PREDICTION METHODS FOR PARAMETRIC ROLLINGUNDER DRIFTING CONDITION AND THEIR VALIDATION-FINAL REPORT OF SCAPE COMMITTEE (PART 3) –Yoshiho IKEDA, Toru KATAYAMA, Tomo FUJIWARA and Abdul MUNIF
Department of Marine System Engineering, Osaka Prefecture University, Sakai, JAPANABSTRACTIn this paper, experimental and theoretical researchworks on heavy parametric rolling in beam waves whichhave been done for these three years by the authors arereviewed, and the recommendations to IMO on the basisof the results of these works are made.
KEY WORDS: Parametric rolling, Beam waves, Large
passenger ship, Dead ship condition, Roll damping, Heave
resonance, Heading angle.
NOMENCLATURE

φ : roll angle
φ0 : roll amplitude
φmax: maximum roll amplitude at the peak of parametricrolling
Hw : wave heightζ 0 : wave amplitude (=Hw / 2 )Z0 : heave amplitude
Tnr : roll natural period
Tnh : heave natural period
Tr : roll period
Th : heave period
Te : encounter wave period (=Th)
Tw : incident wave period
Trp : encounter wave period at the peak of parametric rolling
INTRODUCTION
As well known, occurrence of parametric rolling of ships
without forward speed under drift condition in beam waves
has been pointed out by many researchers, as follows.
Paulling and Rosenberg (1960) pointed out that Froude(1863) observed that ships have unwanted rollcharacteristics if the natural frequencies in heave and roll arein the ratio of 2:1 in beam waves, and they investigatedunstable roll motion produced by nonlinear coupling fromheave motion theoretically and experimentally. Tasai (1965)
noted that small unstable roll motion caused by periodic rollrestoring variation in regular beam waves. Tamiya (1969)
measured unstable roll motion and capsizing of a cylindricalmodel in beam waves and wind, and pointed out that thephenomena can be explained by a Mathieu equation.
Sadakane (1978) studied on the effect of fluctuation of theapparent weight of a ship in regular heavy waves on rollmotion, found that the heavy rolling has harmonic andsubharmonic oscillations by experiments, and noted that thephenomena can be explained by a Mathieu-type instability.
Blocki (1980) reported experimental results of capsizing of acylindrical model of a fishing vessel due to parametricrolling in regular beam waves, and calculated capsizing
probability caused by parametric resonance. Boroday andMorenschildt (1986) carried out an experimental
investigation of the condition giving rise to the cylindricalship models with and without flare parametric rolling in
regular and irregular beam waves, and found large
parametric rolling occurs when the ship has large flare.
Umeda et al. (2002) experimentally showed that parametricrolling for a fishing vessel under drifting condition without
forward velocity occurs in head waves, and Munif andUmeda (2006) concluded that coupling with pitch isessential to numerically explain this type of parametricrolling.
Recently, Ikeda et al. (2005A) experimentally showed thatheavy roll motion with much larger angle than that in 1st
harmonic resonance appears for a modern large passenger
ship with flat stern and large bow flare in beam waves due toparametric rolling. The parametric rolling in beam waves
disappears when the roll damping is large enough. This factsuggests that the roll damping of such ships should bedesigned not to occur heavy parametric rolling in beam
waves as well as to meet the intact stability criteria.

Following the paper, the authors have been carrying out
experimental and theoretical investigations on the parametricroll in beam waves.

seems to saturate to the maximum one (Ikeda et al., 2006).

30

OF

PARAMETRIC

Measurements of roll motions of a model ship werecarried out in the towing tank of Osaka Prefecture University.
The model ship used in the experiments is an 110,000 GT
passenger ship designed for international cooperative
research projects on damage stability for contributing to
IMO regulatory works. The body plan and the principal
particulars are shown in Fig.1 and Table 1, respectively.

T nr / 2
1.02sec

without Bilge Keel
Hw = 0.04m

T nr
2.04sec20
Φ0 (deg)

EXPERIMENTAL
RESULTSROLLING IN BEAM WAVES

T nh
0.740sec

25

15
10
5
0
0

0.5

1
T e (sec)

1.5

2Fig. 2 Measured roll amplitude for naked hull in regularbeam waves of 0.04 m wave height.
12
Ratio １/１

ωroll : roll frequency

10ωe : encounter frequency

ωroll

8

Fig. 1

Ratio １/２

6
4

Body plan of the ship.

2

Table 1 Principle particulars.

