1

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

2

研究組織

氏名 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

委員長 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 2007 年 11 月～

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

会計担当 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

3

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 of Multi-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.

4

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.

5

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 and its 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 the capsizing 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 9th International 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 of a 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 parametric roll 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 of local 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 critical condition 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 of Experiment-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.

6

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 Society of 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 Moment in 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.

7

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 and Role 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 Water on 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.

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

ABSTRACT

This 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) and relevant 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 new generation 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 most fundamental requirements for ship design and operation. Stability criteria of intact stability are enforced by each administration: in Japan the ship stability standard came into force in 1957 and since then no accident due to the lack of intact stability has been reported for passenger ships built with 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 not mandatory 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 mega containerships occasionally suffer parametric rolling resulting in container damage. Thus it is expected to develop rational and mandatory criteria in place of the current recommendatory ones for ensuring internationally consistent 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 five sessions to discuss this matter and then in October 2007 the MSC (Maritime Safety Committee) approved the 2008 IS (Intact Stability) Code, a part of which is to be used as a mandatory 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 is because 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 from the 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 organized sessions 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, this paper attempts to overview activities of the SCAPE committee as well as current problems of the IMO and to present 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 correspond to the GZ curve criteria and the weather criterion, respectively. In 1998, further refinements were made as the resolution 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 the Code, these are mandatory for obtaining the IACS class certificates.

As explained earlier, the IMO started to consider the comprehensive revision of the IS Code at the SLF45 in 2002. Its working group was established under the chairmanship 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 the United States (2002) reported that a post-Panamax containership complying with the IS Code suffered cargo damage due to parametric rolling in head waves. Then Germany, Denmark and Australia proposed to replace the weather criterion with performance-based criteria allowing the use of model experiments and numerical simulation. Italy proposed to modify the empirical coefficients in the weather 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 of conventional ships should be kept. Japan explained that this is because medium-sized passenger ships occasionally suffer dead ship conditions, which result in beam wind and wave condition without forward velocities, due to freak-wave-induced damage to their wheel houses and current weather criterion well ensures stability under such circumstances. Then Japan concluded that preventing capsizing under such dead ship condition has been well realised by the IMO with its weather criterion and is one of fundamental roles for the IMO in the future. This statement obtained the unanimous support from the participants. Then the IMO agreed that a stability criterion for dead ship condition 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. Based on this result, Japan, together with Italy, proposed to allow the use of model experiments for wind-heeling lever and roll 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 with model test guidelines.

At the SLF47 in 2004, Italy and Japan (2004) jointly submitted the draft revision of weather criterion: the draft model test guidelines on wind heeling lever and roll back angle were developed by National Maritime Research Institute and Osaka University, respectively. Italy also developed the draft on roll back angle with the experience of Trieste University. Then the SLF decided to continue to further 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, this revision 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 draft of restructuring the IS Code was developed and was tabled by the intersessional correspondence group coordinated by Mr. C. Mains at the SLF49 in 2006. It consists of three parts: Part A includes the GZ curve criterion and the weather 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 the Code. 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 that session.

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 the area of righting arm curve up to smaller roll angle. Since these are empirical criteria, it is very difficult to rationally revise 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 the 2008 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 the long-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 as parametric 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 group for 1) and Dr. N. Umeda was nominated as the coordinators of 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 well in irregular waves. Second, a piece-wise linear approach seems to be promising for quantifying capsizing probability under dead ship condition. Third, a bifurcation theory can efficiently predict thresholds of surf-riding, which is a prerequisite of broaching, in regular following waves. Germany (2006) systematically conducted numerical simulation in irregular waves for many ships and then proposed a new empirical criterion for the stability variation problem. Italy (2005) proposed a calculation method of capsizing probability based on a linearization procedure that takes 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 new generation 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 equivalent means should be applied. Germany (2007) updated the proposal of a new empirical criterion for all ships and proposed not to use ship-specific operational guidance. As a result 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 framework agreed 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 safety including intact stability, damage stability and fire protection is systematically investigated by several research organisations and universities in Europe. Its ultimate goal is set 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 with probabilistic analyses on Gaussian waves.

As a world-wide independent association of research organisations on ship hydrodynamics, the ITTC established a 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 executed

benchmark testing of numerical codes on intact stability, covering parametric rolling and broaching, by utilising the capsizing model experiments in Japan. (The Specialist Committee on Prediction of Extreme Ship Motions and Capsizing, 2002; The Specialist Committee on Stability in Waves, 2005) These standardisation efforts will be crucial for implementation of the new generation intact stability criteria.

ESTABLISHMENT OF SCAPE ACTIVITIES

In 2005, three learned societies in Japan were merged into a new society named the Japan Society of Naval Architects and Ocean Engineers (JASNAOE). Under the chairmanship of Prof. S. Naito, the JASNAOE decided to have research committees for specified topics and specified terms. The SCAPE committee is one of the research committees that the JASNAOE established in 2005. The purpose of this committee is to contribute to the consideration of new generation intact stability criteria from researchers’ viewpoint. It could help the activities of the Stability 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 to contribute not only to the documents from the Japanese delegation 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 established with members from universities, research institutes and industries 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 Osaka University till Feb. 2007 T. Fujiwara Osaka Prefecture University from Dec.

2007 H. Hashimoto Osaka University Secretary M. Hori Osaka University till Mar. 2006Y. Ikeda Osaka Prefecture University T. Iseki Tokyo Maritime University S. Ishida National Maritime Research

Institute

T. Katayama Osaka Prefecture University N. Kogiso Osaka Prefecture University from Nov.

2007 A. Maki Osaka University from Nov.

2007 A. Matsuda National Research Institute of

Fisheries Engineering

T. Momoki National Research Institute of Fisheries Engineering

from May 2006

A. Munif Osaka Prefecture University till Feb. 2006 Y. Ogawa National Maritime Research

Institute

D. Paroka Osaka University till Mar. 2007H. Taguchi National Maritime Research

Institute

N. Toki Mitsubishi Heavy Industries N. Umeda Osaka University Chairman

OUTCOMES FROM THE ACTIVITIES OF SCAPE COMMITTEE

The SCAPE committee published 43 conference papers at the spring and autumn meetings of the JASNAOE and many papers in the refereed journals and international conferences. Based on these, the outcomes from the SCAPE committee are overviewed. More details can be found in companion reports of this report for individual topics.

For parametric rolling, free-running model experiments of a post-Panamax containership in head and bow waves in the seakeeping and manoeuvring basins of National Research Institute of Fisheries Engineering and National Maritime Research Institute and observed parametric rolling with the roll angle exceeding 20 degrees. It was found that maximum roll angles are almost the same among regular waves, long-crested irregular waves and short-crested irregular waves. Regarding the restoring variation due to waves 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 radiation and diffraction components as well. For the mean of the restoring variation, a formula using a momentum law was developed. These results were compared with the captive model experiments and then better agreements than the existing approach were reported. Taking these aspects into account, numerical simulation of parametric rolling in head waves was executed and compared with the fore-mentioned experiments. While good agreements for the case in moderate regular waves between the calculation and the experiments were reported, some discrepancies can be found for the cases in severe regular waves and irregular waves. For investigating this reason, direct measurement of restoring variation acting on scaled models in irregular waves were executed. It was found, however, that rather acceptable agreement exists also in irregular waves. On the other hand, if we repeat many realisations in model experiments and numerical simulations in irregular waves under the same wave parameters, the scatter of the roll angles 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 of restoring variation exceeds a certain threshold. Thus it is neither Gaussian nor practically-ergodic. Therefore, it is extremely difficult in principle to obtain good agreement among individual realisations so that it is presumed the ensemble average should be used for this comparison. Time-Varying coefficient Vector Auto-Regresive (TVVAR) method, bispectrum analysis and trispectrum analysis were applied to parametric rolling of the post-Panamax containership in head waves for clarifying the nonlinear nature of parametric rolling. Furthermore, model experiment and numerical simulation study were carried out also for a latest PCTC, which is similar to one suffered parametric rolling in the Northern Atlantic, and its parametric rolling in regular and irregular head waves were identified (Hashimoto et al., 2008).

For stability under dead ship condition, a method for calculating capsizing probability in irregular beam wind and waves with Belenky’s piece-wise linear approach (Belenky, 1993) were intensively investigated. Here the restoring arm curve is approximated with piece-wise linear curves and then capsizing probability can be analytically calculated.

Comparing this method with the Monte Carlo simulation, some discrepancies exist in extremely severe sea state. This is 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, acceptable agreement 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 for fishing vessels, and effect of critical angle for down-flooding or cargo shift were taken into account in the calculation 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 a result, 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 not always 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 parametric rolling occurs during model experiments of a large passenger ship without the bilge keels drifting in beam waves. The reason of this is presumed by numerical simulation 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 of surf-riding, which is a prerequisite to broaching. In case of regular following waves, its prediction methods with an uncoupled surge model were already established. In case of regular stern quartering waves, it is essential to take coupling with manoeuvring motions into account. Therefore, a global bifurcation analysis was applied to a surge-sway-yaw-roll mathematical model with a PD auto pilot. As a result, the critical ship speed for surf-riding in stern 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 by systematic 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 the deterministic broaching conditions in regular waves was developed and is well validated with the Monte Carlo simulation (Umeda et al., 2008).

For preventing the above phenomena, devices were proposed and were successfully validated with model experiments. 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 the usefulness of the interim guidelines including its effect on stability assessment (Ogawa et al., 2008). A model testing technique for evaluating amount of trapped water on deck in beam 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 them were reported.

As mentioned above, the outcomes from the SCAPE committee already contributed not only to research progress in ship dynamics but also to the proposals from the Japanese delegation 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 under development.

RECOMMENDATION 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 to unconventional 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, we should bear in mind the fact that most of conventional ships under the current criteria are safely operated and that the current intact stability criteria are quite reliable at least from statistical viewpoint. It is noteworthy that such sufficient safety level also does realised with appropriate operation and it is unrealistic to separate design criteria from operation 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 the criterion 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,

a simplified prediction formula for metacentric height on wave crest is desirable.

It might be practical to use numerical simulations for the performance-based criteria for ships which fail to pass the vulnerability 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 and waves with sufficient statistical confidence and to validate the proposed theory with them. For parametric rolling, it seems to be necessary to calculate ensemble average of many numerical realisations. The first order reliability method (FORM) may provide a way to make this procedure more efficient. (Kogiso & Murotsu, 2008) In addition, the effect of time-varying resistance increase in irregular waves may be important and its effect is now under investigation.

CONCLUDING REMARKS

Responding to the demands of research for new generation intact stability criteria at the IMO, the JASNAOE established the SCAPE committee in 2005 and the 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 was partly carried out as a research activity of Stability Project of 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 Science and 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, (in press).

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, JAPAN

ABSTRACT

Regarding parametric rolling of ships with forward velocity, this paper reports experimental, numerical and analytical studies conducted by the SCAPE committee, together with critical review of theoretical progress on this phenomenon. Here experimental results of a containership 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 compared with several numerical predictions to realize quantitative prediction of parametric rolling. The restoring variation, which is main cause of parametric rolling, is investigated with model experiments and potential theories. Application of time-varying coefficient vector autoregressive model is also attempted. Furthermore, the effect of devices preventing parametric rolling is investigated. Based on these outcomes, the SCAPE committee also proposes draft stability criteria consisting of vulnerability criterion and direct 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 is multiple 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 the underwater hull volume change as the waves pass along the ship. 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 for modern containerships and PCTCs have been reported in recent years. These accidents triggered off a review of the Intact 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 three major capsizing scenarios including parametric rolling as one of roll restoring variation problems. In this stage, a prediction method for parametric rolling with quantitative accuracy is required. For this purpose, all relevant factors of parametric rolling should be systematically investigated to develop a standard numerical simulation technique for the use 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 a container 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 to describe 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 this purpose, prediction of parametric rolling is required to have quantitative accuracy and identify all potential danger both in regular and irregular waves.

Theoretical studies on parametric rolling can be found in the thirties with linear restoring (Watanabe, 1934). Later, in the fifties, linear and nonlinear damping was taken into account (Kerwin, 1955). These studies enable us to discuss parametric rolling with the Mathieu equation. Then, to investigate capsizing, nonlinearity of restoring moment in still water was taken into account. At this stage, nonlinear dynamical system approach including geometrical and analytical 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 et al. (2004) for uncoupled model with realistic modelling of roll restoring variation.

On the other hand, several six degrees-of-freedom models such as Hamamoto and Akiyoshi (1988), De Kat and Paulling (1989), Munif and Umeda. (2000), Matsusiak (2001) have been developed for numerical prediction in time 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 the limited number of initial condition sets, however, there is a possibility to overlook some potential danger because of nonlinearity of the system. In addition, the minimum requirements for the numerical modelling were not established 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 et al., 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 were reported; Ribeiro et al. (2003), Belenky et al. (2003), Neves and Rodriguez (2005), Brunswig et al. (2006), Spanos and Papanikolaou (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 and derived the Fokker-Plank equation as a prerequisite for existence of stationary process of roll. Bulian et al. (2003) experimentally investigated the applicability of the Markov process approach for a Ro-Pax, and confirmed the agreement 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 of parametric rolling in irregular waves. Bulian and Francescutto (2006) proposed a fully analytical approach for the determination of the stochastic stability threshold of parametric rolling, and it overestimates roll motion without a tuning factor. Themelis and Spyrou (2006) proposed a probabilistic assessment of parametric rolling by assuming probability of occurrence of instability is represented by that of encountering the critical or worse wave groups. Belenky et al. (2003) and Belenky (2006) simulated parametric rolling in irregular waves with 50 realisations and numerically 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 parametric rolling. 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 also hydrodynamic 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) with a 1/100 scaled model and NMRI (National Maritime Research Institute) with a 1/76.7 scaled model. This ship is designed by NMRI and her principal particulars and body plan are shown in Table 1 and Fig. 1, respectively. Model experiments were systematically conducted in regular waves, 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 of parametric roll does not always increase with wave steepness. Taguchi et al. (2006A) investigated the effect of initial steady heel on parametric rolling in regular head waves, 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 some extent. 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 with significant wave height and decreases with deviation from pure head wave condition. Effect of ship speed on parametric 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, roll angles both in long-crested and short-crested irregular waves are slightly smaller than that in regular seas. In the region of Froude number greater than 0.1, small parametric

rolling in irregular waves occurs while no parametric roll occurs 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 is experimentally confirmed even in short-crested irregular waves, and those in regular and long-crested irregular waves are more remarkable. Table.1 Principal particulars of the post-Panamax containership

Fig. 1 Body plan of the post-Panamax container ship

0

5

10

15

20

25

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08

wave steepness

ampl

itude

of p

aram

etric

rolli

ng (d

eg.)

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)

χ=180°

0

5

10

15

20

25

0.30 0.40 0.50 0.60 0.70

Te/Tφ

φm

(deg.

)

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)

Fig. 3 Influence of initial steady heel angle on parametric

rolling in regular head waves with Η =11cm (Taguchi et al., 2006A)

T02=1.38s, Fns=0.07

0

5

10

15

20

25

0 5 10 15 20 25

H1/3 (cm)

φ(d

eg.

)

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

Max.（χ=180°）

　〃 （χ=150°）

Fig. 4 Influence of significant wave height on parametric

rolling in long-crested irregular head waves with T02=1.38sec. (Taguchi et al., 2006B)

T02=1.38s, H1/3=12.3cm

0

5

10

15

20

25

135 150 165 180

χ (deg.)

φ(d

eg.

)

1/10　heighest mean （Fns=0.07）

　　　　 　 〃　 　（Fns=0.11）

Max.（Fns=0.07）

　〃　(Fns=0.11)

Fig. 5 Influence of encounter angle on parametric rolling

in long-crested irregular waves with H1/3=12.3cm and T02=1.38 sec. (Taguchi et al., 2006B)

0

5

10

15

20

25

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35Froude number

ampl

itude

of p

aram

etric

rolli

ng (d

eg.)

regular waves

long-crested waves

short-crested waves(cos^4)

short-crested waves(cos^2)

Fig. 6 Influence of ship speed on parametric rolling in

regular and irregular head waves (Hashimoto et al., 2006A)

-30 -20 -10 0 10 20 300

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

elevation of roll angle (degrees)

prob

abili

ty d

ensi

ty

-20 -10 0 10 200

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

elevation of pitch angle (degrees)

prob

abili

ty d

ensi

ty

Fig. 7 Probability density functions of instantaneous roll and pitch angle in short-crested irregular waves (histogram:

experiment; line: Gaussian distribution) (Hashimoto et al., 2006A)

Free running model experiment for a latest PCTC was conducted at Osaka University. Principal particulars are shown in Table.2, and its 1/64 scaled model was used in the experiment. 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

Items Ship NRIFE NMRIlength between perpendiculars: L 283.8m 2.838m 3.700mbreadth: B 42.8m 0.428m 0.558mdepth: D 24.0m 0.24m 0.318mdraught: T 14.0m 0.14m 0.183mblock coefficient: Cb 0.630 0.630 0.630pitch radius of gyration: Kyy/Lpp 0.258 0.247metacentric height: GM 1.06m 0.0106m 0.014mnatural roll period: Tφ 30.3s 3.20s 3.460s

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 parametric rolling for PCTCs are desirable. Table.2 Principal particulars of the pure car and truck carrier

item value length: L 192.0 m breadth: B 32.26 m depth: D 37.0 m mean draught: T 8.18 m block coefficient:Cb 0.54 metacentric height: GM 1.25 m natural roll period: Tφ 22.0 s

parametric rolling in regular head sea

H=0.05 (m)

0

5

10

15

20

25

30

0 0.05 0.1 0.15 0.2 0.25 0.3Fn

max

. rol

l am

plitu

de (d

egre

e)

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

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 constant heel angle of 10 degrees in regular head and following waves at a towing tank and the reaction moment were detected as the restoring moment (Umeda and Hashimoto, 2006; Nakamura et al., 2007). Here the models were free in heave and pitch. Nakamura et al. (2007) conducted a comparative study on the mean and amplitude of metacentric 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 first and second harmonic components. Amplitude of first harmonic GM variation of the post-Panamax containership is much larger than that of the ITTC containership. This is because 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 was found. 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 the effect of surge motion restraint on restoring variation is negligibly small.

Since the estimation of roll restoring variation is important for quantitative parametric roll prediction, a numerical estimateion of the roll restoring variation based on 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 by integrating the wave pressure up to wave surface with taking instantaneous heave and pitch motions calculated by a 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 roll restoring 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 the hydrodynamic components improves the prediction accuracy of restoring variation. This means that parametric roll danger could be underestimated if these components are neglected in the modelling. Fig. 14 shows the comparison of probability density function of GZ between the experiment and 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 is directly used in the calculation. As a result, the amplitude of roll restoring variation in long-crested irregular waves can be 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 of parametric 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 slender body theory. Here steady roll moment is expressed by Kochin function. Comparison of the calculated result with the captive model experiment (Umeda et al., 2005) is shown in Fig. 15. Steady roll moment in head waves decreases with increase of Froude number. From the comparison with captive mode experiment, the calculation can estimate its trend with respect to wave steepness but underestimates the experimental result particularly in high Froude number. This might be because a hydrodynamic lift effect on heeled hull in high speed region becomes significant.

