PREDICTION METHODS FOR PARAMETRIC ROLLING
WITH FORWARD VELOCITY AND THEIR VALIDATION–FINAL REPORT OF SCAPE COMMITTEE (PART 2) Hirotada HASHIMOTO*, Naoya UMEDA*, Yoshitaka OGAWA**,
Harukuni TAGUCHI**, Toshio ISEKI***, Gabriele BULIAN*,
Naoji TOKI****, Shigesuke ISHIDA** and Akihiko MATSUDA******Osaka University, Suita, JAPAN**National Maritime Research Institute, Mitaka, JAPAN***Tokyo University of Marine Science and Technology, Tokyo, JAPAN****Mitsubishi Heavy Industries, Nagasaki, JAPAN*****National Research Institute of Fisheries Engineering, Kamisu, JAPANABSTRACTRegarding parametric rolling of ships with forward
velocity, this paper reports experimental, numerical andanalytical studies conducted by the SCAPE committee,
together with critical review of theoretical progress on
this phenomenon. Here experimental results of acontainership in regular, long-crested and short-crested
irregular head waves, and those of a PCTC (pure car
and truck carriers) are also shown. They are comparedwith several numerical predictions to realize
quantitative prediction of parametric rolling. The
restoring variation, which is main cause of parametricrolling, is investigated with model experiments andpotential theories. Application of time-varying
coefficient vector autoregressive model is also attempted.
Furthermore, the effect of devices preventing
parametric rolling is investigated. Based on theseoutcomes, the SCAPE committee also proposes draft
stability criteria consisting of vulnerability criterion anddirect assessment as a candidate for the new generation
criteria at the IMO (International Maritime
Organization).
KEY WORDS: Parametric Rolling, Restoring Variation,
Regular and Irregular Head Waves, Containership, Pure Car
and Truck Carrier, Non-ergodicity, Spectrum Analysis,
Grim’s Effective Wave, Anti-Rolling Tank
INTRODUCTION
Parametric rolling is roll motion induced by time-varying

restoring arm. Roll frequency of parametric rolling ismultiple of half of the encounter frequency and is nearly
equal to the natural roll frequency. In case the roll frequency
is half of the encounter frequency, this roll motion could be
most significant, and is called as low cycle resonance or
principal resonance. The main reason why the restoring arm
changes with time is that the water plane area and theunderwater hull volume change as the waves pass along theship. Because recent large containerships have exaggerated
bow flare and transom stern, this change can be significant
in longitudinal waves.
As a result, serious accidents of parametric rolling formodern containerships and PCTCs have been reported inrecent years. These accidents triggered off a review of theIntact Stability Code (IS Code) of the IMO, and it has been
discussed to set performance-based criteria as an alternative
to the existing prescribed criteria. The new
performance-based criteria are requested to cover threemajor capsizing scenarios including parametric rolling as
one of roll restoring variation problems. In this stage, aprediction method for parametric rolling with quantitative
accuracy is required. For this purpose, all relevant factors ofparametric rolling should be systematically investigated todevelop a standard numerical simulation technique for theuse of performance-based criteria.
HISTORICAL REVIEW
Although parametric rolling itself is well known among
theoretical researchers, it has been regarded as rather an
exceptional event in actual ocean waves for a long time.
However, a recent model experiment showed that acontainer ship complying with the IS Code of IMO couldsuffer severe parametric rolling even in short-crested
irregular following waves and could capsize in long-crested
irregular following waves (Umeda et al., 1995). Moreover,
severe parametric rolling in head seas at the Pacific Ocean
was reported for a post-Panamax C11 class container ship
(France et al., 2003), and similar incident of head-sea
parametric rolling was reported for a PCTC in the Azores
Islands waters (Hua et al., 2006). The former forced todescribe parametric rolling in the IMO operational guidance,
MSC Circ. 707, and the latter resulted in the review of the
IMO IS Code to open the door to performance-based
criteria instead of existing prescriptive criteria. For thispurpose, prediction of parametric rolling is required to havequantitative accuracy and identify all potential danger bothin regular and irregular waves.
Theoretical studies on parametric rolling can be found in
the thirties with linear restoring (Watanabe, 1934). Later, inthe fifties, linear and nonlinear damping was taken into
account (Kerwin, 1955). These studies enable us to discussparametric rolling with the Mathieu equation. Then, toinvestigate capsizing, nonlinearity of restoring moment instill water was taken into account. At this stage, nonlinear
dynamical system approach including geometrical andanalytical studies is required to identify all potential danger
among co-existing states. Such examples can be found in
Sanchez and Nayfeh (1990), Kan and Taguchi (1992),
Soliman and Thompson (1992) for uncoupled roll models,
Oh et al. (2000) for coupled pitch-roll model and Umeda etal. (2004) for uncoupled model with realistic modelling ofroll restoring variation.
On the other hand, several six degrees-of-freedom models
such as Hamamoto and Akiyoshi (1988), De Kat andPaulling (1989), Munif and Umeda. (2000), Matsusiak(2001) have been developed for numerical prediction intime domain. Here the relationship between wave steepness
and restoring moment is fully taken into account. The works
using these detailed models only show time series from thelimited number of initial condition sets, however, there is apossibility to overlook some potential danger because of
nonlinearity of the system. In addition, the minimum
requirements for the numerical modelling were notestablished well. Thus, the ITTC (International Towing
Tank Conference) conducted a benchmark testing of
numerical codes, which are requested to reproduce
experimental runs of a 150m–long containership (known as
the ITTC Ship A-1) in regular following waves (Umeda etal., 1995), but it was found only one code can successfully
predict the occurrence of parametric rolling and capsizing(Umeda and Renilson, 2001). After the ITTC benchmark
testing, further efforts on numerical modelling werereported; Ribeiro et al. (2003), Belenky et al. (2003), Neves
and Rodriguez (2005), Brunswig et al. (2006), Spanos andPapanikolaou (2006). It is noteworthy here that such recent
efforts mainly focus on head-sea parametric rolling
responding to the recent accidents.
Regarding the parametric rolling in irregular waves,Roberts (1982) applied stochastic averaging technique andderived the Fokker-Plank equation as a prerequisite forexistence of stationary process of roll. Bulian et al. (2003)
experimentally investigated the applicability of the Markov
process approach for a Ro-Pax, and confirmed theagreement is good for a narrow band spectrum. Levadou
and van’t Veer (2006) also showed that the Markov process

