PM-S1-08（社）日本船舶海洋工学会

IMO 復原性基準の機能要件化のための転覆リスク評価法研究委員会

最終報告書

平成 20 年 3 月

1

研究組織

氏名

委員長

会計担当

所属

池田 良穂

大阪府立大学大学院工学研究科

石田 茂資（独）海上技術安全研究所

井関 俊夫

東京海洋大学海洋工学部

梅田 直哉

大阪大学大学院工学研究科

小川 剛孝（独）海上技術安全研究所

片山 徹

大阪府立大学大学院工学研究科

小木曽 望

大阪府立大学大学院工学研究科

田口 晴邦（独）海上技術安全研究所

土岐 直二

三菱重工業（株）長崎研究所

橋本 博公

大阪大学大学院工学研究科

藤原 智

大阪府立大学大学院工学研究科

2007 年 12 月～

堀 正寿

大阪大学大学院工学研究科 （現 今治造船）～2006 年 3 月

牧 敦生

大阪大学大学院工学研究科

2007 年 11 月～

松田 秋彦（独）水産総合研究センター水産工学研究所

桃木 勉（独）水産総合研究センター水産工学研究所

2006 年 5 月～

Abdul Munif

大阪府立大学大学院工学研究科～2006 年 2 月（逝去）Daeng Paroka

大阪大学大学院工学研究科 (現 ハサヌドン大学）～2007 年 3 月

Gabriele Bulian

大阪大学大学院工学研究科 (現 トリエステ大学）～2007 年 2 月

22007 年 11 月～

平成１９年度

発表文献リスト(転覆リスク評価法研究委員会)-下線は当該委員会委員-

船舶復原性国際ワークショップ（ハンブルク）

1) N. Umeda, S. Koga, J. Ueda, E. Maeda, I. Tsukamoto and D. Paroka: Methodology for
Calculating Capsizing Probability for a Ship under Dead Ship Condition, Proceedings of the
9th International Ship Stability Workshop, 2007, pp.1.2.1-1.2.9.
2) T. Fujiwara and Y. Ikeda: Effects of Roll Damping and Heave Motion on Heavy Parametric
Rolling of a Large Passenger Ship in Beam Waves, Proceedings of the 9th International Ship
Stability Workshop, 2007, pp.4.3.1-4.3.9.
3) H. Hashimoto, N. Umeda and G. Sakamoto: Head-Sea Parametric Rolling of a Car Carrier,
Proceedings of the 9th International Ship Stability Workshop, 2007, pp.4.5.1-4.5.7.
4) N. Umeda, M. Shuto and A. Maki: Theoretical Prediction of Broaching Probability for a
Ship in Irregular Astern Seas, Proceedings of the 9th International Ship Stability Workshop,
2007, pp.5.1.1-5.1.7.
5) A. Matsuda, H. Hashimoto and T. Momoki: Non-linear Hydrodinamic Force Measurement
System in Heavy Seas for Broaching Prediction, Proceedings of the 9th International Ship
Stability Workshop, 2007, pp.5.2.1-5.2.5.
6) T. Katayama, M. Kotaki and Y. Ikeda: A Study on the Characteristics of Roll Damping ofMulti-hull Vessels, Proceedings of the 9th International Ship Stability Workshop, 2007,
pp.6.3.1-6.3.6.
本学会春季講演会 OS8（東京）1) 梅田直哉、前田恵里、D. Paroka: 横風横波中の転覆に対するリスクレベルの評価、
日本船舶海洋工学会講演会論文集、第 4 号、2007、pp.163-166.
2) 小川剛孝: 波向と風向の確率分布がデッドシップ状態における船舶の転覆確率に及ぼす影響についての検討、日本船舶海洋工学会講演会論文集、第 4 号、2007、
pp.167-170.
3) 梅田直哉、首藤雅和、牧敦生: 不規則追波中のブローチング発生確率について、日
本船舶海洋工学会講演会論文集、第 4 号、2007、pp.171-174.
4) 松田秋彦、橋本博公、桃木勉、坂本玄太: 大傾斜大波高中の強非線形流体力計測シ
ステムの構築、日本船舶海洋工学会講演会論文集、第 4 号、2007、pp.175-176.
5) 坂本玄太、橋本博公、梅田直哉: 自動車専用運搬船の向波中パラメトリック横揺れ、
日本船舶海洋工学会講演会論文集、第 4 号、2007、pp.177-180
6) 小川剛孝、田口晴邦、塚田吉昭: 向波中における大型コンテナ船のパラメトリック
横揺れの数値シミュレーション、日本船舶海洋工学会講演会論文集、第 4 号、2007、
pp.181-184.
37) 田口晴邦、石田茂資、沢田博史、南真紀子: 向波中のパラメトリック横揺れについて－第３報不規則波中模型実験－、日本船舶海洋工学会講演会論文集、第 4 号、2007、
pp.185-188.
8) 橋本博公、末吉誠、峯垣庄平: パラメトリック横揺れ防止のためのアンチローリン
グタンクの性能推定、日本船舶海洋工学会講演会論文集、第 4 号、2007、pp.189-192.

本学会秋季講演会 OS2（東京）1) 梅田直哉: IMO 性能ベース非損傷時復原性基準策定に向けて、日本船舶海洋工学会
講演会論文集、第 5E 号、2007、pp.19-22.
2) 梅田直哉、塚本泉: 有効波傾斜係数の簡易推定法とその転覆確率に与える影響、日
本船舶海洋工学会講演会論文集、第 5E 号、2007、pp.23-26.
3) 佐藤陽平、田口晴邦、上野道雄、沢田博史: 二次元模型の有効波傾斜係数の水槽試
験、日本船舶海洋工学会講演会論文集、第 5E 号、2007、pp.27-28.
4) 小木曽望、室津義定: 横波および横風を受ける大型客船の転覆確率評価への一次信
頼性法の適用、日本船舶海洋工学会講演会論文集、第 5E 号、2007、pp.29-32.
5) 中村真也、梅田直哉、橋本博公、坂本玄太: 向波中復原力特性変動に与えるコンテ
ナ船の形状影響、日本船舶海洋工学会講演会論文集、第 5E 号、2007、pp.33-36.
6) 小川剛孝: 斜向波中におけるパラメトリック横揺れの推定、日本船舶海洋工学会講演会論文集、第 5E 号、2007、pp.37-40.
7) 小川剛孝、戸澤秀、枚方勝、岡正義: 正面向波及び斜向波中におけるパラメトリッ
ク横揺れが貨物の固縛に及ぼす影響について、日本船舶海洋工学会講演会論文集、第 5E 号、2007、pp.41-42.
8) 池田良穂、片山徹、藤原智: 横波中大振幅パラメトリック横揺れに及ぼす上下揺れの影響、日本船舶海洋工学会講演会論文集、第 5E 号、2007、pp.43-46.
9) 井関俊夫: パラメトリック横揺れの時系列解析について（第２報）、日本船舶海洋工学会講演会論文集、第 5E 号、2007、pp.47-50.
10) H. Hashimoto, F. Stern: An Application of CFD for Advanced Broaching Prediction、Conference Proceedings The Japan Society of Naval Architects and Ocean Engineering, 5E,
2007, pp.51-52.
11) 橋本博公、松田秋彦、山谷悠: 翼型付加物による船舶の転覆防止に関する研究（第一報）、日本船舶海洋工学会講演会論文集、第 5E 号、2007、pp.53-56.

4平成１8 年度

発表文献リスト(転覆リスク評価法研究委員会)-下線は当該委員会委員-

船舶復原性国際会議（リオデジャネイロ）1) S. Ishida, H. Taguchi and H. Sawada: Evaluation of the weather criterion by experiments andits effect to the design of a RoPax ferry, Proceedings of the 9th International Conference on
Stability of Ships and Ocean Vehicles, 1, 2006, pp.9-16.
2) Y. Ogawa, J.O. deKat and S. Ishida: Analytical study on the effect of drift motions on thecapsizing probability under dead ship condition, Proceedings of the 9th International
Conference on Stability of Ships and Ocean Vehicles, 1, 2006, pp.29-36.
3) A. Munif, Y. Ikeda, T. Fujiwara and T. Katayama: Parametric roll resonance of a Large
passenger ship in dead ship condition in all heading angles, Proceedings of the 9thInternational Conference on Stability of Ships and Ocean Vehicles, 1, 2006, pp.81-87.
4) H. Taguchi, S. Ishida, H. Sawada and M. Minami: Model experiment on parametric rolling ofa post-panamax container ship in head waves, Proceedings of the 9th International
Conference on Stability of Ships and Ocean Vehicles, 1, 2006, pp.147-156.
5) H. Hashimoto, N. Umeda and A. Matsuda: Experimental and numerical study on parametricroll of a post-panamax container ship in irregular waves, Proceedings of the 9th International
Conference on Stability of Ships and Ocean Vehicles, 1, 2006, pp.181-190.
6) N. Umeda, M. Hori and H. Hashimoto,: Theoretical prediction of broaching in the light oflocal and global bifurcation analysis, Proceedings of the 9th International Conference on
Stability of Ships and Ocean Vehicles, 1, 2006, pp.353-362.
7) A. Matsuda, H. Hashimoto and N. Umeda: Experimental and theoretical study on criticalcondition of bow-diving, Proceedings of the 9th International Conference on Stability of
Ships and Ocean Vehicles, 1, 2006, pp.455-461.
本学会春季講演会 OS1（大阪）1) D. Paroka, N. Umeda and G. Sakamoto: Effect of Freeboard and Metacentric Height on
Capsizing Probability of Purse Seiners in Irregular Beam Wind and Waves, Conference
Proceedings of the Japan Society of Naval Architects and Ocean Engineers, 2K, 2006, pp.
1-4.
2) N. Umeda, J. Ueda, D. Paroka, G. Bulian and H. Hashimoto: Examination ofExperiment-Supported Weather Criterion with a RoPax Ferry Model, Conference
Proceedings of the Japan Society of Naval Architects and Ocean Engineers, 2K, 2006, pp.
5-8.
3) 池田良穂、片山徹、藤原智: 大型旅客船の航行不能時における安全性に関する実験的研究、日本船舶海洋工学会講演会論文集、第２K 号、2006、pp. 9-12.
54) G. Bulian, N. Umeda and H. Hashimoto: Some Considerations on Grim’s Effective Wave
Concept for Restoring Variations and its Improvement, Conference Proceedings of the Japan
Society of Naval Architects and Ocean Engineers, 2K, 2006, pp. 13-16.
5) 小川剛孝、石田茂資: 波浪中復原力変動についての解析的検討、日本船舶海洋工学
会講演会論文集、第２K 号、2006、pp. 17-18.
6) 梅田直哉、橋本博公、中村真也、ガブリエル・ブリアン: 不規則向波中のパラメト
リック横揺れの数値シミュレーション、日本船舶海洋工学会講演会論文集、第２K
号、2006、pp. 19-22.
7) 井関俊夫: パラメトリック横揺れの時系列解析について、日本船舶海洋工学会講演
会論文集、第２K 号、2006、pp. 23-26.
8) 梅田直哉、牧敦生、橋本博公: 追波、斜め追波中における二軸二舵高速痩せ型船の操縦運動とその制御、日本船舶海洋工学会講演会論文集、第２K 号、2006、pp. 27-30.
本学会秋季講演会 OS11（神戸）1) D. Paroka and N. Umeda: Piece-wise Linear Approach for Calculating Probability of Roll
Angle Exceeding Critical Value in Beam Seas, Conference Proceedings of the Japan Societyof Naval Architects and Ocean Engineers, 3, 2006, pp. 181-184.
2) 梅田直哉、古賀定治、小川剛孝、Daeng Paroka : RoPax フェリーの Dead Ship 状態における復原性予測法、日本船舶海洋工学会講演会論文集、第 3 号、2006、pp. 185-188.
3) 小川剛孝: デッドシップ状態における船舶の転覆確率に及ぼす波と風の相関の影響について、日本船舶海洋工学会講演会論文集、第 3 号、2006、pp. 189-192.
4) 藤原智、片山徹、池田良穂: 横波中パラメトリック横揺れに及ぼす横揺れ減衰力の影響、日本船舶海洋工学会講演会論文集、第 3 号、2006、pp. 193-196.
5) 田口晴邦、石田茂資、沢田博史、南真紀子: 向波中のパラメトリック横揺れについて－第２報定常傾斜の影響－、日本船舶海洋工学会講演会論文集、第 3 号、2006、
pp. 197-200.
6) H. Hashimoto, N. Umeda, G. Sakamoto and G. Bulian: Estimation of Roll Restoring Momentin Long-Crested Irregular Waves, Conference Proceedings of the Japan Society of Naval
Architects and Ocean Engineers, 3, 2006, pp. 201-204.
7) 土岐直二: 技術 Archive 構築の重要性－復原性 Strategy 委員会参加の経験から－、日
本船舶海洋工学会講演会論文集、第 3 号、2006、pp. 205-206.
8) 牧敦生、梅田直哉: ブローチング現象に対する最適制御理論の観点からのアプロー
チ、日本船舶海洋工学会講演会論文集、第 3 号、2006、pp. 207-209.

6

平成１７年度

発表文献リスト(転覆リスク評価法研究委員会)-下線は当該委員会委員-

国際船舶復原性ワークショップ（イスタンブール）

1) N. Umeda, H. Hashimoto, D. Paroka and M. Hori: Recent Developments of Theoretical
Prediction on Capsizes of Intact Ships in Waves. Proceedings of the 8th International Ship
Stability Workshop, 2005, 1.2.1-1.2.10.
2) Y. Ikeda, A. Munif, T. Katayama and T. Fujiwara: Large Passenger Ship in Beam Seas andRole of Bilge keels in its Restraint, Proceedings of the 8th International Ship Stability
Workshop, 2005, 2.2.1-2.2.11.
3) T. Katayama and Y. Ikeda: An Experimental Study of Fundamental Characteristics in Inflow
Velocity from a Damage Opening, Proceedings of the 8th International Ship Stability
Workshop, 2005, 4.2.1-4.2.6.
4) H. Taguchi, S. Ishida and H. Sawada: A Trial Experiment on the IMO Draft Guidelines for
Alternative Assessment of the Weather Criterion, Proceedings of the 8th International Ship
Stability Workshop, 2005, 6.4.1-6.4.9.
本学会秋季講演会 OS3（福岡）1) H. Hashimoto, A. Matsuda and N. Umeda: Model Experiment on Parametric Roll of a
Post-Panamax Container Ship in Short-Crested Irregular Waves, Conference Proceedings of
the Japan Society of Naval Architects and Ocean Engineers, 1, 2005, pp. 71-74.
2) 小川剛孝、石田茂資：波浪中復原力変動振幅及び平均値の推定、日本船舶海洋工学会講演会
論文集、第１号、2005、pp. 75-76.
3) 堀正寿、梅田直哉；大域的分岐としてのブローチング発生条件の推定、日本船舶海洋工学会
講演会論文集、第１号、2005、pp. 77-80.
4) 田口晴邦、石田茂資、沢田博史：Weather Criterion 評価のための標準模型実験法について、
日本船舶海洋工学会講演会論文集、第１号、2005、pp. 81-84.

5) D. Paroka and N. Umeda: Prediction of Capsizing Probability for a Ship with Trapped Wateron Deck Taken into Account, Conference Proceedings of the Japan Society of Naval
Architects and Ocean Engineers, 1, 2005, pp. 85-87.
6) 松田秋彦、高橋秀行、橋本博公：ステレオ計測カメラを用いた横波中の甲板滞留水の計測、
日本船舶海洋工学会講演会論文集、第１号、2005、pp. 89-90.

