the bilges.

Roll decay tests from 14 different test programs for ferries and cruise ships showed that the linear contribution to the roll damping is dominated by the lift of fins, rudders and hull. Wave making effects seem very small, this is plausible because of the relatively long natural period of the vessels 'in this class (over 15 s). Bilge keel drag seems to explain most of the non-linear damping contribution.

Considering the test results from models without stabilizers it was found that the effective lift damping could be represented fairly accurate in the formula suggested by Himeno [Ref 7] Using the overall ship dimensions (instead of the sectiOnal draft used by Himeno) the following estimate

is obtained:

buFmuLLjP VSLT(27tL+0.3(4.1[_))0.15T2(1 -1.4

+aO.7

(KG-I)2)

in which v is the ship speed in knots, T, B and L the ship draft, beam and length between perpendiculars and KG the height of the centre of gravity above the keel. A factor a of 1.4 yielded the best fit. For the presently tested model a value of 37600 kNmslrad per rn/s forward speed

was obtained.

Stabilizer damDinq

The contribution of active fin stabilizers in the above in-house decay tests was estimated by evaluating the contribution of the hydrodynamic angle of attack introduced by the roll velocity of the vessel and the mechanical angle of attack introdUced by the fin actuators. The latter was assumed to be given by the characteristics of a simple PD contr011er with a damping gain. bc (deg fin per deg/s roll velocity) and a restoring gain c(deg fin per deg roll):

in which 4) represents the instantaneous roll angle. With an estimate on the lift slope of the passive and active fins (including the effect of the flap) the following estimate [Dallinga, Ref. 4] was obtained:

1 ₂ racL rF1NcHF acL

bFINS-2rANPU AFINCFHL + _bc

acLpASs &XACT

in which mN and AFIN represent the arm to the centre of gravity of the ship and the fin area, U the forward speed in m/s, c the (frequency dependent) coefficient for carry over effect of the fins to the hull, cHF the coefficient for magnification ofihe passive angle of attack due to the limited bilge radius and CL the lift slope. Values of 1 2 were adopted for both interference coefficients c and

CHF.

In the above, the passive contribution (b = 0) is proportional to the forward speed; the "active" contribution is proportional to the square of forward speed. According to the above estimate the passive damping is around 28400 kNms/rad per rn/s forward speed, which is similar to the lift damping of the hull

The active contribution is around 1720 kNms/rad per (rn/s)2 forward speed per deg/(deg/s) damping gain This implies that at speeds above 7 5 knots and a gain of 10 degldeg/s the active stabilizer damping yields the largest damping contribution.

Stall

Since the present speed range of interest is relatively low the passive angles of attack can become easily relative large For this reason an interesting aspect to explore is the possibility of stalling of the stabilizer fins According to in-house data at relatively low Reynolds numbers stalling of the present type of fins may be .expected around 20 deg. It is understood that at higher Reynolds numbers stalling may be delayed to somewhat larger angles. Adopting 5 deg roll amplitude and a natural frequency of 0.25 rad/s the passive angle of attack exceeds stall at around 3.5 knots. Mechanical fin activity at these speeds might reduce the fin effectivity instead of increasing the damping.

Bilge keel dampinq

The contnbution of the bilge keel (and stabilizer fin) drag to the roll damping at very low speeds is often associated with the energy dissipated by the drag forces. Within this concept the damping is proportional with the roll (velocity) amplitude; the increase is given by:

abBI< ₌ 4 2

p2 lbk hbk (cHFrbk) rbk CDbk

According to work by Ridjanovic [Ref. 9] the effective drag coefficient depends on the amplitUde of the transverse flow and the bilge keel height. The related estimate is:

hbk

CDbk=22.5 +2.4

1trbkcHF4

in which rbk denotes the arm to the centre of rotation axis (the centre of gravity), cHF the magnification of the flow over the bilge, 4) as the roll amplitude (in radians). For 5 deg roll amplitude the effective drag coefficient is around 3 4 for the 30 cm bilge keels and 5 1 for the 80 cm bilge keels. For large roll amplitudes (say 30 deg) these drag coefficients reduce to around 2.6 and 2.8.

At 5 deg roll amplitude the increase of the roll damping becomes 1 .0E6 kNms/rad fOr 30 cm bilge keels and 4.0E6 kNms/rad for the 80 cm bilge keels, Which illustrates the merits of higher bilge keels However, at non-zero speed the damping of both sets of bilge keels was negligible compared to the linear damping components of the hull, fins and rudders at modest and higher forward speed.

Roll decav tests for the present model

The decay tests at zero speed with 30 and 80 cm bilge keels allowed an estimate on the increase of the damping with roll velocity amplitude. The measured increase at 5 deg roll amplitude was around 1 .3E6 kNms/rad per radls roll velocity amplitude for 30 cm bilge keels and 2.6E6 kNms/rad per radIs roll velocity amplitude for: 80 cm bilge keels. Assuming that the bilge keels dominate the non-linear damping the first result matches the ihitial estimate very well, the damping increase of the higher bilge keels is over estimated by theory.