approach can well predict the critical wave height ofparametric rolling in irregular waves. Bulian andFrancescutto (2006) proposed a fully analytical approachfor the determination of the stochastic stability threshold of
parametric rolling, and it overestimates roll motion without
a tuning factor. Themelis and Spyrou (2006) proposed aprobabilistic assessment of parametric rolling by assuming
probability of occurrence of instability is represented by thatof encountering the critical or worse wave groups. Belenky
et al. (2003) and Belenky (2006) simulated parametricrolling in irregular waves with 50 realisations andnumerically confirmed that roll motion is practically
non-ergodic while heave and pitch motions are not.
It is also important to estimate the restoring variation in
longitudinal waves, which is a principal cause of parametricrolling. Paulling (1960) reported that the Froude-Kylov
prediction can explain captive model test results. However,the Froude-Krylov prediction is not always sufficient,particularly for modern ships. To overcome this difficulty,
Nechaev (1989) developed an empirical formula from his
series of captive model experiments. This formula includes
not only the Froude-Krylov component but alsohydrodynamic ones. To evaluate such hydrodynamic
components theoretically, Boroday (1990) applied a strip
theory to a heeled fishing vessel hull, and then good
agreement was found by considering the sum of the
Froude-Krylov and added-mass related terms. It was
expected to pursue this research direction further.
OUTCOMES FROM SCAPE COMMITTEE ACTIVITIES
Experimental Study on Parametric Roll
Free running model experiments for a 6600 TEU
post-Panamax containership were conducted at NRIFE(National Research Institute of Fisheries Engineering) witha 1/100 scaled model and NMRI (National Maritime
Research Institute) with a 1/76.7 scaled model. This ship isdesigned by NMRI and her principal particulars and bodyplan are shown in Table 1 and Fig. 1, respectively. Model
experiments were systematically conducted in regularwaves, long-crested and short-crested irregular waves.
Hashimoto et al. (2006A) investigated the effect of wave
height on parametric rolling in regular head waves, as
shown in Fig. 2. This result indicates that amplitude ofparametric roll does not always increase with wave
steepness. Taguchi et al. (2006A) investigated the effect of
initial steady heel on parametric rolling in regular headwaves, as shown in Fig. 3. Here the initial steady heel angle
is 3 degrees. Occurrence region of parametric roll without
the steady heel is larger than that with steady heel to someextent. Influence of significant wave height and encounter
angle in long-crested irregular waves were examined by
Taguchi et al. (2006B), as shown in Fig. 4 and Fig. 5.
Maximum roll angle of parametric rolling increases withsignificant wave height and decreases with deviation frompure head wave condition. Effect of ship speed onparametric rolling was examined in regular, long-crested,short-crested irregular head waves by Hashimoto et al.
(2006A), as shown in Fig. 6. In smaller Froude number, rollangles both in long-crested and short-crested irregularwaves are slightly smaller than that in regular seas. In theregion of Froude number greater than 0.1, small parametricrolling in irregular waves occurs while no parametric rolloccurs in regular waves.
Fig. 7 shows the probability density function of
instantaneous pitch and roll in short-crested irregular waves(Hashimoto et al., 2006A). Non-Gausssian property isexperimentally confirmed even in short-crested irregularwaves, and those in regular and long-crested irregularwaves are more remarkable.

T02 =1.38s, Fns=0.07

25

1/10 highest mean（χ=180°）〃（χ=150°）Max.（χ=180°）〃 （χ=150°）

φ(deg.)