0

0

2

4

6
ωe

8

10

12

Fig. 3 Comparison between measured roll frequency andencounter frequency.
30
without BK
Tw = 0.85 sec

25
20
Φ0 (deg)

Scale
LOA
LPP
BM
D
Displacement
GM
Tnr
Bilge keel : width
Bilge keel : location

Full Scale
Model
1/1
1/125.32
290 m
2.200 m242.24 m
1.933 m36 m
0.287 m8.4 m
0.067 m53,010 ton 26.98 kg
1.579 m0.0126 m
23 sec2.05 sec
1.1 m
0.0088 m
s.s.3.0－5.0, s.s.5.25－6.0

15
10
5
0

Fig. 2 shows measured roll amplitude of the model ship
without bilge keels in beam waves with regular wave heightof 0.04 m. The peak amplitude of roll resonance at the naturalroll period (Te=2.04 sec) reaches 7 degrees, but the roll
amplitude at encounter period of nearly half of the natural rollperiod (Te=0.85 sec) reaches 25 degrees. As shown in Fig. 3,
the period of the large roll motion is twice of the encounterwave period and almost the same as the natural roll period. It
should be noted that the wave encounter period where thelarge roll appears is near the natural heave period of the ship,
too (Ikeda et al., 2005B).
It is experimentally found that the heavy rolling withtwice period of the wave encounter period strongly depends
on wave height. The maximum amplitude of the heavy rollat each wave height is shown in Fig. 4. The results
demonstrate that the heavy rolling appears just over 30 mm
of wave height, rapidly increases with wave height, andreaches the maximum amplitude that is about 27 degrees. It
should be noted that the roll amplitude does notproportionately increase with increasing wave height but

0

0.02

0.04
H w (m)

0.06

0.08

Fig. 4 Effect of wave height on maximum amplitude ofparametric rolling of the ship in regular beam waves.
EXPERIMENTAL
RESULTSOF
PARAMETRICROLLING IN ALL HEADING ANGLES UNDERDRIFTING CONDITION
To know the characteristics of the parametric rolling inbeam waves, some investigations on parametric rolling inother heading angles to waves are carried out.
In Fig. 5, difference of wave periods when parametricrolling appears in beam and head waves is shown. The
results suggest that parametric rolling appears in different
regions of wave period in beam and head waves. This may
be because of differences of drift speed and amplitude ofstability variation in beam and head waves (Fujiwara et al.,
2006B).

30

180deg head sea
90deg beam sea

25

without Bilge KeelH w = 0.04m

20Φ0 (deg)

heading angle is plotted in a polar diagram as shown in Fig.
8. The results show that amplitudes of the parametric rollingare significant in following and head waves as well as inbeam waves when no bilge keel is attached. In Fig. 9, theeffect of heading angle on parametric rolling of the ship withthe designed bilge keels (BK1+BK2 that is shown in Table
2) is shown in a polar diagram. We can see that in head andfollowing waves parametric rolling occurs even if size ofbilge keels is large enough to erase it in beam waves (Munif
et al., 2006).

15
10
5
0
0

0.5

1

1.5

2

λ/Lpp

Fig. 5 Difference of wave periods for which parametricrolling appears in beam and head waves.

h e a v e (m m )

30
20
10
0
-10

To investigate the effect of heading angle on theparametric rolling of the dead ship, the model are released inhead sea condition (χ=180°) or following sea condition(χ=0°) in regular waves. The time histories of the heave,
pitch, roll and yaw motions are measured and shown in Figs.
6 and 7. From the time histories of the yaw angles, we cansee that, in both cases, the heading angle of the ship slowly
changes to beam sea condition. The results demonstrate thatthe parametric rolling occurs in wide heading angles as wellas in beam waves.

10

30

50

70

90

110

130

150

90

110

130

150

90

110

130

150

7 0 t i m e (s e c ) 9 0

110

130

150

-20
-30

p itc h (d e g )

2 .5
2 .0
1 .5
1 .0
0 .5
0 .0
- 0 .5 1 0

30

50

70

- 1 .0- 1 .5- 2 .0r o ll (d e g )

30
20
10
0

pitch (deg)

10

3

-10

2

-20

1

-30

20

70

120

170

220

70

180
150

-2

120

-3

90

heave (mm)

60

30

30

20

0
10

10
0
-10

50

y a w (d e g )