7) 池田良穂、片山徹、A. Munif, 藤原智：大型客船の横波中大振幅パラメトリック横揺れの計
測、日本船舶海洋工学会講演会論文集、第１号、2005、pp. 91-94.

8) A. Munif, T. Katayama and Y. Ikeda : Numerical Prediction of Parametric Rolling of a Large
Passenger Ship in Beam Seas, Conference Proceedings of the Japan Society of Naval
Architects and Ocean Engineers, 1, 2005, pp. 95-98.
7

CURRENT PROBLEMS IN SHIP INTACT STABILITY AND
ACTIVITY FROM JASNAOE SCAPE COMMITTEE- FINAL REPORT OF SCAPE COMMITTEE (PART 1) Naoya UMEDA
Department of Naval Architecture and Ocean Engineering, Osaka University,
JAPAN

ABSTRACTThis paper reviews current problems in intact
stability criteria, including the development of new
generation intact stability criteria at the IMO(International Maritime Organization), the activities of
the ITTC(International Towing Tank Conference) ‘s
Specialist Committee on Stability in Waves (SiW) andrelevant international researches as the backgrounds of
the SCAPE Committee (Strategic Research Committee
on Estimation Methods of Capsizing Risk for the IMO
New Generation Stability Criteria) in the JASNAOE(Japan Society of Naval Architects and Ocean
Engineers). Then the achievements of the SCAPE
Committee are overviewed. Based on them, the SCAPE
Committee is ready to submit a proposal on the newgeneration intact stability criteria to the IMO via the
RINA (The Royal Institution of Naval Architects) as a
NGO representing the learned societies.
KEY WORDS: Performance-Based Criteria, SCAPE,
Intact Stability Code, Dead Ship Condition, Parametric
Rolling, Broaching.

INTRODUCTION
Intact stability, an ability not to capsize without
damage of enclosed buoyant space, is one of the mostfundamental requirements for ship design and operation.
Stability criteria of intact stability are enforced by eachadministration: in Japan the ship stability standard cameinto force in 1957 and since then no accident due to the lackof intact stability has been reported for passenger ships builtwith this standard. Based on this remarkable achievement,
the weather criterion, as a part of the standard, was adopted
by the IMO in 1985 as a recommendation to the member
states. Until now, the intact stability criteria are notmandatory at least at the level of the IMO.
Recently opinions looking for mandatory intact
stability criteria to be internationally applied, which would
be comparable of mandatory damage stability criteria in the

SOLAS (Safety of Life at Sea) Convention, form the
majority at the IMO. This is partly because latest large
cruise ships and RoPax ferries have high beam-draught
ratios so that the IMO weather criterion could be
excessively stringent and partly because megacontainerships occasionally suffer parametric rolling
resulting in container damage. Thus it is expected todevelop rational and mandatory criteria in place of thecurrent recommendatory ones for ensuring internationallyconsistent safety level against capsizing or cargo damage.
Responding to this situation, the SLF (Sub-Committee on
Stability, Load Lines and on Fishing Vessel Safety) had fivesessions to discuss this matter and then in October 2007 theMSC (Maritime Safety Committee) approved the 2008 IS(Intact Stability) Code, a part of which is to be used as amandatory one from 1 July 2010. The contents of this code,
however, are almost the same as the current
recommendatory requirements, which are mainly based on
empirical backgrounds. On the other hand, the 2008 IS
Code states that rational criteria for dynamic stability
phenomena in waves should be developed in the near future.
Therefore, the IMO started to develop performance-based
intact stability criteria and its target date is 2010. This isbecause the majority of the IMO understands prescriptive
criteria are insufficient to prevent capsizing due to such
dynamic stability phenomena and performance-based
approach allowing the use of latest research in ship
dynamics shall be introduced. As a result, stability research
is now highlighted to respond to this strong demand fromthe regulatory bodies.
Responding to such demand from the IMO, the
JASNAOE established a strategic research committee,
which is named as “Strategic Research Committee on
Estimation Methods of Capsizing Risk for the IMO New
Generation Stability Criteria” (SCAPE) in June 2005.
Eighteen stability experts from research institutes,
universities and industry take part in the SCAPE committee,
and intensively execute researches for this purpose for two
years. Its outcomes had been reported at five organizedsessions during the spring and autumn meetings of the
JASNAOE under the title of “Intact Stability Assessment
towards the IMO New Generation Criteria” for members of

the JASNAOE mainly in Japanese. On this opportunity, thispaper attempts to overview activities of the SCAPE
committee as well as current problems of the IMO and topresent recommendation to be published for the IMO via
the Royal Institution of Naval Architects (RINA) as a NGO
representing the learned societies in the world.

RELEVANT ACTIVITIES AT THE IMO ETC
Stability Criteria of the IMO had been separately
recommended such as the GZ curve criteria (resolution A.
167(ES.IV)) and the weather criterion (resolution. A.
562(14)) and were assembled into a single document as the“Code on Intact Stability for All Types of Ships Covered by
IMO Instruments (IS Code)”, the resolution A. 749(18), in
1993. The sub-chapters 3.1 and 3.2 of the Code correspondto the GZ curve criteria and the weather criterion,
respectively. In 1998, further refinements were made as theresolution MSC.75, which enables the Code to be consistent
with the Torremolinos Convention on Fishing Vessel Safety
and so on. On the other hand, since the UR L2 of the IACS
requests to comply with the sub-chapters 3.1 and 3.2 of theCode, these are mandatory for obtaining the IACS classcertificates.
As explained earlier, the IMO started to consider thecomprehensive revision of the IS Code at the SLF45 in
2002. Its working group was established under thechairmanship of Prof. A. Francescutto. This is because
Germany (2002) and Italy (2002) stated that the weather
criteria could be an excessive design constraint for Ro-Pax
ferries and large passenger ships, respectively. And theUnited States (2002) reported that a post-Panamax
containership complying with the IS Code suffered cargodamage due to parametric rolling in head waves. Then
Germany, Denmark and Australia proposed to replace theweather criterion with performance-based criteria allowing
the use of model experiments and numerical simulation.
Italy proposed to modify the empirical coefficients in theweather criterion. Japan expressed its view that weather
criterion has a role to ensure current safety levels of
conventional ships so that it should be kept but no support
to this view existed at the moment.
Considering the above situation, at the informal
correspondence group meeting in March 2003, Japan(2003A) presented the calculated results for capsizing
probabilities of 75 conventional ships marginally complying
with the IS Code, and stated that safety level ofconventional ships should be kept. Japan explained that thisis because medium-sized passenger ships occasionally
suffer dead ship conditions, which result in beam wind andwave condition without forward velocities, due tofreak-wave-induced damage to their wheel houses andcurrent weather criterion well ensures stability under suchcircumstances. Then Japan concluded that preventing
capsizing under such dead ship condition has been wellrealised by the IMO with its weather criterion and is one offundamental roles for the IMO in the future. This statement
obtained the unanimous support from the participants. Thenthe IMO agreed that a stability criterion for dead shipcondition is indispensable so that the revision of the weather
criterion emerged as a high priority issue at the IMO.

At the SLF 46 held in September 2003, Japan(2003B) presented the systematically executed sample
calculations with proposals from various delegations. Basedon this result, Japan, together with Italy, proposed to allow
the use of model experiments for wind-heeling lever androll back angle in the weather criterion. As a result, the SLF
sub-committee agreed with this proposal and requested
Japan and Italy to draft the revised weather criterion withmodel test guidelines.
At the SLF47 in 2004, Italy and Japan (2004) jointly
submitted the draft revision of weather criterion: the draftmodel test guidelines on wind heeling lever and roll backangle were developed by National Maritime ResearchInstitute and Osaka University, respectively. Italy also
developed the draft on roll back angle with the experienceof Trieste University. Then the SLF decided to continue tofurther consider the model test guidelines.
At the SLF48 in 2005, by harmonising Japanese and
Italian drafts, the interim model test guidelines were
finalised. As a result, the draft revision of the weather
criterion allowing the use of model tests was agreed at the
SLF. With the urgent need of this alternative route, thisrevision was adopted as a separate instrument, i.e,
MSC.1/Circ. 1200, in May 2006.
In parallel to the revision of the weather criterion, a draftof restructuring the IS Code was developed and was tabledby the intersessional correspondence group coordinated by
Mr. C. Mains at the SLF49 in 2006. It consists of threeparts: Part A includes the GZ curve criterion and theweather criterion as mandatory requirements, Part B collects
recommendatory criteria such as criteria for fishing vessels
and containerships and Part C is the explanatory note of theCode. Responding to this draft, France, Norway and Greece
expressed their concern that the criterion for the maximum
stability angle is too stringent for ships having large
beam-draft ratio such as RoPax ferries and large passenger
ships. Thus, the restructured draft was not finalized at thatsession.
To overcome this difficulty, at the SLF50 in May 2007,
Italy, Japan and China proposed to use the Offshore Supply
Vessel Code, multihull criteria in the High Speed Craft
Code and the pontoon criteria, respectively, as alternatives
to the maximum stability angle criterion. All these
proposals intend to relax the requirement for the maximum
stability angle but with more stringent requirements for thearea of righting arm curve up to smaller roll angle. Since
these are empirical criteria, it is very difficult to rationallyrevise them. Thus, the alternative use of the Offshore
Supply Vessel Code was selected among the three because
the USA reported its successful experience in their domestic
standards. Then the draft revision of the IS Code was
finalised at the SLF together with the draft revision of the
SOLAS convention and LL convention for making the part
A of the Code mandatory. Then it was approved, as the2008 IS Code, by the MSC in October 2007.
Other than the above short-term revision of the IS Code,the consideration of the performance-based criteria as thelong-term revision of the Code became serious since 2005
(Germany, 2005). First, three dangerous modes to be dealt
are selected: 1) stability variation problems such asparametric rolling, 2) stability under dead ship condition
and 3) manoeuvring-related problems such as broaching-to.
These are clearly noted in the Part A of the 2008 IS Code.

For each mode, a single splinter group was established: Ms.
H. Cramer was nominated as the coordinator of the groupfor 1) and Dr. N. Umeda was nominated as the coordinatorsof groups 2) and 3). Responding to this situation, Japan(2006A, 2006B, 2006C) reviewed the numerical simulation
methodologies on these three modes and proposed a way to
use them as alternatives to the existing prescriptive criteria.
First, numerical prediction on parametric rolling well agrees
with the model experiments in regular waves but not so wellin irregular waves. Second, a piece-wise linear approachseems to be promising for quantifying capsizing probability
under dead ship condition. Third, a bifurcation theory canefficiently predict thresholds of surf-riding, which is aprerequisite of broaching, in regular following waves.
Germany (2006) systematically conducted numericalsimulation in irregular waves for many ships and thenproposed a new empirical criterion for the stability variation
problem. Italy (2005) proposed a calculation method of
capsizing probability based on a linearization procedure thattakes into account the presence of a critical rolling angle
and, in a simplified way, nonlinearities of restoring
moment.
Based on the above proposals and the relevant
discussion, at the SLF50 in 2007, Japan, the Netherlands
and the USA (2007) jointly submitted a document
describing the framework for developing the newgeneration intact stability criteria. Here the new generation
criteria are expected to be applied to unconventional ships,
not to be new empirical criteria and to be supplemented
with ship-specific operational guidance. A ship is firstly
judged by vulnerability criteria, which are easily used but
are based on physics, and, only if she fails to comply with it,
she is categorised as unconventional and then direct stability
assessment utilising numerical simulation or the equivalentmeans should be applied. Germany (2007) updated the
proposal of a new empirical criterion for all ships andproposed not to use ship-specific operational guidance. As aresult of discussion, significant majority of the working
group supported the proposal from Japan, the Netherlands
and the USA. Therefore, the IMO will develop new
generation intact stability criteria based on the frameworkagreed at the SLF50 with the new target date of the year
2010.
Outside the IMO, the European Commission runs its
FP6 project named SAFEDOR, coordinated by Dr. P.
Sames. Here, for realising risk-based approach, ship safetyincluding intact stability, damage stability and fire
protection is systematically investigated by several researchorganisations and universities in Europe. Its ultimate goal isset to export its results to regulations. For intact stability,
Jensen (2007) applies the First Order Reliability Method(FORM) to the out-crossing problem of parametric rolling
and Themelis and Spyrou (2007) attempts to evaluate
probabilities of dangerous wave condition by combining
deterministic analyses on nonlinear ship motion withprobabilistic analyses on Gaussian waves.
As a world-wide independent association of researchorganisations on ship hydrodynamics, the ITTC establisheda specialist committee on stability in waves since 1996. It
was chaired firstly by Prof. D. Vassalos, then by Dr. J.O. de
Kat and is currently chaired by Dr. N. Umeda. This
committee developed and is updating the recommended
procedure of model tests on intact stability and executedbenchmark testing of numerical codes on intact stability,
covering parametric rolling and broaching, by utilising thecapsizing model experiments in Japan. (The Specialist
Committee on Prediction of Extreme Ship Motions andCapsizing, 2002; The Specialist Committee on Stability inWaves, 2005) These standardisation efforts will be crucial
for implementation of the new generation intact stability
criteria.

ESTABLISHMENT OF SCAPE ACTIVITIESIn 2005, three learned societies in Japan were merged
into a new society named the Japan Society of Naval
Architects and Ocean Engineers (JASNAOE). Under thechairmanship of Prof. S. Naito, the JASNAOE decided tohave research committees for specified topics and specified
terms. The SCAPE committee is one of the researchcommittees that the JASNAOE established in 2005. The
purpose of this committee is to contribute to theconsideration of new generation intact stability criteria fromresearchers’ viewpoint. It could help the activities of theStability Project of Japan Ship Technology Research
Association, which discusses under the chairmanship of Dr.
N. Umeda the response of the Japanese delegation to the
SLF sub-committee. The SCAPE committee plans tocontribute not only to the documents from the Japanesedelegation but also to the documents from the RINA, which
representing the learned societies on naval architecture in
the world, via the JASNAOE. In addition, the SCAPE
committee is expected to support the activities of the SiW
committee of the ITTC. The committee was establishedwith members from universities, research institutes andindustries as listed in Table 1 and is requested to complete
its task by 31 March 2008. The budget of this committee
was supplied by the Japan Society of Promotion of Science,
based on the application of group members, as well as the
JASNAOE.
Table 1 Membership list of the SCAPE committee
G. Bulian
T. Fujiwara

Osaka University
Osaka Prefecture University

H. Hashimoto
M. HoriY. Ikeda
T. Iseki
S. Ishida
T. Katayama
N. Kogiso

Osaka University
Osaka University
Osaka Prefecture University
Tokyo Maritime University
National
Maritime
Research
Institute
Osaka Prefecture University
Osaka Prefecture University

A. MakiOsaka University

A. Matsuda

National Research Institute ofFisheries Engineering
National Research Institute ofFisheries Engineering
Osaka Prefecture University
National
Maritime
Research
Institute
Osaka University
National
Maritime
Research
Institute
Mitsubishi Heavy Industries
Osaka University

T. Momoki
A. Munif
Y. Ogawa
D. Paroka
H. Taguchi
N. TokiN. Umeda

till Feb. 2007
from
Dec.
2007
Secretary
till Mar. 2006

from
2007
from
2007

Nov.
Nov.