20
15
10
5
0
0

5

10

15

20

25

H1/3 (cm)Items
length between perpendiculars: L
breadth: B
depth: D
draught: T
block coefficient: Cb
pitch radius of gyration: Kyy/Lppmetacentric height: GM

Ship
283.8m
42.8m
24.0m
14.0m
0.630
1.06m

NRIFE
2.838m0.428m
0.24m
0.14m
0.630
0.258
0.0106m

NMRI
3.700m0.558m0.318m0.183m0.630
0.247
0.014mnatural roll period: Tφ

30.3s

3.20s

3.460sFig. 4 Influence of significant wave height on parametricrolling in long-crested irregular head waves with
T02=1.38sec. (Taguchi et al., 2006B)
T02 =1.38s, H1/3 =12.3cm

25

1/10 heighest mean （Fns=0.07）〃（Fns=0.11）
Max.（Fns=0.07）〃 (Fns=0.11)20
φ(deg.)

Table.1
Principal particulars of the post-Panamax
containership

15
10
5
0
135

150

165

180

χ (deg.)Fig. 5 Influence of encounter angle on parametric rollingin long-crested irregular waves with H1/3=12.3cm and
T02=1.38 sec. (Taguchi et al., 2006B)

amplitude of parametric rolling (deg.)

25

20

Body plan of the post-Panamax container ship

amplitude of parametric rolling (deg.)

25

20

15

long-crested waves

10short-crested waves(cos^4)
short-crested waves(cos^2)

5

0
15

0

0.05

0.1

0.15 0.2 0.25
Froude number

5

0
0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

Fig. 2 Influence of wave height on parametric rolling in
regular head waves with λ/L=1.6 at ωe=2ωφ(Hashimoto et al., 2006A)

probability density

wave steepness

0.16

0.16

0.14

0.14

0.12

0.12

0.1
0.08
0.06
0.04
0.02

χ=180°
λ/L=1.2(w/o steady heel)=1.2(with steady heel)=1.4(w/o steady heel)=1.4(with steady heel)=1.8(w/o steady heel)=1.8(with steady heel)=2.0(w/o steady heel)=2.0(with steady heel)25

20

φm (deg.)

0.3

0.35

Fig. 6 Influence of ship speed on parametric rolling in
regular and irregular head waves (Hashimoto et al., 2006A)

10

probability density

Fig. 1

regular waves

15

10

0
-30

0.1
0.08
0.06
0.04
0.02

-20
-10
0
10
20
elevation of roll angle (degrees)

30

0
-20

-10
0
10
elevation of pitch angle (degrees)

20

Fig. 7 Probability density functions of instantaneous rolland pitch angle in short-crested irregular waves (histogram:experiment; line: Gaussian distribution)(Hashimoto et al., 2006A)

5
Te /Tφ

0
0.30

0.40

0.50

0.60

0.70

Fig. 3 Influence of initial steady heel angle on parametricrolling in regular head waves with Η =11cm(Taguchi et al., 2006A)

Free running model experiment for a latest PCTC wasconducted at Osaka University. Principal particulars areshown in Table.2, and its 1/64 scaled model was used in theexperiment. Roll amplitude of parametric rolling in regular
head waves is shown in Fig. 8 (Hashimoto et al., 2007A).
Most significant parametric rolling was found from λ/L=1.0
to 1.3. Maximum roll amplitude of parametric rolling withconstant wave height of 0.05m is 23 degrees at Fn=0, which
is at similar level as that of the post-Panamax containership.
PCTCs could be more prone to meet critical wave groups
with their mean wave length is comparable to ship length
because their length is generally smaller than latest
containerships. Therefore, further researches on parametricrolling for PCTCs are desirable.
Table.2
carrier

Principal particulars of the pure car and truck

item
length: L
breadth: B
depth: Dmean draught: T
block coefficient:Cb
metacentric height: GM

value
192.0 m32.26 m37.0 m8.18 m0.54
1.25 mnatural roll period: Tφ

22.0 sparametric rolling in regular head sea
H=0.05 (m)max. roll amplitude (degree)

30
25
20

exp(λ/L=0.8)exp(λ/L=1.0)15exp(λ/L=1.3)
exp(λ/L=1.6)

10
5
0
0

0.05

0.1

0.15

0.2

0.25

0.3

Fn

Fig. 8 Maximum roll amplitude of parametric rolling in
regular head waves with H=0.05m for the PCTC(Hashimoto et al., 2007A)
Theoretical Estimation of Roll Restoring Variation
Roll restoring variation of the post-Panamax
containership and the ITTC Ship A-1 containership, which
has less significant flare as shown in Fig. 9, was measured
in model scale. The models were towed with the constantheel angle of 10 degrees in regular head and following
waves at a towing tank and the reaction moment weredetected as the restoring moment (Umeda and Hashimoto,2006; Nakamura et al., 2007). Here the models were free inheave and pitch. Nakamura et al. (2007) conducted acomparative study on the mean and amplitude ofmetacentric height (GM) variation for these two different
containerships, as shown in Fig. 10 and Fig. 11. Here GM
variation is curve-fitted with mean and amplitudes of firstand second harmonic components. Amplitude of firstharmonic GM variation of the post-Panamax containershipis much larger than that of the ITTC containership. This isbecause the amplitude of restoring variation mainly depends
on the slope of ship side-wall (Hamamoto & Fujino, 1986)
and the modern post-Panamax containership has
exaggerated bow flare and transom stern, which allows
increasing the number of onboard containers. On the other
hand, little difference in the mean of GM variations wasfound. Ogawa et al. (2007A) investigated the effect of wave
height and influence of restraint of surge motion on GM
variation by a model experiment for the post-Panamax