At non-zero ship speed roll decay tests were performed at various speeds and with various fin damping gains. The resulting damping values are compared with the foregoing estimates in Figure 8. The results suggest that the contribution related to fin activity is over estimated by the theoretical estimate.

Fin and bilge keel forces

To obtain an impression of the effectivity of the fins and bilge keels at low speed the forces acting on the individual fins (and their flaps), rudders and bilge keel parts were recorded.

Figure 9 indicates the relation between the instantaneous lift, coefficient (the measured transverse force acting on the fin divided by the product of fin area and dynamic pressure, taking the ship speed as a reference) and the effective angle of attack for the case of a decay test at 10 knots. The latter was the sim of the instantaneous mechanical fin angle and the estimated instantaneous hydrodynamic angle of attacic

The slope of the curve is around 5.5 rad1, which is quite close to the estimated theoretical value for a fin with flap. The results suggest that also at low speeds the fins supply the expected roll

damping.

Figures 10 and 11 indicate the moment generated by the fore and aft 30 cm bilge keel pair. The forces are compared with an estimate based a drag coefficient of 2 and neglecting the hu!l to fin interference. The resulting forces acting on the forward bilge keels are very similar to the empirical estimate indicated in Section 2. The fact that the aft bilge keels deviate more may be attributed to fin-bilge keel interference, which tends to reduce the effectivity of the aft bilge keels at non zero speed.

Knowledge of the instantaneous forces acting on the appendages during a test allows an estimate on the dissipated energy by integrating the product of the roll velocity and the reaction moments. DMding by the integration time yields the mean dissipating power. Figure 12 indicates the result for a decay test at 10 knots. The result demonstrates the large contribution of the stabilizer fins; the contribution of bilge keels and rudders is considerably smaller.

3.3 CORRELATION

According to the literature the GM variations can be given an interpretation in terms of a decrease of the roll damping. If the GM variations exceed a particular threshold the damping becomes negative and large roll angles develop.

Following work of Dunwoody (Refs 5, 6] the damping reduction was calculated from the spectral density of the GM variatiOns at the natural roll frequency for all tests. The related "transfer function" for the GM amplitude per m wave amplitude was derived from hydrostatic calculations in which the ship is in quasi-static equilibrium at a 10 deg heel.

The spectral density follows as the product of the spectral density of the encounter spectrum of the recorded wave (at twice the natural frequency of roll) and the square of the transfer function at the related wave length (which yields an encounter frequency of twice the natural frequency of

roD).

GMa(°')

₂

SGM=( ) Seç(2.0)q)

According to Dunwoody the (non-dimensional) damping reduction follows from:

A=

It92SGM

A 3 4

in which o is the natural frequency in rad/s, k, the effective transverse radius of inertia in m and g the acceleration of gravity.

The total effective roll damping becomes:

(Roll damping from ' (Damping reduction associated

BEFFI

ri

.. .

hull + appendages) with GM variations

in which BCRIT represents the critical damping:

in which represents the ships displacement weight, GM the transverse stability, g the acceleration of gravity and k the transverse radius of inertia in water.

Noteworthy is the fact that the relatively large transverse radius of inertia of the present class of vessels reduces the non-dimensional damping, which increases the sensitivity to GM variations.

Considering the above it shows that if the GM variations are sufficiently large the effective damping may become negative. In this case large roll angles are inevitable.

Results applying the damping estimate derived from decay tests the effective damping was calculated and correlated with the maximum roll response during the tests. Figure 13 shows that, although there is considerable scatter, the theoretical concept based on a one degree of freedom

model is reproduced.

-4. TIME DOMAIN SIMULATIONS IN SIX DEGREES OF FREEDoM

For the time domain calculations use was made of the non-linear, 6 degrees of freedom time domain code FREDYN, which was developed for the Cooperative Research Navies Dynamic

Stability working group to simulate the motion behaviour of a steered ship in moderate to

extreme waves and wind

The mathematical formulation

inciudes physical aspects of

wavemaking and viscous origin in a six degrees of freedom model. It includes rudder, bilge keel and fin effects. The code includes the following force components:

wave excitation forces: determined from integration of the wave-induced pressure in the time domain up to the instantaneous free surface,

diffraction forces: calculated using a linear strip theory approach,

added mass and damping: calcUlated using a linear strip theory approach and converted to retardation functions in the time domain,

linear and non-linear drag (manoeuvring) forces in the transverse plane: estimated using empirical relationships,

resistance and propeller forces: from full scale results, model tests or numerical predictions rudder forces and forces of (active) fins: coefficients from model test data,

roll damping moment: from the Ikeda model, reported by Himeno [Ref. 7].