from
May
2006
till Feb. 2006

till Mar. 2007

Chairman

OUTCOMES FROM THE ACTIVITIES OF SCAPE
COMMITTEEThe SCAPE committee published 43 conference papers
at the spring and autumn meetings of the JASNAOE andmany papers in the refereed journals and international
conferences. Based on these, the outcomes from the SCAPE
committee are overviewed. More details can be found incompanion reports of this report for individual topics.
For parametric rolling, free-running model experimentsof a post-Panamax containership in head and bow waves inthe seakeeping and manoeuvring basins of National
Research Institute of Fisheries Engineering and National
Maritime Research Institute and observed parametric rollingwith the roll angle exceeding 20 degrees. It was found thatmaximum roll angles are almost the same among regularwaves, long-crested irregular waves and short-crested
irregular waves. Regarding the restoring variation due towaves as the cause of parametric rolling, the prediction of
its harmonic component was attempted not only by the
Froude-Krylov component on its own but also the radiationand diffraction components as well. For the mean of therestoring variation, a formula using a momentum law wasdeveloped. These results were compared with the captivemodel experiments and then better agreements than theexisting approach were reported. Taking these aspects into
account, numerical simulation of parametric rolling in headwaves was executed and compared with the fore-mentioned
experiments. While good agreements for the case inmoderate regular waves between the calculation and theexperiments were reported, some discrepancies can befound for the cases in severe regular waves and irregularwaves. For investigating this reason, direct measurement of
restoring variation acting on scaled models in irregularwaves were executed. It was found, however, that rather
acceptable agreement exists also in irregular waves. On theother hand, if we repeat many realisations in modelexperiments and numerical simulations in irregular waves
under the same wave parameters, the scatter of the rollangles could be large. This is because parametric rolling in
irregular waves can be regarded as “practically
non-ergodic”. Parametric rolling appears only if the level ofrestoring variation exceeds a certain threshold. Thus it isneither Gaussian nor practically-ergodic. Therefore, it isextremely difficult in principle to obtain good agreement
among individual realisations so that it is presumed theensemble average should be used for this comparison.
Time-Varying coefficient Vector Auto-Regresive (TVVAR)
method, bispectrum analysis and trispectrum analysis wereapplied to parametric rolling of the post-Panamax
containership in head waves for clarifying the nonlinear
nature of parametric rolling. Furthermore, modelexperiment and numerical simulation study were carried out
also for a latest PCTC, which is similar to one sufferedparametric rolling in the Northern Atlantic, and its
parametric rolling in regular and irregular head waves wereidentified (Hashimoto et al., 2008).
For stability under dead ship condition, a method for
calculating capsizing probability in irregular beam wind andwaves with Belenky’s piece-wise linear approach (Belenky,1993) were intensively investigated. Here the restoring arm
curve is approximated with piece-wise linear curves andthen capsizing probability can be analytically calculated.

Comparing this method with the Monte Carlo simulation,some discrepancies exist in extremely severe sea state. Thisis probably because the Poisson assumption is not
appropriate for the case out-crossing probability is very high.
In moderate sea state, such as the average wind velocity of
26 m/s that the weather criterion assumes, acceptableagreement within the confidence interval was found.
Therefore, the committee concluded the piece-wise linear
approach could be well applied and to extend the method
further. First, since dead ship conditions does not directly
mean beam sea condition, a method for calculating drifting
velocity and attitude was developed and then capsizing
probability under such condition was calculated. Second,
effects of trapped water on deck, which is indispensable forfishing vessels, and effect of critical angle fordown-flooding or cargo shift were taken into account in thecalculation of capsizing probability. Considering these
aspects, annual capsizing risk for ships having large
windage areas, i.e. a PCTC, a RoPax ferry, a large
passenger ship and a containership were calculated. As aresult, it was remarked that shift of down-flooding point for
the PCTC and avoiding severe weather for the RoPax ferry
are effective measures. For purse seiners, their safety level
ensured by the IMO weather criterion on its own is notalways sufficient but that by the water-on-deck criterion of
the Torremolinos Convention seems to be appropriate. So
far, it was assumed in the calculation of long-term capsizing
probability that wind is fully correlated with waves. The
effect of this assumption was examined by calculating
capsizing probability with a joint probability density
function of the average wind velocity, the significant wave
height and the mean wave period (Ogawa et al., 2008).
Moreover, it was found that significant parametricrolling occurs during model experiments of a large
passenger ship without the bilge keels drifting in beam
waves. The reason of this is presumed by numericalsimulation and model experiments to be restoring variation
due to resonant relative heave motions (Ikeda et al., 2008).
For broaching, it is important to predict the threshold ofsurf-riding, which is a prerequisite to broaching. In case ofregular following waves, its prediction methods with an
uncoupled surge model were already established. In case ofregular stern quartering waves, it is essential to takecoupling with manoeuvring motions into account. Therefore,a global bifurcation analysis was applied to asurge-sway-yaw-roll mathematical model with a PD auto
pilot. As a result, the critical ship speed for surf-riding instern quartering waves was obtained as a function of the
auto pilot course. By investigating whether the surf-riding
results in broaching or capsizing was investigated bysystematic numerical simulation in regular waves, areas of
rudders and stabilizing fins as well as control gains were
optimised. For broaching in irregular waves, a stochastic
theory for estimating broaching probability with thedeterministic broaching conditions in regular waves wasdeveloped and is well validated with the Monte Carlosimulation (Umeda et al., 2008).
For preventing the above phenomena, devices were
proposed and were successfully validated with modelexperiments. An anti-rolling tank was proved as a
cost-effective risk control option for preventing container
damage due to parametric rolling of a containership(Hashimoto et al., 2008). In addition, its design method with

a CFD technique was developed. Wings attached to a ship
above calm water surface were proposed for preventing
broaching (Umeda et al., 2008).
Further, model experiments based on the interim model
testing guidelines for the weather criterion were performed
and the reported results enabled us to examine theusefulness of the interim guidelines including its effect onstability assessment (Ogawa et al., 2008). A model testing
technique for evaluating amount of trapped water on deck inbeam waves (Ogawa et al., 2008) and a model test method
for measuring hydrodynamic forces acting on a ship heeled
up to the deck submergence with a forward velocity(Umeda et al., 2008) were improved and results with themwere reported.
As mentioned above, the outcomes from the SCAPE
committee already contributed not only to research progressin ship dynamics but also to the proposals from the Japanesedelegation for the IMO to some extent. Since common sense
based on a linear theory faces difficulty to understand these
nonlinear phenomena, an explanatory note is underdevelopment.

a simplified prediction formula for metacentric height onwave crest is desirable.
It might be practical to use numerical simulations for the
performance-based criteria for ships which fail to pass thevulnerability criteria, because model experiment requires
prohibitively large amount of cost and time as well as many
testing facilities and their staff. For the dead ship condition
and broaching, the methodologies developed by the SCAPE
committee are available. It is desirable to obtain measured
data of capsizing probability under a dead ship condition
from model experiments in artificial irregular wind andwaves with sufficient statistical confidence and to validate
the proposed theory with them. For parametric rolling, it
seems to be necessary to calculate ensemble average ofmany numerical realisations. The first order reliability
method (FORM) may provide a way to make this procedure
more efficient. (Kogiso & Murotsu, 2008) In addition, theeffect of time-varying resistance increase in irregular waves
may be important and its effect is now under investigation.

CONCLUDING REMARKSRECOMMENDATION FOR THE IMO NEW
GENERATION CRITERIA
Based on the research outcomes from the two
year-activities of the SCAPE committee, the committee
would like to invite the IMO to take the following
suggestions into account for developing its new generation
intact stability criteria.
First, the following principles should be adopted: 1) the
new performance-based criteria will be applied tounconventional ships while the conventional ships should
be dealt with the existing prescriptive criteria, 2) no further
empirical criterion should be developed, 3) design criteria
should be supplemented with operational criteria. Although
it is important to rationalise intact stability criteria, weshould bear in mind the fact that most of conventional ships
under the current criteria are safely operated and that thecurrent intact stability criteria are quite reliable at least fromstatistical viewpoint. It is noteworthy that such sufficient
safety level also does realised with appropriate operationand it is unrealistic to separate design criteria fromoperation ones. In the case of unconventional ships,
empirical criterion is not applicable because no empirical
data exists for unconventional ships. If a new criterion stays
empirical, it could rather prevent a design of unconventional
ships.
The
vulnerability
criteria,
which
distinguish
unconventional ship from conventional ones, should be as
simple as possible because they should be applied to all
ships in principle. It is not so good idea to spend much
effort on ships having sufficient stability, such as tankers
and bulk carriers, which do not have real danger against
capsizing. Pursuing this principle, a candidate of thecriterion for dead ship condition could be a weather
criterion with some reasonable prediction of roll damping
and effective wave slope coefficient. For broaching, it could
be sufficient to identify surf-riding thresholds in pure
following waves as the current IMO operational guidance.
For parametric rolling or pure loss of stability on wave crest,

Responding to the demands of research for new
generation intact stability criteria at the IMO, the
JASNAOE established the SCAPE committee in 2005 andthe SCAPE committee has completed their term. Based on
the outcomes of the research on probabilistic stability
assessment methodology for parametric rolling, stability
under dead ship condition and broaching, the committee
provides recommendations to be submitted to the IMO via
the RINA.

ACKNOWLEDGEMENTS
This work was supported by a Grant-in Aid for
Scientific Research of the Japan Society for Promotion of
Science (No. 18360415). The work described here waspartly carried out as a research activity of Stability Projectof Japan Ship Technology Research Association, funded by
the Nippon Foundation. The committee SCAPE itself is run
under the umbrella of the Japan Society of Naval Architects
and Ocean Engineers. The author expresses his sincere
gratitude to the above organisations.

REFERENCES
Belenky, 1993, “A Capsizing Probability Computation
Method”, Journal of Ship Research, Vol. 37, pp. 20-207.
Germany, 2002, “Remarks Concerning the Weather
Criterion”, SLF 45/6/3, IMO.
Germany, 2005, “Report of the Intersessional
Correspondence Group (part 1)”, SLF48/4/1, IMO.
Germany, 2006, “Proposal of a Probabilistic Intact Stability
Criterion”, SLF 49/5/2, IMO.
Germany, 2007, “Proposal on Additional Intact Stability
Regulations”, SLF 50/INF.2, IMO.
Hashimoto, H. et al., 2008, “Prediction Methods for
Parametric Rolling with Forward Velocity and Their
Validation – Final Report of SCAPE Committee (Part2)-“, Proceedings of the 6th Osaka Colloquium on
Seakeeping and Stability of Ships, (in press).
Ikeda, Y. et al., 2008, “Prediction Methods for Parametric
Rolling under Drifting Condition and Their Validation –Final Report of SCAPE Committee (Part 3) –“,Proceedings of the 6th Osaka Colloquium on Seakeeping
and Stability of Ships, (in press).
Italy, 2002, “Weather Criterion for Large Passenger Ships”,
SLF 45/6/5, IMO.
Italy and Japan, 2004, “Proposal on Revision of the
Weather Criterion”, SLF 47/6/16, IMO.
Italy, 2005, “A Modular Methodology for the Estimation of
the Ship Roll Safety under the Action of Stochastic
Wind and Waves”, SLF 48/4/6, IMO.
Japan, 2003A, “Examination of the Weather Criterion”,
SLF46/6/15, IMO.
Japan, 2003B, “Sample Calculations for the Draft Weather
and Wind Criteria”, SLF 46/6/16, IMO.
Japan, 2006A, “Proposal of Methodology of Direct
Assessment for Stability under Dead Ship Condition”,
SLF 49/5/5, IMO.
Japan, 2006B, “A Methodology of Direct Assessment for
Capsizing due to Broaching”, SLF 49/5/6, IMO.
Japan, 2006C, “Remarks of Numerical Modelling for
Parametric Rolling”, SLF 49/5/7, SLF49/5/7/Corr.1,
IMO.
Japan, the Netherlands and the United States, 2007,“Framework for the Development of New Generation
Criteria for Intact Stability”, SLF 50/4/4, IMO.
Jensen, J.J., 2007, “Efficient Estimation of Extreme
Non-Linear Roll Motions Using the First-Order

Reliability Method (FORM), Journal of Marine Scienceand Technology, Vol. 12, pp.191-202.
Kogiso, N. and Murotsu, Y., 2008, “Application of First
Order Reliability Method to Ship Stability – Final
Report of SCAPE Committee (Part 5) –“, Proceedings of
the 6th Osaka Colloquium on Seakeeping and Stability of
Ships, (in press).
Ogawa, Y. et al., 2008, “Prediction Methods for Capsizing
under Dead Ship Condition and Obtained Safety Level –Final Report of SCAPE Committee (Part 4) –“,Proceedings of the 6th Osaka Colloquium on Seakeeping
and Stability of Ships, (in press).
Themelis, N. K. and Spyrou, K.J., 2007, “Probabilistic
Assessment of Ship Stability”, SNAME Marine
Technology
Conference,
Fort
Lauderdale,
pp.18.1-18.19.
The Specialist Committee on Prediction of Extreme Ship
Motions and Capsizing (chaired by D. Vassalos), 2002,“Final Report and Recommendations to the 23rd ITTC”,
Proceedings of the23rd ITTC, Vol. 2, pp. 611-649.
The Specialist Committee on Stability in Waves (chaired by
J.O. de Kat), 2005, “Final Report and Recommendations
to the 24th ITTC”, Proceedings of the24th ITTC, Vol. 2,
pp. 369-408.
Umeda, N. et al., 2008, “Prediction Methods for Broaching
and Their Validation – Final Report of SCAPE
Committee (Part 6) –“, Proceedings of the 6th Osaka
Colloquium on Seakeeping and Stability of Ships, (inpress).
United States, 2002, “Head-Sea Parametric Rolling and Its
Influence on Container Lashing Systems”, SLF 45/6/7,
IMO.

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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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.

REFERENCES
Blocki, W., 1980, “Ship Safety in Connection withParametric Resonance of the Roll”, International
Shipbuilding Progress, 27-306.
Boroday, I. K. and Morenschildt, V. A., 1986, “Stability andParametric Roll of Ships in Waves”, Third International
Conference on Stability of Ships and Ocean Vehicles, pp.
19-26.
Fujiwara, T., Katayama, T., Ikeda, Y. 2006A, “Effects ofRoll Damping on Parametric Rolling in Beam Sea”,
Conference Proceedings of the Japan Society of Naval
Architects and Ocean Engineers, Vol. 3, pp.193-196, (inJapanese).
Fujiwara, T., Munif, A., Katayama, T., Ikeda, Y., 2006B,“Studies on Severe Parametric Roll Resonance on
Modern Ships with Buttock-flow Hull Shape in Dead
Ship Condition”, Proceedings of the 3rd Asia-PacificWorkshop on Marine Hydrodynamics, Shanghai.
Fujiwara, T. and Ikeda, Y., 2007, “Effects of Roll Dampingand Heave Motion on Heavy Parametric Rolling of a
Large Passenger Ship in Beam Waves”, Proceedings ofthe 10th International Ship Stability Workshop,
Hamburg.
Froude, W., 1863, “Remarks on Mr. Scott-Russell’s Paper
on Rolling”, INA
Ikeda, Y., Katayama, T., Munif, A., Fujiwara, T., 2005A,“Experimental Identification of Large Parametric
Rolling of a Modern Large Passenger Ship”, Journal of
the Japan Society of Naval Architects and Ocean
Engineers, Vol. 2.