containership, and the result is shown in Fig. 12. The
experimental result demonstrates that the amplitude of GM
variation linearly increases with wave steepness and theeffect of surge motion restraint on restoring variation is
negligibly small.
Since the estimation of roll restoring variation isimportant for quantitative parametric roll prediction, anumerical estimateion of the roll restoring variation basedon a potential theory was attempted (Umeda et al., 2005). In
the proposed calculation, roll restoring variation is obtained
as the sum of nonlinear Froude-Krylov component byintegrating the wave pressure up to wave surface withtaking instantaneous heave and pitch motions calculated bya linear strip theory into account, and radiation component
induced by vertical motions of a heeled hull and diffraction
component due to an asymmetric hull calculated by a linear
strip theory. In Fig. 13, the measured amplitude of rollrestoring variation is compared with the Froude-Krylov
calculation on its own, and the sum of the Froude-Krylov,
radiation and diffraction components for the post-Panamax
containership (Nakamura et al., 2007). Consideration of thehydrodynamic components improves the prediction
accuracy of restoring variation. This means that parametricroll danger could be underestimated if these components are
neglected in the modelling. Fig. 14 shows the comparison ofprobability density function of GZ between the experimentand the calculation in long-crested irregular waves(Hashimoto et al., 2006B). Here, measured heave and pitch
motions are used for the Froude-Krylov force calculation,and instantaneous wave height distribution along a ship is
obtained by the Fourier analysis with measured data and isdirectly used in the calculation. As a result, the amplitude ofroll restoring variation in long-crested irregular waves canbe well explained as the sum of nonlinear Froude-Krylov
and heel-induced hydrodynamic components.
Since the measured mean of roll restoring variation,
which changes natural roll period in waves, is not negligibly
small, estimation of the mean of roll restoring variation is
also important to predict the occurrence region ofparametric rolling accurately. Ogawa and Ishida (2006)attempted to calculate the mean of roll restoring variation
by applying a momentum theory with Kochin function
obtained by Kashiwagi(1991)’s enhanced unified slenderbody theory. Here steady roll moment is expressed by
Kochin function. Comparison of the calculated result withthe captive model experiment (Umeda et al., 2005) is shownin Fig. 15. Steady roll moment in head waves decreases
with increase of Froude number. From the comparison withcaptive mode experiment, the calculation can estimate its
trend with respect to wave steepness but underestimates theexperimental result particularly in high Froude number.
This might be because a hydrodynamic lift effect on heeledhull in high speed region becomes significant.

Body plan of the ITTC ShipA-1 containership

GM 1st amplitude/Lpp

0.014

ITTC

0.012

post-Panamax

Average of GM variation (Head sea, λ/L=1.0, heel angle =10deg.)

0.01

0.01

0.008
0.006
0.004
0.002
0
0

0.01

0.02

0.03

0.04

0.05

0.06

wave steepness

Fig. 10 Amplitude of 1st harmonic GM variation at
Fn=0.1 and heel angle of 10deg. (Nakamura et al., 2007)
0.014

GM mean/Lpp

post-Panamax

0.007
0.006
0.005
0.004
0.003
0.002
0.001
0
0

0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05 0.055
H/λ

0.008
0.006

0.002
0
0

0.01

0.02

0.03

0.04

0.05

0.06

wave steepness

Mean of GM variation at Fn=0.1 and heel angle of
10deg. (Nakamura et al., 2007)
Heel 10deg (λ/L=1.6)

GM amplitude/Lpp

0.008

0.01

0.004

Fn=0.2
Fn=0.05
Fn=0.2(Surge free)Fn=0.05(Surge free)

0.01
0.009
0.008
0.007
0.006
0.005
0.004
0.003
0.002
0.001
0
0

0.005

0.01

Fn=0.1
Fn=0
Fn=0.1(Surge free)Fn=0(Surge free)

0.015

0.02

0.025

H/λ

Fig. 12 Influence of surge motion on roll restoring
variation in regular head waves with λ/L=1.6(Ogawa et al., 2007A)
post-Panamax containership
0.014
GM 1st amplitude/Lpp

Fn=0.1(Cal.)
Fn=0.2(Cal.)
Fn=0.3(Cal.)
Fn=0.1(Exp. by Umeda et.al.)
Fn=0.2(Exp. by Umeda et.al.)
Fn=0.3(Exp. by Umeda et.al.)