0
-1

30

20

70

120

170

220

170

220

170

220

-20
-30

30

50

Fig. 7 Time histories of motions of the ship without bilge
keel released from following wave condition in regularwaves at Tw = 0.95 sec and Hw= 0.04 m.

roll (deg)
30
20
10
0
-10

20

70

120

-20
-30

yaw (deg)
180
150
120
90
60
30
0

20

70

120

time (sec)Fig. 6 Time histories of motions of the ship
without bilge keel released from head sea condition
in regular waves at Tw=0.95 sec and Hw= 0.04 m.
Using the results of the measurements shown in Figs. 6
and 7, the roll amplitude of the parametric rolling for eachFig. 8 Effect of wave direction on amplitude of parametricrolling of the ship without bilge keel in regular waves.

Fig. 9 Effect of wave direction on amplitudes of rollingand its period of the ship with the designed bilge keels in
regular waves.

height at which parametric rolling appears increases withincreasing the roll damping, and that the roll amplitudes
saturate with increasing wave height, and the saturated roll
amplitudes also depend on the roll damping.
In Fig. 12, measured results of the roll damping for eachbilge keel shown in Table 2 in terms of N coefficient are
shown. N coefficients of the ship increase with increasingarea of bilge keels. It should be noted that N coefficient for abare hull is 0.018 as shown in Fig. 12. The value is not sosmall but near 0.02 which is used for a ship with bilge keels
in the Japanese Stability Code for passenger ships. The fact
may suggest that careful design for bilge keels should bedone to secure for escaping from large parametric rolling inbeam waves (Fujiwara et al., 2006).

30
H w = 0.04m
25

EFFECTS OF ROLL DAMPING
As shown in the previous chapter, parametric rolling inbeam waves significantly depends on the roll damping ofships. For the ship used in our experiments, the designed
bilge keels can erase the parametric rolling in beam waves asshown in Fig. 9.
By changing the roll damping systematically, the effect of
the roll damping on the rolling was investigated. The lengthand attached position of various bilge keels are shown in
Table 2.

Φ0 (deg)

20
15
10
5
0
0

0.2

0.4

0.6

0.8

1

Area of Bilge Keel / Area of Designed Full Bilge Keel

Fig. 10 Effect of area of bilge keels on the maximum
amplitude of parametric rolling in regular beam wave.

Table 2 Size and location of bilge keels.

30

BK location

BK

s.s. no.

LengthArea of Bilge Keel /Area of Designed
Bilge Keel

BK 1

5.25 - 6.00

150 mm

0.273

Mid 1

5.00 - 6.00

200 mm

0.364

Mid 2

4.75 - 6.00

250 mm

0.455

10

Mid 3

4.70 - 6.00

260 mm

0.473

5

Mid 4

4.60 - 6.00

280 mm

0.509

Mid 5

4.50 - 6.00

300 mm

0.545

Mid 6

4.25 - 6.00

350 mm

0.636

BK 2

3.00 - 5.00

400 mm

0.727

BK1 +
BK2

5.25 - 6.00 +3.00 - 5.00

150 mm +400 mm1.000

without BK
BK 1Mid2 BK
Mid3 BK25

Φ0 (deg)

20
15

0
0

20

40
H w (m)60

80

Fig. 11 Effect of roll damping on critical wave height for
occurrence of parametric rolling and on the its maximum
amplitude of the ship in regular beam wave.
0.035
0.03
0.025
0.02
N

In Fig. 10, the maximum roll amplitudes of parametricrolling for different length of bilge keels in regular beam
waves of 0.04 m height are shown. The results demonstratethat the peaks of parametric rolling rapidly decreases withincreasing area of bilge keels, or roll damping, anddisappears at half area of the designed bilge keels.
In order to clarify the effect of the roll damping on thecritical wave height to occur parametric rolling in beam
waves, measurements of ship motions of the model in thecases with BK 1, Mid 2-BK, Mid 3-BK and without bilge
keel are carried out in regular beam waves with various
wave heights. The results of the experiments are shown in
Fig. 11. These results demonstrate that the minimum wave

0.015
0.01
0.005
0

0

0.2

0.4

0.6

0.8

1

Area of Bilge Keel / Area of Designed Full Bilge Keel

Fig. 12

Measured N coefficients for various bilge keels.