Ikeda, Y., Munif, A., Katayama, T., Fujiwara, T., 2005B,“Large Parametric Rolling of a Large Passenger Ship in
Beam Seas and Role of Bilge Keel in Its Restraint”,
Proc. of 8th International Ship Stability Workshop,
Istanbul.
Ikeda, Y., Katayama, T., Fujiwara, T., 2006, “AnExperimental Study on Parametric Rolling of a Large
Passenger Ship at Dead Ship Condition”, Conference
Proceedings of the Japan Society of Naval Architects
and Ocean Engineers, Vol. 2K, pp.9-12, (in Japanese).
Ikeda, Y., Katayama, T., Fujiwara, T., 2007, “Effects ofHeave Motion on Heavy Parametric Rolling in Beam
Waves”, Conference Proceedings of the Japan Society ofNaval Architects and Ocean Engineers, Vol. 5E,
pp.43-46, (in Japanese).
Munif, A., Katayama, T., Ikeda, Y., 2005, “numerical
Prediction of Parametric Rolling of a Large Passenger
Ship in Beam Seas”, Conference Proceedings of theJapan Society of Naval Architects and Ocean Engineers,Vol. 1, pp.95-98.
Munif, A., Ikeda, Y., Fujiwara, T., Katayama, T., 2006,“Parametric roll Resonance of a Large Passenger Ship in
Dead Ship Condition in All Heading Angles”,
Proceedings of 9th International Ship Stability
Workshop, Rio de Janeiro.
Munif, A. and Umeda, N., 2006, “Numerical prediction onparametric roll resonance for a ship having no
significant wave-induced change in hydrostaticallyobtained metacentric height”, International Shipbuilding
Progress 53, 183-203.
Paulling, J. R. and Rosenberg, R.M., 1959, “On Unstable
Ship Motions Resulting from Non-linear Coupling”,
Journal of Ship Research.
Sadakane, H., 1979, “On the Rolling of a Ship on a Billow(Fluctuation of Apparent Ship-Weight)”, Journal of
Kansai Society of Naval Architects, No.169, pp83-93,(in Japanese).
Tamiya, S., 1969, “A Calculation of Non-linear, Nonsymmetric Rolling of Ships”, Conference Proceedingsof the Society of Naval Architects of Japan, pp.85-93, (inJapanese).
Tasai, F., 1965, “Ship Motions in Beam Seas”, Journal of theWest-Japan Society of Naval Architects, Vol. 30,
pp83-105, (in Japanese).
Umeda, N., Iskandar, B. H., Hashimoto, H., Urano, S.,
Matsuda, A., 2002, “Comparison of European and Asian
Trawlers –Stability in Seaways-“, Proceedings of the 2nd
Asia-Pacific Workshop on Marine Hydrodynamics,
Kobe, pp.79-84.

PREDICTION METHODS FOR CAPSIZING UNDER DEAD
SHIP CONDITION AND OBTAINED SAFETY LEVEL- FINAL REPORT OF SCAPE COMMITTEE (PART 4) Yoshitaka OGAWA*, Naoya UMEDA**, Daeng PAROKA**
Harukuni TAGUCHI*, Shigesuke ISHIDA*, Akihiko MATSUDA***,
Hirotada HASHIMOTO**and Gabriele BULIAN*** National Maritime Research Institute, Tokyo, JAPAN** Department of Naval Architecture and Ocean Engineering, Osaka University, Osaka, JAPAN*** National Research Institute of Fisheries Engineering, Kamisu, JAPANABSTRACTRegarding capsizing of ships under dead shipcondition, this paper reports experimental, numericaland analytical studies conducted by the SCAPE
committee, together with critical review of theoreticalprogress on this phenomenon. Here capsizing
probability under dead ship condition is calculated witha piece-wise linear approach. Effects of statisticalcorrelation between wind and waves are examined. Thisapproach is extended to the cases of water on deck andcargo shift. Further, experimental and theoreticaltechniques for relevant coefficients are examined. Basedon these outcomes, it also proposes a methodology fordirect assessment as a candidate for the new generation
criteria at the IMO.
KEY WORDS: Dead Ship Condition; Intact Stability
Code; Beam Winds and Waves; Capsizing Probability,
Piece-Wise Linear Approach, Water on Deck
INTRODUCTION
Stability under dead ship condition is one of threemajor capsizing scenarios in terms of performance-based
criteria, which is requested to be developed for the revisionof the Intact Stability Code (IS Code) by 2010 at theInternational Maritime Organization (IMO). It was expected
to be a probabilistic stability assessment based on physics
by utilizing first principle tools. This is owing to the factthat probabilistic approach can be linked with a risk analysis
and dynamic-based approach enables us to deal with new
ship-types without experience.
Therefore, it is important to establish a method forevaluating capsizing probability under dead ship condition
because its safety level could be a base for other capsizing

scenarios. In case of the dead ship condition, under which
the main propulsion plant, boilers and auxiliaries are not in
operation owing to absence of power, a shipmaster cannot
take any operational countermeasure. This means that
operational factors are not relevant to safety level evaluation.
Although some operational actions such as high-speed
running in following waves can decrease safety level
against capsizing, they can be avoided by operators if thesafety level under dead ship condition is ensured. In otherwords, if perfect operation is taken, the safety level during a
ship’s life-cycle could be comparable to that under deadship condition.
For the dead ship condition, the weather criterion iscurrently implemented in the IS Code and provides asemi-empirical criterion for preventing capsizing in beamwind and waves. The dead ship condition, however, does
not always mean beam wind and waves. If a ship has alongitudinally asymmetric hull form, ship may drift toleeward with a certain heading angle. Therefore, it isimportant to estimate the drifting attitude of a ship and toevaluate its effect on capsizing probability. It is also widely
known that empirical estimation of effective wave slope
coefficient in the weather criterion is often difficult to be
applied to new ship-types such as a RoPax ferry, a large
passenger ship and so forth. Therefore, the IMO (2006)
recently allows us to use model experiments for thispurpose although it is not always feasible. A simplified
prediction method is still desirable if accurate enough.
Based on the above situation, the authors presented amethodology for calculating capsizing probability of a ship
under dead ship condition. In this present methodology,
capsizing probability under dead ship condition is
calculated by means of piece-wise linear approach.
Through this paper, we summarize this methodology andits application of this methodology to the new generation
criteria at the IMO.

First, a prediction method of the annual capsizing
probability in beam seas based on a piece-wise linear
approach was presented.
Second, the methodology is verified with the Monte
Carlo simulation and is extended to the case with cargo shift,down-flooding and water on deck. The effect of several
important factors, such as drifting attitude, drifting velocity,hydrodynamic coefficient estimation and statisticalcorrelation between wind and waves are investigated.
Third, safety level in terms of capsizing probability wasassessed by means of the present approach for passenger
and cargo ships as well as fishing vessels.
HISTORICAL REVIEW
Several methods were developed for estimating
capsizing probability in stationary irregular waves. One is
the numerical experiment known as the Monte Carlosimulation, in which the expected number of capsizing isestimated by repeating numerical simulation in time domain
with different wave realizations (McTaggart and de Kat
2000). If a reliable mathematical model is available, thismethod seems to be easily applicable. However, since
capsizing probability of an existing ship is small, thenumber of numerical runs could be prohibitively large. In
addition, statistical variation of outcomes is not suitable forregulatory purposes. Sheinberg et al. (2006) attempted to
improve this approach by focusing on critical local waves.
Others are theoretical ones, in which capsizing probability
is obtained as the product of the probability of dangerous
condition and the conditional probability of capsizing. Somemethods utilize stationary linear or non-linear theory for theformer and non-linear time-domain simulation ofdeterministic process for the latter (Umeda et al. 1992; Shenand Huang 2000; Bulian and Francescutto, 2004). Shengand Huang (2000) use the Markov process for probability
density of roll and use the deterministic simulation in the
final stage. Bulian and Francescutto (2004) use a simplified
stochastic linearization technique, with a Poisson modellingof the capsize event. Their main drawback is an assumption
for the final deterministic process. Although some studies(Jiang and Troesch, 2000; Falzarno et al., 2001) for
extending the Melnikov approach into random seaways
were carried out, no direct relationship with capsizing
probability was not provided. Avoiding such a drawback,
Belenky (1993) proposed a piece-wise linear approach
where both probabilities are analytically calculated byapproximating the restoring arm curve with piece-wiselinear ones. Because of its fully analytical scheme, thepiece-wise linear approach has been utilized for practical
purpose as the most promising methodology. An approachthat has a series of similarities with the piece-wise linear
method, but that is based on a simplified statisticallinearization technique, has been developed by Bulian andFrancescutto (2004, 2006)
For a ship in irregular beam seas, Belenky derived exact
formulae based on a piece-wise linear approach without
numerical results. Then he extended his method to the casein both beam winds and waves (Belenky 1994) andprovided numerical results with his simplified method(Belenky 1995). Here the formulae relating to the windeffect were not published in detail. Recently, Iskandar et al.
(2001) applied Belenky’s simplified method to estimatedannual capsizing probability of Indonesian Ro-Ro ferry
accidents. Others (Umeda et al. 2002; Munif et al. 2004,
Francescutto et al., 2004) also applied that to estimate thecapsizing probability of fishing vessels and a large
passenger ship.
Although Belenky’s piece-wise linear method is widely
utilized, some unsolved problems remain. Firstly, complete
formulae for covering the wind effect have not been
published. Secondly, numerical results of the exact
formulae have not yet been reported for evaluating theaccuracy of the simplified formula. Thirdly, effects ofseveral elements such as wind have not been examined.
Therefore, Paroka et al. (2006) examined these
problems with deriving the detailed formulae for the case ofbeam winds and waves and calculating with both exact andsimplified formulae for a car carrier, which has a large
windage area.
CAPSIZING PROBABILITY BY MEANS OF THE
PIECE-WISE LINEAR APPROACH
The methodology adopted by the SCAPE committee forcalculating capsizing probability under dead ship conditionis based on the piece-wise linear approach proposed by
Belenky(1993). This is partly because it rigorously takes
acount of nonlinear restoring moment and its efficient
calculation load enables us to obtain annual capsizing
probability by integrating many combinations of wave
height and wave period in long-term wave statistics. As
mentioned above, Belenky’s formulation for beam waves(Belenky, 1993) and its extension to beam wind and waves
case (Belenky, 1994) had not been complete enough so that
Paroka et al. (2006) corrected them. Therefore, in this paper,
the upgraded formulation is overviewed.
kfi(φ)

Steady Wind heeling arm-φv -φCR-φm0
φm0

φCR φv

φ

Fig. 1 Piece-wise linear approximation of righting arm.
The capsizing probability here is calculated with thefollowing nonlinear and uncoupled equation of relative or
absolute rolling angle of a ship under a stochastic wave
excitation and steady wind moment. Usually the ship
motion in beam seas is modeled with equations of coupledmotions in sway and roll with wave radiation forces anddiffraction forces. Watanabe (1938), however, proposed an
one-degree of freedom equation of roll angle, φ , as follows(I xx + A44 )φ&& + B44φ& + WGZ (φ ) = M wind (t ) + M wave (t )(1)where Ixx is momentum of inertia of the ship, A44 is thehydrodynamic coefficient of added inertia, W is the shipdisplacement, B44 is the hydrodynamic coefficient of roll
damping, GZ( φ ) is the righting arm. Mwind (t) is the wind

induced moment consisting of the steady and fluctuating
wind moment and Mwave (t) is the wave exciting moment
based on the Froude-Krylov assumption. This is because theroll diffraction moment and roll radiation moment due tosway can cancel out when the wavelength is sufficiently
longer than the ship breadth. (Tasai, 1969)
The uncoupled equation of the absolute roll motion (1)can be rewritten by dividing by the virtual moment ofinertia as follows:

φ&& + 2αφ& + ω 02 kf i (φ ) = ω 02 (m wind (t ) + m wave (t ))(2).

where kfi( φ )=GZ( φ )/GM, α: roll damping coefficient,ω0 : the natural roll frequency and kf1( φ ) : thenon-dimensional righting arm, mwind (t) : the wind induced
moment consisting of the steady and fluctuating wind
moment and mwave (t) : the wave exciting moment based on
the Froude-Krylov assumption.
As shown in Fig. 1, the restoring moment coefficient inequation (2) is approximated with continuous piece-wisedlines in leeward and windward conditions as follows:

0 < φ < φ m0 range1⎫⎧ kf 0φ
kf i (φ ) = ⎨⎬⎩kf 1 (φ v − φ ) φ m0 < φ range2⎭(3)

P (H 1 / 3 , T01 , Ws , T ) =Pl PT (φ > φm 0 orφ < −φ m 0 )PA ( A > 0; φ > φ m 0 ) +Pw PT (φ > φ m 0 orφ < −φm 0 )PA ( A < 0; φ < −φ m 0 )(6).
Here Pl and PW denote the probability of up-rossing towardleeward and that of down–crossing toward windward.
Under the assumption of Poisson process, PT indicates theprobability of at least one up-crossing or down-crossing at
the border between the first and second range, φ m0 or−φ m0 . PA is the probability of diverging behavior of
absolute value of the roll angle in the second range
including the angle of vanishing stability.
These can be determined as following formulas:
PT (φ > φ m0 orφ < −φ m0 ) = 1 − exp(− (u l + u w )T )Pl =ulul + u w

(8)

uw
Pw =
ul + u w∞

PA ( A > 0;φ > φm0 ) = ∫ f ( A)dA
0

and

(7)

(9)

PA ( A < 0 ;φ < −φm0 ) = ∫ f ( A)dA
0

−∞

kf 0φ
− φ m0 < φ < 0 range1⎫⎧
kf i (φ ) = ⎨(4).
⎬⎩kf 1 (− φ v − φ ) φ < −φ m0 range2⎭
Here, kf0 and kf1 are the slope of the range 1 and range 2,
respectively. φ m0 and −φ m0 are the border between therange 1 and the range 2 in leeward and windward,
respectively. Effects of dividing methods of the righting
arm curve into piece-wise lines and the case using therelative roll angle can be found in Paroka et al. (2006).
Since Equation (2) is linear within each range, it can beanalytically solved without any problems. The bordercondition, however, should be satisfied. Therefore,
capsizing occurs when the roll angle up-croses, φm0, or
down-crosses, -φm0, and then the absolute value of roll angle
increases further. Therefore, capsizing probability can becalculated as the product of the outcrosing probability in therange 1 and the conditional probability of divergence of theabsolute value of roll angle in the range 2. Belenky (1993)
presented the following formula for the probability that aship capsizes when the duration, T, passes in stationary
waves.

P ( H 1 / 3 , T01 , Ws , T ) = 2 PT (φ > φ m 0 ) PA ( A > 0; φ > φ m 0 )(5)
Where H1/3 : the significant wave height, T01: the mean
wave period, WS: the mean wind velocity and T: duration.
Since this formula could exceed 1, Umeda et al. (2004)
proposed the following formula.

where ul and uW denote the expected number of up-crossing
and down-crossing, respectively. f(A) denotes theprobability density function of the coefficient A,, which is afunction of three random variables, namely the initial
angular velocity of roll motion, φ&1 , the initial forced roll, p1,
and the initial forced angular velocity, p&1 , in the second
range.
The probability density function of the A coefficient canbe determined from a joint probability density function ofthese three random variables, which can be calculated byusing the three-dimensional Gaussian probability density(Price and Bishop, 1974).Then, the probability density
function of the coefficient A in leeward can be described asfollows:f ( A) =∞ ∞1∫ ∫ f φ&1 , p& 1 A,φ&1 , p1 , p1 dφ&1 dp1
λ 2 − λ1 − ∞ 0

(

(

) )

(10)

That in windward can be described as follows:f ( A) =∞ 0
1∫ ∫ f φ&1 , p& 1 A,φ&1 , p1 , p1 dφ&1 dp1
λ 2 − λ1 − ∞ − ∞

(

(

) )

(11)

where
λ1,2 = −α ± ω0 2 kf1 + α 2(12).