0.009

Fig. 15 Comparison of mean of GM variation in regular
head waves with λ/L=1.0 and heel angle of 10deg.
(Ogawa and Ishida, 2006)

ITTC

0.012

Fig. 11

Fig. 14 PDF of restoring arm in long-crested irregularwaves with H1/3=0.057m and T01=1.35sec. at Fn=0.1(Hashimoto et al., 2006B)(Average of GM variation)/Lpp

Fig. 9exp
FK + R&D
FK

0.012
0.01
0.008
0.006
0.004
0.002
0
0

0.01

0.02

0.03

0.04

0.05

0.06

wave steepness

Fig. 13 Comparison of amplitude of 1st harmonic GM
variation at Fn=0.1 and heel angle of 10 deg.
(Nakamura et al., 2007)

Numerical Prediction of Parametric Rolling
Numerical models for parametric roll prediction should
be validated not only in regular waves but also in irregularwaves because the relation between roll restoring variation
and wave height is inherently nonlinear. Therefore theGrim’s effective wave concept (Grim, 1961) was adopted
firstly for the prediction of parametric rolling in irregularwaves. The idea of the effective wave concept is that
irregular wave surface around a ship is replaced withequivalent regular wave, which length is equal to ship
length, by a least square method. By introducing theeffective wave, the relationship between the ocean wave
and the effective wave amplitude becomes linear so that therestoring variation can be calculated by a linear
superposition with nonlinear but non-memory relationshipbetween the roll restoring moment and the effective wave
amplitude. In the calculation, roll restoring variation in
irregular waves is calculated as the sum of nonlinear
Froude-Krylov force with heave and pitch obtained byhydrostatic balance for an instantaneous effective wave,
radiation and diffraction components calculated under theassumption that it has a linear relation with wave steepness
and roll angle. By utilising the above modelling, uncoupledroll model was applied to parametric rolling prediction.
Here the roll damping is estimated from a roll decay test
without forward velocity, and long-crested irregular wave isassumed to follow the ITTC spectrum and is expressed as
the sum of 1000 components of sinusoidal waves with theirrandom phases. Fig. 16 shows the comparison of maximum
roll angle of parametric rolling of a post-Panamax
containership between free running model experiment andnumerical simulation (Hashimoto et al., 2006A). Bothcalculations excessively overestimate the maximum rollangle of parametric rolling. Bulian et al. (2006) modified
Grim’s effective wave concept to improve thesteady amplitude and occurrence region of parametric roll.
Fig. 23 shows the comparison of maximum and 1/10 highest
mean of roll angle of parametric rolling in long-crested
irregular waves (Ogawa, 2007). Numerical calculation
results as the ensemble average of 20 realisations
quantitatively agrees with experimental result in 1/10
highest mean of roll angle but not so that in maximum rollangle. It is expected here to identify the crucial factors inthis modelling to realise the agreement with modelexperiments. Furthermore, the application of a CFD
approach based on the RANS equation is now underwaywith the collaboration between Osaka University and the
University of Iowa.
The nonlinear strip method can predict parametric rollingboth in regular and irregular waves with practical accuracy
as mentioned above. By utilising this numerical model,
horizontal and vertical accelerations during parametricrolling were calculated (Ogawa et al., 2007B). The results,
shown in Fig. 24, indicate the gravitational effect onhorizontal acceleration acting on a secured container isdominant.

max. roll angle (degrees)

50
40
FK only30

present
20

experiment

10
0
0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Froude number

s/kHw

Fig. 16 Comparison of maximum roll angle of parametricrolling in long-crested irregular head waves with
H1/3=0.221m and T01=1.32sec. (Hashimoto et al., 2006A)
λ/L=1.2