Fig. 9 Body plan of the ITTC ShipA-1 containership

0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0 0.01 0.02 0.03 0.04 0.05 0.06

wave steepness

GM

1st

am

plitu

de/L

pp

ITTCpost-Panamax

Fig. 10 Amplitude of 1st harmonic GM variation at

Fn=0.1 and heel angle of 10deg. (Nakamura et al., 2007)

0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0 0.01 0.02 0.03 0.04 0.05 0.06

wave steepness

GM

mea

n/Lp

p

ITTCpost-Panamax

Fig. 11 Mean of GM variation at Fn=0.1 and heel angle of

10deg. (Nakamura et al., 2007)

Heel 10deg (λ/L=1.6)

00.0010.0020.0030.0040.0050.0060.0070.0080.009

0.01

0 0.005 0.01 0.015 0.02 0.025H/λ

GM

am

plitu

de/L

pp

Fn=0.2 Fn=0.1Fn=0.05 Fn=0Fn=0.2(Surge free) Fn=0.1(Surge free)Fn=0.05(Surge free) Fn=0(Surge free)

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

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0 0.01 0.02 0.03 0.04 0.05 0.06

wave steepness

GM

1st

am

plitu

de/L

pp

expFK + R&DFK

Fig. 13 Comparison of amplitude of 1st harmonic GM

variation at Fn=0.1 and heel angle of 10 deg. (Nakamura et al., 2007)

Fig. 14 PDF of restoring arm in long-crested irregular

waves with H1/3=0.057m and T01=1.35sec. at Fn=0.1 (Hashimoto et al., 2006B)

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

0

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

0.009

0.01

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

(Ave

rage

of G

M v

aria

tion)

/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.)

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)

Numerical Prediction of Parametric Rolling

Numerical models for parametric roll prediction should be validated not only in regular waves but also in irregular waves because the relation between roll restoring variation and wave height is inherently nonlinear. Therefore the Grim’s effective wave concept (Grim, 1961) was adopted firstly for the prediction of parametric rolling in irregular waves. The idea of the effective wave concept is that irregular wave surface around a ship is replaced with equivalent regular wave, which length is equal to ship length, by a least square method. By introducing the effective wave, the relationship between the ocean wave and the effective wave amplitude becomes linear so that the restoring variation can be calculated by a linear superposition with nonlinear but non-memory relationship between 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 by hydrostatic balance for an instantaneous effective wave, radiation and diffraction components calculated under the assumption that it has a linear relation with wave steepness and roll angle. By utilising the above modelling, uncoupled roll 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 is assumed to follow the ITTC spectrum and is expressed as the sum of 1000 components of sinusoidal waves with their random phases. Fig. 16 shows the comparison of maximum roll angle of parametric rolling of a post-Panamax containership between free running model experiment and numerical simulation (Hashimoto et al., 2006A). Both calculations excessively overestimate the maximum roll angle of parametric rolling. Bulian et al. (2006) modified Grim’s effective wave concept to improve the

approximation accuracy of random wave surface by adjusting its wave crest position as an additional random variable. This modification (named Improved Grim Effective Wave, IGEW) improves prediction of the pitch moment and/or static pitch equilibrium angle, while the roll moment 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 and modelled as a function of Froude number. Taguchi et al. (2006C) reported that modelling with such Froude-Krylov prediction improves the agreement in comparison with the experimental result for the post-Panamax containership in moderate regular head waves as shown in Fig. 17. By utilising the Froude-Krylov component with dynamic heave and 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. Numerical prediction still overestimates maximum roll angle in some cases. Since some discrepancies with the measured restoring variation are found as well, it is still required to improve the estimation 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 shown in Fig. 20. These results indicate that small difference of the estimation of roll restoring variation could result in significant difference of maximum angle of parametric roll in irregular waves. Fig. 21 shows the measured and calculated maximum roll angles of parametric roll for 4 different realisations. Practical non-ergodicity of parametric roll is confirmed both in experiment and calculation. Therefore, how to overcome this practical non-ergodicity of parametric rolling in irregular waves is important for the validation of the code and practical stability assessment. Bulian et al. (2008) investigated this problem further based on 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 as well as the roll restoring. The wave exciting force is calculated by solving the Helmholtz equation, which allows shorter waves. (Ogawa et al., 2007A) Ogawa (2007) conducted numerical simulations in long-crested irregular waves for the post-Panamax containership. In the calculation, irregular waves are expressed as a superposition of 200 sinusoidal waves following the ISSC1964 spectrum. Fig. 22 shows the comparison of the amplitude of parametric rolling in regular head waves between free running model experiment and numerical prediction. The calculation well explains the experimental result of 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 roll angle. It is expected here to identify the crucial factors in this modelling to realise the agreement with model experiments. Furthermore, the application of a CFD approach based on the RANS equation is now underway with the collaboration between Osaka University and the University of Iowa. The nonlinear strip method can predict parametric rolling both in regular and irregular waves with practical accuracy as mentioned above. By utilising this numerical model, horizontal and vertical accelerations during parametric rolling were calculated (Ogawa et al., 2007B). The results, shown in Fig. 24, indicate the gravitational effect on horizontal acceleration acting on a secured container is dominant.

0

10

20

30

40

50

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4Froude number

max

. rol

l ang

le (d

egre

es)

FK only

present

experiment

Fig. 16 Comparison of maximum roll angle of parametric

rolling in long-crested irregular head waves with H1/3=0.221m and T01=1.32sec. (Hashimoto et al., 2006A)

λ/L=1.2

0.0

1.0

2.0

3.0

4.0

5.0

6.0

0.30 0.40 0.50 0.60Te/Tφ

s/kH

w

λ/L=1.6

0.0

1.0

2.0

3.0

4.0

5.0

6.0

0.30 0.40 0.50 0.60Te/Tφ

s/kH

w

λ/L=2.0

0.0

1.0

2.0

3.0

4.0

5.0

6.0

0.30 0.40 0.50 0.60Te/Tφ

s/kH

w

Fig. 17 Influence of encounter period on the amplitude of

parametric rolling in regular head waves with H=0.11m (Taguchi et al., 2006C)

parametric rolling in regular head seaH=0.05 (m)

0

5

10

15

20

25

30

35

40

0 0.05 0.1 0.15 0.2 0.25 0.3Fn

max

. rol

l am

plitu

de (d

egre

e) exp(λ/L=0.8)exp(λ/L=1.0)exp(λ/L=1.3)exp(λ/L=1.6)calc(λ/L=0.8)calc(λ/L=1.0)calc(λ/L=1.3)calc(λ/L=1.6)

Fig. 18 Comparison of maximum roll amplitude of

parametric rolling in regular head waves with H=0.05m (Hashimoto et al., 2007A)

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

0

5

10

15

20

25

30

35

0 0.05 0.1 0.15Fn

max

. rol

l ang

le (d

egre

ss)

cal. with FK on its owncal. with FK and hydrodynamic effectexperiment

Fig. 19 Comparison of maximum roll angle of parametric

rolling in long-crested irregular head waves with H1/3=0.083m and T01=1.22sec. (Hashimoto et al., 2007A)

Fig. 20 Comparison of GZ variation in long-crested irregular waves with H1/3=0.083m and T01=1.22sec.

(Hashimoto et al., 2007A)

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

0

5

10

15

20

25

1 2 3 4realizations

max

imum

roll

angl

e (d

egre

es)

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)

0

5

10

15

20

25

30

0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7Te/TΦ

Φ(d

eg.)

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)

Fig. 22 Comparison of steady amplitude of parametric

rolling in regular head waves with H=8.4m (Ogawa et al., 2007A)

Maximum and 1/10 highest mean of roll angle (χ=180°, T02=12.0sec.)

05

101520253035

0 2 4 6 8 10 12 14 16

H1/3 (m)

Φ(d

eg.)

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

Fig. 23 Comparison of maximum roll angle of parametric

rolling in long-crested irregular head waves with T02=12.0sec. (Ogawa, 2007)

Horizontal Acceleration

(X:S.S.8, Y: C.L., Z: 15m above W.L., Head seas, Fn=0.05)

05

101520253035404550

10 12 14 16 18 20

T(sec.)

aL/gζ

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

Vertical Acceleration (X:5.9m aft from S.S.5, Y: 15m from C.L., Z: 3.9mabove W.L., Head seas, Fn=0.05)

0123456789

10

10 12 14 16 18 20

T(sec.)

aL/gζ

Hw=0.1mHw=6.1mHw=8.4mInertia(Hw=6.1m)Inertia(Hw=8.4m)

Fig. 24 Effect of wave height on the response amplitude

operators of horizontal (above) and vertical (below) accelerations (Ogawa et al., 2007B)

Spectrum Analysis of Parametric Rolling To understand the nonlinear phenomenon of parametric rolling and identify its nature in frequency domain, Iseki (2006) applied the Time-Varying coefficient Vector Auto-Regressive (TVVAR) model to the time series of parametric 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 the estimated bispectra of roll and pitch motions. The peaks of roll 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 harmonic frequencies (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 the frequencies of 1, 3, 5 times as natural roll frequency. By applying the trispectrum analysis, kurtosis of parametric rolling due to higher order harmonic frequencies is demonstrated.

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

0

5

10

15

20

25

1 2 3 4

reallizations

max

imum

roll

angl

e (d

egre

es)

Froude-Krylov componenton its owndynamic component withFroude-Krylov component

Fig. 25 Relative noise contribution to roll angle in regular

head waves with H/λ=1/20, λ/L=1.6 (Iseki, 2006)

0.0

0

0.0

6

0.1

2

0.1

8

0.2

3

0.2

9

0.3

5

0.4

1

0.4

7

0.5

3

0.5

9

0.6

4

0.7

0

0.7

6

0.8

2

0.8

8

0.9

4

1.0

0

0 .00

0.06

0.12

0.18

0.23

0.29

0.35

0.41

0.47

0.53

0.59

0.64

0.70

0.76

0.82

0.88

0.94

1.00

Rolling Bispectrum (Amp.) 1600-2000

1200-1600

800-1200

400-800

0-400

0.0

0

0.0

6

0.1

2

0.1

8

0.2

3

0.2

9

0.3

5

0.4

1

0.4

7

0.5

3

0.5

9

0.6

4

0.7

0

0.7

6

0.8

2

0.8

8

0.9

4

1.0

0

0 .00

0.06

0.12

0.18

0.23

0.29

0.35

0.41

0.47

0.53

0.59

0.64

0.70

0.76

0.82

0.88

0.94

1.00

Pitching Bispectrum (Amp.) 80-100

60-80

40-60

20-40

0-20

Fig. 26 Bispectra of roll and pitch motions in short-crested

irregular head waves with H1/3=0.221m and T01=1.32sec. (Iseki, 2006)

0.0 0.4 0.8 1.2 1.6 2.0-2.0

-1.6

-1.2

-0.8

-0.4

0.0

0.4

0.8

1.2

1.6

2.0

Frequency f1 (Hz)

Freq

uenc

y f 2

(Hz)

Rolling Trispectrum(Amplitude)

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

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)

Prevention Devices of Parametric Rolling For a practical purpose, it is desirable to propose preventing devices of parametric roll without any change of original 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 an anti-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 installing sponsons, 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

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 numerical simulation 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 can simulate 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 of parametric rolling between calculations with damping coefficients obtained by the MPS method and those with a roll 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 a parametric roll prevention device without any model experiments by applying a MPS method.

0

5

10

15

20

25

30

0 20 40 60 80 100water level of ART (%)

max

imum

am

plitu

de o

f par

amet

ric ro

ll(d

egre

e)

Fig. 28 Relation between water level in the ART and

amplitude of parametric rolling in regular head waves at λ/L=1.3 and H/λ=0.03 (Umeda et al, 2008)

-6

-4

-2

0

2

4

6

8

10

0 2 4 6 8 10

time (sec.)

wat

er s

lope

ang

le (d

egre

e)

MPSMPS (fitted)exp.

Fig. 29 Comparison of damping curve of water slope of

ART between model experiment and MPS method (Hashimoto et al, 2007B)

-20

-15

-10

-5

0

5

10

15

20

20 40 60 80 100 120 140

time (sec.)

roll

angl

e (d

eg.)

exp.calc.MPS

Fig. 30 Comparison of roll motion between model

experiment and calculations in regular head waves λ/L=1.6 and H/λ=0.033 (Hashimoto et al, 2007B)

RECOMMENDATION FOR IMO PERFORMANCE- BASED CRITERIA

The IMO requests both the vulnerability criterion and the direct assessment for parametric rolling, by 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. 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 the assumption of uncoupled linear roll equation. Based on this knowledge, occurrence of parametric rolling can be predicted with a ratio between linear roll damping and amplitude of GM variation, which is proportional to roll angle. In addition, amplitude of the parametric rolling can be estimated with approximated analytical methods with nonlinear restoring moment in still water and GM variation due to waves. If we can calculate GM variation in longitudinal regular waves, occurrence and magnitude of parametric 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 parametric rolling with its maximum roll angle exceeding a critical angle in irregular waves. For this purpose, a coupled mathematical model is required including roll, heave and pitch motions in minimum for head seas and an uncoupled roll 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 of realisations because of the possible, in certain conditions, practical non-ergodicity of parametric rolling. CONCLUSIONS

The activity of the SCAPE committee for parametric rolling provides the following conclusions: 1. Influences of wave height, wave period, encounter

angle, ship speed, wave irregularity and initial steady angle on parametric rolling are examined by free running model experiments.

2. Roll restoring variation in regular and irregular head waves are measured and compared with numerical estimations based on potential theories in amplitude and in mean.

3. Numerical prediction methods of parametric rolling are investigated to develop quantitative prediction for realising the performance-based criteria. As a result, estimation of roll restoring variation taking hydrodynamic effects into account is desirable, and

ensemble mean is recommended to avoid the difficulty of possible, in certain conditions, practical non-ergodicity for the assessment of parametric rolling in irregular waves.

4. Effectiveness of the devices for preventing parametric rolling is validated by model experiments, and application of the MPS method to the design of anti-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 activity of 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 their contribution to the works described in this paper. REFERENCES

Belenky, V., Weems, K.M., Lin, W.M. and Paulling, J.R. (2003), “Probabilistic Analysis of Roll Parametric Resonance in Head Seas”, Proceedings of the 8th International Conference on Stability of Ships and Ocean Vehicles, pp.325-340 Belenky, V., Yu, H.C. and Weems, K. (2006), “Numerical Procedures and Practical Experience of Assessment of Parametric Roll of Container Carriers”, Proceedings of the 9th International Conference on Stability of Ships and Ocean Vehicles, pp.119-130 Boroday, I.K. (1990), “Ship Stability in Waves: On the Problem of Righting Moment Estimations for Ships in Oblique Waves”, Proceedings of the 4th International Conference on Stability of Ships and Ocean Vehicles, pp.441-451 Brunswig, J. and Pereira, R. (2006), “Validation of Parametric Roll Motion Predictions for a Modern Containership Design”, Proceedings of the 9th International Conference on Stability of Ships and Ocean Vehicles, pp.157-168 Bulian, G., Francescutto, A. (2006), “On the Effect of Stochastic Variations of Restoring Moment in Long-crested Irregular Longitudinal Sea”, Proceedings of the 9th International Stability of Ships and Ocean Vehicles, pp.131-146 Bulian, G., Francescutto, A. and Lugni, C. (2003), “On the Nonlinear Modeling of Parametric Rolling in Regular and Irregular Waves”, Proceedings of the 8th International Stability of Ships and Ocean Vehicles, pp.305-323

Bulian, G., Umeda, N. and Hashimoto, H. (2006), “Some Consideration on Grim’s Effective Wave Concept for Restoring Variations and Its Improvement”, Conference Proceedings of the Japan Society of Naval Architects and Ocean Engineers, Vol.2K, pp.13-16 Bulian, G., Francescutto, A., Umeda, N. and Hashimoto, H. (2008), “Experimental Investigation on Stochastic Parametric Rolling for a Post-Panamax Containership”, Proceedings of the 6th Osaka Colloquium on Seakeeping and Stability of Ships, (in press) De Kat, J.O. and Paulling, J.R. (1989), “The Simulation of Ship Motions and Capsizing in Severe Seas”, SNAME transactions, Vol.97, pp.139-168 France, W.L. et al. (2003) “An Investigation of Head-Sea Parametric Rolling and Its Influence on Container Lashing Systems”, Marine Technology, 40(1) Grim, O. (1961), “Beitrag zu dem Problem der Sicherheit des Schiffes im Seegang”, Schiff und Hafen 6, pp.490-497 Hamamoto, M. and Akiyoshi, T. (1988) “Study on Ship Motions and Capsizing in Astern Seas -1st Report-”, Journal of the Society of Naval Architects of Japan, Vol.179, pp.77-87 Hamamoto, M. and Fujino, M. (1986), “Capsizing of Ships in Longitudinal Waves”, Proceedings of the Symposium on Safety and Stability of Ships and Offshore Structures, pp. 125-158, (in Japanese) Hashimoto, H., Umeda, N., Matsuda, A. and Nakamura, S. (2006A), “Experimental and Numerical Study on Parametric Roll of a Post-Panamax Containership in Irregular Waves”, Proceedings of the 9th International Stability of Ships and Ocean Vehicles, vol.1, pp.181-190 Hashimoto, H., Umeda, N., Sakamoto, G. and Burian, G. (2006B), “Estimation of Roll Restoring Moment in Long-Crested Irregular Waves”, Conference Proceedings of the Japan Society of Naval Architects and Ocean Engineers, Vol.3, pp.201-204 Hashimoto, H., Umeda, N. and Sakamoto, G. (2007A), “Head-Sea Parametric Rolling of a Car Carrier”, Proceedings of the 9th International Ship Stability Workshop, pp.4.5.1-4.5.7 Hashimoto, H., Sueyoshi, M. and Minegaki, S. (2007B) “An Estimation of the Anti Rolling Tank Performance for the Parametric Roll Prevention”, Conference Proceedings of the Japan Society of Naval Architects and Ocean Engineers, Vol.4, pp.189-192, (in Japanese) Hua, J., Palmquist, M. and Lindgren, G. (2006) “An Analysis of the Parametric Roll Events Measured Onboard the PCTC AIDA”, Proceedings of the 9th International Stability of Ships and Ocean Vehicles, pp.109-118 Iseki, T. (2006), “Time Series Analysis of Parametric Rolling”, Conference Proceedings of the Japan Society of

Naval Architects and Ocean Engineers, Vol.2K, pp.23-26, (in Japanese) Iseki, T. (2007), “Time Series Analysis of Parametric Rolling (2nd Report)”, Conference Proceedings of the Japan Society of Naval Architects and Ocean Engineers, Vol.5E, pp.47-50, (in Japanese) ITTC. (2005), “Predicting the Occurrence and Magnitude of Parametric Rolling”, ITTC - Recommended Procedures and Guidelines, 7.5-02-07-04.3, pp.7-8 Japan, the Netherlands and the United States. (2007), “Framework for the Development of New Generation Criteria for Intact Stability”, SLF50/4/4, IMO Kan, M. and Taguchi, H. (1992), “Capsizing of a Ship in Quartering Seas (Part 4. Chaos and Fractals in Forced Mathieu Type Capsizing Equation)”, Journal of the Society of Naval Architects of Japan, Vol.171, pp.229-244 Kashiwagi, M. (1991), “Calculation Formulas for the Wave-Induced Steady Horizontal Force and Yaw Moment on a Ship with Forward Speed”, Report of Research Institute of Applied Mechanics, Kyushu University, Vol.37, No.107, pp.1-18 Kerwin, J.E. (1955) “Note on Rolling in Longitudinal Waves”, International Shipbuilding Progress, 2(16), pp.597-614 Levadou, M. and van’t Veer, R. (2006), “Parametric Roll and Ship Design”, Proceedings of the 9th International Conference on Stability of Ships and Ocean Vehicles, pp.191-206 Matusiak, J. (2001), “Importance of Memory Effect for Capsizing Prediction”, Proceedings of the 5th International Workshop on Stability and Operational Safety of Ships, pp. 6.3.1-6.3.6 Munif, A. and Umeda, N. (2000), “Modeling Extreme Roll Motions and Capsizing of a Moderate-Speed Ship in Astern Waevs”, Journal of the Society of Naval Architects of Japan, Vol.187 Nakamura, S., Umeda, N., Hashimoto, H. and Sakamoto, G. (2007), “Hull Form Effect on Restoring Variation of Containerships in Head Seas”, Conference Proceedings of the Japan Society of Naval Architects and Ocean Engineers, Vol.5E, pp.33-36, (in Japanese) Nechaev, Y. (1989), “Modelling of Ship Stability in Waves”, Sudostroenie Publishing, (in Russian) Neves, M.A.S. and Rodriguez, C. (2005), “A Non-Linear Mathematical Model of Higher Order for Strong Parametric Resonance of the Roll Motion of Ship in Waves”, Marine Systems and Ocean Technology, Vol.1, No.2, pp.69-81 Ogawa, Y. and Ishida, S. (2006), “The Analytical Examination of the Righting Moment Variation in Waves”, Conference Proceedings of the Japan Society of Naval