Furthermore, Belenky (1993) simplified Equations (10)and (11) by using the fact that roll resonance in the second
range does not exist because of negative restoring slope. As
a result, f(A) can be evaluated with a single integral in placeof a double integral.
For evaluating risk of capsizing, it is necessary to
calculate annual capsizing probability with sea stateVERIFICATION AND EXTENSIONS OF PIECE-WISE
LINEAR APPROACH

In case of a ship has large angle of vanishing stability, theobtained capsizing probability could be very small but theship could capsize owing to cargo shift or down flooding
when the roll angle is larger than her critical angle. In thiscase, the calculation method mentioned above cannot bedirectly applied because angle of vanishing stability cannot
be determined.
Therefore,. Paroka & Umeda. (2006a) extend thepiece-wise linear seas approach to the case having thecritical roll angle. Here the probability of roll angle
exceeding the critical value is defined as the probability ofroll angle of the second range exceeds the critical value
when the roll motion up-crosses or down-crosses the border
between the first and the second ranges of the piece-wiselinear righting arm curve. This can be written as follows:

Comparison with the Monte Carlo simulation

P(H 1 / 3 ,T01 ,Ws ,T ) =statistics of operational water area. The formula of theannual capsizing probability, Pan, can be found in Iskandar
et al. (2000) as follows:()(

Pan = 1 − 1 − P* (T )365×24×3600 / T )

(13)

where
∞∞

P (T ) = ∫ ∫ f (H 1/ 3 , T01 ) ⋅ P(H1/ 3 , T01 , Ws , T )dH1/ 3 dT01 .
*00

Here Ws is assumed to be fully correlated with H1/3

For verifying the piece-wise linear approach, Paroka &Umeda (2006b) calculated capsizing probability of a large
passenger ship in stationary beam wind and waves andcompared it with the Monte Carlo simulation. First, they
confirmed that stochastic scattering of the Monte Carlosimulation results is comparable to its confidence interval
estimated with the assumption of binomial distribution.
Second, it was that the piece-wise linear approach coincides
with the Monte Carlo simulation in moderate sea states,
such as the mean wind velocity of 28 m/s, within theconfidence interval but underestimates it in more severewind velocity as shown in Fig.2. The results of thisunderestimation are that the assumption of the Poissonprocess for outcrossing is not appropriate when the sea stateis extremely severe. In addition, it is impossible for theMonte Carlo simulation to accurately quantify the capsizing
probability in lower wind velocity because the probability is
too small. In conclusion, it is not so easy to verify thepiece-wise linear approach but it seems not to beinappropriate to use the piece-wise linear approach for themean wind velocity of 26 m/s, which the weather criterion
deals with, but there is a room for further improvement.
Wind Velocity (m/s)
28.0

28.5

29.0

29.5

30.0

30.5

31.0

Capsizing probability

1.0E+00

1.0E-01

1.0E-02

1.0E-03

Numerical Experiment
+Err
-Err
Present Method

1.0E-04

Figure 2. Comparison of capsizing probability of thelarge passenger ship between the piece-wise linear
approach and the error tolerance of Monte Carlosimulation. (Paroka & Umeda, 2006b)
Effect of critical angle for down-flooding or cargo shift

Pl PT (φ > φ m0 orφ < −φ m0 )PCR (φ > φ CR ;φ > φ m0 ) + (14)Pw PT (φ > φ m0 orφ < −φ m0 )PCR (φ < −φ CR ;φ < −φ m0 )where PCR is the conditional probability of roll angle
exceeds the critical value, φCR .
The conditional probability of roll angle exceeds thecritical value is calculated with assuming that effect of
exciting moment on the roll motion in the second range isnegligibly small. This assumption is based on the fact that
resonance is impossible in the second range owing to its
negative stiffness. Therefore the roll motion in this range
depends only on the initial condition, which is the same as
the roll motion at the end of the first range. The solution of
the roll motion equation in the second range can be written
as follows:

φ2 (t ) = A exp(λ1t ) + B exp(λ2t ) + (φv − φD )

(15)

where

λ1, 2 = −α ± − α + ω 02 kf1
φD =

(16)

m0

ω 02 kf1
1
A=
φ& − λ {φ − (φ − φ )}(λ1 − λ2 ) 02 2 m0 v D
1B=
λ {φ − (φ − φ )} − φ&02
(λ1 − λ2 ) 1 m0 v D

(17)

[

]

(18)

[

]

(19)

Here φ&02 indicates the initial angular velocity of roll motionin the second range. m0 indicates the non-dimensional
steady wind moment. In order to investigate whether or not
the roll angle exceeds the critical value, the following
iteration procedure is proposed. Firstly the initial angularvelocity of roll motion in the second range with negative A
coefficient in the equation (18) is arbitrarily choose.
Secondly the time of the angular velocity of roll motion
becomes zero is calculated as follows:tm =⎛ Bλ2 ⎞
1⎟ln⎜⎜ −(λ1 − λ2 ) ⎝ Aλ1 ⎟⎠(20)Then the roll angle with the time, tm , can be calculated
by means of the equation (15). If this roll angle is smaller()

P φ&02 > φ&CR =∞2⎧⎪ φ&02 ⎫⎪

&
exp⎨
⎬ dφ
2πD ∫
⎪⎩ 2 D ⎪⎭
q& &
φ

CR

q&

02

(21)

where φ&CR indicate the critical value of the angularvelocity in the second range. The probability of roll angle
exceeding the critical value in a certain exposure time canbe calculated by means of the equation (14) with theconditional probability of roll angle exceeds the criticalvalue is replaced by the equation (21).
Numerical results of the car carrier for several criticalangles ranging from 55 degrees to 75 degrees are shown in
Fig. 3 (Paroka et al., 2006c). The probability of dangerous
condition increases when the critical angle decreases. Thisis because the roll angle can exceed the critical value even if
the roll motion in the second range does not diverge. The
difference disappears when the unstable heel angle owing tosteady wind moment is smaller than the critical value. Whenthe unstable heel angle is smaller than the critical value, theroll angle can exceed the critical value only if the rollmotion in the second range diverges.

trapped on deck have been published (e.g. Belenky, 2003)
however its effect to capsizing probability has not been
investigated.
Paroka & Umeda (2006c) extended the piece-wise linear
approach to the case with trapped water on deck. Here theamount of trapped water was estimated with modelexperiments (Matsuda et al., 2005) by measuring heel andsinkage of the model in irregular beam waves. Water
ingresses and egresses through the bulwark top or freeing
ports and the balance of these can be determined duringb
the experiment. As a result, it was found out that the trappedwater on deck could increase the capsizing probability forthis ship type as shown in Fig. 4. This is because the trappedwater can significantly reduce the mean restoring arm up to
the angle of immersion of the bulwark top, which isapproximated with piece-wise linear curves as a function of
stationary sea state..
As a next step, Paroka & Umeda (2007) incorporated a
numerical model for describing water ingress and egresswith a hydraulic model into the piece-wise linear approach.
As a result, capsizing probability with water on deck can becalculated without a model experiment. Here the dynamic
effect of trapped water on deck such as change in addedmass, damping coefficient and exciting moment are ignored
for simplicity’s means.
GM without Water on Deck (m)
0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

1

Long-term Capsizing Probability

than the critical one, the initial angular velocity is increased
and we repeat these processes. When the maximum rollangle has been larger than the critical value, the iteration is
stopped and the initial angular velocity is set as the criticalangular velocity. The probability of roll angle exceeds thecritical value then can be calculated as the same as theprobability of the initial angular velocity of roll motion
higher than the critical angular velocity. The probability of
the initial angular velocity larger than the critical angularvelocity can be calculated as follows:

1E-5

1E-10

1E-15

1E-20

Without Water on Deck
With Water on Deck

1E-25

1E-30

Fig. 4 Long-term capsizing probability with and withoutwater on deck for the purse seiner operating off Kyusyu.
(Paroka & Umeda, 2006c)
Effect of drifting attitude and velocity

Fig. 3 Dangerous probabilities of the car carrier for different
critical roll angles (Paroka & Umeda., 2006a).
Effect of water on deck
For smaller ships such as fishing vessels, the effect oftrapped water on deck should be taken into account. This isbecause these ships have smaller freeboard but higher
bulwark comparing with their ship size. As a result, whenshipping water occurs, water can be easily trapped bybulwark. Although a certain amount of water can flow out
through freeing ports, the balance between ingress andegress can be trapped on deck. The righting arm candecrease because of the water trapped on deck. Manyexperimental and numerical works about effect of water

The dead ship condition does not always mean beamwind and waves. If a ship has a longitudinally asymmetrichull form, ship may drift to leeward with a certain heading
angle. Therefore, it is important to estimate the drifting
attitude of a ship and to evaluate its effect on capsizing
probability.
X
U

Wind

x
χ
Y

β
y

0
ψ

Wave
y

φ

Y
0'

z
Z

Fig. 5 Co-ordinate systems

(m + m y )v& + (m + mx )ur − m y l yφ&& = YH + YA + YW

(23)(I z + J z )ψ&& = N H + N A + NW
(I x + J x )φ&& − m y l y v& − mx l x ur + 2μφ& + W ⋅ GMφ

(24)
(25)

= KH + KA
whereu,v ; ship speed in x-axis and y-axis,
ψ , φ ; yaw and roll angle,
m ; ship mass,
W ; displacement,
GM ; metacentric height,
mx, my; added mass in x-axis and y-axis,
Ix, Iz; moment of inertia around x-axis and z-axis,
Jx, Jz ; added moment of inertia around x-axis and z-axis,
lx, ly; vertical position of centre of mx,and my
X, Y; external forces in x-axis and y-axis
N, K; yaw and roll moments

Here the subscript H, A and W means hydrodynamic
force/moment, wind force/moment and wave-induced
drifting force/moment.
In this model, a mathematical model for hydrodynamic
forces under low speed manoeuvring motion (Yoshimura,1988), an empirical method for wind forces (e.g. Fujiwara,
2001), and a potential theory for wave-induced drifting
forces (Kashiwagi, 2003) are used.
The Newton method was used to identify equilibria of theequations. As an initial input, the equilibria in the beamwind and waves from starboard and those from port side arecalculated. And then, equilibria for different angle between
the wind and waves, χ , are traced. The calculation aredone in two ways: the first one is that χ changes from 0
degrees to 180 degrees (forward) with the wind direction
fixed, and the second is that χ changes from 180 degrees
to 0 degrees (backward). Stability of equilibrium isevaluated by calculating the eigenvalues of the Jacobean
matrix in the locally linearised system around equilibrium.
The example of results is shown in Fig 6.
These figures indicate that when the relative angle
between wind and waves is 0 degrees, the ship is drifting
leeward in beam wind and wave. Having the relative angle
between wind and waves larger from 0 degrees, the heading
angle gradually increases from the beam wind.
Ogawa et al.(2006b) examined the present method
through the comparison with measured drifting motions. Fig.
7 shows the sway velocity in wind and waves as a functionof wind speed. In this calculation, the significant wave
height and mean wave period were of the same as in theexperiments (H1/3 = 9.5 m, T01 = 10.4 sec). Only the windspeed was varied in the present calculation. It is found thatheading angle (degrees)

(22)

330
300
270
240
210
180
150
120

forward(stable)
forward(unstable)
backward(stable)
backward(unstable)

90
60
30
0
0

30

60

90

120

150

180

angle between wind and waves (degrees)Fig. 6 Heading angle of fixed points as a function of relative
angle between wind and waves (wind speed is 26 m/s, wave
amplitude is 2.85m and wave length ratio is 2.94). (Umeda
et al., 2006b)
Drift speed due to wind & wave
Cal.( wind only)
Cal. (wave only(H1/3=9.5m, T01=12.4sec))Cal.(wind&wave(H1/3=9.5m, T01=12.4sec))Exp.(wind only)Exp.(wave only(H1/3=9.5m, T01=12.4sec))Exp.(wind&wave(H1/3=9.5m, T01=12.4sec))Cal.(Beaufort)

3.5
3
2.5
Drift speed (m/s)(m + m x )u& − (m + m y )vr = X H + X A + X W

360

2
1.5
1
0.5
0
0

5

10

15

20

25

30

35

Wind speed (m/s)

Fig. 7 Relation between the wind speed and the drift speed
in wind and waves. (Ogawa et al., 2006b)
0

20

40

60

80

100

1.00E+00

capsizing probability (case1)

For estimating drifting velocity and attitude of a ship in
regular waves and steady wind, Umeda et al. (2006b,
2007a) proposed a method for identifying equilibria of asurge-sway-yaw-roll model and for examining its local
stability. Equilibria can be estimated by solving 4-DOF
non-linear equations in the following co-ordinate systems
shown in Fig. 5;

the computed drift speed is close to the measured drift speed,
although the drift speed in beam waves only (wind speed =0 in Fig. 7) was underestimated. It is confirmed that present
method is useful for the quantitative computation of thedrift speed under dead ship condition.

1.00E-04
1.00E-08
1.00E-12
1.00E-16
1.00E-20
1.00E-24
1.00E-28
1.00E-32
1.00E-36

angle between wind and waves (degree)

Fig.8 Capsizing probabilities under two coexisting drifting
conditions as functions of relative angle between wind andwaves. (Umeda et al., 2006b)
An example of capsizing probabilities under two
coexisting stable steady drifting states are shown in Fig 8.
Umeda et al. (2006b, 2007a) Here the mean wind velocity is
26 m/s and the duration is one hour. The fully developed
wind waves under this wind velocity are assumed. In thiscalculation, the effect of relative angle to wind is modelled
as the change in lateral windage area, the effect of relative
angle to waves is done as the change in effective wave slope
coefficient and the effect of drifting velocity is done as thechange in the exciting frequency. In addition, the heelinglever due to drift are estimated with the model tests by
Taguchi et al. (2005). It is confirmed that the mostdangerous condition for this subject ship under dead shipcondition is one in beam wind and waves, i.e. χ is 0
degrees. It is clarified that drifting motion obviously
depends on underwater and above-water ship geometry.

simultaneous equation for determining two dimensionalflows, however, a further simplified method is expected to
be developed. For this purpose, Sato et al. (2007) carriedout a series of model tests by means of two dimensionalmodels covering large breadth to draft ratios. They
confirmed that measured effective wave slope coefficient
could be much smaller than that from a strip theory in caseof very large breadth to draft ratios. This fact coincides withthe experiments by Ikeda et al. (1992) for a small
high-speed craft.