6.0
5.0
4.0
3.0
2.0
1.0
0.0

s/kHw

0.30

0.40

0.50

Te /Tφ0.60

λ/L=1.6

6.0
5.0
4.0
3.0
2.0
1.0
0.0
0.30

s/kHw

approximation accuracy of random wave surface byadjusting its wave crest position as an additional randomvariable. This modification (named Improved Grim
Effective Wave, IGEW) improves prediction of the pitch
moment and/or static pitch equilibrium angle, while the rollmoment prediction itself is sufficient even with the original
concept.
By following these studies, the numerical prediction
method was upgraded as follows. Firstly the Froude-Krylov
force is directly calculated around the instantaneous
irregular wave surface with the heave and pitch motions
predicted by a strip theory for an upright condition. Roll
decay test with forward velocities was conducted andmodelled as a function of Froude number. Taguchi et al.
(2006C) reported that modelling with such Froude-Krylov
prediction improves the agreement in comparison with theexperimental result for the post-Panamax containership inmoderate regular head waves as shown in Fig. 17. Byutilising the Froude-Krylov component with dynamic heaveand pitch effect and the radiation and diffraction effect,
Hashimoto et al. (2007A) show the comparison of
maximum roll angle of parametric rolling for the PCTC in
regular head waves as shown in Fig. 18. Numericalprediction still overestimates maximum roll angle in somecases. Since some discrepancies with the measured restoring
variation are found as well, it is still required to improve theestimation accuracy of roll restoring variation for
quantitative prediction of parametric rolling. Fig. 19 shows
the comparison of experimentally and numerically obtained
parametric rolling in long-crested irregular waves(Hashimoto et al., 2007A). Here, model and numerical runs
were repeated with 4 realisations for each case, and the
maximum values in their realisations are plotted. The
comparison of roll restoring variation in irregular waves
between numerical estimation and experiment is also shownin Fig. 20. These results indicate that small difference of theestimation of roll restoring variation could result insignificant difference of maximum angle of parametric rollin irregular waves. Fig. 21 shows the measured andcalculated maximum roll angles of parametric roll for 4
different realisations. Practical non-ergodicity of parametricroll is confirmed both in experiment and calculation.
Therefore, how to overcome this practical non-ergodicity ofparametric rolling in irregular waves is important for thevalidation of the code and practical stability assessment.
Bulian et al. (2008) investigated this problem further basedon more extensive model experiments of the post-Panamax
containership.
A time domain simulation code for coupled sway, heave,roll and pitch motion was applied to parametric rolling
prediction. Here the local added mass and wave-making
damping are calculated for instantaneous water surface aswell as the roll restoring. The wave exciting force iscalculated by solving the Helmholtz equation, which allows
shorter waves. (Ogawa et al., 2007A) Ogawa (2007)
conducted numerical simulations in long-crested irregularwaves for the post-Panamax containership. In thecalculation, irregular waves are expressed as a superpositionof 200 sinusoidal waves following the ISSC1964 spectrum.
Fig. 22 shows the comparison of the amplitude ofparametric rolling in regular head waves between freerunning model experiment and numerical prediction. The
calculation well explains the experimental result of the

0.40

0.50

Te /Tφ0.60

λ/L=2.0

6.0
5.0
4.0
3.0
2.0
1.0
0.0
0.30

0.40

0.50

Te /Tφ0.60

Fig. 17 Influence of encounter period on the amplitude ofparametric rolling in regular head waves with H=0.11m(Taguchi et al., 2006C)

parametric rolling in regular head sea
H=0.05 (m)Maximum and 1/10 highest mean of roll angle (χ=180°, T02=12.0sec.)exp(λ/L=0.8)3530exp(λ/L=1.0)35

exp(λ/L=1.3)3025exp(λ/L=1.6)

25

20

calc(λ/L=0.8)
calc(λ/L=1.0)

15

Φ(deg.)

max. roll amplitude (degree)40

calc(λ/L=1.3)

10

Max.(Exp., Fns=0.07)
1/10 highest (Exp., Fns=0.07)
Max.(Cal., Fn=0.055)
1/10 highest (Cal., Fn=0.055)

20
15
10

calc(λ/L=1.6)

5

5

0

0
0

0.05

0.1

0.15

0.2

0.25

0

0.3

2

4

6

Fig. 18 Comparison of maximum roll amplitude ofparametric rolling in regular head waves with H=0.05m(Hashimoto et al., 2007A)

14

16

Hw=6.1m
Hw=8.4m
Inertia(Hw=6.1m)
Inertia(Hw=8.4m)

45cal. with FK and hydrodynamic effect40

experiment

35

25

aL/gζ

max. roll angle (degress)

12

Horizontal Acceleration(X:S.S.8, Y: C.L., Z: 15m above W.L., Head seas, Fn=0.05)
50cal. with FK on its own

30

10

Fig. 23 Comparison of maximum roll angle of parametricrolling in long-crested irregular head waves with
T02=12.0sec. (Ogawa, 2007)

T01=1.22(sec), H1/3=0.083(m)358
H1/3 (m)

Fn

20
15

30
25
20
15
10

10

5
0

5

10

12

14

0

16

18

20

T(sec.)

0

0.05

0.1

0.15
Vertical Acceleration (X:5.9m aft from S.S.5, Y: 15m from C.L., Z: 3.9m
above W.L., Head seas, Fn=0.05)
Hw=0.1m
Hw=6.1m
Hw=8.4m
Inertia(Hw=6.1m)
Inertia(Hw=8.4m)

Fn

9
8
7
aL/gζ

Fig. 19 Comparison of maximum roll angle of parametricrolling in long-crested irregular head waves with
H1/3=0.083m and T01=1.22sec. (Hashimoto et al., 2007A)

10

6
5
4
3
2
1
0
10

12

14

16

18

20

T(sec.)