Architects and Ocean Engineers, Vol.2K, pp.17-18, (in Japanese) Ogawa, Y. (2007), “Estimation of a Parametric Rolling in Bow Seas”, Conference Proceedings of the Japan Society of Naval Architects and Ocean Engineers, Vol.5E, pp.37-40, (in Japanese) Ogawa, Y., Taguchi, H. and Tsukada, Y. (2007A), “Numerical Simulation of Parametric Rolling of a Large Container ship in Head Seas”, Conference Proceedings of the Japan Society of Naval Architects and Ocean Engineers, Vol. 4, pp.181-184, (in Japanese) Ogawa, Y., Tozawa, S., Hirakata, M. and Oka, M. (2007B), “The effect of a Parametric Rolling on a Securing of a Freight Container in Head Seas and Bow seas”, Conference Proceedings of the Japan Society of Naval Architects and Ocean Engineers, Vol. 5E, pp.41-42, (in Japanese) Oh, I.G., Nayfeh, A.H. and Mook, D.T. (2000), “A Theoretical and Experimental Investigation of Indirectly Excited Roll Motion in Ships”, Philosophical Transactions of the Royal Society of London, 358, pp.1731-1881 Paulling, J.R. (1960), “Transverse Stability of Tuna Clippers”, Fishing Boat of the World, Vol.2, edited by J.O. Traung, Fishing News Limited, pp.489-495 Ribeiro e Silva, S., Santos, T. and Guedes Soares, C. (2003), “Time Domain Simulations of a Coupled Parametrically Excited Roll Response in Regular and Irregular Head Seas”, Proceedings of the 8th International Conference on the Stability of Ships and Ocean Vehicles, pp.349-360 Roberts, J.B. (1982), “Effect of Parametric Excitation of Nonlinear Rolling Motion in Random Seas”, Journal of Ship Research, Vol.26, pp.246-253 Sanchez, N.E. and Nayfeh, A.H. (1990), “Nonlinear Rolling Motions of Ships in Longitudinal Waves”, International Shipbuilding Progress, 37(411), pp.247-272 Soliman, M. and Thompson, J.M.T. (1992), “Indeterminate Sub-Critical Bifurcations in Parametric Resonance”, Proceedings of the Royal Society London, 438, pp.511-518 Spanos, D. and Papanikolaou, A. (2006), “Numerical Simulation of Parametric Roll in Head Seas”, Proceedings of the 9th International Conference on Stability of Ships and Ocean Vehicles, pp.169-180 Taguchi, H., Ishida, S., Sawada, H. and Minami, M. (2006A), “Parametric Rolling of a Ship in Head and Bow Seas Part 2. Effect of Steady Heel”, Conference Proceedings of the Japan Society of Naval Architects and Ocean Engineers, Vol.3, pp.197-200, (in Japanese) Taguchi, H., Ishida, S., Sawada, H. and Minami, M. (2006B), “Parametric Rolling of a Ship in Head and Bow Seas Part 3. A Model Experiment in Irregular Waves”, Conference Proceedings of the Japan Society of Naval

Architects and Ocean Engineers, Vol.4, pp.185-188, (in Japanese) Taguchi, H., Ishida, S., Sawada, H. and Minami, M. (2006C), “Model Experiment on Parametric Rolling of a Post-Panamax Containership in Head Waves”, Proceedings of the 9th International Conference on Stability of Ships and Ocean Vehicles, Vol.1, pp.147-156 Themelis, N. and Spyrou, K.J. (2006), “Probabilistic Assessment of Resonant Instability”, Proceedings of the 9th International Stability of Ships and Ocean Vehicles, pp.37-48 Umeda, N. and Hamamoto, M. et al. (1995), “Model Experiments of Ship Capsize in Astern Seas”, Journal of the Society of Naval Architects of Japan, Vol.177 Umeda, N., Hashimoto, H., Vassalos, D., Urano, S. and Okou, K. (2004), “Nonlinear Dynamics on a Parametric Roll Resonance with Realistic Numerical Modelling”, International Shipbuilding Progress, Vol.51, No.2/3, pp.205-220 Umeda, N. and Renilson, M.R. (2001), “Benchmark Testing of Numerical Prediction on Capsizing of Intact Ships in Following and Quartering Seas”, Proceedings of the 5th International Workshop on Stability and Operational Safety of Ships, pp.6.1.1-6.1.10 Umeda, N., Hashimoto, H., Sakamoto, G. and Urano, S. (2005), “Research on Roll Restoring Variation in Waves”, Conference Proceedings of the Kansai Society of Naval Architects, Vol.24, pp.17-19, (in Japanese) Umeda, N. and Hashimoto, H., (2006) Recent developments of capsizing prediction techniques of intact ships running in waves, Proceedings of the 26th ONR Symposium on Naval Hydrodynamics, Rome, CD. Umeda, N., Hashimoto, H., Minegaki, S. and Matsuda, A. (2007), “Preventing Parametric Roll with Use of Devices and Their Practical Impact”, Proceedings of the 10th International Symposium on Practical Design of Ships and Other Floating Structures, Vol.2, pp.693-698 Umeda, N., Hashimoto, H., Minegaki, S. and Matsuda, A. (2008), “An Investigation of Different Methods for the Prevention of Parametric Rolling”, Journal of Marine Science and Technology, 13(1), (in press) Watanabe, Y. (1934) “On the Dynamic Properties of the Transverse Instability of a Ship Due to Pitching”, Journal of the Society of Naval Architects of Japan, Vol. 53, pp.51-70, (in Japanese)

PREDICTION METHODS FOR PARAMETRIC ROLLING UNDER 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, JAPAN

ABSTRACT

In this paper, experimental and theoretical research works on heavy parametric rolling in beam waves which have been done for these three years by the authors are reviewed, and the recommendations to IMO on the basis of 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

: roll angle : roll amplitude : maximum roll amplitude at the peak of parametric

rolling Hw : wave height

: 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 roll characteristics if the natural frequencies in heave and roll are in the ratio of 2:1 in beam waves, and they investigated unstable roll motion produced by nonlinear coupling from heave motion theoretically and experimentally. Tasai (1965) noted that small unstable roll motion caused by periodic roll restoring variation in regular beam waves. Tamiya (1969) measured unstable roll motion and capsizing of a cylindrical model in beam waves and wind, and pointed out that the phenomena can be explained by a Mathieu equation. Sadakane (1978) studied on the effect of fluctuation of the apparent weight of a ship in regular heavy waves on roll motion, found that the heavy rolling has harmonic and subharmonic oscillations by experiments, and noted that the phenomena can be explained by a Mathieu-type instability. Blocki (1980) reported experimental results of capsizing of a cylindrical model of a fishing vessel due to parametric rolling in regular beam waves, and calculated capsizing probability caused by parametric resonance. Boroday and Morenschildt (1986) carried out an experimental investigation of the condition giving rise to the cylindrical ship 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 parametric rolling for a fishing vessel under drifting condition without forward velocity occurs in head waves, and Munif and Umeda (2006) concluded that coupling with pitch is essential to numerically explain this type of parametric rolling.

Recently, Ikeda et al. (2005A) experimentally showed that heavy 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 to parametric rolling. The parametric rolling in beam waves disappears when the roll damping is large enough. This fact suggests that the roll damping of such ships should be designed not to occur heavy parametric rolling in beam waves as well as to meet the intact stability criteria.

φ

0φmaxφ

0ζ

Following the paper, the authors have been carrying out experimental and theoretical investigations on the parametric roll in beam waves. EXPERIMENTAL RESULTS OF PARAMETRIC ROLLING IN BEAM WAVES

Measurements of roll motions of a model ship were carried 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.

Fig. 1 Body plan of the ship.

Table 1 Principle particulars.

Full Scale Model Scale 1/1 1/125.32 LOA 290 m 2.200 m LPP 242.24 m 1.933 m BM 36 m 0.287 m D 8.4 m 0.067 m Displacement 53,010 ton 26.98 kg GM 1.579 m 0.0126 m Tnr 23 sec 2.05 sec Bilge keel : width 1.1 m 0.0088 m Bilge keel : location s.s.3.0－5.0, s.s.5.25－6.0

Fig. 2 shows measured roll amplitude of the model ship without bilge keels in beam waves with regular wave height of 0.04 m. The peak amplitude of roll resonance at the natural roll period (Te=2.04 sec) reaches 7 degrees, but the roll amplitude at encounter period of nearly half of the natural roll period (Te=0.85 sec) reaches 25 degrees. As shown in Fig. 3, the period of the large roll motion is twice of the encounter wave period and almost the same as the natural roll period. It should be noted that the wave encounter period where the large roll appears is near the natural heave period of the ship, too (Ikeda et al., 2005B).

It is experimentally found that the heavy rolling with twice period of the wave encounter period strongly depends on wave height. The maximum amplitude of the heavy roll at 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, and reaches the maximum amplitude that is about 27 degrees. It should be noted that the roll amplitude does not proportionately increase with increasing wave height but

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

0

5

10

15

20

25

30

0 0.5 1 1.5 2T e (sec)

Φ0

(deg

)

Hw = 0.04mwithout Bilge Keel

T nr / 21.02sec

T nr2.04sec

T nh0.740sec

Fig. 2 Measured roll amplitude for naked hull in regular beam waves of 0.04 m wave height.

0

2

4

6

8

10

12

0 2 4 6 8 10 12ωe

ωro

ll Ratio １/２

Ratio １/１ωroll : roll frequency

ωe : encounter frequency

Fig. 3 Comparison between measured roll frequency and encounter frequency.

0

5

10

15

20

25

30

0 0.02 0.04 0.06 0.08H w (m)

Φ0

(deg

)

without BKTw = 0.85 sec

Fig. 4 Effect of wave height on maximum amplitude of parametric rolling of the ship in regular beam waves. EXPERIMENTAL RESULTS OF PARAMETRIC ROLLING IN ALL HEADING ANGLES UNDER DRIFTING CONDITION

To know the characteristics of the parametric rolling in beam waves, some investigations on parametric rolling in other heading angles to waves are carried out.

In Fig. 5, difference of wave periods when parametric rolling 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 of stability variation in beam and head waves (Fujiwara et al., 2006B).

0

5

10

15

20

25

30

0 0.5 1 1.5 2λ/Lpp

Φ0(d

eg)

180deg head sea 90deg beam sea

H w = 0.04mwithout Bilge Keel

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

To investigate the effect of heading angle on the parametric rolling of the dead ship, the model are released in head 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 can see that, in both cases, the heading angle of the ship slowly changes to beam sea condition. The results demonstrate that the parametric rolling occurs in wide heading angles as well as in beam waves.

pitch (deg)

-3

-2

-1

0

1

2

3

20 70 120 170 220

heave (mm)

-30

-20

-10

0

10

20

30

20 70 120 170 220

roll (deg)

-30

-20

-10

0

10

20

30

20 70 120 170 220

yaw (deg)

0

30

60

90

120

150

180

20 70 120 170 220time (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 each

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

h e a ve (m m )

- 3 0

- 2 0

- 1 0

0

1 0

2 0

3 0

1 0 3 0 5 0 7 0 9 0 1 1 0 1 3 0 1 5 0

p itc h (d e g )

- 2 .0

- 1 .5

- 1 .0

- 0 .5

0 .0

0 .5

1 .0

1 .5

2 .0

2 .5

1 0 3 0 5 0 7 0 9 0 1 1 0 1 3 0 1 5 0

ro l l (d e g )

- 3 0

- 2 0

- 1 0

0

1 0

2 0

3 0

1 0 3 0 5 0 7 0 9 0 1 1 0 1 3 0 1 5 0

y a w (d e g )

0

3 0

6 0

9 0

1 2 0

1 5 0

1 8 0

1 0 3 0 5 0 7 0 9 0 1 1 0 1 3 0 1 5 0t im e (s e c ) Fig. 7 Time histories of motions of the ship without bilge keel released from following wave condition in regular waves at Tw = 0.95 sec and Hw= 0.04 m.

Fig. 8 Effect of wave direction on amplitude of parametric rolling of the ship without bilge keel in regular waves.

Fig. 9 Effect of wave direction on amplitudes of rolling and its period of the ship with the designed bilge keels in regular waves. EFFECTS OF ROLL DAMPING

As shown in the previous chapter, parametric rolling in beam waves significantly depends on the roll damping of ships. For the ship used in our experiments, the designed bilge keels can erase the parametric rolling in beam waves as shown in Fig. 9. By changing the roll damping systematically, the effect of the roll damping on the rolling was investigated. The length and attached position of various bilge keels are shown in Table 2. Table 2 Size and location of bilge keels.

BK location

s.s. no.

BK

Length

Area 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 Mid 3 4.70 - 6.00 260 mm 0.473 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 mm 1.000

In Fig. 10, the maximum roll amplitudes of parametric

rolling for different length of bilge keels in regular beam waves of 0.04 m height are shown. The results demonstrate that the peaks of parametric rolling rapidly decreases with increasing area of bilge keels, or roll damping, and disappears at half area of the designed bilge keels.

In order to clarify the effect of the roll damping on the critical wave height to occur parametric rolling in beam waves, measurements of ship motions of the model in the cases 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

height at which parametric rolling appears increases with increasing 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 each bilge keel shown in Table 2 in terms of N coefficient are shown. N coefficients of the ship increase with increasing area of bilge keels. It should be noted that N coefficient for a bare hull is 0.018 as shown in Fig. 12. The value is not so small 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 be done to secure for escaping from large parametric rolling in beam waves (Fujiwara et al., 2006).

0

5

10

15

20

25

30

0 0.2 0.4 0.6 0.8 1Area of Bilge Keel / Area of Designed Full Bilge Keel

Φ0

(deg

)

H w = 0.04m

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

0

5

10

15

20

25

30

0 20 40 60 80H w (m)

Φ0

(deg

)

without BK

BK 1

Mid2 BK

Mid3 BK

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

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0 0.2 0.4 0.6 0.8 1Area of Bilge Keel / Area of Designed Full Bilge Keel

N

Fig. 12 Measured N coefficients for various bilge keels.

EFFECTS OF HEAVE RESONANCE

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

Tnr (sec) Tnr / 2Tnh Tw (sec) Hw (m) 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

0.04

Effects of the ratio of half of natural roll period to natural heave period, Tnr / 2Tnh on the peak of roll amplitude of the parametric 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 in beam 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.

Measured amplitudes of heave motions and its phase angles from incident waves are shown with calculated results by a strip method (OSM) in Figs. 14-16. The measured heave amplitudes in the both cases shown in Figs. 14 and 15 are very similar to each 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).

0

5

10

15

20

25

30

0 1 2 3T nr / 2T nh

φm

ax (

deg.

)

H w = 0.04mwithout Bilge Keel

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

0

0.5

1

1.5

0 0.5 1 1.5 2 2.5 3T e (sec)

Z 0 / ζ 0

0.787 0.9211.18 1.381.63 1.922.25 Cal. (OSM)

H w = 0.04mwithout Bilge KeelT nh

0.748 sec

T nr / 2T nh

Fig. 14 Heave response curve in the cases when parametric rolling occurs.

0

0.5

1

1.5

0 0.5 1 1.5 2 2.5 3T e (sec)

Z 0 / ζ 0

0.787 0.9211.18 1.381.63 1.922.25 Cal. (OSM)

H w = 0.04mwithout Bilge KeelT nh

0.748 sec

T nr / 2T nh

Fig. 15 Heave response curve in the cases when parametric rolling does not occurs.

-180

-90

0

90

180

0 0.5 1 1.5 2 2.5 3T e (sec)

ε (d

eg.)

0.787 0.9211.18 1.631.92 2.25Cal. (OSM.)

Hw = 0.04mwithout Bilge Keel

Tnh

0.748 sec

T nr / 2T nh

Fig. 16 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 the calculations 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 of stability. 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 the parametric rolling occur when the relative heave motion becomes significant. Furthermore, it occurs even if the time-variation of the draft comparatively small (Ikeda et al., 2007).

-40

-30

-20

-10

0

10

20

30

40

291 292 293 294 295 296

Heave Wave surface Roll

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

0

2

4

6

8

10

12

0.4 0.6 0.8 1 1.2 1.4 1.6T e (sec)

Rel

ativ

e he

ave

mot

ion

(mm

)

0.787 0.9211.18 1.631.92 2.26

H w = 0.04mwithout Bilge Keel

Tnh

0.748 secT nr / 2T nh

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

0

5

10

15

20

25

30

0.4 0.6 0.8 1 1.2 1.4 1.6T e (sec)

Φ0(d

eg)

0

5

10

15

Rel

ativ

e H

eave

Mot

ion

(mm

)

roll amplitude

relative heave motion

T nh0.748 sec

T nr / 20.825 sec T nr / 2T nh = 1.18

Left axis

Right axis

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

0

5

10

15

20

25

30

0.4 0.6 0.8 1 1.2 1.4 1.6T e (sec)

Φ0

(deg

)

0

5

10

15

Rel

ativ

e H

eave

Mot

ion

(mm

)

roll amplitude

relative heave motion

T nh0.748 sec

T nr / 21.195 sec

T nr / 2T nh = 1.63

Left axis

Right axis

Fig. 20 Parametric rolling in beam waves and relative heave motion when Tnr/2Tnh=1.63. THEORETICAL MODELS OF PARAMETRIC ROLLING IN BEAM WAVES

Using a nonlinear 1 DOF model of roll equation, a simulation of the parametric rolling in beam waves is carried out. Nonlinear damping and nonlinear restoring moments are

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

+⎟⎠⎞

⎜⎝⎛ Δ+++++ φωωφγφφβφαφ φ t

GMGM

ecos2

12 23&&&&&&

55523332 φωφω φφ ⎟⎠⎞

⎜⎝⎛ Δ

++⎟⎠⎞

⎜⎝⎛ Δ

+GM

aGMa

GMa

GMa (1)

trkGM

aGMa

ea ωωζφω φφ sin27772 =⎟⎠⎞

⎜⎝⎛ Δ

++

for the polynomial equation up to 7th order of restoring moment. α , β and γ are the roll damping coefficients and , and are the nonlinear coefficient of restoring moment in still water. These coefficients can be determined from the fitting of the GZ curves in still water. GMΔ , 3αΔ ,

5αΔ and 7αΔ are the coefficients of the restoring moment in waves, and they are the function of wave steepness and relative position of ship in waves. These coefficients are calculated 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 of metacentric height in the experiments, or this could be due to the change rolling period, which was identified in the model experiments. But critical wave steepness can be estimated approximately (Munif et al., 2005).