EFFECTS OF HEAVE RESONANCE

1.5

without Bilge KeelH w = 0.04m

1
Z0 / ζ 0

In Fig. 2, it can be seen that large parametric rolling
appears at encounter period between half of natural rollperiod and natural heave period. The results may suggestthat heave resonance may be one of causes of parametricrolling in beam waves. In order to investigate this hypothesis,
measurements of roll motions of the model ship are carriedout for various degree of coincidence between half of naturalroll period and natural heave period, Tnr / 2Tnh. Natural rollperiods of the ship are systematically changed by changing
moment of inertia, as shown in Table 3. This means that thecentre of gravity of the ship keeps at original location.

T nh
0.748 sec

T nr / 2T nh

0.5

0.787
1.18
1.63
2.25

0.921
1.38
1.92
Cal. (OSM)

0
0

0.5

1

1.5
T e (sec)

2

2.5

3

Fig. 14 Heave response curve in the cases when parametricrolling occurs.

Table 3 Conditions of model experiments
1.5

Tnr / 2Tnh

Tw (sec)

1.18

0.787

0.40 ~ 1.18

1.40

0.921

0.45 ~ 1.40

1.65

1.18

0.50 ~ 1.65

2.04

1.38

0.60 ~ 2.30

2.39

1.63

0.70 ~ 2.40

2.92

1.92

1.00 ~ 2.92

3.33

2.25

1.40 ~ 1.70

T nh
0.748 sec

Hw (m)
1

0.04

T nr / 2T nh

0.5

0.787
1.18
1.63
2.25

Measured amplitudes of heave motions and its
phase angles from incident waves are shown withcalculated results by a strip method (OSM) in Figs.
14-16. The measured heave amplitudes in the bothcases shown in Figs. 14 and 15 are very similar toeach other, and do not have any high peaks. The
phase angle shown in Fig. 16 rapidly changes by 2π
near heave resonant period (Fujiwara et al., 2007).
30
without Bilge KeelH w = 0.04m

φmax (deg.)

20
15
10
5
0
1

T nr / 2T nh

23

Fig. 13 Effect of ratio of half of natural roll period tonatural heave period on maximum roll amplitudes at peaksof parametric rolling in beam waves.

0.5

1

1.5
T e (sec)

2

2.5

3

Fig. 15 Heave response curve in the cases when parametricrolling does not occurs.
180

Tnh
0.748 sec

without Bilge Keel
Hw = 0.04m

90
ε (deg.)Effects of the ratio of half of natural roll period to natural
heave period, Tnr / 2Tnh on the peak of roll amplitude of theparametric rolling are shown in Fig. 13. Intuitively,
maximum parametric rolling must appear at when Tnr / 2Tnh
is 1, if heave resonance causes the parametric rolling inbeam waves. The results shown in Fig. 13, however,demonstrate that maximum parametric rolling occurs at
when Tnr / 2Tnh is 1.38. This fact may suggest that heave
resonance only partly causes parametric rolling in beam
waves.

0

0.921
1.38
1.92
Cal. (OSM)

0
0

25

without Bilge KeelH w = 0.04m

Z0 / ζ0Tnr (sec)0
T nr / 2T nh

-90

0.787
1.18
1.92
Cal. (OSM.)-180
0

Fig. 16

0.5

1

1.5
T e (sec)

2

0.921
1.63
2.25

2.5

3

Phase angle of heave motion from incident waves
in the cases when parametric rolling occurs.

An example of time histories of roll, heave and incident
wave are shown in Fig. 17. Using these measured data,
relative heave motion to wave surface, or time variation of
the draft of the ship, can be calculated. The results of thecalculations are shown in Fig. 18. We can see that the
time-variation of the draft is large at the heave resonant
period, and gradually decreases with far from it. The time
variation of the draft in beam waves may cause change ofstability.
Two examples of the time-variation of the draft of the ship
are shown with the amplitudes of parametric rolling in beam
waves in Figs. 19 and 20. These figures demonstrate that theparametric rolling occur when the relative heave motion
becomes significant. Furthermore, it occurs even if thetime-variation of the draft comparatively small (Ikeda et al.,2007).

considered in this model as shown in Eq. (1).