Fig. 24 Effect of wave height on the response amplitudeoperators of horizontal (above) and vertical (below)
accelerations (Ogawa et al., 2007B)

Fig. 20 Comparison of GZ variation in long-crested
irregular waves with H1/3=0.083m and T01=1.22sec.
(Hashimoto et al., 2007A)

maximum roll angle (degrees)

20
15
10
5
0

dynamic component with
Froude-Krylov component

25
20
15
10
5
0

1

2

3

4

1

realizations

2

3

4

reallizations

Fig. 21 Maximum roll angle of parametric rolling in
long-crested irregular waves for 4 realisations (left:
experiment, right: calculation) (Hashimoto et al., 2007A)
Amplitude of parametric roll in regular wave (χ=180°, Hw=8.4m)
30

Cal.(λ/L=1.2)
Cal.(λ/L=1.4)
Cal.(λ/L=2.0)
Exp.(λ/L=1.2)
Exp.(λ/L=1.4)
Exp.(λ/L=2.0)25
20
Φ(deg.)

maximum roll angle (degrees)

Froude-Krylov component
on its own

T01=1.22(sec), H1/3=0.083(m), Fn=0.05

T 01=1.22(sec), H1/3=0.083(m), Fn=0.05
25

15
10
5
0
0.3

0.35

0.4

0.45

0.5
Te/TΦ

0.55

0.6

0.65

0.7

Fig. 22 Comparison of steady amplitude of parametricrolling in regular head waves with H=8.4m(Ogawa et al., 2007A)

Spectrum Analysis of Parametric Rolling
To understand the nonlinear phenomenon of parametricrolling and identify its nature in frequency domain, Iseki(2006) applied the Time-Varying coefficient Vector
Auto-Regressive (TVVAR) model to the time series ofparametric roll obtained by free running model experiment
for the post-Panamax containership. Firstly, the relative
noise contribution to roll angle during parametric rolling in
regular head waves was analysed as shown in Fig. 25. From
the result, the contribution of pitch to roll is negligibly small
in a peak frequency of roll. Secondly a bispectrum analysis
was attempted for the time series of parametric roll in
short-crested irregular waves to clarify the skewness of
non-Gaussian parametric rolling. Fig. 26 shows theestimated bispectra of roll and pitch motions. The peaks ofroll and pitch motions appear on 45 degrees of diagonal line,
which indicates the existence of nonlinearity of each motion
during the parametric rolling. Finally a trispectrum analysis
was attempted to identify the kurtosis of non-Gaussian
parametric rolling due to third and fifth harmonicfrequencies (Iseki, 2007). Fig. 27 shows the analysed
trispectrum of roll motion in regular waves with fixed f3
frequency of 0.32Hz . There are three peaks with thefrequencies of 1, 3, 5 times as natural roll frequency. By
applying the trispectrum analysis, kurtosis of parametricrolling due to higher order harmonic frequencies isdemonstrated.

1.00

Fig. 26

1.00

Pitching Bispectrum (Amp.)

0.94

80-100
60-80
40-60
20-40
0-20

0.88
0.82
0.76
0.70

0.94
0.88
0.82
0.76
0.70

30

1.00

0.94

0.88

0.82

0.00

0.76

0.06

0.00

0.70

0.06
0.64

0.12

0.59

0.18

0.12

0.53

0.18

0.47

0.23

0.41

0.29

0.23

0.35

0.35

0.29

0.29

0.41

0.35

0.23

0.47

0.41

0.18

0.53

0.47

0.12

0.59

0.53

0.06

0.64

0.59

0.00

0.64

1.00

0.94

0.88

0.82

0.76

1600-2000
1200-1600
800-1200
400-800
0-400

0.70

0.64

0.59

0.53

0.47

0.41

0.35

0.29

0.23

0.18

0.12

0.06

0.00

Rolling Bispectrum (Amp.)

maximum amplitude of parametric roll(degree)Fig. 25 Relative noise contribution to roll angle in regular
head waves with H/λ=1/20, λ/L=1.6 (Iseki, 2006)

risk of container damage due to parametric rolling.
However, to estimate the performance of ART, damping
coefficient of water in ART is required in a numericalsimulation of parametric rolling, and a scaled model test of
ART is normally executed. To overcome this annoyance,
Hashimoto et al. (2007B) applied the Moving Particle
Semi-implicit Method (MPS method) to estimate damping
effect of the tank water. As a result, the MPS method cansimulate the free oscillation test and predict the natural
period and damping coefficient of ART quantitatively as
shown in Fig. 29. Fig. 30 shows the comparison ofparametric rolling between calculations with damping
coefficients obtained by the MPS method and those with aroll decay test of a physical model with ART. There is
negligibly small difference between the two calculations.
This comparison shows a possibility to design ART as aparametric roll prevention device without any modelexperiments by applying a MPS method.

Bispectra of roll and pitch motions in short-crested
irregular head waves with H1/3=0.221m and
T01=1.32sec. (Iseki, 2006)

1.6
1.2

-0.4

-1.2
-1.6
1.2

1.6

0
20

40
60
water level of ART (%)

80

100

8

-0.8

0.8

5

10

water slope angle (degree)

0.0

0.4

10

0.8
0.4

0.0

15

0

Frequency f2 (Hz)

6 -8
4 -6
2 -4
0 -2
-2 -0

20

Fig. 28 Relation between water level in the ART andamplitude of parametric rolling in regular head waves at
λ/L=1.3 and H/λ=0.03 (Umeda et al, 2008)

2.0
Rolling Trispectrum(Amplitude)

25

6
4

MPS
MPS (fitted)
exp.