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.5 0.55 0.6 0.65 0.7 0.75 0.8ω e

H/λ

Simulations

Experiments

Area of non parametricresononce

Area of parametricresononce

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 the natural heave period, it is recommended to examine the possibility 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 in beam waves to guarantee the safety in dead ship condition. CONCLUSIONS

The activity of the SCAPE committee for parametric rolling in beam waves under drifting condition provides the following conclusions: 1. Parametric rolling in beam waves can be significant for

3α 5α 7α

a large passenger ship if her bilge keels are removed. 2. If she is equipped with the designed bilge keels, the

parametric rolling in beam waves disappears but that in longitudinal waves does not.

3. When the bilge keel size increases, the critical wave height of parametric rolling in beam waves and its magnitude increases.

4. 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 and roll on its own does not explain the amplitude of parametric rolling in beam waves.

5. The critical wave height for parametric rolling in beam waves can be roughly estimated with a numerical simulation with roll restoring variation in time.

REFERENCES

Blocki, W., 1980, “Ship Safety in Connection with Parametric Resonance of the Roll”, International Shipbuilding Progress, 27-306.

Boroday, I. K. and Morenschildt, V. A., 1986, “Stability and Parametric 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 of Roll Damping on Parametric Rolling in Beam Sea”, Conference Proceedings of the Japan Society of Naval Architects and Ocean Engineers, Vol. 3, pp.193-196, (in Japanese).

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-Pacific Workshop on Marine Hydrodynamics, Shanghai.

Fujiwara, T. and Ikeda, Y., 2007, “Effects of Roll Damping and Heave Motion on Heavy Parametric Rolling of a Large Passenger Ship in Beam Waves”, Proceedings of the 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, “An Experimental 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 of Heave Motion on Heavy Parametric Rolling in Beam Waves”, Conference Proceedings of the Japan Society of Naval 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 the Japan 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 on parametric roll resonance for a ship having no significant wave-induced change in hydrostatically- obtained 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, Non- symmetric Rolling of Ships”, Conference Proceedings of the Society of Naval Architects of Japan, pp.85-93, (in Japanese).

Tasai, F., 1965, “Ship Motions in Beam Seas”, Journal of the West-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, JAPAN

ABSTRACT

Regarding capsizing of ships under dead ship condition, this paper reports experimental, numerical and analytical studies conducted by the SCAPE committee, together with critical review of theoretical progress on this phenomenon. Here capsizing probability under dead ship condition is calculated with a piece-wise linear approach. Effects of statistical correlation between wind and waves are examined. This approach is extended to the cases of water on deck and cargo shift. Further, experimental and theoretical techniques for relevant coefficients are examined. Based on these outcomes, it also proposes a methodology for direct 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 three major capsizing scenarios in terms of performance-based criteria, which is requested to be developed for the revision of the Intact Stability Code (IS Code) by 2010 at the International 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 fact that 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 for evaluating 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 the safety level under dead ship condition is ensured. In other words, if perfect operation is taken, the safety level during a ship’s life-cycle could be comparable to that under dead ship condition.

For the dead ship condition, the weather criterion is currently implemented in the IS Code and provides a semi-empirical criterion for preventing capsizing in beam wind and waves. The dead ship condition, however, does not always mean beam wind and waves. If a ship has a longitudinally asymmetric hull 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. 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 this purpose although it is not always feasible. A simplified prediction method is still desirable if accurate enough.

Based on the above situation, the authors presented a methodology 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 and its 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 statistical correlation between wind and waves are investigated.

Third, safety level in terms of capsizing probability was assessed 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 Carlo simulation, in which the expected number of capsizing is estimated by repeating numerical simulation in time domain with different wave realizations (McTaggart and de Kat 2000). If a reliable mathematical model is available, this method seems to be easily applicable. However, since capsizing probability of an existing ship is small, the number of numerical runs could be prohibitively large. In addition, statistical variation of outcomes is not suitable for regulatory 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. Some methods utilize stationary linear or non-linear theory for the former and non-linear time-domain simulation of deterministic process for the latter (Umeda et al. 1992; Shen and Huang 2000; Bulian and Francescutto, 2004). Sheng and 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 modelling of 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 by approximating the restoring arm curve with piece-wise linear ones. Because of its fully analytical scheme, the piece-wise linear approach has been utilized for practical purpose as the most promising methodology. An approach that has a series of similarities with the piece-wise linear method, but that is based on a simplified statistical linearization technique, has been developed by Bulian and Francescutto (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 case in both beam winds and waves (Belenky 1994) and provided numerical results with his simplified method (Belenky 1995). Here the formulae relating to the wind effect were not published in detail. Recently, Iskandar et al. (2001) applied Belenky’s simplified method to estimated

annual 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 the capsizing 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 the accuracy of the simplified formula. Thirdly, effects of several elements such as wind have not been examined.

Therefore, Paroka et al. (2006) examined these problems with deriving the detailed formulae for the case of beam winds and waves and calculating with both exact and simplified 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 for calculating capsizing probability under dead ship condition is 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)

φφv

-φv

φm0φCR

-φCR -φm0

Steady Wind heeling arm

Fig. 1 Piece-wise linear approximation of righting arm.

The capsizing probability here is calculated with the following 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 coupled motions in sway and roll with wave radiation forces and diffraction forces. Watanabe (1938), however, proposed an one-degree of freedom equation of roll angle, φ , as follows ( ) ( ) ( ) ( )tMtMWGZBAI wavewind4444xx +=+++ φφφ &&& (1) where Ixx is momentum of inertia of the ship, A44 is the hydrodynamic coefficient of added inertia, W is the ship displacement, 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 the roll diffraction moment and roll radiation moment due to sway 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 of inertia as follows:

( ) ( ) ( )( )tmtmkf2 wavewind20i

20 +=++ ωφωφαφ &&& (2).

where kfi( φ )=GZ( φ )/GM, α: roll damping coefficient,

0ω : the natural roll frequency and kf1( φ ) : the non-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 in equation (2) is approximated with continuous piece-wised lines in leeward and windward conditions as follows:

( ) ( ) ⎭⎬⎫

⎩⎨⎧

<<<

−=

2range1range0

kfkf

kf0m

0m

v1

0i φφ

φφφφ

φφ (3)

and

( ) ( ) ⎭⎬⎫

⎩⎨⎧

−<<<−

−−=

2range1range0

kfkf

kf0m

0m

v1

0i φφ

φφφφ

φφ (4).

Here, kf0 and kf1 are the slope of the range 1 and range 2, respectively. 0mφ and 0mφ− are the border between the range 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 the relative roll angle can be found in Paroka et al. (2006).

Since Equation (2) is linear within each range, it can be analytically solved without any problems. The border condition, 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 be calculated as the product of the outcrosing probability in the range 1 and the conditional probability of divergence of the absolute value of roll angle in the range 2. Belenky (1993) presented the following formula for the probability that a ship capsizes when the duration, T, passes in stationary waves.

);0()(2),,,( 00013/1 mAmTs APPTWTHP φφφφ >>>=

(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.

( )( ) ( )( ) ( )000

000

013/1

;0;0

,,,

mAmmTw

mAmmTl

s

APorPPAPorPP

TWTHP

φφφφφφφφφφφφ−<<−<>

+>>−<>=　

(6). Here Pl and PW denote the probability of up-rossing toward leeward and that of down–crossing toward windward. Under the assumption of Poisson process, PT indicates the probability of at least one up-crossing or down-crossing at the border between the first and second range, 0mφ or

0mφ− . 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:

( ) ( )( )Tuuexp1orP wl0m0mT +−−=−<> φφφφ (7)

wl

ww

wl

ll

uuu

P

uuu

P

+=

+=

(8)

( ) ( )

( ) ( )∫

∫

∞−

∞

=−<<

=>>

0

0mA

00mA

dAAf;0AP

dAAf;0AP

φφ

φφ (9)

where ul and uW denote the expected number of up-crossing and down-crossing, respectively. f(A) denotes the probability density function of the coefficient A,, which is a function of three random variables, namely the initial angular velocity of roll motion, 1φ& , the initial forced roll, p1, and the initial forced angular velocity, 1p& , in the second range.

The probability density function of the A coefficient can be determined from a joint probability density function of these three random variables, which can be calculated by using the three-dimensional Gaussian probability density (Price and Bishop, 1974).Then, the probability density function of the coefficient A in leeward can be described as follows:

( ) ( )( ) 110

1111112

dpdp,p,,Ap,f1Af φφφλλ

&&&&∫ ∫−

=∞

∞−

∞ (10)

That in windward can be described as follows:

( ) ( )( ) 110

1111112

dpdp,p,,Ap,f1Af φφφλλ

&&&&∫ ∫−

=∞

∞− ∞− (11)

where 2

12

02,1 αωαλ +±−= kf (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 place of a double integral.

For evaluating risk of capsizing, it is necessary to calculate annual capsizing probability with sea state

statistics of operational water area. The formula of the annual capsizing probability, Pan, can be found in Iskandar et al. (2000) as follows:

( )( )( )Tan TPP

/360024365*11××

−−= (13) where

( ) ( ) ( ) 013/10 0

013/1013/1* ,,,, dTdHTWTHPTHfTP s∫ ∫

∞∞⋅= .

Here Ws is assumed to be fully correlated with H1/3 VERIFICATION AND EXTENSIONS OF PIECE-WISE LINEAR APPROACH

Comparison with the Monte Carlo simulation For verifying the piece-wise linear approach, Paroka & Umeda (2006b) calculated capsizing probability of a large passenger ship in stationary beam wind and waves and compared it with the Monte Carlo simulation. First, they confirmed that stochastic scattering of the Monte Carlo simulation 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 the confidence interval but underestimates it in more severe wind velocity as shown in Fig.2. The results of this underestimation are that the assumption of the Poisson process for outcrossing is not appropriate when the sea state is extremely severe. In addition, it is impossible for the Monte 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 the piece-wise linear approach but it seems not to be inappropriate to use the piece-wise linear approach for the mean wind velocity of 26 m/s, which the weather criterion deals with, but there is a room for further improvement.

1.0E-04

1.0E-03

1.0E-02

1.0E-01

1.0E+00

28.0 28.5 29.0 29.5 30.0 30.5 31.0

Wind Velocity (m/s)

Cap

sizi

ng p

roba

bilit

y

Numerical Experiment+Err-ErrPresent Method

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

In case of a ship has large angle of vanishing stability, the obtained capsizing probability could be very small but the ship could capsize owing to cargo shift or down flooding when the roll angle is larger than her critical angle. In this case, the calculation method mentioned above cannot be directly applied because angle of vanishing stability cannot be determined.

Therefore,. Paroka & Umeda. (2006a) extend the piece-wise linear seas approach to the case having the critical roll angle. Here the probability of roll angle exceeding the critical value is defined as the probability of roll 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-wise linear righting arm curve. This can be written as follows: ( )

( ) ( )( ) ( )0mCRCR0m0mTw

0mCRCR0m0mTl

s013/1

;PorPP;PorPP

T,W,T,HP

φφφφφφφφφφφφφφφφ−<−<−<>

+>>−<>=

(14)

where PCR is the conditional probability of roll angle exceeds the critical value, CRφ .

The conditional probability of roll angle exceeds the critical value is calculated with assuming that effect of exciting moment on the roll motion in the second range is negligibly 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:

( ) ( ) ( ) ( )DvtBtAt φφλλφ −++= 212 expexp (15) where

1202,1 kfωααλ +−±−= (16)

120

0

kfm

D ωφ = (17)

( ) ( ){ }[ ]DvmA φφφλφλλ

−−−−

= 020221

1 & (18)

( ) ( ){ }[ ]020121

1 φφφφλλλ

&−−−−

= DvmB (19)

Here 02φ& indicates the initial angular velocity of roll motion in 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 angular velocity 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:

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛−

−=

1

2

21ln1

λλ

λλ ABtm (20)

Then the roll angle with the time, tm , can be calculated

by means of the equation (15). If this roll angle is smaller

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

( ) ∫∞

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

=>

CR

dDD

Pqq

CR

φ

φφ

πφφ

& &&

&&

&&02

0202 2

exp2

2 (21)

where CRφ& indicate the critical value of the angular velocity in the second range. The probability of roll angle exceeding the critical value in a certain exposure time can be calculated by means of the equation (14) with the conditional probability of roll angle exceeds the critical value is replaced by the equation (21). Numerical results of the car carrier for several critical angles 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. This is 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 to steady wind moment is smaller than the critical value. When the unstable heel angle is smaller than the critical value, the roll angle can exceed the critical value only if the roll motion in the second range diverges.

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 of trapped water on deck should be taken into account. This is because these ships have smaller freeboard but higher bulwark comparing with their ship size. As a result, when shipping water occurs, water can be easily trapped by bulwark. Although a certain amount of water can flow out through freeing ports, the balance between ingress and egress can be trapped on deck. The righting arm can decrease because of the water trapped on deck. Many experimental and numerical works about effect of water

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 the amount of trapped water was estimated with model experiments (Matsuda et al., 2005) by measuring heel and sinkage 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 trapped water on deck could increase the capsizing probability for this ship type as shown in Fig. 4. This is because the trapped water can significantly reduce the mean restoring arm up to the angle of immersion of the bulwark top, which is approximated 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 egress with a hydraulic model into the piece-wise linear approach. As a result, capsizing probability with water on deck can be calculated without a model experiment. Here the dynamic effect of trapped water on deck such as change in added mass, damping coefficient and exciting moment are ignored for simplicity’s means.

1E-30

1E-25

1E-20

1E-15

1E-10

1E-5

1

0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

GM without Water on Deck (m)

Long-

term

Cap

sizi

ng

Pro

babi

lity

Without Water on Deck With Water on Deck

Fig. 4 Long-term capsizing probability with and without water on deck for the purse seiner operating off Kyusyu. (Paroka & Umeda, 2006c) Effect of drifting attitude and velocity

The dead ship condition does not always mean beam wind and waves. If a ship has a longitudinally asymmetric hull 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.

β

U

x

y

X

Y

0

0'

ψ

χ

Wind

Wave

y

z

φ

Y

Z

Fig. 5 Co-ordinate systems

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 a surge-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; ( ) ( ) WAHyx XXXvrmmumm ++=+−+ & (22)

( ) ( ) WAHyyxy YYYlmurmmvmm ++=−+++ φ&&& (23)

( ) WAHzz NNNJI ++=+ ψ&& (24)

( )AH

xxyyxx

KK

GMWurlmvlmJI

+=

⋅++−−+ φφμφ &&&& 2 (25)

where u,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 the equations. As an initial input, the equilibria in the beam wind and waves from starboard and those from port side are calculated. And then, equilibria for different angle between the wind and waves, χ , are traced. The calculation are done 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 is evaluated 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 function of wind speed. In this calculation, the significant wave height and mean wave period were of the same as in the experiments (H1/3 = 9.5 m, T01 = 10.4 sec). Only the wind speed was varied in the present calculation. It is found that

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 the drift speed under dead ship condition.

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)

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

1.00E-36

1.00E-32

1.00E-28

1.00E-24

1.00E-20

1.00E-16

1.00E-12

1.00E-08

1.00E-04

1.00E+00

0 20 40 60 80 100

angle between wind and waves (degree)

cap

sizi

ng

proba

bilit

y (c

ase1)

Fig.8 Capsizing probabilities under two coexisting drifting conditions as functions of relative angle between wind and waves. (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

Drift speed due to wind & wave

0

0.5

1

1.5

2

2.5

3

3.5

0 5 10 15 20 25 30 35Wind speed (m/s)

Drif

t spe

ed (m

/s)

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)

0

30

60

90

120

150

180

210

240

270

300

330

360

0 30 60 90 120 150 180

angle between wind and waves (degrees)

head

ing

angl

e (d

egre

es)

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

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 the change in the exciting frequency. In addition, the heeling lever due to drift are estimated with the model tests by Taguchi et al. (2005). It is confirmed that the most dangerous condition for this subject ship under dead ship condition 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. Effect of hydrodynamic coefficient estimation

It is also widely known that empirical estimation of effective wave slope coefficient in the weather criterion is often difficult to be applied to modern ship-types such as a large 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 heeling lever owing to beam wind by model experiments with the interim 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 the weather criterion. It was clarified that the evaluation of weather criterion considerably differs that with the combination of tests. Umeda et al. (2006a) applied this model 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 and a mechanical guide method. The guide rope method means that 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, a heaving rod and gimbals. Among them, as shown in Fig.9, the looped wire system is less accurate because natural roll period 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 for keeping a beam wave condition. The guideline allows the three different methods for determining the roll angle from the experiment: the direct method, the three step method and the parameter identification. As shown in Fig 10, the difference between the three methods is negligibly small. In conclusion, the interim guidelines can guarantee reasonable accuracy if appropriate constraint is used.

When we calculate capsizing probability with the piece-wise linear approach, the same difficulty exists. This is 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 and time. Therefore, it is desirable to develop an empirical or a theoretical method to accurately predict obtained results from the model test.

It is well known that a strip theory can explain the effective wave slope coefficient in general. (Mizuno, 1973) Since a strip theory requires numerical solution of

simultaneous equation for determining two dimensional flows, however, a further simplified method is expected to be developed. For this purpose, Sato et al. (2007) carried out a series of model tests by means of two dimensional models 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 case of very large breadth to draft ratios. This fact coincides with the experiments by Ikeda et al. (1992) for a small high-speed craft.

Fig. 9 Roll amplitude of the RoPax ferry in regular beam waves measured with different constraining methods (Umeda et al., 2006a).

25.2226.56 26.18

0

5

10

15

20

25

30

direct method three step method parameter identificatin

roll

ampl

itude

(deg

)

Fig. 10 Comparison of estimated roll amplitude by means of three different estimation methods. (Umeda et al., 2006a)

Umeda & Tsukamoto (2007c) investigated several possible simplified versions of theoretical methods and finally 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 the same as those of original transverse ship sections. As shown in Table 1, this simplified formula agrees well with the experiment 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 and experiments results in only small effects on the estimation of the one-hour capsizing probability in stationary beam

s=1/60

0

5

10

15

20

25

0.7 0.8 0.9 1 1.1 1.2 1.3

tuning factor

roll

ampl

itude

(deg

)

Guide RopeLooped Wire SystemMechanical GuideNMRI (Guide Rope)

wind and waves as shown in Fig.11. Therefore, it is recommended 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)

1.0E-191.0E-171.0E-151.0E-131.0E-111.0E-091.0E-071.0E-051.0E-031.0E-01

19 21 23 25 27 29 31 33 35Wind Velocity　[m/ｓ]

Prob

abili

ty

IMO

Strip theory

EXP

simplified Froude-Krylov Fig.11 The effect of the effective wave slope coefficient estimation on the one-hour capsizing probability of the RoPax ferry in stationary beam wind and waves. (Umeda et al., 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 state statistics. Ogawa et al. (2006a,2007) constructed a scatter diagram of wave height, wave period and wind speed by means of the wind and wave database developed from hindcast calculation of wind-wave interactions and examined 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 the probability 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:

( )( ) T/360024365N

T*P11NP×××

⎟⎠⎞⎜

⎝⎛ −−= (26)

where

( ) ( )

( ) s013/1s013/1

0 0 0s013/1

dWdTdHT,W,T,HP

W,T,HfT*P

⋅

∫ ∫ ∫=∞∞∞

(27)

Ogawa et al. (2006a) calculated the annual capsizing

probability of the RO-PAX ferry. Fig. 12 shows her righting arm (GZ) curves. The limiting KG of this ship is governed by the weather criterion so that it was used as a loading condition for the safety assessment. The approximated line of the righting arm curve, of which GM, the area under the GZ curve and the angle of vanishing 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 of capsizing probability are calculated by means of two other wave scatter diagrams. One is the wave scatter diagram normalized for each mean wind velocity (Normalized) and the other is the wave scatter diagram without correlation of wind (No correlation). The wave scatter diagram without correlation with wind was conducted by summing up the present scatter diagram of wave height, wave period and wind speed. Then the capsizing probabilities using two kinds of diagrams is defined as follows: (Normalized)

( ) ( )

( ) 013/1s013/1

0 0s;013/1Ns

dTdHT,W,T,HP

WT,HfW,T*P

⋅

∫ ∫=∞ ∞

(28)

(No correlation)

( ) ( )

( ) 013/1s013/1

0 0013/1NCs

dTdHT,W,T,HP

T,HfW,T*P

⋅

∫ ∫=∞ ∞

(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, the short-term probability in the constant wind and waves is also 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 probability of wind is taken into account in the calculation. The capsizing probability with the normalized scatter diagram is larger than the present method when the mean wind velocity is larger 16 m/s. This indicates that occurrence probability of 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 is probably because unrealistic combinations of wind and waves are included in the calculation.