Non-linearities which are covered are: the effect of large angles (roll, yaw) on excitation forces, rigid body dynamics at large angles, drag forces associated with hull motion, wave orbital velocities and wind and tile integration of the wave-induced pressure up to the free surface. See

Ref 1 and Ref. 2 for more background and applications of FREDYN.

For the present hull cruise ship form time domain simulations with a duration of 30 minutes were performed for a whole range of wave height-wave period combinations in irregular head and following waves at both zero and 5 knots forward speed with active and passive fins. For numerical reasons it was desirable to have a small disturbance in the roll motion. Therefore, simulations started with a small initial

roll angle of 3 degrees and to maintain the small

disturbance, the desired course in the auto-pilot was set on 1 degrees off head seas or following seas. From the results, the limiting wave height for parametric roll was determined and compared to the model test results. As an example, the results for a forward speed of 5 knots are given in Figure 6. The agreement between the calculated results and the model test results seems sufficient to conclude that the described FREDYN code may be used as an engineering tool to determine the probability and extend of parametric rolling in the early design stage.

DESIGN RECOMMENDATIONS.

Considering the nature of the results some design implications are clear more roll damping, smaller stability variations and avoiding "tuning11 between the natural penod of roll and (twice) the typical peak period of storm waves.

Bilge keels

The most obvious measure to increase the roll damping is to mount bilge keels of sufflcient height They not only reduce the nsk of occurrence to some extent, they particularly limit the roll angles in cases in which rolling can not be avoided

Stabilizers

The rolJ damping from the fins may be regarded as the sum of a "passive" contribution and a contribution related to the mechanical fin reaction Since the last contnbution is very small at low speeds an improvement should be sought in the passive contribution (which for the tested design provides 50% of the damping at low speeds) A larger fin area, a higher aspect ratio or mounting more than one fin pairs would certainly enhance the roll damping.

AoIl stabilizing tank

An option which increases the damping at low speed and small roll amplitudes is to mount a U-type anti r011 tank with low internal damping. These syStems have proved very effective for smaller vessels [Ref. 3]. The relatively large response at Small roll angles and the fact that the response does not depend on forward speed provides considerable damping in conditions where stabiJizers and bilge keels are not very effective. Care shoUld be exercised to ascertain the performance of the lightly dampened tank in Off-design operational conditions.

Hull form

Reducing the stability variations would require a reduction of the waterline area of the aft most sections of the ship.

This aspect calls for a complete re-consideration of the design of the aft body because at first sight this seems tO run counter to the current practice to obtain sufficient static stability and good aftbody flow for efficient propulsion.

OPERATIONAL MEASURES.

Apart from careful routing of the ship on board awareness and guidance will be usefUl to avoid large roll angles. A guidance system could be based on a theoretical estimate on the roll damping and the stability variation. The latter could be estimated from instantaneous wave spectra to be derived from the navigational radar.

REFERENCES

DE KAT, J O BROUWER, R, MCTAGGERT, K A, THOMAS, W L:

"Intact Ship Survivability in Extreme Waves New Cntena from a Research and Navy Perspective", STAB'94, May 1994.

DE KAT, J 0: "Irregular Waves and Their Influences on Extreme Ship Motions", 20th Naval Hydrodynamics Symposium, Santa Barbara, August 1994.

DALLINGA, R P: "Passenger Comfort on Board Motor Yachts", HISWA Symposium, Amsterdam, November 1996.

DALLINGA, A P: "Hydromechanic Aspects of the Design Of Fin Stabilizers", RINk Spring Meetings, 1993.

DUNWOODY, A B: "Roll of a Ship in Astern Seas - metacentric height spectra", Journal of Ship. Research, Vol. 33, No. 3, September 1989pp. 221 -228.

DUNWOODY, A B: "Roll of a Ship in Astern Seas - Response tO GM Fluctuations", JoUrnal of Ship Research, Vol. 33, No. 4, December 1989, pp. 284-290.

7

HIMENO, Y "Prediction of Ship Roll Damping - State of the Art", Dept

of Naval

Architecture and Marine Engineering, Report No. 239, September 1981.

LINDEMANN, K, SKOMEDAL: "Modem Huilforms and Parametric Excitation of the Roll Motion", Norwegian Maritime Research No. 2; 1983.

RIDJANOVIC, M: "Drag Coeffiôients of Flat. Plates Oscillating Normally to their Planes", Schiffstechnik, Band 9, Heft 45 1962

Table 1 Main Particulars of the Vessel

Length between perpendiculars m 240.00

Breadth B m 32.20

Draft (even keel) T m 7.75

Displacement weight Tf 37,956.00

Centre of gravity above keel KG m 16.40

Transverse metacentric height GM m 1.50

Longitudinal radius of gyration in air k m 64.80

Transverse radius of inertia in air k m 14.50. (45% 'B)

Natural period of roll

- T,

s 24.80

Fin stabilizers

span a

rn..