Effect of hydrodynamic coefficient estimationroll amplitude (deg)

25
20
15
10
5
0

0.7

0.8

0.9

1

1.1

1.2

1.3

tuning factor
Fig. 9 Roll amplitude of the RoPax ferry in regular beam
waves measured with different constraining methods(Umeda et al., 2006a).
30
25roll amplitude (deg)It is also widely known that empirical estimation ofeffective wave slope coefficient in the weather criterion isoften difficult to be applied to modern ship-types such as alarge passenger ship, a RoPax ferry and so forth. Based on
this background, the IMO allows us to estimate the effective
wave slope and damping coefficients as well as heelinglever owing to beam wind by model experiments with theinterim guidelines (2006).
Taguchi et al. (2005) and Ishida et al. (2006) carried out
almost full set of model tests for a RoPax ferry and verified
the interim guidelines as the alternative assessment of theweather criterion. It was clarified that the evaluation of
weather criterion considerably differs that with the
combination of tests. Umeda et al. (2006a) applied thismodel test approach to the Ro-Pax ferry for further
investigating its feasibility. Based on the interim guidelines,
the model experiments were executed with three different
constraints: a guide rope method, a looped wire system anda mechanical guide method. The guide rope method meansthat the model was controlled by guide ropes. The looped
wire system means that, by utilizing a looped wire attached
to a towing carriage, the yaw motion of the model is fixed
but the sway, heave and roll motions are allowed. The
mechanical guide is the combination of a sub-carriage, aheaving rod and gimbals. Among them, as shown in Fig.9,
the looped wire system is less accurate because natural rollperiod and damping was slightly changed owing to a looped
wire. In case the yaw angle increases with larger wave
steepness, the guide rope method has inherent difficulty forkeeping a beam wave condition. The guideline allows thethree different methods for determining the roll angle fromthe experiment: the direct method, the three step method
and the parameter identification. As shown in Fig 10, thedifference between the three methods is negligibly small. In
conclusion, the interim guidelines can guarantee reasonableaccuracy if appropriate constraint is used.
When we calculate capsizing probability with thepiece-wise linear approach, the same difficulty exists. Thisis because fluid-dynamic coefficients such as effective wave
slope coefficient should be separately estimated in advance.
Therefore it is desirable to execute model experiments
following the IMO interim guidelines. Utilizing model tests,
however, could be not always practical owing to cost andtime. Therefore, it is desirable to develop an empirical or atheoretical method to accurately predict obtained resultsfrom the model test.
It is well known that a strip theory can explain theeffective wave slope coefficient in general. (Mizuno, 1973)
Since a strip theory requires numerical solution of

Guide Rope
Looped Wire System
M echanical Guide
NM RI (Guide Rope)

s=1/60

25.22

26.56

26.18

three step methodparameter identificatin

20
15
10
5
0
direct method

Fig. 10 Comparison of estimated roll amplitude by meansof three different estimation methods. (Umeda et al., 2006a)
Umeda & Tsukamoto (2007c) investigated several
possible simplified versions of theoretical methods andfinally proposed a formula to calculate the effective wave
slope coefficient of the equivalent rectangular sections with
the Froude-Krylov component on its own. Here the local
breath and area of the rectangular section should be thesame as those of original transverse ship sections. As shownin Table 1, this simplified formula agrees well with theexperiment and a strip theory for existing ships having large
windage areas probably because the diffraction component
can almost cancel out the radiation owing to sway and the
local breadth and area are responsible for restoring moment.
Furthermore, they indicate that difference in the effective
wave slope coefficient between the present formula andexperiments results in only small effects on the estimationof the one-hour capsizing probability in stationary beam

wind and waves as shown in Fig.11. Therefore, it isrecommended to use this simplified formula for a regulatory
purpose.
Table.1 Comparison of effective wave slope coefficients
under resonant condition. Here IMO means the formula in
the IMO weather criterion and FK indicates the
Froude-Krylov prediction. (Umeda & Tsukamoto, 2007c)

19

21

23

25

27

29

∞∞∞
P* (T ) = ∫ ∫ ∫ f (H 1 / 3 ,T01 ,Ws )000⋅ P(H 1 / 3 ,T01 ,Ws ,T )dH 1 / 3 dT01 dW s

(27)

Ogawa et al. (2006a) calculated the annual capsizing
probability of the RO-PAX ferry. Fig. 12 shows her rightingarm (GZ) curves. The limiting KG of this ship is governed by theweather criterion so that it was used as a loading condition for thesafety assessment. The approximated line of the righting arm curve,of which GM, the area under the GZ curve and the angle ofvanishing stability are not changed, is also shown in Fig. 12.

Fig. 13 shows the capsizing probabilities within one year
as a function of wind speed. In addition to the capsizing
probability by means of the present method, two kinds ofcapsizing probability are calculated by means of two otherwave scatter diagrams. One is the wave scatter diagram
normalized for each mean wind velocity (Normalized) andthe other is the wave scatter diagram without correlation ofwind (No correlation). The wave scatter diagram without
correlation with wind was conducted by summing up thepresent scatter diagram of wave height, wave period andwind speed. Then the capsizing probabilities using two
kinds of diagrams is defined as follows:

31
33
35
Wind Velocity [m/ｓ]1.0E-01
Probability

1.0E-03
1.0E-05(Normalized)

1.0E-07
1.0E-09∞∞
P* (T ,W s ) = ∫ ∫ f N (H 1 / 3 ,T01 ; W s )1.0E-11
1.0E-13

IMO

1.0E-15

Strip theory

1.0E-17
1.0E-19

0 0⋅ P(H 1 / 3 ,T01 ,W s ,T )dH 1 / 3 dT01

EXP
simplified Froude-Krylov

Fig.11 The effect of the effective wave slope coefficientestimation on the one-hour capsizing probability of the
RoPax ferry in stationary beam wind and waves. (Umeda etal., 2007a)
Effect of correlation between wind and waves
For the provision of the criteria based on
performance-based approaches, the capsizing probability
should be assessed preferably with long-term sea statestatistics. Ogawa et al. (2006a,2007) constructed a scatter
diagram of wave height, wave period and wind speed bymeans of the wind and wave database developed fromhindcast calculation of wind-wave interactions andexamined the effect of the correlation of winds with waves
on the long-term capsizing probability. Table 2 shows
examples of the scatter diagram of wave height, wave
period and wind speed for a water area near Minami Daito
Islands in the Northern Pacific. It is found that theprobability of large wave height and long wave period
increases as the wind speed increases.
The long-term probability of capsizing of a ship within
N years can be described as follows:
PN = 1 − ⎛⎜ 1 − P* (T )⎞⎟
⎝
⎠

where

(N × 365 × 24 × 3600 ) / T

(26)

(28)

(No correlation)∞∞
P* (T ,W s ) = ∫ ∫ f NC (H 1 / 3 ,T01 )0 0⋅ P(H 1 / 3 ,T01 ,W s ,T )dH 1 / 3 dT01

(29)

where fN(H1/3, T01; Ws) denotes the normalized wave scatter
diagram and fNC(H1/3, T01) denotes the wave scatter diagram
without correlation with wind. For comparison sake, theshort-term probability in the constant wind and waves isalso shown in Fig. 13.
It is found that capsizing probability of the RO-PAX
ferry complying with the weather criterion is adequately
small if the statistical correlation between wind and waves
is taken into account. This probability does not depend on
the mean wind velocity because the occurrence probabilityof wind is taken into account in the calculation. The
capsizing probability with the normalized scatter diagram islarger than the present method when the mean wind velocity
is larger 16 m/s. This indicates that occurrence probabilityof such strong wind is not so large in this water area during
a year. The capsizing probability with the no correlation
scatter diagram is larger than the present method. This isprobably because unrealistic combinations of wind andwaves are included in the calculation.
The capsizing probability taken from the stationary sea
state is larger than the present method because theoccurrence probability of such sea state is not large during ayear in this water area. On the other hand, the capsizing
probability from the stationary sea state of the mean windvelocity of 26 m/s is much smaller than the present method.
This suggests that the occurrence probability of sea stateshould be taken into account when we estimate the annual
risk from the short-term capsizing probability.

Capsizing probability (T=3600sec.)
0

5

10

15

20

25

30

35

0.0

Ws=2(m/s) T(sec.)
Hw(m)311.2510.7510.259.759.258.758.257.757.256.756.255.755.254.754.253.753.252.752.251.751.250.750.250-

4-

5-

0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0

0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0

Ws=20(m/s) T(sec.)
Hw(m)5615.250
14.75014.25013.75013.25012.75012.25011.75011.25010.75010.250
9.750
9.250
8.750
8.250
7.750
7.250
6.750
6.250
5.750
5.250
4.750
4.250
3.750
3.250
2.750
2.250
1.750

Ws=40(m/s) T(sec.)
Hw(m)8915.250
14.75014.25013.75013.25012.75012.25011.75011.25010.75010.250
9.750
9.250
8.750
8.250
7.750
7.250

67891011120
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0 0.000297 0.000224
000
0 0.000414 0.000405
00
0 0.000176 0.000589 0.001021 0.000847 3.91E-06
0
0 0.000887 0.001245 0.002002 0.002682 5.05E-05
0 1.1E-05 0.001218 0.004456 0.004191 0.001685 0.000424
0 0.000378 0.001439 0.001877 0.000579 9.28E-05 1.9E-05
0 0.000101 4.15E-05 3.73E-05 3.91E-06 1.86E-07 1.86E-07
0
0
0
0
0
0
0

130
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0

140
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0

150
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0

0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0

789101112131415160
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0 3.73E-07 3.17E-06
0
0
0
0
0
0
0
0
0 3.35E-06 1.86E-07
0
0
0
0
0
0
0
0
0 1.12E-06
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0 7.83E-06 4.84E-06
0
0
0
0
0
0
0
0 6.71E-06 8.24E-05
0
0
0
0
0
0
0
0
0 1.58E-05 0.00011 7.45E-07 3.91E-06
0
0
0
0
0
0 2.61E-06 7.32E-05 6.17E-05 5.78E-06 3.5E-05
00000 1.3E-06 1.01E-05 0.000168 4.84E-06 6.15E-06 1.3E-05
0000
0 9.69E-06 4.45E-05 0.000351
0
0
0
0
0
0
0 2.24E-06 4.49E-05 0.000201 9.82E-05 1.49E-06
0
0
0
0
0
0 4.47E-06 7.84E-05 0.000212 0.000119 7.27E-06 1.45E-05
0000
0 5.33E-05 0.000301 0.000302 0.000128 4.1E-06 3.82E-05
00
0
0 5.03E-06 0.000106 0.00066 1.29E-05 6.28E-05 8.57E-06 6.15E-06
00
0
0 3.78E-05 9.69E-05 0.000248
0 3.73E-06 1.86E-07
00000 5.29E-05 7.81E-05 2.5E-05
0 5.59E-07 4.84E-06
00
0
0 2.96E-05 1.9E-05 3.63E-05
0
0
0
0
0
0
0
0 1.58E-05 2.83E-05 3.06E-05
0
0
0
0
0
0
0
0
0 2.24E-06
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0

100
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0

110
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0

12130
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0 7.45E-07
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0

141516170 1.86E-07
000
0 1.49E-06
0 9.32E-07 5.59E-07
00
0 4.66E-06
00
0
0 9.32E-06
00
0
0 9.5E-06
00
0 5.03E-06 3.17E-06
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0

1.4
Original GZ (kg=10.63m)
Present method

1.2

GZ(m)

1
0.8
0.6
0.4
0.2
0
0

10

20

30

40

50

60

70

80

Angle of Heel(deg.)

Fig. 12 Righting arm curves of the RO-PAX ferry. (Ogawa
et al., 2006a)
The capsizing probability calculation in this paper
except for this section assumes that wind is fully correlated
wave height based on the WMO’s reference data (Umeda etal., 1992). It will be a future task to compare thismethodology with the capsizing probability calculated
taking account of the statistical correlation between wind
and waves. Then the final conclusion whether the statisticalcorrelation should be considered for the performance-based
criteria or not will be discussed.

-5.0
log10(probability)

Table.2 Samples of wave height –period diagram in eachwind speed (Autumn, 1994-2003, Ws=2, 20, 40(m/s)).
(Ogawa et al., 2006a)

-10.0
-15.0
-20.0
-25.0
-30.0
-35.0

Present method
Beaufort(short term)
Normalized
No correlation-40.0

Wind speed(m/s)

Fig. 13 Capsizing probability in one year of the Ro-PAX
ferry under dead ship condition. (Ogawa et al., 2006a)
ASSESSMENT OF SAFETY LEVEL IN TERMS OF THE
PIECE-WISE LINEAR APPROACH
Passenger and cargo ships
Umeda et al. (2007a, 2007b) attempted to evaluate thesafety levels of four passenger and cargo ships by utilizing
the above mentioned capsizing probability calculation.
Their principal dimensions of these ships are shown in
Table 3. The method for calculating capsizing probability
here is the piece-wise linear approach with down-flooding
angle taken into account. The mean wind velocity is
assumed to be fully correlated with the wave height and theeffect of drifting attitude and velocity is ignored. The
restoring arm is approximated to keep the metacentricheight, the angle of vanishing stability and the maximum
restoring arm unchanged.
The calculation results of the annual capsizing
probabilities under the designed conditions in the North
Atlantic are shown in Fig. 14. Here the annual capsizing
probabilities of a LPS (large passenger ship) and acontainership under their designed conditions are smaller
than 10-8 so that the associated risks are assumed to be
negligibly small. In particular, the safety against capsizing
in beam wind and waves for the containership is sufficientalthough her designed metacentric height is small. This isbecause the large freeboard of the containership results insignificant increase of the slope of righting arm curve atlarger heel angle. On the other hand, larger probabilities are
obtained for the Ro-Pax and the PCC (pure car carrier). In
particular, the Ro-Pax has large annual capsizing probability,
although her number of persons onboard is not small,because her natural roll period is small so that resonant
probability increases in ocean waves.
The Ro-Pax ferry investigated here is operated not in the
North Atlantic but in Japan’s limited greater coastal area,
which is water area within about 100 miles from theJapanese coast. (Watanabe and Ogawa, 2000) In addition,
her navigation time ranges only from 6 to 12 hours so that
she can cancel her voyage if bad weather such as a typhoon
is expected. Therefore the calculation of capsizing
probabilities of the case in Japan’s limited greater coastal
area and the case in Japan’s limited greater coastal area with
operational limitation to avoid the significant wave heightgreater than 10 meters are executed and compared with thecase in the North Atlantic, as shown in Fig. 15. Here theaero- and hydrodynamic coefficients were estimated withexperimental data.
The comparison indicates that the effect of water areas is
not so large because worst combinations of significant wave

height and mean wave period exist also in Japan’s limited
greater coastal area. This seems to be reasonable because
typhoons are included in the wave statistics in Japan’s
limited greater coastal area.
Table.3 Principal dimensions of ships under designed
conditions. Here W: displacement, Tφ: natural roll period,
N: number of persons onboard. (Umeda et al., 2007b).
Container
PCC
RO/PAX
Items
LPS
Ship
W (t)
53,010109,225
27,216
14,983
Lpp (m)B (m)d (m)GM (m)

242.25
36.0
8.4
1.58

283.8

192.0

42.8
14.0

Units

new
135 GTold 135
GT

80 GT

39 GT

Lpp

Meters

38.50

34.50

29.00

23.00

BS

Meters

8.10

7.60

6.80

5.90

d

Meters

2.851

2.65

2.25

1.77

FBS

Meters

0.485

0.496

0.484

0.396

Δ

Tons

480.98431.07276.61145.70GM

Meters

1.95

1.65

1.57

1.79

6.06

Tφ

Seconds

5.80

5.87

5.20

3.886

Tz

Seconds

3.413

3.435

3.246

2.218

1.41

hB

Meters

1.50

1.45

1.35

1.30

Af

Meters2

7.05

4.75

3.40

2.60

25.0

8.2

1.06

Items

170.0

32.3

1.25

Table. 4 Principal dimension of the subject ships at fullload departure condition. Here FBS: freeboard, TZ: natural
heave period, hB: height of bulwark and Af: area of freeing
port. (Paroka & Umeda, 2007)

T (s)

23.0

30.3

22.0

17.9

N

4332

33

25

370
0 .8

1.0E+00

0 .7

PROBABILITY

1.0E-04

4.98E-05

3.66E-05

Freeboard (m)0 .6

1.0E-02

0 .5
0 .4
0 .3

N ew 13 5 G T
O ld 13 5 G T
80 G T
39 G T0 .2
0 .1

1.0E-06

0 .0
0 .0

0 .2

0 .4

0 .6

1.0E-10
RO/PAX

1 .0

1 .2

1 .4

1 .6

1 .8

2 .0

LPS

CONTAINER SHIP

PCC

Fig. 16 Marginal lines of purse seiners for the annual
capsizing probability of 10-6. (Paroka & Umeda, 2007)

Fig. 14 Annual Capsizing Probability for passenger andcargo ships in the North Atlantic. (Umeda et al., 2007b)

0.8
0.7
0.6

3.66E-05
1.00E-05

Freeboard (m)1.00E+00

PROBABILITY

0 .8

M e tace ntric H e ight (m )1.0E-08

1.23E-05

0.5
0.4

New 135 GT
Old 135 GT
80 GT
39 GT

0.3
0.2
0.1

1.00E-10

2.92E-12

0.0
0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

M etacentric Height (m)1.00E-15

1.00E-20
NORTH ATLANTIC

JPN(Limited greater coastal area）

JPN(Limited greater coastal
area+Operational limitation）

Fig. 15 Effect of operational conditions on annual capsizing
probability for the RO/PAX. (Umeda et al., 2007b)
On the other hand, the significant reduction in thecapsizing probability is realized by the operational
limitation. This explains that the Ro-Pax ferry is safely
operated in actual situations with the aid of the operational
limitation to avoid bad weather.
Fishing vessels

Fig. 17 Marginal lines of purse seiners for the IMO
weather criterion for fishing vessels. (Paroka & Umeda,
2007)
Paroka & Umeda (2007) estimated the safety levels offour Japanese purse seiners, whose principal dimensions arelisted in Table 4, as one of typical fishing vessels types. Byutilizing the piece-wise linear approach with water trappedon deck taken into account, combinations of metacentricheight and freeboard realizing the annual capsizing
probability is 10-6 are identified as shown in Fig. 16. The
water areas used here are actual fishing grounds within theJapanese EEZ for these vessels so that wave statistics areobtained from the NMRI database and actual fish catchrecords (Ma et al., 2004) with assumption that wind velocity
is fully correlated with the wave height. This diagram
indicates that the designed full departure condition for each
vessel has safety level above 10-6. Combinations ofmetacentric height and freeboard for marginally complying
with IMO weather criterion for fishing vessels and thewater-on-deck criterion in the recommendation of theTorremolinos Convention are shown in Figs.17-18,
respectively. The comparison among these results
demonstrates that the weather criterion for fishing vessels
on its own is not sufficient for ensuring safety of thesefishing vessels because of trapped water on deck but thewater-on-deck criterion in the Torremolinos Convention iscomparable to the current probability-based calculation.