The capsizing probability taken from the stationary sea state is larger than the present method because the occurrence probability of such sea state is not large during a year in this water area. On the other hand, the capsizing probability from the stationary sea state of the mean wind velocity 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. Table.2 Samples of wave height –period diagram in each wind speed (Autumn, 1994-2003, Ws=2, 20, 40(m/s)). (Ogawa et al., 2006a) Ws=2(m/s) T(sec.)Hw(m) 3- 4- 5- 6- 7- 8- 9- 10- 11- 12- 13- 14- 15-11.25- 0 0 0 0 0 0 0 0 0 0 0 0 010.75- 0 0 0 0 0 0 0 0 0 0 0 0 010.25- 0 0 0 0 0 0 0 0 0 0 0 0 09.75- 0 0 0 0 0 0 0 0 0 0 0 0 09.25- 0 0 0 0 0 0 0 0 0 0 0 0 08.75- 0 0 0 0 0 0 0 0 0 0 0 0 08.25- 0 0 0 0 0 0 0 0 0 0 0 0 07.75- 0 0 0 0 0 0 0 0 0 0 0 0 07.25- 0 0 0 0 0 0 0 0 0 0 0 0 06.75- 0 0 0 0 0 0 0 0 0 0 0 0 06.25- 0 0 0 0 0 0 0 0 0 0 0 0 05.75- 0 0 0 0 0 0 0 0 0 0 0 0 05.25- 0 0 0 0 0 0 0 0 0 0 0 0 04.75- 0 0 0 0 0 0 0 0 0 0 0 0 04.25- 0 0 0 0 0 0 0 0 0 0 0 0 03.75- 0 0 0 0 0 0 0 0 0 0 0 0 03.25- 0 0 0 0 0 0 0 0.000297 0.000224 0 0 0 02.75- 0 0 0 0 0 0 0.000414 0.000405 0 0 0 0 02.25- 0 0 0 0 0.000176 0.000589 0.001021 0.000847 3.91E-06 0 0 0 01.75- 0 0 0 0 0.000887 0.001245 0.002002 0.002682 5.05E-05 0 0 0 01.25- 0 0 0 1.1E-05 0.001218 0.004456 0.004191 0.001685 0.000424 0 0 0 00.75- 0 0 0 0.000378 0.001439 0.001877 0.000579 9.28E-05 1.9E-05 0 0 0 00.25- 0 0 0 0.000101 4.15E-05 3.73E-05 3.91E-06 1.86E-07 1.86E-07 0 0 0 00- 0 0 0 0 0 0 0 0 0 0 0 0 0 Ws=20(m/s) T(sec.)Hw(m) 5- 6- 7- 8- 9- 10- 11- 12- 13- 14- 15- 16-15.25- 0 0 0 0 0 0 0 0 0 0 0 014.75- 0 0 0 0 0 0 0 0 0 0 0 014.25- 0 0 0 0 0 0 0 0 0 0 0 013.75- 0 0 0 0 0 0 0 0 0 0 0 013.25- 0 0 0 0 0 0 0 0 0 0 3.73E-07 3.17E-0612.75- 0 0 0 0 0 0 0 0 0 0 3.35E-06 1.86E-0712.25- 0 0 0 0 0 0 0 0 0 0 1.12E-06 011.75- 0 0 0 0 0 0 0 0 0 0 0 011.25- 0 0 0 0 0 0 0 0 0 0 0 010.75- 0 0 0 0 0 0 0 0 0 0 0 010.25- 0 0 0 0 0 0 0 0 0 0 0 09.75- 0 0 0 0 0 0 0 0 7.83E-06 4.84E-06 0 09.25- 0 0 0 0 0 0 0 6.71E-06 8.24E-05 0 0 08.75- 0 0 0 0 0 0 0 1.58E-05 0.00011 7.45E-07 3.91E-06 08.25- 0 0 0 0 0 0 2.61E-06 7.32E-05 6.17E-05 5.78E-06 3.5E-05 07.75- 0 0 0 0 0 1.3E-06 1.01E-05 0.000168 4.84E-06 6.15E-06 1.3E-05 07.25- 0 0 0 0 0 9.69E-06 4.45E-05 0.000351 0 0 0 06.75- 0 0 0 0 2.24E-06 4.49E-05 0.000201 9.82E-05 1.49E-06 0 0 06.25- 0 0 0 0 4.47E-06 7.84E-05 0.000212 0.000119 7.27E-06 1.45E-05 0 05.75- 0 0 0 0 5.33E-05 0.000301 0.000302 0.000128 4.1E-06 3.82E-05 0 05.25- 0 0 0 5.03E-06 0.000106 0.00066 1.29E-05 6.28E-05 8.57E-06 6.15E-06 0 04.75- 0 0 0 3.78E-05 9.69E-05 0.000248 0 3.73E-06 1.86E-07 0 0 04.25- 0 0 0 5.29E-05 7.81E-05 2.5E-05 0 5.59E-07 4.84E-06 0 0 03.75- 0 0 2.96E-05 1.9E-05 3.63E-05 0 0 0 0 0 0 03.25- 0 0 1.58E-05 2.83E-05 3.06E-05 0 0 0 0 0 0 02.75- 0 0 0 2.24E-06 0 0 0 0 0 0 0 02.25- 0 0 0 0 0 0 0 0 0 0 0 01.75- 0 0 0 0 0 0 0 0 0 0 0 0 Ws=40(m/s) T(sec.)Hw(m) 8- 9- 10- 11- 12- 13- 14- 15- 16- 17-15.25- 0 0 0 0 0 0 1.86E-07 0 0 014.75- 0 0 0 0 0 0 1.49E-06 0 9.32E-07 5.59E-0714.25- 0 0 0 0 0 0 0 0 4.66E-06 013.75- 0 0 0 0 0 0 0 0 9.32E-06 013.25- 0 0 0 0 0 0 0 0 9.5E-06 012.75- 0 0 0 0 0 0 0 5.03E-06 3.17E-06 012.25- 0 0 0 0 0 0 0 0 0 011.75- 0 0 0 0 0 0 0 0 0 011.25- 0 0 0 0 7.45E-07 0 0 0 0 010.75- 0 0 0 0 0 0 0 0 0 010.25- 0 0 0 0 0 0 0 0 0 09.75- 0 0 0 0 0 0 0 0 0 09.25- 0 0 0 0 0 0 0 0 0 08.75- 0 0 0 0 0 0 0 0 0 08.25- 0 0 0 0 0 0 0 0 0 07.75- 0 0 0 0 0 0 0 0 0 07.25- 0 0 0 0 0 0 0 0 0 0

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 10 20 30 40 50 60 70 80

Angle of Heel(deg.)

GZ(

m)

Original GZ (kg=10.63m)Present method

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 et al., 1992). It will be a future task to compare this methodology with the capsizing probability calculated taking account of the statistical correlation between wind and waves. Then the final conclusion whether the statistical correlation should be considered for the performance-based criteria or not will be discussed.

Capsizing probability　(T=3600sec.)

-40.0-35.0-30.0-25.0-20.0-15.0-10.0-5.00.0

0 5 10 15 20 25 30 35

Wind speed(m/s)

log 1

0(pr

obab

ility

)

Present methodBeaufort(short term)NormalizedNo correlation

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 the safety 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 the effect of drifting attitude and velocity is ignored. The restoring arm is approximated to keep the metacentric height, 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 a containership 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 sufficient although her designed metacentric height is small. This is because the large freeboard of the containership results in significant increase of the slope of righting arm curve at larger 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 the Japanese 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 height greater than 10 meters are executed and compared with the case in the North Atlantic, as shown in Fig. 15. Here the aero- and hydrodynamic coefficients were estimated with experimental 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).

Items LPS ContainerShip

PCC RO/PAX

W (t) 53,010 109,225 27,216 14,983

Lpp (m) 242.25 283.8 192.0 170.0

B (m) 36.0 42.8 32.3 25.0

d (m) 8.4 14.0 8.2 6.06

GM (m) 1.58 1.06 1.25 1.41

T� (s) 23.0 30.3 22.0 17.9

N 4332 33 25 370

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

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 the capsizing 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

Table. 4 Principal dimension of the subject ships at full

load departure condition. Here FBS: freeboard, TZ: natural heave period, hB: height of bulwark and Af: area of freeing port. (Paroka & Umeda, 2007)

Items Units new

135 GT old 135

GT 80 GT 39 GT

Lpp BS

d

FBS Δ GM Tφ Tz hB Af

MetersMetersMetersMetersTons

MetersSecondsSecondsMetersMeters2

38.50 8.10

2.851 0.485

480.98 1.95 5.80

3.413 1.50 7.05

34.50 7.60 2.65

0.496 431.07

1.65 5.87

3.435 1.45 4.75

29.006.802.25

0.484276.61

1.575.20

3.2461.353.40

23.005.901.77

0.396145.70

1.793.8862.218

1.302.60

0.0 0 .2 0 .4 0 .6 0 .8 1 .0 1 .2 1 .4 1 .6 1 .8 2 .0

0 .0

0 .1

0 .2

0 .3

0 .4

0 .5

0 .6

0 .7

0 .8

Fre

eboar

d (m

)

M etacentric Height (m )

New 135 GT O ld 135 GT 80 GT 39 GT

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

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Fre

eboar

d (

m)

Metacentric Height (m)

New 135 GT Old 135 GT 80 GT 39 GT

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 of

four Japanese purse seiners, whose principal dimensions are listed in Table 4, as one of typical fishing vessels types. By utilizing the piece-wise linear approach with water trapped on deck taken into account, combinations of metacentric height 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 the Japanese EEZ for these vessels so that wave statistics are

3.66E-05

1.23E-05

2.92E-12

1.00E-20

1.00E-15

1.00E-10

1.00E-05

1.00E+00

NORTH ATLANTIC JPN(Limited greater coastal area） JPN(Limited greater coastal

area+Operational limitation）

PRO

BA

BIL

ITY

3.66E-05 4.98E-05

1.0E-10

1.0E-08

1.0E-06

1.0E-04

1.0E-02

1.0E+00

RO/PAX LPS CONTAINER SHIP PCC

PR

OB

AB

ILIT

Y

obtained from the NMRI database and actual fish catch records (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 of metacentric height and freeboard for marginally complying with IMO weather criterion for fishing vessels and the water-on-deck criterion in the recommendation of the Torremolinos 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 these fishing vessels because of trapped water on deck but the water-on-deck criterion in the Torremolinos Convention is comparable to the current probability-based calculation.

0 .0 0 .2 0 .4 0 .6 0 .8 1 .0 1 .2 1 .4 1 .6 1 .8 2 .0

0 .0

0 .1

0 .2

0 .3

0 .4

0 .5

0 .6

0 .7

0 .8

Fre

ebo

ard (

m)

M e ta c en tr ic H e igh t (m )

N e w 1 3 5 G T O ld 1 3 5 G T 8 0 G T 3 9 G T

Fig. 18 Marginal lines of purse seiners for the water-on-deck criterion in the IMO Torremolinos Convention. (Paroka & Umeda, 2007) RECOMMENDATION FOR IMO Performance-BASED CRITERIA

As shown in this paper, it is feasible to calculate annual capsizing probability of a given ship in beam wind and waves by utilizing the piece-wise linear approach presented here. Since a beam wind and wave condition is more dangerous than other drifting attitude, safety level of stability under dead ship condition at least as a relative measure can be estimated with this approach. It is noteworthy here the calculation based on this approach is not time-consuming so that it can be used as a performance-based criterion. In case a simpler approach is also required for a vulnerability criterion, as an existing criterion for stability under dead ship condition the weather criterion can be recommended with refinement with the present achievements such as effective wave slope coefficient and roll damping.

CONCLUSIONS

The activity of the SCAPE committee for stability under dead ship condition provides the following conclusions: 1. A piece-wise linear approach for calculating capsizing

probability in beam and waves is presented. The

calculated probability coincides with the Mote Carlo simulation results within its confidence interval for the sea state relevant to the weather criterion

2. The calculation method was extended to cover effects of flooding angle or cargo shift angle, water on deck, hydrodynamic coefficient estimation, drifting attitude and velocity and the correlation between wind and waves.

3. By utilizing the above extended methods, safety levels of passenger and cargo ships as well as fishing vessels are calculated as annual capsizing probability, and show that current designs ensures reasonable levels of safety against intact capsizing under good operation practice.

ACKNOWLEDGEMENTS

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 thank Drs. 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 in this paper. REFERENCES

Belenky, V.L., (1993): “A Capsizing Probability Computation Method”, Journal of Ship Research, Vol. 37, pp.200-207.

Belenky, V.L. (1994): “Piece-wise Linear Methods for the Probabilistic Stability Assessment for Ship in a Seaway”, Proceedings of the 5th International Conference on Stability of Ship and Ocean Vehicles, Melbourne, 5, pp.13-30.

Belenky, V.L. (1995): “Analysis of Probabilistic Balance of IMO Stability Assessment by Piece-wise Linear Method”, Marine Technology Transaction, 6:195, 5-55.

Belenky, V.L. (2003): “Nonlinear Roll with Water on Deck: Numerical Approach”, Proceedings of the 7th International Conference on Stability of Ship and Ocean Vehicles, Madrid, pp.59-80.

Bulian, G. and Francescutto, A. (2004): “A Simplified Modular Approach for the Prediction of Roll Motion due to the Combined Action of Wind and Waves”, Journal of Engineering for the Maritime Environment, Vol. 218, pp.189-212.

Bulian, G. and Francescutto, A., (2006): “Safety and Operability of Fishing Vessels in Beam and Longitudinal Waves”, International Journal of Small Craft Technology, Vol. 148 Part B2, pp. 1-16.

Francescutto, A., N. Umeda, A. Serra, G. Bulian and D. Paroka, (2004): “Experiment-Supported Weather Criterion and Its Design Impact on Large Passenger Ships”, Proceedings of the 2nd International Maritime Conference on Design for Safety, Sakai, pp.103-113.

Fujiwara, T.(2001): “An Estimation Method of Wind Forces and Moments Acting on Ships”, Mini Symposium on Prediction of Ship Manoeuvring Performance.

Ikeda, Y., Kawahara, Y. and Yokomizo, K.(1992):”A Study of Roll Characteristics of a Hard-Chine Craft ” (in Japanese), Journal of the Kansai Society of Naval Architects, Japan Vol.218, pp.215-227.

IMO (2006): “Interim Guidelines for Alternative Assessment of the Weather Criterion”，MSC.1/Circ. 1200.

Ishida, S., Taguchi, H. and Sawada, H. (2005): “Standard test procedure for Alternative Assessment of the Weather Criterion (in Japanese) ”, Conference Proceedings of the Japan Society of Naval Architects and Ocean Engineers, Vol. 1, pp. 81-84

Iskandar, B. H., N. Umeda and M. Hamamoto (2000): “Capsizing Probability of an Indonesian RoRo Passenger Ship in Irregular Beam Seas”, Journal of the Society of Naval Architects of Japan, 188, pp.183-189.

Jiang, C., Troesch, A. W. and Shaw, S. W. (2000): “Capsize criteria for ship models with memory-dependent hydrodynamics and random excitation”, Philosophical Transactions of the Royal A, Volume 358, Number 1771, pp.1761-1791.

Kashiwagi, M and Ikebuchi, T. (2003): “Performance Prediction System by Means of a Linear Calculation Method”, Proceedings of the 4th JTTC Symposium on Ship Performance at Sea (In Japanese), The Society of Naval Architects of Japan, pp.43-59.

Ma, N., Taguchi, H., Umeda, N., Hirayama, T., Matsuda, A., Amagai, K. and Ishida, S. (2004): “Some Aspects of Fishing Vessel Stability Safety in Japan”, Proceedings of the 2nd International Maritime Conference on Design for Safety, Sakai, pp.127-132.

Matsuda, A., Takahashi, H. and Hashimoto, H. (2005): “Measurement of Trapped Water-on-deck in Beam Waves with Optical Tracking Sensor (in Japanese) ”, Conference Proceedings of the Japan Society of Naval Architects and Ocean Engineers, Vol. 1, pp. 89-90

McTAGGART, K. and De KAT, J. O. (2000): “Capsizing Risk of Intact Frigates in Irregular Seas.” Transaction of the Society of Naval Architects and Marine Engineers, 108, 147-177.

Mizuno, T. (1973): “On the Effective Wave Slope in the Rolling Motion of a Ship among Waves (in Japanese)”, Journal of the Society of Naval Architects of Japan, Vol. 134, pp. 85-102

Munif, A., Hashimoto, H. and UMEDA, N. (2004): “Stability Assessment for a Japanese Purse Seiner in Seaways”, Proceedings of the 2nd Asia-Pacific Workshop on Marine Hydrodynamics, Busan: Pusan National University, 271-275.

Ogawa, Y. (2006a): “The effect of the correlation of winds with waves on the capsizing probability under dead ship condition (in Japanese)”, Conference Proceedings of the Japan Society of Naval Architects and Ocean Engineers, Vol. 3, pp. 189-192

Ogawa, Y. et.al. (2006b): “The effect of drift motion on the capsizing probability under dead ship condition”, Proceedings of 9th International Conference on Stability of Ships and Ocean Vehicles.

Ogawa, Y. (2007): “The study of the effect of distributions of wind directions and wave directions on the capsizing probability under dead ship condition (in Japanese)”, Conference Proceedings of the Japan Society of Naval Architects and Ocean Engineers, Vol. 4, pp. 167-170

Paroka, D., Ohkura, Y. and Umeda, N. (2006):”Analytical Prediction of Capsizing Probability of a Ship in Beam Wind and Waves”, Journal of Ship Research, Vol. 50, No.2, pp. 187-195

Paroka, D., and Umeda, N. (2006a): “Piece-wise Linear Approach for Calculating Probability of Roll Angle Exceeding Critical Value in Beam Seas”, Conference Proceedings of the Japan Society of Naval Architects and Ocean Engineers, Vol. 3, pp. 181-184

Paroka, D. and Umeda, N. (2006b): “Capsizing Probability Prediction for a Large Passenger Ship in Irregular Beam Wind and Waves waves: comparison of analytical and numerical methods”, Journal of Ship Research, Vol. 50, No. 4, pp.371-377.

Paroka, D. and Umeda, N. (2006c): “Prediction of Capsizing Probability for a Ship with Trapped Water on Deck”, Journal of Marine Science and Technology, Vol. 11, No. 4, pp. 237-244.