2
0
-2

0

2

4

8

10

-4

-2.0
2.0

-6

Frequency f1 (Hz)Fig. 27 Trispectrum of roll and pitch motions in regular
head waves with f3=0.32Hz, H/λ=1/20 and λ/L=1.6(Iseki, 2007)

6

time (sec.)

Fig. 29 Comparison of damping curve of water slope ofART between model experiment and MPS method(Hashimoto et al, 2007B)
20
15
10

roll angle (deg.)

Prevention Devices of Parametric Rolling
For a practical purpose, it is desirable to propose
preventing devices of parametric roll without any change oforiginal hull design. Prevention of parametric rolling could
be achieved by reducing the amplitude of restoring moment
variation in longitudinal waves and/or to increase the roll
damping. Floats attached to the ships side (sponsons) to
reduce the amplitude of restoring variation and ananti-rolling tank (ART) to increase the roll damping were
examined by free running model tests for the post-Panamax
containership (Umeda et al., 2008). As a result, maximum
roll amplitude of parametric rolling decreases by installingsponsons, and parametric rolling disappears completely by
installing ART even in severe wave condition as shown in
Fig. 28. Further, a cost-benefit analysis (Umeda et al., 2007)
demonstrates an ART is a cost-effective option to reduce

5
0
-5

20

40

60

80

100

120

140

exp.
calc.
MPS

-10
-15
-20

time (sec.)

Fig. 30 Comparison of roll motion between modelexperiment and calculations in regular head waves λ/L=1.6
and H/λ=0.033 (Hashimoto et al, 2007B)

RECOMMENDATION FOR IMO PERFORMANCEBASED CRITERIAThe IMO requests both the vulnerability criterion and the
direct assessment for parametric rolling, by following theframework proposed by Japan, the Netherlands and theUnited States (2007). Responding to this requirement, the
SCAPE committee recommends the following draft criteria.
Vulnerability criterion
It is important that a vulnerability criterion should be
simple but should guarantee conservative safety level with a
non-empirical approach. Occurrence of parametric rolling
can be predicted by Mathieu’s instability curve under theassumption of uncoupled linear roll equation. Based on thisknowledge, occurrence of parametric rolling can bepredicted with a ratio between linear roll damping andamplitude of GM variation, which is proportional to rollangle. In addition, amplitude of the parametric rolling canbe estimated with approximated analytical methods withnonlinear restoring moment in still water and GM variation
due to waves. If we can calculate GM variation in
longitudinal regular waves, occurrence and magnitude ofparametric rolling can be analytically estimated (ITTC,2005). For simplicity sake, we can assume that the
wavelength is equal to ship length and the representative
wave height can be determined with Grim’s effective wave
concept under the relevant irregular waves.
Direct assessment
If a ship design fails to comply with the vulnerability
criterion, it is expected to apply direct stability assessment.
Here it is necessary to evaluate probability of parametricrolling with its maximum roll angle exceeding a criticalangle in irregular waves. For this purpose, a coupledmathematical model is required including roll, heave andpitch motions in minimum for head seas and an uncoupledroll model is done for following seas. The roll restoring
variation in waves and nonlinear roll damping should be
accurately modelled experimentally or/and theoretically.
Monte Carlo simulation must be run for adequate time
duration and be repeated with sufficient number ofrealisations because of the possible, in certain conditions,practical non-ergodicity of parametric rolling.
CONCLUSIONS
The activity of the SCAPE committee for parametricrolling provides the following conclusions:
1. Influences of wave height, wave period, encounter
angle, ship speed, wave irregularity and initial steadyangle on parametric rolling are examined by freerunning model experiments.
2. Roll restoring variation in regular and irregular headwaves are measured and compared with numericalestimations based on potential theories in amplitude
and in mean.
3. Numerical prediction methods of parametric rollingare investigated to develop quantitative prediction forrealising the performance-based criteria. As a result,estimation of roll restoring variation taking
hydrodynamic effects into account is desirable, and4.

ensemble mean is recommended to avoid the difficultyof possible, in certain conditions, practical
non-ergodicity for the assessment of parametric rollingin irregular waves.
Effectiveness of the devices for preventing parametricrolling is validated by model experiments, andapplication of the MPS method to the design ofanti-rolling tank as a parametric roll prevention device
is proposed.

ACKNOWLEDGEMENTS
This work was supported by Grant-in Aids for Scientific
Research of the Japan Society for Promotion of Science(Nos. 18360415, 18760619 and 17560706) and the US
Office of Naval Research contract No. 0014-06-1-0646
under the administration of Dr. Patrick Purtell. The work
described here was partly carried out as a research activityof SPL project of Japan Ship Technology Research
Association, funded by the Nippon Foundation. The authors
express their sincere gratitude to the above organisations
and thank Prof. M. Sueyoshi, Dr. S. Tozawa, Ms. M.
Minami, Messrs. G. Sakamoto, S. Minegaki, S. Nakamura,
Y. Tsukada, H. Sawada, M. Hirakata and M. Oka for theircontribution to the works described in this paper.
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