2.

3.

calculated probability coincides with the Mote Carlosimulation results within its confidence interval for thesea state relevant to the weather criterion
The calculation method was extended to cover effectsof flooding angle or cargo shift angle, water on deck,hydrodynamic coefficient estimation, drifting attitudeand velocity and the correlation between wind andwaves.
By utilizing the above extended methods, safety levelsof passenger and cargo ships as well as fishing vessels
are calculated as annual capsizing probability, andshow that current designs ensures reasonable levels ofsafety against intact capsizing under good operation
practice.

ACKNOWLEDGEMENTS
0 .8
0 .7

Freeboard (m)

0 .6
0 .5
0 .4
0 .3
0 .2

New 135 GT
O ld 1 3 5 G T
80 GT
39 GT

0 .1
0 .0
0 .0

0 .2

0 .4

0 .6

0 .8

1 .0

1 .2

1 .4

1 .6

1 .8

2 .0

M e t a c e n t r ic H e ig h t ( m )Fig. 18 Marginal lines of purse seiners for thewater-on-deck criterion in the IMO Torremolinos
Convention. (Paroka & Umeda, 2007)
RECOMMENDATION FOR IMO Performance-BASEDCRITERIA
As shown in this paper, it is feasible to calculate annual
capsizing probability of a given ship in beam wind andwaves by utilizing the piece-wise linear approach presented
here. Since a beam wind and wave condition is moredangerous than other drifting attitude, safety level ofstability under dead ship condition at least as a relative
measure can be estimated with this approach. It isnoteworthy here the calculation based on this approach is
not time-consuming so that it can be used as aperformance-based criterion. In case a simpler approach isalso required for a vulnerability criterion, as an existing
criterion for stability under dead ship condition the weather
criterion can be recommended with refinement with thepresent achievements such as effective wave slope
coefficient and roll damping.

CONCLUSIONS
The activity of the SCAPE committee for stability underdead ship condition provides the following conclusions:
1. A piece-wise linear approach for calculating capsizing
probability in beam and waves is presented. The

A part of the present study was carried out in cooperation
with the Japan Ship Technology Research Association
through the part of the Japanese project for the stability
safety (SPL project) that is supported by the Nippon
Foundation. This research was also supported by a Grant-in
Aid for Scientific Research of the Japan Society for
Promotion of Science (No. 18360415). The authors thankDrs. Y. Sato and M. Ueno, Misses E. Maeda and I.
Tsukamoto, Messrs. J. Ueda, S. Koga, H. Sawada and H.
Takahashi for their contribution to the works described inthis paper.
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Umeda, N., Maeda, E. and Hashimoto, H. (2007b):
Capsizing Risk Levels of Ships Under Dead Ship
Condition, Proceedings of the 3rd Annual Conference on
Design for Safety, Berkeley, pp.87-91.
Umeda, N. and Tsukamoto, I. (2007c): “Simplified
Prediction Method for Effective Wave Slope Coefficientand its Effect on Capsizing Probability Calculation (inJapanese)”, Conference Proceedings of the Japan Societyof Naval Architects and Ocean Engineers, Vol. 5, pp.
23-26
Watanabe, Y. (1938): “Some contribution of Theory of
Rolling” Transaction of Institute of Naval Architects, 80,
408-432.
Watanabe, I. and Ogawa, Y. (2000): “Current Research on
Revision of Load Line Regulation”, Symposium on
Current Research on Standards for Manoeuvring, LoadLines and Stability of Ships, The Society of Naval
Architects of Japan.
Yoshimura Y. (1988): Mathematical Model for the
Manoeuvring Ship Motion in Shallow Water (2ndReport) (in Japanese), Journal of the Kansai Society ofNaval Architects, Japan Vol.210, pp.77-84.

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

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 



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% ,         , -   C   -  , -   
&   344'@CC
2   (8       '  φM( . )(t)

Wind
Mwind

t
φ

Wave

φs

0

t

% 1      0t ,  ,-GZ(φ),                Æ        ,        "   "  , "  "     "       " *%   "    , ,   , "        -    " ,@   

GZM 

7th polynomial
approximation

GZ(φs)

φ2

0

φf

φs

φv





  33   
C


 


  C           Dφ (deg)

GZ(φ) by hydrostatic analysis




4

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7  -   * % /       
- -     -    "  , -  *%      -          /    " , -      @'%  Æ             "   " ,    @     (   / - 1        " -         ,  * '* %  /    -      "    "    -  "                 " -        *%          +   ,-   C    "   " ,    @
(8  C   
 
5

%     "   "    -  
" ,    *
 

)




, 

dGZ(φ)= GM
dφ φ=0

 



C 

:

      





 
 

D

=       " ,           @3
' B   B  B 
8 B     ,                 Æ   - *  2((

3(
33



C

%  7  - ,- /           " ,     &   344'@




 

3',        - ,-    Æ 
 ,-    - * % ,- /        "   ,-      " ,@       











   


 

 

/  


35
3D

,    - ,-    Æ     " ,@D  
(::: " 
38
 
((
 ,%              6 ,-      ,-       " ,@  3:':8





243
 

 
 

 C     '  3(  C   )'  3(  C  (3D82 C   (3844  5)2  

32
3:
3)
34

,    ,-        0   1 , E "  *>    ,          ,        -       , 
 "  ,  ,- /      

Rool angle

S ( ) (ω)φ(u, u, tm)

φs0ω1 ω i-1 ω i ω i+1

Transient responseωωn% 3    "    /   "  , 

% 4 /      






 

       C     


















 C      


U1
U2

'(

u*

'3





,           -     " ,   34)2@
3


 
'5
    
 
 


 






½ ¾      ½ 

 ½ 

 ½       '       3   ,      "   "  /      "          ,  *=0.#F#$ 0=#!#%G <?=!&=%#$ G #.%< <!#=#!#%G <%; 
% "        "  -  -  -            "   - ,   6    " ,@0H   ( "

(



I

β O
Safety region

Failure region,              , ,- /      - *       "    "                     -  @C   3     ''  

'D,           "      "   -  * % 1 "       6   * % 6           
% "        34)2 "  6  

Time

φ (U)

Df





tm
steady-state response          

     " ,     /   ,     "       3       '       -     C    3        , * 5  7  '(((7   '((8@0h (U ) = 0

Linearized% 5            
  -             7   7  '(((*%     "   "           
6     ,  ,           , /           -    H(  I  " ,@ 
C     C   '8,   C      /     "             , * D   C      -  "    "  
6  < * '(*%        -          * 8* #  6     -    6 "                       "           -      "         "* %             / * %                     "   *#    1,       1,     34:)           * %
       J     " ,@
3* %    -   "         -      * %     "      "            * #        "    6     < * '8       "    /   < * '(*'* %     "         "        * %     "    1,           1 <-        6   " 16(Nm)2 / (rad/s)

10S(·)(ω)    0 1   '((2 0 1  &
'((2
 !

"###
   
" ! "
 
"$
     
% % " &   '!" 
(!
  
)! "))%  *"()!
   ")$  
)$"# " (#
   " (''"""(    %"(( " * "#'%
% $") "

15

10

Wind
Sam(ω)Wave
Smw(ω)

14

10







  2 "Æ  "  -        - < * '   *

"$" "#%" +$% $' !(( ( ", 1,      34:)@  

       

'2,                 *%             "   -           (* %              * %     /  -         *5* % "         -          "            " ,@    
3
 /  ' 

'


':

$&<# =! <K=0!<.
=    "              -      /  "            + ,  ,-   0 1   0 1   '((2 0 1 & '((2* % 6   "       %
 3* %      - 6  < * '      * ' ,     " -         DD      Æ       %  '*%  , -       C  '48* %  ,   "  ,   ,- "  , -          * 2*%  ,           33    "    * % "        - 
6  < * '5      %  5* %         "  D  DD        -    C     3     33*%      -     "   7                          "  , -       -        (* %   "        6      ,  0 1  E   0 1   '((2*

13

10

0.01

0.1

1

10

ω (rad/s)

% 6 0 ,   " ,  ,- /    "  , -   '4*8  0 1  '((20 1  & '((2  3    "    /   "  ,  " ,  ,- /     "  ,-   '4*8 
,*
,
Ê
Ê


    * 
*   
 ""$ !$(("" "$"! """' %"'"" ! %"$""( "
%"#$""(" $!))""(!
()"% "! ('%"! "#
(!!""%" %((!"
$ "'
$$)""%! !# $"# ")"("
"!" ')("
"$" ''$"
) " ! ( #"' "(" %($#"")")!#"" "%" ) #$"
" ("#"
!" %"%#" "" (!)!"   ! "#$%    "      "   < * '8  58           " -      -            * % 6     /     "   -    "      "    < * '2*                 8((,           -         * #   6  "              -  "         L  (': ,               %,    $  344)*%  -    "        * :* $              ,  -             "  6     /  * %  -  ,   -                          "1,      *<    "         %  D* #  "  "       -  "     "      " ,- /                    , 
 " ,   * #     , "      (3:    (34  " ,        "    (3:4D  -  6   

10

0

2.8

U2

2.6
Reliability index

-1

10

2.4

-2

10

2.2
Limit state function-310h3(u)=0

β4

β3

β2

2.0

-4

10

10

% 72030
40
Iteration

50

60

Safetyregionh4(u)=0

1.8
0

U1

β1

70

-    "     "         /  4  "      
-* 
-* & -  - &
 +"($"!#' +""" ( +""""'$+"%'#$"%%$ +"" """ "(( " )'%""(!%""(""%  +" '(!
"" %"""'!
!   """ '! +""""$$ "%$!
  
+"""(%! """## +""#%)"( )" """((!
"" %$) +""!%
"#)( """%)! +"" #' ""'(""## +"""'$' """%
" """%") +""""+"""$#" """"!!
 """ % """$( """"'$ +"""("#

h2(u)=0
Failure region

h1(u)=0h(u)=0% 9       /   "     "  2.455

Reliability index

01

Reliability index

Limit state function φ - φ (deg)

10

2.450

2.445

2.440
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Trial No.

 ./0/ *1
1.6

40

Magnitude30

φ (deg)

20
10

1.2
1
0.8
0.6
0.4
0.2
0

0

0.09

0.12

0.15

0.17

0.19

0.22

0.25

0.3

0 /* *    1   -*  dφ/dt (deg/s)

-10

0.06

Frequency (rad/s)

0

200

400

600

800

1000

1200

90

Time (sec)% 8         /       * %        /          * )* #  "          58      D((*         /   'D22         -    " ,@ 'D22  :3458  3( ')% -   5(                L  (':*%         /   "     "        " -          *         "               -         ,  * 4*  %  %  % & % %'%                1,         ,        * * 3( Phase (deg)φ (deg)dφ/dt (deg/s)

1.4

60
30
0
-30
-60
-90

0.06

0.09

0.12

0.15

0.17

0.19

0.22

0.25

0.3

Frequency (rad/s)  2  *    1   -*   "  '(           % 1:     ,        " '(     "             *  " '(  38                     "               /*%  -             "  38    * %  ,     "            -     H " I 

 




 

H" I / H" I  H"







/

 





  H " 

" I

" I

10



( '4

5

5(

0

3

,  -E   - 34:4@u"%           " ,@((32D  ((5''53% -         < * ')* ; ,-     " "            L  (':*%              " -         * %              Æ                /          , 1    "  /  .    344' .    3442      7   1 344)
          % ** 3(      ,        " ,     "       "  ,   "        - *%      -           ,                    -   "  < * '3  ''* #  "  "               ,            * #        "     "                    * %       "      " ,     1 "    *

-5

-10
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 % !  -% ? * ' * 335333D8*& $* N   0   . #  .     = -  "  9=.$=< . =0<    " . =0<   0 3O &  5 #%6   ,   6   -' ! %  & %%*PREDICTION METHODS FOR BROACHING AND
THEIR VALIDATION- FINAL REPORT OF SCAPE COMMITTEE (PART 6) Naoya UMEDA*, Hirotada HASHIMOTO*, Atsuo MAKI*, Masatoshi HORI*,
Akihiko MATSUDA** and Tsutomu MOMOKI***Department of Naval Architecture and Ocean Engineering, Osaka University, JAPAN**National Research Institute of Fisheries Engineering, Kamisu, JAPANABSTRACTRegarding broaching associated with surf-riding, thispaper reports experimental, numerical and analyticalstudies conducted by the SCAPE (Strategic ResearchCommittee on Estimation Methods of Capsizing Risk for the
IMO New Generation Stability Criteria) committee,

together with historical review of theoretical progress on
this phenomenon. Here effect of higher-order terms innumerical modelling, extension to novel propulsion
systems and application of global bifurcation theory,
optimal control theory and random process theories are
discussed. Experimental techniques are also investigated.
Based on these outcomes, it also proposes draft stability
criteria consisting of a vulnerability criterion and direct
assessment as a candidate for the new generation
criteria at the IMO (International Maritime
Organization).

KEY WORDS: Broaching, surf-riding, IMO new
generation intact stability criteria, global bifurcation,
broaching probability.

INTRODUCTION
Broaching is one of the three major capsizing scenariosthat the new performance-oriented stability criteria to be
added to the Intact Stability Code at the IMO are requested
to cover (Germany, 2005). This is a phenomenon that a shipcannot keep a constant course despite the maximum steering
effort. If the ship speed is high enough, the centrifugal force
due to this uncontrollable yaw motion could result in
capsizing. This phenomenon often occurs when a ship runs
in following and quartering seas with relatively highforward speed, especially when a ship is surf-ridden. Thus,
this phenomenon is relevant to ships having their Froudenumber of 0.3 or above, such as destroyers, high-speed
RoPax ferries, fishing vessels and so on. Broaching itself

could occur also with slow speed (Oakley et al., 1974) but
such broaching seems not to be a direct threat to shipstability. Thus, this paper focuses on broaching associatedwith surf-riding.