Paroka, D. and Umeda, N. (2007): “The Effect of Freeboard and Metacentric height on Capsizing Probability of Purse Seiners in Irregular Beam Seas”, Journal of Marine Science and Technology, Vol. 12, No. 3, pp. 150-159.

Price, W. G. and Bishop, R.E.D. (1974): “Probabilistic Theory of Ship Dynamics”, London: Chapman and Hall Ltd.

Sato, Y., Taguchi, H., Ueno, M. and Sawada, H. (2007): “An Experimental Study of Effective Wave Slope Coefficient for Two-Dimensional Models (in Japanese)”, Conference Proceedings of the Japan Society of Naval Architects and Ocean Engineers, Vol. 5, pp. 27-28

Shen, D. and Huang, X. (2000): “The Study of Lasting Time Before Capsize of a Ship under Irregular Wave Excitation.” Proceedings of the 7th International Conference on Stability of Ships and Ocean Vehicles, Launceston: The Australian Maritime Engineering CRC Ltd, B: 710-723.

Shienberg, R. et.al. (2006): “Estimating probability of Capsizing for Operator Guidance”, Proceedings of 9th International Conference on Stability of Ships and Ocean Vehicles.

Taguchi, H., Ishida, S. and Sawada, H. (2005): “A Trial Experiment on the IMO Draft Guidelines for Alternative Assessment of the Weather Criterion”, Proceeding of the 8th International Ship Stability Workshop , Istanbul.

Tasai, F. and Takagi, M. (1969): “Theory and calculation method for response in regular waves”, Seakeeping Symposium (in Japanese), The Society of Naval Architects of Japan.

Umeda, N., Ikeda, Y., and Suzuki., S. (1992) : “Risk Analysis Applied to the Capsizing of High-Speed Craft in Beam Seas”, Proceedings of the 5th International Symposium on the Practical Design of Ships and Mobile Units, New Castle upon Tyne, , Vol.2, pp.1131-1145.

Umeda, N., Iskandar, B. H. , Hashimoto, H., Urano, S. and Matsuda, A.(2002): “Comparison of European and Asian Trawlers –Stability in Seaways-“ , Proceedings of Asia Pacific Workshop on Marine Hydrodynamics, Kobe, pp. 79-84.

Umeda, N., Ohkura, Y., Urano, S. and Hori, M. (2004): “Some Remarks on Theoretical Modelling of Intact Stability”, Proceedings of the 7th International Ship Stability Workshop, Shanghai, pp. 85-91.

Umeda, N., Ueda, J., Paroka, D., Bulian, G. and Hashimoto, H. (2006a): “Examination of Experiment-Supported Weather Criterion with a RoPax Ferry Model”, Conference Proceedings of the Japan Society of Naval Architects and Ocean Engineers, Vol. 2, pp. 5-8

Umeda, N., Koga, S., Ogawa, Y. and Paroka, D. (2006b): “Prediction Method for Stability of a RoPax Ferry under Dead Ship Condition (in Japanese)”, Conference Proceedings of the Japan Society of Naval Architects and Ocean Engineers, Vol. 3, pp. 185-188

Umeda, N., Koga, S., Ueda, J., Maeda, E., Tsukamoto, I., and Paroka, D. (2007a): “Methodology for Calculating Capsizing Probability for a Ship under Dead Ship Condition”, Proceedings of the 9th International Ship Stability Workshop, Hamburg, pp. 1.2.1-1.2.19.

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 Coefficient and its Effect on Capsizing Probability Calculation (in Japanese)”, Conference Proceedings of the Japan Society of 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, Load Lines and Stability of Ships, The Society of Naval Architects of Japan.

Yoshimura Y. (1988): Mathematical Model for the Manoeuvring Ship Motion in Shallow Water (2nd Report) (in Japanese), Journal of the Kansai Society of Naval Architects, Japan Vol.210, pp.77-84.

����������� � ��� �� ��������� � ����

�� ���� ���������

� ���� ��� � ���� ������� ���� �� �

������ ����

���������� ������ � ������������ ����� ���� ���� ����������� ������ �����

��� ����� ������

������ ��������� ����� ���� ���� ����������� ������ �����

��������

��� �� ���������� ��� ��������� �� ��� � � � �

�� ���������� ������ ����� �� ��� ���!��� �������

��� ����� "��� ���� �� ��������� "�#��� �� ����

$��� ��� $��� $��� ������� �� ������% ��� �����������

�&����� ������ � ������� � '�"��� ��� ��� ���

� ���� �� ����� ����� $��� ����� ��!�� �� ��� ���

� ��"���� �� �"�� � �("���� ��� ���!����� �� ��� �$�

��� �% ����) ��� ��� ������ �� �� $��� ��� $���

�������� ���� $��� ���"�� � �� ���� ��� ����� � $���

� ��� ���!��� ��������� � ����"����% �� �"�� �"�� �

���� ����"������ �� � �� �� ����� �� �����) ��� ���

������ �"�� � ���� �� $��� ���� � �� ��� ����� �� ��

��"������ ��� ���� ��� ��� �� ��� ����� ���� " ���� �

�����������% �������) ������� �"� �� ��� � ���� � �

��� *��%

+,- .��/�0 ��������� ������� �� ���� ����� ����

����� ��� �� �������� � �� �� ������� �������� ��

����� �� ����� ����� ��� ��� !��� � � � ��"���

#$%���&�%#�$

#� �� � ����� � �� �� " �� �� "������� �� � ������ �

" ���� ������ �&����� '(()�* %��� � ��� ����� � �� �

� ��������� ������� � �� ��� �� " � ���� ���� ���� ����

���� �� ���+�� ���� ,��� ��� ,�-��* .�-��� �� ��

�� ��-� ���� ����� �� ��� � �� ��������� �������

� �� ,��� �� ,�-��������� ���� ��� ,��� �/�� ��� ��

��� ������ �� ,��� �� ������ ���� �������� �� ��

���� �������* ����� ��� 0�1� �� ��� �0�1� �� ���� '((2�

����� �� ����� ���� ����� �� �� �� ����� � �� ������

,��� ����� ���/��� �� " �� ��� ��� �� ��-�� ����

���1�� 3445�* #� ���� ��� 0�1� �� ��� �0�1� ��� &�����

'((2� ������ �� ������� ��� ������� " �� ����� ��

�� �� ,� � �������� �/������ �*

�� �� �� ����� �� 6� �� �������� � �� ��

������ �%�" ����� ����� ��� �� ��� 34)2� ��� ����

,����� ���� �-���� � � "����� ������� � �� �� 6��� "

� �� ��� �������� � ����������* � �� �������� � ��������

" ����-���� ����� ���� �� ���� -��� ��� �� 7��

������� ��� �� ��� �� ��-� ���� ����� � ��,�� ��

����� ���� ��� 7��������� '(((�7 �� ���� '((8�* #� ��

�� ��� �� ������� ��� �� �������� �� � �� " � ������

���� ���� ���� -������� ���� "� ����� ����� ����

�� " � �,� ���� �� ����� � �� ���� ����� ���

� ������� �� ���� 34)2�* ����� ��� 9����� ��� �� ��� ��

��-� ���� ��� �� �� ���� � ����� � �� �� �� � �����

�� ,�-�� �9����� ��� ������ '((2� ��� �� � �������

����� " ����� �� ����� �9������ '((:�* %�� �����

��-� ����� � �� �� ������� " �� ���� ���� ��

�-���� �� " �� ������ ��� �/�� � �� �� ,��� �� �� ����

����� �* ;,�-�� �� �� ���� " �������� �������� �����

,�� � ������ ��������*

#� ��� � �� �� �� 6� �� �������� � �� �� �� �������

�� ��� � �� ��������� ������� � ���� ���� ���� �����

�� ���+�� ���� ,��� ��� ,�-��* #� �� �� ������

�� ����� �� ��� �� �� ��� � �������� �������� �����

�� ���� ��� �� ��� �� ��� ��� " �� ���� � � � ��"���

�� ��-�� ��� ��* %���� �������� ������� �� " �� ����

�������� ���� ������ ����� �0�1� �� ���� '((2� 0�1�

��� &����� '((2� �� �� �/������ �� �� ���� " �� �����

���� �������� ����� ��� �� ����� ��� �� " �� ������

��� �/�� � �� �� ,��� �� ��������� ������� � �� �-���� ��*

%���� �� ��� � " �� ���� � � � ��"��� " �� ���������

����� �� �/����* �������� �� ������� ������ " ��� ���

���� �� ���1��*

���<!#$� �� ��!! ��%#�$ #$ �<=� >#$� =$�

>=?<.

%�� ����� � �� " � ���� ���� ���� ���� ���� �� ����

+�� ���� ,��� ��� ,�-�� �� ����� � �� �� ���* 3 ��� ��

������ �� �� "��,��� ������������� �� � �� � ���

�� �0�1� �� ���� '((2�0�1� ��� &����� '((2�@

���� � ���� A� � �� B�� � ���� � �������� � �������� �3�

,��� ��� �� �� ���� " ��� �� " �� ����� ��� �� �� ���

�������� ������ " ����� ��� ��� �� B�� �� �� �����

������� ���� � � �� �� ���� ����������� � ��� ��� ��

φ

t

φs

φ

0

Wave

M( . )(t)

t0

Wind

tMwind

���% 1 ������ � �� ��� ���� ,��� ��� ,�-��

0

φ (deg)

GZ(φ)

7th polynomialapproximation

GZ(φ) by hydrostatic analysis

φf φv

φ2

dGZ(φ)dφ φ=0

= GM

GZM

φs

GZ(φs)

���% 2 .�-�� � �������� ���/��� �� " �� ��� ���

��

�� ��� ��� ��* ����� �� �� ,����������� ���� � ���

����� �� �� ,�-� �/�� ��� ���� ����� � �� �����

7��- ������ ��* %���� �/�� ��� ���� � �� �������

��-� -��� ��� ��� -��� �� " �� ,��� -���� �*

%�� ��� ��� �� ��-� ��� ����� � ����������

�������� �� ���/��� �� �� �� "��,��� ��-�� ���� ����

������@

��� � ��� � ���� � ���

� � ���� �'�

%�� ������ �� ��� ��� �� ��� �� �� �� ������ "� �� "��

�,��� ���� ���@ ������� � � � (� ���� �� ��/���� -����

�1�� � � � ��� ����� �� ����� " -�������� � ����� �� ��� ��

��,� �� ���* '* %��� ���/��� �� �� -���� ��� �� �� ����

" � � ��� ������� �� ������� � " �� ���� ��-���� "�

� � �� ����� ���� ��� �� ����� " -�������� � ����� � ��

������ ��*

%�� � �������� ����� ������ �� ���+�� �� ���� ,���

-���� � C � �� � ����� "� �� "��,��� � �� ��@

��� �(�8������

C ��� �

��5�

%�� �� ��� ����� �����"� ����� " �� ���� �� ��-�� �� ��

"��,��� � �� ��*

�� �

��

���

���� � �����D�

=� �� ������� ���� � �� "��,��� ������� ��� ���

���� �� ��������@

�� B�� �3

���� � �����'� B� � � B�� B��� �8�

,��� � ��� � ������ � �� ����� ��� ���� �� ������� ��

��� ������ �� ����� �-���*

%�� ,����������� ���� �� �������� �� ����� ����

,��� -���� � C � ��� ����-���� ,��� -���� � ���

�&���� �� ���� 344'�@

�������� � (�8������� �� C �� � ' C � ���� �2�

,��� ���� �� �� �� ����� �� �� �� �� ��������� ��� ��

����� � � �� �� �� ��� ,������ ���� ��� � �� �� ����� "

�� ��� � " �� ,��� "�� "� �� ��� � " �� ������

����� ��� �� "��*

%�� ���� �� " �� ��� � ,��� ,� � �� ,��� "� ����� �

�� ������ �� ��-�� ���� �� �� "��,�@

� ����� ��������

C �� ���������������� �:�

���� �3�������3 �

����

C �

����������)�

� ������� � D�C ��

�

���

�3 � �������

�4�

,���

� � (�(5 �3(�

�� � 2((�

C �

�33�

%�� �����7��- ,�-� �/�� ��� ���� ��� �� ������

�� �� �� �� "��,��� � �� �� �&���� �� ���� 344'�@

�������� � ������ �3'�

,��� ��� ���� �� �� ���� �-� ,�-� ���� ������ ���

� ,�-� ����� ����� �-���* %�� ,�-� �/�� ��� ���� ��� ��

�� ��� �� "� �� ,�-� ���� ���� �� �� "��,�@

������ � ��

��� ���� �35�

� ���� ���

��� ������� �

�

����/�

�� �

��

��3D�

,��� �� ���� �-� ,�-� ���� ������ �� �� ������ ��

"��,�@

�

�������(�::: �" � ��D���

(�( ��,����38�

%�� ��� � ��� � �� �������� �� ����� �� �����6��� ,�-�

����� ���� ��� �� ���� ,�-� ���� ��� �� "��,�@

� � 3:'�:8��

���

� ���

�32�

� �243

� ���

�3:�

����� C �� � � ' � 3(�� C �� � )�' � 3(�� C �

�

� (�3D82 C � � (�3844 �3)�

��� �5�)2����� �34�

,��� �� ���� ,�-� ���� �� ������� �� � ����� ��

����� 0�������1,� �E� "����*

>��� �� ��� � ,��� �� �������� �� �������� ��� ��� ��

,� � �� ���� ��� � ����� ��-�� �� ��� � �� �,� �����

� " �� ,����������� ��� ,�-� �/�� ��� ���� � ��� ��

ω

(ω)

S(

)

ω i-1 ωnω iω1 i+1ω

���% 3 ����� � "� ����� ���/��� �� " �,� �����

��

�������� �� �� "��,��� ����� ���� ���/��� �� ,� � �

����� � "� ������� ��� �� � 3� � � � � �� ��� '� � ���������

���� ���� -�������� ���� C���� �� � 3� � � � � �� �� ��,� ��

���* 5 ��� 7��������� '(((�7 �� ���� '((8�@

������

���� � �����

�����

����

����

������� � �

��

�C���

������

��'(�

�

�����

����

�

����� C��

�������� � ��

��'3�

,��� �� ������� ��� ������ �� �� ��� �� ��� �� ,����

������� ,�-� �/�� ��� ���� �� ����� �-���* � ������ ��

�� ����� ����� " �� �� � "� ����� ������ ��� �� ���

������ �� ����� �� � ��������� ���� ���� -�������@

� � ����C��

��� �� � 3� � � � � �� �''�

,��� ���

��� ��� �� �� � ����� ��-�� �� �

��

�

��� "��,�

������� �� ���� 34)2�@

���

��

3

���� � �����

��

��'5�

���

�

��

� ��������

�

� ����� ��

���

�

��

� ����������

����������

� ����� ��� �� � '� � � � � � � 3�

����

��

� �

����������

� ����� ��

,��� � ����� � "� ������� " ���� �/�� ��� ���� ��

���"���� ����� �� ������ �� ������ �,� ���� �*

�=0.#F#$� 0���=�#!#%G <?=!&=%#�$ �G �#�.%

���<� �<!#=�#!#%G �<%;�� ������

%�� "����� ������� � " �� ����-����� �-�� �� �-����

� �� �� �� ������� � �� �� "����� �-�� ,��� ��� �� ��

�����6�� ��� � �� "��,�@

� � 0�H���� �� � ( " ( � � � � I �'D�

,��� ���� �� �� � ���� � � � "��� �� �� ��� " ���� -���

� ��� ��� �* %��� 1��� " ����� �� ������ �� 6�

������� �����* %�� 6� �� �������� � �� �� ������

�%�" ����� ����� ��� �� ��� 34)2� " ���� 6� �������

00 Time

φs

Roo

l ang

le

tmTransient response steady-state response

φ(u, u, tm)

���% 4 ��/���� �� ������ ����� �� � ������ � � ��

�����

U1

U2

O

Linearized

φ ( )U

( ) = 0h U

Failure region

βu*

Df

Safety region

���% 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�� �� " ��

���� �������� ���� �0�1� �� ���� '((2�0�1� ��� &�����

'((2�

���� ������� �� ! � "�###

��� ������� ��" ! ��"� �� ��� ��"$

� ���� %� %� ��"� &�� ����� '�!"

�� ��� ( � ! � ����� )�! ��"��

�� ���� ))�%� � �����*� "�()!

� ��� "�)$ �� ����� )�$"# ��"��

�� ����(# �� � ��� �� "� (''

"�""( � � ��� ��� ��%"(( ��"��

�� ���*��� "��#'% � ���� %� $") ��"�

����� 2 ������ � " �� ��-�� ���� �������� � ���

�� " ��-�� < * �'� �� �����*

�� �� �� ��

��"$" "��#%" +$�% $' !�(( (

�� "��,� ����1,� � ��� �������� 34:)�@

�����

������������ � ������

���������� ������� �'2�

,��� �� �������� ��� ��� �� �� � �� �� �����*

%�� � �� �� �� �� ����� �� �� �� ��"���� �� ��-��

����� �� ���� �������� � (* %�� � ����� ��� ��

��� �� �� �� ��� ������ � ������ ��� * %�� �������� �

����/ �� �-���� �� �� � �������*

5* %�� "����� ������� � �� �-���� �� �� ���������� ��

���� � � � "��� �� ���� �� ������ ��� � �� �� "��

�,�@

� � ����� �� ��

��

3�'

�/�

�� �

�

'

��� �':�

$&�<�#�=! <K=�0!<.