40
30
20

0
-10

ΔGM
⎞
cos ωet ⎟φ +2GM⎠Δa3 ⎞ 3Δa5 ⎞ 52 ⎛ a32 ⎛ a5++
ωφ ⎜⎟φ + ωφ ⎜⎟φ⎝ GM GM ⎠⎝ GM GM ⎠
Δa ⎞
2⎛ a
2
+ ωφ ⎜ 7 + 7 ⎟φ 7 = ζ a rkωφ sin ωet
GM
GM
⎝
⎠
⎛
⎝

φ&& + 2αφ& + β φ& φ& + γφ&3 + ωφ 2 ⎜1 +

10

291

292

293

294

295

296

-20
-30
Heave

-40

Wave surface

Roll

Fig. 17
Measured time histories of ship motions andwave surface. (Tnr=2.39 sec, Te=1.05 sec)

Relative heave motion (mm)12

T nr / 2T nh

Tnh
0.748 sec

10
8

0.787

0.921

1.18

1.63

1.92

2.26

H w = 0.04m
without Bilge Keel

6
4
2
0
0.4

0.6

0.8

1
1.2
T e (sec)

1.4

1.6

Fig. 18 Relative heave motion to wave surface (=amplitude of time-variation of draft in waves).
T nh
0.748 sec

25

T nr / 2
0.825 sec

Φ0 (deg)

20

15
T nr / 2T nh = 1.18
Left axis
roll amplitude

10

Right axis
relative heave motion

15

5

10

for the polynomial equation up to 7th order of restoring
moment. α , β and γ are the roll damping coefficientsand α 3 , α 5 and α 7 are the nonlinear coefficient of restoring
moment in still water. These coefficients can be determinedfrom the fitting of the GZ curves in still water. ΔGM , Δα 3 ,Δα 5 and Δα 7 are the coefficients of the restoring momentin waves, and they are the function of wave steepness andrelative position of ship in waves. These coefficients arecalculated by Froude-Krylov assumption.
The numerical results, as shown in Fig. 21, demonstrate
the same tendency with the experimental results. However
the numerical results under-estimate the experimental results.
This could be due to change of metacentric height in the
numerical results much lower than the real change ofmetacentric height in the experiments, or this could be dueto the change rolling period, which was identified in themodel experiments. But critical wave steepness can be
estimated approximately (Munif et al., 2005).
0.16

0

0

0.4

0.6

0.8

1
1.2
T e (sec)

1.4

Simulations
Experiments

0.14
0.12

Area of parametricresononce

0.1
H/λ

5

Relative Heave Motion (mm)

30

0.08
0.06

1.6

0.04

Fig. 19 Parametric rolling in beam waves and relative
heave motion when Tnr/2Tnh=1.18.

Area of non parametricresononce

0.02
0
0.5

T nh
0.748 sec

15

T nr / 2
1.195 sec

25

T nr / 2T nh = 1.63
Left axis
roll amplitude

Φ0 (deg)

20

10

Right axis
relative heave motion

15
10

5

5

Relative Heave Motion (mm)

30

0

0

0.4

0.6

0.8

1

1.2

(1)

1.4

1.6

T e (sec)

Fig. 20 Parametric rolling in beam waves and relative
heave motion when Tnr/2Tnh=1.63.
THEORETICAL MODELS OF PARAMETRIC ROLLINGIN BEAM WAVES
Using a nonlinear 1 DOF model of roll equation, asimulation of the parametric rolling in beam waves is carriedout. Nonlinear damping and nonlinear restoring moments are

0.55

0.6

0.65
ωe

0.7

0.75

0.8

Fig. 21 Area of parametric resonance in beam waves
resulted from numerical simulations and experiments.
RECOMMENDATION FOR IMO CRITERIA
For a ship having large slope of side-walls and flat-shallow
stern shape with the natural roll period close to twice as thenatural heave period, it is recommended to examine thepossibility of parametric rolling not only in longitudinal
waves but also in beam waves. An appropriate size of bilge
keels should be designed to erase the parametric rolling inbeam waves to guarantee the safety in dead ship condition.
CONCLUSIONS
The activity of the SCAPE committee for parametric rollingin beam waves under drifting condition provides thefollowing conclusions:
1. Parametric rolling in beam waves can be significant for

2.
3.
4.

5.

a large passenger ship if her bilge keels are removed.
If she is equipped with the designed bilge keels, theparametric rolling in beam waves disappears but that inlongitudinal waves does not.
When the bilge keel size increases, the critical wave
height of parametric rolling in beam waves and its
magnitude increases.
When the relative heave motion in beam waves
becomes significant and the roll damping is not
sufficiently large, parametric rolling in beam waves
could occur. The ratio of natural periods of heave androll on its own does not explain the amplitude ofparametric rolling in beam waves.
The critical wave height for parametric rolling in beam
waves can be roughly estimated with a numericalsimulation with roll restoring variation in time.

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Shipbuilding Progress, 27-306.
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