HISTORICAL REVIEWA physical model experiment is definitely suitable forrealising dangerous phenomena in ship stability because fullscale measurement could be too risky. Du Cane (1957)
executed model experiments of broaching in following
waves at a towing tank. Here he added initial disturbance toa ship model running in the straight course and observed theship response. Nicholson (1974) and Fuwa et al. (1982)
conducted runs of manually-controlled models in stern
quartering waves in seakeeping and manoeuvring basins,and identified broaching associated with surf-riding for a
destroyer and a high-speed fishing craft, respectively. Kan
et al. (1990) and Umeda et al. (1999A) executed
free-running model experiments with auto pilots in stern
quartering waves in seakeeping and manoeuvring basins,
and recorded time histories of capsizing due to broachingfor container ships and fishing vessels. In Umeda’s
experiment, it was confirmed that even a ship complying
with the current IMO IS code can capsize as a result of
broaching.
Davidson (1948) investigated directional stability infollowing waves by using a linear sway-yaw coupled model.
Here he estimated wave-induced forces as the sum of the
Froude-Krylov component and the hydrodynamic lift due towave particle velocity. As a result, it was confirmed thateven a directionally stable ship can be directionally unstable
in a wave downslope. Wahab and Swaan (1964) pursued
this approach further but with the Froude Krylov component
on its own. Eda (1972) applied a coupled surge-sway-yaw
model to this linear stability problem. Hamamoto (1973)
and Renilson (1982) investigated the wave effect on linear
manoeuvring coefficients theoretically and experimentally.
Fujino et al. (1983) improved accuracy in prediction ability

Surf-riding is regarded as a prerequisite of broaching.
Thus it is very important to estimate the surf-riding
threshold. Grim (1951) explained that surf-riding boundary
coincides with the case that a trajectory from an unstableequilibrium of an uncoupled surge model on a wave isconnected to another unstable equilibrium. This is aheteroclinic bifurcation in nonlinear dynamical systemtheory. Makov (1969) demonstrated the validity of Grim’s
statement by using a phase plane analysis. Ananiev (1966)
obtained an analytically approximated solution by applying
a perturbation technique. Umeda and Renilson (1992)
extended the nonlinear dynamical system approach from an
uncoupled surge model to a coupled surge-sway-yaw model.
Spyrou (1995, 1996, 1997) and Umeda (1999) numericallyobtained the heteroclinic bifurcation for an uncoupled surge
model and the coupled surge-sway-yaw-roll model with a
PD autopilot, respectively. Umeda and Vassalos (1996)
obtained the region of unstable periodic motion in stern
quartering waves by applying an averaging method. Spyrou(2001) presented an exact analytical solution of heteroclinic
bifurcation of uncoupled surge model.
The above researches deal with broaching in regularwaves only. Although Rutgerssen and Ottosson (1987) andMotora et al. (1982) executed model experiments inirregular waves in a seakeeping and manoeuvring basin andfull scale measurement at actual sea, respectively, no
probabilistic study on broaching had been reported. Umeda(1990), however, proposed a theoretical method for
calculating the surf-riding probability in irregular waves.
Here the surf-riding probability is calculated with thedeterministic surf-riding threshold and the joint probabilityof local wave height and wave period.

for manoeuvring coefficient in waves with free surface
effects taken into account.
Motora et al. (1982) and Renilson (1982) numericallyintegrated nonlinear equation of surge-sway-yaw motion
and concluded that necessary condition of broaching is thatwave-induced yaw moment exceeds the maximum yaw
moment due to rudder. Umeda and Renilson (1992 and1994) developed a 4 DOF mathematical model based on a
manoeuvring model with a linear wave forces. Assumingthat wave steepness and manoeuvring motions are small, allhigher-order terms, such as interactions due to manoeuvring
and waves, are consistently ignored. Umeda and Hashimoto(2002) applied this model to explain the free-running modelexperiments (Umeda et al., 1999A) so that qualitative
agreement was confirmed. The ITTC specialist committee
on extreme motions and capsizing executed a benchmark
testing study of numerical models with the free-runningmodel test data by Umeda et al. (1999A), and concluded
that some numerical codes can predict the occurrence of
broaching qualitatively (Umeda and Renilson, 2001).
For realizing quantitative prediction, it is necessary to
improve the mathematical model by adding higher orderterms. Umeda et al. (2003) and Hashimoto et al. (2004A)
developed a mathematical model with second-order terms
taken into account, and reported that quantitative prediction
is realized. Here hydrodynamic forces due to interaction
between manoeuvring and waves, hydrodynamic forces due
to large roll angle, nonlinear wave and manoeuvring forces
and so on are estimated with potential theories or captivemodel experiments. Hashimoto et al. (2004B) utilised
fully-captive model experiments in waves with apurpose-built apparatus.

original
model
first
order
model

present model
0.5

nominal Froude number

nominal Froude number

0.5
0.4
0.3
0.2
0.1
0

15

30

45

0.4
0.3
0.2
0.1

auto pilot course (degrees)capsizing due to broaching
capsizing without broachingbroaching without capsizing

stable surf-riding
periodic motion
not identified

0

15

30

45

auto pilot course (degrees)
EXP.(capsizing due to broaching)
EXP.(capsizing without broaching)
EXP.(near stable surf-riding)
EXP.(periodic motion)

Figure 1 Comparison between the experimental results and numericalresults (first order model and present model) for the ITTC Ship A-2 withwave steepness of 0.1 and wavelength to ship length ratio of 1.637(Umeda and Hashimoto, 2006C).

-0.30Roll Moment (kg.f.m)

0.00

0.10

0.20

0.30

0.40

0.50

0.60

-0.35

CFD
EFD

-0.40

-0.45

-0.50

Fn

Figure 2

Comparison between the physical model experiment (EFD) and the RANS solution (CFD) in roll moment
acting on the ONR tumblehome model running in still water with the heel angle of 10 degrees (Hashimoto andStern, 2007).

0.50
capsize without broaching

Nominal Froude Number

0.45
0.40

stable surf-riding

0.35
periodic motion
0.30
surfriding with oscillation

0.25
0.20

not identified

0.15
by heteroclinic bifurcation

0.10
0

5

10

15

20

25

30

35

40

Auto Pilot Course (degrees)

Figure 3

Comparison in motions of the ITTC Ship A-2 running in following waves between numerical simulation
results and the heteroclinic bifurcation (Umeda et al., 2006A).

To avoid broaching, it is effective to increase ship dragby some means. Renilson (1986) investigated the Sea Brake,
which is a kind of sea anchor to add drag to ship resistance.
Umeda et al. (1999B) invented the Anti-Broaching Steering
System (ABSS), in which the rudder deflection limit isallowed to be a large angle, such as 70 degrees, so that therudder drag can deteriorate surf-riding equilibrium under theyaw deviation from the desired course, and successfully
validated it with free-running model experiments. Further,
Umeda et al. (2002) optimised the design of the ABSS byusing numerical simulations.

OUTCOMES FROM SCAPE ACTIVITIESThe two-year activities of the SCAPE committee output
the following fruits for broaching.
Nonlinear modelling
Although Hashimoto et al. (2004A) realised quantitative
prediction by adding many higher-order terms, it isimportant to provide guidelines for simpler but accurateenough modelling. For this purpose Hashimoto and Umeda(2005), Umeda and Hashimoto (2006C) recommended to
add the following three terms to their first-order model as aresult of systematically degrading study.
- nonlinearity of wave-induced surge force,- nonlinearity of sway-roll coupling,- nonlinearity of roll-induced moment in calm water.
Here captive tests in a towing tank are required but not in aseakeeping and manoeuvring basin. Agreement between therecommended mathematical model and the free-runningmodel experiment by Umeda et al. (1999A) is satisfactory as
shown in Figure 1
The above mathematical modelling is developed forsingle-screw and single-rudder ships, while other ships arealso susceptible for broaching. They include twin-screw andtwin-rudder ships as shown in model experiments andwater-jet propelled craft. For twin-screw and twin-rudder
ships Umeda et al. (2006B) proposed a mathematical model
as an extension of the above model and then validated with
Lloyd’s experiment (Renilson, 1982). Furthermore, for a
craft propelled by water-jet Umeda et al. (2005A) provides amathematical model.

For making captive tests easier to obtain some higher
order terms, Matsuda et al. (2007) developed a new captive
testing technique, which allows heave and pitch motions. To
replace such model experiment with numerical calculation,
Hashimoto and Stern (2007) applied a RANS-based CFD
code to heel-induced hydrodynamic forces and showed agood agreement with the model experiment, as shown in
Figure 2.
Bifurcation analysis
It was already established that the critical condition forsurf-riding in regular following and quartering waves is aheteroclinic (or homoclinic) bifurcation. Next step is to
directly evaluate such bifurcation point from mathematicalmodels, especially a 4 DOF model. Hori and Umeda. (2005)
and Umeda et al. (2006A) proposed a theoreticalformulation of this bifurcation problem with the coupledsurge-sway-yaw-roll mathematical model and well validatedwith the direct initial-value simulation as shown in Figure 3.
Here the critical Froude number for surf-riding is determined
for requesting the unstable invariant manifold from a saddle
to reach another saddle within the framework of the Newton
method. Thus the repetition of numerical simulation formany different initial and control values can be avoided.
Above this surf-riding threshold, capsizing due to broachingoccurs if wave steepness is very high. Maki et al. (2007)
further improved this methodology, focusing to an
uncoupled surge model in regular following waves, fromtheoretical and numerical viewpoints so that deterministic
surf-riding threshold hyperplane could be efficiently
estimated.

validated with the Monte Carlo simulation as shown in
Figure 4.
Optimal control theory
In the above theoretical and experimental studies, steering
is modelled as a PD auto pilot. The maximum steering effort,
however, could be different from a PD auto pilot. Therefore,
Maki and Umeda (2006) proposed the application of optimal
control theory. When it is applied, a kind of the bang-bang
control could be obtained as maximum steering effort toprevent broaching.
Prevention device
Other than the Sea brake and the ABSS, a new device forpreventing broaching was proposed by Hashimoto et al.
(2007B). It is a pair of wings attached to bow above a stillwater plane. Once broaching is initiated, a ship deviatedfrom the desired course. This turning results in outward roll.
The wing in the side of roll will be submerged so that yaw
moment towards desired course and the roll moment towards
upright position can be generated. As a result, broaching can
be avoided. The effectiveness of this device was validatedwith a free-running model experiment.

RECOMMENDATION FOR IMO NEW GENERATIONCRITERIA
The IMO requests both the vulnerability criteria and the
direct assessment for broaching, following the framework
proposed by Japan, the Netherlands and the United States(2007). Responding to this requirement, the SCAPE
committee recommends the following draft criteria.

1broaching probability

0.9

theory

0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0

numerical
exp(no.1)

numerical
exp(no.2)

numerical
exp(no.3)

numerical
exp(no.4)

numerical
exp(no.5)

Figure 4 Comparison in broaching probability between
numerical experiments and theory for ahigh-speed fishing craft in irregular stern
quartering waves (Umeda et al., 2007).
Probabilistic assessment
To evaluate danger of broaching in actual seaways, it isessential to evaluate broaching probability or probability ofcapsizing due to broaching in irregular waves. Umeda et al.
(2007) proposed a theoretical calculation method forbroaching probability as an extension of Umeda’s theory forsurf-riding probability (Umeda, 1990). Here the broachingprobability is calculated by integrating the joint probability
density of local wave height and wave period, which was
proposed by Longuet-Higgins (1983), within thedeterministic broaching region. The present theory was well

Vulnerability criterion
It is important that a vulnerability criterion can be easily
used, can guarantee conservative safety level and should bebased on non-empirical approach. The surf-riding thresholdin regular following waves can be substituted to that ofcapsizing due to broaching because surf-riding is aprerequisite to broaching and following waves are mostdangerous for surf-riding. To determine the surf-riding
threshold in regular following waves, an analytical solution
is obviously most suitable because it is easily evaluated but
is still based on theoretical background. Spyrou (2001)
proposed an analytical solution by assuming the balance ofresistance and propeller thrust to be proportional to square ofship velocity. Maki and Umeda (2008) utilised a piece-wiselinear approximation of wave-induced surge force and thendifferent analytical formula for surf-riding threshold in
regular following waves was obtained. Here the balance ofresistance and thrust is approximated to have a linear
relationship with ship velocity. In this procedure, ship
resistance in calm water, propeller thrust and distribution ofsectional hull area are required. They executed comparisonsamong the two analytical formulae, numerically-obtained
bifurcation points and a free-running model experiment sothat they concluded that Maki and Umeda’s analyticalsolution is more accurate than Spyrou’s and almost identicalto the numerically-obtained bifurcation points and theexperiment, as shown in Figure 5.

Direct assessment
If a ship design fails to comply with the vulnerability
criterion, it is expected to apply direct stability assessment tothe subject ship design. It is preferably to evaluate risk levelof the ship design. For this purpose, it is necessary toevaluate probability of capsizing due to broaching. The
SCAPE committee already developed a theoreticalprocedure for broaching probability in irregular stern
quartering waves (Umeda et al., 2007) and an extension ofthis method to probability due to broaching isstraight-forward. First, we should identify deterministic
thresholds for capsizing due to broaching as a function ofwave height and period by heteroclinic bifurcation analysisof the coupled surge-sway-yaw-roll model with an auto pilot
or the repetition of numerical simulation with the samemodel. Second, the probability of capsizing due to broachingis calculated by integrating the joint probability density of
local wave height and wave period, which was proposed by
Longuet-Higgins (1983), within the deterministic capsizing
region mentioned before. In this procedure, ship resistancein calm water, propeller thrust, hull form offset, calm-water
manoeuvring coefficients, roll damping coefficient andrestoring moment are required at least.

0.5
0.45
0.4
0.35

Fn

0.3
0.25

Numerical Bifurcation Analysis
Spyrou's Formula

0.2

Present Method

0.15
0.1
0.05
0
1

1.2

1.4

1.6

1.8

2

λ/L

Figure 5 The surf-riding thresholds in following waves obtained by model experiments, numerical bifurcation analysis andanalytical bifurcation analysis for the ITTC Ship A-2 with the wave steepness of 0.08 (Maki and Umeda, 2008).

CONCLUDING REMARKS
The activity of the SCAPE committee for broachingprovides the following concluding remarks:
1. For quantitative prediction in regular waves, anenhanced system-based model is proposed andvalidated with the free-running model experiments.
2. For estimating the deterministic threshold of surf-riding
in stern quartering waves as a heteroclinic bifurcation,an equation set was theoretically formulated and wasvalidated with the direct numerical simulation.
3. For calculating broaching probability in irregularwaves, a theory was formulated and was validated with
the Monte Carlo simulation.
4. A device for preventing broaching was proposed andwas validated with model experiments.

ACKNOWLEDGEMENTS
This work was supported by Grant-in Aids for
Scientific Research of the Japan Society for Promotion of
Science (Nos. 18360415, 18360426 and 18760619) 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 activity of
SPL project of Japan Ship Technology Research Association,funded by the Nippon Foundation. The authors express theirsincere gratitude to the above organisations.
The authors thank Prof. F. Stern, Messrs. G. Sakamoto,
M. Shuto and S. Sanya for their contribution to the works
described in this paper.

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