=������ �� " �� ���� �� ��� � �� ��������� ���

����� � �� ��-�� ��� �� �� �������� �/����� " � ���� ����

���� ���� ���� ���� �� ���+�� ���� ,��� ��� ,�-��

� ����� �� 0�1� �� ��� �0�1� �� ���� '((2� 0�1� ���

&����� '((2�* %�� �����6�� �� " �� ���� �� ��� �� �� %��

��� 3* %�� ��� ��� �� ��-� ��6��� �� < * �'� �� ����� � ��

�� ���* '� ,��� �� ����� " -�������� � ����� � �� �� � ���

DD ������ ��� �� ������ � �� ��� �� �� %���� '*

%�� ���� ,��� -���� � �� ���� �� �� ����� C � �

'4�8�����* %�� �,� ���� � " �� ,��� ��� �� ,�-� "

��� ���� ,��� -���� � �� ����� � �� �� ���* 2*

%�� �,� ���� � �� ����� ���� �� � � 33 ����� � "��

����� ��� �* %�� "� ����� ��� �� ��������� -������

��6��� �� < * �'5� �� ��� �� �� %���� 5* %�� �������� � �� ���

��� �� "� D� � DD � ��������� ���� ���� -��������

����

�� C�

��

��� �� � 3� � � � � 33�*

%�� �� ������ �� ���������� �-���� �� �� �� "� ��

�� ������7� � �� �� ���� �� ��� ��� ���� �� �� ��

��� ��� �� ����� �� �� �� � �������� �� ����� � " ��

���� ,��� -���� � ��� �� �� ������ -���� � �� �� ��

� � � (* %�� ������� " �� ������ �������� �� ��6��� ��

������� ,� � 0�1� �� ����E ���� � �0�1� �� ���� '((2�*

1013

1014

1015

1016

0.01 0.1 1 10ω (rad/s)

S(·)

(ω)

(N

m)2 /

(rad

/s)

Sam(ω) Smw(ω)WaveWind

���% 6 0,� ���� � " ,����������� ��� ,�-� �/�� ���

���� � " ���� ,��� -���� �� '4*8 ��� �0�1� �� ����

'((2�0�1� ��� &����� '((2�

����� 3 ����� � "� ����� ���/��� �� " �,� ���� �

" ,����������� ��� ,�-� �/�� �� ���� � " ���� ,���

-���� �� '4*8 ���

,��* ,���

��

�� ����� � ��

�� ����� �

���*��� �

������� ���*��� �

�������

� "�"$ !�$((��"�� "� " $�"! ��"�

"�"' %�"�'��"�� "� ! %�"$"��"��

( "�� %�"#$��"�� "�(" $�!))��"��

% "��! (��'%��"�� "�(! �()���"��

! "��# �(!!��"�� "�%" %�((!��"��

$ "��' �$$)��"�� "�%! !�# $��"��

# "� �)"(��"�� "�!" '�)(���"��

) "� ! (�� #��"�� "�$" ��''$��"��

' "�(" %�($#��"�� "�)" �)!#��"��

�" "�%" )� #$��"�� ���" (�"#���"��

�� ��"" (�!)!��"�� ��!" %�"%#��"��

��������� � ! "#$�

%�� ���� �� ����� �� " �� ���� � � � "��� ��� < * �'8�

�� �� 58 ������ �� �� ������ ��� �� ����� " -������

��� � ����� � �-�� �������� ��� ����� � ������ �� ��

����* %�� 6�� � �������� ���/��� �� �� ���� " �����

�� �-���� �� " �� ���� � � � "��� �� �� < * �'2�* �� ��

��������� ��������� �� ��� ���� �� �� � � 8((�����

,��� �� ��������� �� �� � � ���� -��� �� � ��� �� ���

���� * # �� ��6��� "� �� � ��� ������ ��� �� �� ��

�� �� -���� " �� ��������� ������� � �� L� � (�':� ,���

���� ������ �� ����� �� �� ����� ������� %,�� �

��� ��� ��� $�������� 344)�*

%�� ��-������ ��� � " �� ���� �� ����� � �� ��

���* :* $������� ������ �� ����� �� ���� � ��� ���

��, ��-������ �� �� � � ��� ���� �� ������ �� �����

���� � " �� 6�� � �������� ���/��� ��* %�� ��-��

����� ,��� �� ���-�� �� �� ������ ����� ���� ������ ��

� �� ��� ��� ������� �������� �� �� ��� ��� "

���1,� ��������� ���� ��*

<��� ����� � " �� � ����� ������ ��� �� ��� �� ��

%���� D* # �� "��� "� �� ����� � -���� " ���� ����

�� � "� ����� �� �� ���� " ,�-� �/�� ��� ���� �

�� ��������� ������� � �� ���������� �� ������� ,� �

�� " ,����������� ���� * #� ���� ��� �� �,� "��

�������� ����������� �� � (�3:� ��� �� � (�34������ "

�� ,����������� ���� ���� �� �� ��� �� �����"��

������ �� � (�3:4D������� ��-� � �����6��� ���� � ��

10-4

10-3

10-2

10-1

100

101

1.8

2.0

2.2

2.4

2.6

2.8

0 10 20 30 40 50 60 70

Limit state function

Reliability index

Lim

it st

ate

func

tion

φ0 -

φ (d

eg)

Reliability index

Iteration

���% 7 ��-������ ��� � " ���� � � � "��� �� ��� ����

����� � ����/

����� 4 ����� � " ������ ��� � ����� �� ����

� -��* � -��* &� -��� � -��� &�

� +"�($�� "�!#'� +"�"" (� +"�"""'$

+"�%'#$ "�%%�$ +"�"� " "�" "(

( ������� "� )'% "�"(!% "�"(""

% ������� +"� '(! "�" %" "�"�'!

! ������ ��� "�"" '! +"�"""�$

$ "��%$! ���� +"�""(%! "�""�#

# +"�"#%) "�( )" "�""((! "�" %$

) +"��"!% "��#)( "�""%)! +"�" �#

' "�"'�( "�"## +"�""'$' "�""%��

�" "�""%") +"�""" +"�""$#" "�"""!!

�� "�""� % "�""�$( "�"""'$ +"�""("#

-10

0

10

20

30

40

0 200 400 600 800 1000 1200

φ (deg)

dφ/dt (deg/s)

φ (d

eg)

dφ/

dt (d

eg/s

)

Time (sec)

���% 8 ��� ����� ������ ������ ��� �/�� � ��

�������� �* %�� �� ������ ��� � ��� �� ������ ���

�/�� � �� �� ����� � �� �� ���* )* # �� "��� �� �� �� �����

������ �� ���� ���� � 58 ������ ���� � D((�����*

�� �� � ����� �������� � ����/ � � '�D22� �� �������

��� ������� � �� �-���� �� �� "��,�@

� � ���'�D22� � :�3458 � 3(�� �')�

%��� -���� �� 5( ���� ������ ��� �� � ����� �� �� �� �

��� ������ ��� L� � (�':*

%�� ���������� ������ �� �� �� ����� ���/��� �� "

�� ���� � � � "��� �� �� � ��� ���� " �-���� ��� �� ����

������ ������� �* �� ��� �� ���� � � � "��� �� �� ������

��� �� � ���� ������� ��� ��-� ��� ���� ������ ��� �

�� ��,� �� ���* 4*

�������� ��%��� � ���% ��� %�&��% %'%��� � ���

��� ���� ������ ��� � �� ������� �� ���1,� ���������

���� ��� � � ��� ,� � ������ ��� ��� ��� �* ���* 3( ���

U1

U2

h(u)=0

β1

β2β4β3

Safetyregion

Failure regionh1(u)=0

h2(u)=0

h3(u)=0

h4(u)=0

���% 9 ��� ���� ��� ���/��� �� " ���� � � � "��� ��

2.440

2.445

2.450

2.455

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

Rel

iabi

lity

inde

x

Trial No.��� .�/��0�/��� ��*�1

00.20.40.60.8

11.21.41.6

0.06 0.09 0.12 0.15 0.17 0.19 0.22 0.25 0.3

Mag

nitu

de

Frequency (rad/s)

�0� ���/���*� *����� ���� �1 ������ -��* �������

-90

-60

-30

0

30

60

90

0.06 0.09 0.12 0.15 0.17 0.19 0.22 0.25 0.3

Phas

e (d

eg)

Frequency (rad/s)� � �2��� *����� ���� �1 ������ -��* �������

���% 1: ������ ��� � � ����� "� '( ������ ����� ��

� � ��� ��� �

��,� �� �������� � ������� " '( �������� ���� � " ���

���� ����� �� ������ ��� ��� ��� �* �� " '( ���� 38

���� �� ������ ��� � �� � ����� ��� ���� ��� ��� ����

�� ���� ��� ���� "� �� ����� ����� ��� ���� �� ����

�������� � ����/*

%�� ��������� �-�� �� ������ �� � ����� �� �� ����� ���

" �� 38 ������ ��� �* %���� �� �,� ��� ���� ���� � "

�� ��������� ������� � � � � � �� �� �-���� �� �� �

"��,��� �� ��-���E� ���� ��� ��-���� 34:4�@

� � �H"�I �

�����

��/

��������H"�I ��������

�H"� "�I� (

������ �'4�

�� �

�����

�H"�I ������

��/���������� �����

�H"� "�I �5(�

%���� �� ��������� ������� � �� ������ �� "��,�@

(�(32D � � � (�(5'' �53�

%�� -���� �� ���� ��� �� � ����� �� < * �')�* ;,�-�� �

�� � ��� "� "� �� �� �� � ��� ������ ��� L� � (�':*

%��� ������ �� �� ���� �� ������ ��� � �� ��������

" �-���� ��� �� ��������� ������� �* %��� ���� ��� ���

����� �� � �� � ���� ��� �� ����� ��� ���� ������

��� � �������� ���� �� ���� �� ���/��� �������� ���

������ �� ����� �� ,1 ������ ��"��� ���/����

�� �.�� ��� �� ��� 344'� .�� ��� �� ��� 3442��

� ���� �� ������ ��� 7�������� ��� ��1������� 344)�

� ����� � ����� �� �� �� %�����*

���* 3( ��� ��� ��� ��, �� ����� ��� ��� ����� �����

" �,� ���� ����� � "� ����� ������ � " �� ,����

������� ���� " �� � ����� ������ ��� �� ����� �-���*

%��� ������ � ��-� ���� ���� � �� �������� �� ,���

�� ����� ��� ����� � ���

����� ��� �� ����� ����� � �� �-����

� �� "� < �* �'3� ��� �''�* # �� "��� "� ��� ���� ��

��� �� ��� ���� ������ ��� � ,� � ���� ���� ���� ��������

� � �������* #� ���� ��� ���� ����� � "� ����� ������

" �� ��� ���� ������ ��� � ��� �� ���� ���� ����� ��� ��

������ ����� �����* %��� ���� ������� �� �� ����� " ��

���� � � � ��"��� ,��� �� � 1��� " �� ��������*

(���� %���� %�&!���

%�� ��-�� ��� �� " �� ���� � � � ��"��� ��� � �� ��

" �� �� ��� �� "�� ����"� ������ "�� � "

�� ��������� ������� � �� ,��� �� �� ��-������ " ��

��� ���� ������ ��� ��������* � �/������ �� ��� ,� �

����� � ����� ��C��� � �� �� ����� ���� ���� �� ����

" �� ������ ��� ����� �� ���� � C��� � ���� �� ������ ��

������� � " �� ������ ��� �� ������� �� ������� ��� ,���

����� �� ���� ,�-� ������� ���� *

� �� ������ �� ����� " �� ���� � � � ��"��� �� ��-���

��� �� �� �� ,����������� ����� �� ����� ��� , ����

��� " �� ���� -�������* %�� ��-�� ��� �� �� ��"���

" ������ ������ �� ��� C���� � 5� D� 8� " ����� � ,���

"� ����� ������ �@ �� � (�3'������� �� � (�38�������

��� �� � (�3:������* %�� ����� " �� ���� � � � ��"��� ��

�� ��� �� ���� � ��� ������ �� ,� � ) � 3((( �������*

�� � �� ����% !& � ��� %��� !&������' #" �� ��

������ ����� ������� ��� � �� ����� ��� " �� ����

���� ���

����

�� C��

������ �� ����� " �� ���� � � � ��"���

,��� �� � ����� �� ��MC�� �����* %�� ���� � " �� ��� ����� �

"� ����� ������ � ���� � � � � ��� �� ��,� �� ���* 33* #�

�� 6���� �� ��� �� �� ��"� � ���� �� ������ �� �� � ���

���� � ��� �� �� �� "����� ���� �� � ���� �* %�� �����" �� ���� � � � ��"��� ��� �� ����� �� ��� �� "� ��

������* # �� "��� �� �� ���� � � � ��"��� �� �����

�� ����� ��� �� �� ����� �� ������ " � ������ "��

����� ������ *

-10

-5

0

5

10

-10 -5 0 5 10

u 3

u3

��� �� � "� ���*���

-10

-5

0

5

10

-10 -5 0 5 10u 4

u4

�0� �� � "�!���*���

-12

-8

-4

0

4

8

12

-12 -8 -4 0 4 8 12

u 5

u5

� � �� � "�#���*���

���% 11 <� ��� �� " ���� � � � ��"��� " ���� C���� �� �

5� D� 8�

�� � �� ����% !& � ����&��� !&������' � �� ����%

%�� ,��� �/�� �� ���� ,��� ��-� � ��� "� �� , ��"�

"��� "� ����� ������ �� ���� C���� �� �� C� ��� �� � *�* %��

����� ����� ��� " �� ���� �� ��-�� �� "��,�@

��

����� C��

�� � �

����� C��

��5'�

>��� �� , ���� -������� �� ����� �� � " �� "�

-�������@ ���� � ��� ���� C� ��� �C��� � ��� ��� �C��� C� ��� �� ����� ���

��� ��� " �� �/�� ��� ���� �� ��-�� �� "��,�@�������������������

������ � � ��� �������� � � ��C� �����C��� � � ��� �����C��� � � ��C� ��

�� � *� �55�

-10

-5

0

5

10

-10 -5 0 5 10

u 4

u3

��� ���� ���

-10

-5

0

5

10

-10 -5 0 5 10

u 4

u3

�0� ���� &���

-10

-5

0

5

10

-10 -5 0 5 10

u 5

u3

� � ���� ���

-10

-5

0

5

10

-10 -5 0 5 10

u 5

u3

�*� �&��� ���

���% 12 <� ��� �� " ���� � � � ��"��� ,� � ������ ����

�� � "� ����� ������

#" �� ����� ������ ����� ������� � �� ����� ����� ����

�� ���� � � � ��"��� ,��� �� � ������� ����� ,� � ����� �

���� �� ����*

$������� �/������ �� ��-�� "� �� �����6��� "��

����� ������ � �� ��,� �� ���* 3'* %�� ���� � � � ���

"��� �� ����� � ������ ,��� ���� ����� ����� � "

���� ������ * � �� �� ������ �� " �� ������ �

�� ����� ��� � �� ������ �� � ������� �� �� ���� � � �

��"��� ����� � " �� ����� ��"��� ,� � ����� �� ��

���*

%��� ���� ,��� �� ���"�� " ��-������ �� ����� ����

���� ������ ��� �������� ���� �� " �-���� ��� �� ����

������ ������� � ,� � ���� �������*

��$�!&.#�$.

#� ��� � ���� �� 6� �� �������� � �� �� ������ ��

������� �� ��� � �� ��������� ������� � ���� ���� ����

���� �� ���+�� ���� ,��� ��� ,�-��* %���� � ���

������ �/������ � �� ����6�� �� �� ���� �� ����������

�� �-���� �� " �� ���� ��������� ������� � �� �����

"� ����� ����� ��� �� ���/��� �� " �,� ���� �* <��

��������� � �� ���"�� � ��� � ������ ��� ��6� ��

������ ��� �/�� � ��*

;,�-�� �� ������� " �� ��������� ������� � �� ���*

%��� ���� ������� �� ��� ���� � � � "��� �� �� � ����

������ � �� � ��� ��� ���� ������ ��� �* %���� ���

������ �/������ � � �� ����6�� �� ��� �� ��� ���� ���

���� ��� � �� ��-� ���� ���� ���� �������� � ������� ��� ��

���� ��� �� ��-� �� ����� ��� " ���� ����� � "� �����

������ ,� � ���� ���� ���� ����� ��� �� �� ������

����� ������*

%���� �� ����� " �� ���� � � � "��� �� �� ��-�� ��� ��*

# �� "��� �� �� ���� � � � ��"��� " �� ���� ����� �

"� ����� ������ �� ����� �����* �� �� �� �����

�� " �� ������ ����� � "� ����� ������ � �� �����

������� �����* %��� ���� �� ���"�� ��-��� �� �����

��� ���� ������ ��� �������� ���� ��� �� "� ��*

#� �� �� ��� � �� ��������� ������� � �/�� ��� � ��

� ���� ��-��� �� ����� ��� ���� ������ ��� ������

��� ���� �� �� ������� �� " ��� ���� *

=�7$�>!<��<�<$%.

%��� ,1 ,�� ���� �� �� ��� ��� =�� " .���� �6�

������� " �� 9���� .��� � " 0� �� " .������

�$* 3)52(D38�* %�� �� �� ,��� �/���� ��� �������

� �� 0"* &����� ���1� &��-��� � " ��1��� -�������

����� � � ��� � ���*

�<�<�<$�<.

�����1�� ?* !* �3445�� N= ��������� 0������ � ���� ��

�� �� ��O� + �&��� ! ���� $�%��&��� ?�* 5:� $* 5�

��* '((�'(:*

�� 7��������� =* ��� ��1������� %* �344)�� N��� ���� ���

���� 0�� � �� ��� ��� .�������� ��������� �O� ��&��,

��&�� ��!��'� ?�* '(� $* 3� ��* 5:�D4*

�� 7��������� =* �'(((�� N%�� ���� � " �����?����

��� ��� .�� ��� �� ���� ��� .���O� �& -�-���%���

.������&��� ��������%� ?�* 38� $* 3� ��* )3�4(

�� ��-���� �* �34:4�� N$�, ��������� � ����� " . ���

��� .�� ���O� + �&��� ! ��&����&�� ��������%� ?�* :�

$* D� ��* D85�D:'

9������ 9* 9* �'((:�� N<���� <� ��� �� " </ ��� $��

����� ��� � ��� ����� �� ��� ��� ��������� �

�� �� ������O� + �&��� ! ��&��� ������� ��� ����,

� � �'� ?�* 3'� $* D� ��* 343�'('

9������ 9* 9*� ��� ������ 9* �'((2�� N</ ��� ������� 0�����

��� " 9��1��� &�� � �� .���� ��� . ���� �� >�-��

�� ����O� �& -�-���%��� .������&��� ��������%� ?�* '3�

$* D� ��* 55(�55:

7���� $*� $�1��,�� .*� ��� �� ��� G* �'(('�� N%������

��� =��� �� =������ ��������� � =������� " !����

�� �� ����� � 0�� �O� ����&��� ������� $�%��&�� ����&,

���� ���� ?�* )� $* 5� ��* 45�3((

7� ;*� �� 7��������� =*� ��� ��+���� 7* �'((8�� N���

���� 0�� </�� � �� " $������ ����� ?��� ���O�

�& -�-���%��� .������&��� ��������%� ?�* '(� $* '�

��* 352�3D:

������� ;* �*� 7��1� .*� ��� !���� $* �* �34)2�� ���� �%

! ��&����&�� ��!��'� 0�� ����;���

�� ��� � �* ��� $�������� %* �344)�� N�������

%,�� �@ = 2'5������������� < ����� ��� �� &��"�

0���������� $���� ����� O� ��� �&��%���� �%

� � ������ ��� � �����& �������� �� ?�* )� $* 3� ��

5M5(

0�1�� �*� ��1�,�� G*� ��� &����� $* �'((2�� N=���� ��

��� 0���� �� " ��������� 0������ � " � .��� �� ����

>��� ��� >�-��O� + �&��� ! ���� $�%��&��� ?�* 8(�

$* '� ��* 3):�348

0�1�� �* ��� &����� $* �'((2�� N��������� 0������ �

0���� �� " � !��� 0������� .��� �� #����� ����

>��� ��� >�-��@ ������� " =���� ���� ��� $����

���� �� ���O� + �&��� ! ���� $�%��&��� ?�* 8(� $* D�

��* 5:3�5::

���1,� �� �*� ��� �������� �* �34:)�� N. �� ��� ��������� �

���� ������� ����� !�� .� ������O� � �����&% �

��&����&�%� ?�* 4� $* 8� ��* D)4�D4D

.��� .*� ��� �� ��� G* �344'�� N��������� � <-���� ��

�� ��� " .�� ��� ,� � �����/ !��� . � ��O� $���,

�-����' ��� #�����/��� � ! ��&����&�� �'%���% 012� ���1�

,� �� �* ��� %�" ����� ������ 0* ����*�� .������?�����

��* 5'8�55)

.��� .*� ��� �� �� G*� �3442�� N��������� � =������� &����

� $���� $� ,1O� +��. ����&���� ��� + �&���3 ��&��% ��

?�* D(� $* 5� ��* 'D'�'D2

%�" ����� ������ 0* ��� �� ��� G* �34)2�� ��������� � !

��&����&�� �'%���% $����-����' ��� &'� .������?����

&����� $*� #1���� G*� ��� .���1�� .* �344'�� N���1 =�����

��� =������ �� ��������� " ;����.���� ��" �� ����

.���O� �& �� 4�� ����&���� ��� �'�� %��� � ��� �&�������

��%��� ! ���� ��� � -��� ���%� ?�* '� ��* 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, JAPAN

ABSTRACT

Regarding broaching associated with surf-riding, this paper reports experimental, numerical and analytical studies conducted by the SCAPE (Strategic Research Committee 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 in numerical 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 scenarios that 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 ship cannot 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 high forward speed, especially when a ship is surf-ridden. Thus, this phenomenon is relevant to ships having their Froude number 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 ship stability. Thus, this paper focuses on broaching associated with surf-riding. HISTORICAL REVIEW

A physical model experiment is definitely suitable for realising dangerous phenomena in ship stability because full scale 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 to a ship model running in the straight course and observed the ship 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 broaching for 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 in following 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 to wave particle velocity. As a result, it was confirmed that even 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

for manoeuvring coefficient in waves with free surface effects taken into account.