5.25

chord c m 2.63

thickness t m 0.63

arpa A m 13.81

flap area ratiO ce/c % 20.00

flap angle ratio control gains damping term 5/a b deg/deg deg/(deg/s) 2.00 10.00

restoring term c deg/deg 2.00

Bilge keels forward part length 'bkfwd m 42.00 height aftpart length hbkd 'bkaft m m

0.30 & 0.80

36.00

Table 2 Maximum roll angles

Table 3 Treshhold wave height

Peakperiod[s] 12.2 14.1 Heading [deg.] 1 180 0 180 0 Speed [knots] 0 5 0 5 0 5 0 5 Sign. wave height [m] Bilge keel height Em] 2.75 0.3 0.9 11.2 1.1 0.8 3.75 0.3 14.6 17.9 17.9 3.2 0.7 16.5

08

103 99 5.0 0.3 23.3 24.2 24.8 3.6 288

08

166 113 190 7.5 0.3 33.0 12.2 33.6 38.8 44.0 19.5 42.0 2.4

08

248 318 320

66

10 0.3 42.0 14.6 0.8 Heading [deg.] Speed [knots] Peak period [s]

Sign. wave height

[m] 3OcmBK 8OcmBK 180 0 12.2 -2.75 14.1 -3.75 180 5 12.2

-5

14.1 -4.75 0 0 12.2 -2.0 2,75 14.1 -2.75 3.25 0 5 122

-275

14.1 -7.0

/

COG '

16

Fig I General Arrangement and Small Scale Body Plan

/7

7

BASElINE

seline

WAVE 3 m rrc deg ROLL deg FIN ANGLE dog 15.0 15. RUD ANGLE 15.00-0 $

I

4

6 8 10

Significant Wave i-ieigbt [m]

50

40

30

20

10 0

'14.lsTp

A' 180 deg

rii

$ Tp, Odeg

z

:

_A2.2sTp,

_sip,

/'i80deg..

180:_4

Significant Wave Hei lit. Em]

Fig 4 Maximum roll angles at 5 knots, 30 cm bilge keels, active fins effect of heading and wave period

50

40

10

Significant Wave Height

Imi

Fig 5 MaximUm roll angles at 0 knots in following seas, active fins, effect of bilge keel height 10

>14

-

13-14

-

12-13

I-I

-10-il

Ui

I

9-10

Ui

_8.9

7.8

6-7

I-z

5-6

c_)

-3-4

z

0

-Cl)

12

0-1

Measured - - -

0 knots 5 knots Calculated_______ 2 10

59 104

82

2

3712513570

22

35

46

23

6 Head

'

FoIIowing1

I

LOWER CARIBBEAN WAVE STATISTICS Head

3

jFoIIowihg

4-5

67

89

10-11

1213

5-6

7-8

9-10

11-12

>13

ZERO CROSSING PERIOD (s)

Wave length/ship length &/L [in/rn]

1*106

9* io5

8*105

7*105

6*105

5*105

4*105

3*105

2*105

j* o5

0

4*105

_2*105

!3* iOS

.4*105

.5*105

0

Roll Damping

calculated

,

-_d

---

-H15 deg/dg/s ro

rate

/

/

I

1O dgid

eg/s.

I

Passive fin

(0 -d g/degis)

de&de s

bc

2.5

.

5

7.5

10 ...5

15

17.5

20

Forward speed in knots

Fig 8 Estimated and measured roll damping for various fin roll rate control gains

/

Decay Test

Bc = 5 deg

at 10 knots

per

degls

II I

SB

-15

-10

-5

0 5 10 15

Instantaneous Effective. Angle of Attack [deg]

1000

Moment

generated by

Aft. Bilge Keels

0

[kNm]

Fig 10 Aft bilge keel forces at 10 knots, 30 cm bilge keels

1000

Moment

generated by

Fwd Bilge Keels

0

[INm]

--1000

-4

-2

0

2

Roll velocity Ideg/si

L100

-4

-2

0

2

Roll velocity [deg/si

Decay Test at

Fins = 5deg

blots

per deg/s

Estimate based

/

\%

on cD

2,

4

0

40

20

0

Fig 12 Roll damping dissipation power

60

Large GM

variations

Fins

\

S.-Rudders

N

Mt Bilge Keels

20

40

Time [SI

Decay Test at 10 knots

Bc

5 deg per deWs

60

Mean Power

Dissipation

[kWJ

250

200

150

100

50

0

-50

/

-2

-1.5

-1

-0.5

0

0.5

Effective non-dimensional damping BEFF E-]

Fig 13 Maximum roll angles as a function of effective roll damping, all tests

N

1wd $Jlge Keels

DO

High roll