Motora et al. (1982) and Renilson (1982) numerically integrated nonlinear equation of surge-sway-yaw motion and concluded that necessary condition of broaching is that wave-induced yaw moment exceeds the maximum yaw moment due to rudder. Umeda and Renilson (1992 and 1994) developed a 4 DOF mathematical model based on a manoeuvring model with a linear wave forces. Assuming that wave steepness and manoeuvring motions are small, all higher-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 model experiments (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-running model 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 order terms. 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 captive model experiments. Hashimoto et al. (2004B) utilised fully-captive model experiments in waves with a purpose-built apparatus.

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 unstable equilibrium of an uncoupled surge model on a wave is connected to another unstable equilibrium. This is a heteroclinic bifurcation in nonlinear dynamical system theory. 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) numerically obtained 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 regular waves only. Although Rutgerssen and Ottosson (1987) and Motora et al. (1982) executed model experiments in irregular waves in a seakeeping and manoeuvring basin and full 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 the deterministic surf-riding threshold and the joint probability of local wave height and wave period.

0 15 30 450.1

0.2

0.3

0.4

0.5

auto pilot course (degrees)

nom

inal

Fro

ude

num

ber

capsizing due to broachingcapsizing without broachingbroaching without capsizing

stable surf-ridingperiodic motionnot identified

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

present model

0 15 30 450.1

0.2

0.3

0.4

0.5

auto pilot course (degrees)

nom

inal

Fro

ude

num

ber

original model

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

first order model

-0.50

-0.45

-0.40

-0.35

-0.300.00 0.10 0.20 0.30 0.40 0.50 0.60

FnRo

ll Mom

ent (

kg.f.

m)

CFDEFD

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 and Stern, 2007).

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

0.50

0 5 10 15 20 25 30 35 40

Auto Pilot Course (degrees)

Nom

inal

Fro

ude

Num

ber

capsize without broaching

stable surf-riding

periodic motion

surfriding with oscillation

not identified

by heteroclinic bifurcation

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 drag by 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 is allowed to be a large angle, such as 70 degrees, so that the rudder drag can deteriorate surf-riding equilibrium under the yaw 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 by using numerical simulations.

OUTCOMES FROM SCAPE ACTIVITIES

The 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 is important to provide guidelines for simpler but accurate

enough 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 a result 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 a seakeeping and manoeuvring basin. Agreement between the recommended mathematical model and the free-running model experiment by Umeda et al. (1999A) is satisfactory as shown in Figure 1

The above mathematical modelling is developed for single-screw and single-rudder ships, while other ships are also susceptible for broaching. They include twin-screw and twin-rudder ships as shown in model experiments and water-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 a mathematical 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 a good agreement with the model experiment, as shown in Figure 2.

Bifurcation analysis

It was already established that the critical condition for surf-riding in regular following and quartering waves is a heteroclinic (or homoclinic) bifurcation. Next step is to directly evaluate such bifurcation point from mathematical models, especially a 4 DOF model. Hori and Umeda. (2005) and Umeda et al. (2006A) proposed a theoretical formulation of this bifurcation problem with the coupled surge-sway-yaw-roll mathematical model and well validated with 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 for many different initial and control values can be avoided. Above this surf-riding threshold, capsizing due to broaching occurs if wave steepness is very high. Maki et al. (2007) further improved this methodology, focusing to an uncoupled surge model in regular following waves, from theoretical and numerical viewpoints so that deterministic surf-riding threshold hyperplane could be efficiently estimated.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

numericalexp(no.1)

numericalexp(no.2)

numericalexp(no.3)

numericalexp(no.4)

numericalexp(no.5)

broa

chin

g pr

obab

ility

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

numericalexp(no.1)

numericalexp(no.2)

numericalexp(no.3)

numericalexp(no.4)

numericalexp(no.5)

broa

chin

g pr

obab

ility

Figure 4 Comparison in broaching probability between

numerical experiments and theory for a high-speed fishing craft in irregular stern quartering waves (Umeda et al., 2007).

Probabilistic assessment

To evaluate danger of broaching in actual seaways, it is essential to evaluate broaching probability or probability of capsizing due to broaching in irregular waves. Umeda et al. (2007) proposed a theoretical calculation method for broaching probability as an extension of Umeda’s theory for surf-riding probability (Umeda, 1990). Here the broaching probability is calculated by integrating the joint probability density of local wave height and wave period, which was proposed by Longuet-Higgins (1983), within the deterministic broaching region. The present theory was well

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 to prevent broaching. Prevention device

Other than the Sea brake and the ABSS, a new device for preventing broaching was proposed by Hashimoto et al. (2007B). It is a pair of wings attached to bow above a still water plane. Once broaching is initiated, a ship deviated from 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 validated with a free-running model experiment. RECOMMENDATION FOR IMO NEW GENERATION CRITERIA

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. Vulnerability criterion

It is important that a vulnerability criterion can be easily used, can guarantee conservative safety level and should be based on non-empirical approach. The surf-riding threshold in regular following waves can be substituted to that of capsizing due to broaching because surf-riding is a prerequisite to broaching and following waves are most dangerous 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 of resistance and propeller thrust to be proportional to square of ship velocity. Maki and Umeda (2008) utilised a piece-wise linear approximation of wave-induced surge force and then different analytical formula for surf-riding threshold in regular following waves was obtained. Here the balance of resistance and thrust is approximated to have a linear relationship with ship velocity. In this procedure, ship resistance in calm water, propeller thrust and distribution of sectional hull area are required. They executed comparisons among the two analytical formulae, numerically-obtained bifurcation points and a free-running model experiment so that they concluded that Maki and Umeda’s analytical solution is more accurate than Spyrou’s and almost identical to the numerically-obtained bifurcation points and the experiment, as shown in Figure 5.

theorytheorytheory

Direct assessment If a ship design fails to comply with the vulnerability

criterion, it is expected to apply direct stability assessment to the subject ship design. It is preferably to evaluate risk level of the ship design. For this purpose, it is necessary to evaluate probability of capsizing due to broaching. The SCAPE committee already developed a theoretical procedure for broaching probability in irregular stern quartering waves (Umeda et al., 2007) and an extension of this method to probability due to broaching is straight-forward. First, we should identify deterministic thresholds for capsizing due to broaching as a function of

wave height and period by heteroclinic bifurcation analysis of the coupled surge-sway-yaw-roll model with an auto pilot or the repetition of numerical simulation with the same model. Second, the probability of capsizing due to broaching is 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 resistance in calm water, propeller thrust, hull form offset, calm-water manoeuvring coefficients, roll damping coefficient and restoring moment are required at least.

Figure 5 The surf-riding thresholds in following waves obtained by model experiments, numerical bifurcation analysis and analytical 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 broaching provides the following concluding remarks: 1. For quantitative prediction in regular waves, an

enhanced system-based model is proposed and validated 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 was validated with the direct numerical simulation.

3. For calculating broaching probability in irregular waves, a theory was formulated and was validated with the Monte Carlo simulation.

4. A device for preventing broaching was proposed and was 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 their sincere 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.

REFERENCES

Ananiev, D.M., 1966, “On Surf-Riding in Following Seas”, Transaction of Krylov Society, Vol. 13, pp.169-176, (in Russian).

Du Cane, P. 1957, “Model Evaluation of Four High Speed Hull Forms in Following and Head Sea Conditions”, Proceedings of the Symposium on the Behaviour of Ships in a Seaway, NSMB, Wageningen, pp.737-755.

Davidson, K.S.M. 1948, “A Note on the Steering of Ships in Following Seas”, Proceedings of International Congress of Applied Mechanics, London, pp.554-556.

Eda, H., 1972, “Directional Stability and Control of Ships in Waves”, Journal of Ship Research, Vol. 16, pp.205-218.

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

1 1.2 1.4 1.6 1.8 2λ / L

Fn

Numerical Bifurcation AnalysisSpyrou's FormulaPresent Method

Fujino, M., K. Yamasaki and Ishii, Y. 1983, ”On the Stability Derivatives of a Ship Travelling in Following Waves”, Journal of Society of Naval Architects of Japan, Vol. 152, pp.167-179.

Fuwa, T., Sugai, K et al. 1982, ”An Experimental Study on Broaching of a Small High Speed Craft, Papers of Ship Research Institute, No. 66, p.1-40.

Germany, 2005, “Report of the Intersessional Correspondence Group (part 1)”, SLF48/4/1, IMO.

Grim, O., 1951, “Das Schiff in von Achtern Auflaufender See“ , Jahrbuch Schiffbautechnishe Gesellschaft, 4 Band, pp.264-287.

Hamamoto, M., 1973, “On the Hydrodynamic Derivatives for the Directional Stability of Ships in Following Sea (Part 2)”, Journal of the Naval Architects of Japan, Vol. 133, pp.133-142, (in Japanese).

Hashimoto, H., Umeda, N. and Matsuda, A., 2004Ａ , “Importance of Several Nonlinear Factors on Broaching Prediction”, Journal of Marine Science of Technology, Vol. 9, pp. 80-93.

Hashimoto, H., Umeda, N. and Matsuda, A., 2004B, “Model Experiment on Heel-Induced Hydrodynamic Forces in Waves for Broaching Prediction”, Proceedings of the 7th International Ship Stability Workshop, Shanghai Jiao Tong University, pp.144-155.

Hashimoto, H. and Umeda, N., 2005, "Investigation on the Quantitative Prediction of Broaching", Conference Proceedings of the Kansai Society of Naval Architects, Vol. 24, pp. 5-8, (in Japanese).

Hashimoto, H. and Stern, F, 2007A, “An Application of CFD for Advanced Broaching Prediction”, Conference Proceedings of the Japan Society of Naval Architects and Ocean Engineers, Vol. 5E, pp.51-52.

Hashimoto, H., Matsuda, A. and Sanya, Y., 2007B, “Investigation on the Wing-Type Appendage for Ship Capsizing Prevention”, Journal of the Japan Society of Naval Architects and Ocean Engineers, Vol.6, (in press), (in Japanese).

Hori, M. and Umeda, N., 2005, “Global Bifurcation Analysis on Surf-Riding Threshold of a Ship in Quartering Seas“, Conference Proceedings of the Japan Society of Naval Architects and Ocean Engineers, Vol. 1, pp.77-80, (in Japanese).

Japan, the Netherlands and the United States, 2007, “Framework for the Development of New Generation Criteria for Intact Stability”, SLF50/4/4, IMO.

Kan, M, Saruta, T., Taguchi, H. Et al., 1990, ”Capsizing of a Ship in Quartering Seas (Part 1), Journal of the Society of Naval Architects of Japan, Vol. 167, pp.81-90, (in Japanese).

Longuet-Higgins, M.S., 1983, “On the Joint Distribution of Wave Periods and Amplitudes in a Random Wave Field”, Proceedings of the Royal Society of London, Series A389, pp. 241-258.

Maki, A. and Umeda, N., 2006, “Approach for Broaching Phenomenon in the Light of Optical Control Theory”, Conference Proceedings of the Japan Society of Naval Architects and Ocean Engineers, Vol. 3E, pp.207-208, (in Japanese).

Maki, A., Umeda, N. and Hori, M., 2007, “Prediction of Global Bifurcation Points as Surf-Riding Threshold in

Following Seas”, Journal of the Japan Society of Naval Architects and Ocean Engineers, Vol., 5, p. 205-215, (in Japanese).

Maki, A. and Umeda, N., 2008, “Analytical Predictions of Surf-Riding Thresholds and Their Experimental Validation”, Proceedings of the 10th International Ship Stability Workshop, Daejeon, (to be appeared).

Makov, Y. 1969, “Some Results of Theoretical Analysis of Surf-Riding in Following Seas”, Transaction of the Krylov Society, Vol. 126, pp,124-128, (in Russian).

Matsuda, A., Hashimoto, H. and Momoki, T., 2007, “Non-linear Hydrodynamic Force Measurement System in Heavy Seas for Broaching Prediction”, Proceedings of the 9th International Ship Stability Workshop, Hamburg, pp. 5.2.5.-5.2.5.

Motora, S., Fujino, M. and Fuwa, T. 1982, “On the Mechanism of Broaching-To Phenomena”, Proceedings of the 2nd International Conference on Stability and Ocean Vehicles, The Society of Naval Architects of Japan, Tokyo, pp.535-550.

Nicholson, K. 1974, “Some Parametric Model Experiments to Investigate Broaching-To”, in Bishop RED, Price WG (eds) The dynamics of Marine Vehicles and Structure in Waves. Mechanical Engineering Publ, London, pp. 160-166.

Oakley, O.H. Paulling, J.R and Wood, P.D. 1974, “Ship Motions and Capsizing in Astern Seas”, Proceedings of the 10th Naval Hydrodynamics Symposium, Cambridge, IV-1, pp.1-51.

Renilson, M.R. 1982, “An Investigation into the Factors Affecting the Likelihood of Broaching-to in Following Seas”, Proceedings of the 2nd International Conference on Stability and Ocean Vehicles, The Society of Naval Architects of Japan, Tokyo, pp.551-564.

Renilson, M.R. 1986, “The Seabrake – A Device for Assisting in Prevention of Broaching-To”, Proceedings of the 3rd International Conference on Stability of Ships and Ocean Vehicles, Gdansk Technical University, Vol. 2, pp.75-80.

Rutgersson, O and Ottosson, P. 1987, “Model Tests and Computer Simulations – An Effective Combination for Investigation of Broaching Phenomena”, Transaction of the Society of Naval Architects and Marine Engineerings, Vol. 95, pp.263-281.

Spyrou, K.J., 1995, “Surf-Riding, Yaw Instability and Large Heeling of Ships in Following/ Quartering Waves”, Schiffstechnik, Vol. 42, pp. 103-112.

Spyrou, K.J., 1996, “Dynamic Instability in Quartering Waves: the Behaviour of a Ship During Broaching”, Journal of Ship Research, Vol. 40, No. 1, pp.46-59.

Spyrou, K.J., 1997, “Dynamic Instability in Quartering Waves Part III: Nonlinear Effects on Periodic Motions”, Journal of Ship Research, Vol. 41, No. 3, pp.210-223.

Spyrou, K.J., 2001, “Exact Analytical Solutions for Asymmetric Surging and Surf-Riding”, Proceedings of the 5th International Workshop on Stability and Operational Safety of Ships, University of Trieste, pp.4.4.1-3.

Umeda, N., 1990, “Probabilistic Study on Surf-riding of a Ship in Irregular Following Seas”, Proceedings of the 4th International Conference on Stability of Ships and Ocean Vehicles, University Federico II of Naples, pp.336-343.

Umeda,N. and Renilson, M.R. 1992, “Broaching - A Dynamic Analysis of Yaw Behaviour of a Vessel in a Following Sea”, in Manoeuvring and Control of Marine Craft (Wilson,P.A. eds.), Computational Mechanics Publications (Southampton), pp.533-543.

Umeda, N. and Renilson, M.R., 1994 “Broaching of a Fishing Vessel in Following and Quartering Seas”, Proceedings of the 5th International Conference on Stability of Ships and Ocean Vehicles, Florida Institute, Melbourne, Vol.3, pp.115-132.

Umeda, N. and D. Vassalos, 1996, “Non-linear Periodic Motions of a Ship Running in Following and Quartering Seas”, Journal of the Society of Naval Architects of Japan, Vol.179, pp.89-101.

Umeda, N., 1999, “Nonlinear Dynamics on Ship Capsizing due to Broaching in Following and Quartering Seas”, Journal of Marine Science of Technology, Vol.4, pp.16-26.

Umeda, N., Matsuda, A., Hamamoto, M. and Suzuki, S., 1999A, “Stability Assessment for Intact Ships in the Light of Model Experiments”, Journal of Marine Science of Technology, Vol.4, pp.45-57.

Umeda, N., A. Matsuda and M. Takagi, 1999B, “Model Experiment on Anti-Broaching Steering System”, Journal of the Society of Naval Architects of Japan, Vol. 185, May, pp.41-48.

Umeda, N. and Renilson, M.R., 2001, “Benchmark Testing of Numerical Prediction on Capsizing of Intact Ships in Following and Quartering Seas,”, Proceedings of the 5th International Workshop on Stability and Operational Safety of Ships, University of Trieste, pp.6.1.1-10.

Umeda, N. and Hashimoto, 2002, “Qualitative Aspects of Nonlinear Ship Motions in Following and Quartering Seas with High Forward Velocity”, Journal of Marine Science and Technology, Vol., 6, pp. 111-121.

Umeda, N., Urano, S. et al., 2002 “Anti-Broaching Steering System –Comparison between Experiment and Calculation-“, Proceedings of Asia Pacific Workshop on Marine Hydrodynamics, The Kansai Society of Naval Architects, Osaka, pp. 85-90.

Umeda, N., H. Hashimoto and A. Matsuda, 2003, “Broaching Prediction in the Light of an Enhanced

Mathematical Model with Higher Order Terms Taken into Account”, Journal of Marine Science and Technology, Vol. 7, pp. 145-155.

Umeda, N., Ohkura, Y., Urano, S., Hori, M. and Hashimoto, H., 2004, “Some Remarks on Theoretical Modelling of Intact Stability”, Proceedings of the 7th International Ship Stability Workshop, Shanghai Jiao Tong University, pp.85-91.

Umeda, N., Hashimoto, H. and Shuto, M., 2005A, “Broaching Prediction of a Waterjet-propelled Craft”, Proceedings of the Kansai Society of Naval Architects, No. 24, pp. 9-12, (in Japanese).

Umeda, N., Hashimoto, H., Paroka, D. and Hori, M., 2005B, “Recent Developments of Theoretical Prediction on Capsizes of Intact Ships in Waves”, Proceedings of the 8th International Ship Stability Workshop, Istanbul Technical University, pp.1.2.1-1.2.10.

Umeda, N., Hori, M. and Hashimoto, H., 2006A, “Theoretical Prediction of Broaching in the Light of Local and Global Bifurcation Analysis”, Proceeding the 9th International Conference on Stability of Ships and Ocean Vehicles, Federal University of Rio de Janeiro., Vol.1, pp.353-362.

Umeda, N., Maki, A. and Hashimoto, H. 2006B, “Manoeuvring and Control of a High-Speed Slender Vessel with Twin Screws and Twin Rudders in Following and Quartering Seas”，Journal of the Japan Society of Naval Architects and Ocean Engineers, Vol. 4, pp.155-164, (in Japanese).

Umeda, N. and Hashimoto, H., 2006C, “Recent Developments of Capsizing Prediction Techniques of Intact Ships Running in Waves”, Proceedings of the 26th ONR Symposium on Naval Hydrodynamics, INSEAN, Rome, CD.

Umeda, N., M. Shuto and A. Maki, 2007, “Theoretical Prediction of Broaching Probability for a Ship in Irregular Astern Seas”, Proceedings of the 9th International Ship Stability Workshop, Germanisher Lloyd, Hamburg, pp. 5.1.1-.5.1.7.

Wahab, R. and Swaan, W.A. 1964, “Coursekeeping and Broaching of Ships in Following Seas”, Journal of Ship Research, Vol. 7, No. 4, pp.1-15.